Class 1 d | S \ Book ,K f- ££ffi¥RIGHT DEPOSm THE I MECHANICAL ENGINEER'S POCKET-BOOK. A REFERENCE-BOOK OF RULES, TABLES, DATA, AND FORMULAE, FOR THE USE OF ENGINEERS, MECHANICS, AND STUDENTS. 8 1189J WILLIAM KENT, A.M., M.E., Consulting Engineer, Member Amer. Soc'y Mechl. Engrs. and Amer Inst. Mining Engrs. FIRST THOUSAND. FIRST EDITION. NEW YORK : JOHN WILEY & SONS, 53 East Tenth Street. 1895. *> <^w *\%K Copyright, 1895, BY WILLIAM KENT. #' ROBERT DRUMMOND, ELECTROTYPER ANP PRINTER, NEW YORK. PBEFACE. More than twenty years ago the author began to follow the advice given by Nystrom : " Every engineeer should make his own pocket-book, as he proceeds in study and practice, to suit his particular business." The manuscript pocket-book thus begun, however, soon gave place to more modern means for disposing of the accumulation of engi- neering facts and figures, viz., the index rerum, the scrap- book, the collection of indexed envelopes, portfolios and boxes, the card catalogue, etc. Four years ago, at the re- quest of the publishers, the labor was begun of selecting from this accumulated mass such matter as pertained to mechanical engineering, and of condensing, digesting, and arranging it in form for publication. In addition to this, a careful examination was made of the transactions of engi- neering societies, and of the most important recent works on mechanical engineering, in order to fill gaps that might be left in the original collection, and insure that no impor- tant facts had been overlooked. Some ideas have been kept in mind during the prepara- tion of the Pocket-book that will, it is believed, cause it to differ from other works of its class. In the first place it was considered that the field of mechanical engineering was so great, and the literature of the subject so vast, that as little space as possible should be given to subjects which especially belong to civil engineering. While the mechan- ical engineer must continually deal with problems which belong properly to civil engineering, this latter branch is so well covered by Trautwine's " Civil Engineer's Pocket- book " that any attempt to treat it exhaustively would not only fill no " long-felt want," but would occupy space which should be given to mechanical engineering. IV PREFACE. Another idea prominently kept in view by the author has been that he would not assume the position of an " au- thority " in giving rules and formulae for designing, but only that of compiler, giving not only the name of the originator of the rule, where it was known, but also the volume and page from which it was taken, so that its derivation may be traced when desired. When different formulas for the same problem have been found they have been given in contrast, and in many cases examples have been calculated by each to show the difference between them. In some cases these differences are quite remark- able, as will be seen under Safety-valves and Crank-pins. Occasionally the study of these differences has led to the author's devising a new formula, in which case the deriva- tion of the formula is given. Much attention has been paid to the abstracting of data of experiments from recent periodical literature, and numer- ous references to other data are given. In this respect the present work will be found to differ from other Pocket- books. The author desires to express his obligation to the many persons who have assisted him in the preparation of the work, to manufacturers who have furnished their cata- logues and given permission for the use of their tables, and to many engineers who have contributed original data and tables. The names of these persons are mentioned in their proper places in the text, and in all cases it has been endeavored to give credit to whom credit is due. The thanks of the author are also due to the following gentle- men who have given assistance in revising manuscript or proofs of the sections named : Prof. De Volson Wood, mechanics and turbines ; Mr. Frank Richards, compressed air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C. Humphreys, illuminating gas ; Mr. Albert E. Mitchell, locomotives ; Prof. James E. Denton, refrigerating-ma- chinery ; Messrs. Joseph Wetzler and Thomas W. Varley, electrical engineering ; and Mr. Walter S. Dix, for valu- able contributions on several subjects, and suggestions as to their treatment. Wm. Kent. Passaic, N. J., April, 1895. CONTENTS. (For Alphabetical Index see page 1075.) MATHEMATICS. Arithmetic. PAGE Arithmetical and Algebraical Signs 1 Greatest Common Divisor 2 Least Common Multiple 2 Fractions 2 Decimals 3 Table. Decimal Equivalents of Fractions of One Inch 3 Table. Products of Fractions expressed in Decimals 4 Compound or Denominate Numbers 5 Reduction Descending and Ascending 5 Ratio and Proportion 5 Involution, or Powers of Numbers 6 Table. First Nine Powers of the First Nine Numbers 7 Table. First Forty Powers of 2 7 Evolution. Square Root 7 Cube Root _ 8 Alligation 10 Permutation 10 Combination 10 Arithmetical Progression 11 Geometrical Progression 11 Interest 13 Discount 13 Compound Interest 14 Compound Interest Table, 3, 4, 5, and 6 per cent 14 Equation of Payments 14 Partial Payments . 15 Annuities 16 Tables of Amount, Present Values, etc., of Annuities 16 "Weights and Measures. Long Measure Old Land Measure Nautical Measure , Square Measure , Solid or Cubic Measure Liquid Measure The Miners' Inch Apothecaries' Fluid Measure Dry Measure Shipping Measure Avoirdupois Weight Troy Weight Apothecaries' Weight , To Weigh Correctly on an Incorrect Balance , Circular Measure au Measure of time 20 PAGE Board and Timber Measure 20 Table. Contents in Feet of Joists, Scantlings, and Timber 20 French or Metric Measures 21 British and French Equi valents 21 Metric Conversion Tables . . 23 Compound Units. of Pressure and Weight 27 of Water, Weight, and Bulk 27 of Work, Power, and Duty 27 of Velocity 27 of Pressure per unit area 27 Wire and Sheet Metal Gauges 28 Twist-drill and Steel-wire Gauges. 28 Music-wire Gauge 29 Circular- mil Wire Gauge 30 New U. S. Standard Wire and Sheet Gauge, 1893 30 Algebra. Addition. Multiplication, etc 33 Powers of Num bers 33 Parentheses, Division 34 Simple Equations and Problems 34 Equations containing two or more Unknown Quantities 3D Elimination 35 Quadratic Equations 35 Theory of Exponents 36 Binomial Theorem 36 Geometrical Problems of Construction 37 of Straight Lines 37 ofAngles 38 of Circles 39 of Triangles 41 of Squares and Polygons 42 of the Ellipse 45 of the Parabola 48 of the Hyperbola 49 of the Cycloid 49 of the Tractrix or Schiele Anti-friction Curve 50 of the Spiral 50 of the Catenary -. , - 51 of the Involute 52 Geometrical Propositions 53 Mensuration, Plane Surfaces. Quadrilateral, Parallelogram, etc 54 Trapezium and Trapezoid 54 Triangles .... 54 Polygons. Table of Polygons 55 Irregular Figures 55 Properties of the Circle 57 Values of n and its Multiples, etc 57 Relations of arc, chord, etc 58 Relations of circle to inscribed square, etc 58 Sectors and Segments 59 Circular Ring 59 The Ellipse 59 The Helix 60 The Spiral 60 Mensuration, Solid Bodies. Prism 60 Pyramid 60 Wedge 61 The Prismoidal Formula 62 Rectangular Prismoid 61 Cylinder 61 Cone CI CONTENTS. \11 PAGE Sphere 61 Spherical Triangle 61 Spherical Polygon 61 Spherical Zone 62 Spherical Segment 62 Spheroid or Ellipsoid , 63 Polyedron 62 Cylindrical Ring 62 Solids of Revolution 62 Spindles 63 Frustrum of a Spheroid 63 Parabolic Conoid 64 Volume of a Cask 64 Irregular Solids : 64 Plane Trigonometry. Solution of Plane Triangles 65 Sine, Tangent, Secant, etc 65 Signs of the Trigonometric Functions 66 Trigonometrical Formulae 66 Solution of Plane Right-angled Triangles 68 Solution of Oblique-angled Triangles 68 Analytical Geometry. Ordinates and Abscissas 69 Equations of a Straight Line, Intersections, etc 69 Equations of the Circle , 70 Equations of the Ellipse , 70 Equations of the Parabola 70 Equations of the Hyperbola 70 Logarithmic Curves 71 Differential Calculus. Definitions 72 Differentials of Algebraic Functions 72 Formulae for Differentiating 73 Partial Differentials 73 Integrals , 73 Formulae for Integration 74 Integration between Limits 74 Quadrature of a Plane Surface. 74 Quadrature of Surfaces of Revolution 75 Cubature of Volumes of Revolution 75 Second, Third, etc., Differentials 75 Maclaurin's and Taylor's Theorems „ 76 Maxima and Minima 76 Differential of an Exponential Function 77 Logarithms 77 Differential Forms which have Known Integrals 78 Exponential Functions 78 Circular Functions 78 The Cycloid 79 Integral Calculus , 79 Mathematical Tables. Reciprocals of Numbers 1 to 2000 80 Squares, Cubes, Square Roots, and Cube Roots from 0.1 to 1600 86 Squares and Cubes of Decimals 101 Fifth Roots and Fifth Powers 102 Circumferences and Areas of Circles, Diameters 1 to 1000 103 Circumferences and Areas of Circles, Advancing by Eighths from B V to 100 108 Decimals of a Foot Equivalent to Inches and Fractions of an Inch 112 Circumferences of Circles in Feet and Inches, from 1 inch to 32 feet 11 inches in diameter 113 Lengths of Circular Arcs, Degrees Given 114 Lengths of Circular Arcs, Height of Arc Given 115 Areas of the Segments of a Circle , 116 CONTENTS. PARE Spheres 118 Contents of Pipes and Cylinders, Cubic Feet and Gallons 120 Cylindrical Vessels, Tanks, Cisterns, etc 121 Gallons in a Number of Cubic Feet 122 Cubic Feet in a Number of Gallons 122 Square Feet in Plates 3 to 32 feet long and 1 inch wide 123 Capacities of Rectangular Tanks in Gallons 125 Number of Barrels in Cylindrical Cisterns and Tanks. 126 Logarithms . 127 Table of Logarithms 129 Hyperbolic Logarithms 156 Natural Trigonometrical Functions 159 Logarithmic Trigonometrical Functions 162 MATERIALS. Chemical Elements 163 Specific Gravity and Weight of Materials 163 Metals, Properties of 164 The Hydrometer 165 Aluminum 166 Antimony 166 Bismuth 166 Cadmium 167 Copper 167 Gold 167 Iridium 167 Iron , 167 Lead 167 Magnesium 168 Manganese 168 Mercury 168 Nickel 168 Platinum 168 Silver 168 Tin 168 Zinc 168 Miscellaneous Materials. Order- of Malleability, etc., of Metals 169 Formulas and Table for Calculating' Weight of Rods, Plates, etc 169 Measures and Weights of Various Materials 169 Commercial Sizes of Iron Bars 170 Weights of Iron Bars 171 of Flat Rolled Iron .* 172 of Iron and Steel Sheets 174 of Plate Iron 175 of Steel Blooms 176 of Structural Shapes 177 Sizes and Weights of Carnegie Deck Beams 177 - " •' " Steel Channels 178 " ZBars 178 " " Pencoyd Steel Angles 179 Tees ." 180 " " " Channels 180 " " Roofing Materials 181 " " Terra-cotta 181 " " Tiles: 181 " " Tin Plates ...181 " " Slates 183 " " Pine Shingles 183 " " Sky-light Glass ...184 Weights of Various Roof-coverings 184 Cast-iron Pipes or Columns.. , 185 " " " '' 12-ft. lengths 186 '• " Pipe fittings 187 " " " Water and Gas-pipe 188 " and thickness of Cast-iron Pipes 189 Safe Pressures on Cast Iron Pipe 189 CONTENTS. Sheet-iron Hydraulic Pipe 191 Standard Pipe Flanges 192 Pipe Flanges and Cast-iron Pipe 193 Standard Sizes of Wrought-iron Pipe 194 Wrought-iron Welded Tubes 196 Riveted Iron Pipes 197 Weight of Iron for Riveted Pipe 197 Spiral Riveted Pipe 198 Seamless Brass Tubing 198, 199 Coiled Pipes 199 Brass, Copper, and Zinc Tubing 200 Lead and Tin-lined Lead Pipe 201 Weight of Copper and Brass Wire and Plates 202 " Round Bolt Copper 203 " Sheet and Bar Brass 203 Composition of Rolled Brass 203 Sizes of Shot 204 Screw-thread, U. S. Standard 204 Limit-gauges for Screw-threads 205 Size of Iron for Standard Bolts 20G Sizes of Screw-threads for Bolts and Taps 207 Set Screws and Tap Screws 208 Standard Machine Screws 209 Sizes and Weights of Nuts .- 209 Weight of Bolts with Heads 210 Track Bolts 210 Weights of Nuts and Bolt-heads 211 Rivets 211 Sizes of Turnbuckles 211 Washers 212 Track Spikes 212 Railway Spikes 212 Boat Spikes 212 Wrought Spikes 213 Wire Spikes. 213 CutNails 213 Wire Nails 214, 215 Iron Wire, Size, Strength, etc 216 Galvanized lion Telegraph Wire 217 Tests of Telegraph Wire 217 Copper Wire Table, B. W. Gauge 218 " " " Edison or Circular Mil Gauge 219 " B.&S.Gauge 220 Insulated Wire 221 Copper Telegraph Wire 221 Electric Cables 221, 222 Galvanized Steel-wire Strand 223 Steel-wire Cables for Vessels 223 Specifications for Galvanized Iron Wire 224 Strength of Piano Wire , 224 Plough-steel Wire 224 Wires of different metals , 225 Specifications for Copper Wire 225 Cable-traction Ropes 226 Wire Ropes 226, 227 Plough-steel Ropes 227, 228 Galvanized Iron Wire Rope 228 Steel Hawsers 223, 229 Flat Wire Ropes , 2-,'9 Galvanized Steel Cables 230 Strength of Chains and Ropes 230 Notes on use of Wire Rope 231 Locked Wire Rope 231 Crane Chains 232 Weights of Logs, Lumber, etc 232 Sizes of Fire Brick 233 Fire Clay, Aualysis , 234 Magnesia Bricks 235 Asbestos.... 235 X CONTENTS. Strength of Materials. PAGE Stress and Strain 236 Elastic Limit 236 Yield Point 237 Modulus of Elasticity 237 Resilience 238 Elastic Limit and Ultimate Stress 238 Repeated Stresses 238 Repeated Sh< >eks 240 Stresses due to Sudden Shocks 241 Increasing Tensile Strength of Bars by Twisting 241 Tensile Strength 242 Measurement of Elongation 243 Shapes of Test Specimens 243 Compressive Strength 244 Columns, Pillars, or Struts 246 Hodgkinson's Formula 246 Gordon's Formula 247 Mom ent of Inertia 247 Radius of Gyration 247 Elements of Usual Sections 248 Solid Cast-iron Columns 250 Hollow Cast-iron Columns 250 Wrought-iron Columns 251 Safe load of Cast-iron Columns 253 Eccentric loading of Columns 255 Built Columns 256 Phcenix Columns 257 Working Formulas for Struts 259 Merriman's Formula for Columns 260 Working Strains in Bridge Members 262 Working Stresses for Steel 263 Resistance of Hollow Cylinders to Collapse 264 Collapsing Pressure of Tubes or Flues. 265 Formula for Corrugated Furnaces 266 Transverse Strength 266 Formulas for Flexure of Beams 267 Safe Loads on Steel Beams 269 Elastic Resilience 270 Beams of Uniform Strength 271 Properties of Rolled Structural Shapes 272 Spacing of I Beams 273 Properties of Steel I Beams 274 " " " Channels , 275 " ZBars 276 Iron Beams and Channels 277 Trenton Angle Bars 279 Tee Bars 280 Size of Beams for Floors 280 Flooring Material 281 Tie Rods for Brick Arches 281 Torsional Strength 281 Elastic Resistance to Torsion 282 Combined Stresses 282 Stress due to Temperature 283 Strength of Flat Plates 283 Strength of Unstayed Flat Surfaces 284 Unbraced Heads of Boilers 285 Thickness of Flat Cast-iron Plates 286 Strength of Stayed Surfaces 286 Spherical Shells and Domed Heads 286 Stresses in Steel Plating under Water Pressure 287 Thick Hollow Cylinders under Tension — 287 Thin Cylinders under Tension 289 Hollow Copper Balls 289 Holding Power of Nails, Spikes, Bolts, and Screws 289 Cut versus Wire Nails 290 Strength of Wrought-iron Bolts 292 CONTENTS. XI PAGE Initial Strain on Bolts 292 Stand Pipes and their Design 292 Riveted Steel Water-pipes 295 Mannesmann Tubes 296 Kirkaldy's Tests of Materials 296 Cast Iron 296 Iron Castings 297 Iron Bars, Forgings, etc 297 Steel Rails and Tires 298 Steel Axles, Shafts, Spring Steel 299 Riveted Joints 299 Welds 300 Copper, Brass, Bronze, etc 300 Wire, Wire-rope 301 Ropes, Hemp, and Cotton 301 Belting, Canvas . 302 Stones, Brick, Cement 302 Tensile Strength of Wire « 303 Watertown Testing-machine Tests 303 Riveted Joints 303 W rough t-iron Bars, Compression Tests 304 Steel Eye-bars 304 Wrought-iron Columns 305 Cold Drawn Steel 305 American Woods 306 Shearing Strength of Iron and Steel , 306 Holding Power of Boiler-tubes 307 Chains, Weight, Proof Test, etc 307 Wrought-iron Chain Cables 308 Strength of Glass 308 Copper at High Temperatures 309 Strength of Timber 309 Expansion of Timber 311 Shearing Strength of Woods 312 Strength of Brick, Stone, etc , 312 " Flagging 313 " " Lime and Cement Mortar 313 Moduli of Elasticity of Various Materials 314 Factors of Safety 314 Properties of Cork 316 Vulcanized India-rubber 316 Xylolith or Woodstone 316 Aluminum, Properties and Uses 317 Alloys. Alloys of Copper and Tin, Bronze 319 Copper and Zinc, Brass 321 Variation in Strength of Bronze 321 Copper-tin-zinc Alloys 322 Liquation or Separation of Metals 323 Alloys used in Brass Foundries 325 Copper-nickel Alloys 326 Copper-zinc-iron Alloys 326 Tobin Bronze 327 Phosphor Bronze 327 Aluminum Bronze 328 Aluminum Brass 329 Caution as to Strength of Alloys 329 Aluminum hardened 330 Alloys of Aluminum, Silicon, and Iron 330 Tungsten-aluminum Alloys 331 Aluminum-tin Alloys 331 Manganese Alloys t 331 Manganese Bronze 331 German Silver ? 332 Alloys of Bismuth 332 Fusible Alloys 333 Bearing Metal Alloys 333 CONTENTS. PAGE Alloys containing Antimony 336 White-metal Alloys 336 Type-metal 336 Babbitt metals 336 Solders 338 Ropes and Chains. Strength of Hemp, Iron, and Steel Ropes 33^ Flat Ropes , 339 Working Load of Ropes and Chains 339 Strength of Ropes and Chain Cables 340 Rope for Hoisting or Transmission 340 Cordage, Technical terms of 341 Splicing of Ropes 341 Coal Hoisting 343 Manila Cordage, Weight, etc 344 Knots, how to make 344 Splicing Wire Ropes 346 Springs. Laminated Steel Springs 347 Helical Steel Springs 347 Carrying Capacity of Springs 349 Elliptical Springs 352 Phosphor-bronze Springs 352 Springs to Resist Torsional Force 352 Helical Springs for Cars, etc 353 Riveted Joints. Fairbairn's Experiments , 354 Loss of Strength by Punching 354 Strength of Perforated Plates 354 Hand vs. Hydraulic Riveting 355 Formulae for Pitch of Rivets 357 Proportions of Joints 358 Efficiencies of Joints 359 Diameter of Rivets 360 Strength of Riveted Joints 361 . Riveting Pressures 362 Shearing Resistance of Rivet Iron 363 Iron and Steel. Classification of Iron and Steel 364 Grading of Pis: Iron 365 Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 365 Tests of Cast Iron 369 Chemistry of Foundry Iron 370 Analyses of Castings 373 Strength of Cast Iron 374 Specifications for Cast Iron 374 Mixture of Cast Iron with Steel 375 Bessemerized Cast Iron 375 Bad Cast Iron 375 Malleable Cast Iron 375 Wrought Iron 377 Chemistry <»f Wrought Iron 377 Influence of Rolling on Wrought Iron 377 Specifications for Wrought Iron 378 Stay-holt Iron 379 Formula} for Unit Strains in Structures 379 Permissible Stresses in Structures 381 Proportioning Materials in Memphis Bridge 382 Tenacity of Iron at High Temperatures 382 Effect of Cold on Strength of Iron 383 Expansion of Iron by Heat 385 Durability of Cast Iron 385 Corrosion of Iron and Steel 386 Manganese Plating of Iron 387 CONTENTS. Non-oxidizing Process of Annealing 387 Painting Wood and Iron Structures. . 388 Qualities of Paints 389 Steel. Relation between Chem. and Phys. Properties 389 Variation in Strength 391 Open-hearth 392 Bessemer 392 Hardening Soft Steel 393 Effect of Cold Rolling 393 Comparison of Full-sized and Small Pieces 393 Treatment of Structural Steel 394 Influence of Annealing upon Magnetic Capacity 396 Specifications for Steel 397 Boiler, Ship and Tank Plates 399 Steel for Springs, Axles, etc 400 May Carbon be Burned out of Steel? . . 402 Recalescence of Steel 402 Effect of Nicking a Bar 402 Electric Conductivity 403 Specific Gravity -. 403 Occasional Failures 403 Segregation in Ingots 404 Earliest Uses for Structures 405 Steel Castings 405 Manganese Steel 407 Nickel Steel 407 Aluminum Steel , 409 Chrome Steel 409 Tungsten Steel , 409 Compressed Steel 410 Crucible Steel 410 Effect of Heat on Grain „ 412 " " Hammei-ing, etc 412 Heating and Forging 412 Tempering Steel . 413 MECHANICS. Force, Unit of Force 415 Inertia 415 Newton's Laws of Motion 415 Resolution of Forces 415 Parallelogram of Forces 416 Moment of a Force 416 Statical Moment, Stability 417 Stability of a Dam 417 Parallel Forces ., 417 Couples 418 Equilibrium of Forces 418 Centre of Gravity 418 Moment of Inertia 419 Centre of Gyration •. 420 Radi us of Gyration .... 420 Centre of Oscillation 421 Centre of Percussion 422 The Pendulum 422 Conical Pendulum 423 Centrifugal Force 423 Acceleration , 423 Falling Bodies.. 424 Value of g 424 Angular Velocity . . 425 Height due to Velocity 425 Parallelogram of Velocities 426 Mass 427 Force of Acceleration . . 427 Motion on Inclined Planes 428 Momentum . ..... 428 XIV CONTENTS. PAGE Vis Viva 428 Work, Foot-pound 428 Power, Horse-power 429 Energy 429 Work of Acceleration 430 Force of a Blow , 430 Impact of Bodies 431 Energy of Recoil of Guns 431 Conservation of Energy 432 Perpetual Motion 432 Efficiency of a Machine 432 Amiual-power, Man-power 433 Work of a Horse 434 Man-wheel 434 Horse-gin 434 Resistance of Vehicles 435 Elements of Machines. The Lever 435 The Bent Lever 436 The Moving Strut 436 The Toggle-joint 436 The Inclined Plane 437 The Wedge 437 The Screw 437 The Cam 438 The Pulley 438 Differential Pullev 439 Differential Windlass 439 Differential Screw 439 Wheel and Axle ... 439 Toothed-wheel Gearing 439 Endless Screw 440 Stresses in Framed Structures. Cranes and Derricks 440 Shear Poles and Guys 442 King Pest Truss or Bridge 442 Queen Post Truss 442 Burr Truss 443 Pratt or Whipple Truss 443 Howe Truss 445 Warren Girder 445 Roof Truss 446 HEAT. Thermometers and Pyrometers 448 Centigrade and Fahrenheit degrees compared 449 Copper-ball Pyrometer 451 Thermo-electric Pyrometer 451 Temperatures in Furnaces 451 Wiborgh Air Pyrometer 453 Seegers Fire-clay Pyrometer 453 Mesur6 and Nouel's Pyrometer 453 Uehling and Steinhart's Pyrometer 453 Air-thermometer 454 High Temperatures judged by Color 454 Boiling-points of Substances 455 Melting-points 455 Unit of Heat 455 Mechanical Equivalent of Heat 456 Heat of Combustion 456 Specific Heat 457 Latent Heat of Fusion 459, 461 Expansion by Heat, 460 Absolute Temperature 461 A bsolute Zero 461 CONTENTS. XV PAGE Latent Heat 461 Latent Heat of Evaporation 462 Total Heat of Evaporation 462 Evaporation and Drying 462 Evaporation from Reservoirs 463 Evaporation by the Multiple System 463 Resistance to Boiling 463 Manufacture of Salt 464 Solubility of Salt and Sulphate of Lime 464 Salt Contents of Brines 464 Concentration of Sugar Solutions 465 Evaporating by Exhaust Steam 465 Drying in Vacuum 466 Radiation of Heat 467 Conduction and Convection of Heat 468 Steam-pipe Coverings 470 Rate of External Conduction 471 Transmission through Plates 473 " in Condenser Tubes 473 " " Cast-iron Plates 474 " from Air or Gases to Water . . 474 " from Steam or Hot Water to Air 475 " through Walls of Buildings 478 Thermodynamics 478 PHYSICAL PROPERTIES OF GASES. Expansion of Gases 479 Boyle and Marriotte's Law 479 Law of Charles, Avogadro's Law 479 Saturation Point of Vapors 480 Law of Gaseous Pressure 480 Flow of Gases 480 Absorption by Liquids 480 AIR. Properties of Air 481 Air-manometer 481 Pressure at Different Altitudes 481 Bai-ometric Pressures 482 Levelling by the Barometer and by Boiling Water 482 To find Difference in Altitude 483 Moisture in Atmosphere 483 Weight of Air and Mixtures of Air and Vapor 484 Specific Heat of Air 484 Flow of Air. Flow of Air through Orifices 484 Flow of Air in Pipes 485 Effect of Bends in Pipe 488 Flow of Compressed Air 488 Tables of Flow of Air 489 Anemometer Measurements 491 Equalization of Pipes 492 Loss of Pressure in Pipes -. 493 Wind. Force of the Wind ; 493 Wind Pressure in Storms 495 Windmills 495 Capacity of Windmills 497 Economy of Windmills 498 Electric Power from Windmills 499 Compressed Air. Heatiner of Air by Compression 499 Loss of Energy in Compressed Air 499 Volumes and Pressures 500 XVI CONTENTS. PAGE Loss due to Excess of Pressure 501 Horse-power Required for Compression 5(1 Table for Adiabatic Compression 5G2 Mean Effective Pressures 502 Mean and Terminal Pressures . . 503 Air-compressors . . 503 Practical Results 505 Efficiency of Compressed-air Engines 506 Requirements of Rock-drills 506 Popp Compressed-air System 507 Small Compressed-air Motors 507 Efficiency of Air-heating Stoves 507 Efficiency of Compressed-air Transmission 50* Shops Operated by Compressed Air 509 Pneumatic Postal Transmission 509 Mekarski Compressed-air Tramways 509 Compressed Air Working Pumps in Mines 511 Fans and Blowers. Centrifugal Fans 511 Rest Proportions of Fans 512 Pressure due to Velocity 513 Experiments with Blowers 514 Quantity of Air Delivered 514 Efficiency of Fans and Positive Blowers 516 Capacity of Fans and Blowers 517 Table of Centrifugal Fans 518 Engines, Fans, and Steam-coils for the Blower System of Heating 519 Sturtevant Steel Pressure-blower 519 Diameter of Blast-pipes 519 Centrifugal Ventilators for Mines 521 Experiments on Mine Ventilators 522 DiskFans 524 Air Removed by Exhaust Wheel. 525 Efficiency of Disk Fans 525 Positive Rotary Blowers 526 Blowing Engines . . . . . . 526 Steam-jet Blowers 527 Steam-jet for Ventilation 527 HEATING AND VENTILATION. Ventilation 528 Quantity of Air Discharged through a Ventilating Duct 530 Artificial Cooling of Air 531 Mine-ventilation 531 Friction of Air in Underground Passages 531 Equivalent Orifices 533 Relative Efficiency of Fans and Heated Chimneys 533 Heating and Ventilating of Large Buildings 534 Rules for Computing Radiating Surfaces 536 Overhead Steam-pipes 537 Indirect Heating-surface 537 Boiler Heating-surface Required 538 Proportion of Grate-surface to Radiator-surface 538 st cam consumption in Car-heating 538 Diameters of Steam Supply Mains 539 Registers and Cold-air Ducts 539 Physical Properties of Steam and Condensed Water 540 Size of Steam-pipes for Heating 540 Heating a Greenhouse by Steam 541 Heating a Greenhouse by Hot Water 543 Hot-water Heating 542 Law of Velocity of Flow 542 Proportions of Radiating Surfaces to Cubic Capacities 543 Diameter of Main and Branch Pipes 543 Rules for Hot- water Heating 544 Arrangements of Mains 544 CONTENTS. XV11 PAGE Blower System of Heating and Ventilating. . 545 Experiments with Radiators . . . 545 Heating a Building to 70° F... 545 Heating bj r Electricity 546 WATER. Expansion of Water — 547 Weight of Water at different temperatures ■ 547 Pressure of Water due to its Weight 549 Head Corresponding to Pressures 549 Buoyancy 550 Boiling-point 550 Freezing-point 550 Sea-water 549,550 Ice and Snow 550 Specific Heat of Water 550 Compressibility of Water 551 Impurities of Water 551 Causes of Incrustation 551. Means for Preventing Incrustation 552 Analyses of Boiler-scale 552 Hardness of Water 553 Purifying Feed-water 554 Softening Hard Water 555 Hydraulics. Flow of Water. Fomulae for Discharge through Orifices 555 Flow of Water from Orifices 555 Flow in Open and Closed Channels 557 General Formulae for Flow 557 Table Fall of_Feet per mile, etc 558 Values of S/r for Circular Pipes 559 Kutter's Formula . . 559 Molesworth's Formula 562 Bazin's Formula 563 D'Arcy's Formula 563 Older Formulae 564 Velocity of Water in Open Channels 564 Mean, Surface and Bottom Velocities 564 Safe Bottom and Mean Velocities 565 Resistance of Soil to Erosion 565 Abrading and Transporting Power of Water 565 Grade of Sewers 566 Relations of Diameter of Pipe to Quantity discharged 566 Flow of Water in a 20-inch Pipe 566 Velocities in Smooth Cast-iron Water-pipes 567 Table of Flow of Water in Circular Pipes 568-573 Loss of Head 573 Frici ional Heads at given rates of discharge 577 Effect of Bend and Curves 578 Hydraulic Grade-line 578 Flow of Water in House-service Pipes 578 Air-bound Pipes 579 VerticalJets , 579 Water Delivered through Meters 579 Fire Streams 579 Friction Losses in Hose 580 Head and Pressure Losses by Friction . . 580 Loss of Pressure in smooth 214-inch Hose . , 580 Rated capacity of Steam Fire-engines 580 Pressures required to throw water through Nozzles 581 The Siphon 581 Measurement of Flowing Water 582 Piezometer 582 Pitot Tube Gauge . 583 The Venturi Meter 583 Measurement of Discharge by means of Nozzles 584 -Will CONTENTS. PAGE Flow through Rectangular Orifices 584 Measurement of an Open Stream 584 Miners' 1 Inch Measurements 585 Flow of Water over Weirs 586 Francis's Formula for Weirs 586 Weir Table 587 Baziu*s Experiments 587 Water-power. Power of a Fall of Water 588 Horse-power of a Running Stream 589 Current Motors 589 Horse-power of Water Flowing in a Tube 589 Maximum Efficiency of a Long Conduit 589 Mill-power 589 Value of Water-power 590 The Power of Ocean Waves 599 Utilization of Tidal Power 600 Turbine Wheels. Proportions of Turbines 591 Tests of Turbines 596 Dimensions of Turbines 597 The Pelton Water-wheel 597 Pumps. Theoretical capacity of a pump 601 Depth of Suction 602 Amount oi W r ater raised by a Single-acting Lift-pump 602 Proportioning the Steam cylinder of a Direct-acting Pump 602 Speed of Water through Pipes and Pump -passages 602 Sizes of Direct-acting Pumps 603 The Deane Pump 603 Efficiency of Small Pumps 603 The Wor thington Duplex Pump 604 Speed of Piston 605 Speed of Water through Valves 605 Boiler feed Pumps 605 Pump Valves 606 Centrifugal Pumps 606 Lawrence Centrifugal Pumps 607 Efficiency of Centrifugal and Reciprocating Pumps 608 Vanes of Centrifugal Pumps 609 The Centrifugal Pump used as a Suction Dredge 609 Duty Trials of Pumping Engines 609 Leakage Tests of Pumps 611 Vacuum Pumps 612 The Pulsometer 612 The Jet Pump 614 The Injector 614 Air-lift Pump 6l4 The Hydraulic Ram 614 Quantity of Water Delivered by the Hydraulic Ram 615 Hydraulic Pressure Transmission. Energy of Water under Pressure 616 Efficiency of Apparatus 616 Hydraulic Presses 617 Hydraulic Power in London 617 Hydraulic Riveting Machines . 618 Hydraulic Forging 618 The Aiken Intensifier 019 Hydraulic Engine 619 FUEL. Theory of Combustion 620 Total Heat of Combustion 621 CONTENTS. XIX PAGE Analyses of Gases of Combustion , — 622 Temperature of the Fire 622 Classification of Solid Fuel 623 Classification of Coals 624 Analyses of Coals „ , 624 Western Lignites 631 Analyses of Foreign Coals 631 Nixon's Navigation Coal 632 Sampling Coal for Analyses 632 Relative Value of Fine Sizes 632 Pressed Fuel 632 Relative Value of Steam Coals 633 Approximate Heating Value of Coals 634 Kind of Furnace Adapted for Different Coals 635 Downward-draught Furnaces 635 Calorimetric Tests of American Coals 636 Evaporative Power of Bituminous Coals 636 Weathering of Coal 637 Coke 637 Experiments in Coking 637 Coal Washing 63S Recovery of By-products in Coke manufacture 638 Making Hard Coke 638 Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 Products of the Distillation of Coal 639 Wood as Fuel 639 Heating Value of Wood , 639 Composition of Wood 640 Charcoal , 640 Yield of Charcoal from a Cord of Wood 641 Consumption of Charcoal in Blast Furnaces 641 Absorption of Water and of Gases by Charcoal 641 Composition of Charcoals 642 Miscellaneous Solid Fuels 642 Dust-fuel— Dust Explosions i 642 Peat or Turf 643 Sawdust as Fuel 643 Horse-manure as Fuel 643 Wet Tan-bark as Fuel 643 Straw as Fuel 643 Bagasse as Fuel in Sugar Manufacture , 643 Petroleum. Products of Distillation 645 Lima Petroleum 645 Value of Petroleum as Fuel 645 Oil vs. Coal as Fuel 646 Fuel Gas. Carbon Gas 646 Anthracite Gas , 647 Bituminous Gas , 647 Water Gas 648 Prod ucer-gas from One Ton of Coal 649 Natural Gas in Ohio and Indiana 649 Combustion of Producer-gas 650 Use of Steam in Producers 650 Gas Fuel for Small Furnaces 651 Illuminating Gas. Coal-gas 651 Water-gas 652 Analyses of Water-gas and Coal gas 653 Calorific Equivalents of Constituents 654 Efficiency of a Water-gas Plant 654 Space Required for a Water-gas Plant 650 Fuel-value of Illuminating-gas 656 XX CONTENTS. PAGE Flow of Gas in Pipes 657 Service for Lamps 658 STEAM. Temperature and Pressure 659 Total Heat 659 Latent Heat of Steam 659 Latent Heat of Volume 660 Specific Heat of Saturated Steam 660 Density and Volume 660 Superheated Steam „ 661 Regnault's Experiments 661 Table of the Properties of Steam 662 Flow of Steam. Napier's Approximate Rule 669 Flow of Steam in Pipes 669 Loss of Pressure Due to Radiation 671 Resistance to Flow by Bends 672 Sizes of Steam-pipes for Stationary Engines 673 Sizes of Steam-pipes for Marine Engines 674 Steani Pipes. Bursting-tests of Copper Steam-pipes 674 Thickness of Copper Steam-pipes.. 675 Reinforcing Steam-pipes 675 Wire-wound Steam-pipes 675 Riveted Steel Steam pipes 675 Valves in Steam-pipes 675 Flanges for Steam-pipe 676 The Steam Loop 676 Loss from an Uncovered Steam-pipe 676 THE STEAM BOILER. The Horse- power of a Steam -boiler 677 Measures for Comparing the Duty of Boilers 678 Steam-boiler Proportions 678 Heating-surface 678 Horse-power, Builders' Rating 679 Grate-surface 680 Areas of Flues 680 Air-passages Through Grate-bars * 681 Performance of Boilers 681 Conditions which Secure Economy 682 Efficiency of a Boiler 683 Tests of Steam-boilers 685 Boilers at the Centennial Exhibition 685 Tests of Tubulous Boilers 686 High Rates of Evaporation — 687 Economy Effected by Heating the Air 687 Results of Tests with Different Coals 688 Maximum Boiler Efficiency with Cumberland Coal 689 Boilers Using Waste Gases 689 Boilers for Blast Furnaces 689 Rules for Conducting Boiler Tests 690 Table of Factors of Evaporation 695 Strength of Steam-boilers. Rules for Construction 700 Shell-plate Formulae 701 Rules for Flat Plates 701 Furnace Formulae 702 Material for Stays 703 Loads allowed on Stays 703 Girders 703 Rules for Construction of Boilers in Merchant Vessels in U. S 705 CONTENTS. XXI PAGE U. S. Rule for Allowable Pressures 706 Safe-working Pressures 707 Rules Governing Inspection of Boilers in Philadelphia 708 Flues and Tubes for Steam Boilers 709 Flat-stayed Surfaces 709 Diameter of Stay-bolts.. 710 Strength of Stays ." 710 Stay-bolts in Curved Surfaces 710 Boiler Attachments, Furnaces, etc. Fusible Plugs 710 Steam Domes 711 Height of Furnace 711 Mechanical Stokers 711 The Hawley Down-draught Furnace 712 Under-feed Stokers 712 Smoke Prevention , < 712 Gas-fired Steam-boilers 714 Forced Combustion 714 Fuel Economizers 715 Incrustation and Scale 716 Boiler-scale Compounds 717 Removal of Hard Scale 718 Corrosion in Marine Boilers 719 Use of Zinc 720 Effect of Deposit on Flues 720 Dangerous Boilers 720 Safety Valves. Rules for Area of Safety-valves , 721 Spring-loaded Safety-valves 724 The Injectoi*. Equation of the Injector 725 Performance of Injectors 726 Boiler-feeding Pumps 726 Feed -water Heaters. Strains Caused by Cold Feed-water „ 727 Steam Separators. Efficiency of Steam Separators 728 Determination of Moisture in Steam. Coil Calorimeter 729 Throttling Calorimeters. 729 Separating Calorimeters 730 Identification of Dry Steam 730 Usual Amount of Moisture in Steam 731 Chimneys. Chimney Draught Theory 731 Force or Intensity of Draught 732 Rate of Combustion Due to Height of Chimney 733 High Chimneys not Necessary : 734 Heights of Chimneys Required for Different Fuels 734 Table of Size of Chimneys 734 Protection of Chimney from Lightning 736 Some Tall Brick Chimneys 737 Stability of Chimneys 738 Weak Chimneys 739 Steel Chimneys ... 740 Sheet-iron Chimneys 741 THE STEAM ENGINE. Expansion of Steam 742 Mean and Terminal Absolute Pressures 743 XXll CONTENTS. PAGE Calculation of Mean Effective Pressure 744 Work of Steam in a Single Cylinder. . . 746 Measures for Comparing the Duty of Engines 748 Efficiency, Thermal Units per Minute 749 Real Ratio of Expansion 750 Effect of Compression 751 clearance in Low and High Speed Engines 751 Cylinder- condensation 752 Water-consumption of Automatic Cut-off Engines 753 Experiments on Cylinder-condensation 753 Indicator Diagrams 754 Indicated Horse-power 755 Rules for Estimating Horse-power 755 Horse-power Constant "756 Errors of Indicators 756 Table of Engine Constants 756 To Draw Clearance on Indicator-diagram 759 To Draw Hyperbolic Curve on Indicator-diagram 759 Theoretical Water Consumption . . 700 Leakage of Steam 761 Compound Engines. Advantages of Compounding 762 Woolf and Receiver Types of Engines 762 Combined Diagrams 764 Proportions of Cylinders inCompound Engines 7(55 Receiver Space 766 Formula for Calculating Work of Steam 767 Calculation of Diameters of Cylinders 768 Triple-expansion Engines 769 Proportions of Cylinders 769 Annular Ring Method 769 Rule for Proportioning Cylinders . 771 Types of Three-stage Expansion Engines 771 Sequence of Cranks 772 Velocity of Steam Through Passages 772 Quadruple Expansion Engines 772 Diameters of Cylinders of Marine Engines 773 Progress in Steam-engines 773 A Double-tandem Triple-expansion Engine 773 Principal Engines, World's Columbian Exhibition, 1893 774 Steam Engine Economy. Economic Performance of Steam Engines — 775 Feed-water Consumption of Different Types 775 Sizes and Calculated Performances of Vertical High-speed Engines 777 Most Economical Point of Cut-off 777 Type of Engine Used when Exhaust-steam is used for Heating 780 Comparison of Compound and Single cylinder Engines 780 Two-cylinder and Three-cylinder Engines 781 Effect of Water in Steam on Efficiency . . 781 Relative Commercial Economy of Compound and Triple-expansion Engines 781 Triple-expansion Pumping-engines 782 Test of a Triple-expansion Engine with and without Jackets 783 Relative Economy of Engines under Variable Loads 783 Efficiency of Non-condensing Compound Engines 784 Economy of Engines under Varying Loads 784 Steam Consumption of Various Sizes 785 Steam Consumption in Small Engines 786 Steam Consumption at Various Speeds 786 Limitation of Engine Speed 787 Influence of the Steam Jacket 7S7 Counterbalancing Engines 788 Preventing Vibrations of Engines 789 Foundations Embedded in Air 789 Cost of Coal for Steam-power 789 COKTEKTS. Storing Steam Heat. ............. 789 Cost of Steam-power 790 Rotary Steam-engines. Steam Turbines 791 The Tower Spherical Engine 792 Dimensions of Parts of Engines. Cylinder , 792 Clearance of Piston 792 Thickness of Cylinder 792 Cylinder Heads 794 Cylinder-head Bolts 795 The Piston 795 Piston Packing-rings 796 Fit of Piston-rod .. : 790 Diameter of Piston-rods 797 Piston-rod Guides 798 The Connecting-rod 799 Connecting-rod Ends 800 Tapered Connecting-rods » 801 The Crank-pin 801 Crosshead-pin or Wrist-pin 804 The Crank-arm 805 The Shaft, Twisting Resistance ... . 806 Resistance to Bending 808 Equivalent Twisting Moment 808 Fly-wheel Shafts 809 Length of Shaft-bearings 810 Crank-shafts with Centre-crank and Double-crank Arms 813 Crank-shaft with two Cranks Coupled at 90°.... 814 Valve-stem or Valve-rod 815 Size of Slot-link 815 The Eccentric 816 The Eccentric-rod 816 Reversing-gear ., 816 Engine-frames or Bed-plates , 817 Fly-wheels. Weight of Fly-wheels 817 Centrifugal Force in Fly-wheels 820 Arms of Fly-wheels and Pulleys 820 Diameters for Various Speeds 821 Strains in the Rims 822 Thickness of Rims S23 A Wooden Rim Fly-wheel 824 Wire-wound Fly-wheels 824 The Slide-valve. Definitions, Lap, Lead, etc 824 Sweet's Valve-diagram ,. 826 The Zeuner Valve-diagram 827 Port Opening 828 Lead 829 Inside Lead 829 Ratio of Lap and of Port-opening to Valve-travel 829 Crank Angles for Connecting-rods of Different Lengths 830 Relative Motions of Crosshead and Crank 831 Periods of Admission or Cut-off for Various Laps and Travels 831 Diagram for Port-opening, Cut-off, and Lap 832 • Piston-valves 834 Setting the Valves of an Engine 834 To put an Engine on its Centre 834 Link-motion 834 Governors. Pendulum or Fly-ball Governors 836 To Change the Speed of an Engine 837 XXIV CONTENTS. PAGE Fly-wheel or Shaft-governors 83S Calculation of Springs for Shaft-governors 838 Condensers, Air-pumps, Circulating-pumps, etc. The Jet Condenser 839 Ejector Condensers 840 The Surface Condenser . 8-10 Condenser Tubes 840 Tube-plates 841 Spacing of Tubes 841 Quantity of Cooling Water 841 Air-pump 841 Area through Valve-seats 842 Circulating-pump 843 Feed-pumps for Marine-engines 843 An Evaporative Surface Condenser 844 Continuous Use of Condensing Water 844 Increase of Power by Condensers 846 Evaporators and Distillers 847 GAS, PETROLEUM, AND HOT-AIR ENGINES. Gas-engines 847 Efficiency of the Gas-engine 848 Tests of the Simplex Gas Engine 848 A 320-H.P. Gas-engine , 848 Test of an Otto Gas-engine 849 Temperatures and Pressures Developed 849 Test of the Clerk Gas-engine 849 Combustion of the Gas in the Otto Engine 849 Use of Carburetted Air in Gas-engines 849 The Otto Gasoline-engine 850 The Priest-man Petroleum-engine 850 Test of a 5-H.P. Priestman Petroleum-engine 850 Naptha-engines 851 Hot-air or Caloric-engines : 851 Test of a Hot-air Engine 851 LOCOMOTIVES. Efficiency of Locomotives and Resistance of Trains 851 Inertia and Resistance at Increasing Speeds 853 Efficiency of the Mechanism of a Locomotive 854 Size of Locomotive Cylinders 854 Size of Locomotive Boilers 855 Qualities Essential for a Free-steaming Locomotive 855 Wootten's Locomotive 855 Grate-surface, Smoke-stacks, and Exhaust-nozzles for Locomotives 855 Exhaust Nozzles 856 Fire-brick Arches 856 Size, Weight, Tractive Power, etc 856 Leading American Types 858 Steam Distribution for High Speed 858 Speed of Railway Train's 859 Dimensions of Some American Locomotives 859-862 Indicated Water Consumption 862 Locomotive Testing Apparatus 863 Waste of Fuel in Locomotives 863 Advantages of Compounding 863 Counterbalancing Locomotives 864 Maximum Safe Load on Steel Rails . . 865 .Narrow-guage Railways 865 Petroleum-burning Locomotives 865 Fireless Locomotives 866 SHAFTING. Diameters Resist Torsional Strain 867 Deflection of Shafting 868 Horse-power Transmitted by Shafting 869 Table fur Laying Out Shafting 871 CONTENTS. XXV PULLEYS. PAGE Proportions of Pulleys 873 Convexity of Pulleys 874 Cone or Step Pulleys 874 BELTING. Theory of Belts and Bands 876 Centrifugal Tension 876 Belting Practice, Formulae for Belting 877 Horse-power of a Belt one inch wide 878 A. F. Nagle's Formula 878 Width of Belt for Given Horse-power , . . . , . . . 879 Taylor's Rules for Belting. . . 880 Notes on Belting .■ 882 Lacing of Belts 883 Setting a Belt on Quarter-twist . . . . . 883 To Find the Length of Belt 884 To Find the Angle of the Arc of Contact 884 To Find the Length of Belt when Closely Rolled 884 To Find the Approximate Weight of Belts 884 Relations of the Size and Speeds of Driving and Driven Pulleys 884 Evils of Tight Belts 885 Sag of Belts 885 Arrangements of Belts and Pulleys 885 Care of Belts , 886 Strength of Belting 886 Adhesion, Independent of Diameter. 886 Endless Belts 886 Belt Data 886 Belt Dressing 887 Cement for Cloth or Leather : 887 Rubber Belting 887 GEARING. Pitch, Pitch-circle, etc 887 Diametral and Circular Pitch 888 Chordal Pitch 889 Diameter of Pitch-line of Wheels from 10 to 100 Teeth 889 Proportions of Teeth 889 Proportion of Gear-wheels 891 Width of Teeth 891 Rules for Calculating the Speed of Gears and Pulleys 891 Milling Cutters for Interchangeable Gears 892 Forms of the Teeth. The Cycloidal Tooth 892 The Involute Tooth 894 Approximation by Circular Arcs .896 Stepped Gears 897 Twisted Teeth 897 Spiral Gears 897 Worm Gearing 897 Teeth of Bevel-wheels ... 898 Annular and Differential Gearing 898 Efficiency of Gearing 899 Strength of Gear Teeth. Various Formulae for Strength 900 Comparison ot Formulae — , 903 Maximum Speed of Gearing „„. , 905 A Heavy Machine-cut Spur-gear 905 Frictional Gearing 905 Frictional Grooved Gearing 906 HOISTING. Weight and Strength of Cordage 906 Working Strength of Blocks 900 CONTENTS. Efficiency of Chain-blocks 907 Proportions of Hooks 907 Power of Hoisting Engines 908 Effect of Slack Pope on Strain in Hoisting 908 Limit of Depth for Hoisting 908 Large Hoisting Records 908 Pneumatic Hoisting 909 Counterbalancing of Winding-engines 909 Belt Conveyors 911 Bands for Carrying Grain 911 Cranes. Classification of Cranes 911 Position of the Inclined Brace in a Jib Crane '. 912 A Large Travelling-crane 912 A 150-ton Pillar Crane 912 Compressed-air Travelling Cranes 912 Wire-rope Haulage. Self-acting Inclined Plane 913 Simple Engine Plane 913 Tail-rope System 913 Endless Rope System 914 Wire-rope Tramways 914 Suspension Cableways and Cable Hoists 915 Stress in Hoisting-ropes on Inclined Planes 915 Tension Required to Prevent Wire Slipping on Drums 916 Taper Ropes of Uniform Tensile Strength 916 Effect of Various Sized Drums on the Life of Wire Ropes 917 WIRE-ROPE TRANSMISSION. The Driving Wheels 918 Horse-power of Wire-rope Transmission 919 Durability of Wire Ropes 919 Inclined Transmissions 919 The Wire-rope Catenary 919 Diameter and Weight of Pulleys for Wire-rope 921 Table of Transmission of Power by Wire Ropes 921 Long-distance Transmissions 921 ROPE DRIVING. Formulae for Rope Driving 922 Horse-power of Transmission at Various Speeds 924 Sag of the Rope Between Pulleys 925 Tension on the Slack Part of the Rope 925 Miscellaneous Notes on Rope-driving 926 FRICTION AND LUBRICATION. Coefficient of Friction 928 Rolling Friction 988 Friction of Solids 928 Friction of Rest 928 Laws of Unlubricated Friction 928 Friction of Sliding Steel Tires 928 Coefficient of Polling Friction 929 Laws of Fluid Friction 929 Angles of Repose 929 Friction of Motion 929 Coefficient of Friction of Journal 930 Experiments on Friction of a Journal 931 ( ioefficients of Friction of Journal with Oil Bath 932 ( ^efficients of Friction of Motion and of Rest 932 Value of Anti-friction Metals 932 Cast-iron for Bearings 938 Friction of Metal Under Steam-pressure 933 Moi in's Laws of Friction , 933 CONTENTS. XXVll PAGE Laws of Friction of well-lubricated Journals 934 Allowable Pressures on Bearing-surface 935 Oil-pressure in a Bearing 937 Friction of Car-journal Brasses 937 Experiments on Overheating of Bearings 938 Moment of Friction and Work of Friction 938 Pivot Bearings 939 The Schiele Curve 939 Friction of a Flat Pivot-bearing 939 Mercury-bath Pivot 940 Ball Bearings 940 Friction Rollers 940 Bearings for Very High Rotative Speed 941 Friction of Steam-engines.. 941 Distribution of the Friction of Engines 941 Lubrication. Durability of Lubricants 942 Qualifications of Lubricants 943 Amount of Oil to run an Engine :•,, 943 Examination of Oils 944 Penna. R. R. Specifications 944 Solid Lubricants 945 Graphite, Soapstone, Metaline 945 THE FOUNDRY. Cupola Practice 946 Charging a Cupola 948 / Charges in Stove Foundries 949 / Results of Increased Driving 949 Pressure Blowers 950 j Loss of Iron in Melting 950 Use of Softeners 950 Shrinkage of Castings 951 Weight of Castings from Weight of Pattern 952 Moulding Sand 952 Foundry Ladles 952 \ THE MACHINE SHOP. Speed of Cutting Tools 953 Table of Cutting Speeds 954 Speed of Turret Lathes 954 Forms of Cutting Tools 955 Rule for Gearing Lathes , 955 Change-gears for Lathes 956 Metric Screw-threads. . 956 Setting the Taper in a Lathe 956 Speed of Drilling Holes 956 Speed of Twist-drills 957 Milling Cutters 957 Speed of Cutters . .. 95S Results with Milling-machines 959 Milling with or Against Feed l 900 Milling-machine vs. Planer 960 Power Required for Machine Tools 960 Heavy Work on a Planer 960 Horse-power to run Lathes 961 Power used by Machine Tools 963 Power Required to Drive Machinery 964 Power used in Machine-shops 965 Abrasive Processes. The Cold Saw 966 Reese's Fusing-disk 966 Cutting Stone with Wire. 966 The Sand-blast 966 Emery-wheels 967-969 Grindstones 968-970 XXVI 11 CONTENTS. Various Tools and Processes. PAGE Taps for Machine-screws 970 Tap Drills 971 Taper Bolts, Pins, Reamers, etc 972 Punches, Dies, Presses 972 Clearance Between Punch and Die 972 Size of Blanks for Drawing-press 973 Pressure of Drop-press 973 Flow of Metals 973 Forcing and Shrinking Fits 973 Efficiency of Screws 974 Powell's Screw-thread 975 Proportioning Parts of Machine 975 Keys for Gearing, etc 975 Holding-power of Set-screws 977 Holding-power of Keys.. 978 DYNAMOMETERS. Traction Dynamometers 978 The Prony Brake. 978 The Alden Dynamometer 979 Capacity of Friction-brakes 980 Transmission Dynamometers 980 ICE MAKING OR REFRIGERATING MACHINES. Operations of a Refrigerator-machine 981 Pressures, etc., of Available Liquids 982 Ice-melting Effect 983 Ether-machines 983 Air-machines. 983 Ammonia Compression-machines 983 Ammonia Absorption-machines 984 Sulphur-dioxide Machines. 985 Performance of Ammonia Compression-machines 986 Economy of Ammonia Compression-machines 987 Machines Using Vapor of Water 988 Efficiency of a Refrigerating machine 988 Test Trials of Ref rigerating-machines 990 Temperature Range. , 991 Metering the Ammonia 992 Properties of Sulphur Dioxide and Ammonia Gas 992 Properties of Brine used to absorb Refrigerating Effect 994 Chloride-of-calciurn Solution 994 Actual Performances of Refrigerating Machines. Performance of a 75-ton Refrigerating-machine. 994, 998 Cylinder-heating 997 Tests of Ammonia Absorption-machine 997 Ammonia Compression-machine, Results of Tests 999 Means for Applying the Cold 999 Artificial Ice-manufacture. Test of the New York Hygeia Ice-making Plant 1000 MARINE ENGINEERING. Rules for Measuring Dimensions and Obtaining Tonnage of Vessels 1001 The Displacement of a Vessel 1001 Coefficient of Fineness 1002 Coefficient of Water-lines . . 1002 Resistance of Ships 1002 Coefficient of Performance of Vessels 1003 Defects of the Common Formula for Resistance. . . 1003 Rankine's Formula 1003 Dr. Kirk's Method 1004 To find the I. H. P. from the Wetted Surface 1005 E. R. Mumford's Method 100G Relative Horse-power required for different Speeds of Vessels 1000 CONTENTS. XXIX PAGE Resistance per Horse-power for different Speeds 100(5 Results of Trials of Steam-vessels of Various Sizes 1007 Speed on Canals, 1008 Results of Progressive Speed-trials in Typical Vessels 1008 Estimated Displacement, Horse-power, etc., of Steam-vessels of Various Sizes 1009 The Screw-propeller. Size of Screw 1010 Propeller Coefficients 1011 Efficiency of the Propeller 1012 Pitch-ratio and Slip for Screws of Standard Form 1012 Results of Recent Researches 1013 The Paddle-wheel. Paddle-wheel with Radial Floats , 1013 Feathering Paddle-wheels 1013 Efficiency of Paddle-wheels, 1014 Jet-propulsion. Reaction of a Jet 1015 Recent Practice in Marine Engines. Forced Draught 1015 Boilers.. 1015 Piston-valves 1016 Steam-pipes 1016 Auxiliary Supply of Fresh-water Evaporators 1016 Weir's Feed-water Heater 1016 Passenger Steamers fitted with Twin-screws 1017 Comparative Results of Working of Marine-engine, 1872, 1881, and 1891.. 1017 Weight of Three-stage Expansion-engines 1017 Particulars of Three-stage Expansion-engines 1018 CONSTRUCTION OF BUILDINGS Walls of Warehouses, Stores, Factories, and Stables 1019 Strength of Floors, Roofs, and Supports 1019 Columns and Posts 1019,1022 Fireproof Buildings , 1020 Iron and Steel Columns , 1020 Lintels. Bearings, and Supports 1020 Strains on Girders and Rivets 1020 Maximum Load on Floors 1021 Strength of Floors 1021 Safe Distributed Loads on Southern-pine Beams 1023 ELECTRICAL ENGINEERING. Standards of Measurement. C. G. S. System of Physical Measurement 1024 Practical Units used in Electrical Calculations 1024 Relations of Various Units 1025 Equivalent Electrical and Mechanical Units 1026 Analogies between Flow of Water and Electricity 1027 Analogy between the Ampere and Miner's Inch 1027 Electrical Resistance. Laws of Electrical Resistance 1028 Equivalent Conductors . 1028 Electrical Conductivity of Different Metals and Alloys 1028 Relative Conductivity of Different Metals 1029 Conductors and Insulators 1029 Resistance Varies with Temperature 1029 Annealing 1029 Standard of Resistance of Copper Wire 1030 Electric Currents. Ohm's Law 1030 Divided Circuits . . . io31 CONTENTS. Conductors in Series 10; Internal Resistance 10: Joint Resistance of Two Branches 10: KirchhofTs Laws ....... 10: Power of the Circuit 10: Heat Generated by a Current 10: Heating of Conductors 10.' Heating of Wires of Cables 10' Copper-wire Table 1034, 10, Heating of Coils 10; Fusion of Wires 10J Electric Transmission. Section of Wire required for a Given Current 10, c Constant Pressure 101: Three-wire Feeder 10.- Short-circuiting 10> Economy of Electric Transmission 10; Table of Electrical Horse-powers 10-) Wiring Formulae for Incandescent Lighting 104 Wire Table for 100 and 500 Volt Circuits 104 Cost of Copper for Long-distance Transmission 104 Graphical Method of Calculating Leads '. 104 Weight of Copper for Long-distance Transmission 104 Efficiency of Long-distance Transmission 104 Efficiency of a Combined Engine and Dynamo 104 Electrical Efficiency of a Generator and Motor 104 Efficiency of an Electrical Pumping Plant 104 Electric Railways. Test of a Street Railway Plant 105 Proportioning Boiler, Engine, and Generator for Power Stations 105 Electric Lighting. Quantity of Energy Required to Produce Light 105 Life of Incandescent Lamps 105 Life and Efficienc}- Tests of Lamps - 105 Street Lighting 105.' Lighting-power of Arc-lamps 105: Candle-power of the Arc light 105'. Electric Welding 105: Electric Heaters 105< Electric Accumulators or Storage-batteries. Use of Storage-batteries in Power and Light Stations. 105( Working Current of a Storage-cell — 105G Electro-chemical Equivalents 1051 Electrolysis 1051 Electro-magnets. Units of Electro-magnetic Measurement 105? Lines of Loops of Force 10.".'. Strength of an Electro-magnet 105J1 Force in the Gap between Two Poles of a Magnet 106C The Magnetic Circuit 1060 Determining the Polarity of Electro-magnets 1000 Dynamo-Electric Machines. Kinds of Dynamo-electric Machines as regards Manner of Winding.. . . 1061 ( !urrent ( ie'nerated by a Dynamo-electric Machine 1061 Torque of an Armature 1062 Electro-motive Force of the Armature Circuit 100:2 Si rength of the Magnetic Field 1063 Application to Designing of Dynamos 1064 Permeability '066 Permissible Amperage for Magnets with Cotton-covered Wire 10(16, HK>K Formula- <>f Efficiency of Dynamos 1060 The Electric Motor 10™ NAMES AND ABBREVIATIONS OF PERIODICALS AND TEXT-BOOKS FREQUENTLY REFERRED TO IN THIS WORK. Am. Mach. American Machinist. Bull. I. & S. A. Bulletin of the American Iron and Steel Association (Philadelphia). Burr's Elasticity and Resistance of Materials. Clark, R, T. D. D. K. Clark's Rules, Tables, and Data for Mechanical En- gineers. Clark, S. E. D. K. Clark's Treatise on the Steam-engine. Engg. Engineering (London). Eng. News. Engineering News. Engr. The Engineer (London). Fairbairn's Useful Information for Engineers. Flynn's Irrigation Canals and Flow of Water. Jour. A. C. I. W. Journal of American Charcoal Iron Workers' Association. Jour. F. I. Journal of the Franklin Institute. Kapp's Electric Transmission of Energy. Merriman's Strength of Materials. Lanza's Applied Mechanics. Proc. Inst. C. E. Proceedings Institution of Civil Engineers (London). Proc. Inst. M. E. Proceedings Institution of Mechanical Engineers (Lon- don). Peabody's Thermodynamics, Proceedings Engineers' Club of Philadelphia. Rankine, S. E. Rankine'sThe Steam Engine and other Prime Movers. Rankine's Machinery and Millwork. Rankine, R. T. D. Rankine's Rules, Tables, and Data. Reports of U. S. Test Board. Reports of U. S. Testing Machine at Watertown, Massachusetts. Rontgen's Thermodynamics. Seaton's Manual of Marine Engineering. Hamilton Smith, Jr.'s Hydraulics. The Stevens Indicator. Thompson's Dynamo-electric Machinery. Thurston's Manual of the Steam Engine. Thurston's Materials of Engineering. Trans. A. I. E. E. Transactions American Institute of Electrical Engineers. Trans. A. I. M. E. Transactions American Institute of Mining Engineers. Trans. A. S. C E. Transactions American Society of Civil Engineers. Trans. A. S. M. E. Transactions American Soc'ty of Mechanical Engineers . Trautwine's Civil Engineer's Pocket Book. The Locomotive (Hartford, Connecticut). Unwin's Elements of Machine Design. Weisbach's Mechanics of Engineering. Wood's Resistance of Materials, Wood's Thermodynamics. xxxi MATHEMATICS. Arithmetical and Algebraical Signs and Abbreviations. Z. angle. L right angle. _L perpendicular to. sin., sine. cos., cosine. tang., or tan., tangent. sec, secant. versin., versed sine. cot., cotangent. cosec, cosecant. covers., co- versed sine. In Algebra, the first letters of the ^alphabet, a, b, c, d, etc., are gener- ally used to denote known quantities, and the last letters, iv, x, y, z, etc., unknown quantities. Abbreviations and Symbols com- monly used. d, differential (in calculus). /, integral (in calculus). / , integral between limits a and 6. A, delta, difference. 2. sigma, sign of summation. >7r, pi, ratio of circumference of circle to diameter = 3. 14159. g, acceleration due to gravity = 32.16 ft. per sec. Abbreviations frequently used in this Book. L., 1., length in feet and inches. B., b., breadth in feet and inches. D., d., depth or diameter. H., h., height, feet and inches. T., t., thickness or temperature. V., v., velocity. F., force, or factor of safety, f., coefficient of friction. E., coefficient of elasticity. R., r., radius. W., w., weight. P., p., pressure or load. H.P., horse-power. I.H.P., indicated horse-power. B.H.P., brake horse-power. h. p., high pressure, i. p., intermediate pressure. 1. p., low pressure. A.W. G., American Wire Gauge (Brown & Sharpe). B.W.G., Birmingham Wire Gauge. r. p. m., or revs, per miu., revolutions per minute. ~r plus (addition). + positive. — minus (subtraction). — negative. ± plus or minus. T minus or plus. = equals. X multiplied by. al or a.b = a x 6. -5- divided by. / divided by. b or a-b = a/b = a -s- b. 2 2 .2 = — : .002= — -. 10' 1000 V square root. V cube root. V 4th root. is to, :: so is, : to (proportion). 2 : 4 :: 3 : 6, as 2 is to 4 so is 3 to 6. ratio; divided by. 2 : 4, ratio of 2 to 4 = 2/4. therefore. > greater than. < less than. D square . round. ° degrees, arc or thermometer. ' minutes or feet. " seconds or inches. /// "' accents to distinguish letters, as a', a", a'". a l « 2 - <*3. «?,' a c- rea( i a SUD 1> « sub b, etc. ° C ) [ ] \ vincula, denoting that the numbers enclosed are to be taken together ; as, (a + b)c = 4 + 3 x 5 = 35. a 2 a 3 , a squared, a cubed. u n a raised to the nth power. al = v<**i a * = v~" 3 - a-i = -,a- 2 = ~ a a 2 10 9 = 10 to the 9th power = 1,000,000,- 000. sin. a = the sine of a. sin.— 1 a = the arc whose sine is a. 1 sin. a- 1 = — sin. a. log. = logarithm. log. or hyp. log. = hyperbolic loga- rithm. MATHEMATICS. ARITHMETIC. The user of this book is supposed to have had a training in arithmetic as well as in elementary algebra. Only those rules are given here which are apt to be easily forgotten. GREATEST COMMON MEASURE, OR GREATEST COMMON DIVISOR OF TWO NUMBERS. Rule.- Divide the greater number by the less ; then divide the divisor by the remainder, and so on, dividing always the last divisor by the last remainder, until there is no remainder, and the last divisor is the greatest common measure required. LEAST COMMON MUL.TIFUE OF TWO OR MORE NUMBERS. Rule.— Divide the given numbers by any number that will divide the greatest number of them without a remainder, and set the quotients with the undivided numbers in a line beneath. Divide the second line as before, and so on, until there are no two numbers that can be divided ; then the continued product of the divisors and last quotients will give the multiple required. FRACTIONS. To reduce a common fraction to its lowest terms.— Divide both terms by their greatest common divisor: j| = f To change an improper fraction to a mixed number.— Divide the numerator by the denominator; the quotient is the whole number, and the remainder placed over the denominator is the fraction: - 3 5 9 = 9|. To change a mixed number to an improper fraction.— Multiply the whole number by the denominator of the fraction; to the prod- uct add the numerator; place the sum over the denominator: ]| = % 5 '. To express a whole number in the form of a fraction with a given denominator.— Multiply the whole number by the given denominator, and place the product over that denominator: 13 = 5 3 9 . To reduce a compound to a simple fraction, also to multiply fractions.— Multiply the numerators together for a new numerator and the denominators together for a new denominator: 2 c 4 8 . 2 4 8- § of- = -, also § Xg = - g . To reduce a complex to a simple fraction,— The numerator and denominator must each first be given the form of a simple fraction; then multiply the numerator of the upper fraction by the denominator of the lower for the new numerator, and the denominator of the upper by the numerator of the lower for the new denominator: 1 = I = 1 - 1 H | 12" 2' To divide fractions.— Reduce both to the form of simple fractions, invert the divisor, and proceed as in multiplication: 3 ' 3 3 ' 3 3 4 12* Cancellation of fractions.— In compound or multiplied fractions, divide any numerator and any denominator by any number which will divide them both without remainder, striking out the numbers thus divided and setting down the quotients in their stead. To reduce fractions to a common denominator.— Reduce each traction to the form of a simple ft action; then multiply each numera- DECIMALS. 6 tor by all the denominators except its own for the new numerators, and all the denominators together for the common denominator: 1 1 3 _ 21 14 '18 2 1 3' 7~ 42' 42' 42' To add fractions.— Reduce them to a common denominator, then add the numerators and place their sum over the common denominator: 3 - 21 + 14 + 18 _ 53 _ 42 ~~ 42 " To subtract fractions. — Reduce them to a common denominator, subtract the numerators and place the difference over the common denomi- nator: 1 _ 3 _ 7-6 _ J_ 2 7 ~ 14 14" DECIMALS. To add decimals.— Set down the figures so that the decimal points are one above the other, then proceed as in simple addition: 18.75 -f- .012 = 18.762. To subtract decimals.— Set down the figures so that the decimal points are one above the other, then proceed as in simple subtraction: 18.75 - .012 = 18.738. To multiply decimals.— Multiply as in multiplication of whole numbers, then point off as many decimal places as there are in multiplier and multiplicand taken together: 1.5 X .02 = .030 = .03. To divide decimals.— Divide as in whole numbers, and point off in the quotient as many decimal places as those in the dividend exceed those in the divisor. Ciphers must be added to the dividend to make its decimal places at least equal those in the divisor, and as many more as it is desired to have in the quotient: 1.5 -=- .25 = 6. 0.1 -=- 0.3 = 0.10000 -*- 0.3 = 0.3333 + Decimal Equivalents of Fractions of One Incb. 1-64 1-32 3-64 1-16 3-32 7-64 1-8 9-64 5-32 11-64 3-16 13-64 7-32 15-64 1-4 015625 17-64 03125 9-32 046875 19-64 0625 £-16 078125 21-64 09375 11-32 109375 23-64 .125 3-8 .140625 2.i-64 .15625 13-32 .171875 27-64 .1875 7-16 .203125 29-64 .21875 15-32 .234375 31-64 .25 1-2 .265625 33-64 .28125 17-32 .296875 35-64 .3125 9-16 .328125 37-64 .34375 19-32 .359375 39-64 .375 5-8 .390625 41-64 .40625 21-32 .421875 43-64 .4375 11-16 .453125 45-64 .46875 23-32 .484375 47-64 .50 3-4 .515625 49-64 .53125 25-32 .546875 51-64 .5625 13-16 .578125 53-64 .59375 27-32 .609375 55-64 625 7-8 .640625 57-64 .65625 29-32 .671875 59-64 .6875 15-16 .703125 61-64 .71875 31-32 .734375 63-64 .75 1 .796875 .8125 .828125 .84375 .859375 .875 .890625 .90625 .921875 .9375 .953125 .96875 .984375 To convert a common fraction into a decimal.— Divide the numerator by the denominator, adding to the numerator as many ciphers prefixed by a decimal point as are n^c^ssary to give the number of decimal places desired in the result: % = 1.00U0h-3 = 0.3333 -f. To convert a decimal into a common fraction.— Set down the decimal as a numerator, and place as the denominator 1 with as many ciphers annexed as there are decimal places in the numerator; erase the ARITHMETIC. t— 1 ! © i T+H X' Hob l " a! CO echo © L ^ °R 3 CO ro 8 coN & © '-2 g H» >o IK 5! 00 Hrl "^ (O © $ co © C5 © io|qo o? 2 « S IO © o!<° 55 $ H ^ t io in R CO CO 3 £ QD © 8 HjM CM co so ^. to Tt $ ^ T* ,_, in m w 3 H^ c? CJ CO CO - • "*! "W © ^ in m ^ ^ M]CO ^ o; « *! CO © CO 00 CO J 50 t- ?' SR (8 00 ■<* S 8 $ « •°|H w ,_, ^ m ... nt -f s 1-4^ o © O Si c* w S lO r> ,_, ejw IO © © ■M ^ w © K g ft; o o © © © r* - ,_. ,_, -* Hw o o o © © © o • o o ■m no ,_, ns ,_ ft g *> o © o o © © • 8 S 8 § W © j: © IS © 52 s IN O £ © t-H © t-h n O* © © CO © © *- o ^ h» «£ h« -* CCJQO * He* -H lOfr> -(•-- *** H« rH COMPOUND NUMBERS. decimal poiut in the numerator, and reduce the fraction thus formed to its lowest terms: M 25 1 ooco 3333 1 ■ 25 = 100 = 4 ; - 3333 = 1000() = 3' nearly - To reduce a recurring decimal to a common fraction.— Subtract the decimal figures that do not recur from the whole decimal in- cluding one set of recurring figures; set down the remainder as the numer- ator of the fraction, and as many nines as there are recurring figures, fol- lowed by as many ciphers as there are non-recurring figures, in the denom- inator. Thus: .79054054, the recurring figures being 054. Subtract 79 78975 . . . t ... . . ,117 n = (reduced to its lowest terms) — . COMPOUND OR DENOMINATE NUMBERS. Reduction descending - .— To reduce a compound number to a lower denomination. Multiply the number by as many units of the lower denomi- nation as makes one of the higher. 3 yards to inches: 3 X 36 = 108 inches. .04 square feet to square inches: .04 X 144 = 5.76 sq. in. If the given number is in more than one denomination proceed in steps from the highest denomination to the next lower, and so on to the lowest, adding in the units of each denomination as the oper .tion proceeds. 3 yds. 1 ft. 7 in. to inches: 3x3 = 9, + 1 = 10, 10 X 12 = 120, -f 7 = 127 in. Reduction ascending.— To express a number of a lower denomi- nation in terms of a higher, divide the number by the numb r of units of the lower denomination contained in one of the next higher; the quotient is in the higher denomination, and the remainder, if any, in the lower. 127 inches to higher denomination. 127-5-12 = 10 feet + 7 inches; 10 feet -=-3 = 3 yards + 1 foot. Ans. 3 yds. 1 ft. 7 in. To express the result in decimals of the higher denomination, divide the given number by the number of units of the given denomination contained in one of the required denomination, carrying the result to as many places of decimals as may be desired. 127 inches to yards: 127 -s- 36 = 3J| = 3.5277 + yards. RATIO AND PROPORTION. Ratio is the relation of one number to another, as obtained by dividing one by the other. Ratio of 2 to 4, or 2 : 4 = 2/4 = 1/2. Ratio of 4 to 2, or 4 : 2 = 2. Proportion is the equality of two ratios. Ratio of 2 to 4 equals ratio of 3 to 6, 2/4 = 3/6: expressed thus, 2 : 4 : : 3 : 6; read, 2 is to 4 as 3 is to 6. The first and fourth terms are called the extremes or outer terms, the second and third the means or inner terms. The product of the means equals the product of the extremes: 2 : 4 : : 3 : 6; 2 X 6 = 12; 3 X 4 = 12. Hence, given the first three terms to find the fourth, multiply the second and third terms together and divide by the first. 4X3 2 : 4 : : 3 : what number ? Ans. — - — = 6. ARITHMETIC. Algebraic expression of proportion.— a : b : : c : d; r = -,,ad . be, be , ad ad = be; from which a = — : d= — ; b = — ; c = -=—. x 8 = 4. Single Rule of Three ; or, finding the fourth term of a proportion when three terms are given.— Rule, as above, when the terms are stated in their proper order, multiply the second by the third and divide by the first. The difficulty is to state the terms in their proper order. The term which is of the same kind as the required or fourth term is made the third : the first and second must be like each other in kind and denomination. To deter-, mine which is to be made second and which first requires a little reasoning. If an inspection of the problem shows that the answer should be greater than the third term, then the greater of the other two given terms should be made the second term — otherwise the first. Thus, 3 men remove 54 cubic feet of rock in a day; how many men will remove in the same time 10 cubic yards ? The answer is to be men— make men third term; the answer is to be more than three men, therefore make the greater quantity, 10 cubic yards, the second term ; but as it is not the same denomination as the other term it must be reduced, = 270 cubic feet. The proportion is then stated: 54 : 270 : : 3 : x (the required number) ; 'A The problem is more complicated if we increase the number of given terms. Thus, in the above question, substitute for the words " in the same time " the words " iu 3 days." First solve it as above, as if the work were to be done in the same time; then make another proportion, stating it thus: If 15 men do it in the same time, it will take fewer men to do it in 3 days; make 1 day the 2d term and 3 days the first term. 3 : 1 : : 15 men : 5 men. Compound Proportion, or Double Rule of Three.— By this rule are solved questions like the one just given, in which two or more stat- ings are required by the single rule of three. In it as in the single rule, there is one third term, which is of the same kind and denomination as the fourth or required term, but there may be two or more first and second terms. Set down the third term, take each pair of terms of the same kind separately, and arrange them as first and second by the same reasoning as is adopted in the single rule of three, making the greater of the pair the second if this pair considered alone should require the answer to be greater. Set down all the first terms one under the other, and likewise all the second terms. Multiply all the first terms together and all the second terms together. Multiply the product of all the second terms by the third term, and divide this product by the product of all the first terms. Example: If 3 men remove 4 cubic yards in one day, working 12 hours a day, how many men working 10 hours a day will remove 20 cubic yards in 3 days ? Yards 4 : 201 Davs 3 : 1 : : 3 men. Hours 10 : 12 | Products 120 : 240 : : 3 : 6 men. Ans. To abbreviate by cancellation, any one of the first terms may cancel either the third or any of the second terms; thus. 3 in first cancels 3 in third, making it 1, 10 cancels into 20 making the latter 2, which into 4 makes it 2, which into 12 makes it 0, and the figures remaining are only 1 : 6 : : 1 : 6. INVOLUTION, OR POWERS OF NUMBERS. Involution is the continued multiplication of a number by itself a given number of times. The number is called the root, or first power, and the products are called powers. The second power is called the square and POWERS OF LUMBERS. the third power the cube. The operation may be indicated without being performed by writing a small figure called the index or exponent to the right of and a little above the root; thus, 3 3 = cube of 3, = 27. To multiply two or more powers of the same number, add their exponents; thus, 22 X 2 3 ' = 2 5 , or 4 X 8 = 32 = 2 5 . To divide two powers of the same number, subtract their exponents; thus, -22 = 2! =2; 22-4-2" = i ~ 2* ' - -. The exponent may thus be nega- tive. 2 3 ■+- 2 3 = 2° = 1, whence the zero power of any number = 1. The first power of a number is the number itself. The exponent may be frac- tional, as 2*, 2s, which means that the root is to be raised to a power whose exponent is the numerator of the fraction, and the root whose sign is the denominator is to be extracted (see Evolution). The exponent may be a decimal, as 2 ' 5 , 2 1-5 ; read, two to the five-tenths power, two to the one and five-tenths power. These powers are solved by means of Logarithms (which see,). First Nine Powers of the First Nine Numbers. 1st 2d 3d 4th 5th 6th 7th 8th 9th Pow'r Pow'r Power. Power. Power. Power. Power. Power. Power. 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 2187 6561 19683 4 16 64 256 1024 4096 16384 65536 262144 5 25 125 625 3125 15625 78125 390625 1953125 6 36 216 1296 7776 46656 279936 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 134217728 9 81 729 6551 59049 531441 4782969 43046721 387420489 The First Forty Powers of 2. © 1 3 © 6 3 J 33 > Ph > P- > Ph 1 9 512 18 262144 27 2 10 1024 19 524288 28 4 11 2048 20 1048576 29 8 12 4096 21 2097152 30 16 13 8192 22 4194304 31 32 14 16384 23 8388608 32 64 15 32768 24 16777216 33 128 IS 65536 25 33554432 34 256 17 131072 26 67108864 35 ©* 3 © is o £* Ph 134217728 36 268435456 37 536870912 38 1073741824 39 2147483048 40 4294967296 8589934592 17179869184 34350738368 68719476736 137438953472 549755813888 1099511627776 EVOLUTION. Evolution is the finding of the root (or extracting the root) of any number the power of which is given. The sign \/ indicates that the square root is to be extracted: \' V V^ the cube root, 4th root, nth root. A. fractional exponent with 1 for the numerator of the fraction is also used to indicate that the operation of extracting the root is to be performed ; thus, 2*, 2^= V2,.V2. When the power of a number is indicated, the involution not bein^ per- formed, the extraction of any root of that power may also be indicated by 8 ARITHMETIC. dividing the index of the power by the index of the root, indicating the division by a fraction. Thus, extract the square root of the 6th power of 2: 4/06 = 2* = 2 ? = 2 U = 8. The Gth power of 2, as in the table above, is G4 ; |/64 = 8. Difficult problems in evolution are performed by logarithms, but the square root and the cube root may be extracted directly according to the rules given below. The 4th root is the square root of the square root. The Gth root is the cube root of the square root, or the square root of the cube root ; l he 9th root is the cube root of the cube root ; etc. To Extract the Square Root.— Point off the given number into periods of two places each, beginning: with units. If there are decimals, point these off likewise, beginning at the decimal point, and supplying as many ciphers as may be needed. Find the greatest number whose square is less than the first left-hand period, and place it as the first figure in the quotient. Subtract its square from the left-hand period, and to the remainder annex the two figures of the second period for a dividend. Double the first figure of the quotient for a partial divisor; find how many times the latter is contained in the dividend exclusive of the right-hand figure, and set the figure representing that number of times as the second figure in the quotient, and annex it to the right of the partial divisor, forming the complete divisor. Multiply this divisor by the second figure in the quotient and subtract the product from the divi- dend. To the remainder bring down the next period and proceed as before, in each case doubling the figures in the root alread}* found to cbtain the trial divisor. Should the product of the second figure in the root by the completed divisor be greater than the dividend, erase the second figure both from the quotient and from the divisor, and substitute the next smaller figure, or one small enough to make the product of the second figure by the divisor less than or equal to the dividend. 3.1415926536|1.77245 + 271274 1 189 34712515 | 2429 3542 18692 7084 35444 ! 160865 1141776 fir To extract the square root of a fraction, extract the root of numerator /i 2 and denominator separately. A / - = -, or first convert the fraction into a imal 'f/|= V- decimal, {/ = i/.4444 + = .6G66 + . To Extract the Cube Root.— Point off the number into periods of 3 figures each, beginning at the right hand, or unit's place. Point off deci- mals in periods of 3 figures from the decimal point. Find the greatest cube that does not exceed the left-hand period ; write its root as the first figure in the required root Subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. Square the first figure of the root ; multiply bv 300, and divide the product into the dividend for a trial divisor ; write the quotient after the first figure Of I lie root as a trial second figure. ( lomplete the divisor by adding to 3C0 times the square of the first figure, 30 times the product of the first by the second figure, and the square ofthe second figure. Multiply this divisor by the second figure: subtract the product from the remainder. (Should the product be greater than the remainder, the last figure of the root and the complete divisor are too large ; CUBE ROOT. substitute for the last figure the next smaller number, and correct the trial divisor accordingly.) To the remainder bring down the next period, and proceed as before to find the third figure of the root— that is, square the two figures of the root already found; multiply by 300 for a trial divisor, etc. If at any time the trial divisor is less than the dividend, bring down an- other period of 3 figures, and place in the root and proceed. The cube root of a number will contain as many figures as there are periods of 3 in the number. Shorter Methods of Extracting the Cube Root.— 1. From Wentworth's Algebra: 1,881,365,963,625 |12345 22 881 72* 300 60 -I 364 [ 641 153365 = 43200 3 = 10801 3 2 = 91 44289 (■ 132867 1089J 20498963 123 2 — 45387001 123 x 4 = 14760 1 4* = 16] 4553476 }- 18213904 6J_ 2285059625 300 x 1234 2 = 456826 30 x 1234 x 5 = 185100 5 2 = 25 457011925 2285059625 After the first two figures of the root are found the next trial divisor is found by bringing down the sum of the 60 and 4 obtained in completing the preceding divisor , then adding the three lines connected by the brace, and annexing two ciphers. This method shortens the work in long examples, as is seen in the case of the last two trial divisors, saving the labor of squaring 123 and 1234. A further shortening of the work is made by obtaining the last two figures of the root by division, the divisor employed being three times the square of the part of the root already found; thus, after finding the first three figures: 3 x 1232 = 45387|20498963|45.1 + ~181548 234416 226935 74813 The error due to the remainder is not sufficient to change the fifth figure of the root. 2. By Prof. H. A. Wood (Stevens Indicator, July, 1890): I. Having separated the number into periods of three figures each, count- ing from the right, divide by the square of the nearest root of the first period, or first two periods ; the nearest root is the trial root. II. To the quotient obtained add twice the trial root, and divide by 3. This gives the root, or first approximation. III. By using the first approximate root as a new trial root, and proceed- ing as before, a nearer approximation is obtained, which process may be repeated until the root has been extracted, or the approximation carried as far as desired. 10 ARITHMETIC. Example.— Required the cube root of 20. The nearest cube to 20 is 3 3 2 = 9) 20.0 2.2 6_ 3)8.1 2.7 IstT. R. 2.7» = 7.29) 20.000 2.743 5.4 3)8.143 2.714, 1st ap. cube root. 2.7142 = 7.365796)2 0.0000000 2.7152534 5.428 3) 8.1432534 2.7144178 2d ap. cube root. Remark. — In the example it will be observed that the second term, or first two figures of the root, were obtained by using for trial root the root of the first period. Using, in like manner, these two terms for trial root, we obtained four terms of the root ; and these four terms for trial root gave seven figures of the root correct. In that example the last figure should be 7. Should we take these eight figures for trial root we should obtain at least fifteen figures of the root correct. To Extract a Higher Root than tlie Cube.— The fourth root is the square root of the square root ; the sixth root is the cube root of the square root or the square root of the cube root. Other roots are most con- veniently found by the use of logarithms. ALLIGATION shows the value of a mixture of different ingredients when the quantity and value of each is known. Let the ingredients be a, b, c, d, etc. , and their respective values per unit w, x, y, z, etc. A = the sum of the quantities = a + & + c-f d, etc. P = mean value or price per unit of A. AP = aw -f bx + cy -f- dz, etc. _ aw -\- bx -f cy -\- dz - — • PERMUTATION shows in how many positions any number of things may be arranged in a row; thus, the letters a, b, c may be arranged in six positions, viz. abc, acb, cab, cba, bac, bca. Rule.— Multiply together all the numbers used in counting the things; thus, permutations of 1,2, and 3 = 1x2x3 = 6. In how many positions can 9 things in a row be placed ? 1X2X3X4X5X6X7X8X9 = 362880. COMBINATION shows how many arrangements of a few things may be made out of a greater number. Rule : Set down that figure which indicates the greater number, and after it a series of figures diminishing by 1, until as many are set down as the number of the few things to be taken in each combination. Then beginning under the last one set down said number of few things ; then going backward set down a series diminishing by 1 until arriving under the first of the upper numbers. Multiply together all the upper numbers to form one product, and all the lower numbers to form another; divide the upper product by the lower one. GEOMETRICAL PROGRESSION. 11 How many combinations of 9 things can be made, taking 3 in each com- bination ? 1X2X3 ARITHMETICAL. PROGRESSION, in a series of numbers, is a progressive increase or decrease in each succes- sive number by the addition or subtraction of tbe same amount at each step, as 1, 2, 3, 4, 5, etc., or 15, 12, 9, 6, etc. The numbers are called terms, and the equal increase or decrease the difference. Examples in arithmetical pro- gression may be solved by the following formulae : Let a = first term, I = last term, d = common difference, n — number of terms, s = sum of the terms: I = + (n - l)d, = -\d±^2ds + (a-\dY 2s ~ n ' s (11 — l)d ~n + 2 = g ?i[2a + (n - i)d], l~\-a Z 2 _ ft 2 ~ 2 + 2d ' n = d + «) 5 . = ^ n l- 21 - ( n - !)<*]• = I — (n - l)d, s (n - l)d ~ n ~ 2 ' -\*W-( l + \dY -2ds, 2 ' 11 I — a = n — l' 2(s - an) "~ n(n — 1)' Z 2 - a 2 2(wZ - s) ~ 2s - I — a — ?i(n. — 1) I — a , ' d - 2a ± |/(2a - cZ) 2 + 8cZs *— +. 1 ' 2d 2s 2l-\-d ± 4/(21 + d) 2 - 8cZs GEOMETRICAL. PROGRESSION, in a series of numbers, is a progressive increase or decrease in each suc- cessive number by the same multiplier or divisor at each step, as 1, 2, 4, 8, 16. etc., or 243, 81, 27, 9, etc. The common multiplier is called the ratio. Let a = first term, I = last term, r = ratio or constant multiplier, n = number of terms, in = any term, as 1st, 2d, etc., s = sum of the terms: I = ar «-., = "+"■-■>* . =fr=i>^, r r n — 1 log I = log a + (n — 1) log r, l(s - l) n ~ 1 - a(s - a) n ~ 1 = 0. m = ar m lm log m = log a -f (m - 1) log r. s = a ^' n ~ *> = rl — a ■ n ~\/] n - " ~\Z a n i r n _ z VZ - 1/a r n - '' n x ARITHMETIC (r - l).s r n - 1 ' log a = log I - (n - 1) log r s — a lo „ r _ log * - log a ?i — 1 - f r + g "" = 0. a a . l og * - l og a , j logr ' log Z — log a ■ s-l ' l +; - z - "log( a) - log (s - Z) + 1, log [a + (?' - l)s] - log a log r log Z - log [Ir - (r - l)s] log i + 1- Population of the United States. (A problem in geometrical progression.,) Increase in 10 Annual Increase, STear. Population. Years, per cent. per cei 1860 31,443,321 1870 39,818,449* 26.63 2.39 1880 50.155,783 25.96 2.33 1890 62,622,250 24.86 2.25 1895 Est. 69,733,000 Est. 2.174 1900 " 77,652,000 Est. 24.0 " 2.174 Estimated Population in Each Tear from 1860 to 1899. (Based on the above rates of increase, in even thousands.) I860.... 31,443 1870.... 39,818 1880.... 50,156 1890.... 62.622 1861 .... 32,195 1871.... 40,748 1881.... 51,281 I 1891.... 63.984 1862.... 32,964 1872.... 41,699 1882 ... 52,433 1892... 65.375 1863.... 33,752 1873. .. 42,673 1883.... 53.610 1893.... 66,797 1864.... 34,558 1874... 43,670 1884 ... 54,813 1894... 68,249 1865.... 35,384 1875.... 44,690 1885.... 56.043 1895.... 69.733 1806.... 36.229 1876.... 45.373 1886 ... 57,301 1 1896... 71,219 1867.... 37,095 1877 .. 46,800 1887.... 58.588 1897.... 72.799 1868.... 37,981 1878 ... 47,893 1888.... 59,903 1898 ... 74,382 1869.... 38,889 1879 .... 49,011 1889... 61,247 1899.... 75,999 The above table has been calculated by logarithms, as follows : logr : log Z - log a -T- (n - 1), log m — log a + (»"• — 1) log r Pop. 1870 . " I860.. 39.si8449 1og = 31,443321 log = 7.6000S41 7.4975288 diff. = .1025553 1 = 10, diff. ■+- 10 = .01025553 add log for 1860 7.497528S :10gZ : log a : log r, : log a log for 1861 = add again log for 1862 7.51803986 No. = 32,964 . 7.50778433 No. .01025553 Compound interest is a form of geometrical progression; the ratio being 1 plus the percentage. * Corrected by addition of 1.260.078, estimated error of the census of 1870, Census Bulletin' No. 1G, Dee. 12. 1890. DISCOUNT. 13 INTEREST AND DISCOUNT. Interest is money paid for the use of money for a given time; the fac tors are : p, the sum loaned, or the principal: t, the time iu years; ?•, the rate of interest; i, the amount of interest for the given rate and time; a = p + i = the amount of the principal with interest at the end of the time. Formulae : i = interest = principal X time X rate per cent = i — *-—• - amount = principal + intere lOOi = rate = — — ; pt 100; _ ptr tr ~ a 100 ; lOOi pr ' 100' ph\ ~ 100' = principal = -^- = a - -. time = If the rate is expressed decimally as a per cent,— thus, 6 per cent = .06,— the formulas become i = prt; a — p(\ -4- rf): r = — ; t = — • ; p — -- = — — - Rules for finding Interest.— Multiply the principal by the rate per annum divided by 100, and by the time in years and fractions of a year. T , " , ' principal x rate per annum If the time is given in days, interest = ~~o"^ 77^ — • In banks interest is sometimes calculated on the basis of 360 days to a year, or 12 months of 30 days each. Short rules for interest at 6 per cent, when 360 days are taken as 1 year: Multiply the principal by number of days and divide by 6000. Multiply the principal by number of months and divide by 200. The interest of 1 dollar for one month is y% cent. Interest of 100 Dollars for Different Times and Rates. Time. 2% Z% 4% h% 6% 8% 10* 1 year $2.00 $3.00 $4.00 $5.00 $6.00 $8.00 $10.00 1 month .16§ .25 .83* .41§ .50 .66f .83i 1 day = 3^ year .0055| .0083J .01 11 J .013Sf .0166§ .0222| .0277| 1 day = 3 i 5 year .005179 .008219 .010959 .013699 .016438 .0219178 .0273973 Discount i^ interest deducted for payment of money before it is due. True discount is the difference between the amount of a debt pay- able at a future date without interest and its present worth. The present worth is that sum which put at interest at the legal rate will amount to the debt when it is due. To find the present worth of an amount due at future date, divide the amount by the amount of $1 placed at interest for the given time. The dis- count equals the amount minus the present worth. What discount should be allowed on $103 paid six months before it is due, interest being 6 per cent per annum ? ■ = $100 present worth, discount = 3.00. 1 + 1 X .06 X - Bank discount is the amount deducted by a bank as interest on money loaned on promissory notes. It is interest calculated not on the act- ual sum loaned, but on the gross amount of the note, from which the dis- count is deducted in advance. It is also calculated on the basis of 360 days in the year, and for 3 (in some banks 4) days more than the time specified in the note. These are called days of grace, and the note is not payable till the last of these days. 14 ARITHMETIC. What discount will be deducted by a bank in discounting a note for $103 payable 6 months hence ? Six months = 182 days, add 3 days grace = 185 . ,103 X 185 days -6uTio— = $3 - 1 ' 6 - Compound Interest.— In compound interest the interest is added to the principal at the end of each year, (or shorter period if agreed upon). Letp = the principal, r = the rate expressed decimally, n = no of years, and a the amount : : amount = p (1 + r) n ; r = rate s/-p- p = principal, = , no of years = ?i, log a — log p '' log (1 + r) Compound Interest Table. (Value of one dollar at compound interest, compounded yearly, at 3, 4, 5, and 6 per cent, from 1 to 50 years.) 1 Z% 4% b% G% 03 S% 4% 5* 6£ £ s i 1.03 1.04 1.05 1.06 16 1.6047 1.8730 2.1829 2.5403 2 1.0609 1.0816 1.1025 1.1236 17 1.6528 1.9479 2.2920 2.6928 3 1.0927 1.1249 1.1576 1.1910 18 1.7024 2.0258 2.4066 2.8543 4 1.1255 1.1699 1.2155 1.2625 19 1.7535 2.1068 2 5269 3.0256 5 1.1593 1.2166 1.2763 1.3382 20 1.8061 2.1911 2.6533 3.2071 6 1.1941 1.2653 1.3401 1.4185 21 1.8603 2.2787 2.7859 3.3995 7 1.2299 1.3159 1.4071 1.5036 22 1.9161 2.3699 2.9252 3.6035 8 1.2668 1.3686 1.4774 1.5938 23 1.9736 2.4647 3.0715 3.8197 9 1.3048 1.4233 1.5513 1.6895 24 2.0328 2.5633 3.2251 4 0487 10 1.3439 1.4802 1.6289 1.7908 25 2.0937 2.6658 3.3864 4.2919 11 1.3842 1.5394 1.7103 1.8983 30 2.4272 3.2434 4.3219 5 7435 12 1.4258 1.6010 1.7058 2.0122 35 2.8138 3.9160 5.5166 7.6861 13 1.4685 1.6651 1.8856 2.1329 40 3.2620 4.8009 7 0100 10.2858 14 1.5126 1.7317 1.9799 2.2609 45 3.7815 5.8410 8.9850 13.7646 15 1.5580 1.8009 2.0789 2.3965 50 4.3838 7.1064 11.6792 18.4190 At compound interest at 3 per cent money will double itself in 23J4 years, at 4 per cent in 17% years, at 5 per cent in 14.2 j r ears, and at per cent in 11.9 years. EQUATION OF PAYMENTS. By equation of payments we find the equivalent or average time in which one payment should be made to cancel a number of obligations due at dif- ferent dates; also the number of days upon which to calculate interest or discount upon a gross sum which is composed of several smaller sums pay- able at different dates. Rule.— Multiply each item by the time of its maturity in days from a fixed date, taken as a standard, and divide the sum of the products by the sum of the items: the result is the average time in days from the standard date. A owes B $100 due in 30 days. $200 due in 60 days, and $300 due in 90 clays. In how many days may the whole be paid in one sum of $600 ? 100 x 30 + 200 x 60 + 300 x 90 = 42,000 ; 42,000 h- 600 = 70 days, am. A owes B $100, $200, and $300, which amounts are overdue respectively 30, 60. and 90 days. If lie now pays the whole amount. $600, how many days* interest should he pay on that sum ? Ans. 70 days. ANNUITIES. 15 PARTIAL. PAYMENTS. To compute interest on notes and bonds when partial payments have been made: United States Rule.— Find the amount of the principal to the time of the first payment, and, subtracting the payment from it, find the amount of the remainder as a new principal to the time of the next payment. If the payment is less than the interest, find the amount of the principal to the time when the sum of the payments equals or exceeds the interest due, and subtract the sum of the payments from this amount. Proceed in this manner till the time of settlement. Note.— The principles upon which the preceding rule is founded are: 1st. That payments must be applied first to discharge accrued interest, and then the remainder, if any, toward the discharge of the principal. 2d. That only unpaid principal can draw interest. Mercantile Method.— When partial payments are made on short notes or interest accounts, business men commonly employ the following method : Find the amount of the whole debt to the time of settlement ; also find the amount of each payment from the time it was made to the time of set- tlement. Subtract the amount of payments from the amount of the debt; the remainder will be the balance due. ANNUITIES. An Annuity is a fixed sum of money paid yearly, or at other equal times agreed upon. The values of annuities are calculated by the principles of compound interest. 1. Let i denote interest on $1 for a year, then at the end of a year the amount will be 1 + i. At the end of n years it will be (1 + i) n . 2. The sum which in n years will amount to 1 is or (l + i)~ n , or the (1 -f i) n present value of 1 due in n years. 3. The amount of an annuity of 1 in any number of years n is 4. The present value of an annuity of 1 for any number of years n is I_ ■(!. + <)-» q + «) n -i 5. The annuity which 1 will purchase for any number of years n is 6. The annuity which would amount to 1 in n years is (l+i) w -l Amounts, Present Values, etc., at 5% Interest. Years (1) (2) (3) (4) (5) (6) (1 + 9" a + ir n (1 + t)» - 1 l-d+i)- 1 * i i * t l-(l-fz)-" (l + i)»--l 1 1.05 .952381 1. .952381 1.05 1. 2 1.1025 .907029 2.05 1.859410 .537805 .487805 3 1.157625 .863838 3.1525 2.723248 .367209 .317209 4 1.215506 .822702 4.310125 3.545951 .282012 .232012 5 1.276282 .783526 5.525631 4.329477 .230975 .180975 6 1.340096 .746215 6.801913 5.075692 .197017 .147018 7 1.407100 .710681 8.142008 5.786373 .172820 .122820 8 1.477455 .676839 9.549109 6.463213 .154722 .104722 9 1.551328 .644609 11.026564 7.107822 .140690 .090690 10 1.628895 .613913 12.577893 7.721735 .129505 .079505 16 ARITHMETIC. tb 000005 ec M CO Of L- i-ioocc OS cococoom tjmcc i- O 00 i.- CD Ci Of t- 0? X lO Of Oi I- X in m Tf -rr co co co c< o* i- Of XSCcPCO Of co oi co x COtOtlrir- O CD CO t- £> sssss Scttsgig O J- Of CO so fflWOOllT " it H 1 - t- OJ o*- TT CO Of S i- mtooic woff.t- fc- CVCOi-COOJ co wm-f oi i- m c - a 1 OOiOliO e© CO -MM -st cjimcoo nocxi- !go-*o5in Of x » co -t t- CO O x so £ SO i-iCOGOOCO OiOtOtDC OOmCOOI- 10 10^01- i- o oj x o octi-t co: t- m 05 m i-i co x i- i-i cr. o tt x coinxo o c c H o i-< so m i-i n oj io ■* it CiO>CpXm ©JOC5 00 1- t»nlOO«3 MO!t--P-P a)-*-l»l' co co in in rt< tt co co co o? 1-1 1-1 1-1 is CO Cf Tf CD Oi r- J> X TP -J< O -cc 05 <■>* tt co in ** tp o* c coox ^fco = 4< eg c © ,_,-j~CDOf QOOSiOffl 0OO)©Ht- tONlCiO Oilft — Oil~ c.^KXin ofi-HOaoot- scorning -sp tp co M Of »-. n .-i -C0 W*COQCN icoet-ffl oi t - x o> tj< o m o> t- x Of CO OS GO -P O Of GO' {- 00 O ice CO CO OS ift 0? Oi I- t- i-H CO CO O CO Oi of co x m eo i-i oi CO' i - i- co m in \p Tr^fcococj of 1-1 1-< t-i tp co Of 1-1 t-i — -i k> CI coincoooc osomTPo- t---^COl-0 TtOffiOOlf -JOlOl-O CO CO t' r- r Oi co d ih »n ~ ?_•?•? GO O CO 'r^ Oi CO L- •^lnoirro cocoooox i-cominm Tt'cfTj'eocN i-i l- -s* — Oi Of i-i i-i i-i n 3? «1 x •*»< -p -<* so a n co « - aimtt-o co t- o -* t- SSSolS = eoioosec •**cbof i-Hi- W-c*OOiC co i-i o co at. oj co o in i- i, co co m ir T? TT 5< CO Mrl^Hrt 01 O rr tP CO M — XXOS-p gXCOCTO Oi CD l- i-i O OftOONM t- O CO O O 9 2 o» w 5 as «r TP CO Of .- .- WlOr-O- CO i-i O C5 X CO t- — so o xmof oo ■V TT Tf rr CO co oi m co i-i Of i-« i-i — i-i H m oj m to co O t- CD i- IT 0>— 0»C0GC in Ai so oo sc NOXlOJ in o m i— of co o in oi x c? t» ~ cV. u- TCOCI-- CO — OOSOC -r x o? t- o- ^ % "5 ^ CO it ococo — • Of Of i-i i-i i-i S5 5 ^° co -r m cc l> ' * S ec NMOJC CO omc WEIGHTS AND MEASURES. 17 TABLES FOR CALCULATING SINKING-FUNDS AND PRESENT VALUES. Engineers and others connected with municipal work and industrial enter- prises often find it necessary to calculate payments to sinking-funds which will provide a sum of money sufficient to pay off a bond issue or other debt at the end of a given period, or to determine the present value of certain annual charges. The accompanying tables were computed by Mr. John W. Hill, of Cincinnati, Eng'g News, Jan. 25, 1894. Table I (opposite page) shows the annual sum at various rates of interest required to net $1000 in from 2 to 50 years, and Table II shows the present value at various rates of interest of an annual charge of $1000 for from 5 to 50 years, at five-year intervals and for 100 years. Table II. Capitalization of Annuity of SI 000 for from 5 to 100 Years. Rate of Interest, per cent. 23^ 3^ VA 5 5^ 4,389.91 4,329.45 4,268.09 7,912.67 7,721.73 7,537.54 10,739.42 10,379.53 10,037.48 13,007.88 12,462.13 11,950.26 14,828.12 14,093.86 13,413.82 16,288.77 15,372.36 14,533.63 17,460.89 16,374.36 15,390.48 18,401.49 17,159.01 16.044.92 19,156.24 17,773.99 16,547.65 19,761 93 18,255.86 16,931.97 21,949.21 19,847.90 18,095.83 ■20,930 23,145 2c, 103 "",368. 36,614 4,579 8,530 11,937 14,877, 17,413, 19,600. 24,518 25,729 31,598 ,514.92 4,451.68 ,316.45 8,110.74 ,517.23 11,118.06 ,218.is!l3,590.21 ,481.28,15,621.93 ,891.85,17,291.86 ,000.43 18,664.37 ,354.83 19,792.65 ,495.23 20,719.89 ,455.21 21,482.08 ,655.36,24,504.96 WEIGHTS AND MEASURES. Long Measure.— Measures of Length. 12 inches = 1 foot. 3 feet = 1 yard. 5> yards, or 16£ feet = 1 rod, pole, or perch. 40 poles, or 220 yards = 1 furlong. 8 furlongs, or 1760 yards, or 5280 feet = 1 mile. 3 miles = league. Additional measures of length in occasional use : 1000 mils = 1 inch; 4 inches = 1 hand ; 9 inches = 1 span ; 2| feet = 1 military pace ; 2 yards = 1 fathom. Old Land Measure.— 7.92 inches = 1 link; 100 links, or 66 feet, or 4 poles = 1 chain; 10 chains = 1 furlong; 8 furlongs = 1 mile; 10 square chains = 1 acre. Nautical Measure. 6080.26 feet, or 1.15156 stat- ute miles 3 nautical miles 60 nautical miles, or 6 statute miles 360 degrees Y = 1 nautical mile, or knot.* ~ 1 league. i = .1 degree (at the equator). = circumference of the earth at the equator. * The British Admiralty takes the round figure of 6080 ft. which is the length of the " measured mile" used in trials of vessels. The value varies from 6080.26 to 6088.44 ft. according to different measures of the earth's di- ameter. There is a difference of opinion among writers as to the use of the word " knot " to mean length or a distance— some holding that it should be 18 ARITHMETIC. Square Measure.— Measures of Surface. 144 square inches, or 183.35 circular inches : 1 square foot. 9 square feet = l square yard. 30^ square yards, or 272J square feet = 1 square rod, pole, or perch. 40 square poles = 1 rood. 4 roods, or 10 sq. chains, or 160 sq. ) poles, or 4840 sq. yards, or 43560 V = 1 acre, sq. feet, ) 640 acres = 1 square mile. An acre equals a square whose side is 208.71 feet. A circular inch is the area of a circle 1 inch in diameter = 0.7854 square inch. 1 square inch = 1.2732 circular inches. A circular mil is the area of a circle 1 mil, or .001 inch in diameter. 1000 2 or 1,000,000 circular mils = 1 circular inch. 1 square inch = 1,273,239 circular mils. The mil, and circular mil are used in electrical calculations involving the diameter and area of wires. Solid or Cubic Measure.— Measures of Volume. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 1 cord of wood = a pile, 4x4x8 feet = 128 cubic feet. 1 perch of masonry = 16| X 1| X 1 foot = 24f cubic feet. Liquid Measure. 4 gills = 1 pint. 2 pints = 1 quart. 4 quarts - 1 o-Allon i U - S - 231 cubic inches - - l gallon -j Eng 277 2U cubic inches# 31J gallons = 1 barrel. 42 gallons = 1 tierce. 2 barrels, or 63 gallons — 1 hogshead. 84 gallons, or 2 tierces = 1 puncheon. 2 hogsheads or 126 gallons = 1 pipe or butt. ' 2 pipes, or 3 puncheons = 1 tun. The U. S. gallon contains 231 cubic iuches; 7.4805 gallons = 1 cubic foot. A cylinder 7 in. diam. and 6 in. high contains 1 gallon, very nearly, or 230.9 cubic inches. The British Imperial gallon contains 277.274 cubic inches = 1.20032 U. S. gallon. The Miner's Inch.— (Western U. S. for measuring flow of a stream of water). The term Miner's Inch is more or less indefinite, for the reason that Cali- fornia water companies do not all use the same head above the centre of the aperture, and the inch varies from 1.36 to 1.73 cubic feet per minute each; but the most common measurement is through an aperture 2 inches high and whatever length is required, and through a plank 1£ inches thick. The lower edge of the aperture should be 2 inches above the bottom of the measuring-box, and the plank 5 inches high above the aperture, thus mak- ing a 6-inch head above the centre of the stream. Each square inch of this opening represents a miner's inch, which is equal to a flow of lb cubic feet per minute. Apothecaries' Fluid Measure. 60 minims = 1 fluid drachm. 8 drachms, or 437* grains, or 1.732 cubic inches = 1 fluid ounce. Dry Measure, U. S. 2 pints = 1 quart. 8 quarts = 1 peck. 4 pecks = 1 bushel. used only to denote a rate of speed. The length between knots on the log line is T £oo 43.055 4.784 9.884 5 = 0.7750 53.819 5.980 12.355 6 = 0.9300 64.583 7.176 14.826 7 = 1.0850 75.347 8.372 17.297 8 = 1.2400 86.111 9.568 19.768 9 = 1.3950 96.874 10.764 22.239 CUBIC. Cubic Centi- metres to Cubic Inches. Cubic Deci- metres to Cubic Inches. Cubic Metres to Cubic Feet. Cubic Metres to Cubic Yards. 1 = 0.0610 61.023 35.314 1.308 2 = 0.1220 122.047 70.629 2.616 3 = 0.1831 183.070 105.943 S.924 4 = 0.2441 244.093 141.258 5.232 5 = 0.3051 305.117 176.572 6.540 6 = 0.3661 366.140 211.887 7.848 7 = 0.4272 427.163 247.201 9.156 8 = 0.4882 488.187 282.516 10.464 9 = 0.5492 549.210 317.830 11.771 CAPACITY. Millilitres or Cubic Centi- litres to Fluid Drachms. Centilitres to Fluid Ounces. Litres to Quarts. Dekalitres to Gallons. Hektolitres to Bushels. 1 = 0.27 0.338 1.0567 2.6417 2.8375 2 = 0.54 0.676 2.1134 5.2834 5.6750 3 = 0.81 1.014 3.1700 7.9251 8.5125 4 = 1.08 1.352 4.2267 10.5668 11.3500 5 - 1.35 1.691 5.2834 13.2085 14.1875 6 = 1.62 2.029 6.3401 15.8502 17.0250 7 = 1.89 2.368 7.3968 18.4919 19.8625 8 = 2.16 2.706 8.4534 21.1336 22.7000 9 = 2.43 3.043 9.5101 23.7753 25 5375 26 AKITHMETIC. WEIGHT. Milligrammes to Grains. Kilogrammes to Grains. Hectogrammes (100 grammes) to Ounces Av. Kilogrammes to Pounds Avoirdupois. 1 = 0.01543 15432.36 3.5274 2.20462 2 = 0.03086 30864.71 7.0548 4.40924 3 = 0.04630 46297.07 10.5822 6.61386 4 = 0.06173 61729.43 14.1096 8.81849 5 = 0.07716 77161.78 17.6370 11.02311 6 = 0.09259 92594.14 21.1644 13.22773 7 — 0.10803 108026.49 24.6918 15.43235 8 = 0.12346 123458.85 28.2192 17.63697 9 = 0.13889 138891.21 31.7466 19.84159 WEIGHT— (Continued). Quintals to Pounds Av. Milliers or Tonnes to Pounds Av. Grammes to Ounces, Troy. 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 220.46 440.92 661.38 881.84 1102.30 1322.76 1543.22 1763.68 1984.14 2204.6 4409.2 6613.8 8818.4 11023.0 13227.6 15432.2 17636.8 19841.4 0.03215 0.06430 0.09645 0.12860 0.16075 0.19290 0.22505 0.25721 0.28936 1 The only authorized material standard of customary length is the Troughton scale belonging to this office, whose length at 59°. 62 Fahr. con- forms to the British standard. The yard in use in the United States is there- fore equal to the British yard. The only authorized material standard of customary weight is the Troy pound of the mint. It is of brass of unknown density, and therefore not suitable for a standard of mass. It was derived from the British standard Troy pound of 1758 by direct comparison. The British Avoirdupois pound was also derived from the latter, and contains 7000 grains Troy. The grain Troy is therefore the same as the grain Avoirdupois, and the pound Avoirdupois in use in the United States is equal to the British pound Avoirdupois. The metric system was legalized in the United States in 1866. By the concurrent action of the principal governments of the world an International Bureau of Weights and Measures has been established near Paris. The International Standard Metre is derived from the Metre des Archives, and its length is defined by the distance between two lines at 0° Centigrade, on a platinum-iridium bar deposited at the International Bureau. The International Standard Kilogramme is a mass of platinum-iridium deposited at the same place, and its weight in vacuo is the same as that of the Kilogramme des Archives. Copies of these international standards are deposited in the office of standard weights and measures of the U. S. Coast and Geodetic Survey. The litre is equal to a cubic decimetre of water, and it is measured by the quantity of distilled water which, at its maximum density, will counterpoise the standard kilogramme in a vacuum; the volume of such a quantity of water being, as nearly as has been ascertained, equal to a cubic decimetre, WEIGHTS AND MEASURES — COMPOUND UNITS. 27 COMPOUND UNITS. Measures of Pressure and Weight. 1 lb. per square inch. ' 144 lbs. per square foot. 2.0355 ins. of mercury at 32° F. 2.0416 " " " " 62° F. 2.309 ft. of water at 62° F. 27.71 ins. " " " 62° F. f 2116.3 lbs. per square foot. I 33.947 ft. of water at 62° F. 1 atmosphere (14.7 lbs. per sq. in.). = -j 30 ins. of mercury at 62° F. | 29.922 ins. of mercury at 32° F. 1.760 millimetres of mercury at 32° F. {.0361 lb. per square inch. 5.196 lbs. " " foot. .0736 in. of mercury at 62° F. ., . , „ . . nnn tt, J 5.2021 lbs. per square foot. 1 inch of water at 32° F. = -J 036]25 lb £ ^ „ inch ( .433 lb. per square inch. 1 foot of water at 62° F. =■{ 62.355 lbs. " " foot. ( .883 in. of mercury at 62° F. f .49 lb. per square inch. • • i. * *■ «oo I? J 70.56 lbs. " " foot. 1 inch of mercury at 62° F. = \ im ft of water at 62 o F- [13.98 ins. " : ' " 62° F. W eight of One Cubic Foot of Pure Water. At 32° F. (freezing-point) 62.418 lbs. " 39.1° F. (maximum density) 62.425 " " 62° F. (standard temperature) 62.355 " " 212° F. (boiling-point, under 1 atmosphere) 59.76 " American gallon = 231 cubic ins. of water at 62° F. = 8.3356 lbs. British " =277.274" " " " " " = 10 lbs. Measures of Work, Poorer, and. Duty. Work.— The sustained exertion of pressure through space. Unit of work.— One foot-pound, i.e., a pressure of one pound exerted through a space of one foot. Horse-power.- The rate of work. Unit of horse-power = 33,000 ft.- lbs. per minute, or 550 ft.-lbs. per second = 1,980,000 ft.-lbs. per hour. Heat unit = heat required to raise 1 lb. of water 1° F. (from 39° to 40°). 33000 Horse-power expressed in heat units = ~^?r = 42.416 heat units per min- ute = .707 heat unit per second = 2545 heat units per hour. iVii f f , i „ Q „ tt T > ™„ h™,. f 1,980.000 ft.-lbs. per lb. of fuel. 1 lb. of fuel per H. I\ per hour= } £ m heat unitg ^ „ 1,000,000 ft.-lbs. per lb. of fuel = 1.98 lbs. of fuel per H. P. per hour. 5280 22 Velocity.— Feet per second = ^x = 15 * mi,es per hour. Gross tons per mile = ^—z = — lbs. per yard (single rail.) French and British Equivalents of Weight and Press- ure per Unit of Area. French. British. 1 gramme per square millimetre ""= 1.422 lbs. per square inch. 1 kilogramme per square " =1422.32 " " " 1 " " " centimetre = 14.223 " " " " 1.0335 kilogrammes per square centimetre I _ ..„ >< u « ,. (1 atmosphere) j 0.70308 kilogramme per square centimetre = 1 lb. per square inch. 28 ARITHMETIC. WIRE AND SHEET-METAL. GAUGES COMPARED. "o Ot3 & C C M ^E. 05 British Imperial Standard dard or Plate Steel. dard 1893.) a a 'Sees CO Roebling'i Washbu & Moen Gauge ° 05 03 £ Wire Gauge. (Legal Standard in Great Britain since March 1, 1884.) U. S. Stan. Gauge f Sheet and '. Iron and S (Legal Stan since July 1, it es inch. inch. inch. inch. inch. millira. inch. 0000000 .49 .500 12.7 .5 7/0 000000 .46 .464 11.78 .469 6/0 00000 .43 .45 .432 10.97 .438 5/0 0000 .454 .46 .393 .40 .4 10.16 .406 4/0 000 .425 .40964 .362 .36 .372 9.45 .375 3/0 00 .38 .3648 .331 .33 .348 8.84 .344 2/0 .34 .32486 .307 .305 .324 8.23 .313 1 .3 . .2893 .283 .285 .3 7.62 .281 1 2 .284 .25763 .263 .265 .276 7.01 .266 2 3 .259 .22942 .244 .245 .252 6.4 .25 3 4 .238 .20431 .225 .225 .232 5.89 .231 4 5 .22 .18194 .207 .205 .212 5.38 .219 5 6 .203 .16202 .192 .19 .192 4.88 .203 6 7 .18 .144-28 .177 .175 .176 4.47 .188 7 8 .165 .12849 .162 .16 .16 4.06 .172 8 9 .148 .11443 .148 .145 .144 3-66 .156 9 10 .134 .10189 .135 .13 .128 3-26 .141 10 11 .12 .09074 .12 .1175 .116 2.95 .125 11 12 .109 .08081 .105 .105 .104 2-64 .109 12 13 .095 .07196 .092 .0925 .092 2-34 .094 13 14 .083 .06408 .08 .08 .08 2-03 .078 14 15 .072 .05707 .072 .07 .072 1.83 .07 15 16 .065 .05082 .063 .061 .064 1.63 .0625 16 17 .058 .04526 .054 .0525 056 1.42 .0563 17 18 .049 .0403 .047 .045 .048 1.22 .05 18 19 .042 .03589 .041 .04 .04 1.01 .0438 19 20 .035 .03196 .035 .035 .036 .91 .0375 20 21 .032 .02846 .032 .031 .032 .81 .0344 21 22 .028 .02535 .028 .028 .028 .71 .0313 22 23 .025 .02257 .025 .025 .024 .61 .0281 23 24 .022 .0201 .023 .0225 .022 .56 .025 24 25 .02 .0179 .02 .02 .02 .51 .0219 25 26 .018 .01594 .018 .018 .018 .45 .0188 26 27 .016 .01419 .017 .017 .0164 .42 .0172 27 28 .014 .01264 .016 .016 .0148 .38 .0156 28 29 .013 .01126 .015 .015 .0136 .35 .0141 29 30 .012 .01002 .014 .014 .0124 .31 .0125 30 31 .01 .00893 .0135 .013 .0116 .29 .0109 31 32 .009 .00795 .013 .012 .0108 .27 .0101 32 33 .008 .00708 .011 .011 .01 .25 .0094 33 34 .007 .0063 .01 .01 .0092 .23 .0086 34 35 .005 .00561 .0095 .0095 .0084 .21 .0078 35 36 004 .005 .009 .009 .0076 .19 .007 36 37 .00445 .0085 .0085 .0068 .17 .0066 37 38 .00396 .008 .008 .006 .15 .0063 38 39 .0g353 .0075 .0075 .0052 .13 39 40 .00314 .007 .007 .0048 .12 40 41 .0044 .11 41 42 .004 .10 42 43 .0036 .09 43 44 .0032 .08 44 45 .0028 .07 45 46 .0024 .06 46 47 .002 .05 47 48 .0016 .04 48 49 .0012 .03 49 50 .001 .025 50 WIRE GAUGE TABLES. 29 EDISON, OR CIRCULAR Mil. GAUGE, FOR ELEC- TRICAL WIRES. Gauge Num- ber. Circular Mils. Diam- eter in Mils. 3 3,000 54.78 5 5,000 70.72 8 8,000 89.45 12 12,000 109.55 15 15,000 122.48 20 20,000 141.43 25 25,000 158.12 30 30,000 173.21 35 35,000 187.09 40 40,000 200.00 45 45,000 212.14 50 50,000 223.61 55 55,000 234.53 60 60,000 244.95 65 65,000 254.96 Gauge Num- ber. Circular Mils. Diam- eter in Mils. 70 70,000 264.58 75 75,000 273.87 80 80,000 282.85 85 85,000 291.55 90 90,000 300.00 95 95,000 308.23 100 100,000 316.23 110 110,000 331.67 120 120,000 346.42 130 130,000 360.56 140 140,000 374.17 150 150,000 387.30 160 160.000 400.00 170 170,000 412.32 180 180,000 424.27 Gauge Num- ber. Circular Mils. Diam- eter in Mils. 190 190,000 435.89 200 200,000 447.22 220 220,000 469.05 240 240.000 489.90 260 260,000 509.91 280 280,000 529.16 300 300,000 547.73 320 320,000 565.69 340 340,000 583.10 360 360,000 600.00 TWIST DRILL AND STEEL WIRE GAUGE. (Morse Twist Drill and Machine Co.) No. Size. inch. 1 .2280 2 .2210 3 .2130 4 .2090 5 .2055 6 .2040 7 .2010 8 .1990 9 .1960 10 .1935 11 .1910 12 .1890 13 .1850 14 .1820 15 .1800 No. Size. No. Size. inch. inch. 16 .1770 31 .1200 17 .1730 32 .1160 18 .1695 33 .1130 19 .1660 34 .1110 20 .1610 35 .1100 21 .1590 36 .1065 22 .1570 37 .1040 23 .1540 38 .1015 24 .1520 39 .0995 25 .1495 40 .0980 26 .1470 41 .0960 27 .1440 42 .0985 28 .1405 43 .0890 29 .1360 44 .0860 30 .1285 45 .0820 No. Size. inch. 46 .0810 47 .0785 48 .0760 49 .0730 50 .0700 51 .0670 52 .0635 53 .0595 54 .0550 55 .0520 56 .0465 57 .0430 58 .0420 59 .0410 60 .0400 STEEL MUSIC-WIRE GAUGE. (Washburn & Moen Mfg. Co.) No. Size. No. Size. • No. Size. No. Size. inch. inch. inch. inch. 12 .0295 17 .0378 21 .0461 25 .0585 13 .0311 18 .0395 22 .0481 26 .0626 14 .0325 19 .0414 23 .0506 27 .0663 15 .0313 20 .043 24 .0547 28 .0719 16 .0359 30 ARITHMETIC. THE EDISON OR CIRCULAR OTIL WIRE GAUGE. (For table of copper wires by this'gauge, giving weights, electrical resist- ances, etc., see Copper Wire.) Mr. C. J. Field {Stevens Indicator, July, 1887) thus describes the origin of the Edison gauge: The Edison company experienced inconvenience and loss by not having a wide enough range nor sufficient number of sizes in the existing gauges. This was felt more particularly in the central-station work in making- electrical determinations for the street system. They were compelled to make use of two of the existing gauges at least, thereby introducing a complication that was liable to lead to mistakes by the contractors and linemen. In the incandescent system an even distribution throughout the entire system and a uniform pressure at the point of delivery are obtained by cal- culating for a given maximum percentage of loss from the potential as delivered from the dynamo. In carrying this out, on account of lack of regular sizes, it was often necessary to use larger sizes than the occasion demanded, and even to assume new sizes for large underground conductors. It was also found that nearly all manufacturers based their calculation for. the conductivity of their wire on a variety of units, and that not one used the latest unit as adopted by the British Association and determined from Dr. Matthiessen's experiments ; and as this was the unit employed in the manufacture of the Edison lamps, there was a further reason for construct- ing a new gauge. The engineering department of the Edison company, knowing the requirements, have designed a gauge that has the widest range obtainable and a large number of sizes which increase in a regular and uniform manner. The basis of the graduation is the sectional area, and the number of the wire corresponds. A wire of 100,000 circular mils area is No. 100 ; a wire of one half the size will be No. 50 ; twice the size No. 200. In the older gauges, as the number increased the size decreased. With this gauge, however, the number increases with the wire, and the number multiplied by 1000 will yive the circular -nils. The weight per mil-foot, 0.00000302705 pounds, agrees with a specific gravity of 8.889, which is the latest figure given for copper. The ampere capacity which is given was deduced from experiments made in the com- pany's laboratory, and is based on a rise of temperature of 50° F. in the wire. In 1893 Mr. Field writes, concerning gauges in use by electrical engineers: The B. and S. gauge seems to be in general use for the smaller sizes, up to 100,000 c. m., and in some cases a little larger. From between one and two hundred thousand circular mils upwards, the Edison gauge or its equivalent is practically in use, and there is a general tendency to designate all sizes above this in circular mils, specifying a wire as 200,000, 400,000, 500,- 000, or 1,000,000 c. m. In the electrical business there is a large use of copper ware and rod and other materials of these large sizes, and in ordering them, speaking of them, specifying, and in every other use, the general method is to simply specify the circular milage. I think it is going to be the only system in the future for the designation of wires, and the attaining of it means practically the adoption of the Edison gauge or the method and basis of this gauge as the correct one for wire sizes. THE U. S. STANDARD GAUGE FOR SHEET AND PLATE IRON AND STEEL, 1893. The Committee on Coinage, Weights, and Measures of the House of Representatives in 1893, in introducing the bill establishing the new sheet and plate gauge, made a report from which we take the following : The purpose of this bill is to establish an authoritative standard gauge for the measurement of sheet and plate iron. There is in this country no uniform or standard gauge, and the same numbers in different gauges represent different thicknesses of sheets or plates. This has given rise to much misunderstanding and friction between employers and workmen and mistakes and fraud between dealers and con- sumers. The practice of describing the different thicknesses of sheet and plate iron by gauge numbers has been so long established and become so uni- versal both here and in Great Britain that it is not deemed advisable to change this mode of designation; but these descriptive gauge numbers GAUGE FOR SHEET AND PLATE IRON AND STEEL. 31 IT. S. STANDARD GAUGE FOR SHEET AND PLATE IRON AND STEEL., 1893. o II S3 ; is .5 a> g o x a. 2 a Js ° ^ a Approximate Thickness iu Decimal Parts of an Inch. III £ 5 a S Oo' H .S f £ 1 Weight per Square Foot in Ounces Avoirdupois. Weight per Square Foot in Pounds Avoirdupois. «"£ a A 9 & .L r -=- S3 -g 2 ft! 03.2 Weight per Square Meter in Pounds Avoirdupois. 0000000 1-2 0.5 12.7 320 20. 9.072 97.65 215.28 000000 15-32 0.46S75 11.90625 300 18.75 8.505 91.55 201.82 00000 7-16 0.4375 11.1125 280 17.50 7.938 85.44 188.37 0000 13-32 0.40625 10.31875 260 16.25 7.371 79.33 174.91 000 3-8 0.375 9.525 240 15. 6.804 73.24 161.46 oa 11-32 0.34375 8.73125 220 13.75 6.237 67.13 148.00 5-16 0.3125 7.9375 200 12.50 5.67 61.03 134.55 1 9-32 0.28125 7.14375 180 11.25 5.103 54.93 121.09 2 17-64 0.265625 6.746875 170 10.625 4.819 51.88 114.37 3 1-4 0.25 6.35 160 10. 4.536 48.82 107.64 4 15-64 0.234375 5.953125 150 9.375 4.252 45.77 100.91 5 7-32 0.21875 5.55625 140 8.75 3.969 42.72 94.18 6 13-64 0.203125 5.159375 130 8.125 3.685 39.67 87.45 7 3-16 0.1875 4.7625 120 7.5 3.402 36.62 80.72 8 11-64 0.171875 4.365625 110 6.875 3.118 33.57 74.00 9 5-32 0.15625 3.96875 100 6.25 2.835 30.52 67.27 10 9-64 0.140625 3.571875 90 5.625 2.552 27.46 60.55 11 1-8 0.125 3.175 80 5. 2.268 24.41 53.82 12 7-64 0.109375 2.778125 70 4.375 1.984 21.36 47.09 13 3-32 0.09375 2.38125 60 3.75 1.701 18.31 40.36 14 5-64 0.078125 1.984375 50 3.125 1.417 15.26 33.64 15 9-128 0.0703125 1.7859375 45 2.8125 1.276 13.73 30.27 16 1-16 0.0625 1.5875 40 2.5 1.134 12.21 26.91 17 9-160 0.05625 1.42875 36 2.25 1.021 10.99 24.22 18 1-20 0.05 1.27 32 2. 0.9072 9.765 21.53 19 7-160 0.04375 1.11125 28 1.75 0.7938 8.544 18.84 20 3-80 0.0375 0.9525 24 1.50 0.6804 7.324 16.15 21 11-320 0.034375 0.873125 22 1.375 0.6237 6.713 14.80 22 1-32 0.03125 0.793750 20 1.25 0.567 6.103 13 46 23 9-320 0.028125 0.714375 18 1.125 0.5103 5.493 12.11 24 1-40 0.025 0.635 16 1. 0.4536 4.882 10.76 25 7-320 0.021875 0.555625 14 0.875 0.3969 4.272 9.42 26 3-160 0.01875 0.47625 12 0.75 0.3402 3.662 8.07 27 11-640 0.0171875 0.4365625 11 0.6875 0.3119 3.357 7.40 28 1-64 0.015625 0.396875 10 0.625 0.2835 3.052 6.73 29 9-640 0.0140625 0.3571875 9 0.5625 0.2551 2.746 6.05 30 1--80 0.0125 0.3175 8 0.5 0.2268 2.441 5.38 31 7-640 0.0109375 0.2778125 7 0.4375 0.1984 2.136 4.71 32 13-1280 0.01015625 0.25796875 ^ 0.40625 0.1843 1.983 4.37 33 3-320 0.009375 0.238125 6 0.375 0.1701 1.831 4.04 34 11-1280 00859375 0.21828125 5^ 0.34375 0.1559 1.678 3 70 35 5-640 0.0078125 0.1984375 5 0.3125 0.1417 1.526 3.36 36 9-1280 0.00703125 0.17859375 $A 0.28125 ).1270 1.373 3.03 37 17-2560 0.006640625 0.168671875 m 0.265625 0.1205 1.297 2.87 38 1-160 0.00625 0.15875 4 0.25 0.1134 1.221 2.69 32 MATHEMATICS. ought to have the same meaning and significance at all times and under all circumstances. To accomplish this and furnish a legal guide in the collection of govern- ment duties, the United States should establish a legal standard gauge. None of the existing gauge-tables or scales exactly meet the requirements of accuracy and convenience, nor rest on a systematic basis; but the one submitted by your committee is believed to fully meet these requirements. It is based on the fact that a cubic foot of iron weighs 480 pounds. This is the same basis on which the Imperial gauge of Great Britain rests, and also the New Birmingham and Amalgamated Association gauges. A sheet of iron 1 foot square and 1 inch thick weighs 40 pounds, or 640 ounces, and 1 ounce in weight should be 1/640 inch thick. The scale has been arranged so that each descriptive number represents a certain number of ounces in weight, and an equal number of six hundred and fortieths of an inch in thickness, and the weights, and hence the thicknesses, have been arranged in a regular series of gradations. A micrometer for measuring the thickness of sheets and plates can be constructed to indicate six hun- dred and fortieths of an inch as easily as one thousandths, and thus the measurement of a sheet of iron will give the thickness in six hundred and fortieths of an inch and in weight in ounces at the same time. It is probable that the adoption of this gauge will gradually lead to the abandonment of the numbers and to the use of the number of ounces in weight per square foot as the descriptive terms of the different thicknesses of sheet and plate iron. It will become as easy to order a 20-ounce sheet as a No. 22, or a 10 ounce as a No. 28; and this will cause a more general and intelligent comprehension of just what is being contracted for, and the opportunity for mistake or fraud growing out of an uncertainty of designa- tion will be removed. A natural consequence also will be the substitution of such weight desig- nation for the arbitrary methods now in vogue of describing tin and terne plates as IC, IX, IXX, DC, DX, etc. The law establishing the new gauge enacts as follows : That for the purpose of securing uniformity, the following is established as the only standard gauge for sheet and plate iron and steel in the United States of America, namely : And on and after July 1, 1893, the same and no other shall be used in determining duties and taxes levied by the United States of America on sheet and plate iron and steel. Sec. 2. That the Secretary of the Treasury is authorized and required to prepare suitable standards in accordance herewith. Sec. 3. That in the practical use and application of the standard gauge hereby established a variation of 2\£ per cent either way may be allowed. 33 ALGEBRA. Addition.— Add a and 6. Ans. a-\-b. Add a, b, and — c. Ans. a-\-b — c. Add 2a and — 3a. Ans. — a. Add 2ab, — Sab, — c, — 3c. Ans. — ab — 4c. Subtraction. — Subtract a from b. Ans. 6 — a. Subtract — a from — 6. Ans. — b + «• Subtract 6 -f- c from a. Ans. a — & — c. Subtract 3a 2 6- 9c from 4a 2 6 + c. Ans. a 2 b -f- 10c. Rule: Change the signs of the subtrahend and proceed as in addition. Multiplication.— Multiply a by b. Ans. ab. Multiply ab by a 4- 6. Ans. a 2 b + a6 2 . Multiply a 4- 6 by a + 6. Ans. (a 4- &)(a 4- b) = a 2 4- 2a6 + 6 2 . Multiply — a by — 6. Ans. ab. Multiply - a by b. Ans. — a&. Like signs give plus, unlike signs minus. Powers of numbers, — The product of two or more powers of any number is the number with an exponent equal to the sum of the powers: a 2 x a 3 = a 5 ; a 2 6 2 xa6 = a 3 6 3 ; - lab x 2ac = - 14 a 2 6c. To multiply a polynomial by a monomial, multiply each term of the poly- nomial by the monomial and add the partial products: (6a — 36) x 3c = 18ac -36c. To multiply two polynomials, multiply each term of one factor by each term of the other and add the partial products: (5a — 66) x (3a — 46) = 15a 2 - 38a6 + 246 2 . The square of the sum of two numbers = sum of their squares 4- twice their product. The square of the difference of two numbers = the sum of their squares — twice their product. The product of the sum and difference of two numbers = the difference of their squares: (a 4- 6) 2 = a 2 4- 2a6 4- 6 2 ; (a - 6) 2 = a - 2a6 4- 6 2 ; (a 4- 6) x (a- 6) = a 2 -& 2 . The square of half the sums of two quantities is equal to their product plus ( a _j_ \ 2 / a - 6 \ 2 — - — ) = a6 4- ( — T~ ) The square of the sum of two quantities is equal to four times their prod- ucts, plus the square of their difference: (a 4 6j 2 — 4a6 4- (a — 6) 2 The sum of the squares of two quantities equals twice their product, plus the square of their difference: a 2 4- 6 2 = 2a6 + (a — 6j 2 . The square of a trinomial = the square of each term + twice the product of each term by each of the terms that follow it: (a -\-b -\-c)' 2 = a 2 4"6 2 4" c 2 4- 2ab 4- 2ac + 26c ; (a - 6 - c) 2 = a 2 + 6 2 4- c 2 - 2a6 - 2ac + 26c. The square of (any number 4- J^>) — square of the number + the number 4- 14; = the number X (the number + 1) 4- M; (a + ^) 2 = a 2 4-«4-M, =a(a 4-D + M- (4^) 2 =4 2 + 4 + M=4 x 5 + 14 = 2 0^. The product of any number 4- }/k by an .Y other number 4- V& = product of the numbers + half their sum 4- 14. (a 4- V>) X 64- H) = ab -\- J^(a+ 6)4- J4. 41/3 X 6^ = 4 X 6 4- 1/ 2 (4 4- 6) + 14 = 24 + 5 4- U = 29J4- Square, cube, 4tn power, etc., of a binomial a 4 6. (a 4- 6) 2 = a 2 + 2a6 4- 6 2 ; (a + 6) 3 = a 3 + 3a 2 6 4 3a6 2 + b 3 ; (a + 6) 4 = a 4 4- 4a 3 6 + 6a 2 6 2 + 4ab 3 + 6*. In each case the number of terms is one greater than the exponent of the power to winch the binomial is raised. 2. In the first term the exponent of a is the same as the exponent of the power to which the binomial is raised, aud it decreases by 1 in each succeed- ing term. 3. 6 appears in the second term with the exponent 1, and its exponent increases by 1 in each succeeding term. 4. The coefficient of the first term is 1. 5. The coefficient of the second term is the exponent of the power to which the binomial is raised. 6. The coefficient of each succeeding term is found from the next pre- ceding term by multiplying its coefficient by the exponent of a, and divid- ing the product by a number greater by 1 than the exponent of 6. (See Binomial Theorem, below.) 34 ALGEBRA. Parentheses.— When a parenthesis is preceded by a plus sign it may be removed without changing the value of the expression: a -f- b 4- (a + b) — 2a + 2b. When a parenthesis is preceded by a minus sign it may be removed if we change the signs of all the terms within the parenthesis: 1 — (a — b — c) = 1 — a -4- b + c. When a parenthesis is within a parenthesis remove the inner one first: a - \b -(d-e) }.] = a- [6_ Jc-d + e}] = a -■ [b - c +'d - e] = a - b + c - d + e. A multiplication sign, X, has the effect of a parenthesis, in that the oper- ation indicated by it must be performed before the operations of addition or subtraction, a -f 6 X a + b = a -4- ab -4- b; while (a -\- b) X (a -\- b) — a 2 + 2ab + 6 2 , and (a -f 6) X a + 6 = a 2 -f ab + 6. Division.— -The quotient is positive when the dividend and divisor have Jike signs, and negative when they have unlike signs: abc -*■■& = ac; abc -. b = — ac. To divide a monomial by a monomial, write the dividend over the divisor with a line between them. If the expressions have common factors, remove the common factors: a?bx ax a 4 a 3 1 _ 2 a 2 bx-r-aby= — r — = — ; — = a; — = — = a aby y a 3 a 5 a 2 To divide a polynomial by a monomial, divide each term of the polynomial by the monomial: (8ab — Y2ac) -=- 4a = 2b — 3c. To divide a polynomial by a polynomial, arrange both dividend and divi- sor in the order of the ascending or descending powers of some common letter, and keep this arrangement throughout the operation. Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient. Multiply all the terms of the divisor by the first term of the quotient and subtract the product from the dividend. If there be a remainder, consider it as a new dividend and proceed as before: (a 2 — ft 2 ) -*- (a -f- b). a 2 - o 2 | a + ft. cfl + ab\ a_-_b. - ab - 6 2 . - ab - V *. The difference of two equal odd powers of any two numbers is divisible by their difference and also by their sum: (a 3 - 6 3 ) ^ ( a _ 5) = a 2 + ab _|_ 52 . (a 3 _ 53) -h (a + &) = a 2 - a& + 6 2 . The difference of two equal even powers of two numbers is divisible by their difference and also by their sum: (a 2 — 6 2 ) -h (a — b) — a + b. The sum of two equal even powers of two numbers is not divisible by either the difference or the sum of the numbers; but when the exponent of each of the two equal powers is composed of an odd and an even factor, the sum of the given power is divisible by the sum of the powers expressed by the even factor. Thus x & -j- y 6 is not divisible by x + y or byx — y, but is divisible by x 2 -\- y 2 . Simple equations. — An equation is a statement of equality between two expressions; as, a -\- b = c -{- d. A simple equation, or equation of the first degree, is one which contains only the first power of the unknown quantity. If equal changes be made (by addition, subtraction, multiplication, or division) in both sides of an equation, the results will be equal. Any term may be changed from one side of an equation to another, pro- vided its sign be changed: a -f- b = c + d; a — c -\- d — b. To solve an equation having one unknown quantity, transpose all the terms involving the unknown quantity to one side of the equation, and all the other terms to the other side; combine like terms,.and divide both sides by the coefficient of the unknown quantity. Solve 8a; - 29 = 26 - 3x. 8x + Sx = 29 + 26; 11a; = 55; x = 5, ans. Simple algebraic problems containing one unknown quantity are solved by making x = the unknown quantity, and stating the conditions of the problem in the form of an algebraic equation, and then solving the equa- tion. What two numbers are those whose sum is 48 and difference 14 v Let x — the smaller number, x + 14 the greater, x -j- x -4- 14 = 48. 2a; = 34, x — 17; x -\- 14 = 31, ans. Find a number whose treble exceeds 50 as much as its double falls short of 40. Let x = the number. 3a; - 50 = 40 - 2a;; 5a; = 90; x - 18, ans. Prov^ ing, 54 - 50 = 40 - 36, • ALGEBRA. 35 liquations containing two unknown quantities.— If one equation contains two unknown quantities, x and y, an indefinite u umber of pairs of values of x and y may be found that will satisfy the equation, but if a second equation be given only one pair of values can be found that will satisfy both equations. Simultaneous equations, or those that may be satis- fied by the same values of the unknown quantities, are solved by combining the equations so as to obtain a single equation containing only one unknown quantity. This process is called elimination. Elimination by addition or subtraction. — Multiply the equation by such numbers as will make the coefficients of one of the unknown quanti- ties equal in the resulting equation. Add or subtract the resulting equa- tions according as they have unlike or like signs. bolve j 4x - 5y = 3. Subtract: 4x - 5y = 3 lly = .11; y = 1/ Substituting value of y in first equation, 2x + 3 = 7: x = 2. Elimination by substitution.— From one of the equations obtain the value of one of the unknown quantities in terms of the other. Substitu- tute for this unknown quantity its value in the other equation and reduce the resulting equations. '. j 2x + 3y = 8. (1). From (1) we find x = — ^—^ . Solve i „ , „ r /()1 2 t 3a; -f 7y = 7. (2). Substitute this value in (2): 3^ - : i— ) + 7y — 7; = 24 - 9y + Uy = 14, whence y — — 2. Substitute this value in (1): 2x — 6 = 8; x = 7. Elimination by comparison. — From each equation obtain the value of one of the unknown quantities in terms of the other. Form an equation from these equal values, and reduce this equation. I 3a; - 4y = 7. Equating these values of x, ~^- — - — — — ; 19^ = — 19; y = : — 1. Substitute this value of y in (1): 2a; + 9 = 11; x = 1. If three simultaneous equations are given containing three unknown quantities, one of the unknown quantities must be eliminated between two pairs of the equations: then a second between the two resulting equations. Quadratic equations. — A quadratic equation contains the square of the unknown quantity, but no higher power. A pure quadratic contains the square only; an affected quadratic both the square and the first power. To solve a pure quadratic, collect the unknown quantities on one side, and the known quantities on the other; divide by the coefficient of the un- known quantity and extract the square root of each side of the resulting equation. Solve 3a; 2 - 15 = 0. 3a; 2 = 15; a; 2 = 5; x = V5 A root like ^5, winch is indicated, but which can be found only approxi- mately, is called a surd. Solve 3a; 2 + 15 = 0. 3a; 2 = - 15; a; 2 = - 5; x = V- 5. The square root of — 5 cannot be found even approximately, for the square of any number positive or negative is positive; therefore a root which is in- dicated, but cannot be found even approximately, is called imaginary. To solve an affected quadratic. — 1. Convert the equation into the form a 2 * 2 ± 2abx = c, multiplying or dividing the equation if necessary, so as to make the coefficient of x 2 a square number. 2. Complete the square of the first member of the equation, so as to con- vert it to the form of a^x 2 ± 2abx -f- 6 2 , which is the square of the binomial ax ± b, as follows: add to each side of the equation the square of the quo- tient obtained by dividing the second term by twice the square root of the first term. 3, Extract the square root of each side of the resulting equation. Solve 3.x 2 — 4x = 32. To make the coefficient of a; 2 a square number, multiply by 3: 9a* 2 - 12a; = 96; 12a; -f- (2 x 3a;) = 2; 2 2 = 4. Complete the square: 9a; 2 - 12a; -\- 4 — 100. Extract the root: 3a; — 2 = ± 36 ALGEBRA. 10, whence x - 4 or - 2 2/3. The square root of 100 is either -j- 10 of - 10, since tlie square of — 10 as well as -j- 10 2 = 100. Problems involving quadratic equations have apparently two solutions, as a quadratic has two roots. Sometimes both will be true solutions, but gen- erally one only will be a solution and the other be inconsistent with the conditions of the problem. The sum of the squares of two consecutive positive numbers is 481. Find the numbers. Let x = one number, x + 1 the other, re 2 + (x + l) 2 = 481. 2a; 2 -f 2x + 1 = 481. x 2 + x — 240. Completing the square, a; 2 + x-\- 0.25 = 240.25. Extracting the root we obtain a* -j- 0.5 = ± 15.5; x = 15 or — 16. The positive root gives for ihe numbers 15 and 16. The negative root — 16 is inconsistent with the conditions of the problem. Quadratic equations containing two unknown quantities require different methods for their ^plution, according to the form of the equations. For these methods reference must be made to works on algebra. Theory of exponents.— ya when n is a positive integer is one of n equal factors of a. y a m means a is to be raised to the with power and the nth root extracted. {ya) '"means that the nth root of a is to be taken and the result raised to the mtli power. n — h - m m y a m = \ya ) = a n . When the exponent is a fraction, the numera- tor indicates a power, and the denominator a root, a? — y a = a 3 ; as = |/a3 = a 1 " 5 - To extract the root of a quantity raised to an indicated power, divide the exponent by the index of the required root; as, ty~a™ = a "Z "' 4/a 6 = J = a 2 . Subtracting 1 from the exponent of a is equivalent to dividing by a : a 2 -i = a J = a; a* - 1 = a = - = 1 ; a - 1 = a - 1 = - ; a- 1 -* = a ~' 2 = ~ 2 A number with a negative exponent denotes the reciprocal of the number with the corresponding positive exponent. • ' A factor under the radical sign whose root can be taken may, by having the root taken, be removed from under the radical sign: y/~(tfb = -f/o 2 x |/6 = a y'b. A factor outside the radical sign may be raised to the corresponding power and plact-d under it: \/\ = \/% - v^l = 1 ** \/i = 1 ^ Binomial Theorem.— To obtain any power, as the nth, of an ex- pression of the form x + a , , sn n , «-_i . n (n - !) aM ~ 2 ^.. 1 n (n - (a + x) n = a n -f na x -\ — x* -\ g g The following laws hold for any term in the expansion of (a -j- x) n . The exponent of x is less by one than the number of terms. The exponent of a is n minus the exponent of x. The last factor of the numerator is greater by one than the exponent of a. The last factor of the denominator is the same as the exponent of x. In the rth term the exponent of x will be r — 1. The exponent of a will be n — (r — 1). or n — r -4- 1. The last factor of the numerator will be n — r -\- 2. The last factor of the denominator will be = r — 1. Hence the rth term = "° l "^^ ~ 8) ' ' '^'iT^' + ^ a»-r+l x r-l GEOMETRICAL PROBLEMS. 3? GEOMETRICAL PROBLEMS. f -hr+- Y~ Fig 5. c D 1 1 1 1 1 . To bisect a straight line, or an arc of a circle (Fig. l).— From the ends A, B. as centres, de- scribe arcs intersecting at C and D, and draw a line through C and D which will bisect the line at E or the arc at F. 2. To draw a perpendicular to a straight line, or a radial line to a circular arc— Same as in Problem 1. C D is perpendicular to the line A B, and also radial to the arc. 3. To draw a perpendicular to a straight line from a given point in that line (Fig. a).— With any radius, from the given point A in the line B C, cut the line at B and C. With a longer radius describe arcs from B and C, cutting each other at D, and draw the perpendicular D ^4. 4. From the end A of a gievn line -1 J> to erect a perpendic- ular A JE (Fig. 3).— From any centre F, above A D, describe a circle passing through the given point A, and cut- ting the given line at D. Draw D F and produce it to cut the circle at E, and draw the perpendicular A E. Second Method (Fig. 4).— From the given point A set off a distance A E equal to three parts, by any scale ; and on the centres A and E, with radii of four and five parts respectively, describe arcs intersecting at C. Draw the perpendicular A C. Note.— This method is most useful oh very large scales, where straight edges are inapplicable. Any multiples of the numbers 3, 4, 5 may be taken with the same effect as 6, 8, 10, or 9, 12, 15. 5 . To draw a perpendicular to a straight line from any point without it (Fig. 5.)— From the point A y with a sufficient radius cut the given line at F and G, and from these points describe arcs cut- ting at E. Draw the perpendicular A E. 6. To draw a straight line parallel to a given line, at a given distance apart (Fig. 6). — From the centres A, B, in the given liue, with the given distance as radius, describe arcs C, D, aud draw the par- allel lines C D touching the arcs. 38 GEOMETRICAL PROBLEMS. Q 7. To divide a straight line Into a number of equal parts (Fig. 7).— To divide the line A B into, say, five parts, draw the line A C at an angle from .4; set off five equal parts; draw B 5 and draw parallels to it from the other points of division in A C. These parallels divide A B as required. No'iE.— By a similar process a line may be divided into a number of un- equal parts; setting off divisions on A C, proportional by a scale to the re- quired divisions, and drawing parallel cutting A B. The triangles ^411, A22, A&i, etc., are similar triangles. 8. Upon a straight line to draw an angle equal to a giveu angle (tig K).— Let A be lite given aiijrie and b G the line. From the point A with any radius describe the arc D E. From F with the same radius describe 1 H. Set off the arc / H equal to D E, and draw F H. The angle Fis equal to A, as required. 9. To draw angles of 60° and. 30° (Fig. 9).— From F, with any radius F I, describe an arc I J?; and from I, with the same radius, cut the arc at H and draw F H to form the required angle IF H. Draw the perpendicular H Kto the base line to form the angle of 30° F H K. 10. To draw an angle of 45° (Fig. 10).— Set off the distance F I; draw the perpendicular 1 H equal to IF, and join HF to form the angle at F. The angle at H is also 45°. 11. To bisect an angle (Fig. 11).— Let A C Bbe the angle; with G as a centre draw an arc cutting the sides at A, B. From A and B as centres, describe arcs cutting each other at D. Draw C D, dividing the angle into two equal parts. 12. Through two given points to describe an arc of a circle with a given radius (Fig. 12).— From the points A and B as centres, with the given radius, de- scribe arcs cutting at C; and from C with the same radius describe an arc A B. GEOMETRICAL PROBLEMS. 39 ■V 13. To find the centre of a circle or of an arc of a circle (Fig. 13). — Select three points, A, B, C, in the circumference, Avell apart; with the same radius describe arcs from these three points, cutting each other, and draw the two lines, D E, F G, through their intersections. The point 0, where they cut, is the centre of the circle or arc. To describe a circle passing through three given points. — Len A, B, C be the given points, and proceed as in last problem to find the centre 0, from which the circle may be described. 14. To describe an arc of a circle passing through three given points when the centre is not available (Fig. 14). —From the extreme points A, B, as centres, describe arcs A H, B G. Through the third point draw A E, B F, cutting the arcs. Divide A F and B E into any num- ber of equal parts, and set off a series of equal parts of the same length on the upper portions of the arcs beyond the points E F. Draw straight lines, B L, B M, etc., to the divisions in A F, and A I, A K, etc., to the divisions in E G. The successive intersections N, 0, etc., of these lines are points in the circle required between the given points A and C. which may be drawn in ; similarly the remaining part of the curve B C may be described. (See also Problem 54.) 15. To draw a tangent to a circle from a given point in the circumference (Fig. 15). —Through the given point A, draw the radial line A C, and a perpendicular to it, F G, which is the tangent re- quired. 16. To draw tangents to a circle from a point without it (Fig. 16).— From A, with the radius A C, describe an arc B CD, and from C. with a radius equal to the diameter of the circle, cut the arc at B D. Join B C, CD, cutting the circle at E F y and draw A E, A F, the tangents. Note.— When a tangent is already drawn, the exact point of contact may be found by drawing a perpendicular to it from the centre. 17. Between two inclined lines to draw a series of cir- cles touching these lines and touching each other (Fig. 1?). -Bisect the inclination of the given lines A B, C D, by the line N 0. From v point P in this line draw the perpendicular P B to the line A B, and 40 GEOMETRICAL PROBLEMS. A on P describe the circle B D, touching the lines and cutting: the centre line at E. From J? draw EF perpendicular to the centre line, cutting A B at F, and from F describe an arc E G, cut- ting A B at G. Draw G H parallel to B P, giving H, the centre of the next circle, to be described with the radius H E, and so on for the next circle IN. Inversely, the largest circle may be described first, and the smaller ones in succession. This problem is of fre- quent use in scroll-work. 18. Between two inclined lines to draw a circular seg- ment tangent to the lines and passing through a point F on the line^ C which bisects the angle of the lines (Fig. 18). — Through .Fdraw D A at right angles to F G ; bisect the angles A and D, as in Problem 11, by lines cutting at C, and from Cwith radius C Fdva,w the arc H F G required. 19. To draw a circular arc that will he tangent to two given lines A B and C D in- clined to one another, one tangential point E being given (Fig. 19).— Draw the centre line G F. From E draw E F at right to angles A B ; then F is the centre of the circle required. 20. To describe a circular arc joining two circles, and touching one of them at a given point (Fig. 20). — To join the circles A B, F G, by an arc touching one of them at F, draw the radius E F, and produce it both ways. Set off F H equal to the radius A C of the other circle; join C H and bisect it with the perpendicular LI, cutting E F at I. On the centre I, with radius IF, de- scribe the arc F A as required. 21. To draw a circle with a given radius It that will be tangent to two given circles A and Ji (Fig. 21).— From centre of circle A with radius equal B plus radius of A, and from centre of B with radius equal to R + radius of B, draw two arcs cutting each ot her in C, which will be the centre of the circle re- quired. 22. To construct an equi- lateral triangle, the sides being given (Fig. 22).— On the ends of out- side, A, B, with A B as radius, describe arcs cutting at C, and draw AC,CB, GEOMETRICAL PROBLEMS. 41 23. To construct a triangle of unequal sides (Fig. 23).— On either end of the base A D, with the side B as radius, describe an arc; and with the side C as radius, on the other end of the base as a centre, cut the arc at E. Join A E, D E. 24. To construct a square on a given straight line A B (Fig. 24). — At ..4 erect a perpendicular A C, as in Problem 4. Lay off A D equal to A B ; from D and B as centres with radius equal A B, describe arcs cutting each other in E. Join D E and BE. 25. To construct a rect- angle with given base M F and height E H (Fig. 25).— On the base E Fdvaw the perpendiculars EH, F G equal to the height, and join Q H. 26. To describe a circle about a triangle (Fig. 26).— Bisect two sides A B, A C of the tri- angle at E F, and from these points draw perpendiculars cutting at K. On the centre K, with the radius K A, draw the circle ABC. 27. To inscribe a circle in a triangle (Fig. 27).— Bisect two of the angles A, C, of the triangle by lines cutting at D ; from D draw a per- pendicular D Eto any side, and with D E as radius describe a circle. When the triangle is equilateral, draw a perpendicular from one of the angles to the opposite side, and from the. side set off one third of the per- pendicular. 28. To describe a circle about a square, and to in- scribe a square in a circle (Fig. 28). — To describe the circle, draw the diagonals A B, C D of the square, cut- ting at E. On the centre E, with the radius A E, describe the circle. To inscribe the square.— Draw the two diameters, A B, CD, at right angles, and join the points A, B, C D, to form the square. Note.— In the same way a circle may be described about a rectangle. 42 GEOMETRICAL PROBLEMS. 29* To inscribe a circle in a square (Fig. 29).— To inscribe the circle, draw the diagonals A B, C D of the square, cutting at E; draw the perpendicular E F to one side, and with the radius E F describe the circle. 30. To describe a square about a circle (Fig. 30).— Draw two diameters A B, CD at right angles. With the radius of the circle and A, B, C and D as centres, draw the four half circles which cross one another in the corners of the square. 31. To inscribe a pentagon in a circle (Fig. 31).— Draw diam- eters AC, B D at right angles, cutting at o. Bisect A o at E. and from E, with radius E B, cut A C at F ; from B, with radius B F, cut the circumfer- ence at G, H, and with the same radius step round the circle to Zand K\ join the points so found to form the penta- gon. 32. To construct a penta- gon on a given line A & (Fig. 32).— From B erect a perpendicular B G half the length of A B; join A C and prolong it to D, making CD — B C. Then B D is the radius of the circle circumscribing the pentagon. From A and B as centres, with B D as radius, draw arcs cutting each other in O, which is the centre of the circle. 33. To construct a hexagon upon a given straight line (Fig. 33).— From A and B, the ends of the given line, with radius A B, de- scribe arcs cutting at g ; from g, with the radius g A, describe a circle ; with the same radius set off the arcs A G, G F, and B D, D E. Join the points so found to form the hexagon. The side of a hexagon = radius of its circum- scribed circle. 34. To inscribe a hexagon in a circle (Fig. 34).— Draw a diam- eter A CB. From A and B as centres, with the radius of the circle A C, cut the circumference at D, E, F, G, and draw A D,D E, etc., to form the hexa- gon. The radius of the circle is equal to the side of the hexagon ; therefore the points D, E, etc., may also be found by stepping the radius six times round the circle. The angle between the diameter and the sides of a hexagon and also the exterior angle between a side and an adjacent side prolonged is 60 degrees; therefore a hexagon may conveniently be drawn by the use of a 60-degree triangle. L GEOMETRICAL PROBLEMS. 43 35. To describe a hexagon about a circle (Fig. 35).— Draw a diameter A D B, and with the radius A D, on the centre A, cut the circum- ference at C ; join A C, and bisect it with the radius D E ; through E draw FG, parallel to A C, cutting the diam- eter at F, and with the radius D ^de- scribe the circumscribing circled if. Within this circle describe a hexagon by the preceding problem. A more convenient method is by use of a 60- degree triangle. Four of the sides make angles of 60 degrees with the diameter, and the other two are par- allel to the diameter. 36. To describe an octagon on a given straight line (Fig. 36).— Produce the given line A B both ways, and draw perpendiculars A E, B F; bisect the external angles A and B by the lines A H, B C, which make equal to A B. Draw C D and H G par- allel to A E, and equal to A B ; from the centres G, D, with the radius A B, cut the perpendiculars at E, F, and draw S F to complete the octagon. 37. To convert a square into an octagon (Fig. 37).— Draw the diagonals of the square cutting at e ; from the corners A, B, C, D, with ieas radius, describe arcs cutting the sides at gn, fk, hm, and ol, and join the points so found to form the octagon. Adjacent sides of an octa- gon make an angle of 135 degrees. 38. To inscribe an octagon in a circle (Fig. 38).— Draw two diameters, A C, B D at right angles; bisect the arcs A B, B C, etc., at ef, etc., and join A e, e B, etc., to form the octagon. 39. To describe an octagon about a circle (Fig. 39).— Describe a square about the given circle A B ; draw perpendiculars h k, etc., to the diagonals, touching the circle to form the octagoD. 40. To describe a polygon of any number of sides upon a given straight line (Fig. 40).— Produce the given line A B, and on A, 44 GEOMETRICAL PROBLEMS. with the radius A B, describe a semi- circle; divide the semi-circumference into as many equal parts as there are to be sides in the polygon— say, in this example, five sides. Draw lines from A through the divisional points D, 6, and c, omitting one point a ; and on the centres B, D, with the radius AB, cut A b at E and A c at F. Draw D E, E F, F B to complete the polygon. 41. To inscribe a circle w it lain a polygon (Figs. 41, 42).— When the polygon has an even number of sides (Fig. 41), bisect two opposite sides at A audi?; draw AB. and bisect it at C by a diagonal D E, and with the radius C A describe the circle. When the number of sides is odd (Fig. 42), bisect two of the sides at A and B, and draw lines A E, B D to the opposite angles, intersecting at C ; from C, with the radius C A, describe the circle. 42. To describe a circle without a polygon (Figs. 41. 42). — Find the centre (J as before, and with the radius C D describe the circle. 43. To inscribe a polygon of any number of sides with- in a circle (Fig. 43).— Draw the diameter A B and through the centre E draw the perpendicular EC, cutting the circle at F. Divide E F into four equal parts, and set off three parts equal to those from F to G. Divide the diameter A B into as many equal parts as the polygon is to have sides ; and from C draw C D, through the second point of division, cutting the circle at D. Then A D is equal to one side of the polygon, and by stepping round the circumference with the length A D the polygon may be com- pleted. TABLE OF POLYGONAL ANGLES. Number Angle Number Angle Number Angle of Sides. at Centre. of Sides. at Centre. of Sides. at Centre. No. Degrees. No. Degrees. No. Degrees. 3 120 9 40 15 24 4 90 10 36 16 22£ 5 72 11 32 T 8 T 17 21 & 6 60 i 12 30 18 20 7 51f i 13 2?A 19 19 8 45 !- H 25f 20 18 GEOMETRICAL PROBLEMS. 45 In this table the angle at the centre is found by dividing 360 degrees, the number of degrees in a circle, by the number of sides in the polygon; and by setting off round the centre of the circle a succession of angles by means of the protractor, equal to the angle in the table due to a given number of sides, the radii so drawn will divide the circumference into the same number of parts. 44. To describe an ellipse when the length and breadth are given (Fig. 44).— A B, transverse axis; C D, conjugate axis; F G, foci. The sum of the distances from C to i^and G, also the sum of the distances from F and G to any other point in the curve, is equal to the transverse axis. From the centre C, with A E as radius, cut the axis A B at i^and G, the foci ; fix a couple of pins into the axis at F and G, and loop on a thread or cord upon them equal in length to the axis A B, so as when stretched to reach to the extremity C of the con- jugate axis, as shown in dot-lining. Place a pencil inside the cord as at H, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins F, G, and so describe the ellipse. Note.— This method is employed in setting off elliptical garden - plots, 2d Method (Fig. 45). — Along the straight edge of a slip of stiff paper mark off a distance a c equal to A C, half the transverse axis; and from the same point a distance a b equal to C D, half the conjugate axis. Place the slip so as to bring the point b on the line A B of the transverse axis, and the point c on the line D E ; and set off on the drawing the position of the point a. Shifting the slip so that the point b travels on the transverse axis, and the point c on the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. 3d Method (Fig. 46).— The action of the preceding method may be em- bodied so as to afford the means of describing a large curve continuously by means of a bar m fc, with steel points m, 1, fc, riveted into brass slides adjusted to the length of the semi- axis and fixed with set-screws. A rectangular cross E G, with guiding- slots is placed, coinciding with the two axes of the ellipse A C and B H. By sliding the points k, I in the slots, and carrying round the point m, the curve may be continuously described. A pen or pencil may be fixed at m. 4th Method (Fig. 47).— Bisect the transverse axis at C. and through C draw the perpendicular D E, making C D and C E each equal to half the conjugate axis. From D or E, with the radius A C, cut the transverse axis at F, F', for the foci. Divide A C into a number of parts at the 46 GEOMETRICAL PROBLEMS. G ;'' D "N ' x > i\\ aV h !' ' ! c | \\ •] avd^'i ' /"• 'A X^QL/ \j^r*/ &x!?"~~ E \i / If" B Fig. 48. points 1, 2, 3, etc. With the radius A I on F and .F 1 ' as centres, describe arcs, and with the radius B I on the same centres cut these arcs as shown. Repeat the operation for the other divisions of the transverse axis. The series of intersections thus made are points in the curve, through which the curve may be traced. 5th Method (Fig. 48).— On the two axes A B, D E as diameters, on cent re C, describe circles; from a number of points a, 6, etc., in the circumference A FB, draw radii cutting the inner circle at a', b', etc. From a, b, etc., draw perpendiculars to AB; and from a', b', etc., draw parallels to A B, cut- ting the respective perpendiculars at n, o. etc. The intersections are points in the curve, through which the curve may be traced. 6th Method (Fig. 49). — When the transverse and conjugate diameters are given, A B, C D, draw the tangent EF parallel to AB. Produce CD, and on the centre G with the radius of half A B, describe a semicircle HDK; from the centre G draw any number of straight lines to the points E, r, etc., in the line E F, cutting the circumference at I, m, n, etc.; from the centre O of the ellipse draw straight lines to the points E, r, etc. ; and from the points I, m, n, etc., draw parallels to G C, cutting the lines E, O r, etc., at L, M, N, etc. These are points in the circumference of the ellipse, and the curve may be traced through them. Points in the other half of the ellipse are formed by ex- tending the intersecting lines as indi- cated in the figure. 45. To describe an ellipse approximately by means of circular arcs.— First.— With arcs of two radii (Fig. 50).— Find the differ- ence of the two axes, and set it off from the centre O to a and c on O A and O C ; draw ac, and set off half a c to d ; draw d i parallel to a c; set off O e equal to O d; join e i, and draw the parallels e m, d m. From m, with radius m C, describe an arc through C ; and from i describe an arc through D; from d and e describe arcs through A and B. The four arcs form the ellipse approximately. Note.— This method does not apply satisfactorily when the conjugate axis is less than two thirds of the trans- verse axis. 2d Method (by Carl G. Barth, Fig. 51). —In Fig. 51 a b is the major and c d the minor axis of the ellipse to be approximated. Lay off b e equal to the semi-minor axis c O, and use a e as radius for the arc at each extremity of the minor axis. Bisect e o sit f and lay off e g equal to ef, and use g b as radius for the arc at each extremity of the major axis. Fig. 49. ^~~c \ c \ I /h a 'V\ J V" \ / V m B D Fig. 50. Fig. 51. GEOMETRICAL PROBLEMS. 47 £^~? X^K"""! s'\ Fig J i ) The method is not considered applicable for cases in which the minor axis is less than two thirds of the major. , 3d Method : With arcs of three radii (Fig. 52).— On the transverse axis A B draw the rectangle B G on the height 0(7; to the diagonal A C draw the perpendicular G H D\ set off O K ,^\ \^ equal to O C, and describe a semi- IjjK. y\ \ / X sy circle on A K, and produce OCtoL; // v, V N / ,Jp\ \j set off O If equal to C L, and from D - ~^— A— , — !- — i- '-'—i — _-J describe an arc with radius D M ; from A, with radius O L, cut this arc at a. Thus the five centres D, a, b, H, H' are found, from which the arcs are described to form the ellipse. Note. — This process works well for nearly all proportions of ellipses. It is employed in striking out vaults and stone bridges. 4th Method (by F. R. Honey, Figs. 53 and 54).— Three radii are employed. With the shortest radius describe the two arcs which pass through the ver- tices of the major axis, with the longest the two arcs which pass through the vertices of the minor axis, and with the third radius the four arcs which connect the former. A. simple method of determining the radii of curvature is illustrated in Fig. 53. Draw the straight lines a /and a c, forming any angle at a. With a as a centre, and with radii a b and ac, re- spectively, equal to the semi- ininor and semi-major axes, draw the arcs b e and c d. Join ed, and through b and c r speetively draw b g and c / parallel to e d, intersecting a c at g, and af at/; af is the radius of curvature at the ver- tex of the minor axis; and a g the radius of curvature at the vertex of the major axis. Lay off d h (Fig. 53) equal to one eighth of b d. Join e h, and draw c k and o l parallel to e h Take a k for the longest radius (=B),al for the shortest ramus (= r) and the arithmetical mean, or one half the sum of the semi-axes, as'follows- US (= p) ' and e,n P lo y these radii for the eight-centred oval Let a b and c d (Fig. 54) be the major and minor axes. Lay. off a e equal to r, and af equal to p; also lay off c g equal to B, and c h equal to p. With g as a, centre and g h as a radius, draw the arc h k; with the centre e and radius e f draw the arc fk, intersecting/ifc at k. Draw the line g k and produce it, making g I equal to B. Draw k e and produce it, making k m equal to p. With the centre g and radius g c (= B) draw the arc c I ; with the centre k and radius k I (= p) draw the arc I m, and with the centre e and radius e m (= r) draw the arc m a- The remainder of the work is symmetrical with respect to the axes. 48 GEOMETRICAL PROBLEMS. K i L A E G nL. F v 71/ o \ V o \ o \ % D B b 'a c Fig. 55. ' 46. The Parabola. —A parabola (D A C, Fig. 55) is a curve such that every point in the curve is equally- distant from the directrix KL and the focus F. The focus lies in the axis A B drawn from the vertex or head of the curve A, so as to divide the figure into two equal parts. The vertex A ' is equidistant from the directrix and the focus, or A e — A F. Any line parallel to the axis is a diameter. A straight line, as E G or DC, drawn across the figure at right angles to the axis is a double ordinate, and either half of it is an ordinate. The ordinate to the axis E F G, drawn through the focus, is called the parameter of the axis. A segment of the axis, reckoned from the vertex, is an abscissa of the axis, and it is an abscissa of the ordi- nate drawn from the base of the ab- scissa. Thus, A B is an abscissa of the ordinate B C. Abscissae of a parabola are as the squares of their ordinates. To describe a parabola when an abscissa and its ordi- nate are given (Fig. 55).— Bisect the given ordinate B Cat a. draw A a, and then a b perpendicular to it, meeting the axis at b. Set off A e. A F, each equal to B b; and draw K e L perpendicular to the axis. Then K L is the directrix and F is the focus. Through F and any number of points, o, o, etc., in the axis, draw doubJe ordinates, n o n, etc , and from the centre F, with the radii Fe, o e, etc., cut the respective ordinates at E, G, n, n, etc. The curve may be traced through these points as shown. 2d Method : By means of a square and a cord (Fig. 56).— Place a straight- edge to the directrix EN, and apply to it a square LEG. Fasten to the end G one end of a thread or cord equal in length to the edge E G, and attach the other end to the focus F ; slide the square along the straight- edge, holding the cord taut against the eiige of the square by a pencil D, by which the curve is described. ; b a B a b c d Fig. 57. Zd Method : When the height and the base are given (Fig. 57). — Let A B be the given axis, and CD a double ordinate or b»se; to describe a para- bola of which the vertex is at A. Through A draw EF parallel to CD, and through C and D draw C E and D F parallel to the axis. Divide B C and B D into any number of equal parts, say five, at a, b, etc., and divide C E and DF'iuto the same number of parts. Through the points a, 6, c, d in the base CD on each side of the axis draw perpendiculars, and through a,b,c, din C E and D F draw lines to the vertex A, cutting the perpendicu- lars at e, /, g, h. These are points in the parabola, and the curve C AD may be traced as shown, passing through them. GEOMETRICAL PROBLEMS. 49 Fig. 59. 47. The Hyperbola (Fig. 58).— A hyperbola is a plane curve, such that the difference of the distances from any point of it to two fixed points is equal to a given distance. The fixed points are called the foci. To construct a hyperbola. —Let F' and F be the foci, and F' F the distance between them. Take a ruler longer than the distance F' F, and fasten one of its extremities at the focus F'. At the other extremity, H, attach a thread of such a length that the length of the ruler shall exceed the length of the thread by a given distance A B. Attach the other ex- tremity of the thread at the focus F. Press a pencil, P, against the ruler, and keep the thread constantly tense, while the ruler is turned around F' as a centre. The point of the pencil will describe one branch of the curve. 2d Method: By points (Fig. 59).— From the focus F' lay off a distance F' N equal to the transverse axis, or distance between the two branches of the curve, and take any other distance, as F'H, greater than F'N. With F" as a centre and F'H as a radius describe the arc of a circle. Then with .Fas a centre and N H as a radius describe an arc intersecting the arc before described at p and q. These will be points of the hyperbola, for F' q — Fq is equal to the trans- verse axis A B. If, with F as a centre and F' H as a radius, an arc be described, and a second arc be described with F' as a centre and N H as a radius, two points in the other branch of the curve will be determined. Hence, by changing the centres, each pair of radii will determine two points in each branch. The Equilateral Hyperbola.— The transverse axis of a hyperbola is the distance, on a line joining the foci, between the two branches of the curve. The conjugate axis is a line perpendicular to the transverse axis, drawn from its centre, and of such a length that the diagonal of the rect- angle of the transverse and conjugate axes is equal to the distance between the foci. The diagonals of this rectangle, indefinitely prolonged, are the asymptotes of the hyperbola, lines which the curve continually approaches, but touches only at an infinite distance. If these asymptotes are perpen- dicular to each other, the hyperbola is called a rectangular or equilateral hyperbola. It is a property of this hyperbola that if the asymptotes are taken as axes of a rectangular system of coordinates (see Analytical Geom- etry), the product of the abscissa and ordinate of any point in the curve is equal to the product of the abscissa and ordinate of any other point ; or, if p is the ordinate of any point and v its abscissa, and p x and v } are the ordi- nate and abscissa of any other point, pv—pi v x ; or pv = a constant. 48. The Cycloid (Fig. 60).— If a circle Ad be rolled along a straight line ^4 6, any point of the circumference as A will describe a curve, which is called a cycloid. The circle is called the generating circle, and A the generat- ing point. To draw a cycloid. — Divide the circumference of the generating circle into an even number of equal parts, as A 1, 12, etc., and set off these distances on the base. Through the points 1, 2, 3, etc., on the circle draw horizontal lines, and on them set off distances la — Al, 26 = A2, 'ic = AS, etc. The points A, a, b, c, etc., will be points in the cycloid, through which draw the curve. 50 GEOMETRICAL PROBLEMS. 49. The Epicycloid (Fig. 61) is generated by a point D in one circle D C rolling upon the circumference of another circle A C B, instead of on a flat surface or line; the former being the generating circle, and the latter the fundamental circle. The generat- ing circle is shown in four positions, in which the generating point is succes- sively marked D, D', D", £>'". A D'" B is the epicycloid. 50. The Hypocycloid (Fig. 62) is generated by a point in the gener- ating circle rolling on the inside of ihe fundamental circle. When the generating circle = radius of the other circle, the hypocycloid becomes a straight line. 51. The Tractrix or Schiele's anti-friction curve (Fig. 63).— R is the radius of the shaft, C, 1,2, etc., the axis. From O set off on R a small distance, oa; with radius R and centre a cut the axis at 1, join a 1, and set off a like small distance ab; from b with radius R cut axis at 2, join b 2, and so on, thus finding points o, a, 6, c, d, etc., through which the curve is to be drawn. Fig. 63. 52. The Spiral.— The spiral is a curve described by a point which moves along a straight line according to any given law, the line at the same time having a uniform angular motion. The line is called the radius vector. If the radius vector increases directly as the measuring angle, the spires, or parts described in each revolution, thus gradually increasing their dis- tance from each other, the curve is known as the spiral of Archimedes (Fig. 64). This curve is commonly used for cams. To, describe it draw the radius vector in several different directions around the centre, with equal angles between them; set off the distances 1, 2, 3, 4, etc., corresponding to the scale upon which the curve is drawn, as shown in Fig. 64. In the common spiral (Fig. 64) the pitch is uniform; that is, the spires are equidistant. Such a spiral is made by rolling up a belt of uniform thickness. To construct a spiral with four centres (Fig. 65).— Given the pitch of the spiral, construct a square about the centre, with the sum of the four sides equal to the pitch. Prolong the sides in one direction as shown; the corners are the centres for each arc of the external angles, forming a quadrant of a spire. Fig. 65. GEOMETRICAL PROBLEMS. 51 Fig. 66. 53. To find the diameter of a circle into which a certain number of rings will fit on its inside (Fig. 66).— For instance, what is the diameter of a circle into which twelve ^-h>ch rings will fit. as per sketch ? Assume that we have found the diameter of the required circle, and have drawn the rings inside of it. Join the centres of the rings by straight lines, as shown : we then obtain a regular polygon with 12 sides, each side being equal to the di- ameter of a given ring. We have now to find the diameter of a circle cir- cumscribed about this polygon, and add the diameter of one ring to it; the sum will be the diameter of the circle into which the rings will fit. Through the centres A and D of two adjacent rings draw the radii CA and CD; since the polygon has twelve sides the angle A C D = 30° and A C B = 15°. One half of the side A D is equal to A B. We now give the following pro- portion : The sine of the angle AC B is to AB as 1 is to the required ra- dius. From this we get the following rule : Divide A B by the sine of the angle A C B ; the quotient will be the radius of the circumscribed circle ; add to the corresponding diameter the diameter of one rine: ; the sum will be the required diameter FG. 54. To describe an arc of a circle which is too large to be drawn by a beam compass, by means of points in the arc, radius being given.— Suppose the radius is 20 feet and it is desired to obtain five points in an arc whose half chord is 4 feet. Draw a line equal to the half chord, full size, or on a smaller scale if more con- venient, and erect a perpendicular at one end, thus making rectangular axes of coordinates. Erect perpendiculars at points 1, 2, 3, and 4 feet from the first perpendicular. Find values of y in the formula of the circle. a* 2 + 2/ 2 = B 2 by substituting for x the values 0, 1, 2, 3, and 4, etc., and for H 2 the_square of _the r adiu s, or 400. The values will he y — V R 2 — x 2 — ^400, ^399, ^396, V 391, V 384; = 20, 19.975, 19.90, 19.774, 19.596. Subtract the smallest, or 19.596, leaving 0.404, 0.379, 0.304, 0.178, feet. Lay off these distances on the five perpendiculars, as ordinates from the half chord, and the positions of five points on the arc will be found. Through these the curve may be drawn. (See also Problem 14.) 55. The Catenary is the curve assumed by a, perfectly flexible chord when its ends are fastened at two points, the weight of a unit length being constant. The equation of the catenary is Fig. 67. in which e is the base of the Naperian system of log- arithms. To plot the catenary.— Let o (Fig. 67) be the origin of coordinates. Assigning to a any value as 3, the equation becomes = ?le3 (* + .-.-). To find the lowest point of the curve. q/ o -o\ o Vxxtx = 0\.-.y=ile fe J = |(1 y- 1) = 3. 52 GEOMETRICAL PROBLEMS. Then put x = 1 ; (1.396 + 0.717) = 3.17. -(1.948 4- 0.513) = l Put x = 3, 4, 5, etc., etc., and find the corresponding values of y. For each value of y we obtain two symmetrical points, as for example p and p 1 . In this way, by making a successively equal to 2, 3, 4, 5, 6, 7, and 8, the curves of Fig. 68 were plotted. In each case the distance from the origin to the lowest point of the curve is equal to a ; for putting x = o, the general equation reduces to y — a. For values of a — 6, 7, and 8 the catenary closely approaches the parabola. For derivation of the equation of the catenary see Bowser's Analytic Mechanics. For comparison of the catenary with the parabola, see article by F. It. Honey, Amer. Machinist, Feb. 1, 1894. 56. The Involute is a name given to the curve which is formed by the end of a string which is unwound from a cylinder and kept taut ; con- sequently the string as it is unwound will always lie in the direction of a tangent to the cylinder. To describe the involute of any given circle, Fig. 68, take any point A on its circum- ference, draw a diameter A B, and from B draw B b perpendicular to AB. Make Bb equal in length to half the circumference of the circle. Divide Bb and the semi-circumference into the same number of equal parts, say six. From each point of division 1, 2, 3, etc., on the circumference draw lines to the centre C of the circle. Then draw 1 a perpendicular to C 1 ; 2a, 2 perpendicular to C2; and so on. Make 1 a equal to b />, ; 2 q a equal to b 6 2 ; 3 a 3 equal to. ft b 3 ; and so on. by a curve ; this curve will be the Fig. 68. , etc., Join the 'points A, a x \ > required involute. 57. Method of plotting angles without using a protrac- tor. --The radius of a circle whose circumference is 360 is 57.3 (more ac- curately 57.296). Striking a semicircle with a radius 57.3 by any scale, spacers set to 10 by the same scale will divide the arc into 18 spaces of 10° each, and intermediates can be measured indirectly at the rate of 1 by scale for each 1°, or interpolated by eye according to the degree of accuracy required. The following table shows the chords to the above-mentioned radius, for every 10 degrees from 0° up to 110°. By means of one of these, Angle, 10°. 20°. 40° 50° Chord. , 0.999 . 9.988 . 19.899 . 29.658 . 39.192 . 48.429 Angle. 60° . . . 70°... 80°... 90°... 100°... 110°... Chord. 57.296 . 65.727 . 73.658 . 81.029 . 87.782 a 10° point is fixed upon the paper next less than the required angle, and the remainder is laid off at the rate of 1 by scale for each degree. GEOMETRICAL PROPOSITIONS. 53 GEOMETRICAL PROPOSITIONS. In a right-angled triangle the square on the hypothenuse is equal to the sum of the squares on the other two sides. If a triangle is equilateral, it is equiangular, and vice versa. If a straight line from the vertex of an isosceles triangle bisects the base, it bisects the vertical angle and is perpendicular to the base. If one side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite angles. If two triangles are mutually equiangular, they are similar and their corresponding sides are proportional. If the sides of a polygon are produced in the same order, the sum of the exterior angles equals four right angles. In a quadrilateral, the sum of the interior angles equals four right angles. In a parallelogram, the opposite sides are equal ; the opposite angles are equal; it is bisected by its diagonal; and its diagonals bisect each other. If three points are not in the same straight line, a circle may be passed through them. If two arcs are intercepted on the same circle, they are proportional to the corresponding angles at the centre. If two arcs are similar, they are proportional to their radii. The areas of two circles are proportiona"l to the squares of their radii. If a radius is perpendicular to a chord, it bisects the chord and it bisects the arc subtended by the chord. A straight line tangent to a circle meets it in only one point, and it is perpendicular to the radius drawn to that point. If from a point without a circle tangents are drawn to touch the circle, there are but two; they are equal, and they make equal angles with the chord joining the tangent points. Jf two lines are parallel chords or a tangent and parallel chord, they intercept equal arcs of a circle. If an angle at the circumference of a circle, between two chords, is sub- tended by the same arc as an angle at the centre, between two radii, the angle at the circumference is equal to half the angle at the centre. If a triangle is inscribed in a semicircle, it is right-angled. If an angle is formed by a tangent and chord, it is measured by one half of the arc intercepted by the chord; that is, it is equal to half the angle at the centre subtended by the chord. If two chords intersect each other in a circle, the rectangle of the seg- ments of the one equals the rectangle of the segments of the other. And if one chord is a diameter and the other perpendicular to it, the rectangle of the segments of the diameter is equal to the square on half the other chord, and the half chord is a mean proportional between the seg- ments of the diameter. 54 MENSURATION. MENSURATION. PLANE SURFACES. Quadrilateral.— A four-sided figure. Parallelogram.— A quadrilateral with opposite sides parallel. Varieties.— Square : four sides equal, all angles right angles. Rectangle: opposite sides equal, all angles right angles. Rhombus: four sides equal, opposite angles equal, angles not right angles. Rhomboid: opposite sides equal, opposite angles equal, angles not right angles. Trapezium.— A quadrilateral with unequal sides. Trapezoid. — A quadrilateral with only one pai* of opposite sides parallel. Diagonal of a square = |/2x side' 2 = 1.4142 x side. Diagonal of a rectangle = ^/product of two adjacent sides. Area of any parallelogram = base x altitude. Area of rhombus or rhomboid = product of two adjacent sides X sine of angle included between them. Area of a trapezium = half the product of the diagonal by the sum of the perpendiculars let fall on it from opposite angles. Area of a trapezoid = product of half the sum of the two parallel sides by the perpendicular distance between them. To find the area of any quadrilateral figure.— Divide ihe quadrilateral into two triangles; the sum of the areas of the triangles is the area. Or, multiply half the product of the two diagonals by the sine of the angle at their intersection. To find the area of a quadrilateral inscribed in a circle. —From halt' the sum of the four sides subtract each side severally; multi- ply the four remainders together; the square root of the product is the area. Triangle.— A three-sided plane figure. Varieties.— Right-angled, having one right angle; obtuse-angled, having one obtuse angle ; isosceles, having two equal angles and two equal sidet.; equilateral, having three equal sides and equal angles. The sum of the three angles of every triangle — 180°. The two acute angles of a right-angled triangle are complements of each other. Hypothenuse of a right-angled triangle, the side opposite the right angle. = |/sum of the squares of the other two sides. To find the area of a triangle : Rule 1. Multiply the base by half the altitude. Rule 2.- Multiply half the product of two sides by the sine of the included angle. Rule 3. From half the sum of the three sides subtract each side severally ; multiply together the half sum and the three remainders, and extract the square root of the product. The area of an equilateral triangle is equal to one fourth the square of one of its sides multiplied by the square root of 3, = ~ , a being the side; or 4 a 2 X .433013. Hypothenuse and one side of right-angled triangle given, to find other side, Required side = ^hyp 2 — given side 2 . If the two sides are equal, side = hyp -f- 1.4142; or hyp X .7071. Area of a triangle given, to find base: Base = twice area -5- perpendicular height. Area of a triangle given, to find height: Height = twice area -*- base. Two sides and base given, to find perpendicular height (in a triangle in which both of the angles at the base are acute). Rule.— As the base is to the sum of the sides, so is the difference of the sides to the difference of the divisions of the base made by drawing the per- pendicular. Half this difference being added to or subtracted from half the base will give the two divisions thereof. As each side and its opposite PLAtfE SURFACES. 55 division of the base constitutes a right-angled triangle, the perpendicular is ascertained by the rule perpendicular = Vhyp 2 — base 2 . Polygon. — A plane figure having three or more sides. Regular or irregular, according as the sides or angles are equal or unequal. Polygons are named from the number of their sides and angles. To find the area of an Irregular polygon.— Draw diagonals dividing the polygon into triangles, and find the sum of the areas of these triangles. To find the area of a regular polygon : Rule.— Multiply the length of a side by the perpendicular distance to the centre; multiply the product by the number of sides, and divide it by 2. Or, multiply half the perimeter by the perpendicular let fall from the centre on one of the sides. The perpendicular from the centre is equal to half of one of the sides of the polygon multiplied by the cotangent of the angle subtended by the half side. The angle at the centre = 360° divided by the number of sides. TABLE OF REGULAR POLYGONS. Radius of Cir- cumscribed t_j 8 Ci •cle. .c-* °1§ < bo 5 II a ^ >> 11 £-• CO CD T3 a £2 CD 5 02 o cd 02 g ll II cw&2 Si" O a e3 530 -3 18 g^O to 11 a3 C-r-> £ fc < Ph 02 H M < < 3 Triangle .4330127 2. .5773 .2887 1.732 120° 60° 4 Square 1. 1.414 .7071 .5 1.4142 90 90 5 Pentagon 1.7204774 1.238 .8506 .6882 1.1756 72 108 6 Hexagon 2 5980762 1.156 1. .866 1. 60 120 7 Heptagon 3.6339124 1.11 1.1524 1.0383 .8677 5126' 128 4-7 8 Octagon 4.8284271 1.083 1.3066 1.2071 .7653 45 135 9 Nonagon 6.1818242 1.064 1.4619 1.3737 .684 40 140 10 Decagon 7.6942088 1.051 1.618 1.5388 .618 36 144 11 Undecagon 9.3656399 1.042 1.7747 1.7028 .5634 32 43' 147 3-11 12 Dodecagon 11.1961524 1.037 1.9319 1.866 .5176 30 150 To find the area of a regular polygon, when the length of a side only is given : Rule. — Multiply the square of the side by the multiplier opposite to the name of the polygon in the table. To find the area of an ir- regular figure (Fig. 69).— Draw or- di nates across its breadth at equal distances apart, the first and the last ordinate each being one half space from the ends of the figure. Find the average breadth by adding together the lengths of these lines included be- tween the boundaries of the figure, and divide by the number of the lines added ; multiply this mean breadth by the length. The greater the number of lines the nearer the approximation. 2 3 ~J - -Length Fig. 69. In a, figure of very ^regular outline, as an indicator-diagram from a high- speed steam-engine, mean lines may be substituted for the actual lines of the figure, being so traced as to intersect the undulations, so that the total area of the spaces cut off may be compensated by that of the extra spaces in- closed. 56 MENSURATlOtf. 2d Method: The Trapezoidal Rule. — Divide the figure into any suffi- cient number of equal parts; acid half the sum of the two end ordinates to the sum of all the other ordinates; divide by the number of spaces (that is, one less than the number of ordinates) to obtain the mean ordinate, and multiply this by the length to obtain the area. 3d Method : Simpson's Rule.— Divide the length of the figure into any even number of equal parts, at the common distance D apart, and draw or- dinates through the points of division to touch the boundary lines. Add together the first and last ordinates and call the sum A ; add together the even ordinates and call the sum B; add together the odd ordinates, except the first and last, and call the sum C. Then, area of the figure = 4th Method : Durand's Rule.— Add together 4/10 the sum of the first and last ordinates, 1 1/10 the sum of the second and the next to the last (or the penultimates), and the sum of all the intermediate ordinates. Multiply the sum thus gained by the common distance between the ordinates to obtain the area, or divide this sum by the number of spaces to obtain the mean or- dinate. Prof. Durand describes the method of obtaining his rule in Engineering News, Jan. 18, 1894. He claims that it is more accurate than Simpson's rule, and practically as simple as the trapezoidal rule. He thus describes its ap- plication for approximate integration of differential equations. Any defi- nite integral may be represented graphically by an area. Thus, let -/' u dx be an integral in which u is some function of x, either known or admitting of computation Or measurement. Any curve plotted with x as abscissa and u as ordinate will then represent the variation of u with x, and the area be- tween such curve and the axis Xwill represent the integral in question, no matter how simple or complex may be the real nature of the function u. Substituting in the rule as above given the word " volume " for "area " and the word " section " for " ordinate," it becomes applicable to the deter- mination of volumes from equidistant sections as well as of areas from equidistant ordinates. Having approximately obtained an area by the trapezoidal rule, the area by Durand's rule may be found by adding algebraically to the sum of the ordinates used in the trapezoidal rule (that is, half the sum of the end ordi- nates + sum of the other ordinates) 1/10 of (sum of penultimates — sum of first and last) and multiplying by the common distance between the other ordinates. 5th Method.— Draw the figure on cross-section paper. Count the number of squares that are entirely included within the boundary; then estimate the fractional parts of squares that are cut by the boundary, add together these fractions, and add the sum to the number of whole squares. The result is the area in units of the dimensions of the squares. The finer the ruling of the cross-section paper the more accurate the result. 6th Method.— Use a planimeter. 7th Method.— -With a chemical balance, sensitive to one milligram, draw the figure on paper of uniform thickness and cut it out carefully; weigh the piece cut out, and compare its weight with the weight per square inch of the paper as tested by weighing a piece of rectangular shape. THE CIRCLE. 57 THE CIRCLE. Circumference = diameter x 3.1416, nearly; more accurately, 3.14159265359. 22 355 Approximations, — = 3.143; — — = 3.1415929. The ratio of circum. to diam. is represented by the symbol Multiples of 7T. 1tt = 3.14159265359 2tt = 6.28318530718 3tt= 9.42477796077 4tt = 12.56637001436 577 = 15.70796326795 for =18.84955592154 7ir = 21.99114857513 8n = 25.13274122872 9tt = 28.27433388231 Ratio of diam. to circumference = (called Pi). Multiples of n -. r = .7853982 Reciprocal of -n = 1 .2732 Multiples of -. — = .31831 n — = .63662 IT — = .95493 -=1.27324 — = 1.59155 — = 1.90986 x 2 = 1.5707963 x 3 = 2.3561945 x 4 = 3.1415927 x 5 = 3.9269908 x 6 = 4.7123890 x 7 = 5.4977871 x 8 = 6.2831853 x 9 = 7.0685835 ■eciprocal of n = 0.3183099. 0.2617! — = 2.22817 - = 2.54648 10 _ 12 = 1.570796 = 1.047197 12 - = 0.101321 7T 2 Vn = 1.772453 - A ™ • -VTHTTr. _ Diameter Versed sine = ^— ; = ~(D - V D*- Cd*) l(D + VD* - Cd 2 ), if F is greater than radius. -V. «».-^. Half the chord of the arc is a mean proportional between the versed sine and diameter minus versed sine : \cd = Vv x {D- V). Length of a Circular Arc— Huyghens's Approximation. Let C represent the length of the chord of the arc and c the length of the chord of half the arc; the length of the arc r 8c- C L = ~S~' Professor Williamson shows that when the arc subtends an angle of 30°, the radius being 100,000 feet (nearly 19 miles), the error by this formula is about two inches, or 1/600000 part of the radius. When the length of the arc is equal to the radius, i.e., when it subtends an angle of 57°. 3, the error is less than 1/7680 part of the radius. Therefore, if the radius is 100.000 feet, the error is less than ' = 13 feet. The error increases rapidly with the increase of the angle subtended. In the measurement of an arc which is described with a short radius the error is so small that it may be neglected. Describing an arc with a radius of 12 inches subtending an angle of 30°, the error is 1/50000 of an inch. For 57°. 3 the error is less than 0".0015. In order to measure an arc when it subtends a large angle, bisect it and measure each half as before— in this case making B = length of the chord of half the arc, and b — length of the chord of one fourth the arc ; then T 166 - 2B L = 8~ " Relation ot the Circle to its Equal, Inscribed, and Cir- cumscribed Squares. Diameter of circle x .88623 ( _ - , - , Circumference of circle x .28209 f - side ot equal squaie. Circumference of circle x 1.1281 = perimeter of equal square, THE ELLIPSE. 59 Diameter of circle x .7071 ) Circumference of circle x .22508 >• = side of inscribed square. Area of circle x .90031-f- diameter ) Area of circle x 1.2732 = area of circumscribed square. Area of circle x .63662 = area of inscribed square. Side of square x 1.4142 = diam. of circumscribed circle. " " x 4.4428 = circum. " " " " " x 1.1284 = diam. of equal circle. " " x 3.5449 = circum. " " Perimeter of square x 0.88623 = " " " Square inches x 1.2732 = circular inches. Sectors and Segments.- To find the area of a sector of a circle. Rule 1. Multiply the arc of the sector by half its radius. Rule 2. As 360 is to the number of degrees in the arc, so is the area of the circle to the area of the sector. Rule 3. Multiply the number of degrees in the arc by the square of the radius and by .008727. To find the area of a segment of a circle: Find the area of the sector which has the same arc, and also the area of the triangle formed by the chord of the segment and the radii of the sector. Then take the sum of these areas, if the segment is greater than a semi- circle, but take their difference if it is less. Another Method: Area of segment =-- (arc — sin A) in which A is the central angle, R the radius, and arc the length of arc to radius 1. To find the area of a segment of a circle when its chord and height or versed sine only are given. First find radius, as follows : 1 rsquare of half the chord , . ...,~] radius = - [_ 5^ + height J. 2. Find the angle subtended by the arc, as follows: ^ = sine radius of half the angle. Take the corresponding angle from a table of sines, and double it to get the angle of the arc. 3. Find area of the sector of which the segment is a part ; „ . , degrees of arc area of sector = area of circle x -— . ooU 4. Subtract area of triangle under the segment: ~ vl «,^«o — height of segment). The remainder is the area of the segment. When the chord, arc, and diameter are given, to find the area. From the length of the arc subtract the length of the chord. Multiply the remainder by the radius or one-half diameter; to the product add the chord multiplied by the height, and divide the sum by 2. Another rule: Multiply the chord by the height and this product by .6834 plus one tenth of the square of the height divided by the radius. To find the chord: From the diameter subtract the height; multiply the remainder by four times the height and extract the square root. When the chords of the arc and of half the arc and the versed sine are given: To the chord of the arc add four thirds of the chord of half the arc; multiply the sum by the versed sine and the product by .40426 (approximate). Circular Ring. — To find the area of a ring included between the cir- cumferences of two concentric circles: Take the difference between the areas of the two circles; or, subtract the square of the less radius from the square of the greater, and multiply their difference by 3.14159. The area of the greater circle is equal to irR*; and the area of the smaller, nr 2 . Their difference, or the area of the ring, is 7r(i? 2 - r 2 ). The Ellipse.— Area of an ellipse = product of its semi-axes x 3.14159 = product of its axes x .785398. 1416^ + Tlie Ellipse.— Circumference (approximate) = 3.1416 V — ^ — , D and d being the two axes. Trautwine gives the following as more accurate: When the longer axis D is not more than five times the length of the shorter axis, d, 60 MENSURATION". Circumfei ence = 3.1416 ^ When D is more than 5d, the divisor 8 .8 is divisors : D d 6, 7, 8, 9, 10, 12, 14, Divisor = 9 9.2, 9.3, 9.35, 9.4, 9.5, 9.G, _ ( D- d) 2 8.8 is to be replaced by the following 16, 18, 20, 30, 40, 50. 9.68, 9.75, 9.8, 9.92, 9.98, 10. / w 2 n* n 6 \ Reuleaux gives : Circumference = n (a + b)yi -f T+ fi4+or fi +--« )» in which n = — r— , , a and b being the semi-axes. a 4-6 Area of a segment of an ellipse the base of which is parallel to one of the axes of the ellipse. Divide the height of the segment by the axis of which it is part, and find the area of a circular segment, in a table of circu- lar segments, of which the height is equal to the quotient; multiply the area thus found by the product of the two axes of the ellipse. Cycloid.— A curve generated by the rolling of a circle on a plane. Length of a cycloidal curve = 4 X diameter of the generating circle. Length of the base = circumference of the generating circle. Area of a cycloid = 3 X area of generating circle. Helix (Screw).— A line generated by the progressive rotation of a point arouud an axis and equidistant from its centre. Length of a helix.— To the square of the circumference described by the generating'-point add the square of the distance advanced in one revolution, and take the square root of their sum multiplied by the number of revolu- tions of the generating point. Or, y(c 2 + h 2 )n = length, n being number of revolutions. Spirals.— Lines generated by the progressive rotation of a point around a fixed axis, with a constantly increasing distance from the axis. A plane spiral is when the point rotates in one plane. A conical spiral is when the point rotates around an axis at a progressing distance from its centre, and advancing in the direction of theaxis, as around a cone. Length of a plane spiral line. — When the distance between the coils is uniform. Rule.— Add together the greater and less diameters; divide their sum by 2; multiply the quotient by 3.1416, and again by the number of revolutions. Or, take the mean of the length of the greater and less circumferences and multiply it by the number of revolutions. Or, length = mi — - — , d and d' being the inner and outer diameters. Length of a conical spiral line.— Add together the greater and less diam- eters; divide their sum by 2 and multiply the quotient by 3.1416. To the square of the product of this circumference and the number of revolutions of the spiral add the square of the height of its axis and take the square root of the sum. Or, length = |/(™^^) 2 + ft 2 . SOLID BODIES. The Prism.— To find the surface of a right prism : Multiply the perim- eter of the base by the altitude for the convex surface. To this add the areas of the two ends when the entire surface is required. Volume of a prism = area of its base X its altitude. The pyramid.— Convex surface of a regular pyramid = perimeter of its base X half the slant height. To this add area of the base if the whole surface is required. Volume of a pyramid = area of base X one third of the altitude. SOLID BODIES. 61 To find the surface of a frustum of a regular pyramid : Multiply half the slant height by the sum of the perimeters of the two bases for the convex surface. To this add the areas of the two bases when the entire surface is required. To find the volume of a frustum of a pyramid : Add together the areas of the two bases and a mean proportional between them, and multiply the sum by one third of the altitude. (Mean proportional between two numbers = square root of their product.) Wedge.— 'A wedge is a solid bounded by five planes, viz.: a rectangular base, two trapezoids, or two rectangles, meeting in an edge, and two tri- angular ends. The altitude is the perpendicular drawn from an}' point in the edge to the plane of the base. To find the volume of a wedge : Add the length of the edge to twice the length of the base, and multiply the sum by one sixth of the product of the height of the wedge and the breadth of the base. Rectangular prismoid.— A rectangular prismoid is a solid bounded by six planes, of which the two bases are rectangles, having their corre- sponding sides parallel, and the four upright sides of the solids are trape- zoids. To find the volume of a rectangular prismoid: Add together the areas of the two bases and four times the area of a parallel section equally distant from the bases, and multiply the sum by one sixth of the altitude. Cylinder.— Convex surface of a cylinder = perimeter of base X altitude. To this add the areas of the two ends when the entire surface is required. Volume of a cylinder = area of base X altitude. Cone.— Convex surface of a cone = circumference of base X half the slant side. To this add the area of the base when the entire surface is required. Volume of a cone = area of base X k altitude. To find the surface of a frustum of a cone : Multiply half the side by the sum of the circumferences of the two bases for the convex surface; to this add the areas of the two bases when the entire surface is required. To find the volume of a, frustum of a cone : Add together the areas of the two bases and a mean proportional between them, and multiply the sum by one third of the altitude. Spliere.— To find the surface of a sphere : "Multiply the diameter by the ciicumference of a great circle; or, multiply the square of the diameter by 3.14159. Surface of sphere = 4 x area of its great circle. " " " = convex surface of its circumscribing cylinder. Surfaces of spheres are to each other as the squares of their diameters. To find the volume of a sphere : Multiply the surface by one third of the radius; or, multiply the cube of the diameter by 1/Qn; that is, by 0.5236. Value of |rr to 10 decimal places = .5235987756. The volume of a sphere = 2/3 the volume of its circumscribing cylinder. Volumes of spheres are to each other as the cubes of their diameters. Spherical triangle.— To find the area of a spherical triangle : Com- pute the surface of the quadrantal triangle, or one eighth of the surface of the sphere. From the sum of the three angles subtract two right angles; divide the remainder by 90, and multiply the quotient by the area of the quadrantal triangle. Spherical polygon. —To find the area of a spherical polygon : Com- pute the surface of the quadrantal triangle. From the sum of all the angles subtract the product of two right angles by the number of sides less two; divide the remainder by 90 and multiply the quotient by the area of the quadrantal triangle. The prismoid.— The prismoid is a solid having parallel end areas, and may be composed of any combination of prisms, cylinders, wedges, pyra- mids, or cones or frustums of the same, whose bases and apices lie in the end areas. Inasmuch as cylinders and cones are but special forms of prisms and pyramids, and w T arped surface solids may be divided into elementary forms of them, and since frustums may also be subdivided into the elementary forms, it is sufficient to say that all prismoids may be decomposed into prisms, wedges, and pyramids. If a formula can be found which is equally applicable to all of these forms, then it will apply to any combination of them. Such a formula is called 62 MENSURATION. The Prismoidal Formula. Let A = area of the base of a prism, wedge, or pyramid; A], A 2 , Am — the two end and the middle areas of a prismoid, or of any of its elementary solids; h = altitude of the prismoid or elementary solid; V = its volume; V = J ^U 1 +4Am^A 2 ) i For a prism A x , Am and A 2 are equal, = A ; V — - X 6A = hA. b For a wedge with parallel ends, A 2 = 0, Am — -A 1 ; V — ^(A. x + 2A X ) — — • For a cone or pyramid, A 2 = 0, ^l»i = '-zA\\ V = 7.(^*1 +^i) = ir- The prismoidal formula is a rigid formula for all prismoids. The only approximation involved in its use is in the assumption that the given solid may be generated by a right line .moving over the boundaries of the end areas. The area of the middle section is never the mean of the two end areas if the prismoid contains any pyramids or cones among its elementary forms. When the three sections are similar in form the dimensions of the middle area are always the means of the corresponding end dimensions. This fact often enables the dimensions, and hence the area of the middle section, to be computed from the end areas. Polyedrons.— A polyedron is a solid bounded by plane polygons. A regular polyedron is one whose sides are all equal regular polygons. To find the surface of a regular polyedron..— Multiply the area of one of the faces by the number of faces ; or, multiply the square of one of the edges by the surface of a similar solid whose edge is unity. A Table of the Regular Polyedrons whose Edges are Unity. Names. No. of Faces. Surface. Volume. Tetraedron 4 1.7320508 0.1178513 Hexaedron. 6 6.0000000 1 .0000000 Octaedron 8 3.4641016 0.4714045 Dodecaedron 12 20.6457288 7.6631189 Icosaedron 20 8.6602540 2.1816950 To find the volume of a regular polyedron.— Multiply the surface by one third of the perpendicular let fall from the centre on one of the faces ; or, multiply the cube of one of the edges by the solidity of a similar polyedron whose edge is unity. Solid of revolution.— The volume of any solid of revolution is equal to the product of the area of its generating surface by the length of the path of the centre of gravity of that surface. The convex surface of any solid of revolution is equal to the product of the perimeter of its generating surface by the length of path of its centre of gravity. Cylindrical ring.— Let d = outer diameter ; d' — inner diameter ; 1 1 1 ■= (d — d') = thickness = t ; - n P* = sectional area ; - (d -\- d') = mean diam- eter = M ; nt = circumference of section ; nM — mean circumference of ring; surface = n t X n M; = - ^ (d 2 - d' 2 ); = 9.86965 1 M; = 2.46741 (cZ 2 -d' 2 ); volume = InP Mn; = 2AQ74WM. 4 Spherical zone.— Surface of a spherical zone or segment of a sphere = its altiiude x the circumference of a great circle of the sphere. A great circle is one whose plane passes through the centre of the sphere. Volume of a zone of a sphere. — To the sum of the squnres of the radii of the ends add one third of the square of the height ; multiply the sum by the height and by 1.5708. Spherical segment.— Volume of a spherical segment with one base.— SOLID BODIES. 63 Multiply half the height of the segment by the area of the base, and the cube of the height by .5236 and add the two products. Or, from three times the diameter of the sphere subtract twice the height of the segment; multi- ply the difference by the square of the height and by .5236. Or, to three times the square of the radius of the base of the segment add the square of its height, and multiply the sum by the height and by .5236. Spheroid or ellipsoid.— When the revolution of the spheroid is about the transverse diameter it is prolate, and when about the conjugate it is oblate. Convex surface of a segment of a spheroid.— Square the diameters of the spheroid, and take the square root of half their sum ; then, as the diameter from which the segment is cut is to this root so is the height of the segment to the proportionate height of the segment to the mean diameter. Multiply the product of the other diameter and 3.1416 by the proportionate height. Convex surface of a frustum or zone of a spheroid.— Proceed as by previous rule for the surface of a segment, and obtain the proportionate height of the frustum. Multiply the product of the diameter parallel to the base of the frustum and 3.1416 by the proportionate height of the frustum. Volume of a spheroid is equal to the product of the square of the revolving axis by the fixed axis and by .5236. The volume of a spheroid is two thirds of that of the circumscribing cylinder. Volume of a segment of a spheroid.— 1. When the base is parallel to the revolving axis, multiply the difference between three times the fixed axis and twice the height of the segment, by the square of the height and by .5236. Multiply the product by the square of the revolving axis, and divide by the square of the fixed axis. 2. When the base is perpendicular to the revolving axis, multiply the difference between three times the revolving axis and twice the height of the segment by the square of the height and by .5236. Multiply the product by the length of the fixed axis, and divide by the length of the revolving axis. Volume of the middle frustum of a spheroid.— 1. When the ends are circular, or parallel to the revolving axis : To twice the square of the middle diameter add the square of the diameter of one end ; multiply the sum by the length of the frustum and by .2618. 2. AVhen the ends are elliptical, or perpendicular to the revolving axis: To twice the product of the transverse and conjugate diameters of the middle section add the product of the transverse and conjugate diameters of one end ; multiply the sum by the length of the frustum and by .2618. Spindles.— Figures generated by the revolution of a plane area, when the curve is revolved about a chord perpendicular to its axis, or about its double ordinate. They are designated by the name of the arc or curve from which they are generated, as Circular, Elliptic, Parabolic, etc., etc. Convex surface of a circular spindle, zone, or segment of it — Rule: Mul- tiply the length by the radius of the revolving arc; multiply this arc by the central distance, or distance between the centre of the spindle and centre of the revolving arc ; subtract this product from the former, double the remainder, and multiply it by 3.1416 Volume of a circxdar spindle. — Multiply the central distance by half the area of the revolving segment; subtract the product from one third of the cube of half the length, and multiply the remainder by 12 5664. Volume of frustum or zone of a circxdar spiudle.— From the square of half the length of the whole spindle take one third of the square of half the length of. the frustum, and multiply the remainder by the said half length of the frustum ; multiply the central distance by the revolving area which generates the frustum ; subtract this product from the former, and multi- ply the remainder by 6.2832. Volume of a segment of a circxdar sj)indle. — Subtract the length of the segment from the half length of the spindle ; double the remainder and ascertain the volume of a middle frustum of this length ; subtract the result from the volume of the whole spindle and halve the remainder. Volnxne of a cycloidal spindle = five eighths of the volume of the circum- scribing cylinder. — Multiply the product of the square of twice the diameter of the generating circle and 3.927 by its circumference, and divide this pro- duct by 8. Parabolic conoid.— Volume of a parabolic conoid (generated by the revolution of a parabola on its axis).— Multiply the area of the base by half, the height. 64 MEISSUKATION". Or multiply the square of the diameter of the base by the height and by .3927. Volume of a frustum of a parabolic conoid.— Multiply half the sum of the areas of the two ends by the height. Volume of a parabolic spindle (generated by the revolution of a parabola on its base).— Multiply the square of the middle diameter by the length and by .4189. The volume of a parabolic spindle is to that of a cylinder of the same height and diameter as 8 to 15. Volume of the middle frustum of a parabolic spindle.— Add together 8 times the square of the maximum diameter, 3 times the square of the end diameter, and 4 times the product of the diameters. Multiply the sum by the length of the frustum and by .05236. This rule is applicable for calculating the content of casks of parabolic form. Casks. — To find the volume of a cash of any form. — Add together 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of the diameters. Multiply the sum by the length, and divide by 31,773 for the content in Imperial gallons, or by 26,470 for U. S. gallons. This rule was framed by Dr. Hutton, on the supposition that the middle third of the length of the cask was a frustum of a parabolic spindle, and each outer third was a frustum of a cone. To find the idlage of a cask, the quantity of liquor in it when it is not full. 1. For a lying cask : Divide the number of wet or dry inches by the bung diameter in inches. If the quotient is less than .5, deduct from it one fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part of the excess above .5. Multiply the remainder or the sum by the whole content of the cask. The product is the quantity of liquor in the cask, in gallons, when the dividend is ivet inches; or the empty space, if dry inches. 2. For a standing cask : Divide the number of wet or dry inches by the length of the cask. If the quotient exceeds .5, add to it one tenth of its excess above .5; if less than .5, subtract from it one tenth of what it wants of .5. Multiply the sum or the remainder by the whole content of the cask. The product is the quantity of liquor in the cask, when the dividend is wet inches; or the empty space, if dry inches. Volume of cask (approximate) U. S. gallons = square of mean diam. X length in inches X .0034. Mean diam. — half the sum of the bung and head diams. Volume of an irregular solid.— Suppose it divided into parts, resembling prisms or other bodies measurable by preceding rules. Find the content of each part; the sum of the contents is the cubic contents of the solid. The content of a small part is found nearly by multiplying half the sum of the areas of each end by the perpendicular distance between them. The contents of small irregular solids may sometimes be found by im- mersing them under water in a prismatic or cylindrical vessel, and observ- ing the amount by which the level of the water descends when the solid is withdrawn. The sectional area of the vessel being multiplied by the descent of the level gives the cubic contents. Or, weigh the solid in air and in water; the difference is the weight of water it displaces. Divide the weight in pounds by 62.4 to obtain volume in cubic feet, or multiply it by 27.? to obtain the volume in cubic inches. When the solid is very large and a great degree of accuracy is not requisite, measure its length, breadth, and depth in several < hTerent places, and take the mean of the measurement for each dimension, and multiply the three means together. When the surface of the solid is very extensive it is better to divide it into triangles, to find the area of each triangle, and to multiply it by the mean depth of the triangle for the contents of each triangular portion; the contents of the triangular sections are to be added together. The mean depth of a triangular section is obtained by measuring the depth at each angle, adding together the. three measurements, and taking one third of the sum. PLANE TRIGONOMETRY. 65 PLANE TRIGONOMETRY. Trigonometrical Functions. Every triangle has six parts — three angles and three sides. When any three of these parts are given, provided one of them is a side, the other parts may be determined. By the solution of a triangle is meant the deter- mination of the unknown parts of a triangle when certain parts are given. The complement of an angle or are is what remains after subtracting the angle or arc from 90°. In general, if we represent any arc by A, its complement is 90° — A. Hence the complement of an arc that exceeds 90° is negative. - Since the two acute angles of a right-angled triangle are together equal to a right angle, each of them is the complement of the other. The supplement of an angle or arc is what remains after subtracting the angle or arc from 180°. If A is an arc its supplement is 180° — A. The sup- plement of an arc that exceeds 180° is negative. The sum of the three angles of a triangle is equal to 180°. Either angle is the supplement of the other two. In a right-angled triangle, the right angle being equal to 90°, each of the acute angles is the complement of the other. In all right-angled triangles having the same acute angle, the sides have to each other the same ratio. These ratios have received special names, as follows: If A is one of the acute angles, a the opposite side, b the adjacent side, and c the hypothenuse. The sine of the angle A is the quotient of the opposite side divided by the a hypothenuse. Sin. A = -.■ The tangent of the angle A is the quotient of the opposite side divided by a * the adjacent side. Tang. A = t- The secant of the angle A is the quotient of the hypothenuse divided by c the adjacent side. Sec. A = t' The cosine, cotangent, and cosecant of an angle are respec- tively the sine, -tangent, and secant of the complement of that angle. The terms sine, cosine, etc., are called trigonometrical functions. In a circle whose radius is unity, the sine of .an arc, or of the angle at the centre measured by that arc, is the perpendicular let fall from one extrem- ity of the arc upon the diameter passing through the other extremity. The tangent of an arc is the line which touches the circle at one extrem- ity of the arc, and is limited by the diameter {produced) passing through the other extremity. - The secant of an arc is that part of the produced diameter which is intercepted betireen the centre and the tangent. The versed, sine of an arc is that part of the diameter intercepted between the extremity of the arc and the foot of the sine. • In a circle whose radius is not unity, the trigonometric functions of an arc I will be equal to the lines here defined, divided by the radius of the circle. If. I C A (Fig. 70) is an angle in the first quadrant, and C F— radius, mu . f+u . FG _ CG KF The sine of the angle — Tang. I A z Rad.' Cosec. = i "Rad." Rad, Secant =R^uT Cot - = CL GA Rad." Versin ' = Rad/ If radius is 1, then Rad. in the denominator is omitted, and sine = F G, etc. The sine of an arc = half the chord of twice the arc. The sine of the supplement of the arc is the same as that of the arc itself. Sine of arc B D F = F G = sin arc -FM. I 66 PLANE TRIGONOMETRY. The tangent of the supplement is equal to the tangent of the arc, but with a contrary sign. Tang. B D F — B M. The secant of the supplement is equal to the secant of the arc, but with a contrary sign. Sec. B D F — CM. Signs of the functions in the four quadrants.— If we divide a circle iuto four quadrants by a vertical and a horizontal diame- ter, the upper right-hand quadrant is called the first, the upper left the sec- ond, the lower left the third, and the lower right the fourth. The signs of the functions in the four quadrants are as follows: First quad. Second quad. Third quad. Fourth quad. Sine and cosecant, + + — — Cosine and secant, -j- — — 4- Tangent and cotangent, + — 4- — The values of the functions are as follows for the angles specified: Angle Sine Cosine Tangent Cotangent . Secant Cosecant . . . "Versed sine „ • „ 30 45 60 90 120 135 150 180 270 . 1 1 v 3 1 Vs 1 1 2 V~i 2 2 V2 2 1 V3 1 1 1 1 V3 2 Vg 2 2 V2 2 Vs 1 v 3 1 00 -*s -1 1 Vs c, ao V'6 f3 ~vl - 1 - Vs cc 1 GO vi 2 V% V~2 2 2 V-3 00 1 2 ~V2 2 13 2 -1 ) sin \{C - D). (12) Equation (9) may be enunciated thus: The sum of the sines of any two angles is equal to twice the sine of half the sum of the angles multiplied by the cosine of half their difference. These formulae enable us to transform a sum or difference into a product. The sum of the sines of two angles is to their difference as the tangent of half the sum. of those angles is to the tangent of half their difference. sin A + sin B = * ** \u + B)oo*^A - B) = t ^jA + B) sin A- sin B 2 C08jfA + B)sin±{A -B) tan\(A - B) (13) The sum of the cosines of two angles is to their difference as the cotan- gent of half the sum of those angles is to the tangent of half their difference. cos A + c 2 cos \{A + B) cos \(A - B) cot \{A + B) cos B - cos A a gin 1 {A + B) gin l u _ B) tftn 1 {A _ B - (14) The sine of the sum of two angles is to the sine of their difference as the sum of the tangents of those angles is to the difference of the tangents. sin ( A + B) _ tan A + tan B sin {A — B) tan A — tan B" 1 ' ' (15) sin (A + B) cos A cos B sin (A - B) cos A cos B cos (A + B) cos A cos i? cos (A - B) — tan A + tan £; = tan A - tan B; = 1 — tan ^4 tan i?; = 1 + fan A tan i?; tan (A + B) = tan (.4 - B) = cot C4 -f- B) = cot U - 5) = tan ^ 4- tan ff 1 - tan A tan 5' tan A - tan i? _ 1 4- tan A tan B' cot ^4 cot B — 1 . cot-B-f cot4 ' cot ,4 cot ff 4- 1 cot B — cot A ' 68 PLANE TRIGONOMETRY. Solution of Plane Right-angled Triangles. Let A and B be the two acute angles and C the right angle, and a, b, and c the sides opposite these angles, respectively, then we have 1. sin A = cos B — - ; c 3. tan A = cot B = T ; 6 2. cos A = sin B = -; 4. cot A = tan 5 = - . 1. In any plane right-angled triangle the sine of either of the acute angles is equal to the quotient of the opposite leg divided by the hypothenuse. 2. The cosine of either of the acute angles is equal to the quotient of the adjacent leg divided by the hypothenuse. 3. The tangent of either of the acute angles is equal to the quotient of the opposite leg divided by the adjacent leg. 4. The cotangent of either of the acute angles is equal to the quotient of the adjacent leg divided by the opposite leg. 5. The square of the hypothenuse equals the sum of the squares of the other two sides. Solution of Oblique-angled Triangles. The following propositions are proved in works on plane trigonometry. In any plane triangle — Theorem 1. The sines of the angles are proportional to the opposite sides. Theorem 2. The sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their differ- ence. Theorem 3. If from any angle of a triangle a perpendicular be drawn to the opposite side or base, the whole base will be to the sum of the other two sides as the difference of those two sides is to the difference of the segments of the base. Case I. Given two angles and a side, to find the third angle and the other two sides. 1. The third angle = 180° — sum of the two angles. 2. The sides may be found by the following proportion : The sine of the angle opposite the given side is to the sine of the angle op- posite the required side as the given side is to the required side. Case II. Given two sides and an angle opposite one of them, to find the third side and the remaining angles. The side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the sine of the required angle. The third angle is found by subtracting the sum of the other two. from 180°, and the third side is found as in Case I. Case III. Given two sides and the included angle, to find the third side and the remaining angles. The sum of the required angles is found by subtracting the given angle from 180°. The difference of the required angles is then found by Theorem II. Half the difference added to half the sum gives the greater angle, and half the difference subtracted from half the sum gives the less angle. The third side is then found by Theorem I. Another method : Given the sides c, 6, and the included angled, to find the remaining side n and the remaining angles B and C. From either of the unknown angles, as B, draw a perpendicular B e to the opposite side. Then Ae = c cos A, i? e = c sin .4, e C -— b — Ae, Be h- e C — tan C. Or, in other words, solve Be, Ae and B e C as right-angled triangles. Case IV. Given the three sides, to find the angles. Let fall a perpendicular upon the longest side from the opposite angle, dividing the given triangle into two right-angled triangles. The two seg- ments of the base may be found by Theorem III. There will then be given the hypothenuse and one side of a right-angled triangle, to find the angles. For areas of triangles, see Mensuration. ANALYTICAL GEOMETRY. 69 ANALYTICAL GEOMETRY. Analytical geometry is that branch of Mathematics which has for its object the determination of the forms and magnitudes of geometrical magnitudes by means of analysis. Or di nates and abscissas.— In analytical geometry two intersecting Y lines YY', XX' are used as coordinate axes, XX' being the axis of abscissas or axis of X, and YY' the axis of ordinates or axis of Y. A. the intersection, is called the origin of co- ordinates. The distance of any point P from the axis of Y measured parallel to the axis of X is called the abscissa of the point, as AD or CP, Fig. 71. Its distance from the axis of X. measured parallel to the axis of Y, is called the ordinate, as AC or PD. The abscissa and ordinate taken together are called the coor- dinates of the point P. The angle of intersec- tion is usually taken as a right angle, in which case the axes of Xand Y are called rectangu- lar coordinates. The abscissa of a point is designated by the letter x and the ordinate by y. ^ The equations of a point are the equations which express the distances of thp point from the axis. Thus x = a, y = b are the equations of the point P. Equations referred to rectangular coordinates.— The equa- tion uf a line expresses the relation which exists between the coordinates of every puint of the line. Equation of a straight line, y = ax ± b, in which a is the tangent of the angle the liue makes with the axis of X, and b the distance above A in which the line cuts the axis of Y. Every equation of the first degree between two variables is the equation of a straight line, as Ay -j- Bx 4- C = 0, which can be reduced to the form y = ax ± b. Equation of the distance between two points: P /c / 7 /a D Y' Fig. 1. D = \\x" - a;') 2 + W - V') 2 , in which x'y', x"y" are the coordinates of the two points. Equation of a line passing through a given point: y — ?j' — a(x — x'), in which x'y' are the coordinates of the given point, a, the tangent of the angle the line makes with the axis of x, being undetermined, since any num- ber of lines may be drawn through a given point. Equation of a line passing through two given points: y-y' = y x „ ~ v , {x - x'). Equation of a line parallel to a given line and through a given point: y - y 1 - a(x - x'). Equation of an angle V included between two given lines: . yT a' — a tang V— , , & 1 + a'a in which a and a' are the tangents of the angles the lines make with the axis of abscissas. If the lines are at right angles to each other tang V = a>, and 1 -f a'a = 0. Equation of an intersection of two lines, whose equations are y = ax + &, and y = a'x -f- b\ b — b' , ab' — a'b x = • , and y = — — . a — a' a — a' 70 ANALYTICAL GEOMETRY. Equation of a perpendicular from a given point to a given line: y — y' — (x — x'). Equation of the length of the perpendicular P: p _ y' - ax' - b Vl X o« "The circle.— Equation of a circle, the origin of coordinates being at the centre, and radius = R : a: 2 -f ?/ 2 = iJ 2 . If the origin is at the left extremity of the diameter, on the axis of X: 2/ 2 = 2Rx - a; 2 . If the origin is at any point, and the coordinates of the centre are a-'//' : (x - a;') 2 + (y- yV = R 2 . Equation of a tangent to a circle, the coordinates of the point of tangency being x"y" and the origin at the centre, yy" + xx" - .R 2 . The ellipse. —Equation of an ellipse, referred to rectangular coordi- nates with axis at the centre: A i y i 4. 52^2 _ A 1 B 2 % in which A is half the transverse axis and B half the conjugate axis. Equation of the ellipse when the origin is at the vertex of the transverse axis : ?/ 2 = ^Ax - a,- 2 ). The eccentricity of an ellipse is the distance from the centre to either focus, divided by the semi-transverse axis, or Va* - # 2 e = A— The parameter of an ellipse is the double ordinate passing through the focus. It is a third proportional to the transverse axis and its conjugate, or Any ordinate of a circle circumscribing an ellipse is to the corresponding ordinate of the ellipse as the semi-transverse axis to the semi-conjugate. Any ordinate of a circle inscribed in an ellipse is to the corresponding ordi- nate of the ellipse as the semi-conjugate axis to the semi-transverse. Equation of the tangent to an ellipse, origin of axes at the centre : • A*yy" + B^xx" = A^B*, y"x" being the coordinates of the point of tangency. Equation of the normal, passing through the point of tangency, and per- pendicular to the tangent: y - y" xx -Efrfa - x )• The normal bisects the angle of the two lines drawn from the point of tangency to the foci. The lines drawn from the foci make equal angles with the tangrent. Tlae parabola. —Equation of the parabola referred to rectangular coordinates, the origin being at the vertex of its axis, y" 1 — 2px, in which 2p is the parameter or double ordinate through the focus. ANALYTICAL GEOMETRY. 71 The parameter is a third proportional to any abscissa and its corresponding ordinate, or x :y ::y :2p. Equation of the tangent: yy" = p(x + x"), y''x" being coordinates of the point of tangency. Equation of the normal: y - y" xx — —(x — x"). The sub-normal, or projection of the normal on the axis, is constant, and equal to half the parameter. The tangent at any point makes equal angles with the axis and with the line drawn from the point of tangency to the focus. The hyperbola.— Equation of the hyperbola referred to rectangular coordinates, origin at the centre: A 2 y 2 - B 2 x 2 = - AW 2 , in which A is the semi-transverse axis and B the semi-conjugate axis. Equation when the origin is at the vertex of the transverse axis: Conjugate and equilateral hyperbolas. — If on the conjugate axis, as a transverse, and a focal distance equal to VA 2 -+- B 2 , we construct the two branches of a hyperbola, the two hyperbolas thus constructed are called conjugate hyperbolas. If the transverse and conjugate axes are equal, the hyperbolas are called equilateral, in which case y 2 — x 2 — — A 2 when A is the transverse axis, and x 2 — y 2 = — B 2 when B is the trans- verse axis. The parameter of the transverse axis is a third proportional to the trans- verse axis and its conjugate. 2 A : 2B : : 2B : parameter. The tangent to a hyperbola bisects the angle of the two lines drawn from the point of tangency to the foci. The asymptotes of a hyperbola are the diagonals of the rectangle described on the axes, indefinitely produced in both directions. In an equilateral hyperbola the asymptotes make equal angles with the transverse axis, and are at right angles to each other. The asymptotes continually approach the hyperbola, and become tangent to it at an infinite distance from the centre. Conic sections.— Every equation of the second degree between two variables will represent either a circle, an ellipse, a parabola or a hyperbola. These curves are those which are obtained by intersecting the surface of a cone by planes, and for this reason they are called conic sections. Logarithmic curve.— A logarithmic curve is one in which one of the coordinates of any point is the logarithm of the other. The coordinate axis to which the lines denoting the logarithms are parallel is called the axis of logarithms., and the other the axis of numbers. If y is the axis of logarithms and x the axis of numbers, the equation of the curve is y = log x. If the base of a system of logarithms is a, we have a v = x, in which y is the logarithm of x. Each system of logarithms will give a different logarithmic curve. If y — 0, x - 1. Hence every logarithmic curve will intersect the axis of numbers at a distance from the origin equal to 1, 72 DIFFERENTIAL CALCULUS. DIFFERENTIAL CALCULUS. The differential of a variable quantity is the difference between any two of its consecutive values; hence it is indefinitely small. It is expressed by writing d before the quantity, as dx, which is read differential of x. The term -,- is called the differential coefficient of y regarded as a func- tion of x. The differential of a function is equal to its differential coefficient mul- tiplied by the differential of the independent variable; thus, -i-dx = dy. The limit of a variable quantity is that value to which it continually approaches, so as at last to differ from it by less than any assignable quan- tity. The differential coefficient is the limit of the ratio of the increment of the independent variable to the increment of the function. The differential of a constant quantity is equal to 0. The differential of a product of a constant by a variable is equal to the constant multiplied by the differential of the variable. If u — Av, du — Adv. In any curve whose equation is y=f(x), the differential coefficient rr' = tan a: hence, the rate of increase of the function, or the ascension of dx the curve at any point, is equal to the tangent of the angle which the tangent line makes with the angle of abscissas. All the operations of the Differential Calculus comprise but two objects: 1. To find the rate of change in a function when it passes from one state of value to another, consecutive with it. 2. To find the actual change in the function : The rate of change is the differential coefficient, and the actual change the function. Differentials of algebraic functions.— The differential of the sum or difference of any number of functions, dependent on the same variable, is equal to the sum or difference of their differentials .taken sepa- rately : If u — y -f- z — w, du — dy + dz — div. The differential of a product of two functions dependent on the same variable is equal to the sum of the products of each by the differential of the other : , . . , , , d(uv) du . dv d(uv) = vdu + udv. = . v J , : uv u v The differential of the product of any number of functions is equal to the sum of the products which arise by multiplying the differential of each function by the product of all the others: d(uts) = tsdu -f- usdt 4- utds. The differential of a fraction equals the denominator into the differential of the numerator minus the numerator into the differential of the denom- inator, divided by the square of the denominator : -G)= If the denominator is constant, dv — 0, and dt = — — = — > -u 2 v If the numerator is constant, du — 0, and dt xx — The differential of the square root of a quantity is equal to the differen- tial of the quantity divided by twice the square root of the quantity: If v == tt% or v — Vu, dv = -; = -u~ ?du. 2 Yu ^ DIFFERENTIAL CALCULUS. 73 The differential of any power of a function is equal to the exponent multi- plied by the function raised to a power less one, multiplied by the differen- tial of the function, d(ii n ) = nu n ~ 1 du. Formulas for differentiating algebraic functions. /x\ _ ydx - xdy 1. d(a) = 0. 2. d (ax) — adx. 3.d(x-\-y) = dx + dy. 4. d (x — y) = dx — dy. 5. d (xy) = xdy + ydx. 6. d | 7. d (x m ) = mx m ' dx s. div*) = -^; 9.d\x 7 = - - dx. To find the differential of the form u = (a -\~ bx n ) '■ Multiply the exponent of the parenthesis into the exponent of the varia- ble within the parenthesis, into the coefficient of the variable, into the bi- nomial raised to a power less 1, into the variable within the parenthesis raised to a power less 1, into the differential of the variable. du = d(a + bx n ) m = mnb(a + bx n ) m " l x n ~ 1 dx. To find the rate of change for a given value of the variable : Find the differential coefficient, and substitute the value of the variable in the second member of the equation. Example.— If tc is the side of a cube and u its volume, u = x 3 , -=- : '— 3ic 2 . dx Hence the rate of change in the volume is three times the square of the ige. If the edge is denoted by 1, the rate of change is 3: Application. The coefficient of expansion by heat of the volume of a body is three times the linear coefficient of expansion. Thus if the side of a cube expands .001 inch, its volume expands .003 cubic inch. 1.001 3 = 1.003003001. A partial differential coefficient is the differential coefficient of a function of two or more variables under the supposition that only one of them has changed its value. A partial differential is the differential of a function of two or more vari- ables under the supposition that only one of them has changed its value. The total differential of a function of any number of variables is equal to the sum of the partial differentials. If u — f(xy), the partial differentials are -rdx, ~rdy. If u = x" 2 + V 3 — z, die — —dx + -—dy + -z-dz; = 2xdx-\- Zy^dy — dz. ax ay dz Integrals.— An integral is a functional expression derived from a differential. Integration is the operation of finding the primitive function from the differential function. It is indicated by the sign /, which is read u the integral of." Thus / 2xdx = x* : read, the integral of 2xdx equals x 2 . To integrate an expression of the form mx m ~~ 1 dx or x m dx, add 1 to the exponent of the variable, atid divide by the new exponent and by the differ- ential of the variable: f 3x 2 dx = x 3 . (Applicable in all cases except when - 1. For lx dx see formula 2 page 78.) The integral of the product of a constant by the differential of a vari- able is equal to the constant multiplied by the integral of the differential: fax m dx — afx m dx = a — — , x m + a - J m 4- 1 The integral of the algebraic sum of any number of differentials is equal to the algebraic sum of their integrals: du = 2ax*dx - bydy - z*dz; fdu = %ix 3 - ^y* - -*. Since the differential of a constant is 0, a constant connected with a vari- able by the sign + or — disappears in the differentiation ; thus d(a + x m ) = ix m = mx m ~ dx. Hence in integrating a differential expression we must 74 DIFFERENTIAL CALCULUS. annex to the integral obtained a constant represented by C to compensate for the term which may have been lost in differentiation. Thus if we have dy = adx\ fdy = afdx. Integrating, y = ax ± C. The constant 0, which is added to the first integral, must have such a value as to render the functional equation true for every possible value that may be attributed to the variable. Hence, after having found the first integral equation and added the constant C, if we then make the variable equal to zero, the value which the function assumes will be the true value of C. An indefinite integral is the first integral obtained before the value of the constant C is determined. A particular integral is the integral after the value of C has been found. A definite integral is the integral corresponding to a given value of the variable. Integration between limits. — Having found the indefinite inte- gral and the particular integral, the next step is to find the definite integral, and then the definite integral between given limits of the variable. The integral of a function, taken between two limits, indicated by given values of x, is equal to the difference of the definite integrals correspond- ing to those limits. The expression I dy = a I dx is read: Integral of the differential of y, taken between the limits x' and x' , the least limit, or the limit corresponding to the subtractive integral, being placed below. Integrate du = 9x'*dx between the limits x = 1 and x = 3, u being equal to 81 when x = 0. fdu = f9x'*dx = 3x 3 + C; C = 81 when x = 0, then t/x -■ du = 3(3) 3 + 81, minus 3(1) 3 + 81 = Integration of particular forms. To integrate a differential of the form du — (a + bx n ) m x n ' 1 dx. 1. If there is a constant factor, place it without the sign of the integral, and omit the power of the variable without the parenthesis and the differ- ential; 2. Augment the exponent of the parenthesis by 1, and then divide this quantity, with the exponent so increased, by the exponent of the paren- thesis, into the exponent of the variable within the parenthesis, into the co- efficient of the variable. Whence fdu = ^ + ^ m+1 = C. J (m + \)nb The differential of an arc is the hypothenuse of a right-angle triangle ol which the base is dx and the perpendicular dy. If z is an arc, dz = Vdx* + dy* z =f Vdx^-^-dy 1 . Quadrature of a plane figure. The differential of the area of a plane surf ace is equal to the ordinate into the differential of the abscissa. ds — ydx. To apply the principle enunciated in the last equation, in finding the area of any particular plane surface : Find the value of y in terms of x, from the equation of the bounding line; substitute this value in the differential equation, and then integrate between the required limits of x. Area of tlie parabola.— Find the area of any portion of the com- mon parabola whose equation is 2/2 — 2px; whence y = ^2px. DIFFERENTIAL CALCULUS. 75 Substituting this value of y in the differential equation ds — ydx gives r r , ,- r x 2 i/%p 3 I ds= I y->pxdx = y-2p I x^dx = — - — xs -f C; 2/i/2px x x 2 or, s = r 3 = 3 *y+C. Tf we estimate the area from the principal vertex, x = 0. y = 0, and C — 0; and denoting the particular integral by s', s' — - xy. That is, the area of any portion of the parabola, estimated from the ver- tex, is equal to % of the rectangle of the abscissa and ordinate of the extreme point. The curve is therefore qnadrable. Quadrature of surfaces of revolution. —The differential of a surface of revolution is equal to the circumference of a circle perpendicular to the axis into the differential of the arc of the meridian curve. ds = 2iry^/dx 2 + dy*; in which y is the radius of a circle of the bounding surface in a plane per- pendicular to the axis of revolution, and x is the abscissa, or distance of the plane from the origin of coordinate axes. Therefore, to find the volume of any surface of revolution: Find the value of y and dy from the equation of the meridian curve in terms of x and dx, then substitute these values in the differential equation, and integrate between the proper limits of x. By application of this rule we may find : The curved surface of a cylinder equals the product of the circumference of the base into the altitude*. The convex surface of a cone equals the product of the circumference of the base into half the slant height. The surface of a sphere is equal to the area of four great circles, or equal to the curved surface of the circumscribing cylinder. Cubature of volumes of revolution.— A volume of revolution is a volume generated by the revolution of a plane figure about a fixed line I called the axis. I If we denote the volume by V, dV = ny 2 dx. I The area of a circle described by any ordinate y is try 7 ", hence the differ- I ential of a volume of revolution is equal to the area of a circle perpendicular I to the axis into the differential of the axis. I The differential of a volume generated by the revolution of a plane figure J about the axis of Y is irx 2 dy. I To find the value of F"for any given volume of revolution : I Find the value of y 2 in terms of x from the equation of the meridian I curve, substitute this value in the differential equation, and then integrate I between the required limits of x. I By application of this rule we may find : t The volume of a cylinder is equal to the area of the base multiplied by the I altitude. The volume of a cone is equal to the area of the base into one third the : altitude. The volume of a prolate spheroid and of an oblate spheroid (formed by the revolution of an ellipse around its transverse and its conjugate axis re- spectively) are each equal to two thirds of the circumscribing cylinder. If the axes are equal, the spheroid becomes a sphere and its volume = ^jt.R 2 x D — - irD 3 ; B being radius and D diameter. 6 o The volume of a paraboloid is equal to half the cylinder having the same base and altitude. The volume of a pyramid equals the area of the base multiplied by one third the altitude. Second, third* etc., diiferentials.— The differential coefficient being a function of the independent variable, it may be differentiated, and we thus obtain the second differential coefficient: Dividing by dx, we have for the second differential coeffi- ,/du\ c d \dx) = i 76 DIFFERENTIAL CALCULUS. cient — j, which is read: second differential of u divided by the square of the differential of x (or dx squared). d 3 u The third differential coefficient — -= is read: third differential of u divided ax 6 by dx cubed. The differentials of the different orders are obtained by multiplying the differential coefficients by the corresponding powers of dx: thus — dx 3 = third differential of u. Sign of the first differential coefficient.— If we have a curve whose equation is y = fx, referred to rectangular coordinates, the curve will recede from the axis of X when -r~ is positive, and approach the axis when it is negative, when the curve lies within the first angle of the coordinate axes. For all angles and every relation of y and x the curve will recede from the axis of X when the ordinate and first differential co- efficient have the same sign, and approach it when they have different signs. If the tangent of the curve becomes parallel to the axis of X at any point -p = 0. If the tangent becomes perpendicular to the axis of X at any . . dy point — -co. Sign of the second differential coefficient.— The second dif- ferential coefficient has the same sign as the ordinate when the curve is convex toward the axis of abscissa and a contrary sign when it is concave. Maclaurin's Theorem.— For developing into a series any function of a single variable as u — A-\- Bx + Cx 2 -f- Dx 3 -J- Ex*, etc., in which A, B, C, etc., are independent of x: u = (u) +(-r) x + r-*\z-*) x + 1 o q V^—W « 3 + etc. x = q \dx/ x = 1 .2^-dx 2/ x = 1 • 2 . 3^dx 3 / x = In applying the formula, omit the expressions x = 0, although the coeffi- cients are always found under this hypothesis. Examples : (a + x) m = a m -\- ma m «i l 111 _ n Ini — 9\ ... .. ' 6 x 3 + etc. 1 _ J x . , a -\-x ~ a a 2 a 3 a 4 ' a » + i ' Taylor's Theorem.— For developing into a series any function of the sum oV difference of two independent variables, as u' — fix '± y): , du , dHi y* , d 3 u y 3 , '., in which u is what u' becomes when y = 0, — is what -, becomes when dx dx y = 0, etc. Maxima and minima.— To find the maximum or minimum value of a function of a single variable: 1. Find the first differential coefficient of the function, place it equal to 0, and determine the roots of the equation. 2. Find the second differential coefficient, and substitute each real root, in succession, for the variable in the second member of the equation. Each root which gives a negative result will correspond to a maximum value of the function, and each which gives a positive result will correspond to a minimum value. Example.— To find the value of x which will render the function y a maximum or minimum in the equation of the circle, y 2 -f- x 2 = B' 2 ; dy x . . x . . . ■f- = ; making = gives x = 0, dx y y DIFFERENTIAL CALCULUS. 77 d 2 y x 2 4- v^ The second differential coefficient is: — - = -^~- dx y 3 d 2 y 1 When x = U, y = R; hence -— - = — — , which being negative, y is a maxi- mum for R positive. In applying the rule to practical examples we first find an expression for the function which is to be made a maximum or minimum. 2. If in such expression a constant quantity is found as a factor, it may be omitted in the operation ; for the product will be a maximum or a mini- mum when the variable factor is a maximum or a minimum. 3. Any value of the independent variable which renders a function a max- imum or a minimum will render any power or root of that funciiou a maximum or minimum; hence we may square both members of an equa- tion to free it of radicals before differentiating. By these rules we may find: The maximum rectangle which can be inscribed in a triangle is one whose altitude is half the altitude of the triangle. The altitude of the maximum cylinder which can be inscribed in a cone is one third the altitude of the cone. The surface of a cylindrical vessel of a given volume, open at the top, is a minimum when the altitude equals half the diameter. The altitude of a cylinder inscribed in a sphere when its convex surface is a maximum is r |/2. r = radius. The altitude of a cylinder inscribed in a sphere when the volume is a 2r maximum is ~7. ' Differential of an exponential function. If u - a x (1) then du = da x = a x k dx, (2) in which A; is a constant dependent on a. l The relation between a and 7c is a k = e; whence a — e k , ..... (3) in which e = 2.7182818 . . . the base of the Naperian system of logarithms. Logarithms.— The logarithms in the Naperian system are denoted by Z, Nap. log or hyperbolic log, hyp. log, or log e ; and in the common system always by log. k — Nap. log a, log a — k log e (4) The common logarithm of e, = log 2.7182818 . . . = .4342945 . .- . , is called the modulus of the common system, and is denoted by M. Hence, if we have the Naperian logar thm of a number we can find the common logarithm of the same number by multiplying by the modulus. Reciprocally, Nap. log - com. log x 2 3025851. If in equation (4) we make a — 10, we have 1 = k log e, or - = log e = M. . That is, the modulus of the common system is equal to 1, divided by the Naperian logarithm of the common base. From equation (2) we have du da x , , — = = kdx. u a x If we make a = 10, the base of the common system, x = log u, and ,,, , , du 1 du d(log u) = dx — — x - = — x M . u k u That is, the differential of a common logarithm of a quantity is equal to the diffei-ential of the quantity divided by the quantity, into the modulus. If we make a — e, the base of the Naperian system, x becomes the Nape- 78 DLFFEBENTIAL CALCULUS. rian logarithm of u, and k becomes 1 (see equation (3)); hence M = 1, and d(Nap. log u) = dx = — ; = — . a x u That is, the differential of a Naperian logarithm of a quantity is equal to the differential of the quantity divided by the quantity; and in the Naperian system the modulus is 1. Since k is the Naperian logarithm of a, du = a x I a dx. That is, the differential of a function of the form a x is equal to the function, into the Naperian logarithm of the base a, into the differential of the exponent. It* we have a differential in a fractional form, in which the numerator is the differential of the denominator, the integral is the Naperian logarithm of the denominator. Integrals of fractional differentials of other forms are given below: Differential forms which have known integrals; ex- ponential functions. (I — Nap. log.) /■ a x I a dx — a x -f- C; J ~ = J dxx~ 1 = lx + C; I (xy x ~ 1 dy + y x ly x dx) = y* + C; /dx , ,. = Kx + \/x* ± a 2 ) 4- C; 4/a; 2 ± a 2 dx , / , \ l{x ± a + |/x 2 ± 2ax) + C; / 2adx 2 -x* /' 2 adx a;2- tt 2 j/rr 2 ± 2ax / 2adx _ / i/a* + x* - a\ #|/a 2 + x* \|/aa+ls» + a/ / 2adx -i( a ~ \/<* r ~^\ /x ~ 2 dx /l -(- |/F V x + x~ 2 ~ ~ \ « Circular functions.— Let 2 denote an arc in the first quadrant, y its sine, a; its cosine, v its versed sine, and t its tangent; and the following nota- tion be employed to designate an arc by any one of its functions, viz., sin -1 y denotes an arc of which y is the sine cos a; " " " " " re is the cosine, tan -1 t " " " " " t is the tangent DIFFERENTIAL CALCULUS. (read "arc whose sine is y," etc.), — we have the following differential forms which have known integrals (r = radius): cos z dz — sin z 4- C; f /— dx -\ \ n — • = cos x x +- C\ |/l - X 2 f sin zdz — cos z-\- C\ Q" 1 y + C; -. ver-sin 1 v + C; \/%v - v 2 /rdy - -i — — =sin l y -f- C; y,. 2 _ ^2 / — = cos x+ C; J |A- 2 - x 2 I sin z dz = ver-sin z -\- C\ — — = tan z + C; cos 2 z / rd v tfitrv + jfl = versin _1 v + C > /Mi* =tan- 1 ^0; J 4/a 2 - w 2 a -7 = cos" 1 - 4- C; 4/a 2 - W 2 » e/ j/ita /*/2aw — u 2 / adu a 2 4 « 2 = ver-sin ~" * — \- C: a : tan x - 4- G. The cycloid.— If a circle he rolled along a straight line, any point of the circumference, as P, will describe a curve which is called a cycloid. The circle is called the generating circle, and _Pthe generating point. The transcendeutal equation of the cycloid is and the differential equation is dx - 1 y - \/*ry ydx \/'Zry - y 2 . The area of the cycloid is equal to three times the area of the generating circle. The surface described by the arc of a cycloid when revolved about its base is equal to 64 thirds of the generating circle. The volume of the solid generated bv revolving a cycloid about its base is equal to five eighths of the circumscribing cylinder. Integral calculus. — In the integral calculus we have to return from the differential to the function from which it was derived A number of differential expressions are given above, each of which has a known in- tegral corresponding to it, and which being differentiated, will produce the ' given differential. In all classes of functions any differential expression may be integrated when it is reduced to one of the known forms; and the operations of the integral calculus consist mainly in making such transformations of given differential expressions as shall reduce them to equivalent ones w r hose in- tegrals are known. For methods of making these transformations reference must be made to .00145985 75( .00133333 15 .00122699 880 .00113636 5 .00105820 i 5 .00145773 1 .00133156 16 .00122549 1 .00113507 6 .00105708 • .00145560 .00132979 17 .00122399 2 .00113379 .00105597 J 1 .00145319 .0013-2802 18 .00122249 3 .00113250 8 .00105485 ( ) .00145137 4 .00132626 19 .00122100 4 .00113122 9 .00105374 69( ) .00144927 .00132450 820 .00121951 5 .00112991 950 .00105263 ] .00144718 i .00132275 1 .00121803 6 .00112867 1 .00105152 > .00144509 .00132100 2 .00121654 7 .00112740 2 .00105042 5 .00144300 £ .00131926 3 .00121507 8 .00112613 3 .00104932 ' .00144092 i .00131752 4 .00121359 9 .00112486 4 .00104822 j .00143885 7GC .00131579 5 .00121212 S90 .00112360 5 .00104712 i .00143678 1 .00131406 6 .90121065 1 .00112233 6 .00104602 .00143472 .00131234 7 .001-20919 2 .00112108 .00104493 8 .00143266 2 .00131062 8 .00120773 3 .00111982 8 .00104384 £ .00143061 4 .00130890 9 .00120627 4 .00111857 9 .00104275 700 .00142857 5 .00130719 . 830 .00120482 5 .00111732 960 .00104167 1 .00142653 1 .00130548 1 .00120337 6 .00111607 1 .00104058 2 .00142450 7 .00130378 .00120192 7 .00111483 2 .00103950 3 .00142247 8 .00130208 3 .00120048 8 .00111359 3 .00103842 4 .00142045 9 .00130039 4 .00119904 9 .00111235 4 .00103734 ft .nni41844 770 .00129870 5 .00119760 900 .00111111 5 .00103627 RECIPROCALS OF NUMBERS. 83 No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 966 .00103520 1031 .000969932 1096 .000912409 116! '.000861328 1226 .000815661 7 ,00103413 2 .000968992 7 .000911577 2'' .000860585 7 .000814996 8 .00103306 3 .000988054 8 .000910747 3 .000859845 8 .000814332 9 .00103199 4 .000967118 9 .000909918 41.000859106 9 .000813670 970 .00103093 5 .000966184 1100 .000909091 5 '.000858369 12 .000813008 1 .00102987 6 .000965251 1 .000908265 6 .000857633 1 .000812348 2 .00102881 7 .000964320 2 000907441 7 1.000856898 2 .000811688 3 .00102775 8 .000963391 3 .000906618 81.000856164 3 .000811030 4 .C01 02669 9 .000962464 4 .000905797 9 .000855432 4 .000810373 5 .00102564 1040 .000961538 5 .000904977 1170 .000854701 5 .000809717 6 .00102459 1 .000960615 6 .000904159 1 .000853971 6 .000809061 7 .00102354 2 .000959693 7 .000903342 2 .000853242 7 .000808407 8 .00102250 3 .000958774 8 .000902527 3 .000852515 8 .000807754 9 .00102145 4 .000957854 9 .000901713 4 .000851789 9 .000807102 980 .00102041 5 .000956938 1110 .000900901 5 .090851064 1240 .000806452 I .00101937 6 .000956023 11 000900090 6 .000850340 1 .000805802 2 .00101833 7 .000955110 12 .000899281 7 .000849618 2 .000805153 3 .00101729 8 .000954198 13 .000898473 8 .000848896 3 .000804505 4 .00101626 9 .000953289 14 .000897666 9 .000848176 •4 .000803858 5 .00101523 1050 .000952381 15 .000896861 1180 .000847457 5 .000803213 6 .00101420 1 .000951475 16 .000896057 1 .000846740 6 .000802568 7 .00101317 2 .000950570 17 .000895255 2 .000846024 7 .000801925 8 .00101215 3 .000949668 18 .000894454 3 .000S45308 8 .000801282 9 .00101112 4 .000948767 19 .000893655 4 .000844595 9 .000800640 990 .00101010 5 .000947867 1120 .000892857 5 .000843882 1250 .000800000 1 .00100908 6 .000946970 1 .000892061 6 .000843170 1 .000799360 2 .00100806 7 .000946074 2 .000891266 7 .000842460 2 .000798722 3 .00100705 8 .000945180 3 .000890472 8 .000841751 3 .000798085 4 .00100604 9 .000944287 4 .000889680 9 .000841043 4 .000797448 5 .00100502 1060 .000943396 5 .000888889 1190 .000840336 5 000796813 6 .00100102 1 .000942507 6 .000888099 1 .000839631 6 .000796178 7 .00100301 2 .000941620 7 .000887311 2 .000838926 7 000795545 8 .00100200 3 .000940734 8 .000886525 3 .000838222 8 .000794913 9 .00100100' 4 .000939850 9 .000885740 4 000837521 9 .000794281 1000 .00100000 5 .000938967 1130 .000884956 5 .000836820 .000793651 1 .000999001 6 000938086 1 .000884173 6 .000836120 1 .000793021 2 .000998004 7 .000937207 2 .000883392 7 .000835422 2 .000792393 3 .000997009 8 .000936330 3 .000882612 8 .000834724 3 .000791766 4 .000996016 9 .000935454 4 .000881834 9 .000834028 4 .000791139 5 .000995025 1070 .000934579 5 .000881057 1200 .600833333 5 .000790514 6 .000994036 1 .000933707 e .000880282 1 000832639 6 .000789889 7 .000993049 2 .000932836 7 .000879508 2 .000831947 7 .000789266 8 .000992063 3 .000931966 8 .000878735 3 .000831255 8 .000788643 9 .000991080 4 .000931099 9 .000877963 4 .000830565 9 .000788022 1010 .000990099 5 .000930233 1140 .000877193 5 .000829875 1270 .000787402 11 .000989120 6 .000929368 1 .000876424 6 .000829187 1 .000786782 12 .000988142 7 .000928505 2 .000875657 7 .000828500 2 .000786163 13 .000987167 8 .000927644 I .000874891 8 .000827815 3 .000785546 14 .000986193 £ .000926784 4 .000874126 9 .000827130 4 .000784929 15 .000985222 1080 .000925926 5 .000873362 1210 .000826446 5 .000784314 16 .000984252 1 .000925069 6 .000872600 11 .000825764 6 .000783699 17 .000983284 2 .000924214 7 .000871840 12 .000825082 7 .000783085 18 .000982318 3 .000923361 8 .000871080 13 .000824402 8 .000782473 19 .000981354 4 .000922509 9 090870322 14 .000823723 9 .000781861 1020 .000980392 5 .000921659 1150 .000869565 15 .000823045 1280 .000781250 1 .000979432 6 .000920810 1 .000868810 16 .000822368 1 .000780640 2 .000978474 7 .000919963 2 .000868056 17 .000821693 2 .000780031 3 .000977517 8 .000919118 3 .000867303 18 000821018 3 .000779423 4 .000976562 9 000918274 4 .000866551 19 .000820344 4 .000778816 5 .000975610 1090 .000917431 5 .000865801 1220 .000819672 5 .000778210 6 .000974659 1 .000916590 6 .000865052 1 .000819001 6 .000777605 7 .000973710 2 .000915751 7 .000864304 2 .000818331 7 .000777001 8 .000972763 3 .000914913 ,8 .000863558 3 .000817661 8 000776397 9 .000971817 4 .000914077 9 .000862813 4 .000816993 9 .000775795 1030 .000970874 5 .000913242 1160 .000862009 5 .000816326 ! .000775194 84 MATHEMATICAL TABLES. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 1291 .000774593 1356 .000737463 1421 '.000703730 1486 .000672948 1551 .000644745 2 .000773994 7 000736920 2 .000703235 7 .000672495 2 .000644330 3 .000773395 8 .000736377 3 .000702741 8 .000672043 3 .000643915 4 .000772797 9 .000735835 4 .900702247 9 .000671592 4 .000643501 5 .000772. '01 1360 .000735294 5 .000701754 1490 .000671141 5 .000643087 6 .000771605 1 .000734754 6 .000701262 1 .000670691 6 .000642673 7 .000771010 2 .000734214 7 .000700771 2 .000670241 7 .000642261 8 .000770416 3 .000733676 8 .0U07 00280 3 .000669792 8 .000641848 9 .000769823 4 .000733138 9 .000699790 4 .000669344 9 .000641437 1300 .000769231 5 .00073 .'601 1430 .000699301 5 .000668896 1560 .000641026 1 .000768639 6 .000732064 1. 000698812 6 .000668449 1 .000640615 2 .000768049 7 .000731529 2 .000698324 7 .000668003 2 .000640205 3 .000767459 8 .000730994 3 .000097837 8 .000667557 3 .000639795 4 .000766871 9 .000730460 4 .000697350 9 .000667111 4 .000639386 5 .000766283 1370 .000729927 5 .000696864 1500 .000666667 5 .000638978 6 .000765697 1 .000729395 6 '.000696379 1 .000666223 6 .000638570 7 .000765111 2 .000728863 7;. 000695894 2 .000665779 7 .000638162 8 .000764526 3 .000728332 8 ! . 000695410 3 .000655336 8 .000637755 9 .000763942 4 .000727802 9 .000694927 4 .000664894 9 .000637319 1310 .000763359 5 .000727273 1440 .000694444 5 .000664452 1570 .000636943 11 .000762776 6 .000726744 1 .0006 6 .000664011 1 .000636537 12 .000762195; 7 .000726216: 2 .000693481 7 .000663570 2 .000636132 13 .000761615 8 .000725689 3 .000693001 8 .000663130 3 .000635728 14 .000761035: 9 .000725163 4 .000692521 9 .000662691 4 .000635324 15 .000760456 1380 .000724638 5 .000692041 1510 .000662252 5 .000634921 16 .000759878 1 .000724113 6.000691563 11 .000661813 6 .000634518 17 .000759301 2 .000723589 71.000691085 12 .000661376 7 .000634115 18 .000758725; 3 .000723066 8 . 0006 90608 13 .000660939 8 .000633714 19 .000758150 4 .000722543 9,-000690131 14 .000660502 9 .000633312 1320 .000757576 5 .000722022 14501.000689655 15 .000660066 1580 .000632911 1 .000757002 6 .000721501 1 .000689180 16 .000659631 1 .000632511 2 .000756430 7 .000720980 2 .000688705 17 .000659196 2 .000632111 3 .000755858 8 .000720461 3 .000688231 18 .000658761 3 .000631712 4 .000755287 9 .000719942 4 .000687758 19 .000658328 4 .000631313 5 .000754717 1390 .000719424 5 .000687285 1520 .000657895 5 .000630915 6 .000754148 1 .000718907 6 .000686813 1 .000657462 6 .000630517 7 .000753579 2 000718391 7 .000686341 2 .000657030 7 .000630120 8 .000753012 3 .000717875 8 .000685871 3 .000656598 8 .000629723 9 .000752445 4 .000717360 9 .000685401 4 .000656168 9!. 000629327 .000751880 5 .000716846 1460 .000684932 5 .000655738 1590 .000628931 1 .000751315 6 .000716332 1 .000684463 6 .000655308 ] .000628536 2 .000750750 7 .000715820 2 .000683994 7 .000654879 2 .000628141 3 .000750187 8 .000715308 3 .000683527 8 .000654450 3 .000627746 4 .000749625 9 .000714796 4 .000683060 9 .000654022 4 .000627353 5 .000749064 1400 .000714286 5 .000682594 1530 .000653595 5 .000626959 6 .000748503 1 .000713776 61.000682128 1 .000653168 6 .000626566 7 .000747943 2 .000713267 7 .000681663 2 .000652742 7 .000626174 8 .0007473S4 3 .000712758 81 .000681199 3 .000652316 8 .000625782 9 . 000746826 4 .000712251 9; .000680735 4 .000651890 9 .000625391 .0007462691 5 .000711744 1470|. 000680272 5 .000651466! 1600 .000625000 1 .000745712' 6 .000711238 1 .000679810 6 .000651042; 2 .000624219 2 .000745156; 7 .000710732 2i .000679348 7 .0006506181 4 .000623441 3 .009744602 8 .000710227 3 . 000678887 8 .000650195' 6 .000622665 4 .000744048 9 .000709723 4;. 000678426 9 .000649773 8 .000621890 5 .000743494! 1410 .000709220 5 .000677966 1540 . 000649351 : 1610 .000621118 6 .000742942 11 .000708717 6 .000677507 1 .000648929 2 .000620347 7 .000742390 12 .000708215 7 .000677048 2 . 000648508 ! 4 .000619578 8 .000741840 13 .000707714 81.000676590 3 .000648088 6 .000618812 9 .000741290 14 .000707214 91.000676132 4 .000647668 8 .000618047 1350 . 000740741 ; 15 .000706714 1480, 000675676 5 .000647249 1620 .000617284 1 .000740192 16 .000706215 1. 00' »ii75219 6 .000646830 2 .000616523 2 000739645 17 .000705716 2 .000674 "64 7 .000646412 4 .000615763 3 .000739098' 18 .000705219 3 .000674309 8 . 0001545995 6 .000615006 4 .000738552 19 .000704722 4.000673s:,4 9 .000645578 8 .000614250 5 .0007380071 1420 .000704225 5 .000673*01 1550 .000615161 1630 .000613497 RECIPROCALS OF NUMBERS. 85 No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 1632 .000612745 1706 .000586166 1780 .000561798 1854 .000539374 1928 .000518672 4 .000611995 8 .000585480 2 .000561167 6 .000538793 1930 .000518135 6 .000611247 1710 .000584795 4 .000560538 8 .000538213 2 .000517599 8 .000610500 12 .000584112 6 .000559910 1860 .000537634 4 .000517063 1640 .000609756 14 .000583430 8 .000559284 2 .000537057 6 .000516528 2 .000609013 16 .000582750 17 90 .000558659 4 .000536480 8 .000515996 4 .000608272 18 .000582072 2 .000558035 6 .000535905 1940 .000515464 6 .000607533 1720 .000581395 4 .000557413 8 .000535332 2 .000514933 8 .000606796 2 .000580720 6:. 000556793 1870 .000534759 4 .000514403 165U .000606061 4 .000580046 8 .000556174 2 .000534188 6 .000513874 2 .000605327 6 .000579374 18 00 .000555556 4 .000533618 8 .000513347 4 .000604595 8 .000578704 2 .000554939 6 .000533049 1950 .000512820 6 .000603865 1730 .000578035 4 .000554324 8 .000532481 2 .000512295 8 .000603136 2 .000577367 6 .000553710 1880 .000531915 4 .000511770 161 .000602410 4 .000576701 8 . 000553097 2 .000531350 6 .000511247 2 .000601685 6 .000576037 18101.000552486 4 .000530785 8 .000510725 4 .000600962 8 .000575374 12 .000551876 6 .000530222 1960 .000510204. 6 .000600240 1740 .000574713 14 .000551268 8 .000529661 2 .000509684 8 .000599520 2 .000574053! 16 .000550661 1890 .000529100 4 .000509165 1670 .000598802 4 .000573394 18 .000550055 2 .000528541 6 .000508647 2 .000598086 6 .000572737 1820 .000549451 4 .000527983 8 .000508130 4 .000597371 8 .000572082 2 .000548848 6 .000527426 1970 .000507614 6 .000596658 1750 .000571429 4 .000548246 8 .000526870 2 .000507099 8 .000595947 2 .000570776 6 .000547645 1900 .000526316 4 .600506585 ig-;o .000595238 4 .000570125! 8 .000547046 2 .000525762 6 .000506073 ' 2 .000594530 6 .000569476' 1830 .000546448 4 .0005^5210 8 .000505561 4 .000593824 8 .000568828 2 .000545851 6 .000524659 1980 .000505051 6 .000593120 1760 .000568182 4 .000545253 8 .000524109 2 .000504541 8 .000592417 2 .000567537 61.000544662, 1910 .000523560 4 .000504032 .000591716 4 .000566893 8 .000544069 12 .000523012 6 .000503524 2 .000591017 6 .000566251 1840 .000543478 14 .000522466 8 .000503018 4 .000590319 8 .000565611 2 .000542888 16 .000521920 1990 .000502513 6 .000589622 1770 .000564972 4 .000542299 18 .000521376 2 .000502008 8 .000588928 2 .000564334 6 .000541711 1920 .000520833 4 .000501504 1700 .000588235 4 .000563698] 8 .000541125 2 .000520291 6 .000501002 2 .000587544 6 .000563063 1850 .000540540 4 .000519750 8 .000500501 4 .000586854 8 .000562430! 2. 000539957 6 .000519211 2000 000500000 "Use of reciprocals,,— Reciprocals may be conveniently used to facili- tate computations in long division. Instead of dividing as usual, multiply the dividend by the reciprocal of the divisor. The method is especially useful when many different dividends are required to be divided by the same divisor. In this case find the reciprocal of the divisor, and make a . small table of its multiples up to 9 times, and use this as a multiplication- i table instead of actually performing the multiplication in each case. Example.— 9871 and several other numbers are to be divided by 1638. The reciprocal of 1638 is .000610500. Multiples of the reciprocal : 1. .0006105 2. .0012210 3. .0018315 4. .0024420 5. .0030525 6. .0036630 7. .0042735 8. .0048840 9. .0054945 10. .0061050 The table of multiples is made by continuous addition of 6105. The tenth line is written to check the accuracy of the addition, but it is not afterwards used. Operation: Dividend 9871 Take from table 1 . . .0006105 0.042735 00.48840 005.4945 6.0262455 Quotient. . Correct quotient by direct division 6.0262515 The result will generally be correct to as many figures as there are signifi- cant figures in the reciprocal, less one, and the error of the next figure will in general not exceed one. In the above example the reciprocal has six sig- nificant figures, 610500, and the result is correct to five places of figures. 86 MATHEMATICAL TABLES. SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS OF NUMBERS FROM .1 TO 1600. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. .1 .01 .001 .3162 .4642 3.1 9.61 29.791 1.761 1.458 .15 .0225 .0034 .3873 .5313 .2 10.24 32.768 1.789 1.474 .2 .04 .008 .4472 .5848 .3 10.89 35.937 1.817 1.489 .25 .0625 .0156 .500 .6300 .4 11.56 39.304 1.844 1.504 .3 .09 027 .5477 .6694 .5 12.25 42.875 1.871 1.518 .35 .1225 .0429 .5916 .7047 .6 12.96 46.656 1.897 1.533 .4 .16 .064 .6325 .7368 7 13.69 50.653 1.924 1.547 .45 .2025 .0911 .6708 .7663 !8 14.44 54.872 1.949 1.560 .5 .25 .125 .7071 .7937 .9 15.21 59.319 1.975 1.574 .55 .3025 .1664 .7416 .8193 4. 16. 64. 2. 1.5874 .6 .36 .216 .7746 .8434 .1 16.81 68.921 2.025 1.601 .65 .4225 .2746 .8062 .8062 .2 17.64 74.088 2.049 1.613 .7 .49 .343 .8367 .8879 .3 18.49 79.507 2.074 1.626 .75 .5625 .4219 .8600 .9086 .4 19.36 85.184 2.098 1.639 .8 .64 .512 .8944 .9283 .5 20.25 91.125 2.121 1.651 .85 .7225 .6141 .9219 .9473 .6 21.16 97.336 2.145 1.663 .9 .81 .729 .9487 .9655 7 22.09 103.823 2.168 1.675 .95 .9025 .8574 .9747 .9830 '.8 23.04 110.592 2.191 1.687 1 1. 1. 1. 1. .9 24.01 117.649 2.214 1.698 1.05 1.1025 1.158 1.025 1.016 5. 25. 125. 2.2361 1.7100 1.1 1.21 1.331 1.049 1.032 .1 26.01 132.651 2.258 1.721 1.15 1.3225 1.521 1.072 1.048 .2 27.04 140.608 2.280 1.732 1.2 1.44 1.728 1.095 1.063 .3 28.09 148.877 2.302 1.744 1.25 1.5625 1.953 1.118 1.077 .4 29.16 157.464 2.324 1.754 1.3 1.69 2.197 1.140 1.091 .5 30.25 166.375 2.345 1.765 1.35 1.8225 2.460 1.162 1.105 .6 31.36 175.616 2.366 1.776 1.4 1.96 2.744 1 . 183 1.119 7 32.49 185.193 2.387 1.786 1.45 2.1025 3.049 1.204 1.132 '.8 33.64 195.112 2.408 1.797 1.5 2.25 3.375 1.2247 1 . 1447 .9 34.81 205.379 2.429 1.807 1.55 2.4025 3.724 1.245 1.157 6. 36. 216. 2.4495 1.8171 1.6 2.56 4.096 1.265 1.170 .1 37.21 226.981 2.470 1.827 1.05 2.7225 4.492 1.285 1.182 .2 38.44 238.328 2.490 1.837 1.7 2.89 4.913 1.304 1.193 .S 39.69 250.047 2.510 1.847 1.75 3.0625 5.359 1.323 1.205 A 40.96 262.144 2.530 1.857 1.8 3.24 5.832 1.342 1.216 .5 42.25 274.625 2.550 1.866 1.85 3.4225 6.332 1.360 1.228 .6 43.56 287.496 2.569 1.S76 1.9 3.61 6.859 1.378 1.239 7 44.89 300.763 2.588 1.885 1.95 3.8025 7.415 1.396 1.249 !8 46.24 314.432 2.608 1.895 2. 4. 8. 1.4142 1.2599 .9 47.61 328.509 2.627 1.904 .1 4.41 9.261 1.449 1.281 7. 49. 343. 2.6458 1.9129 .2 4.84 10.648 1.483 1.301 .1 50.41 357.911 2.665 1.922 .3 5.29 12.167 1.517 1.320 .2 51.84 373.248 2.683 1.931 A 5.76 13.824 1.549 1.339 .3 53.29 389.017 2.702 1.940 .5 6.25 15.625 1.581 1.357 .4 54.76 405.224 2.720 1.949 .6 6.76 17.576 1.612 1.375 .5 56.25 421.875 2.739 1.957 7 7.29 19.683 1.643 1.392 .6 57.76 438.976 2.757 1.966 '.8 7.84 21.952 1.673 1.409 .7 59.29 456.533 2.775 1.975 .9 8.41 24.389 1.703 1.426 .8 60.84 474.552 2.793 1.983 3. 9. 27. 1.7321 1.4422 .9 62.41 493.039 2.811 1.992 SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 8? No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 8. 64. 512. 2.8284 2. 45 2025 91125 6.7082 3.5569 .1 65.61 531.441 2.846 2.008 46 2116 97336 6.7823 3.5830 .2 ' 67.24 551.368 2.864 2.017 47 2209 103823 6.8557 3.6088 .3 68.89 571.787 2.881 2 025 48 2304 110592 6.9282 3.6342 .4 70.56 592.704 2.898 2.033 49 2401 117649 7. 3.6593 .5 72.25 614.125 2.915 2.041 50 2500 125000 7.0711 3.6840 .6 73.96 636.056 2.933 2.049 51 2601 132651 7.1414 3.7084 .7 75 69 658.503 2.950 2.057 52 2704 140608 7.2111 3.7325 .8 77^44 681.472 2.966 2.065 53 2809 148877 7.2801 3.7563 .9 79.21 704.969 2.983 2.072 54 2916 157464 7.3485 3.7798 9. 81. 729. 3. 2.0801 55 3025 166375 7.4162 3.8030 .1 82.81 753.571 3.017 2.088 56 3136 175616 7.4833 3.825^) .52 84.64 778.688 3.033 2.095 57 3249 185193 7.5498 3.84S5 .3 86.49 804.357 3.050 2.103 58 3364 195112 7.6158 3.8709 .4 88.36 830.581 3.066 2.110 59 3481 205379 7.6811 3.8930 .5 90.25 857.375 3.082 2.118 60 3600 216000 7.7460 3.9149 .6 92.16 884.736 3.098 2.125 61 3721 226981 7.8102 3.9365 .7 94.09 912.673 3.114 2 133 62 3844 238328 7.8740 3.9579 .8 96.04 941.192 3.130 2.140 63 3969 250047 7.9373 3.9791 .9 98.01 970.299 3.146 2.147 64 4096 262144 8. 4. 10 100 1000 3.1623 2.1544 65 4225 274625 8.0623 4.0207 11 121 1331 3.3166 2.2240 66 4356 287496 8.1240 4.0412 12 144 1728 3.4641 2.2894 67 4189 300763 8.1854 4.0615 13 169 2197 3.6056 2.3513 68 4624 314432 8.2462 4.0S17 14 196 2744 3.7417 2.4101 69 4761 328509 8.3066 4.1016 15 225 3375 3.8730 2.4662 70 4900 343000 8.3666 4.1213 16 256 4096 4. 2.5198 71 5041 357911 8.4261 4.1408 17 289 4913 4.1231 2.5713 72 5184 373248 8.4853 4.1602 18 324 5832 4.2426 2.6207 73 5329 389017 8.5440 4.1793 19 361 6859 4.3589 2.6684 74 5476 405224 8.6023 4.1983 20 400 8000 4.4721 2.7144 75 5625 421875 8.6603 4.2172 21 441 9261 4.5826 2 7589 76 5776 438976 8.7178 4.2358 22 484 10648 4.6904 2.8020 77 5929 456533 8.7750 4.2543 23 529 12167 4.7958 2.8439 78 6084 474552 8.8318 4.2727 24 576 13824 4.8990 2.8845 79 6241 493039 8.8882 4.2908 25 625 15625 5. 2.9240 80 6400 512000 8.9443 4.3089 26 676 . 17576 5.0990 2.9625 81 6561 531441 9. 4.3267 27 729 19683 5.1962 3. 82 6724 551368 9.0554 4.3445 28 784 21952 5.2915 3 0366 83 6889 571787 9.1104 4.3621 29 841 24389 5.3852 3.0723 84 7056 592704 9.1652 4.3795 30 900 27000 5.4772 3.1072 85 7225 614125 9.2195 4.3968 31 961 29791 5.5678 3.1414 86 7396 636056 9.2736 4.4140 32 1024 32768 5.6569 3.1748 87 7569 658503 9 3276 4.4310 33 1089 35937 5.7446 3.2075 88 7744 OKI 472 9.3808 4.4480 34 1156 39304 5.8310 3.2396 89 7921 704969 9.4340 4.4647 35 1225 42875 5.9161 3.2711 90 8100 729000 .9.4868 4.4814 36 1296 46656 6. 3.3019 91 8281 753571 9.5394 4.4979 37 1369 50653 6.0828 3.3322 92 8464 778688 9.5917 4.5144 38 1444 54872 6.1644 3.3620 93 8649 804357 9 6437 4.5307 39 1521 59319 6.2450 3.3912 94 8836 830584 9.6954 4.5468 40 1600 64000 6.3246 3 4200 95 9025 857375 9 7468 4.5629 41 1681 68921 6.4031 3.4482 96 9216 884736 9.7980 4.5789 42 1764 . 74088 6.4807 3.4760 97 9409 912673 9.8489 4.5947 43 1849 79507 6.5574 3.5034 98 9604 941192 9.8995 4.6104 44 1936 85184 6.6332 3.5303 99 9801 970299 9.9499 4.6261 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 100 10000 1000000 10. 4.6416 155 240.25 3723875 12.4499 5.3717 101 10201 1030301 10.0199 4.6570 156 24336 3796416 12.49UI 5.3832 10:2 10404 1061208 10.0995 4.6723 157 24649 3869893 I2.53IN 5.3947 103 10609 1092727 10.1489 4.6875 158 24964 3944312 12.569b 5.4061 104 10816 1124864 10.1980 4.7027 159 25281 4019679 12.6095 5.4175 105 11025 1157625 10.2470 4.7177 160 25600 4096000 12.6491 5.4288 106 112 6 1191016 10.2956 4.7326 161 25921 4173281 12.6886 5.4401 107 11449 1225043 10.3141 4.7475 162 26244 4251528 12.7279 5 4514 108 11664 1259712 10.3923 4.7622 163 26569 4330747 12.7671 5.4626 109 11881 1295029 10.4403 4.7769 164 26896 4410944 12.8062 5.4737 110 12100 1331000 10.4881 4.7914 165 27225 4492125 12.8452 5.4848 111 12321 1367631 10.5357 4.8059 166 27556 4574296 12.8841 5.4959 112 12544 1404928 10.5S30 4; 8203 167 27889 4657463 12.9228 5.5069 113 12769 1442897 10.6301 4.8346 168 28224 4741632 12.9615 5.5178 114 12996 1481544 10.6771 4:8488 169 28561 4826809 13.0000 5.5288 115 13225 1520875 10.7238 4.8629 170 28900 4913000 13.0384 5.5397 110 13456 1560896 10.7703 4.8770 171 29241 5000211 13.0767 5.5505 117 13689 1601613 10.816? 4.8910 172 29584 5088448 13.1149 5.5613 118 13924 1613032 10.K62K 4.9049 173 29929 5177717 13.1529 5.5721 119 14161 1685159 10.9087 4.9187 174 30276 5268024 13.1909 5.5828 120 14400 1728000 10.9545 4.9324 175 30625 5359375 13.2288 5.5934 121 14641 1771561 11.0000 4.9461 176 30976 5451776 13.2665 5.6041 122 14884 1815848 11.0454 4.9597 177 31329 5545233 13.3041 5.6147 123 15129 1860867 11.0905 4.9732 178 31684 5639752 13.3417 5.6252 124 15376 1906624 11.1355 4.9866 179 32041 5735339 13.3791 5.6357 125 15625 1953125 11.1803 5,0000 180 32400 5832000 13.4164 5.6462 126 15876 2000376 11.2250 5.0133 181 32761 5929741 13.4536 5.6567 127 16129 2048383 11.2694 5.0265 182 33124 602S568 13.4907 5.6671 128 16384 2097152 11.3137 5.0397 183 33489 6128487 13.5277 5.6774 129 16641 2146689 11.3578 5.0528 184 33856 6229504 13.5617 5.6877 130 16900 2197000 11.4018 5.0658 185 34225 6331625 13.6015 5.6980 131 17161 2248091 11.4455 5.0788 186 34596 6434856 13.6382 5.7083 - 17424 2299968 11.4891 5.0916 187 34969 6539203 13. oris 5.7185 133 17689 2352637 11.5326 5.1045 188 35344 6644672 13.7113 5.7287 134 17956 2406104 11.5758 5.1172 189 35721 6751269 13.7477 5.7388 135 18225 2460375 11.6190 5.1299 190 36100 6859000 13.7840 5.7489 136 18496 2515456 11.6619 5.1426 191 36481 6967871 13.8203 5.7590 137 18769 2571353 11.7047 5.1551 192 36864 7077888 13.8564 5.7690 138 19044 2628072 11.7473 5.1676 193 37249 7189057 13.8924 5 7790 139 19321 2685619 11.7898 5.1801 194 37636 7301384 13.9284 5.7890 140 19600 2744000 11.8322 5.1925 195 38025 7414875 13.9642 5.7989 141 19881 2803221 11.8743 5.2048 196 38416 7529536 14.0000 5.8088 112 20164 2863288 11.9164 5.2171 197 38809 7645373 14.0357 5.8186 143 20449 2924207 11.9583 5.2293 198 39204 7762392 14.0712 5.8285 144 20736 2985984 12.0000 5.2415 199 39601 7880599 14.1067 5.8383 145 21025 3048625 12.0416 5.2536 200 40000 8000000 14.1421 5.8480 146 21316 3112136 12.0S30 5.2656 201 40401 8120601 14.1774 5.8578 147 21609 3176523 12.1244 5.2776 202 40804 8242408 14.2127 5.8675 148 21904 3241792 12.1655 5.2896 203 41209 8365127 14.2478 5.8771 149 22201 3307949 12.2066 5.3015 204 41616 8489664 14.2829 5.8868 150 22500 3375000 12.2474 5.3133 205 42025 8615125 14.3178 5.8964 151 22801 3442951 12.2882 5.1251 206 42436 8741816 14.3527 5 9059 152 23104 3511803 12.32SS 5.3368 207 42849 8869743 14.3875 5.9155 153 23409 358 1 577 12. 3693 5.3485 208 43264 8998912 14.4222 5.9250 154 23716 3652264 12 4097 5.3601 .209 43681 9129329 14.4568 5.9345 SQUARES, CUBES, SQUARE AND CUBE ROOTS. 89 No. Square. 2 10 44100 an 44521 212 44944 213 45369 214 45796 215 46225 216 46656 217 47089 218 47524 219 47961 250 48400 221 48841 222 49284 228 49729 224 50176 225 50625 226 51076 227 51529 228 51984 229 52441 280 52900 231 53861 282 53824 288 54289 284 54756 285 55225 236 55696 28? 56169 288 56644 289 57121 240 57600 241 58081 242 58564 248 59049 244 59536 245 60025 246 60516 247 61009 24 S 61504 249 62001 250 62500 251 63001 252 63504 25 5 64009 251 64516 255 65025 256 65536 257 66049 258 66564 259 67081 260 67600 21 i1 68121 282 68644 2(18 69169 264 69696 Cube. Sq. Root. Cube Root. 9261000 14.4914 5.9439 9393931 14.5258 5.9533 9528128 14.5602 5.9627 9663597 14.5945 5.9721 9800344 14.6287 5,9814 9938375 14.6629 5.9907 10077696 14.6969 6.0000 10218313 14.7309 6 0092 10360232 14.7648 6.0185 10503459 14.7986 6 0277 10648000 14.8324 6.0368 10793861 14.8661 6.0459 10941048 14.8997 6.0550 11089567 : « •■ 6.0641 11239424 14.9666 6.0732 11390625 15.0000 6.0822 11543176 15.0333 6.0912 11697083 15.0665 6.1002 11852352 15.099? 6.1091 12008989 15.1327 6.1180 12167000 15.1658 6.1269 12326391 15.1987 6.1358 12487168 15.2315 6.1446 12649337 15.2643 6.1534 12812904 15.2971 6,1622 12977875 15.3297 6.1710 13144256 15.3623 6.179? 13312053 15 3948 6.1885 13481272 15.4272 6.1972 13651919 15.4596 6.2058 13824000 15.4919 6.2145 13997521 15.5242 6.2231 14172488 15.5563 6.2317 14348907 15.5885 6.2403 14526784 15.6205 6.2488 14706125 15.6525 6.2573 14886936 15.6844 6.2658 15069223 15.7162 6.2743 15252992 15.7480 6.2828 15438249 15.7797 6.2912 15625000 15.8114 6.2996 15813251 15.8430 6.3080 16003008 15.8745 6.3164 16194277 15.9060 6.3247 16387064 15.9374 6.3330 16581375 15.9687 6.3413 16777216 16.0000 6.3496 16974593 16.0312 6.3579 17173512- 16.0624 6.3661 17373979 16.0935 6.3743 17576000 16.1245 6 3825 17779581 16.1555 6.390? 17984728 16.1864 6.3988 18191447 16.2173 6.4070 18399744 16.2481 6.4151 No. Square. Cube. 265 70225 1S609625 266 70756 18821096 267 71289 19034163 268 71824 19248832 269 72361 19465109 270 72900 19683000 271 73441 19902511 272 73984 20123648 273 74529 20346417 274 75076 20570824 275. 75625 20796875 276 76176 21024576 277 76729 21253933 278 77284 21484952 279 77841 21717639 280 78400 21952000 281 78961 22188041 282 79524 22425768 283 80089 22665187 284 80656 22906304 285 81225 23149125 286 81796 23393656 287 82369 23639903 288 82944 23887872 289 83521 24137569 290 84100 24389000 291 84681 24642171 292 85264 24897088 293 85849 25153757 294 86436 25412184 295 87025 25672375 296 87616 25934336 297 88209 26198073 298 88804 26463592 299 89401 26730899 300 90000 27000000 301 90601 27270901 302 91204 27543608 303 91809 27818127 304 92416 28094464 305 93025 28372625 305 93636 28652616 307 94249 28934443 308 94864 29218112 309 95481 29503629 310 96100 29791000 311 96721 30080231 312 9?'344 30371328 313 97969 30664297 314 98596 30959144 315 99225 31255875 316 99856 31554496 317 100489 31855013 318 101124 32157432 319 101761 32461759 Sq. Cube Root. Root. 16.2788 6.4232 16.3095 6.4312 16.3401 6.4393 16.8707 6.4473 16.4012 6.4553 16 431? 6.4633 16.4621 6.4713 16.4924 6.4792 16.5227 6.4872 16.5529 6.4951 16.5831 6.5030 16.6132 6.5108 16.6433 6.5187 16.6733 6.5265 16.7033 6.5343 16.7332 6.5421 16.7631 6.5499 16.7929 6.5577 16.8226 6.5654 16.8523 6.5731 16.8819 6.5808 16.9115 6.5885 16.9411 6.5962 16.9706 6.6039 17.0000 6.6115 17.0294 6.6191 17.0587 6.6267 17.0880 6.6343 17.1172 6.6419 17.1464 6.6494 17.1756 6.6569 17.2047 6.6644 17.2337 6.6719 17.2627 6.6794 17.2916 6.6869 17.3205 6.6943 17.3494 6.7018 17.3781 6.7092 17.4069 6.7166 17 4356 6.7240 17.4642 6.7313 17.4929 6.7387 17.5214 6.7460 17.5499 0.7533 17.5784 6.7606 17.6068 6.7679 17.6352 6.7752 17.6635 6.7824 17.6918 6.7897 17.7200 6.7969 17.7482 6.8041 17.7764 6.8113 17.8045 6.8185 17.8326 6.8256 17.8606 6.8328 90 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 320 102400 32768000 17.8885 6.8399 375 140625 52734375 19.3649 7.2112 321 103041 33076161 17.9165 6.8470 376 141376 53157376 19.3907 7.2177 322 103684 33386248 17.9444 6.8541 377 142129 53582633 19.4165 7.2240 323 104329 33698267 17.9722 6.8612 37 S 142884 54010152 19.4422 7.2304 324 104976 34012224 18.0000 6.8683 379 143641 54439939 19.4679 7.2368 325 105625 34328125 18.0278 6.8753 380 144400 54872000 19.4936 7.2432 326 106276 34645976 18.0555 6.8824 381 145161 55306341 19.5192 7.2495 327 106929 34965783 18.0831 6.8894 3S2 145924 55742968 19.5448 7.2558 328 107584 35287552 18.1108 6.8964 383 146689 56181887 16.5704 7.2622 329 108241 35611289 18.1384 6.9034 384 147456 56623104 19.5959 7.2685 330 108900 35937000 18.1659 6.9104 385 148225 57066625 19.6214 7.2748 331 109561 36264691 18.1934 6.9174 148996 57512456 19.6469 7.2811 332 110224 36594368 18.2209 6.9244 149769 57960603 19.6723 7.2874 333 110889 36926037 18.2483 6.9313 3 150544 58411072 19.6977 7.2936 334 111556 37259704 18.2757 6.9382 389 151321 58863869 19.7231 7.2999 335 112225 37595375 18.3030 6.9451 390 152100 59319000 19.7484 7.3061 336 112896 37933056 18.3303 6.9521 391 152881 59776471 19.7737 7.3124 337' 113569 38272753 18.3576 6.9589 192 153664 60236288 19.7990 7.3186 33S 114244 38614472 18.3848 6.9658 154449 60698457 19.8242 7.3248 339 114921 38958219 18.4120 6.9727 394 155236 61162984 19.8494 7.3310 340 115600 39304000 18.4391 6.9795 395 156025 61629875 19.8746 7.3372 341 116281 39651821 18.4662 6.9864 J9I 156816 62099136 19.8997 7.3434 342 116964 40001688 18.4932 6 9932 397 157609 62570773 19.9249 7.3496 343 117649 40353607 18.5203 7.0000 39S 158404 63044792 19.9499 7.3558 344 118336 40707584 18.5472 7.0068 399 159201 63521199 19.9750 7.3619 315 119025 41063625 18.5742 7.0136 400 160000 64000000 20 0000 7.3681 346 119716 41421736 18.6011 7.0203 401 160801 64481201 20 0250 7.3742 347 120409 41781923 18.6279 7.0271 402 161604 64964808 20.0499 7.3803 348 121104 42144192 18.6548 7.0338 403 162409 65450827 20 0749 7.3864 349 121801 42508549 18.6815 7.0406 404 163216 65939264 20.0998 7.3925 350 122500 42875000 18.7083 7.0473 405 164025 66430125 20.1246 7.3986 351 123201 43243551 18.7350 7.0540 406 164836 66923416 20.1494 7.4047 352 123904 43614208 18.7617 7.0607 407 165649 67419143 20.1742 7.4108 353 124609 43986977 1S.7S-3 7.0674 40S 166464 67917312 20.1990 7.4169 354 125316 44361864 18.8149 7.0740 409 167281 68417929 20.2237 7.4229 355 126025 44738875 18.8414 7.0807 410 168100 68921000 20.2485 7.4290 3 6 126736 45118016 18.8680 7.0873 411 168921 69426531 20.2731 7.4350 357 127449 45499293 18.8944 7.0940 412 169744 69934528 20.2978 7.4410 358 128164 45882712 18 9209 7.1006 413 170569 70444997 20.3224 7.4470 359 128881 46268279 18.9473 7.1072 414 171396 70957944 20.3470 7.4530 360 129600 46656000 18.9737 7.1138 415 172225 71473375 20.3715 7.4590 361 130321 47045881 19.0000 7.1204 416 173056 71991296 20.3961 7.4650 362 131044 47437928 19.0263 7.1269 417 173889 72511713 20.4206 7.4710 363 131769 47832147 17.0526 7.1335 418 174724 73034632 20.4450 7.4770 364 132496 48228544 19.0788 7.1400 419 175561 73560059 20.4695 7.4829 365 133225 48627125 19.1050 7.1466 420 176400 74088000 20.4939 7.4889 366 133956 49027896 19.1311 7.1531 421 177241 74618461 20.5183 7.4948 367 134689 49430863 19.1572 7.1596 422 178084 75151448 20.5426 7.5007 368 135424 49836032 19.1833 7.1661 423 178929 75686967 20.5670 7.5067 309 136161 50243409 19.2094 7.1726 424 179776 76225024 20.5913 7.5126 370 136900 50653000 19.2354 7.1791 425 180625 76765625 20.6155 7.5185 371 137641 51064811 19.2614 7.1855 426 181476 77308776 20.6398 7.5244 372 138384 51478848 19.2873 7.1920 427 182329 77854483 20.6640 7.5302 S73 139129 51895117 19.3132 7.1984 428 183184 78402752 20 6882 7.5361 374 139876 52313624 19.3391 7.2048 429 184041 78953589 20.7123 7.5420 SQUARES, CUBES, SQUARE AKB CUBE ROOTS. 91 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube' Root. 430 184900 79507000 20.7364 7.5478 485 235225 114084125 22.0227 7.8568 431 185761 80062991 20.7605 7.5537 486 236196 114791256 22.0454 7.8622 432 186624 80621568 20.7846 7.5595 487 237169 115501303 22.0681 7.8676 433 1S7489 SI 182737 20.80S7 7.5654 488 238144 116214272 22.0907 7.8730 434 188356 81746504 20.8327 7.5712 489 239121 116930169 22.1133 7.8784 435 189225 82312875 20.8567 7.5770 490 240100 117649000 22.1359 7.8837 436 190096 82881856 20.8806 7.5828 491 241081 118370771 22.1585 7.8891 437 190969 83453453 20.9045 7.5886 492 242064 119095488 22.1811 7.8944 438 191844 84027672 20.9284 7.5944 492 243049 119823157 22.2036 7.8998 439 192721 84604519 20.9523 7.6001 494 244036 120553784 22.2261 7.9051 440 193600 85184000 20.9762 7.6059 495 245025 121287375 22.2486 7.9105 441 194481 85766121 21.0000 7.6117 496 246016 122023936 22.2711 7.9158 442 195364 86350888 21.0238 7. 6174 497 247009 122763473 ■J2.2!)85 7.9211 443 196249 86938307 21.0476 7.6232 498 248004 123505992 22.3159 7.9264 444 197136 87528384 21.0713 7.6289 499 249001 124251499 22 3383 7.9317 445 198025 88121120 21.0950 7.6346 500 250000 125000000 22.3607 7.9370 446 198916 88716536 21.1187 7.6403 501 251001 125751501 22.3830 7.9423 447 199809 89314623 21.1424 7.6460 502 252004 126506008 22.4054 7.9476 448 200704 89915392 21.1660 7.6517 503 253009 127263527 22.4277 7.9528 449 201601 90518849 21.1896 7.6574 504 254016 128024064 22.4499 7.9581 450 202500 91125000 21.2132 7.6631 505 255025 128787625 22.4722 7.9634 451 203401 91733851 21.2368 7.6688 506 256036 129554216 22.4944 7.9686 452 204304 92345408 21.2603 7.6744 507 257049 130323843 22.5167 7.9739 453 205209 92959677 21.2838 7.6800 508 258064 131096512 22.5389 7.9791 454 206116 93576664 21.3073 7.6857 509 259081 131872229 22.5610 7.9843 455 207025 94196375 21.3307 7.6914 510 260100 132651000 22.5832 7.9896 456 207936 94818816 21.3542 7.6970 511 261121 133432831 22.6053 7.9948 457 208849 95443993 21.3776 7.7026 512 262144 134217728 22.6274 8.0000 458 209764 96071912 21 4009 7.7082 513 263169 135005697 22.6195 8.0052 459 210681 96702579 21.4243 7.7138 514 264196 135796744 22.6716 8.0104 460 211600 97336000 21.4476 7.7194 515 265225 136590875 22.6936 8.0156 461 212521 97972181 21.4709 7.7250 516 266256 137388096 22.7156 8.0208 462 213444 98611128 21.4942 7.7306 517 267289 138188413 22.7376 8.0260 463 214369 99252847 21.5174 7.7362 518 26S324 138991832 22.7596 8.0311 464 215296 99897344 21.5407 7.7418 519 269361 139798359 22.7816 8.0363 465 216225 100544625 21.5639 7.7473 520 270400 140608000 22.8035 8.0415 466 217156 101194696 21.5870 7.7529 521 271441 141420761 22.8254 8.0466 467 218089 101847563 21.6102 7.7584 522 272484 142236648 22.8473 8.0517 468 219024 102503232 21.6333 7.7639 523 273529 143055667 22.8692 8.0569 469 219961 103161709 21.6564 7.7695 524 274576 443877824 22.8910 8.0620 470 220900 103823000 21.6795 7.7750 525 275625 ' 144703 125 22.9129 8.0671 471 221841 104487111 21.7025 7.7805 526 276676 445531576 22.9317 8.0723 472 222784 105154048 21.7256 8.7860 527 277729 1146363183 22.1)0(15 8.0774 473 223729 105823817 21.7486 7.7915 528 278784 147 197952 22.9783 8.0825 474 224676 106496424 21.7715 7.7970 529 279841 148035889 23.0000 8.0876 475 225625 107171875 21.7945 7.8025 530 280900 148877000 23.0217 8.0927 476 226576 107850176 21.8174 7.8079 531 281961 i 149721291 23.0434 8.0978 477 227529 108531333 21.S403 7.8134 532 283024 150568768 23.0651 8.1028 478 228484 109215352 21.8632 7.8188 533 284089 ! 151419437 23.0868 8.1079 479 229441 109902239 21.8861 7.8243 534 285156 |152273304 23.1084 8.1130 480 230400 110592000 21.9089 7.8297 535 286225 153130375 23.130' 8.1180 481 231361 111284641 21.9317 7.8352 536 287296 1153990656 23.1517 8.1231 482 232324 111980168 21.9545 7.8406 537 288369 1 1548541 53 23.1733 8.1281 483 233.'89 112678587 21.9773 7.8460 538 289444 J155720872 23.1948 8.1332 484 234256 113379904 22.0000 7.8514 539 290521 156590819 23.2164 8.1382 92 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 540 291600 157464000 23.2379 8.1433 595 354025 210644875 24.3926 8.4108 541 292681 158340421 >:, :;/)! 8.1483 596 355216 211708736 24.4131 8.4155 542 293764 159220088 23.2809 8.1533 597 356409 212776173 24.4336 8.4202 543 294849 160103007 23.3(124 8.1583 598 357604 213847192 24.4540 8.4249 544 295936 160989184 23.3238 8.1633 599 358801 214921799 24.4745 8.4296 545 297025 161878625 23.3452 8.1683 6C0 360000 216000000 24.4949 8 4343 546 298116 162771336 23.3666 8.1733 601 361201 217081801 24.5153 8.4390 547 299209 163667323 23.3880 8.1783 602 362404 218167208 24.5357 8.4437 548 300304 164566592 23.4094 8.1833 603 363609 219256227 24.5561 8.4484 549 301401 165469149 23.4307 8.1882 604 364816 220348864 24.5764 8.4530 550 302500 166375000 23.4521 8.1932 605 366025 221445125 24.5967 8.4577 551 30360! 167284151 23.4734 8.1982 606 367236 222545016 21.6171 8.4623 552 304704 168196608 23.4947 8.2031 607 30S449 223648543 24.6374 8.4670 553 305809 169112377 23.5160 8.2081 608 369664 224755712 24.6577 8.4716 554 306916 170031464 23.537.2 8.2130 609 370881 225866529 24.6779 8.4763 555 308025 170953875 23.5584 8.2180 610 372100 226981000 24 6982 8.4809 556 309136 171879616 23.571)7 8.2229 611 373321 228099131 24.71S4 8.4856 557 310249 172808693 23.6008 8.2278 612 374544 229220928 24 7386 8.4902 558 311364 173741112 23.6220 8.2327 613 375769 230346397 24.7588 8.4948 559 312481 174676879 23.6432 8.2377 614 376996 231475544 24.7790 8.4994 560 313600 175616000 23.6643 8.2426 615 378225 232608375 24.7992 8.5040 561 314721 176558481 !>:::.! 8.2475 616 379456 233744896 24.8193 8.5086 562 315844 177504328 23.7065 8.2524 617 380689 234885113 24.8395 8.5132 563 316969 178453547 23.7276 8.2573 618 381924 236029032 24.8596 8.5178 564 318096 179406144 23.7487 8.2621 619 383161 237176659 24.8797 8.5224 565 319225 180362125 23.7697 8.2670 620 384400 238328000 24.8998 8.5270 566 320356 181321496 23.7908 8.2719 621 385641 239483061 24.9199 8.5316 567 321489 182284263 23.8118 8.2768 622 386884 240641848 24.9399 8.5362 568 322624 183250432 23.8328 8.2816 623 388129 241804367 24.9600 8.5408 569 323761 184220009 23.8537 8.2865 624 389376 242970624 24.9800 8.5453 570 324900 185193000 23.8747 8.2913 625 390625 244140625 25.0000 8.5499 571 326041 186169411 23.8956 8.2962 626 391876 245314376 25.0200 8.5544 572 327184 187149248 23.9165 8.3010 627 393129 246491883 25.0400 8.5590 573 328329 188132517 23.9374 8.3059 628 394384 247673152 25.0599 8.5635 574 329476 189119224 23.9583 8.3107 629 395641 248858189 25.0799 8.5681 575- 330625 190109375 23.9792 8.3155 630 396900 250047000 25.0998 8.57'26 576 331776 191102976 24.0000 8.3203 631 398161 251239591 25.1197 8.5772 577 332929 192100033 24.0208 8.3251 632 390424 252435968 25.1396 8.5817 .578 334084 193100552 24.0416 8.3300 633 400689 253636137 25.1595 8.5862 579 335241 194104539 24.0624 8.3348 634 401956 254840104 25.1794 8.5907 580 336400 195112000 24.0832 8.3396 635 403225 256047875 25.1992 8.5952 581 337561 196122941 24.1039 8.3443 636 404496 257259456 25.2190 8.5997 582 338724 197137368 24.1247 8.3491 637 405769 258474853 8.6043 583 339889 198155287 24.1454 8.3539 638 407044 259694072 25.2587 S.G0SS 584 341056 199176704 24.1661 8.3587 639 408321 260917119 25.2784 8.6132 585 342225 200201625 24.1868 8.3634 640 409600 262144000 25.2982 8.6177 586 343396 201230056 24.2074 8.3682 641 410881 263374721 25.3180 8.6222 587 344569 202262003 24.2281 8.3730 642 412164 264609288 25.3377 8.6267 588 345744 203297472 24.2487 8.3777 643 413449 265847707 25.3574 8.6312 589 346921 204336469 24.2693 8.3825 644 414736 267089984 25.3772 8.6357 590 348100 205379000 24.2899 8.3872 645 416025 268336125 25.3969 8.6401 591 349281 206425071 24 3105 8.3919 646 417316 269586136 25.4165 8 6446 592 350464 207474688 24.3311 8.3967 647 418609 270840023 25.4362 8.6490 593 351649 208527857 24.3516 8.4014 648 419904 27'2097792 25.4558 8.6535 594 352836 209584584 24.3721 8.4061 649 421201 273359449 25.4755 8.6579 SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 93 -No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 650 422500 274625000 25.4951 8.6624 705 497025 350402625 26.5518 8.9001 651 423801 275894451 25.5147 8.6668 706 498436 351895816 26.5707 8.9043 652 425104 277167808 25. 53 1 3 8.6713 707 499849 353393243 26.5895 8.9085 653 426409 278445077 25.5539 8.6757 708 501264 154894912 26.6083 8.9127 654 427716 279726264 25.5734 8.6801 709 502681 356400829 26.6271 8.9169 655 429025 281011375 25.5930 8.6845 710 504100 357911000 26.6458 8.9211 656 430336 282300416 25.6125 8.6890 711 505521 359425431 26.6646 8.9253 657 431649 283593393 25.6320 8.6934 712 506944 360944128 26.6833 8.9295 658 432964 284890312 25.6515 8 6978 713 508369 362467097 26.7021 8.9337 659 434281 286191179 25.6710 8.7022 714 509796 363994344 26.7208 8.9378 660 435600 287496000 25.6905 8.7066 715 511225 365525875 26.7395 8.9420 661 436921 288804781 25.71)911 8.7110 716 512656 367061696 -.6.7582 8.9462 662 438244 290117528 25 7294 8.7154 717 514089 368601813 26.7769 8.9503 603 439569 291434247 25.74S8 8.7198 718 515524 370146232 •.'6.7955 8.9545 664 440896 292754944 25.7682 8.7241 719 516961 371694959 26.8142 8.9587 665 442225 294079625 25.7876 8.7285 720 518400 373248000 26.8328 8.9628 666 443556 295408296 25.8070 8.7329 721 519841 374805361 26.S514 8.9670 667 444889 296740963 25.S263 8.7373 722 521284 376367048 26.8701 8.9711 668 4462-24 298077632 25.8457 8.7416 723 522729 377933067 26.8887 8.9752 669 447561 299418309 25.8650 8.7460 724 524176 379503424 26.9072 8.9794 670 448900 300763000 25.8844 8.7503 725 525625 381078125 26.9258 8.9835 671 450241 302111711 25.9037 8.7547 726 527076 382657176 26.9444 8.9876 672 451584 303464448 25.9230 8.7590 727 528529 384240583 26.9629 8.9918 673 452929 304821217 25.9122 8.7634 728 5299S4 385828352 26.9815 8.9959 674 454276 306182024 25.9615 8.7677 729 531441 387420489 27.0000 9.0000 675 455625 307546875 25.9808 8.7721 730 532900 389017000 27 0185 9 0041 676 456976 308915776 26.0000 8.7764 731 534361 390617891 27.0370 9.0082 677 458329 310288733 26.0192 8.7807 732 535824 392223168 27.0555 9.0123 678 459684 311665752 26.0384 8.7850 733 537289 393832837 27.0740 9.0164 679 461041 313046839 26.0576 8.7893 734 538756 395446904 27.0924 9.0205 6S0 462400 314432000 26.0768 8.7937 735 540225 397065375 27.1109 9.0246 681 463761 315821241 . O'.itii; 8.7980 736 541696 398688256 27.1293 9.0287 682 465124 317214568 26.1151 8.8023 737 543169 400315553 27.1477 9.0328 683 466489 318611987 26.1343 8.8066 738 544644 401947272 27.1662 9.0369 684 467856 320013504 26.1534 8.8109 739 546121 403583419 27.1846 9.0410 685 469225 321419125 26.1725 8.8152 740 547600 405221000 27.2029 9.0450 686 470596 322828856 26.1916 8.8194 741 549801 406869021 27.221o 9.0491 687 471969 324242703 26.2107 8.8237 742 550564 408518488 27.2397 9.0532 688 473344 325660672 26.22!H 8.8280 743 552049 410172407 27.2581 9.0572 689 474721 327082769 26.2488 8.8323 744 553536 411830784 27.2764 9.0613 600 476100 328509000 26.2679 8.8366 745 555025 413493625 27.2947 9.0654 691 477481 329939371 26.2869 8.8408 746 556516 415160936 27.3130 9.0694 692 478864 331373888 26.3059 8.8451 747 558009 416832723 27.3313 9.0735 693 480249 332812557 .; :;• ' 8.8493 748 559504 418508992 27.3496 9.0775 694 481636 334255384 26.3439 8.8536 749 561001 420189749 27.3679 9.0816 695 483025 335702375 26.3629 8.8578 750 562500 421875000 27.3861 9.0856 696 484416 337153536 26.3818 8.8621 751 564001 423564751 27.4044 9.0896 697 485809 338608873 26.4008 8.8663 752 565504 425259008 27.4226 9.0937 698 487204 340068392 23.4197 8.8706 753 567009 426957777 27.4408 9.0977 699 488601 341532099 26.4386 8.8748 754 568516 428661064 27.4591 9.1017 700 490000 343000000 26.4575 8.8790 755 570025 430368875 27.4773 9.1057 701 491401 344472101 26.4764 8.8833 756 571536 432081216 27.4955 9.1098 702 492804 345948408 26.4953 8. 8875 1 757 573049 433798093 27.5136 9.1138 r m 494209 347428927 26.5141 8.8917 1 758 574564 435519512 27.5318 9.1178 :m 4956 10 318:)lo664 26.5330 8.89591759 576081 437245479 27.5500 9.1218 94 MATHEMATICAL TABLES. No. Square. 760 577600 761 579121 702 580644 763 582169 764 583696 765 585225 766 586756 767 588289 7'6S 589824 769 591361 770 592900 771 594441 77-2 595984 778 597529 774 599076 775 600625 776 602176 777 603729 778 605284 779 606841 780 608400 781 609961 782 611524 788 613089 784 614656 785 616225 786 617796 787 619369 788 620944 789 622521 790 624100 791 625681 792 627264 798 628849 794 630436 795 632025 796 633616 797 635209 798 636804 799 638401 800 640000 SOI 641601 802 643204 808 644809 804 646416 805 648025 80(1 649636 807 651249 SOS 652S64 809 654481 810 656100 811 657721 812 659344 81.-, 660969 814 662596 438976000 440711081 442450728 444194947 445943744 447697125 449455096 451217663 452! 454756609 456533000 458314011 460099648 461889917 463684824 465484375 2' 4672S8576 2' 469097433 2' 470910952 2' 472729139 474552000 476379541 478211768 480048687 481890304 Sq. Root. 48373662: 485587656 487443403 489303872 491169069 493039000 494913671 496793088 498677257 500566184 502459875 504358336 506261573 508169592 510082399 512000000 513922401 515849608 517781627 519718464 521660125 523606616 525557943 527514112 529475129 28.4429 53144100o'28.4605 533411731 28.4781 5358S7828 28.4956 537867797 28.5132 539353144 28.5807 5681 5862 6043 6225 .6405 .6586 .676' 17.7489 7669 78 M) 8029 8209 0285 9464 '7.96 18 Cube Root. 28.0000 28.0179' 28.0357 28.0585 28.0713 28. 0891 j 28.1069, 28.1247 28.1425 28.1603 28.1780 2S.1957 28.2135 28.2812 28.2489 28.2843 28.3019 3373 28.3549 28.407 28.4253 9.1258 9.1298 9.1338 9.1537 9.1577 9.1617 9.1657 9.1696 9.1736 9.1775 9.1815 9.1855 9.1894 9.1933 9.1973 9.2012 9.2052 9.2091 9.2130 9^2209 9.2248 9.2287 9 2326 9.2365 9.2404 9.2443 9.2482 9.2521 9.2560 9.267 9.2716 9.2754 9.2793 9.2 9.2870 9.3025 9.3063 9.3102 9.3140 9.3179 9.3217 9.3255 9.3294 9.3332 9.8370 No. Square, 815 664225 816 665856 817 i 667489 818 669121 819' 670761 820 ' 672400 821 1 674041 822 : 675684 678976 680625 682276 683929 685584 687241 700569 702244 703921 705600 707281 708964 710649 712336 714025 715716 717409 719104 720801 722500 724201 725904 727609 731025 732736 781119 736164 739600 741321 751689 753424 755161 Sq. Cube Root. Root. 541343375 543338496 545338513 547343432 549353259 551368000 553387661 555412248 557441767 559476224 561515625 563559976 565609283 567663552 569722789 571787000 573856191 575930368 578009537 580093704 582182875 584277056 586376253 5884804' 590589719' 592704000 594823321 5969476S8 599077107 601211584 28.5482 28.5657 28.5832 28.6007 28.6182 28.6356 28.6531 28.6705 28.7402 28.7576 28.7750 28.79$ 28.8097 28.8271 28.8444 28.8617 28.8791 28.9137 28.9310 28.9482 28.9655 29.0000 29.0175 29.0345 29.0517 603351125 29. ( 605495736 29.0861 607645423 29.1O 6098001 92 j 29. 1204 611960049 29. 1376 614125000 616295051 618470208 620650477 622835864 625026375 627222016 629422793 631628712 .1548 .1719 29.1890 29.2062 29.2233 29.2404 29.2575 2746 29.2916 636056000 638277381 640503928 642735647 644972544 647214625 649461896 651714363 653972032 656234909 29.3258 29 342* 29.359* 29.3769 29.3939 4109 29.427' .4449 29.4618 29.47* SQUARES, CUBES, SQUARE AND CUBE ROOTS. 95 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. S70 756900 658503000 29.4958 9.5464 925 855625 791453125 30.4138 9.7435 871 758641 660776311 29.5127 9.5501 926 857 170 794022776 730.4302 9.7470 87:2 760384 663054848 29.5296 9.5537 927 859329 796597983 30.4467 9.7505 878 762129 665338617 29.5466 9.5574 928 861184 7'99 178752 30.4631 9 7540 874 763876 667627624 29.5635 9.5610 929 863041 801765089 30.4795 9.7575 875 765625 669921875 29.5804 9.5647 930 864900 804357000 30.4959 9.7610 876 767376 672221376 29.5973 9.5683 931 866761 806954491 [30. 5123 9.7645 877 769129 674526133 29.6142 9.5719 932 868624 809557568,30.5287 9.76S0 878 7708S4 676836152 29.6311 9.5756 933 870489 812166237 30.5450 9.7715 879 772641 679151439 29.6479 9.5792 934 872356 814780504 30.5614 9.7750 880 774400 681472000 29.6648 9.5828 935 874225 817400375 30.5778 9.7785 881 776161 683797841 29.6816 9.5865 936 876096 820025856 30.5941 9.7819 882 777924 686128968 29.6985 9.5901 937 877969 822656953 30.6105 9.7S54 88.3 779689 688465387 29.7153 9.5937 938 879844 825293672:30.6268 9.7889 884 781456 690807104 29.7321 9.5973 939 881721 827936019 30.6431 9.7924 885 783225 693154125 29.7489 9.6010 940 883600 830584000 30.6594 9.7959 886 784996 695506456 29.7658 9.6046 941 J 885481 833237621 30.6757 9.7993 SS7 786769 697864103 29.7825 9.6082 942; 887364 835896888 30.6920 9.8028 888 788544 700227072 29.7993 9.6118 943 i 889249 838561807 30.7083 9.8063 889 790321 702595369 29.8161 9.6154 944 891136 841232384 30.7246 9.8097 890 792100 704969000 29.8329 9.6190 945' 893025 84390S625 30.7409 9.8132 891 ■93881 707347971 29.8496 9.6226 946! 894916 846590536 30.7571 9.8167 89-2 795664 709732288 29.8664 9.6262 947 896809 849278123 30.7734 9.8201 8! 13 797449 712121957 29.SS31 9.6298 948 898704 851971892 30.7S1MI 9.8236 894 799236 714516984 29.8998 9.6334 949 900601 854670349 30.S058 9.8270 895 801025 716917375 29.9166 9.6370 950 902500 857375000 30.8221 9.8305 890 802816 719323136 29.9333 9.6406 951 904401 860085351 j 30. 8383 9.8339 897 804609 721734273 29.9500 9.6442 952 906304 862801408 30.8545 9.8374 898 806404 724150792 29 . 9666 9.6477 953 908209 865523177 30.8707 9.8408 899 808201 726572699 29.9833 9 6513 954 910116 868250664 30.8869 9.8443 900 810000 729000000 30 0000 9.6549 955 912025 870983875 30.9031 9.8477 901 811801 731432701 30.0167 9.6585 956 913936 87'3722816 30.9192 9.8511 902 813604 733870S08 30.0333 9.6620 957 915849 876467493 30.9354 9.8546 903 815409 736314327 30.0500 9.6656 958 917764 879217912 30.9516 9.8580 901 817216 738763264 30.0066 9.6692 959 919681 881974079 30.9677 9.8614 905 819025 741217625 30.0832 9.6727 960 921600 884736000 30.9839 9.8648 900 820836 743677416 30.0998 9.8763 961 923521 887503681 31.0000 9.8683 907 822649 746142643 30.1164 9.6799 962 925444 890277128 31.0161 9.8717 908 824464 748613312 30.1330 9.6S34 963 927369 893056347 31.0322 9.8751 909 826281 751089429 30.1496 9.6870 964 929296 895841344 31.0483 9.8785 910 828100 753571000 30.1662 9.6905 965 931225 898632125 31.0644 9.8819 911 829921 756058031 30.1828 9.6941 966 933156 901428696 31.0805 9.8854 912 831744 758550528 30.1993 9.6976 967 935089 904231063 31.0966 9.8888 913 833569 761048497 30.2159 9.7012 968 937024 907039232 31.1127 9.8922 914 835396 763551944 30.2324 9.7'047 969 938961 909853209 31.1288 9.8956 915 837225 766060875 30.2490 9.7082 970 940900 912673000 31.1448 9.8990 916 839056 768575296 30.2655 9.7118 971 942841 915498611 31.1609 9.9024 91? 840889 771095213 30.2820 9.7153 972 9447S4 918330048 31.1769 9.9058 918 842724 773620632 30.2985 9.7188 973 946729 921167317 31.1929 9.9092 919 844561 776151559 30.3150 9.7224 974 948676 924010424 31.2090 9.9126 920 846400 778688000 30.3315 9.7259 975 950625 926859375 31.2250 9.9160 921 848241 781229961 30,3480 9.7294 976 952576 929714176 31.2410 9.9194 922 850084 783777448 9.7329 977 954529 932574833 31.2570 9 9227 923 851929 786330467 30.3809 9.7364| 978 956484 935141352 31.2730 9.9261 924 853776 788889024 30.3974 9.7400! 979 958441 938313739 31.2890 9.9295 96 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube. Root. No. Square. Cube. Sq. Root. Cube Root. 980 960400 941192000 31.3050 9.9329 1035 1071225 1108717875 32.1714 10.1153 981 962361 944076141 31.32(i! 9.9363 1036 1073296 1111934656 32.1870 10.1186 982 964324 946966168 31.3369 9.9396 1037 1075369 1115157653 32.202.') 10.1218 983 966289 949862087 31.3528 9.9430 1038 1077444 1118386872 32.2180 10.1251 984 968256 952763904 31.3688 9.9464 1039 1079521 1121622319 32.2335 10.1283 985 970225 955671625 31.3847 9.9497 1040 1081600 1124864000 32.2490 10.1316 980 972196 958585256 31.4006 9.9531 1041 1083681 1128111921 10.1348 1M 974169 961504803 31.4166 9.9565 1042 1085764 1131366088 32.2800 10.1381 988 976144 964430272 31 . 4325 9.9598 1043 1087S49 1134626507 32.2955 10.1413 989 978121 967361669 31.4484 9.9632 1044 1089936 1137893184 32.3110 10.1446 990 980100 970299000 31.4643 9.9666 1045 1092025 1141166125 32.3265 10.1478 991 982081 973242271 31.4802 9.9699 1046 1094116 1144445336 32.3419 10.1510. 99',' 984064 976191488 31.4960 9.9733 1047 1096209 1147730823 32.3574 10.1543 993 986049 979146657 31.5119 9.9766 104S 1098304 1151022592 32.3728 10.1575 994 988036 982107784 31.5278 9.9800 1049 1100401 1154320649 32.3883 10.1607 995 990025 985074875 31.5436 9.9833 1050 1102500 1157625000 32.4037 10.1640 !)9(i 992016 988047936 31.5595 9.9866 1051 1104601 1160935651 32.4191 10.1672 997 994009 991026973 31.5753 9.9900 1052 1106704 1164252608 32.4345 10.1704 998 996004 994011992 31.5911 9 9933 1053 1108809 1167575877 32.4500 10.1736 999 998001 997002999 31.6070 9.9967 1054 1110916 1170905464 32.4654 10.1769 1000 1000000 1000000000 31.6228 10.0000 1055 1113025 1174241375 32.4808 10.1801 1001 1002001 1003003001 31.6386 10.0033 1056 1115136 1177583616 32.4962 10.1833 1004004 1006012008 31.6544 10.0067 1057 1117249 1180932193 32.5115 10.1865 1003 1006009 1009027027 31.6702 10.0100 1058 1119364 1184287112 32.5269 10.1897 1004 1008016 1012048064 31.6860 10.0133 1059 1121481 1187648379 32.5423 10.1929 1005 1010025 1015075125 31.7017 10.0166 1060 1123600 1191016000 32.5576 10.1961 H 1012036 1018108216 31.7175 10.0200 1061 1125721 1194389981 32.5730 10.1993 li 1014049 1021147343 31.7333 10 0233 1062 1127844 1197770328 32.5883 10.2025 1008 1016064 1024192512 31.7490 10.0266 1063 1129969 1201157047 32.6036 10.2057 1009 1018081 1027243729 31.7648 10.0299 1064 1132096 1204550144 32.6190 10.2089 1010 1020100 1030301000 31.7805 10.0332 1065 1134225 1207949625 32.6343 10.2121 1011 1022121 1033364331 31.7962 10.0365 1066 1136356 1211355496 32.6497 10.2153 1012 1024144 1036433728 31.8119 10.0398 1067 1138489 1214767763 32.6650 10.2185 1013 1026169 1039509197 31.8277 10.0431 1068 1140624 1218186432 32.6803 10.2217 1014 1028196 1042590744 31.8434 10.0465 1069 1142761 1221611509 32.6956 10.2249 1015 1030225 1045678375 31.8591 10.0498 1070 114490Q 1147041 1225043000 32.7109 10.2281 1016 1032251; 1048772096 31.8748 10.0531 1071 1228480911 32.7261 10.2313 1017 1034289 1051871913 31.8904 10.0563 1072 1149184 1231925248 32.7414 10.2345 1018 1036324 1054977832 31.9061 10.0596 1073 1151329 1235376017 32.7567 10.2376 1019 1038361 1058089859 31.9218 10.0629 1074 1153476 1238833224 32.7719 10.2408 1020 1040400 1061208000 31.9374 10.0662 1075 1155625 1242296875 32.7872 10.2440 1021 1042441 1064332261 31.9531 10.0695 1076 1157776 1245766976 32.8024 10.2472 1044484 1067462648 31.9687 10.0728 1077 1159929 1249243533 32.8177 10.2503 1046529 1070599167 31.9844 10.0761 1078 1162084 1252726552 32.8329 10.2535 1024 1048576 1073741824 32.0000 10.0794 1079 1164241 1256216039 32.8481 10.2567 1025 1050625 1076890625 32.0156 10.0826 1080 1166400 1259712000 32.8634 10.2599 1052676 1080045576 32.0312 10.0859 1081 1168561 1263214441 32.8786 10.2630 1054729 1083206683 32.0468 10.0892 1082 1170724 1266723368 32.8938 10.2662 1056784 1086373952 32.0624 10.0925 10S3 1172889 1270238787 32.9090 10.2693 1029 1058841 1089547389 32.0780 10.0957 1084 1175056 1273760704 32.9242 10.2725 1030 1060900 1092727000 32.0936 10.0990 1085 1177225 1277289125 32.9393 10.2757 103! 1062961 1095912791 32.1092 10.1023 1086 1179396 1280824056 10.2788 1032 1065024 1099104768 32.1248 10.1055 1087 1181569 1284365503 32.9697 10.2820 1033 1067089 1102302937 32.1403 10.1088 1088 1183744 1287913472 32.9848 10.2851 1034 1069156 1105507304 32.1559 10.1121 1089 1185921 1291467969 33.00001 10.2883 SQUABES, CUBES, SQUABE AKD CUBE BOOTS. 97 No. Square. Cube. Sq. Boot. Cube Boot. No. Square. Cube. Sq. Boot. Cube Boot. 1090 1188100 1295029000 33.0151 10.2914 1145 1311025 1501123625 33.8378 10.4617 11)91 1190281 129S596571 33.0303 10.2946 1146 1313316 1505060136 10.4647 • 1192464 1302170688 33.0454 10.2977 1147 1315609 1509003523 33.8674 10.4678 1093 1194649 1305751357 33.0606 10.3009 1148 1317904 1512953792 33.8821 10.4708 1094 1196836 1309338584 33.0757 10.3040 1149 1320201 1516910949 33.8969 10.4739 1095 1199025 1312932375 33.0908 10.3071 1150 1322500 1520875000 33.9116 10.4769 1096 1201216 1316532736 33.1059 10.3103 1151 1324801 1524845951 33.9264 10.4799 1097 1203409 1320139673 33.1210 10.3134 1152 1327104 1528823808 33.9411 10.4830 1098 1205604 1323753192 33.1361 10.3165 1153 1329409 1532808577 33 . 9559 10.4860 1099 1207801 1327373299 33.1512 10.3197 1154 1331716 1536800264 33.9706 10.4890 1100 1210000 1331000000 33.1662 10.3228 1155 1334025 1540798875 33.9853 10.4921 1101 1212201 1334633301 33.1813 10.3259 1156 1336336 1544804416 34.0000 10.4951 110:2 1214404 1338273208 33.1964 10.3290 1157 1338649 1548816893 34.0147 10.4981 1103 1216609 1341919727 33.2114 10.3322 1158 1340964 1552836312 34.0294 10.5011 1104 1218816 1345572864 33.2264 10.3353 1159 1343281 1556862879 34.0441 10.5042 1105 1221025 1349232625 33.2415 10.3384 1160 1345600 1560896000 34.0588 10.5072 1106 1223236 1352899016 33.2566- 10.3415 1161 1347921 1564936281 34.0735 10.5102 1107 1225449 1356572043 33.2716 10.3447 1162 1350244 1568983528 34.0881 10.5132 11 OS 1227664 1360251712 33.2866 10.3478 1163 1352569 1573037747 34.1028 10.5162 1109 1229881 1363938029 33.3017 10.3509 1164 1354896 1577098944 34.1174 10.5192 1110 1232100 1367631000 33.3167 10.3540 1165 1357225 1581167125 34.1321 10.5223 1111 1234321 1371330631 10.3571 1166 1359556 1585242296 34.1467 10.5253 1112 1236544 1375036928 33.3407 10.3602 1167 1361889 1589324463 34.1614 10.5283 1113 1238769 1378749897133.3617 10.3633 1168 1364224 1593413632 34.1760 10.5313 in* 1240996 1382469544 33.3766 10.3664 1169 1366561 1597509809 34.1906 10.5343 1115 1243225 1386195875 33.3916 10 3695 1170 1368900 1601613000 34.2053 10.5373 1116 1245456 1389928896 33.4066 10.3726 1171 1371241 1605723211 34.2199 10.5403 1117 1247689 13931568613 33.4215 10.3757 1172 1373584 1609840448 34.2345 10.5433 HIS 1249924 1397415032 33.4365 10.3788 1173 1375929 1613964717 34.2491 10.5463 1119 1252161 1401168159 33.4515 10.3819 1174 1378276 1618096024 34.2637 10.5493 1120 1254400 1404928000 33.4664 10.3850 1175 1380625 1622234375 34.2783 10.5523 1121 1256641 1403694561 33.4813 10.3881 1176 1382976 1626379776 34.2929 10.5553 1 122 1258884 1412467848 33.4963 10.3912 1177 1385329 1630532233 34.3074 10.5583 1123 1261129 1416247867 33.5112 10.3943 1178 1387684 1634691752 34.3220 10.5612 1124 1263376 1420034624 33.5261 10.3973 1179 1390041 163S858339 34.3366 10.5642 1125 1265625 1423828125 33.5410 10.4004 1180 1392400 1643032000 34.3511 10.5672 1 126 1267876 1427628376 33.5559 10.4035 1181 1394761 1647212741 34.3657 10.5702 1127 1270129 1431435383 33.5708 10.4066 1182 1397124 1651400568 34.3802 10.5732 12723S4 1435249152 33.5857 10.4097 1183 1399489 1655595487 34.3948 10 5762 1129 1274641 1439069689 33.6006 10.4127 1184 1401856 1659797504 34.4093 10.5791 1130 1276900 1442897000 33.6155 10.4158 1185 1404225 1664006625 34.4238 10.5821 1131 1279161 1446731091 10.4189 1186 1406596 1668222856 34.4384 10.5851 1132 1281424 1450571968 33.6452 10.4219 1187 1408969 1672446203 34.4529 10.5881 1283689 1454419637 33.6601 10.4250 1188 1411344 1676676672 34.4674 10.5910 1134 1285956 1458274104 33.6749 10.4281 1189 1413721 1680914269 34.4819 10.5940 1135 1288225 1462135375 33.689S 10.4311 1190 1416100 1685159000 34.4964 10.5970 1136 1290496 1466003456 33.7046 10.4342 1191 1418481 1689410871 34.5109 10.6000 1137 1292769 1469878353 33.7174 10.4373 1192 1420864 1693669888 34.5254 10.6029 1138 1295044 1473760072 33.7342 10.4404 1193 1423249 1697936057 34.539S 10.6059 1139 1297321 1477648619 33.7491 10.4434 1194 1425636 1702209384 34.5543 10.6088 1140 1299600 1481544000 33.7639 10.4464 1195 1428025 1706489875 34.5688 10.6118 1141 1301881 1485446221 33.7787 10.4495 1196 1430416 1710777536 34.5832 10.6148 114.2 1304164 1489355288 33.7935 10.4525 1197 1432809 1715072373 34.5077 10 6177 1143 1306449 1493271207 33. 8083! 10. 4556 1198 1435204 1719374392 34.6121 10.6207 1144 1308736 1497193984 33.8231 10.4586 1199 ,437601 1723683599 34.6266 10.6236 98 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root, Cube Root. 1200 1440000 1 1728000000 34.6410 10.6266 1201 1442401 1732323601 34.6554 10.6295 1202 1444804 1736654408 34.6699 10.6325 1203 1447209 1740992427 34.6843 10.6354 1204 1449616 1745337664 34.6987 10.6384 1205 1452025 1749690125 34.7131 10.6413 120(1 1454436 1754049816 34.7275 10.6443 1207 1456849 1758416743 34.7419 10.6472 1208 1459264 1762790912 34.7563 10.6501 1209 1461681 1767172329 34.7707 10.6530 1210 1464100 1771561000 34.7851 10.6560 1211 1466521 1775956931 34.7994 10.8590 1212 1468944 1780360128 34.8138 10.6619 1213 1471369 1784770597 34.8281 10.6648 1214 1473796 1789188344 34.8425 10.6678 1215 1476225 1793613375 34.8569 10.6707 1216 1478656 1798045696 34.8712 10.6736 1217 1481089 1802485313 34.8855 10.6765 1218 1483524 1806932232 34.8999 10.6795 1219 1485961 1811386459 34.9142 10.6824 1220 1488400 1815848000 34.9285 10.6853 1221 1490841 1820316861 34.9428 10.6882 1222 1493284 1824793048 31.9571 10.6911 1223 1495729 1829276567 34.9714 10.6940 1224 1498176 1833767424 34.9857 10.6970 1225 1500625 1838265625 35.0000 10.6999 1226 1503076 1842771176 35.0143 10.7028 1227 1505529 1847284083 35.0286 10.7057 1228 1507984 1851804352 35.0428 10.7086 1229 1510441 1856331989 35.0571 10.7115 1230 1512900 1860867000 35.0714 10.7144 1231 1515361 1S65409391 35.0856 10.7173 1232 1517824 1869959168 35.0999 10.7202 1233 1520289 1874516337 35.1141 10.7231 1234 1522756 187 9080904 35.1283 10.7260 1235 1525225 1883652875 35.1426 10.7289 1236 1527696 1888232256 35.1568 10.7318 1237 1530169 1892819053 35.1710 10.7347 1238 1532644 1897413272 35.1852 10.7376 1239 1535121 1902014919 35.1994 10.7405 1240 1537600 1906624000 35.2136 10.7434 1241 1540081 1911240521 35.2278 10.7463 1242 1542564 1915864488 35.2-120 10.7491 1243 1545049 1920495907 35.2562 10.7520 1244 154? 536 1925134784 35.2704 10.7549 1245 1550025 1929781125 35.2846 10.7578 1216 1552516 1934434936 35.29s? 10.7607 1247 1555009 1939096223 35.3129 10.7635 121S 1557504 1943764992 35.3270 10.7664 1249 1560001 1948441249 35.3412 10.7693 1250 1562500 1953125000 35.3553 10.7722 1251 1505001 1957816251 35.3695 10.7750 1252 1567504 1962515008 35.3836 10.7779 1253 1570009 1967221277 35.3977 10.7808 125 4 1572516 1971935064 35.4119 10.7837 No. Square. Cube. Sq. Root. Cube Root. 1255 1575025 1976656375 35.4260 10.7865 1256 1577536 1981385216 35.4401 10.7894 1257 1580049 1986121593 35.4542 10.7922 1258 1582564 1900S05512 35.4683 10.7951 1259 1585081 1995616979 35.4824 10.7980 1260 1587600 2000376030.35.4965 10.8008 1201 ,1590121 2005142581 35.5106 10.8037 1262 1592644 2009916728 35.5246 10.8065 1263 1595169 2014698447 35.5387 10.8094 1264 1597696 2019487744 35.5528 10.8122 1265 1600225 2024284625 35.5668 10.8151 1266 1602756 2029089096 35.5809 10.8179 1267 1605289 2033901163 10 8208 1268 1607824 2038720S32 35.6090 10.8236 1269 1610361 2043548109 35.6230 10.8265 1270 1612900 2048383000 35.6371 10.8293 1271 1615441 2053225511 35.6511 10.8322 1272 1617984 2058075648 35.6651 10.8350 1273 1620529 2062933417 35.6791 10 8378 1274 1623076 2067798824 35.6931 10.8407 1275 1625625 2072671875 35.7071 10.8435 1276 1628176 2077552576 35.7211 10.8463 1277 1630729 2082440933 35.7351 10.8492 1278 1633284 2087336952 35.7491 10.8520 1279 1635841 2092240639 35.7631 10.8548 1280 1638400 2097152000 35.7771 10.8577 12S1 1640961 2102071041 35.7911 10.8605 1 282 1643524 2106997768 35.8050 10.8633 1646089 2111932187 35.8190 10.8661 1284 1648656 2116874304 35.8329 10.8690 1285 1651225 2121824125 35.8469 10.8718 1653796 2126781656 35.8608 10.8746 12S7 1656369 2131746903 35.8748 10.8774 12SS 1658944 2136719872 35.8887 10.8802 3289 1661521 2141700569 35.9026 10.8831 1290 1664100 2146689000 35.9166 10.8859 1291 1666681 2151685171 35.9305 10.8887 1202 1669264 21566890S8 35.9444 10.8915 1293 1671849 2161700757 35.9583 10.8943 1294 1674136 2166720184 35.9722 10.8971 1295 1677025 2171747375 35.9861 10.8999 1296 1679616 2176782336 36.0000 10.9027 1207 1682209 2181825073 36.0139 10.9055 12 '.is 1684804 2186875592 36.0278 10.9083 1209 1687401 2191933899 36.0416 10.9111 1300 1690000 2197000000 36.0555 10.9139 1301 1692601 2202073901 36.0694 10.9167 1302 1605201 2207155608 36.0832 10.9195 1303 1697809 2212245127 36.0971 10.9223 1304 1700416 2217342464 36.1109 10.9251 1305 1703025 2222447625 36.1248 10.9279 1306 1705636 2227560616 36.1386 10.9307 1307 170824912232681443 36.1525 10.9335 130S 17108642237810112 36.1663 10.9363 1300 1713481 2242946629 36.1801 10.9391 SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 09 No. Square. 1716100 2248091000 17187212253243231 1721344 225840332S 1723969 2263571297 1720596 2268747144 i 1729225 22' 1731856 2279122496 1734489 2284322013 1737124 2289529432 17397612294744759 1742400 2299968000 1745041 1747684 1750329 1752971 1755625 17582' 1760929 1763584 1766241 1768900 1771561 1774224 1776 1782225 1784 1787 1790244 1792921 1795600 1798281 1800964 1803649 1806336 1809025 1811716 1814409 1817104 1819801 1822500 1825201 1827904 1830609 231568526"; 2320940224 2326203125 2331473976 2336752783 2342039552 2347334289 2352637000 2357947 2363266368 2373927704 2379270375 2384621056 2389979753 2395346472 2400721219 2406104000 2411494821 1841449 1844164 1846881 1849600 1852321 1855044 1857769 1860496 i. 191 36.20' 36.2215 36.2353 36.2491 2629 36.276' 2905 36.3043 36.3180 36.3318 6.3456 30.3593 36.3731 242230060' 2427715584 2433138625 2438569736 2444008923 2449456192 2454911549 2460375000 2465846551 2471326208 2476813977 2482309864 2487813875 2493326016 2498846293 2504374712 250991127 2515456000 2521008881 2526569928 253213914 2537716544 Sq. Root. 36.4005 36.4143 36.4280 36.4417 36.4555 36.4692 36.4829 36.4966 36.5103 36.5240 36.5377 36.5513 36.5650 5787 36.6197 36.6333 36.6469 36.6742 36.6879 36.7015 36.7151 36.7287 36.7423 36.7560 36.8103 5.8239 Cube Root, 10.9418 10.9446 10.9474 10.9502 10.9530 10.9557 10.9585 10.9613 10.9640 10.9668 10.9696 10.9721 10.9752 10.9779 10.9807 10.9834 10.9862 10.9890 10.9917 10.9945 10.9972 11.0000 11.0028 11.0055 11.0083 11.0110 11.0138 11.0165 11 0193 11.0220 11.0247 11.0275 11.0302 11.0330 11.0357 11.0384 11.0412 11.0439 11.0466 11.0494 11.0521 11.0548 11.0575 11.0603 11.0630 11.0657 11.0684 11.0712 11.0739 11.0766 8782 11.0793 36.8917 11.0820 36.9053 11.0847 36.9188.11.0875 36.9324 11.0902 No. Square. Cube. Sq. Root. Cube Root. 1365 1863225 2543302125 36.9459 11.0929 1300 1865956 2548895896 36.9594 11.0956 1367 1868689 2554497863 36.9730 11.0983 1368 1871424 2560108032 36.9865 11.1010 1369 1874161 2565726409. 37.0000 11.1037 1370 1876900 2571353000 37.0135 11.1064 1371 1879641 2576987811 37.0270 11.1091 1372 1882384 2582630848 37.0405 11.1118 1373 1885129 2588282117 37.0540 11.1145 1374 1887876 2593941624 37.0675 11. 1172 1375 1890625 2599609375 37.0810 11.1199 1376 1893376 2605285376 37 0945 11.1226 1377 1896129 2610969633 37.1080 11.1253 137S 1898884 2616662152 37.1214 11.1280 1379 1901641 2622362939 37.1349 11.1307 1380 1904400 2628072000 37.1484 11.1334 1381 1907161 2633789341 37.1618 11.1361 1382 1909924 2639514968 37.1753 11.1387 13S3 1912689 2645248887 37.1887 11.1414 1384 1915456 2650991104 37.2021 11.1441 1385 1918225 2656741625 37.2156 11.1468 1386 1920996 2662500456 37.2290 11.1495 1387 1923769 2668267603 37.2424 11.1522 1388 1926544 2674043072 37.2559 11.1548 1389 1929321 2679826869 37.2693 11.1575 1390 1932100 2685619000 37.2827 11.1602 1391 1934881 2691419471 37.2961 11.1629 1392 1937664 2697228288 37.3095 11.1655 1393 1940449 2703045457 37.3229 11.1682 1394 1943236 2708870984 37.3363 11.1709 1395 1946025 2714704875 37.3497 11.1736 1948816 2720547136 37.3631 11.1762 1397 1951609 2726397773 37.3765 11.1789 1398 1954404 2732256792 37.3898 11.1816 1399 1957201 2738124199 37.4032 11.1842 1400 1960000 2744000000 37.4166 11.1869 1401 1962801 2749884201 37.4299 11.1896 1402 1905604 2755776808 37.4433 11.1922 1403 1968409 2761677827 37.4566 11.1949 1404 1971216 2767587264 37.4700 11.1975 1405 1974025 2773505125 37.4833 11.2002 1406 1976836 2779431416 37.4967 11.2028 1407 1979649 2785366143 37.5100 11.2055 1408 1982464 2791309312 37.5233 11.2082 1409 1985281 2797260929 37.5366 11.2108 1410 1988100 2803221000 37.5500 11.2135 1411 1990921 28091S9531 37.5633 11.2161 1412 1993744 2815166528 37.5766 11.2188 1413 1996569 2821151997 37.5899 11.2214 1414 1999396 2827145944 37.6032 11.2240 1415 2002225 2833148375 37.6165 11.2267 1416 2005056 2839159296 37.6298 11 2293 1417 2007889 2845178713 37.6431 11.2320 1418 2010724 2851206632 37.6563 11.2346 1419 2013561 2857243059 37.6696 11.2373 100 MATHEMATICAL TABLES. No. Square Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 1420 2016400 2863288000 37.6829 11.2399 1475 2175625 3209046875 38.4057 11.3832 1421 2019241 2869341461 37.6962 11.2425 1476 2178576 3215578176 38.4187 11.3858 2022084 2875403448 37.7094 11.2452 1477 2181529 3222118333 38.4318 11.3883 2024929 2881473967 37.7227 11.2478 1478 2184484 3228667352 38.4448 11.3909 1424 2027776 2887553024.37.7359 11.2505 1479 2187441 3235225239 38.4578 11.3935 1425 2030625 2893640625 37.7492 11.2531 1480 2190400 3241792000 38.4708 11.3960 2033476 2899736776 s 37. 7624 11.2557 1481 2193361 3248367641 38.4838 11.3986 2036329 2905841483 37.7757 11.2583 1482 2196324 3254952168 38.4968 11.4012 2039184 2911954752 37.7889 11.2610 1483 2199289 3261545587 38.5097 11.4037 1429 2042041 2918076589-37.8021 11.2636 1484 2202256 3268147904 38.5227 11.4063 1430 2044900 2924207000 37.8153 11.2662 1485 2205225 3274759125 38.5357 11.4089 1431 2047761 2930345991 '37.8286 11.2689 1486 2208196 3281379256 38.5487 11.4114 2050624 2936493568 37.8418 2942040:37 37.8550 11.2715 1487 2211169 3288008303 38.5616 11.4140 2053489 11.2741 1488 2214144 3294646272 38.5746 11.4165 1434 2056356 2948814504 37.8682 11.2767 1489 2217121 3301293169 38.5876 11.4191 1435 2059225 2954987875 37.8814 11.2793 1490 2220100 3307949000 38.6005 11.4216 2062096 2961169856 37.8946 11.2820 1491 2223081 3314613771 38.6135 11.4242 2064969 2967360453 37.9078 11.2846 1492 2220004 3321287488 38.6264 11.4268 143K 2067844 2973559672 37.9210 11.2872 1493 2229049 3327970157 38.6394 11.42(13 1439 2070721 2979767519:87.9342 11.2898 1494 2232036 3634661784 38.6523 11.4319 1440 2073600 2985984000 37.9473 11.2924 1495 2235025 3341362375 38.6652 11.4344 1441 2076481 2992209121 37.9605 11.2950 1496 2238016 3348071936 38.6782 11.4370 2079364 2998442888 37.9737 11.2977 1497 2241009 3354790473 38.6911 11.4395 2082249 3004685307 37.9868 11.3003 1498 2244004 3361517992 38.7040 11.4421 1444 2085136 3010936384 38.0000,11.3029 1499 2247001 3368254499 38.7169 11.4446 1445 2088025 3017196125 38.0132 11.3055 1500 2250000 3375000000 38.7298 11.4471 2090916 3023464536 38.0263 11.3081 1501 2253001 3381754501 38.7427 11.4497 1447 2093809 3029741623 38.0395 11.3107 1502 2256004 3388518008 38.7556 11.4522 2096704 3036027392 38.0526 11.3133 1503 2259009 3395290527 38.7685 11.4548 2099601 3042321849 38.0657 11.3159 1504 2262016 3402072064 38.7814 11.4573 1450 2102500 3048625000 38.0789 11.3185 1505 2265025 3408862625 38.7943 11.4598 1451 2105401 3054936851 38.0920 11.3211 1506 2268036 3415662216 38.8072 11.4624 2108304 3061257408 38.1051 11.3237 1507 2271049 3422470843 38.8201 11.4649 2111209 3067586677 38.1182 11.3263 1508 2274064 3429288512 11.4675 1454 2114116 3073924664 38.1314 11.3289 1509 2277081 3436115229 38.8458 11.4700 1455 2117025 3080271375 38.1445 11.3315 1510 2280100 3442951000 38.8587 11.4725 1450 2119936 3086626816 38. 1576; 11. 3341 1511 2283121 3449795831 38.8716 11.4751 1 457 2122849 3092990993 38. 1707 111. 3367 1512 2286144 3456649728 38 8844 11.4776 1458 2125764 3099363912 38.1838 11.3393 1513 2289169 3463512697 38.8973 11.4801 1459 2128681 3105745579 38.1969 11.3419 1514 2292196 3470384744 38.9102 11.4826 1 4 oo 2131600 3112136000 38.2099 11.3445 1515 2295225 3477265875 38.9230 11.4852 1401 21 845.' 1 3118535181 11.3471 1516 2298256 3484156096 38.9358 11.4877 1462 2137444 3124943128 38.2361 11.3496 1517 2301289 3491055413 38.9487 11.4902 2140369 3131359847 38.2492 11.3522 1518 2304324 3597963832 38.9615 11.4927 1464 2143296 3137785344 38.2623 11.3548 1519 2307361 3504881359 38.9744 11.4953 14(35 2146225 3144219625 38.2753 11.3574 1520 2310400 3511808000 38.9872 11.4978 2149156 3150662696 3S.2884 11.3600 1521 28134 41 3518743761 39.0000 11.5003 2152089 3157114563 38.3014 11.3626 1522 2316484 3525688648 39.0128 11.5028 2155024 3163575232 3^.81 15 11.3652 1523 2319529 3532642667 39.0256 11 5054 1409 2157961 3170044709 38.3275 11.3677 1524 2322576 3539605824 39.0384 11.5079 1470 2160900 3176523000 38.3406 11.3703 1525 2325625 3546578125 39.0512 11.5104 1471 2163841 3183010111 38.3536 11.3729 1526 2328676 3553559576 39.0640 11.5129 1472 2166784 3189506048 38.3667 11.3755 1527 2881720 3560550183 39.0768 11.5154 1473 21 OUT-.'!) 3196010817 8s . 8797 11.3780 1528 2334784 3567549952 39.0896 11.5179 1474 2172076 3202524424 38.3927 11.3806 1529 2337841 3574558889 39.1024 11.5204 SQUARES, CUBES, SQUARE AHD CUBE ROOTS. 101 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 1530 2340900 358157700( 39.1152 11.5230 1565 2449225 3833037125 39.5601 11.6102 1531 2343961 3588604291 39.1280 11.5255 1566 2452356 3840389496 39.5727 11.6126 1532 2347024 359564076* 39.1408 11.5280 1567 2455489 3847751263 39.5854 11.6151 1533 2350089 360268643" 39.1535 11.5305 1568 2458624 3855123432 39.5980 11.6176 1534 2353156 360974130- 39.1663 11.5330 1569 2461761; 3862503009 39.6106 11.6200 1535 2356225 3616S0537c 39.1791 11.5355 1570 2464900 3869893000 39.6232 11.6225 1536 2359296 3623878656 i 39. 1918 11.5380 1571 2468041 3877292411 39.6358 11.6250 1537 2362369 3630961 153' 39. 2046 11.5405 1572 2471184 3884701248 39.6485 11.6274 1538 2365444 363805287; 39.2173 11.5430 1573 24743293892119517 39.6611 11.6299 1539 2368521 36451538H 39.2301 11.5455 1574 2477476 3899547224 39.6737 11.6324 1540 2371600 365226400C 39.2428 11.5480 1575 2480625 '3906984375 39.6863 11.6348 1541 2374681 3659383421 39.2556 11.5505 1576 2483776 3914430976 39.6989 11.6373 1542 2377761 36665 1208* 39.2683 11.5530 1577 2486929 3921887033 39.7115 11.6398 1543 2380S49 367365000" 39.2810 11.5555 1578 2490084 3929352552 39.7240 11.6422 1544 2383936 3680797184 39.2938 11.5580 1579 2493241 3936827539 39.7366 11.6447 1545 2387025 368795362c 39.3065 11.5605 1580 2496400 3944312000 39.7492 11.6471 2390116 36951 19336 39.3192 11.5630 1581 2499561 3651805941 39.7618 11.6496 1547 2393209 370229432: 39.3319 11.5655 1582 2502724 3959309368 39.7744 11.6520 1548 2396304 370947859; 39.3446 11.5680 1583 2505889 3966822287 39.7869 11.6545 1549 2399401 371667214S 39.3573 11.5705 1584 2509056 3974344704 39.7995 11.6570 1550 2402500 372387500C 39.3700 11.5729 1585 2512225 3981876625 39.8121 11.6594 1551 2405601 3731087151 39.3827 11.5754 1586 2515396 3989418056 39.8246 11.6619 155-2 2408704 373830860* 39.3954 11.5779 1587 25185693996969003 39.8372 11.6643 1553 2411809 374553937" 39.4081 11.5804 1588 2521744 4004529472 39.8497 11.6668 1554 2414916 3752779464 39.4208 11.5829 1589 25249214012099469 39.8623 11.6692 1 555 2418025 376002887c 39.4335 11.5854 1590 2528100 4019679000 39.8748 11.6717 5; 2421136 3767287616 39.4462 11.5879 1591 2531281 14027268071 39.8873 11.6741 1 557 2424249 3774555693 39.4588 11.5903 1592 2534464 '4034866688 39.8999 11.6765 2427364 3781833112 39.4715 11.5928 1593 2537049 ! 4042474857 39. 9124 11. 6790 1559 2430481 37891 1987C 39.4842 11.5953 1594 25408364050092584 39.9249,11.6814 1560 2433600 37964 1600( 39 4968 11.5978 1595 2544025 4057719875 39.9375 11.7839 1561 2436721 380372148 39.5095 11.6003 1596 2547216 4065356736 39.9500 11.6863 1564 2439844 381103632* 39.5221 11.6027 1597 2550409 4073003173 39.9625 11.6888 1563 2442969 381836054' 39.5348 11.6052 1598 2553604 4080659192 39.9750,11.6912 1564 2446096 382569414 39.5474 11.6077 1599 1600 25568014088324799 2560000 4096000000 39.9875 11.6936 40.0000 11.6961 SQUARES AND CUBES OF DECIMALS. No. Square. Cube. No. Square. Cube. No Square. Cube. .1 .01 .001 .01 .0001 .000 001 .001 .00 00 01 .000 000 001 .2 .04 .008 .02 .0004 .000 008 00 00 04 .000 000 008 .3 .09 .027 .03 .0009 .000 027 00 00 09 .000 000 027 .4 .16 .064 .04 .0016 .000 064 .004 00 00 16 .000 000 064 .5 .25 .125 .05 .0025 .000 125 .00.: 00 00 25 .000 000 125 .6 .36 .216 .06 .0036 .000 216 00 00 36 .000 000 216 7 .49 .343 .07 .0049 .000 343 00 00 49 .000 000 343 .8 .64 .512 .08 .0064 .000 512 .00* 00 00 64 .000 000 512 .9 .81 .729 .09 .0081 .000 729 00 00 81 .000 000 729 1.0 1.00 1.000 .10 .0100 .001 000 .OK 00 01 00 .000 001 000 1.2 1.44 1.728 .12 .0144 .001 728 .01; 00 01 44 .000 001 728 Note that the square has twice as many decimal places, and the cube three times as many decimal places, as the root. 102 MATHEMATICAL TABLES. FIFTH ROOTS AND FIFTH POWERS. (Abridged from Trautwine.) u o ° Power. £« .10 .000010 .15 .000075 .20 .000320 .25 .000977 .30 .002430 .35 .005252 .40 .010240 .45 .018453 .50 .031250 .55 .050328 .60 .077760 .65 .116029 .70 .168070 .75 .237305 .80 .327680 .85 .443705 .90 .590490 .95 .773781 1.00 1.00000 1.05 1.27628 1.10 1.61051 1.15 2.01135 1.20 2.48832 1.25 3.05178 1.30 3.71293 1.35 4.48403 1.40 5.37824 1.45 6.40973 1.50 7.59375 1.55 8.94661 1.60 10.4858 1.65 12.2298 1.70 14.1986 1.75 16.3141 1.80 18.8957 1.85 21.6700 1.90 24.7610 1.95 28.1951 2.00 32.0C00 2.05 36.2051 2.10 40.8410 2.15 45.9401 2.20 51.5363 2.25 57.6650 2.30 64.3634 2.35 71.6703 2.40 79.6262 2.45 88.2735 2.50 97.6562 2.55 107.820 2.60 118.814 2.70 143.489 2.80 172.104 2.90 205.111 3.00 243.000 3.10 286.292 3.20 335.541 3.30 391.354 3.40 454.354 3.50 525.219 3.60 604.662 693.440 792.352 902.242 1024.00 1158.56 1306.91 1470.08 1649.16 1845.28 2059.63 2293.45 2548.04 2824.75 3125.00 3450.25 3802.04 4181.95 4591.65 5032.84 5507.32 6016.92 6563.57 7149.24 7776.00 8445.96 9161.33 9924.37 10737 11603 12523 13501 14539 15640 16807 18042 19349 20731 22190 23730 25355 27068 28872 30771 32768 34868 37074 39390 41821 44371 47043 49842 52773 55841 59049 62403 65908 69569 81537 85873 17.0 17.8 18.0 18.2 18.4 18. G 18 19.0 19.2 19.4 19.6 lli.S 20.0 20.2 20.4 20.6 20. 21.0 21.2 21.4 21.6 90392 95099 100000 110408 121665 138823 146933 161051 176234 192541 210131 228776 24SN82 270271 2981(33 317580 343597 371293 400746 43.2040 465259 500490 537824 577353 619174 710082 759375 811368 866171 1048576 1115771 1186367 1260493 1338278 1419857 1505366 1594947 1688742 1786899 1889568 2109061 2226203 2348493 2476099 2609193 2747949 2892547 3043168 3200000 3363232 3533059 3709677 4084101 4282322 4488166 4701850 21.8 22.0 22.2 22.4 22.6 22.8 28.0 '8.2 28.4 23.6 28 . 8 24.0 24.2 24.4 24.6' 24. 8 ! 25. 1 25. 2' 25.4 25.6 25.8. .0 .2 26.4 .6 20.8 27.0 27\4 27.6 27.8 28.0 4923597 5153632 58!i2180 5639493 5895793 6161327 6436343 6721093 7015834 7 32(1825 7630332 7962624 8299976 8641-666 '.I0IWI7S 9381200 9765625 10162550 10572278 10995116 11431377 11881376 12345437 12823886 13317055 13825281 14348907 15443752 16015681 16604430 17210368 17833868 18475309 19135075 19813557 20511149 21228253 21965275 22722628 23500728 24300000 26393634 28629151 31013642 33554432 36259082 , ov 39135393 J 87 42191410 45435424 52521875 56382167 60466176 64783487 69343957 74157715 79235168 84587005 90224199 9615801.2 CIRCUMFERENCES AND AREAS OF CIRCLES. 103 CIRCUMFERENCES AND AREAS OF CIRCL.ES. Diam. Circum. Area. 1 3.1416 0.7854 2 6.2832 3.1416 3 9.4248 7.0686 4 12.5664 12.5664 5 15.7080 19.635 6 18.850 28.274 7 21.991 38.485 8 25.133 50.266 9 28.274 63.617 10 31.416 78.540 11 34.558 95.033 12 37.699 113.10 13 40.841 132.73 14 43.982 153.94 15 47.124 176.71 16 50.265 201.06 17 53.407 226.98 18 56.549 254.47 19 59.690 283.53 20 62.832 314.16 21 65.973 346.36 22 69.115 380.13 23 72.257 415.48 24 75.398 452.39 25 78.540 490.87 26 81.681 530.93 27 84.823 572.56 28 87.965 615.75 29 91.106 660.52 30 94.248 706.86 31 97.389 754.77 32 100.53 804.25 33 103.67 855.30 34 106.81 907.92 35 109.96 962.11 36 113.10 1017.88 37 116.24 1075.21 38 119.38 1134.11 39 122.52 1194.59 40 125.66 1256.64 41 128.81 1320.25 42 131.95 1385.44 43 135.09 1452.20 44 138.23 1520.53 45 141.37 1590.43 46 144.51 1661.90 47 147.65 1734.94 48 150.80 1809.56 49 153 94 1885.74 50 157.08 1963.50 51 160.22 2042.82 52 163.36 2123.72 53 166.50 2206.18 54 169.65 2290.22 55 172.79 2375.83 56 175.93 2463.01 57 179.07 2551.76 58 182.21 2642.08 59 185.35 2733.97 60 188.50 2827.43 61 191.64 2922.47 62 194.78 3019.07 63 197.92 3117.25 64 201.06 3210.99 Diam. Circum. Area. 65 204.20 3318.31 66 207.34 3421.19 67 210.49 3525.65 68 213.63 3631.68 69 216.77 3739.28 70 219.91 3848.45 71 223 05 3959.19 72 226.19 4071.50 73 229.34 4185.39 74 232.48 4300 84 75 235.62 4417.86 76 238.76 4536.46 77 241.90 4656.63 78 245.04 4778.36 79 248.19 4901.67 80 251.33 5026.55 81 254.47 5153.00 82 257.61 5281.02 83 260.75 5410.61 84 263.89 5541.77 85 267.04 5674 50 86 270.18 5808.80 87' 273.32 5944.68 88 276.46 6082.12 89. 279.60 6221.14 90 282.74 6361.73 91 285.88 6503.88 92 289.03 6647.61 93 292.17 6792.91 94 295.31 6939.78 95 298.45 7088.22 96 301.59 7238.23 97 304.13 7389.81 98 307.88 7542.96 99 311.02 7697.69 100 314.16 7853.98 101 317.30 8011.85 102 320.44 8171.28 103 323.58 8332.29 104 326.73 8.494.87 105 329 87 8659.01 106 333.01 8824.73 107 336.15 8992.02 108 339.29 9160.88 109 342.43 9331.32 110 345.58 9503.32 111 348.72 9676.89 112 351.86 9852.03 113 355.00 10028.75 114 358.14 10207.03 115 361.28 10386 89 116 364.42 10568.32 117 367.57 10751.32 118 370.71 10935.88 119 373.85 11122.02 120 376.99 11309.13 121 380.13 11499.01 122 383.27 11689.87 123 386.42 11882.29 124 389.56 12076.28 125 392.70 12271.85 126 395.84 12468.98 127 398.98 12667.69 128 402.12 12867.96 Diam. Circum. Area. 129 405.27 13069.81 130 408.41 13273.23 131 411.55 13478.22 132 414.69 13684 78 133 417.83 13892.91 134 420.97 14102.61 135 424.12 14313.88 136 427.26 14526.72 137 430.40 14741.14 138 433.54 14957.12 139 436.68 15174.68 140 439.82 15393.80 141 442.96 15614.50 142 446.11 15836.77 143 449.25 16060.61 144 452.39 16286.02 145 455.53 16513.00 146 458.67 16741.55 147 461.81 16971.67 148 464.96 17203.36 149 468.10 17436.62 150 471.24 17671.46 151 474.38 17907 86 152 477.52 18145.84 153 480.66 18385.39 154 483.81 18626.50 155 486.95 18869.19 156 490.09 19113.45 157 493.23 19359.28 158 496.37 19606.68 159 499.51 19855.65 160 502.65 20106.19 161 505.80 20358.31 162 508.94 20611.99 163 512.08 20867.24 164 515.22 21124.07 165 518.36 21382.46 166 521.50 21642.43 167 524.65 21903 97 168 527.79 22167 08 169 530.93 22431.76 170 534.07 22698.01 171 537.21 22965.83 172 540.35 23235:22 173 543.50 23506.18 174 546.64 23778.71 175 549.78 24052.82 176 552.92 24328.49 177 556.06 24605.74 178 559.20 24884.56 179 562.35 25164.94 180 565.49 25446 90 181 568.63 25730.43 182 571.77 26015.53 183 574.91 26302.20 184 578.05 26590.44 185 581.19 26880.25 186 584.34 27171.63 187 587.48 27464.59 188 590.62 27759.11 189 593.76 28055.21 190 596.90 28352.87 191 600.04 28652.11 192 603.19 28952.92 104 MATHEMATICAL TABLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 193 606.33 29255.30 260 816.81 53092.92 327 1027.30 83981.84 194 609.47 29559.25 261 819.96 53502.11 328 1030.44 84496.28 195 612.61 29864.77 262 823.10 53912.87 329 1033.58 ' 85012.28 196 615.75 30171.86 263 826.24 54325.21 330 1036.73 85529.86 197 618.89 30480.52 264 829.38 54739.11 331 1039.87 86049.01 198 622.04 30790.75 265 832.52 55154.59 332 1043.01 86569.73 199 625.18 31102.55 266 835.66 55571.63 333 1046.15 87092.02 200 (528.32 31415.93 267 838.81 55990.25 334 1049.29 87615.88 201 631.46 31730.87 268 841.95 56410.44 335 1052.43 88141.31 202 634.60 32047.39 269 845.09 56832.20 336 1055.58 88668.31 203 637.74 32365.47 270 848.23 57255.53 337 1058.72 89196.88 204 '640.88 32685.13 271 851.37 57680.43 338 1061.86 89727.03 205 644.03 33006.36 272 854.51 58106.90 339 1065.00 90258.74 206 647.17 33329.16 273 857.65 58534.94 340 1068.14 90792. 1)3 207 650.31 33653.53 274 860.80 58964.55 341 1071.28 91326.88 208 653.45 33979.47 275 863.94 59395.74 342 1074.42 91863.31 209 656.59 34306.98 276 867.08 59828.49 343 1077.57 92401.31 210 659.73 34636.06 277 870.22 60262.82 344 1080.71 92940.88 211 662.88 34966.71 278 873.36 60698.71 345 1083.85 93482.02 212 666.02 35298.94 279 876.50 61136.18 346 1086.99 94024.73 213 669.16 35632.73 280 879.65 61575.22 347 1090.13 94569.01 214 672.30 35968.09 281 882.79 62015.82 348 1093.27 95114.86 215 675.44 36305.03 282 885.93 62458.00 349 1096.42 95662.28 216 678.58 36643.54 283 889.07 62901.75 350 1099.56 96211.28 217 681.73 36983.61 284 892.21 63347.07 351 1102.70 96761.84 218 684.87 37325.26 285 895.35 63793.97 352 1105.84 97313.97 219 688.01 37668.48 286 898.50 64242.43 353 1108.98 97867.68 220 091.15 38013.27 287 901.64 64692.46 354 1112.12 98422.96 221 694.29 38359.63 288 904.78 65144.07 355 1115.27 98979.80 222 697.43 38707.56 289 907.92 65597.24 356 1118.41 - 99538.22 223 700.58 39057.07 290 911.06 66051.99 357 1121.55 100098.21 224 703.72 39408.14 291 914.20 66508.30 358 1124.69 100659.77 225 706.86 39760.78 292 917.35 669C6.19 359 1127.83 101222.90 226 710.00 40115.00 293 920.49 67425.65 360 1130.97 101787.60 227 713.14 40470.78 294 923.63 67886.68 361 1134.11 102353.87 228 716.28 40828.14 295 926.77 68349.28 362 1137.26 102921. 7'2 229 719.42 41187.07 296 929.91 68813.45 363 1140.40 103491.13 230 722.57 41547.56 297 933.05 69279.19 364 1143.54 104062.12 231 725.71 41909.63 298 936.19 69746.50 365 1146.68 104634.67 232 728.85 42273.27 299 939.34 70215.38 366 1149.82 105208.80 233 731.99 42638.48 300 942.48 70685.83 367 1152.96 105784.49 234 735.13 43005.26 301 945.62 71157.86 368 1156.11 106361.76 235 738.27 43373.61 303 948.76 71631.45 369 1159.25 106940.60 236 741.42 43743.54 303 951.90 72106.62 370 1162.39 107521.01 237 744.56 44115.03 304 955.04 72583.36 371 1165.53 108102.99 238 747.70 44488.09 305 958.19 73061.66 372 1168.67 108686.54 239 750.84 44862.73 306 961.33 73541.54 373 1171.81 109271.66 240 753.98 45238.93 307 964.47 74022.99 374 1174.96 109858.35 241 757.12 45616.71 308 967.61 74506.01 375 1178.10 110446.62 242 760.27 45996.06 309 970.75 74990.60 376 1181.24 111036.45 243 763.41 46376.98 310 973.89 75476.76 377 1184.38 111627.86 244 766.55 46759.47 311 977.04 75964.50 378 1187.52 112220.83 245 769.69 47143.52 312 980.18 76453.80 379 1190.66 112815.38 246 772.83 47529.16 313 983.32 76944.67 380 1193.81 113411.49 247 775.97 47916.36 314 986.46 77437.12 381 1196.95 114009.18 248 779.11 48305.13 315 989.60 77931.13 382 1200.09 114608.44 249 782.26 48695.47 316 992.74 78426.72 383 1203.23 115209.27 250 785.40 49087.39 317 995.88 78923.88 384 1206.37 115811.67 251 788.54 49480.87 318 999.03 79422.60 385 1209.51 116415.64 252 791.68 49875.92 319 1002.17 79922.90 386 1212.65 117021.18 253 794.82 50272.55 320 1005.31 80424.77 387 1215.80 117628.30 254 797.96 50670.75 321 1008.45 80928.21 388 1218.94 118236.98 255 801.11 51070.52 322 1011.59 81433 22 389 1222.08 118847.24 256 804.25 51471.85 323 1014.73 81939.80 390 1225.22 119459.06 257 807.39 51874.76 324 1017.88 82447.96 391 1228.36 1 20072. 4ft 258 810.53 52279.24 325 1021.02 82957.68 392 1231.50 12068742 259 813.67 52685.29 326 1024.16 83468.98 393 1234.65 121303.96 CIRCUMFERENCES AND AREAS OF CIRCLES. 105 1237 1240.93 1244.07 1247.21 1250.35 1253.50 1256.64 1266 1269.20 1272.35 1275.49 1278.63 1281.77 1284.91 1288.05 1291.19 1294.34 1297.48 1300.6.2 1303.76 1306.90 1310.04 1313.19 1316.33 1319.47 1322.61 1325.75 1328.89 1332.04 1335.18 1338.32 1341.46 1344.60 1347.74 1350.88 1354.03 1357.17 1360.31 1363.45 1366.59 1369.73 1372.88 1376.02 1379.16 1382.30 1385.44 1388.58 1391.73 1391.87 1398.01 1401.15 1404.29 1407.43 1410.58 1413.72 1416 1420.00 1423.14 1426.28 1429.42 1432.5' 1435.7 1438.85 1441.99 1445.13 121922.0' 122541.7; 123163.00 123785.82 124410.21 125036.17 125663.71 126292.81 126923.48 127555.73 128189.55 128824 "" 129161.89 130100.42 130740.52 131382.19 132025.43 132670.24 133316.63 133964.58 134614.10 135265.20 135917.86 136572.10 137227.91 137885.29 138544.24 139204.76 139866.85 140530.51 141195.74 141862.54 142530.92 143200.86 143872.38 144545.46 145220.12 145896.35 146574 15 147253.52 147934.46 148616.97 149301.05 149986.70 150673.93 151362.72 152053.08 152745.02 153438 53 154133.60 154830.25 155528.47 156228.26 156929.62 157632.55 158337.06 159043.13 159750.77 160459.99 161170.77 161883.13 162597.05 163312.55 164029.6) 164748.26 165468.47 166190.25 Diam. Circum. Area. 461 1448.27 166913.60 462 1451.42 167638 53 463 1454.56 16S365.02 464 1457.70 169093.08 465 1460.84 169822.72 466 1463.98 170553.92 467 1467.12 171286.70 468 1470.27 172021.05 469 1473 41 172756.97 470 1476.55 173494.45 471 1479.69 174233.51 472 1482.83 174974.14 473 1485.97 175716.35 474 1489.11 176460 12 475 1492.26 177205.46 476 1195.40 177952.37 477 1498.54 178700.86 478 1501.68 179450.91 479 1504.82 180202.54 480 1507.96 180955.74 481 1511.11 181710.50 482 1514.25 182466.84 483 1517.39 183224.75 484 1520.53 183984.23 485 1523.67 184745.28 486 1526.81 185507.90 487 1529.96 186272.10 488 1533.10 187037.86 489 1536.24 187805.19 490 1539.38 188574.10 491 1542.52 189344.57 492 1545.66 190116.62 493 1548.81 190890.24 494 1551.95 191665.43 495 1555 09 192442.18 496 1558.23 193220.51 497 1561.37 194000.41 498 1564.51 194781.89 499 1567.65 195564.93 500 1570.80 196349.54 501 1573.94 197135.72 502 1577.08 197923.48 503 1580.22 198712.80 504 1583.36 199503.70 505 1586 50 200296.1? 506 1589.65 201090.20 507 1592.79 201885.81 508 1595.93 202682.99 509 1599.07 203481.74 510 1602.21 204282.06 511 1605.35 205083.95 512 1608.50 205887.42 513 1611.64 206692.45 514 1614.78 207499.05 515 1617.92 208307.23 516 1621.06 209116.97 517 1624.20 209928.29 518 1627.34 210741.18 519 1630.49 211555.63 520 1633.63 212371.66 521 1636.77 213189.26 522 1639.91 214008.43 523 1643.05 214829.17 524 1646.19 215651.49 525 1649.34 216475.37 526 1652.48 217300.82 527 1655.62 218127.85 Diam. Circum. Area. 528 1658.76 218956.44 529 1661.90 219786.61 530 1665.04 220618.34 531 1668.19 221451.65 532 1671.33 222286.53 533 1674.47 223122.98 534 1677.61 223961.00 535 16S0.75 224800.59 536 1683.89 225641.75 537 1687.04 226484.48 538 1690.18 227328.79 539 1693.32 228174.66 540 1696.46 229022.10 541 1699.60 229871.12 542 1702.74 230721.71 543 1705.88 231573.86 544 1709.03 232427.59 545 1712.17 233282.89 546 1715.31 234139.76 • 547 1718.45 234998.20 548 1721.59 235858.21 549 1724.73 236719.79 550 1727.88 237582.94 551 1731.02 238447.6? 552 1734.16 239313.96 553 1737.30 240181.83 554 1740.44 241051.26 555 1743.58 241922.27 556 1746.73 242794. 85 557 1749.87 243668.99 55S 1753.01 244544.71 559 1756.15 245422 00 560 1759.29 246300.86 561 1762.43 247181.30 562 1765.58 248063.30 563 176S.72 248946.87 564 1771.86 249832.01 565 1775.00 250718,73 566 1778.14 251607.01 56? 1781.28 252496.87 568 1784.42 253388.30 569 1787.57 254281.29 570 1790.71 255175.86 571 1793.85 256072.00 572 1796.99 256969.71 573 1800.13 257868.99 574 1803.27 258769.85 575 1806.42 259672.27 576 1809.56 260576.26 577 1812.70 261481.83 578 1815 84 262388.96 579 1818.98 263297.67 580 1822.12 264207.91 581 1825.27 265119.79 582 1828.41 266033.21 583 1831.55 266948.20 584 1834.69 267864.76 585 1837.83 268782.89 586 1840.97 269702.59 587 1844.11 270623.86 588 1847.26 271546.70 589 1850.40 272471.12 590 1853.54 273397.10 591 1856.68 274324.06 592 1859.82 275253.78 593 1862.96 276184.48 594 1866.11 277116.75 106 MATHEMATICAL TABLES. Diain. Circum. Area. Diam Circum. Area. Diam- Circum. Area. 595 1869.25 278050.58 603 2082.88 345236.69 731 2296.50 419686.15 596 1872.39 278985.99 664 2086.02 346278.91 732 2299.65 420835.19 597 1875.53 279922.97 665 2089.16 347322.70 733 2302.79 421985.79 598 1878.67 280861.52 666 2092.30 348368.07 734 2305.93 423137.97 599 1881.81 281801 65 667 2095.44 349415.00 735 2309.07 424291.72 600 1884.96 282743.34 668 2098.58 350463.51 736 2312.21 425447.04 601 1888.10 283686.60 609 2101.73 351513.59 737 2315.35 426603.94 602 1891.24 284631.44 670 2104.87 352565.24 738 2318.50 427762.40 603 1894.38 285577.84 671 2108.01 353618.45 739 2321.64 428922.43 604 1897.52 286525.82 672 2111.15 354673 24 740 2324.78 430084.03 605 1900.66 287475.36 673 2114.29 355729.60 741 2327.92 431247.21 606 1903.81 288426.48 674 2117.43 356787.54 742 2331.06 432411.95 607 1906.95 289379.17 675 2120.58 357847.04 743 2334.20 433578.27 608 1910.09 290333.43 676 2123.72 358908.11 744 2337.34 434746.16 609 1913.23 291289.26 677 2126.86 359970.75 745 2340.49 435915.62 610 1916.37 292246.66 678 2130.00 361034.97 746 2343.03 437086.64 611 1919 51 293205.63 679 2133.14 362100.75 747 2346.77 438259.24 612 1922.65 294166.17 680 2136.28 363168.11 748 2349.91 439433.41 613 1925.80 295128.28 681 2139.42 364237.04 749 2353.05 440609 16 614 1928.94 296091.97 682 2142.57 365307.54 750 2356.19 441786.47 615 1932.08 297057.22 683 2145.71 360379.60 751 2359.34 442965.35 616 1935.22 298024.05 684 2148.85 367453.24 752 2362.48 444145.80 617 1938.36 298992.44 685 2151.99 368528.45 753 2365.02 445327.83 618 1941.50 299962.41 686 2155.13 369005.23 754 2368.76 446511.42 619 1944.65 300933.95 687 2158.27 370683.59 755 2371.90 447696.59 620 1947.79 301907.05 688 2161.42 371763.51 756 2375.04 448883.32 621 1950.93 302881.73 689 2164.56 372845.00 757 2378.19 450071.63 622 1954.07 303857.98 690 2167.70 373928.07 758 2381.33 451261.51 623 1957.21 304835.80 691 2170.84 375012.70 759 2384.47 452452.96 624 1960.35 305815.20 692 2173.98 376098.91 760 2387.61 453645.98 625 1963.50 306796.16 693 2177.12 377186.68 761 2390.75 454840.57 626 1966.64 307778.69 694 2180.27 378276.03 762 2393.89 456036.73 627 1969.78 308762.79 695 2183.41 379366.95 763 2397.04 457234.46 628 1972.92 309748.47 696 2186.55 380459.44 764 2400.18 458433.77 629 1976.06 310735.71 697 2189.69 381553.50 765 2403.32 459034.64 630 1979.20 311724.53 698 2192.83 382649.13 766 2406.46 460837.08 631 1982.35 312714.92 699 2195.97 383746.33 767 2409.60 462041.10 632 1985.49 313706.88 700 2199.11 384845.10 768 2412.74 463246.69 633 1988.63 314700.40 701 2202.26 385945.44 769 2415.88 464453.84 634 1991.77 315695.50 702 2205.40 387047.36 770 2419.03 465662.57 635 1994.91 316692.17 703 2208.54 388150.84 771 2422.17 466872.87 636 1998.05 317690.42 704 2211.68 389255.90 772 2425.31 468084.74 637 2001.19 318690.23 705 2214.82 390362.52 773 2428.45 469298.18 638 2004.34 319691.61 706 2217.96 391470.72 774 2431.59 470513.19 639 2007.48 320694.56 707 2221.11 392580.49 775 2434.73 471729.77 640 2010.62 321699.09 708 2224 25 393091.82 776 2437.88 472947.92 641 2013.76 322705.18 709 2227.39 394804.73 777 2441.02 474167.05 642 2016.90 323712.85 710 2230.53 395919.21 778 2444.16 475388.94 643 2020.04 324722.09 711 2233.07 397035 26 779 2447.30 476611.81 644 2023.19 325732.89 712 2236.81 398152.89 780 2450.44 477836.24 645 20-20.33 326745.27 713 2239.96 399272.08 781 2453.58 479062.25 646 2029.47 327759.22 714 2243.10 400392.84 782 2456.73 480289.83 647 2032.61 328774.74 715 2240.24 401515.18 783 2459.87 481518.97 648 2035.75 329791.83 716 2249.38 402639.08 784 2463.01 482749.69 649 2038.89 330810.49 717 2252.52 403764.56 785 2466.15 483981.98 650 2042.04 331830.72 718 2255.66 404891.60 786 2469.29 485215.84 651 2045.18 332852.53 719 2258.81 406020.22 787 2472.43 486451.28 652 2048.32 333875.90 720 2261.95 407150.41 788 2475.58 487688.28 653 2051.46 334900.85 721 2265.09 408282.17 789 2478.72 488926.85 654 2054.60 335927.36 722 2268.23 409415.50 790 2481.86 490166.99 655 2057.74 336955.45 723 2271.37 410550.40 791 2485.00 491408.71 656 2060.88 337985.10 724 2274.51 411686.87 792 2488.14 492651.99 657 2064.03 339016.33 725 2277.05 412824.91 793 2491.28 493896.85 658 2007.17 340049.13 726 2280.80 413964.52 794 2494.42 495143.28 659 2070.31 341083.50 727 22S3.94 415105.71 795 2497.57 496391.27 660 2073.45 342119.44 728 2287.08 416248.46 796 2500.71 497640.84 661 2076.59 343156.95 729 2290.22 417392.79 797 2503.85 498891.98 602 2079.73 344196.03 730 2293.36 418538.68 798 2506.99 500144.69 CIRCUMFERENCES AND AREAS OF CIRCLES. 107 Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 799 2510.13 501398.97 867 2723.76 590375.16 935 2937.39 686614.71 800 2513.27 502654.82 868 2726.90 591737.83 936 2940.53 688084.19 801 2516.42 503912.25 869 2730.04 593102 06 937 2! 143.67 689555.24 802 2519.56 505171.24 S70 2733.19 594467.87 938 2946 81 691027.86 803 2522.70 506431.80 871 2736.33 595835.25 939 2949.96 692502.05 804 2525.84 507693.94 872 2739.47 597204.20 940 2953.10 693977.82 805 2528.98 508957.64 873 2742.61 598574.72 941 2956.24 695455.15 806 2532.12 510222.92 874 2745.75 599946.81 942 2959.38 696934.06 807 2535.27 511489.77 875 2748.89 601320.47 943 2962.52 698414.53 808 2538.41 512758.19 876 2752.04 602695.70 944 2965.66 699896.58 S09 2541.55 514028.18 877 2755.18 604072.50 945 2968.81 701380.19 810 2544.69 515299.74 878 2758.32 605450.88 946 2971.95 702865.38 811 2547.83 516572.87 879 2761.46 606830.82 947 2975.09 704352.14 812 2550.97 517847.57 880 2764.60 608212.34 948 2>)78.23 705840.47 813 2554.11 519123.84 881 2767.74 609595.42 949 2981.37 707330.37 814 2557.26 520401.68 882 2770.88 610980.08 950 2984.51 708821.84 815 2560.40 521681.10 883 2774.03 612366.31 951 2987.65 710314.88 816 2563.54 522962.08 884 2777.17 613754.11 952 2990.80 711809.50 817 2566.68 524244.63 885 2780.31 615143.48 953 2993.94 713305.68 818 2569.82 525528.76 886 2783.45 616534.42 954 2997.08 714803.43 819 2572.96 526814.46 887 2786.59 617926.93 955 3000.22 716302.76 820 2576.11 528101.73 888 2789.73 619321.01 956 3003.36 717803.66 821 2579.25 529390.56 889 2792.88 620716.66 957 3006.50 719306.12 822 2582.39 530680.97 890 2 71)0.02 622113.89 958 3009.65 720810.16 823 2585.53 531972.95 891 2799.16 623512.68 959 3012.79 722315.77 824 2588.67 533266.50 892 2802.30 624913.04 960 3015.93 723822.95 825 2591.81 534561.62 893 2805.44 626314.98 961 3019.07 725331.70 826 2594.96 535858.32 894 2808.58 627718.49 962 3022.21 726842.02 827 2598.10 537156.58 895 2811.73 629123.56 963 3025.35 728353.91 828 2601.24 538456 41 896 2814.87 630530.21 964 3028.50 729867.37 829 2604.38 539757.82 897 2818.01 631938.43 965 3031.64 731382.40 830 2607.52 541060.79 898 2821.15 633348.22 966 3034.78 732899.01 831 2610.66 542365.34 899 2824.29 634759.58 967 3037.92 734417.18 832 2613.81 543671.46 900 2827.43 636172.51 968 3041.06 735936.93 833 2616.95 544979.15 901 2830.58 637587.01 969 3044.20 737458.24 834 2620.09 546288.40 902 2833.72 639003.09 970 3047.34 738981.13 835 2023.23 547599.23 903 2836.86 640420.73 971 3050.49 740505.59 836 2626.37 548911.63 904 2840.00 641839.95 972 3053.63 742031.62 S37 2629.51 550225.61 905 2843.14 643260.73 973 3056.77 743559.22 838 2032,65 551541.15 906 2846.28 644683.09 974 3059.91 745088.39 839 2635.80 552858.26 907 2849.42 646107.01 975 3063.05 746619.13 840 2638.94 554176.94 908 2852.57 647532.51 976 3066.19 748151.44 : 841 2642.08 555497.20 909 2855.71 648959.58 977 3069.34 749685.32 842 2645.22 556819.02 910 2858.85 6503S8.22 978 3072.48 751220.78 843 2648 36 558142.42 911 2861.99 651818.43 979 3075.62 752757.80 844 2651.50 559467.39 912 2865.13 653250.21 980 3078.76 754296.40 845 2654.65 560793.92 913 2868.27 654683.56 981 3081.90 755836.56 846 2657.79 562122.03 914 2871.42 656118.48 982 3085.04 757378.30 847 2660.93 563451.71 915 2874.56 657554.98 983 3088.19 758921.61 848 2664.07 564782.96 916 2877.70 658993.04 984 3091.33 760466.48 849 2667.21 566115.78 917 2880.84 660432.68 985 3094.47 762012.93 850 2670.35 567450.17 918 2883.98 661873.88 986 3097.61 763560.95 851 2673.50 568786.14 919 2887.12 663316 66 987 3100.75 765110.54 852 2676.64 570123.67 920 2890.27 664761.01 988 3103.89 766661.70 853 2679.78 571462.77 921 2893.41 666206.92 989 3107.04 768214.44 854 2682.92 572803.45 922 2896.55 667654.41 990 3110.18 769768.74 855 2686.06 574145.69 923 2899.69 669103.47 991 3113.32 771324.61 856 2689.20 575489.51 924 2902.83 670554.10 992 3116.46 772882.06 857 2692.34 576834.90 925 2905.97 672006.30 993 3119.60 774441.07 858 2695.49 578181.85 926 2909.11 673460.08 994 3122.74 776001.66 859 2698.63 579530.38 927 2912.26 674915.42 995 3125.88 777563.82 860 2701.77 580880.48 928 2915.40 676372.33 996 3129.03 779127.54 861 2704.91 582232.15 929 2918.54 677830.82 997 3132.17 780692.84 862 2708.05 583585.39 930 2921.68 679290.87 998 3135.31 782259.71 863 2711.19 584940.20 931 2924.82 680752.50 999 3138.45 783828.15 864 2714.34 586296.59 932 2927.96 682215.69 1000 3141.59 785398.16 865 2717.48 587654.54 933 2931.11 683680.46 866 2720 62 589014.07 934 2934.25 685146.80 108 MATHEMATICAL TABLES. CIRCUMFERENCES AND AREAS OF CIRCLES Advancing l>y Eighths. Diam. Circum. Area. Diam. Circum. Area. Diain. Circum. Area, j 1/64 .04909 .00019 2 Va 7.4613 4.4301 6 Va 19.242 29.465 1 1/32 .09818 .00077 7/16 7.6576 4.6664 Va 19.635 30.680 3/64 .14726 .00173 H 7.8540 4.9087 Va 20.028 31.919 1/16 .19635 .00307 9/16 8.0503 5.1572 Vz 20.420 33.183 3/32 .29452 .00690 Va 8.2467 5.4119 % 20.813 34.472 H .39270 .01227 11/16 8.4430 5.6727 H 21 206 35.785 5/32 .49087 .01917 Va 8.6394 5.9396 Va 21.598 37.122 3/16 .58905 .02761 13/16 8.8357 6.2126 7. 21.991 38.485 7/32 .68722 .03758 % 9.0321 6.4918 H 22.384 39.871 15/16 9.2284 6.7771 H 22.776 41.282 H .78540 .01909 Va 23.169 42.718 9/32 .88357 .06213 3. 9.4248 7.0686 H 23.562 44.179 5/16 .98175 .07670 1/16 9.6211 7.3662 Va 23.955 45 664 11/32 1.0799 .09281 Va 9.8175 7.6699 H 24.347 47.173 Va 1.1781 .11045 3/16 10.014 7.9798 Va 24.740 48.707 13/32 1.2763 .12962 H 10.210 8.2958 8. 25.133 50.265 7/16 1.3744 .15033 5/16 10.407 8.6179 H 25.525 51.849 15/32 1.4726 . 17257 % 10.603 8.9462 Va 25.918 53.456 7/16 10.799 9.2806 % 26.311 55.088 M 1 5708 .19635 y* 10.996 9.6211 % 26.704 56.745 17/32 1.6690 .22160 9/16 11.192 9.9678 Va 27.096 58.426 9/16 1.7671 .24850 Va 11.388 10.321 ¥a 27.489 60.132 19/32 Va 1.8653 .27688 11/16 11.585 10.680 Va 27.882 61.862 1.9635 .30680 u 11.781 11.045 9. 28.274 63.617 21/32 2.0617 .33824 13/16 11.977 11.416 Va 28.667 65.397 11/16 2.1598 .37122 Va 12.174 11.793 X A 29.060 67.201 23/32 2.2580 .40574 15/16 12.370 12.177 Va 29.452 69.029 4. 12.566 12.566 y* 29.845 70.SS2 Va 25/32 2.3562 .44179 1/16 12.763 12.962 Va 30.238 72.760 2.4544 .47937 H 12.959 13.364 8 30.631 74.662 13/16 2.5525 .51849 3/16 13.155 13.772 31.023 76.589 27/32 2.6507 .55914 H 13.352 14.186 10. 31.416 78.540 % 2.7489 .60132 5/16 13.548 14.607 H 31.809 80.516 29/32 2.8471 .64504 Va 13.744 15.033 % 32.201 82.516 15/16 2.9452 .69029 7/16 13.941 15.466 32.594 84.541 31/82 3.0434 .73708 % 14.137 15.904 Va 32.987 86.590 9/16 14.334 16.349 33.379 88.664 1. 3.1416 .7854 Va 14.530 16.800 Va 33.772 90.763 1/16 3.3379 .8866 11/16 14.726 17.257 Va 34.165 92.886 Va 3.5343 .9940 Va 14.923 17.728 11. 34.558 95.033 3/16 3.7306 1.1075 13/16 15.119 18.190 Va 34.950 97.205 k 3.9270 1.2272 Va 15.315 18.665 Va i 35.343 99.402 5/16 4.1233 1.3530 15/16 15 512 19.147 35.736 101.62 % 4.3197 1.4849 5. 15.708 19.635 36.128 103.87 7/16 4.5160 1.6230 1/16 15.904 20.129 Va 36.521 106.14 ^ 4.7124 1.7671 H 16.101 20.629 Va 36.914 108.43 9/16 4.9087 1.9175 3/16 16.297 21.135 Vs 37.306 110.75 Va 5.1051 2.0739 H 16.493 21.648 12. 37.699 113.10 11/16 5.3014 2.2365 5/16 16.690 22.166 Va 38.092 115.47 % 5.4978 2.4053 Va 16.886 22.691 Va 38.485 117.86 13/16 5.6941 2.5S02 7/16 17.082 23.221 Va 38.877 120.28 Va 5.8905 2.7612 M 17.279 23.758 Vz 39.270 122.72 15/16 6.0868 2.9483 9/16 17.475 24.301 Va 39.663 125.19 Va 17.671 24.850 Va 40.055 127.68 o 6.2832 3.1416 11/16 17.868 ' 25.406 Va 40.448 130.19 1/16 6.4795 3.3410 n 18.064 25.967 13. 40.841 132.73 % 6.6759 3.5466 13-16 18.261 26.535 Va 41.233 135.30 3/16 6.8722 3.7583 % 18.457 27.109 Va 41.626 137.89 Va 7.0686 3.9761 15-16 18.653 27.688 Va 42.019 140.50 5/16 7.2649 4.2000 6. 18.850 28.274 K 42.412 143.14 CIRCUMFERENCES AND AREAS OF CIRCLES. 109 Diam. Circum . Area. Diam. Circum. Area. Diam. Circum. Area. 13% 42.804 145.80 21% 68.722 375.83 30% 94.640 712.76 Va 43.197 148.49 32. 69.115 380.13 Va 95.033 728.69 % 43.590 151.20 % 69.508 384.46 Vs 95.426 724.64 14 43.982 153.94 Va 69.900 388.82 V* 95.819 730.62 Vs 44.375 156.70 Vs 70.293 393.20 Vs 96.211 736.62 Va 44.768 159.48 Vq 70.686 397.61 Va 96.604 742.64 Vs 45.160 162.30 Vs 71.079 402.04 Vs 96.997 748.69 % 45 . 553 165.13 Va 71.471 406.49 31. 97.389 754.77 45.946 167.99 Vs 71.864 410.97 Vs 97.782 760.87 % 46.338 170.87 23. 72.257 415.48 Va 98.175 766.99 Vs 46.731 173.78 Vs 72.649 420.00 98.567 773.14 15. 47.124 176.71 Va 73.042 424.56 V% Vs 98.960 779.31 % 47.517 179.67 Vs 73.435 429.13 99.353 785.51 Va 47.909 182.65 Vi 73.827 433.74 1 99.746 791.73 % 48.302 185.66 Vs 74.220 438.36 100.138 797.98 V* 48.695 188.69 Va 74.613 443.01 32. 100.531 804.25 Vs 49.087 191.75 Vs 75.006 447.69 ~ Vs 100.924 810.54 Va 49.480 194.83 24 75.398 452.39 Va 101.316 816.86 Vs 49.873 197.93 Vs 75.791 457.11 Vs 101.709 823.21 16 50.265 201.06 Va 76.184 461.86 Vz 102.102 829.58 % 50.658 204.22 Vs 76.576 466.64 Vs 102.494 835.97 Va 51.051 207.39 Vz 76.969 471.44 ¥ 102.887 842.39 % 51.414 210.60 Vs 77.362 476.26 103.280 848.83 H 51.836 213.82 Va 77.754 481.11 33 8 103.673 855.30 Vs 52.229 217.08 Vs 78.147 485.98 Vs 104.065 861.79 K 52.622 220.35 25. 78.540 490.87 Va 104.458 868.31 Vs 53.014 223.65 Vs 78.933 495.79 Vs 104.851 874.85 17 53.407 226.98 Va 79.325 500.74 Vk 105.243 881.41 % 53.800 230.33 Vs 79.718 505.71 Vs 105.636 888.00 Va 54.192 233.71 Mi 80.111 510.71 Va 106.029 894.62 54.585 237.10 Vs 80.503 515.72 Vs 106.421 901.26 vl 04.978 240.53 Va 80.896 520.77 34 106.814 907.92 55.371 243.98 Vs 81.289 525.84 Vs 107.207 914.61 Va 55 . 763 247.45 26 81.681 530.93 1 107.600 921.32 Vs 56.156 250.95 Vs 82.074 536.05 107.992 928.06 18 56.549 254.47 Va 82.467 541.19 Vz 108.385 934.82 Vs 56.941 258.02 Vs 82 860 546.35 108.778 941.61 Va 57.334 261.59 ri 83.252 551 . 55 Va 109.170 948.42 Vs 57.727 265.18 83.645 556.76 Vs 109.563 955.25 Vz 58.119 268.80 Va 84.038 562.00 35. 109.956 962.11 Vs 58.512 272.45 Vs 84.430 567.27 Vs 110.348 969.00 Va 58.905 276.12 27 84.823 572.56 Va 110.741 975.91 Vs 59.298 279.81 Vs 85.216 577.87 Vs 111.134 982.84 19. 59.690 283.53 Va 85.608 583.21 H 111.527 989.80 H 60.083 287.27 Vs 86.001 588.57 Vs 111.919 996.78 Va 60.476 291.04 V2 86.394 593.96 Va 112.312 1003.8 Vs 60.868 294.83 Vs 86.786 599.37 Vs 112.705 1010.8 % 61.261 298.65 Va 87.179 604.81 36. 113.097 1017.9 Vs 61.654 302.49 Vs 87.572 610.27 Vs 113.490 1025.0 Va 62.046 306.35 28 87.965 615.75 Va 113.883 1032.1 Vs 62.439 310.24 Vs 88.357 621.26 Vs 114.275 1039.2 20. 62.832 314.16 Va 88.750 626 80 V* 114.668 1046.3 % 63.225 318.10 Vs 89.143 632.36 Vs 115.061 1053.5 % 63.617 322.06 H 89.535 637.94 Va 115.454 1060.7 64.010 326.05 89.928 643.55 Vs 115.846 1068.0 Vz 64.403 330.06 Va 90.321 649.18 37 116.239 1075.2 % 64.795 334.10 Vs 90.713 654.84 Vs 116.632 1082.5 Va 65.188 338.16 29 91.106 660.52 Va 117.024 1089.8 Vs 65.581 342.25 Vs 91 .499 666.23 Vs 117.417 1097.1 21. 65.973 346.36 Vs 91.892 671.96 Vz 117.810 1104.5 % 66.366 350.50 92.284 677.71 118.202 1111.8 Va 66.759 354.66 Vs 92.677 683.49 Va 118.596 1119.2 Vs 67.152 358.84 93.070 689.30 Vs 118.988 1126.7 H 67.544 363.05 Va 93.462 695.13 38. 119.381 1134.1 Vs 67.937 367.28 Vs 93.855 700.98 Vs 119.773 1141.6 ■ Va 68.330 371.54 30. 94.248 706 86 Va 120.166 1149.1 110 MATHEMATICAL TABLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. s$% 120.559 1156.6 46% 146.477 1707.4 54% 172.395 2365.0 y a 120.951 1164.2 Va 146.869 1716.5 55. 172.788 2375.8 % 121.344 1171.7 Va 147.262 1725.7 Va 173.180 2386.6 u 121.737 1179.3 47. 147.655 1734.9 Va 173.573 2397.5 Va 122.129 1186.9 Va 148.048 1744.2 Va 173.966 2408.3 39. 122.522 1194.6 Va 148.440 1753.5 3 174.358 2419.2 v& 122.915 1202.3 148.833 1762.7 174.751 2430.1 v± 128.308 1210.0 Mi 149.226 1772.1 Va 175.144 2441.1 v& 123.700 1217.7 Va 149.618 1781.4 Va 175.536 2452.0 g 124.093 1225.4 Va 150.011 1790.8 56. 175.929 2463.0 124.486 1233.2 Va 150.404 1800.1 Va 176.322 2474.0 % 124.878 1241.0 48. 150.796 1809.6 Va 176.715 2185.0 % 125.271 1248.8 Va 151.189 1819.0 Va 177.107 2196.1 40. 125.664 1256.6 ' Va 151.582 1828.5 It 177.500 2507.2 Va 126.056 1264.5 Va 151.975 1837.9 177.893 2518.3 Va 126.449 1272.4 II 152.367 1847.5 Va 178.285 2529.4 Va 126.842 12S0.3 152.760 1857.0 Va 178.678 2540.6 t 127.235 1288.2 Va 153.153 1866.5 57. 179.071 2551.8 127.627 1296.2 Va 153.545 1876.1 Va' 179.463 2563.0 H 128.020 1304.2 49. 153.938 1885.7 Va 179.856 2574.2 Va 128.413 1312.2 Va 154.331 1895.4 Va 180.249 2585.4 41. 128.805 1320.3 Va 154.723 1905.0 Vz 180.642 2596.7 Va 129.198 1328.3 Va 155.116 1914.7 Va 181.034 2608.0 Va 129.591 1336.4 V* 155.509 1924.4 Va 181.427 2619.4 a 129.983 1344.5 Va 155.902 1934.2 Va 181.820 2630.7 130.376 1352.7 Va 156.294 1943.9 58. 182.212 2642.1 % 130.769 1360.8 Va 156 687 1953.7 Va 182.605 2653.5 % 131.161 1369.0 50. 157.080 1963.5 Va 182.998 2664.9 % 131.554 1377.2 Va 157.472 1973.3 Va 183.390 2676.4 42. 131.947 1385.4 Va 157.865 1983.2 V2 183.783 2687.8 Va 132.340 1393.7 Va 158.258 1993.1 Va 184.176 2699.3 a 132.732 1402.0 Mi 158.650 2003.0 Va 184.569 2710.9 133.125 1410.3 Va 159.043 2012.9 Va 184.961 2722.4 l A 133.518 1418.6 Va 159.436 2022.8 59. 185.354 2734.0 Va 133.910 1427.0 Va 159.829 2032.8 Va 185.747 2745.6 Va 134.303 1435.4 51 160.221 2042.8 Va 186.139 2757.2 Va 134.696 1443.8 Va 160.614 2052.8 Va 186.532 2768.8 43. 135.088 1452.2 Va 161.007 2062.9 Va 1S6.925 2780.5 Va 135.481 1460.7 161.399 2073.0 187.317 2792.2 Va 135.874 1469.1 4 Va 161.792 2083.1 Va 187.710 2803.9 Va 136.267 1477.6 162.185 2093.2 Va 188.103 2815.7 l A 136.659 1486.2 Va 162.577 2103.3 60. 188.496 2827.4 Va 137.052 1494.7 Va 162.970 2113.5 Va 188.888 2839.2 Va 137.445 1503.3 53. 163.363 2123.7 Va 189.281 2851.0 Va 137.837 1511.9 Va 163.756 2133.9 Va 189.674 2862.9 44 138.230 1520.5 Va 164.148 2144.2 Mi 190.066 2874.8 X A 138.023 1529.2 164.541 2154.5 Va 190.459 2886.6 Va 139.015 1537.9 II 164.934 2164.8 Va 190.852 2898.6 Vs. 139.408 1546.6 165.326 2175.1 Va 191.244 2910.5 Vz 139.801 1555.3 % 165.719 2185.4 61. 191.637 2922.5 Va 140.194 1564.0 Va 166.112 2195.8 Va 192.030 2934.5 Va 140.586 1572.8 53. 166.504 2206.2 Va 192.423 2946.5 Va 140.979 1581.6 Va 166.897 2216.6 Va 192.815 2958.5 45 141.372 1590.4 Va 167.290 2227.0 Vz 193.208 2970.6 Va 141.764 1599.3 Va 167.683 2237.5 Va 193.601 2982.7 Va 142.157 1608.2 H 168.075 2248.0 Va 193.993 2994.8 Va 142.550 1617.0 Va 168.468 2258.5 % 194.386 3006 9 Mi 142.942 1626.0 Va 168.861 2269.1 62. 194.779 3019.1 Va 143.335 1634.9 % 169.253 2279.6 Va 195.171 3031.3 Va 143.728 1643.9 54. 169.646 2290.2 Va 195.564 3043.5 Va 144.121 1652.9 Va 170.039 2300.8 Va 195.957 3055.7 46 144.513 1661.9 Va 170.431 2311.5 V% 196.350 3068.0 Va 144.906 1670.9 170.824 2322.1 Va 196.742 3080.3 Va 145.299 1680.0 m\ 171.217 2332.8 Va 197.135 3092.6 Va 145.691 1689.1 Va 171.609 2343.5 Va 197.528 3104.9 y* 146.084 1698.2 Va -172.002 2354.3 63 197.920 3117.2- CIRCUMFERENCES AND AREAS OF CIRCLES. ill Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 63^ 198.313 3129.6 71% 224.231 4001.1 79% 250.149 4979.5 Va 198.706 3142.0 Vz 224.624 4015.2 Va 250 542 4995.2 % 199.098 3154.5 Vs 225.017 4029.2 Vs 250.935 5010.9 - Vz 199.491 3166.9 u 225.409 4043.3 80 251.3-27 5026.5 % 199.884 3179.4 Vs 225.802 4057.4 Vs 251.720 5042.3 u 200.277 3191.9 72. 226.195 4071.5 Va 252.113 5058.0 Vs 200.669 3204.4 Vs 226.587 4085.7 Vs 252.506 5073.8 64. 201.062 3217.0 Va 226.980 4099.8 252.898 5089.6 Vs 201.455 3229.6 Vs 227.373 4114.0 % 253.291 5105.4 Va 201.847 3242.2 227.765 4128.2 Va 253.684 5121.2 Vs 202.240 3254.8 % 228.158 4142.5 Vs 254.076 5137.1 Vz 202.633 3267.5 8 228.551 4156.8 81. 254.469 5153.0 Vs 203.025 3280.1 228.944 4171.1 Vs 254.862 5168.9 Va 203.418 3292.8 73 229.336 4185.4 Va 255.254 5184.9 Vs 203.811 3305.6 Vs 229.729 4199.7 Vs 255.647 5200.8 65. 204.204 3318.3 Va 230.122 4214.1 g 256.040 5216.8 Vs 204.596 3331.1 Vs 230.514 4228.5 256.433 5232.8 Va 204.989 3343.9 Vz 230.907 4242.9 S 256.825 5248.9 Vs 205.382 3356.7 Vs 231.300 4257.4 257.218 5264.9 Vz 205.774 3369.6 g 231.692 4271.8 82. 257.611 5281.0 % 206.167 3382.4 232.085 4286.3 % 258.003 5297.1 206.560 3395.3 74. 232.478 4300.8 Va 258.396 5313.3 % 206.952 3408.2 Vs 232.871 4315.4 Vs 258.789 5329.4 66. 207.345 3421.2 Va 233.263 4329.9 Vz 259.181 5345.6 Vs 207.738 3434.2 Vs 233 656 4344.5 Vs 259.574 5361.8 Va 208.131 3447.2 Vz 234.019 4359.2 M 259.967 5378.1 Vs 208.523 3460.2 Vs 234.441 4373.8 % 260.359 5394.3 Vs 208.916 3473.2 1 234.834 4388.5 83. 260.752 5410.6 209.309 3486.3 235.227 4403.1 Vs 261.145 5426.9 Va 209.701 3499.4 75. 235.619 4417.9 Va 261.538 5443.3 Vs 210.094 3512.5 Vs 236.012 4432.6 Vs 261.930 5459.6 67. 210.487 3525 7 Va 236 . 405 4447.4 Vz 262.323 5476.0 % 210.879 3538.8 Vs 236.798 4462.2 Vs 262.716 5492.4 Va 211.272 3552.0 Vz 237.190 4477.0 M 263.108 5508.8 Vs 211.665 3565.2 Vs 237.583 4491.8 Vs 263.501 5525.3 H 212.058 3578.5 M 237.976 4506.7 84. 263.894 5541.8 Vs 212.450 3591.7 Vs 238.368 4521.5 Vs 264.286 5558.3 Va 212.843 3ti05.0 76. 238.761 4536.5 Va 264.679 5574.8 Vs 213.236 3618.3 Vs 239.154 4551.4 Vs 265.072 5591.4 68. 213.628 3631.7 Va 239.546 4566.4 Vz 265.465 5607.9 Vs 214.021 3645.0 Vs 239.939 4581.3 Vs 265.857 5624.5 Va 214.414 3658.4 Vz 240.332 4596.3 Va 266.250 5641.2 Vs 214.806 3671.8 Vs 240.725 4611.4 Vs 266.643 5657.8 Vz 215.199 3685.3 Va 241.117 4626.4 85. 267.035 5674.5 % 215.592 3698.7 Vs 241.510 4641.5 Vs 267.428 5691.2 Va 215.984 3712.2 77. 241.903 4656.6 I 267.821 5707.9 % 216.377 3725.7 Vs 242.295 4671.8 268.213 5724.7 69. 216.770 3739.3 Va 242.688 4686.9 Vz 268.606 5741.5 Vs 217.163 3752.8 Vs 243.081 4702.1 Vs 268.999 5758.3 Va 217.555 3766.4 Vz 243.473 4717.3 S 269.392 5775.1 M 217.948 3780.0 Vs 243.866 4732.5 269.784 5791.9 Vz 218.341 3793.7 Va 244.259 4747.8 86. 270.177 5808.8 Vs 218.738 3807.3 Vs 244 . 652 4763.1 Vs 270.570 5825.7 Va 219.126 3821.0 78. 245.044 4778.4 8 270.962 5842.6 Vs 219.519 3834.7 Vs 245 437 4793.7 271.355 5859.6 70. 219.911 3S48.5 % 245.830 4809.0 II 271.748 5876.5 Vs 220.304 3862.2 246.222 4824.4 272.140 5893.5 Va 220.697 3876.0 Vz 246.615 4839 8 M 272.533 5910.6 Vs 221.090 3889.8 Vs 247.008 4855.2 Vs 272.926 5927.6 Vz 221.482 3903.6 Va 247.400 4870.7 87. 273.319 5944.7 Vs 221.875 3917.5 Vs 247.793 4886. 2 Vs 273.711 5961.8 Va 222.268 3931.4 79. 248.186 4901.7 Va 274.104 5978.9 Vs 222.660 3945.3 Vs 248.579 4917.2 Vs 274.497 5996.0 71. 223.053 3959.2 Va 248.971 4932.7 Vz 274.889 6013.2 Vs 223.446 3973.1 Vs 249.364 4948.3 Vs 275.282 6030.4 Va 223.838 3987.1 Vz 249.757 4963.9 Va 275.675 6047.6 112 MATHEMATICAL TABLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 87% 276,067 6064.9 92. 289.027 6647.6 96^ 301.986 7257.1 88. 276.460 6082.1 Va 289.419 6665.7 Va 302.378 7276.0 H 276.853 6099.4 Va 289.812 6683.8 302.771 7294.9 U 277.246 6116.7 Va 290.205 6701.9 y% 303.164 7313.8 277.638 6134.1 Va 290.597 6720.1 Va 303.556 7332.8 M& 278.031 6151.4 290.990 6738.2 Va 303.949 7351.8 % 278.424 6168.8 % 291.383 6756.4 Va 304.342 7370.8 % 278.816 6186.2 % 291.775 6774.7 97. 304.734 7389.8 % 279.209 6203.7 93. 292.168 6792.9 Va 305.127 7408.9 89. 279.602 6221.1 Va 292.561 6811.2 Va 305.520 7428.0 Va 279.994 6238.6 H 292.954 6829.5 % 305.913 7447.1 8 280.387 6256.1 Va 293.346 6847.8 V% 306.305 7466.2 280.780 6273.7 y* 293.739 6866.1 Va 306.698 7485.3 M 281.173 6291.2 % 294.132 6884.5 Va 307.091 7504.5 281.565 6-08.8 n 294.524 6902.9 Va 307.483 7523.7 S 281.958 6326.4 294.917 6921.3 98 307.876 7543.0 282.351 6344.1 94 295.310 6939.8 Va 308.269 7562.2 90. 282.743 6361.7 y& 295.702 0958.2 Va 308.661 7581.5 Va 283.136 6379.4 Va 296.095 6976.7 % 309.054 7600.8 a 283.529 6397.1 Va 296.488 6995.3 8 309.447 7620.1 283.921 6414.9 M 296.881 7013.8 309.840 7639.5 Ps 284.314 6432.6 Va 297.273 703^.4 Va 310.232 7658.9 284.707 6450.4 s 297.666 7051.0 Va 310.625 7678.3 % 285.100 6468.2 298.059 7069.6 99. 311.018 7697.7 285.492 6486.0 95. 298.451 7088.2 Va 311.410 7717.1 91. 285.885 6503.9 n 298.844 7106.9 Va 311.803 7736.6 Va 286.278 6521.8 299.237 7125.6 Va 312.196 7756.1 M 286.670 6539.7 % 299.629 7144.3 8 312.588 7775.6 % 287.063 6557.6 Vt 300.022 7163.0 312.981 7795.2 i^ 287.456 6575.5 Va 300.415 7181.8 S 313.374 7814.8 % 287.848 6593.5 Va 300.807 7200.6 313.767 7834.4 1 288.241 6611.5 Va 301.200 7219.4 100. 314.159 7854.0 288.634 6629.6 96. 301.593 7238.2 DECIMALS OF A FOOT EQUIVALENT TO INCHES AND FRACTIONS OF AN INCH. Inches. Va Va Va X Va Va Va .01042 .02083 .03125 .04166 05208 .06250 07292 1 .0833 .0937 .1042 .1146 .1250 1354 .1459 1563 2 .1667 .1771 .1875 .1979 .2083 2188 .2292 2396 3 .2500 .2604 .2708 .2813 .2917 3021 .3125 3229 4 .3333 .3437 .3542 .3646 .3750 3854 .3958 4063 5 .4167 .4271 .4375 .4479 .4583 4688 .4792 4896 6 .5000 .5104 .5208 .5313 .5417 5521 .5625 5729 7 .5833 .5937 .6042 .6146 .6250 6354 .6459 6563 8 .6667 .6771 .6875 .6979 .7083 7188 .7292 7396 9 .7500 .7604 .7708 .7813 .7917 8021 .8125 8229 10 .8333 .8437 .8542 .8646 .8750 8854 .8958 9063 11 .9167 .9271 .9375 .9479 .9583 9688 .9792 9896 CIRCUMFERENCES OE CIRCLES. 113 fH««C'*>0!ONXaOi-iNCOtJiiO«ONCOfflO'-iWM^iO!D!-XO!Ori« nriT-irti-.T-innHrtO(«M««(MMNIl«MMM c -^sc ££ ^X^^^^ SUV „* S!S££M:&S HOOWMiOOWOOn^iOCQOQ.OnMiOOaUOn.rtSJiOOOOOT-irtft^ -innWMMMWM^I OTHTft-OMfflOWtOO DrHiOQOH^NOMNOM 3i.»Nt-0D00X03C501OO bSSS^SSSSSS ^^S&fcXSS&S «^&SXR3S£:5? assess H(»fflOOO(MiOi-fflOOMCQini-XOOil»lOL-COOOM«lJiOi.'a)OOr H-*!ot»o)rnoc'!'«'oi.>a^oe('*ioi.-aoco!'*OfOjp05 0J»OCJ(?JinODT-l^i_-— i Tj< t~ © « « S« ?< CO CO ) ■* T)"* lO iO 'C O ffl 5D l- I- i' I- 00 X 00 Oi ffl OS o 3s ej ic co i-i SN^iOi-ffioowMmt'ffioowmiot-xoorimiot-QOoOrHmiojD HON' j*rt^i»OW!0 0)M!DOll»«X^iOXHrM-OM!OOMt0 01Win»Nin»i. u — jioioK5»to»!.-i-i.'i«xxx®o'.0!0 i-ii-;i-l^(7JO?C>JCOCO« d^scssOT?:* ^ss^^s:^:*: ^^x^^;s? ^^ HOJ'-iois^ i "ni-OJHo:jnnoi--o)Oo«*ini»aooweoioi-oioo«M ;H Wt-OM©Oi«1005«iOXi-lTfNrHifNOM©OJMOaOT10X'HiOXT [it nr-iT-iT-io.NSjrowin'^'VTi'iOiniotDtocsoi.-t-t-xxxaaac H«DXO!i-'r^NTTtCI.-05iHT-(0)iJl) , tOi.-OiT-lOO)'*lOi'01HO fa -rt^^w^jojcoco » co © en c? in ao — 'iox-ntNOM[-oco« 35 c> in go —I •<* £- o 0©i~l-£>GOCOGOaiC7»OS© rf^ste^ ;osx^:£:sss3$§s ^^^xs;^ ii-iWrfffll-OliHrt Oa>Wi0050JiOX-Hrtoco«oci^jioo5'r?ioQO i-i i-i i-i oj snj L~J>GOaOOOC»0505© i-tc*ec^in«ot-cooO'-io>eo-<#»ncof-Goc»Oi-i(rjcoTt'in«D!>GOC50'-ic nrtnrinnrirlT-ir(«W««««INNM«!i3eO0 114 MATHEMATICAL TABLES. LENGTHS OF CIRCULAR ARCS. (Degrees being given. Radius of Circle =1.) Formula.— Length of arc — — X radius X number of degrees. Rule. — Multiply the factor in table for any given number of degrees by the radius. Example. — Given a curve of a radius of 55 feet and an angle of 78° 20'. What is the length of same in feet ? Factor from table for 78° 1.3613568 Factor from table for 20' .0058178 Factor 1.3671746 1.3671746 X 55 = 75.19 feet. . Degrees. Minutes. 1 .0174533 61 1.0646508 121 2.1118484 1 .0002909 2 .0349066 62 1.0821041 122 2.1293017 2 .0005818 3 .0523599 63 1.0995574 123 2.1467550 3 .0008727 4 .0698132 64 1.1170107 124 2.1642083 4 .0011636 5 .0872665 65 1.1344640 125 2.1816616 5 .0014544 6 .1047198 66 1.1519173 126 2.1991149 6 .0017453 7 .1221730 67 1.1693706 127 2.2165682 7 .0020362 8 .1396263 68 1.1868239 128 2.2340214 8 .0023271 9 .1570796 69 1.2042772 129 2.2514747 9 .0026180 10 .1745329 70 1.2217305 130 2.2689280 10 .0029089 11 .1919862 71 1.2391838 131 2.2863813 11 .0031998 12 .2094395 72 1.2566371 132 2.3038346 12 .0034907 13 .2268928 73 1.2740904 133 2.3212879 13 .0037815 14 .2443461 74 1.2915436 134 2.3387412 14 .0040724 15 .2617994 75 1.3089969 135 2.3561945 15 .0043633 16 .2792527 76 1.3264502 136 2.3736478 16 .0046542 17 .2967060 77 1.3439035 137 2.3911011 17 .0049451 18 .3141593 78 1.3613568 138 2.4085544 18 .0052360 19 .3316126 79 1.3788101 139 2.4260077 19 .0055269 20 .3490659 80 1.3962634 140 2.4434610 20 .0058178 21 .3665191 81 1.4137167 141 2.4609142 21 .0061087 22 .3839724 82 1.4311700 142 2.4783675 22 .0063995 23 .4014257 83 1.4486233 143 2.4958208 23 .0066904 24 .4188790 84 1.4660766 144 2.5132741 24 .0069813 25 .4363323 85 1.4835299 145 2.5307274 25 .0072722 26 .4537856 86 1.5009832 146 2.5481807 26 .0075631 27 .4712389 87 1.5184364 147 2.5656340 .. "27 .0078540 28 .4886922 88 1.5358897 148 2.5830873 28 .0081449 29 .5061455 89 1.5533430 " 149 2.6005406 29 .0084358 30 .5235988 90 1.5707963 150 2.6179939 30 .0087266 31 .5410521 - ' 91 1.5882496 151 2.6354472 31 .0090175 32 .5585054 92 1.6057029 152 2.6529005 32 .0093084 33 .5759587 93 1.6231562 153 2.6703538 33 .0095993 34 .5934119 94 1.6406095 154 2.6878070 34 .0098902 35 .6108052 95 1.6580628 155 2.7052603 35 .0101811 36 .6283185 96 1.6755161 156 2.7227136 36 .0104720 37 .6457718 97 1.6929694 157 2.7401669 37 .0107629 38 .6632251 98 1.7104227 158 2.7576202 38 .0110538 39 .6806784 99 1.7278760 159 2.7750735 39 .0113446 40 .6981317 100 1.7453293 160 2.7925268 40 .0116355 41 .7155850 101 1.7627825 161 2.8099801 41 .0119264 42 .7330383 102 1.7802358 162 2.8274334 42 .0122173 43 .7504916 103 1.7976891 163 2.8448867 43 .0125082 44 .7679449 104 1.8151424 164 2.8623400 44 .0127991 45 .7853982 105 1.8325957 165 2.8797933 45 .0130900 46 .8028515 106 1.8500490 166 2.8972466 46 .0133809 47 .8203047 107 1.8675023 167 2.9146999 47 .0136717 48 .8377580 108 1.8S49556 168 2.9321531 48 .0139626 49 .8552113 109 1.9024089 169 2.9496064 49 .0142535 50 .8726646 110 1.9198622 170 29670597 50 .0145444 51 .8901179 111 1.9373155 171 2.9845130 51 .0148353 52 .9075712 112 1.9547688 172 3.0019663 52 .0151262 53 .9250245 113 1.9722221 173 3.0194196 53 .0154171 54 .9424778 114 1.9896753 174 3.0308729 54 .0157080 55 .9599311 115 2.0071286 175 3.0543262 55 .0159989 56 .9773844 116 2.0245819 176 3.0717795 56 .0162897 57 .9948377 117 2 0420352 177 3.0892328 57 .0165806 58 1.0122910 118 2.0594885 178 3.1066861 58 .0168715 59 1.0297443 119 2.0769418 179 3.1241394 59 .0171624 60 1.0471976 120 2.0943951 180 3.1415927 60 .0174533 LENGTHS OF CIRCULAR ARCS. 115 LENGTHS OF CIRCXJL.AR ARCS. (Diameter = 1. Given the Chord and Height of the Arc.) Rule for Use op the Table.— Divide the height by the chord. Find in the column of heights the number equal to this quotient. Take out the corre- sponding number from the column of lengths. Multiply this last number by the length of the given chord; the product will be length of the arc. ' If the arc is greater than a semicircle, first find the diameter from the formula, Diam. = (square of half chord -4- rise) + rise; the formula is true whether the arc exceeds a semicircle or not. Then find the circumference. From the diameter subtract the given height of arc, the remainder will be height of the smaller arc of the circle; find its length according to the rule, and subtract it from the circumference. Hgts. Lgths. ! Hgts. Lgths. 1.05896 Hgts. Lgths. Hgts. Lgths. Hgts. Lgths. .001 1. 00002 1 .15 .238 1.14480 .326 1.26288 .414 1.40788 .005 1.00007' .152 1.06051 .24 1.14714 .328 1 . 26588 .416 1.41145 .01 1.00027 .154 1.06209 .242 1.14951 .33 1.26892 .418 1.41503 .015 1.00061 .156 1.06368 .244 1.15189 .332 1.27196 .42 1.41861 .02 1.00107 .158 1.06530 .246 1.15428 .334 1.27502 .422 1.42221 .0-25 1.00167 .16 1.06693 .248 1.15670 .336 1.27810 .424 1.42583 .03 1.00240 .162 1.06858 .25 1.15912 .338 1.28118 .426 1.42945 .035 1.00327 .164 1.07025 .252 1.16156 .34 1.28428 .428 1.43309 .04 1.00426 .166 1.07194 .254 1.16402 .342 1.28739 .43 1.43673 .045 1.00539 .168 1.07365 .256 1.16650 .344 1.29052 .432 1.44039 .05 1.00665 .17 1.07537 .258 1.16899 .346 1.29366 .434 1.44405 .055 1.00805 .172 1.07711 .26 1.17150 .348 1.29681 .436 1.44773 .06 1.009571 .174 1.07888 .262 1.17403 .35 1.29997 .438 1.45142 .065 1.01123 .176 1.08066 .264 1.17657 .352 1.30315 .44 1.45512 .07 1.01302 .178 1.08246 .266 1.17912 .354 1.30634 .442 1.45883 .075 1.01493 .18 1.08428 .268 1.18169 .356 1.30954 .444 1.46255 .08 1.01698 .182 1.08611 .27 1.18429 .358 1.31276 .446 1.46628 .085 1.01916 .184 1.08797 .272 1.18689 .36 1.31599 .448 1.47002 .09 1.02146 .186 1.08984 .274 1.18951 .362 1.31923 .45 1.47377 .095 1.02389 .188 1.09174 .276 1.19214 .364 1.32249 .452 1.47753 .10 1.02646 .19 1.09365 .278 1.19479 .366 1.32577 .454 1.4S131 .102 1.02752 .192 1.09557 .28 1.19746 .368 1.32905 .456 1.48509 .104 1.02860 .194 1.09752 .282 1.20014 .37 1.33234 .458 1.48889 .106 1.02970 .196 1.09949 .284 1.20284 .372 1.33564 .46 1.49269 .108 1.03082 .198 1.10147 .286 1.20555 .374 1.33896 .462 1.49651 .11 1.03196 .20 1.10347 .288 1.20827 .376 1.34229 .464 1.50033 .112 1.03312 .202 1.10548 .29 L 21 102 .378 1.34563 .466 1.50416 .114 1.03430 .204 1.10752 .292 1.21377 .38 1.34899 .468 1.50800 .116 1.03551 .206 1.10958 .294 1.21654 .382 1.35237 .47 1.51185 .118 1.03672 .208 1.11165 .296 1.21933 .384 1.35575 .472 1.51571 .12 1.03797 .21 1.11374 .298 1.22213 .386 1.35914 .474 1.51958 .122 1.03923 .212 1.11584 .30 1.22495 .388 1.36254 .476 1.52346 .124 1.04051 .214 1.11796 .302 1.22778 .39 1.36596 .478 1.52736 .126 1.04181 .216 1.12011 .304 1.23063 .392 1.36939 .48 1.53126 .128 1.04313 .218 1.12225 .306 1.23349 .394 1.37283 .482 1.53518 .13 1.04447 .22 1.12444 .308 1.23636 .396 1.37528 .484 1.53910 .132 1.04584 .222 1.12664 .31 1.23926 .398 1.37974 .486 1.54302 .134 1.04722 .224 1.12885 .312 1.24216 .40 1.38322 .488 1.54696 .136 1.04862 .226 1.13108 .314 1.24507 .402 1.38671 .49 1.55091 .138 1.05003 .228 1.13331 .316 1.24801 .404 1.39021 .492 1.55487 .14 1.05147 .23 1.13557 .318 1.25095 .406 1.39372 .494 1.55854 .142 1.05293 .232 1.13785 .32 1.25391 .408 1.39724 .496 1.56282 .144 1.05441 .234 1.14015 .322 1.25689 .41 1.40077 .498 1.56681 .146 1.05591 .236 1.14247 .324 1.25988 .412 1.40432 .5 1.57080 .148 1 1.05743 116 MATHEMATICAL TABLES. AREAS OF THE SEGMENTS OF A CIRCL.E. (Diameter = 1 ; Rise or Versed Sine in parts of Diameter being given.) Rule for Use of the Table,— Divide the rise or height of the segment by the diameter to obtain the versed sine. Multiply the area in the table cor- responding to this versed sine by the square of the diameter. If the segment exceeds a semicircle its area is area of circle— area of seg- ment whose rise is (diam. of circle— rise of giveu segment). Given chord and rise, to find diameter. Diam. = (square of half chord -*- rise) 4- rise. The half chord is a mean proportional between the two parts into which the chord divides the diameter which is perpendicular to it. Versed Verged Versed Versed Versed Sine. Area. Sine. Area. Area. Sine. Area. Sine. Area. .001 .00004 .054 .01646 .107 .04514 .16 .08111 .213 .12235 .002 .00012 .055 .01691 .108 .04576 .161 .08185 .214 .12317 .003 .00022 .056 .01737 .109 .04638 .162 .08258 .215 .12399 .004 .00034 .057 .01783 .11 .04701 .163 .08332 .216 .12481 .005 .00047 .058 .01830 .111 .04763 .164 .08406 .217 .12563 .006 .00062 .059 .01877 .112 .04826 .165 .08480 .218 .12646 .007 .00078 .06 .01924 .113 .04889 .166 .08554 .219 .12729 .008 .00095 .061 .01972 .114 .04953 .167 .08629 .22 .12811 .009 .00113 .062 .02020 .115 .05016 .168 .08704 .221 .12894 .01 .00133 .063 .02068 .116 .05080 .169 .08779 .222 .12977 .011 .00153 .064 .02117 .117 .05145 .17 .08854 .223 .13060 .012 .00175 .065 .02166 .118 .05209 .171 .08929 .224 .13144 .013 .00197 .066 .02215 .119 .05274 .172 .09004 .225 .13227 .014 .0022 .067 .02265 .12 .05338 .173 .09080 .226 .13311 .015 .00244 .068 .02315 .121 .05404 .174 .09155 .227 .13395 .016 .00268 .069 .02366 .122 .05469 .175 .09231 .228 .13478 .017 .00294 .07 .02417 .123 .05535 .176 .09307 .229 .13562 .018 .0032 .071 .02468 .124 .05600 .177 .09384 .23 .13646 .019 .00347 .072 .02520 .125 .05666 .178 .09460 .231 .13731 .02 .00375 .073 .02571 .126 .05733 .179 .09537 .232 .13815 .021 .00403 .074 .02624 .127 .05799 .18 .09613 .233 .13900 .022 .00432 .075 .02676 .128 .05866 .181 .09690 .234 .13984 .023 .00462 .076 .02729 .129 .05933 .182 .09767 .235 . 14069 .024 .00492 .077 .02782 .13 .06000 .183 .09845 .236 .14154 .025 .00523 .078 .02836 .131 .06067 .184 .09922 .237 . 14239 .026 .00555 .079 .02889 .132 .06135 .185 . 10000 .238 .14324 .027 .00587 .08 .02943 .133 .06203 .186 .10077 .239 .14409 .028 .00619 .081 .02998 .134 .06271 .187 .10155 .24 .14494 .029 .00653 .082 .03053 .135 .06339 .188 .10233 .241 . 14580 .03 .00687 .083 .03108 .136 .06407 .189 .10312 .242 .14666 .031 .00721 .084 .03163 .137 .06476 .19 . 10390 .243 .14751 .032 .00756 .085 .03219 .138 .06545 .191 .10169 .244 .14837 .033 .00791 .086 .03275 .139 .06614 .192 .10547 .245 .14923 .034 .00827 .087 .03331 .14 .06683 .193 .10626 .246 .15009 .035 .00864 .08S .03387 .141 .06753 .194 .10705 .247 .15095 .036 .00901 .089 .03444 .142 .06822 .195 .10784 .248 .15182 .037 .00938 .09 .03501 .143 .06892 .196 . 10864 .249 .15268 .038 .00976 .091 .03559 .144 .06963 .197 .10943 .25 .15355 .039 .01015 .092 .03616 .145 .07033 .198 .11023 .251 .15441 .04 .01054 .093 .03674 .146 .07103 .199 .11102 .252 .15528 .041 .01093 .094 .03732 .147 .07174 .2 .11182 .253 .15615 .042 .01133 .095 .03791 .148 .07245 .201 .11262 .254 .15702 .043 .01173 .096 .03850 .149 .07316 .202 .11343 .255 .15789 .044 .01214 .097 .03909 .15 .07387 .203 .11423 .256 . 15876 .045 .01255 .098 .03968 .151 .07459 .204 .11504 .257 .15964 .046 .01297 .099 .01028 .152 .07531 .205 .11584 .258 .16051 .047 .01339 .1 .04087 .153 .07603 .206 .11665 .259 .16139 .048 .01382 .101 .04148 .154 .07675 .207 .11746 .26 .16226 .049 .01425 .102 .04208 .155 .07747 .208 .11827 .261 .16314 .05 .01468 .103 .04269 .156 .07819 .209 .11908 .262 .16402 .051 .01512 .104 .04330 .157 .07892 .21 .11990 .263 .16490 .052 .01556 .105 .04391 .158 .07965 .211 .12071 .264 .16578 .053 .01601 .106 .04452 .159 .08038 .212 .12153 .265 .16666 AREAS OF THE SEGMENTS OF A CIRCLE. 117 Versed *,,. Versed Sine. A ... Versed Sine. Area. Versed Sine. A ,„. Sine. ,„,. .266 16755 .313 .21015 .36 .25455 407 .30024 .454 .34676 .267 16843 .314 .21108 .361 .25551 .408 .30122 .455 .34776 .268 16932 .315 .21201 .362 .25647 .409 .30220 .456 .34876 .269 17020 .316 .21294 .363 .25743 .41 .30319 .457 .34975 .27 17109 .317 .21387 .364 .25839 .411 .30417 .458 .35075 .271 17198 .318 .21480 .365 .25936 412 .30516 .459 .35175 .272 17287 .319 .21573 .366 .26032 .413 .30614 .46 .35274 .273 17376 .32 .21667 .367 .26128 .414 .30712 .461 .35374 .274 17465 .321 .21760 .368 .26225 .415 .30811 .462 .35474 .275 17554 .322 .21853 .369 .26321 .416 .30910 .463 .35573 .276 17644 .323 .21947 .37 .26418 .417 .31008 .464 .35673 277 17733 .324 .22040 .371 .26514 .418 .31107 .465 .35773 .278 17823 .325 .22134 .372 .26611 .419 .31205 .466 .35873 .27£ 17912 .326 .22228 . 73 .26708 .42 .31304 .467 .35972 .28 18002 .327 .22322 .374 .26805 .421 .31403 .468 .36072 .281 18092 .328 .22415 375 .26901 .422 .31502 .469 .36172 .282 18182 .329 .22509 i376 .26998 .423 .31600 .47 .36272 .283 18272 .S3 .22603 .377 .27095 .424 .31699 .471 .36372 .284 18362 .331 .22697 .378 .27192 .425 .31798 .472 .36471 .285 18152 .332 .22792 .379 .27289 .426 .31&97 .473 .36571 .286 18542 .333 .22886 .38 .27386 .427 .31996 .474 .36671 .287 18633 .334 .22980 .381 .27483 .428 .32095 .475 .36771 .288 18723 .335 .23074 .382 .27580 .429 .32194 .476 .36871 .289 18814 .336 .23169 .383 .27678 .43 .32293 .477 .36971 .29 18905 .337 .23263 .384 .27775 .431 .32392 .478 .37071 .291 18996 .338 .23358 .385 .27872 .432 .32491 .479 .37171 .292 19086 .339 .23453 .386 .27969 .433 .32590 .48 .37270 .293 19177 .34 .23547 .387 .28067 .434 .32689 .481 .37370 .294 19268 .341 .23642 .388 .28164 .435 .32788 .482 .37470 .295 19360 .342 .23737 .389 .28262 .436 .32887 .483 .37570 .296 19451 : .343 .23832 .39 .28359 .437 .32987 .484 .37670 .297 19542 .344 .23927 .391 .28457 .438 .33086 .485 .37770 .298 19634 .345 .24022 .392 .28554 .439 .33185 .486 .37870 .299 19725 .346 .24117 .393 .28652 .44 .33284 .487 .37970 .3 19817 .347 2421 2 .394 .28750 .441 .33384 .488 .38070 .301 19908 .348 >24307 .395 .28848 .442 .33483 .489 .38170 .302 20000 .349 .24403 .396 .28945 .443 .33582 .49 .38270 .303 20092 .35 .24498 .397 .29043 .444 .33682 .491 .38370 .304 20184 1 .351 .24593 .398 .29141 .445 .33781 .492 .38470 .305 20276 .352 .24689 .399 .29239 .446 .33880 .493 .38570 .306 20368 .353 .24784 .4 .29337 .447 .33980 .494 .38670 .307 20460 .354 .24880 .401 .29435 .448 .34079 .495 .38770 .308 20553 .355 .24976 .402 .29533 .449 .34179 .496 .38870 .309 20645 .356 .25071 .403 .29631 .45 .34278 .497 .38970 .31 20738 .357 .25167 .404 .29729 .451 .34378 .498 .39070 .311 20830 .358 .25263 .405 .29827 .452 .34477 .499 .39170 .312 20923 .359 .25359 .406 .29926 .453 .34577 .5 .39270 For rules for finding the area of a segment see Mensuration, page 59. 118 MATHEMATICAL TABLES. SPHERES. (Some errors of 1 in the last figure only. From Trautwine.) Diam. Sur- face. Solid- ity. Diam. Sur- face. Solid- ity. Diam. Sur- face. Solid- ity. 1-32 .00307 .00002 3 Ya 5-16 33.183 17.974 9 Va 306.36 504.21 1-16 .01227 .00013 34.472 19.031 10. 314.16 523.60 3-32 .02761 .00043 % 35.784 20.129 Ya 322.06 543 48 Ya .04909 .00102 7-16 37.122 21.268 I 330.06 563.86 5-32 .07670 .00200 »J 38.484 22.449 338.16 584.74 3-16 .11045 .00345 39.872 23.674 346.36 606.13 7-32 .15033 .00548 % 41.283 24.942 % 354.66 628.04 H .19635 .00818 11-16 42.719 26.254 Ya 363.05 650.46 9-32 .24851 .01165 Ya 44.179 27.611 Va 371.54 673.42 5-16 .30680 .01598 13-16 45.664 29.016 11. 380.13 696.91 11-32 .37123 .02127 Va 47.173 30.466 Ya 388.83 720.95 Ya .44179 .02761 15-16 48.708 31.965 Ya 397.61 745.51 13-32 .51848 .03511 4. 50.265 33.510 Va 406.49 770.64 7-16 .60132 .04385 Ya 53.456 36.751 II 415.48 796.33 15-32 .69028 .05393 Ya 56.745 40.195 424.50 822.58 y* .78540 .06545 Ya 60.133 43.847 Va 433.73 849.40 9-16 .99403 .09319 Yz 63.617 47.713 Va 443.01 876.79 % 1.2272 .12783 % 67.201 51.801 12. 452.39 904.78 11-16 1.4849 .17014 Ya 70.883 56.116 Ya 471.44 962.52 Ya 1.7671 .22089 Va 74.663 60.663 490.87 1022.7 13-16 2.0739 .28084 5. 78.540 65.450 Ya 510.71 1085.3 Va 15-16 2.4053 .35077 H 82.516 70.482 13. 530.93 1150.3 2.7611 .43143 H 86.591 75.767 Ya 551.55 1218.0 1. 3.1416 .52360 Ya 90.763 81.308 Y* 572.55 1288.3 1-16 3.5466 .62804 H 95.033 87.113 Ya 593.95 1361.2 Ya 3.9761 .74551 % 99.401 93.189 14. 615.75 1436.8 3-16 4.4301 .87681 H 103.87 99.541 Ya 637.95 1515.1 Ya 4.9088 1.0227 % 108.44 106.18 660.52 1596.3 5-16 5.4119 1.1839 6. 113.10 113.10 % 683.49 1680.3 % 5.9396 1.3611 Ya 117.8? 120.31 15. 706 85 1767.2 7-16 6.4919 1.5553 Ya 122.72 127.83 Ya 730.63 1857.0 Y2 9-16 7.0686 1.7671 % 127.68 135.66 Vk 754.77 1949.8 7.6699 1.9974 Y2 132.73 143.79 Va 779.32 2045.7 % 8.2957 2.2468 % 137.89 152.25 16. 804.25 2144.7 11-16 8.9461 2.5161 Ya 143.14 161.03 Ya 829.57 2246.8 Ya 13-16 9.6211 2.8062 Va 148.49 170.14 Ya 855.29 2352.1 10.321 3.1177 7. 153.94 179.59 Va 881.42 2460.6 Va 11.044 3.4514 Ys 159.49 189.39 17. 907.93 2572.4 15-16 11.793 3.8083 Ya 165.13 199.53 Ya 934.83 2687.6 2. 12.566 4.1888 Ya 170.87 210.03 Ya 962.12 2806.2 1-16 13.364 4.5939 Yi 176.71 220.89 Va 989.80 2928.2 Ya 14.186 5.0243 % 182.66 232.13 18. 1017.9 3053.6 3-16 15.033 5.4809 Ya 188.69 243.73 Ya 1046.4 3182.6 Va 15.904 5.9641 Va 194.83 255.72 8 1075.2 3315.3 5-16 16.800 6.4751 8. 201.06 268.08 Ya 1104.5 3451.5 % 17.721 7.0144 Ys 207.39 280.85 19. 1134.1 3591.4 7-16 18.666 7.5829 Ya 213.82 294.01 Ya 1164.2 3735.0 Vi 19.635 8.1813 % 220.36 307.58 A 1194.6 3882.5 9-16 20.629 8.8103 K 226.98 321.56 Ya 1225.4 4033.7 % 21.648 9.4708 233.71 335.95 20. 1256.7 4188.8 11-16 22.691 10.164 Ya 240.53 350.77 Va 1288.3 4347.8 u 23.758 10.889 Va 247.45 360.02 Y* 1320.3 4510.9 13-16" 24.850 11.649 9. 254.47 381.70 Ya 1352.7 4677.9 Va 25.967 12.443 Ys 261.59 397.83 21. 1385.5 4849.1 15-16 27.109 13.272 8 268.81 414.41 Ya 1418.6 5024.3 3. 28.274 14.137 270.12 431.44 A 1452.2 5203.7 1-16 29.465 15.039 Ya 283.53 448.92 Ya 1486.2 5387.4 % 30.680 15.979 % 291.04 466.87 22. 1520.5 5575.3 3-16 31.919 16.957 u 298.65 485.31 Ya 1555.3 5767.6 SPHERES. SPHERES— (Contained.) 119 Diam. Sur- face. Solid- ity. Diam. Sur- face. Solid- ity. Diam. Sur- face. Solid- ity. *$ 1590.4 5964.1 40 Yz 5153.1 34783 70 Yz 15615 183471 1626.0 6165.2 41. 5281 . 1 36087 71 15837 187402 23. 1661.9 6370.6 Mi 5410.7 37423 Yz 16061 191389 H 1698.2 6580.6 42. 5541.9 38792 72 16286 195,133 y% 1735.0 6795.2 Yz 5674.5 40194 Yz 16513 199532 % 1772.1 7014.3 43. 5808.8 41630 73 16742 203689 24. 1809.6 7238.2 }4 5944.7 43099 Yz 16972 207903 Ya 1847.5 7466.7 44. 6082.1 44602 74 17204 212175 1 1885.8 7700.1 Yz 6221.2 46141 Yz 17437 216505 1924.4 7938.3 45. 6361.7 47713 75 17672 220894 25. 1963.5 8181.3 Yz 6503.9 49321 Yz 17908 225341 i 2002.9 8429.2 46. 6647.6 50965 76 18146 229848 2042.8 8682.0 Yz 6792.9 52645 Yz 18386 234414 M 2083.0 8939.9 47. 6939.9 54362 77 18626 239041 26. 2123.7 9202.8 % 7088.3 56115 Yz 18869 243728 J4 2164.7 9470.8 48. 7238.3 57906 7S 19114 248475 ^ 2206.2 9744.0 % 7389.9 59734 Yz 19360 253284 94 2248.0 10022 49. 7543.1 61601 79 19607 258155 27. 2290.2 10306 H 7697.7 63506 Yz 19856 263088 Ya 2332.8 10595 50. 7854.0 65450 80 20106 268083 y* 2375.8 10889 Yz 8011.8 67433 Yz 20358 273141 H 2419.2 11189 51. 8171.2 69456 81 20612 278263 28. 2463.0 11494 y% 8332.3 71519 Yz 20867 283447 S 2507.2 11805 52. 8494.8 73622 82 21124 288696 2551.8 12121 Yz 8658.9 75767 Yz 21382 294010 % 2596.7 12443 53. 8824.8 77952 83 21642 299388 29. 2642.1 12770 Yz 8992.0 80178 Yi 21904 304831 M 2687.8 13103 54. 9160.8 82448 84 22167 310340 *6 2734.0 13442 Ya 9331.2 84760 Yz 22432 315915 H 2780.5 13787 55. 9503.2 87114 85 22698 321556 30. 2827.4 14137 Yi 9676.8 89511 Yz 22966 327264 J4 2874.8 14494 56. 9852.0 91953 80 23235 333039 2922.5 14856 H 10029 94438 Y-2 23506 338882 M 2970.6 15224 57. 10207 96967 87 23779 344792 31. 3019.1 15599 Yz 10387 99541 Yz 24053 350771 M 3068.0 15979 58 10568 102161 88 24328 356819 % 3117.3 16366 k 10751 104826 Yz 24606 362935 Y\ 3166.9 16758 59 10936 107536 89 24885 369122 32. 3217.0 17157 % 11122 110294 Yz 25165 375378 a 3267.4 17563 60 11310 113098 90 25447 381704 3318.3 17974 "" y* 11499 115949 Yz 25730 388102 % 3369.6 18392 61 11690 118847 91 26016 394570 33. 3421.2 18817 X 11882 121794 Yz 26302 401109 H 3473.3 19248 62 12076 124789 92 26590 407721 14 3525.7 19685 Yz 12272 127832 Yz 26880 414405 U 3578.5 20129 63. 12469 130925 93 27172 421161 34. 3631.7 20580 Yz 12668 134067 Yz 27464 427991 » 3685.3 21037 64. 12868 137259 94 27759 434894 » 3739.3 21501 Yz 13070 140501 Yz 28055 441871 35. 3848.5 22449 65. 13273 143794 95 28353 448920 ^ 3959.2 23425 y> 13478 147138 Yz 28652 456047 36. 4071.5 24429 66 13685 150533 96 28953 463248 ^ 4185.5 25461 Yz 13893 153980 Yz 29255 470524 37. 4300.9 26522 67. 14103 157480 97 29559 477874 H 4417.9 27612 Yz 14314 161032 Yz 29865 485302 38. 4536.5 28731 6S 14527 164637 98 30172 492808 ^ 4656.7 29880 Yz 14741 168295 Yz 30481 500388 39. 4778.4 31059 69. 14957 172007 99 30791 508047 3£ 4901.7 32270 Yz 15175 175774 Yz 31103 515785 40. 5026.5 33510 70. 15394 179595 100 31416 523598 120 MATHEMATICAL TABLES. CONTENTS IN CUBIC FEET AND U. S. GALLONS OF PIPES AND CYLINDERS OF VARIOUS DIAMETERS AND ONE FOOT IN LENGTH. 1 gallon = 231 cubic inches. 1 cubic foot - 7.4805 gallons. For 1 Foot in For 1 Foot in For 1 Foot in a Length. a Length. a Length. 3 & §1 5 Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. & a> 11 C3hH s Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. 5 Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. H .0003 .0025 m .2485 1.859 19 1.969 14.73 5-16 .0005 .004 .2673 1.999 19^ 2.074 15.51 % .0008 .0057 k .2867 2.145 20 2.182 16.32 7-16 .001 .0078 .3068 2.295 20^ 2.292 17.15 H ,0014 .0102 Wa .3276 2.45 21 2.405 17.99 9-16 .0017 .0129 8 .3491 2.611 21^2 2.521 18.86 % .0021 .0159 m .3712 2.777 22 2.640 19.75 11-16 .0026 .0193 sy 2 .3941 2.948 22^ 2.761 20.66 % .0031 .0230 8% .4176 3.125 23 2.885 21.58 13-16 .0036 .0269 9 .4418 3.305 23^ 3.012 22.53 Vs .0042 .0312 m .4667 3.491 24 3.142 23.50 15-16 .0048 .0359 9^ .4922 3.682 25 3.409 25.50 1 .0055 .0408 9% .5185 3.879 26 3.687 27.58 M .0085 .0638 10 .5454 4.08 27 3.976 29.74 1H .0123 .0918 10M .5730 4.286 28 4.276 31.99 m .0167 .1249 10^ .6013 4.498 29 4.587 34.31 2 .0218 .1632 10% .6303 4.715 30 4.909 36.72 .0276 .2066 11 .66 4.937 31 5.241 39.21 .0341 .2550 .6903 5.164 32 5.585 41.7S *4 .0412 .3085 .7213 5.396 33 5.940 44.43 3 .0491 .3672 u% .7530 5.633 34 6.305 47.16 3M .0576 .4309 12 .7854 5.875 35 6.681 49.98 .0668 .4998 12J6 .8522 6.375 36 7.069 52.88 m .0767 .5738 13 .9218 6.895 37 7.467 55.86 4 .0873 .6528 13J^ .994 7.436 38 7.876 58.92 QA .0985 .7369 14 1.069 7.997 39 8.296 62.06 V/z .1134 .8263 14J4 1 147 8.578 40 8.727 65.28 4% .1231 .9206 15 1.227 9.180 41 9.168 68.58 5 .1364 1.020 15J4 1.310 9.801 42 9.621 71.97 m .1503 1.125 16 1.396 10.44 43 10.085 75.44 5V 2 .1650 1.234 16^ 1.485 11.11 44 10.559 78.99 Wa. .1803 1.349 17 1.576 11.79 45 11.045 82.62 6 .1963 1.469 17^ 1.670 12.49 46 11.541 86.33 6M .2131 1.594 18 1.768 13.22 47 12.048 90.13 6^ .2304 1.724 18V£ 1.867 13.96 48 12.566 94.00 To find the capacity of pipes greater than the largest given in the table, look in the table for a pipe of one half the given size, and multiply its capac- ity by 4; or one of one third its size, and multiply its capacity by 9, etc. To find the weight of water in any of the given sizes multiply the capacity in cubic feet by 62)4 or the gallons by 8*4 or, if a closer approximation is required, by the weight of a cubic foot of water at the actual temperature in the pipe. Given the dimensions of a cylinder in inches, to find its capacity in U. S. gallons: Square the diameter, multiply by the length and by .0034. If d '= d 2 X 7854 X I diameter, I — length, gallons = ^ = .0034d 2 l. CAPACITY OF CYLINDKICAL VESSELS. 121 CYLINDRICAL. VESSELS, TANKS, CISTERNS, ETC. Diameter in Feet and Inches, Area in Square Feet, and U. S. Gallons Capacity for One Foot in Depth. 1 gallon = 231 cubic inches = ° U „ 1q 05 °° = 0.13368 cubic feet. Diani. Area. Gals. Diam. Area. Gals. Diam. Area. Gals. Ft. In. Sq. ft. lfoot depth. Ft. In. Sq. ft. lfoot depth. Ft. In. Sq. ft. 1 foot depth. 1 .785 5.87 5 8 25.22 188.66 19 283.53 2120.9 1 1 .922 6.89 5 9 25.97 194.25 19 3 291.04 2177.1 1 2 1.069 8.00 5 10 26.73 199.92 19 6 298.65 2234.0 1 3 1.227 9.18 5 11 27.49 205.67 19 9 306.35 2291.7 1 4 1.396 10.44 6 28.27 211.51 20 314.16 2350.1 1 5 1.576 11.79 6 3 30.68 229.50 20 3 322.06 2409.2 1 6 1.767 13.22 6 6 33.18 248.23 20 6 330.06 2469.1 1 7 1.969 14.73 6 9 35.78 267.69 20 9 338.16 2529.6 1 8 2.182 16.32 7 38.48 287.88 21 346.36 2591.0 1 9 2.405 17.99 7 3 41.28 308.81 21 3 354.66 2653.0 1 10 2.640 19.75 7 6 44.18 330.43 21 6 363.05 2715.8 1 11 2.885 21.58 7 9 47.17 352.88 21 9 371.54 2779.3 2 3.142 23.50 8 50.27 376.01 22 380.13 2843.6 2 1 3.409 25.50 8 3 53.46 399.88 22 3 388.82 2908.6 2 2 3.687 27.58 8 6 56.75 424.48 22 6 397.61 2974.3 2- 3 3.976 29.74 8 9 60.13 449.82 22 9 406.49 3040.8 2 4 4.276 31.99 9 63.62 475.89 23 415.48 3108.0 2 5 4.587 34.31 9 3 67.20 502.70 23 3 424.56 3175.9 2 6 4.909 36.72 9 6 70.88 530 24 23 6 433.74 3244.6 2 7 5.241 39.21 9 9 74.66 558.51 23 9 443.01 3314.0 2 8 5.585 41.78 10 78.54 587.52 24 452.39 3384.1 2 9 5.940 44.43 10 3 82.52 617.26 24 3 461.86 3455.0 2 10 6.305 47.16 10 6 86.59 647.74 24 6 471.44 3526.6 2 11 6.681 49.98 10 9 90.76 678.95 24 9 481.11 3598.9 3 7.069 52.88 11 95.03 710.90 25 490.87 3672.0 3 1 7.467 55.86 11 3 99.40 743.58 25 3 500.74 3745.8 3 2 7.876 58.92 11 6 103.87 776.99 25 6 510.71 3820.3 3 3 8.296 62.06 11 9 108.43 811.14 25 9 520.77 3895.6 3 4 8.727 65.28 12 113.10 846.03 26 530.93 3971.6 3 5 9.168 68.58 12 3 117.86 881.65 26 3 541.19 4048.4 3 6 9.621 71.97 12 6 122.72 918.00 26 6 551.55 4125.9 3 7 10.085 75.44 12 9 127.68 955.09 26 9 562.00 4204.1 3 8 10.559 78.99 13 132.73 992.91 27 572.56 4283.0 3 9 11.045 82.62 13 3 137.89 1031.5 27 3 583.21 4362.7 3 10 11.541 86.33 13 6 143.14 1070.8 27 6 593.96 4443.1 3 11 12.048 90.13 13 9 148.49 1110.8 27 9 604.81 4524.3 4 12.566 94.00 14 153.94 1151.5 28 615.75 4606.2 4 1 13.095 97.96 14 3 159.48 1193.0 28 3 626.80 4688.8 4 2 13.635 102.00 14 6 165.13 1235.3 28 6 637.94 4712.1 4 3 14.186 106.12 14 9 170.87 1278.2 28 9 649.18 4856.2 4 4 14.748 110.32 15 176.71 1321.9 29 660.52 4941.0 4 5 15.321 114.61 15 3 182 65 1366.4 29 3 671.96 5026.6 4 6 15.90 118.97 15 6 188.69 1411.5 29 6 683.49 5112.9 4 7 16.50 123.42 15 9 194.83 1457.4 29 9 695.13 5199.9 4 8 17.10 127.95 16 201.06 1504.1 30 706.86 5287.7 4 9 17.72 132.56 16 3 207.39 1551.4 30 3 718.69 5376.2 4 10 18.35 137.25 16 6 213 82 1599.5 30 6 730.62 5465.4 4 11 18.99 142.02 16 9 220.35 1648.4 30 9 742.64 5555.4 5 39.63 146.88 17 226.98 1697.9 31 754.77 5646.1 5 1 20.29 151.82 17 3 23'171 1748.2 31 3 766.99 5737.5 5 2 20.97 156.83 17 6 240.53 1799.3 31 6 779.31 5829.7 5 3 21.65 161.93 17 9 247.45 1851.1 31 9 791.73 5922.6 5 4 22.34 167.12 18 254.47 1903.6 32 804.25 6016.2 5 5 23.04 172.38 18 3 261.59 1956.8 32 3 816.86 6110.6 5 6 23.76 177.72 18 6 268.80 2010.8 32 6 829.58 6205.7' 5 7 24.48 183.15 18 9 276.12 2065.5 32 9 842.39 6301.5 122 MATHEMATICAL TABLES. GALLONS AND CUBIC FEET. United States Gallons in a given Number of Cubic Feet. 1 cubic foot = 7.480519 U. S. gallons; 1 gallon = 231 cu. in. = .13368056 cu. ft. Cubic Ft. Gallons. Cubic Ft. Gallons. Cubic Ft. Gallons. 0.1 02 0.3 0.4 0.5 0.75 1.50 2.24 2.99 3.74 50 60 70 80 90 374.0 448.8 523.6 598.4 673.2 8,000 9,000 10,000 20,000 30,000 59,844.2 67,324.7 74,805.2 149,610.4 224,415.6 0.6 0.7 0.8 0.9 1 4.49 5.24 5.98 6.73 7.48 100 200 300 400 500 748.0 1,496.1 2,244.2 2,992.2 3,740.3 40,000 50,000 60,000 70,000 80,000 299,220.8 374,025.9 448,831.1 523,636.3 598,441.5 2 3 4 5 6 14.96 22.44 29.92 37.40 44.88 600 700 800 900 1,000 4,488.3 5,236.4 5,984.4 6,732.5 7,480.5 90,000 100,000 200,000 300,000 400,000 673,246.7 748,051.9 1,496,103.8 2,244,155.7 2,992,207.6 7 8 9 10 20 52.36 59.84 67.32 74.80 149.6 2,000 3,000 4,000 5,000 6,000 14,961.0 22,441.6 29,922.1 37,402.6 44,883.1 500,000 600,000 700,000 800,000 900,000 3,740,259.5 4.488,311.4 5,236,363.3 5,984,415.2 6,732,467.1 30 40 224.4 299.2 7,000 52,363.6 1,000,000 7,480,519.0 Cubic Feet in a given Number of Gallons. Gallons. Cubic Ft. Gallons. Cubic Ft. Gallons. Cubic Ft. 1 2 3 4 5 6 8 9 10 .134 .267 .401 .535 .668 .802 .936 1.069 1.203 1.337 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 133.681 267.361 401.042 534.722 668.403 802.083 935.764 1,060.444 1,203.125 1,336.806 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000 9,000,000 10,000,000 133,680.6 267,361.1 - 401,041.7 534,722.2 668,402.8 802,083.3 935,763.9 1,069,444.4 1,203,125.0 1,336,805.6 DUMBER OF SQUARE FEET IN" PLATES. 123 NUMBER OF SQUARE FEET IN PLATES 3 TO 32 FEET LONG, AND 1 INCH WIDE. For other widths, multiply by the width in inches. 1 sq. in. = .0069f sq. ft. Ft. and in. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. 3. 36 .25 7.10 94 .6528 13. 8 152 1.056 1 37 .2569 11 95 .6597 9 153 1.063 2 38 .2639 8. 96 .6667 10 154 1.069 3 39 .2708 1 97 .6736 11 155 1.076 4 40 .2778 2 98 .6806 13. 156 1.083 5 41 !2847 3 99 .6875 1 157 1.09 6 42 .2917 4 100 .6944 2 158 1.097 7 43 .2986 5 101 .7014 3 159 1.104 8 44 .3056 • 6 102 .7083 4 160 1.114 9 45 .3125 7 103 .7153 5 161 1.118 10 46 .3194 8 104 .7222 6 162 1.125 11 47 .3264 9 105 .7292 7 163 1.132 4. 48 .3333 10 106 .7361 8 164 1.139 1 49 .3403 11 107 .7431 9 165 1.146 2 50 .3472 9. 108 .75 10 166 1.153 3 51 .3542 1 109 .7569 11 167 1.159 4 52 .3611 2 no .7639 14.0 168 1.167 5 53 .3681 3 in .7708 1 169 1.174 6 54 .375 4 112 .7778 2 170 1.181 7 55 .3819 5 113 .7847 3 171 1.188 8 56 .3889 6 114 .7917 4 172 1.194 9 57 .3958 7 115 .7986 5 173 1.201 10 58 .4028 8 116 .8056 6 174 1.208 11 59 .4097 9 117 .8125 7 175 1.215 5. 60 .4167 10 118 .8194 8 176 1.222 1 61 .4236 11 119 .8264 9 177 1.229 2 62 .4306 10.0 120 .8333 10 178 1.236 3 63 .4375 1 121 .8403 11 179 1.243 4 64 .4444 2 122 .8472 15.0 180 1.25 5 65 .4514 3 123 .8542 1 181 1.257 6 66 .4583 4 124 .8611 2 182 1.264 7 67 .4653 5 125 .8681 3 183 1.271 8 68 .4722 6 126 .875 4 184 1.278 9 69 .4792 7 127 .8819 5 185 1.285 10 70 .4861 8 128 .8889 6 186 1.292 11 71 .4931 9 129 .8958 7 187 1.299 6. 72 .5 10 130 .9028 8 188 1.306 1 73 .5069 11 131 .9097 9 189 1.313 2 74 .5139 11.0 132 9167 10 190 1.319 3 75 .5208 1 133 .9236 11 191 1.326 4 76 .5278 2 134 .9306 16.0 192 1.333 5 77 .5347 3 135 .9375 1 193 1.34 6 78 .5417 4 136 .9444 2 194 1.347 7 79 .5486 5 137 .9514 3 195 1.354 8 80 .5556 6 138 .9583 4 196 1 361 9 81 .5625 7 139 .9653 5 197 1.368 10 82 .5694 8 140 .9722 6 198 1.375 11 83 .5764 9 141 .9792 7 199 1.382 1. 84 .5834 10 142 .9861 8 200 1.389 1 85 .5903 11 143 .9931 9 201 1.396 2 86 .5972 12.0 144 1.000 10 202 1.403 3 87 .6042 1 145 1.007 11 203 1.41 4 88 .6111 2 146 1.014 17.0 204 1.417 5 89 .6181 3 147 1.021 1 205 1.424 6 90 .625 4 148 1.028 2 206 1.431 7 91 .6319 5 149 1.035 3 207 1.438 8 92 .6389 6 150 1.042 4 208 1.444 9 93 .6458 7 151 1.049 5 209 1.451 124 MATHEMATICAL TABLES. SQUARE FEET IN PliATES- -(Continued.) Ft. and Ins. Long. Ins. Square Ft. and Ins. Square Feet. Ft. and Ins. Square Feet. Long. Feet. Long. Long. Long. Long. 17.6 210 1.458 22.5 269 1.868 27.4 328 2.278 7 211 1.465 6 270 1.875 5 329 2.285 8 212 1.472 7 271 1.882 6 330 2.292 9 213 1.479 8 272 1.889 7 331 2.299 10 214 1.486 9 273 1.896 8 332 2.306 11 215 1.493 10 274 1.903 9 333 2.313 18.0 216 1.5 11 275 1.91 10 334 2.319 1 217 1.507 33. 276 1.917 11 335 2.326 2 218 1.514 1 277 1.924 28.0 336 2.333 3 219 1.521 2 278 1.931 1 337 2.34 4 220 1.528 3 279 1.938 2 338 2.347 5 221 1.535 4 280 1.944 3 339 2.354 6 222 1.542 5 281 1.951 4 340 2.361 7 223 1.549 6 282 1.958 5 341 2.368 8 224 1.556 7 283 1.965 6 342 2.375 9 225 1.563 8 284 1.972 7 343 2.382 226 1.569 9 285 1.979 8 344 2.389 11 227 1.576 10 286 1.986 9 345 2.396 19.0 228 1.583 11 287 1.993 10 346 2.403 1 229 1.59 24.0 288 2 11 347 2.41 2 230 1.597 1 289 2.007 29.0 348 2.417 3 231 1.604 2 290 2.014 1 349 2.424 4 232 1.611 3 291 2.021 2 350 2.431 5 233 1.618 4 292 2.028 3 351 2.438 6 234 1.625 5 293 2.035 4 352 2.444 7 235 1.632 6 294 2.042 5 353 2.451 8 236 1.639 7 295 2.049 6 354 2.458 9 237 1.645 8 296 2.056 7 355 2.465 10 238 1.653 9 297 2.063 8 356 2.472 11 239 1.659 10 298 2.069 9 357 2.479 20.0 240 1.667 11 299 2.076 10 358 2.486 1 241 1.674 25.0 300 2.083 11 359 2.493 2 242 1.681 1 301 2.09 30.0 360 2.5 3 243 1.688 2 302 2.097 1 361 2.507 4 244 1.694 3 303 2.104 2 362 2.514 5 245 1.701 4 304 2.111 3 363 2.521 6 246 1.708 5 305 2.118 4 364 2.528 7 247 1.715 6 306 2.125 5 365 2.535 8 .248 1.722 7 307 2.132 6 366 2.542 9 249 1.729 8 308 2.139 7 367 2.549 10 250 1.736 9 309 2.146 8 368 2.556 11 251 1.743 10 310 2.153 9 369 2.563 21.0 252 1.75 11 311 2.16 10 370 2.569 1 253 1.757 26.0 312 2.167 11 371 2.576 2 254 1.764 1 313 2.174 31.0 372 2.583 3 255 1.771 2 314 2.181 1 373 2.59 4 256 1.778 3 315 2.188 2 374 2.597 5 257 1.785 4 316 2.194 3 375 2.604 6 258 1.792 5 317 2.201 4 376 2.611 7 259 1.799 6 318 2.208 5 377 2.618 8 260 1.806 7 319 2.215 6 378 2.625 9 261 1.813 8 320 2.222 7 379 2.632 10 262 1.819 9 321 2.229 8 380 2.639 11 263 1.826 10 322 2.236 9 381 2.646 22.0 264 1.833 11 323 2.243 10 382 2 653 1 265 1.84 27.0 324 2.25 11 383 2.66 2 266 1.847 1 325 2.257 32.0 384 2.667 3 267 1.854 . 2 326 2.264 1 385 2.674 4 268 1.861 1 3 337 2.271 2 386 2.681 CAPACITY OF KECTAtfGtJLAft TAtfES. 125 CAPACITIES OF RECTANGULAR TANKS IN U. S. GALLONS, FOR EACH FOOT IN DEPTH. 1 cubic foot = 7.4805 U. S. gallons. Width Length of Tank. of Tank. feet. 2 ft. in. 2 6 feet. 3 44.88 56.10 67.32 ft. in. 3 6 feet. 4 ft. in. 4 6 feet. 5 ft. in. 5 6 82.29 102.86 123.43 144.00 164.57 185.14 205.71 226.28 feet. 6 ft. in. 6 6 feet. 1 ft, in. 2 2 G 3 29.92 37.40 46.75 52.36 65.45 78.54 91.64 59.84 74.80 89.77 104.73 119.69 67.32 84.16 100.99 117.82 134.65 151.48 74.81 93.51 112.21 130.91 149.61 168.31 187.01 89.77 112.21 157.09 179.53 201.97 224.41 246.86 269.30 97.25 121.56 145.87 170.18 194.49 218.80 243.11 267.43 291.74 316.05 104.73 130.91 157.09 3 6 183.27 4 4 6 209.45 235.63 5 261.82 5 6 288.00 6 314 18 6 6 340.36 366.54 Width Length of Tank. of Tank. ft. in. 7 6 feet. 8 ft. in. 8 6 feet. 9 ft. in. 9 6 feet. 10 ft. in. 10 6 feet. 11 ft. in. 11 6 feet. 12 ft. in. 2 2 6 3 3 6 4 4 6 5 5 6 6 6 6 7 6 8 8 6 112.21 140.26 168.31 196.36 224.41 252.47 280.52 308.57 336.62 364.67 392.72 420.78 119.69 149.61 179.53 209.45 239.37 269.30 299.22 329.14 359.06 388.98 418.91 448.83 478.75 127.17 158.96 190.75 254.34 286.13 317.92 349.71 381.50 413.30 445 09 476.88 508.67 540.46 134.65 168.31 202.97 235.63 269.30 302.96 336.62 370.28 403.94 437.60 471 27 504.93 538.59 572.25 605.92 142 13 177.66 233.19 248.73 284.26 319.79 390.85 426.39 461.92 497.45 568.51 604.05 639.58 675.11 149.61 187.01 224.41 261.82 299.22 336.62 374.03 411.43 448.83 486.23 523.64 561 04 598.44 685.84 673.25 710.65 748.05 157.09 196.36 235.63 274.90 314.18 353.45 392.72 432.00 471.27 510.54 549.81 589.08 628.36 667 63 706.90 746.17 785.45 824.73 164.57 205.71 246.86 288.00 329.14 370.28 411.43 452.57 493.71 534.85 575.99 617.14 - 699.42 740.56 781.71 822.86 864.00 905.14 172.05 215.06 258.07 301.09 344.10 387.11 430.13 473.14 516.15 559.16 602.18 645.19 688.20 731.21 774.23 817.24 860.26 903.26 946.27 989.29 179 53 224.41 269.30 314.18 359.06 403.94 448.83 493.71 538.59 583.47 628 36 673.24 718.12 763.00 9 807 89 9 6 852 77 10 897.66 10 6 942 56 11 987 43 11 6 1032.3 12 1077 2 12G MATHEMATICAL TABLES. NUMBER OF BARRELS (31 1-2 GAL.L.ONS) IN CISTERNS AND TANKS. = 4.21094 cubic feet. Reciprocal = Depth Diameter in Feet. Feet. 5 6 7 8 9 10 11 12 13 14 1 4.663 6.714 9.139 11.937 15.10S 18.652 22.569 26.859 31.522 36.557 5 23.3 33.6 45.7 59.7 75.5 93.3 112.8 134.3 157.6 182.8 6 28.0 40.3 54.8 71.6 90.6 111.9 135.4 161.2 189.1 219 3 7 32.6 47.0 64.0 83.6 105.8 130.6 158.0 188.0 220.7 255.9 8 37.3 53.7 73.1 95.5 120.9 149.2 180.6 214.9 252.2 292.5 9 42.0 60.4 82.3 107.4 136.0 167.9 203.1 241.7 283.7 329.0 10 46.6 67.1 91.4 119.4 151.1 186.5 225.7 268.6 315.2 365.6 11 51.3 73.9 100.5 131.3 166.2 205.2 248 3 295.4 346.7 402.1 12 56.0 80.6 109.7 143.2 181.3 223.8 270.8 322.3 378.3 438.7 13 60.6 87.3 118.8 155.2 196.4 242.5 293.4 349.2 409.8 475.2 14 65.3 94.0 127.9 167.1 211.5 261.1 316.0 376.0 441.3 511.8 15 69.9 100.7 137.1 179.1 226.6 289.8 338.5 402.9 472.8 548.4 16 74.6 107.4 146.2 191.0 241.7 298.4 361.1 429.7 504.4 5S4.9 17 79.3 114.1 155.4 202.9 256.8 317.1 383.7 456.6 535.9 621 .5 18 83.9 120.9 164.5 214.9 271.9 335.7 406.2 483.5 567.4 658.0 19 88.6 127.6 173.6 226.8 287.1 354.4 428.8 510.3 598.9 694.6 20 93 3 134.3 182.8 238.7 302.2 373.0 451.4 537.2 630.4 731.1 Depth Diameter in Feet. Feet. 15 16 17 18 19 20 21 22 1 41.966 47.748 53.903 60.431 67.332 74.606 82.253 90.273 5 209.8 238.7 269.5 302.2 336.7 373.0 411.3 451.4 6 251.8 286.5 323.4 362 6 404.0 447.6 493.5 541.6 7 293.8 334.2 377.3 423 471.3 522.2 575.8 631.9 8 335.7 382.0 431.2 483.4 538.7 596.8 658.0 722.2 9 377.7 429.7 485 1 543.9 606.0 671.5 740.3 812.5 10 419.7 477.5 539.0 604.3 673.3 746.1 822.5 902.7 11 461.6 525.2 592.9 664.7 740.7 820.7 904.8 993.0 12 503.6 573.0 646.8 725.2 808.0 895.3 987.0 1083.3 13 545.6 620.7 700.7 785.6 875.3 969.9 1069.3 1173.5 14 587.5 668.5 754.6 846.0 942.6 1044.5 1151.5 1263.8 15 629.5 716.2 808.5 906.5 1010.0 1119.1 1233.8 1354.1 16 671.5 764.0 862.4 966.9 1077.3 1193.7 1316.0 1444.4 17 713.4 811.7 916.4 1027.3 1144.6 1268.3 1398.3 1534.5 18 755.4 859.5 970.3 1087.8 1212.0 1342.9 1480.6 1624.9 19 797.4 907.2 1024.2 1148.2 1279.3 1417.5 1562.8 1715.2 20 839.3 955.0 1078.1 1208.6 1346.6 1492.1 1645.1 1805.5 LOGARITHMS. 127 NUMBER OF BARRELS (31 1-2 GALLONS) IN CISTERNS AND TANKS.— Contained. Depth Diameter in Feet. in Feet. 23 24 25 26 27 28 29 30 1 98.666 107.482 116.571 120.083 135.968 146.226 157.858 167.86? 5 493.3 537.2 582.9 630.4 679.8 731.1 784.3 839.3 6 592.0 644.6 699.4 756.5 815.8 877.4 941.1 1007.2 7 690.7 752.0 816.0 882.6 951.8 1023.6 1098.0 1175.0 8 789.3 859.5 932.6 1008.7 1087.7 1169.8 1254.9 1342.9 9 888.0 966.9 1049.1 1134.7 1223.7 1316.0 1411.7 1510.8 10 986.7 1074.3 1165.7 1260.8 1359.7 1462.2 1568.6 1678.6 11 1085.3 1181.8 1282.3 1386.9 1495.6 1608.5 1725.4 1846.5 12 1184.0 1289.2 1398.8 1513.0 1631.6 1754.7 1882.3 2014.4 13 1282.7 1396.6 1515.4 1639.1 1767.6 1900.9 2039.2 2182.2 14 1381.3 1504 1632.0 1765.2 1903.6 2047.2 2196.0 2350.1 15 1480.0 1611.5 1748.6 1891.2 2039.5 2193.4 2352.9 2517.9 16 1578.7 1718.9 1865.1 2017.3 2175.5 2339.6 2509.7 2685.8 17 1677.3 1826.3 1981.7 2143.4 2311.5 2485.8 2666.6 2853.7 18 1776.0 1933.8 2098.3 2269.5 2447.4 2632.0 2823.4 3021.5 19 1874.7 2041.2 2214.8 2395.6 2583.4 2778.3 2980.3 3189.4 20 1973.3 2148.6- 2321.4 2521.7 2719.4 2924.5 3137.2 3357.3 LOGARITHMS. Logarithms (abbreviation log).— The log of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number. The fixed number is called the base. Thus if the base is 10, the log of 1000 is 3, for 10 3 = 1000. There are two systems of logs in general use, the common, in which the base is 10, and the Naperian, or hyperbolic, in which the base is 2.718281828 .... The Naperian base is commonly de- noted by e, as in the equation e y = x, in which y is the Nap. log of x. In any system of logs, the log of 1 is 0; the log of the base, taken in that system, is 1. In any system the base of which is greater than 1, the logs of all numbers greater than 1 are positive and the logs of all numbers less than 1 are negative. The modulus of any system is equal to the reciprocal of the Naperian log of the base of that system. The modulus of the Naperian system is 1, that of the common system is .4342945. The log of a number in any system equals the modulus of that system X the Naperian log of the number. The hyperbolic or Naperian log of any number equals the common log X 2.30 5851. Every log consists of two parts, an entire part called the characteristic, or index, and the decimal part, or mantissa. The mantissa only is given in the usual tables of common logs, with the decimal point omitted. The charac- teristic is found by a simple rule, viz., it is one less than the number of figures to the left of the decimal point in the number whose log is to be found. Thus the characteristic of numbers from 1 to 9.99 + is 0, from 10 to 99.99 + is 1, from 100 to 999 -f- is 3, from .1 to .99 + is - 1, from .01 to .099 4- is - 2, etc. Thus log of 2000 is 3.30103; log of .2 is - 1.30103; " " 200 " 2.30103; " " .02 " - 2.30103; " " 20 "1.30103; " " .002 "- 3.30103; ** " 2 " 0.30103; *' " ,0002 " - 4.30103,- 128 MATHEMATICAL TABLES. The minus sign is frequently written above the characteristic thus: log .002 = 3.30103. The characteristic only is negative, the decimal part, or mantissa, being always positive. When a log consists of a negative index and a positive mantissa, it is usual to write the negative sign over the index, or else to add 10 to the index, and to indicate the subtraction of 10 from the resulting logarithm. Thus log .2 = T.30103, and this may be written 9.30103 - 10. In tables of logarithmic sines, etc., the — 10 is generally omitted, as being understood. Rules for use of the table of Logarithms. -To find the log of any whole number.— For 1 to 100 inclusive the log is given complete in the small table on page 129. For 100 to 999 inclusive the decimal part of the log is given opposite the given number in the column headed in the table (including the two figures to the left, making six figures). Prefix the characteristic, or index, 2. For 1000 to 9999 inclusive : The last four figures of the log are found opposite the first three figures of the given number and in the vertical column headed with the fourth figure of the given number ; prefix the two figures under column 0, and the index, which is 3. For numbers over 10,000 having five or more digits : Find the decimal part of the log for the first four digits as above, multiply the difference figure in the last column by the remaining digit or digits, and divide by 10 if there be only one digit more, by 100 if there be two more, and so on ; add the quotient to the log of the first four digits and prefix the index, which is 4 if there are five digits, 5 if there are six digits, and so on. The table of pro- portional parts may be used, as shown below. To find the log of a decimal fraction or of a whole number and a decimal.— First find the log of the quantity as if there were no decimal point, then prefix the index according to rule ; the index is one less than the number of figures to the left of the decimal point. Required log of 3.141593. log of 3.141 =0.497068. Diff . = 138 From proportional parts 5 = 690 " " 09 = 1242 003 = 041 log 3.141593 0.4971498 To find the number corresponding to a given log.— Find in the table the log nearest to the decimal part of the given log and take the first four digits of the required number from the column N and the top or foot of the column containing the log which is the next less than the given log:. To find the 5th and 6th digits subtract the log in the table from the given log, multiply the difference by 100, and divide by the figure in the Diff. column opposite the log ; annex the quotient to the four digits already found, and place the decimal point according to the rule ; the number of figures to the left of the decimal point is one greater than the index. Find number corresponding to the log 0.497150 Next lowest log in table corresponds to 3141 497068 Diff. = 82 Tabular diff. = 138; 82 ~ 138 = .59 + The index being 0, the nvimber is therefore 3.14159 -4-. To multiply two numbers by the use of logarithms.— Add together the logs of the two numbers, and find the number whose log is the sum. To divide two numbers.— Subtract the log of the less from the log of the greater, and find the number whose log is the difference. To raise a number to any given power.— Multiply the log of the number by the exponent of the power, and find the number whose log is the product. To find any root of a given number.— Divide the log of the number by the index of the root. The quotient is the log of the root. To find the reciprocal of a number.— Subtract the decimal part of the log of the number from 0, add 1 to the index and change the sign of the index, The result is the log of the reciprocal. LOGARITHMS. 129 Required the reciprocal of 3.141593. Log of 3.141593, as found above 0.4971498 Subtract decimal part from gives _0.5028502 Add 1 to the index, and changing sign of the index gives. . l .5028502 which is the log of 0.31831 . To find the fourth term of a proportion by logarithms. —Add the logarithms of the second and third terms, aud from their sum subtract the logarithm of t'ne first term. When one logarithm is to be subtracted from another, it may be more convenient to convert the subtraction into an addition, which may be done by first subtracting th^ given logarithm from 10, adding the difference tc the other logarithm, and afterwards rejecting the 10. The difference between a given logarithm and 10 is called its arithmetical complement, or cologarithm. To subtract one logarithm from another is the same as to add its comple- ment and then reject 10 from the result. For a — b = 10 — b -j- a — 10. To work a proportion, then, by logarithms, add the complement of the logarithm of the first term to the logarithms of the second and third terms. The characteristic must afterwards be diminished by 10. Example in logarithms with a negative index. —Solve by which means divide 526 by 1011 and raise the quotient logarithms (^ m j to the 2.45 power. log 526 = log 1011 = log of quotient = Multiply by 2.7 3.004751 716235 2.45 - 2.581175 - 2.8 64940 - 1.43 2470 - 1.30 477575 = .20173, Ans. In multiplying - 1.7 by 5, we say: 5 x 7 - 35, 3 to carry; 5 x — 1 = — 5 less + 3 carried — — 2. In adding -2-j-8-f-3 + l carried from previous column, we say: 1 -j- 3 + 8 = 12, minus 2 = 10, set down and carry 1 ; 1 -fc- 4 — 2 = 3. Logarithms of Numbers from 1 to 100. Log. 0.000000 0.301030 0.477121 0.602060 0.698970 0.778151 0. 0. 0.954243 1.000000 1.041393 1.079181 1.113943 1.146128 1.176091 1.204120 1.230449 1.255273 1.278754 1. N. Log. N. Log. N. Log. 21 1.322219 41 1.612784 61 1.785330 22 1.342423 42 1.623249 62 1 792392 82 23 1.361728 43 1.633468 63 1 799341 83 24 1.380211 44 1.643453 64 1 806180 84 25 1.397940 45 1.653213 65 1 812913 85 26 1.414973 46 1.662758 66 1 819544 86 | 27 1.431364 47 1.672098 67 1 826075 87 28 1.447158 48 1.681241 68 1 832509 88 29 1.462398 49 1.690196 69 1 838849 89 30 1.477121 50 1.698970 70 1 845098 90 31 1.491362 51 1.707570 71 1 851258 91 32 1.505150 52 1.716003 72 1 857332 92 33 1.518514 53 1.724276 73 1 863323 93 34 1.531,479 54 1.732394 74 1 869232 94 35 1.544068 55 1.740363 75 1 875061 95 36 1.556303 56 1.748188 76 1 880814 96 37 1.568202 57 1.755875 77 1 886491 97 38 1.579784 58 1.763428 78 1 892095 98 39 1.591065 j 59 1.770852 79 1 897627 99 40 1.602060 1 60 1.778151 80 1 903090 100 1.908485 1.913814 1.919078 1.924279 1.929419 1.934498 1.939519 1.944483 1.949390 1.954243 1.959041 1.9i - 1.973128 177724 1.982271 56772 1.9! l.< 2.000000 130 LOGARITHMS OF NUMBERS. No. 100 L. 000.] [No. 109 L. 040. N. 1 2 8 4 5 6 7 2598 3029 6894 7321 1147 1 1570 5360 5779 9532 . 9947 8 9 Diff. 100 1 2 000000 4321 8600 0434 4751 9026 0868 5181 1301 5609 1734 6038 2166 6466 3461 7748 3891 8174 432 428 0300 4521 8700 0724 4940 9116 1993 6197 2415 6616 424 420 3 4 012837 7033 3259 7451 3680 7868 4100 8284 0361 4486 8571 2619 6629 0775 4896 8978 416 412 408 5 6 7 021189 5306 9384 1603 5715 9789 2016 6125 2428 6533 2841 6942 3252 7350 3664 7757 1812 5830 9811 4075 8164 0195 4227 8223 0600 4628 8620 1004 5029 9017 1408 5430 9414 2216 6230 3021 7028 0998 404 400 8 9 033424 7426 04 3826 7825 0207 0602 397 Proportional Parts. Diff. 434 422 421 420 419 418 417 416 415 414 413 412 411 410 409 408 407 406 405 404 403 402 401 400 1 43.4 43.3 43.2 43.1 43.0 42.9 42.8 42.7 42.6 42.5 42.4 42.3 42.2 42.1 42.0 41.9 41.8 41.7 41.6 41.5 41.4 41.3 41.2 41.1 41.0 40.9 40.8 40.7 40.6 40.5 40.4 40.3 40.2 40.1 40.0 39.5 86.4 86.2 86.0 85.8 85.2 85.0 84.8 84.6 84.4 84.2 84.0 83.2 83.0 82.2 82.0 81.8 81.6 81.4 81.2 81.0 79.2 79.0 129.6 129.3 129.0 128.7 128.4 128.1 127.8 127.5 127.2 126.9 126.6 126.3 126.0 125.7 125.4 125.1 124.8 124.5 124.2 123.9 123.6 123.3 123.0 122.7 122.4 122.1 121.8 121.5 121.2 120.9 120.6 120.3 120.0 119.7 119.4 119.1 118.8 118.5 173.6 173.2 172.8 172.4 172.0 171.6 171.2 170.8 170.4 170.0 168.8 168.4 168.0 167.6 167.2 166.8 166.4 166.0 165.6 165.2 164.8 164.4 164.0 163.6 163.2 162.8 162.4 162.0 161.6 161.2 160.8 160.4 160.0 159.6 159.2 158.8 158.4 158.0 5 6 7 8 217.0 260.4 303.8 347.2 216.5 259.8 303.1 346.4 216.0 259.2 302.4 345.6 215.5 258.6 301.7 344.8 215.0 258.0 301.0 344.0 214.5 257.4 300.3 343.2 214.0 256.8 299.6 342.4 213.5 256.2 298.9 341.6 213.0 255.6 298.2 340.8 212.5 255.0 297.5 340.0 212.0 254.4 296.8 339.2 211.5 253.8 296.1 338.4 211.0 253.2 295.4 337.6 210.5 252.6 294.7 336.8 210.0 252.0 294.0 336.0 209.5 251.4 293.3 335.2 209.0 250.8 292.6 334.4 208.5 250.2 291.9 333.6 208.0 249.6 291.2 332.8 207.5 249.0 290.5 332.0 207.0 248.4 289.8 331.2 206.5 247.8 289.1 330.4 206.0 247.2 288.4 329.6 205.5 246.6 287.7 328.8 205.0 246.0 287.0 328.0 204.5 245.4 286.3 327.2 204.0 244.8 285.6 326.4 203.5 244.2 284.9 325.6 203.0 243 6 284.2 324.8 202.5 243.0 283.5 324.0 202.0 242.4 282.8 323.2 201.5 241.8 282.1 322.4 201.0 241 2 281.4 321.6 200.5 240.6 280.7 320.8 200.0 240.0 280.0 320.0 199.5 239.4 279.3 319.2 199.0 238.8 278.6 318.4 198.5 238.2 277.9 317.6 198.0 237.0 277.2 316.8 197.5 237.0 276.5 316 ; LOGARITHMS OP tttJMBERS. 131 No. 110 L. 041.] [No 119 L. 078. N. 1 2 S 4 • 3362 7275 1153 4996 8805 6 3755 7664 1538 5378 9185 7 8 9 Diff. 110 1 2 041393 5323 9218 1787 5714 9606 2182 6105 9993 2576 6495 2969 6885 4148 8053 4540 8442 4932 8830 393 390 0380 4230 8046 0766 4613 8426 1924 5760 9563 2309 6142 9942 2694 6524 386 383 3 4 053078 6905 3463 7286 3846 7666 0320 4083 7815 379 376 373 5 6 7 060698 4458 8186 1075 4832 8557 1452 5206 8928 1829 5580 9298 2206 5953 9668 2582 6326 2958 6699 3333 7071 3709 7443 0038 3718 7368 0407 4085 7731 0776 4451 8094 1145 4816 8457 1514 5182 8819 370 366 363 8 9 071882 5547 2250 5912 2617 6276 2985 6640 3352 7004 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 395 39.5 79.0 118.5 158.0 197.5 237.0 276.5 316.0 355.5 394 39.4 78.8 118.2 157.6 197.0 236.4 275.8 315.2 j 354.6 393 39.3 78.6 117.9 157.2 196.5 235.8 275.1 314.4 353.7 392 39.2 78.4 117.6 156.8 196.0 235.2 274.4 313.6 352.8 391 39.1 78.2 117.3 156.4 195.5 234.6 273.7 312.8 351.9 390 39 78.0 117.0 156.0 195.0 234.0 273.0 312.0 351.0 389 38.9 77.8 116.7 155.6 194.5 233.4 272.3 311.2 350.1 388 38.8 77.6 116.4 155.2 194.0 232.8 271.6 310.4 349.2 387 38.7 77.4 116.1 154.8 193.5 232.2 270.9 309.6 348.3 386 38.6 77.2 115.8 154.4 193.0 231.6 270.2 308.8 347.4 385 38.5 77.0 115.5 154.0 192.5 231.0 269.5 308.0 346.5 384 38.4 76.8 115.2 153.6 192.0 230.4 268.8 307.2 345.6 383 38.3 76.6 114.9 153.2 191.5 229.8 268.1 306.4 344.7 382 38.2 76.4. 114.6 152.8 191.0 229.2 267.4 305.6 343.8 381 38.1 76.2 114.3 152.4 190.5 228.6 266.7 304.8 342.9 380 38.0 76.0 114.0 152.0 190.0 228.0 266.0 304.0 342.0 379 37.9 75.8 113.7 151.6 189.5 227.4 265.3 303.2 341.1 378 37.8 75.6 113.4 151.2 189.0 226.8 264.6 302.4 340.2 377 37.7 75.4 113.1 150.8 188.5 226.2 263.9 301.6 339.3 376 37.6 75.2 112.8 150.4 188.0 225.6 263.2 300.8 338.4 375 37.5 75.0 112.5 150.0 187.5 225.0 262.5 300.0 337.5 374 37.4 74.8 112.2 149.6 187.0 224.4 261.8 299.2 336.6 373 37.3 74.6 111.9 149.2 186.5 223.8 261.1 298.4 335.7 372 37.2 74.4 111.6 148.8 186.0 223.2 260.4 297.6 334.8 371 37.1 74.2 111.3 148.4 185.5 222.6 259.7 296.8 333.9 370 37.0 74.0 111.0 148.0 185.0 222.0 259.0 296.0 333.0 369 36.9 73.8 110.7 147.6 184.5 221.4 258.3 295.2 332.1 368 36.8 73.6 110.4 147.2 184.0 220.8 257.6 294.4 331.2 367 36.7 73.4 110.1 146.8 183.5 220.2 256.9 293.6 330.3 366 36.6 73.2 109.8 146.4 183.0 219.6 256.2 292.8 329.4 365 36.5 73.0 109.5 146.0 182.5 219.0 255.7 292.0 328.5 364 36.4 72.8 109.2 145.6 182.0 218.4 254.8 291.2 327.6 363 36.3 72.6 108.9 145.2 181.5 217.8 254.1 290.4 326.7 362 36.2 72.4 108.6 144.8 181.0 217.2 253.4 289.6 325.8 361 36.1 72.2 108.3 144.4 180.5 216.6 252.7 288.8 324.9 360 36.0 72.0 108.0 144.0 180.0 216.0 252.0 288.0 324.0 359 35.9 71.8 107.7 143.6 179.5 215.4 251.3 287.2 323.1 a58 35.8 71.6 107.4 143.2 179.0 214.8 250.6 286.4 322.2 357 35.7 71.4 107.1 142.8 178.5 214.2 249.9 285.6 321.3 356 35 .'6 71.2 106.8 142.4 178.0 213.6 249.2 284.8 320.4 132 LOGARITHMS OF KUMBERS. No. 120 L. 079.] [N 0. 134 L. 130. N. 1 1 2 3 i 4 II 6 i 6 7 8 9 Diff. 079181 1 9543 9904 120 0266 3861 7426 0626 4219 7781 1 0987 4576 8136 1667 5169 8644 1347 4934 8490 1707 5291 8845 2067 5647 9198 2426 6004 9552 360 1 2 3 4 5 082785 6360 9905 093422 6910 3144 6716 3503 7071 357 355 0258 3772 7257 0611 4122 7604 0963 4471 7951 1315 4820 8298 2018 5518 8990 2370 5866 9335 2721 6215 9681 3071 6562 0026 3462 6871 352 349 346 343 341 6 8 9 130 1 100371 3804 7210 0715 4146 7549 1059 4487 7888 1403 4828 8227 1747 5169 8565 2091 5510 8903 2434 5851 9241 2777 6191 9579 3119 6531 9916 0253 3609 6940 338 335 333 110590 3943 7271 0926 4277 7603 1263 4611 7934 1599 4944 8265 1934 5278 8595 2270 5611 8926 2605 5943 9256 2940 6276 9586 3275 6608 9915 0245 3525 6781 330 328 325 2 3 4 120574 3852 7105 13 0903 4178 7429 1231 4504 7753 1560 4830 8076 1888 5156 8399 2216 5481 8722 2544 5806 9045 2871 6131 9368 3198 6456 9690 0012 323 Proportional Parts. Diff. 1 2 3 4 5 6 7 355 35.5 71.0 106.5 142.0 177.5 213.0 248.5 354 35.4 70.8 106.2 141.6 177.0 212.4 247.8 353 35.3 70.6 105.9 141.2 176.5 211.8 247.1 .352 35.2 70.4 105.6 140.8 176.0 211.2 246.4 351 35.1 70.2 105.3 140.4 175.5 210.6 245.7 350 35.0 70.0 105.0 140.0 175.0 210.0 245.0 349 34.9 69.8 104.7 139.6 174.5 209.4 244.3 848 34.8 69.6 104.4 139.2 174.0 208.8 243.6 847 34.7 69.4 104.1 138.8 173.5 208.2 242.9 346 34.6 69.2 103.8 138.4 173.0 207.6 242.2 345 34.5 69.0 103.5 138.0 172.5 207.0 241.5 344 34.4 68.8 103.2 137.6 172.0 206.4 240.8 343 34.3 68.6 102.9 137.2 171.5 205.8 240.1 342 34.2 68.4 102.6 136.8 171.0 205.2 239.4 341 34.1 68.2 102.3 136.4 170.5 204.6 238.7 340 34.0 68.0 102.0 136.0 170.0 204.0 238.0 339 33.9 67.8 101.7 135.6 169.5 203.4 237.3 338 33.8 67.6 101.4 135.2 169.0 202.8 236.6 337 33.7 67.4 101.1 134.8 168.5 202.2 235.9 336 33.6 67.2 100.8 134.4 168.0 201.6 235.2 335 33.5 67.0 100.5 134.0 167.5 201.0 234.5 334 33.4 66.8 100.2 133.6 167.0 200.4 233.8 333 33.3 66.6 99.9 133.2 166.5 199.8 233.1 332 33.2 66.4 99.6 132.8 166.0 199.2 232.4 331 33.1 66.2 99.3 132.4 165.5 198.6 231.7 330 33.0 66.0 99.0 132.0 165.0 198.0 231.0 829 32.9 65.8 98.7 131.6 164.5 197.4 230.3 828 32 8 65.6 98.4 131.2 164.0 196.8 229.6 327 32.7 65.4 98.1 130.8 163.5 196.2 228.9 326 32.6 65.2 97.8 130.4 163.0 195.6 228.2 325 32.5 65.0 97.5 130.0 162.5 195.0 227.5 824 32.4 64.8 97.2 129.6 162.0 194.4 226.8 323 32.3 64.6 96.9 129.2 161.5 193.8 226.1 322 32.2 64.4 96.6 128.8 161.0 193.2 225.4 284.0 283.2 282.4 281.6 278.4 277.6 276.8 276.0 310.5 275.2 309.6 274.4 308.7 273.6 307.8 272.8 306.9 272.0 306.0 271.2 305.1 270.4 304.2 269.6 303.3 268.8 302.4 268.0 301.5 267.2 300.6 266.4 299.7 265.6 298.8 264.8 297.9 264.0 297.0 263.2 296.1 262.4 295.2 261.6 294.3 260.8 293.4 260.0 292.5 259.2 291.6 25S.4 290.7 257.6 289.8 LOGARITHMS OP NUMBERS. 133 No. 135 L. 130.] [No. 149 L. 175. N. 1 2 3 4 5 ! 1 6 7 8 9 Diff. 135 i 130334 0655 0977 1298 1619 1939 ! 2260 2580 2900 3219 321 6 3539 3858 ! 4177 4496 4814 5133 5451 5769 6086 6403 318 7 ! 6721 8 | 9879 7037 | 7354 7671 7987 ! 8303 ! 8618 8934 9249 9564 316 0194 0508 ftftaa 1136 1450 1763 2076 2389 2702 314 9 143015 3327 3639 3951 4263 4574 4885 5196 i 5507 5818 311 140 6128 6438 6748 7058 7367 ' 7676 : 7985 8294 8603 8911 309 1 | 9219 9527 9835 0142 3205 0449 ' 0756 | 3815 1063 4120 1370 4424 1676 1982 307 2 152288 2594 2900 3510 4728 5032 305 3 5336 5640 5943 1 6246 6549 6852 7154 7457 7759 8061 303 4 8362 8664 8965 9266 9567 9868 0168 3161 0469 3460 0769 3758 1068 301 5 \ 161368 1667 1967 2266 2564 2863 4055 299 I 6 1 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 I 7 731? 7613 7908 8203 8497 8792 1726 1 9086 2019 9380 9674 2603 9968 2895 295 293 8 170262 0555 0848 1141 1434 2311 9 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 Proportio nal Pa aTS. Diff. 1 2 3 4 5 6 7 8 9 321 32.1 64.2 96.3 128.4 160.5 192.6 224.7 256.8 288.9 320 32.0 64.0 96.0 128.0 160.0 192.0 224.0 256.0 288.0 319 31.9 63.8 95.7 127.6 159.5 191.4 223.3 255.2 28Z.1 286.2 318 31.8 63.6 95.4 127.2 159.0 190.8 222.6 254.4 317 31.7 63.4 95.1 126.8 158.5 190.2 221.9 253.6 285.3 316 31.6 63.2 94.8 126.4 158.0 189.6 221.2 252.8 284.4 315 31.5 63.0 94.5 126.0 157.5 189.0 220.5 252.0 283.5 314 31.4 62.8 94.2 125.6 157.0 188.4 219.8 251.2 282.6 313 31.3 62.6 93.9 125.2 156.5 187.8 219.1 250.4 281.7 312 31.2 62.4 93.6 124.8 156.0 187.2 218.4 249.6 280.8 311 31.X 62.2 93.3 124.4 155.5 186.6 217.7 248.8 279.9 310 31.0 62.0 93.0 124.0 155.0 186.0 217.0 248.0 279.0 309 30.9 61.8 92.7 123.6 154.5 185.4 216.3 247.2 278.1 308 30.8 61.6 92.4 123.2 154.0 184.8 215.6 246.4 277.2 307 30.7 61.4 92.1 122.8 153.5 184.2 214.9 245.6 276.3 306 30.6 61.2 91.8 122.4 153.0 183.6 214.2 244.8 275.4 1 305 30.5 61.0 91.5 122.0 152.5 183.0 213.5 244.0 274,5 304 30.4 60.8 91.2 121.6 152.0 182.4 212.8 243.2 273.6 303 30.3 60.6 90.9 121.2 151.5 181.8 212.1 242.4 272.7 302 30.2 60.4 90.6 120.8 151.0 181.2 211.4 241.6 271.8 301 30.1 60.2 90.3 120.4 150.5 180.6 210.7 240.8 270.9 300 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 299 29.9 59.8 89.7 119.6 149.5 179.4 209.3 239.2 269.1 298 29.8 59.6 89.4 119.2 149.0 178.8 208.6 238.4 268.2 297 29.7 59.4 89.1 118.8 148.5 178.2 207.9 237.6 267.3 296 29.6 59.2 88.8 118.4 148.0 177.6 207.2 236.8 266.4 295 29.5 59.0 88.5 118.0 147.5 177.0 206.5 236.0 265.5 294 29.4 58.8 88.2 117.6 147.0 176.4 205.8 235.2 264.6 293 29.3 58.6 87.9 117.2 146.5 175.8 205.1 ! 234.4 263.7 292 29.2 58.4 87.6 116.8 146.0 175.2 204.4 233.6 262.8 1 291 29.1 58.2 87.3 116.4 145.5 174.6 203.7 232.8 261.9 290 29.0 58.0 87.0 116.0 145.0 174.0 203.0 232.0 261.0 2S9 28.9 57.8 86.7 115.6 144.5 173.4 202.3 231.2 260.1 . 288 28.8 57.6 86.4 115.2 144.0 172.8 201.6 230.4 259.2 ! 287 28.7 57.4 86.1 114.8 143.5 172.2 200.9 229.6 258.3 i 286 28.6 57.2 85.8 114.4 143.0 171.6 200.2 228.8 257.4 134 LOGARITHMS OF KUMBKRS. No. 150 L. 176.1 [N o. 169 L. 230. N. 12 3 4 5 6 7 8 9 Diff. 150 176091 6381 l 6070 : 6959 7248 7536 7825 8113 8401 8689 289 1 8977 9264 i 9552 i 9839 ! A1 s>« 0413 3270 0699 3555 0986 3839 1272 4123 1558 4407 287 285 2 181844 2129 2415 ! 2700 ; 2985 3 4691 4975 5259 J 5542 j 5825 6108 6391 6674 6956 7239 283 4 5 7521 7803 HlMiA fiXKfi Mftl7 8928 9209 9490 9771 0051 281 279 190332 0612 0892 ! 1171 ! 1451 j 1730 2010 2289 2567 6 3125 3403 3681 ! 3959 4237 4514 4792 5069 5346 i 5623 878 7 5900 6176 6453 6729 i 7005 7281 7556 7832 8107 i 8382 276 8 8657 8932 9206 9481 ] 9755 1 \ 0029 | 2761 0303 3033 0577 3305 0850 1124 3577 3848 274. 272 9 201397 1670 1943 2216 ■ 2488 160 4120 4391 4663 4934 i 5204 5475 5746 6016 6286 | 6556 ■271 1 6826 7096 7365 7634 j 7904 8173 8441 8710 8979 ! 9247 269 2 9515 9783 ' 0051 2720 0319 j 0586 2986 , 3252 ! 0853 3518 1121 37'83 1388 4049 1654 ! 1921 4314 ' 4579 267 266 3 212188 2454 4 4844 5109 5373 5638 5902 6166 6430 6694 6957 j 7221 264 5 6 7484 7747 8010 8273 8536 8798 1414 9060 9323 9585 j 9846 2196 ! 2456 262 220108 0370 0631 0892 1153 1675 1936 261 7 2716 2976 3236 3496 3755 4015 4274 4533 4792 ■ 5051 259 8 5309 5568 5826 6084 6342 6600 6858 7115 7372 1 7630 258 9 7887 8144 8400 8657 8913 9170 9426 9682 9938 ! 23 i 0193 256 Proportional Parts. Dift. 1 2 3 4 5 6 7 8 9 285 28.5 57.0 85.5 114.0 142.5 171.0 199.5 228.0 256.5 284 28.4 56.8 85.2 113.6 142.0 170.4 198.8 227.2 255.6 283 28.3 56.6 84.9 113.2 141.5 169.8 198.1 226.4 254.7 282 28.2 56.4 84.6 112.8 141.0 169.2 197.4 225.6 253.8 281 ! 28.1 56.2 84.3 112 4 140.5 168.6 196.7 224.8 252.9 280 28.0 56.0 84.0 112.0 140.0 168.0 196.0 224.0 252.0 279 27.9 55.8 83.7 111.6 139.5 167.4 195.3 223.2 251.1 278 27.8 55.6 83.4 111.2 139.0 166.8 194.6 ! 222.4 250.2 277 27.7 55.4 83.1 110.8 138.5 166.2 193.9 221.6 249.3 276 27.6 55.2 82.8 110.4 138.0 165.6 193.2 220.8 248.4 275 27.5 55.0 82.5 110.0 137.5 165.0 192.5 220.0 247.5 274 27.4 54.8 82.2 109.6 137.0 164.4 191.8 219.2 246.6 273 27.3 54.6 81.9 109.2 136.5 163.8 191.1 218.4 245.7 272 27.2 54.4 81.6 108.8 136.0 163.2 190.4 217.6 244.8 271 27.1 54.2 81.3 108.4 135.5 162.6 189.7 216.8 243.9 270 27.0 54.0 81.0 108.0 135.0 162.0 189.0 216.0 243.0 269 26.9 53.8 80.7 107.6 134.5 161.4 188.3 215.2 242.1 268 26.8 53.6 80.4 107.2 134.0 160.8 187.6 214.4 241.2 267 26.7 53.4 80.1 106.8 133.5 i 160.2 186.9 213.6 240.3 266 26.6 53.2 79.8 106.4 133.0 159.6 186.2 212.8 239.4 265 26.5 53.0 79.5 106.0 132.5 159.0 185.5 212.0 238.5 264 26.4 52.8 79.2 105.6 132.0 158.4 184.8 I 211.2 237.6 263 26.3 52.6 78.9 105.2 131.5 157.8 184.1 : 210.4 236.7 262 26.2 52.4 78.6 104.8 131.0 157.2 183.4 ! 209.6 235.8 261 26.1 52.2 78.3 104.4 130.5 156.6 182.7 j 208.8 234.9 260 26.0 52.0 78.0 104.0 130.0 156.0 182.0 : 208.0 234.0 259 25.9 51.8 77.7 103.6 129.5 155.4 181.3 207.2 233.1 258 25.8 51.6 77.4 103.2 129.0 1 154.8 180.6 i 206.4 232.2 257 25.7 51.4 77.1 102.8 128.5 154.2 179.9 205.6 231.3 256 25.6 51.2 76.8 102.4 128.0 1 153.6 179.2 j 204.8 230.4 255 25.5 51.0 76.5 102.0 1&7.5 153.0 178.5 j 204.0 229.5 LOGARITHMS OP NUMBERS. 135 No. 170 L. 230.] [N o. 189 L. 278. N. 1 2 3 4 6 6 7 8 9 Diff. 170 1 3 230449 2996 5528 8046 0704 3250 5781 8297 0960 3504 6033 8548 1215 3757 6285 8799 1470 4011 6537 9049 1724 4264 6789 9299 1979 4517 7041 9550 2234 4770 7292 9800 24S8 5023 7544 2742 5276 7795 255 253 252 0050 2541 5019 7482 9932 0300 2790 5266 7728 250 249 248 246 4 5 C 240549 3038 5513 7973 0799 3286 5759 8219 1048 3534 6006 8464 1297 3782 6252 8709 1546 4030 6499 8954 1795 4277 6745 9198 2044 4525 6991 9443 2293 4772 7237 9687 ' 0176 2610 5031 7439 9833 245 243 242 241 339 8 9 ISO 1 250420 2853 5273 7679 0664 3096 5514 7918 0908 3338 5755 8158 1151 3580 5996 8398 1395 3822 6237 8637 1638 4064 6477 8877 1881 4306 6718 9116 2125 4548 6958 9355 2368 4790 7198 9594 2 3 4 5 6 260071 2451 4818 7172 9513 0310 2688 5054 7406 9746 0548 2925 5290 7641 9980 0787 3162 5525 7875 1025 3399 5761 8110 1263 3636 5996 8344 1501 3873 6232 8578 1739 4109 6467 8812 1976 -1346 6702 9046 2214 4582 6937 9279 238 237 235 234 0213 2538 4850 7151 0446 2770 ! 5081 7380 0679 3001 5311 7609 0912 3233 5542 7838 1144 3464 5772 8067 1377 3696 6002 8296 1609 3927 6232 8525 233 232 230 229 8 9 271842 4158 6462 2074 4389 6692 2306 4620 6921 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 255 25.5 51.0 76.5 102.0 127.5 153.0 175.5 204.0 229.5 254 25.4 50.8 76.2 101.6 127.0 152.4 177.8 203.2 228.6 253 25.3 50.6 75.9 101.2 126.5 151.8 177.1 202.4 227.7 252 25.2 50.4 75.6 100.8 126.0 151.2 176.4 201.6 226.8 251 25.1 50.2 75.3 100.4 125.5 150.6 175.7 200.8 225.9 250 25 50.0 75.0 100.0 125.0 150.0 175.0 200.0 225.0 249 24.9 49.8 74.7 99.6 124.5 149.4 174.3 199.2 224.1 248 24.8 49.6 74.4 99.2 124.0 148.8 173.6 198.4 223.2 247 24.7 49.4 74.1 98.8 123.5 148.2 172.9 197.6 222.3 246 24.6 49.2 73.8 98.4 123.0 147.6 172.2 196.8 221.4 245 24.5 49.0 73.5 98.0 122.5 147.0 171.5 196.0 220.5 244 24.4 48.8 73.2 97.6 122.0 146.4 170.8 195.2 219.6 243 24.3 48.6 72.9 97.2 121.5 145.8 170.1 194.4 218.7 242 24.2 48.4 72.6 96.8 121.0 145.2 169.4 193.6 217.8 241 24.1 48.2 72.3 96.4 120.5 144.6 168.7 192.8 216.9 240 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 239 23.9 47.8 71.7 95.6 119.5 143.4 167.3 191.2 215.1 238 23.8 47.6 71.4 95.2 119.0 142.8 166.6 190.4 214.2 237 23.7 47.4 71.1 94.8 118.5 142.2 165.9 189.6 213.3 236 23.6 47.2 ~ 70.8 94.4 118.0 141.6 165.2 188.8 212.4 235 23.5 47.0 70.5 94.0 117.5 141.0 164.5 188.0 211.5 234 23.4 46.8 70.2 93.6 117.0 140.4 163.8 187.2 210.6 233 23.3 46.6 69.9 93.2 116.5 139.8 163.1 186.4 209.7 232 23.2 46.4 69.6 92.8 116.0 139.2 162.4 185.6 208.8 231 23.1 46.2 69.3 ' 92.4 115.5 138.6 161.7 1&4.8 207.9 230 23.0 46.0 69 92.0 115.0 138.0 161.0 184.0 207.0 229 22.9 45.8 68.7 91.6 114.5 137.4 160.3 ias.2 206.1 228 22.8 45.6 68.4 91.2 114.0 136.8 159.6 182.4 205.2 227 22.7 45.4 68.1 90.8 113.5 136.2 158.9 181.6 204.3 226 22.6 45.2 67.8 90.4 113.0 135.6 158 2 180.8 203.4 136 LOGABITHMS OF HUMBEKS. No. 190 L. 278.] [No. 214 L. 332 iu N. 1 2 3 4 5 6 7 8 9 Diff. Is, 190 278754 8982 9211 9439 9667 9895 ' fi 0123 2396 0351 2622 0578 2849 0806 3075 228 . 227 1 281033 1261 1488 1715 1942 2169 t 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 8 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 4 5 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 222 .'0 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 1 6 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 221 :i 7 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 •I 9 8853 9071 9289 9507 9725 9943 0161 2331 0378 2547 0595 2764 0813 2980 218 217 200 301030 1247 1464 1681 1898 2114 i 1 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 2 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 :l 8 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 4 9630 9843 :)' 0056 2177 0268 2389 0481 2600 0693 2812 0906 3023 1118 3234 1330 3445 212 211 5 311754 1966 3656 I fi 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 \ 8 9 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 207 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 fi 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 1 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 : 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 SI 8787 8991 9194 9398 9601 9805 0008 2034 0211 2236 203 202 4 330414 0617 0819 1022 1225 1427 1630 1832 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 Dil 225 22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0 202. f ;i 224 22.4 44.8 67.2 89 fi 112.0 134.4 156.8 179.2 201J 223 22.3 44.6 66.9 89 2 111.5 133.8 156.1 178.4 200." 222 22.2 44.4 66.6 88 8 111.0 133.2 155.4 177.6 199.* 221 22.1 44.2 66.3 88 4 110.5 132.6 154.7 176.8 198. 1 220 22.0 44.0 66.0 88 110.0 132.0 154.0 176.0 198.1 219 21.9 43.8 65.7 87 6 109.5 131.4 153.3 175.2 197.1 218 21.8 43.6 65.4 87 2 109.0 130.8 152.6 174.4 196.J i. 217 21.7 43.4 65.1 86 8 108.5 130.2 151.9 173.6 195.,' 216 21.6 43.2 64.8 HIS 4 108.0 129.6 151.2 172.8 194 ^ I 1 ;, 215 21.5 43.0 64.5 86 107.5 129.0 150.5 172.0 193.f 1 214 21.4 42.8 64.2 85 6 107.0 128.4 149.8 171.2 193.f 1 213 21.3 42.6 63.9 85 2 106.5 127.8 149.1 170.4 191. 1 1: 212 21.2 42.4 63.6 84 8 106.0 127.2 148.4 169.6 190.* II 211 21.1 42.2 63.3 84 4 105.5 126.6 147.7 168.8 189,t \< 210 21.0 42.0 63.0 84 105.0 126.0 147.0 168.0 189.1 li. 209 20.9 41.8 62.7 83 6 104.5 125.4 146.3 167.2 188.1 1 208 20.8 41.6 62.4 83 2 104.0 124.8 145.6 166 4 187.2 11 207 20.7 41.4 62.1 82 8 103.5 124.2 144.9 165.6 1864 1: 206 20.6 41.2 61.8 82 4 103.0 123.6 144.2 164.8 185.4 l;>: 205 20.5 4d.O C1.5 82 102.5 123 143.5 164.0 184.5 llv 204 20.4 40.8 61.2 81 H 102.0 122.4 142.8 163.2 183. b 1M 203 20.3 40.6 60.9 81 2 101.5 121.8 142.1 162.4 182.7 181.8 |:~ 202 20.2 40.4 60.6 ,0 8 101.0 121.2 141.4 161.6 1,: LOGARITHMS OF NUMBERS. m fro. 215 L. 332.] [No. 239 L. 380. N. 1 2 3 4 5 6 7 8 9 Diff. 315 332438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 6 4454 4055 4856 50&7 5257 5458 5658 5859 6059 6260 201 r 6460 6600 6860 7060 7260 7459 7659 7858 8058 8257 200 8 8456 8656 8855 9054 9253 9451 9650 9849 0047 2028 0246 2225 199 198 9 340444 0642 0841 1039 1237 1435 1632 1830 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 197 1 4392 4589 4785 4981 5178 5374 5570 5766 5962 6157 196 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 | 3 4 8305 8500 8694 8889 9083 9278 1216 9472 1410 9666 9860 1796 194 193 0054 1989 350248 0442 0636 0829 1023 1603 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 192 7 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 9 9835 0025 1917 0215 2105 0404 2294 0593 2482 0783 2671 0972 2859 1161 3048 1350 3236 1539 3424 189 188 230 361728 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 0143 1991 0328 2175 0513 2360 0698 2544 0883 2728 185 184 5 371068 1253 1437 1622 1806 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 38 0030 181 Proportio NAL Pi RTS. Diff. 1 2 3 4 5 6 7 8 9 ! oqo 20.2 40.4 60.6 80.8 101.0 121.2 141.4 161.6 181.8 201 20.1 40.2 60.3 80.4 100.5 120.6 140.7 160.8 180.9 200 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 199 19.9 39.8 59.7 79.6 99.5 119.4 139.3 159.2 179.1 198 19.8 39.6 59.4 79.2 99.0 118.8 138.6 158.4 178.2 197 19.7 S9.4 59.1 78.8 98.5 118.2 137.9 157.6 177.3 196 19.6 39.2 58.8 78.4 98.0 117.6 137.2 156.8 176.4 195 19.5 39.0 58.5 78.0 97.5 117.0 136.5 156.0 175.5 194 19.4 38.8 58.2 77.6 97.0 116.4 135.8 155.2 174.6 193 19.3 38.6 57.9 77.2 96.5 115.8 135.1 154.4 173.7 192 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172.8 191 19.1 38.2 57.3 76.4 95.5 114.6 133.7 152.8 171.9 190 19.0 38.0 57.0 76.0 95.0 114.0 133.0 152.0 171.0 189 1 18.9 37.8 56.7 75.6 94.5 113.4 132.3 151.2 170.1 188 18.8 37.6 56.4 75.2 94.0 112.8 131.6 150.4 169.2 187 is. r 37 4 56.1 74.8 93.5 112.2 130.9 149.6 168.3 186 18.6 37.3 55.8 74.4 93.0 111.6 130.2 148.8 167.4 185 18.5 37.0 55.5 74.0 92.5 111.0 129.5 148.0 166.5 184 18.4 36.8 55.2 73.6 92.0 110.4 128.8 147.2 165.6 183 18.3 36.6 54.9 73.2 91.5 109.8 128.1 146.4 164.7 182 18.2 36.4 54.6 72.8 91.0 109.2 127.4 145.6 163.8 181 18.1 36.2 54.3 72.4 90.5 108.6 126.7 144.8 162.9 180 18.0 36.0 54.0 72.0 90.0 108.0 126.0 144.0 162.0 179 17.9 35.8 53.7 71.6 89.5 107.4 125.3 143.2 161.1 138 LOGAKITHMS OP J5fuMBEft& r No. 240 L. 380.] ^ [No. 269 L. 431. :; N. 1 2 3 4 5 6 J 8 9 Diff. i 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 181 u 1 2017 2:97 2377 2557 2737 2917 3097 3277 3456 3636 180 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 3 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 ! 4 7390 7568 7746 7924 8101 8279 8456 8634 8811 8989 178 ; 5 9166 9343 9520 9698 9875 0051 1817 0228 1993 0405 2169 0582 2345 0759 2521 177 176 6 390935 1112 1288 1464 1641 7 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 8 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 7940 8114 8287 8461 8634 8808 8981 9154 9328 9501 173 i 1 9674 9847 0020 1745 0192 1917 0365 2089 0538 2261 0711 2433 0883 2605 1056 2777 1228 2949 2 401401 1573 172 3 3121 3292 3464 3635 .3807 3978 4149 4320 4492 4663 171 4 4834 5005 5176 5346 5517 5688 f>858 6029 6199 6370 171 5 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 i 6 8240 8410 8579 , 8749 8918 9087 9257 9426 9595 9764 169 : 9933 0102 1788 0609 2293 0777 2461 0946 2629 1114 2796 1283 2964 1451 3132 8 411620 1956 2124 168 9 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 167 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 167 1 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 2 3 8301 9956 8467 0121 8633 0286 8798 8964 0616 91*29 9295 9460 9625 9791 165 : 165 0451 0781 0945 1110 1275 1439 4 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 5371 5534 5697 5S60 6023 6186 6349 163 7 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 9 9752 43 9914 0075 | 0236 0398 1 0559 0720 0881 1042 1203 161 Proportional Parts. I Diff 1 2 3 4 5 6 7 8 9 178 17.8 35.6 53.4 71.2 89.0 106.8 124.6 142.4 160.2 177 17.7 35.4 53.1 70.8 88.5 106.2 123.9 141.6 159.3 176 17.6 35.2 52.8 70.4 88.0 105.6 123.2 140.8 158.4 175 17.5 35.0 52.5 70.0 87.5 105.0 122.5 140.0 157.5 174 17.4 34.8 52.2 69.6 87.0 104.4 121.8 139.2 156.6 173 17.3 34.6 51.9 69.2 86.5 103.8 121.1 138.4 155.7: 172 17.2 34.4 51.6 68.8 86.0 103.2 120.4 137.6 154.8 171 17.1 34.2 51.3 68.4 85.5 102.6 119.7 136.8 153.9 170 17.0 34.0 51.0 68.0 85.0 102.0 119.0 136.0 153.0 169 16.9 33.8 50.7 67.6 84.5 101.4 118.3 135.2 152.1 168 16.8 33.6 50.4 67.2 84.0 100.8 117.6 134.4 151.2 167 16.7 33.4 50.1 66.8 83.5 100.2 116.9 133.6 150.3 166 16.6 33.2 49.8 66.4 83.0 99.6 116.2 132.8 149.4 165 16.5 33.0 49.5 66.0 82.5 99.0 115.5 132.0 148.5 164 16.4 32.8 49.2 65.6 82.0 98.4 114.8 131.2 147.6 163 16.3 32.6 48.9 65.2 81.5 97.8 114.1 130.4 146.7 162 16.2 32.4 48.5 64.8 81.0 97.2 113.4 129.6 145.8 161 16.1 32.2 48.3 64.4 80.5 96.6 112.7 128.8 144.9 LOGARITHMS OF NUMBERS. 139 No. 270 L. 431.] #> [No. 299 L. 476. 6163 7751 4045 5604 7158 8706 4845 6366 7882 471292 2756 4216 5671 1525 1685 3130 I 3290 47" 6481 8067 9491 1066 2637 4201 5760 7313 ! I ! 1846 3450 5048 6640 8226 4357 5915 0403 1940 3471 4997 6518 8033 9543 1048 2548 4042 5532 7016 8495 5150 6670 8184 0116 1585 9170 4:340 5829 7312 8790 4653 6107 2007 3610 5207 6799 8384 .9964 1538 3106 4669 6226 7778 9324 0865 2400 5454 6973 8487 1499 2997 4490 5977 7460 0410 1878 3341 4799 6252 2167 I 3770 I 5367 i 6957 8542 ! 0122 1695 3263 4825 947t 1018 2553 4082 5606 7125 0146 1649 3146 4639 6126 1852 3419 4981 6537 1172 2706 4235 5758 7276 1799 3296 4788 6274 7756 0557 2025 3487 4944 6397 0704 2171 7 8 9 2488 2649 2809 4090 4219 4409 5685 5844 6004 7275 7433 7592 8859 0437 9017 9175 0594 0752 2009 2166 2323 3576 3732 3889 5137 5293 5449 6692 6848 7003 8242 8397 8552 9787 9941 0095 1633 1326 1479 2859 3012 3165 4387 4540 4692 5910 6062 6214 7428 7579 '7731 8940 9091 9242 0447 0597 0748 1948 2098 2248 3445 3594 3744 4936 5085 52:34 6423 6571 6719 7904 8052 8200 9380 9527 9675 0851 0998 1145 2318 2464 2610 3779 3925 4071 5235 5381 5526 6687 6832 6976 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 161 16.1 32.2 48.3 64.4 80.5 96.6 112.7 128.8 144.9 160 16.0 32.0 48.0 64.0 80.0 96.0 112.0 128.0 144.0 159 15.9 31.8 47.7 63.6 79.5 95.4 111.3 127.2 143.1 158 15.8 31.6 47.4 63.2 79.0 94.8 110.6 126.4 142.2 157 15.7 31.4 47.1 62.8 78.5 94.2 109.9 125.6 141.3 156 15.6 31.2 46.8 62.4 78.0 93.6 109.2 124.8 140.4 155 15.5 31.0 46.5 62.0 77.5 93.0 108.5 124.0 139.5 154 15.4 30.8 46.2 61.6 77.0 92.4 107.8 123.2 138.6 153 15.3 30.6 45.9 61.2 76.5 91.8 107.1 122.4 137.7 152 15.2 30.4 45.6 60.8 76.0 91.2 106.4 121.6 136.8 151 15.1 30.2 45.3 60.4 75.5 90.6 105.7 120.8 135.9 150 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 149 14.9 29.8 44.7 59.6 74.5 89.4 104.3 119.2 134.1 148 14.8 29.6 44.4 59.2 74.0 88.8 103.6 118.4 133.2 147 14.7 29.4 44.1 58.8 73.5 88.2 102.9 117.6 132.3 146 14.6 29.2 43.8 58.4 73.0 87.6 102.2 116.8 131.4 145 14.5 29.0 43.5 58.0 72.5 87.0 101.5 116.0 130.5 144 14.4 28.8 43.2 57.6 72.0 86.4 100.8 115.2 129.6 143 14.3 28.6 42.9 57.2 71.5 85.8 100.1 114.4 128.7 142 14.2 28.4 42.6 56.8 71.0 85.2 99.4 113.6 127.8 141 14.1 28.2 42.3 56.4 70.5 84.6 98.7 112.8 126.9 140 14.0 28.0 42.0 56.0 70.0 84.0 98.0 112.0 126.0 140 LOGARITHMS OF NUMBERS. No. 300 L. 477.] [No. 339 L. 531. 477121 8566 480007 1443 2874 4300 5721 7138 8551 2760 4155 5544 6930 8311 9687 501059 2427 3791 5150 6505 7856 3218 4548 5874 7196 8514 9828 521138 2444 3746 5045 8917 530200 (711 0151 1586 3016 4442 5863 1502 2900 4294 5(583 7068 8448 9824 5286 6640 7991 0679 2017 3351 4681 8646 9959 2575 3876 5174 6469 7759 9015 0294 1729 3159 4585 01 )().■} 7421 8833 0239 1642 3040 4433 5*22 7206 1872 3302 4727 6147 7563 8974 0380 1782 3179 4572 5960 7344 8724 0099 1470 2837 4199 5557 6911 0947 2284 3617 4946 6271 7592 0221 1530 2835 4136 5434 6727 0520 1922 3319 4711 6099 7483 1740 1081 2418 3750 5079 6403 7724 9040 0353 1661 8145 0456 0584 0712 0840 0968 7844 7989 9287 9431 0725 2159 3587 5011 6430 7845 3458 4850 0374 1744 3109 4471 5828 7181 8530 9874 1215 2551 5211 6535 7855 0484 17! 12 8274 9559 8133 8278 8422 9575 9719 3730 5153 6572 1012 2445 3872 5295 6714 8127 9537 4015 5437 6855 0801 2201 77'59 9137 3246 4607 5964 7316 0009 2684 4016 5344 6668 7987 0941 2341 5128 6515 0615 7114 8402 96b'7 0143 1482 2818 4149 5476 6800 8119 9434 0745 2053 3356 4656 5951 7243 8531 9815 5267 6653 4878 6234 0277 1616 2951 4282 0876 2183 1096 5014 6370 7721 0411 1750 3084 4415 5741 7064 8382 1007 2314 3616 4915 6210 7501 8788 0072 1351 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 139 13.9 27.8 41.7 55.6 69.5 83.4 97.3 111.2 125.1 138 13.8 27.6 41.4 55.2 69.0 82.8 96.6 110.4 124.2 137 13.7 27.4 41.1 54.8 68.5 82.2 95.9 109.6 123.3 136 13.6 27.2 40.8 54.4 68.0 81.6 95.2 108.8 122.4 135 13.5 27.0 40.5 54.0 67.5 81.0 94.5 108.0 121.5 134 13.4 26.8 40.2 53.6 67.0 80.4 93.8 107.2 120.6 133 13.3 26.6 39.9 53.2 66.5 79.8 93.1 106.4 119.7 132 13.2 26.4 39.6 52.8 66.0 79.2 92.4 105.6 118.8 131 13.1 26.2 89.3 52.4 65.5 78.6 91.7 104.8 117.9 130 13.0 26.0 39.0 52.0 65.0 78.0 91.0 104.0 117.0 129 12.9 25.8 38.7 51.6 64.5 77.4 90.3 103.2 116.1 128 12.8 25.6 38.4 51.2 64.0 76.8 89.6 102.4 115.2 127 12 7 25.4 38.1 50.8 63.5 76.2 88.9 101.6 114.3 LOGARITHMS OF NUMBERS. 141 I No. 340 L. 531.] [No. 379 L. 579. 531479 2754 4026 5294 6558 7819 9076 540329 1579 5307 6543 7775 550228 1450 5094 6303 7507 8709 9907 3481 4666 5848 7026 8202 9374 6341 7492 0351 1572 2790 4004 5215 6423 7627 8829 0026 1221 2412 3600 4784 5966 7144 8319 9491 0660 4147 5303 6457 7607 8754 4280 5547 6811 0473 1694 2911 4126 5336 6544 7748 0146 1340 2531 3718 4903 6084 7262 8436 0776 1942 3104 4263 5419 6572 7722 3 1862 3136 4407 5674 6937 8197 9452 0595 1816 3033 4247 5457 6664 7868 9068 0265 1459 2650 3837 5021 6202 7379 8554 9725 2058 3220 4379 5534 6687 1990 2117 3264 3391 4534 4661 5800 5927 7063 j 7189 8322 : 8448 9578 I 9703 0955 2203 3447 2078 3323 4564 j 5802 7036 8267 9494 0717 1938 3155 6785 7988 9188 0385 1578 2769 3955 5139 6320 7497 8671 9842 5925 7159 0840 2060 3276 4489 5699 6905 8108 0504 1698 2887 4074 5257 6437 7614 1010 1126 2174 2291 3336 3452 4494 4610 5650 5765 6802 6917 7951 8086 9097 9212 2245 3518 4787 6053 7315 8574 1080 2327 3571 4812 6049 8512 9739 0962 2181 9428 0624 1817 3006 4192 5376 6555 0076 1243 2407 8181 9326 4914 3519 4731 5940 7146 8349 9548 0743 1936 3125 4311 5494 6673 7849 9023 0193 1359 7147 8295 9441 2500 3772 5041 1206 2425 3640 : 3244 4429 5612 6791 7967 9140 0309 1476 2639 3800 4957 6111 7262 8410 9555 2627 3899 5167 7693 0982 2174 4548 5730 1592 2755 3915 5072 6226 7377 Proportional Parts. Diff. » 2 3 4 5 6 7 8 9 128 12.8 25.6 38.4 51.2 64.0 76.8 89.6 102.4 115.2 127 12.7 25.4 38.1 50.8 63.5 76.2 88.9 101.6 114.3 126 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 125 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 112.5 124 12.4 24.8 37.2 49.6 62.0 74.4 86.8 99.2 111.6 123 12.3 24.6 36.9 49.2 61.5 73.8 86.1 98.4 110.7 122 12.2 24.4 36.6 48.8 61.0 73.2 85.4 97.6 109.8 121 12.1 24.2 36.3 48.4 60.5 72.6 84.7 96.8 108.9 120 12.0 24.0 36.0 48.0 60.0 72.0 84.0 96.0 108.0 119 11.9 23.8 35.7 47.6 59.5 71.4 83.3 95.2 107.1 142 LOGARITHMS OF NUMBERS. No. 380. L. 579.]^ [No. 414 L. 617. N. 1 2 3 4 5 6 7 8 9 Diff. 380 1 579784 9898 0012 1153 , 0126 1267 0241 1381 j 0355 0469 ' 0583 | 0697 0811 1950 114 580925 1039 1495 1 1608 '< 1722 I 1836 2 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 3 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 4 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 5 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 fi 6587 6700 6812 6925 7037 ! 7149 7262 7374 7486 7599 7 7711 7823 7935 8047 8160 ! 8272 8384 8496 8608 8720 112 8 8832 8944 9056 9167 '9279 9391 9503 9615 9726 9838 9 9950 0284 1399 0842 0061 1176 0173 1287 0396 1510 0619 1 0730 390 591065 i 1621 1732 1843 1955 2066 1 2177 2288 2399 2510 2621 2732 2843 i 2954 3064 3175 111 2 3286 3397 3508 3618 3729 3840 3950 1 4061 4171 4282 3 4393 4503 4614 4724 4834 I 4945 5055 j 5165 5276 5386 4 5496 5606 5717 5827 5937 ! 6047 6157 6267 6377 6487 5 6597 6707 6817 6927 7037 | 7146 7256 j 7366 7476 7586 110 6 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 7 8791 8900 9009 9119 9228 ! 9337 9446 . 9556 9665 9774 8 9883 9992 0101 0210 0319 i 0428 0537 0646 0755 0864 9 600973 1082 1191 1299 1408 1517 1625 1734 1843 1951 400 2060 2169 2277 2386 2494 2603 2711 2819 2928 3036 1 3144 3253 3361 3469 3577 3686 3794 3902 4010 4118 108 2 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 3 , 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 4 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 5 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 6 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 7 9594 9701 9808 9914 0128 1192 0341 1405 0447 1511 8 610660 0767 0873 0979 1086 1298 1617 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 2784 2890 2996 3102 3207 3313 3419 • 3525 3630 3736 1 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 2 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 3 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 4 i 7000 7105 7210 7315 | 7420 7525 7629 7734 7839 7943 Proportional. Parts. Diff . 1 2 3 4 5 6 7 8 9 118 11.8 23.6 35.4 47.2 59.0 70.8 82.6 94.4 106.2 117 11.7 23.4 35.1 46.8 58.5 70.2 81.9 93.6 105.3 116 11.6 23.2 34.8 46.4 58.0 69 6 81.2 92.8 104.4 115 11.5 23.0 34.5 46.0 57.5 69.0 80.5 92.0 103.5 114 11.4 22.8 34.2 45.6 57.0 68.4 79.8 91.2 102.6 113 11.3 22.6 33.9 45.2 56.5 67.8 79.1 90.4 101.7 112 11.2 22.4 33.6 44.8 56.0 67.2 78.4 89.6 100.8 111 11.1 22.2 33.3 44.4 55.5 66.6 77.7 88.8 99.9 110 11.0 22.0 33.0 44.0 55.0 66.0 77.0 88.0 99.0 109 10.9 21.8 32.7 43.6 54.5 65.4 76.3 87.2 98.1 108 10.8 21.6 32.4 43.2 54.0 64.8 75.6 86.4 97.2 107 10.7 21.4 32.1 42.8 53.5 64.2 74.9 85.6 96.3 106 10.6 21.2 31.8 42.4 53.0 63.6 74.2 84.8 95.4 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93.6 LOGARITHMS OF KtJMBERS. 143 No. 415 L. 618.] [No. 459 L. 662 N. 1 2 3 4 5 6 7 8 9 Diff. 415 6 618048 9093 8153 9198 8257 9302 8362 9406 8466 9511 8571 9615 8676 9719 0760 1799 2835 3869 4901 5929 6956 7980 9002 8780 9824 8884 9928 0968 2007 3042 4076 5107 6135 7161 8185 9206 8989 0032 1072 2110 3146 4179 5210 6238 7263 8287 9308 105 r 8 9 420 1 2 4 5 6 620136 1176 2214 3249 4282 5312 6340 7366 8389 9410 0240 1280 2318 3353 4385 5415 6443 7468 8491 9512 0344 "1384 2421 3456 4488 5518 6546 7571 8593 9613 0448 1488 2525 3559 4591 5621 6648 7673 8695 9715 0552 1592 2628 3663 4695 5724 6751 8797 9817 0656 1695 2732 3766 4798 5827 6853 7878 8900 9919 0936 1951 2963 3973 4981 5986 6989 7990 8988 9984 0978 1970 2959 3946 4931 5913 6894 7872 8848 9821 0864 1903 2939 3973 5004 6032 7058 8082 9104 104 103 102 0021 1038 2052 3064 4074 5081 6087 7089 8090 9088 0123 1139 2153 3165 4175 5182 6187 7189 8190 9188 0183 1177 2168 3156 4143 5127 6110 7089 8067 9043 0224 1241 2255 3266 4276 5283 6287 7290 8290 9287 0283 1276 2267 3255 4242 5226 6208 7187 8165 9140 0326 1342 2356 3367 4376 5383 6388 7390 8389 9387 8 9 430 1 2 3 4 5 6 630428 1444 2457 3468 4477 5484 6488 7490 8489 9486 0530 1545 2559 3569 4578 5584 6588 7590 8589 9586 0631 1647 2660 3670 4679 5685 6688 7690 8689 9686 0733 1748 2761 3771 4779 5785 6789 7790 8789 9785 0835 1849 2862 3872 4880 5886 6889 7890 8888 9885 101 100 0084 1077 2069 3058 4044 5029 6011 6992 7969 8945 9919 ~0890 1859 2826 3791 4754 5715 6673 7629 8584 9536 0382 1375 2366 3354 4340 5324 6306 7285 8262 9237 0210 1181 2150 3116 4080 5042 6002 6960 7916 8870 9821 7 8 9 440 1 2 3 4 5 6 640481 1474 2465 3453 4439 5422 6404 7383 8360 9335 0581 1573 2563 3551 4537 5521 6502 7481 8458 9432 0680 1672 2662 3650 4636 5619 6600 7579 8555 9530 0779 1771 2761 3749 4734 5717 6698 7676 8653 9627 0879 1871 2860 3847 4832 5815 8750 9724 99 98 0016 0987 1956 2923 3888 4850 5810 6769 7725 8679 9631 0113 1084 2053 3019 3984 4946 5906 6864 7820 8774 9726 7 8 9 450 1 2 3 4 5 6 650308 1278 2246 3213 4177 5138 6098 7056 8011 8965 9916 0405 1375 2343 3309 4273 5235 6194 7152 8107 9060 0502 1472 2440 3405 4369 5331 6290 7247 8202 9155 0599 1569 2536 3502 4465 5427 6386 7343 8298 9250 0696 1666 2633 3598 4562 5523 6482 7438 8393 9346 0793 1762 2730 3695 4658 5619 6577 7534 8488 9441 97 96 0011 0960 1907 0106 1055 2002 0201 1150 2096 0296 1245 2191 0391 1339 2286 0486 1434 2380 0581 1529 2475 0676 1623 2569 0771 1718 2663 8 9 660865 1813 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 : 94.5 104 10.4 20.8 31.2 41.6 52.0 62.4 72 8 83.2 93.6 103 10.3 20.6 30.9 41.2 51.5 61.8 72 1 82.4 92.7 102 10.2 20.4 30.6 40.8 51.0 61.2 71.4 81.6 91.8 101 10.1 20.2 30.3 40.4 50.5 60.6 70 7 80.8 90.9 100 10.0 20.0 30.0 40.0 50.0 60.0 70 80.0 90.0 99 9.9 19.8 29.7 39.6 49.5 59.4 69.3 79.2 89.1 144 LOGARITHMS OF NUMBERS. No. 460 L. 662.] fNo. 499 T. 698 N 460 1 2 8 4 5 6 3324 7 8 9 Diff. 662758 2852 2947 3041 3135 3230 3418 3512 1 3701 3795 3889 3983 4078 4172 4266 4360 4454 2 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 3 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 4 6518 6612 6705 6799 6892 6986 7079 7173 7266 5 7453 7546 7640 7733 7826 7920 8013 8106 8199 6 8386 8479 8572 8665 8759 8852 8945 9038 9131 9224 7 8 9317 9410 9503 9596 9689 9782 9875 9967 0895 0060 0988 0153 670246 0339 0431 0524 0617 0710 0802 93 9 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 1 3021 3113 3205 3297 3390 3482 3574 3666 3758 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 4 4861 5778 i 4953 5870 5045 5962 5137 6053 5228 6145 5320 6236 5412 6328 5503 6419 5595 6511 5687 6602 6694 6785 6876 6968 7059 7151 7242 7333 7424 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 9337 7 8 9 8518 9428 8609 9519 8700 9610 8791 9700 8882 9791 8973 9882 9064 9973 9155 9246 91 680336 0426 0517 0607 0698 0789 0879 0063 0970 0154 1060 0245 1151 480 1 2 1241 2145 3047 1332 2235 3137 1422 2326 3227 1513 2416 3317 1603 2506 3407 1693 2596 3497 1784 2686 3587 1874 2777 3677 1964 2867 3767 2055 2957 3857 4756 5652 6547 90 3 4 5 6 3947 4845 5742 6636 4037 4935 5831 6726 4127 5025 5921 6815 4217 5114 6010 6904 4307 1 5204 6100 6994 4396 5294 6189 7083 4486 5383 6279 7172 4576 5473 6368 7261 4666 5563 6458 7351 7 8 9 7529 8420 9309 7618 8509 9398 7707 8598 9486 7796 8687 9575 .7886 8776 9664 7975 8865 9753 8064 8953 9841 8153 9042 9930 - 8242 9131 8331 9220 89 [)()19 0107 490 1 2 3 4 5 6 8 9 690196 1081 1965 2847 3727 4605 5482 6356 7229 8100 0285 1170 2053 0373 1258 2142 0462 1347 2230 0550 : 1435 2318 ; 0639 1524 2406 0728 1612 2494 0816 1700 2583 0905 1789 ^671 0993 1877 2759 2935 3815 4693 5569 6444 7317 8188 3023 3903 4731 5657 6531 7404 8275 3111 3991 4868 5744 6618 7491 3362 3199 4078 4956 5832 ,' 6706 7578 1 8449 3287 4166 5044 5919 6793 7665 8535 3375 4254 5131 6007 6880 7752 8622 3463 4342 5219 6094 6968 7839 ' 8709 I 3551 4430 5307 3182 ?055 "926 S796 3639 4517 5394 6269 7142 8014 8883 88 87 Proportional, Parts. Diff. 1 2 3 4 5 6 1 7 8 9 1 98 97 96 95 94 93 92 91 90 89 88 87 i 86 1 9.8 9.7 9.6 9.5 9.4 9.3 9.2 9.1 9.0 8.9 8.8 8.7 [ 8.6 | 19.6 19.4 19.2 19.0 18.8 18.6 18.4 18.2 18.0 17.8 17:6 17.4 17.2 29.4 29.1 28.8 28.5 28.2 27.9 27.6 27.3 27.0 26.7 26.4 26.1 25.8 39.2 38.8 38.4 38.0 37.6 37.2 36.8 36.4 36.0 35.6 35.2 t 34.-8 I 1 34.4 49.0 48.5 48.0 47.5 47.0 46.5 46.0 45.5 45.0 44.5 44.0 1 43 /5 I 43.0 1 58.8 58.2 57.6 57.0 56.4 55.8 55.2 54.6 54.0 53.4 52.8 "52.2 51.6 68.6 67.9 67.2 66.5 65.8 65.1 64.4 63.7 63.0 62.3 61.6 | -60. "9 60.2 78.4 77.6 76.8 76.0 75.2 74.4 73.6 72.8 72.0 71.2 70.4 69.6 68.8 88.2 87.3 86.4 85.5 84.6 83.7 82.8 81.9 81.0 80.1 79.2 1 78."3 i 77.4 | LOGARITHMS OF LUMBERS. 145 No. 500 L. 698.] ^ [No. 544 L. 736. N. 1 2 3 4 6 6 7 8 9 Diff. 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 1 9838 9924 0011 0877 0098 0963 0184 1050 0444 1309 0531 1395 0617 1482 f 2 700704 0790 1136 1222 3 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 4 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 5 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 7 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 9 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 510 7570 7655 7740 7826 7911 7996 8081 8166 8251 8336 85 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 0033 0879 3 710117 0202 0287 0371 0456 0540 0625 0710 0794 4 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 7 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 9 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 1 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 83 3 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 4 9331 9414 9497 9580 9663 9745 9828 9911 9994 0077 0903 5 720159 0242 0325 0407 0490 0573 0655 0738 0821 6 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 9 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 4276 4358 4440 4522 4604 4685 4767 4849 4931 5013 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 2 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 6 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 7 9974 0055 0863 0136 0944 0217 1024 0298 1105 0378 1186 0459 1266 0540 1347 0621 1428 0702 1508 8 730782 y 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 540 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 1 8197 3278 3358 3438 3518 3598 3679 3759 3839 3919 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 4 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 Proportional Parts. Diff. 1 8.7 2 3 4 5 e J r 8 9 87 17.4 26 1 34.8 43.5 52.2 60.9 69.6 78.3 86 8.6 17.2 25 8 34.4 ' 43.0 51.6 60.2 68.8 77.4 85 8.5 17.0 25 5 14.0 42.5 51.0 59.5 68.0 76.5 84 8.4 16.8 25 2 33.6 42.0 50.4 58.8 67.2 75.6 146 LOGARITHMS OF NUMBEEg, No. 545 L. 736.] [No. 584 L. 787. N. | 1 2 3 4 5 6 7 8 j 9 Diff. 545 736397 6476 6556 6635 6715 6795 6874 6954 7034 7113 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 7 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 8 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 9 9572 9651 9731 9810 9889 9968 0047 0836 0126 0915 0205 0994 0284 1073 79 550 740363 0442 0521 0600 0678 0757 1 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 3 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 4 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 6 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 ' 7 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 8 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 2 9736 9814 9891 9968 0045 0817 0123 0894 0200 0971 0277 1048 0354 1125 0431 1202 3 750508 0586 0663 0740 4 1279 1356 1433 1510 1587 1604 1741 1818 1895 1972 77 ; 5 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 6 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 8 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 9 5112 5189 5265 5341 5417 5494 5570 5646 . 5722 5799 570 5875 5951 6027 6103 6180 6256 6332 6408 6484 6560 1 6636 6712 6788 6864 6940 | 7016 7092 7168 7244 7320 76 2 7396 7472 7548 7624 7700 i 7r?5 7851 7927 8003 8079 3 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 4 8912 8988 9063 9139 9214 | 9290 9366 9441 9517 9592 5 9668 9743 9819 9894 9970 1 0045 0799 0121 0875 0196 0950 0272 1025 0347 1101 6 760422 0498 0573 0649 0724 7 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 9 2679 2754 2829 2904 2978 ! 3053 3128 3203 3278 3353 580 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 1 4176 4251 4326 4400 4475 j 4550 4624 4699 4774 4848 2 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 3 5669 5743 5818 5892 5966 6041 6115 6190 6264 6338 4 6413 6487 6562 6636 6710 j 6785 6859 6933 7007 7082 Proportional Parts. Difl. 1 2 3 4 5 6 7 8 9 83 8.3 16.6 24.9 33.2 41.5 49.8 58.1 66.4 74.7 82 8.2 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.8 81 8.1 16.2 24.3 32.4 40.5 48.6 56.7 64.8 72.9 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 79 7.9 15.8 23.7 31.6 39.5 47.4 55.3 63.2 7i.i : 78 7.8 15.6 23.4 31.2 39.0 46.8 54.6 62.4 70.2 1 77 7.7 15.4 23.1 30.8 38.5 46.2 53.9 61.6 69.3 76 7.6 15.2 • 22.8 30.4 38.0 45.6 53.2 60.8 68.4 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 74 7.4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 63.6 LOGARITHMS OF NUMBERS. 147 No. 585 L. 767.] [No. 629 L. ' 770115 0852 1587 4517 5246 5974 6701 7427 8151 8874 9596 7803lf 1037 1755 2473 3189 3904 4617 6751 7460 3790 4488 5185 5880 6574 7268 7960 8651 7230 ! 7304 7972 ! 8046 8712 I 8786 9451 ; 9525 0189 | 0263 0926 0999 1661 1734 2395 i 2468 3128 3201 3860 ■ 3933 4590 ; 4663 5319 i 5392 6047 I 6120 6774 ! 6S46 7379 8120 7499 8224 1109 1827 2544 5401 6112 - 9019 9741 2462 3162 3860 4558 5254 5949 6644 7337 8029 8720 0461 1181 1899 2616 3332 4046 4760 5472 6183 6893 7602 8310 9016 9722 1129 1831 2532 3231 3930 4627 5324 6019 1073 1808 2542 3274 4006 4736 5465 6193 6919 7644 3403 4118 4831 5543 6254 1199 1901 7475 8167 8564 9303 1514 2248 2981 3713 4444 5173 5902 6629 7354 9524 9245 0965 1684 2401 3117 3832 4546 5259 5970 0215 0918 1620 2322 3022 3721 4418 5115 5811 6505 8582 9272 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 75 7.5 15.0 22.5 30.0 37.5 . 45.0 52.5 60.0 67.5 74 7.4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 66.6 73 7.3 14.6 21.9 29.2 36.5 43.8 51.1 58.4 65.7 72 7.2 14.4 21.6 28.8 36.0 43.2 50.4 57.6 64.8 71 7.1 14.2 21.3 28.4 .35.5 42.6 49.7 56.8 63.9 70 7.0 14.0 21.0 28.0 35.0 42.0 49.0 48.3 56.0 63.0 69 6.9 13.8 20.7 27.6 34.5 41.4 55.2 62.1 148 LOGARITHMS OF NUMBERS. No. 680 L. 799.] [No. 674 L. 829. N. 1 2 3 4 5 6 7 8 9 Diff. 630 1 799341 9409 9478 9547 9616 9685 9754 9823 9892 9961 1 800029 0098 0167 0236 0305 0373 0442 0511 0580 0648 2 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 4 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 8 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 640 806180 6248 6316 6384 6451 6519 6587 6655 6723 6790 1 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 2 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 3 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 5 9560 9896 0031 0703 0098 0770 0165 0837 6 810233 0300 0367 0434 0501 0569 0636 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 9 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 650 2913 2980 3047 3114 3181 3247 3314 3381 3448 3514 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 5 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 6 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 8 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 9 660 8885 9544 8951 9610 9017 9676 9083 9741 9149 9807 9215 9873 9281 9939 9346 9412 9478 0004 0661 0070 0727 0136 0792 1 820201 0267 0333 0399 0464 0530 0595 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 7 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 68 6.8 13.6 20 4 27.2 34.0 40.8 47.6 54.4 61.2 67 6.7 13.4 20 1 26.8 33.5 40.2 46.9 53.6 60.3 66 6.6 13.2 19 8 26.4 33.0 39.6 46.2 52.8 59 4 65 6.5 13.0 19 5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 IS. 8 19 2 25.6 32.0 3S.4 44.8 51.2 57. G LOGARITHMS OF NUMBERS. 149 No. 675 L. 829.] [No. 719 L. 857. N. 1 i 2 'i 4 $ 6 7 8 9818 9 Diff. 675 829304 9368 9432 9497 9561 9625 9690 9754 9882 6 9947 0011 0075 0139 0204 0268 0332 0396 0460 0525 7 830589 0653 0717 0781 0845 , 0909 0973 1037 1102 1166 8 1230 1294 1358 1422 1486 j 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 ! 2189 2253 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 1 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 2 ' 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 6 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 9 1 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 8849 8912 8975 9038 9101 9164 9227 9289 9352 9415 1 9478 9541 9604 9667 9729 9792 9855 9918 9981 0043 0671 2 840106 0169 0232 0294 0357 0420 0482 0545 0608 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 5 1985 2047 2110- 2172 2235 2297 2360 2422 2484 2547 6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 7 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 8 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 9 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 700 5098 5160 5222 5284 5346 5408 5470 5532 5594 5656 62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 7 8 9419 9481 9542 9604 9665 9726 9788 9849 0462 9911 0524 9972 0585 850033 0095 0156 0217 0279 0340 0401 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 6 | 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 8 i 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 9 i 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 63 6.3 12.6 18.9 25.2 31.5 37.8 44.1 50.4 56.7 62 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 55.8 61 6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54 9 60 6.0 12.0 18.0 24.0 30.0 ! 36.0 42.0 48.0 54.0 150 LOGARITHMS OF NUMBERS. No. 720 L. 857.] [No. 764 L. 883. N. > I * 3 4 6 6 7 8 9 Diff. 720 857332 7393 7453 7513 7574 7634 7694 7755 7815 7875 1 7935 7995 8056 8116 8176 1 8236 8297 8357 8417 8477 2 8537 8597 8657 8718 8778 ! 8838 8898 8958 9018 9078 3 9138 9198 9258 9318 9379 i 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 0038 0637 0098 0697 0158 0757 0218 0817 0278 0877 5 860338 0398 0458 0518 0578 i 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 i 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 9 2728 2787 2847 2906 2966 3025 3085 3114 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 5 6287 6346 6405 6465 6524 6583 C642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 91,3 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 1 9818 9877 9935 9994 0053 0638 0111 0696 0170 0755 0228 0813 0287 0872 0345 0930 2 870404 0462 0521 0579 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2008 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 6 2739 2797 2855 2913 2972 3030 3088 8146 3204 3262 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 8 | 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 ! 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 750 5061 5119 5177 5235 5293 5351 5409 5466 5524 5582 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 3 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 8 9669 9726 9784 9841 9898 9956 0013 0585 0070 0642 0127 0699 0185 0756 9 880242 0299 0356 0413 0471 0528 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.2 53.1 58 5.8 11.6 17.4 23.2 29.0 34.8 40.6 46.4 52.2 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.2 | 16.8 22.4 28.0 33.6 39.2 44.8 50.4 LOGARITHMS OF NUMBERS. 151 No. 765 L. 883.] [No. 809 L. 908. N. 1 2 3 4 5 6 7 8 9 Diff. 765 883661 3718 3775 3832 3888 3945 4002 4059 4115 4172 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 7 4^95 4852 4909 4965 5022 5078 5135 5192 5248 5305 8 5361 5418 5474 5531 5587 5644 5700 5757 5813 i 5870 9 5926 5983 6039 6096 6152 6209' 6265 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 6998 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 56 6 9862 9918 9974 0030 0589 0086 0645 0141 0700 0197 0756 0253 0812 0309 0868 0365 0924 7 890421 0477 0533 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 8 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 3 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 4 9821 9875 9930 9985 0039 0586 0094 0640 0149 0695 0203 0749 0258 0804 0312 0859 5 900367 0422 0476 0531 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 800 3090 3144 3199 3253 3307 3361 3416 3470 3524 3578 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 6 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 7 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 Proportional Parts. Diff. 1 2 3 4 5 J 6 7 8 9 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 55 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 LOGARITHMS OF NUMBERS. No. 810 L. 908.] [No. 854 L. 931. N. 1 2 3 4 I 5 8753 "• ! 7 8 ! 9 8967 Diff. 810 908485 8539 8592 8646 8699 8807 8860 8914 1 1 9021 9074 9128 9181 9235 | 9289 9342 9396 9449 9503 2 9556 9610 9663 9716 9770 ! 9823 9877 9930 9984 0037 0571 3 910091 0144 0197 0251 0304 0358 0411 0464 0518 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 5 1158 1211 1264 1317 1371 ! 1424 1477 1530 1584 163? 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 7 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 8 2753 2806 2859 2913 2966 31)19 3072 3125 3178 3231 9 3284 3337 3390 3443 3496 | 3549 3602 3655 3708 3761 53 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 2 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 3 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 4 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 5 6454 6507 6559 6612 6664 6717 6770 1822 6875 6927 6 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 i 9340 9392 9444 9496 9549 1 9601 9653 9706 9758 9810 9862 9914 9967 0019 0541 0071 0593 2 920123 0176 0228 0280 0332 0384 0436 0489 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 7 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 9 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 2 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 3 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 7 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 8 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 850 9419 9470 9521 9572 9623 i 9674 9725 9776 9827 9879 51 1 9930 9981 0032 0542 0083 0592 0134 0643 i 0185 1 0694 0236 0745 0287 0796 0338 0389 2 930440 0491 0847 0898 3 0949 1000 1051 1102 1153 ! 1204 1254 1305 1356 1407 4 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 Proportional Parts. Diff. 1 2 3 4 5 6 7 8 9 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 52 5.2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 51 5.1 10.2 15.3 20.4 25.5 30.6. 35.7 40.8 45.9 50 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 LOGAEITHMS OF NUMBEKS. 153 No. 855 L. 931.1 N. [No. 899 L. 954. 4498 5003 5507 6011 6514 7016 7518 8019 9519 940018 0516 1014 1511 2008 2504 3000 3495 4976 5469 5961 6452 6943 7434 7924 8413 0851 1338 2792 3276 3760 3538 4044 4549 5054 5558 6061 6564 7066 7568 8069 8570 9070 1064 1561 2058 2&54 3049 3544 4038 4532 5025 5518 6010 6501 4094 4599 5104 5608 6111 6614 7116 7618 8119 8620 9120 9619 0414 0900 1386 1872 2356 2841 9975 0462 1435 1920 2118 2626 3133 3639 4145 4650 5154 5658 8670 9170 2677 . , 3183 | 3234 3690 i 3740 4195 | 4246 ' 4751 5255 5759 6262 6765 ^roo 5205 5709 6212 6715 7217 7718 8219 8720 0168 0218 0666 0716 1163 1213 1660 1710 2157 2207 2653 2702 3148 3198 3643 3692 4137 4186 4631 4680 5124 5173 5616 5665 6108 6157 6600 6649 7090 7139 7581 7630 8070 8119 8560 8608 9048 9097 9536 9585 0024 0073 0511 0560 0997 1046 1483 1532 1969 2017 2453 2502 2938 2986 3421 3470 3905 3953 8770 9270 0267 0765 1263 1760 2256 2752 3247 3742 4236 4729 5222 5715 6207 8657 9146 0121 0608 1095 1580 2066 2550 3034 3518 4001 2271 2778 3285 3791 4296 4801 5306 6815 7317 7819 0317 0815 1313 2801 3297 3791 4285 4?79 5272 5764 6256 6747 7238 7728 8217 8706 9195 3841 4347 5356 5860 8370 8870 1362 1859 2355 5321 5813 8755 9244 0170 0657 1143 1629 2114 2599' 0219 0706 1192 1677 2163 2647 3131 3615 4902 5406 5910 6413 6916 7418 7919 8420 0417 0915 1412 1909 2405 2901 4877 5370 0267 0754 1240 4049 4098 ' 4146 3437 3943 4448 6966 7468 8470 8970 0467 0964 1462' 1958 2455 2950 3445 0316 3711 4194 Proportional Parts. Diff. 1 5.1 5.0 ! 4.9 ! 4.8 2 3 4 5 6 7 8 9 51 50 49 48 10.2 10.0 9.8 9.6 15.3 15.0 14.7 14.4 — -r 20.4 20.0 19.6 19.2 25.5 25.0 24.5 24.0 30.6 30.0 29.4 28.8 35.7 35.0 34.3 33.6 40.8 40.0 39.2 38.4 45.9 45.0 44.1 43.2 154 LOGARITHMS OF NUMBERS. No 900 L. 954.1 [No. 944 L. 975. N. 1 2 3 4 5 6 7 4580 8 9 Diff. 900 954243 4291 4339 4387 4435 4484 4532 4628 4677 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 5 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 6 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 7 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 9 8564 8612 8659 .8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 1 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 2 9995 0042 0090 0566 0138 0613 0185 0661 0233 0280 0756 0328 0376 0423 3 960471 0518 0709 0804 0851 0899 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 6 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 9 * 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 1 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 2 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 4 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 5 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 6 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 8 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 3 9882 9928 9975 0021 0486 0068 0533 0114 0579 0161 0626 0207 0672 0254 0719 0300 4 970347 0393 0440 0765 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 3543 1 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 • Pro PORTIO i *al Parts. DifE 1 2 3 4 5 6 7 8 9 47 4.7 9.4 14 1 8.8 23.5 28.2 32.9 37.6 42.3 I 46 4.6 9.2 | ]3 8 8.4 23.0 27.6 32.2 36.8 41.4 (. LOGARITHMS OF KUMBERS. 155 { No. 945 L. 975.] 1 [No. 989 L. 995. N. 1 i 1 55 4 5 6 r 8 9 Diff. 945 975432 5478 5524 '0 5616 5662 5707 5753 5799 5845 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 8 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 7724 7769 7815 7861 7906 7952 7998 8043 8089 8135 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 4 5 9548 980003 9594 0049 9639 9685 9730 9776 9821 9867 9912 9958 0094 0140 0185 0231 0276 0322 0367 0412 6 (1458 0503 0549 0594 0640 0685 0730 0776 0821 0867 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 8 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 6 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 7 9895 9939 9983 0028 0472 0072 0516 0117 0561 0161 0605 0206 0650 0250 0694 0294 0738 8 990339 0383 0428 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448^ 1492 1536 1580 1625 1 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 4 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 6 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 8 4757 4801 4845 4889 4i)33 4977 5021 5065 5108 5152 9 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 Proportional, Parts. Diff. 1 2 3 4 5 6 7 8 9 46 4.6 9.2 '■ 13.8 ] 8.4 23.0 27.6 32 .2 36.8 41.4 45 4.5 9.0 j 13.5 ] 8.0 22.5 27.0 31 .5 36.0 40.5 '| 44 4.4 8.8 13.2 7.6 22.0 26.4 30 .8 35.2 39. e 1 43 k, 4.3 8.6 I 12.9 ] 7.2 21.5 25.8 30 .1 34.4 38.7 J56 MATHEMATICAL TABLES. No. }90 L. 995.] [N o. 999 L. 999. N. 1 2 3 4 5 6 7 8 9 Diff. 990 995635 5679 5723 5767 5811 5854 5898 5942 5986 6030 1 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 2 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 3 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 4 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 5 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 7 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 8 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 9 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 HYPERBOLIC LOGARITHMS. No. Log. No. Log. No. Log. No. Log. No. Log. 1.01 .0099 1.45 .3716 1.89 .6366 2.33 .8458 2.77 1.0188 1.02 .0198 1.46 .3784 1.90 .6419 2.34 .8502 2.78 1.0225 1.03 .0296 1.47 .3853 1.91 .6471 2.35 .8544 2.79 1.0260 1.04 .0392 1.48 .3920 1.92 .6523 2.36 .8587 2.80 1.0296 1.05 .0488 1.49 .3988 1.93 .6575 2.37 .8629 2.81 1.0332 1.06 .0583 1.50 .4055 1.94 .6627 2.38 .8671 2.82 1.0367 1.07 .0677 1.51 .4121 1.95 .6678 2.39 .8713 2.83 1.0403 1.08 .0770 1.52 .4187 1.96 .6729 2.40 .8755 2.S4 1.0438 1.09 .0862 1.53 .4253 1.97 .6780 2.41 .8796 2.85 1.0473 1.10 .0953 1.54 .4318 1.98 .6831 2.42 .8838 2.86 1.0508 1.11 .1044 1.55 .4383 1.99 .6881 2.43 .8879 2.87 1.0543 1.12 .1133 1.56 .4447 2.00 .6931 2.44 .8920 2.88 1.0578 1.13 .1222 1.57 .4511 2.01 .6981 2.45 .8961 2.89 1.0613 1.14 .1310 1.58 .4574 2.02 .7031 2.46 .9002 2.90 1.0647 1.15 .1398 1.59 .4637 2.03 .7080 2.47 .9042 2.91 1.0682 1.16 .1484 1.60 .4700 2.04 .7129 2.48 .9083 2.92 1.0716 1.17 .1570 1.61 .4762 2.05 .7178 2.49 .9123 2.93 1.0750 1.18 .1655 1.62 .4824 2.06 .7227 2.50 .9163 2.94 1.0784 1.19 .1740 1.63 .4886 2.07 .7275 2.51 .9203 2.95 1.0813 1.20 .1823 1.64 .4947 2.08 .7324 2.52 .9243 2.96 1.0852 1.21 .1906 1.65 .5008 2.09 .7372 2.53 .9282 2.97 1.0886 1.22 .1988 1.66 .5068 2.10 .7419 2.54 .9322 2.98 1.0919 1.23 .2070 1.67 .5128 2.11 .7467 2.55 .9361 2.99 1.0953 1.24 .2151 1.68 .5188 2.12 .7514 2.56 .9400 3.00 1.0986 1.25 .2231 1.69 .5247 2.13 .7561 2.57 .9439 3.01 1.1019 1.26 .2311 1.70 .5306 2.14 .7608 2.58 .9478 3.02 1.1053 1.27 .2390 1.71 .5365 2.15 .7655 2.59 .9517 3.03 1.1086 1.28 .2469 1.72 .5423 2.13 .7701 2.60 .9555 3.04 1.1119 1.29 .2546 1.73 .5481 2.17 .7747 2.61 .9594 3.05 1.1151 1.30 .2624 1.74 .5539 2.18 .7793 2.62 .9632 3.06 1.1184 1.31 .2700 1.75 .5596 2.19 .7839 2.63 .9670 3.07 1.1217 1.32 .2776 1.76 .5653 2.20 .7885 2.64 .9708 3.08 1.1249 1.33 .2852 1.77 .5710 2.21 .7930 2.65 .9746 3.09 1.1282 1.34 .2927 1.78 .5766 2.22 .7975 2.66 .9783 3.10 1.1314 1.35 .3001 1.79 .5822 2^23 .8020 2.67 .9821 3.11 1.1346 1.36 .3075 1.80 .5878 2.24 .8065 2.68 .9858 3.12 1.1378 1.37 .3148 1.81 .5933 2.25 .8109 2.69 .9895 3.13 1.1410 1.38 .3221 1.82 .5988 2.26 .8154 2.70 .9933 3.14 1.1442 1.39 .3293 1.83 .6043 2.27 .8198 2.71 .9969 3.15 1.1474 1.40 .3365 1.84 .6098 2.28 .8242 2.72 1.0006 3 16 1.1506 1.41 .3436 1.85 .6152 2.29 .8286 2.73 1.0043 3.17 1.1537 1.42 .3507 1.86 .U206 2.30 .8329 2.74 1.0080 3.18 1.1569 1.43 .3577 1.87 .6259 2.31 .8372 2.75 1.0116 3.19 1.1600 1.44 .3646 1.88 .6313 2.32 .8416 2.76 1.0152 3.20 1.1632 HYPERBOLIC LOGARITHMS. 157 No. Log. No. Log. No. Log. No. Log. No. Log. 3.21 1 . 1663 3.87 1.3533 4.53 1.5107 5.19 1.6467 5.85 1.7664 3.22 1.1694 3.88 1.3558 4.54 1.5129 5.20 1.6487 5.86 1 .7681 3.23 1.1725 3.89 1.3584 4.55 1.5151 5.21 1.6506 5.87 1.7699 3.24 1.1756 3.90 1.3610 4.56 1.5173 5.22 1.6525 5.88 1.7716 3.25 1.1787 3.91 1.3635 4.57 1.5195 5.23 1.6514 5.89 1.7733 3.26 1.1817 3.92 1.3661 4.58 1.5217 5.24 1.6563 5.90 1.7750 3.27 1.1848 3.93 1.3686 4.59 1.5239 5.25 1.6582 5.91 1.7766 3.28 1.1878 3.94 1.3712 4.60 1.5261 5.26 1.6601 5.92 1.7783 3.29 1.1909 3.95 1.3737 4.61 1.5282 5.27 1.6620 5.93 1.7800 * 3.30 1.1939 3.96 1.3762 4.62 1.5304 5.28 1.6639 5.94 1.7817 3.31 1 1969 3.97 1.3788 4.63 1.5326 5.29 1.6658 5.95 1.7834 3.32 1.1999 3.98 1.3813 4.64 1.5347 5.30 1.6677 5.96 1.7851 3.33 1.2030 3.99 1.3838 4.65 1.5369 5.31 1.6696 5.97 1.7867 3.34 1.2060 4.00 1.3863 4.66 1.5390 5.32 1.6715 5.98 1.7884 3.35 1.2090 4.01 1.3888 4.67 1.5412 5.33 1.6734 5.99 1.7901 3.36 1.2119 4.02 1.3913 4.68 1.5433 5.34 1.6752 6.00 1.7918 3.37 1.2149 4.03 1.3938 4.69 1.5454 5.35 1.6771 6.01 1.7934 3! 38 1.2179 4.04 1.3962 4.70 1.5476 5.36 1.6790 6.02 1.7951 3.39 1.2208 4.05 1.3987 4.71 1.5497 5.37 1.6808 6.03 1.7967 3.40 1.2238 4.06 1.4012 4.72 1.5518 5.38 1.6827 6.04 1.7984 3.41 1.2267 4.07 1.4036 4.73 1.5539 5.39 1.6845 6.05 1.8001 3.42 1.2296 4.08 1.4061 4.74 1.5560 5.40 1.6864 6.06 1.8017 3.43 1.2326 4.09 1.4085 4.75 1.5581 5.41 1.6882 6.07 1.8034 3.44 1.2355 4.10 1.4110 4.76 1.5602 5.42 1.6901 6.08 1.8050 3.45 1.2384 4.11 1.4134 4.77 1.5623 5.43 1.6919 6.09 1.8066 3.46 1.2413 4.12 1 .4159 4.78 1.5644 5.44 1.6938 6.10 1.8083 3.47 1.2442 4.13 1.4183 4.79 1.5665 5.45 1.6956 6.11 1.8099 3.48 1.2470 4.14 1.4207 4.80 1.5686 5.46 1.6974 6.12 1.8116 3.49 1.2499 4.15 1.4231 4.81 1.5707 5.47 1.6993 6.13 1.8132 3.50 1.2528 4.16 1.4255 4.82 1.5728 5.48 1.7011 6.14 1.8148 3.51 1.2556 4.17 1.4279 4.83 1.5748 5.49 1.7029 6.15 1.8165 3.52 1.2585 4.18 1.4303 4.84 1.5769 5.50 1.7047 6.16 1.8181 3.53 1.2613 4.19 1.4327 1 4.85 1.5790 5.51 1.7066 6.17 1.8197 3.54 1.2641 4.20 1.4351 4.86 1.5810 5.52 1.7084 6.18 1.8213 3.55 1.2669 4.21 1.4375 . 4.87 1.5831 5.53 1.7102 6.19 1 .8229 3.56 1.2698 4.22 1.4398 4.88 1.5851 5.54 1.7120 6.20 1.8245 3.57 1.2726 4.23 1.4422 4.89 1.5872 5.55 1.7138 6.21 1.8262 3.58 1.2754 4.24 1.4446 4.90 1.5892 5.56 1.7156 6.22 1.8278 3.59 1.2782 4.25 1.4469 4.91 1.5913 5.57 1.7174 6.23 1.8294 3.60 1.2809 4.26 1.4493 4.92 1.5933 5.58 1.7192 6.24 1.8310 3.61 1.2837 4.27 1.4516 4.93 1.5953 5.59 1.7210 6.25 1.8326 3.62 1.2865 4.28 1.4540 4.94 1.5974 5.60 1.7228 6.26 1.8342 3.63 1.2892 4.29 1.4563 4.95 1.5994 5.61 1.7246 6.27 1.8358 3.64 1.2920 4.30 1.4586 4.96 1.6014 5.62 1.7263 6.28 1.8374 3.65 1.2917 4.31 1.4609 4.97 1.6034 5.63 1.7281 1 6.29 1.8390 3.66 1.2975 4.32 1.4633 4.98 1.6054 5.64 1.7299 ! 6.30 1.8405 3.67 1.3002 4.33 1 .4656 4.99 1.6074 5.65 1.7317 ! 6.31 1.8421 3.68 1.3029 4.34 1.4679 5.00 1.6094 5.66 1.7334 6.32 1.8437 3.69 1.3056 4.35 1.4702 5.01 1.6114 5.67 1.7352 ! 6.33 1.8453 3.70 1.3083 4.36 1.4725 5.02 1.6134 5.68 1.7370 6.34 1.8469 3.71 1.3110 4.37 1.4748 5.03 1.6154 5.69 1.7387 6.35 1.8485 3.72 1.3137 4.38 1.4770 5.04 1.6174 5.70 1.7405 6.36 1.8500 3.73 1.3164 4.39 1.4793 5.05 1.6194 5.71 1.7422 | 637 1.8516 3.74 1.3191 4.40 1.4816 5.06 1.6214 5.72 1.7440 6.38 1.8532 3.75 1.3218 4.41 1.4839 5.07 1.6233 5.73 1.7457 6.39 1.8547 3.76 1.3244 4.42 1.4861 5.08 1.6253 5.74 1.7475 ! 6.40 1.8563 3.77 1 .3271 4.43 1.4884 5.09 1.6273 5.75 1.7492 6.41 1 8579 3.78 L3297 4.44 1.4907 5.10 1.6292 5.76 1.7509 6.42 1.8594 3.79 1.3324 4.45 1.4929 5.11 1.6312 5.77 1.7527 6.43 1.8610 3.80 1.3350 4.46 1.4951 5.12 1.6332 5.78 1.7544 6.44 1.8625 3.81 1.3376 4.47 1.4974 5.13 1.6351 5.79 1.7561 6.45 1.8641 3.82 1.3403 4.48 1.4996 5.14 1.6371 5.80 1.7579 6.46 1.8(156 3.83 1.3429 4.49 1.5019 1 5.15 1.6390 5.81 1.7596 6.47 1.8672 3.84 1.3455 4.50 1.5041 1 5.16 1.6409 5.82 1.7613 6.48 1.8687 3.85 1.3481 4.51 1.5063 j 5.17 1.6429 5.83 1.7630 6.49 1.8703 3.86 1.3507 4.52 1.5085 1 5.18 1.6148 5.84 1.7647 1 6.50 1.8718 158 MATHEMATICAL TABLES. No. Log. No. Log. No. Log. No. Log. No. Log. 6.51 1.8733 7.15 1.9671 7.79 2.0528 8.66 2.1587 9.94 2.2966 6.52 1.8749 7.16 1.9685 7.80 2.0541 8.68 2.1610 9.96 2.2986 6.53 1.8764 7.17 1.9699 7.81 2.0554 8.70 2.1633 9.98 2.3006 6.54 1.8779 7.18 1.9713 7.82 2.0567 8.72 2.1656 10.00 2.3026 6.55 1.8795 7.19 1.9727 7.83 2.0580 8.74 2.1679 10.25 2.3279 6.56 1.8810 7.20 1.9741 7.84 2.0592 8.76 2.1702 10.50 2.3513 6.57 1.8825 7.21 1.9754 7.85 2.0605 8.78 2.1725 10.75 2.3749 6.58 1.8840 7.22 1.9769 7.86 2.0618 8.80 2.1748 11.00 2.3979 6.59 1.8856 7.23 1.9782 7.87 2.0631 8.82 2.1770 11.25 2.4201 6.60 1.8871 7.24 1.9796 7.88 2.0643 8.84 2.1793 11.50 2.4430 6.61 1.8886 7.25 1.9810 7.89 2.0656 8.86 2.1815 11.75 2.4636 6.62 1.8901 7.26 1.9824 7.90 2.0669 8.88 2.1838 12.00 2.4849 6.63 1.8916 7.27 1.9838 7.91 2.0681 8.90 2.1861 12.25 2.5052 6.64 1.8931 7.28 1.9851 7.92 2.0694 8.92 2.1883 12.50 2.5262 6.65 1.8946 7.29 1.9865 7.93 2.0707 8.94 2.1905 12.75 2.5455 6.66 1.8961 7.30 1.9879 7.94 2.0719 8.96 2.1928 13.00 2.5649 6.67 1.8976 7.31 1.9892 7.95 2.0732 8.98 2.1950 13.25 2.5840 6.68 1.8991 7.32 1.9906 7.96 2.0744 9.00 2.1972 13.50 2.6027 6.69 1.9006 7.33 1.9920 7.97 2.0757 9.02 2.1994 13.75 2.6211 6.70 1.9021 7.34 1.9933 7.98 2.0769 9.04 2.2017 14.00 2.6391 6.71 1.9036 7.35 1.9947 7.99 2.0782 9.06 2.2039 14.25 2.6567 6.72 1.9051 7.36 1.9961 8.00 2.0794 9.08 2.2061 14.50 2.6740 6.73 1.9066 7.37 1.9974 8.01 2.0807 9 10 2.2083 14.75 2.6913 6.74 1.9081 7.38 1.9988 8.02 2.0819 9.12 2.2105 15.00 2.7081 6.75 1.9095 7.39 2.0001 8.03 2.0832 9.14 2.2127 15.50 2.7408 6.76 1.9110 7.40 2.0015 8.04 2.0844 9.16 2.2148 16.00 2.7726 6.77 1.9125 7.41 2-0028 8.05 2.0857 9.18 2.2170 16.50 2.8034 6.78 1.9140 7.42 2.0041 8.06 2.0869 9.20 2.2192 17.00 2.8332 6.79 1.9155 7.43 2.0055 8-07 2.0882 9.22 2.2214 17.50 2.8621 6.80 1.9169 7.44 2.0069 8-08 2.0894 9.24 2.2235 18.00 2.8904 6.81 1.9184 7.45 2.0082 8-09 2.0906 9.26 2.2257 18.50 2.9173 6.82 1.9199 7.46 2-0096 8-10 2.0919 9.28 2.2279 19.00 2.9444 6.83 1.9213 7.47 2.0108 8-11 2.0931 9.30 2.2300 19.50 2.9703 6.84 1.9228 7.48 2.0122 8-12 2.0943 9.32 2.2322 20.00 2.9957 6.85 1.9242 7.49 2.0136 8.13 2.0956 9.34 2.2343 21 3.0445 6.86 1.9257 7.50 2.0149 8.14 2.0968 9.36 2.2364 22 3.0910 6.87 1.9272 7.51 2.0162 8.15 2.0980 9.38 2.2386 23 3.1355 6.88 1.9286 7.52 2.0176 8.16 2.0992 9.40 2.2407 24 3.1781 6.89 1.9301 7.53 2.0189 8.17 2.1005 9.42 2.2428 25 3.2189 6.90 1.9315 7.54 2.0202 8.18 2.1017 9.44 2.2450 26 3.2581 6.91 1.9330 7.55 2.0215 8.19 2.1029 9.46 2.2471 27 3.2958 6.92 1.9314 7.56 2.0229 8-20 2.1041 9.48 2.2492 28 3.3322 6.93 1.9359 7.57 2.0242 8-22 2.1066 9.50 2.2513 29 3.3673 6.94 1.9373 7.58 2.0255 8.24 2.1090 9.52 2.2534 30 3.4012 8.95 1.9387 7.59 2.0268 8.26 2.1114 9.54 2.2555 31 3.4340 6.96 1.9102 7.60 2.0281 8.28 2.1138 9.56 2.2576 32 3.4657 6.97 1.9416 7.61 2.0295 8.30 2.1163 9.58 2.2597 33 3.4965 6.98 1.9430 7.62 2.0308 8.32 2.1187 9.60 2.2618 34 3.5263 6.99 1.9445 7.63 2.0321 8.34 2.1211 9.62 2.2638 35 3.5553 7.00 1.9459 7.64 2.0334 8.36 2.1235 9.64 2.2659 36 3.5835 7.01 1.9473 7.65 2.0347 8.38 2.1258 9.66 2.2680 37 3.6109 7.02 1.9488 7.66 2.0360 8.40 2.1282 9.68 2.2701 38 3.6376 7.03 1.9502 7.67 2.0373 8.42 2.1306 9.70 2.2721 39 3.6636 7.04 1.9516 7.68 2.0386 8.44 2.1330 9.72 2.2742 40 3.6889 7.05 1.9530 7.69 2.0399 8.46 2.1353 9.74 2.2762 41 3.7136 7.06 1.9544 7.70 2.0412 8.48 2.1377 9.76 2.2783 42 3.7377 7.07 1.9559 7.71 2.0425 8.50 2.1401 9.78 2.2803 43 3.7612 7.08 1.9573 7.72 2.0438 8.52 2.1424 9.80 2.2824 44 3.7842 7.09 1.9587 7.73 2.0451 8.54 2.1448 9.82 2.2844 45 3.8067 7.10 1.9601 7.74 2.0464 8.56 2.1471 9.84 2.2865 46 3.8286 7.11 1.9615 7.75 2.0477 8.58 2.1494 9.86 2.2885 47 3.8501 7.12 1.9629 7.76 2.0490 8.60 2.1518 9.88 2.2905 48 3.8712 7.13 1.9643 7.77 2.0503 8.62 2.1541 9.90 2.2925 49 3.8918 7.14 1.9657 7.78 2.0516 8.64 2.1564 9.92 2.2946 50 3.9120 NATURAL- TRIGONOMETRICAL FUNCTIONS. 159 NATURAL TRIGONOMETRICAL FUNCTIONS. o M. Sine. Co-Vers. Cosec. | Tang. Cotan. | Secant. Ver. Sin.j Cosine. 00000 L.0O0O Infinite 00000 Infinite ] 1.0000 .00000 1.0000 90 15 00436 .99564 229.18 00136 229.18 ! 1.0000 .00001 .99999 45 30 00873 .99127 114.59 00K73 114.59 I 1.0000 .00001 .99996 s 30 45 01309 .98691 76.397 01309 76.390 1.0001 .00009 . 99991 J 15 1 01745 .98255 57.299 01745 57.290 1.0001 .00015 .99985 89 15 .02181 .97819 15.840 02182 45.829 1.0002 .00024 .99976' 45 .30 .02618 .973S2 38.202 02618 38.188 1.0003 .00034 .99966 30 45 .03051 .96946 32.746 03055 32.730 1.0005 .00047 .99953 15 2 .03190 .96510 28.654 03492 28.636 1.0006 .00061 .99939 88 15 .03926 .96074 25.471 03929 25.452 1.0008 .00077 .99923 45 30 .04362 .95638 22.926 04366 22.904 1.0009 .00095 .99905, 30 45 .04798 .95202 20.843 04803 20.819 1.0011 .00115 .99885! 15 3 .05234 .94766 19.107 05211 19.081 1.0014 .00137 .99863 87 15 .05669 .94331 17.639 05678 17.611 1.0016 .00161 .99839 45 30 .06105 .93895 16.380 06116 16.350 1.0019 .00187 .99813 30 45 .06510 .93460 15.290 06551 15.257 • 1.0021 .00214 .99786 15 4 .06976 .93024 14.336 06993 14.301 j 1.0024 .00244 .99756 86 15 .07411 .92589 13.494 07131 13.457 ! 1.0028 .00275 .99725 45 30 .07846 .92154 12.745 07870 12.706 ! 1.0031 .00308 .99692 30 45 .08281 .91719 12.076 08309 12.035 1.0034 .00313 .99656 15 5 .08716 .91284 11.174 08719 11.430 1.0038 .00381 .99619 85 15 .09150 .90850 10.929 09189 10.883 i 1.0012 .00420 .99580 45 30 .09585 .90415 10.433 09629 10.385 1.0046 .00460 .99540 30 45 .10019 .89981 9.9812 10069 9.9310 1.0051 .00503 .99497 15 6 .10153 .89547 9.5668 10510 9.5144 1.0055 .00548 .99152 84 15 .10887 .89113 9.1855 10952 9.1309 1.0060 .00594 .99406 45 30 .11320 .88680 8.8337 11393 8.7769 1.0065 .00643 .99357 30 45 .11754 .88246 8.5079 11836 8.4190 1.0070 .00693 .93307 15 1 .12187 .87813 8.2055 12278 8.1443 1.0075 .00745 .99255 83 15 .12620 .87380 7.9240 12722 7.8606 1.0081 .00800 .99200 45 30 . 13053 .86947 7.6613 .13165 7.5958 1.0086 .00856 .99144 30 45 .13185 .86515 7.4156 .13609 7.3479 1.0092 .00913 .99086 15 8 .13917 .86083 7.1853 .14054 7.1154 1.0098 .00973 .99027 82 15 .14349 .85651 6.9690 .11199 6.S969 1.0105 .01035 .98965 45 30 .14781 .85219 6.7655 .141)15 6.6912 1.0111 .01098 .98902 30 45 .15212 .84788 6.5736 .15391 6.4971 1.0118 .01164 .98836 15 9 .15613 .84357 6.3924 .15838 6.3138 1.0125 .01231 .98769 81 15 .16074 .83926 6.2211 .16286 6.1402 1.0132 .01300 .98700 45 30 .16505 .83495 6.0589 .16731 5.9758 1.0139 .01371 .98629 30 45 .16935 .83065 5.9049 •17183 5.8197 1.0147 .01444 .98556 15 10 !•■'. ' .82635 5.7588 •17633 5.6713 1.0154 .01519 .98481 80 15 .17794 .82206 5.6198 .18083 5.5301 1.0162 .01596 .98104 45 30 .18224 .81776 5.4874 •18534 5.3955 1.0170 .01675 .98325 30 45 .18652 .81348 5.3612 .18986 5.2672 1.0179 .01755 .98245 15 11 .19081 .80919 5.2408 •19438 5.1446 1.0187 .01837 .98163 79 15 .19509 .80491 5.1258 •19891 5.0273 1.0196 .01921 .98079 45 30 .19937 .80063 5.0158 .20315 4.9152 1.0205 .02008 .97992 30 45 .20364 .79636 4.9106 •20800 4.8077 1.0214 .02095 .97905 15 12 .20791 .7920P 4.8097 .21256 4.7046 1.0223 .02185 .97815 7S 15 .21218 .7878; 4.7130 •21712 4.6057 1.0233 .02277 .97723 45 30 .21644 .78356 4.6202 •22169 4.5107 1.0243 .02370 .97630 30 45 .2207C .7793C 4.5311 •22628 4.4191 1.0253 .02466 .97534 15 13 .22495 .7750c 4.4154 •23087 4.3315 1.0263 .02563 .97137 77 15 .2292C .7708( ) 4.3630 •23547 4.2468 1.0273 .02662 .97338 45 30 .2334c .7665? ) 4.2837 •2400S 4.1653 1.0284 .02763 .97237 30 45 .2376C .7623 4.2072 •24171 4.0867 1.0295 .02866 .97134 15 14 .2419; .7580* ] 4.1336 .2493? 4.010E 1.0306 .0297C .97030 76 15 .2461f .7538 5 4.0625 . 3.937c 1.0317 .03077 .96923 45 30 .2503* .7496 I 3.9939 .2586- 3.8667 1.032S .0318c .96815 30 45 .2546( ) .7454 ) 3.9277 .26325 3.7981 1.034 . 0329c .98705 15 15 .2588; 5 .7411 * 3.8637 .2679." 1 3.732C 1.0353 .03407 .96593 75 Cosine . Ver. Sin . Secant. Cotan | Tang. Cosec. Co-Vers Sine. ° M. From 75° to SO° read from pottom of table upwards, 160 MATHEMATICAL TABLES. • M. Sine. Co-Vers. c™,. Tang. Cotan. Secant. Ver. Sin. c.„„. 15 .25882 .74118 3.8637 .26795 3.7320 1.0353 .03407 .96693 75 15 .26303 .73697 3.8018 .27263 3.6680 1.0365 .03521 .96479 45 30 .26724 .73276 3.7420 .27732 3.6059 1.0377 .03637 .96363 30 45 .27144 .72856 3.6840 .28203 3.5457 1.0390 .03754 .96246 15 16 .27564 .72436 3.6280 .28674 3.4874 1.0403 .03874 .96126 74 15 .27983 .72017 3.5736 .29147 3.4308 1.0416 .03995 .96005 45 30 .28402 .71598 3.5209 .29621 3.3759 1 .0429 .04118 .95882 30 45 .28820 .71180 3.4699 .30096 3.3226 1.0443 .04243 .95757 15 17 .29237 .70763 3.4203 .30573 3.2709 1.0457 .04370 .95630 73 15 .21)654 .70346 3.3722 .31051 3.2205 1.0471 .04498 .95502 45 30 .30070 .69929 3.3255 .31530 3.1716 1.0485 .04628 .95372 30 45 .30486 .69514 3.2801 .32010 3.1240 1.0500 .04760 .95240 15 18 .30902 .69098 3.2361 .32492 3.0777 1.0515 .04894 .95106 72 15 .31316 .68684 3.1932 .32975 3.0326 1.0530 .05030 .94970 45 30 .31730 .68270 3.1515 .33459 2.9887 1.0545 .05168 .94832 30 45 .32144 .67856 3.1110 .33945 2.9459 1.0560 .05307 .94693 15 19 .32557 .67443 3.0715 .34433 2.9042 1.0576 .05448 .94552 71 15 .32969 .67031 3.0331 .34921 2.8636 1.0592 .05591 .94409 45 30 .33381 .66619 2.9957 .35412 2.8239 1.0608 .05736 .94264 30 45 .33792 .66208 2.9593 .35904 2.7852 1.0625 .05882 .94118 15 20 .34202 .65798 2.9238 .36397 2-7475 1 .0642 .06031 .93969 70 15 .34612 .65388 2.8892 .36892 2.7106 1.0659 .06181 .93819 45 30 .35021 .64979 2.8554 .37388 2-6746 1.0676 .06333 .93667 30 45 .35429 .64571 2.8225 .37887 2-6395 1.0694 .06486 .93514 15 21 .35837 .64163 2.7904 .38386 2.6051 1.0711 .06642 .93358 69 15 .36244 .63756 2.7591 .38888 2-5715 1.0729 .06799 .93201 45 30 .36650 .63350 2.7285 .39391 2-5386 1.0748 .06958 .93042 30 45 .37056 .62944 2.6986 .39896 2-5065 1.0766 .07119 .92881 15 22 .37461 .62539 2.6695 .40403 2-4751 1.0785 .07282 .92718 68 15 .37865 .62135 2.6410 .40911 2-4443 1.0804 .07446 .92554 45 30 .38268 .61732 2.6131 .41421 2-4142 1.0824 .07612 .92388 30 45 .38671 .61329 2.5859 .41933 2-3847 1.0844 .07780 .92220 15 23 .39073 .60927 2.5593 .42447 2-3559 1.0864 .07950 .92050 67 15 .39474 .60526 2.5333 .42963 2-3276 1.0884 .08121 .91879 45 30 .39875 .60125 2.5078 .43481 2-2998 1.0904 .08294 .91706 30 45 .40275 .59725 2.4829 .44001 2 2727 1.0925 .08469 .91531 15 24 .40674 .59326 2.4586 .44523 2-2460 1.0946 .08645 .91355 66 15 .41072 .58928 2.4348 .45047 2.2199 1.0968 .08824 .91176 45 30 .41469 .58531 2.4114 .45573 2.1943 1.0989 .09004 .90996 30 45 41866 .58134 2.3886 .46101 2.1692 1.1011 .09186 .90814 15 25 .-12262 .57738 2.3662 .46631 2.1445 1.1034 .09369 .90631 65 15 .42657 .57343 2.3443 .47163 2.1203 1.1056 .09554 .90446 45 30 .43051 .56949 2.3228 .47697 2.0965 1.1079 .09741 .90259 30 45 .43445 .56555 2.3018 .48234 2.0732 1.1102 .09930 .90070 15 26 .43837 .56163 2.2812 .48773 2.0503 1.1126 . 10121 .89879 64 15 .44229 .55771 2.2610 .49314 2.0278 1.1150 .10313 .89687 45 30 .44620 .55380 2.2412 .49858 2.0057 1.1174 . 10507 .89493 30 45 .45010 .54990 2.2217 .50404 1.9840 1.1198 .10702 .89298 15 27 .45399 .54601 2.2027 .50952 1.9626 1.1223 .10899 .89101 63 15 .45787 .54213 2.1840 .51503 1.9416 1.1248 .11098 .88902 45 30 .46175 .53825 2.1657 .52057 1.9210 1.1274 .11299 .88701 30 45 .46561 .53439 2.1477 .52612 1.9007 1.1300 .11501 .88499 15 28 .46947 .53053 2.1300 .53171 1.8807 1.1326 .11705 .88295 62 15 .47332 .52668 2.1127 ; 1.8611 1.1352 .11911 .88089 45 30 .47716 .52284 2.0957 .54295 1.8418 1.1379 .12118 .87882 30 45 .48099 .51901 2.0790 .54862 1.8228 1.1406 .12327 .87673 15 29 .48481 .51519 2.0627 .55431 1.8040 1.1433 .12538 .87462 61 15 .48862 .51138 2.0466 .56003 1.7856 1.1461 .12750 .87250 45 30 .49242 .50758 2.0308 .56577 1.7675 1 . 1490 .12964 .87036 30 45 .49622 .50378 2.0152 .57155 1.7496 1.1518 .13180 .868:20 15 30 .50000 .50000 2.0000 .57735 1.7320 1.1547 .13397 .86603 60 Cosine. Ver. Sin. Sec,.,. Cotan. Tang. o«. Co-Vers. Sine. • M. From 60° to 75° read from bottom of table upwards. NATURAL TRIGONOMETRICAL FUNCTIONS. 161 ■• M. Sine. Co-Vers. C.,.c. Tang. Cotan. Secant. Ver. Sin. Cosine. 30 .50000 .50000 2.0000 .57735 1.7320 1.1547 .13397 .86603 GO 15 .50377 .49623 1.9850 .58318 1.7147 1.1576 .13616 .86384 45 30 .50754 .49246 1.9703 .58904 1.6977 1.1606 .13837 .86163 30 45 .51129 .48871 1.9558 .59494 1.6808 1.1636 . 14059 .85941 15 31 .51504 .48496 1.9416 .60086 1.6643 1.1666 .14283 .85717 59 15 .51877 .48123 1.9276 .60681 1.6479 1.1697 .14509 .85491 45 30 .52250 .47750 1.9139 .61280 1.6319 1.1728 .14736 .85264 30 45 .52621 .47379 1.9004 .61882 1.6160 1.1760 .14965 .85035 15 32 .52992 .47008 1.8871 .62487 1.6003 1.1792 .15195 .84805 58 15 .53361 .46639 1.8740 .63095 1.5849 1.1824 .15427 .84573 45 30 .53730 .46270 1.8612 .63707 1.5697 1.1857 .15661 .84339 30 45 .54097 .45903 1.8485 .64322 1.5547 1.1890 .15896 . 84104 15 33 .54464 .45536 1.8361 .64941 1.5399 1 1924 . 16133 .83867 57 15 .54829 .45171 1.8238 .65563 1.5253 1.1958 .16371 .83629 45 30 .55194 .44806 1.8118 .66188 1.5108 1.1992 .16611 .83389 30 45 .55557 .44443 1.7999 .66818 1.4966 1.2027 .16853 .83147 15 34 .55919 .44081 1.7883 .67451 1.4826 1.2062 .17096 .82904 56 15 .56280 .43720 1.7768 .68087 1.4687 1.2098 .17341 .82659 45 30 .56641 .43359 1.7655 .68728 1.4550 1.2134 .17587 .82413 30 45 .57000 .43000 1.7544 .69372 1.4415 1.2171 .17835 .82165 15 35 .57358 .42642 1.7434 .70021 1.4281 1.2208 .18085 .81915 55 15 .57715 .42285 1.7327 .70673 1.4150 1.2245 .18336 .81664 45 30 .58070 .41930 t.7220 .71329 1.4019 1.2283 .18588 .81412 30 45 .58425 .41575 1.7116 .71990 1.3891 1.2322 .18843 .81157 15 36 .58779 .41221 1.7013 .72654 1.3764 1.2361 .19098 .80902 54 15 .59131 .40869 1.6912 1.3638 1.2400 .19356 .80644 45 30 .59482 .40518 1.6812 1.3514 1.2440 .19614 .80386 30 45 .59832 .40168 1.6713 .74673 1.3392 1.2480 .19875 .80125 15 37 .60181 .39819 1.6616 .75355 1.3270 1.2521 .20136 .79864 53 15 .60529 .39471 1.G521 .76042 1.3151 1.2563 .20400 .79600 45 30 .60876 .39124 1.G427 .76733 1.3032 1.2605 .20665 .79335 30 45 .61222 .38778 1.G334 .77428 1.2915 1.2647 .20931 .79069 15 38 .61566 .38434 1.G243 .78129 1.2799 1.2690 .21199 .78801 52 15 .61909 .38091 1.0153 .78834 1.2685 1.2734 .21468 .78532 45 30 .62251 .37749 1.6064 .79543 1.2572 1.2778 .21739 .78261 3" 45 - .37408 1.5976 .80258 1.2460 1.2822 .22012 .77988 15 39 .37068 1. 589.1 .80978 1.2349 1.2868 .22285 .77715 51 15 .63271 .36729 1.5805 .81703 1.2239 1.2913 .22561 .77439 45 30 .63608 .36392 1.5721 .82434 1.2131 1.2960 .22838 .77162 30 45 .63944 .36056 1.5639 .83169 1.2024 1.3007 .23116 .76884 15 40 .64279 .35721 1.5557 .83910 1.1918 1.3054 .23396 .76604 50 15 .64612 .353S8 1.5477 .84656 1.1812 1.3102 .23677 .76323 45 30 .64945 . 35055 1.5398 .85408 1.1708 1.3151 .23959 .76041 30 45 .65276 .34724 1.5320 .86165 1.1606 1.3200 .24244 .75756 15 41 .65606 .34394 1 5242 .86929 1.1504 1.3250 .24529 .75471 49 15 .65935 .34065 1.5166 .87698 1.1403 1.3301 .24816 .75184 45 30 .66262 .33738 1 5092 .88472 1.1303 1.3352 .25104 .74896 30 45 .66588 .33412 1.5018 1.1204 1.3404 .25394 .74606 15 42 .66913 .33087 1.4945 .90040 1.1106 1.3456 .25686 .74314 48 15 .67237 .32763 1.4873 .90834 1.1009 1.3509 .25978 .74022 45 30 .67559 .32441 1.4802 .91633 1 0913 1.3563 .26272 .73728 30 45 .67880 .32120 1.4732 .92439 1.0818 1.3618 .26568 .73432 15 43 .68200 .31800 1.4663 93251 1.0724 1.3673 .26865 .73135 47 15 .68518 .31482 1.4595 .94071 1.0630 1.3729 .27163 .72837 45 30 .68835 .31165 1.4527 .94896 1.0538 1 .3786 .27463 .72537 30 45 .69151 .30849 1.4461 .95729 1.0446 1.3843 .27764 .72236 15 44 .69466 .30534 1.4396 .96569 1.0355 1.3902 .28066 .71934 46 15 .69779 .30221 1.4331 .97416 1 .0265 1.3961 .28370 .71630 45 30 .70091 .29909 1.4267 .98270 1.0176 1.4020 .28675 .71325 30 45 .70401 .29599 1.4204 .99131 1.0088 1.4081 .28981 .71019 15 45 .70711 .29289 Ver. Sin. 1.4142 1.0000 1 .0000 1.4142 .29289 .70711 Sine. 45 c„,«. Secant. Cotan. Tang. Cosec. Co-Vers. M. ! Fro m45 ° to 6 J° read fron i ootttc in of i aole i ipwa rds. 162 MATHEMATICAL TABLES. LOGARITHMIC SINES, ETC. Deg. Sine. Cosec. Versin. Tangent. Cotan. Covers. Secant. Cosine. Deg. In.Neg. 8.24186 Infinite. In.Neg. In.Nea:. Infinite. 10.00000 10.00000 10.00000 90 1 11.75814 6.18271 8.24192 11.75808 9.99235 10.00007 9.99993 89 2 8.54282 11.45718 6.78474 8.54308 11.45692 9.98457 10.00026 9.99974 88 ' 3 8.71880 11.28120 7.13687 8.71940 11.28060 9.97665 10.00060 9.99940 87 4 8.84358 11.15642 7.38667 8.84464 11.15536 9.96860 10.00106 9.99894 86 5 8.94030 11.05970 7.58039 8.94195 11.05805 9.96040 10.00166 9.99834 85 6 9.01923 10.98077 7.73863 9.02162 10.97838 9.95205 10.00239 9.99761 84 7 9.08589 10.91411 7.87238 9.08914 10.91086 9.94356 10.00325 9.99675 83 8 9.14356 10.85644 7.98820 9.14780 10.85220 9.93492 10.00425 9.99575 82 i 9 9.19433 10.80567 8.09032 9.19971 10.80029 9.92612 10.00538 9.99462 81 10 9.23967 10.76033 8.18162 9.24632 10.75368 9.91717 10.00665 9.99335 80 11 9.28060 10.71940 8.26418 9.28865 10.71135 9.90805 10.00805 9.99195 79 ! 12 9.31788 10.68212 8.33950 9.32747 10.67253 9.89877 10.00960 9.99040 78 13 9.35209 10.64791 8.40875 9.36336 10.63664 9.88933 10.01128 9.98872 77 14 9.38368 10.61632 8.47282 9.39677 10.60323 9.87971 10.01310 9.98690 76 15 9.41300 10.58700 8.53243 9.42805 10.57195 9.86992 10.01506 9.98494 75 16 9.44034 10.55966 8.58814 9.45750 10.54250 9.85996 10.01716 9.98284 74 17 9.46594 10.53406 8.64043 9.48534 10.51466 9.84981 10.01940 9.98060 73 18 9.48998 10.51002 8.68969 9.51178 10.48822 9.83947 10.02179 9.97821 72 19 9.51264 10.48736 8.73625 9.53697 10.46303 9.82894 10.02433 9.97567 71 20 9.53405 10.46595 8.78037 9.56107 10.43893 9.81821 10.02701 9.97299 70 21 9.55433 10.44567 8.82230 9.58418 10.41582 9.80729 10.02985 9.97015 69 22 9.57358 10.42642 8.86223 9.60641 10.39359 9.79615 10.03283 9.96717 68 23 9.59188 10.40812 8.90034 9.62785 10.37215 9.78481 10.03597 9.96403 67 24 9.60931 10.39069 8.93679 9.64858 10.35142 9.77325 10.03927 9.96073 66 25 9.62595 10.37405 8.97170 9.66867 10.33133 9.7614t' 10.04272 9.95728 65 26 9.64184 10.35816 9.00521 9.68818 10.31182 9.74945 10.04634 9.95366 64 27 9.65705 10.34295 9.03740 9.70717 10.29283 9.73720 10.05012 9.94988 63 28 9.67161 10.32839 9.06838 9.72567 10.27433 9.72471 10.05407 9.94593 62 29 9.68557 10.31443 9.09823 9.74375 10.25625 9.71197 10.05818 9.94182 61 30 9.69897 10.30103 9.12702 9.76144 10.23856 9.69897 10.06247 9.93753 60 31 9.71184 10.28S16 9.15483 9.77877 10.22123 9.68571 10.06693 9.93307 59 32 9.72421 10.27579 9.18171 9.79579 10.20421 9.67217 10.07158 9.92842 58 33 9.73611 10.26389 9.20771 9.81252 10.18748 9.65836 10.07641 9.92359 57 i 34 9.74756 10.25244 9.23290 9.82899 10.17101 9.64425 10.08143 9.91857 56 35 9.75859 10.24141 9.25731 9.84523 10.15477 9.62984 10.08664 9.91336 55 36 9.76922 10.23078 9.28099 9.86126 10.13874 9.61512 10.09204 9.90796 54 37 9.77946 10.22054 9.30398 9.87711 10.12289 9.60008 10.09765 9.90235 53 38 9.78934 10.21066 9.32631 9.89281 10.10719 9.58471 10.10347 9.89653 52 39 9.79887 10.20113 9.34802 9.90837 10.09163 9.56900 10.10950 9.89050 51 40 9.80807 10.19193 9.36913 9.92381 10.07619 9.55293 10.11575 9.88425 50 41 9.81694 10.18306 9.38968 9.93916 10.06084 9.53648 10.12222 9.87778 49 42 9.82551 10.17449 9.40969 9.95444 10.04556 9.51966 10.12893 9.87107 48 43 9.88378 10.16622 9.42918 9.96966 10.03034 9.50243 10.13587 9.86413 47 44 9.84177 10.15823 9.44818 9.98484 10.01516 9.48479 10.14307 9.85693 46 45 9.84949 10.15052 9.46671 10.00000 10.00000 9.46671 10.15052 9.84949 t 45 Cosine. Secant. Covers. Cotan. Tangent. Versin. Cosec. Sine. t From 45° to 90° rea< 1 from bottoi a of tal Me npi vards. i i 1 • 1 SPECIFIC GRAVITY. 163 MATERIALS. THE CHEMICAL ELEMENTS. Tl»e Common Elements (42). o o Name. sit 0-53 11 Name. -i_j § s Name. 053 <£ % <£ Al Aluminum 27.1 F Fluorine 19. Pd Palladium 106. Sb Antimony 120. Au Gold 196.2 P Phosphorus 30.96 As Arsenic 75. H Hydrogen 1. Pt Platinum 195. Ba Barium 137. I Iodine 126.6 K Potassium 39.03 Bi Bismuth 208. Ir Iridium 193. Si Silicon 28.4 B Boron 10.9 Fe Iron 56. Ag Silver 107.7 Br Bromine 79.8 Pb Lead 206.4 Na Sodium 23. Cd Cadmium 111.8 Li Lithium 7.01 Sr Strontium 87.4 Ca Calcium 40. Mg Magnesium 24. S Sulphur 32. C Carbon 12. Mn Manganese 55. Sn Tin 118. CI Chlorine 35.4 Hg Mercury 199.8 Ti Titanium 50. Cr Chromium 52.3 Ni Nickel 58.3 W Tungsten 184. Co Cobalt 59. N Nitrogen 14. Va Vanadium 51.2 Cu Copper 63.2 Oxygen 15.96 Zn Zinc 65. The atomic weights of many of the elements vary in the decimal place as given by different authorities. Tlie Rare Elements (27). Beryllium, Be. Caesium, Cs. Cerium, Ce. Didymium, D. Erbium, E. Gallium, Ga. Glucinum, G. Indium, In. Lanthanum, La. Molybdenum, Mo. Niobium, Nb. Osmium, Os. Germanium, Ge. Rhodium, R. Rubidium, Rb. Ruthenium, Ru. Samarium, Sm. Scandium, Sc. Selenium, Se. Tantalum, Ta. Tellurium, Te. Thallium, Tl. Thorium, Th. Uranium, U. Ytterbium, Yr. Yttrium, Y. Zirconium, Zr. SPECIFIC GRAVITY. The specific gravity of a substance is its weight as compared with the weight of an equal bulk of pure water. To find, the specific gravity of a substance. W = weight of body in air; 10 = weight of body submerged in water. W Specific gravity = — . W — io If the substance be lighter than the water, sink it by means of a heavier substance, and deduct the weight of the heavier substance. Specific-gravity determinations are usually referred to the standard of the weight of water at 62° F., 62.355 lbs. per cubic foot. Some experimenters have used 60° F. as the standard, and others 32° and 39.1° F. There is no general agreement. Given sp. gr. referred to water at 39.1° F., to reduce it to the standard of 62° F. multiply it by 1.00112. Given sp. gr. referred to water at 62° F.. to find weight per cubic foot mul- tiply by 62.355. Given weight per cubic foot, to find sp. gr. multiply by 0.016037. Given sp. gr., to rind weight per cubic inch multiply by .036085, 164 MATERIALS. Weight and Specific Gravity of Metals. Aluminum Antimony Bismuth Brass: Copper + Zinc 1 80 20 i 70 30 }-.. 60 40 | 50 50 J Bionze-j Tinj 5 to 20| Cadmium Calcium Chromium Cobalt Gold, pure Copper , Iridium Iron, Cast " Wrought Lead Manganese Magnesium ( 32< Mercury •< 60' (212- Nickel Platinum Potassium Silver Sodium Steel Tin : Titanium Tungsten Zinc Specific Gravity. Range accord- ing to several Authorities. 2.56 6.66 9.74 to 2.71 to 6.86 to 9.90 8.52 to 8.96 8.6 to 8.7 1.58 5.0 8.5 to 8.6 19.245 to 19.361 8.69 to 8.92 to 23. to 7.48 to 7.9 to 11.44 7.4 11.07 7. to 8. 13.60 to 1.75 to 13.62 13.58 13.37 to 13.38 8.279 to 8.93 20.33 to 22.07 0.865 10.474 to 10.511 0.97 7.69* to 7.932t 7.291 to 7." 5.3 17. to 17. 6.86 to 7. Specific Grav- ity. Approx. Weight Mean Value, used in Calculation of Cubic Foot, lbs. Weight. 2.67 166.5 6.76 421.6 9.82 612.4 f8.60 536.3 J 8.40 |8.36 523.8 521.3 L8.20 511.4 8.853 552. 8.65 539. 19.258 1200.9 8.853 552. 1396. 7.218 450. . 7.70 480. 11.38 709.7 8. 499. 1.75 109. 13.62 849.3 13.58 846.8 13.38 834.4 8.8 548.7 21.5 1347.0 10.505 655.1 7.854 489.6 7.350 458.3 7.00 436.5 Weight per Cubic Inch, lbs. .0963 .3103 .3031 .3017 .3195 .3121 .6949 .3195 .8076 .2601 .2779 .4106 .2887 .0641 .4915 .4900 .4828 .3175 .7758 .3791 .2834 .2652 * Hard and burned. t Very pure and soft. The sp. gr. decreases as the carbon is increased. In the first column of figures the lowest are usually those of cast metals. which are more or less porous; the highest are of metals finely rolled or drawn into wire. Specific Gravity of Liquids at 60° F. Acid, Muriatic 1.200 " Nitric 1.217 " Sulphuric 1.849 Alcohol, pure 794 95 per cent 816 50 " " 934 Ammonia, 27.9 per cent 891 Bromine 2.97 Carbon disulphide 1.26 Ether, Sulphuric 72 Oil, Linseed 94 Oil, Olive 92 " Palm 97 " Petroleum 78 to .88 " Rape 92 " Turpentine 87 Whale.. Tar Vinegar.. Water .92 Compression of the following Fluids under a Pressure of 15 lbs. per Square Inch. Water 00004663 I Ether 00006158 Alcohol 0000216 I Mercury 00000265 SPECIFIC GRAVITY. 165 The Hydrometer. The hydrometer is an instrument for determining the density of liquids. It is usually made of glass, and consists of three parts: (1) the upper part, a graduated stem or fine tube of uniform diameter; (2) a bulb, or enlarge- ment of the tube, containing air ; and (3) a small bulb at the bottom, con- taining shot or mercury which causes the instrument to float in a vertical position. The graduations are figures representing either specific gravities, or the numbers of an arbitrary scale, as in Beaume's, Tvvaddell"s, Beck's, and other hydrometers. . There is a tendency to discard all hydrometers with arbitrary scales and to use only those which read in terms of the specific gravity directly. Beaume's Hydrometer and Specific Gravities Compared, Liquids Heavier than Water, sp. gr. Liquids Lighter than Water, sp. gr. 60 § Liquids Li Heavier Li than t Water, W sp. gr. si quids ghter nan ater, ). gr. Liquids Heavier than Water, sp. gr. Liquids Lighter than Water, sp. gr. 1.000 1.007 1.013 1.020 1.027 1.034 1.041 1.048 1.056 1.063 1.070 1.078 1.086 1.094 1.101 1.109 1.118 1.126 1.134 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 1.143 1.152 1.160 1.169 1.178 1.188 1.197 1.206 1.216 1.226 1.236 1.246 1.256 1.267 1.277 1.288 1.299 1.310 1.322 942 936 930 924 918 913 907 901 896 890 885 880 874 869 864 859 854 849 844 38 39 40 41 42 44 46 48 50 52 54 56 58 60 65 70 75 76 1.333 1.345 1.357 1.369 1.382 1.407 1.434 1.462 1.490 1.520 1.551 1.583 1.617 1.652 1.747 1.854 1.974 2.000 .839 1 .834 .830 3 4 .825 .820 5 .811 6 7 .802 .794 8 .785 9 10 11 12 13 14 15 '"l'.bbb" .993 .986 .980 .973 .967 .960 .954 .948 ;768 .760 .753 .745 16 17 18 Specific Gravity and "Weight of Wood, Alder Apple Ash Bamboo.. . Beech Birch Box, Cedar Cherry Chestnut . . . Cork Cypress Dogwood . . . Ebony Elm .. Fir. Gum Hackmatack Hemlock . . Hickory Holly .60 to .1 .31 to .< .62 to .1 .56 to .' .91tol.: .49 to .' .61 to .' .46 to J .24 Avge. ).80 .68 .76 .13tol 1.55 to .48 to .84 to 1 .59 .36 to .69 to , .65 1.12 .62 .66 .56 .24 .53 .76 1.23 .61 Weight Hornbeam. . . Juniper Larch....' Lignum vitse Linden Locust Mahogany. .. Maple Mulberry — Oak, Live " White.. " Red.... Pine, White . " Yellow. Poplar Spruce Sycamore Teak Walnut Willow Avge, .76 .56 .56 3 1.00 .76 .56 .56 .65 to 1. .604 .56 to 1.06 .81 .57 to .79 .68 .56 to .90 .73 .96 to 1.26 1.11 .69 to .86 .77 .73 to .75 .74 .35 to .55 .45 .46 to .76 .61 .38 to .58 .48 .40 to .50 .59 to .62 .60 .66 to .98 .82 .50 to .67 .58 .49 to .59 166 MATERIALS. Weight and Specific Gravity of Stones, Brick, Cement, etc. Pounds per Cubic Foot. Specific Gravity. Asphaltum Brick, Soft 87 100 112 125 135 140 to 150 100 112 60 78 120 to 150 120 to 140 72 to 80 90 to 110 250 156 to 172 180 to 196 160 to 170 100 to 120 130 to 150 200 to 220 50 to 55 170 to 200 150 160 to 180 140 to 160 140 to 180 90 to 100 72 74 to 80 165 90 to 110 140 to 150 170 to 180 135 to 200 170 to 200 110 to 120 166 to 175 1.39 1.6 Common Hard 1.79 2.0 " Pressed " Fire 2.16 2.24 to 2.4 1.6 1.79 .96 1.25 Clay 1.92 to 2.4 1.92 to 2.24 1.15 to 1.28 rammed Emery 1.44 to 1.76 4. 2.5 to 2.75 " flint Gneiss 1 2.88 to 3.14 2.56 to 2.72 Granite j 1.6 to 1.92 Gypsum 2.08 to 2.4 3.2 to 3.52 Lime, quick, in bulk .8 to .88 2.72 to 3.2 2.4 2.56 to 2.88 2.24 to 2.56 2.24 to 2.88 Mortar Pitch 1.44 to 1.6 1.15 1.18 to 1.28 2.64 Sand 1.44 to 1.76 2.24 to 2.4 Slate 2.72 to 2.88 2.16 to 3.4 Trap 2.72 to 3.4 Tile 1.76 to 1.92 2.65 to 2.8 PROPERTIES OF THE USEFUL. METALS. Aluminum, Al.— Atomic weight 27.1. Specific gravity 2.6 to 2.7. The lightest of all the useful metals except magnesium. A soft, ductile, malleable metal, of a white color, approaching silver, but with a bluish cast. Very non-corrosive. Tenacity about one third that of wrought-iron. For- merly a rare metal, but since 1890 its production and use have greatly in- creased on account of the discovery of cheap processes for reducing it from the ore. Melts at about 1160° F. For further description see Aluminum, under Strength of Materials. Antimony (Stibium), Sb.— At. wt. 120. Sp. gr. 6.7 to 6.8. A brittle metal of a bluish-white color and highly crystalline or laminated structure. Melts at 842° F. Heated in the open air it burns with a bluish-white flame. Its chief use is for the manufacture of certain alloys, as type-metal (anti- mony 1, lead 4), britannia (antimony 1, tin 9), and various anti-friction metals (see Alloys). Cubical expansion by heat from 32° to 212° F., 0.0070. Specific heat .050. Bismuth, Bi.— At. wt. 20S. Bismuth is of a peculiar light reddish color, highly crystalline, and so brittle that it can readily be pulverized. It melts at 510° F., and boils at about 2300° F. Sp. gr. 9.823 at 54° F., and 10.055 just above the melting-point. Specific heat about .0301 at ordinary PROPERTIES OF THE USEFUL METALS. 167 temperatures. Coefficient of cubical expansion from 32° to 212° , 0.0040. Con- ductivity for heat about 1/56 and for electricity only about 1/80 of that of silver. Its tensile strength is about 6400 lbs. per square inch. Bismuth ex- pands in cooling, and Tribe has shown that this expansion does not take place until after solidification. Bismuth is the most diamagnetic element known, a sphere of it being repelled by a magnet; and on account of its marked thermo-electric properties it is much used in laboratories in the construction of delicate thermopiles. In the arts bismuth is used chiefly in the preparation of alloys. Cadmium, Cd.— At. wt. 112. Sp. gr. 8.6 to 8.7. A bluish-white metal, lustrous, with a fibrous fracture. Melts below 500° F. and volatilizes at about 680° F. It is used as an ingredient in some fusible alloys with lead, tin, and bismuth. Cubical expansion from 32° to 212° F., 0.0094. Copper, Cu.— At. wt. 63.2. Sp. gr. 8.81 to 8.95. Fuses at about 1930° F. Distinguished from all other metals by its reddish color. Very ductile and malleable, and its tenacity is next to iron. Tensile strength 20.000 to 30,000 lbs. per square inch. Heat conductivity 73.6$ of that of silver, and su- perior to that of other metals. Electric conductivity equal to that of gold and silver. Expansion by heat from 32° to 212° F., 0.0051 of its volume. Specific heat .093. (See Copper under Strength of Materials; also Alloys.) Gold (Aurum), An..— At. wt. 197. Sp. gr., when pure and pressed in a die, 19.34. Melts at about 1915° F. The most malleable and ductile of all metals. One ounce Troy may be beaten so as to cover 160 sq. ft. of surface. The average thickness of gold-leaf is 1/282000 of an inch, or 100 sq. ft. per ounce. One grain may be drawn into a wire 500 ft. in length. The ductil- ity is destroyed by the presence of 1/2000 part of lead, bismuth, or antimony. Gold is hardened by the addition of silver or of copper. In U. S. gold coin there are 90 parts gold and 10 parts of alloy, which is chiefly copper with a little silver. By jewelers the fineness of gold is expressed in carats, pure gold being 24 carats, three fourths fine 18 carats, etc. Iridium.— Iridium is one of the rarer metals. It has a white lustre, re- sembling that of steel; its hardness is about equal to that of the ruby; in the cold it is quite brittle, but at a white heat it is somewhat malleable. It is one of the heaviest of metals, having a specific gravity of 22.38. When heated in the air to a red heat the metal is very slowly oxidized. It is insol- uble in all single acids, but is very slightly soluble in aqua regia after being heated in the state of fine powder for many hours. In a massive state, how- ever, aqua regia does not attack it. Iridium is extremely infusible. With the heat of the oxyhydrogen or electric furnaces, a globule of very small size may be melted. Mr. John Holland found that by heating the ore in a Hessian crucible to a white heat and adding to it phosphorus, and continuing the heating for a few minutes, he could obtain a perfect fusion of the metal, which could be poured out and cast into almost any desired shape. This material was about as hard as the natural grains of iridium, and contained, according co two determina- tions, 7.52$ and 7.74$ of phosphorus. By heating the metal in a bed of lime the phosphorus could be completely removed. In this operation the metal is first heated in an ordinary furnace at a white heat, and finally, after no more phosphorus makes it's appearance, it is removed and placed in an electric furnace with a lime crucible, and there heated until the last traces of phosphorus are removed; the metal which then remains will resist as much heat without fusion as the native metal. For uses of iridium, methods of manufacturing it, etc., see paper by W. D. Dudley on the "Iridium Industry," Trans. A. I. M. E. 1884. Iron (Ferrum), Fe.— At. wt. 56. Sp. gr.: Cast, 6.85 to 7.48; Wrought, 7.4 to 7.9. Pure iron is extremely infusible, its melting point being above 3000° F.,but its fusibility increases with the addition of carbon, cast iron fus- ing about 2500° F. Conductivity for heat 11.9, and for electricity 12 to 14.8, silver being 100. Expansion in bulk by heat: cast iron .0033, and wrought iron .0035, from 32° to 212° F. Specific heat: cast iron .1298, wrought iron .1138, steel .1165. Cast iron exposed to continued heat becomes permanently ex- panded 1)4, to -3 per cent of its length. Grate-bars should therefore be allowed about 4 per cent play. (For other properties see Iron and Steel under Strength of Materials.) Lead (Plumbum), Pb.— At. wt. 206.4. Sp. gr. 11.07 to 11.44 by different authorities. Melts at about 625° F., softens and becomes pasty at about 617° F. If broken by a sudden blow when just below the melting-point it is quite brittle and the fracture appears crystalline. Lead is very malleable and ductile, but its tenacity js such that it can be drawn into wire with great 168 MATERIALS. difficulty. Tensile strength, 1600 to 2400 lbs. per square inch. Its elasticity is very low, and the metal flows under very slight strain. Lead dissolves to some extent in pure water, but water containing- carbonates or sulphates forms over it a film of insoluble salt which prevents further action. (For alloys of lead see Alloys.) Magnesium, Mg<- At. wt. 24. Sp. gr. 1.69 to 1.75. Silver-white, brilliant, malleable, and ductile. It is one of the lightest of metals, weighing only about two thirds as much as aluminum. In the form of filings, wire, or thin ribbons it is highly combustible, burning with a light of dazzling brilliancy, useful for signal-lights and for flash-lights for photographers. It is nearly non-corrosive, a thin film of carbonate of magnesia forming on ex- posure to damp air, which protects it from further corrosion. It may be alloyed with aluminum, 5 per cent Mg added to Al giving about as much in- crease of strength and hardness as 10 per cent of copper. Cubical expansion by heat 0.0083, from 32° to 212° F. Melts at 1200° F. Specific heat .25. Manganese, Mn.-At. wt. 55. Sp. gr. 7 to 8. The pure metal is not used in tne arts, but alloys of manganese and iron, called spiegeleisen when containing below 25 per cent of manganese, and ferro-manganese when con- taining from 25 to 90 per cent, are used in the manufacture of steel. Metallic manganese oxidizes rapidly in the air, and its function in steel manufacture is to remove the oxygen from the bath of steel whether it exists as oxide of iron or as occluded gas. Mercury (Hydrargyrum), Hg,- At. wt. 199.8. A silver- white metal, liquid at temperatures above— 39° F., and boils at 680° F. Unchangeable as gold, silver, and platinum in the atmosphere at ordinary temperatures, but oxidizes to the red oxide when near its boiling-point. Sp.gr.: when liquid 13.58 to 13.59, when frozen 14.4 to 14.5. Easily tarnished by sulphur fumes, also by dust, from which it may be freed by straining through a cloth. No metal except iron or platinum should be allowed to touch mercury. The smallest portions of tin, lead, zinc, and even copper to a less extent, cause it to tarnish and lose its perfect liquidity. Coefficient of cubical expansion from 32° to 212° F. .0182; per deg. .000101. Nickel, Ni.— At. wt. 58.3. Sp. gr. 8.27 to 8.93. A silvery-white metal with a strong lustre, not tarnishing on exposure to the air. Ductile, hard, and as tenacious as iron. It is attracted to the magnet and may be made magnetic like iron. Nickel is very difficult of fusion, melting at about 3000° F. Chiefly used in alloys with copper, as german-silver, nickel-silver, etc., and recently in the manufacture of steel to increase its hardness and strength, also for nickel-plating. Cubical expansion from 32° to 212° F., 0.0038. Specific heat .109. Platinum, Pt.— At. wt. 195. A whitish steel-gray metal, malleable, very ductile, and as unalterable by ordinary agencies as gold. When fused and refined it is as soft as copper. Sp. gr. 21.15. It is fusible only by the oxyhydrogen blowpipe or in strong electric currents. When combined with iridium it forms an alloy of great hardness, which has been used for gun- vents and for standard weights and measures. The most important uses of platinum in the arts are for vessels for chemical laboratories and manufac- tories, and" for the connecting wires in incandescent electric lamps. Cubical expansion from 32° to 212° F., 0.0027, less than that of any other metal ex- cept the rare metals, and almost the same as glass. Silver (Argentum), Ag.- At. wt. 107.7. Sp. gr. 10.1 to 11.1, according to condition and purity. It is the whitest of the metals, very malleable and ductile, and in hardness intermediate between gold and copper. Melts at about 1750° F. Specific heat .056. Cubical expansion from 32° to 212° F., 0.0058. As a conductor of electricity it is equal to copper. As a conductor of heat it is superior to all other metals. Tin (Stannum) Sn.— At. wt. 118. Sp. gr. 7.293. White, lustrous, soft, malleable, of little strength, tenacity about 3500 lbs. per square inch. Fuses at 442° F. Not sensibly volatile when melted at ordinary heats. Heat con- ductivity 14.5, electric conductivitj r 12.4; silver being 100 in each case. Expansion of volume by heat .0069 from 32° to 212° F. Specific heat .055. Its chief uses are for coating of sheet-iron (called tin plate) aDd for making alloys with copper and other metals. Zinc, Zn,- At. wt. 65. Sp. gr. 7.14. Melts at 780° F. Volatilizes and burns in the air when melted, with bluish-white fumes of zinc oxide. It is ductile and malleable, but to a much less extent than copper, and its tenacity, about 5000 to 6000 lbs. per square inch, is about one tenth that of wrought iron. It is practically non-corrosive in the atmosphere, a thin film of car- bonate of zinc forming upon it. Cubical expansion between 32° and 212° F., MEASURES AND WEIGHTS OF VARIOUS MATERIALS. 169 0.0088. Specific heat .096. Electric conductivity 29, heat conductivity 36, silver being 100. Its principal uses are for coating iron surfaces, called " galvanizing," and for making brass and other alloys. Table Showing the Order of Malleability. Ductility. Tenacity. Infusibility Gold Platinum Iron Platinum Silver Silver Copper Iron Aluminum Iron Aluminum Copper Gold Copper Copper Gold Platinum Tin Silver Silver Lead Aluminum Zinc Aluminum Zinc Zinc Gold Zinc Platinum Tin Tin Lead Iron Lead Lead Tin FORMULA AND TABLE FOR CALCULATING THE WEIGHT OF RODS, BARS, PLATES, TUBES, AND SPHERES OF DIFFERENT MATERIALS. Notation : b = breadth, t = thickness, s = side of square, d = external diameter, d 1 = internal diameter, all in inches. Sectional areas : of square bars — s 2 ; of flat bars = bt ; of round rods = .7854d 2 ; f tubes = .7854(d 2 - dj 2 ) = 3.1416(d*- * 2 ). Volume of 1 foot in length : of square bars = 12s 2 ; of flat bars = 126* ; of round bars = 9.4248d 2 ; of tubes = 9.4248(d 2 - d 3 2 ) = 37.6992(d* - * 2 ), in cubic inches. Weight per foot length = volume x weight per cubic inch of the material. Weight of a sphere = diam. 3 X .5236 X weight per cubic incl Material. 1 1 3 If J Us °ftfl t per cubic , lbs. re Weights, ught Iron ■d n gf . 285 1 m 0, O Us C fit Cast iron 7.218 450. 37.5 3^s 2 3%bt .260415-16 2.454d 2 .1363d 3 Wrought Iron. . . . 7.7 480. 40. 3& 2 3Hbt .2779|1. 2.618d 2 .1455d s Steel 7.854 8.855 489.6 552. 40.8 46. 3.4s 2 3.833s 2 SAbt 3.8336* .2833 1.02 .31951.15 2.670d 2 3. Olid 2 .1484d 3 Copper & Bronze | (copper and tin) j .1673d 3 ** rass \ 35 Zinc Lead 8.393 523.2 43.6 3.633s 2 3.6336* .30291.09 2.854d 2 .1586d 3 11.38 709.6 59.1 4.93s 2 4.936* .4106 1.48 3.870d 2 .2150d s Aluminum 2.67 166.5 13.9 1.16s 2 1.166* .09630.347 0.908d 2 .0504d 3 Glass 2.62 163.4 13.6 1.13s 2 1.136t .09450.34 0.891d 2 .0495d 3 Pine Wood, dry . . . 0.481 30.0 2.5 0.21s 2 0.216* .01741-16 0.164d 2 .0091d 3 For tubes use the coefficient of d 2 in ninth column, as for rods, and multiply it into (d 2 — dj 2 ); or take four times this coefficient and multiply it into (dt - * 2 ). For hollow spheres use the coefficient of d 3 in the last column and multiply ic into (d 3 — d l 3 ). MEASURES AND WEIGHTS OF VARIOUS MATERIALS (APPROXIMATE). Brickwork.— Brickwork is estimated by the thousand, and for various thicknesses of wall runs as follows: 8J4-in. wall, or 1 brick in thickness, 14 bricks per superficial foot. mi " " V \\i " " " 21 " 17 " " " 2 " " " 28 " 21^ « «-« «« 2% " " " 35 " An ordinary brick measures about 814. X 4 X 2 inches, which is equal to 60 cubic inches, or 26 2 bricl ts to < i cub c foot. The a perage wei ?ht is 4} ^lbs. m MATERIALS. Fuel.— A bushel of bituminous coal weighs 76 pounds and contains 2688 cubic inches = 1.554 cubic feet. 29.47 bushels = 1 gross ton. A bushel of coke weighs 40 lbs. (35 to 42 lbs.). One acre of bituminous coal contains 1600 tons of 2240 lbs. per foot of thickness of coal worked. 15 to 25 per cent must be deducted for waste in mining. 44.8 cubic feet bituminous coal when broken down = 1 ton, 2240 lbs. 42.3 " " anthracite " " " " - 1 ton, 2240 lbs. 123 " " of charcoal = 1 ton, 2240 lbs. 70.9 " " "coke ... = 1 ton, 2240 lbs. cubic foot of anthracite coal =50 to 55 lbs. " " " bituminous" = 45 to 55 lbs. " " Cumberland coal = 53 lbs. " " Cannel coal = 50.3 lbs. " " charcoal (hardwood) = 18.5 lbs. (pine) =18 lbs. A bushel of charcoal.— In 1881 the American Charcoal-Iron Work- ers' Association adopted for use in its official publications for the standard bushel of charcoal 2748 cubic inches, or 20 pounds. A ton of charcoal is to be taken at 2000 pounds. This figure of 20 pounds to the bushel was taken as a fair average of different bushels used throughout the country, and it has since been established by law in some States. Ores, Earths, etc. 13 cubic feet of ordinary gold or silver ore, in mine =1 ton = 2000 lbs. 20 " " " broken quartz ,..= 1 ton = 2000 lbs. 18 feet of gravel in bank =1 ton. 27 cubic feet of gravel when dry,... , =1 ton. 25 " " " sand = 1 ton. 18 " " " earth in bank = 1 ton. 27 " " " " when dry .= 1 ton. 17 " " " clay = 1 ton. Cement.— English Portland, sp. gr. 1.25 to 1.51, per bbl 400 to 430 lbs. Rosendale, U. S., a struck bushel 62 to 70 lbs. Liime.— A struck bushel 72 to 75 lbs. Grain.— A struck bushel of wheat = 60 lbs.; of corn = 56 lbs. ; of oats = 30 lbs. Salt.— A struck bushel of salt, coarse, Syracuse, N. Y. = 56 lbs.; Turk's Island = 76 to 80 lbs. Weight of Earth Filling. (From Howe's " Retaining Walls.") Average weight in lbs. per cubic foot. Earth, common loam, loose 72 to 80 " shaken 82 to 92 " " " rammed moderately 90 to 100 Gravel 90 to 106 Sand 90 to 106 Soft flowing mud 104 to 120 Sand, perfectly wet 118 to 129 COMMERCIAL SIZES OF IRON BARS. Flats. Width. Thickness. Width. Thickness. Width. Thickness. % %to % m ^tol^ 4 Mto2 Vsto % 2 % to m V A to\% Htoiy 8 Qi M to 2 Mto2 1 Ys to 15/16 m 5 m fctol 2% 5^ ^to2 i% ^tol^ 2% 3/16 to W* 6 Mto2 Mtol^ y> to iy 8 H to 1^ w* J4to2 y 6 to\y 4 m 7 14 to 2 i% ^tolj^ 3 ^to2 Wz ^to2 m 3/16 to V/% Wz 14 to 2 WEIGHTS OF WROUGHT IRON BARS. 171 Rounds : M to 1% inches, advancing by 16ths, and 1% to 5 inches by 8ths. Squares : 5/16 to 1J4 inches, advancing by 16ths, and V/± to 3 inches by 8th s. Half rounds: 7/16, J^, %, 11/16, %, 1, \%, 1*4, 1)4, 1%, 3 inches. Hexagons : ' 6 Ato\y% inches, advancing by 8ths. Ovals : % X U, % X 5/16, % X %, % X 7/16 inch. Half ovals : ^ X H, % X 5/33, % x 3/16, % x 7/32, \y a X ^, 1% X %, 1% X % inch. Round-edge flats : 1% X ^, 1M X %, W& X % inch. Rands : >^ to \% inches, advancing by 8ths, 7 to 16 B. W. gauge. 134 to 5 inches, advancing by 4ths, 7 to 16 gauge up to 3 inches, 4 to 14 gauge, 3J4 to 5 inches. WEIGHTS OF SQUARE AND ROUND RARS OF WROUUHT IRON IN POUNDS PER LINEAL FOOT. Iron weighing 480 lbs. per cubic foot. For steel add £ per cent. htof are Bar Foot g- htof nd Bar Foot . . M © a &C3 t-l>i>J>l>-«Oi»«O 1 WOO«l0 01M(OOT)iQO'-iWfflmSC!00'*ODrt 10.05 MSOO 0!C0Q0(.-OOl0in-»MC0Wr-inOO05Q0Xl-«3«l0lDTP03WWi-(-.OO # >* QOi-«o^co^HOQO£-»n^cOi-iOQOJ>»0'*^>'-a5cccs»nTf>o?i--iCJCccoioco QD^SDlO^iMWOOSOOt-ffllOTliWr-iOOJOOl-lO^MffJi-iOOSl-ffllOtCC H THO!M'*»Oa5!.»!.-»OJOrHO)COTMniOCONX050^WCOMT)'l050N» © *' Mt-Omt-OCO!>OMNO«NOmNOtOt-OOTi>OCONOMt-OCON COC010MrtOX«-iOCO® i-iN«j^ioio®NXO!OOrt!Mm^iniofflt.(B»oC"ijJmTiiinicffl It CO CO<»TPCOi--l05i^inCOi--i05CO!OTf<0?OCOtD^COT-05t-inCO-i--i05COCOTfOJO i>ineOi^Oi©^C>i©COiOCOi^CTSL-lOOJ©CC©^i^C&L--»OeO©ao«OTttowosfflsoojffl fie TH«wco^'!)nno50i>a)a)0500Ti««m'*'*iotD!Dj>a)a!o>ooTH OiWffi ©^ CO OJlOQOOC010GOOfOlOGOOC010000COlOQOOCO»OCOOC010000 0?iOCOO ®WO)lOiHi>MOtD«COlOnNMO®«a)lCrHt-mOtOmX10HN050 i-Hi-tWMm^ioioo^oooajcsooi-iwojaiw^iointofONajXfflo 5 s M i Wi-ii>e»ai'*oinr-ii..ma)Tf'O"0rHi.-MM^O(CrHt-MC»^'Offlni.»m §2 H 1 iOOlOO®i-iOr«ONl-«NN«)M(»MO'Jiffl'*OiOOiOOlOn!B^(B THrtM0}MK)Tf'*10intD©i-i>XX0l0iOOHTHNC0Wt)i'^i010t0!0 1 «0MiHa3^THa3inMC»tDC00!t0MO!>'*rHa)Ttt7-ia)»0MC»ffl«l05SDWO *os'*a)Mxwi-«50i-i!ooinoioo>tosna)roi-«Nin£>ai»CSlC>OOr-lrt«OTCr;M'*'*10 0? rHCOiOJ>oooc>ieoioi-cooc>?coint-ooo?coiOi>coO(Nco nia)wOt--COOOOOC75«3C>?0510^COT}<-r-ll>MQOXXOOJ05000Hrt on 1 SMcoiotoooosOHM^iotDGOfflonnTtnocooooJO^M^intoajao m(ooiwioxHina)rt-*i»o«i«oo«®awina)i-iiOXr-Tft.oM(eo s T^r^^ciOJOiooccfOTjirj<-^ioin»oio«5«5?oj>i>£:-coooooai050JO 1 OWMT|iOfflO!QO'«^l-0!WiONO«10XO«3 - v-,^-lr-li--lCJC>J(>J(?*COOOCOCOThTfi"*^10Jr5lO»0«Cl<»tt>t-J>J>£-CCaO - O^^CO-^in«Ol-GOQ0050i-HC>?COCO-^iOCO{-!;-CC'050--CJCOCO-<*10«)t- W'tCXOWTPtaXOWlONOlT-.COlO^ClrHMini.-OCt'^CXOMTr© i^i^i^i^^OJOJcicjw'coirooscomTjiTji^Ttio^ioioicco^ocd nss£ 7 ^7^7^73^ ^7 ^7 ^7 7 3?7 :TT ^7 ^7 ^ ^ ^ ■^-tcotf3£>O5*-<00JLOi-ieoiOi-oiT-i-i-i»-i ' WEIGHTS OF FLAT WROUGHT IRON. 173 g8S8§8§8S8S8§8g8888S8888 a* in i> o ©i 10 1> o oi 10 1-' o o m' ©' m'diopgo hhhWWMWWMK 05COOOl'!Din-H»KI«^003COOO{-incOWOQO(-lfl«! WiOQOr-i^i-OMeoaWfflt-OMCWQOTrOmrtC-M s 00!>inM«OOONiOMC>OCOt-iO£OONCCO£-COOt z 0>-iwM^ioioffli-'aje>oOrt«»o®oooiH!iiio(o o S£^^^^*lNC*o«cococoT*Tr.ioininco<» OOKiWOKiSMOKiOMOOOiBMOiOOiOOinOiOO Qot-wiowci-OQONfflioejMTHOt-iowot-iawo OS t-iMm!>0SHMC!OQ0O(»*t000OM!>HlO»«!0O 0WMONT)i^00i0Na«O05C0(NtDC jOrft l-iOWOQOfflrf HOSN^OlOl-iOWOO^OJiOOi T^ccin^ooo©>Tt«o HM^(di»OJO«3iO^QOOiHMi(:cOr<5JNOMttio -©*cjcooo-*ococ smaow^NfflrtiooincB^oomwHOC iij'*ioi>aJdi-f. ONODOOOSOiOrHMTMOtOOOCSO -KNCNJOiC-JCOCOCOCO-q h w cm s-i a o? co co « H ©i ©? ©? ©i CO CO « m«:awma)i-!'cddrt««iiO(DNcocNioNo'o(inNd STHrti-n-iiHiirKSNiHMweijmmv ;>©ascsoio»05cocccococct-£--t-i~« gmio^Tceo c^^ior-S^coinJ>o5^-coint-6iT--iico5co£^r T^oJcd^»nt^cooio-r-ico-*in«5t--oiT-ico5dod'i-ico"inQ6 lftC5-*Q0COa0©}t-*-'«O©iO©^ owHooiooa -iWNWCJWNWC? JSOSOOOONOOlOTtiTfMCOW^'-iOO SS8 [ CM CO '93 CO Tii>-'in<»WQOw®0'»t , Qocot-inwo)OQ( «WW«CJMM ^co^" I 88£8 ° ggSSi? S 2 174 MATERIALS. WEIGHT OF IRON AND STEEL SHEETS. Weights per Square Foot. (For weights by new U. S. Standard Gauge, see page 31.) Thickness by Birmingham Gauge. Thickness by American (Brown and Sharpe's) Gauge. Thick- Thick- No. of ness in Iron. Steel. No. of ness in Iron. Steel. Gauge. Inches. Gauge. Inches. 0000 .454 18.16 18.52 0000 .46 18.40 18.77 000 .425 17.00 17.34 000 .4096 16.38 16.71 00 .38 15.20 15.30 00 .3648 14.59 14.88 .34 13.60 13.87 .3249 13.00 13.26 1 .3 12.00 12.24 1 .2893 11.57 11.80 2 .284 11.36 11.59 2 .2576 10.30 10.51 3 .259 10.36 10.57 3 .2294 9.18 9.36 4 .238 9.52 9.71 4 .2043 8.17 8.34 5 .22 8.80 8.98 5 .1819 7.28 7.42 6 .203 8.12 8.28 6 .1620 6.48 6.61 7 .18 7.20 7.34 7 .1443 5.77 5.89 8 .165 6.60 6.73 8 .1285 5.14 5.24 9 .148 5.92 6.04 9 .1144 4.58 4.67 10 .134 5.36 5.47 10 .1019 4.08 4.16 11 .12 4.80 4.90 11 .0907 3.63 3.70 12 .109 4.36 4.45 12 .0808 3.23 3.30 13 .095 3.80 3.88 13 .0720 2.88 2.94 14 .083 3.32 3.39 14 .0641 2.56 2.62 15 .072 2.88 2.94 15 .0571 2.28 2.33 16 .065 2.60 2.65 16 .0508 2.03 2.07 17 .058 2.32 2.37 17 .0453 1.81 1.85 18 .049 1.96 2.00 18 .0403 1.61 1.64 19 .042 1.68 1.71 19 .0359 1.44 1.46 20 .035 1.40 1.43 20 .0320 1.28 1.31 21 .032 1.28 1.31 21 .0285 1.14 1.16 22 .028 1.12 1.14 22 .0253 1.01 1.03 23 .025 1.00 1.02 23 .0226 .904 .922 24 .022 .88 .898 24 .0201 .804 .820 25 .02 .80 .816 25 .0179 .716 .730 26 .018 .72 .734 26 .0159 .636 .649 27 .016 .64 .653 27. .0142 .568 .579 28 .014 .56 .571 28 .0126 .504 .514 29 .013 .52 .530 29 .0113 .452 .461 30 .012 .48 .490 30 .0100 .400 .408 31 .01 .40 .408 31 .0089 .356 .363 32 .009 .36 .367 32 .0080 .320 .326 33 .008 .32 .326 33 .0071 .284 .290 34 .007 .28 .286 34 .0063 .252 .257 35 .005 .20 .204 35 .0056 .224 .228 Iron. Steel. Specific gravity 7.7 7.854 Weight per cubic foot 480. 489 . 6 " " inch .2778 .2833 As there are many gauges in use differing from each other, and even the thicknesses of a certain specified gauge, as the Birmingham, are not assumed the same by all manufacturers, orders for sheets and wires should always state the weight per square foot, or the thickness in thousandths of an inch. WEIGHT OF PLATE IROiN". 175 - 0«t-OWt-OMt-OMNOMt-OCON , OM«OM!OOM!OOM»CM»OM(OOi.-WON»Ot"030t'KOl-030 OMSO»©t-i-t-a)QOXOic»aoOi-iM9!Mv*ioffl«oN9oao)0 so OMioooomoNomiooootomooowio iotoi.-ooo>-iwMi050(»xO'-'OjMi050^0MinxominoooroffioooMin NOMWowffloiiiocoT-nfioo^TiiJ-omofflsuBmiHi-MOfflWWiniHt- SO TP Tji -<3i to W »Q JO SO-SO SO J> £* t- 00 00 QO OS Ol O O t-i t-i 0» CO CO TP in «p SO SO C- 00 00 ^ o«WiOi-oo»-miot'OiOrtMiOffflocoi- offl.xi-oicio^cowrtoorooo^toioioniHoooMcconoQOt-mMWO WM'*Tl(^i'*iOiOiO«OSO«Ot-l'i-t-lX)(X)QOO>e»Oi-i'- '. ■ ■. - 1 iCiOtB?QOCOC35-*OiOOCOT-it-C-» M00WTI'TP'*Tl"IOIOiOl0(0®tDl-i>l-L-00000S0!OOi-ir.NKiMTl''*K:i0(0 « ooooooooooooooooooooooo o«K!t»o«m!-05imNO«iONONiooiooinoiocicoiooiootf)0 eoM«icO'»'!t-*-*mioiOK:®ffl«noi>t.»Nxa?c»caoOrH-oiC(mcoTr'woNMONKi CO"MK"*iOlOS3NCOroOOrt«ro^lO!OXOriMiOfflXO«Mmt-a)0 lOi'OrHMinNoirteoiot-OMTftoxowfflOiooMt-^ioOTfxfMffloiD wejWMMsjMM^^iTf^ioiOioioioefflasi-^t'OOWfflosoooT-n-iww m CO 2 oooioMootiioNOooowoooieMOooiooiooinortOioojoo lOCONnOKt-ffllOnWi-iOaOi-ffiinmWOl-iOUOl-inMOL'lOMOOOK) O a Mtsooooii»if;Na-o3io!oa)OMTf(oomi-"inw«650MNTHir;anj ««NwcomK)Mm«'*Tfi^'*TTioiOioio®tDSi!>t»c-a)»cscsc»ooOT^ a $ ONmONWOi-COONMOt-MONMOWNOWNQMC-OCONOCON o»«o®woewofflwo«MO!ewow(Oon(oo»fflon»owoo go 0) a o i-i co m to cc"o THMintoxdrtcoofflx'o'cofflocotBOMcodcotodMtod -^ CM O! D> 0«ini-05>COiOCfflO*OiOOiOOiCCiOO i0!Ol-Qt)O«WWin!0t--00OHWMOffl£-O5!i0l-O«i0l-OWi0l>OWiO 1 -irti-i^(c^^o^cococococococoTrTjiTj'Tfoioiniococococos>.i.~t- 3 12.50 13.54 14.58 15.63 16.67 17.71 18.75 19.79 20.83 21.88 22.92 23.96 25.00 26.04 27.08 28.13 29.17 30.21 31.25 33.33 35.42 37.50 39.59 41.67 43.75 45.84 47.92 50.00 52.08 54.17 56.25 58.33 60.42 62.50 35 OMNOWNOWNOMNOCCNOCONONMONCOONWONMOC-WO o»oiow"0oo©iowrto»©iomi-i0!0»oo»oomo®woono 00»-i01CO'*10 10tO(>QDaOO.-«M'*10CSa)Ori'05lOtOQOOHSi3 10®000 to owiftxowiONOwoxowifloooMinoiooiooiooiooioOinowo inr-it-coo»«a)io^t.MOB5)X>nr,i-oniai-ooiio£-osjiot-OMio t-OOM-aOOrHi-iMMM^lOlOOCONOOCOOi-iOjmiOfflC-OOOweJMlOfflN » o«»«t-»o«mioMJooMmioNQOO»t-o«Noeot-omt-o»NO 0*»«COOl005m£-r-nOOTXC>)(OOlO«IHOXtDmMiHOX®OCOnO mwins>«£'t-!>ooa)0505000rtr-«ww'*ioffltoi.-xopp«2' w ^-sf. 7 OrtrfOTOT^inWNQOXaO-'WWm'tiOl-XOCJWiOl'XOOMiOl-XO iot-o>THmiot-ai-noiO£-o}!'*(oxo«fflCinoni-r.iooTtiix)M»oio e}CiO!McomP3CO'*^"*'*ininioioioo©!ONc-£-Qoa)0!OiOOOT-iiH8}OJ a rtHT-i«i-ii-iTHH«wc)otww««wc)MCOKiM«Tt 5.75 35 69 104 138 173 207 242 276 311 345 380 x 4 5.11 31 61 92 123 153 184 215 246 276 307 338 4 x 4 4.54 27 55 82 109 136 164 191 218 246 272 300 x 3^ 3.97 24 48 72 96 119 143 167 181 215 238 262 x 3 3.40 20 41 61 82 102 122 143 163 184 204 224 3^x zy Q 3.48 21 42 63 84 104 125 146 167 188 209 230 x 3 2.98 18 36 54 72 89 107 125 143 161 179 197 3x3 2.56 15 31 46 61 77 92 108 123 138 154 169 SIZES AKD WEIGHTS OF STKtJCTlJRAL SHAPES. 177 SIZES AND WEIGHTS OF STRUCTURAL. SHAPES. Minimum and Maximum "Weights and Dimensions oi Carnegie I-Beams. STEEL BEAMS. Section Weight per Foot, in lbs. Flange Width. Web Thickness. Increase of Web and Flanges for a£g 03 03.5 fipq Min. Max. Min. Max. Min. Max. crease of weight. B 1 24 80.00 100.00 6,95 7.20 .50 .75 .0123 B 2 20 80.00 100.00 7 00 7.30 .60 .90 .015 B 3 20 64.00 75.00 6.25 6.41 .50 .66 .015 B 4 15 80.00 100.00 ^.41 6.79 .77 1.16 .020 B 5 15 60.00 75.00 6.04 6.34 .54 .84 .020 B 6 15 50.00 59.00 5.75 5.93 .45 .63 .020 *B 7 15 41.00 49.00 5.50 5.66 .40 .56 .020 B 8 12 40.00 56.70 5.50 5.91 .39 .80 .025 *B 9 12 32.00 39.00 5.25 5.42 .35 .52 .025 BIO 10 33.00 40.00 5.00 5.21 .37 .58 .029 Bll 10 25.50 32.00 4.75 4.94 .32 .51 .029 B12 9 27.00 33.00 4.75 4.95 .31 .51 .033 B13 9 21.00 26.00 4.50 4.66 .27 .43 .033 B14 8 22.00 27.00 4.50 4.68 .27 .45 .037 B15 8 18.00 21.70 4.25 4.39 .25 .39 .037 B16 7 20.00 22.00 4.25 4.33 .27 .35 .042 B17 7 15.50 19.00 4.00 4.15 .23 .38 .042 B18 6 16.00 20.00 3.63 3.83 .26 .46 .049 B19 6 13.00 15.00 3.50 3.60 .23 .34 .049 B20 5 13.00 16.00 3.13 3.31 .26 .44 .059 B21 5 10.00 12.00 3.00 3.12 .22 .33 .059 B22 4 10.00 13.00 2.75 2.97 .24 .46 .074 B23 4 7.50 9.00 2.63 2.74 .20 .31 .074 B24 4 6.00 8.00 2.18 2.33 .18 .33 .074 Iron. Steel. Given weight in pounds per foot, to find sectional area-=- 3J^ 3.4 " " " " " x 0.3 .2941 Given sectional area, to find weight in lbs. per foot x Zy§ 3.4 " " " " " "lbs. per yard x 10 10.2 Maximum and Minimum Weights and Dimensions of Carnegie Deck Beams. Section Index. Depth of Beam, inches. Weight per Foot, lbs. Flange Width. Web Thickness. Increase of Web and Flanges per lb. in- crease of weight. Min. Max. Min. Max. Min. Max. B100 B101 B102 B103 B105 10 9 8 7 6 27.23 26.52 20.15 18.10 15.30 35.70 30.60 24.48 23.46 18.36 5.25 4.94 5.00 4.87 4.38 5.50 5.07 5.16 5.10 4.53 .38 .44 .31 .31 .28 .63 .57 .47 .54 .43 .029 .032 .037 .042 .049 178 MATERIALS. Weights and Dimensions of Carnegie Steel Channels. Increase Depth Weight per Foot, in lbs. Flange Width. Web Thickness. of Web and Sec- tion of Chan- nel, in Flanges for each lb. in- Index inches. Min. Max. Min. Max. Min. Max. crease of weight. CI 15 32.00 51 00 3.40 3.78 .40 .78 .020 C2 12 20.00 30.25 2.90 3.15 .30 .55 .025 C3 10 15.25 23.75 2.66 2.91 .26 .51 .029 C4 9 12.75 20.50 2.44 2.69 .24 .49 .033 C5 8 10.00 17.25 2.20 2.47 .20 .47 .037 C6 7 8.50 14.50 2.00 2.25 .20 .45 .042 C7 6 7.00 12.00 1.89 2.14 .19 .44 .049 C8 5 6.00 10.25 1.78 2.03 .18 .43 .059 C9 4 5.00 8.25 1.67 1.91 .17 .41 .074 Weights and Dimensions of Carnegie 'Z-Bars. Size. Weight. Section Thickness of Metal. Index. Flange. Web. Flange. Iron. Steel. Z 1 Va 3 y 2 6 3 H 15.3 15.6 ** 7-16 3 9-16 6 1-16 3 9-16 18.0 18.3 " V2 3 Va 6 H 3 % 20.6 21.0 Z 2 9-16 3 \b 6 3 y* 22.3 22.7 '• Va 3 9-16 6 1-16 3 9-16 24.9 25.4 " 11-16 3 Va 6 % 3 V 8 27.5 28.0 Z 3 % 3 M 6 3 H 3 9-16 28.8 29.3 " 13-16 3 9-16 6 1-16 31.3 32.0 " Va 3 Va 6 % 3 % 33.9 34.6 Z 4 5-16 3 y A 5 3 M 11.3 11.6 " Va 3 5-16 5 1-16 3 5-16 13.7 13.9 " 7-16 3 Va 5 y 8 3 % 16.0 16.4 Z 5 14 3 Va 5 3 H 17.5 17.8 44 9-16 3 5-16 5 1-16 3 5-16 19.8 20.2 " Va 3 % 5 y 8 in 22.1 22.6 Z 6 11-16 3 X 5 23.2 23.7 " H 3 5-16 5 1-16 3 5-16 25.5 26.0 >t 13-16 3 % 5 y a 3 % 27.8 28.3 Z 7 H 3 1-16 4 3 1-16 8.0 8.2 5-1 6 3 H 4 1-16 3 H 10.1 10.3 " % 3 3-16 4 % 3 3-16 12.2 12.4 Z 8 r-i6 3 1-16 4 3 1-16 13.5 13.8 ** M 3 % 4 1-16 3 H 15.5 15.8 •* 9-16 3 3-16 4 H 3 3-16 17.6 17.9 Z 9 Va 3 1-16 4 3 1-16 18.5 18.9 " 11-16 3 X 4 1-16 3 y 8 20.5 20.9 " H 3 3-16 4 14 3 3-16 22.5 22.9 Z10 H 2 11-16 3 2 11-16 6.6 6.7 " 5-16 2 % 3 1-16 2 % 8.3 8.4 Zll % 2 11-16 3 2 11-16 9.5 9.7 " 7-16 2 % 3 1-16 2 % 11.2 11.4 Z12 Y* 2 11-16 3 2 11-16 12.3 12.5 9-16 2 % 3 1-16 2 % 13.9 14.2 SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 179 Pencoyd Steel Angles. EVEN LEGS. t Size in Inches. Approximate Weight in Pounds per Foot for Various Thicknesses in Inches. o o % 3-16 M 5-16 .375 7-16 Vo 9-16 % 11-16 % % 1 'A .125 .1875 .25 .3125 .4375 .50 .5625 .625 .6875 .75 .875 1.00 120 6 x6 14.8 17.3 19.9 22.3 24.9 26.5 29.1 34.2 39.3 121 5 x5 14.3 16.4 18.5 20.7 22.8 29 2 33 4 122 4 x4 8.2 9.8 11.3 13.0 14.6 16.1 177 19 3 123 3^x3^ 7.1 8.6 10.0 11.4 12.8 14.2 124 3 x3 4.9 6.0 7.1 8.3 9.4 10.5 11.6 125 '2% x 2?4 4.5 5.6 6.7 7.8 8.9 126 2^x2^ 3.1 4.1 5.1 6.1 7.1 127 •2U*m 2.7 3.6 4.5 5.4 128 2 x2 2.44 3.3 4.1 4.9 12SJ iMxiM 2.14 2.9 3.6 4.4 130 l^xl^ 1.16 1.80 2.4 3.0 3.6 131 m*m 1.02 1.53 2.04 132 1 xl 0.82 1.16 1.53 UNEVEN LEGS. 6 Approximate Weight in Pounds per Foot for Various cc Size Thicknesses in Inches. '°.2 Inches. H 3-16 H 5-16 % 7-16 % 9-16 % 11-16 H 7,4 1 £ .125 .1875 .25 .3125 .375 .43/5 .50 .5625 .625 .6875 .75 .875 1.00 154 7 x3^ 17.0 18.9 20.9 22.8 24.8 28 6 32.5 152 6^x4 12.9 15.0 17.1 19.3 21.4 23.6 25,7 30.0 34.3 140 6 x4 12.2 14.4 16.4 18.6 21 22.8 24.9 29.1 33.3 151 6 xSy 2 11.5 13.6 15.6 17.6 19.7 21,7 23 8 27 8 31.9 153 5^x3V6 11.0 12.8 14.6 16.4 141 5 x4 11.0 12.8 14.6 16.4 18 2 20 21 8 142 5 x3^ 8.7 10.3 12.0 13.6 15.2 16.8 18 5 143 5 x3 8.2 9.7 11.2 12.8 14.3 15.8 17 3 18 9 144 4^x3 7.7 9.2 10.6 12.1 13.6 15.0 16 5 18 145 4 x3K 3 7.7 9.2 10.6 12.1 13.6 15.0 16.5 18 146 4 x3 7.1 S.6 10.0 11.4 12 8 14 2 147 3^x3 6.6 7.9 9.2 10.5 11.8 13.1 150 3^x2^ 4.9 6.0 7.1 8.3 9.4 159 3^x2 4.5 5.6 6.7 148 3 x2^ 4.5 5.6 6.7 7.8 8 9 149 3 x2 4.1 5.1 6.1 7.1 8 2 155 2^x2 2.7 3.6 4.5 5.4 6.3 7,2 156 2^x1^ 2.24 a ! 3.8 4.6 157 2 xlM 1.94 2.7 3.3 4.0 180 MATERIALS. Pencoyd Tees. EVEN TEES. UNEVEN TEES. Weight per Weight per Foot. Size Foot. Size Chart Chart Number. Number. Inches. Iron. Steel. Iron. Steel. 70 4 x4 12.40 12.65 107 5 x4 14.70 15.00 71 &A x Wz 10.17 10.37 106 5 x3i^ 16.13 16.46 72 3 x3 8.33 8.50 93 5 x2 9-16 11.03 11 25 82 3 x3 6.43 6.56 92 5 x2^ 10.23 10.44 83 7.53 7.68 90 4^x314 14.83 15.13 84 2^x2^ 2^x2^ 4.83 4.93 109 4 x4^ 13.23 13.50 73 6.50 6.63 91 4 x3^ 13.93 14.21 74 2^x2^ 5.73 5.85 94 4 x3 8.63 8 81 2U * Wa 3.90 3.98 95 4 x3 8.37 8.53 76 2J4x2J4 3.93 4.01 96 4 x2 6.43 6.56 77 2 x2 3.47 3.54 97 3 x3^ 9.37 9.55 78 2.37 2.41 98 3 x2^ 7.93 8.09 79 2.00 2.04 110 3 x2^ 5.87 5.98 80 ty\*ty± 1.50 1.53 111 3 x2^ 6.87 7.00 81 1 xl 1.03 1.05 117 3 *2y z 5.00 5.10 85 4 x4 10.98 11.19 99 3 xlK 2 3.73 3.81 105 2^| xl% 2^x114 7.13 7.28 104 6.53 6.66 100 3.03 3.09 108 2M x 9-16 2.20 2.24 101 2 xl^ 2.90 2.96 112 2 xl 1-16 2.07 2.11 102 2 xl 2.33 2.38 103 2 x 9-16 2.03 2.07 116 iMxiM 3.47 3.54 113 lMxl 1-16 1.87 1.90 114 l^x 15-16 1.37 1.39 115 lJ4x 15-16 1.13 1.16 118 3 x2i/ 2 5.92 6.03 119 m**% 5.63 5.74 Pencoyd Car-Builders' Channels , Iron. CG.GrQ « s 1 CO he a; <°s [m"" Approximate Weight in Pounds per *-P-i -§ £$ *S Foot for Each Thickness of 3 a Web, in Inches. T3 J § O £ a 12 g»| 3^ PI Safe § S.-^fc CD 5-16 ¥a 7-16 k 9-16 % 55 13 H 29.5 29.5 32.2 34.9 37.6 40.3 .023 54 12 3 9-32 22.4 23.6 26.1 28.6 31.1 33.6 .025 33U 10^ 10^ 7-16 23.6 23.6 25.8 .029 33 5-16 17.6 17.6 19.8 .029 Pencoyd Car- Builders' Channels , Steel. 55 33 13 12 10^ 10^ We, 3 \Wh l2i/ 2 7-16 5-16 30.1 22.8 24.1 17.9 24.1 17.9 30.1 26.6 20.2 32.9 29.2 24.1 35.6 31.7 26.3 38.4 34.3 41.1 .022 .024 .028 .028 SIZES AND WEIGHTS OF ROOFING MATERIALS. 181 SIZES AND WEIGHTS OF ROOFING MATERIALS. Corrugated Iron (Phoenix Iron Co.). BLACK IRON. GALVANIZED IRON. Thick- ness in Indies. Weight in Lbs. per Sq. Ft., Flat. Weight iu Lbs. per Sq. Ft. on Roof. Flat. Weight in Lbs. per Sq. Ft., on Roof. Corrugated Weight in Lbs. per Sq. Ft., Flat. Weight in Lbs. per Sq. Ft. on Roof. Flat. Weight in Lbs. per Sq. Ft., on Roof. Corrugated 0.065 0.049 0.035 0.028 0.022 0.018 2.61 1.97 1.40 1.12 0.88 0.72 3.03 2.29 1.63 1.31 1.03 0.84 3.37 2.54 1.82 1.45 1.14 0.93 3.00 2.37 1.75 1.31 1.06 0.94 3.50 2.76 2.03 1.53 1.24 1.09 3.88 3.07 2.26 1.71 1.37 1.21 The above table is calculated for the ordinary size of sheet, which is from 2 to 2J/2 feet wide, and from 6 to 8 feet long, allowing 4 inches lap in length and 2)/2 inches iu width of sheet. The galvanizing of sheet iron adds about one- third of a pound to its weight per square foot. In corrugated iron made by the Keystone Bridge Co., the corrugations are 2.42V long, measured on the straight line; they require a length of iron of 2.725'' to make one corrugation, and the depth of corrugation is 21-32". One corrugation is allowed for lap in the width of the sheet and 6" in the length, for the usual pitch of roof of two to one. Sheets can be corrugated of any length not exceeding ten feet. The most advantageous width is 30^j", which (allowing \Q' for irregularities) will make eleven corrugations = 30", or, making allowance for laps, will cover 24J4" of the surface of the roof. By actual trial it was found that corrugated iron No. 20, spanning 6 feet, will begin to give a permanent deflection for a load of 30 lbs. per square foot, and that it will collapse with a load of 60 lbs. per square foot. The distance between centres of purlins should therefore not exceed 6 feet, and, prefer- ably, be less than this. Terra-Cotta. Porous terra-cotta roofing 3" thick weighs 16 lbs. per square foot and 2" thick, 12 lbs. per square foot. Ceiling made of the same material 2" thick weighs 11 lbs. per square foot. Tiles. Flat tiles 6J4" X 10^" X %" weigh from 1480 to 1850 lbs. per square of roof, the lap being one-half the length of the tile. Tiles tvith grooves and fillets weigh from 740 to 925 lbs. per square of roof. Pan-tiles U}£" X 10^" laid 10" to the weather, weigh 850 lbs. per square. Tin. The usual sizes for roofing tin are 14" X 20" and 20" X 28". Without allowance for lap or waste, tin roofing weighs from 50 to 62 lbs. per square. Tin on the roof weighs from 62 to 75 lbs. per square. Roofing plates or terne plates (steel plates coated with an alloy of tin and lead) are made only in IC and IX thicknesses (27 and 29 Birmingham gauge). "Coke" and "charcoal" tin plates, old names used when iron made with coke and charcoal was used for the tinned plate, are still used in the trade, although steel plates have been substituted for iron ; a coke plate now commonly meaning one made of Bessemer steel, and a charcoal plate one of open-hearth steel. The thickness of the tin coating on the plates varies with different " brands. 11 For valuable information on Tin Roofing, see circulars of Merchant & Co., Philadelphia. 182 MATERIALS. TIN PLATES. (TINNED SHEET STEEL.) Standard Stock Sizes, with Number of Sheets and Net Weight per Box. B. W. Thickness. Size. Sheets. Net Weight lbs. B. W. Thickness. Size. Sheets. Net Weight 29 IC 10x14 225 108 29 IC 10x20 225 160 27 IX lOx 14 225 135 27 IX 10x20 225 195 26 IXX 10x14 225 160 26 IXX 10x20 225 222 29 IC 12x12 225 110 29 IC 11x22 225 190 27 IX 12x12 225 138 27 IX 11 x22 225 235 26 IXX 12x12 225 165 26 IXX 11x22 225 275 29 IC 14x20 112 108 29 IC 12 x 24 112 110 27 IX 14x20 112 135 27 IX 12x24 112 138 26 IXX 14 x 20 112 160 26 IXX 12x24 112 165 25 IXXX 14x20 112 180 29 IC 13 x 26 112 132 24^ IXXXX 14x20 112 200 27 IX 13x26 112 162 29 IC 20x28 112 216 26 IXX 13x26 112 192 27 IX 20 x 28 112 270 29 IC 14x22 112 120 26 IXX 20 x 28 112 320 27 IX 14x22 112 148 25 IXXX 20x28 56 180 26 IXX 14x22 112 174 24^ IXXXX 20x28 56 200 29 IC 14x24 112 130 29 IC 13x13 225 132 27 IX 14x24 112 161 27 IX 13x13 225 162 26 IXX 14x24 112 190 26 IXX 13x13 225 192 29 IC 14x28 112 155 29 IC 14x14 225 155 27 IX 14 x 28 112 193 27 IX 14x14 225 193 26 IXX 14x28 112 230 26 IXX 14x14 225 230 29 IC 14x31 112 178 29 IC 15x15 225 178 27 IX 14x31 112 210 27 IX 15x15 225 218 26 IXX 14x31 112 240 26 IXX 15x15 225 260 27 IX 14x56 56 185 29 IC 16x16 225 200 26 IXX 14 x 56 56 220 27 IX 16x16 225 248 27 IX 14x60 56 200 26 IXX 16x16 225 290 26 IXX 14x60 56 240 29 IC 17x17 225 230 29 IC 15x21 112 120 27 IX 17x17 225 289 27 IX 15 x 21 112 152 26 IXX 17x17 225 340 26 IXX 15x21 112 176 29 IC 18x18 112 138 29 IC 16x19 112 120 27 IX 18x18 112 158 27 IX 16x19 112 147 26 IXX 18x18 112 178 26 IXX 16x19 112 170 29 IC 20x20 112 160 29 IC 16x20 112 127 27 IX 20x20 112 195 27 IX 16x20 112 154 26 IXX 20x20 112 222 26 IXX 16x20 112 180 29 IC 22x22 112 190 29 IC 16x22 112 138 27 IX 22x22 112 235 27 IX 16x22 112 170 26 IXX 22x22 112 275 26 IXX 16x22 112 200 29 IC 24x24 112 220 27 IX 24x24 112 276 26 IXX 24x24 112 330 B. W. Thickness. Size. Sheets. Net Weight lbs. B. W. Thickness. Size. Sheets. Net Weight lbs. 28 DC 12^x17 100 94 23 DXXX 15x21 100 244 25 DX 12^x17 100 122 22 DXXXX 15x21 100 275 24 DXX 12^x17 100 143 28 DC 17x25 50 94 23 DXXX 121^x17 100 164 25 DX 17x25 50 122 22 DXXXX 12^x17 100 185 24 DXX 17x25 50 143 28 DC 15x21 100 130 23 DXXX 17x25 50 164 25 DX 15x21 100 180 22 DXXXX 17x25 50 185 24 DXX 15x21 100 213 Terne Plates, 112J|2^J^ 801! i a box 1 Tagger's Tin and Iron, 36 and . per box. 12 lbs.; IX 140 " " " 28, IC, 224 lbs., 1X280 " " " B. W. G., 10 x 14 and 14 x 20. 112 lbs. per box. SIZES AtfD WEIGHTS OF ROOEIXG MATERIALS. 183 Slate. Number and superficial area of slate required for one square of roof. (1 square ~ 1U0 square feet.) Dimensions in Inches. Number per Square. Superficial Area in Sq. Ft. Dimensions in Inches. Number per Square. Superficial Area in Sq. Ft. 6x12 7x12 8x12 533 457 400 355 374 327 291 261 246 221 213 192 267 12x18 10x20 11x20 12x20 14x20 16x20 12x22 14x22 12x24 14x24 16 x 24 14x26 16x26 160 169 154 141 121 137 126 108 114 98 86 89 78 240 235 9x12 7x14 8x14 9x 14 '""254" 231 10x14 8x16 9x16 246 228 10x16 9x18 10x18 240"' 225 As slate is usually laid, the number of square feet of roof covered by one slate can be obtained from the following formula : width X (length — 3 inches) _ the number of square feet of roof covered. Weight of slate of various lengths and thicknesses required for one square of roof : Length Weight in Pounds per Square for the Thickness. Inches. %" 3-16" Ya!' %" H" %" H" 1" 12 483 724 967 1450 1936 2419 2902 3872 14 460 688 920 1379 1842 2301 2760 3683 16 445 667 890 1336 1784 2229 2670 3567 18 434 650 869 1303 1740 2174 2607 3480 20 425 637 851 1276 1704 2129 2553 3408 22 418 626 836 1254 1675 2093 2508 3350 24 412 61? 825 1238 1653 2066 2478 3306 26 407 610 815 1222 1631 2039 2445 3263 The weights given above are based on the number of slate required for one square of roof, taking the weight of a cubic foot of slate at 175 pounds. Pine Shingles. Number and weight of pine shingles required to cover one square of Number of Inches Exposed to Weather. Number of Shingles per Square of Roof. Weight in Pounds of Shingle on One-square of Roofs. Remarks. 4 ft ft 900 800 720 655 600 216 192 173 157 144 The number of shingles per square is for common gable-roofs. For hip- roofs add five per cent, to these figures. The weights per square are based on the number per square. 184 MATERIALS. Skylight Glass. ■ The weights of various sizes and thicknesses of fluted or rough plate-glass required for one square of roof. Dimensions in Inches. Thickness in Inches. Area in Square Feet. Weight in Lbs. per Square Of Roof. 12x48 15x60 20x100 94x156 3-16 % 3.997 6.246 13.880 101.768 250 350 500 700 In the above table no allowance is made for lap. If ordinary window-glass is used, singrle thick glass (about 1-16") will weigh about 82 lbs. per square, and double thick glass (about %") will weigh about 164 lbs. per square, no allowance being made for lap. A box of ordinary window-glass contains as nearly 50 square feet as the size of the panes will admit of. Panes of any size are made to order by the manufacturers, but a great variety of sizes are usually kept in stock, ranging from 6x8 inches to 36 x 60 inches. APPROXIMATE WEIGHTS OF VARIOUS ROOF- COVERINGS. For preliminary estimates the weights of various roof coverings may be taken as tabulated below: ■pj aTT ,„ Weight in Lbs. per JName> Square of Roof. Cast-iron plates (%" thick) 1500 Copper 80-125 Felt and asphalt , 100 Felt and gravel 800-1000 Iron, corrugated 100-375 Iron, galvanized, flat 100- 350 Lath and plaster 900-1000 Sheathing, pine, 1" thick yellow, northern . . 300 " •' " " southern.. 400 Spruce, 1" thick 200 Sheathing, chestnut or maple, 1 " thick 400 " ash, hickory, or oak, 1" thick 500 Sheet iron (1-16" thick) 300 " " " and laths 500 Shingles, pine 200 Slates (W thick) 900 Skylights (glass 3-16" to Yq" thick) 250-700 Sheet lead 500- 800 Thatch 650 Tin 70-125 Tiles, flat . . 1500-2000 (grooves and fillets) 700-1000 " pan 1000 " with mortar 2000-3000 Zinc 100-200 WEIGHT OF CAST-IRON" PIPES OR COLUMNS. 185 WEIGHT OF CAST-IRON PIPES OR COLUMNS. In libs, per Lineal Foot. Cast iron = 450 lbs. per cubic foot. Bore. Thick. of Metal. Weight per Foot. Bore. Thick. of Metal. Weight per Foot. Bore. Thick. of Metal. Weight per Foot. Ins. Ins. Lbs. Ins. Ins. Lbs. Ins. Ins. Lbs. 3 % 12.4 10 H 79.2 22 H 167.5 H 17.2 10K H 54.0 % 196.5 22.2 % 68 2 23 S 174.9 ®A I 14.3 H 82.8 205.1 19.6 11 H 56.5 l 235.6 % 25.3 % 71.3 24 S 182.2 4 % 16.1 % 86.5 213.7 % 22.1 11^2 1 58.9 l 245.4 % 28.4 74.4 25 s 189.6 *H % 17.9 % 90.2 222.3 H 24.5 12 1 61.3 1 255.3 % 31.5 77.5 26 n 197.0 5 % 19.8 *A 93.9 % 230.9 u 27.0 12^ 63.8 i 265.1 % 34.4 % 80.5 27 H 204.3 5H 21.6 29.4 13 V 97.6 66.3 l 239.4 274.9 % 37.6 % 83.6 28 S 211.7 6 % 23.5 % 101.2 248.1 14 31.8 14 % 71.2 l 284.7 % 40.7 % 89.7 29 s 219.1 6^ % 25.3 % 108.6 256.6 Yz 34.4 15 95.9 1 294.5 % 43.7 % 116.0 30 % 265.2 7 % 27.1 % 136.4 1 304.3 M 36.8 16 % 102.0 m 343.7 Vs 46.8 M 123.3 31 % 273.8 Wz ¥s 29.0 if 145.0 l 314.2 8 39.3 17 108.2 i^ 354.8 49.9 % 130.7 32 % 282.4 8 % 30.8 % 153.6 i 324.0 41.7 18 % 114.3 m 365.8 % 52.9 H 138.1 33 % 291.0 8^ Vz 44.2 % 162.1 i 333.8 % 56.0 19 % 120.4 m 376.9 s 68.1 I 145.4 34 % 299.6 9 46.6 170.7 l 343.7 % 59.1 20 ft 126.6 i/^j 388.0 % 71.8 152.8 35 % 308.1 Wz H 49.1 7 4 179.3 i 353.4 % 62.1 21 132.7 m 399.0 M 75.5 8 160.1 36 % 316.6 10 51.5 187.9 i 363.1 % 65.2 22 % 138.8 Wh 410.0 The weight of the two flanges may be reckoned = weight of one foot. 186 MATERIALS. WEIGHTS OF CAST-IRON PIPE TO LA¥ 12 FEET LENGTH. Weights are Gross Weights, including Hub. (Calculated by F. H. Lewis.) Thickness. Inside Diameter. Inches. Equiv. Decimals. 4" 6" 8" 10'/ 12" 14" 16" 18" 20" 13^32 7-16 15-32 17-32 9-16 19-32 Vs 11-16 .375 .40625 .4375 .4687 .5 .53125 .5625 .59375 .625 .6875 .75 .8125 .875 .9375 1. 1.125 1.25 1.375 209 228 247 266 286 306 327 304 331 358 386 414 442 470 498 400 435 470 505 541 577 613 649 686 581 624 668 712 756 801 845 935 1026 692 : 744 795 846 899 951 1003 1110 1216 1324 1432 804 863 922 983 1043 1103 1163 1285 1408 1531 1656 1783 1909 1050 1118 1186 1254 1322 1460 1598 1738 1879 2021 2163 1177 1253 1329 1405 1481 1635 1789 1945 2101 2259 2418 2738 3062 1640 H 13 16 1810 1980 % 15 16 2152 2324 2498 1% 2672 3024 3380 Thickness. Inside Diameter. Inches. Equiv. Decimals. 22" 24" 27" 30" 33" 36" 42" 48" 60" % .625 1799 11-16 .6875 1985 2160 2422 % .75 2171 2362 2648 2934 3221 3507 13-16 .8125 2359 2565 2875 3186 3496 3806 4426 % .875 2547 27G9 3103 3437 3771. 4105 4773 5442 15-16 .9375 2737 2975 3332 3690 4048 4406 5122 5839 1. 2927 3180 3562 3942 4325 4708 5472 6236 m 1 125 . 3310 3598 4027 4456 4886 5316 6176 7034 m 1.25 3698 4016 4492 4970 5447 5924 6880 7833 9742 1.375 4439 4964 5491 6015 6540 7591 8640 10740 1.5 5439 6012 6584 7158 8303 9447 11738 i% 1.625 1.75 1.875 2. 2.25 2.5 2.75 6539 7159 7737 7782 8405 9022 9742 10468 11197 10260 11076 11898 12725 14385 12744 M 2 13750 14763 15776 2 l A 2U 17821 19880 21956 CAST-IEOK" PIPE FITTINGS. 187 CAST-IRON PIPE FITTINGS. Approximate Weight. Addyston Pipe and Steel Co., Cincinnati, Ohio. Size in Weight Size in Weight Size in 1 Weight Inches, | in Lbs. Size in 1 Weight Inches. in Lbs. Inches. in Lbs. Inches. | in Lbs. CROSSES. TEES. SLEEVES. REDUCERS. 2 40 8x3 220 6 65 10x4 128 3 104 10 390 8 86 12x10 278 3x2 90 10x8 330 10 140 12x8 254 4 150 10x6 312 12 176 12x6 250 4x3 114 10x4 292 14 208 12x4 250 4x2 110 10x3 290 16 340 14x12 475 6 200 12 565 20 500 14x10 430 6x4 150 12x10 510 24 710 14x8 340 6x3 150 12x8 492 30 965 14x6 285 8 325 12x6 484 36 1500 16x12 475 8x6 265 12x4 460 16x10 435 8x4 265 14x12 650 90° ELBOWS. 20x16 690 8x3 225 510 14x10 14x8 650 575 20x14 20x12 575 10 2 14 540 10x8 415 14x6 545 3 34 20x8 300 10x6 388 14x4 525 4 48 24x20 745 10x4 338 14x3 490 6 110 30x24 1305 10x3 350 16 790 8 145 30x18 1385 12 700 16x14 850 10 225 36x30 1730 12x10 650 16x12 825 12 370 12x8 615 16x10 890 14 450 ANGLE REDUC- 12x6 540 16x8 755 16 525 ERS FOR GAS. 12x4 525 495 16x6 16x4 630 655 20 24 900 1400 12x3 6x4 1 95 14x10 750 635 20 20x16 1375 1115 Y s or 45° BENDS. 6x3 | 80 14x8 14x6 570 1025 1070 20x12 20x10 20x8 1025 1090 900 S PIPES. 16 3 4 30 65 16x14 4 1 90 16x12 1025 20x6 875 6 85 6 1 190 16x10 1010 20x4 845 8 160 16x8 825 21x10 1465 10 190 PLUGS. 16x6 700 650 24 24 x 12 1875 1425 12 16 290 510 16x4 2 2 20 1790 24x8 1375 20 740 3 5 20x12 1370 24x6 1375 24 1425 4 8 20x10 1225 30 3025 30 2000 6 12 20x8 1000 30x24 2640 8 26 20x6 1000 30x20 2200 1-16 or 22W> BENDS. 10 46 20x4 1000 30x12 2035 12 66 24 2190 2020 30 x 10 30x6 2050 1825 14 16 70 24x20 6 150 100 24x6 1340 *36 5140 8 155 20 150 30x20 2635 36x30 4200 10 165 24 185 30x12 2250 36 x 12 4050 12 260 30 370 30x8 1995 45° BR^ PIP] lNCH :s. 16 500 TEES. 24 30 1280 1735 CAPS. 2 28 3 90 3 15 3 76 6x6x4 145 REDUCERS. 4 25 3x2 76 100 8 8x6 300 290 6 8 60 4 3x2 35 75 4x3 90 24 2765 4x3 42 10 100 4x2 87 150 24 x 24 x 20 30 2145 4170 4x2 6x4 40 95 12 120 6 6x4 130 125 120 266 36 10300 6x3 8x6 8x4 8x3 80 126 116 116 DRIP BOXES. 6x3 SLEEVES. 6x2 4 8 235 8 2 10 355 8x6 252 3 20 10x8 212 10 760 8x4 222 4 44 10x6 150 20 1420 188 MATERIALS. WEIGHTS OF CAST-IRON WATER- AND GAS-PIPE. (Addyston Pipe and Steel Co., Cincinnati, Ohio.) .5 <8 Standard Water-Pipe. .9 t Standard Gas-Pipe. 02 hH Per Foot. Thick- ness. Per Length. Per Foot. Thick- ness. Per Length. 2 7 5-16 63 2 6 u 48 3 3 15 17 8 180 204 3 12M 5-16 150 4 22 y* 264 4 17 % 204 6 33 H 396 6 30 7-16 360 8 42 y a 504 8 40 7-16 480 10 60 9-16 720 10 50 7-16 600 12 75 9-16 900 12 70 Yz 840 14 117 ¥4 1400 14 84 9-16 1000 16 125 H 1500 16 100 9-16 1200 18 167 Va 2000 18 134 11-16 1600 20 200 15-16 2400 20 150 11-16 1800 24 250 1 3000 24 184 % 2200 30 350 m 4200 30 250 8 3000 36 475 1% 5700 36 350 4200 42 600 w& 7200 42 383 % 4600 48 775 V4 9300 48 542 Ws 6500 60 1330 2 15960 W 900 1% 10800 THICKNESS OF CAST-IRON PIPES. P. H. Baermann, in a paper read before the Engineers' Clnb of Phila- delphia in 1882, gave twenty different formulas for determining the thick- ness of cast-iron water-pipes under pressure. The formulas are of three classes! 1. Depending upon the diameter only. 2. Those depending upon the diameter and head, and which add a con- stant. 3. Those depending upon the diameter and head, contain an additive or subtractive term depending upon the diameter, and add a constant. The more modern formulas are of the third class, and are as follows: t = .OOOOHhd 4- .Old + .36 Shedd, No. 1. t = .00006/id + .0133d + .296 Warren Foundry, No. 2. t = .000058/id + .0152d + .312 Francis, No. 3. t= .000048/id-f .013d + .32 Dupuit, No. 4. t = .00004M + .1 Vd + .15 Box, No. 5. t = .000135/id-f .4 - .OOlld Whitman, No. 6. t = .00006(/i 4- 2S0d) 4- .333 - .0033d Fanning, No. 7. t = .00015/id + .25 - '.0052d Meggs, No. 8. In which t = thickness in inches, h = head in feet, d =» diameter in inches. Rankine, "Civil Engineering," p. 721, says: "Cast-iron pipes should be made of a soft and tough quality of iron. Great attention should be made to moulding them correctly, so that the thickness may be exactly uniform all round. Each pipe should be tested for air-bubbles and flaws by ringing it with a hammer, and for strength by exposing it to double the intended greatest working pressure." The rule for computing the thickness of a pipe to resist a given working pressure is t = -|-, where r is the radius in inches, p the pressure in pounds per square inch, and / the tenacity of the iron per square inch. When/ = 18000, and a factor of safety of 5 is used, the above expressed in terms of d and h becomes "There are limitations, however, arising from difficulties in casting, and by the strain produced by shocks, which cause the thickness to be made greater than that given by the above formula." THICKNESS OF CAST-IROX PIPE. 189 Thickness of Metal and Weight per Length for Different Sizes of Cast-iron Pipes under Various Heads of Water. (Warren Foundry and Machine Co.) 50 100 150 200 250 300 I "t. Head. Ft. Head. Ft. Head. Ft. Head. Ft. Head. Ft. Head. Size. 1 5 ^ W . « w . A w . ia 80 _J & 05 A j'cS S"S> $ S'Sd o) ce S-!p : tefl c-^ bio B 1n tc = a tea ted 1 5^ £^ ' S J %& & ^ 2^ 1^ ^J "ota •s^ * 3«H ^fc &u & - ,£3«M t^j- Iq^ &u l*n b H O P. ^o ft HO ft H<-> & H° ft H<-> ft 3 344 144 .858 149 .362 153 .371 157 .380 161 .390 166 4 361 197 .878 204 .385 211 .397 218 .409 226 .421 235 5 254 .398 265 .408 275 .423 286 .438 298 .453 309 6 :m 815 .411 330 .42S 345 .447 361 .465 377 .483 393 8 422 445 .450 475 .474 502 498 529 .522 557 .546 584 10 459 600 .489 641 .519 682 .549 723 .579 766 .609 808 12 491 768 .527 826 .563 885 .599 944 .635 1004 .671 1064 14 952 .566 1031 .608 1111 .650 1191 .692 1272 .734 1352 16 557 1152 .604 1253 .652 1360 .700 1463 .748 1568 .796 1673 18 1870 .643 1500 .697 1630 .751 1761 .805 1894 .859 2026 20 1608 .682 1763 .742 1924 .802 2086 .862 2248 .922 2412 24 : 2120 .759 2349 .831 2580 .903 2811 .975 3045 1.047 3279 30 v 3020 .875 3376 .965 3735 1.055 4095 1.145 4458 1.235 4822 36 Hi 4070 .990 4581 1.098 5096 1.206 5613 1.314 6133 1.422 6656 42 ISO 5265 1.106 5958 1.232 6657 1 358 7360 1.484 8070 1.610 8804 48 1 078 6616 1.222 7521 1.366 8431 1.510 9340 1.654 10269 1.798 11195 All pipe cast vertically in dry sand; the 3 to 12 inch in lengths of 12 feet, all larger sizes in lengths of 12 feet 4 inches. Safe Pressures and Equivalent Heads of Water for Cast- iron Pipe of Different Sizes and Thicknesses. (Calculated by F. H. Lewis, from Fanning's Formula.) Size of Pipe. Thick- ness. 4" 6" 8" 10" 12" 14" 16" 18" 20" : = -- .2 5 5 5 5 — T u — T Si 11 11 ^.5 11 ii •a si w II a 7-16 1-2 9-16 5-8 112 330 516 774 49 199 X74 112 2v, 458 631 18 74 130 186 42 171 300 429 a 89 132 177 224 101 205 304 408 516 24 02 99 187 174 212 249 55 148 228 310 401 488 574 42 74 100 138 170 202 234 200 97 170 244 310 392 405 538 012 56 84 112 140 108 190 224 129 194 387 452 510 41 00 91 110 141 160 191 210 95 152 210 207 382 440 497 51 74 90 119 141 104 209 250 118 3 4 170 13 16 ] 7-8 "74 3"5 878 118 481 580 190 MATERIALS. Safe Pressures , etc., for Cast-iron Pipe. -(Continued.) Size of Pipe. 22" 24" 27" 30" 33" 36" 42 48" 60" Thick- ness. 9$, a - ■ a a,* fl a I = d ■ - ii PS ■±£ a 9% a - ffl : i,'- '■■ z u H {■ c Ph n s* - 1| ¥ 11-16 40 9?! ::o r>9 19 64 3-4 60 49 113 36 S3 24 55 13-16 so 6S 157 JW 120 90 7-8 101 si; (ill 159 54 124 42 97 32 74 15-16 1",1 III;-) '-{42 85 196 69 159 55 127 44 101 1 149! 327 286 10" 235 S4 194 69 159 57 131 3S ss '24 55 1 1-8 41 y MY 1 135 311 114 263 96 "21 S" IS! 5J) ii 99 1 1-4 224 511; >i 11)9 144 332 124 :: 107 247 si 187 ' 143 34 78 1 3-8 237 .-in; 202 465 174 401 151 34 S 13" K 103 237 .81 49 113 1 1-2 236 544 204 470 17S 410 i.v; 124 286 99 64 147 1 5-8 "34 53S "(if, 47" IS" 4H 14; ia 1 3-4 "33 537 207 477 167 385 1 7-8 181 "" " 2 211 484 174 2 1-8 193 355 424 482 2 3-4 214 Note.— The absolute safe static pressure which may be put upon pipe is given by the formula P = -jr X -=-, in ±J 5 which formula P is the pressure per square inch; T, the thickness of the shell; S, the ultimate strength per square inch of the metal in tension; and D, the inside diameter of the pipe. In the tables S is taken as 18000 pounds per square inch, with a working strain of one fifth this amount or 3600 pounds per square inch. The formula for the 7200 T absolute safe static pressure then is: P = — — . It is, however, usual to allow for "water-ram" by in- creasing the thickness enough to provide for 100 pounds additional static pressure, and, to insure sufficient metal for good casting and for wear and tear, a further increase equal to .333 yl — 7™)- The expression for the thickness then becomes: "Or I). T = [p±mD + 100// 100. 7200 and for safe working pressure P=^-Bsa(.- : , The additional section provided as above represents an increased value under static pressure for the different sizes of pipe as follows (see table in margin). So that to test the pipes up to one fifth of the ultimate strength of the material, the pressures in the marginal table should be added to the pressure-values given in the table above. Size of Lbs. Pipe. 4" 676 6 476 8 316 10 316 12 276 14 248 16 226 18 209 20 196 22 185 24 176 27 165 30 156 33 149 36 143 42 J 33 48 126 60 116 SHEET-IRON" HYDRAULIC PIPE. 191 SHEET-IRON HYDRAULIC PIPE. (Pelt on Water-Wheel Co.) Weight per foot, with safe head for various sizes of double-riveted pipe. 'o yjJ? 6X) -cS^ o £^ o w° to ■"i-b u o • <8 c § ^3 700 sy 2 if m 6J^ 4 P 2530 4 .486 ^ 800 9 it 2¥s 7*4 4 % 2100 *% .498 H 900 9J4 ii 2% 7% 8 3 1430 5 .525 ¥ 1000 K 10 11 2H 2% 8 3 1630 6 .563 1060 K 11 8 3 2360 7 .60 J2 1120 12^ IPs 2% 10% 8 3200 8 .639 ff 1»H 2% 11% 8 4190 9 .678 IS mo 15 m 2% 13 12 3610 10 .713 M 1380 16 ii 3 B 3 uy 4 12 2970 12 .79 18 1470 19 irf 3J4 uy 2 12 4280 14 .864 % A 21 18% 12 1 4280 15 .904 if 1600 22M m 35l 20 16 1 3660 16 .946 l 1600 23^ l/s 3% 214 16 l 4210 18 1.02 1A IK 1690 A 25 lf f 3^ 22% 16 - 4540 20 1.09 1780 27^ 3*S 25 5 4490 22 1.18 h% 29^ lie 3% 27M 4320 24 1.25 m 1920 31^ 32 1^ m 3% 4 29M 29^ 31^ 31% U4 5130 26 1.30 l r 5 s 1980 1% 2040 H 33% 34J4 m 2 3% 4^ 5030 28 1.38 !4 36 36^ m 2A 4 414 33^ 34 6 5000 30 1.48 1U 2000 38 38% i 1 ^ 2^ 4 4% 35^ 36 28 4590 36 1.71 1% '1920 44^ 45% 1% 2% 1% 2% 44 4v s 42 42% 1% 5790 42 1.87 2 12100 51 52% *m 59^ 4% 5% 48^ 49y,\m 5700 48 2.17 2J4 '2130 2 m 434 5% 54% 56 144 6090 Notes. — Sizes up to 24 inches are designed for 200 lbs. or less. Sizes from 24 to 48 inches are divided into two scale's, one for 200 lbs., the other for less. The sizes of bolts given are for high pressure. For medium pressures the diameters are J^-inch less for pipes 2 to 20 inches diameter inclusive, and J4 inch less for larger sizes, except 48-inch pipe, for which the size of bolt is 1% inches. When two lines of figures occur under one heading, the single columns up to 24 inches are for both medium and high pressures. Beginning with 24 inches, the left-hand columns are for medium and the right-hand lines are for high pressures. The sudden increase in diameters at 16 inches is due to the possible inser- tion of wrought-iron pipe, making with a nearly constant width of gasket a greater diameter desirable. When wrought-iron pipe is used, if thinner flanges than those given are sufficient, it is proposed that bosses be used to bring the nuts up to the standard lengths. This avoids the use of a reinforcement around the pipe. Figures in the third, fourth, fifth, and last columns refer only to pipe for 200 lbs. pressure. In drilling valve flanges a vertical line parallel to the spindles should be midway between two holes on the upper side of the flanges. CAST-IRON PIPE AND PIPE FLANGES. 193 DIMENSIONS OF PIPE FLANGES AND CAST-IRON PIPES. (J. E. Codman, Engineers 1 Club of Philadelphia, 1889.) fe * f-> *-« u ^ 'n GO ^ Thickness O3 Diameter of bolt-circle = 1.092D + 2.566. Diameter of bolt = 0.011D -f 0.73. Number of bolts = 0.78D + 2.56. PIPE FLANGES FOR HIGH STEAM-PRESSURE. (Chapman Valve Mfg. Co.) Size of Diameter Number of Diameter Diameter of Length of Pipe. of Flange. Bolts. of Bolts. Bolt Circle. Pipe-Thread. Inches. Inches. Inches. Inches. Inches. Inches. 2^ 7^ 6 Vs 5V 8 1% 3 9 6 6% Ws 3^ 9 7 % 7J4 1 7-16 4 10 8 % m 1 9-16 ' 4)* 10^ 8 i 8 m 1 11-16 5 11 9 m 1 13-16 6 13 10 10% m 7 14 12 Vs n% 1 15-16 8 15 12 % 13 9 16 13 Vs 14 2 10 17^ 15 Vs 1534 i«M 2V S 12 20 18 % m \ 14 23 18 1 2014 2v 2 1 15 23^ 18 1 21M 2% 194 MATERIALS. STANDARD SIZES, ETC., OF WROUGHT-IRON PIPE. For Water, Gas, or Steam, (Briggs Standard.) Diameter of Tube. 02 "cS a| H Ins. £ a a SocS « A o5 a o v £ 9 v Length of Pipe per Sq. Ft. of Inside Sur- face. Length of Pipe per Sq. Ft. of Outside Surface. * . a eg OS . a aj 1*1 ° a 52 o 2 Ins. $+6 30* Ins. Ins. Ins. Ins. Feet. Feet. Ins. Ins. Ys .270 .405 .068 .848 1.272 14.15 9.44 .0572 .129 Vi .364 .540 .088 1.144 1.696 10.50 7.075 .1041 .229 % .494 .675 .091 1.552 2.121 7.67 5.657 .1916 .358 y* .623 .840 .109 1.957 2.652 6.13 4.502 .3048 .554 S .824 1.050 .113 2.589 3.299 4.635 3.637 .5333 .866 l 1.048 1.315 .134 3.292 4.134 3.679 2.903 .8627 1.357 im 1.380 1.660 .140 4.335 5.215 2.768 2.301 1.496 2.164 m 1.610 1.900 .145 5.061 5.969 2.371 2.01 2.038 2.835 2 2.067 2.375 .154 6.494 7.461 1.848 1.611 3.355 4.430 %H 2.468 2.875 .204 7.754 9.032 1.547 1.328 4.783 6.491 3 3.067 3.500 .217 9.636 10.996 1.245 1.091 7.388 9.621 Wz 3.548 4.000 .226 11.146 12.566 1.077 .955 9.887 12.566 4 4.026 4.500 .237 12.648 14.137 .949 .849 12.730 15.904 4^ 4.508 5.000 .246 14.153 15.708 .848 .765 15.939 19.635 5 5.045 5.563 .259 15.849 17.475 .757 .629 19.990 24.299 6 6.065 6.625 .2S0 19.054 20 813 .63 .577 28.889 34.471 7 7.023 7.6?5 .301 22.063 23.954 .544 .505 38.737 45.663 8 7.982 8.625 .322 25.076 27.096 .478 .444 50.039 58.426 *9 9.000 9.688 .344 28.277 30.433 .425 .394 63.633 73.715 10 10.019 10.750 .366 31.475 33.772 .381 .355 78.838 90.762 * By the action of the Manufacturers of Wrought-iron Pipe and Boiler Tubes, at a meeting held in New York, May 9, 1889, a change in size of actual outside diameter of 9-inch pipe was adopted, making the latter 9.625 instead of 9.688 inches, as given in the table of Briggs' standard pipe diameters. For discussion of the Briggs Standard of Wrought-iron Pipe Dimensions, see Report of the Committee of the A. S. M. E. in " Standard Pipe and Pipe Threads," 1886. Trans., Vol. VIII, p. 29. The figures in the next to the last column are derived from the formula Z>-(0.05Z> + 1.9) X 1 in which D = outside diameter of the tubes, and n the number of threads to the inch. The figures in the last column are derived from the formula 0.8— x 2 -f d, or 1.6 \- d, in which d is the diameter at the bottom of the ti n thread at the end of the pipe. Having the taper, length of full-threaded portion, and the sizes at bottom and top of thread at the end of the pipe, as given in the table, taps and dies can be made to secure these points correctly, the length of the imperfect threaded portions on the pipe, and the length the tap is run into the fittings beyond the point at which the size is as given, or, in other words, beyond the end of the pipe, having no effect upon the standard. The angle of the thread is 60°, and it is slightly rounded off at top and bottom, so that, instead of its depth being equal to its pitch, as is the case with a full V-thread, it is 4/5 the pitch, or equal to 0.8— , n being the number of threads per inch. WROUGHT-IRON PIPE. 195 Sizes, etc., of Wrought-iron Pipe— (Continued.) Sizes, etc. Screwed Ends. .5t3 CD «m cO o hi $ a i§ .so c 05O r O Oj tf G O-u . tit! £ eter of torn of ead at of Pipe. eter of of ead at of Pipe. o£tj i-5 ,Sfo5 r*& H*s |e& EDO Inch. im Diarr Top Thr End Inch. Feet. Lbs. Lbs. No. Inches. Inches. M 2500. .243 .0006 .005 27 .19 .334 .393 *4 1385. .422 .0026 .021 18 .29 .433 .522 % 751.5 .561 .0057 .047 18 30 .567 .656 u 472.4 .845 .0102 .085 14 .39 .701 .815 3 4 270. 1.126 .0230 .190 14 .40 .911 1.025 l 166.9 1.670 .0408 .349 11^ .51 1.144 1.283 J8 96.25 2.258 .0638 .527 ll^S .54 1.488 1.627 70.65 2.694 .0918 .760 11^ .55 1.727 1.866 42.36 3.667 .1632 l.:-56 ny 2 i£ 8 2.2 2.339 2^ 30.11 5.773 .2550 2.116 8 ?B9 2.62 2.82 3 19.49 7.547 .3673 3.049 8 .95 3.241 3.441 3*6 14.56 9.055 .4998 4.155 8 1.00 3.738 3.938 4 11.31 10.728 .6528 5.405 8 1.05 4.235 4.435 43^ 9.03 12.492 .8263 6.851 8 1.10 4.732 4.932 5 7.20 14.564 1.020 8.500 8 1.16 5.291 5.491 6 4.98 18.767 1.469 12.312 8 1.26 6.346 6.546 7 3.72 23.410 1.999 16.662 8 1.36 7.34 7.54 8 2.88 28.348 2.611 21.750 8 1.46 8.334 8.534 9 2.26 34.077 3.300 27.500 8 1.57 9.39 9.59 10 1.80 40.641 4.081 34.000 8 1.68 10.445 10.645 Taper of conical tube ends, 1 in 32 to axis of tube = % inch to the foot total taper. 1 inch and below are butt-welded, and proved to 300 pounds per square inch hydraulic pressure. V/± inch and above are lap-welded, and proved to 500 pounds per square inch hydraulic pressure. SIZES ABOVE 10 INCHES. (Morris, Tasker & Co., Limited.) ^ aj . 6 5 a "3 ° cS O) 3 a 53 a O -CO "^53 "1 O .02 ^o3'S o 3 ° S o o & < < H H H t^ M -1 O J in. in. in. in. in. in. i-q.in. sq. in. ft. ft. ft. lbs. 11 11.224 12 .388 35.26 37.70 98.94 113.10 .340 .318 1.455 47.73 12 12.180 13 .41 38.26 40.81 116.54 132.73 .313 .293 1.235 54.66 13 13.136 14 .432 41.27 43.98 134.58 153.94 .290 .273 1.069 61.94 14 14.092 15 .454 44.27 47.12 155.97 176.72 .271 .254 .923 70.01 15 15.048 16 .476 47.27 50.27 177.87 201.06 .254 .238 .809 78.27 16 16.004 17 .498 50.28 53.41 201.16 225.98 .238 .225 .715 87.12 17 16.960 18 .520 53.28 56.55 225.91 254.47 .225 .212 .638 96.38 18 17.916 19 .542 56.28 59.69 252.10 283.53 .213 .201 .571 106.07 19 18.S72 20 .564 59.29 62.83 279.72 314.16 .202 .191 .515 116.21 20 19.828 21 .586 62.29 65. 97 308.77 346.36 .192 .183 .466 126.76 i 196 MATERIALS. WROUGHT-IRON WELDED TUBES, EXTRA STRONG. Standard Dimensions. Thickness, Actual Inside Actual Inside Nominal Double Diameter, Diameter, Diameter. Diameter. Strong. Extra Extra Double Extra Strong. Strong. Strong. Inches. Inches. Inches. Inches. Inches. Inches. H 0.405 0.100 0.205 H 0.54 0.123 0.294 % 0.675 0.127 0.421 V*. 0.84 0.149 0.298 0.542 0.244 ¥a 1.05 0.157 0.314 0.736 0.422 l 1.315 0.182 0.364 0.951 0.587 m 1.66 0.194 0.388 1.272 0.884 v& 1.9 0.203 0.406 1.494 1.088 2 2.375 0.221 ' 0.442 1.933 1.491 &A 2.875 0.280 0.560 2.315 1.755 3 3.5 0.304 0.608 2.892 2.284 3^ 4.0 • 0.321 0.642 3.358 2.716 4 4.5 0.341 0.682 3.818 3.136 STANDARD SIZES, ETC., OF IiAP-WEL.DED CHAR- COAL-IRON BOILER-TUBES. (Morris, Tasker & Co., Limited). s s S 2 5g oa Internal External ■So i t*0 3®£ '4 "3 3 V. to 3=2 3 e Area. Area. _ CD ill gaol ga§ "Sort H M m H° W a *A J Ins. Ins. Ins. Ins. Ins. Sq. In. Sq.Ft Sq. In. Sq.Ft Ft. . Ft. Ft. Lbs. 1 .856 .072 2.689 3.142 .575 .004 .785 .0055 4.460 3.819 4.139 .708 1 1-4 1.106 .072 3.474 3.927 .960 .0067 1.227 .0085 3.455 3.056 3.255 .9 1 1-2 1.334 .083 4.191 4.712 1.396 .0097 1.767 .0123 2.863 2.547 2.705 1.25 1 3-4 1.560 .095 4.901 5.498 1.911 .0133 2.405 .0167 2.448 2.183 2.315 1.665 2 1.804 .098 5.667 0.888 2.556 .0177 3.142 .0218 2.118 1.909 2.013 1.981 2 1-4 2.054 .098 6.484 7.069 3.314 .0280 3.976 .0276 1.850 1.698 1.774 2.238 2 1-2 2.283 .109 7.172 7.854 4.094 .0284 4.909 .0311 1.673 1.528 1.600 2.755 2 3-4 2.538 .109 7.957 8.639 5.039 .035 5.940 .0412 1.508 1.390 1.449 3.045 3 2.783 .109 8.743 9.425 6.083 .0422 7.069 .0491 1.373 1.273 1.323 8 888 3 1-4 3.012 .119 9.462 10.210 7.125 .0495 8.296 .0576 1.268 1.175 1.221 3 1-2 .119 10.248 10.995 8.357 .058 9.621 .0668 1.171 1.091 1.131 4 8272 3 3-4 8.512 .119 11.033 11.781 9.687 .0673 11.045 .0767 1.088 1.018 1.053 4.590 4 3.741 .130 11.753 12.566 10.1192 .0763 12.566 .0872 1.023 .955 .989 5.82 4 1-2 4.241 .130 13.323 14.137 14.126 ,0981 15.904 .1104 .901 .849 .875 0.01 5 4.7-1) .140 14.818 15.708 17.497 . 1215 19.035 .1364 .809 .764 .786 7.226 5.099 .151 17.904 18.849 25.509 .1771 28.274 .1963 .670 .637 .653 9.846 7 6.657 .172 20.914 21.991 34.; Si )5 .2417 38.484 .2678, .574 .545 .560 12.485 8 7.636 .182 23.989 25.132 45.795 .318 50.265 .8491 .500 .478 .489 15.109 9 8.615 .193 27.055 28.274 58.291 .4048 63.617 .4418 .444 .424 .434 18.002 10 9.578 .214 30.074 31.416 71.975 .4998 78.540 .5454 .399 .382 .391 22.19 11 10.560 .22 33.175 34 557 87.479 .6075 95.033 .6001 .361 .347 .354 25.489 12 11.54.'; .229 36.26 137.699 103.749 .7207 113.097 .7854 .330 .318 .324 28.516 13 12.524 .238 39.345 40.840 123. 1N7 .8554 182.782 .9213 .305 .293 .299 14 13.504 .248 42.414 43.982 143.189 .9948 153 938 1.009 .272 .277 20.271 15 14.1X2 .259 45.496 47.124 164.718 1.1438 176.715 1.2272 '.263 .254 .258 40 012 16 .271 48.562 50.265 187.667 1.3032 201.002 1.188 .247 .238 .242 45.199 17 16.432 .284 51.662 53.407 212.227 1.4738 226.980 1.5702 .232 .224 .228 49.908 18 .292 54.714 56.548 238.224 1.6543 254.469 1.7071 .219 .212 .215 54.816 19 " .3 57.805 59.690 265.903 1.8465 283.529 1.909 .207 .200 .203 59.479 20 19.360 .32 60.821 62.832 294.373 2.0443 314.159 2.1817 .197 .190 .193 00.705 21 20.320 .34 63.837 65.973 324.311 2.2522 346.361 2.4053 .188 .181 .184 78.404 In estimating tl e effective ^team-heating or boiler sui face of tubes, t ne surface in contact with air o be taken. For heating- liq • gases of co mbustion (wh ether internal or exte •nal to t he tubes) is to uids by stea m, superheat ng- steam, or transft rring h eat from one liouid or aas to ai lother, the i lean surface c f the tubes is o be tal :en. 1 RIVETED IRON PIPE. 197 To find the square feet of surface, S, in a tube of a given length, L, in feet, and diameter, d, in inches, multiply the length in feet by the diameter in inches and by .2618. Or, S = 3 - 1416dL = ,2618dL. For the diameters in the table below, multiply the length in feet by the figures given opposite the diameter. Inches, Diameter. Square Feet per Foot Length. Inches, Diameter. Square Feet per Foot Length. Inches, Diameter. Square Feet per Foot Length. 3 m m m 2 .0654 .1309 .1963 .2618 .3272 .3927 .4581 .5236 m 3 m ■ m m 4 1 5890 6545 7199 7854 8508 9163 9817 0472 5 6 7 8 9 10 11 12 1.3090 1.5708 1.8326 2.0944 2.3562 2.6180 2.8798 3.1416 RIVETED IRON PIPE. (Abendroth & Root Mfg. Co.) Sheets punched and rolled, ready for riveting, are packed in convenient form for shipment. The following table shows the iron and rivets required for punched and formed sheets. Number Square Feet of Iron I'S'S'i'S'S Number Square Feet of Iron o'g'S'3'S'S required to make 100 Lineal required to make 100 Lineal fcsSg,Sg Feet Punched and Formed xiinate vets 1 1 t requ 100 Li Punc Fori ts. Feet Punched and Formed Sheets when put together. Sheets when put together. •R^«S $ Diam- Width of Square Feet. gtfa^-aS Diam- Width of Square Feet. eter in Inches. Lap in Inches. < eter in Inches. Lap in Inches. So tfcSfc 0JO2 < 3 1 90 1,600 14 m 397 2,800 4 1 116 1,700 15 m 423 2,900 5 V4 150 1,800 16 m 452 3,000 6 m 178 1,900 18 m 506 3,200 7 m 206 2,000 20 562 3,500 8 Wz 234 2,200 22 1/^2 617 3,700 9 m 258 2,300 24 m 670 3,900 10 VA 289 2,400 26 m 725 4,100 11 \y^ 314 2,500 28 779 4,400 12 W* 343 2,600 30 iH 836 4,600 13 m 369 2,700 36 m 998 5,200 WEIGHT OF ONE SQUARE FOOT OF SHEET-IRON FOR RIVETED PIPE. Thickness by the Rirmingham Wire -Gauge. No. of Gauge. Thick- ness in Decimals of an Inch. Weight in lbs , Black. Weight in lbs., Galvan- ized. No. of Gauge. Thick- ness in Decimals of an Inch. Weight in lbs., Black. Weight in lbs., Galvan- ized. 26 24 20 .018 .022 .028 .035 !S8 1.12 1.40 .94 1.13 1.38 1.69 18 16 14 12 .049 .065 .083 .109 1.97 2.61 3.33 4.37 2.19 2.82 3.52 4.50 198 MATERIALS. SPIRAL RIVETED PIPE. (Abendroth & Root Mfg. Co.) Thickness. Diam- Approximate Weight Approximate Burst- eter, in lbs. per foot in ing Pressure in lbs. B. W. G. No. Inches. Inches. Length. per sq. in. 26 .018 3 to 6 llbs.= 24 .022 3 to 12 J " =y B of diam. in ins. 22 .028 3 to 14 " =.4 20 .035 3 to 24 " =.5 2700 lbs.-4-diam. in ins. 18 .049 3 to 24 " =.6 8600 " h- " 10 .065 6 to 24 " =.8 4800 " -r- " 14 .083 8 to 24 1 " =1.1 " (5400 " -- " " The above are black pipes. Galvanized weighs from 10 to 30 per cent heavier. Double Galvanized Spiral Riveted Flanged Pressure Pipe, tested, to 150 lbs. hydraulic pressure. Inside diameters, inches Thickness, B. W. G Nominal weight per foot, lbs. . . 3 20 *-4 4 20 3 51 6 7 8, 9 10 11 12 13 14 20,18 18 18118 16 16 16 16 14 A\ 5 6 7| 8 11 12 14 15 20 15 14 22 16 IN 20 14 14 14 24 29 34 DIMENSIONS OF SPIRAL, PIPE FITTINGS. Dimensions in Inches. Diameter Inside Diameter. Outside Diameter Flanges. Number Bolt Holes Diameter Bolt Holes. Circles on which Bolt Holes are Drilled. Sizes of Bolts. ins. 3 6 4 H m 7-16x1% 4 7 8 y z 5 15-16 7-16x134 7-16 x \% 5 8 8 H 6 15-16 6 8% 8 % m 7 10 8 Va 9 y*xm 8 11 8 Va 10 9 13 8 Va y 2 x2 10 14 8 Va }4x2 11 15 12 Va y**2 12 16 12 Va 1414 y 2 x2 13 17 12 Va 15M y 2 x2 14 17% 12 Va 16M 17 7-16 y^zy* 15 19 12 Va %*2}4 16 21 3-16 12 Va 19^ Hx2*4 18 23 y A 16 11-16 21J4 20 25 Ys 1.6 11-16 2sy 8 y*x2y 2 SEAMLESS BRASS TUBE. IRON-PIPE SIZES. (Randolph & Clowes). (For actual dimensions see tables of Wrought-iron Pipe.) Weight Nom- Weight Nom- Weight Nom- Weight per inal per inal per inal per Foot, lbs. Size. Foot, lbs. Size. Foot, lbs. Size. Foot, lbs. ya .266 H 1.228 2 4. 4 11.719 \ .461 1 1.837 2^ 6.323 5 15.935 .617 m 2.468 3 8.266 6 20.690 y* .925 M 3.045 m 9.878 8 26.286 29.881 BEASS TUBIKG ; COILED PIPES. 199 SEAMLESS' DRAWN BRASS-TUBING. (Randolph & Clowes, Waterbury, Conn.) Outside diameter 3-16 to 7% inches. Thickness of walls 8 to 25 Stubbs' Gauge, length 12 feet. The following are the standard sizes: SEAMLESS DRAWN BRASS-TUBING. Outside Diam- Length Feet. Stubbs' or Old Outside Diam- Length Feet. Stubbs' or Old Outside Diam- Length Feet. Stubbs' or Old eter. Gauge. eter. Gauge. eter. Gauge. Ya 12 20 m 12 14 2% 12 11 5-16 12 19 m 12 14 2M 12 11 ¥s 12 19 m 12 13 3 12 11 8 12 18 m 12 13 &A 12 11 % 12 18 1 13-16 12 13 12 11 % 12 17 m 12 12 4 10 to 12 11 13-16 12 17 1 15-16 12 12 5 10 to 12 11 % 12 17 2 12 12 5H 10 to 12 11 15-16 12 17 m 12 12 Wz 10 to 12 11 1 12 16 2U 12 12 m 10 to 12 11 % 12 16 Ws 12 12 6 10 to 12 11 12 15 zyk 12 11 COILED PIPES. (National Pipe-bending Co., New Haven, Conn.) COILS OF STEEL OR IRON PIPE ; WELDED LENGTHS. Butt-welded Pipe. Lap- welded Pipe. Size of pipe Inches Least outside diameter of coil contain- ing 25 feet of pipe and less. . . Inches Least outside diameter of coils over 25 feet and not over 200 feet Inches H 2 6 % *A 7 V2 % 4 SA 1 6 9 • 1M 8 11 1% 12 14 2 18 18 COILS OF SEAMLESS DRAWN BRASS AND COPPER TUBING. Size of tube, outside diameter Ins. Least outside diam- eter of coils — Ins. M 1'4 1A Welded solid drawn-steel tubes, imported by P. S. Justice & Co., Phila- delphia, are made in sizes from y% to 4J-£ inches external diameter, varying by J^ths, and with thickness of walls from 1-16 to 11-16 inches. The maxi- mum length is 15 feet, 200 MATERIALS. WEIGHT OF BRASS, COPPER, AND ZINC TUBING* Per Foot. Thickness by Brown & Sharped Gauge. Copper, Brass, No. 17. Brass, No. 20. Lightning-rod Tube, No. 23. Inch. Lbs. Inch. Lbs. Inch. Lbs. Va .107 Vb .032 y 2 .162 5-16 .157 3-16 .039 9-16 .176 % .185 Va .063 % .186 7-16 .234 5-16 .106 11-16 .211 Vk .266 % .126 H .229 9-11 .318 7-16 .158 % Va .3-33 .377 H 9-16 .189 .208 Zinc, No. 20. % .462 % .220 .542 .675 8 .252 .284 W* ^ .161 m .740 l .378 % .185 Wz .915 m m .500 Va Vs .234 Wa .980 .580 .272 2 1.90 1 .311 21^ 1.506 Wa .380 3 2.188 m .452 LEAD PIPE IN LENGTHS OF 10 FEET. In. 3-8 Thick. 5-16 Thick. Va Thick. 3-16 Thick. 2y a 3 m 4 M 5 lb. oz. 17 20 ' 22 25 31 lb. oz. 14 16 18 21 lb. oz. 11 12 15 16 18 20 lb. oz. 8 9 9 8 12 8 14 LEAD WASTE-PIPE. 1}4 in., 2 lbs. per foot. I %% in., 4 lbs. per foot. 2 " 3 and 4 lbs. per foot. 4 "5, 6, and 8 lbs. 3 " 3^ and 5 lbs. per foot. | Ay 2 " 6 and 8 lbs. 5 in. 8, 10, and 12 lbs. LEAD AND TIN TUBING. V% inch. 34 inch. SHEET LEAD. Weight per square foot, 2)4, 3, 3J4 4, i}4, 5, 6, 8, 9, 10 lbs. and upwards. Other weights rolled to order. BLOCK-TIN PIPE. % in , 4}4 6^, and 8 oz. per foot. }4 " 6, 73^, and 10 oz. " % " 8 and 10 oz. " Y± " 10 and 12 oz. " 1 in., 15, and 18 oz. per foot. Wa " 1}4 and l^lbs. " 1)4 " 2 and 2V£ lbs. 2 il 2U and 3 lbs. " LEAD PIPE. 201 LEAD AND TIN-L.INED LEAD PIPE. (Tatham & Bros., New York.) , Weight per Foot and Rod. 'IS 6 Weight per 1^ £ Foot and Rod. II "3 »3 3^ s %in. E 7 lbs. per rod 1 in. E iy lbs. per foot 10 " D 10 oz. per foot 6 " D 2 " " 11 " C 12 " 8 " C 2y 2 " 14 " B 1 lb. 12 " B sy 4 " 17 " A M " 16 " A 4 « « 21 " AA \y « 19 " AA 4% " 24 " AAA. m « 27 " AAA 6 " 30 7-16 in. 13 oz. 1?4 in- E 2 " 10 " 1 lb. D 2% " 12 y 2 m. E 9 lbs. per rod 7 " C 3 " 14 D % lb. per foot 9 " B 4% " 16 " C 1 " 11 '■ A 19 " B Wa " 13 " AA Wa " '• 6% " 25 " \y " m " " AAA " A 16 l^in. E 3 " 12 " AA 2 " 19 D sy 2 " 14 " 2^ <; 23 " C 4M " 17 AAA 3 " 25 " B 5 " 19 %}n. . E 12 " per rod 8 " A 6^ " 23 D 1 " per foot 9 " AA 8 " 27 " C m ;; 13 " AAA 9 " " B 16 l%in. C 4 « 13 " A 2y 2 " 20 B 5 " " 17 " AA m « 22 " A 6^ " 21 " AAA 3^ " 25 " AA 8^ " 27 %in.- E 1 •' per foot 8 2 in. C 4.% " 15 D 1M " 10 " B 6 " 18 " C 2^ " 12 " A 7 " 22 " B 16 " AA 9 " 27 " A 3 " 20 " AAA 11% " " AA 31^ " 23 " AAA 4% " 30 WEIGHT OF L.EAD PIPE WHICH SHOULD BE USED FOR A GIVEN HEAD OF WATER. (Tatham & Bros., New York.) Head or Pressure per sq. inch. Calibre and Weight per Foot. of Feet Fall. Letter. %inch. y% inch. % inch. %inch. 1 inch. Mm. 30 ft, 50 ft. 75 ft. 100 ft. 150 ft. 200 ft. 15 lbs. 25 lbs. 38 lbs. 50 lbs. 75 lbs. 100 lbs. D C B A AA AAA 10 oz. 12 oz. 1 lb. 1)4 lbs. 1*6 lbs. \% lbs. JOS. 1 lb. 1% lbs. 1M lbs. 2 lbs. 3 lbs. 1 lb. \y 2 lbs. 2 lbs. 2}4 lbs. 2% lbs. Sy lbs. \y± lbs. 1H lbs. 2J4 lbs. 3 lbs. m lbs. 4% lbs. 2 lbs. 2y lbs. zy 4 lbs. 4 lbs. 4% lbs. 6 lbs. 214 lbs. 3 lbs. 3% lbs. 4% 1. s. 6 lbs. Wa lbs. To find the thickness of lead pipe required when the head of water is given. (Chadwick Lead Works). Rule.— Multiply the head in feet by size of pipe wanted, expressed deci- mally, and divide by 750; the quotient will give thickness required, in one- hundredths of an inch. Example.— Required thickness of half -inch pipe for a head of 25 feet. 25 X 0.50 -i- 750 = 0.16 inch. 202 MATERIALS. ■yjCiGOCOCOCOOOOTI o^iio«OH50 5ior.o)ontoo^o ■"J'<*Wi-iOOOi-i-ffliOlO'«C ^-^-it-h^^h^hOOOOOOOC oooooocoooooooooc wNMf lOtONCOCRO'HWK * lO CO J> X; OS C otcictQtOiCtcii£ S88£2 » os t- oo ■* oo its th co o le 0050i--(0505005TlC3500QO'*T-iC3SJ>lO"C) 5l-QOOM( - m ~ ~ — ; — . — . w BTHMMWi-ir"-! nt-lft«S'COOiOjT-ll-<7- .iooo»ncoo5eO'*'OJco-i-iQO-<*eoirjj>iATj(coc t* Tf< 00 CO H>*^M»MMMM«THrt"ririOOOOOOOOOO OOOOrl«Mt) m SV> 7-16 14 15-16 9 VA 6 2 5-16 3 5-16 V/\ H 13 1 8 m 5K m 4 m m 9-16 12 1 1-16 7 m 5 m 4 m 3 % 11 m 7 Wa 5 2% 4 4 3 11-16 11 U. S. OR SELLERS SYSTEM OF SCREW-THREADS. 205 Screw-Tlireads, Whit worth (English) Standard. a ,£3 a o a" A o s "o i ft s Q S P Pn s Oh Q s 4 20 96 n 1 8 m 5 3 % 5-16 18 11-16 n XYa 7 Ws m, 34 m % 16 13-16 10 m 7 2 4H 3?4 m 7-16 14 10 6 24 4 m 3 % 12 Vs 9 Ws 6 zy* 4 4 3 9-16 12 15-16 9 5 m 3J/ 2 u. S. OR SEINERS SYSTEM OF SCREW-THREADS. BOLTS AND THREADS. HEX. NUTS AND HEADS. ^ « ^ °,2 .. , 4a W g s ft 5 2 H o3 5 ■g M.S rA < g.s aJEi 2 < a . ■£ ° a* in 3 oil C <» EH Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. 4 20 .185 .0062 .049 .027 ^ 7-16 37-64 4 3-16 ■ 7-10 5-16 18 .240 .0074 .077 .045 19-32 17-32 11-16 5-16 4 10-12 % 16 .294 .0078 .110 .068 11-16 % 51-64 % 5-16 63-64 7-16 14 .344 .0089 .150 .093 25-32 23-32 9-10 7-16 % 17-64 % 13 .400 .0096 .196 .126 % 13-16 1 ^ 7-16 1 15-64 9-16 12 .454 .0104 .249 .162 31-32 29-32 M 9-16 y 2 1 23-64 % 11 .507 .0113 .307 .202 11-16 1 1 7-32 % 9-16 m % 10 .620 .0125 .442 .302 1M 13-16 17-16 M 11-16 1 49-64 Vs 9 .731 .0138 .601 .420 17-16 W* 1 21-32 % 13-16 2 1-32 1 8 .837 .0156 .785 .550 1% 19-16 W% 1 15-16 2 19-64 1^ 7 .940 .0178 .994 .694 1 13-16 194 23-32 1? ^ 1 1-16 2 9-16 14 7 1.065 .0178 1.227 .893 2 1 15-16 2 5-16 l 1 4 13-16 2 53-64 1^ 6 1.160 .0208 1.485 1.057 2 3-16 m 2 17-32 \\ 1 5-16 3 3-32 1^1 6 1.284 .0208 1.767 1.295 2% 2 5-16 m 13 4 17-16 3 23-64 5Vo 1.3S9 .0227 2.074 1.515 2 9-16 2^ 2 31-32 If I 19-16 3% P 5 1.491 .0250 2.405 1.746 2^ 2 11-16 3 3-16 Is i 1 11-16 3 57-64 5 1.616 .0250 2.761 2.051 2 15-16 2% 3 13-32; r 4 1 13-16 4 5-32 1% 1.712 .0277 3.142 2.302 3^ 3 1-16 3% 12 ' 1 15-16 4 27-64 24 4 l a 1.962 .0277 3.976 3.023 Wz 3 7-16 4 1-16 2J4 2 3-16 4 61-64 ^ 4 2.176 .0312 4.909 3.719 3% 3 13-16 4y 2 Wz 2 7-16 5 31-C4 2% 4 2.426 .0312 5.940 4.620 44 4% 4 3-16 4 29-32 2% 211-16 6 3 3^3 2.629 .0357 7.069 5.428 4 9-16 5% 3 2 15-16 6 17-32 3^1 3^ 2.S71 .0357 8.296 6.510 5 4 15-16 5 13-16 314 3 3-16 1 7 1-16 34 3.100 .0384 9.621 7.548 5% 5 5-16 6 7-64 3J^ 3 7-16 7 39-61 3% 3 3.317 .0113 11.045 8.641 Wa 5 11-16 6 21-32 3% 3 11-161 8^ 4 3 3.567 .0413 12.566 9.963 $8 6 1-16 7 3-32 4 3 15-16 8 41-64 44 •- >r .s 3.798 .0435 14.186 11.329 6^2 6 7-16 7 9-16 44 4 3-16 1 9 3-16 4 l A •-• ;i 4 4.028 .0454 15.004 12.753 6^ 6 13-16 7 31-32 4^ 4 7-16 1 9% 4% 2% 4.256 .0476 17.721 14.226 -4 7 3-16 8 13-32 4% 4 11-16 10J4 5 2>2 4.480 .0500 19.635 15.763 7 9 16 8 27-32 5 4 15-16 10 49-64 54 '■ihi 4.730 .0500 2l.l',4S 17.572 8 7 15-16 9 9-32 54 5 3-16 11 23-64 5^ 4.953 .0526 23.758 19.267 8% 8 5-16 9 23-32 5^ 5 7-16 uy 8 5M 5.203 .0526 25.967 21.262 8^ 8 11-16 10 5-32 5% 5 11-16 12% 6 %Y4, 5.423 .0555 1 28.274 23.098 W& 9 1-16 10 19-32 6 5 15-16 12 15-16 lilMIT GAUGES FOR IRON FOR SCREW THREADS. In adopting the Sellers, or Franklin Institute, or United States Standard. as it is variously called, a difficulty arose from tbe fact that it is the habit of iron manufacturers to make iron over- size, and as there are no over-size 206 MATERIALS. screws in the Sellers system, if iron is too large it is necessary to cut it away with the dies. So great is this difficulty, that the practice of making taps and dies over-size has become very general. If the Sellers system is adopted it is essential that iron should be obtained of the correct size, or very nearly so. Of course no high degree of precision is possible in rolling iron, and when exact sizes were demanded, the question arose how much allowable variationjthere should be from the true size. It was proposed to make limit- gauges for inspecting iron with two openings, one larger and the other smaller than the standard size, and then specify that the iron should enter the large end and not enter the small one. The following table of dimen- sions for the limit-gauges was recommended by the Master Car-Builders' Association and adopted by letter ballot in 1883. Size of Size of Size of Size of Size of Large Small Differ- Size of Large Small Differ- Iron. End of End of ence. Iron. End of End of ence. Gauge. Gauge. Gauge. Gauge. 14 in. 0.2550 0.2450 0.010 96 in. 0.6330 0.6170 0.016 5-16 0.3180 0.3070 0.011 H 0.7585 0.7415 0.017 % 0.3810 0.3690 .0.012 % 0.8840 0.8660 0.018 7-16 0.4440 0.4310 9.013 1 1.0095 0.9905 0.019 % 0.5070 0.4930 0.014 m 1.1350 1.1150 0.020 9-16 0.5700 0.5550 0.015 m 1.2605 1.2395 0.021 Caliper gauges with the above dimensions, and standard reference gauges for testing them are made by the Pratt & Whitney Co. THE MAXIMUM VARIATION IN SIZE OF ROUGH IRON FOR U. S. STANUARD ROUTS. Am. Mach., May 12, 1892. By the adoption of the Sellers or U. S. Standard thread taps and dies keep their size much longer in use when flatted in accordance with this system than when sharp, though it has been found advisable in practice in most cases to make the taps of somewhat larger outside diameter than the nom- inal size, thus carrying the threads further towards the V-shape and giving corresponding clearance to the tops of the threads when in the nuts or tapped holes. Makers of taps and dies often have calls for taps and dies, U. S. Standard, " for rough iron." An examination of rough iron will show that much of it is rolled out of round to an amount exceeding the limit of variation in size allowed. In view of this it may be desirable to know what the extreme variation in iron may be, consistent with the maintenance of U. S. Standard threads, i.e., threads which are standard when measured upon the angles, the only place where it seems advisable to have them fit closely. Mr. Chas. A. Bauer, the general manager of the Warder. Bushnell & Glessner Co., at Springfield, Ohio, in 1884 adopted a plan which may be stated as follows : All bolts, whether cut from rough or finished stock, are standard size at the bottom and at the sides or angles of the threads, the variation for fit of the nut and allowance for wear of taps being made in the machine taps. Nuts are punched with holes of such size as to give 85 per cent of a full thread, expe- rience showing that the metal of wrought nuts will then crowd into the threads of the taps sufficiently to give practically a full thread, while if punched smaller some of the metal will be cut out by the tap at the bottom of the threads, which is of course undesirable. Machine taps are made enough larger than the nominal to bring the tops of the threads up sharp, plus the amount allowed for fit and wear of taps. This allows the iron to be enough above the nominal diameter to bring the threads up full (sharp) at top, while if it is small the only effect is to give a flat at top of tli reads ; neither condition affecting the actual size of the thread at the point at which it is intended to bear. Limit gauges are furnished to the mills, by which the iron is rolled, the maximum size being shown in the third column of the table. The minimum diameter is not given, the tendency in rolling being nearly always to exceed the nominal diameter. In making the taps the threaded portion is turned to the size given in the eighth column of the table, which gives 6 to 7 thousandths of an inch allow- ance for fit and wear of tap. Just above the threaded portion of the tap a SIZES OF SCREW-THREADS FOR BOLTS AND TAPS. 207 place is turned to the size given in the ninth column, these sizes being the same as those of the regular U. S. Standard bolt, at the bottom of the thread, plus the amount allowed for fit and wear of tap ; or, in other words, d' = U. S. Standard d + (D' - D). Gauges like the one in the cut, Fig. 72, are furnished for this sizing. In finishing the threads of the tap a tool Fig. 72. is used which has a removable cutter finished accurately to gauge by grind- ing, this tool being correct U. S. Standard as to angle, and flat at the point. It is fed in and the threads chased until the flat point just touches the por- tion of the tap which has been turned to size d'. Care having been taken with the form of the tool, with its grinding on the top face (a fixture being provided for this to insure its being ground properly), and also with the set- ting of the tool properly in the lathe, the result is that the threads of the tap are correctly sized without further attention. « It is evident that one of the points of advantage of the Sellers system is sacrificed, i.e., instead of the taps being flatted at the top of the threads they are sharp, and are consequently not so durable as they otherwise would be ; but practically this disadvantage is not found to be serious, and is far overbalanced by the greater ease of getting iron within the prescribed limits ; while any rough bolt when reduced in size at the top of the threads, by filing or otherwise, will fit a hole tapped with the U. S. Standard hand taps, thus affording proof that the two kinds of bolts or screws niade for the two different kind* of work are practically interchangeable. By this system \" iron can be .005'' smaller or .0108" larger than the nominal diameter, or, in other words, it may have a total variation of .0158", while \\" iron can be .0105" smaller or .0309" larger than nominal— a total variation of .0414"— and within these limits it is found practicable to procure the iron. STANDARD SIZES OF SCREW-THREADS FOR BOIiTS AND TAPS. (Chas. A. Bauer.) 1 2 3 4 5 6 7 8 9 10 A u D d h / D'-D D' d{ H Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. 5-16 20 .2608 .1855 .0379 .0062 .006 .2668 .1915 .2024 IS .3245 .2403 .0421 .0070 .006 .3305 .2463 .2589 % 16 .3885 .2938 .0474 .0078 .006 .3945 .2998 .3139 7-16 14 .4530 .3447 .0541 .0089 .006 .4590 .3507 .3670 % 13 .5166 .4000 .0582 .0096 .006 .5226 .4060 .4236 9-16 12 .5805 .4543 .0631 .0104 .007 .5875 .4613 .4802 3 A 11 .6447 .5069 .0689 .0114 .007 .6517 .5139 .5346 4 10 .7717 .6201 .0758 .0125 .007 .7787 .6271 .6499 9 .8991 .7307 .0842 .0139 .007 .9061 .7377 .7630 i 8 1.0271 .8376 .0947 .0156 .007 1.0341 .8446 .8731 m 7 1.1559 .9394 .1083 .0179 .007 1.1629 .9464 .9789 7 1.2809 1.0644 .1083 .0179 .007 1.2879 1.0714 1.1039 A = nominal diameter of bolt. D = actual diameter of bolt. d = diameter of bolt at bottom of thread. 71 = number of threads per inch. / = flat of bottom of thread. h = depth of thread. D' and d' = diameters of tap. H — hole in nut before tapping. D d h f=i H= D> ' n 1.29904 .7577 D-d ■ TV - .85(2/i.) 208 MATERIALS. STANDARD SET-SCREWS AND CAP-SCREWS. American, Hartford, and Worcester Machine-Screw Companies. (Compiled by W. S. Dix.) Diameter of Screw. . . . Threads per Inch Size of Tap Drill* (A) No. 43 (B) 3-16 24 No. 30 (C) Va 20 No. 5 (D) 5-16 18 17-64 (E) ¥s 16 21-64 (F) 7-16 14 % (G) 12 27-64 Diameter of Screw.... Threads per Inch Size of Tap Drill*.... (H) 9-16 12 31-64 (1) % 11 17-32 (J) ft 21-32 (K) V i 49-64 (L) 1 8 % (M) 7 63-64 (N) m Set Screws. Hex. Head Cap-screws. Sq. Head Cap-screws, S sort Diam Long Diam. Lengths (under Short Diam. of Head. Long Diam, of Head. Lengths (under Short Diam. of Head. Long Diam. of Head. Lengths (under of Head of Head Head). Head). Head). (C) V A .35 %to3 7-16 .51 % to 3 %to3M % .53 %to 3 %to3M M to 3^ (D) 5-16 .44 34 to 3J4 y* .58 7-16 62 (E) % .53 %to3^ 9-16 .65 % to 3y 2 y 2 71 (F) 7-16 .62 34 to 3% 9|to4 n .72 34 to Wa 9-16 so Mto3% (G) ^ .71 % .87 %to4 % 89 U to 4 (H) 9-16 .80 % to 4y 4 %to4y 2 13-16 .94 H to 4y 4 11-16 98 ¥a to 414 (1) % .89 Va 1.01 1 to 4% U 1 06 1 to4V£ 14 to 4Ya (J) Va 1.06 1 to 4% 1 1.15 Wa to 4% \y 2 to 5 Wa to 5 2 to 5 Vs 1 24 (K) % 1.24 1M to 5 V/a 1.30 m 1 00 \y± to 5 (L) 1 1.42 \\b to 5 13^ to 5 1M 1.45 m 1 77 m to 5 (M) iy 8 1.60 1.59 1 1)5 2 to 5 (N) m 1.77 2 to 5 m 1.73 2 to 5 m 2 13 234 to 5 Round and Filister Head Cap-screws. Flat Head Cap-screws. Button-head Cap- screws. Diam. of Head. Lengths (under Head). Diam. of Head. Lengths (including Head). Diam. of Head. Lengths (under Head). (A) 3-16 Mto2J^ y± %-to 1% 7-32 (.225) MtolM (B) H M to 234 % 34 to 2 5-16 34 to 2 (O % Mto3 15-32 34 to 24 %to2% % to 3 7-16 Mto2M M to 2^ (D) 7-16 M to 314 it 9-16 (E) 9-16 %to3^ % to 3M % M to 2M % to 3 (F) % (G) M 13-16 1 to 3 ¥a M to 4 Va 14 to 3 13-16 1 to 3 (H) 13-16 1 to 4J4 Wa to 4^ Wz to 4% 1 l^to3 l%to3 15-16 1J4 to 3 0) % 1% 1 l^to3 (J) 1 2 to 3 1M 1M to 3 (K) iy 8 1M to 5 2 to 5 (L) 14 * For cast iron. Threads are U. S. Standard. Cap-screws are threaded 34 length up to and including 1" diam. x 4" long, and }/& length above. Lengths increase by 4" each regular size between the limits given. Lengths of heads, except flat and button, equal diam. of screws. The angle of the cone of the flat-head screw is 76°, the sides making angles of 52° with the top. STANDARD MACHINE SCREWS. 209 STANDARD MACHINE SCREWS. (Am. Screw Co.'s Catalogue, 1883, 1892.) No. Threads per Diam. of Diam. of Flat Head. Diam. of Round Head. Diam. of Filister Head. Lengths. Inch. Body. From To 2 56 .0842 .1631 .1544 .1332 3-16 k 3 48 .0973 .1894 .1786 .1545 3-16 % 4 32, 36, 40 .1105 .2158 .2028 .1747 3-16 % 5 32, 36, 40 .1236 .2421 .2270 .1985 3-16 Vs 6 30, 32 .1368 .2684 .2512 .2175 3-16 7 30,32 .1500 .2947 .2754 .2392 M m 8 30,32 .1631 .3210 .2936 .2610 J4 n 9 24, 30, 32 ■ .1763 .3474 .3238 .2805 H 10 24, 30, 32 .1894 .3737 .3480 .3035 H M 12 20, 24 .2158 .4263 .3922 .3445 % m 14 20, 24 .2421 .4790 .4364 .3885 3 A 2 16 16, 18, 20 .2684 .5316 .4866 .4300 % m 18 16, 18 .2947 .5842 .5248 .4710 Y2 %\£ 20 16, 18 .3210 .63CS .5690 .5200 Vi m 22 16, 18 .3474 .6894 .6106 .5557 % 3 24 14, 16 .3737 .7420 .6522 .6005 y* 3 26 14, 16 .4000 .7420 .6938 .6425 3 28 14, 16 .4263 .7946 .7354 .6920 3 30 14, 16 .4520 - .8473 .7770 .7240 i 3 Lengths vary by 16ths from 3-16 to y>, by Sths from y> to 1*4, by 4ths from 1*4 to 3. SIZES AND WEIGHTS OF SQUARE AND HEXAGONAL NUTS. United States Standard Sizes. Chamfered and trimmed. Punched to suit U. S. Standard Taps. £ 6< S-J8 Square. Hexagon. a o M o a s is a o tCtJ 2% o a s o o C to § to % M V\ 13-64 11-16 9-16 7270 .0138 7615 .0131 5-16 19-32 5-16 H 13-16 11-16 4700 .0231 5200 .0192 % 11-16 % 19-64 1 13-16 2350 .0426 3000 .0333 7-16 25-32 7-16 11-32 Ws, % 1630 .0613 2000 .050 H % y, 25-64 1 1120 .0893 1430 .070 9-16 31-32 9-16 29-64 1^ 890 .1124 1100 .091 % 1 1-16 % 33-64 i% m 640 .156 740 .135 « \V± '*4 39-64 m 1 7-16 380 .263 450 .222 1 7-16 % 47-64 2 1-16 1 11-16 280 .357 309 .324 i% 1 53-64 2 5-16 1% 170 .588 216 .463 ■?] 4 1 13-16 1^ 59-64 2 9-16 2 1-16 130 .769 148 .676 1 A 2 W A 1 1-16 2 13-16 2 5-16 96 1.04 111 .901 y. A 2 3-16 \% 1 5-32 W& 2% 70 1.43 85 1.18 1 /» 2% % 1 9-32 3% m 58 1.72 68 1.47 2 9-16 1 13-32 3% 2 15-16 44 2.27 56 1.79 2% W A n Ws 3 3-16 34 2.94 40 2.50 y /r 2 15-16 m 4y 8 3% 30 3.33 37 2.70 2 %y& 2 1 23-32 4 7-16 3% 23 4.35 29 3.45 2J4 Wz 2M 1 15-16 4 15-16 4 1-16 19 5.26 ' 21 4.76 Wh 2Yo 2 3-16 5^ 4^ 12 8.33 15 6.67 2% 4k 23/j 2 7-16 6 4 15-16 9 11.11 11 9.09 3 4% 3 2% Wz 5 5-16 m 13.64 Wfl 11.76 210 MATERIALS, WEIGHTS OF 100 BOL.TS WITH SQUARE HEADS AND NUTS. (Hoopes & Townsend'. > List.) Length un- Diameter of Bolts. der Head to Point. J4 in. 5-16 in. %in. 7-16 in. ^in. %va. %in. %\n. lin. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. M 4.00 7.00 10.50 15.20 22.50 39.50 63.00 m 4.35 7.50 11.25 16.30 23.82 41.62 66.00 2 4.75 8.00 12.00 17.40 25.15 43.75 69.00 109.00 163 &A 5.15 8.50 12.75 18.50 26.47 45.88 72.00 113.25 169 w/z 5.50 9.00 13.50 19.60 27.80 48.00 75.00 117.50 174 m 5.75 9.50 14.25 20.70 29.12 50.12 78.00 121.75 180 3 6.25 10.00 15.00 21.80 30.45 52.25 81.00 126.00 185 3^ 7.00 11.00 16.50 24.00 33.10 56.50 87.00 134.25 196 4 7.75 12.00 18.00 26.20 35.75 60.75 93.10 142.50 207 m 8.50 13.00 19.50 28.40 38.40 65.00 99.05 151.00 218 5 9.25 14.00 21.00 30.60 41.05 69.25 105.20 159.55 229 5^ 10.00 15.00 22.50 32.80 43.70 73.50 111.25 168.00 240 6 10.75 16.00 24.00 35.00 46.35 77.75 117.30 176.60 251 6^ 25.50 37.20 49.00 82.00 123.35 185.00 262 7 27.00 39.40 51.65 86.25 129.40 193.65 WS V* 28.50 30.00 41.60 43.80 46.00 54.30 59.60 64.90 90.50 94.75 103.25 135.00 141.50 153.60 202.00 210.70 227.75 284 295 9 317 10 48.20 50.40 52.60 70.20 75.50 80.80 86.10 91.40 96.70 102.00 107.30 112.60 117.90 123.20 5.45 111.75 120.25 128.75 137.25 145.75 154.25 162.75 171.00 179.50 188.00 206.50 8.52 165.70 177.80 189.90 202.00 214.10 226.20 238.30 250.40 262.60 274.70 286.80 12.27 224.80 261.85 278.90 295.95 313.00 330.05 347.10 364.15 381.20 398.25 415.30 16.70 339 11 360 12 382 13 404 14 426 15 448 16 470 17 492 18 514 19 536 20 558 Per inch additional. [■1.87 2.13 3.07 4.18 21.82 TRACK BOLTS. Witli United States Standard Hexagon Nuts. Rails used. Bolts. Nuts. No. in Keg, 200 lbs. Kegs per Mile. r %x4^£ m 230 6.3 %x4 m 240 6. 45to851bs..J M*3^ m VA 254 260 5.7 5.5 1 m 266 5.4 I Mx3 1M 283 5.1 r %x3^ 1 1-16 375 4. 30 to 40 lbs...-! %x3 ¥8*m 1 1-16 1 1-16 410 435 3.7 3.3 1 %*zy2 1 1-16 465 3.1 r ^x3 Vs 715 2. 20to301bs..J Vs Vs 760 800 2. 2. I % 820 2. NUTS AND BOLT-HEADS — RIVETS. 211 WEIGHTS OF NUTS AND BOLT-HEADS, IN POUNDS. For Calculating the Weight of Longer Dolts. Diameter of Bolt, in Inches. X A % K % H % Weight of hexagon nut and head. Weight of square nut and head . . .017 .021 .057 .069 .128 .164 .267 .320 .43 .55 .13 .88 Diameter of Bolt, in Inches. 1 m m m 2 17 21 3 Weight of hexagon nut and head. Weight of square nut and head . . 1.10 1.31 2.14 2.56 3.78 4.42 5.6 7.0 8.75 10.5 28. S 36.4 NUMBER OF RIVETS IN 100 POUNDS. Lengths. 96 in. 7-16 in. ^in. 9-16 in. %iu. 11-16 in. Mill. %m. % 1965 1848 1692 1419 1335 1222 1092 1027 940 944 846 763 665 597 538 l 450 !tf 1512 1437 1092 1036 840 797 726 691 512 487 415 389 356 „ 228 Wb 1368 988 760 653 460 370 329 211 m m 1300 949 730 624 440 357 280 180 1260 924 711 596 420 340 271 174 m m 1200 900 693 553 390 325 262 169 1156 840 648 532 375 312 257 165 2 1100 789 608 511 360 297 243 156 m 1031 744 573 502 354 289 237 152 2H 999 721 555 491 347 280 232 149 945 682 525 475 335 260 220 141 hi 900 650 500 443 312 242 208 133 3 828 598 460 411 290 224 197 127 779 562 433 379 267 212 180 115 ' 743 536 413 352 248 201 169 108 m 715 513 395 341 241 192 160 102 4 326 312 230 220 184 177 158 150 99 4M 96 298 210 171 146 94 4% 284 200 166 138 89 5 270 256 244 233 190 180 172 164 161 156 151 145 135 130 124 120 87 84 80 6 223 157 140 115 74 a 213 150 138 111 71 207 146 134 107 69 6M 203 143 129 104 67 7 198 140 125 100 64 turnbuckli.es. Turnbuckles with right and left threads are made of standard sizes. B = H= length of Fig. 73. diameter of bolt, = 6 finches in all sizes of turnbuckle. tapped heads = 1}4B. L — length = 6 inches -f 3 B. 212 MATEKIALS. SIZES OF WASHERS. Diameter in inches. Size of Hole, in inches. Thickness, Birmingham Wire-gauge. Bolt in inches. No. in 100 lbs. 1 5-16 No. 16 M 29,300 % ' 16 5-16 18,000 7-16 ' 14 % 7,600 Wz 9-16 ' 11 Mi 3.300 m y& ' 11 9-16 2,180 lit 11-16 ' 11 % 2,350 13-16 ' 11 H 1,680 2 31-32 ' 10 % 1,140 M m ' 8 l 580 J H « 8 m 470 3 Wb ' 7 m m 360 3 m ' 6 360 TRACK SPIKES. Rails used. Spikes. Number in Keg, 200 lbs. Kegs per Mile, Ties 24 in. between Centres. 45 to 85 5^x9-16 380 30 40 " 52 5 x 9-16 400 27 35 " 40 5 x}4 490 22 24 " 35 Qb^Y* 550 20 24 " 30 4^x7-16 725 15 18 " 24 4 x 7-16 820 13 16 " 20 3^x% 1250 9 14 " 16 3 x% 2^x3^ 1350 8 8 " 12 1550 7 8 " 10 2^x5-16 22G0 5 STREET RAILWAY SPIKES. Spikes. Number in Keg, 200 lbs. Kegs per Mile, Ties 24 in. between Centres. 5^x9-16 5 xy 2 4J^x7-16 400 575 800 30 19 13 BOAT SPIKES. Number in Keg of 200 lbs. Length. H 5-16 % \k 2375 2050 1825 5 " 1230 1175 990 880 940 800 650 600 525 475 6 " 7 " 450 375 8 " 335 9 " 300 10 " 275 spikes; cut nails. 213 WROUGHT SPIKES. Number of Nails in Keg of 1 50 Pounds. Size. Hin. 5-16 in. 36 in. 7-16 in. y 2 in. 2250 1890 1650 1464 1380 1292 1161 VA I 4 " 1208 1135 1064 930 868 662 635 573 4y 2 ;; 5 " 742 570 482 455 424 391 6 " 8 " '.'..'.'. 9 " 10 " .... 445 384 300 270 249 236 306 256 240 222 11 " 203 12 " 180 WIRE SPIKES. Size. Approx. Size of Wire Nails. Ap. No. in 1 lb. Size. Approx. Size of Wire Nails. Ap. No. in 1 lb. lOd Spike 16d " 20d " 30d " ...... 40d " ...... 50d " 3 in. No. 7 sy 2 " " 6 4 " " 5 4 y « « 4 5 " " 3 by, " » 2 50 35 26 20 15 12 60d Spike 6^ in/; 8 " " '.'..'.'.'. 9 " " ...... 6 in. No. 1 6^ " " 1 7 " " 8 " " 00 9 " " 00 10 9 5 4^ LENGTH AND NUMRER OF CUT NAII.S TO THE POUND. Size. to a 3 a o s s o o 5 U a si '£, "a a si o *d m o o V O 53 o % Min. ." M 2 VA m J* zy A 4 i*. 5y 2 6 800 500 376 224 180 % 2d 800 480 288 200 168 124 88 70 58 44 34 23 18 14 10 8 95 74 62 53 46 42 3S 33 20 '84' 64 48 36 30 24 20 16 iioo 720 523 410 268 188 146 130 102 76 62 54 1000 760 368 3d "398 130' 96 82 68 4d 5d 6d 224 126 98 75 65 55 40 27 7d 8d 128 110 91 71 54 40 33 27 9d lOd oq 12d 16d oo 20d oy 2 8 30d 40d 50d 60d.. ....... 6 214 MATERIALS. NWM TfiO«3i.-C005©e\l ~ ;o 10 »n co co oj S£2£ ;t-»rtXr JrHrHMl-lOWQUO^ :g3 •Suiqsiai.j paqa^g pire q^ootus uouiuioq peq-reg •sp'Bjg pm? sp'BN uoraraoo •saqouj 'q^Saeq > i- CO OS CO i> SKMOQOOM » co co 10 co co a* ^010iOCOXf«05C< 3Oi000MOQ0!0fflC0«Ni- T-n-lT-ii-ii-irie!«MOJK)COMTjiTtT-iON« noiwiooai- OiO»»OMl-C( X>X>-13 o 03 a j> c s'S b£ ? u 03 I P!°! Silas S- 03 O f- 1 ^ 03 c § S e: .2 &CO g M $ M » e «..« 03 P. 3 C ' p. bC 3 _. += (- !ON««Offl«OOOS5 CW^lOCOt-QOOS^^NMOOOiCC Jiffl»t'QOO)«MOftfflO)0)« 0WM5OWMC H-ffiOOI^l'Ot -CJJJM-*lOt»»rHlf3C DrHTtlO05<0C0C000TfNC0{«i(>C0fflOffl» -n-lT^««MW*10i>05riinOi>inOO!50 ... . . ■ - . . . . .... : rnt»OW*!DQ05J«OWOt00CTjiO00Q0 •H'O-OlTPrtfflWiMr-IHiOQOHWCOOaDM ■«CIMMtfTliffl(X10'*£-5!rti-MlOCCT •MNtO»W(OOOffl!>«OOt- ■ossaeifflwijiioi-u-ioioeo ■ttiaiou'KWOMinrtOoow 216 MATERIALS. SIZE, WEIGHT, LENGTH, AND STRENGTH OF IRON WIRE. (Trenton Iron Co.) Tensile Strength (Ap- Diam. Area of proximate) of Charcoal No. by- in Deci- Section in Decimals of One Inch. Feet to Weight of Iron Wire in Pounds. Wire mals of the One Mile Gauge. One Inch. Pound. in pounds. Bright. Annealed. 00000 .450 .15904 1.863 2833.248 12598 9449 0000 .400 .12566 2.358 2238.878 9955 7466 000 .360 .10179 2.911 1813.574 8124 6091 00 .330 .08553 3.465 1523.861 6880 5160 .305 .07306 4.057 1301.678 5926 4445 1 .285 .06379 4.645 1136.678 5226 3920 2 .265 .05515 5.374 982 555 4570 3425 3 .245 .04714 6.286 839.942 3948 2960 4 .225 .03976 7.454 708.365 3374 2530 5 .205 .03301 8.976 588.139 2839 2130 6 .190 .02835 10.453 505.084 2476 1860 7 .175 .02405 12.322 428.472 2136 1600 8 .160 .02011 14.736 358.3008 1813 1360 9 .145 .01651 17.950 294.1488 1507 1130 10 .130 .01327 22.333 236.4384 1233 925 11 .1175 .01084 27.340 193.1424 1010 758 12 .105 .00866 34.219 154.2816 810 607 13 .0925 .00672 44 092 119.7504 631 473 14 .080 .00503 58.916 89.6016 474 356 15 .070 .00385 76.984 68.5872 372 280 16 .061 .00292 101.488 52.0080 292 220 17 .0525 .00216 137.174 38.4912 222 165 18 .045 .00159 186.335 28.3378 169 127 19 .040 .0012566 235.084 22.3872 137 103 20 .035 .0009621 308.079 17.1389 107 80 21 .031 .0007547 392.772 13.4429 22 .028 .0006157 481.234 10.9718 5o 2 z : ,- -.: ■■■. ■■.:. _ _ " -:-■..■-.: - -. -■■ ■■.-. - - - • • : ~ z ' z ; _ : : ~ - -- "- ■ 7 r;i*N«CiOO(-^Ci'HC^N'MC:lC«OM01lOOO>X :•• - - r - ■-" '.-' ' - r ■--.-■- - »-#OHt-«-«- - :-rn«HH KKOOiOOOOMWeN S^C*00«t5«01'«KlrtHOOOOOOOOOOOO< a t-»io ■* is in to >n ■* th»kno! ccotoffiMMiOrtO^cOHODiionooinoo ■■...■:.- - - ;: v. .:. -■■:.■.■.■■:■■■• ■ -■ ' - :■■--■-::■■..- Sl-HOl-dHS-tiSl-H:: . -'. - ' • HMNHHOO - - Z- Z ZZ Z Z z 777 7 - : - : : ": 7 7 . ": 7 -. : - ; CiO-*CC»3N(Ni-It-It-HOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO -*m -* o> co co loco-* os to co 03 us co e» « us oj oo in e? © eo to •* po e« oscot-m^ S^«eoOTC»CS O ■ ■:■ ■" ■■:■'"■ :■- , — " -.-.-■;■.■.■■- ■-..-..::- - - -■:::-«■•.: ■-.:..,. . •'-■;- - : - ■ - ■ : . -: ■ .. - : . - ' - - - - .:.■."■ : . t : .:■ '. :. r .. ' .... ..;::.:.-. :: — ■: ■ -. ■ — ■ -. r. ■ ...:.■' '.--..:..■,■ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOrHr- ft £- £\&V2 i> ft ^ HPQw in. lbs. lbs. in. lbs. lbs. No. 8 H 52 8,320 No.15 M 10 1,600 9 15-32 42 6,720 16 7-32 8 1,280 10 7-16 36 5,720 17 3-16 6 960 11 % 29 4,640 18 11-64 4 3-10 688 12 5-16 21 3,360 19 9-64 3 3-10 528 13 9^32 16 2,560 20 & 2 4-10 384 14 17-64 12 1,920 21 3-32 2 - 320 For special purposes these strands can be made of 50 to 100 per cent greater tensile strength. When used to run over sheaves or pulleys the use of soft-iron stock is advisable. FLEXIBLE STEEL-WIRE CABLES FOR VESSELS. (Trenton Iron Co., 1886.) With numerous disadvantages, the system of working ships' anchors with chain cables is still in vogue. A heavy chain cable contributes to the hold- ing-power of the anchor, and the facility of increasing that resistance by paying out the cable is prized as an advantage. The requisite holding- power is obtained, however, by the combined action of a comparatively light anchor and a correspondingly great mass of chain of little service in proportion to its weight or to the weight of the anchor. If the weight and size of the anchor were increased so as to give the greatest holding-power required, and it were attached by means of a light wire cable, the combined weight of the cable and anchor would be much less than the total weight of the chain and anchor, and the facility of handling would be much greater. English shipbuilders have taken the initiative in this direction, and many of the largest and most serviceable vessels afloat are fitted with steel-wire cables. They have given complete satisfaction. The Trenton Iron Co.'s cables are made of crucible cast-steel wire, and guaranteed to fulfil Lloyd's requirements. They are composed of 72 wires subdivided into six strands of twelve wires each. In order to obtain great flexibility, hempen centres are introduced in the strands as well as in the completed cable. FLEXIBLE STEEL-WIRE HAWSERS. These hawsers are extensively used, They are made with six strands of twelve wires each, hemp centres being inserted in the individual strands as well as in the completed rope. The material employed is crucible cast steel, galvanized, and guaranteed to fulfil Lloyd's requirements. They are only one third the weight of hempen hawsers ; and are sufficiently pliable to work round any bitts to which hempen rope of equivalent strength can be applied. 13-inch tarred Russian hemp hawser weighs about 39 lbs. per fathom. 10-inch white manila hawser weighs about 20 lbs. per fathom. lj^-inch stud chain weighs about (58 lbs. per fathom. 4-inch galvanized steel hawser weighs about 12 lbs. per fathom. Each of the above named has about the same tensile strength. 224 MATERIALS. SPECIFIC ATIONS FOR GALVANIZED IRON WIRE. Issued by the British Postal Telegraph Authorities. Weight per Mile. Diameter. Tests for Strength and Ductility. fe 73 2 2 2 S^fa £<5 V a a 02 -d bt- == a a 'S d a S a ft §s> a a Allowed. eg a c3 Allowed. m" r-* — SH 50 © a H- •4-1° o a d bD i o a 6 •™ct_|.S .aw ts OQ fe SD~ K &C+- 3 fe Ph ix 53 S 1 a a .a a a a a I a a ' S a a '3 a a 1 a a a a a a J a a M o S a a a a M o fa a a a a a a 1 a 3.SP lbs. lbs. lbs. mils. mils. mils. lbs. lbs. lbs. ohms. 800 767 833 242 237 247 2480 15 2550 14 2620 13 6.75 5400 600 571 «29 209 204 214 17 1910 16 1960 15 9.00 5400 450 424 477 181 176 186 10 19 18 1460 17 12.00 5400 400 377 424 171 166 176 21 , 20 1300 19 13.50 5400 200 190 213 121 118 125 620 30 638 28 655 26 27.00 5400 STRENGTH OF PIANO-WIRE. The average strength of English piano- wire is given as follows by Web- ster, Horsfals & Lean : Numbers Equivalents in Fractions Ultimate Numbers Equivalents in Fractions Ultimate. in Music- Tensile in Music- Tensile wire of Inches in Strength in wire of inches in Strength in Gauge. Diameters. Pounds. Gauge. Diameters. Pounds. 12 .029 225 18 .041 395 13 .031 250 19 .043 425 14 .033 285 20 .045 500 15 .035 305 21 .047 540 16 .037 340 22 .052 650 17 .039 360 These strengths range from 300,000 to 340,000 lbs. per sq. in. The compo- sition of this wire is as follows: Carbon, 0.570; silicon, 0.090; sulphur, 0.011; phosphorus, 0.018; manganese, 0.425. "PIiOUGH"-STEEIi WIRE. The term "plough," given in England to steel wire of high quality, was derived from the fact that such wire is used for the construction of ropes used for ploughing purposes. It is to be hoped that the term will not be used in this country, as it tends to confusion of terms. Plough-steel is known here in some steel-works as the quality of plate steel used for the mould-boards of ploughs, for which a very ordinary grade is good enough. Experiments by Dr. Percy on the English plough-steel (so-called) gave the following results: Specific gravity, 7.814 ; carbon, 0.828 per cent; manga- nese, 0.587 per cent; silicon, 0.143 per cent; sulphur, 0.009 per cent; phos- phorus, nil; copper, 0.030 per cent. No traces of chromium, titanium, or tungsten were found. The breaking strains of the wire were as follows: Diameter, inch 093 .132 .159 .191 Pounds per sq. inch 344,960 257,600 224,000 201,600 The elongation was only from 0.75 to 1.1 per cent. SPECIFICATIONS FOR HARD-DRAWN" COPPER WIRE. 225 WIRES OF DIFFERENT METALS AND AL.L.OYS. (J. Bucknall Smith's Treatise on Wire.) Brass Wire is commonly composed of an alloy of 1 3/4 to 2 parts of copper to 1 part of zinc. The tensile strength ranges from 20 to 40 tons per square inch, increasing with the percentage of zinc in the alloy. German or Nickel Silver, an alloy of copper, zinc, and nickel, is practically brass whitened by the addition of nickel. It has been drawn into wire as fine as .002" diam. Platinum wire may be drawn into the finest sizes. On account of its high price its use is practically confined to special scientific instruments and electrical appliances in which resistances to high temperature, oxygen, and acids are essential. It expands less than other metals when heated, which property permits its being sealed in glass without fear of cracking. It is therefore used in incandescent electric lamps. Phosphor-bronze Wire contains from 2 to 6 per cent of tin and from 2V to y% per cent of phosphorus. The presence of phosphorus is detri- mental to electric conductivity. " Delta-metal " wire is made from an alloy of copper, iron, and zinc. Its strength ranges from 45 to 62 tons per square inch. It is used for some kinds of wire rope, also for wire-gauze. It is not subject to deposits of ver- digris. It has great toughness, even when its tensile strength is over 60 tons per square inch. Aluminum Wire. — Specific gravity .268. Tensile strength only about 10 tons per square inch. It has been drawn as fine as 11,400 yards to the ounce, or .042 grains per yard. Aluminum Bronze, 90 copper, 10 aluminum, has high strength and ductility; is inoxidizable, sonorous. Its electric conductivity is 12.6 per cent of that of pure copper. Silicon Bronze, patented in 1882 by L. Weiler of Paris, is made as follows: Fluosilicate of potash, pounded glass, chloride of sodium and cal- cium, carbonate of soda and lime, are heated in a plumbago crucible, and after the reaction takes place the contents are thrown into the molten bronze to be treated. Silicon-bronze wire has a conductivity of from 40 to 98 per cent of that of copper wire and four times more than that of iron, while its tensile strength is nearly that of steel, or 28 to 55 tons per square inch of section. The conductivity decreases as the tensile strength in- creases. Wire whose conductivity equals 95 per cent of that of pure copper gives a tensile strength of 28 tons per square inch, but when its conductivity is 34 per cent of pure copper, its strength is 50 tons per square inch. It is being largely used for telegraph wires. It has great resistance to oxidation. Ordinary Drawn and Annealed Copper Wire has a strength of from 15 to 20 tons per square inch. SPECIFICATIONS FOR HARD-DRAWN COPPER WIRE. The British Post Office authorities require that hard-drawn copper wire supplied to them shall be of the lengths, sizes, weights, strengths, and con- ductivities as set forth in the annexed table. Weight per Statute Approximate Equiva- SB ° J S°f BO Mile. len t Diameter. 11 a w a * £§ log p-a S p a s a id a p a 5 eg $02 a 9 -2 1 a c.22 | 113J4 650 20 4.53 50 400 390 410 158 155J^ 160J4 1300 10 2.27 50 226 MATERIALS. WIRE ROPES. List adopted by manufacturers in 1892. See pamphlets of Trenton Iron Co., John A. Roebling's Sons Co., and other makers. Pliable Hoisting Rope. With 6 strands of 19 wires each. IRON. c «H a- S O o^ a «5 S-© .S ° o o ® "3 0> & J 0J si ft «5»S a o £s» g> g a s ^ 03 ftcco 1 2^ 6% 8.00 74 15 14 13 2 2 6 6.30 65 13 13 12 3 1% 5^ 5.25 54 11 12 10 4 1% 5 4.10 44 9 11 8^ 5 M 4% 3.65 39 8 10 7^ 5% Ws 4% 3.00 33 6^ 9^ 7 6 ij| 4 2.50 27 5J^ 8^3 6V6 7 3^ 2.00 20 4 6 8 l m 1.58 16 3 6^ 5M 4^ 9 % 2% 1.20 11.50 2^ 51^ 10 % 2J4 0.88 8.64 1% 4% 4 10M % 2 0.60 5.13 ■a 3% m. 10^ 9-16 1% 0.44 4.27 3^ 2H 10% K m 0.35 3.48 3 m 10a 7-16 m 0.29 3.00 % 2% 2 10^ % m 0.26 2.50 Ya 2^ i» CAST STEEL. 1 2M 2 M mi 6 ®/ 2 8.00 155 31 sy 2 8 2 6.30 5.25 125 106 25 21 3 7^ 6^4 5% 4 m 5 4.10 86 17 15 5 M 4% 3.65 77 15 14 5^ 4% 3.00 63 12 13 5^ 6 m 4 2.50 52 10 12 5 7 m 3^ 2.00 42 8 11 4y 9 8 3^ 1.58 33 6 9^ 4 9 % 2% 1.20 25 5 w 2 3^ 10 H 2y 4 0.88 18 sy 2 7 3 10J4 % 2 0.60 12 zy 2 5% 2M 10^ 9-16 1% 0.44 9 w 2 5 18 im X i« 0.35 7 1 4y 2 10a 7-16 1% 0.29 sy 2 *y 2 % m IM 10% % VA 0.26 H *y 2 1 Cable-Traction Ropes. According to English practice, cable-traction ropes, of about 3}4 in. in circumference, are commonly constructed with six strands of seven or fif- teen wires, the lays in the strands varying from, say, 3 in. to 3^ in., and the lays in the ropes from, say, 7^ in. to 9 in. In the United States, however, strands of nineteen wires are generally preferred as being more flexible; but, on the other hand, the smaller external wires wear out more rapidly. The Market street Street Railway Company, San Francisco, has used ropes 1J4 in- hi diameter, composed of six strands of nineteen steel wires, weighing 2^ lbs. per foot, the longest continuous length being 24,125 ft. The Chicago City Railroad Company has employed cables of identical construction, the longest length being 27,700 ft. On the New York and Brooktyn Bridge cable- railway steel ropes of 11,500 ft, long, containing 114 wires, have been used, WIRE EOPES. 227 Transmission and Standing; Rope. With 6 strands of 7 wires each. a &c«w N S a oo5 ® "eg = 08 2. © 3 a be 1 ** fig °M .2 a) t8 s g3 u '5o--£ go G.03O C ©S-C3 ■CO H s O > e.?*j PQ-" £^ w 5 =tfw g 11 m 4% 4^ 3.37 36 9 10 13 12 m 2.77 30 Wz 9 12 13 m 3% 2.28 25 6J4 8^ 10% 9v| 14 m 3% 1.82 20 5 7*6 15 i 3 1.50 16 4 6^ 8^ 16 Vs 2% 1.12 12.3 3 5% 7^ 17 u 2% 0.88 8.8 m - 4% 4^ 63^ 18 11-16 2^ 0.70 7.6 2 6 19 % 1% 0.57 5.8 1^ 4 514 20 9-16 Ws 0.41 4.1 1 3J4 4^ 21 M W& 0.31 2.83 3 2M 4 22 7-16 VA 0.23 2.13 2^ 3M 23 % Ui 0.19 1.65 2^ 2% 24 5-16 9-32 l % 0.16 0.125 1.38 1.03 2 1% 2^ 2M 25 CAST STEEL. 11 m m 3.37 62 13 13 8^ 12 a* 4U 2.77 52 10 12 8 13 m 3% 2.28 44 9 11 7M 14 1H 1.82 36 ~y% 10 6M 15 3 1.50 30 6 9 5% 16 % 2% 1.12 22 4^ 8 5 17 H m 0.88 17 m 7 4V4 18 11-16 2Y 8 0.70 14 3 6 4 19 Vs m 0.57 11 2M 5M 4M 3^ 20 9-16 1% 0.41 8 m 3 21 Y* m 0.31 6 i« 4 2^ 22 23 7-16 % i*t 0.23 0.19 4^ 4 JM ft f4 24 5-16 1 0.16 3 s 2% 1% 25 9-32 % 0.12 2 2J4 1^ Plough-Steel Rope. Wire ropes of very high tensile strength, which are ordinarily called "Plough-steel Ropes," are made of a high grade of crucible steel, which, when put in the form of wire, will bear a strain of from 100 to 150 tons per square inch. Where it is necessary to use very Ions or very heavy ropes, a reduction of the dead weight of ropes becomes a matter of serious consideration. It is advisable to reduce all bends to a minimum, and to use somewhat larger drums or sheaves than are suitable for an ordinary crucible rope hav- ing a strength of 60 to 80 tons per square inch. Before using Plough-steel Ropes it is best to have advice on the subject of adaptability. 228 MATERIALS. Plough-Steel Rope. With 6 strands of 19 wires each. Trade Diameter in Weight per foot in pounds. Breaking Strain in Proper Work- Min. Size of Drum or Number. inches. tons of 2000 lbs. ing Load. Sheave in feet. 1 2*4 8.00 240 46 9 2 2 6.30 189 37 8 3 m 5.25 157 31 7^ 4 4.10 123 25 6 5 V4 3.65 110 22 &A 5^ Ws 3.00 90 18 m 6 1M 2.50 75 15 5 7 V4 2.00 60 12 4*4 8 1.58 47 9 414 m 9 Va 1.20 37 7 10 I 0.88 27 5 w% 10J4 0.60 18 m 3 ioy 2 9-16 0.44 13 2 2Y2 10% ¥2 0.35 10 M 2 With 7 Wires to the Strand. 15 1 1.50 45 9 % 16 1 1.12 33 ®A 5 17 *0.88 25 qa 4 18 11-16 0.70 21 4 3^ 19 % 0.57 16 3M 3 20 9-16 0.41 12 2 21 y* 0.31 9 IJUs 22 7-16 0.23 5 a 2 23 % 0.19 4 i^ Galvanized Iron Wire Rope. For Ships' Rigging and Guys for Derricks. CHARCOAL ROPE. Cir. of Break- Weight per Fathom in pounds. Cir. of Break- Circum- ference in inches. Weight per Fath- om in pounds. new Manila Rope of equal ing Strain in tons of 2000 Circum- ference in inches new Manila Rope of equal ing Strain in tons of 2000 Strength. pounds Strength. pounds 5^ 26^ 11 43 2% &A 5 9 5^ 24^ 10^ 40 m *H m 8 5 22 10 35 2 m *A 7 m 21 W2 33 m %A 3M 5 *a 19 9 30 1^ 2 3 m m 16^ m 26 m m m za 4 14M 8 23 m \ *A 2J4 m 12% m 20 1 &A 2 w* 10-M &A 16 % a m 1 m W* 6 14 3 A VA ¥4 3 8 5M 514 12 ¥s m % m m 10 A y* m % WIRE ROPES. 229 Galvanized Cast-steel Yacht Rigging. Circum- ference in inches. Weight per Fath- om in pounds. Cir. of new Manilla Rope of equal Strength. Break- ing Strain in tons of 2000 pounds Circum- ference in inches Weight per Fathom in pounds. Cir. of new Manilla Rope of equal Strength. Break- ing Strain in tons of 2000 pounds 4 3^ 3 m it 10% 8 m 4^ 13 11 {$£ 8 66 43 32 2? 22 18 2 Wa m i 3^ 2 m m ey 2 by A Wa 414 3% 3 14 10 8 6^ 5^ 3^ Steel Hawsers, For Mooring, Sea, and Lake Towing. Circumfer- ence. Breaking Strength. Size of Manilla Haw- ser of equal Strength. Circumfer- ence. Breaking Strength. Size of Manilla Haw- ser of equal Strength. Inches. 2y 2 Tons. 15 18 22 Inches. 7 Inches. 3^ 4 Tons. 29 35 Inches. 9 10 Steel Flat Ropes. (J. A. Roebling*s Sons Co.) Steel-wire Flat Ropes are composed of a number of strands, alternately twisted to the right and left, laid alongside of each other, and sewed together with soft iron wires, These ropes are used at times in place of round ropes in the shafts uf mines. They wind upon themselves on a narrow winding- drum, which takes up less room than one necessary for a round rope. The soft-iron sewing-wires wear out sooner than the steel strands, and then it becomes necessary to sew the rope with new iron wires. Width and Thickness Weight per foot in Strength in pounds. Width and Thickness Weight per foot in Strength in pounds. in inches. pounds. in inches. pounds. %x2 1.19 35,700 ^x3 2.38 71,400 %x2^ 1.86 55.800 %*3}4 2.97 89,000 %x3 2.00 60,000 3^x4 3.30 99,000 %*m 2.50 75,000 14x4^2 4.00 120,000 %x4 2.86 85,800 1^x5 4.27 128,000 %x4V 2 3.12 93,600 y^y* 4.82 144,600 3 / 8 x5 3.40 100,000 ^x6 5.10 153,000 9$x5« 3.90 110,000 ^x7 5.90 177,000 For safe working load allow from one fifth to one seventh of the breaking stress. " Lang Lay" Rope. In wire rope, as ordinarily made, the component strands are laid up into rope in a direction opposite to that in which the wires are laid into strands; that is, if the wires in the strands are laid from right to left, the strands are laid into rope from left to right. In the " Lang Lay,' 1 sometimes known as "Universal Lay," the wires are laid into strands and the strands into rope in the same direction; that is, if the wire is laid in the strands from right to left, the strands are also laid into rope from right to left. Its use has beeu found desirable under certain conditions and for certain purposes, mostly for haulage plants, inclined planes, and street railway cables, although it has also been used for vertical hoists in mines, etc. Its advantages are that 230 MATERIALS. GALVANIZED STEEL CABLES. For Suspension Bridges, (Roebling's. * 5 ^o o i A ,3 o 58 .go* o a &i a o a CD 02 <" 0) 02 «m *£ ^ 02 CO ft '£ ®2« ft ■J ft -2 1 +3 e O .bC '5 r- O S g c 3 £.5 ft A 5 P-Sft £ S p .2ft £ 5 £ 2% 220 13 214 155 8.64 1% 1*6 95 5.6 2W 200 11.3 2 110 6.5 75 4 35 m 180 10 1% 100 5.8 1W 65 3.7 COMPARATIVE STRENGTHS OF FLEXIBLE GAL- VANIZED STEEL-WIRE HAWSERS, With Chain Cable, Tarred Russian Hemp, and White Manila Ropes. (Trenton Iron Co.) Patent Flexible Tarred Rus- White Steel-wire Hawsers Chain Cable sian Hemp Manilla and Cables. Rope. Ropes. a 6fl oA . ft u 73 £ R S c c3 Pa 2 £ S 2 3 fa o ft n 9) 3-n ° op fa s, 1 2 So to fa J-l » ft o3 « fa U si 53 "5.9 P 0) b w a a U To c3 c3 1>S o5 N U 2 fa 6 33 to '53 ea £ fa 6 1 ea fa 1 : M 1% 6 2% 1W 2W 1M 2 11.-, 1 214 7W w 14 4W 6 n% 3 2U 3 1% 2% 1% 4 9 4 3W y. 3V 4 2 *W i-'i 51/, 1(W 9-16 17 5W TJ4 5 6 5 4 8 5 2% 12 53/ ( 8 7 5 4W 7% 21 -j 3% 9 18*6 10-16 21 7 9W 10 9 5% 6 low 2U 4>., 12 15 13 nw pM 7 12^4 2% 51,, 15 16W 11-16 2b 8W 12% 16 14 V t% 15 3 7 18 18 12-16 30 10W 15W 9 19 :6U Wo 18 8 22 19^ 13-16 35 11% 15 8-10 17 8-10 10 23 20 m 13 22% 9 26 SI 15-16 48 23 7-10 11 28 9 !4Vo 25 4 12 33 24 1 54 18 27 12 33 29 10 18 31W 38W 15 39 27 1% 68 22% 34W 13 39 34 11 22 5 64 30 1 17-32 113 37W 55W 15 56 50 123/, 51 >k es 74 33 1% 47W 66W 17 67 60 13W 8bW 62 6 33 88 36 m km; 55W 77W 1!) 84 72 15 42 ',8W > l <> 37 102 39 1 15-16 mi 94W 21 106 89 7 41 116 42 2 1-16 76W 107 1-10 23 123 106 7W 47 130 45 2 3-16 86W 120W 24 134 115 8 53 150 48 2 5-16 96J4 134% 25 146 125 WIKE HOPES. 231 it is somewhat more flexible than rope of the same diameter and composed of the same number of wires laid up in the ordinary manner; and (especi- ally) that owing to the fact that the wires are laid more axially in the rope, longer surfaces of the wire are exposed to wear, and the endurance of the rope is thereby increased. (Trenton Iron Co.) Notes on the Use of Wire Rope. (J. A. Roebling's Sons Co.) Two kinds of wire rope are manufactured. The most pliable variety con- tains nineteen wires in the strand, and is generally used for hoisting and running rope. The ropes with twelve wires and seven wires in the strand are stiffer, and are better adapted for standing rope, guys, and rigging. Or- ders should state the use of the rope, and advice will be given. Ropes are made up to three inches in diameter, upon application. For safe working load, allow one fifth to one seventh of the ultimate strength, according to speed, so as to get good wear from the rope. When substituting wire rope for hemp rope, it is good economy to allow for the former the same weight per foot which experience has approved for the latter. Wire rope is as pliable as new hemp rope of the same strength; the for- mer will therefore run over the same-sized sheaves and pulleys as the latter. But the greater the diameter of the sheaves, pulleys, or drums, the longer wire rope will last. The minimum size of drum is given in the table. Experience has demonstrated that the wear increases with the speed. It is, therefore, better to increase the load than the speed. Wire rope is manufactured either with a wire or a hemp centre. The lat- ter is more pliable than the former, and will wear better where there is short bending. Orders should specify what kind of centre is wanted. Wire rope must not be coiled or uncoiled like hemp rope. When mounted on a reel, the latter should be mounted on a spindle or flat turn-table to pay off the rope. When forwarded in a small coil, without reel, roll it over the ground like a wheel, and run off the rope in that way. All untwisting or kiukiug must be avoided. To preserve wire rope, apply raw linseed-oil with a piece of sheepskin, wool inside; or mix the oil with equal parts of Spanish brown or lamp-black. To preserve wire rope under water or under ground, take mineral or vege- table tar, and add one bushel of fresh-slacked lime to one barrel of tar, which will neutralize the acid. Boil it well, and saturate the rope with the- hot tar. To give the mixture body, add some sawdust. In no case should galvanized rope be used for running rope. One day's use scrapes off the coating of zinc, and rusting proceeds with twice the rapidity. The grooves of cast-iron pulleys and sheaves should be filled with well- seasoned blocks of hard wood, set on end, to be renewed when worn out. This end-wood -will save wear and increase adhesion. The smaller pulleys or rollers which support the ropes on inclined planes should be constructed on the same plan. When large sheaves run with very great velocity, the grooves should be lined with leather, set on end, or with India rubber. This is done in the case of sheaves used in the transmission of power between distant points by means of rope, which frequently runs at the rate of 4000 feet per minute. Steel ropes are taking the place of iron ropes, where it is a special object to combine lightness with strength. But in substituting a steel rope for an iron running rope, the object in view should be to gain an increased wear from the rope rather than to reduce the size. Locked Wire Rope. Fig. 74 shows what is known as the Patent Locked Wire Rope, made by the Trenton Iron Co. It is claimed to wear two to three times as long as an ordinary wire rope of equal diameter and of like material. Sizes made are from %, to \y^ inches diameter- 232 MATERIALS. CRANE CHAINS. (Pencoycl Iron Works.) "D.B. G " Special Crane. Crane. © " © T3 be . ctf oi -^ >> O . O a S ° 71 J © o 3 § & 33 © S* 3 o Pi .53 h © l^ 5 Oh ^ «j o j wedge (or bullhead). 9x4^ wide x 2J^ to 2 in. thick, tapering lengthwise. A . 98 bricks to circle 5 ft. inside diam. '\ Arch \ No. 2 wedge 9x4^x2^ to \y 2 in. thick. ) 4 60 bricks to circle 2% ft. inside diam. /9x4>^x(2Xi%/ No. 1 arch 9x4J^x2^ to 2 in. thick, f " / tapering breadthwise. ' 72 bricks to circle 4 ft. inside diam. No. 2 arch 9 x 4}4 x 2]4 to V& \ 42 bricks to circle 2 ft. inside diam. ..iSkew\ No. 1 skew 9 to 7 x 4}4 to 2J^. \ Bevel on one end. „ } No. 2 skew 9x2j^x4^to2^. <§:7) x 4V^ x 2>^/ Equal bevel on both edges. ' No.3skew.... 9 x 2% x 4% to iy 2 . Taper on one edge. — — \ 24 inch circle 8% to 5% x 4% x 2%. \____ \ Edges curved, 9 bricks line a 24-inch circle. f " ) 36-inch circle 8% to 6^ x 4}4 x 2^. / 9 x 2kx av-^A 13 bricks line a 36-inch circle. J k X2 /2 j 48-inch circle 8^ to 714 x 4^ x 2%. * ' 17 bricks line a 48-inch circle. 13J^-inch straight 13^ x 2% x 6. 13^-inch key No. 1 13j| x 2% x 6 to 5 inch. Skew~ — \ 90 bricks turn a 12-ft. circle. 1 133^-inch key No. 2 13^ x 2^ x 6 to 4% inch. , • , 52 bricks turn a 6-ft. circle. ~ X2 l X2 x v Bridge wall, No. 1 13x6^x6. Bridge wall, No. 2 13X 6j| x 3. 6 in. Ci rc i e Mill tile 18, 20, or 24 x 6 x 3. ££ \ Stock-hole tiles 18, 20, or 24'x 9 x 4. 18-inch block 18x9x6. Flat back 9x6x2^. Flat back arch 9 x 6 x 3J^ to 2%. 22-inch radius, 56 bricks to circle. Locomotive tile 32 x 10 x 3. 34x10x3. Cupola 40x10x3. Tiles, slabs, and blocks, various sizes 12 to 30 inches long, 8 to 30 inches wide, 2 to 6 inches thick. Cupola brick, 4 and 6 inches high, 4 and 6 inches radial width, to line shells 23 to 66 in diameter. A 9-inch straight brick weighs 7 lbs. and contains 100 cubic inches. (=120 lbs. per cubic foot. Specific gravity 1.93.) One cubic foot of wall requires i7 9-inch bricks, one cubic yard requires 460. Where keys, wedges, and other " shapes " are used, add 10 per cent iu estimating the number required. 234 MATERIALS. One ton of fire-clay should be sufficient to lay 3000 ordinary bricks. To secure the best results, fire-bricks should be laid in the same clay from which they are manufactured. It should be used as a thin paste, and not as mor- tar. The thinner the joint the better the furnace wall. In ordering bricks the service for which they are required should be stated. NUMBER OF FIRE-BRICK REQUIRED FOR VARIOUS CIRCL.ES. KEY BRICKS. ARCH BRICKS. WEDGE BRICKS. s s o to 25 17 9 d to 6 to 6 to 1 o Eh 6 to 6 to OS 3 o E-i 6 to d to OS O E-< ft. in. I 6 25 30 34 38 42 46 51 55 59 63 67 71 76 80 84 88 92 97 101 105 109 113 117 2 13 25 38 32 25 19 13 6 10 21 32 42 53 63 58 52 47 42 37 31 2G 21 16 11 5 9 19 29 38 47 57 66 76 85 94 104 113 113 42 31 21 10 42 49 57 64 72 80 87 95 102 110 117 125 132 140 147 155 162 170 177 185 193 2 6 18 3G 54 72 72 72 72 72 72 7:2 72 72 72 72 72 72 72 ""8" 15 23 30 38 45 53 60 68 75 83 90 98 105 113 121 GO 48 3G 24 12 60 3 3 6 4 4 6 5 5 6 6 6 6 7 7 6 8 8 6 9 9 6 10 10 G 11 11 6 12 20 40 59 79 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 15 23 30 38 46 53 61 68 76 83 91 98 106 68 76 83 91 98 106 113 121 128 136 144 151 159 166 174 181 189 196 304 12 6 For larger circles than 12 feet use 113 No. 1 Key, and as many 9-inch brick as may be needed in addition. ANALYSES OF IT. SAVAGE FIRE-CLAY, (1) (2) (3) (4) 1871 1877. 1878. 1885. Mass - ^TavJoT CteSfoScal (2 samples) Institute of N e^ ay T e r °i v Sm-v?v of D'"- Otto Technology. pr £ e £ g™*^ PemfsylJania. Wuth " 50.457 56.80 Silica 44.395 56.15 35.904 30.08 Alumina 33.558 33.295 1.15 Titanic acid 1.530 1.504 1.12 Peroxide iron . 1.080 0.59 0.133 ... . Lime trace 0.17 0.018 Magnesia 0.108 0.115 trace 0.80 Potash (alkalies) 0.247 12.744 10.50 Water and inorg. matter. 14.575 9.68 100.760 100.450 100:493 100.000 MAGNESIA BRICKS. 235 MAGNESIA BRICKS. "Foreign Abstracts " of the Institution of Civil Engineers, 1893, gives a paper by C. Bischof on the production of magnesia bricks. The material most in favor at present is the magnesite of Styria, which, although less pure considered as a source of magnesia than the Greek, has the property of fritting at a high temperature without melting. The composition of the two substances, in the natural and burnt states, is as follows: Magnesite. Styrian. Greek. Carbonate of magnesia 90.0 to 96.0$ 94.46$ " lime 0.5 to 2.0 4.49 " iron 3.0 to 6.0 FeO 0.08 Silica 1.0 0.52 Manganous oxide 0.5 Water 0.54 Burnt Magnesite. Magnesia ... 77.6 82.46-95.36 Lime 7.3 0.83—10.92 Alumina and ferric oxide 13.0 0.56— 3.54 Silica 1.2 0.73—7.98 At a red heat magnesium carbonate is decomposed into carbonic acid and caustic magnesia, which resembles lime in becoming hydrated and recar- bonated when exposed to the air, and possesses a certain plasticity, so that it can be moulded when subjected to a heavy pressure. By long-continued or stronger heating the material becomes dead-burnt, giving a form of mag- nesia of high density, sp. gr. 3.8, as compared with 3.0 in the plastic form, which is unalterable in the air but devoid of plasticity. A mixture of two volumes of dead-burnt with one of plastic magnesia can be moulded into bricks which contract but little in firing. Other binding materials that have been used are: clay up to 10 or 15 per cent; gas -tar, perfectly freed from water, soda,, silica, vinegar as a solution of magnesium acetate which is readily decomposed by heat, and carbolates of alkalies or lime. Among . magnesium compounds a weak solution of magnesium chloride may also be used. For setting the bricks lightly burnt, caustic magnesia, with a small proportion of silica to render it less refractory, is recommended. The strength of the bricks may be increased by adding iron, either as oxide or silicate. If a porous product is required, sawdust or starch may be added to the mixture. When dead-burnt magnesia is used alone, soda is said to be the best binding material. See also papers by A. E. Hunt, Trans. A. I. M. E., xvi, 720, and by T. Egles- ton, Traus. A. I. M. E., xiv. 458. Asbestos.— J. T. Donald, Eng. and M. Jour., June 27, 1891. Analysis. Canadian. Italian. Broughton. Templeton. Silica.. 40.30$ 40.57$ 40.52?$ Magnesia 43.37 41.50 42.05 Ferrous oxide 87 2.81 1.97 Alumina 2.27 .90 2.10 Water 13.72 13.55 13.46 100.53 99.33 100.10 Chemical analysis throws light upon an important point in connection with asbestos, i.e., the cause of the harshness of the fibre of some varieties. Asbestos is principally a hydrous silicate of magnesia, i.e.. silicate of mag- nesia combined with water. When harsh fibre is analyzed it is found to contain less water than the soft fibre. In fibre of very fine quality from Black Lake analysis showed 14.38$ of water, while a harsh-fibred sample gave only 11.70$. If soft fibre be heated to a temperature that will drive off a portion of the combined water, there results a substance so brittle that it may be crumbled between thumb and finger. There is evidently some con- nection between the consistency of the fibre and the amount of water in its composition. 236 STRENGTH OF MATERIALS. STRENGTH OF MATERIALS. Stress and Strain.— There is much confusion among writers on strength of materials as to the definition of these terms. An external force applied to a body, so as to pull it apart, is resisted by an internal force, or resistance, and the action of these forces causes a displacement of the mole- cules, or deformation. By some writers the external force is called a stress, and the internal force a strain; others call the external force a strain, and the internal force a stress: this confusion of terms is not of importance, as the words stress and strain are quite commonly used synonymously, but the use of the word strain to mean molecular displacement, deformation, or dis- tortion, as is the custom of some, is a corruption of the language. See En- gineering NeioSy June 23, 1892. Definitions by leading authorities are given below. Stress.— A. stress is a force which acts in the interior of a body, and re- sists the external forces which tend to change its shape. A deformation is the amount of change of shape of a body caused by the stress. The word sti'ain is often used as synonymous with stress and sometimes it is also used to designate the deformation. (Merriman.) The force by which the molecules of a body resist a strain at any point is called the stress at that point. The summation of the displacements of the molecules of a body for a given point is called the distortion or strain at the point considered. (Burr). Stresses are the forces which are applied to bodies to bring into action their elastic and cohesive properties. These forces cause alterations of the forms of the bodies upon which they act. Strain is a name given to the kind of alteration produced by the stresses. The distinction between stress and strain is not always observed, one being used for the other. (Wood.) Stresses are of different kinds, viz. : tensile, compressive, transverse, tor- sional, and shearing stresses. A tensile stress, or pull, is a force tending to elougate a piece. A com- pressive stress, or push, is a force tending to shorten it. A travsverse stress tends to bend it. A torsional stress tends to twist it. A shearing stress tends to force one part of it to slide over the adjacent part. Tensile, compressive, and shearing stresses are called simple stresses. Transverse stress is compounded of tensile and compressive stresses, and torsional of tensile and shearing stresses. To these five varieties of stresses might be added tearing stress, which is either tensile or shearing, but in which the resistance of different portions of the material are brought into play in detail, or one after the other, in- stead of simultaneously, as in the simple stresses. Effects of Stresses.— The following general laws for cases of simple tension or compression have been established by experiment. (Merriman): 1. When a small stress is applied to a body, a small deformation is pro- duced, and on the removal of the stress the body springs back to its original form. For small stresses, then, materials may be regarded as perfectly elastic. 2. Under small stresses the deformations are approximately proportional to the forces or stresses which produce them, and also approximately pro- portional to the length of the bar or body. 3. When the stress is great enough a deformation is produced which is partly permanent, that is, the body does not spring back entirely to its original form on removal of the stress. This permanent part is termed a set. In such cases the deformations are not proportional to the stress. 4. When the stress is greater still the deformation rapidly increases and the body finally ruptures. 5. A sudden stress, or shock, is more injurious than a steady stress or than a stress gradually applied. Elastic liiniit.— The elastic limit is defined as that point at which the deformations cease to be proportional to the stresses, or, the point at which the rate of stretch (or other, deformation) begins to increase. It is also defined as the point at which the first permanent set becomes visible. The last definition is not considered as good as the first, as it is found that with some materials a set occurs with any load, no matter how small, and that with others a set which might be called permanent vanishes with lapse of time, and as it is impossible to get the point of first set without removing STRESS AND STRAIN. 237 the whole load after each increase of load, which is frequently inconvenient. The elastic limit, defined, however, as the point at which the extensions be- gin to increase at a higher ratio than the applied stresses, usually corresponds Very nearly with the point of first measurable permanent set. Yield-point.— The term yield-point has recently been introduced into the literature of the strength of materials. It is defined as that point at which the rate of stretch suddenly increases rapidly. The difference be- tween the elastic limit, strictly defined as the point at which the rate of stretch begins to increase, and the yield-point, at which the rate increases suddenly, may in some cases be considerable. This difference, however, will not be discovered in short test-pieces unless the readings of elongations are 1 OOOi of an inch. In using a coarser instrument, such as calipers reading to 1/100 of an inch, the elastic limit and the yield-point will appear to be simultane- ous. Unfortunately for precision of language, the term yield-point was not introduced until long after the term elastic limit had been almost univer- sally adopted to signify the same physical fact which is now defined by the term yield-point, that is, not the point at which the first change in rate, observable only by a microscope, occurs, but that later point (more or less indefinite as to its precise position) at which the increase is great enough to be seen by the naked eye. A most convenient method of determining the point at which a sudden increase of rate of stretch occurs in short speci- mens, when a testing-machine in which the pulling is done by screws is used, is to note the weight on the beam at the instant that the beam " drops. " During the earlier portion of the test, as the extension is steadilj r increased by the uniform but slow rotation of the screws, the poise is moved steadily along the beam to keep it in equipoise; suddenly a point is reached at which the beam drops, and will not rise until the elongation has been considerably increased by the further rotation of the screws, the advancing of the poise meanwhile being suspended. This point corresponds practically to the point at which the rate of elongation suddenly increases, and to the point at which an appreciable permanent set is first found. It is also the point which has hitherto been called in practice and in text-books the elastic limit, and it will probably continue to be so called, although the use of the newer term "yield-point " for it, and the restriction of the term elastic limit to mean the earlier point at which the rate of stretch begins to increase, as determin- able only by micrometric measurements, is more precise and scientific. In tables of strength of materials hereafter given, the term elastic limit is used in its customary meaning, the point at which the rate of stress has be- gun to increase, as observable by ordinary instruments or by the drop of the beam. With this definition it is practically synonymous with yield- point. Coefficient (or Modulus) of Elasticity.— This is a term express- ing the relation between the amount of extension or compression of a mate- rial and the load producing that extension or compression. It may be defined as the load per unit of section divided by the extension per unit of length; or the reciprocal of the fraction expressing the elonga- tion per inch of length, divided by the pounds per square inch of section producing that elongation. Let P be the applied load, k the sectional area of the piece, I the length of the part extended, A the amount of the extension, and _£Jthe coefficient of elasticity. Then p — — the load on a unit of section ; k j = the elongation of a unit of length. k ' l~ kk' The coefficient of elasticity is sometimes defined as the figure expressing the load which would be necessary to elongate a piece of one square inch section to double its original length, provided the piece would not break, and the ratio of extension to the force producing it remained constant. This definition follows from the formula above given, thus: Iffc=one square inch, I and ^ each = one inch, then E — P. Within the elastic limit, when the deformations are proportional to the 238 STRENGTH OF MATERIALS. stresses, the coefficient of elasticity is constant, but beyond the elastic limit it decreases rapidly. In cast iron there is generally no apparent limit of elasticity, the deforma- tions increasing at a faster rate than the stresses, and a permanent set being produced by small loads. The coefficient of elasticity therefore is not con- stant during any portion of a test, but grows smaller as the load increases. The same is true in the case of timber. In wrought iron and steel, however, there is a well-defined elastic limit, and the coefficient of elasticity within that limit is nearly constant. Resilience, or Work of Resistance of a Material.— Within the elastic limit, the resistance increasing uniformly from zero stress to the stress at the elastic limit, the work done by a load applied gradually is equal to one half the product of the final stress by the extension or other deforma- tion. Beyond the elastic limit, the extensions increasing more rapidly than the loads, and the strain diagram approximating a parabolic form, the work is approximately equal to two thirds the product of the maximum stress by the extension. The amount of work required to break a bar, measured usually in inch- pounds, is called its resilience; the work required to strain it to the elastic limit is called its elastic resilience. Under a load applied suddenly the momentary elastic distortion is equal to twice that caused by the same load applied gradually. When a solid material is exposed to percussive stress, as when a weight falls upon a beam transversely, the work of resistance is measured by the product of the weight into the total fall. Elevation of Ultimate Resistance and Elastic Limit.— It was first observed by Pruf. R. H. Thurston, and Commander L. A. Beards- lee, U. S. N., independently, in 1873, that if wrought iron be subjected to a stress beyond its elastic limit, but not beyond its ultimate resistance, and then allovvtd to "rest 1 ' for a definite interval of time, a considerable in- crease of elastic limit and ultimate resistance may be experienced. In other words, the application of stress and subsequent " rest " increases the resist- ance of wrought iron. This " rest " may be an entire release from stress or a simple holding the test-piece at a given intensity of stress. Commander Beardslee prepared twelve specimens and subjected them to an intensity of stress equal to the ultimate resistance of the material, with- out breaking the specimens. These were then allowed to rest, entirely free from stress, from 24 to 30 hours, after which period they were again stressed until broken. The gain in ultimate resistance by the rest was found to vary from 4.4 to 17 per cent. This elevation of elastic and ultimate resistance appears to be peculiar to iron and steel: it has not been found in other metals. Relation of the Elastic liimit to Endurance under Re» peated Stresses (condensed from Engineering, August 7, 1891). — When engineers first began to test materials, it was soon recognized that if a specimen was loaded beyond a certain point it did not recover its origi- nal dimensions on removing the load, but took a permanent set; this point was called the elastic limit. Since below this point a bar appeared to recover completely its original form and dimensions on removing the load, it ap- peared obvious that it had not been injured by the load, and hence the work- ing load might be deduced from the elastic limit by using a small factor of safety. Experience showed, however, that in many cases a bar would not carry safely a stress anywhere near the elastic limit of the material as determined by these experiments, and the whole theory of any connection between the elastic limit of a bar and its working load became almost discredited, and engineers employed the ultimate strength only in deducing the safe working load to which their structures might be subjected. Still, as experience accu- mulated it was observed that a higher factor of safety was required for a live load than for a dead one. In 1871 Wohler published the results of a number of experiments on bars of iron and steel subjected to live loads. In these experiments the stresses were put on and removed from the specimens without impact, but it was, nevertheless, found that the breaking stress of the materials was in every case much below the statical breaking load. Thus, a bar of Krupp 1 s axle steel having a tenacity of 49 tons per square inch broke with a stress of 28.6 tons per square inch, when the load was completely removed and replaced without impact 170,000 times. These experiments were made on a large STRESS AND STRAIN". 239 number of different brands of iron and steel, and the results were concor- dant in showing that a bar would break with an alternating stress of only, say, one third the statical breaking strength of the material, if the repetitions of stress were sufficiently numerous. At the same time, however, it ap- peared from the general trend of the experiments that a bar would stand an indefinite number of alternations of stress, provided the stress was kept below the limit. Prof. Bauschinger defines the elastic limit as the point at which stress ceases to be sensibly proportional to strain, the latter being measured with a mirror apparatus reading to ^-^th of a millimetre, or about in. This limit is always below the yield-point, and may on occasion be zero. On loading a bar above the yield-point, this point rises with the stress, and the rise continues for weeks, months, and possibly for years if the bar is left at rest under its load. On the other hand, when a bar is loaded beyond its true elastic limit, but below its yield-point, this limit rises, but reaches a maxi- mum as the yield-point, is approached, and then falls rapidly, reaching even to zero. On leaving the bar at rest under a stress exceeding that of its primitive breaking-down point the elastic limit begins to rise again, and may, if left a sufficient time, rise to a point much exceeding its previous value. This property of the elastic limit of changing with the history of a bar has done more to discredit it than auything else, nevertheless it now seems as if it, owing to this very property, were once more to take its former place in the estimation of engineers, and this time with fixity of tenure. It had long been known that the limit of elasticity might be raised, as we have said, to almost any point within the breaking load of a bar. Thus, in some experi- ments by Professor Styffe, the elastic limit of a puddled-steel bar was raised 16,000 lbs. by subjecting the bar to a load exceeding its primitive elastic limit. A bar has two limits of elasticity, one for tension and one for compression. Bauschinger loaded a number of bars in tension until stress ceased to be sensibly proportional to strain. The load was then removed and the bar tested in compression until the elastic limit in this direction had been ex- ceeded. This process raises the elastic limit in compression, as would be found on testing the bar in compression a second time. In place of this, however, it was now again tested in tension, when it was found that the artificial raising of the limit in compression had lowered that in tension be- low its previous value. By repeating the process of alternately testing in . tension and compression, the two limits took up points at equal distances from the line of no load, both in tension and compression. These limits Bauschinger calls natural elastic limits of the bar, which for wrought iron correspond to a stress of about 8^ tons per square inch, but this is practically the limiting load to which a bar of the same material can be strained alter- nately in tension and compression, without breaking when the loading is repeated sufficiently often, as determined by Wohler's method. As received from the rolls the elastic limit of the bar in tension is above the natural elastic limit of the bar as defined by Bauschinger, having been artificially raised by the deformations to which it has been subjected in the process of manufacture. Hence, when subjected to alternating stresses, the limit in tension is immediately lowered, while that in compression is raised until they both correspond to equal loads. Hence, in Wohler's ex- periments, in which the bars broke at loads nominally below the elastic limits of the material, there is every reason for concluding that the loads were really greater than true elastic limits of the material. This is con- firmed by tests on the connecting-rods of engines, which of course work under alternating stresses of equal intensity. Careful experiments on old rods show that the elastic limit in compression is the same as that in ten- sion, and that both are far below the tension elastic limit of the material as received from the rolls. The common opinion that straining a metal beyond its elastic limit injures it appears to be untrue. It is not the mere straining of a metal beyond one elastic limit that injures it, but the straining, many times repeated, be3 7 ond its two elastic limits. Sir Benjamin Baker has shown that in bending a shell plate for a boiler the metal is of necessity strained beyond its elastic limit, so that stresses of as much as ? tons to 15 tons per square inch may obtain in it as it comes from the rolls, and unless the plate is annealed, these stresses will still exist after it has been built into the boiler. In such a case, however, when exposed to the additional stress due to the pressure inside 240 STRENGTH OF MATERIALS. the boiler, the overstrained portions of the plate will relieve themselves by stretching and taking a permanent set, so that probably after a year's work- ing very little difference could be detected in the stresses in a plate built in- to the boiler as it came from the bending rolls, and in one which had been annealed, before riveting into place, and the first, in spite of its having been strained beyond its elastic limits, and not subsequently annealed, would be as strong as the other. Resistance of Metals to Repeated Shocks. More than twelve years were spent by Wohler at the instance of the Prus- sian Government in experimenting upon the resistance of iron and steel to repeated stresses. The results of his experiments are expressed in what is known as Wohler's law, which is given in the following words in Dubois's translation of Weyrauch: " Rupture may be caused not only by a steady load which exceeds the carrying strength, but also by repeated applications of stresses, none of which are equal to the carrying strength. The differences of these stresses are measures of the disturbance of continuity, in so far as by their increase the minimum stress which is still necessary for rupture diminishes." A practical illustration of the meaning of the first portion of this law may I e given thus: If 50,000 pounds once applied will just break a bar of iron or steel, a stress very much less than 50,000 pounds will break it if repeated sufficiently often. This is fully confirmed by the experiments of Fairbairn and Spangenberg, as well as those of Wohler; and, as is remarked by Weyrauch, it may be considered as along-known result of common experience. It pai-tially ac- counts for what Mr. Holley has called the " intrinsically ridiculous factor of safety of six." Another "long-known result of experience " is the fact that rupture may be caused by a succession of shocks or impacts, none of which alone would be sufficient to cause it. Iron axles, the piston-rods of steam hammers, and other pieces of metal subject to continuously repeated shocks, invariably break after a certain length of service. They have a " life " which is lim- ited. Several years ago Fairbairn wrote: " We know that in some cases wrought iron subjected to continuous vibration assumes a crystalline structure, and that the cohesive powers are much deteriorated, but we are ignorant of the causes of this change." We are still ignorant, not only of the causes of this change, but of the conditions under which it takes place. Who knows whether wrought iron subjected to very slight continuous vibration will en- dure forever? or whether to insure final rupture each of the continuous small shocks must amount at least to a certain percentage of single heavy shock (both measured in foot pounds), which would cause rupture with one applica- tion ? Wohler found in testing iron by repeated stresses (not impacts) that in one case 400,000 applications of a stress of 500 centners to the square inch caused rupture, while a similar bar remained sound after 48,000,000 applica- tions of a stress of 300 centners to the square inch (1 centner = 110.2 lbs.). Who knows whether or not a similar law holds true in regard to repeated shocks ? Suppose that a bar of iron would break under a single impact of 1000 foot-pounds, how many times would it be likely to bear the repetition of 100 foot-pounds, or would it be safe to allow it to remain for fifty years subjected to a continual succession of blows of even 10 foot-pounds each ? Mr. William Metcalf published in the Metallurgical Rev ieiv, Dec. 1877, the results of some tests of the life of steel of different percentages of carbon under impact. Some small steel pitmans were made, the specifications for which required that the unloaded machine should run 4}4 hours at the rate of 1200 revolutions per minute before breaking. The steel was all of uniform quality, except as to carbon. Here are the results; The .30 C. ran 1 h. 21 m. Heated and bent before breaking. .49 0. " lh. 28 m., " " " .43 C. "4 h, 57 m. Broke without heating. .65 C. " 3 h. 50 m. Broke at weld where imperfect. .80C. " 5h. 40 m. .84 C. " 18 h. .87 C. Broke in weld near the end. .96 C. Ran 4.55 m., and the machine broke down. Some other experiments by Mr. Metcalf confirmed his conclusion, viz., STRESS AND STRAIK. 241 that high-carbon steel was belter adapted to resist repeated shocks and vi- brations than low-carbon steel. These results, however, would scarcely be sufficient to induce any en- gineer to use .84 carbon steel in a car-axle or a bridge-rod. Further experi- ments are needed to confirm or overthrow them. (See description of proposed apparatus for such an investigation in the author's paper iu Trans. A. L M. E., vol. viii , p. 76, from which the above extract is taken.) Stresses Produced by Suddenly Applied Forces and Shocks. (Mansfield Merriman, R. R. <& Eng. Jour., Dec. 1889.) Let P be the weight which is dropped from a height h upon the end of a bar, and let y be the maximum elongation which is produced. The work performed by the falling weight, then, is W='P(h + y), and this must equal the internal work of the resisting molecular stresses. The stress in the bar, which is at first 0, increases up to a certain limit Q, which is greater than P; and if the elastic limit be not exceeded the elonga- tion increases uniformly with the stress, so that the internal work is equal to the mean stress 1/2Q multiplied by the total elongation y, or W=1/2Qij. Whence, neglecting the work that may be dissipated in heat, l/2Qy = Ph + Py. If e be the elongation due to the static load P, within the elastic limit y = — e ; whence ?=p(iV i+2 ")' (1) which gives the momentary maximum stress. Substituting this value of Q, there results / = e (l-f|/l+2^), (2) which is the value of the momentary maximum elongation. A shock results when the force P, before its action on the bar, is moving with velocity, as is the case when a weight P falls from a height h. The above formulas show that this height h may be small if e is a small quan- tity, and yet very great stresses and deformations be produced. For in- stance, let h = 4e, then Q = 4P and y = 4e ; also let h = 12e, then Q = 6P and y = 6 e. Or take a wrought-iron bar 1 in. square and 5 ft. long: under a steady load of 5000 lbs. this will be compressed about 0.0012 in., supposing that no lateral flexure occurs; but if a weight of 5000 lbs. drops upon its end from the small height of 0.0048 in. there will be produced the stress of 20,000 lbs. A suddenly applied force is one which acts with the uniform intensity P upon the end of the bar, but which has no velocity before acting upon it. This corresponds to the case of h — in the above formulas, and gives Q = 2P and y = 26 for the maximum stress and maximum deformation. Prob- ably the action of a rapidly-moving train upon a bridge produces stresses of this character. Increasing the Tensile Strength of Iron Bars by Twist- ing them.— Ernest L. Ransome of San Francisco has obtained an English Patent, No. 16221 of 1888, for an " improvement in strengthening and testing wrought metal and steel rods or bars, consisting in twisting the same in a cold state. . . . Any defect in the lamination of the metal which would otherwise be concealed is revealed by twisting, and imperfections are shown at once. The treatment may be applied to bolts, suspension-rods or bars subjected to tensile strength of any description." Results of tests of this process were reported by Lieutenant F. P. Gilmore, U. S. N.,in a paper read before the Technical Society of the Pacific Coast, published in the Transactions of the Society for the month of December, 1888. Tests were also made in 1889 in the University of California. The exper- iments include trials with thirty-nine bars, twenty-nine of which were va- 242 STREHGTH OF MATERIALS. riously twisted, from three-eighths of one turn to six turns per foot. The test-pieces were cut from one and the same bar, and accurately measured and numbered. From each lot two pieces without twist were tested for ten- sile strength and ductility. One group of each set was twisted until the pieces broke, as a guide for the amount of twist to be given those to be tested for tensile strain. The following is the result of one set of Lieut. Gilmore's tests, on iron bars 8 in. long, .719 in. diameter. No. of Conditions. Twists Twists Tensile Tensile Gain per Bars. Turns. per ft. Strength. per sq. in. cent. 2 Not twisted. 22,000 54,180 2 Twisted cold. y 2 % 23,900 59,020 9 2 " " i m 25,800 63,500 17 2 " " 2 3 26,300 64,750 19 1 "* m 26,400 65,000 20 TENSILE STRENGTH. The following data are usually obtained in testing by tension in a testing- machine a sample of a material of construction : The load and the amount of extension at the elastic limit. The maximum load applied before rupture. The elongation of the piece, measured between gauge-marks placed a stated distance apart before the test; and the reduction of area at the point of fracture. The load at the elastic limit and the maximum load are recorded in pounds per square inch of the original area. The elongation is recorded as a per- centage of the stated length between the gauge-marks, and the reduction area as a percentage of the original area. The coefficient of elasticity is cal- culated from the ratio the extension within the elastic limit per inch of length bears to the load per square inch producing that extension. On account of the difficulty of making accurate measurements of the frac- tured area of a test-piece, and of the fact that elongation is more valuable than reduction of area as a measure of ductility and of resilience or work of resistance before rupture, modern experimenters are abandoning the custom of reporting reduction of area. The " strength per square inch of fractured section " formerly frequently used in reporting tests is now almost entirely abandoned. The data now calculated from the results of a tensile test for commercial purposes are: 1. Tensile strength in pounds per square incli of original area. 2. Elongation per cent of a stated length between gauge-marks, usually 8 inches. 3. Elastic limit in pounds per square inch of original area. The short or grooved test specimen gives with most metals, especially with wrought iron and steel, an apparent tensile strength much higher than the real strength. This form of test-piece is now almost entirely aban- doned. The following results of the tests of six specimens from the same 1*4" steel bar illustrate the apparent elevation of elastic limit and the changes in other properties due to change in length of stems which were turned down , in each specimen to .798" diameter. (Jas. E. Howard, Eng. Congress 1893, Section G.) Description of Stem. Elastic Limit, Lbs. per Sq. In. Tensile Strength, Contraction of Lbs. per Sq. In. Area, per cent. 1.00" long .50 " .25 " Semicircular groove, A" radius Semicircular groove. Ys," radius V-sbaped groove 64,900 65,320 68,000 86,000, about 90,000, about 94,400 97,800 102,420 134,960 117,000 Indeterminate. TENSILE STRENGTH. 243 Tests plate made by the author in 1879 of straight and grooved test-pieces of boiler-plate steel cut from the same gave the following results : 5 straight pieces, 56,605 to 59,012 lbs. T. S. Aver. 57,566 lbs. . , 4 grooved " 64,341 to 67,400 " " " 65,452" Excess of the short or grooved specimen, 21 per cent, or 12,114 lbs. Measurement of Elongation.— In order to be able to compare records of elongation, it is necessary not only to have a uniform length of section between gauge-marks (say 8 inches), but to adopt a uniform method of measuring the elongation to compensate for the difference between the apparent elongation when the piece breaks near one of the gauge-marks, and when it breaks midway between them. The following method is rec- ommended (Trans. A. S. M. E., vol. xi., p. 622): Mark on the specimen divisions of 1/2 inch each. After fracture measure from the point of fracture the length of 8 of the marked spaces on each fractured portion (or 7 -f- on one side and 8 + on the other if the fracture is not at one of the marks). The sum of these measurements, less 8 inches, is the elongation of 8 inches of the original length. If the fracture is so near one end of the specimen that 7 + spaces are not left on the shorter portion, then take the measurement of as many spaces (with the fractional part next to the fracture) as are left, and for the spaces lacking add the measurement of as many corresponding spaces of the longer portion as are necessary to make the 7-f- spaces. Shapes of Specimens for Tensile Tests.— The shapes shown in Fig. 74 were recommended by the author in 1882 when he was connected f< 16-to-20" — »} b ~ ^3 No. 1. Square or flat bar, as 1 m im rolled. m — 16-to-20- -=^ _ _ - -==^§11 \%0 No - 2 - Round bar, as rolled. H ; 16 "' t0 20 ~ • T^^l m No. 3. Standard shape for 1 ' I I H flats or squares. Fillets 14 1 r u g" u" a m inch radius. f* - 16-'to-20'' >| =gg^g^=g = - " I \j|| ^°- 4- Standard shape for 1 ^ jj ^ ^ ^^ ^^) rounds. Fillets ]4 in - radius. U .jj-to-i-2- * No. 5. Government shape for i- ^^ > gj m marine boiler-plates of iron. I r - ^f|| || Not recommended for other ~Z\~M~ tests, as results are generally I l \*~ in error. Fig. 75. with the Pittsburgh Testing Laboratory. They are now in most general use, the earlier forms, with 5 inches or less in length between shoulders, being almost entirely abandoned. Precautions Required in making Tensile Tests.— The testing-machine itself should be tested, to determine whether its weighing apparatus is accurate, and whether it is so made and adjusted that in the test of a properly made specimen the line of strain of the testing-machine is absolutely in line with the axis of the specimen. The specimen should be so shaped that it will not give an incorrect record of strength. It should be of uniform minimum section for not less than five inches of its length. Regard must be had to the time occupied in making tests of certain mate- rials. Wrought iron and soft steel can be made to show a higher than their actual apparent strength by keeping them under strain for a great length of time. Tn testing soft alloys, copper, tin, zinc, and the like, which flow under con- stant strain their highest apparent strength is obtained by testing them rapidly. In recording tests of such materials the length of time occupied in the test should be stated. 244: STRENGTH OF MATERIALS. For very accurate measurements of elongation, corresponding to incre- ments of load during the tests, the electric contact micrometer, described in Trans. A. S. M. E., vol. vi., p. 479, will be found convenient. When read- ings of elongation are then taken during the test, a strain diagram may be plotted from the reading, which is useful in comparing the qualities of dif- ferent specimens. Such strain diagrams are made automatically by the new Olsen testing-machine, described in Jour. Frank. Inst. 1891. The coefficient of elasticity should be deduced from measurement ob- served between fixed increments of load per unit section, say between 2000 and 12,000 pounds per square inch or between 1000 and 11,000 pounds instead of between and 10,000 pounds. COMPRESSIVE STRENGTH. What is meant by the term "compressive strength " has not yet been settled by the authorities, and there exists more confusion in regard to this term than in regard to any other used by writers on strength of materials. The reason of this may be easily explained. The effect of a compressive stress upon a material varies with the nature of the material, and with the shape and size of the specimen tested. While the effect of a tensile stress is to produce rupture or separation of particles in the direction of the line of strain, the effect of a compressive stress on a piece of material may be either to cause it to fly into splinters, to separate into two or more wedge-shaped pieces and fly apart, to bulge, buckle, or bend, or to flatten out and utterly re- sist rupture or separation of particles. A piece of speculum metal under compressive stress will exhibit no change of appearance until rupture takes place, and then it will fly to pieces as suddenly as if blown apart by gun- powder. A piece of cast iron or of stone will generally split into wedge- shaped fragments. A piece of wrought iron will buckle or bend. A piece of wood or zinc may bulge, but its action will depend upon its shape and size. A piece of lead will flatten out and resist compression till the last degree; that is, the more it is compressed the greater becomes its resistance. Air and other gaseous bodies are compressible to any extent as long as they retain the gaseous condition. Water not confined in a vessel is com- pressed by its own weight to the thickness of a mere film, while when con- fined in a vessel it is almost incompressible. It is probable, although it has not been determined experimentally, that solid bodies when confined are at least as incompressible as water. When they are not confined, the effect of a compressive stress is not only to shorten them, but also to increase their lateral dimensions or bulge them. Lateral strains are therefore induced by compressive stresses. The weight per square inch of original section required to produce any given amount or percentage of shortening of any material is not a constant quantity, but varies with both the length and the sectional area, with the shape of this sectional area, and with the relation of the area to the length. The " compressive strength" of a material, if this term be supposed to mean the weight in pounds per square inch necessary to cause rupture, may vary with every size and shape of specimen experimented upon. Still more diffi- cult would it be to state what is the " compressive strength " of a material which does not rupture at all, but flattens out. Suppose we are testing a cylinder of a soft metal like lead, two inches in length and one inch in diam- eter, a certain weight will shorten it one per cent, another weight ten per cent, another fifty per cent, but no weight that we can place upon it will rupture it, for it will flatten out to a thin sheet. What, then, is its compres- sive strength ? Again, a similar cylinder of soft wrought iron would prob- ably compress a few per cent, bulging evenly all around ; it would then com- mence to bend, but at first the bend would be imperceptible to the eye and too small to be measured. Soon this bend would be great enough' to be noticed, and finally the piece might be bent nearly double, or otherwise dis- torted. What is the " compressive strength " of this piece of iron ? Is it the weight per square inch which compresses the piece one per cent or five per cent, that which causes the first bending (impossible to be discovered;, or that which causes a perceptible bend ? As showing the confusion concerning the definitions of compressive strength, the following statements from different authorities on the strength of wrought iron are of interest. Wood's Resistance of Materials states, " comparatively few experiments have been made to determine how much wrought iron will sustain at the point of crushing. Hodgkinson gives 65,000, Rondulet 70,800, Weisbach 7'2,000 COMPKESSIVE STRENGTH. 245 Rankine 30,000 to 40,000. It is generally assumed that wrought iron will resist about two thirds as much crushing as to tension, but the experiments fail to give a very definite ratio." Mr. Whipple, in his treatise on bridge-building, states that a bar of good wrought iron will sustain a tensile strain of about 60,000 pounds per square inch, and a compressive strain, in pieces of a length not exceeding twice the least diameter, of about 90,000 pounds. The following values, said to be deduced from the experiments of Major Wade, Hodgkinson, and Capt. Meigs, are given by Haswell : American wrought iron 127,720 lbs. " " " (mean) 85,500 " English " } 65 ' 200 " Stoney states that the strength of short pillars of any given material, all having the same diameter, does not vary much, provided the length of the piece is not less than one and does not exceed four or five diameters, and that the weight which will just crush a short prism whose base equals one square inch, and whose height is not less than 1 to 1J4 and does not exceed 4 or 5 diameters, is called the crushing strength of the material. It would be well if experimenters would all agree upon some such definition of the term " crushing strength," and insist that all experiments which are made for the purpose of testing the relative values of different materials in com- pression be made on specimens of exactly the same shape and size. An arbitrary size and shape should be assumed and agreed upon for this pur- pose. The size mentioned by Stoney is definite as regards area of section, viz., one square inch, but is indefinite as regards length, viz., from one to five diameters. In some metals a specimen five diameters long would bend, and give a much lower apparent strength than a specimen having a length of one diameter. The words " will just crush " are also indefinite for ductile materials, in which the resistance increases without limit if the piece tested does not bend. In such cases the weight which causes a certain percentage of compression, as five, ten, or fifty per cent, should be assumed as the crushing strength. For future experiments on crushing strength three things are desirable : First, an arbitrary standard shape and size of test specimen for comparison of all materials. Secondly, a standard limit of compression for ductile materials, which shall be considered equivalent to fracture in brittle mate- rials. Thirdly, an accurate knowledge of the relation of the crushing strength of a specimen of standard shape and size to the crushing strength of specimens of all other shapes and sizes. The latter can onlj' be secured by a very extensive and accurate series of experiments upon all kinds of materials, and on specimens of a great number of different shapes and sizes. The author proposes, as a standard shape and size, for a compressive test specimen for all metals, a cylinder one inch in length, and one half square inch in sectional area, or 0.798 inch diameter; and for the limit of compres- sion equivalent to fracture, ten per cent of the original length. The term "compressive strength," or "compressive strength of standard specimen," would then mean the weight per square inch required to fracture by Com- pressive stress a cylinder one inch long and 0.798 inch diameter,* or to reduce its length to 0.9 inch if fracture does not take place before that reduc- tion in length is reached. If such a standard, or any standard size whatever, had been used by the earlier authorities on the strength of materials, we never would have had such discrepancies in their statements in regard to the compressive strength of wrought iron as those given above. The reasons why this particular size is recommended are : that the sectional area, one-half square inch, is as large as can be taken in the ordinary test- ing-machines of 100,000 pounds capacity, to include all the ordinary metals of construction, cast and wrought iron, and the softer steels; and that the length, one inch, is convenient for calculation of percentage of compression. If the length were made two inches, many materials would bend in testing, and give incorrect results. Even in cast iron Hodgkinson found as the mean of several experiments on various grades, tested in specimens % inch in height, a compressive strength per square inch of 94,730 pounds, while the mean of the same number of specimens of the same irons tested in pieces \y% inches in height was only F8,800 pounds. The best size and shape of standard specimen should, however, be settled upon only after consultation and agreement among several authorities. 246 STKENGTH OF MATERIALS. The Committee on Standard Tests of the American Society of Mechanical Engineers say (vol. xi., p. 624) : '"Although compression tests nave heretofore been made on diminutive sample pieces, it is highly desirable that tests be also made on long pieces from 10 to 20 diameters in length, corresponding more nearly with actual practice, in order that elastic strain and change of shape may be determined by using proper measuring apparatus. The elastic limit, modulus or coefficient of elasticity, maximum and ulti- mate resistances, should be determined, as well as the increase of section at various points, viz., at bearing surfaces and at crippling point. The use of long compression-test pieces is recommended, because the in- vestigation of short cubes or cylinders has led to no direct application of the constants obtained by their use in computation of actual structures, which have always been and are now designed according to empirical for- mulae obtained from a few tests of long columns." COLUMNS, PILL.ARS, OR STRUTS. Hodgkinson's Formula for Columns. P = crushing weight in pounds; d = exterior diameter in inches; d x — in- terior diameter in inches; L = length in feet. Kind of Column. Both ends rounded, the Both ends flat, the length of the column length of the column exceeding 15 times exceeding 30 times its diameter. its diameter. Solid cylindrical col- j umns of cast iron ) Hollow cylindrical col- ) umns of cast iron j Solid cylindrical col- ) umns of wrought iron, f Solid square pillar of ) Dantzic oak (dry) ) Solid square pillar of ) red deal (dry) [ P = 33,5 ,^ 3-7 W3-76_d i3 -76 #3.65 ,/3-55 _ rl 3-55 P = 99,320- ~* P = 299,600^- d 4 P = 24,540^ P = 17,510 L* The above formulae apply only in cases in which the length is so great that the column breaks by bending and not by simple crushing. If the column be shorter than that given in the table, and more than four or five times its diameter, the strength is found by the following formula : W : PCX P f MG'iT in which P= the value given by the preceding formulae, K= the transverse section of the column in square inches, C = the ultimate compressive resis- tance of the material, and W = the crushing strength of the column. Hodgkinsou's experiments were made upon comparatively short columns, the greatest length of cast-iron columns being W% inches, of wrought iron 90% inches. The following are some of his conclusions: 1. In all long pillars of the same dimensions, when the force is applied in the direction of the axis, the strength of one which has flat ends is about three times as great as one with rounded ends. 2. The strength of a pillar with one end rounded and the other flat is an arithmetical mean between the two given in the pieced ing case of the same dimensions. 3. The strength of a pillar having both ends firmly fixed is the same as one of half the length with both ends rounded. 4. The strength of a pillar is not increased more than one seventh by en- larging it at the middle. MOMiM OF IKEilTiA Atfi) HADIUS OF GYRATION. Ml Gordon's formulae deduced from Hodgkinson's experiments are more generally used than Hodgkinson's own. They are: Columns with both ends fixed or fiat, P = — - — - ; Columns with one end flat, the other end round, P = - ; l + 1.8a^ Columns with both ends round, or hinged, P = — - 2 ; S = area of cross-section in inches; P — ultimate resistance of column, in pounds; / = crushing strength of the material in lbs. per square inch; , ,. - ..'.-.'., „ Moment of inertia r = least radius of gyration, in inches,?' 2 = — — 5 : ; area of section I = length of column in inches; a — a coefficient depending upon the material; / and a are usually taken as constants; they are really empirical variables, dependent upon the dimensions and character of the column as well as upon the material. (Burr.) For solid wrought-iion columns, values commonly taken are: / = 36,000 to 40 ' 000;a = 3^ tO 40^00- For solid cast-iron columns, / = 80,000, a = ^-rr^. fin oon For hollow cast-iron columns, fixed ends, p = ^, I = length and 1+800- d = diameter in the same unit, and p = strength in lbs. per square inch. Sir Benjamin Baker gives, For mild steel, / = 67,000 lbs., a = — * . For strong steel, / = 114,000 lbs., a = j^ Q Mr. Burr considers these only loose approximations for the ultimate resis- tances. MOMENT OF INERTIA AND RADIUS OF GYRATION. The moment of inertia of a section is the sum of the products of each elementary area of the section into the square of its distance from an assumed axis of rotation, as the neutral axis. The radius of gyration of the section equals the square root of the quotient of the moment of inertia divided by the area of the section. If E = radius of gyration, 1= moment of inertia and A = area, *=;# l=- The moments of inertia of various sections are as follows; d = diameter, or outside diameter; d t = inside diameter; 6 = breadth; h = depth;*!, h u inside breadth and diameter; - Solid rectangle I = l/126/i 3 ; Hollow rectangle I = 1/12(67* 3 - Mi 3 ) ; Solid square 1= 1/126 4 ; Hollow square ..J= 1/12(6* - 6, 4 ); Solid cylinder I = l/647rd 4 ; Hollow cylinder I = \/Mir(d* - d^). Moments of Inertia and Radius of Gyration for Various Sections, and their Use in tbe Formulas for Strength of Girders and Columns.-The strength of sections to resist strains, either as girders or as columns, depends not only on the area but also on the form of the section, and the property of the section which forms the basis of the constants used in the formulas for strength of girders and columns to express the effect of the form, is its moment of inertia about its neutral axis. Thus the moment of resistance of any section to transverse bending 248 STKEKGTS OF MATERIALS. is its moment of inertia divided by the distance from the neutral axis to the fibres farthest removed from that axis; or ,„. ,' . , Moment of inertia ,, I Moment of resistance = 7- - — • —. M = -. Distance ot extreme fibre from axis y Moment of Inertia of Compound Shapes. (Peneoyd Iron Works.)— The moment of inertia of any section about any axis is equal to the I about a parallel axis passing through its centre of gravity -f- (the area of the section X the square of the distance between the axes). By this rule, the moments of inertia or radii of gyration of any single sec- tions being known, corresponding values may be obtained for any combina- tion of these sections. Radius of Gyration of Compound Shapes.— In the case of a pair of any shape without a web the value of B can always be found with- out considering the moment of inertia. The radius of gyration for any section around an axis parallel to another axis passing through its centre of gravity is found as follows: Let r = radius of gyration around axis through centre of gravity; R = radius of gyration around another axis parallel to above; d = distance be- tween axes: R = \/d? + r*. When r is small, R may be taken as equal to d without material error. Graphical Method for Finding Radius of Gyration.— Ben j. F. La Hue, Eng. News, Feb. 2, 1893, gives a short graphical method for finding the radius of gyration of hollow, cylindrical, and rectangular col- umns, as follows: For cylindrical columns: Lay off to a scale of 4 (or 40) a right-angled triangle, in which the base equals the outer diameter, and the altitude equals the inner diameter of the column, or vice versa. The hypothenuse, measured to a scale of unity (or 10), will be the radius of gyration sought. This depends upon the formula 'W Mom. of Inertia ^D 2 + d 2 Area 4 in which A = area and D = diameter of outer circle, a = area and d = dia- meter of inner circle, and G = radius of gyration. ^D' 1 -f- d 2 is the expres- sion for the hypothenuse of a right-angled triangle, in which D and d are the base and altitude. The sectional area of a hollow round column is .7854(D 2 — d 2 ). By con- structing a right-angled triangle in which D equal s the hypothenuse and d equals the altitude, the base will equal |/D 2 - d\ Calling the value of this expression for the base B, the area will equal .7854.B 2 . Value of G for square columns: Lay off as before, but using a scale of 10, a right-angled triangle of which the base equals D or the side of the outer square, and the altitude equals d, the side of the inner square. With a scale of 3 measure the hypothenuse, which will be, approximately, the radius of gyration. This process for square columns gives an excess of slightly more than 4%. By deducting 4% from the result, a close approximation will be obtained. A very close result is also obtained by measuring the hypothenuse with the same scale by which the base and altitude were laid off, and multiplying by the decimal 0.29; more exactly, the decimal is 0.28867. The formula is ■/ Mom. of inertia 1 ,_ \/D* + d\ = 0-28867 y D -x + d a This may also be applied to any rectangular column by using the lesser diameters of an unsupported column, and the greater diameters if the col- umn is supported in the direction of its least dimensions. ELEMENTS OF TJSUAX SECTIONS. Moments refer to horizontal axis through centre of gravity. This table is intended for convenient application where extreme accuracy is not impor- tant. Some of the terms are only approximate; those marked * are correct. Values for radius of gyration in flanged beams apply to standard minimum sections only. A = area of section; b = breadth; h = depth; D = diameter. ELEMENTS OF USUAL SECTIONS. 249 Shape of Section. Solid Rect- angle. Hollow Rect- angle. Moment of Inertia. 12 b h»-bjh^ 12 AD* * 16 Moment of Resistance, bhs-bji^ 6/i AD* 8 Square of Least Radius of Gyration. h* -f hi- 12 2)2 4 16 Least Radius of Gyration. h±h i 4.89 f-Hri Hollow Circle. A, area of large section : a, area of small section D* + d* " 5.64 F=6 Solid Triangle. 6/i3 36 The least of of the two h* b* r 8 ° r 24 The least of the two : h b — or — .24 4.9 Even Angle. Ah* 10.2 Ah 7.2 62 25 9 Uneven Angle Ah* 9.5 Ah 6.5 lZQl* + &2) 2.6(/t 4- 6) JE3 it* 19 9.5 h* 22.5 4.74 ^2 11.1 Ah 8 ^2 6.66 Ah Ah* 7.34 Ah 3.67 ta 62 22.5 4.74 6J 21 6 4.58 62 12.5 6 3.54 62 36.5 Distance of base from centre of gravity, solid triangle, ^; even angle, ~ • uneven angle, ^-i.; even tee, ^-5; deck beam, — -; all other shapes given in o.O Q,tJ 4-Q the table, --or -. 250 STRENGTH OF MATERIALS. Solid Cast-iron Columns. Hurst giv.'S the following table, based on Hodgkinson's formula (tons of 2240 lbs.). The figures are the safe load or ^ of the breaking weight in tons, for solid columns, ends flat and fixed. ■38. Length of Column in Feet. £"5 a a 6. 8. 10. 12. 14. 16. 18. 20. 25. Wz .82 .50 .34 .25 .19 .15 .13 .11 .07 1% 1.43 .87 .60 .44 .34 .27 .22 .18 .13 ,_£__ 2.34 - 1.41 .97 .71 .55 .44 .36 .30 .20 2J4 3.52 2.16 1.48 1.08 .83 .67 .54 .46 .31 2p 2 5.15 3.16 2.16 1.58 1.22 .97 .80 .66 .56 2'M 7.26 4.45 3.05 2.23 1.72 1.37 1.12 .94 .64 3 9.93 6.09 4.17 3.06 2.35 1.87 1.53 1.28 .88 3^ 17.29 10.60 7.26 5.32 4.10 3.26 2.67 2.23 1.53 4 27.96 17.15 11.73 8.61 6.62 5.28 4.32 3.61 2.47 #£ 42.73 26.20 17.93 13.15 10.12 8.07 6.60 5.52 3.78 5 2 62.44 38.29 26.20 19.22 14.79 11.79 9.65 8.06 5.52 5^ 88.00 53.97 36.93 27.09 20.84 16.61 13.60 31.37 7.78 6 120.4 73.82 50.51 37.05 28.51 22.72 18.60 15.55 10.64 6^ 160.6 98.47 67.38 49.43 38.03 30.31 24.81 20.74 14.19 209.7 128.6 87.98 64.53 49.66 39.57 32.33 27.08 18.53 ^ 268.8 164.8 112.8 82.73 63.66 50.73 41.53 34.72 23.76 8 339.1 207.9 142.3 104.4 80.31 64.00 52.39 43.80 29.97 - 8J^ 421.8 258.6 177.0 129.8 99.90 79.61 65.16 54.48 37.28 9 518.2 317.7 217.4 159.5 122.7 97.80 80.05 66.92 45.80 9^ 629.5 386.0 264.2 193.8 149.1 118.8 97.25 81.70 55.64 10 757.2 464.3 317.7 233.1 179.3 142.9 117.0 97.79 66.92 1<% 902.6 553.5 378.7 277.8 213.8 170.3 139.4 116.6 79.77 11 1067.1 654.4 447.8 328.5 252.7 201.4 164.9 137.8 94.31 -UM 1252.3 767.9 525.5 385.4 296.6 236.4 193.5 161.7 110.7 12 1459.6 895.1 612.5 449.3 345.7 275.5 225.5 188.5 129.0 The correction for short columns should be applied where the length is less than 30 diameters. SC Strength in tons of short columns j = 106' + %C ' 49 S being the strength for long columns given in the above table, and C times the sectional area Of the metal in inches. Hollow Columns.— The strength nearly equals the difference be- tween that of two solid columns the diameters, of which are equal to the external and internal diameters of the hollow one. Ultimate Strength of Hollow, Cylindrical Wrought and Cast-iron Columns, when fixed at the ends. (Pottsville Iron and Steel Co.) Computed by Gordon's formula, p - C p = Ultimate strength in lbs. per square inch; J 40,000 lbs. for wrought-iron; I "!0,0001bs. for cast-iron; / = C = 1/3000 for wrought-iron, and 1/800 for cast-iron. COLUMNS, PILLARS, OR STRUTS. 80000 251 For cast-iron, p = - ^ 1 For wrought-iron, p - S00\h) 40000 ^3000\hf HOLLOW CYLINDRICAL COLUMNS. Ratio Maximum Load per sq. in. Safe Load per square inch. of Length to Diameter. 1 h Cast Iron. Wrought Iron. Cast Iron, Factor of 6. Wrought Iron, Factor of 4. 8 74075 39164 12346 9791 10 71110 38710 11851 9677 12 67796 38168 11299 9542 14 64256 37546 10709 9386 16 60606 36854 10101 9213 18 56938 36100 9489 9025 20 53332 35294 8889 8823 22 49845 34442 8307 8610 24 46510 33556 7751 8389 26 43360 32642 7226 8161 28 40404 31712 6734 7928 30 37646 30768 6274 7692 32 35088 29820 5848 7455 34 32718 28874 5453 7218 36 30584 27932 5097 6983 38 28520 27002 4753 6750 40 26666 26086 4444 6522 42 24962 25188 4160 6297 44 23396 24310 3899 6077 46 21946 23454 3658 5863 48 20618 22620 3436 5655 50 19392 21818 3262 5454 52 18282 21036 3047 5259 54 17222 20284 2870 5071 56 16260 19556 2710 4889 58 15368 18856 2561 4714 60 14544 18180 2424 4545 Ultimate Strength of Wrought-iron Columns. p = ultimate strength per square inch; I = length of column in inches; r — least radius of gyration in inches. . . . 40000 For square end-bearings, r> — 1 + 000 \r J For one pin and one square bearing, p = For two pin-bearings, 40000 40000 1 -f KXX>\r/ 30000' 40000 1 + -1-/1Y 2 20000\rj For safe working load on these columns use a factor of 4 when used in buildings, or when subjected to dead load only; but when used in bridges the factor should be 5. 252 STRENGTH OF MATERIALS. WROUGHT-IRON COLUMNS. Ultimate Strength in lbs. Safe Strength in lbs. per 1 per square inch. I r square inch— Factor of 5. r Square Ends. Pin and Square End. Pin Ends. Square Ends. Pin and Square End. Pin Ends. 10 39944 39866 39800 10 7989 7973 7960 15 39776 39702 39554 15 7955 7940 7911 20 39604 39472 39214 20 7921 7894 7843 25 39384 39182 38788 25 7877 7836 7758 30 39118 38834 38278 30 7821 7767 7656 35 38810 38430 37690 35 7762 7686 7538 40 38460 37974 37036 40 7692 7595 7407 45 38072 37470 36322 45 7614 7494 7264 50 37646 36928 35525 50 7529 7386 7105 55 37186 36336 34744 55 7437 7267 6949 60 36697 35714 33898 60 7339 7143 6780 65 36182 34478 33024 65 7236 6896 6605 TO 35634 34384 32128 70 7127 6877 6426 75 35076 33682 31218 75 7015 6736 6244 80 34482 32966 30288 80 6896 6593 6058 85 33883 32236 29384 85 6777 6447 5877 90 33264 31496 28470 90 6653 6299 5694 95 32636 30750 27562 95 6527 6150 5512 100 32000 30000 26666 100 6400 6000 5333 105 31357 29250 25786 105 6271 5850 5157 maximum Permissible Stresses in columns used in buildings. (Building Ordinances of City of Chicago, 1893.) Maximum permissible loads : For cast-iron round columns : 8 = 10000a 1+1 I = length of column in inches; d = diameter of column in inches; a = area of column in square inches. For cast-iron rectangular columns; _ 10000a I and a as before ; " — 72 * d = least horizontal dimension of column. For riveted or other forms of wrought-iron columns: _ 12000a I = and a as before; ^ - p • r = least radius of gyration in inches. 1Jr 36000r 2 For riveted or other steel columns, if less than 60r in length: 60? 8 = 17,000 - I and r as before. If more than 60r in length: S = 13,500a. a as before. For wooden posts: S = - 1 + 1 a = area of post in square inches; d = least side of rectangular post in inches ; I = length of post in inches; \ 600 for white or Norway pine ; c= < 800 for oak; ( 900 for long-leaf yellow pine. HOLLOW CYLINDRICAL CAST IROK COLUMNS. 253 SAFF LOAD OF HOLLOW CYLINDRICAL, CAST-IRON COLUMNS. (New Jersey Steel Iron Co.) (One fifth the breaking weight.) The following tables give the safe load in tons of 2,000 lbs., for columns having capitals and bases accurately turned to a true plane, and having a perfectly fair bearing on these surfaces. In the case of columns having turned ends, but set only with the degree of care usual in ordinary building, only one half of these loads should be taken; and for columns not turned at all, or having rounded ends, one third of these amounts should be taken for the safe load. Columns having one end accurately turned to a true plane, and the other rounded, may be loaded to two thirds the amount given in the tables. Safe Load, in Tons of 2000 lbs. for Cast-iron Columns with Turned Capitals and Rases. Outside Outside Outside Outside Diameter, Diameter, • Diameter, Diameter, * 3 inches. Thickness in a 3 inches. z 4 inches. 4 inches. £ Thickness in Thickness in Thickness in "Sx inches. ix PI a 17 inches. bx = S3 inches. ;x = 17 inches. 1. H M i H 3.0 H 3.6 1 3.9 Hi 24.9 32.9 1 38.3 41.7 Y2 7.0 H 8.9 1 10.1 m 7 13 8 15.917.2 10.7 8 10 9 13.0 14.0 IS 2.8 3.3 3.5 8 21.7 2S.4 33.0 35.8 is 6.4 8.1 9.1 9.7 9 8.9 10.7 11.4 19 2.5 3.0 3.2 9 19.0 24.8 28. 7 31.0 19 5.8 7.4 8,3 8.8 10 7 5 8.9 9.fi 20 2.3 2.7 2.9 10 17.4 22.0 24.9 26.3 20 5.3 6 8 7.6 8.1 11 6 4 7.6 8.1 81 2.1 2.5 2.7 11 14.8 18.7 21.1 22.4 21 4.9 6.2 7 7 5 12 5.4 6.6 7.0 22 1.9 2.3 2.5 12 12.7 16.2 18.2 19.3 22 4.6 5 8 6.5 6.9 13 4 R 5.7 6.1 23 1.8 2.1 2.3 13 11.1 14.1 15.9 16.8 23 4.2 5.3 6.0 6 4 14 4.2 5.0 5.4 24 1.7 2.0 2.1 14 9.8 12.4 14.0 14.9 24 3.9 5 5 6 5 9 IS 3 7 4.5 4.8 25 1.6 1.9 2.0 15 8.7 11.1 12.5 13.2 25 3.7 4.6 5.2 5 5 16 3.4 4.0 4.3 16 7.8 9.9 11.3 11.8 a Outside Diameter, 5 inches. Outside Diameter, 6 inches. Outside Diameter, 7 inches. t, Thickness in inches. Thickness in inches. Thickness in inches. J 39.5 n 53.8 1 m M 1 m W* Ya i 150.7 m 7 65.0 73.3 77.3 95.5 110.3 122.1 102.4 128.7 169.4 8 35.1 47.6 57.3 64.4 69.7 85.7 98.7 108.8 93 6 117.0 136 9 153.5 9 31.3 42.3 50.7 56.8 62.8 77.1 88.5 97.3 85.6 106.7 124.6 139 3 10 38.0 37.7 45.1 50.4 56.9 69.6 79.6 87.4 78.4 97.5 113.5 1?fi 6 11 35 3 33.8 40 3 44.9 51.6 63.0 71.9 78.7 71.8 89.2 103.6 115.3 12 33.7 30.5 36.2 40.3 46.9 57.2 65.2 71.2 66.0 81.7 94.8 105.3 13 31.0 27.6 32.2 35.2 42.9 52.1 59.3 64.6 60.7 75.1 87 96.5 14 18.5 24.3 28.3 31.0 39.3 47.6 54.1 58.9 56.0 69.2 80.0 88.6 15 16.5 21.6 25.2 27.6 36.8 43.9 49.0 52.6 51.8 63.9 73 8 81 6 16 14.8 19.4 22.6 24.7 33.0 39.4 44.0 47.2 48.1 59.2 68.2 75.4 17 13.3 17.5 20.4 22.3 29.8 35.5 39.7 42.5 44.6 54.9 63.2 69 8 18 12.1 15.9 18.5 20.2 27.0 32.2 36.0 38.6 42.0 50.9 57.8 63.0 19 11.0 14.5 16.9 18.4 24.6 29.4 32.8 35.2 38.3 46.4 52 7 57.4 .20 10.1 13.3 15.4 16.9 22.6 26.9 30 1 32.3 35.1 42.5 48.3 53,6 21 9.3 12.2 14.2 15.5 20.8 24.8 27 7 29.7 32.3 39.1 44 5 48 4 22 8.6 11.3 13.1 14.4 19.2 22.9 25.6 27.4 29.8 36.2 41 1 44.7 23 8.0 10.5 12.2 13.3 17.8 21.2 23.7 25.4 27.7 33.5 38.1 41.5 24 7.4 9.7 11.3 12.4 16.6 19.7 22.1 23.7 25.7 31.2 35.4 38,6 25 6.9 9.1 10.6 11.5 15.4 18.4 20.6 22.1 24.0 29.1 33.1 36.0 254 STRENGTH OE MATERIALS. Safe Load, in Tons of 2000 lbs. for Cast-iron Columns witli Turned Capitals and Bases. Outside Diameter, Outside Diameter. Outside Diameter, 8 inches. 9 inches. 10 inches. Thickness in inches. Thickness in inches. Thickness in inches. J H 128 8 1 Wa m H 1 m M M 1 m m 7 162.fi 193 219.5 154.8 197.7 236.6 271.4 181.6 233.4 280.9 324 2 8 118 7 150.1 177.7 201.6 144.7 184.5 220.2 252.0 171.1 219.5 263.8 303 9 9 109.8 138.5 163.6 185.2 135.0 171.8 204.7 233.9 160.9 206.2 247.3 284 5 10 101. B 127,8 150.7 170.2 126.0 160.0 190.3 217.0 151.2 193.4 231.6 266.0 11 94 118.0 139.0 156.7 117.5 149.0 177.0 201.4 142.0 181.4 216.9 248.7 12 87 109.2 128. 2 144.3 109.6 138.8 164.5 187.0 133.4 170.1 203.1 232.6 13 80 7 101 1 118.5 133.2 102.4 129.4 153.2 173.9 125.3 159.6 190.3 217.7 14 75 93 8 109.8 95.7 120.8 142.8 161.9 117.8 149.8 178.4 203 . S IS 69 8 87 1 101.9 114.2 89.5 112.9 133.3 150.9 110.8 140.7 167.5 191.1 Hi 65.0 81 1 94.7 106.1 83.9 105.7 124.6 140.9 104.3 132.4 157.3 179.3 17 60.7 75.7 88.3 98.7 78.7 99.0 116.7 131.8 98.3 124.6 148.0 168.5 18 56.8 70.7 82.4 92.1 86.1 73.9 69.6 92.9 87.4 109.4 102.7 123.5 115.9 92.7 87.5 117.4 139.3 158.5 110.8 131.3 20 51.1 62.7 72.1 79.5 65.5 82.3 96.7 108.9 82.7 104.6 124.0 140.8 21 47 57; 7 66\4 73.2 61.8 75.5 91.0 102.6 78.3 99.0 117.2 133.0 43.5 53.3 61.3 67.6 58.4 73.2 85.9 96.7 74.2 93.7 110.9 125.8 :>H 40.3 49.4 56.8 62.7 55.9 69.3 80.4 89.5 70.4 88.9 105.1 119.1 37.5 46.0 52.9 58.3 52.0 64.4 74.8 83.3 66.9 84.3 99.7 112.9 35.0 42.9 49.3 54.4 48.5 60.1 69.8 77.7 64.9 81.0 94.2 106.3 Outside Diameter, Outside Diameter, Outside Diameter. a 11 inches. 12 inches. 13 inches. Thickness in inches. Thickness in inches. Thickness in inches. c J 1 269.4 M 377.6 .2 1 1M m 2 1 m IK 2 7 325,9 469.5 305.3 370.8 431.7 540.9 341.5 414.4 485.7 612.7 H 255 1 30S 1 356.8 442.2 290.9 352.8 410.2 512.8 327.0 396.3 464.1 583.9 9 241.2 290 8 336,3 415.6 276.6 335.0 389.1 485.0 312.4 378.4 442.5 555.5 10 227.8 274.2 316.7 390.3 262.7 317.7 368.6 458.3 298.0 360.6 421.3 527.8 11 214.9 258.4 298.1 366.3 249.2 301.0 348.8 432.9 284.0 343.4 400.6 501.1 12 202.7 243.5 280.5 343.9 236.3 285.1 330.0 408.6 270.5 326.7 380.8 475 3 13 191.2 229.4 264.0 322.8 223.9 270.0 312.2 385.7 257.5 310.8 361.8 450.7 14 180.5 216.2 248.5 303.3 212.3 255.6 295.3 364.1 245.0 295.5 343.7 427.4 15 170.3 203.9 234.1 285.1 201.2 242.1 279.4 343.9 233.2 281.1 326.5 405.4 16 160.9 192.4 220.7 268.3 190.8 229.4 264.5 325.0 222.0 267.3 310.3 384.6 17 152 1 181.7 208.2 252.7 181.1 217.5 250.6 307.4 211.3 254.4 295.0 365.1 18 143.9 171.7 196.7 238.3 171.9 206.3 237.5 290.9 201.3 242.1 280.5 316.7 19 136.2 162.5 185.9 225.0 163.3 195.8 225.3 275.6 191.8 230.6 267.0 329.5 20 129.1 153 9 176.0 212.6 155.2 186.0 213.9 261.3 182.8 219.7 254.2 313.3 21 122.4 145.9 166.7 201.2 147.7 176.9 203.2 247.9 174.4 209.5 242.2 298.2 22 116,3 138.4 158.1 190.6 140.6 168.3 193.3 235.5 166.5 199.9 230.9 284.0 23 110.5 131.5 150.1 180.7 134.0 160.3 184.0 224.0 159.0 190.8 220.4 270.7 24 105 2 125 1 142.7 171.6 127.8 152.8 175.3 213.2 152.0 182.3 210.4 258.3 25 100.2 119.1 135 7 163.1 122.0 145.8 167.1 203.1 145.4 174.3 201.0 246.6 ECCENTRIC LOADING OF COLUMNS. 255 Safe Load of Cast-iron Columns- (Continued). Outside Diameter, Outside Diameter, Outside Diameter, c 14 inches. 15 inches. 16 inches. tc Thickness in inches. Thickness in inches. Thickness in inches. c 1 m M 2 684.6 1 1M 1^ 2 1 m 648.0 2 7 877.7 461.1 539.9 413.7 506.1 594.0 756.7 449.8 551.1 828.6 8 363.1 442.8 518.0 655.9 399.3 487.9 572.2 727.7 435.3 532.8 626.3 799.8 9 848.5 424.4 -196.3 627.0 384.4 469.5 550.1 698.4 420.5 514.4 604.1 770.4 10 888.8 406.3 474.6 598.5 369.7 451.0 528.2 669.3 405.6 496.0 581.8 740.9 11 819.4 388.5 453.4 570.7 355.1 433.0 506.3 640.9 390.6 477.4 559.8 711.7 12 305.4 371.1 432.6 543.6 340.6 415.0 485.0 612.8 376.0 459.3 538.0 683.4 IS 291.8 354.3 412.7 517.7 326.6 397.6 464.5 585.9 361.6 441.2 516.7 655,1 14 278.8 338.2 393.6 493.0 313.0 380.7 444.4 559.7 347.6 423.8 495.9 628 15 266 2 322.7 375.3 469.4 299.9 364.5 425.2 534.9 333.9 406.9 475.9 601 8 16 254.3 308.0 357.9 446.9 287.2 348.9 406.7 510.9 320.7 390.6 456.6 576.6 17 242.9 294.0 341 .4 425.7 275.1 334.0 389.1 488.1 308.0 374.9 438.0 552.5 18 232.0 280.6 325.6 405.5 263.6 319.7 372.2 466.5 295.8 359.9 420.1 529.4 19 221.7 268.0 310.8 386.5 252.5 306.2 356.2 445.9 284.1 345.4 403.0 507.3 20 212.0 256.1 296.7 368.6 242.0 293.3 341.0 426.3 272.9 331.6 386.8 486.3 21 202.7 244.7 283.5 351.8 232.0 281.0 326.5 407.8 262.1 318.4 371.2 466 2 194.0 284.0 270.9 335.9 222.5 269.3 312.8 390.3 251.9 305.9 356.4 447 2 185.7 224.0 259.1 320.9 213.4 258.3 299.8 373.7 242.2 293.9 342.3 429.1 24 177.9 214.4 248.0 306.8 204.9 247.8 287.5 358.1 232.9 282.5 328.8 411,9 25 170.5 205.4 237.5 294.1 196.7 237.8 275.9 343.2 224.0 271.6 316.1 395.6 ECCENTRIC LOADING OF COLUMNS. In a given rectangular cross-section, such as a masonry joint under press- ure, the stress will be distributed uniformly over the section only when the resultant passes through the centre of the section ; any deviation from such a central position will bring a maximum unit pressure to one edge and a minimum to the other; when the distance of the resultant from one edge is one third of the entire width of the joint, the pressure at the nearer edge is twice the mean pressure, while that at the farther edge is zero, and that when the resultant approaches still nearer to the edge the pressure at the farther edge becomes less than zero; in fact becomes a tension, if the material (mortar, etc., there is capable of resisting tension. Or, if, as usual in masonry joints, the material is practically incapable of resisting tension, the pressure at the nearer edge, when the resultant approaches it nearer than one third of the width, increases very rapidly and dangerously, becom- ing- theoretically infinite when the resultant reaches the edge. With a given position of the resultant relatively to one edge of the joint or section, a similar redistribution of the pressures throughout the section may be brought about by simply adding to t or diminishing the width of the section. Let P = the total pressure on any section of a bar of uniform thickness. = the width of that section = the area of the section, when thickness : 1. : the mean unit pressure on the section. M — the maximum unit pressure on the section. m — the minimum unit pressure on the section. d = the eccentricity of the resultant — its distance from the centre of the section. Thenilf = p (l4~ ) and m = p (l - ^). When d is greater than l/6w, the resultant in that case being less than one third of the width from one edge, p becomes negative. (J. C. Traut- wine, Jr., Engineering News, Nov, 23, 1893.) 256 STRENGTH OF MATERIALS. BUILT COLUMNS. From experiments by T. D. Lovett, discussed by Burr, the values of /and a in several cases are determined, giving empirical forms of Gordon's for- mula as follows: p = pounds crushing strength per square inch of section, I = length of column in inches, r — radius of gyration in inches. Keystone Columns. 39,500 (1) ! + ? Flat Ends. Square Columns. 39,000 . Phoenix Columns. American Bridge Co. Columns. F <8) 1 + i<*> 35,000 r* Flat Ends, Swelled. i+s ! + f 2 (7) 1 + 17,000 r 2 ' 22,700 r Pin Ends, Swelled. Round Ends. r<8> i + 12,500 1 (10) l + o 36,000 (ID 1 + 11,500 ? With great variations of stress a factor of safety of as high as 6 or 8 may be used, or it may be as low as 3 or 4, if the condition of stress is uniform or essentially so. Burr gives the following general principles which govern the resistance of built columns : The material should be disposed as far as possible from the neutral axis of the cross-section, thereby increasing r; There should be no initial' internal stress; The individual portions of the column should be mutually supporting; The individual portions of the column should be so firmly secured to each other that no relative motion can take place, in order that the column may fail as a whole, thus maintaining the original value of r. Stoney says: "When the length of a rectangular wrought-iron tubular column does not exceed 30 times its least breadth, it fails by the bulging or buckling of a short portion of the plates, not by the flexure of the pillar as a whole." In Trans. A. S. C. E., Oct. 1880, are given the following formulae for the ultimate resistance of wrought-iron columns designed by C. Shajer Smith : BUILT COLUMNS. 257 Flat Ends. Square Phoenix American Bridge Common Column. Column. Co. Column. Column. (15) -1 w (18) 1 i J_ J? 1 i J_ l l ,,JLi! X4.JL.* "^5820 d 2 ^~4500 d 2 '^3750 d 2 "^OO d 2 One Pin End. _^ (18) _i2|^ (16> _i^ <19) -Sf^ , 1- *~3000 d a 1 + 2250 d 2 1 + 2250 d 2 1 + 1500 d 2 Two Pin Ends. 37,500 n - { 36,600 (1?) 36,500 ^ 36,500 l-4-_ i_l__L_± l + _i_ ± 1J_J_ ± M900 d 2 M500 d 2 ^1750 d 2 ^1200 d 2 The " common " column consists of two channels, opposite, with flanges outward, with a plate on one side and a lattice on the other. The formula for " square " columns may be used without much error for the common-chord section composed of two channel-bars and plates, with the axis of the pin passing through the centre of gravity of the cross- section. (Burr). Compression members composed of two channels connected by zigzag bracing may be treated by formulae 4 and 5, using / = 36,000 instead of 39,000. Experiments on full-sized Phoenix columns in 1873 showed a close agree- ment of the results with formulas 6-8. Experiments on full-sized Phoenix columns on the Watertovvn testing-machine in 1881 showed considerable dis- crepancies when the value of I h- r became comparatively small. The fol- lowing modified form of Gordon's formula gave tolerable results through the whole range of experiments : 40,000 (l-f y) Phoenix columns, flat end, p = --^ — (24) 1 + 50,000 r 2 Plotting results of three series of experiments on Phoenix columns, a more simple formula than Gordon's is reached as follows : Phoenix columns, flat ends, p = 39,640 - p = 64,700 - 4600 \ 1 when I -h r is less than 30. Dimensions of Phoenix Columns, (Phoenix Iron Co.) The dimensions are subject to slight variations, which are unavoidable in rolling iron shapes. The weights of columns given are those of the 4, 6, or 8 segments of which they are composed. The rivet-heads add from 2 to 5 per cent to the weights given. Rivets are spaced 3, 4, or 6 inches apart from centre to centre, and somewhat more closely at the ends than towards the centre of the column. Q columns have 8 segments, .£7 columns 6 segments, O, B 2 , B l , and A have 4 segments. Least radius of gyration - D X -3636, 258 Phoenix Columns. One Segment. Diameters in inches. One Column. Safe Load in net tons a CO i-ao r 7 = 80 - = 100 r - = 120 r 1 = 140 - = 160 5,000 6,000 7,000 8,000 9,000 10,000 11.000 5,040 6,055 7,080 8,100 9,130 10,160 11.200 12,240 13,280 5,170 6,240 7,330 8,430 9,550 10,680 11,750 13,000 14,180 5,390 6,560 7,780 9,040 10,340 11,680 13,070 14,500 15,990 5,730 7,090 8,530 10,060 11,690 13,440 15,310 17,320 19,480 6,250 7,890 9,720 11,660 14,060 16,670 19,640 23,080 6,980 9,090 11,610 14,640 18,380 23,090 8,220 11,330 15,510 21,460 10.250 15,560 24,720 i2;ooo 13,000 STRENGTH OF WROUGHT IRON AND STEEL COLUMNS. 261 II.— Wrought-iron Columns with Fixed Ends. Unit- load. Maximum Compressive Unit-stress C. — or B. A H° £=*> 1 = 60 V 7=M - = 100 - = 120 -=140 r - = 160 r 6,000 6,010 6,060 6,130 6,240 6,380 6,570 6,800 7,090 7,000 7,020 7,080 7,180 7,330 7,530 7,780 8,110 8,530 8,000 8,025 8,100 8,240 8,430 8,700 9,040 9,490 10,060 9,000 9,030 9,130 9,300 9,550 9,890 10,340 10,930 11,690 10,000 10,040 10,160 10,370 10,710 11,110 11,680 12,440 13,440 11,000 11,050 11,200 11,450 11,830 12,360 13,070 14,020 15,310 12,000 12,060 12,240 12,540 13,000 13,640 14,510 15,690 17,320 13,000 13,070 13,280 13.640 14,210 14,940 15,990 17,440 19,480 14,000 14,080 14,320 14,740 15,380 16,280 17,530 19,290 21,820 III.- -Steel Columns with Round Ends. Unit- load. Maximum Compressive Unit-stress C. p - A or B. A 7 = 2 ° | = 40 7 = 60 1 = 80 -=100 V 1 = 120 - = 140 r , — = 160 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 6,050 7,070 8,090 9,110 10,130 11,160 12.200 13,330 14,250 6,200 7,270 8,380 9,450 10,560 11,690 12,820 13,970 15,130 6,470 7,650 8,770 10,090 11,360 12.670 14,020 15,400 16,830 6,880 8,230 9,650 11,140 12,710 14 370 7,500 9,130 10,870 12,850 15,000 17,370 8,430 10,540 12,990 15.850 19,230 23,300 28,300 9,870 12,900 16,760 20,930 28,850 12,300 17,400 24,590 16,130 20.000 18,000 | 22,940 19,960 1 26.250 IV.— Steel Columns with Fixed Ends. Unit- load. Maximum Compressive Unit-stress C. p ~otB. A r 1 = 40 r 1 = 60 r - = 80 r - = 100 — =120 r -=140 r — = 160 7,000 7.020 7,070 7,150 7,270 7,430 7,650 7,900 8,230 8,000 8.020 8,090 8,200 8,380 8,570 8,770 9,200 9,650 9,000 9,030 9,110 9,250 9,450 9,730 10,090 10,550 11,140 10,000 10,030 10,130 10,310 10,560 10,910 11,360 11,810 12,710 11,000 11,040 11,160 11,380 11,690 12,110 12,670 13,410 14,370 12,000 12.050 12,200 12,450 12,820 13.330 14,020 14,930 16,130 13,000 13,060 13,230 13,530 13,970 14,580 15,400 16,500 17.990 14,000 14,070 14,250 14,610 15,130 15,850 16,830 18,150 19,960 15,000 15,080 15,310 15,710 16,310 17,140 18,290 19,870 22,060 The design of the cross-section of a column to carry a given load with maximum unit-stress C may be made by assuming dimensions, and then 262 STRENGTH OF MATEEIALS. computing C by formula (1). If the agreement between the specified and computed values is not sufficiently close, new dimensions must be chosen, and the computation be repeated. By the use of the above tables the work will be shortened. The formula (1) may be put in another form which in some cases will ab- breviate the numerical work. For B substitute its value P-f- A, and for Ar"* write I, the least moment of inertia of the cross-section; then P nPl% x -^ = % < 3 > in which I and r* are to be determined. For example, let it be required to find the size of a square oak column with fixed ends when loaded with 24.000 lbs. and 16 ft, long, so that the maximum compressive stress C shall be 1000 lbs. per square inch. Here /= 24,000, C = 1000, n = y 4 , tt 2 = 10, E = 1,500,000, I = 16 x 12, and (3) be- comes I - 24r 2 = 14.75. Now let x be the side of the square; then I=- and ,* = -, so that the equation reduces to x* - 24a: 2 = 177, from which a 2 is found to be 29.92 sq. in., and the side x = 5.47 in. Thus the unit-load B is about 802 lbs. per square inch. WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. Theodore Cooper gives the following in his Bridge Specifications : Compression members shall be so proportioned that the maximum load shall in no case cause a greater strain than that determined by the follow- ing formula : p 8000 „ ■r = — for square-end compression members ; 1 + 40,000r 2 P = ~ for compression members with one pin and one square end ; ^ 30,000r 2 P= „ — for compression members with pin-bearings; 1-j ~ 20,000r 2 (These values may be increased in bridges over 150 ft. span. See Cooper's Specifications.) P = the allowed compression per square inch of cross-section; I = the length of compression member, in inches; r = the least radius of gyration of the section in inches. No compression member, however, shall have a length exceeding 45 times its least width. The Phoenix Bridge Company give the following : The greatest working stresses in wrought-iron compression members of spans 150 feet in length and under shall be the following: Flat Ends. Pin Ends. Phoenix column P = = — P = — T 50,000r 2 T 30,000r a T *.*■ i i t> 800 ° « 7800 Latticed or common column P = — — - P = — — H i-j - T 40,000r 2 ^ 30,000r 2 Angle-iron struts P = 9000 - 30- P = 9000 - 34 - WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. 263 Upper chords shall be proportioned by the flat-end formula. A mean between flat-end and pin-end results shall be used for one pin end and one flat end. Lateral and trausverse struts shall be designed by taking working stresses equal to one and four tenths those given by the preceding formulae. Working Stresses allowed in Bridge Tension Members. (Theodore Cooper's Specifications.) All parts of the structure shall be so proportioned that the maximum * loads shall in no case cause a greater tension than the following (except in spans exceeding 150 feet) : Pounds per sq. in. Ou lateral bracing 15,000 On solid rolled beams, used as cross floor-beams and stringers. 9,000 On bottom chords and main diagonals (forged eye-bars) 10,000 On bottom chords and main diagonals (plates or shapes), net section 8,000 On counter rods and long verticals (forged eye-bars) 8,000 On counter and long verticals (plates or shapes), net section.. 6,500 On bottom flange of riveted cross-girders, net section 8,000 On bottom flange of riveted longitudinal plate girders over 20 ft. long, net section 8,000 On bottom flange of riveted longitudinal plate girders under 20 ft. long, net section 7,000 On floor-beam hangers, and other similar members liable to sudden loading (bar iron with forged ends) 6,000 On floor beam hangers, and other similar members liable to sudden loading (plates or shapes), net section 5,000 Members subject to alternate strains of tension and compression shall be proportioned to resist each kind of strain. Both of the strains shall, how- ever, be considered as increased by an amount equal to 8/10 of the least of the two strains, for determining the sectional area by the above allowed strains. The Phoenix Bridge Company specify : The greatest working stresses in all wrought-iron tensile members of railway spans 150 feet in length and under, shall be as follows: Pounds per sq. in. In, counter web members 8,000 In long verticals 8,000 In main-web and lower-chord members (eye-bars) 10,000 In suspension loops 7,000 In suspension plates (net section) 7,000 In tension members of lateral and transverse bracing 15,000 In counter rods and long verticals of lattice girders (net sec- tion) 7,000 In lower chords and main tension members of lattice girders (net section) 8,000 In bottom flange of plate girders (net section) 8,000 In bottom flange of rolled beams 8,000 In angle-iron lateral ties (net section) 12,000 In spans over 150 feet in length, the greatest working tensile stresses per square inch of wrought iron, lower-chord and end main-web eye-bars, shall be total stress\ V- max. total stress / whenever this quantity exceeds 10,000. Working Stresses for Steel. The greatest allowed working stresses for steel tension members, for spans of 200 feet in length and less, shall be as follows ; 264 STRENGTH OF MATERIALS. Pounds per sq. in. In counter web members 10,500 In long verticals 10,000 In all main-web and lower-chord eye-bars 13,200 In plate hangers (neb section) 9,000 In tension members of lateral and transverse bracing 19,000 In steel-angle lateral ties (net section) 15,000 For spans over 200 feet in length the greatest allowed working stresses per square inch, in lower-chord and end main-web eye-bars, shall be taken at / mm^otalst^X V max. total stress / max. total stress s whenever this quantity exceeds 13,200. The greatest allowable stress in the main-web eye-bars nearest the centre of such spans shall be taken at 13,200 pounds per square inch ; and those for the intermediate eye-bars shall be found by direct interpolation between the preceding values. The greatest allowable working stresses in steel plate and lattice girders and rolled beams shall be taken as follows : Pounds per sq. in. Upper flange of plate girders (gross section) 10,000 Lower flange of plate girders (net section) 10,000 In counters and long verticals of lattice girders (net section) . . 9,000 In lower chords and main diagonals of lattice girders (net section) 10,000 In bottom flanges of rolled beams 10,000 In top flanges of rolled beams 10,000 RESISTANCE OF HOLLOW CYLINDERS TO COLLAPSE. Fairbairn's empirical formula {Phil. Trans. 1858) is p- 9,675,600^, . . . . . .(1) where p = pressure in lbs. per square inch, t = thickness of cylinder, d = diameter, and I — length, all in inches ; or, p = 806,600 tj-z-i if L is in feet (2) He recommends the simpler formula p = 9,675,600^ ( 3 ) as sufficiently accurate for practical purposes, for tubes of considerable diameter and length. The diameters of Fairbairn's experimental tubes were 4", 6", 8", 10", and 12", and their lengths, between the cast-iron ends, ranged between 19 inches and 60 inches. His formula (3) has been generally accepted as the basis of rules for ascertaining the strength of boiler flues. In some cases, however, limits are fixed to its application by a supplementary formula. Lloyd's Register contains the following formula for the strength of circular boiler-flues, viz., "^ « The English Board of Trade prescribes the following formula for circular flues, when the longitudinal joints are welded, or made with riveted butt- straps, viz., P - M' 000 *" (5) r -(L + Dd- • ; (5) For lap-joints and for inferior workmanship the numerical factor may be reduced as low as 60,000, RESISTANCE OF fiOLLOW CYLINDERS TO COLLAPSE. 265 The rules of Lloyd's Register, as well as those of the Board of Trade, pre- scribe further, that in uo case the value of P must exceed the amount given by the following equation, viz.* P=*£ <6> In formulae (4), (5), (6) P is the highest working pressure in pounds per square inch, t and d are the thickness and diameter in inches, L is the length of the flue in feet measured between the strengthening rings, in case it is fitted with such. Formula (4) is the same as formula (3), with a factor of safety of 9. In formula (5) the length L is increased by 1 ; the influence which this addition has on the value of P is, of course, greater for short tubes than for long ones. Nystrom has deduced from Fairbairn's experiments the following formula for the collapsing strength of flues : 4Tt* P= ^£' = t* (J 266 STRENGTH OF MATERIALS. Instances of collapsed flues of Cornish and Lancashire boilers collated by Clark, showed that the resistance to collapse of flues of %-inch plates, 18 to 43 feet long, and 30 to 50 inches diameter, varied as the J, 75 power of the diameter. Thus, for diameters of 30 35 40 45 50 inches, the collapsing pressures were 76 58 45 37 30 lbs. per sq. in; for 7-16-inch plates the collapsing pressures were 60 49 42 " " " For collapsing pressures of plain iron flue-tubes of Cornish and Lanca- shire steam-boilers, Clark gives: _ 200,000^ P = collapsing pressure, in pounds per square inch; t = thickness of the plates of the furnace tube, in inches. d = internal diameter of the furnace tube, in inches. For short lengths the longitudinal tensile resistance may be effective in augmenting the resistance to collapse. Flues efficiently fortified by flange- joints or hoops at intervals of 3 feet may be enabled to resist from 50 lbs. to 60 lbs. or 70 lbs. pressure per square inch more than plain tubes, accord- ing to the thickness of the plates. Strength of Small Tubes.— The collapsing resistance of solid- drawn tubes of small diameter, and from .134 inch to .109 inch in thickness, has been tested experimentally by Messrs. J. Russell & Sons. The results for wrought-iron tubes varied from 14.33 to 20.07 tons per square-inch sec- tion of the metal, averaging 18.20 tons, as against 17.57 to 24.28 tons, averag- ing 22.40 tons, for the bursting pressure. (For strength of Segmental Crowns of Furnaces and Cylinders see Clark, S. E., vol. i, pp.- 649-651 and pp. 627, 628.) Formula for Corrugated Furnaces (Eng^g, July 24, 1891, p. 102).— As the result of a series of experiments on the resistance to collapse of Fox's corrugated furnaces, the Board of Trade and Lloyd's Registry altered their formulae for these furnaces in 1891 as follows: Board of Trade formula is altered from i a ,5ooxr =w , fto i4.oooxr = trp T = thickness in inches; D = mean diameter of furnace; WP = working pressure in pounds per square inch. Lloyd's formula is altered from 1000 x(T*) 1884 X(r») p D D ~~ T = thickness in sixteenths of an inch; D = greatest diameter of furnace; WP = working pressure in pounds per square inch. TRANSVERSE STRENGTH. In transverse tests the strength of bars of rectangular section is found to vary directly as the breadth of the specimen tested, as the square of its depth, and inversely as its length. The deflection under any load varies as the cube of the length, and inversely as the breadth and as the cube of the depth. Represented algebraically, if S = the strength and D the deflection, I the length, b the breadth, and d the depth, a "• bd * a r, Z 3 S varies as ~r and D varies as —ts> I bd 3 For the purpose of reducing the strength of pieces of various sizes to a common standard, the term modulus of rupture (represented by R) is used. Its value is obtained by experiment on a bar of rectangular section TRANSVERSE STRENGTH. 267 supported at the ends and loaded in the middle and substituting numerical values in the following formula : 3 PI E - «6d5' in which P — the breaking load in pounds, I = the length in inches, 6 the breadth, and d the depth. The modulus of rupture is sometimes defined as the strain at the instant of rupture upon a unit of the section which is most remote from the neutral axis on the side which first ruptures. This definition, however, is based upon a theory which is yet in dispute among authorities, and it is better to define it as a numerical value, or experimental constant, found by the ap- plication of the formula above given. 1 rom the above formula, making I 12 inches, and b and d each 1 inch, it follows that the modulus of rupture is 18 times the load required to break a bar one inch square, supported at two points one foot apart, the load being applied in the middle. „ ™ . . „ , , t , span in feet X load at middle in lbs. Coefficient of transverse strength = _— — ^r-. — : — : — — : — - breadth in inches x (depth m inches) 2 . Fundamental Formulae for Flexure of Beams (Merriman). Resisting shear = vertical shear; Resisting moment = bending moment; Sum of tensile stresses = sum of compressive stresses; Resisting shear = algebraic sum of all the vertical components of the in- ternal stresses at any section of the beam. Tf A be the area of the section and Ss the shearing unit stress, then resist- ing shear = ASs; and if the vertical shear = V, then V = ASs.' The vertical shear is the algebraic sum of all the external vertical forces on one side of the section considered. It is equal to the reaction of one sup- port, considered as a force acting upward, minus the sum of all the vertical downward forces acting between the support and the section. The resisting moment = algebraic sum of all the moments of the inter- nal horizontal stresses at any section with reference to a point in that sec- tion, = ' — , in which S — the horizontal unit stress, tensile or compressive as the case may be, upon the fibre most remote from the neutral axis, c = the shortest distance from that fibre to said axis, and I = the moment of inertia of the cross-section with reference to that axis. The bending moment M is the algebraic sum of the moment of the ex- ternal forces on one side of the section with reference to a point in that sec- tion = moment of the reaction of one support minus sum of moments of loads between the support and the section considered. The bending moment is a compound quantity = product of a force by the distance of its point of application from the section considered, the distance being measured on a line drawn from the section perpendicular to the direction of the action of the force. Concerning the above formula, Prof. Merriman, Eng. News, July 21, 1894, says: The formula just quoted is true when the unit-stress S on the part of the beam farthest from the neutral axis is within the elastic limit of the material. It is not true when this limit is exceeded, because then the neutral axis does not pass through the centre of gravity of the cross-section, and because also the different longitudinal stresses are not proportional to their distances from that axis, these two requirements being involved in the de- duction of the formula. But in all cases of design the permissible unit- stresses should not exceed the elastic limit, and hence the formula applies rationally, without regarding the ultimate strength of the material or any of the circumstances regarding rupture. Indeed so great reliance is placed upon this formula that the practice of testing beams by rupture has been almost entirely abandoned, and the allowable unit-stresses are mainly de- rived from tensile and compressive tests. 268 STRENGTH OF MATERIALS. s.® a /TN o © < 0) Q i-i ICO th 100 5;N ^|^ .oico " N -ii-i§! ml nlSS © += © 02 a SI O.gj, SI g|« gjo g|o {jj|« 5h Sh Siv o S © II II II II ll II II ii II II W s el b; s s S th ICO (Ah Oh S c^ §0Q th |CM i-ii-* th.oo -f. thioo T-ti«o th |; H ICO s §0 JV^ V TH 100 eg 0? g co ico m "IS S o o 53 m || ill di ill S s|| CO 1,0* 1-* ia|£ + H5S ||?.'S1l ^ Bq co ^ W th;^ I" X s o % § © > ji Jh ^ a t: c o '1 E "3 c xr -d ^ © £ c c i © TJ « a • a * d » J J c S3 | c a J ] ) 1 a » '% j 5S % pq c cS ~- 'a 3 T i © P CO ' d % ! \ a : £ § * a £ 1 ce IS cj i? « cS & H 72 a CC 02 P- 02 o: Bh a APPROXIMATE SAFE LOADS IN LBS. ON STEEL BEAMS. 269 Formulae for Transverse Strength of Beams.— Referring to table on pieeeding page, P = load at middle; W= total load, distributed uniformly; I = length, o = breadth, d = depth, in inches; E = modulus of elasticity; R ■= modulus of rupture, or stress per square inch of extreme fibre; / = moment of inertia; c = distance between neutral axis and extreme fibre. For breaking load of circular section, replace bd 2 by 0.59<2 3 . For good wrought iron the value of R is about 80,000, for steel about 120,000, the percentage of carbon apparently having no influence. (Thurston, lion and Steel, p. 491). For cast iron the value of R varies greatly according to quality. Thurston found 45,740 and 67,980 in No. 2 and No. 4 cast iron, respectively. For beams fixed at both ends and loaded in the middle, Barlow, by experi- ment, found the maximum moment of stress = 1/6PI instead of %Pl, the result given by theory. Prof. Wood (Resist. Matls. p. 155) says of this case: The phenomena are of too complex a character to admit of a thorough and exact analysis, and it is probably safer to accept the results of Mr. Barlow in practice than to depend upon theoretical results. APPROXIMATE GREATEST SAFE LOADS IN LBS. ON STEEL. BEAMS. (Pencoyd Iron Works.) Based on fibre strains of 16,800 lbs. for steel. (For iron the loads should be one sixth less, corresponding to a fibre strain of 14,000 lbs. per square inch). L = length in feet between supports; a = interior area in square A — sectional area of beam in square inches; inches; d = interior depth in inches. D = depth of beam in inches w = working load in net tons. Shape of Section. Greatest Safe Load in Pounds Deflection in Inches. Load in Middle. Load Distributed. Load in " Middle. Load Distributed. Solid Rect- angle. 940.4 D L 1880.4D L wL 3 32AD* ivL 3 52 AD* HollowRect- 940( AD -ad) L 1880(AD-ad) L wU wL 3 angle. 32(AD 2 -ad' 2 ) 52(AD*-ad*) Solid Cylin der. 700AD L 1400 AD L ivL 3 24AD* wL 3 38AD'* Hollow 700( AD- ad) L UOO(AD-ad) L tvL 3 wL 3 24(AD2-ad*) 38(AD*-ad*) Even-legged Angle or Tee. 930AD moAD L tvL 3 32AD* wL 3 L 52AD* Channel or Z bar. 1600AD L 3200AD L wL 3 53AD 2 ivL 3 85AD 2 Deck Beam. 1450 AD L 2900,4 D L wL 3 50AD* ivL 3 80AD* I Beam. 1780AD L mQAD L wL 3 58 A D 2 wL 3 93,4 D 2 I II III IV V 270 STRENGTH OF MATERIALS. The above formulae for the strength and stiffness of rolled beams of va- rious sections are intended for convenient application in cases where strict accuracy is not required. The rules for rectangular and circular sections are correct, while those for the flanged sections are approximate, and limited in their application to the standard shapes as given in the Pencoyd tables. When the section of any beam is increased above the standard minimum dimensions, the flanges re- maining unaltered, and the web alone being thickened, the tendency will be for the load as found by the rules to be in excess of the actual; but within the limits that it is possible to vary any section in the rolling, the rules will apply without any serious inaccuracy. The calculated safe loads will be approximately one half of loads that would injure the elasticity of the materials. The rules for deflection apply to any load below the elastic limit, or less than double the greatest safe load by the rules. If the beams are long without lateral support, reduce the loads for the ratios of width to span as follows : Length of Beam. 20 times flange width. 30 " 40 " « « 50 " 60 ■" Proportion of Calculated Load forming Greatest Safe Load. . Whole calculated load. 9-10 8-10 " 7-10 6-10 5-10 These rules apply to beams supported at each end. For beams supported otherwise, alter the coefficients of the table as described below, referring to the respective columns indicated by number. Changes of Coefficients for Special Forms of Beams. Kind of Beam. Fixed at one end, loaded at the other. One fourth of the coeffi- cient, col. II. Coefficient for Safe Load. One sixteenth of the co- efficient of col. IV. Coefficient for Deflec tion. Fixed at one end, load evenly distributed. One fourth of the coeffi- cient of col. III. Five forty-eighths of the coefficient of col. V. Both ends rigidly fixed, or a continuous beam, with a load in middle. Twice the coefficient of col. II. Four times the coeffi- cient of col. IV. Both ends rigidly fixed, or a continuous beam, with load evenly dis- tributed. One and one-half times the coefficient of col. III. Five times the coefficient of col. V. ELASTIC RESILIENCE. In a rectangular beam tested by transverse stress, supported at the ends and loaded in the middle, 2 Rbd* r ~ 3 I 5 _ 1 PV A _ 4 Eod 3 ' in which, if P is the load in pounds at the elastic limit, R = the modulus of transverse strength, or the strain on the extreme fibre, at the elastic limit, E = modulus of elasticity, A = deflection, L b, and d = length, breadth, and depth in inches. Substituting for P in (2) its value in (1), we have A-- — 6 Ed • BEAMS OF UNIFORM STRENGTH THROUGHOUT LENGTH. 271 ELEVATION. The elastic resilience = half the product of the load and deflection = J^jPA, and the elastic resilience per cubic inch _ 1 PA ~ 2 Ibd' Substituting the values of P and A, this reduces to elastic resilience per 1 P 2 cubic inch = ^j-j™. which is independent of the dimensions; and therefore the elastic resilience per cubic inch for transverse strain may be used as a modulus expressing one valuable qualit}' of a material. Similarly for tension: Let P = tensile stress in pounds per square inch at the elastic limit; e = elongation per unit of length at the elastic limit; E = modulus of elasticity = P -f- e; whence e = P 4- E. 1 P 2 Then elastic resilience per cubic inch = J^Pe = 5-^'. BEAMS OF UNIFORM STRENGTH THROUGHOUT THEIR LENGTH. The section is supposed in all cases to be rectangular throughout. The beams shown in plan are of uniform depth throughout. Those shown in elevation are of uniform breadth throughout. B = breadth of beam. D = depth of beam. Fixed at one end, loaded at the other; curve parabola, vertex at loaded end ; PP 2 proportional to distance from loaded end. The beam may be reversed, so that the up- per edge is parabolic, or both edges may be parabolic. Fixed at one end, loaded at the other; triangle, apex at loaded end; PP 2 propor- tional to the distance from the loaded end. Fixed at one end; load distributed; tri- angle, apex at unsupported end; BD' 1 pro- portional to square of distance from unsup- ported end. Fixed at one end; load distributed ; curves two parabolas, vertices touching each other at unsupported end; PP 2 proportional to distance from unsupported end. Supported at both ends; load at any one point; two parabolas, vertices at the points of support, bases at point loaded ; BD 2 pro- portional to distance from nearest point of support. The upper edge or both edges may also be parabolic. Supported at both ends; load at any one point; two triangles, apices at points of sup- port, bases at point loaded; BD 2 propor- tional to distance from the nearest point of support. Supported at both ends; load distributed; curves two parabolas, vertices at the middle of the beam; bases centre line of beam; PP 2 proportional to product of distances from points of support. Supported at both ends; load distributed; curve semi-ellipse; BD 2 proportional to the product of the distances from the points of support. PASE S? ELEVATION. |9| £72 STRENGTH OF MATERIALS. PROPERTIES OF ROLLED STRUCTURAL SHAPES. Explanation of Tables of the Properties of Carnegie I Beams, Channels, and Z Bars. The tables of I beams are calculated for the minimum weight to which each pattern can be rolled. The tables of channels are calculated for the minimum and maximum weights of the various shapes, while the properties of Z bars are given for thicknesses differing by 1/16 inch. Columns 11 and 13, in the tables fori beams and channels, give coefficients by the help of which the safe uniformly-distributed load may readily be de- termined. To do this, divide the coefficient given by the span or distance between suppoi-ts in feet. If the weight of the section is intermediate be- tween the minimum and maximum weights given, add to the coefficient for the minimum weight the value given in columns 12 or 14 (for one pound increase of weight), multiplied by the number of pounds the section is heavier than the minimum. If a section is to be selected (as will usually be the case) intended to carry a certain load, for a length of span already determined on, ascertain the coefficient which this load and span will require, and refer to the table for a section having a coefficient of this value. The coefficient is obtained by multiplying the load, in pounds uniformly distributed, by the span length in feet. In case the load is not tmiformly distributed, but is concentrated at the middle of the span, multiply the load by 2 and then consider it as uni- formly distributed. The deflection will be 8/10 of the deflection for the latter' load. For other cases of loading obtain the bending moment in foot-pounds; this multiplied by 8 will give the coefficient required. If the loads are quiescent, the coefficients for a fibre strain of 16,000 lbs. per square inch for steel and 12,000 lbs. for iron may be used ; but if moving loads are to be provided for, the coefficients for 12,500 and 10,000 lbs., respectively, should be taken. Inasmuch as the effects of impact may be very considerable (the strains produced in an unyielding, inelastic material by a load suddenly applied being double those produced by the same load in a quiescent state), it will sometimes be advisable to use still smaller fibre strains than those given in the tables. In such cases the co- efficients can readily be determined by proportion. Thus, for a fiber strain of 8000 lbs. per square inch the coefficient will equal the coefficient for 10,000 lbs. fibre strain, from the table, multiplied by 8/10. The moments of resistance given in column 9 are used to determine the fibre strain per square inch in a beam, or other shape, subjected to bending or transverse strains, by dividing the same into the bending moment expressed in inch-pounds. For Carnegie Z bars, complete tables of moments of inertia, moments of resistance, radii of gyration, and values of the coefficients (C) are given for thicknesses varying by 1/16 inch. These coefficients may be applied, as explained above, for cases where the Z bars are subjected to transverse loading, as, for example, in the case of roof-purlins. For more complete and detailed information concerning structural shapes, consult the pocket-books and circulars issued by the manufacturers. PROPERTIES OF ROLLED STRUCTURAL SHAPES. 2?3 a h . a 2 ! 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TRENTON IRON BEAMS AND CHANNELS. (New Jersey Steel and Iron Co.) *■• »j ,JD u . „ .0 -1 a . "Irf JD 03 « a §0 . |& II o &> o a fl 53 .5 > 2^M d cc^ a c§^ a ei_i a £4 °.S v a <» S ac Erf a © a I S m a"" a a o— ' O 1 5 fl I Beams. Channels. 20 272 m 11-16 1,320,000 15 190 m M 625,000 20 200 6 H 990,000 15 120 4 H 401,000 K% 200 5% .6 748,000 mi 140 4 11-16 381,000 15 3-16 150 5 ^ 551,000 1214 mi 70 3 .33 200.100 15^ 125 5 .42 460,000 60 2% % 134,750 12 5-16 170 1 .6 511,000 10 48 2y 2 5-16 102,000 12^ 125 .47 377,000 9 70 m 7-16 146,000 12 120 5j| .39 375,000 9 50 2^ .33 104,000 12 96 .32 306,000 8 45 2^ .26 88,950 10^ 135 5 .47 360,000 8 33 2.2 .20 65,800 10^ 105 4« % 286.000 7 36 2^ H 62,000 « 90 4^ 5-16 250,000 7 25y 2 2 .20 39,500 9 125 4V 2 .57 268,000 6 45 2^ .40 58,300 9 85 iy 2 % 199.000 6 33 2M .28 45,700 9 70 4 .3 167,000 6 22^ 1% .18 33,680 8 80 4J^ % 168,000 5 19 1% .20 22,800 8 65 4 .3 135,000 4 16H 1M .20 15,700 7 55 SH .3 101,000 172,000 3 15 ifc .20 10,500 6 120 5J4 6 90 5 % 132,000 De ck Be* ims. 6 6 50 3^ 3 3 76,800 40 M 62,600 5 40 3 5-16 49,100 8 65 4^2 % 91,800 5 30 37 m 3 M 5-16 38,700 36,800 7 55 4J^ 5-16 63,500 4 4 30 2% J4 30,100 4 18 1 2 3-16 1 18,000 Trenton Beams and Channels. To find which beam, supported at both ends, will be required to support with safety a given uniformly distributed load: Multiply the load in pounds by the span in feet, and take the beam whose " Coefficient for Strength " is nearest to and exceeds the number so found. The weight of the beam itself should be included in the load. The deflection in inches, for such distributed load, will be found by divid- ing the square of the span taken in feet, by 70 times the depth of the beam, taken in inches, for iron beams, and by 52.5 times the depth for steel. Example.— Which beam will be required to support a uniformly distrib- uted load of 12 tons (= 24,000 lbs.) on a span of 15 feet ? 24,000 X 15 = 360,000, which is less than the coefficient of the 1214-inch 125- lb. iron beam. The weight of the beam itself would be 625 lbs., which, ad- ded to the load and multiplied by the span, would still give a product less than the coefficient; thus, 24,625 X 15 = 369,375. The deflection will be 15 X 15 _ ■ , -■ „ , = 0.26 inch. 70 X Vl\i The safe distributed load for each beam can be found by dividing the co- efficient by the span in feet, and subtracting the weight of the beam. 'TkEXTOtf AKGLfi-BAftS. 279 When the load is concentrated entirely at the centre of the span, one half of this amount must be taken. The beams must be secured against yielding sideways, or the safe loads will be much less. For beams used with plastered ceilings, the deflection allowed should not exceed 1/80 inch per foot of span, to avoid cracking of the plaster. TRENTON ANGLE-BARS. Size of Approximate Weight, in pounds per yard, for Coeff. for Transverse Strength. Bar. each thickness in inches. 7/16 V> 9/16 % 11/16 % 13/16 % Thinnest Bar. 6 x6 ft*. 5 64.3 71.1 77.8 84.4 91.0 97.3 36,900 lbs. 4^x4^2 fe° 42\5 716 47 5 52.3 9/16 57.2 61.9 11/16 18,000 " % 4 x4 28.6 33 1 37.5 41.8 46.1 50.5 54.4 12,184 " 3^x3^ 24.8 28.7 5/16 32.5 % 36.2 7/16 39.8 Yo, 43.4 9/16 9,200 " % 11/16 3 x3 14.4 17.7 21.1 24 4 27.5 30.6 33.6 36.5 4,611 " 5/16 % 13/32 7/16 15/32 % 17/32 9/1& 2Mx2% 16.2 19.2 20.7 22.2 23.6 25.0 26.3 27.7 4,710 " M 5/16 11/32 % 13/32 7/16 15/32 M, 2^x2^ 11.9 14.7 16.0 17.3 18.6 20.0 21.2 22.5 3,156 " M 9/32 5/16 11/32 % 13/32 7/16 2J4x2M 10.6 11.9 13.1 14.3 15.5 16.8 17.8 2,530 " 7/32 % 9/32 5/16 11/33 H 2 x2 8.3 9.4 10.4 11.5 12.6 13 6 1,752 " 3/16 7/32 Va 9/32 5/16 11/ ,2 % m*m 6.21 7.18 8.13 9 05 9.96 10 8 11.7 1,150 " 1%*1}4 5.27 6.09 6.88 7.64 8.40 9.13 832 " % 5/32 3/16 7/32 l A 1 xl 2.97 2.34 3.66 2.88 4 34 3.40 4.99 3.91 5.63 4 38 393 " 246 " %x % 2.03 1.72 3.48 2.09 2.93 2.46 186 '-' 133 " Uneven Legs. 5 x3J, 4^x3 4 x3 3^x3 3^x2^ 3^x1^ 3 x2^ 3 x2 7/16 41.8 35.3 30.9 40.0 35.0 9/16 53.1 44.7 39 % 58.6 49.2 43.0 11/16 64.0 53.7 46. S 69.4 58.1 50.6 % 30.5 26.7 14.4 5/16 20.9 24.8 7/16 28.7 32.5 9/16 36.2 % 39.8 11/16 43.4 19.3 17.7 23.0 21.1 26.5 24.4 30.0 27.5 33.4 30.6 36.7 33.6 40 36.5 13.1 H 11.9 5/16 16.2 7/32 10.4 il/32 17.7 % 11.9 19.2 9/32 13.3 13/3*2 20.7 5/16 14.6 7/16 20.0 9/16 27.7 22.5 { 30,680, ) 14,750, j 18,353, I 9,651, j 14,580, 1 7,020, j 9,850, 1 5,871, j 5,515, m 1,148, 4,490, 2Y2" 230 STRENGTH 0£ MATERIALS. TRENTON TEE BARS. Designation of Bar. Table. Leg. 4" x4" 3^"x3^" 2^"x2^"" 2" x 2" v 2H" . 2" <2y 2 " <2" «3" Approximate Weight, in pounds per yard, for each thickness in inches. 5-16" 5-16" 7-1" 5-32" 7 lbs. I 4 ' 5-16" 14.6 lbs. 5.6 Y 2 " 37.5 lbs. \4t" 27.5 ." %" 17.3 " 5-16" 11.5 " ' ' 5.5 ' 3-16' W 35.0 %" 17.3 Coefficient for Transverse Strength. Thinnest Bar. 15,800 ibs. 10,550 6,680 3,850 3,087 1.970 1,033 596 268 6,344 2,540 6,404 6,173 1,355 604 457 421 SIZE OF BEAMS, AND THEIR DISTANCE APART, Suitable for Floors having Loads per square foot from 100 lbs. to 300 lbs. (New Jersey Steel and Iron Co.) Load per ft. Load per ft. Load r Load per Load per sq. ft. sq. sq. sq. sq. ft. 250 lbs. .ja- 100 lbs. 150 lbs. 200 lbs. 300 lbs. il ^ ^ ^ _^ a s a is H If be '53 If .a bD '53 ©1 1 a p « c3 £ft 5 g o3 .2SO£ -aJ ta N ft Si 2 C 0," £ T3 o3 e8-H T3 «3 N ft *3 ° O m O CO O CO s CO P CO P in. lb. feet in. lb. feet in. lb. feet in. lb. feet in. lb. feet H 4 30 4.6 4 30 3.1 5 30 3.0 6 40 3.9 6 40 3 2 5 30 5,9 5 30 4.0 6 40 4.8 6 501 4.7 6 50J 3.0 5 50 3 9 i»i 5 30 3.8 6 40 4.1 6 40 3.0 7 55 3 3 5 40 4.8 6 50 5.0 6 50 3.7 7 551 4.0 8 65 4 4 iij 6 40 4.2 6 50 3.4 7 55 3.4 8 65| 3.6 8 65 a 6 50 5.2 7 55 4.6 8 65 4.5 9 70 4.5 9 70 3.8 14{ 7 55 5.0 7 55 3.3 8 65 3.3 9 70 3.3 9 85 3 3 8 65 6.7 8 65 4.5 9 70 4.1 10^ 90, 5.0 10^ 90 4 2 16 \ 8 65 5.0 8 65 3.3 9 85 3.7 10*6 90, 3.8 1<% 105 3.6 9 70 6.3 9 70 4.2 10V£ 90 4.7 10^ 105, 4.3 I214 125 4,8 H 9 70 4.9 9 85 3.9 10}^ 105 4.2 10^ 105; 3.4 12*4 125! 4.5 10^ 135 3 6 9 85 5.9 10J6 90 4.9 12 96 4.6 12J4 125 3 7 20 -j 10J^ 90 6.0 10^ 105 12J4 125 4.5 10^ 105 3.4 1234 125 3.6 I214 125 3 6.0 12i4 125 4.5 12J4 170 4.9 15 150 4.4 22 -j 10fc5 90 4.9 12 96 4.0 1214 125 3.7 12J4 1251 3.0 1234 170 8.3 lOfc 105 5.6 lSi/4 125 4.9 15 125 4.5 15 125! 3.6 15 150 3.6 u\ 12 96 5.0 I2J4 125 4.1 12J4 125 3.0 1214 170, 3.3 [5 150 3.0 1234 125 6.1 15 125 5.0 15 150 4.5 15 150, 3.6 [5 200 4.1 26--J 12J4 125 5.1 15 425 4.3 15 150 3.8 15 150, 3.0 i5 200 3.5 15 150 15 150 5.1 4.3 15 200 15 200 5.2 4.4 15 200 4.2 15 200! 3.5 20 200 20 200 4.7 28] 15 125 5 5 3 9 15 200 15 150 5.9 3.7 20 200 15 200 6.0 3.8 20 200! 4.8 20 200 4.1 20 272 20 200 5 3 30 1 15 150 5.6 3.4 15 .200 5.1 20 200 5.2 20 2T2 5.5 20 272 4 6 1 i TORSIONAL STRENGTH. 281 FLOORING MATERIAL. For fire-proof flooring, the space between the floor-beams may be spanned with brick arches, or with hollow brick made especially for the purpose, the latter being much lighter than ordinary brick. Arches 4 inches deep of solid brick weigh about 70 lbs. per square foot, including the concrete levelling material, and substantial floors are thus made up to 6 feet span of arch, or much greater span if the skew backs at the springing of the arch are made deeper, the rise of the arch being prefer- ably not less than 1/10 of the span. Hollow brick for floors are usually in depth about % of the span, and. are used up to, and even exceeding, spans of 10 feet. The weight of the latter material will vary from 20 lbs. per square foot for 3-foot spans up to 60 lbs. per square foot for spans of 10 feet. Full particulars of this construction are given by the manufacturers. For supporting brick floors the beams should be securely tied with rods to resist the lateral pressure. In the following cases the loads, in addition to the weight of the floor itself, may be assumed as: For street bridges for general public traffic 80 lbs. per sq. ft. For floors of dwellings 40 lbs. " " For churches, theatres, and ball-rooms 80 lbs. " " For hay-lofts 80 lbs. " " For storage of grain 100 lbs. " " For warehouses and general merchandise 250 lbs. " " For factories . 200 to 400 lbs. " " For snow thirty inches deep . ... 16 lbs. " " For maximum pressure of wind 50 lbs. " " For brick walls 112 lbs. per cu. ft. For masonry walls 116-144 lbs. " " Roofs, allowing thirty pounds per square foot for wind and snow: For corrugated iron laid directly on the purlins. . . 37 lbs. per sq. ft. For corrugated iron laid on boards 40 lbs. " " For slate nailed to laths 43 lbs. " " For slate nailed on boards 46 lbs. " " If plastered below the rafters, the weight will be about ten pounds per square foot additional. TIE-RODS FOR REAMS SUPPORTING BRICK ARCHES. The horizontal thrust of brick arches is as follows: 1 5TFS 2 -^—^ — = pressure in pounds, per lineal foot of arch: W — load in pounds, per square foot; 8 — span of arch in feet; B = rise in inches. Place the tie-rods as low through the webs of the beams as possible and spaced so that the pressure of arches as obtained above will not produce a greater stress than 15,000 lbs. per square inch of the least section of the bolt. TORSIONAL STRENGTH. Let a horizontal shaft of diameter — d be fixed at one end, and at the other or free end, at a distance = I from the fixed end, let there be fixed a horizontal lever arm with a weight — P acting at a distance = a from the axis of the shaft so as to twist it; then Pa = moment of the applied force. Resisting moment = twisting moment = — , in which S = unit shearing resistance, J = polar moment of inertia of the section with respect to the axis, and c = distance of the most remote fibre from the axis, in a cross- section. For a circle with diameter d, ^--55-; c = %d; 32 ' s; <* = i/- 282 STRENGTH OF MATERIALS. For hollow shafts of external diameter d and internal diameter d lv d = 3 / 5 -l-Pa For a square whose side = d, J=±; c = 4VU; f =Pa =1 ^!=0. 28M » S . For a rectangle whose sides are 6 and d, j=^ + ™. c = y 2 Vi* + d>; *U J b = < M ' + '»9g . 12 12 c 6 |/62 + d 2 The above formulae are based on the supposition that the shearing resist- ance at any point of the cross-section is proportional to its distance from the axis; but this is true only within the elastic limit. In materials capable of flow, while the particles near the axis are strained within the elastic limit those at some distance within the circumference may be strained nearly to the Ultimate resistance, so that the total resistance is something greater than that calculated by the formulae. (See Thurston. " Matls. of Eng.," Part II. p. 527.) Saint Venant finds for square shafts Pa = 0.281d 3 S (Rankine, "Mach. and Millwork," p. 504). For working strength, however, the for- mulae may be used, with S taken at the safe working unit resistance. The ultimate torsional shearing resistance Maximum tensile (or compressive) unit stress. Combined Flexure and Torsion.— If S — greatest unit stress due to flexure alone, and 3 24 r 2 where r = radius of the plate. Dr. Grashof gives a formula from which we have the following rule: < 2 X 72,000 This formula of Grashof's has been adopted by Professor Unwin in his "Elements of Machine Design." These formulae by Rankine and Grashof may be regarded as being practically the same. On trying to make the rules given by these authorities agree with the results of his experience of the strength of unstayed 'flat ends of cylindrical boilers and domes that had given way after long use, Mr. Wilson was led to believe that the above rules give the breaking strength much lower than it STRENGTH OF FLAT PLATES. 285 actually is. He describes a number of experiments made by Mr. Nichols of Kirkstall, which gave results varying widely from each other, as the method of supporting the edges of the plate was varied, and also varying widely from the calculated bursting pressures, the actual results being in all cases very much the higher. Some conclusions drawn from these results are: 1. Although the bursting pressure has been found to be so high, boiler- makers must be warned against attaching any importance to this, since the plates deflected almost as soon as any pressure was put upon them and sprang back again on the pressure being taken off. This springing of the plate in the course of time inevitably results in grooving or channelling, which, especially when aided by the action of the corrosive acids in the water or steam, will in time reduce the thickness of the plate, and bring about the destruction of an unstayed surface at a very low pressure. 2. Since flat plates commence to deflect at very low pressures, they should never be used without stays; but it is better to dish the plates when they are not stayed by flues, tubes, etc. 3. Against the commonly accepted opinion that the limit of elasticity should never be reached in testing a boiler or other structure, these experi- ments show that an exception should be made in the case of an unstayed flat end-plate of a boiler, which will be safer when it has assumed a perma- nent set that will prevent its becoming grooved by the continual variation of pressure in working. The hydraulic pressure in this case simply does what should have been done before the plate was fixed, that is, dishes it. 4. These experiments appear to show that the mode of attaching by flange or by an inside or outside angle-iron exerts an important influence on the manner in which the plate is strained by the pressure. When the plate is secured to an angle-iron, the stretching under pressure is, to a certain extent, concentrated at the line of rivet-holes, and the plate par- takes rather of a beam supported than fixed round the edge. Instead of the strength increasing as the square of the thickness, when the plate is attached by an an^le-iron, it is probable that the strength does not increase even directly as the thickness, since the plate gives way simply by stretching at the rivet-holes, and the thicker the plate, the less uniformly is the strain borne by the different layers of which the plate may be considered to be made up. When the plate is flanged, the flange becomes compressed by the pressure against the body of the plate, and near the rim, as shown by the contrary flexure, the inside of the plate is stretched more than the outside, and it may be by a kind of shearing action that the plate gives way along the line where the crushing and stretching meet. 5. These tests appear to show that the rules deduced from the theoretical investigations of Lame, Rankine, and Grashof are not confirmed by experi- ment, and are therefore not trustworthy. Unbraced Wrought-iron Heads of Boilers, etc. {The Loco- motive, Feb. 1890). — Few experiments have been made on the strength of flat heads, and our knowledge of them comes largely from theory. Experi- ments have been made on small plates 1-16 of an inch thick, yet the data so obtained cannot be considered satisfactory when we consider the far thicker heads that are used in practice, although the results agreed well with Ran- kine's formula. Mr. Nichols has made experiments on larger heads, and from them he has deduced the following rule: " To find the proper thick- ness for a flat unstayed head, multiply the area of the head by the pressure per square inch that it is to bear safely, and multiply this by the desired factor of safety (say 8); then divide the product by ten times the tensile strength of the material used for the head." His rule for finding the burst- ing pressure when the dimensions of the head are given is: " Multiply the thickness of the end-plate in inches by ten times the tensile strength of the material used, and divide the product by the area of the head in inches." In Mr. Nichols's experiments the average tensile strength of the iron used for the heads was 44,800 pounds. The results he obtained are given below, with the calculated pressure, by his rule, for comparison. 1. An unstayed flat boiler-head is 34^ inches in diameter and 9-16 inch thick. What is its bursting pressure? The area of a circle 34J^ inches in diameter is 935 square inches; then 9-1(5 X 44,800 x 10 = 252,000, and 252,000 -*- 935 = 270 pounds, the calculated bursting pressure. The head actually burst at 280 pounds. 2. Head 34^ inches in diameter and % inch thick. The area = 935 square inches; then, % x 44,800 x 10 = 168.000, and 168,000 -h 935 = 180 pounds, calculated bursting pressure. This head actually burst at 200 pounds. 286 BTEEKGTH 0£ MATERIALS. 3. Head 26J4 inches in diameter, and % inch thick. The area 541 square inches. Then, % x 41,800 x 10 = 168,000, and 168,000 -h 541 = 311 pounds. This head burst at 370 pounds. 4. Head 28J^ inches in diameter and % inch thick. The area = 638 square inches; then, % x 44,800 X 10 = 168,000, and 168,000 -f- 638 = 263 pounds. The actual bursting pressure was 300 pounds. In the third experiment, the amount the plate bulged under different pressures was as follows : At pounds per sq. in.... 10 20 40 80 120 140 170 200 Plate bulged 1/32 1/16 Y 8 U % Yn % % The pressure was now reduced to zero, " and the end sprang back 3-16 inch, leaving it with a permanent set of 9-16 inch. The pressure of 200 lbs. was again applied on 36 separate occasions during an interval of five days, the bulging and permanent sec being noted on each occasion, but without any appreciable difference from that noted above. The experiments described were confined to plates not widely different in their dimensions, so that Mr. Nichols's rule cannot be relied upon for heads that depart much from the proportions given in the examples. Thickness of Flat Cast-iron Plates to resist Bursting Pressures. — In Church's Life of Ericsson is found the following letter: '* My dear Sir: The proper thickness of a square cast-iron plate will be ob- tained by following: Multiply the side in feet (or decimals of a foot) by J4 of the pressure in pounds and divide by 850 times the side in inches; the quotient is the square of the thickness in inches. " Example.— A plate 5 feet or 60 inches square, with a pressure of 30 lbs. per square inch. " Thickness - 5 X %* jf?* 3 ° - 2.64. a/2M = 1.62 inches. 85(1 X w ' " For a circular plate, multiply 11-14 of the diameter in feet by J4 of the pressure on the plate in pounds. Divide by 850 times 11-14 of the diameter in inches. [Extract the square root.] " Example.— Plate 5 feet diameter, pressure 30 lbs. per square inch. " Area 2827 X — = 8 -^^ = 21,202; diam. 60 x — = 47.1; 5 x — = 3.92. 4 4 14 14 3.92 X 21,202 = 83,811 8.50 X 4.71 = 41,035 = " A great mathematician would cover half a dozen sheets with figures to solve this problem." Strength of Stayed Surfaces.— A flat plate of thickness t is sup- ported uniformly by stays whose distance from centre to centre is a, uniform load p lbs. per square inch. Each stay supports pa* lbs. The greatest stress on the plate is . 2a2 /TT . f= 9 JiP- (Unwin). SPHERICAL SHELLS AND DOMED BOILER-HEADS. To find the Thickness of a Spherical Shell to resist a given Pressure.— Let d = diameter in inches, and p the internal press- ure per square inch. The total pressure which tends to produce rupture around the great circle will be \fcrd' l p. Let S = safe tensile stress per square inch, and t the thickness of metal in inches; then the resistance to the pressure will be irdtS. Since the resistance must be equal to the pressure. Y^d^p = ndtS. Whence t = ||-. The same rule is used for finding the thickness of a hemispherical head to a cylinder, as of a cylindrical boiler. Thickness of a Domed Head of a boiler.— If S = safe tensile stress per square inch, d = diameter of the shell in inches, and t = thickness of the shell, t = pd -s- 2S ; but the thickness of a hemispherical head of the same diameter is t = pd-r-4S. Hence if we make the radius of curvature of a domed head equal to the diameter of the boiler, we shall have t = — = — , or the thickness of such a domed head will be equal to the thick- 4S 2S ness of the shell. THICK CYLINDERS UNDER TENSION. 287 Stresses in Steel Plating due to Water-pressure, as in plating of vessels and bulkheads (Engineering, May 22, 1891, page 629). Mr. J. A. Yates has made calculations of the stresses to which steel plates are subjected by external water-pressure, and arrives at the following con- clusions : Assume 2a inches to be the distance between the frames or other rigid supports, and let d represent the depth in feet, below the surface of the water, of the plate under consideration, t = thickness of plate in inches, D the deflection from a straight line under pressure in inches, and P — stress per square inch of section. For outer bottom and ballast- tank plating, a = 4 , D should not be greater than .05 — , and — not greater than 2 to 3 tons ; while for bulkheads, etc., a = 2352-, D should not be greater than .1—, and — not greater than 7 tons. To illustrate the application of these formulae the following cases have been taken : For Outer Bottom, etc. For Bulkheads, etc. Thick- Depth Spacing of Thick- Depth of Water. Maximum Spac- ness of below Frames should ness of ing of Rigid Plating. Water. not exceed Plating Stiffeners. ft. in. in. ft. ft. in. H 20 About 21 \ 20 9 10 10 " 42 20 7 4 §1 18 " 18 10 14 8 % 9 " 36 y* 20 4 10 8 10 " 20 H 10 9 8 5 " 40 % 10 4 10 It would appear that the course which should be followed in stiffening bulkheads is to fit substantially rigid stiffening frames at comparatively wide intervals, and only work such light angles between as are necessary for making a fair job of the bulkhead, THICK HOLLOW CYLINDERS UNDER TENSION. Burr, " Elasticity and Resistance of Materials," p. 36, gives . t — thickness; r = interior radius ; ( A + A 5 1 I ft = maximum allowable hoop tension at the t = r < \T~-z). — * \ ' interior of the cylinder; { p p = intensity of interior pressure. Merriman gives s = unit stress at inner edge of the annulus; r = interior radius ; t = thickness ; I = length. (1) The total interior pressure which tends to rupture the cylinder is 2rl — p. If p be the unit pressure, then p = ^rqTf f rom wmcn one of tn e quantities s, p, r, or t can be found when the other three are given. P)t . t _ rp P' t = - 288 STRENGTH OF MATERIALS. In eq. (1), if t be neglected in comparison with r, it reduces to 2slt, which is the same as the formula for thin cylinders. If t = r, it becomes sit, or only half the resistance of the thin cylinder. The formulas given by Burr and by Merriman are quite different, as will be seen by the following example : Let maximum unit stress at the inner edge of the annulus = 8000 lbs. per square inch, radius of cylinder = 4 inches, interior pressure = 4000 lbs. per square inch. Required the thickness. BrBurr, t^K^^)* - . \ = ^ - .,- I.H. inches. ~ ™ . 4 X 4000 , . . By Mernman, t = mQ _ mQ = 4 inches. Limit to Useful Thickness of Hollow Cylinders (Eng'g, Jan. 4, 1884).— Professor Barlow lays down the law of the resisting powers of thick cylinders as follows : " In a homogeneous cylinder, if the metal is incompressible, the tension on every concentric layer, caused by an internal pressure, varies inversely as the square of its distance from the centre. 1 ' Suppose a twelve-inch gun to have walls 15 inches thick. Pressure on exterior _ 6*_ _ _ Pressure on interior ~ 21 2 ~~ So that if the stress on the interior is 12J4 tons per square inch, the stress on the exterior is only 1 ton. Let s = the stress on the inner layer, and s, that at a distance x from the axis ; r = internal radius, R = external radius. r 2 5, : s : : r 2 : a; 2 , or S* = s — . 1 x* The whole stress on a section 1 inch long, extending from the interior to R — r the exterior surface, is £ a . o w C w a 5* . ax CD £ a © be £-2^3 fi"" 1 &-~ _Q.« jQ" H XoSO 43 "3 AM £J3 © CO 8 ST o o 1 1" o c? o m © & © lbs. lbs. lbs. lbs. lbs. lbs. y* 13 .38 .12 350 460 580 810 1160 5800 9-16 12 .44 .15 450 600 750 1050 1500 7500 % 11 .49 .19 560 750 930 1310 1870 9000 u 10 .60 .28 750 1130 1410 1980 2830 14000 Vs 9 .71 .39 1180 1570 1970 2760 3940 19000 l 8 .81 .52 1550 2070 2600 3630 5180 25000 m 7 .91 .65 1950 2600 3250 4560 6510 30000 m 7 1.04 .84 2520 3360 4200 5900 8410 39000 Ws 6 1.12 1.00 3000 4000 5000 7000 10000 46000 w% 6 1.25 1.23 3680 4910 6140 8600 12280 56000 w% 5^ 1.35 1.44 4300 5740 7180 10000 14360 65000 m Ws 5 1.45 1.65 4950 6600 8250 11560 16510 74000 5 1.57 1.95 5840 7800 9800 13640 19500 85000 2 4^ 1.66 2.18 6540 8720 10900 15260 21800 95000 m 4^ 1.92 2.88 8650 11530 14400 20180 28800 125000 n 4 2.12 3.55 10640 14200 17730 24830 35500 150000 4 2.37 4.43 13290 17720 22150 31000 44300 186000 3 &A 2.57 5.20 15580 20770 26000 36360 52000 213000 &A 3^ 3.04 7.25 21760 29000 36260 50760 72500 290000 4 3 3.50 9.62 28860 38500 48100 67350 96200 385000 When it is known what load is to be put upon a bolt, and the judgment of the engineer has determined what stress is safe to put upon the iron, look down in the proper column of said stress until the required load is found. The area at the bottom of the thread will give the equivalent area of a flat bar to that of the bolt. Effect of Initial Strain in Bolts.— Suppose that bolts are used to connect two parts of a machine and that they are screwed up tightly be- fore the effective load comes on the connected parts. Let P 3 = the initial tension on a bolt due to screwing up, and P 2 = the load afterwards added. The greatest load may vary but little from Pj or P 2 , according as the former or the latter is greater, or it may approach the value P x -f- P^ de- pending upon the relative rigidity of the bolts and of the parts connected. Where rigid flanges are bolted together, metal to metal, it is probable that the extension of the bolts with any additional tension relieves the initial tension, and that the total tension is Pj or P 2 , but in cases where elastic packing, as india rubber, is interposed, the extension of the bolts may very little affect the initial tension, and the total strain may be nearly P x + P 2 . Since the latter assumption is more unfavorable to the resistance of the bolt, this, contingency should usually be provided for. (See Unwin, "Ele- ments of Machine Design " for demonstration.) STAND-PIPES AND THEIR DESIGN. (Freeman C. Coffin, New England Water Works Assoc, Eng. News, March 16, 1893.) See also papers by A. H. Howland, Eng. Club of Phil. 1887; B. F. Stephens, Amer. Water Works Assoc, Eng. News, Oct. 6 and 13, 1888; W. Kiersted, Rensselaer Soc of Civil Eng., Eng'g Record, April 25 and May 2, 1891, and W. D. Pence, Eng. Neivs, April and May, 1894. The question of diameter is almost entirely independent of that of height. The efficient capacity must be measured by the length from the high-water line to a point below which it is undesirable to draw the water on account of loss of pressure for fire-supply, whether that point is the actual bottom of the stand-pipe or above it. This allowable fluctuation ought not to exceed 50 ft., in most cases. This makes the diameter dependent upon two condi- STAND-PIPES AND THEIR DESIGN. 293 tions, the first of which is the amount of the consumption during the ordi- nary interval between the stopping and starling of the pumps. This should never draw the water below a point that will give a good fire stream and leave a margin for still further draught for fires. The second condition is the maximum number of fire streams and their size which it is considered necessary to provide for, and the maximum length of time which they are liable to have to run before the pumps can be relied upon to reinforce them. Another reason for making the diameter large is to provide for stability against wind-pressure when empty. The following table gives the height of stand-pipes beyond which they are not safe against wind pressures of 40 and 50 lbs. per square foot. The area of surface taken is the height multiplied by one half the diameter. Heights of Stand-pipe that will Resist Wind-pressure l>y its Weight alone, when Empty. Diameter, Wind, 40 lbs. Wind, 50 lbs. feet. per sq. ft. per sq. ft. 20 45 35 25 ... 70 55 30 150 80 35 160 To have the above degree of stability the stand-pipes must be designed with the outside angle-iron at the bottom connection. Any form of anchorage that depends upon connections with the sid3 plates near the bottom is unsafe. By suitable guys the wind-pressure is re- sisted by tension in the guys, and the stand-pipe is relieved from wind strains that tend to overthrow it. The guys should be attached to a band of angle or other shaped iron that completely encircles the tank, and rests upon some sort of bracket or projection, and not be riveted to the tank. They should be anchored at a distance from the base equal to the height of the point at which they are attached, if possible. The best plan is to build the stand-pipe of such diameter that it will resist the wind by its own stability. Thickness of the Side Plates. The pressure on the sides is outward, and due alone to the weight of the water, or pressure per square inch, and increases in direct ratio to the height, and also to the diameter. The strain upon a section 1 inch in height at any point is the total strain at that point divided bj r two — for each side is supposed to bear the strain equally. The total pressure at any point is equal to the diameter in inches, multiplied by the pressure per square inch, due to the height at that point. It may be expressed as follows: H = height in feet, and/ = factor of safety; d — diameter in inches; p — pressure in lbs. per square inch; .434 = p for 1 ft. in height; s = tensile strength of material per square inch; T — thickness of plate. Then the total strain on each side per vertical inch _ AMHd _pd_ .AMHdf _ pdf_ ~ 2 ' ~ 2 ' ~ 2s ~ 2s ' Mr. Coffin takes/ = 5. not counting reduction of strength of joint, equiv- alent to an actual factor of safety of 3 if the strength of the riveted joint is taken as 60 per cent of that of the plate. The amount of the wind strain per square inch of metal at any joint can be found by the following formula, in which H = height of stand-pipe in feet above joint; T = thickness of plate in inches; p — wind-pressure per square foot: W — wind-pressure per foot in height above joint; W = Dp where D is the diameter in feet; m — average leverage or movement about neutral axis or central points in the circumference; or, m = sine of 45°, or .707 times the radius in feet. 294 STRENGTH OF MATERIALS. Then the strain per square inch of plate (Hrv)~ circ. in ft. X mT Mr. Coffin gives a number of diagrams useful in the design of stand-pipes, together with a number of instances of failures, with discussion of their probable causes. Mr. Kiersted's paper contains the following : Among the most prominent strains a stand-pipe has to bear are: that due to the static pressure of the water, that due to the overturning effect of the wind on an empty stand- pipe, and that due to the collapsing effect, on the upper rings, of violent wind storms. For the thickness of metal to withstand safely the static pressure of water, let t — thickness of the plate iron in inches; H = height of stand-pipe in feet; D = diameter of stand-pipe in feet. Then, assuming: a tensile strength of 48.000 lbs. per square inch, a factor of safety of 4, and efficiency of double-riveted lap-joint equalling 0.6 of the strength of the solid plate, (-.MJoaurxi* *=4S«; d.bD which will give safe heights for thicknesses up to % to % of an inch. The same formula may also apply for greater heights and thicknesses within practical limits, if the joint efficiency be increased by triple riveting. The conditions for the severest overturning wind strains exist when the stand-pipe is empty. Formula for Rand-pressure of 50 pounds per square foot, when d = diameter of stand-pipe in inches; x = any unknown' height of stand-pipe; x = VWndt = 15.85 VdT. The following table is calculated by these formulae. The stand-pipe is intended to be self-sustaining; that is, without guys or stiffeners. Heights of Stand-pipes for Various Diameters and Thicknesses of Plates. Thickness of Diameters in Feet. Plate in Frac- tions of an Inch. 5 6 7 8 9 10 12 14 15 16 18 20 25 3-16 7 32 50 55 CO 70 75 SO 85 55 60 65 55 05 75 90 100 110 115 125 130 50 00 70 85 100 115 120 130 135 145 150 35 50 55 70 85 100 115 130 145 155 1G5 40 50 60 75 85 100 110 120 135 145 100 ' 40 45 55 70 80 90 100 115 125 135 150 160 "40 50 65 75 85 95 105 120 130 140 150 160 "35 45 55 05 75 85 95 105 115 125 135 145 155 35 40 50 00 70 80 85 95 105 110 120 130 140 4-16 65 75 SO 90 95 70 SO 90 95 100 75 85 95 100 110 115 25 5 16 35 6 16 40 7-16 S--16 45 55 9 16 60 10-16 65 11-16 75 12-16 13 16 80 qo 14 16 95 15 16 105 16-16 no Heights to nearest 5 feet. Rings are to build 5 feet vertically. Failures of Stand-pipes have been numerous in recent years. A list showing 23 important failures inside of nine years is given in a paper by Prof. W. D. Pence, Eng'g News, April 5, 12, 19 and 26, May 3, 10 and 24, and June 7. 1894. His discussion of the probable causes of the failures is most valuable. WROUGHT-IRON AND STEEL WATER-PIPES. 295 Kenneth Allen, Engineers Club of Philadelphia, 1886, gives the following rules for thickness of plates for stand pipes. Assume: Wrought iron plate T. S. 48,000 pounds in direction of fibre, and T. S. 4 r »,000 pounds across the fibre. Strength of single riveted joint .4 that of the plate, and of double riveted joint, .7 that of the plate ; wind pressure = 50 pounds per square foot ; safety factor = 3. Let /(. = total height in feet ; r — outer radius in feet ; r' = inner radius in feet ; p — pressure per square inch ; t = thickness in inches ; d — outer diameter in feet. Then for pipe filled and longitudinal seams double riveted t -. pr x 12 48,000 X.7XH hcl . : 4301' and for pipe empty and lateral seams, single riveted, we have by equating I moments : 3 X 2f (|)» = 144 X 6000 (r 4 - ?•"») , whence r 4 — r' 4 = 27144" Table showing required Thickness of Bottom Plate. Diameter. Height in Feet. 5 feet, 10 feet. 15 feet. 20 feet. 25 feet. 30 feet. 50 t 7-64* H * 11-64* 15-64 19-64 23-64 60 +11-64* 9-64* 7-32 9-32 23-64 27-64 70 + 7-32 11-64* 9-32 21-64 13-32 31-64 80 +19-64 3-16 % 15-32 9-16 90 t % 7-32 5-16 27-64 17-32 % 100 +29-64 +15-64 23-64 15-32 37-64 45-64 •125 +23-64 7-16 37-64 47-64 Vs 150 +33-64 17-32 45-64 % 1 3-64 175 +11-16 39-64 13-16 1 1-32 1 7-32 200 +29-32 45-64 15-16 1 11-64 1 25-64 * The minimum thickness should = 3-16". N.B.— Dimensions marked + determined by wind-pressure. Water Tower at Xonkers, N. Y.— This tower, with a pipe 122 feet high and 20 feet diameter, is described in Engineering Neivs, May 18, 1892. The thickness of the lower rings is 11-16 of an inch, based on a tensile strength of 60,000 lbs. per square inch of metal, allowing 65$ for the strength of riveted joints, using a factor of safety of 3}^ and adding a constant of % inch. The plates diminish in thickness by 1-16 inch to the last four plates at the top, which are J4 i'icli thick. The contract for steel requires an elastic limit of at least 33,000 lbs. per square inch ; an ultimate tensile strength of from 56,000 to 66,000 lbs. per square inch ; an elongation in 8 inches of at least 20%, and a reduction of area of at least 45%. The inspection of the work was made by the Pittsburgh Testing Laboratory. According to their report the actual conditions de- veloped were as follows : Elastic limit from 34,020 to 39,420 ; the tensile strength from 58,330 to 65,390 ; the elongation in 8 inches from 22J4 to 32% ; reduction in area from 52.72 to 71.32% ; 17 plates out of 141 were rejected in the inspection. WBOUGHT'IRON AND STEEL, WATER-PIPES. Riveted Steel Water-pipes {Engineering News, Oct, 11, 1890, and Aug. 1, 1891.)— The use of riveted wrought-iron pipe has been common in the' Pacific States for many years, the largest being a 44-inch conduit in connection with the works of' the Spring Valley Water Co., which supplies San Francisco. The use of wrought iron and steel pipe has been neces- sary in the West, owing to the extremely high pressures to be withstood and the difficulties of transportation. As an example : In connection with 296 STRENGTH OF MATERIALS. the water supply of Virginia City and Gold Hill, Nev., there was laid in 1872 an llj^-inch riveted wrought-iron pipe, a part of which is under a head of 1720 feet". Irs the East, the most important example of the use of riveted steel water pipe is that of the East Jersey Water Co., which supplies the city of Newark. The contract provided for a maximum high service supply of 25,000,000 gal- lons daily. In this case 21 miles of 48-inch pipe was laid, some of it under 340 feet head. The plates from which the pipe is made are about 13 feet long by 7 feet wide, open-hearth steel. Four plates are used to make one section of pipe about 27 feet long. The pipe is riveted longitudinally with a double row, and at the end joints with a single row of rivets of varying diameter, corresponding to the thickness of the steel plates. Before being rolled into the trench, two of the 27-feet lengths are riveted together, thus diminishing still further the number of joints to be made in the trench and the extra excavation to give room for jointing. All changes in the grade of the pipe- line are made by 10° curves and all changes in line by 2^£, 5, 7J^ and 10° curves. To lay on curved lines a standard bevel was used, and the different curves are secured by varying the number of beveled joints used on a certain length of pipe. The thickness of the plates varies with the pressure, but only three thick- nesses are used, J4, 5-16, and % inches, the pipe made of these thicknesses having a weight of 160, 185, and 225 lbs. per foot, respectively. At the works all the pipe was tested to pressure \y% times that to which it is to be sub- jected when in place. Mamiesmaim Tubes for High Pressures.— At the Mannes- maiin Works at Komotau, Hungary, more than 600 tons or 25 miles of 3-iuch and 4-inch tubes averaging J4 inch in thickness have been successfully tested to a pressure of 2000 lbs. per square inch. These tubes were intended for a high-pressure water-main in a Chilian nitrate district. This great tensile strength is probably due to the fact that, in addition to being: much more worked than most metal, the fibres of the metal run spirally, as has been proved by microscopic examination. While cast-iron tubes will hardly stand more than 200 lbs. per square inch, and welded tubes are not safe above 1000 lbs. per square inch, the Mannesmann tube easily withstands 2000 lbs. per square inch. The length up to which they can be readily made is shown by the fact that a coil of 3-inch tube 70 feet long was made recently. For description of the process of making Mannesmann tubes see Trans. A. I. M. E , vol. xix., 384. STRENGTH OF VARIOUS M ATERIALS. EXTRACTS FROM KIRKALDY'S TESTS. The recent publication, in a book by W. G. Kirkaldy, of the results of many thousand tests made during a quarter of a century by his father, David Kir- kaldy, has made an important contribution to our knowledge concerning the range of variation in strength of numerous materials. A condensed abstract of these results was published in'the American Machinist, May 11 and 18, 1893, from which the following still further condensed extracts are taken: The figures for tensile and compressive strength, or, as Kirkalds* calls them, pulling and thrusting stress, are given in pounds per square inch of original section, and for bending strength in pounds of actual stress or pounds per BD 2 (breadth x square of depth) for length of 36 inches between supports. The contraction of area is given as a percentage of the original area, and the extension as a percentage in a length of 10 inches, except when otherwise stated. The abbreviations T. S., E. L., Contr., and Ext. are used for the sake of brevity, to represent tensile strength, elastic limit, and per- centages of contraction of area, and elongation, respectively. Cast Iron.— 44 tests: T. S. 15.468 to 28,740 pounds; 17 of these were un- sound, the strength ranging from 15,468 to 24,357 pounds. Average of all, 23,805 pounds. Thrusting stress, specimens 2 inches long, 1.34 to 1.5 in. diameter: 43 tests, all sound, 94,352 to 131,912; one, unsound, 93,759; average of all, 113,825. Bending stress, bars about 1 in. wide by 2 in deep, cast on edge. Ulti- mate stress 2876 to 3854; stress per BD 2 — 725 to 892; average, 820. Average modulus of rupture, R, = stress per BD 2 X length, = 29,520. Ultimate de- flection, .29 to .40 in.; average .34 inch. Other tests of cast iron, 460 tests, 16 lots from various sources, gave re-. EXTRACTS FROM KIRK ALB Y*S TESTS. 29? suits with total range as follows: Pulling stress, 12,688 to 33,616 pounds; thrusting stress, 66,363 to 175,950 pounds; bending stress, per i>'D 2 , 505 to 1128 pounds; modulus of rupture, R, 18,180 to 40,608. Ultimate deflection, .21 to .45 inch. The specimen which was the highest in thrusting stress was also the high- est in bending, and showed the greatest deflection, but its tensile strength was only 26,502. The specimen with the highest tensile strength had a thrusting stress of 143,939, and a bending strength, per 7?D 2 , of 979 pounds with 0.41 deflection. The specimen lowest in T. S. was also lowest in thrusting and bending, but gave .38 deflection. The specimen which gave .21 deflection had T. S., 19,188: thrusting. 101.281; and bending, 561. Iron Castings.— 69 tests; tensile strength, 10,416 to 31,652; thrusting stress, ultimate per square inch, 53,502 to 132,031. Channel Irons.— Tests of 18 pieces cut from channel irons. T. S. 40,693 to 53,141 pounds per square inch; contr. of area from 3.9 to 32.5 %. Ext. in 10 in. from 2.1 to 22.5 %. The fractures ranged all the way from 100 % fibrous to 100^ crystalline. The highest T. S., 53,141, with 8.1 % contr. and 5.3 % ext., was 100 $ crystalline; the lowest T. S.,- 40,693, with 3.9 contr. and 2.1 % ext., was 75 % crystalline. All the fibrous irons showed from 12.2 to R.5* ext., 17.3 to 32.5 contr., and T. S. from 43,426 to 49,615. The fibrous irons are therefore of medium tensile strength and high ductility. The crystalline irons are of variable T. S., highest to lowest, and low ductility. Lowmoor Iron Bars.— Three rolled bars 2% inches diameter; ten- sile tests: elastic, 23,200 to 24,200; ultimate, 50,875 to 51,905; contraction, 44.4 to 42.5; extension, 29.2 to 24.3. Three hammered bars, 414 inches diameter, elastic 25,100 to 24,200; ultimate, 46,S10 to 49,223; contraction, 20.7 to 46.5-; extension, 10.8 to 31.6. Fractures of all, 100 per cent fibrous. In the ham- mered bars the lowest T. S. was accompanied by lowest ductility. Iron Bars, Various.— Of a lot of 80 bars of various sizes, some rolled and some hammered (the above Lowmoor bars included) the lowest T. S. (except one) 40,808 pounds per square inch, was shown by the Swedish "hoop L " bar 314 inches diameter, rolled. Its elastic limit was 19,150 pounds; contraction 68.7 % and extension 37.7$ in 10 inches. It was also the most ductile of all the bars tested, and was 100 % fibrous. The highest T. S., 60,780 pounds, with elastic limit, 29,400; contr., 36.6; and ext., 24.3 %, was shown by a " Farnley " 2-inch bar, rolled. It was also 100 % fibrous. The lowest ductility 2M% contr., and 4.1 % ext., was shown by a 3%-inch hammered bar, without brand. It also had the lowest T. S., 40,278 pounds, but rather high elastic limit, 25,700 pounds. Its fracture was 95 % crystal- line. Thus of the two bars showing the lowest T. S., one was the most duc- tile and the other the least ductile in the whole series of 80 bars. Generally, high ductility is accompanied by low tensile strength, as in the Swedish bars, but the Farnley bars showed a combination of high ductility and high tensile strength. Locomotive Forgings, Iron.— 17 tests: average, E. L., 30,420; T. S., 50.521; contr., 36.5: ext. in 10 inches, 23.8. Broken Anchor Forgings, Iron.— 4 tests: average, E. L., 23,825; T. S , 40,083; contr., 3.0; ext. in 10 inches, 3.8. ■ Kirkaldy places these two irons in contrast to show the difference between good and bad work. The broken anchor material, he says, is of a most treacherous character, and a disgrace to any manufacturer. Iron Plate Girder. —Tensile tests of pieces cut from a riveted iron girder after twenty years 1 service in a railway bridge. Top plate, average of 3 tests, E. L., 26,600; T. S., 40,806; contr. 16 1; ext. in 10 inches, 7.8. Bottom plate, average of 3 tests, E. L., 31,200; T. S., 44,288; contr., 13.3; ext. in 10 inches, 6.3. Web-plate, average of 3 tests, E. L., 28,000; T. S., 45,902; contr., 15 9; ext. in 10 inches, 8.9. Fractures all fibrous. The results of 30 lists from different parts of the girder prove- that the iron has undergone «o change during twenty years of use. Steel Plates.— Six plates 100 inches long, 2 inches wide, thickness vari- ous, .36 to .97 inch T. S., 55,485 to 60,805; E. L., 29,600 to 33,200; contr., 52.9 to 59.5; ext.. 17.05 to 18.57. Steel Bridge Links.— 40 links from Hammersmith Bridge, 1886. 298 STREHGTH -OF MATERIALS. 02 Eh" £ H u o o 1 a Fracture. c 67.294 60,753 75,936 64,044 63,745 65,980 63,980 38,294 36,030 44,166 32,441 38,118 36,792 39,017 u.bf. 30.1 31.2 34.7 52.8 40.8 6.0 14.11$ 15.51 12.42 13.43 15.46 17.78 6.62 15 30 100 35 70$ Highest T. S. and E. L Lowest E. L Greatest Contraction Greatest Extension Least Contr. and Ext 85 70 65 100 The ratio of elastic to ultimate strength ranged from 50.6 to 65.2 per cent; average, 56.9 per cent. Extension in lengths of 100 inches. At 10,000 lbs. per sq. in., .018 to .024; mean, .0^0 inch; at 20.000 lbs. per sq. in. .049 to .063; mean, .055 inch; at 30,000 lbs. per sq. in., .083 to .100; mean, .090; set at 30,000 pounds per sq. in., to .002; mean, 0. The mean extension between 10,000 to 30,000 lbs. per sq. in. increased regu- larly at the rate of .007 inch for each 2000 lbs. per sq. in. increment of strain. This corresponds to a modulus of elasticity of 28,571, 4J9. The least increase of extension for an increase of load of 20,000 lbs. per sq. in., .065 inch, cor- responds to a modulus of elasticity of 30,769,231, and the greatest, .076 inch, to a modulus of 26,315,789. Steel Rails.— Bending tests, 5 feet between supports, 11 tests of flange rails 72 pounds per yard, 4.63 inches high. Elastic stress. Ultimate stress. Deflection at 50,000 Ultimate Pounds. 34,200 32,000 Pounds. 60,960 56,740 Pounds. 3.24 ins. 3.76 " 3.53 " Hardest. Softest . . Mean .... All uncracked at 8 inches deflection. Pulling tests of pieces cut from same rails. Mean results. Elastic Ultimate Contraction of Stress. Pounds. area of frac- per sq. in. per sq. in. ture. Top of rails 44,200 83,110 19.9* Botton of rails 40,900 77,820 30.9$ Deflection. 3 ins. Extension in 10 ins. 13.5* 22.8* Steel Tires.— Tensile tests of specimens cut from steel tires. Krupp Steel.— 262 Tests. Highest. . Mean Lowest . . E. L. 69,250 52,869 41,700 T. S. 119,079 104,112 90,523 Contr. 31.9 29.5 45.5 Vickers, Sons & Co.— 70 Tests. Highest. . Mean Lowest.. E. L. 58,600 51,066 43,700 T. S. 120,789 101,264 Contr. 11.8 14.-7 Ext. in 5 inches. 18.1 19.7 23.7 Ext. in 5 inches. 8.4 12.4 16.0 Note the correspondence between Krupp's and Vickers , steels as to ten- sile strength and elastic limit, and their great difference in contraction and elongation. The fractures of the Krupp steel averaged 22 per cent silky, 78 per cent granular; of the Vicker steel, 7 per cent silky, 93 per cent granu- lar. EXTRACTS FROM KIRKALDY^S TESTS. 299 Steel Axles. — Tensile tests of specimens cut froni steel axles- Patent Shaft and Axle Tree Co. — 157 Tests. Ext. in E. L. T. S. Contr. 5 inches. Highest 49,800 99,009 81.1 16.0 Mean... 36,267 72,099 33.0 23.6 Lowest 31,800 61,382 34.8 25.3 Vickers, Sons & Co.— 125 Tests. Ext. in E. L. T. S. Contr. 5 inches. Highest 42,600 83,701 18.9 13.2 Mean 37,618 70,572 41.6 27.5 Lowest 30,250 56,388 49.0 37.2 The average fracture of Patent Shaft and Axle Tree Co. steel was 33 per cent silky, 67 per cent granular. The average fracture of Vickers' steel was 88 per cent silky, 12 per cent granular. Tensile tests of specimens cut from locomotive crank axles. Vickers'.— 82 Tests, 1879. Ext. in E. L. T. S. Contr. 5 inches. Highest 26,700 68,057 28.3 18.4 Mean 24,146 57,922 32.9 24.0 Lowest 21,700 50,195 52.7 36.2 Vickers'.— 78 Tests, 1884. Ext. in E. L. T. S. Contr. 5 inches. Highest 27,600 64,873 27.0 20.8 Mean 23,573 56,207 32.7 25.9 Lowest 17,600 47,695 35.0 27.2 Fried. Krupp.— 43 Tests, 1889. Ext. in E. L. T. S. Contr. 5 inches. Highest 31,650 66,868 48.6 35.6 Mean 29,491 61,774' 47.7 32.3 Lowest 21,950 55,172 55.3 35.6 Steel Propeller Shafts.— Tensile tests of pieces cut from two shafts, mean of four tests each. Hollow shaft, Whitworth. T. S.. 61,290; E. L., 30,575; contr., 52.8; ext. in 10 inches, 28 6. Solid Shaft, Vickers', T. S., 46,870; E. L. 20,425; contr., 44.4; ext. in 10 inches, 30.7. Thrusting tests, Whitworth, ultimate, 56,201; elastic, 29,300; set at 30,000 lbs., 0.18 per cent; set at 40,000 lbs., 2.04 per cent; set at 50,000 lbs., 3.82 per cent. Thrusting tests, Vickers', ultimate, 44,602; elastic, 22,250; set at 30,000 lbs., 2.29 per cent; set at 40,000 lbs., 4.69 per cent. Shearing strength of the Whitworth shaft, mean of four tests, was 40,654 lbs. per square inch, or 66.3 per cent of the pulling- stress. Specific gravity of the Whitworth steel. 7.867: of the Vickers', 7.856. Spring Steel.— Untempered, 6 tests, average, E. L., 67,916; T. S., 115,668; contr., 37.8; ext. in 10 inches, 16.6. Spring steel untempered. 15 tests, average, E. L., 38,785; T. S., 69,496; contr., 19.1 ; ext. in 10 inches, 29 8. 'These two lots were shipped for the same purpose, viz., railway carnage leaf springs. Steel Castings.— 44 tests, E. L., 31,816 to 35,567; T. S., 54,928 to 63,840; contr., 1.67 to i5.8; ext., 1.45 to 15.1. Note the great variation in ductility. The steel of the highest strength was also the most ductile. Riveted Joints, Pulling Tests of Riveted Steel Plates, Triple Riveted I^ap Joints, Machine Riveted, Holes J>rilled. Plates, width and thickn ss, inches : 13.50 X .25 13.00 X .51 11.75 X .78 12.25 X 1.01 14.00 X .77 Plates, gross sectional area square inches : 3.375 6.63 9.165 12.372 10.780 Stress, total, pounds : 199,320 332,640 423,180 528,000 455,210 300 STRENGTH OF MATERIALS. Stress per square inch of gross area, joint : 59,058 50,172 46,173 42,696 42,227 Stress per square inch of plates, solid : 70,765 65,300 64,050 62,280 68,045 Ratio of strength of joint to solid plate : 83.46 76.83 72.09 68.55 62.06 Ratio net area of plate to gross : 73.4 65.5 62.7 , 64.7 72.9 Where fractured : plate at plate at plate at plate at rivets holes. holes. holes. holes. sheared. Rivets, diameter, area and number : .45, .159, 24 .64, .321, 21 .95, .708, 12 1.08, .918, 12 .95, .708, 12 Rivets, total area : 3.816 6.741 8.496 10.992 8.496 Strength of Welds.— Tensile tests to determine ratio of strength of weld to solid bar. Iron Tie Bars.— 28 Tests. Strength of solid bars varied from 43,201 to 57,065 lbs. Strenth of welded bars varied from 17,816 to 44,586 lbs. Ratio of weld to solid varied from 37.0to79.1$ Iron Plates.— 7 Tests. Strength of solid plate from 44,851 to 47,481 lbs. Strength of welded plate from 26,442 to 38,931 lbs. Ratio of weld to solid 57.7 to 83.9$ Chain Links.— 216 Tests. Strength of solid bar from 49,122 to 57,875 lbs. Strength of welded bar from 39,575 to 48,824 lbs. Ratio of weld to solid 72.1 to 95.4$ Iron Bars.— Hand and Electric Machine Welded. 32 tests, solid iron, average 52,444 17 " electri- welded, average 46,836 ratio 89.1$ 19 " hand " " 46,899 " 89.3$ Steel Bars and Plates.— 14 Tests. Strength of solid 54,226 to 64,580 Strength of weld 28,553 to 46,019 Ratio weld to solid 52.6 to 82.1$ The ratio of weld to solid in all the tests ranging from 37.0 to 95.4 is proof of the great variation of workmanship in welding. Cast Copper.— 4 tests, average, E. L., 5900; T. S., 24,781; ccntr., 24.5; ext., 21.8. Copper Plates.— As rolled, 22 tests, .26 to .75 in. thick; E. L.,9766 to 18,650; T. S., 30,99.; to 34,281 ; contr., 31.1 to 57.6; ext., 39.9 to 52.2. The va- riation in elastic limit is due to difference in the heat at which the plates were finished. Annealing reduces the T. S. only about 1000 pounds, but the E. L. from 3000 to 7000 pounds. Another series, .38 to .52 thick; 148 tests, T. S., 29,099 to 31,924; contr., 28.7 to 56.7; ext. in 10 inches, 28.1 to 41.8. Note the uniformity in tensile strpngth. Drawn Copper.— 74 tests (0.88 to 1.08 inch diameter); T. S., 31,634 to 40,557; contr., 37.5 to 64.1; ext. in 10 inches, 5.8 to 48.2. Bronze from a Propeller Blade.— Means of two tests each from centre and edge. Central portion (sp. gr. 8.320). E. L., 7550; T. S., 26,312; contr., 25.4; ext. in 10 inches, 32.8. Edge portion (sp. gr. 8550). E. L., 8950; T. S., 35,960; contr., 37.8; ext. in 10 inches, 47.9. Cast German Silver.— 10 tests: E. L., 13,400 to 29,100; T. S., 23,714 to 46,540; contr., 3.2 to 21.5; ext. in 10 inches, 0.6 to 10.2. Thin Sheet Metal.— Tensile Strength. German silver, 2 lots 75.816 to 87,129 Bronze, 4 lots 73,380 to 92,086 Brass. 2 lots , 44,398 to 58,188 Copper, 9 lots.... 30,470 to 48,450 Iron, 13 lots, lengthway 44,331 to 59,484 Iron. 13 lots, crossway 39,838 to 57,350 Steel, 6 lots 49,253 to 78,251 Steel, 6 lots, crossway 55,948 to 80,799 EXTRACTS FROM KIRKALD1 S TESTS. 301 "Wire.— Tensile Strength. German silver, 5 lots 81 ,735 to 92,224 Bronze, 1 lot . . 78,049 Brass, as drawn, 4 lots 81,114 to 98,578 Copper, as drawn, 3 lots 37,607 to 46,494 Copper annealed, 3 lots 34,936 to 45,210 Copper (another lot), 4 lots 35,052 to 62,190 Copper (extension 36.4 to 0.6$). Iron,81ots 59,246 to 97,908 Iron (extension 15.1 to 0.7$). Steel, Slots 103,272 to 318,823 The Steel of 318,823 T. S. was .047 inch diam., and had an extension of only 0.3 per cent; that of 103,272 T. S, was .107 inch diam. and had an extension of 2.2 per cent. One lot of .044 inch diam. had 267,114 T. S., and 5.2 per cent extension. Wire Ropes. Selected Tests Showing Range of "Variation. of o Strands. *+« 4> gj A B u O ®5 Description. a! 5 II Kb o p -2.2 2 © Hemp Core. e3 60 • a §£ 5oq Galvanized 7.70 53.00 6 19 .1563 Main 339,780 Ungalvanized 7.00 53.10 7 19 .1495 Main and Strands 314,860 Ungalvanized 6.38 42.50 7 19 .1347 Wire Core 295,920 Galvanized.. 7.10 37.57 « 30 .1004 Main and Strands 272,750 Ungalvanized 6.18 40.46 7 19 .1302 Wire Core 268,470 Ungalvanized 6.19 40.33 7 19 .1316 Wire Core 221,820 Galvanized.. 4.92 20.86 6 30 .0728 Main and Strands 190,890 Galvanized 5.36 18.94 fi 12 .1104 Main and Strands 136,550 Galvanized 4.82 21.50 6 7 .1693 Main 129,710 Ungalvanized 3.65 12.21 6 19 .0755 Main 110,180 Ungalvanized 3.50 12.65 7 7 .122 Wire Core 101,440 Ungalvanized 3 . 82 14.12 fi 7 .135 Main 98,670 Galvanized 4.11 11.35 6 12 .080 Main and Strands 75,110 Galvanized 3.31 7.27 6 12 .068 Main and Strands 55,095 Ungalvanized 3.02 8.62 6 7 .105 Main 49,555 Ungalvanized Galvanized 2.68 6.26 H « .0963 Main and Strands 41,205 2.87 5.43 6 12 .0560 Main and Strands 38,555 Galvanized 2.46 3.85 (J 12 .0472 Main and Strands 28,075 Ungalvanized...,. 1.75 2.80 6 7 .0619 Main 24,552 Galvanized 2.04 2.72 6 12 .0378 Main and Strands 20,415 Galvanized 1.76 1.85 6 12 .0305 Main 14,634 Hemp Ropes, Untarred.— 15 tests of ropes from 1.53 to 6.90 inches circumference, weighing 0.42 to 7.77 pounds per fathom, showed an ultim- ate strength of from 1670 to 33,808 pounds, the strength per fathom weight varying; from 2872 to 5534 pounds. Hemp Ropes, Tarred. —15 tests of ropes from 1.44 to 7.12 inches circumference, weighing from 0.38 to 10.39 pounds per fathom, showed an ultimate strength of from 1046 to 31.549 pounds, the strength per fathom weight varying from 1767 to 5149 pounds. Cotton Ropes.— 5 ropes, 2.48 to 6.51 inches circumference, 1.08 to 8.17 pounds per fathom. Strength 3089 to 23,258 pounds, or 2474 to 3346 pounds per fathom weight. Manila Ropes.— 35 tests: 1.19 to 8.90 inches circumference, 0.20 to 11.40 pounds per fathom. Strength 1280 to 65,550 pounds, or 3003 to 7394 pounds per fathom weight. 302 STRENGTH OF MATERIALS. Belting. No. of Tensile strength lots. per square inch. 11 Leather, single, ordinary tanned 3248 to 4824 4 Leather, single, Helvetia 5631 to 5944 7 Leather, double, ordinary tanned 2160 to 3572 8 Leather, double Helvetia 4078 to 5412 6 Cotton, solid woven 5648 to 8869 14 Cotton, folded, stitched 4570 to 7750 1 Flax, solid, woven 9946 1 Flax, folded, stitched 6389 6 Hair, solid, woven 3852 to 5159 2 Rubber, solid, woven 4271 to 4343 Canvas.— 35 lots: Strength, lengthwise, 113 to 408 pounds per inch; crossways, 191 to 468 pounds per inch. The grades are numbered 1 to 6, but the weights are not given. The strengths vary considerably, even in the same number. Marbles.— Crushing strength of various marbles. 38 tests, 8 kinds. Specimens were 6-inch cubes, or columns 4 to 6 inches diameter, and 6 and 12 inches high. Range 7542 to 13,720 pounds per square inch. Granite.— Crushing strength, 17 tests; square columns 4X4 and 6x4, 4 to 24 inches high, 3 kinds. Crushing strength ranges 10,026 to 13,271 pounds per square inch. (Very uniform.) Stones.— (Probably sandstone, local names only given.) 11 kinds, 42 tests, 6x6, columns 12, 18 and 24 inches high. Crushing strength ranges from 2105 to 12,122. The strength of the column 24 inches long is generally from 10 to 20 per cent less than that of the 6-inch cube. Stones.— (Probably sandstone) tested for London & Northwestern Rail- way. 16 lots, 3 to 6 tests in a lot. Mean results of each lot ranged from 3785 to 11.956 pounds. The variation is chiefly due to the stones being from different lots. The different specimens in each lot gave results which gen- erally agreed within 30 per cent. Bricks.— Crushing strength, 8 lots; 6 tests in each lot; mean results ranged from 1835 to 9209 pounds per square inch. The maximum variation in the specimens of one lot was over 100 per cent of the lowest. In the most uniform lot the variation was less than 20 per cent. Wood..— Transverse and Thrusting Tests. Sizes abt. in square. Span inches. Ultimate Stress. 8 = LW 4BD* 45,856 1096 to to 80,520 1403 37,948 657 to to 54,152 790 32,856 1505 to to 39,084 1779 23,624 1190 to to 26,952 1372 Pitch pine Dantzic fir English oak American white oak uy 2 to 12^ 12 to 13 4^X 12 4Y 2 X 12 120 120 5438 2478 3423 2473 Demerara greenheart, 9 tests (thrusting) 8169 to 10,785 Oregon pine, 2 tests 5888 and 7284 Honduras mahogany, 1 test 6769 Tobasco mahogany, 1 test 5978 Norway spruce, 2 tests 5259 and 5494 American yellow pine, 2 tests 3875 and 3993 English ash, 1 test 3025 Portland Cement.— (Austrian.) Cross-sections of specimens 2 x 2^ inches for pulling testsonly; cubes, 3x3 inches for thrusting tests; weight, MISCELLANEOUS TESTS OF MATERIALS. 303 98.8 pounds per imperial bushel; residue, 0.7 per cent with sieve 2500 meshes per square inch: 88.8 per cent by volume of water required for mixing; time of setting, 7' days; 10 tests to each lot. The mean results in lbs. per sq. in. were as follows: Cement Cement 1 Cement, ' 1 Cement, 1 Cement, alone, alone, 2 Sand, 3 Sand, 4 Sand, Age. Pulling. Thrusting. Thrusting. Thrusting. Thrusting. 10 days 376 2910 893 407 228 20 days 420 3342 1023 494 275 30 days 451 3724 1172 594 338 Portland Cement.- Various samples pulling tests, 2 x 2% inches cross-section, all aged 10 days, 180 tests; ranges 87 to 643 pounds per square inch. TENSILE STRENGTH OF WIRE. (From J. Bucknall Smith's Treatise on Wire.) Tons per sq. Pounds per in. sectional sq. in. sec- area, tional area. Black or annealed iron wire 25 56,000 Bright hard drawn 35 78,400 Bessemer, steel wire 40 89,600 Mild Siemens-Martin steel wire.... 60 134,000 High carbon ditto (or " impi'oved ") 80 179,200 Crucible cast-steel ,l improved " wire 100 224,000 "Improved" cast-steel "plough" 120 268,800 Special qualities of tempered and improved cast- steel wire may attain 150 to 170 336,000 to 380,800 MISCELLANEOUS TESTS OF MATERIALS. Reports of Work of the Watertown Testing-machine in 1883. TESTS OF RIVETED JOINTS, IRON AND STEEL PLATES. 6 .. •^ 6 a3 •B 2 a a H > 2> V "£.2 1 fi ©o . S'S-g 3 P-l 0J z. > 6 Tensile Streng Joint in Net St tion of Plate p square inch, pounds. w .2 o '5 1-5 « a © •Sfi o & H * % 11-16 u 10% 6 m 39,300 47,180 47.0 X * % 11-16 n wy 2 6 i% 41,000 47,180 49.0 t * Vk % 13-16 10 5 2 35,650 44,615 45.6 % * M U 13-16 10 5 2 35,150 44,615 44.9 % * Ys 11-16 H 10 5 2 46,360 47,180 59.9 § * 11-16 H 10 5 2 46.875 47,180 60.5 § * M& H 13-16 10 5 2 46,400 44,615 59.4 § * * % 13-16 10 5 2 46,140 44,615 59.2 § * l 1 1-16 10^ 4 m 44,260 44,635 57.2 § * % l 1 1-16 ioy 2 4 2% 42,350 44,635 54.9 § * % *H 1 3-16 11.9 4 2.9 42,310 46.590 52.1 § * % 1*4 1 3-16 11.9 4 2.9 41,920 46,590 51.7 § * Ys M 13-16 10^ 6 m 61,270 53.330 59.5 X t % M 13-16 10^ 6 m 60,830 53,330 59.1 X t y* 15-16 1 10 5 47,530 57,215 40.2 X t y* 15-16 1 10 5 2 49,840 57,215 42.3 % t % 11-16 S 10 5 2 62,770 53,330 71.7 § t Vs 11-16 10 5 2 61,210 53,330 69.8 § t y* 15-16 1 10 5 2 68,920 57,215 57.1 § t 14 15-16 1 10 5 2 66,710 57,215 55.0 § t % 1 1 1-16 9V£ 4 2% 62,180 52,445 63.4 § t % 1 1 1-16 Wz 4 2% 62,590 52,445 63.8 § t u m 1 3-16 10 4 2y 2 54,650 51,545 54 § + % m 1 3-16 10 4 2y 2 54,200 51,545 53.4 § X Lap-joint. 5 Butt-joint. 304 STRENGTH OP MATERIALS. The efficiency of the joints is found by dividing the maximum tensile stress on the gross sectional area of plate by the tensile strength of the material. COMPRESSION TESTS OF 3 X 3 INCH WROUGHT-IRON BARS. Tested with Two Pin Ends, Pins V/z inch in Diameter. Tested with One Flat and One Pin Length, inches. Ultimate Com- pressive Strength pounds per square inch. Tested with Two Flat Ends, Ulti- mate Compressive Strength, pounds per square inch. End, Ultimate Compressive Strength, pounds per square inch. $ 28,260 (31,990 J 26,310 | 26,640 j 24.030 j 25,380 j 20,660 1 20,200 j 16,520 1 17,840 I 13,010 \ 15,700 90 j 26,780 1 25,580 J 23,010 1 22,450 j 25,120 120... | 25,190 j 22,450 1 21,870 150 Tested with two pin- j ends. Length o f bars -< 120 inches. I 2^ Diameter of Pins. inch inches. Ult. Comp. Str., per sq. in., lbs. 16,250 17,740 21,400 22,210 TENSILE TEST OF SIX STEEL EYE-BARS. COMPARED WITH SMALL TEST INGOTS. The steel was made by the Cambria Iron Company, and the eye-bar heads made by Keystone Bridge Company by upsetting and hammering. All the bars were made from one ingot. Two test pieces, %-inch round, rolled from a test-ingot, gave elastic limit 48,040 and 42,210 pounds; tensile strength, 73,150 and 69,470 pounds, and elongation in 8 inches, 22.4 and 25.6 per cent, respectively. The ingot from which the eye-bars were made was 14 inches square, rolled to billet, 7x6 inches. The eye-bars were rolled to Q% X 1 inch. Cnemical tests gave carbon .27 to .30; manganese, .64 to .73; phosphorus, .074 to .098. Elongation per cent, in Gauged Length. 15.8) 6.96 8.6 12.3 12.0 16.4 13.9 The average tensile strength of the %-inch test pieces was 71,310 lbs., that of the eye-bars 67,230 lbs., a decrease of 5.7$. The average elastic limit of the test pieces was 45,150 lbs., that of the eye-bars 36,402 lbs., a decrease of 19.4$. The elastic limit of the test pieces was 63.3^ of the ultimate strength, that of the eye-bars 54.2% of the ultimate strength. Gauged Elastic Tensile Length, limit, lbs. strength per inches. per sq. in. sq. in., lbs. 160 37,480 67,800 160 36,650 64,000 160 71,560 200 37,600 68,720 200 35,810 65,850 200 33,230 64,410 200 37,640 68,290 MISCELLANEOUS TESTS OE MATERIALS. 305 COMPRESSION OF WROUGHT-IRON COLUMNS, LATTICED BOX AND SOLID WEB. ALL TESTED WITH PIN ENDS. Columns made of 6 inch channel, solid web.. 8-inch channels, with 5-16-in. continuous plates 5-16-inch continuous plates and angles. Width of plates, 12 in., 1 in. and 7.35 in. 7-16-inch continuous plates and angles. Plates 12 in. wide . 8-inch channels, latticed 8-inch channels, latticed, swelled sides. 10 " " " 10 " " " 10-inch channels, latticed, swelled sides. * 10-inch channels, latticed one side; con- tinuous plate one side 1 10 inch channels, latticed one side; con- tinuous plate one side e3 ^ Live Load » Dead Load. Greatest> Mean . Iron and steel 3 6 from 6 to 40 Timber 4 to 5 8 to 10 Masonry ,. 4 8 FACTORS OF SAFETY. 315 The great factor of safety, 40, is for shafts in millwork which transmit very variable efforts. Unwin gives the following "factors of safety which have been adopted in certain cases for different materials." They " include an allowance for ordinary contingencies." T)pad ' Live Load - > T oad In Temporary In Permanent In Structures • Structures. Structures, subj. to Shocks. Wrought iron and steel. 3 4 4 to 5 10 Cast iron 3 4 5 10 Timber 4 10 Brickwork 6 .... Masonry 20 .... 20 to 30 Unwin says says that " these numbers fairly represent practice based on experience in many actual cases, but they are not very trustworthy." Prof. Wood in his " Resistance of Materials " says : " In regard to the margin that should be left for safety, much depends upon the character of the loading. If the load is simply a dead weight, the margin may be com- paratively small; but if the structure is to be subjected to percussive forces or shocks, the margin should be comparatively large on account of the indeterminate effect produced by the force. In machines which are sub- jected to a constant jar while in use, it is very difficult to determine the proper margin which is consistent with economy and safety. Indeed, in such cases, economy as well as safety generally consists in making them excessively strong, as a single breakage may cost much more than the extra material necessary to fully insure safety." For discussion of the resistance of materials to repeated stresses and shocks, see pages 238 to 240. Instead of using factors of safety it is becoming customary in designing to fix a certain number of pounds per square inch as the maximum stress which will be allowed on a piece. Thus, in designing a boiler, instead of naming a factor of safety of 6 for the plates and 10 for the stay-bolts, the ultimate tensile strength of the steel being from 50,000 to 60,0001bs. persq. in., an allowable working stress of 10,000 lbs. per sq. in. on the plates and 6000 lbs. per sq. in. on the stay-bolts may be specified instead. So also in Merriman's formula for columns (see page 260) the dimensions of a column are calculated after assuming a maximum allowable compressive stress per square inch on the concave side of the column. The factors for masonry under dead load as given by Rankine and by Unwin, viz., 4 and 20, show a remarkable difference, which may possibly be explained as follows : If the actual crushing strength of a pier of masonry is known from direct experiment, then a factor of safety of 4 is sufficient for a pier of the same size and quality under a steady load; but if the crushing strength is merely assumed from figures given by the authorities (such as the crush- ing strength of pressed brick, quoted above from Howe's Retaining Walls, 40 to 300 tons per square foot, average 170 tons), then a factor of safety of 20 may be none too great. In this case the factor of safety is really a " factor of ignorance." The selection of the proper factor of safety or the proper maximum unit stress for any given case is a matter to be largely determined by the judg- ment of the engineer and by experience. No definite rules can be given. The customary or advisable factors in many particular cases will be found where these cases are considered throughout this book. In general the following circumstances are to be taken into account in the selection of a factor : 1. When the ultimate strength of the material is known within narrow limits, as in the case of structural steel when tests of samples have been made, when the load is entirely a steady one of a known amount, and there is no reason to fear the deterioration of the metal by corrosion, the lowest factor that should be adopted is 3. 2. When the circumstances of 1 are modified by a portion of the load being variable, as in floors of warehouses, the factor should be not less than 4. 3. When the whole load, or nearly the whole, is apt to be alternately put on and taken off, as in suspension rods of floors of bridges, the factor should be 5 or 6. 4. When the stresses are reversed in direction from tension to compres- sion, as in some bridge diagonals and parts of machines, the factor should be not less than 6. 316 STRENGTH OF MATERIALS. 5. When the piece is subjected to repeated shocks, the factor should be not less than 10. 6. When the piece is subject to deterioi'ation from corrosion the section should be sufficiently increased to allow for a definite amount of corrosion before the piece be so far weakened by it as to require removal. 7. When the strength of the material, or the amount of the load, or both are uncertain, the factor should be increased by an allowance sufficient to cover the amount of the uncertainty. 8. When the strains are of a complex character and of uncertain amount, such as those in the crank-shaf t of a reversing engine, a very high factor is necessary, possibly even as high as 40, the figure given by Rankine for shafts in millwork. THE MECHANICAL PROPERTIES OF CORK. Cork possesses qualities which distinguish it from all other solid or liquid bodies, namely, its power of altering its volume in a very marked degree in consequence of change of pressure. It consists, practically, of an aggrega- tion of minute air-vessels, having thin, water-tight, and very strong walls, and hence, if compressed, the resistance to compression rises in a manner more like the resistance of gases than the resistance of an elastic solid such as a spring. In a spring the pressure increases in proportion to the dis- tance to which the spring is compressed, but with gases the pressure in- creases in a much more rapid manner; that is, inversely as the volume which the gas is made to occupy. But from the permeability of cork to air, it is evident that, if subjected to pressure in one direction only, it will gradually part with its occluded air by effusion, that is, by its passage through the porous walls of the cells in which it is contained. The gaseous part of cork constitutes 53$ of its bulk. Its elasticity has not only a very considerable range, but it is very persistent. Thus in the better kind of corks used in bottling the corks expand the instant they escape from the bottles. This expansion may amount to an increase of volume of 75%, even after the corks have been kept in a state of compression in the bottles for ten years. If the cork be steeped in hot water, the volume continues to increase till it attains nearly three times that which it occupied in the neck of the bottle. When cork is subjected to pressure a certain amount of permanent defor- mation or "permanent set" takes place very quickly. This property is common to all solid elastic substances when strained beyond their elastic limits, but with cork the limits are comparatively low. Besides the perma- nent set, there is a certain amount of sluggish elasticity— that is, cork on; being released from pressure springs back a certain amount at once, but the complete recovery takes an appreciable time. Cork which had been compressed and released in water many thousand times had not changed its molecular structure in the least, and had contin- ued perfectly serviceable. Cork which has been kept under a pressure of three atmospheres for many weeks appears to have shrunk to from 80$ to 85% of its original volume.— Van NostrancVs Eng'g Mag. 1886, xxxv. 307. TESTS OF VULCANIZED INDIA-RUBBER. Lieutenant L. Vladomiroff, a Russian naval officer, has recently carried out a series of tests at the St. Petersburg Technical Institute with a view to establishing rules for estimating the quality of vulcanized india-rubber. The following, in brief, are the conclusions arrived at, recourse being had to physical properties, since chemical analysis did not give any reliable re- sult: 1. India-rubber should not give the least sign of superficial cracking when bent to an angle of 180 degrees after five hours of exposure in a closed air-bath to a temperature of 125° C. The test-pieces should be 2.4 inches thick. 2. Rubber that does not contain more than half its weight of metal- lic oxides should stretch to five times its length without breaking. 3. Rub- ber free from all foreign matter, except the sulphur used in vulcanizing it, should stretch to at least seven times its length without rupture. 4. The extension measured immediately after rupture should not exceed \2% of the original length, with given dimensions. 5. Suppleness may be determined by measuring the percentage of ash formed in incineration. This may form the basis for deciding between different grades of rubber for certain pur- poses. 6. Vulcanized rubber should not harden under cold. These rules have been adopted for the Russian navy.— Iron Age, June 15, 1893. XTLOLITH, OR WOODSTONE is a material invented in 1883, but only lately introduced to the trade by Otto Serrig & Co., of Pottschappel, near Dresden. It is made of magnesia ALITMIHUM — ITS PROPERTIES AHD USES. 317 cement, or calcined magnesite, mixed with sawdust and saturated with a solution of chloride of calcium. This pasty mass is spread out into sheets and submitted to a pressure of about, 1000 ibs. to the square inch, and then simply dried in the air. Specific gravity 1.553. The fractured surface shows a uniform close grain of a yellow color. It has a tensioual resistance when dry of 100 lbs. per square inch, and when wet about 66 lbs. When immersed in water for 12 hours it takes up 2.1$ of its weight, and 3.8$ when immersed 216 hours. When treated for several days with hydrochloric acid it loses 2.3$ in weight, and shows no loss of weight under boiling in water, brine, soda-lye, and solution of sulphates of iron, of copper, and of ammonium. In hardness the material stands between feldspar and quartz, and as a non-conductor of heat it ranks between asbestos and cork. It stands fire well, and at a red heat it is rendered brittle and crumbles at the edges, but retains its general form and cohesion. This xylolith is sup- plied in sheets from *4 in - to iy Q in. thick, and up to one metre square. It is extensively used in Germany for floors in railway stations, hospitals, etc., and for decks of vessels. It can be sawed, bored, and shaped with ordinary woodworking tools. Putty in the joints and a good coat of paint make it entirely water-proof. It is sold in Germany for flooring at about 7 cents per square foot, and the cost of laying adds about 4 cents more.— Eng'g News, July 28, 1892, and July 27, 1893. ALUMINUM-ITS PROPERTIES AND USES. (By Alfred E. Hunt, Pres't of the Pittsburgh Reduction Co.) The specific gravity of pure aluminum in a cast state is 2.58 ; in rolled bars of large section it is 2 6 ; in very thin sheets subjected to high com- pression under chilled rolls, it is as much as 2.7. Taking the weight of a given bulk of cast aluminum as 1, wrought iron is 2.90 times heavier ; struc- tural steel, 2.95 times ; copper, 3.60 ; ordinary high brass, 3.45. Most wood suitable for use in structures has about one third the weight of aluminum, which weighs 0.092 lb. to the cubic inch. Pure aluminum is practically not acted upon by boiling water or steam. Carbonic oxide or hydrogen sulphide does not act upon it at any tempera- ture under 600° F. It is not acted upon by most organic secretions. Hydrochloric acid is the best solvent for aluminum, and strong solutions of caustic alkalies readily dissolve it. Ammonia has a slight solvent action, and concentrated sulphuric acid dissolves aluminum upon heating, with evolution of sulphurous acid gas. Dilute sulphuric acid acts but slowly on the metal, though the presence of any chlorides in the solution allow rapid decomposition. Nitric acid, either concentrated or dilute, has very little action upon the metal, and sulphur has no action unless the metal is at a red heat. Sea-water has very little effect on aluminum. Strips of the metal placed on the sides of a wooden ship corroded less than 1/1000 inch after six months' exposure to sea- water, corroding less than copper sheets similarly In malleability pure aluminum is only exceeded by gold and silver. In ductility it stands seventh in the series, being exceeded by gold, silver, platinum, iron, very soft steel, and copper. Sheets of aluminum have been rolled down to a thickness of 0.0005 inch, and beaten into leaf nearly as thin as gold leaf. The metal is most malleable at a temperature of between 400° and 600° F., and at this temperature it can be drawn down between rolls with nearly as much draught upon it as with heated steel. It has also been drawn down into the very finest wire. By the Mannesmann process aluminum tubes have been made in Germany. Aluminum stands very high in the series as an electro-positive metal, and contact with other metals should be avoided, as it would establish a galvanic couple. The electrical conductivity of aluminum is only surpassed by pure copper, silver, and gold. With silver taken at 100 the electrical conductivity of aluminum is 54.20 ; that of gold on the same scale is 78; zinc is 29.90; iron is only 16, and platinum 10.60. Pure aluminum has no polarity, and the metal in the market is absolutely non-magnetic. Sound castings can be made of aluminum in either dry or " green " sand moulds, or in metal "chills.'" It must not be heated much beyond its melting-point, and must be poured with care, owing to the ready absorption of occluded gases and air. The shrinkage in cooling is 17/64 inch per foot, or a little more than ordinary brass. It should be melted in plumbago crucibles, and the metal becomes molten at a temperature of 1120° F. ac- cording to Professor Roberts-Austen, or at 1300° F. according to Richards. 318 STRENGTH OP MATERIALS. The coefficient of linear expansion, as tested on %-ineh round aluminum rods, is 0.00002295 per degree centigrade between the freezing and boiling point of water. The mean specific heat of aluminum is higher than that of any other metal, excepting only magnesium and the alkali metals. From zero to the melting-point it is 0.2185; water being taken as 1, and the latent heat of fusion at 28.5 heat units. The coefficient of thermal conductivity of unannealed aluminum is 37.96; of annealed aluminum, 38.37. As a conductor of heat aluminum ranks fourth, being exceeded only by silver, copper, and gold. Aluminum, under tension, and section for section, is about as strong as cast iron. The tensile strength of aluminum is increased by cold rolling or cold forging, and there are alloys which add considerably to the tensile strength without increasing the specific gravity to over 3 or'3.25. The strength of commercial aluminum is given in the following table as the result of many tests : Elastic Limit Ultimate Strength Percentage per sq. in. in per sq. in. in of Reducfn Form. Tension, Tension, of Area in lbs. lbs. Tension. Castings 6,500 15,000 15 Sheet 12,000 24,000 35 Wire 16,000-30,000 30,000-65,000 60 Bars 14,000 28,000 40 The elastic limit per square inch under compression in cylinders, with length twice the diameter, is 3500. The ultimate strength per square inch under compression in cylinders of same form is 12,000. The modulus of elasticity of cast aluminum is about 11,000,000. It is rather an open metal in its texture, and for cylinders to stand pressure an increase in thickness must be given to allow for this porosity. Its maximum shearing stress in castings is about 12,000, and in forgings about 16,000, or about that of pure copper. Pure aluminum is too soft and lacking in tensile strength and rigidity for many purposes. Valuable alloys are now being made which seem to give great promise for the future. They are alloys containing from 2% to 7$ or 8$ of copper, manganese, iron, and nickel. As nickel is one of the principal constituents, these alloys have the trade name of " Nickel-aluminum." Plates and bars of this nickel alloy have a tensile strength of from 40,000 to 50,000 pounds per square inch, an elastic limit of 55$ to 60$ of the ultimate ten- sile strength, an elongation of 20$ in 2 inches, and a reduction of area of 25$. This metal is especially capable of withstanding the punishment and distortion to which structural material is ordinarily subjected. Nickel- aluminum alloys have as much resilience and spring as the very hardest of hard-drawn brass. Their specific gravity is about 2.80 to 2.85, where pure aluminum has a specific gravity of 2.72. In castings, more of the hardening elements are necessary in order to give the maximum stiffness and rigidity, together with the strength and ductility of the metal; the favorite alloy material being zinc, iron, manganese, and copper. Tin added to the alloy reduces the shrinkage, and alloys of alumi- num and tin can he made which have less shrinkage than cast iron. The tensile strength of hardened aluminum-alloy castings is from 20,000 to 25,000 pounds per square inch. Alloys of aluminum and copper form two series, both valuable. The first is aluminum-bronze, containing from 5$ to 11^>$ of aluminum; and the second is copper-hardened aluminum, containing from 2% to 15$ of copper. Aluminum-bronze is a very dense, fine-grained, aud strong alloy, having good ductility as compared with tensile strength. The 10$ bronze in forged bars will give 100,000 lbs. tensile strength per square inch, with 60,000 lbs. elastic limit per square inch, and 10$ elongation in 8 inches. The 5$ to 7J^$ bronze has a specific gravity of 8 to 8.30, as compared with 7.50 for the 10$ to \\y>% bronze, a tensile strength of 70,000 to 80,000 lbs., an elastic limit of 40,000 lbs. per square inch, and an elongation of 30$ in 8 inches. Aluminum is used by steel manufacturers to prevent the retention of the occluded gases in the steel, and thereby produce a solid ingot. The propor- tions of the dose range from y% lb. to several pounds of aluminum per ton of steel. Aluminum is also used in giving extra fluidity to steel used in castings, making them sharper and sounder. Added to cast iron, aluminum causes the iron to be softer, free from shrinkage, and lessens the tendency to "chill.'" With the exception of lead and mercury, aluminum unites with all metals, ALLOYS. 319 though it unites with antimony with great difficulty. A small percentage of silver whitens and hardens the metal, and gives it added strength; and this alloy is especially applicable to the manufacture of fine instruments and apparatus. The following alloys have been found recently to be useful in the arts: Nickel-aluminum, composed of 20 parts nickel to 8 of aluminum; rosine, made of 40 parts nickel, 10 parts silver, 30 parts aluminum, and 20 parts tin, for jewellers'' work; mettaline, made of 35 parts cobalt, 25 parts aluminum, 10 parts iron, and 30 parts copper. The aluminum-bourbounz metal, shown at the Paris Exposition of 1889, has a specific gravity of 2.9 to 2.96, and can be cast in very solid shapes, as it has very little shrinkage. From analysis the following composition is deduced: Aluminum, 85. 74$; tin, 12.94^; silicon, 1.32$; iron, none. The metal can be readily electrically welded, but soldering is still not sat- isfactory. The high heat conductivity of the aluminum withdraws the heat of the molten solder so rapidly that it " freezes " before it can flow suffi- ciently. A German solder said to give good results is made of 80# tin to 20# zinc, using a flux composed of 80 parts stearic acid, 10 parts chloride of zinc, and 10 parts of chloride of tin. Pure tin, fusing at 250° C, has also been used as a solder. The use of chloride of silver as a flux has been patented, and used with ordinary soft solder has given some success. A pure nickel soldering-bit should be used, as it does not discolor aluminum as copper bits do. ALLOYS. ALLOYS OF COPPER AND TIN. (Extract from Report of U. S. Test Board.*) Mean Com- position by tcfl c ~ Eg W u £& §.3 o p, 05 CD §»« CD~ 3 > S-p cos H 5 o c. c * '+3?' a3 "gffl.9 Q d .Sep. jq Bronze with 10^ tin 72,053 Bronze with 8^ tin 73,958 Bronze with 6% tin 77,656 AI.L.OYS OF COPPER, TIN, AND ZINC. (Report of U. S. Test Board, Vol. II, 1881.) No. Analysis, Original Mixture. Transverse Strength. Tensile Strength per square inch. Elongation per cent in 5 inches. in Report. Modulus Deflec- Cu. Sn. Zn. of Rupture tion, ins. A. B. A. B. 72 90 5 5 41,334 2.63 23,660 30,740 2.34 9.68 5 88.14 1.86 10 31,986 3 67 3-',000 33,000 17.6 19.5 70 85 5 ; 10 44,457 2.85 28,840 28,560 6.80 5.28 71 85 10 5 62,470 2.56 35,680 36,000 2.51 2.25 89 85 12:5 2.5 62,405 2.83 34,500 32,800 1.29 2.79 88 82.5 12.5 5 69,960 1.61 36,000 34,000 .86 .92 77 82.5 15 2.5 69,045 1.09 33,600 33,800 .68 67 80 5 15 42,618 3.88 37,560 32,300 ii!6" 3.59 68 80 10 10 67,117 2.45 32.830 31, "950 1.57 1.67 69 80 15 5 54,476 .44 32,350 30,760 .55 .44 86 77.5 10 12.5 63,849 1.19 35,500 36,000 1.00 1.00 87 77.5 12.5 10 61.705 .71 36,000 32,500 .72 .59 63 75 5 20 55,355 2.91 33,140 34,960 2.50 3.19 85 75 7.5 17.5 62,607 1.39 33,700 39,300 1.56 1.33 64 75 10 15 58,345 .73 35,320 34,000 1.13 1.25 65 75 15 10 51.109 .31 35,440 28,000 .59 .54 66 75 20 5 40; 235 .21 23,140 27,660 .43 83 72.5 7.5 20 51,839 2.86 32,700 34,800 3.73 "3! 78" 84 72.5 10 17.5 53,230 .74 30,000 30,000 .48 .49 59 70 5 25 57,349 1.37 38,000 32,940 2.06 .99 82 70 7.5 22.5 48,836 .36 38,000 32,400 .84 .40 60 70 10 20 36,520 .18 33,140 26,300 .31 6t 70 15 15 37,924 .20 33,440 27.800 .25 62 70 20 10 15,126 .08 17,000 12,900 .03 81 67.5 2.5 30 58,343 2.91 34,720 45.850 7.27 "3I69" 74 67.5 5 27.5 55,976 .49 34,000 34,460 1.06 .43 75 67.5 7.5 25 46,875 .32 29,500 30,000 .36 .26 80 65 2.5 32.5 56,949 2.36 41,350 38.300 3.26 3.02 55 65 5 30 51 ,369 .56 37,140 36,000 1.21 .61 56 65 10 25 27,075 .14 25.720 22.500 .15 .19 57 65 65 62.5 15 20 2.5 20 15 35 13,591 11,932 69,255 .07 .05 2.34 6,820 3,765 44,400 7,231 2,665 45,000 58 79 "2". 15;"! *2!i9" 78 60 2.5 37.5 69,508 1.46 57,400 52,900 4.87 3.02 52 60 5 35 46,076 .28 41,160 38,330 .39 .40 53 60 10 30 24,699 .13 21,780 21,240 .15 54 60 15 25 18,248 .09 18,020 12,400 12 58.22 2.30 39.48 95,623 1.99 66,500 67,600 "k'.\h" "3.15* 3 58.75 8.75 32.5 35,752 .18 Broke before t est ; ver y brittle 4 57.5 55 21.25 0.5 21.25 44.5 2,752 72,308 .02 3.05 725 68,900 1,300 68,900 73 "9.43"' "2. 88*" 50 55 5 40 38,174 .22 27,400 30,500 .46 .43 51 55 10 35 28,258 .14 25,460 18,500 .29 .10 49 50 5 45 20,814 .11 23,000 31,300 .66 .45 The transverse tests were made in bars 1 in. square, 22 in. between sup- ports. The tensile tests were made on bars 0.798 in. diam. turned from the two halves of the transverse-test bar, one half being marked A and the other B, ALLOYS OF COPPER, TIN, AND ZINC. 323 Ancient Bronzes.— The usual composition of ancient bronze was the same as that of modern gun-metal— 90 copper, 10 tin; but the proportion of tin varies from 5$ to 15$, and in some cases lead has been found. Some an- cient Egyptian tools contained 88 copper, 12 tin. Strength of the Copper-zinc Alloys.— The alloys containing less than 15$ of zinc by original mixture were generally defective. The bars were full of blow-holes, and the metal showed signs of oxidation. To insure good castings it appears that copper-zinc alloys should contain more than 15$ of zinc. From No. 2 to No. 8 inclusive, 16.98 to 80.06$ zinc the bars show a remark- able similarity in all their properties. They have all nearly the same strength and ductility, the latter decreasing slightly as zinc increases, and are nearly alike in color and appearance. Between Nos. 8 and 10, 30.06 and 36.36$ zinc, the strength by all methods of test rapidly increases. Between No. 10 and No. 15, 36.36 and 50.14^ zinc, there is another group, distinguished by high strength and diminished ductility. The alloy of maximum tensile, transverse and torsional strength contains about 41$ of zinc. The alloys containing less than 55$ of zinc are all yellow metals. Beyond 55$ the color changes to white, and the alloy becomes weak and brittle. Be- tween 70$ and pure zinc the color is bluish gray, the brittleness decreases and the strength increases, but not to such a degree as to make them useful for constructive purposes. Difference hetween Composition hy Mixture and hy Analysis.— There is in every case a smaller percentage of zinc in the average analysis than in the original mixture, and a larger percentage of copper. The loss of zinc is variable, but in general averages from 1 to 2$. Liquation or Separation of the Metals.— In several of the bars a considerable amount of liquation took place, analysis showing a difference in composition of the two ends of the bar. In such cases the change in composition was gradual from one end of the bar to the other, the upper end in general containing the higher percentage of copper. A notable instance was bar No. 13, in the above table, turnings from the upper end containing 40.36^ of zinc, and from the lower end 48.52$. Specific Gravity. — The specific gravity follows a definite law. varying with the composition, and decreasing with the addition of zinc. From the plotted curve of specific gravities the following mean values are taken: Per cent zinc 10 20 30 40 50 60 70 80 90 100. Specific gravity 8.80 8.72 8.60 8.40 8.36 8.20 8.00 7.72 7.40 7.20 7.14. Graphic Representation of the L 1 part Copper ) .2 2. % ,a The first brass on the above list is an extremely tough metal with low elastic limit, made purposely so as to " upset " easily. The other, which is called Aluminum-brass No 2, is very hard. We have not in this country or in England any official standard by which to judge of the physical characteristics of cast metals. There are two con- ditions that are absolutely necessary to be known before we can make a fair comparison of different materials: namely, whether the casting was made in dry or green sand or in a chill, and whether it was attached to a larger casting or cast by itself. It has also been found that chill-castings give higher results than sand-castings, and that bars cast by themselves purposely for testing almost invariably run higher than test-bars attached to castings. It is also a fact that bars cut out from castings are generally weaker than bars cast alone, (E. H. Cowles.) Caution as to Reported Strength of Alloys.— The same variation in strength which has been fouud in tests of gun-metal (copper and tin) noted above, must be expected in tests of aluminum bronze and in fact of all alloys. They are exceedingly subject to variation in density and in grain, caused by differences in method of molding and casting, tempera- ture of pouring, size and shape of casting, depth of " sinking head," etc. 330 Alloys. Aluminum Hardened by Addition of Copper Rolled Sheets .04 Inch Thick. (The Engineer, Jau 2, 1891.) Tensile Strength Al. Cu. Sp. Gr. Sp. Gr. in pounds per Per cent. Per cent. Calculated. Determined. square inch. 100 2.67 26,535 98 2 2.78 2.71 43,563 96 4 2.90 2.77 44,130 94 6 3.02 2.82 54,773 92 8 3.14 2.85 50,374 Tests of Aluminum Alloys. (Engineer Harris, U. S. N., Trans. A. I. M. E., vol. xviii.) Composition. Tensile Strength, per sq. in. lbs. Elastic Limit, lbs. per sq.in. Elonga- tion, per ct. Reduc- tion of Area, per ct. Cop- per. Alumi- num. Silicon. Zinc. Iron. 91.50* 88.50 91.50 90.00 6.50* 9.33 6.50 9.00 3.33 3.33 6.50 6.50 9.33 6.50 1.75* 1.66 1.75 1.00 0.33 0.33 1.75 0.50 1.66 0.50 0.25* 0.50 0.25 60,700 66,000 67,600 72,830 82,200 70,400 59,100 53,000 69,930 46,530 18,000 27,000 24,000 33,000 60,000 55,000 19,000 19,000 33,000 17,000 23.2 3.8 13. 2.40 2.33 0.4 15.1 6.2 1.33 7.8 30.7 7.8 21.62 5.78 63.00 63.00 91.50 93.00 33.33* 33.33 "6!25"' 9.88 4.33 23.59 15.5 88.50 92.00 0.50 3.30 19.19 For comparison with the above 6 tests of " Navy Yard Bronze," Cu 88, Sn 10, Zn 2, are given in which the T. S. ranges from 18,000 to 24,590, E. L. from 10,000 to 13,000, El. 2.5 to 5.8*. Red. 4.7 to 10.89. Alloys of Aluminum, Silicon and Iron. M. and E. Bernard have succeeded in obtaining through electrolysis, by treating directly and without previous purification, the aluminum earths (red and white bauxites) the following : Alloys such asferro-aluminum, ferro-silicon-aluminum and silicon-alumi- num, where the proportion of silicon may exceed 10% which are employed in the metallurgy of iron for refining steel and cast-iron. Also silicon-aluminum, where the proportion of silicon does not exceed 10%, which may be employed in mechanical constructions in a rolled or hammered condition, in place of steel, on account of their great resistance, especially where the lightness of the piece in construction constitutes one of the main conditions of success. The following analyses are given: 1. Alloys applied to the metallurgy of iron, the refining of steel and cast iron: Types. Aluminum. Iron. Silicon. Manganese. No. 1 70* 25* 5* 0* No.2 ... 70 20 10 No. 3 70 15 15 No.4 70 10 20 No. 5 70 10 10 10 No. 6 70 trace 20 10 2. Mechanical alloys: Types. Aluminum. Silicon. Iron. No. 1 92* 6.75* 1.25* No.2 90 9.25 0.75 No.3 90 10.00 trace. Up to this time it has been thought that silicon was rather injurious when alloyed with aluminum. From numerous experiences it has been demon- strated that it gives to aluminum some remarkable properties of resistance; the best results were with alloys where the proportion of iron was very low, and the proportion of silicon in the neighborhood of 10*. Above that pro- ALLOYS OF MANGANESE AND COPPER. 331 portion the alloy becomes ciTstalline and can no longer be employed. The density of the alloys of silicon is approximately the same as that of alumi- num. — La Metallurgie, 1892. Tungsten and Aluminum.— Mr. Leinhardt Mannesmann says that the 'addition of a little tungsten to pure aluminum or its alloys communi- cates a remarkable resistance to the action of cold and hot water, salt water and other re-agents. Wh-n the proportion of tungsten is sufficient the alloys offer great resistance to tensile strains. Aluminum and Tin.— M. Bourbouze has compounded an alloy of aluminum and tin, by fusing together 100 parts of the former with 10 parts of tbe latter. This alloy is paler than aluminum, and has a specific gravity of 2.85. The alloy is not as easily attacked by several reagents as alumi- num is, and it can also be worked more readily. Another advantage is that it can be soldered as easily as bronze, without further preliminary prepara- tions. Aluminum-Antimony Alloys.— Dr. C. R. Alder Wright describes some aluminum-antimony alloys in a communication read before the Society of Chemical Industry. The results of his researches do not disclose the existence of a commercially useful alloy of these two metals, and have greater scientific than practical interest. A remarkable point is that the alloy with the chemical composition Al Sb has a higher melting point than either aluminum or antimony alone, and that when aluminum is added to pure antimony the melting-point goes up from that of antimony (450° C.) to a certain temperature rather above that of silver (1000° C). ALLOYS OF MANGANESE AND COPPER, Various Manganese Alloys.— E. H. Cowles, in Trans. A. I. M. E., vol. xviii, p. 495, states that as the result of numerous experiments on mixtures of the several metals, copper, zinc, tin. lead, aluminum, iron, and manganese, and the metalloid silicon, and experiments upon the same in ascertaining tensile strength, ductility, color, etc., the most important determinations appear to be about as follows : 1. That pure metallic manganese exerts a bleaching effect upon copper more radical in its action even than nickel. In other words, it was found that ~18]4% of manganese present in copper produces as white a color in the resulting alloy as 25$ of nickel would do, this being the amount of each required to remove the last trace of red. 2. That upwards of 20$ or 25% of manganese may be added to copper with- out reducing its ductility, although doubling its tensile streugth and chang- ing its color. 3. That manganese, copper, and zinc when melted together and poured into moulds behave very much like the most " yeasty " German silver, producing an ingot which is a mass of blow-holes, and which swells up above the mould before cooling. 4. That the alloy of manganese and copper by itself is very easily oxidized. 5. That the addition of 1.25$ of aluminum to a manganese-copper alloy converts it from one of the most refractory of metals in the casting process into a metal of superior casting qualities, and the noncorrodibility of which is in many instances greater than that of either German or nickel silver. A "silver-bronze '" alloy especially designed for rods, sheets, and wire has the following composition : Manganese, 18; aluminum, 1.20; silicon, 0.5 ; zinc, 13; and copper, 67.5$. It has a tensile strength of about 57,000 pounds on small bars, and 20% elongation. It has been rolled into thin plate and drawn into wire .008 inch in diameter. A test of the electrical conductivity of this wire (of size No. 32) shows its resistance to be 41.44 times that of pure copper. This is far lower conductivity than that of German silver. Manganese Bronze. (F. L. Garrison, Jour. F. I., 1891.)— This alloy has been used extensively for casting propeller-blades. Tests of some made by B. H. Cramp & Co., of Philadelphia, gave an average elastic limit of 30,000 pounds per square inch, tensile strength of about 60,000 pounds per square inch, with an elongation of 8% to 10$ in sand castings. When rolled, the elastic limit is about 80,000 pounds per square inch, tensile strength 95,000 to 106,000 pounds per square inch, with an elongation of 12$ to 15$. Compression tests made at United States Navy Department from the metal in the pouring-gate of propeller-hub of U. S. S. Maine gave in two tests a crushing stress of 126,450 and 135,750 lbs. per sq. in. The specimens w r ere 1 inch high by 0.7 X 0.7 inch in cross-section = 0.49 square inch. Both speci- 332 ALLOYS. mens gave way by shearing, on a plane making an angle of nearly 45° with the direction of stress. A test on a specimen 1 X 1 X 1 inch was made from a piece of the same pouring-gate. Under stress of 150,000 pounds it was flattened to 0.72 inch high by about 1J4 X 134 inches, hut without rupture or any sign of distress. One of the great objections to the use of manganese bronze, or in fact any alloy except iron or steel, for the propellers of iron ships is on account of the galvanic action set up between the propeller and the stern-posts. This difficulty has in great measure been overcome by putting strips of rolled zinc around the propeller apertures in the stern. frames. The following analysis of Parsons' manganese bronze No. 2 was made from a chip from the propeller of Mr. W. K. Vanderbilt's yacht Alva. Copper i 88.644 Zinc 1 .570 Tin 8.700 Iron 0.720 Lead : 0.295 Phosphorus trace 99.923 It will be observed there is no manganese present and the amount of zinc is very small. E. H. Cowles, Trans. A. I. M. E., vol. xviii, says : Manganese bronze, so called, is in reality a manganese brass, for zinc instead of tin is the chief element added to the copper. Mr. P. M. Parsons, the proprietor of this brand of metal, has claimed for it a tensile strength of from 24 to 28 tons on small bars when cast in sand. Mr. W. C. Wallace states that brass-founders of high repute in England will not admit that manganese bronze has more than from 12 to 17 tons tensile strength. Mr. Horace See found tensile strength of 45,000 pounds, and from 6$ to 12J^$ elongation. GERMAN-SILVER AND OTHER NICKEL ALLOYS. Copper. Nickel. Zinc. Chinese packfong 40.4 31.6 6.5 parts, tutenag 8 3 6.5 " German silver , 2 1 1 " " " (cheaper) 8 2 3.5 " " " (closely resembles sil). 8 .3 3.5 " For analyses of some German-silvers see page 326. German Silver.— The composition of German silver is a very uncertain thing and depends largely on the honesty of the manufacturer and the price the purchaser is willing to pay. It is composed of copper, zinc, and nickel in varying propor lions. The best varieties contain from 18$ to 25$ of nickel and from 20$ to 30$ of zinc, the remainder being copper. The more expensive nickel silver contains from 25$ to 33$ of nickel and from 75$ to 66$ of copper. The nickel is used as a whitening element; it also strengthens the alloy and renders it harder and more non-corrodible than the brass made without it, of copper and zinc. Of all troublesome alloys to handle in the foundry or rolling-mill, German silver is the worst. It is unmanageable and refractory at every step in its transition from the crude elements into rods, sheets, or wire. (E. H. Cowles, Trans. A. I. M. E., vol. xviii. p. 494.) ALLOYS OF BISMUTH. By adding a small amount of bismuth to lead that metal may be hard- ened and toughened. An alloy consisting of three parts of lead and two of bismuth has ten times the hardness and twenty times the tenacity of lead. The alloys of bismuth with both tin and lead are extremely fusible, and take fine impressions of casts and moulds. An alloy of one part bismuth, two parts tin, and one part lead is used by pewter-workers as a soft solder, and by soap-makers for moulds. An alloy of five parts bismuth, two parts tin, and three parts lead melts at 199° F , and is somewhat used for ster- eotyping, and for metallic writing-pencils. Thorpe gives the following proportions for the better-known fusible metals: BEARING-METAL ALLOYS. 333 Name of Alloy. Bismuth. Lead. Tin. Cad- mium Mer- cury. Melting- point. 50 50 50 50 50 50 50 31.25 28.10 25.00 25.00 25.00 26.90 20.55 18.75 24.10 25.00 25.00 12.50 12.78 21.10 202° F. 203° " 201° " D'Arcet's with mercury. Wood's 12 ".50 10.40 14.03 250.6 113° " 149° " 149° " Guthrie's l ' Entectic ' ' . . . " Very low. 1 ' The action of heat upon some of these alloys is remarkable. Thus, Lipo- witz's alloy, which solidifies at 149° Fah., contracts very rapidly at first, as it cools from this point. As the cooling goes on the contraction becomes slower and slower, until the temperature falls to 101.3° Fah. From this point the alloy expands as it cools, until the temperature falls to about 77° Fah., after which it again contracts, so that at 32° F. a bar of the alloy has the same length as at 115° F. Alloys of bismuth have been used for making fusible plugs for boilers, but it is found that they are altered by the continued action of heat, so that one cannot rely upon them to melt at the proper temperature. Pure Banca tin is used by the U. S. Government for fusible plugs. FUSIBLE ALLOYS. (From various sources.) Sir Isaac Newton's, bismuth 5, lead 3, tin 2, melts at 212° F. Rose's, bismuth 2, lead 1, tin 1, melts at 200 " Wood's, cadmium 1, bismuth 4, lead 2. tin 1, melts at 165 " Guthrie's, cadmium 13.29, bismuth 47.38, lead 19.36, tin 19.97, melts at. 160 " Lead 3, tin 5, bismuth 8, melts at , . 208 " Lead 1, tin 3, bismuth 5, melts at 212 " Lead 1, tin 4, bismuth 5, melts at 240 " Tin 1, bismuth 1, melts at , 286 " Lead 2, tin 3, melts at 334 " Tin 2, bismuth 1, melts at 336 " Lead 1, tin 2, melts at 360 " Tin 8, bismuth 1, melts at 392 " Lead 2, tin 1, melts at 475 " Lead 1, tin 1, melts at 466 " Lead 1, tin 3, melts at 334 " Tin 3, bismuth 1, melts at 392 " Lead 1, bismuth 1, melts at 257 " Lead 1, Tin 1, bismuth 4, melts at 201 " Lead 5, tin 3, bismuth 8," melts at 202 " Tin 3, bismuth 5, melts at , 202 " BEARING-METAL ALLOYS. (G B. Dudley, Jour. F. I., Feb. and March, 1892.) Alloys are used as bearings in place of wrought iron, cast iron, or steel, partly because wear and friction are believed to be more rapid when two metals of the same kind work together, partly because the soft metals are more easily worked and got into proper shape, and partly because it is de- sirable to use a soft metal which will take the wear rather than a hard metal, which will wear the journal more rapidly. A good bearing-metal must have five characteristics: (1) It must be strong enough to carry the load without distortion. Pressures on car-journals are frequently as high as 350 to 400 lbs. per square inch. (2) A good bearing-metal should not heat readily. The old copper-tin bearing, made of seven parts copper to one part tin, is more apt to heat than some other allovs. In general, research seems to show that the harder the bearing-metal, the more likely it is to heat. (3) Good bearing-metal should work well in the foundry. Oxidatiou while melting causes spongy castings. It can be prevented by a liberal use of powdered charcoal while melting. The addition of \% to 2% of zinc or a small amount of phosphorus greatly aids in the production of sound cast- ings. This is a principal element of value in phosphor- brgnz§, 334 (4) Good bearing-metals should show small friction. It is true that friction is almost wholly a question of the lubricant used; but the metal of the bear- ing has certainly some influence. (5) Other things being equal, the best bearing-metal is that which wears slowest. The principal constituents of bearing-metal alloys are copper, tin, lead, zinc, antimony, iron, and aluminum. The following table gives the constitu- ents of most of the prominent bearing-metals as analyzed at the Pennsyl- vania Railroad laboratory at Altoona. Analyses of Bearing-metal Alloys. Cop- per. Tin. Lead. Zinc Camelia metal Anti-friction metal White metal Car-brass lining Salgee anti-friction Graphite bearing-metal Antimonial lead Carbon bronze Cornish bronze Delta metal ♦Magnolia metal American anti-friction metal.. Tobin bronze Graney bronze Damascus bronze Manganese bronze Ajax metal Anti-friction metal Harrington bronze Car-box metal Hard lead Phosphor-bronze Ex. B. metal 70.21 1.61 trace 9 91 14.38 75.47 77.83 92.39 trace 59.00 75.80 76.41 90.52 81.24 2.K 9.20 10.60 9.58 10.98 87.92 84 1.15 67.73 80.69 14.5' 12.40 5.10 83.55 78.44 0.31 15.06 12.52 12.08 15.10 16.73 18.83 0.55 trace ? (1) trace 0.98 38.40 16.45 19.60 (2) trace(3) 0.07 trace(4) 0.65 0.11 0.97 8.00 7.2^ 88.32 84.33 94.40 9.61 15.00 .(5) • (6) 42j trace • (7) • (8) Other constituents: (1) No graphite. (5) No manganese. (2) Possible trace of carbon. (6) Phosphorus or arsenic, 0.37. (3) Trace of phosphorus. (7) Phosphorus, 0.94. (4) Possible trace of bismuth. (8) Phosphorus, 0.20. * Dr. H. C. Torrey says this analysis is erroneous and that Magnolia metal always contains tin. As an example of the influence of minute changes in an alloy, the Har- rington bronze, which consists of a minute proportion of iron in a copper- zinc alloy, showed after rolling a tensile strength of 75,000 lbs. and 20$ elon- gation in 2 inches. In experimenting on this subject on the Pennsylvania Railroad, a certain number of the bearings were made of a standard bearing-metal, and the same number were made of the metal to be tested. These bearings were placed on opposite ends of the same axle, one side of the car having the standard bearings, the other the experimental. Before going into service the bearings were carefully weighed, and after a. sufficient time they were again weighed. The standard bearing-metal used is the " S bearing-metal " of the Phos- phor-bronze Smelting Co. It contains about 79.70$ copper, 9.50$ lead. 10$ tin, and 0.80$ phosphorus. A large number of experiments have shown that the loss of weight of a bearing of this metal is 1 lb. to each 18,000 to 25,000 miles travelled. Besides the measurement of wear, observations were made on the frequency of " hot boxes " with the different metals. The results of the tests for wear, so far as given, are condensed into the following table: BEARIKG-METAL ALLOYS. 335 Composition. Rate Metal. . -*• — -^ of Copper. Tin. Lead. Phos. Arsenic. Wear. Standard 79.70 10.00 9.50 0.80 — 100 Copper-tin 87.50 12.50 .... .... 148 Copper-tin, second experiment, same metal 153 Copper-tin, third experiment, same metal. .... .147 Arsenic-bronze 89.20 10.00 ... .... 0.80 142 Arsenic-bronze 79.20 10.00 7.00 .... 0.80 115 Arsenic-bronze 79.70 10.00 9.50 .... 0.80 101 "K"bronze 77.00 10.50 12.50 .... .... 92 " K " bronze, second experiment, same metal : 92.7 Alloy"B" 77.00 8.00 15.00 .... .... 86.5 The old copper-tin alloy of 7 to 1 lias repeatedly proved its inferiority to the phosphor-bronze metal. Many more of the copper-tin bearings heated than of the phosphor-bronze. The showing of these tests was so satisfac- tory that phosphor-bronze was adopted as the standard bearing-metal of the Pennsylvania R.R , and was used for a long time. The experiments, however, were continued. It was found that arsenic practically takes the place of phosphorus in a copper-tin alloy, and three tests were made with arsenic- bronzes as noted above. As the proportion to lead is increased to correspond with the standard, the durability increases as well. In view of these results the "K " bronze was tried, in which neither phosphorus nor arsenic were used, and in which the lead was increased above the proportion in the standard phosphor-bronze. The result was that the metal wore 7.30% slower than the phosphor-bronze. No trouble from heating was experienced with the " K " bronze more than with the standard. Dr. Dudley continues: At about this time we began to find evidences that wear of bearing-metal alloys varied in accordance with the following law: " That alloy which has the greatest power of distortion without rupture (resilience), will best resist wear." It was now attempted to design an alloy in accordance with this law, taking first the proportions of copper and tin, 9V£ parts copper to 1 of tin was settled on by experiment as the standard, although some evidence since that time tends to show that 12 or possibly 15 parts copper to 1 of tin might have been better. The influence of lead on this copper-tin alloy seems to be much the same as a still further diminution of tin. However, the tendency of the metal to yield under pressure increases as the amount of tin is diminished, and the amount of the lead increased, so a limit is set to the use of lead. A certain amount of tin is also necessary to keep the lead alloyed with the copper. Bearings were c ist of the metal noted in the table as alloy " B," and it wore 13.5% slower than the standard phosphor-bronze. This metal is now the standard bearing-metal of the Pennsylvania Railroad, being slightly changed in composition to allow the use of phosphor-bronze scrap. The formula adopted is: Copper, 105 lbs.; phosphor-bronze, 60 lbs. ; tin, 9% lbs.; lead, 25J4 lbs. By using ordinary care in the foundry, keeping the metal well covered with charcoal during: the melting, no trouble is found in casting good bearings with this metal. The copper and the phosphor-bronze can be put in the pot before putting it in the melting-hole. The tin and lead should be added after the pot is taken from the fire. It is not known whether the use of a little zinc, or possibly some other combination, might not give still better results. For the present, however, this alloy is considered to fulfil the various conditions required for good bearing-metal better than any other alloy. The phosphor-bronze had an ultimate tensile strength of 30,000 lbs., with 6% elongation, whereas the alloy " B " had 24,000 lbs. tensile strength and 11% elongation. (For other bearing-metals, see Alloys containing antimony, on next page. S36 ALLOYS CONTAINING ANTIMONY. Various Analyses of Babbitt Metal and other Alloys containing Antimony. Tin. 50 =89.3 96 =88.9 85.7 81.9 81.0 70.5 22 45.5 89.3 85 Copper Antimony. Zinc. Lead. Bismuth. 1 1.8 4 3.7 1.0 "2"" 4 10 1.5 1.8 5 5 parts 8.9perct. 8 parts 7.4 per ct. 10.1 16.2 16. 25.5 62. 13. 7.1 10. Harder Babbitt | for bearings* j 2.9 1.9 1. »i « " 6. "Babbitt" 40.0 Plate pewter. . White metal. . . 1 ft Bearings on Ger. locomotives. * It is mixed as follows: Twelve parts of copper are first melted and then 36 parts of tin are added; 24 parts of antimony are put in, and then 36 parts of tin, the temperature being lowered as soon as the copper is melted in order not to oxidize the tin and antimony, the surface of the bath being protected from contact with the air. The alloy thus made is subsequently remelted in the proportion of 50 parts of alloy to 100 tin. (Joshua Rose.) White-metal Alloys.— The following alloys are used as lining metals by the Eastern Railroad of France (1890): Number. 1 2 3 Lead. 65 70 Antimony. 25 11.12 20 8 Tin. 83.33 10 12 Copper. 5.55 4 80 No. 1 is used for lining cross-head slides, rod-brasses and axle-bearings: No. 2 for lining axle-bearings and connecting-rod brasses of heavy engines; No. 3 for lining eccentric straps and for bronze slide-valves; and No. 4 for metallic rod-packing. Some o£ the best-known white-metal alloys are the following (Circular of Hoveler & Dieckhaus, London, 1893): Tin. Antimony. 1. Parsons' 2. Richards' 3. Babbitt's 4. Fentons' 5. French Navy.. 6. German Navy . . 70 55 16 85 1 15 18 7^ Lead. 2 10^ Copper. Zinc. 2 27 4% 79 87^ " There are engineers who object to white metal containing lead or zinc. This is, however, a prejudice quite unfounded, inasmuch as lead and zinc often have properties of great use in white alloys." It is a further fact that an "easy liquid" alloy must not contain more than 18$ of antimony, which is an invaluable ingredient of white metal for improving its hardness; but in no case must it exceed that margin, as this would reduce the plasticity of the compound and make it brittle. Hardest alloy of tin and lead: 6 tin, 4 lead. Hardest of all tin alloys (?): 74 tin, 18 antimony, 8 copper. Alloy for thin open-work, ornamental castings: Lead 2, antimony 1. White metal for patterns: Lead 10, bismuth 6, antimony 2, common brass 8, tin 10. Type-metal is made of various proportions of lead and antimony, from 17$ to 20% antimony according to the hardness desired. Babbitt Metals. (C. R. Tompkins, Mechanical Netvs, Jan. 1891.) The practice of lining journal-boxes with a metal that is sufficiently fusi- ble to be melted in a common ladle is not always so much for the purpose of securing anti-friction properties as for the convenience and cheapness of forming a perfect bearing in line with the shaft without the necessity of ALLOYS CONTAINING ANTIMONY. 337 boring them. Boxes that are bored, no matter how accurate, require great care in fitting and attaching them to the frame or other parts of a machine. It is not good practice, however, to use the shaft for the purpose of cast- ing the bearings, especially if the shaft be steel, for the reason that the hot metal is apt to spring it; the better plan is to use a mandrel of the same size or a trifle larger for this purpose. For slow-running journals, where the load is moderate, almost any metal that may be conveniently melted and will run free will answer the purpose. For wearing properties, with a moderate speed, there is probably nothing superior to pure zinc, but when not combined with some other metal it shrinks so much in cooling that it cannot be held firmly in the recess, and soon works loose; and it lacks those anti-friction properties which are necessary in order to stand high speed. For line-shafting, and all work where the speed is not over 300 or 400 r. p. m., an alloy of 8 parts zinc and 2 parts block-tin will not only wear longer than any composition of this class, but will successfully resist the force of a heavy load. The tin counteracts the shrinkage, so that the metal, if not overheated, will firmly adhere to the box until it is worn out. But this mixture does not possess sufficient anti-friction properties to warrant its use in fast-running journals. Among all the soft metals in use there are none that possess greater anti- . friction properties than pure lead; but lead alone is impracticable, for it is so soft that it cannot be retained in the recess. But when by any process lead can be sufficiently hardened to be retained in the boxes without materially injuring its anti-friction properties, there is no metal that will wear longer in light fast-running journals. With most of the best and most popular anti-friction metals in use and sold under the name of the Babbitt metal, the basis is lead. Lead and antimony have the property of combining with each other in all proportions without impairing the anti-friction properties of either. The antimony hardens the lead, and when mixed in the proportion of 80 parts lead by weight with 20 parts antimony, no other known composition of metals possesses greater anti-friction or wearing properties, or will stand a higher speed without heat or abrasion. It runs free in its melted state, has no shrinkage, and is better adapted to light high-speeded machinery than any other known metal. Care, however, should be manifested in using it, and it should never be heated beyond a temperature that will scorch a dry pine stick. Many different compositions are sold under the name of Babbitt metal. Some are good, but more are worthless; while but very little genuine Babbitt metal is sold that is made strictly according to the original formula. Most of the metals sold under that name are the refuse of type-foundries and other smelting-works, melted and cast into fancy ingots with special brands, and sold under the name of Babbitt metal. It is difficult at the present time to determine the exact formulas used by the original Babbitt, the inventor of the recessed box, as a number of differ, ent formulas are given for that composition. Tin, copper, and antimony were the ingredients, and from the best sources of information the original proportions were as follows : Another writer gives: 50partstin = 89. 3# 83. 3# 2parts copper = 3.6% 8.3% 4 parts antimony = t.\% 8.3% The copper was first melted, and the antimony added first and then about ten or fifteen pounds of tin, the whole kept at a dull-red heat and constantly stirred until the metals were thoroughly incorporated, after which the balance of the tin was added, and after being thoroughly stirred again it was then cast into ingots. When the copper is thoroughly melted, and before the antimony is added, a handful of powdered charcoal should be thrown into the crucible to form a flux, in order to exclude the air and pre- vent the antimony from vaporizing; otherwise much of it will escape in the form of a vapor and consequently be wasted. This metal, when carefully prepared, is probably one of the best metals in use for lining boxes that are subjected to a heavy weight and wear; but for light fast-running journals the copper renders it more susceptible to friction, and it is more liable to heat than the metal composed of lead and antimony in the proportions just given. 338 STRENGTH OF MATERIALS. SOLDERS. Common solders, equal parts tin and lead ; fine solder, 2 tin to 1 lead ; cheap solder, 2 lead, 1 tin. Fusing-point of tin- lead alloys: Tin 1 to lead 25 . . " 1 " " 10.. 1 . .334° F. ..340 . 356 .558° F. Tin 1^ to lead ..541 " 2 ' ..511 " 3 3 482 " 4 2 441 " 5 " 1 " " 1 370 " 6 Common pewter contains 4 lead to 1 tin. Gold solder: 14 parts gold, 6 silver, 4 copper. Gold solder for 14-carat gold: 25 parts gold, 25 silver, 12^ brass, 1 zinc. Silver solder: Yellow brass 70 parts, zinc 7, tin 11^. Another: Silver 145 parts, brass (3 copper, 1 zinc) 73, zinc 4. German-silver solder: Copper 38, zinc 54, nickel 8. Novel's solders for aluminum: Tin 100 parts, lead 5; melts at 536° to 572° F. " 100 " zinc 5; " 536 to 612 "1000 " copper 10 to 15; " 662 to 842 "1000 " nickel 10 to 15; " 662 to 842 Novel's solder for aluminum bronze: Tin 900 parts, copper 100, bismuth 2 to 3. It is claimed that this solder is also suitable for joining aluminum to copper, brass, zinc, iron, or nickel. ROPES AND CABLES. STRENGTH OF ROPES. (A S. Newell & Co., Birkenhead. Klein's Translation of Weisbach, vol. iii, part 1, sec. 2.) \ Hemp. Iron. Steel. Tensile Weight Weight Weight Strength. Girth. per Fathom. Girth. per Fathom. Girth. per Fathom. Inches. Pounds. Inches. Pounds. Inches. Pounds. Gross tons. 2% 2 1 1 2 m M 1 1 3 m 4 m 2 4 m 2^ m w% 5 m 5 3 6 2 Wz \% 2 7 5^ 7 m 4 m 2^ 8 9 6 9 2y* 5 ■m 3 10 11 6^2 10 m 6 2 m 12 m §y% 2^ 4 13 7 12 7 2J4 4^ 14 3 Wa 15 7^ 14 3^ m m 8 2% 5 16 17 8 16 9 m 5^ 18 m 10 2% 6 20 m 18 3% m 11 12 2% 6^ 22 24 w* 22 m 13 3J4 8 26 10 26 4 14 28 11 30 m 15 m, 9 30 4^ 16 18 m 10 32 36 12 34 m 20 m 12 40 STRENGTH OF ROPES. Flat Ropes. 339 Hemp. Iron. Steel. Weight Weight. Weight Tensile Strength. Girth. per Fathom. Girth. per Fathom . Girth. per Fathom . Inches. Pounds. Inches. Pounds. Inches. Pounds. Gross tons. 4 xl^ 20 2y 4 xy% 11 20 5 x \y A 5V£ x 1% 24 y&*% 13 23 26 m*% 15 27 m * w* 28 3 x% 16 2 x^ 10 28 6 x 1^ 30 m* 5 /8 18 2yi*y 2 11 32 7 xl% 36 3y 2 *y 8 20 2y^y 2 12 36 m*m 40 3^x11/16 22 2y 2 xy 2 13 40 sy 2 x 2y 4 45 4 x 11/16 25 2%x% 15 45 9 x 2i/ 2 50 4^x34 28 3 x% 16 50 9^x2% 55 4^x?4 32 3^x3^ 18 56 10 x 2^ 60 4%xH 34 3^x3^ 20 60 Working Load, Diameter, and Weight of Ropes and Chains. (Klein's Weisbach, vol. iii, part 1, sec. 2, p. 561.) Hemp ropes: d = diam. of rope. Wire rope: d = diam. of wire, n -— number of wires, G = weight per running foot, k = permissible load in pounds per square inch of section, P — permissible load on rope or chain. Oval chains : d = diam of iron used ; inside dimensions of oval l.5d and 2.6d. Each link is a piece of chain 2.6d long. G = weight of a single link == 2.10d 3 lbs. ; G = weight per running foot = 9.73d 2 lbs. Hempen Rope. Wire Rope. Dry and Un tarred. Wet or Tarred. k (lbs.) = d (ins.) = P(lbs.) = G (lbs ) = 1420 0.03 VP 1120d 2 = 2855(? 1.28cZ2 = 0.00035P 1160 0.033 VP 916d2 = 1975G 1.54d2 == o.0005P 17000 0.0087 \f- v n 13350nd 2 = 4590G 2.91wd 2 =0.000218P Open-link Chain. Stud-link Chain. k (lbs. d (ins P(lbs G(lbs ) = ) = ) = ) = 8500 0.0087 VP 13350d 2 = V660G 9.73d 2 = 0.000737P 11400 0.0076 \/P 17800^2 = 1660# 10.65d 2 = 0.0006P Stud Chains 4/3 times as strong as open-link variety. [This is contrary to the statements of Capt. Beardslee, U. S. N., in the report of the U. S. Test Board. He holds that the open link is stronger than the studded link. See p. 308 ante]. 340 STRENGTH OF MATERIALS. STRENGTH AND WEIGHT OF WIRE ROPE, HEMPEN ROPE, AND CHAIN CABLES. (Klein's Weisbach.) Breaking Load in tons of 2240 lbs. Kind of Cable. Girth of Wire Rope and of Hemp Rope Diameter of Iron of Chain, inches. Weight of One Foot In length. Pounds. 1 Ton ( Wire Rope < Hemp Rope ( Chain ( Wire Rope ■I Hemp Rope { Chain I Wire Rope •< Hemp Rope ( Chain Wire Rope •< Hemp Rope ( Chain ( Wire Rope ■< Hemp Rope ( Chain ( Wire Rope < Hemp Rope ( Chain i Wire Rope •< Hemp Rope ( Chain | Wire Rope ■\ Hemp Rope ( Chain i Wire Rope •< Hemp Rope ( Chain ( Wire Rope ■< Hemp Rope ( Chain 1.0 2.0 H 2.0 5.0 y z 2.5 7.0 11/16 3.0 8.0 13/16 3.5 9.0 29/32 4.0 10.0 31/32 4.5 11.0 1.1/16 5.0 12.5 1.3/16 5.5 14.0 1.5/16 6.0 5.0 1.7/16 0.125 0.177 8 Tons 0.500 0.438 978 12 Tons 16 Tons 2.667 0.753 2.036 4.502 1.136 2.365 20 Tons 24 Tons 6.169 1.546 3.225 7.674 2.043 4.166 30 Tons .. 36 Tons 8.836 2.725 5.000 10.335 3.723 5 940 44 Tons 54 Tons 13.01 4.50 6.94 16.00 5.67 7.92 19.16 Length sufficient to provide the maximum working stress : Hempen rope, dry and untarred 2855 feet. " " wet or tarred 1975 *' Wire rope 4590 " Open-link chain 1360 " Stud chain 1660 " Sometimes, when the depths are very great, ropes are given approximately the form of a body of uniform strength, by making them of separate pieces, whose diameters diminish towards the lower end. It is evident that by this means the tensions in the fibres caused by the rope's own weight can be considerably diminished. Rope for Hoisting or Transmission. Manila Rope (C. W. Hunt Company, New York.)— Rope used for hoisting or for trans- mission of power is subjected to a very severe test. Ordinary rope chafes and grinds to powder in the centre, while the exterior may look as though it was little worn. In bending a rope over a sheave, the strands and the yarns of these strands slide a small distance upon each other, causing friction, and wear the rope internally. The " Stevedore " rope used by the C. W. Hunt Co. is made by lubricating the fibres with plumbago, mixed with sufficient tallow to hold it in position. This lubricates the yarns of the rope, and prevents internal chafing and wear. After running a short time the exterior of the rope gets compressed and coated with the lubricant. In manufacturing rope, the fibres are first spun into a yarn, this yarn being twisted in a direction called "right hand." From 20 to 80 of these yarns, depending on the size of the rope, are then put together and twisted in the opposite direction, or "left hand," into a strand. Three of these STRENGTH OF ROPES. 341 strands, for a 3-strand, or four for a 4-strand rope, are then twisted together, the twist being again in the "right hand " direction. When the strand is twisted, it untwists each of the threads, and when the three strands are twisted together into rope, it untwists the strands, but again twists up the threads. It is this opposite twist that keeps the rope in its proper form. When a weight is hung on the end of a rope, the tendency is for the rope to untwist, and become longer. In untwisting the rope, it would twist the threads up, and the weight will revolve until the strain of the untwisting strands just equals the strain of the threads being twisted tighter. In making a rope it is impossible to make these strains exactly balance each other. It is this fact that makes it necessary to take out the "turns" in a new rope, that is, untwist it when it is put at work. The proper twist that should be put in the threads has been ascertained approx- imately by experience. The amount of work that the rope will do varies greatly. It depends not only on the quality of the fibre and the method of laying up the rope, but also on the kind of weather when the rope is used, the blocks or sheaves over which it is run, and the strain in proportion to the strain put upon the rope. The principal wear comes in practice from defective or badly set sheaves, from excess of load and exposure to storms. The loads put upon the rope should not exceed those given in the tables, for the most economical wear. The indications of excessive load will be the twist coming out of the rope, or one of the strands slipping out of its proper position. A certain amount of twist comes out in using it the first day or two, but after that the rope should remain substantially the same. If it does not, the load is too great for the durability of the rope. If the rope wears on the outside, and is good on the inside, it shows that it has been chafed in running over the pulleys or sheaves. If the blocks are very small, it will increase the sliding of the strands and threads, and result in a more rapid internal wear. Rope made for hoisting and for rope transmission is usually made with four strands, as experience has shown this to be the most serviceable. The strength and weight of " stevedore " rope is estimated as follows: Breaking strength in pounds = 720 (circumference in inches) 2 ; Weight in pounds per foot = .032 (circumference in inches) 2 . The Technical Words relating to Cordage most frequently heard are: Yarn . —Fibres twisted together. Thread.— Two or more small yams twisted together. String.— The same as a thread but a little larger yarns. Strand.— Two or more large yams twisted together. Cord. — Several threads twisted together. Rope. — Several strands twisted together. Hawser.— A rope of three strands. Shroud-Laid.— A rope of four strands. ' Cable —Three hawsers twisted together. Yarns are laid up left-handed into strands. Strands are laid up right-handed into rope. Hawsers are laid up left-handed into a cable. A rope is : Laid by twisting strands together in making the rope. Spliced by joining to another rope by interweaving the strands. Whipped.— By winding a string around the end to prevent untwisting. Served.— When covered by winding a yarn continuously and tightly around it. Parceled.— By wrapping with canvas. Seized. — When two parts are bound together by a yarn, thread or string. Payed.— When painted, tarred or greased to resist wet. Haul. — To pull on a rope. Taut. — Drawn tight or strained. Splicing of Ropes. — The splice in a transmission rope is not only the weakest part of the rope but is the first part to fail when the rope is worn out. If the rope is larger at the splice, the projecting part will wear on the pulleys and the rope fail from the cutting off of the strands. The following directions are given for splicing a 4-strand rope. The engravings show each successive operation in splicing a \% inch manila rope. Each engraving was made from a full-size specimen. 342 STRENGTH OE MATERIALS. Fig. 81. Splicing of Ropes. SPLICING OF ROPES. 343 Tie a piece of twine, 9 and 10, around the rope to be spliced, about 6 feet from each end. Then unlay the strands of each end back to the twine. Butt the ropes together and twist each corresponding pair of strands loosely, to keep them from being tangled, as shown in Fig. 78. The twine 10 is now cut. and the strand 8 unlaid and strand 7 carefully laid in its place for a distance of four and a half feet from the junction. The strand (3 is next unlaid about one and a half feet and strand 5 laid in its place. The ends of the cores are now cut off so they just meet. Unlay strand 1 four and a half feet, laying strand 2 in its place. Unlay strand 3 one and a half feet, laying in strand 4. Cut all the strands off to a length of about twenty inches, for convenience in manipulation. The rope now assumes the form shown in Fig. 79 with the meeting points of the strands three feet apart. Each pair of strands is successively subjected to the following operation: From the point of meeting of the strands 8 and 7, unlay each one three turns; split both the strand 8 and the strand 7 in halves as far back as they are now unlaid and " whip " the end of each half strand with a small piece of twine. The half of the strand 7 is now laid in three turns and the half of 8 also laid in three turns. The half strands no.w meet and are tied in a simple knot, 11, Fig. 80, making the rope at this point its original size. The rope is now opened with a marlin spike and the half strand of 7 worked around the half strand of 8 by passing the end of the half strand 7 through the rope, as shown in the engraving, drawn taut and again worked around this half strand until it reaches the half strand 13 that was not laid in. This half strand 13 is now split, and the half strand 7 drawn through the opening thus made, and then tucked under the two adjacent sti'ands, as shown in Fig. 81. The other half of the strand 8 is now wound around the other half strand 7 in the same manner. After each pair of strands has been treated in this manner, the ends are cut off at 12, leaving them about four inches long. After a few days' wear they will draw into the body of the rope or wear off. so that the locality of the splice can scarcely be detected. Coal Hoisting. (C. W. Hunt Co.).— The amount of coal that can be hoisted with a rope varies greatly. Under the ordinary conditions of use a rope hoists from 5000 to 8000 tons. Where the circumstances are more favorable, the amounts run up frequently to 12,000 or 15,000 tons, occasion- ally to 20,000 and in one case 32,400 tons to a single fall. When ahoisting rope is first put in use, it is likely from the strain put upon it to twist up when the block is loosened from the tub. This occurs in the first day or two only. The rope should then be taken down and the "turns " taken out of the rope. When put up again the rope should give no further trouble until worn out. It is necessary that the rope should be much larger than is needed to bear the strain from the load. Practical experience for many years has substantially settled the most economical size of rope to be used which is given in the table below. Hoisting ropes are not spliced, as it is difficult to make a splice that will not pull out while running over the sheaves, and the increased wear to be obtained in this way is very small. Coal is usually hoisted with what is commonly called a " double whip; " that is, with a running block that is attached to the tub which reduces the strain on the rope to approximately one half the weight of the load hoisted. The following table gives the usual sizes of hoisting rope and the proper working strain: Stevedore Hoisting-rope. C. W. Hunt Co. Circumference of the rope in ins. 4 4^ Proper Working Nominal size of Approximate Strain on the Rope Coal tubs. Double Weight of a Coil, in lbs. whip. in lbs. 350 1/6 to 1/5 tons. 360 500 1/5 =a :: 480 650 M 650 800 1000 S ::« :: 830 960 Hoisting rope is ordered by circumference, transmission rope by diameter. 344 STRENGTH OF MATERIALS. Weight and Strength of Manila Cordage. Dodge Manufacturing Co. o ^ 11 o g a Q P. a q T3 a 3 Sg ■Si a 0) P« c o a Size. Dia in inche <»fa 5 en fe w nil 1*8 «3 O .9 .S3 5 55 " c » £ S s* fa £ 1 00"° A fa 3/16 12 540 50' 1 5/16 310 16,000 1' W H 18 780 33' 4" 1% 346 18,062 1 8 5/16 24 1,000 25 v/% 390 20,250 1 6 % 30 1,280 20 1 9/16 435 22,500 1 5 7/16 37 1,562 17 8 1% 480 25,000 1 3 M 46 2,250 13 1« 581 30,250 1 9/16 65 3,062 9 3 2 678 36,000 10% % 80 4,000 7 6 m 797 42,250 9 % 13/16 98 5,000 6 2 l A 920 49,000 6% 120 6,250 5 2y 2 1,106 56,250 m % 142 7,500 4 3 2% 1,265 64,000 5^ 1 170 9,000 3 6 Ws 1,420 72,250 5 1 1/16 200 10,500 3 3 1,572 81,000 &A v/& 230 12,250 2 7 M 1,760 90,250 4 M 271 14,000 2 3 33/ 8 1,951 100,000 3^ T. Spencer Miller (Enc/g Neivs, Dec. 6, 1890) gives the following table of breaking strength of mauila rope, which he considers more reliable than the strength computed by Mr. Hunt's formula, Breaking strength = 720 X (circumference in inches) 2 . Mr. Miller's formula is: Breaking weight lbs. = circumference 2 X a coefficient which varies from 900 for \#' to 700 for 2" diameter rope, as shown in the table. Diam. in. Circum- ference. in. Ultimate Strength. lbs. Coeffi- cient. Diam. in. Circum- ference, in. Ultimate Strength. lbs. Coeffi- cient. 2 2H 2,000 3,250 4,000 6,000 7,000 9,350 900 845 820 790 780 765 m m m 2 5 6 10,000 13,000 15,000 18,200 21,750 25,000 760 745 735 725 712 700 For rope-driving Mr. Hunt recommends that the working strain should not exceed 1/20 of the ultimate breaking strain. For further data on ropes see " Rope-driving." Knots. — A great number of knots have been devised of which a few only are illustrated, but those selected are the most frequently used. In the cuts. Fig. 82, they are shown open, or before being drawn taut, in order to show the position of the parts. The names usually given to them are: A. Bight of a rope. B. Simple or Overhand knot. C. Figure 8 knot. D. Double knot. E. Boat knot. F. Bowline, first step. G. Bowline, second step. H. Bowline completed. I. Square or reef knot. J. Sheet bend or weaver's knot. K. Sheet bend with a toggle. L. Carrick bend. M. Stevedore knot completed. N. Stevedore knot commenced. O. Slip knot. P. Flemish loop. Q. ~ R. Chain knot with toggle. Half-hitch. Timber-hitch. Clove hitch. Rolling-hitch. Timber-hitch and half-hitch. W. Blackwall-hitch. X. Fisherman's bend. Round turn and half-hitch. Wall knot commenced. " " completed. Wall knot crown commenced. " " " completed. U. Y. Z. A A. BB. CC. 345 The principle of a knot is that no two parts, which would move in the same direction if the rope were to slip, should lay along side of and touch- ing each other. The bowline is one of the most useful knots, it will not slip, and after being strained is easily untied. Commence by making a bight in the rope, then put the end through the bight and under" the standing part as shown in O, then pass the end again through the bight, and haul tight. The square or reef knot must not be mistaken for the " granny " knot that slips under a strain. Knots H, K and M are easily untied after being under strain. The knot M is useful when the rope passes through an eye and is held by the knot, as it will not slip and is easily untied after being strained. ABC E Fig. 82.— Knots. The timber hitch S looks as though it would give way, but it will not; the greater the strain the tighter it will hold. The wall knot looks complicated, but is easily made by proceeding as follows: Form a bight with strand 1 and pass the strand 2 around the end of it, and the strand 3 round the end of 2 and then through the bight of 1 as shown in the cut Z. Haul the ends taut when the appearance is as shown in AA. The end of the strand 1 is now laid over the centre of the knot, strand 2 laid over 1 and 3 over 2, when the end of 3 is passed through the bight of 1 as shown in BB. Haul all the strands taut as shown in CC. 346 STRENGTH OF MATERIALS. To Splice a Wire Rope.— The tools required will be a small marline spike, nipping cutters, and either clamps or a small hernp-rope sling with which to wrap around and untwist the rope. If a bench-vise is accessible it will be found convenient. In splicing rope, a certain length is used up in making the splice. An allowance of not less than 16 feet for ^ inch rope, and proportionately longer for larger sizes, must be added to the length of an endless rope in ordering. Having measured, carefully, the length the rope should be after splic- ing, and marked the points M and M', Fig. 83, unlay the strands from each end E and E' to M and M' and cut off the centre at M and M', and then: (1). Interlock the six unlaid strands of each end alternately and draw them together so that the points M and M' meet, as in Fig. 84. (2). Unlay a strand from one end, and following the unlay closely, lay into the seam or groove it opens, the strand opposite it belonging to the other end of the rope, until within a length equal to three or four times the length of one lay of the rope, and cut the other strand to about the same length from the point of meeting as at A, Fig. 85. (3). Unlay the adjacent strand in the opposite direction, and following the unlay closely, lay in its place the corresponding opposite strand, cutting the ends as described before at B, Fig. 85. There are now four strands laid in place terminating at A and B, with the eight remaining at M M\ as in Fig. 85. It will be well after laying each pair of strands to tie them temporarily at the points A and B. Pursue the same course with the remaining four pairs of opposite strands, NT Fig. 86. Fig. 87. Splicing Wire Rope. stopping each pair about eight or ten turns of the rope short of the preced- ing pair, and cutting the ends as before. We now have all the strands laid in their proper places with their respect- ive ends passing each other, as in Fig. 86. All methods of rope-splicing are identical to this point: their variety con- sists in the method of tucking the ends. The one given below is the one most generally practiced. Clamp the rope either in a vise at a point to the left of A, Fig. 86, and by a hand-clamp applied near A, open up the rope by untwisting sufficiently to cut the core at A, and seizing it with the nippers, let an assistant draw it out slowly, you following it closely, crowding the strand in its place until it is all laid in. Cut the core where the strand ends, and push the end back into its place. Remove the clamps and let the rope close together around it. Draw out the core in the opposite direction and lay the other strand in the centre of the rope, in the same manner. Repeat the operation at the five remaining points, and hammer the rope lightly at the points where the ends pass each other at A, A, B, B, etc.. with small wooden mallets, and the splice is complete, as shown in Fig. 87. If a clamp and vise are not obtainable, two rope slings and short wooden levers may be used to untwist and open up the rope. A rope spliced as above will be nearly as strong as the original rope and smooth everywhere. After running a few days, the splice, if well made, cannot be found except by close examination. The above instructions have been adopted by the leading rope manufac- turers of America, HELICAL STEEL SPRINGS. 347 SPRINGS. Definitions. —A spiral spring is one which is wound around a fixed point or centre, and continually receding from it like a watch spring. A helical spring is one which is wound around an arbor, and at the same time advancing like the thread of a screw. An elliptical or laminated spring is made of flat bars, plates, or "leaves, 11 of regularly varying lengths, super- posed one upon the other. Laminated Steel Springs.— Clark (Rules, Tables and Data) gives the following from his work on Baihvay Machinenj, 1855: 1.66L 3 . _ bthi m = 1.66£s . ~ bthi ' S ~~ 11. 3L ; n ~ Abt* ' A. = elasticity, or deflection, in sixteenths of an inch per ton of load, . s = working strength, or load, in tons (2340 lbs.), L = span, when loaded, in inches, b = breadth of plates, in inches, taken as uniform, t = thickness of plates, in sixteenths of an inch, n = number of plates. Note.— The span and the elasticity are those clue to the spring when weighted. 2 When extra thick back and short plates are used, they must be replaced by an equivalent number of plates of the ruling thickness, prior to the em- ployment of the first two formulae. This is found by multiplying the num- ber of extra thick plates by the cube of their thickness, and dividing by the cube of the ruling thickness. Conversely, the number of plates of the ruling thickness given by the third formula, required to be deducted and replaced by a given number of extra thick plates, are found by the same calculation. 3. It is assumed that the plates are similarty and regularly formed, and that they are of uniform breadth, and but slightly taper at the ends. Reuleaux's Constructor gives for semi-elliptic springs: _ Snbh* , . 6P1 3 P= -6T and f=Enbh*' S = max. direct fibre-strain in plate; b — width of plates; n = number of plates in spring; h = thickness of plates; I = one half lengtii of spring; / = deflection of end of spring; P = load on one end of spring; E — modulus of direct elasticity. The above formula for deflection can be relied upon where all the plates of the spring are regularly shortened; but in semi-elliptic springs, as used, there are generally several plates extending the full length of the spring, and the proportion of these long plates to the whole number is usually about 5 5P1 3 one fourth. In such cases / = ' , .. . (G-. R. Henderson, Trans. A. S, M. E., Enbh 3 ' ' vol. xvi.) In order to compare the formulae of Reuleaux and Clark we may make the following substitutions in the latter: s in tons — Pin lbs. -=- 1120; as = 16/; L = 21; t — 16/i; then 1.66 X 823 x P Pi 3 A s = 1Q f = Amnv ii->fw,,;w,3 ' whence / - " 4096 X 1120 X «6/i 8 ' J 5,527,133' which corresponds with Reuleaux's formula for deflection if in the latter we take E = 33,162,800. P 256n67ia , _ 12.687n6fc a Also s = -rr^r = -t— ., whence P = , 1120 11.3 X 21 I which corresponds with Reuleaux's formula for working load when Sin the latter is taken at 76,120. The value of Eis usually taken at 30,000,000 and S at S0,000, in which case Reuleaux's formulae become _ 13,333h6Zi 2 . . PI 3 P — — — ■ and / 5,000,000ii.6/t3' Helical Steel Springs. — Clark quotes the following from the report on Safety Valves (Trans. Inst. Engrs. and Shipbuilders in Scotland, 1874-5): „ d 3 X w 348 SPRINGS. E = compression or extension of one coil in inches, d = diameter from centre to centre of steel bar constituting the spring, in inches, w = weight applied, in pounds, D = diameter, or side of the square, of the steel bar, in sixteenths of an inch, C = a constant, which may be taken as 22 for round steel and 30 for square steel. Note.— The deflection Efor one coil is to be multiplied by the number of free coils, to obtain the total deflection for a given spring. The relation between the safe load, size of steel, and diameter of coil, may be taken for practical purposes as follows: w —, for round steel; 3 V wd r, , , for square steel. Rankine's Machinery and Millwork, p. 390, gives the following: W _ cd* _ ,196/ri' . _ 12.566n/r 2 . v ~ 64nr 3 ' 1 ~ r ' *• ~ cd ' —- 1 = greatest safe sudden load. In which d is the diameter of wire in inches; c a co-efficient of transverse elasticity of wire, say 10,500,000 to 12,000,000 for charcoal iron wire and steel; r radius to centre of wire in coil; n effective number of coils; / greatest safe shearing stress, say 30,000; W any load not exceeding greatest safe load; v corresponding extension or compression; W 2 greatest safe load; and v 1 greatest safe steady extension or compression. If the wire is square, of the dimensions d x d, the load for a given deflec- tion is greater than for a round wire of the diameter d in the ratio of 2.81 to 1.96 or of 1.43 to 1, or of 10 to 7, nearly. Wilson Hartnell (Proc. Inst. M. E., 1882, p. 426), says: The size of a spiral spring may be calculated from the formula on page 304 of " Rankine's Use- ful Rules and Tables"; but the experience with Salter's springs has shown that the safe limit of stress is more than twice as great as there given, namely 60,000 to 70,000 lbs. per square inch of section with % inch wire, and about 50,000 with \& inch wire. Hence the work that can be done by springs of wire is four or five times as great as Rankine allows. For % inch wire and under, _, . , ' . .. 12,000 X (diam. of wire) 3 Maximum load in lbs. = ~ ^ ^ : — - ; Mean radius of springs ^ . . . . ., . ' „ . . . . 180,000 x (diam.)4 Weight in lbs. to deflect spring 1 in. = — - — y - / J NQ . & Number of coils X (rad.) 3 The work in foot-pounds that can be stored up in a spiral spring would lift it above 50 ft. In a few rough experiments made with Salter's springs the coefficient of rigiditv was noticed to be 12,600,000 to 13,700,000 with \i inch wire; 11,000,000 for 11/32 inch; and 10,600,000 to 10,900,000 for % inch wire. Helical Springs.— J. Begtrup, in the American Machinist of Aug. 18, 1892, gives formulas for the deflection and carrying capacity of helical springs of round and square steel, as follow: W = .3927-^^ ™^ ,>, )• for round steel. F ~ 8 Ed* ' TT=.471 Sd3 y for square steel. HELICAL SPRINGS. 349 W = carrying capacity in pounds, S = greatest tensile stress per square inch of material, d =' diameter of steel, D = outside diameter of coil, F — deflection of one coil, E — torsional modulus of elasticity, P = load in pounds. From these formulas the following table has been calculated by Mr. Beg- trup. A spring being made of an elastic material, and of such shape as to allow a great amount of deflection, will not be affected by sudden shocks or blows to the same extent as a rigid body, and a factor of safety very much less than for rigid constructions may be used; HOW TO USE THE TABLE. When designing a spring for continuous work, as a car spring, use a greater factor of safety than in the table; for intermittent working, as in a steam-engine governor or safety valve, use figures given in table; for square steel multiply line W by 1.2 and line F by .59. Example 1. — How much will a spring of %" round steel and 3" outside diameter carry with safety ? In the line headed D we find 3, and right un- derneath 473, which is the weight it will carry with safety. How many coils must this spring have so as to deflect 3" with a load of 400 pounds ? Assum- ing a modulus of elasticity of 12 millions we find in the centre line headed F the figure .0610; this is deflection of one coil for a load of 100 pounds; therefore .061 X 4 = .244" is deflection of one coil for 400 pounds load, and 3 -7- .244 = 12J/2 is the number of coils wanted. This spring will therefore be 4%" long when closed, counting working coils only, and stretch to 1%" . Example 2. — A spring 3J4" outside diameter of 7/16" steel is wound close; how much can it be extended without exceeding the limit of safety ? We find maximum safe load for this spring to be 702 pounds, and deflection of one coil for 100 pounds load .0405 inches; therefore 7.02 x .0405 = .284" is the greatest admissible opening between coils. We may thus, without know- ing the load, ascertain whether a spring is overloaded or not. Carrying Capacity and Deflection of Helical Springs of Round Steel. d = diameter of steel. D = outside diameter of coil. W = safe working load in pounds— tensile stress not exceeding 60,000 pounds per square inch. F = deflection by a load of 100 pounds of one coil, and a modulus of elasti- city of 10, 12 and 14 millions respectively. The ultimat e carrying capacity will be about twice the safe load. w • T) .25 .50 .75 1.00 1.25 1.50 1.75 2.00 o" -1 W 35 15 9 7 5 4.5 3.8 3.3 III \ .0276 .3588 1.433 3.562 7.250 12.88 20.85 31.57 f\ .0236 .3075 1.228 3.053 6.214 11.04 17.87 27.06 } .0197 .2562 1.023 2.544 5.178 9.200 14.89 22.55 - .50 .75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 §**-i W 107 65 46 36 29 25 22 19 17 [ .0206 .0937 .2556 .5412 .9856 1.624 2.492 3.625 5.056 f] .0176 .0804 .2191 .4639 .8448 1.392 2.136 3.107 4.334 I D .0147 .0670 .182 .3866 .7010 1.160 2.00 1.780 2.589 2.50 3.612 5 75 1.00 1.25 1.50 1.75 2.25 2 75 3.00 OO j.," W 241 167 128 104 88 75 66 59 53 49 k 1 .0137 .0408 .0907 .1703 .2866 .4466 .6571 .9249 1.256 1 660 f\ .0118 .0350 .0778 .1460 .2457 .3828 .5632 .7928 1 077 1 423 ^Q ( .0098 .0292 .0648 .1217 .2048 .3190 .4693 .6607 .8975 1.186 D 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 T W 368 294 245 210 184 164 147 134 123 113 II 1 .0199 .0389 .0672 .1067 .1593 .2270 .3109 .4139 .5375 6835 v\ .0171 .0333 .0576 .0914 .1365 .1944 .2665 .3548 .4607 .5859 I .0142 .0278 .0480 .0762 .1137 .1610 .2221 .2957 .3839 .4883 350 SPEINGS. Carrying Capacity and Deflection of Helical Springs of Round Steel.— (Continued). 5 D 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 2 W 605 500 426 371 329 295 267 245 226 209 195 in ( .0136 .0242 .0392 .0593 .0854 .1187 .1583 .2066 .2640 .3312 .4089 II f\ .0117 .0207 .0336 .0508 .0732 .1012 .1357 .1771 .2263 .2839 .3505 •8 1 .0097 .0173 .0280 .0424 .0610 3.00 .0853 .1131 .1476 .1886 .2366 .2921 2.00 2.25 2.50 2.75 3.25 3.50 3.75 4.00 4.25 4.50 II W 765 663 589 523 473 433 398 368 343 321 30! \ .0169 .0259 .0377 .0528 .0711 .0935 .1200 .1513 .1874 .2290 .2761 f\ .0145 .0222 .0323 .0452 .0610 .0801 .1029 .1297 .1006 . 1963 .2367 is 1 D .0120 .0185 .0269 .0376 .0508 .0668 3.25 .0858 .1081 .1338 .1635 .1972 - 2.00 2.25 2:50 2.75 3.00 3.50 3.75 4.00 4 50 5.00 2 W 1263 1089 957 853 770 702 644 596 544 486 432 j> i .0081 .0126 .0186 .0262 .0357 .0472 .0617 .0772 .0960 .1423 .2016 II f\ .0069 .0108 .0160 .0225 .0306 .0405 .0529 .0661 .0823 .1220 1728 'e \ .0058 .0090 .0133 .0187 .0255 .0337 .0441 .0551 .0686 .1017 .1440 D 2.00 2.25 2.50 2.75 3.00 3 25 3.50 3.75 4.00 4.50 5.00 & W 1963 1683 1472 1309 1178 1071 982 906 841 736 654 [ .0042 .0067 .0099 .0141 .0194 .0259 .0336 .0427 .0534 .0796 .1134 II f\ .0036 .0057 .0085 .0121 .0167 .0222 .0288 .0366 .0457 .0683 .0972 tt ( .0030 .0048 .0071 .0101 .0139 .0185 .0240 .0305 .0381 .0569 5.00 .0810 - 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 5.50 2 W 2163 1916 1720 1560 1427 1315 1220 1137 1065 945 849 OS I .0056 .0081 .0112 .0151 .0197 .0252 .0316 .0390 .0474 .0679 .0935 II f\ .0048 .0070 .0096 .01.29 .0169 .0216 .0271 .0334 .0406 0582 .0801 ■s } D .0040 .0058 .0080 .0108 .0141 .0180 .0225 .0278 .0339 .0485 .0668 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 5.00 5.50 S5 W 3068 2707 2422 2191 2001 1841 1704 1587 1484 1315 1180 1 .0034 .0049 .0068 .0092 .0121 .0155 .0196 .0243 .0297 .0427 .0591 II F\ .0029 .0042 .0058 .0079 .0104 .0133 .0168 .0208 .0254 .0366 .0506 •8 I n .0024 .0035 .0049 .0066 .0086 .0111 .0140 .0173 .0212 5.00 .0305 .0422 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.50 6.00 w 3311 2988 2723 2500 2311 2151 2009 1885 1776 1591 1441 IIS [ .0043 .0058 .0077 .0100 .0127 .0157 .0193 .0233 .0279 .0388 .0522 ■«!-• jrl .0037 .0050 .0066 .0086 .0108 .0135 .0165 .0200 .0239 .0333 .0447 J_ .0030 .0042 .0055 .0071 .0090 .0112 .0138 4.50 .0167 4.75 .0199 .0277 5.50 .0373 3.00 3.25 3.50 3 75 4.00 4.25 5.00 6.00 S W 4418 3976 3615 3313 3058 2840 2651 2485 2339 2093 1893 ( .0028 .0038 .0051 .0066 .0084 .0105 .0129 .0157 .0189 .0264 .0356 F \ .0024 0033 .0044 .0057 .0072 .0090 .0111 .0135 .0162 .0226 0305 "8 ( .0020 .0027 .0036 .0047 .0060 .0075 .0093 .0113 .0135 .0188 .0254 D 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 6.00 6.50 £ W 6013 5490 5051 4676 4354 4073 3826 3607 3413 3080 2806 A .0021 .0027 .0035 .0045 .0055 .0067 .0081 .0097 .0115 .0156 .0207 ll .0018 .0024 .0030 .0038 .0047 .0058 .0070 .0083 .0098 .0134 .0177 -8 i .0015 .0020 .0025 .0032 .0039 .0048 .0058 .0069 .0082 .0112 .0148 D 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 6.00 6.50 ^ W 9425 8568 7854 7250 6732 6283 5890 5544 5236 4712 4284 II { .0012 .0016 .0021 .0026 .0033 .0041 .0049 .0059 0071 .0097 .0129 f\ .0010 .0014 .0018 .0023 .0028 .0035 .0043 .0051 .0061 .0083 .0111 I .0008 .0011 .0015 .0019 .0023 .0029 .0035 .0043 .0051 .0069 .0092 The formulae for deflection or compression given by Clark, Hartnell, and Begtrup, although very different in form, show a substantial agreement when reduced to the same form. Let d = diameter of wire in inches, D l = mean diameter of coil, n the number of coils, to the applied weight in pounds, and C a coefficient, then HELICAL SPRINGS. 351 Cd* ' Cd* Compression or extension of one coil _ Weight in pounds to cause comp. or ext. of 1 in. _ The coefficient C reduced from Hartnell's formula is 8 X 180,000 =1,440,000; according to Clark, 16 4 X 22 = 1,441.792, and according to Begtrup (using 12,000,000 for the torsional modulus of elasticity) = 12,000,000 -*- 8 = 1,500,000. Rankine's formula for greatest safe extension, v x = — '- — ■ may take the form v t = '' ™ * ■ if we use 30,000 and 12,000,000 as the values for / and c respectively. The several formulas for safe load given above may be thus compared, letting d = diameter of wire, and D x = mean diameter of coil, Rankine, .196/d3 T _ 3(d X 16)3 .39275^3 TT W = — * — ; Clark, W = ^—^ — - ; Begtrup, W = ~— ; Hartnell, 12000d 3 1 W — — . Substituting for / the value 30,000 given by Rankine, and for d 3 i73 S, 60,000 as given by Begtrup, we have W = 11,760 — Rankine; 12,288 j- d 3 AS 1 x Clark; 23,562— Begtrup; 24,000 j- Hartnell. Taking from the Pennsylvania Railroad specifications the capacity when closed of the following springs, in which d — diameter of wire, D diameter outside of coil, D x = D — d, c capacity, H height when free, and h height when closed, all in inches. = \\i -Di = 1J4 c = 400 H = 9 h = 6 3 2\i 1,900 8 5 5M 5 2,100 7 4U 5 4 8,100 ltfU 8 8 m 10,000 9 5% 4% 3M 16,000 4% S% d 3 and substituting the values of c in the formula c — W = x — we find x, the d 3 X coefficient of — to be respectively 32,000; 38,000; 32,400; 24,888; 34,560; 42,140, average 34,000. d 3 Taking 12,000 as the coefficient of —according to Rankine and Clark for safe load, and 24,000 as the coefficient according to Begtrup and Hartnell, we have for the safe load on these springs, as we take one or the other co- efficient, T. 8. K. D. I. C. Rankine and Clark 150 600 1,012 3,000 3.750 5,400 lbs. Hartnell 300 1,200 2,024 6,000 7^500 10,800 " Capacity when closed, as above 400 1,900 2,100 8,100 10,000 16,000 " J. W. Cloud (Trans. A. S. M. E., v. 173) gives the following: „ Snd* _ . 32 PRH p =Tqr and /= -^^ ; P = load on spring; 8 — maximum shearing fibre-strain in bar; d = diameter of steel of which spring is made ; B = l'adius of centre of coil; I = length of bar before coiling; G = modulus of shearing elasticity; / = deflection of spring under load. Mr. Cloud takes S = 80,000 and G ~ 12,600,000. The stress in a helical spring is almost wholly one of torsion. For method of deriving the formulas for springs from torsional formula see Mr. Cloud's paper, above quoted. No. T. d = H S. H K. M D. l I. m a m 352 .ELLIPTICAL SPRINGS, SIZES, AND PROOF TESTS. Pennsylvania Railroad Specifications, 1889. a "3 8* Tests. S.g o*§ fl O w 05 Class. ■ ■§ "* •a p.y To stand ins. High. With Load of lbs. J 5 m ( 3% between bands. 4800 A, Triple 40 11*4 3 x% 3 8 (3M ;; 5500 A. p. t.* 6650 C, Quadruple.. 40 l5i/ 2 3 x% 3 1! , 8000 A. p. t.* A Triple 36 11% 3 x% 3 11 i (i 6000 8000 ( 5 bet. centre of eye E, Single 40 sin. 3 x% 3x11/32 •< and top of leaf. (3 2^ between bands. When free 2350 11,800 F, Triple 35 11% 3 x% 3x11/32 G, Double 32 % 3 x% 3 W :: « When free 8000 if, Double 36 9^ 3 x% 4 {1* - :: 5400 6000 ft- j Double, I *' I 6 plates j 22 10% 3^x3^ 4^x11/32 13/16 " 13,800 T \ Double, j ■** 1 7 plates j 22 10% 3^x% 4^x11/32 13/16 " (4 15,600 8000 Jlf, Quadruple.. 40 16H 3 x% 3 I 3 2 10,000 A. p. t.* * A. p. t., auxiliary plates touching. PHOSPHOR-BRONZE SPRINGS. Wilfred Lewis (Engineers' Club, Philadelphia, 1887) made some tests with phosphor-bronze wire, .12 in. diameter, coiled in the form of a spiral spring, 134 in. diameter from centre to centre, making 52 coils. This spring was loaded gradually up to a tension of 30 lbs., but as the load was removed it became evident that a permanent set had taken place. Such a spring of steel, according to the practice of the P. R. R., might be used for 40 lbs. A weight of 21 lbs. was then suspended so as to allow a small amount of vibration, and the length measured from day to day. In 30 hours the spring lengthened from 20% inches to 213/g inches, and in 200 hours to 21J4 inches. It was concluded tliat 21 lbs. was too great for durability, and that probably 10 lbs. was as much as could be depended upon with safety. For a given load the extension of the bronze spring was just double the extension of a similar steel spring, that is, for the same extension the steel spring is twice as strong. SPRINGS TO RESIST TORSIONAL FORCE. (Reuleaux's Constructor.) Flat spiral or helical spring. .; P = ^ ~ ; / = R# = 12 , Round helical spring P = 6 R Sn d 3 - R&. Ebh 3 ' MPIR* ' [v E d 4 ' it G d 4 SPR*l b* + h* G b 3 h 3 ' Flat bar, in torsion P - P = force applied at end of radius or lever-arm R; & — angular motion at end of radius R; S = permissible maximum stress, = 4/5 of permissible stress in flexure; E — modulus of elasticity in tension; G — torsional modu- lus, = 2/5 E\ I — developed length of spiral, or length of bar; d — diameter of wire; b = breadth of flat bar; h = thickness, HELICAL SPRINGS FOR CARS AND LOCOMOTIVES. 353 Ma Sao < r H a 2* ©3 ©X ©o as ©£ ^ a co M S ^ : : ^ : ^ . CO CO • coo to t- J> rf *> *^ fr- . CD 50 CO . t> i- lO TtiOO »o to •^"aj i^ts^ls et cc cj cj c3 . 03 nj cScisSaScSa; •OO 'OOOO ' 1- O O Of O o • i-i so • C> ©J CO lO OOO 000 000 000 800 000 000 000 000 500 700 000 500 •TH^CO^ t)i e e to o • t- t- i- CO CO CO i-" QO 00 o u 00 t-i jo co co iO -* to co od m e flo«xo!ioat-xoooc» o cs >c os -* c !3 . ^^ co m eo^io !> ««r.MM t-i 1 DCSOSCSOOCOcoOCO ~ ■ ~ CO " i- — CO O -T X 0> ?> 'X) OS CO " o-*05Cicoi-coirtincoioj>-tf 3 co e* eo ouo ! - o to h ?; o r %iococsc5eocooo£>t-»oiOTPiom!>-*« D tO CO CO « O S^SaCSOSSS S ^^'"S^^ •3 '« -4' MS C?c5 dp 354 RIVETED JOINTS. RIVETED JOINTS. Fairhairn's Experiments. (From Report of Committee on Riveted Joints, Proc. Inst. M. E., April, 1881.) The earliest published experiments on riveted joints are contained in the memoir by Sir VV. Fairbairn in the Transactions of the Royal Society. Mak- ing certain empirical allowances, he adopted the following ratios as ex- pressing the relative strength of riveted joints: Solid plate 100 Double-riveted joint 70 Single-riveted joint 56 These well-known ratios are quoted in most treatises on riveting, and are still sometimes referred to as having a considerable authority. It is singular, however, that Sir W. Fairbairn does not appear to have been aware that the proportion of metal punched out in the line of fracture ought to be different in properly designed double and single riveted joints. These celebrated ratios would therefore appear to rest on a very unsatisfactory analysis of the experiments on which they were based. Loss of Strength in Punched Plates.— A report by Mr. W. Parker and Mr. John, made in 1878 to Lloyd's Committee, on the effect of punching and drilling, showed that thin steel plates lost comparatively little from punching, but that in thick plates the loss was very considerable. The following table gives the results for plates punched and not annealed or reamed: Thickness of Material of Loss of Tenacity, Plates. Plates. per cent. V A Steel 8 % " 18 ^ " 26 M " 33 % Iron 18 to 23 The effect of increasing the size of the hole in the die-block is shown in the following table: Total Taper of Hole Material of Loss of Tenacity due to in Plate, inches. Plates. Punching, per cent. 1-16 Steel 17.8 Vs " 12.3 34 " (Hole ragged) 24.5 The plates were from 0.675 to 0.712 inch thick. When %-in. punched holes were reamed out to 1% in. diameter, the loss of tenacity disappeared, and the plates carried as high a stress as drilled plates. Annealing also restores to punched plates their original tenacity. Strength of Perforated Plates. (P. D. Bennett, Eng'g, Feb. 12, 1886, p. 155.) Tests were made to determine the relative effect produced upon tensile strength of a flat bar of iron or steel: 1. By a %-inch hole drilled to the re- quired size; 2, by a hole punched J^ inch smaller and then drilled to the size of the first hole; and, 3, by a hole punched in the bar to the size of the drilled bar. The relative results in strength per square inch of original area were as follows: 1. Iron. Unperforated bar 1.000 Perforated by drilling 1 . 029 " " punching and drilling. 1.030 " " punching only 0.795 In tests 2 and 4 the holes were filled with rivets driven by hydraulic pres- sure. The increase of strength per square inch caused by drilling is a phe- nomenon of similar nature to that of the increased strength of a grooved bar over that of a straight bar of sectional area equal to the smallest section of the grooved bar, Mr. Bennett's tests on an iron bar 0,84 in. diameter, 10 in. 2. 3. 4. Iron . Steel. Steel. 1.000 1.000 1.000 1.012 1.068 1.103 1.008 1.059 1.110 0.894 0.935 0.927 EFFICIENCY OF RIVETING BY DIFFERENT METHODS. o5o long, and a similar bar turned to 0.84 in. diameter at one point only, showed that the relative strength of the latter to the former was 1.323 to i.000. Riveted Joints.— Drilling versus Punching; of Holes. The Report of the Research Committee of the Institution of Mechanical Engineers, on Riveted Joints (1881), and records of investigations by Prof. A. B. W. Kennedy (1881, 1882, and 1885), summarize the existing information regarding the comparative effects of punching and drilling upon iron and steel plates. From an examination of the voluminous tables given in Pro- fessor Un win's Report, the results of the greatest number of the experi- ments made on iron and steel plates lead to the general conclusion that, while thin plates, even of steel, do not suffer very much from punching, yet in those of J^-inch thickness and upwards the loss of tenacity due to punch- ing ranges from 10$ to 23$ in iron plates, and from 11$ to 33$ in the case of mild steel. In drilled plates there is no appreciable loss of strength. It is possible to remove the bad effects of punching by subsequent reaming or annealing; but the speed at which work is turned out in these days is not favorable to multiplied operations, and such additional treatment is seldom practised. The introduction of a practicable method of drilling the plating of ships and other structures, after it has been bent and shaped, is a matter of great importance. If even a portion of the deterioration of tenacity can be prevented, a much stronger structure results from the same material and the same scantling. This has been fully recognized in the modern English practice (1887) of the construction of steam-boilers with steel plates; punch- ing in such cases being almost entirely abolished, and all rivet-holes being drilled after the plates have been bent to the desired form. Comparative Efficiency of Riveting done by Different Methods. The Reports of Professors Unwin and Kennedy to the Institution of Me- chanical Engineers (Proc. 1881, 1882, and 1885) tend to establish the four fol- lowing points: 1. That the shearing resistance of rivets is not highest in joints riveted by means of the greatest pressure; 2. That the ultimate strength of joints is not affected to an appreciable extent by the mode of riveting; and, therefore, 3. That very great pressure upon the rivets in riveting is not the indispen- sable requirement that it has been sometimes supposed to be; 4. That the most serious defect of hand-riveted as compared with machine- riveted work consists in the fact that in hand -riveted joints visible slip commences at a comparatively small load, thus giving such joints a low value as regards tightness, and possibly also rendering them liable to failure under sudden strains after slip has once commenced. The following figures of mean results, taken from Prof. Kennedy's tables (Proceedings 1885, pp. 218-225), give a comparative view of hand and hy- draulic riveting, as regards their ultimate strengths in joints, and the periods at which in both cases visible slip commenced. Total Breaking Load. Load at which Visible Slip began. Hand-riveting. Hydraulic Rivet- ing. Hand-riveting. Hydraulic Rivet- ing. Tons. 86.01 '82". 16 U9.2 m.6 Tons. 85.75 77.00 82.70 78.58 145.5 140.2 183.1 183.7 Tons. 21.7 25^6 3L7 25^6 Tons. 47.5 35.0 53.7 54.0 49.7 46.7 56.0 In these figures hand-riveting appears to be r ither better than hydraulic riveting, as far as regards ultimate strength of joint; but is very much in- ferior to hydraulic work, in view of the small proportion of load borne by it before visible slip commenced. 356 RIVETED JOIffTS. Some of the Conclusions of the Committee of Research on Riveted Joints. (Proc. Inst. M. E., Apl. 1885.) The conclusions all refer to joints made in soft steel plate with steel rivets, the holes all drilled, and the plates in their natural state (unannealed). In every case the rivet or shearing area has been assumed to be that of the holes, not the nominal (or real) area of the rivets themselves. Also, the strength of the metal in the joint has been compared with that of strips cut from the same plates, and not merely with nominally similar material. The metal between the rivet-holes has a considerably greater tensile re- sistance per square inch than the imperforated metal. This excess tenacity amounted to more than 20$, both in %-inch and %-inch plates, when the pitch of the rivet was about 1.9 diameters. In other cases %-inch plate gave an excess of 15$ at fracture with a pitch of 2 diameters, of 10$ with a pitch of 3.6 diameters, and of 6.6$, with a pitch of 3.9 diameters; and ^-inch plate gave 7.8$ excess with a pitch of 2.8 diameters. In single-riveted joints it may be taken that about 22 tons per square inch is the shearing resistance of rivet steel, when the pressure on the rivets does not exceed about 40 tons per square inch. In double-riveted joints, with rivets of about % inch diameter, most of the experiments gave about 24 tons per square inch as the shearing resistance, but the joints in one series went at 22 tons. The ratio of shearing resistance to tenacity is not constant, but diminishes very markedly and not very irregularly as the tenacity increases. The size of the rivet heads and ends plays a most important part in the strength of the joints— at any rate in the case of single-riveted joints. An increase of about one third in the weight of the rivets (all this increase, of course, going to the heads and ends) was found to add about 8J^$ to the resistance of the joint, the plates remaining unbroken at the full shearing resistance of 22 tons per square inch, instead of tearing at a shearing stress of only a little over 20 tons. The additional strength is probably due to the prevention of the distortion of the plates by the great tensile stress in the rivets. The intensity of bearing pressure on the rivet exercises, with joints propor- tioned in the ordinary way, a very important influence on their strength. So long as it does not exceed 40 tons per square inch (measured on the pro- jected area of the rivets), it does not seem to affect their strength ; but pres- sures of 50 to 55 tons per square inch seem to cause the rivets to shear in most cases at stresses varying from 16 to 18 tons per square inch. For or- dinary joints, which are to be made equally strong in plate and in rivets, the bearing pressure should therefore probably not exceed 42 or 43 tons per square inch. For double-riveted butt-joints perhaps, as will be noted later, a higher pressure may be allowed, as the shearing stress may probably not be more than 16 or 18 tons per square inch when the plate tears. A margin (or net distance from outside of holes to edge of plate) equal to the diameter of the drilled hole has been found sufficient in all cases hitherto tried. To attain the maximum strength of a joint, the breadth of lap must be such as to prevent it from breaking zigzag. It has been found that the net metal measured zigzag should be from 30$ to 35$ in excess of that measured straight across, in order to insure a straight fracture. This corresponds to a diagonal pitch of 2/3 p -\- d/S, if p be the straight pitch and d the diam- eter of the rivet-hole. Visible slip or "give" occurs always in a riveted joint at a point very much below its breaking load, and by no means proportional to that load. A collation of the results obtained in measuring the slip indicates that it de- pends upon the number and size of the rivets in the joint, rather than upon anything else ; and that it is tolerably constant for a given size of rivet in a given type of joint. The loads per rivet at which a joint will commence to slip visibly are approximately as follows : Diameter of Rivet. Type of Joint. Riveting. Slipping Load per Rivet. % inch % " H " 1 inch 1 " 1 " Single-riveted Double-riveted Double- riveted Single-riveted Double-riveted Double-riveted Hand Hand Machine Hand Hand Machine 2.5 tons 3.0 to 3.5 tons 7 tons 3.2 tons 4.3 tons 8 to 10 tons DOUBLE-RIVETED LAP-JOINTS. 35? To find the probable load at which a joint of any breadth will commence to slip, multiply the number of rivets in the given breadth by the proper figure taken from the last column of the table above. It will be understood that the above figures are not given as exact; but they represent very well the results of the experiments. The experiments point to simple rules for the proportioning of joints of maximum strength. Assuming that a bearing pressure of 43 tons per square inch may be allowed on the rivet, and that the excess tenacity of the plate is 10% of its original strength, the following table gives the values of the ratios of diameter d of hole to thickness t of plate (d -+- t), and of pitch p to diam- eter of hole (p-=-cZ) in joints of maximum strength in %-inch plate. For Single-riveted Plates. Original Tenacity of Plate. Shearing Resistance of Rivets. Tons per 28 Lbs. per 67,200 67,200 62,720 Tons per sq. in. 24 24 Lbs. per . in. 49,200 49,200 53,760 53,760 Ratio. d-i-t Ratio. p -t-d 2.48 2.48 2.28 2.28 2.30 2.40 2.27 2.36 Ratio. Plate Area 0.667 0.785 0.713 0.690 This table shows that the diameter of the hole (not the diameter of the rivet) should be 2% times the thickness of the plate, and the pitch of the rivets 2% times the diameter of the hole. Also, it makes the mean plate area 71% of the rivet area. If a smaller rivet be used than that here specified, the joint will not be of uniform, and therefore not of maximum, strength; but with any other size of rivet the best result will be got by use of the pitch obtained from the simple formula p=a— + d, where, as before, d is the diameter of the hole. The value of the constant a in this equation is as follows: For 30-ton plate and 22-ton rivets, a = 0.524 " 28 " 22 " " 0.558 " 30 " 24 " " 0.570 " 28 " 24 " " 0.606 d" 2 Or, in the mean, the pitch p = 0.56 —r- -\- d. It should be noticed that with too small rivets this gives pitches often con- siderably smaller in proportion than 2% times the diameter. For double-riveted, lap-joints a similar calculation to that given above, but with a somewhat smaller allowance for excess tenacity, on account of the large distance between the rivet-holes, shows that for joints of maximum strength the ratio of diameter to thickness should remain pre- cisely as in single-riveted joints; while the ratio of pitch to diameter of hole should be 3.64 tor 30-ton plates and 22 or 24 ton rivets, and 3.82 for 28-ton plates with the same i ivets. Here, still more than in the former case, it is likely that thf prescribed size of rivet may often be inconveniently large. In this case the diameter of rivet should be taken as large as possible; and the strongest joint for a given thickness of plate and diameter of hole can then be obtained by using the pitch given by the equation p = a-j+d, where the values of the constant a for different strengths of plates and rivets may be taken as follows: 358 RIVETED JOINTS. + d. Table of Proportions of DouMe-riveted Lap-joints, in which p = Original tenacity Thickness of of Plate, Plate. Tons per sq. in. Shearing Resist- ance of Rivets. Tons per sq. in. 24 24 22 22 24 24 22 22 Value of Con- stant. a 1.15 1.22 1.05 1.12 1.17 1.25 1.07 1.14 Practically, having assumed the rivet diameter as large as possible, can fix the pitch as follows, for any thickness of plate from '% to % inch: For 30-ton plate and 24-ton rivets { , " 28 " " " 22 " " \ J rt 2 p - 1.06 -- + d In double-riveted butt-joints it is impossible to develop the full shearing resistance of the joint without getting excessive bearing pressure, because the shearing area is doubled without increasing the area on which the pressure acts. Considering only the plate resistance and the bearing pressure, and taking this latter as 45 tons per square inch, the best pitch would be about 4 times the diameter of the hole. We may probably say with some certainty that a pressure of from 45 to 50 tons per square inch on the rivets will cause shearing to take place at from 16 to 18 tons per square inch. Working out the equations as before, but allowing excess strength of only 5% on account of the large pitch, we find that the proportions of double- riveted butt-joints of maximum strength, under given conditions, are those of the following table: Double-riveted Butt-joints. Ratio P d 3.85 4.06 4.03 4.27 4.20 Practically, therefore, it may be said that we get a double-riveted butt-joint of maximum strength by making the diameter of hole about 1.8 times the thickness of the plate, and making the pitch 4.1 times the diameter of the hole. The proportions just given belong to joints of maximum strength. But in a boiler the one part of the joint, the plate, is much more affected by time than the other part, the rivets. It is therefore not unreasonable to estimate the percentage by which the plates might be weakened by corrosion, etc., before the boiler would be unfit for use at its proper steam-pressure, and to add correspondingly to the plate area. Probably the best thing to do in this case is to proportion the joint, not for the actual thickness of plate, but for a nominal thickness less than the actual by the assumed percentage. In this case the joint will be approximately one of uniform strength by the time it has reached its filial workable condition ; up to which time the joint as a whole will not really have been weakened, the corrosion only gradually bringing the strength of the plates down to that of rivets. riginal Ten- Shearing Re- Bearing acity sistance Pres- Ratio of Plate, of Rivets, sure, Tons per Tons per Tons per d sq. in. sq. in. sq. in. T 30 16 45 1.80 28 16 45 1.80 30 18 48 1.70 28 18 48 1.70 30 16 50 2.00 28 16 50 2.00 KIVETED JOINTS. 359 Efficiencies of Joints. The average results of experiments by the committee gave: For double- riveted lap-joints in %-iuch plates, efficiencies ranging from 67.1$ to 81. 2$. For double-riveted butt-joints (in double shear) 61.4$ to 71.3$. These low re- sults were probably due to the use of very soft steel in the rivets. For single- riveted lap-joints of various dimensions the efficiencies varied from 54.8$ to 60.8$. The experiments showed that the shearing resistance of steel did not in- crease nearly so fast as its tensile resistance. With very soft steel, for instance, of only 26 tons tenacity, the shearing resistance was about 80$ of the tensile resistance, whereas with very hard steel of 52 tons tenacity the shearing resistance was only somewhere about 65$ of the tensile resistance. Proportions of Pitch and Overlap of Plates to Diameter of Rivet-Hole and Thickness of Plate. (Prof. A. B. W. Kennedy, Proc. Inst. M. E„ April, 1885.) t = thickness of plate; d — diameter of rivet (actual) in parallel hole; p — pitch of rivets, centre to centre; s = space between lines of rivets; [I — overlap of plate. The pitch is as wide as is allowable without imparing the tightness of the joint under steam. For siugle-riveted lap-joints in the circular seams of boilers which have double-riveted longitudinal lap joints, d = t x 2.25; p — d x 2.25 =h5 (nearly); l=t x Q. For double-riveted lap-joints: d = 2.25*; p = 8t; s = 4.5*; I = 10.5*. Single-riveted Joints. Double-riveted Joints. * d P I * d V s i 3-16 7-16 15-16 W8 3-16 7-16 V4 Vs 2 H 9-16 m 1% Va 9-16 2 13-16 m 5-16 11-16 19-16 m 5-16 11-16 2^ M % 13-16 M 2M % 13-16 3 m 4 7-16 1 2 3-16 2% 7-16 1 W* 2 4% Y2 UH V& 3 H % 4 2M 5% 9-16 1J4 2 13-16 W& 9-16 V4 2y 2 With these proportions and good workmanship there need be no fear of leakage of steam through the riveted joint. The net diagonal area, or area of plate, along a zigzag line of fracture should not be less than 30$ in excess of the net area straight across the joint, and 35$ is better. Mr. Theodore Cooper (E. R. Gazette, Aug. 22, 1890) referring to Prof. Ken- nedy's statement quoted above, gives as a sufficiently approximate rule for the proper pitch between the rows in staggered riveting, one half of the pitch of the rivets in a row plus one quarter the diameter of a rivet-hole. Apparent Excess in Strength of Perforated over Unper- forated Plates. (Proc. Inst. M. E., October, 1888.) The metal between the rivet-holes has a considerably greater tensile re- sistance per square inch than the imperforated metal. This excess tenacity amounted to more than 20$, both in %-inch and %-inch plates, when the pitch of the rivets was about 1.9 diameters. In other cases %-inch plate gave an excess of 15$ at fracture with a pitch of 2 diameters, of 10$ with a pitch of 3.6 diameters, and of 6.6$ with a pitch of 3.9 diameters; and %-inch plate gave 7.8$ excess with a pitch of 2.8 diameters, 360 RIVETED JOINTS. (1) The "excess strength due to perforation " is increased by anything which tends to make the stress in the plate uniform, and to diminish the effect of the narrow strip of metal at the edge of the specimen. (2) It is diminished by increase in the ratio of p/d, of pitch to diameter of hole, so that in this respect it becomes less as the efficiency of the joint increases (3) It is diminished by any increase in hardness of the plate. (4) For a given ratio p/d, of pitch to diameter of hole, it is also apparently diminished as the thickness of the plate is increased. The ratio of pitch to thickness of plate does not seem to affect this matter directly, at least within the limits of the experiments. Test of Double-riveted L 1 .761 .758 % m 11/16 .755 .738 9/16 2¥a 13/16 .701 .690 % 3^ % .754 .760 9/16 3 Vs .714 .708 % z% 13/16 .762 .776 9/16 m 15/16 .727 .722 % m % .777 .788 9/16 m 1 .745 .733 7/16 Ws 11/16 .714 .711 9/16 \y A 1 1/16 .742 .750 H. De B. Parsons (R. R. & Eng. Journal, 1890) holds that it is an error to assume that the shearing strength of the rivet is equal to the tensile strength. Also, referring to the apparent excess in strength of perforated over unper- forated plates, he claims that on account of the difficulty in properly match- ing the holes, and of the stress caused by forcing, as is too often the case in practice, this additional strength cannot *be trusted much more than that of friction. Adopting the sizes of iron rivets as generally used in American practice for steel plates from V| to 1 inch thick: the tensile strength of the plates as 60,000 lbs.; the shearing strength of the rivets as 40,000 for single-shear and 35,500 for double - shear, Mr. Parsons calculates the following table of pitches, so that the strength of the rivets against shearing will be approxi- mately equal to that of the plate to tear between rivet-holes. The diameter of the rivets has in all cases been taken at 1/16 in. larger than the nominal size, as the rivet is assumed to fill the hole under the power riveter. J&iveted Joints. Lap or Butt with Single Welt— Steel Plates and Iron Rivets. Thickness of Plates. Diameter of Rivets. 1 1/8 Single. 1 3/16 1 11/16 1 11/16 1% 2 3/16 Ws 2 11/16 m 2 7/16 2Vs 2 7/16 Efficiency. Single. Double. 55.7% 70. 0# 52.7 68.6 49.0 65.9 43.6 60.4 42.0 59.5 38.6 55.4 38.1 54.9 362 RIVETED JOINTS. Calculated Efficiencies Steel Plates and Steel Rivets.— The differences between the calculated efficiencies given in the two tables above are notable. Those given by Mr. Ruggles are probably too high, since he assumes the shearing strength of the rivets equal to the tensile"strength of the plates. Those given by Mr. Parsons are probably lower than will be obtained in practice, since the figure he adopts for shearing strength is rather low, and he makes no allowance for excess of strength of the perfo- rated over the imperforated plate. The following table has been calculated by th author on the assumptions that the excess strength of the perforated plate is 10$, and that the shearing strength of the rivets per square inch is four fifths of the tensile strength of the plate. If t = thickness of plate, d = diameter of rivet-hole, p = pitch, and T — tensile strength per square inch, then for single-riveted plates (p - d)t x i.ior = T&X „d* The coefficients .571 and 1.142 agree closely with the averages of those given in the report of the committee of the institution of Mechanical En- gineers, quoted on pages 357 and 358, ante. Diam. Pitch. Efficiency. yi Diam. Pitch. Efficiency. a of Rivet- bio o> &J0 2-5 si 2$ a of Rivet- c > o > '£, > 2 > H ft s xg Q s H S | fi s "a °s in. in. in. in. % % in. in. in. in. % i 3/16 7/16 1.020 1.603 57.1 72.7 l A U 1.392 2.035 46.1 63.1 " l A 1.261 2.023 60.5 75.3 % 1.749 2.624 50.0 66.6 H y* 1.071 1.642 53.3 69*6 i - 1 2.142 3.284 53.3 70.0 9/16 1.285 2.008 56.2 72.0 " V& 2.570 4.016 56.2 72.0 5/16 9/16 1.137 1.712 50.5 67.1 9/16 I •1.321 1.892 43.2 60.3 " % 1.339 2.053 53.3 69.5 " 1.652 2. -129 47.0 64.0 " 11/16 1.551 2.415 55.7 71.5 '■ 1 2.015 3.030 50.4 67.0 % % 1.218 1.810 48.7 65.5 " i% 2.410 3.6'.M 53.3 69.5 % 1.607 2.463 53.3 69.5 " 134 2.836 4.422 55.9 71.5 " Vs 2.011 3.206 57.1 72 7 % % 1.264 1.778 40.7 57.8 7/16 % 1.136 1.647 45.0 62.0 Vs 1.575 2.274 44.4 61.5 " H 1.484 2.218 49.5 66.2 " 1 1.914 2.827 47.7 64.6 " % 1.869 2.864 53.2 69.4 " 1U 2.281 3.438 50.7 67.3 l 2.305 3.610 56.6 72.3 m 2,678 4.105 53.3 69.5 Riveting Pressure Required for Rridge and Roller Work. (Wilfred Lewis, Engineers' Club of Philadelphia, Nov., 1893.) A number of %-inch rivets were subjected to pressures between 10.000 and 60.000 lbs. At 10,000 lbs. the rivet swelled and filled the hole without forming a head. At 20,000 lbs. the head was formed and the plates were slightly pinched. At 30.000 lbs. the rivet was well set. At 40,000 lbs. the metal in the plate surrounding the rivet began to stretch, and the stretching became more and more apparent as the pressure was increased to 50,000 and 60,000 lbs. From these experiments the conclusion might be drawn that the pres- sure required for cold riveting was about 300,000 lbs. per square inch of rivet section. In hot riveting, until recently there was never any call for a pres- sure exceeding 60,000 lbs., but now pressures as high as 150,000 lbs. are not uncommon, and even 300,000 lbs. have been contemplated as desirable. SHEARING RESISTANCE OF RIVET IRON AND STEEL. 363 Apparent Shearing Resistance of Rivet Iron and Steel. (Proc. Inst. M. E., 1879, Engineering, Feb. 20, 1880.) The true shearing resistance of the rivets cannot be ascertained from experiments on riveted joints (1), because the uniform distribution of the load to all the rivets cannot be insured; (2) because of the friction of the plates, which has the effect of increasing the apparent resistance to shear- ing in an element uncertain in amount. Probably in the case of single- riveted joints the shearing resistance is not much affected by the friction; Ultimate Shearing Stress Tons per sq in. Lbs. per sq. in. Iron, single shear (12 bars). . 24.15 54.096 \ rs -\ ay .- h . a " double shear (8 bars). . 22.10 49.504 j ^ iarke - kt 22.62 50.669 Barnaby. 22.30 49.952 Rankine. " %-in. rivets 23.05 to 25.57 51.632 to 57.277 ) " %-in. rivets 24.32 to 27.94 54.477 to 62.362 V Riley. " mean value 25.0 56.000 j " %-in. rivets 19.01 42.582 Greig and Eyth. Steel 17 to 26 38.080 to 58.240 Parker. Landore steel, %-\n. rivets.. 31.67 to 33.69 70.941 to 75.466 ) " %-xa. rivets... 30.45 to 35.73 68 . 208 to 80 . 035 - Riley. " " mean\alue.. 33.3 74.592 ) Brown's steel 22.18 49.683 Greig and Eyth. Fairbairn's experiments show that a rivet is 6*^ weaker in a drilled than in a punched hole. By rounding the edge of the rivet-hole the apparent shearing resistance is increased 12%. Mr. Maynard found the rivets 4% weaker in drilled holes than in punched holes. But these results were obtained with riveted joints, and not by direct experiments on shearing. There is a good deal of difficulty in determining the true diameter of a punched hole, and it is doubtful whether in these experiments the diameter was very accurately ascertained. Messrs. Greig and Eyth's experiments also indicate a greater resistance of the rivets in punched holes than in drilled holes. If, as appears above, the apparent shearing resistance is less for double than for single shear, it is probably due to unequal distribution of the stress on the two rivet sections. The shearing resistance of a bar, when sheared in circumstances which prevent friction, is usually less than the tenacity of the bar. The following results show the decrease : Tenacity of Bar. Shearing Resistance. Harkort, iron - Lavalley, iron... Greig and Eyth, iron.. " " steel. 16.5 20.2 19.0 22.1 0.62 0.79 0.85 0.77 In Wohler's researches (in 1870) the shearing strength of iron was found to be four-fifths of the tenacity. Later researches of Bauscbinger confirm this result generally, but they show that for iron the ratio of the shearing resistance and tenacity depends on the direction of the stress relatively to the direction of rolling. The above ratio is valid only if the shear is in a plane perpendicular to the direction of rolling, and if the tension is applied parallel to the direction of rolling. The shearing resistance in a plane parallel to the direction of rolling is different from that in a plane perpen- dicular to that direction, and again differs according as the plane of shear is perpendicular or parallel to the breadth of the bar. In the former case the resistance is 18 to 20$ greater than in a plane perpendicular to the fibres, or is equal to the tenacity. In the latter case it is only half as great as in a plane perpendicular to the fibres. 364 IRON AND STEEL. IRON AND STEEL. CLASSIFICATION OF IRON AND STEEL. J ol^ d T3 B a 1 1 ^> 1- 0) 3 ft-a 3 a § .2 g erg t i £ a-~ as a ^ £ a 2 -- 1 Ot 5 -' a> S s fl ?ll CD 2 1 o w Obtained by dir ess from ores, Ian, Chenot, a i* process irons. Obtained by ir process from c as finery-hea puddled irons. O | i £ e £3:5*1 oil £ ftO o 2 .c c3 ^3 o PS j _£" s CO s Crucible Besseme and Open-he steels. Mitis.* c3 o s e TS 1 1 -23 6% E-T.g J2 ,-SS % g d a 1 O s En 03 (2) Mai cast iron tained fro 1 by ann in oxides. h O 8 e 1 O a o S -d bfi a> a i& oj g H o .o u,~£ O .2 s O o fo> ft 02 eg > . * a 5 02 '2"° ® III!" 02 r« O y ft«2 a ft3 W &JC -S 2-2-2 - «- ^-c'H "t a^ a-BSS, .2,2 c^ 73 a^ 02 ^ ^a-a 'If fl ^S03 M § fl | r > *2h2 sua o o> o £o?£ CAST IKON". 365 CAST IRON. Grading of Pig Iron.— Pig iron is commonly graded according to its fracture, the number of grades varying in different districts. In Eastern Pennsylvania the principal grades recognized are known as No. 1 and 2 foundry, gray forge or No. 3, mottled or No. 4, and white or No. 5. Inter- mediate grades are sometimes made, as No. 2 X, between No. 1 and No. 2, and special names are given to irons more highly silicized than No. 1, as No. 1 X, silver-gray, and soft. Charcoal foundry pig iron is graded by num- bers 1 to 5, but the quality is very different from the corresponding num- bers in anthracite and coke pig. Southern coke pig iron is graded into ten or more grades. Grading by fracture is a fairly satisfactory method of grading irons made from uniform ore mixtures and fuel, but is unreliable as a means of determining quality of irons produced in different sections or from different ores. Grading by chemical analysis, in the latter case, is the only satisfactory method. The following analyses of the five standard grades of northern foundry and mill pig irons are given by J. M. Hartman (Bull. I. & S.A., Feb., 1892): No. 1. No. 2. No. 3. No. 4. No. 4 B. No. 5. Iron 92.37 92.31 94.66 94.48 94.08 94.68 Graphitic carbon . . 3.52 2.99 2.50 2.02 2.02 Combined carbon.. .13 .37 1.52 1.98 1.43 3.83 Silicon 2.44 2.52 .72 .56 .92 .41 Phosphorus 1.25 1.08 .26 .19 .04 .04 Sulphur 02 .02 trace .08 .04 .02 Manganese 28 .72 .34 .67 2.02 .98 Characteristics of These Irons. No. 1. Gray. — A large, dark, open-grain iron, softest of all the numbers and used exclusively in the foundry. Tensile strength low. Elastic limit low. Fracture rough. Turns soft and tough. No. 2. Gray.— A mixed large and small dark grain, harder than No. 1 iron, and used exclusively in the foundry. Tensile strength and elastic limit higher than No. 1. Fracture less rough than No. 1. Turns harder, less tough, and more brittle than No. 1. No. 3. Gray.— Small, gray, close grain, harder than No. 2 iron, used either in the rolling-mill or foundry. Tensile strength and elastic limit higher than No. 2. Turns hard, less tough, and more brittle than No. 2. No. 4. Mottled.— White background, dotted closely with small black spots of graphitic carbon; little or no grain. Used exclusively in the rolling-mill. Tensile strength and elastic limit lower than No. 3. Turns with difficulty; less tough and more brittle than No. 3. The manganese in the B pig iron replaces part of the combined carbon, making the iron harder and closing the grain, notwithstanding the lower combined carbon. No. 5. White. — Smooth, white fracture, no grain, used exclusively in the rolling mill. Tensile strength and elastic limit much lower than No. 4. Too hard to turn and more brittle than No. 4. Southern pig irons are graded as follows, beginning with the highest in silicon: Nos. 1 and 2 silvery, Nos. 1 and 2 soft, all containing over 3$ of silicon; Nos. 1, 2, and 3 foundry, respectively about 2 ?5#, 2 5^ and 2% silicon; No. 1 mill, or "foundry forge;" No. 2 mill, or gray forge; mottled; white. Good charcoal chilling iron for car w T heels contains, as a rule, 0.56 to 0.95 silicon, 0.08 to 0.90 manganese, 0.05 to 0.75 phosphorus. The following is an analysis of a remarkably strong car wheel: Si, 0.734; Mn, 0.438; P. 0.428. S, 0.08» Graphitic C. 3.083; Combined C, 1,247; Copper, 0.029. The chill was very hard— *4 in - deep at root of flange, y% in. deep on tread. A good ordnance iron analyzed: Si. 0.30; Graphitic C, 2.20; Combined C, 1.70; P, 0.44; Mn, 3 55 (?). Its specific gravity was 7.22 and tenacity 31,734 lbs. per sq. in Influence of Silicon, Phosphorus, Sulphur, and Man- ganese upon Cast Iron. — W. J. Keep, of Detroit, in several papers (Trans. A. I. M. E., 1889 to 1893), discusses the influence of various chemical elements on the quality of cast iron. From these the following notes have been condensed: Silicon.— Pig iron contains all the carbon that it could absorb during its reduction in the blast-furnace. Carbon exists in cast iron in two distinct forms. In chemical unjon, as " combined " carbon, it cannot be discerned, except as it may increase the whiteness of the fracture, in so-called white 366 IRON AND STEEL. iron. Carbon mechanically mixed with the iron as graphite is visible, vary- ing in color from gray to black, while the fracture of the iron ranges from a light to a very dark gray. Silicon will expel carbon, if the iron, when melted, contains all the car- bon tbat it can hold and a portion of silicon be added. Prof. Turner concludes from his tests that the amount of silicon producing the maximum strength is about 1.80$. But this is only true when a white base is used. If an iron is used as a base which will produce a sound casting to begin with, each addition of silicon will decrease strength. Silicon itself is a weakening agent. Variations in the percentage of silicon added to a pig iron will not insure a given strength or physical structure, but these results will depend upon the physical properties of the original iron. After enough silicon has been added to cause solid castings, any further addition and consequent increase of graphite weakens the casting. The softness and strength given to castings by a suitable addition of silicon is, by a further increase of silicon, changed to stiffness, brittleness, and weakness. As strength decreases from increase of graphite and decrease of combined carbon, deflection increases; or, in other words, bending is increased by graphite. When no more graphite can form and silicon still increases, de- flection diminishes, showing that high silicon not only weakens iron, but makes it stiff. This stiffness is not the same strength-stiffness which is caused by compact iron and combined carbon. It is a brittle- stiffness. In pig irons which received their silicon while in the blast-furnace the graphite more easily separates, and the shrinkage is less than in any mix- ture. As silicon increases, shrinkage also increases. Silicon of itself in- creases shrinkage, though by reason of its action upon the carbon in ordi- nary practice it is truly said that silicon "takes the shrinkage out of cast- iron." The slower a casting crystallizes, the greater will be the quantity of graphite formed within it. Silicon of itself, however small the quantity present, hardens cast-iron; but the decrease of hardness from the change of the combined carbon to graphite, caused by the silicon, is so much more rapid than the hardening produced by the increase of silicon, that the total effect is to decrease hard- ness, until the silicon reaches from 3 to 5%. As practical foundry-work does not call for more than 3% of silicon, the ordinary use of silicon does reduce the hardness of castings; but this is pro- duced through its influence on the carbon, and not its direct influence on the iron. When the change from combined to graphite carbon has ceased to dimin- ish hardness, say at from 2/ to b% of silicon, the hardening by the silicon itself becomes more and more apparent as the silicon increases. Shrinkage and hardness are almost exactly proportional. When silicon varies, and other elements do not vary materially, castings with low shrink- age are soft; as shrinkage increases, the castings grow hard in almost, if not exactly, the same proportion. For ordinary foundry-practice the scale of shrinkage may be made also the scale of hardness, provided variations in sulphur, and phosphorus especially, are not present to complicate the re- sult. The term " chilling " irons is generally applied to such as, cooled slowly, would be gray, but cooled suddenly, become white either to a depth suffi- cient for practical utilization {e.g., in car-wheels) or so far as to be detrimen- tal. Many irons chill more or less in contact with the cold surface of the mould in which they are cast, especially if they are thin. Sometimes this is a valuable quality, but for general foundry purposes it is desirable to have all parts of a casting an even gray. Silicon exerts a powerful influence upon this property of irons, partially or entirely removing their capacity of chilling. When silicon is mixed with irons previously low in silicon the fluidity is increased. It is not the percentage of silicon, but the state of the carbon and the action of silicon through other elements, which causes the iron to be fiuid. Silicon irons have always had the reputation of imparting fluidity to other irons. This conies, no doubt, from the fact that up to 3% or 4% they increase the quantity of graphite in the resulting casting. From the statement of Prof. Turner, that the maximum strength occurs with just such a percentage of silicon, and his statement that a founder can, with silicon, produce just the quality of iron that he may need, and from his naming the composition of what he calls a typical foundry-iron, some INFLUENCE OP SILICON, ETC., UPON CAST IRON. 367 founders have inferred that if they knew the percentages of silicon in their irons and in their ferro-silicon, they need only mix so as to get 2% of silicon in order to obtain, always and with certainty, the maximum strength. The solution of the problem is not so simple. Each of the irons which the foun- der uses will have peculiar tendencies, given them in the blast-furnace, which will exert their influence in the most unexpected ways. However, a white iron which will invariably give porous and brittle castings can be made solid and strong by the addition of silicon; a further addition of sili- con will turn the iron gray; and as the grayness increases the iron will grow weaker. Excessive silicon will again lighten the grain and cause a hard and brittle as well as a very weak iron. The only softening and shrinkage-les- sening influence of silicon is exerted during the time when graphite is being produced, and silicon of itself is not a softener or a lessener of shrinkage; but through its influence on carbon, and only during a certain stage, does it produce these effects. Phosphorus.— While phosphorus of itself, in whatever quantity present, weakens cast-iron, yet in quantities less than 1.5% its influence is n t suffi- ciently great to overbalance other beneficial effects, which are exerted before the percentage reaches 1%. Probably no element of itself weakens cast iron as much as phosphorus, especially when present in large quantities. Shrinkage is decreased when phosphorus is increased. All high-phosphorus pig irons have low shrinkage. Phosphorus does not ordinarily harden cast iron, probably for the reason that it does not increase combined carbon. The fluidity of the metal is slightly increased by phosphorus, but not to any such great extent as has been ascribed to it. The property of remaining long in the fluid state must not be confounded with fluidity, for it is not the measure of its ability to make sharp castings, or to run into the very thin parts of a mould. Generally speaking, the state- ment is justified that, to some extent, phosphorus prolongs the fluidity of the iron while it is filling the mould. The old Scotch irons contained about 1% of phosphorus. The foundry-irons which are most sought for for small and thiu castings in the Eastern States contain, as a general thing, over 1% of phosphorus. Certain irons which contain from 4% to 7% silicon have been so much used on account of their ability to soften other irons that they have come to be known as " softeners " and as lesseners of shrinkage. These irons are valu- able as carriers of silicon ; but the irons which are sold most as softeners and shrinkage-lesseners are those containing from 1% to 2% of phosphorus. We must therefore ascribe the reputation of some of them largely to the phosphorus and not wholly to the silicon which they contain? From y%% to \% of phosphorus will do all that can be done in a beneficial way, and all above that amount weakens the irou, without corresponding benefit. It is not necessary to search for phosphorus-irons. Most irons contain more than is needed, and the care should be to keep it within limits. Sulphur.— Only a small percentage of sulphur can be made to remain in carbonized iron, and it is difficult to introduce sulphur into gray cast iron or into any carbonized iron, although gray cast iron often takes from the fuel as much more sulphur as the iron originally contained. Percentages of sulphur that could be retained by gray cast iron cannot materially injure the iron except through an increase of shrinkage. The higher the carbon, or the higher the silicon, the smaller will be the influence exerted by sulphur. The influence of sulphur on all ca^t iron is to drive out carbon and silicon and to increase chill, to increase shrinkage, and, as a general thing, to decrease strength ; but if in practice sulphur will not enter such iron, we shall not have any cause to fear this tendency. In every-day work, however, it is found at times that iron which was gray when put into the cupola comes out white, with increased shrinkage and chill, and often with decreased strength. This is caused by decreased silicon, and can be remedied by an increase of silicon. Mr. Keep's opinion concerning the influence of sulphur, quoted above, is disagreed with by J. B. Nau (Iron Age, March 29, 1894). He says : "Sulphur, in whatever shape it may be present, has a deleterious influence on the iron. It has the tendency to render the iron white by the influence it exercises on the combination between carbon and iron. Pig iron contain- ing a certain percentage of it becomes porous and full of holes, and castings made from sulphurous iron are of inferior quality. This happens especially when the element is present in notable quantities. With foundry-iron con- taining as high as 0.1% of sulphur, castings of greater strength may be ob- 368 IRON AND STEEL. tained than when no sulphur is present. Thus, in some tests on this element quoted by R. Akerman, it is stated that in the foundry-iron from Finspong, used in the manufacture of cannons, a percentage of 0.1$ to 0.14$ of sulphur in the iron increased its strength to a considerable extent. The percentage of sulphur found originally in the iron put in the cupola is liable to be further increased by part of the sulphur that is invariably found in the coke used. It is seldom that a coke with a small percentage of sulphur is found, whereas coke containing 1$ of it and over is very common. With such a fuel in the cupola, if no special precautions are resorted to, the percentage of sulphur in the metal will in most cases be increased." That the sulphur contents of pig iron may be increased by the sulphur contained in the coke used, is shown by some experiments in the cupola, reported by Mr. Nau. Seven consecutive heats were made. The sulphur content of the coke was 1$, and 11.7$ of fuel was added to the charge. Before melting, the silicon ranged from 0.320 to 0.830 in the seven heats ; after melting, it was from 0.110 to 0.534, the loss in melting being from .100 to .375. The sulphur before melting was from .076 to .090, and after melting from .132 to .174, a gain from .044 to .098. From the results the following conclusions were drawn : 1. In all the charges, without exception, sulphur increased in the pig iron after its passage through the cupola. In some cases this increase more than doubled the original amount of sulphur found in the pig iron. 2. The increase of the sulphur contents in the iron follows the elimination of a greater amount of silicon from that same iron. A larger amount of limestone added to these charges would have produced a more basic cinder, and undoubtedly less sulphur would have been incorporated in the iron. 3. This coke contained 1$ of sulphur, and if all its sulphur had passed into the iron there would have been an average increase of 0.12 of sulphur for the seven charges, while the real iDcrease in the pig iron amounted to only 0.081. This shows that two thirds of the sulphur of the coke was taken up by the iron in its passage through the cupola. Manganese.— Manganese is a nearly white metal, having about the same appearance when fractured as white cast iron. Its specific gravity is about 8, while that of white cast iron, reasonably free from impurities, is but a little above 7.5. As produced commercially, it is combined with iron, and with small percentages of silicon, phosphorus, and sulphur. It is generally produced in the blast-furnace. If the manganese is under 40$, with the remainder mostly iron, and silicon not over 0.50$, the alloy is called spiegeleisen, and the fracture will show flat reflecting surfaces, from which it takes its name. With manganese above 50$, the iron alloy is called ferro-manganese. As manganese increases beyond 50$, the mass cracks in cooling, and when it approaches 98$ the mass crumbles or falls in small pieces. Manganese combines with iron in almost any proportion, but if an iron containing manganese is remelted, more or less of the manganese will escape by volatilization, and by oxidation with other elements present in the iron. If sulphur be present, some of the manganese will be likely to unite with it and escape, thus reducing the amount of both elements in the casting. Cast iron, when free from manganese, cannot hold more than 4.50$ of car- bon, and 3.50$ is as much as is generally present ; but as manganese increases, carbon also increases, until we often find it in spiegel as high as 5$, and in ferro-manganese as high as 6$. This effect on capacity to hold carbon is peculiar to manganese. Manganese renders cast iron less plastic and more brittle. Manganese increases the shrinkage of cast iron. An increase of 1$ raised the shrinkage 26$. Judging from some test records, manganese does not influence chill at ad; but other tests show that with a given percentage of silicon the carbon may be a little more inclined to remain in the combined form, and therefore the chill may be a little deeper. Hence, to cause the chill to be the same, it would seem that the percentage of silicon should be a little higher with manganese than without it. An increase of 1$ of manganese increased the hardness 40$. If a hard chill is required, manganese gives it by adding hardness to the whole casting. J. B. Nau {Iron Age, March 29, 1894), discussing the influence of manga- nese on cast iron, says: Manganese favors the combination between carbon and iron. Its influ- ence, when present in sufficiently large quantities, is even great enough nor, only to keep the carbon which would be naturally found in pig iron com- TESTS OF CAST IROK. 369 bined, but it increases the capacity of iron to retain larger amounts of car- bon and to retain it all in the combined state. Manganese iron is often used for foundry purposes when some chill and harduess of surface is required in the casting. For the rolls of steel-rail mills we always put into the mixture a large amount of mauganiferous iron, and the rolls so obtained alwaj r s presented the desired hardness of surface and in general a mottled structure on the outside. The inside, which al- ways cooled much slower, was gray iron. One of the standard mixtures that invariably gave good results was the following: 50$ of foundry iron with 1.3$ silicon and 1.5$ manganese; 35$ of foundry iron with 1$ silicon and 1.5$ manganese; 15% steel (rail ends) with about 0.35$ to 0.40$ carbon. The roll resulting from this mixture contained about 1% of silicon and 1% of manganese. Another mixture, which differed but little from the preceding, was as follows: 45% foundry iron with about 1.3$ silicon and 1.5$ manganese; 30$ foundry iron with about 1% silicon and 1.5$ manganese; 10$ white or mottled iron with about 0.5$ to 0.6$ Si. and 1.2$ Mn. 15$ Bessemer steel-rail ends with about 0.35$ to 0.40$ C. and 0.6$ to 1$ Mn. The pis: iron used in the preceding mixtures contained also invariably from 1.5$ to 1.8$ of phosphorus, so that the rolls obtained therefrom carried about 1.3$ to 1.4$ of that element. The last mixture used produced rolls containing on the average 0.8$ to 1$ of silicon and 1$ of manganese. When- ever we tried to make those rolls from a mixture containing but 0.2$ to 0.3$ manganese our rolls were invariably of inferior quality, grayer, and con- sequently softer. Manganese iron cannot be used indiscriminately for foundry purposes. When greater softness is required in the castings man- ganese has to be avoided, but when hardness to a certain extent has to be obtained manganese iron can be used with advantage. Manganese decreases the magnetism of the iron. This characteristic in- creases with the percentage of manganese that enters into the composition of the iron. The iron loses all its magnetism when manganese reaches 25$ of its composition. This peculiarity has been made use of by French metallurgists to draw a clear line between spiegel and ferro-manganese. When the pig contains less than 25$ of manganese it is classified as spiegel, and when it contains more than 25 it$ is classified as ferro-manganese. For this reason manganese iron has to be avoided in castings of dynamo fields and other pieces belonging to electric machinery, where magnetic conduc- tibility is one of the first considerations. Irregular Distribution of Silicon in Pig Iron.— J. W. Thomas (Iron Age, Nov. 12, 1891) finds in analyzing samples taken from every other bed of a cast of pig iron that the. silicon varies considerably, the iron coming first from the furnace having generally the highest percentage. In one series of tests the silicon decreased from 2.040 to 1.713 from the first bed to the eleventh. In another case the third bed had 1.260 Si., the seventh 1.718, and the eleventh 1.101. He also finds that the silicon varies in each pig, be- ing higher at the point than at the butt. Some of his figures are: point of pig 2.328 Si., butt of same 2.157; point of pig 1.834, butt of same 1.787. Some Tests of Cast Iron. (G. Lanza, Trans. A. S. M. E., x., 187.)— The chemical analyses were as follows: Gun Iron, Common Iron, per cent. per cent. Total carbon 3.51 Graphite 2.80 Sulphur 0.133 0.173 Phosphorus 0.155 0.413 Silicon 1.140 1.89 The test specimens were 26 inches long and square in section; those tested with the skin on being very nearly one inch square, and those tested with the skin removed being cast nearly one and one quarter inches square, and afterwards planed down to one inch square. Tensile Elastic ^dubis Strength. Limit. °^^ Unplaned common. 20.200 to 23,000 T. S. Av. = 22,066 6,500 13,194,233 Planed common.... 20,300 to 20,800 " " =20,520 5,833 11,943,953 Unplaned gun 27,000 to 28,775 " " =28,175 11,000 16,130,300 Planed gun 29,500 to 31,000 " " = 30,500 8,500 15,932,880 370 IttOH AND STEEL. The el istic limit is not clearly defined in cast iron, the elongations increas- ing faster than the increase of the loads from the beginning of the test. The modulus of elasticity is therefore variable, decreasing as the loads in- crease. For example, the following results of a test of common cast iron, reported by Prof. Lanza: T hs r>er sn in Elongation in Sets, Modulus of L,bs. per sq. in. 13 4 inches in Elasticity. 1000 .0004 18,217,400 2000 .0013 16,777,700 3000 .0024 14,085,400 4000 .0036 13,101,200 5000 .0048 12,809,200 6000 .0061 .0000 12,319,300 8000 .0088 .0001 11,600.800 10000 .0119 .0001 10,930 500 12000 .0162 .0007 9,714,200 CHEMISTRY OF FOUNDRY IRONS. (C. A. Meissner, Columbia College Qly, 1890; Iron Age, 1890.) Silicon is a very important element in foundry irons. Its tendency when not above 2^% is to cause the carbon to separate out as graphite, giving the casting the desired benefits of graphitic iron. Between 2\4>% and S}4% silicon is best adapted for iron carrying a fair proportion of low silicon scrap and close iron, for ordinarily no mixture should run below 1}4,% silicon to get good castings. From 3$ to h% silicon, as occurs in silvery iron, will carry heavy amounts of scrap. Castings are liable to be brittle, however, if not handled carefully as regards proportion uf scrap used. From 1% #to 2% silicon is best adapted for machine work; will give strong clean castings if nor, much scrap is used with it. Below \% silicon seems suited for drills and castings that have to stand great variations in temperature. Silicon has the effect of making castings fluid, strong, and open-grained ; also sound, by its tendency to separate the graphite from the total carbon, and consequent slight expansion of the iron on cooling, causing It to fill out thoroughly. Phosphorus, when high, has a tendency to make iron fluid, retain its heat longer, thereby helping to fill out all small spaces in casting. It makes iron brittle, however, when above %% in castings. It is excellent when high to use in a mixture of low-phosphorus irons, up to 1%% giving good results, but, as said before, the casting should be below %%. It has a strong tendency when above 1% in pig to make the iron less graphitic, pre- venting the separation of graphite. Sulphur in open iron seldom bothers the founder, as it is seldom present to any extent. The conditions causing open iron in the furnace cause low sulphur. A little manganese is an excellent antidote against sulphur in the furnace. Irons above 1% manganese seldom have any sulphur of any con- sequence. Graphite is the all-important factor in foundry irons; unless this is present in sufficient amount in the castiug, the latter will be liable to be poor. Graphite causes iron to slightly expand on cooling, makes it soft, tough and fluid. (The statement as to expansion on cooling is denied by VV. J. Keep.) Relation of the Appearance of Fracture to the Chemical Composition. — S. H. Chauvenet says. when run [from the blast-fur- nace] the lower bed is almost always close-grain, but shows practically the same analysis as the large grain in the rest of the cast. If the iron runs rapidly, the lower bed may have as large grain as any in the cast. If the iron runs rapidly for, say, six beds and some obstruction in the tap-hole causes the seventh bed to fill up slowly and sluggishly, this bed may be close-grain, although the eighth bed, if the obstruction is removed, will be open-grain. Neither the graphitic carbon nor the silicon seems to have any influence on the fracture in these cases, since by analysis the graphite and silicon is the same in each. The question naturally arises whether it would not be better to be guided by the analysis than by the fracture. The frac- ture is a guide, but it is not an infallible guide. Should not the open- and the close-grain iron from the same cast be numbered under the same grade when they have the same analysis ? Mr. Meissner had many analyses made for the comparison of fracture CHEMISTRY OF FOUNDRY IRONS. 371 with analysis, and unless the condition of furnace, whether the iron ran fast or slow, and from what part of pig bed the sample is taken, are known, the fracture is often very misleading. Take the following analyses: A. B. C. D. E. F. 4.315 0.008 3.010 4.818 0.008 2.757 4.270 0.007 2.680 3.328 0.033 2.243 3.869 0.006 3.070 0.108 3.861 Sulphur Graphitic car.. 0.006 3.100 0.096 A. Very close-grain iron, dark color, by fracture, gray forge. B. Open-grain, dark color, by fracture. No. 1. C. Very close-grain, by fracture, gray forge. D. Medium-grain, by fracture, No. 2, but much brighter and more open than A, C, or F. E. Very large, open-grain, dark color, by fracture, No. 1. F. Very close-grain, by fracture, gray forge. By comparing analyses A and B, or E and F, it appears that the close- grain iron is in each case the highest in graphitic carbon. Comparing A and E, the graphite is about the same, but the close-grain is highest in silicon. Analyses of Foundry Irons. (C. A. Meissner.) Scotch Irons. Name. Grade. Silicon. Phos- phorus. Manga- nese. Sul- phur. Graph- ite. Comb. Carton Summerlee 1 1 1 2 1 1 1 1 2 2.70 2.47 3.44 2.70 2.15 ■ 2.59 1.70 3.03 4.00 0.545 0.760 1.000 0.810 0.618 0.840 1.100 1.200 0.900 1.80 2.51 1.70 2.90 2.80 1.70 1.83 2.85 3.41 0.01 0.015 0.015 0.02 0.025 0.010 0.008 0.010 3.09 2.00 3.76 3.75 3.50 1.78 0.25 0.80 21 3.75 0.40 Glengarnock Glengarnock said to carry % scrap 0.90 Description of Samples.— No. 1. Well known Ohio Scotch iron, almost silvery, but carries two-thirds scrag: made from part black-band ore. Very successful brand The high silicon gives it its scrap-carrying capacity. No. 2. Brier Hill Scotch castings, made at scale works; castings demand- ing more fluidity than strength. 372 IRON AND STEEL. No. 3. Formerly a famous Ohio Scotch brand, not now in the market Made mainly from black-band ore. No. 4. A good Ohio Scotch, very soft and fluid; made from black-band ore-mixture. Nos. 5a and 56. Brier Hill Scotch iron and casting; made for stove pur- poses; 350 lbs. of iron used to 150 lbs. scrap gave very soft fluid iron; worked well. No. 6a. Shows comparison between Summerlee (Scotch) (6a) and Brier Hill Scotch (6b). Drillings came from a Cleveland foundry, which found both irons closely alike in physical and working quality. No. 7. One of the best southern brands, very hard to compete with, owing to its general qualities and great regularity of grade and general working. Description of Samples.— No. 8. A famous Southern brand noted for fine machine castings. No. 9. Also a Southern brand, a very good machine iron. Nos. 10a and 106. Formerly one of the best known Ohio brands. Does not shrink; is very fluid and strong. Foundries having used this have reported very favorably on it. No. 11. Iron from Brier Hill Co., made to imitate No. 3 ; was stronger than No. 3; did not pull castings; was fluid and soft. No. 12. Copy of a very strong English machine iron. No. 13. A Pennsylvania iron, very tough and soft. This is partially Besse- mer iron, which accounts for strength, while high silicon makes it soft. No. 14. Castings made from Brier Hill Co.'s machine brand for scale works, very satisfactory, strong, soft and fluid. No. 15. Castings made from Brier Hill Co.'s one half machine brand, one half Scotch brand, for scale works, castings desired to be of fair strength, but very fluid and soft. No. 16a. Brier Hill machine brand made to compete with No. 3. No. 166. Castings (clothes-hooks) from same, said to have worked badly, castings being white and irregular. Analysis proved that some other iron too high in manganese had been used, and probably not well mixed. No. 17. A Pennsylvania iron, no shrinkage, excellent machine iron, soft and strong. No. 18. A very good quality Northern charcoal iron. "Standard Grades" of the Brier Hill Iron and Coal Company. Brier Hill Scotch Iron. — Standard Analysis, Grade Nos. 1 and 2. Silicon 2.00 to 3.00 Phosphorus 0.50to0.75 Manganese 2.00 to 2.50 Used successfully for scales, mowing-machines, agricultural implements, novelty hardware, sounding-boards, stoves, and heavy work requiring no special strength. CHEMISTRY OF FOUNDRY IRONS. 373 Brier Hill Silvery Iron.— Standard Analysis, Grade No. 1. Silicon 3.50 to 5.50 Phosphorus 1.00 to 1.50 Manganese 2.00 to 2.25 Used successfully for hollow-ware, car-wheels, etc., stoves, bumpers, and similar work, with heavy amounts of scrap in all cases. Should be mainly used where fluidity and no great strength is required, especially for heavy work. When used with scrap or close pig low in phosphorus, castings of considerable strength and great fluidity can be made Fairly Heavy Machine Iron.— Standard Analysis, Grade No. 1. Silicon 1.75 to 2.50 Phosphorus 0.50to0.60 Manganese 1.20 to 1.40 The best iron for machinery, wagon-boxes, agricultural implements, pump-works, hardware specialties, lathes, stoves, etc., where no large amounts of scrap are to be carried, and where strength, combined with great fluidity and softness, are desired. Should not have much scrap with it. Regular Machine Iron. — Standard Analysis, Grade Nos. 1 and 2. Silicon.... 1.50to2.00 Phosphorus 0. 30 to .50 Manganese 0.80 to 1.00 Used for hardware, lawn-mowers, mower and reaper works, oil-well machinery, drills, fine machinery, stoves, etc. Excellent for all small fine castings requiring fair fluidity, softness, and mainly strength. Cannot be well used alone for lai-ge castings, but gives good results on same when used with above mentioned heavy machine grade; also when used with the Scotch ia right proportion. Will carry but little scrap, and should be used alone for good strong castings. For Axles and Materials Requiring Great Strength, Grade No. 2. Silicon. 1.50 Phosphorus 0.200 and less. Manganese 0.80 This gave excellent results. A good neutral iron for guns, etc., will run about as follows: Silicon 1.00 Phosphorus 0.25 Sulphur 0.20 Manganese none. It should be open No. 1 iron. This gives a very tough, elastic metal. More sulphur would make tough but decrease elasticity. For fine castings demanding elegance of design but no strength, phos- phorus to 3.00$ is good. Can also stand 1.50$ to 2.00$ manganese. For work of a hard, abrasive character manganese can run 2.00$ in casting. Analyses of Castings. Sample No. Silicon. Phos- phorus. 1.400 0.351 0.327 0.577 0.742 1.208 0.418 1.280 0.879 0.408 0.660 1.439 0.900 0.980 Manganese Sulphur. Graphite. Comb. Carbon. 31 2.50 0.85 1.53 1.84 2.20 2.50 2.80 3.10 3.30 2.88 4.50 3.43 2.68 1.90 2.20 0.92 1.08 1.04 1.10 1.16 0.54 1.14 0.80 1.10 0.78 0.90 1.30 1.20 32 0.030 0.040 33 34a 3.10 0.58 346 34c 356 35c 35cZ 35e 0.025 37a 876 374 IRON AND STEEL. No. 31. Sewing-machine casting, said to be very fluid and good casting. This is an odd analysis. I should say it would have been too hard and brit- tle, yet no complaint was made. No. 32. Very good machine casting, strong, soft, no shrinkage. No. 33. Drilliugs from an annealer-box that stood the heat very well. No. 34a. Drillings from door-hinge, very strong and soft. No. 346. Drilliugs from clothes-hooks,' tough and soft, stood severe ham- mering. No. 34c. Drillings from window-blind hinge, broke off suddenly at light strain. Too high phosphorus. No. 35a. Casting for heavy ladle support, very strong. Nos 356 and 35c. Broke after short usage. Phosphorus too high. Car- bumpers. No. 35o\ Elbow for steam heater, very tough and strong. No. 36. Cog-wheels, very good, shows absolutely no shrinkage. No. 37. Heater top network, requiring fluidity but no strength. No. 37a. Gray part of above. No. 376. White, honeycombed part of above. Probably bad mixing and got chilled suddenly. STRENGTH OF CAST IRON. Rankine gives the following figures: Various qualities, T. S 13,400 to 29,000, average 16,500 Compressive strength 82,000 to 145,000, " 112,000 Modulus of elasticity 14,000,000 to 22,900,000, " 17,000,000 Specific Gravity and Strength. (Major Wade, 1856.) Third-class guns: Sp. Gr. 7.087, T. S. 20,148. Another lot: least Sp. Gr. 7.163, T. S. 22,402. Second-class guns: Sp. Gr. 7.154, T. S. 24,767. Another lot : mean Sp. Gr. 7.302, T. S. 27,232. First class guns: Sp. Gr. 7.204, T. S. 28,805. Another lot: greatest Sp. Gr. 7.402, T. S. 31,027. Strength of Charcoal Pig Iron. -Pig iron made from Salisbury ores, in furnaces at Wassaic and Millerton, N. Y., has shown over 40,000 lbs. T. S. per square inch, one sample giving 42,281 lbs. Muirkirk, Md., iron tested at the Washington Navy Yard showed: average for No. 2 iron, 21,601 lbs. ; No. 3, 23,959 lbs. ; No. 4, 41,329 lbs. ; average density of No. 4, 7.336 (J. C. I. W., v. p. 44.) Nos. 3 and 4 charcoal pig iron from Chapinville, Conn., showed a tensile strength per square inch of from 34,761 lbs. to 41,882 lbs. Charcoal pig iron from i Shelby, Ala. (tests made in August, 1891), showed a strength of 34,800 lbs. for No. 3; No. 4, 39,675 lbs.; No. 5, 46,450 lbs.; and a mixture of equal parts of Nos. 2, 3, 4. and 5, 41.470 lbs. (Bull. I. & 8. A.) Variation of Density and Tenacity of Gun-irons.— An in- crease of density invariably follows the rapid cooling of cast iron, and as a general rule the tenacity is increased by the same means. The tenacity generally increases quite uniformly with the density, until the latter ascends to some given point; after which an increased density is accompanied by a diminished tenacity. The turning-point of density at which the best qualities of gun-iron attain their maximum tenacity appears to be about 7.30. At this point of density, or near it, whether in proof-bars or gun-heads, the tenacity is greatest. As the density of iron is increased its liquidity when melted is diminished. This causes it to congeal quickly, and to form cavities in the interior of the casting. (Pamphlet of Builders 1 Iron Foundrv, 1893.) Specifications for Cast Iron tor the World's Fair Build- ings, 1892. — Except where chilled iron is specified, all castings shall be of tough gray iron, free from injurious cold-shuts or blow-holes, true to pattern, and of a workmanlike finish. Sample pieces 1 in. square, cast from the same heat of metal in sand moulds, shall be capable of sustaining on a clear span of 4 feet 6 inches a central load of 500 lbs. when tested in the rough bar. Specifications for Tests of Cast Iron in 12" B. Ii. Mortars* (Pamphlet of Builders Iron Foundry, 1893.)— Charcoal Gun Iron.— The tensile strength of the metal must average at each end at least 30,000 lbs. per square inch ; no specimen to be over 37,000 lbs. per square inch ; but one specimen from each end may be as low as 28,000 lbs. per square inch. The MALLEABLE CAST IROH. 375 long extension specimens will not be considered in making up these aver- ages, but must show a good elongation and an ultimate strength, for each specimen, of not less than 24,000 lbs. The density of the metal must be such as to indicate that the metal has been sufficiently refined, but not carried so high as to impair the other qualities. Specifications for Grading Pig Iron for Car Wheels by Chill Tests made at the Furnace. (Penna. R. R. Specifications, 1883.)— The chill cup is to be filled, even full, at about the middle of every cast from the furnace. The test-piece so made will be 7}4 inches long, 3^ inches wide, and 1% inches thick, and is to be broken across the centre when entirely cold. The depth of chill will be shown on the bottom of the test- piece, and is to be measured by the clean white portion to the point where gray specks begin to show in the white. The grades are to be by eighths of an inch, viz., %, %, %, ^, %%, %, etc., until the iron is mottled; the lowest grade being % of an inch in depth of chill. The pigs of each cast are to be marked with the depth of chill shown by its test-piece, and each grade is to be kept by itself at the furnace and in forwarding. Mixture of Cast Iron with Steel.— Car wheels are sometimes made from a mixture of charcoal iron, anthracite iron, and Bessemer steel. The following shows the tensile strength of a number of tests of wheel mixtures, the average tensile strength of the charcoal iron used being 22,000 lbs.: lbs. per sq. in. Charcoal iron with 2)4% steel 22,467 " 8%* steel 26,733 " " " 6}4% steel and 6}4% anthracite 24,400 " " " 714% steel and iy>% anthracite 28,150 " 2y%% steel, 2%% wro't iron, and G*4% anth. .. 25,550 41 " " 5 % steel, 5% wro't iron, and 10 % anth 26,500 (Jour. C. I. W., hi. p. 184.) Cast Iron Partially Bessemerized.— Car wheels made of par- tially Bessemerized iron (blown in a Bessemer converter for 2>)4 minutes), chilled in a chill-test mould over an inch deep, just as a test of cold-blast charcoal iron for car wheels would chill. Car wheels made of this blown iron have run 250.000 miles. [Jour. C. I. W., vi. p. 77.) Bad Cast Iron.— On October 15, 1891, the cast-iron fly-wheel of a large pair of Corliss engines belonging to the Amoskeag Mfg. Co., of Manchester, N. H., exploded from centrifugal force. The fly-wheel was 30 feet diam- eter and 110 inches face, with one set of 12 arms, and weighed 116,000 lbs. After the accident, the rim castings, as well as the ends of the arms, were found to be full of flaws, caused chiefly by the drawing and shrinking of the metal. Specimens of the metal were tested for tensile strength, and varied from 15,000 lbs. per square inch in sound pieces to 1000 lbs. in spongy ones. None of these flaws showed on the surface, and a rigid examination of the parts before they were erected failed to give any cause to suspect their true nature. Experiments were carried on for some time after the accident in the Amoskeag Company's foundry in attempting to duplicate the flaws, but with no success in approaching the badness of these castings. MALLEABLE CAST IRON. Malleableized cast iron, or malleable iron castings, are castings made of ordinary cast iron which have been subjected to a process of decarboni- zation, which results in the production of a crude wrought iron. Handles, latches, and other similar articles, cheap harness mountings, plowshares, iron handles for tools, wheels, and pinions, and many small parts of ma- chinery, are made of malleable cast iron. For such pieces charcoal cast iron of the best quality (or other iron of similar chemical composition), should be selected. Coke irons low in silicon and sulphur have been used in place of charcoal irons. The castings are made in the usual way, and are then imbedded in oxide of iron, in the form, usually, of hematite ore, or in per- oxide of manganese, and exposed to a full red-heat for a sufficient length of time, to insure the nearly complete removal of the carbon. This decarboniza- tion is conducted in cast-iron boxes, in which the articles, if small, are packed in alternate layers with the decarbonizing material. The largest pieces require the longest time. The fire is quickly raised to the maximum temperature, but at the close of the process the furnace is cooled very slowly. The operation requires from three to five days with ordinary small castings, and may take two weeks for large pieces. 376 iROtf AKi) STEEL. Rules for Use of Malleable Castings, by Committee of Master Carbuilders' Ass'n, 1890. 1. Never run abruptly from a heavy to a light section. 2. As the strength of malleable cast iron lies in the skin, expose as much surface as possible. A star-shaped section is the strongest possible from which a casting can be made. For brackets use a number of thin ribs instead of one thick one. 3. Avoid all round sections; practice has demonstrated this to be the weakest form. Avoid sharp angles. 4. Shrinkage generally in castings will be 3/16 in. per foot. Strength, of Malleable Cast Iron.— Experiments on the strength of malleable cast iron, made in 1891 by a committee of the Master Car- builders' Association. The strength of this metal varies with the thickness, as the following results on specimens from 34 in. to iy 2 in. in thickness show: Dimensions. Tensile Strength. Elongation. Elastic Limit. in. in. lb. per sq. in. percent in 4 in. lb. per sq. in. 1.52 by .25 34,700 2 21,100 1.52 " .39 33,700 2 15,200 1.53 " .5 32,800 2 17,000 1.53 " .64 32,100 2 19,400 2. " .78 25,100 M 15,400 1.54 " .88 33,600 m 19,300 1.06 " 1.02 30,600 i 17,600 1.28 " 1.3 27,400 i 1.52 " 1.54 28,200 1H The low ductility of the metal is worthy of notice. The committee gives the following table of the comparative tensile resistance and ductility of malleable cast iron, as compared with other materials: Cast iron Malleable cast iron Wrought iron Steel castings Ultimate Strength, lb. per sq. in Comparative Strength ; Cast Iron = 1. 20,000 32,000 50,000 60,000 Elongation Per Cent in 4 in. 0.35 2.00 20.00 10.00 Comparative Ductility; Malleable Cast Iron = 1. 0.17 1 10 Another series of tests, reported to the Association in 1892, gave the following: Thick- ness. Width. Area. Elastic Limit. Ultimate Strength. Elongation in 8 in. in. in. sq. in. lb. per sq. lb. per sq. in. percent. .271 2.81 .7615 23.520 32,620 1.5 .293 2.78 .8145 22,650 28,160 .6 .39 2.82 1.698 20,595 32,060 1.5 .41 2.79 1.144 20,230 28,850 1.0 .529 2.76 1.46 19,520 27,875 1.1 .661 2.81 1.857 18,840 25,700 .7 .8 2.76 2.208 18.390 25,120 1.1 1.025 2.82 2.890 18,220 28,720 1.5 1.117 2.81 3.138 17,050 25,510 1.3 1.021 2.82 2.879 18,410 26,950 1.3 . WROUGHT IRON". 377 WROUGHT IRON. Influence of Chemical Composition on the Properties of Wrought Iron. (Beaidslee on Wrought Iron and Chain Cables. Abridgement by W. Kent. Wiley & Sons, 1879.)— A series of 2000 tests of specimens from 14 brands of wrought iron, most of them of high repute, was made in 1877 by Capt. L. A. Beaidslee, U.S.N., of the United States Testing Board. Forty-two chemical analyses were made of these irons, with a view to determine what influence the chemical composition had upon the strength, ductility, and welding power. From the report of these tests by A. L. Holley the following figures are taken : Average Tensile Strength. Chemical Composition. Brand. S. P. Si. C. Mn. Slag. L P B J O C 66,598 54,363 52,764 51,754 51,134 50,765 trace j 0.009 1 0.001 0.008 j 0.003 jO.005 (0.004 \ 0.005 0.007 j 0.065 I 0.084 0.250 0.095 0.231 0.140 0.291 0.067 0.078 0.169 0.080 0.105 0.182 0.028 0.156 0.182 0.321 0.065 0.073 0.154 0.212 0.512 0.033 0.066 0.015 0.027 0.051 0.045 0.042 0.042 0.005 0.029 0.033 0.009 0.017 trace 0.053 0.007 0.005 0.021 0.192 0.452 0.848 1.214 "6!678" 1.724 1.168 0.974 Where two analyses are given they are the extremes of two or more ana- lyses of the brand. Where one is given it is the only analysis. Brand L should be classed as a puddled steel. Order op Qualities Graded from No. 1 to No. 19. rand. ■rensue Strength. reduction of Area. Elongation. Welding Power. L 1 18 19 most imperfect. P 6 6 3 badly. B 12 16 15 best. J 16 19 18 rather badly. O 18 1 4 very good. C bi . 19 12 16 The reduction of area varied from 54.2 to 25.9 per cent, and the elonga- tion from 29..9 to 8.3 per cent. Brand O, the purest iron of the series, ranked No. 18 in tensile strength, but was one of the most ductile; brand B, fquite impure, was below the average both in strength and ductility, but was the best in welding power ; P, also quite impure, was one of the best in every respect except welding, while L, the highest in strength, was not the most pure, it had the least ductility, and its welding power was most imperfect. The evidence of the influence of chemical composition upon quality, therefore, is quite contra- dictory and confusing. The irons differing remarkably in their mechanical properties, it was found that a much more marked influence upon their qualities was caused by different treatment in rolling than by differences in composition. In regard to slag Mr. Holley says : " It appears that the smallest and most worked iron often has the most slag. It is hence reasonable to con- clude that an iron may be dirty and yet thoroughly condensed." In his summary of " What is learned from chemical analysis," he says : " So far, it may appear that little of use to the makers or users of wrought iron has been learned. . . . The character of steel can be surely pred- icated on the analyses of the materials; that of wrought iron is altered by subtle and unobserved causes " Influence of Reduction in Rolling from Pile to Rar on the Strength of "Wrought Iron.— The tensile strength of the irons used in Beardslee's tests ranged from 46,000 to 62,700 lbs. per sq. in., brand L, which was really a steel, not being considered. Some specimens of L gave figures as high as 70,000 lbs. The amount of reduction of sectional 378 IRON AND STEEL. 4 3 2 1 % Ya 80 80 72 25 9 3 15.7 8.83 4.36 3.14 2.17 1.6 46,322 47,761 48,280 51,128 52,275 59,585 23,430 26,400 31,892 36,467 39,126 area in rolling the bars has a notable influence on the strength and elastic limit; the greater the reduction from pile to bar the higher the strength. The following are a few figures from tests of one of the brands : Size of bar, in. diam.: Area of pile, sq. in.: Bar per cent of pile : Tensile strength, lb.: Elastic limit, lb.: Specifications for Wrought Iron (F. H. Lewis, Engineers' Club of Philadelphia, 1891).— 1. All wrought iron must be tough, ductile, fibrous, and of uniform quality for each class, straight, smooth, free from cinder- pockets, flaws, buckles, blisters, and injurious cracks along the edges, and must have a workmanlike finish. No specific process- or provision of manufacture will be demanded, provided the material fulfils the require- ments of these specifications. 2. The tensile strength, limit of elasticity, and ductility shall be deter- mined from a standard test-piece not less than J4 inch thick, cut from the full-sized bar, and planed or turned parallel. The area of cross-section shall not be less than Y 2 square inch. The elongation shall be measured after breaking on an original length of 8 inches. 3. The tests shall show not less than the following results: For bar iron in tension For shape iron For plates under 36 in. wide. For plates over 36 in. wide . . Ultimate Strength, lbs per sq. inch. 50,000 48,000 48,000 46,000 Limit of Elasticity, lbs. per sq. inch. 26,000 26,000 26,000 25,000 Elongation in 8 inches, per cent. 12 10 4. When full-sized tension members are tested to prove the strength of their connections, a reduction in their ultimate strength of (500 x width of bar) pounds per square inch will be allowed. 5. All iron shall bend, cold, 180 degrees around a curve whose diameter is twice the thickness of piece for bar iron, and three times the thickness for plates and shapes. 6. Iron which is to be worked hot in the manufacture must be capable of bending sharply to a right angle at a working heat without sign of fracture. 7. Specimens of tensile iron upon being nicked on one side and bent shall show a fracture nearly all fibrous. 8. All rivet iron must be tough and soft, and be capable of bending cold until the sides are in close contact without sign of fracture on the convex side of the curve. Pennsylvania Railroad Specifications for merchant Bar Iron or Steel.— Miscellaneous merchant bar iron or steel for which no special specifications defining shapes and uses are issued, should have a tensile strength of 50,000 to 55,000 lbs. per square inch and an elongation of 20$ in a section originally 2 inches long. No iron or steel will be accepted under this specification if tensile strength falls below 48,000 lbs. or goes above 60,000 lbs. per square inch, nor if elon- gation is less than 15$ in 2 inches, nor if it shows a granular fracture cover- ing more than 50$ of the fractured surface, nor if it shows any difficulty in welding. In preparing test-pieces from round or rectangular bars, they will be turned or shaped so that the tested sections may be the central portion of the bar, in all sizes up to 1% inches in any diametrical or side measurement. In larger sizes test-pieces will be made to fall about half-way from centre to circumference. Bars of iron y% in. thick or less, or tortured forms of iron, such as angle, tee or channel bars" will be accepted if tensile strength is above 45,000 lbs. and elongation above \2%; but the testing of such sizes and sections is optional, FORMULAE FOR UNIT STRAINS FOR IROls' A j\ r D STEEL. 379 Specifications for Wrought Iron for the World's Fair Buildings. [Eny\j News, March 26. 1892.)— All iron to be used in the tensile members of open trusses, laterals, pins and bolts, except plate iron over 8 inches wide, and shaped iron, must show by the standard test-pieces a tensile strength in lbs. per square inch of : 7,000 X area of original bar in sq. in. circumference of original harm inches' with an elastic limit not less than half the strength given by this formula, and an elongation of 20^ in 8 in. Plate iron 24 inches wide and under, and more than 8 inches wide, must show by the standard test-pieces a tensile strength of 48,000 lbs. per sq. in. with an elastic limit not less than 26,000 lbs. per square inch, and an elon- gation of not less than 12%. All plates over 24 inches in width must have a tensile strength not less than 46,000 lbs., with an elastic limit not less than 26,000 lbs. per square inch. Plates from 24 inches to 36 inches in width must have au elongation of not less than 10$; those from 38 inches to 48 inches in width, 8$; over 48 inches in width, 5%. All shaped iron, flanges of beams and channels, and other iron not herein- before specified, must show by the standard test-pieces a tensile strength in lbs. per square inch of : 7,000 X area of original bar 1 circumference of original bar' with an elastic limit of not less than half the strength given by this formula, and an elongation of 15% for bars % inch and less in thickness, and of 12$ for bars of greater thickness. For webs of beams and channels, specifications for plates will apply. All rivet iron must be tough and soft, and pieces of the full diameter of the rivet must be capable of bending cold, until the sides are in close contact, without sign of fracture on the convex side of the curve. Stay-bolt Iron.— Mr. Vauclain, of the Baldwin Locomotive Works, at a meeting of the American Railway Master Mechanics' Association, in 1892, says: Many advocate the softest iron in the market as the best for stay-bolts. He believed in an iron as hard as was consistent with heading the bolt nicely. The higher the tensile strength of the iron, the more vibra- tions it will stand, for it is not so easily strained beyond the yield-point. The Baldwin specifications for stay-bolt iron call for a tensile strength of 50,000 to 52,000 lbs. per square inch, the upper figure being preferred, and the lower being insisted upon as the minimum. FORfflULi: FOK UNIT STRAINS FOR. IRON AND STEEL IN STRUCTURES. (F. H. Lewis, Engineers' (Jlub of Philadelphia, 1891.) The following formulae for unit strains per square inch of net sectional area shall be used in determining the allowable working stress in each mem- ber of the structure. (For definitions of soft and medium steel see Specifi- cations for Steel.) Tension Members. Floor-beam hangers or suspenders, forged bars Counter-ties Suspenders, hangei and counters, riveted members, net sec- tion Solid rolled beams , Riveted truss members and tension flanges of girders, net sec- tion Forged eyebars Lateral or cross-sec tion rods Wrought Iron. Will not be used 6000 5000 8000 7000(l + ™Hi) V max./ Will not be used Soft Steel. Will not be used 5500 8000 8% greater than iron Will not be used Medium Steel. 7000 7000 7000 Will not be used 9000(1+™^) V max. ' 9000(l+™) V max./ /For eyebars\ V only, 17,000 ) 380 IRON AND STEEL. Shearing. On pins and shop rivets On field rivets In webs of girders.. Wrought Iron. 6000 4800 Will not be used 6600 5200 5000 Medium Steel. 7200 Will not be used 6000 Bearing. Wrought Iron. Soft Steel. Medium Steel. On projected semi- intrados of main-pin holes 12,000 13,200 14,500 On projected semi-in- trados of rivet-holes* 12,000 13,200 14,500 On lateral pins 15,000 16,500 18,000 Of bed-plates on ma- sonry 250 lbs. per sq. in. * Excepting that in pin-connected members taking alternate stresses, the bearing stress must not exceed 9000 lbs. for iron or steel. Bending. On extreme fibres of pins when centres of bearings are considered as points of application of strains: Wrought Iron, 15,000. Soft Steel, 16,000. Medium Steel, 17,000. Compression Members. Chord sections : Flat ends One flat and one pin end.. Chords with pin ends and all end-posts All trestle-posts Intermediate posts Lateral struts, and com- pression in collision struts, stiff suspenders and stiff chords. . . Wrought Iron. 7 ooo(i + i^)- J \ max./ 7000 (l-f^)- 35 * V max./ 7000 (l+i™-)- 40 V max./ 700 o(l + ^)-35 \ max./ 10,500 - 50 - Soft Steel. Wo greater than Medium Steel. 20£ greater than In which formulae I ~ length of compression member in inches, and r = least radius of gyration of member in inches. No compression member shall have a length exceeding 45 times its least width, and no post should be used in which I -+-r exceeds 125. Members S ubject to Alternate Tension and Compression. Wrought Iron. Soft Steel. Medium Steel. For compression only. . For the greatest stress . Use the formulae above ^Jf « max. lesser ^ 70001 1 - „ r— \ 2 max. greater^ 20$ greater than iron Use the formula giving the greatest area of section. The compression flanges of beams and plate girders shall have the same cross-section as the tension flanges. FORMULAE FOB UNIT STRAINS FOR IRON AND STEEL. 381 W. H. Burr, discussing the formulae proposed by Mr. Lewis, says: " Taking the results of experiments as a whole, I am constrained to believe that they indicate at least 15$ increase of resistance for soft-steel columns over those of wrought iron, with from 20% to 25% for medium steel, rather than 10$ and 20$ respectively. " The high capacity of soft steel for enduring torture fits it eminently for alternate and combined stresses, and for that reason I would give it 15$ increase over iron, with about 22% for medium steel. "Shearing tests on steel seem to show that 15$ and 22$ increases, for the two grades respectively, are amply justified. " I should not hesitate to assign 15$ and 22$ increases over values for iron for bearing and bending of soft and medium steel as being within the safe limits of experience. Provision should also be made for increasing pin- shearing, bending and bearing stresses for increasing ratios of fixed to mov- ing loads " Maximum Permissible Stresses in Structural Materials used in Buildings. (Building Ordinances of the City of Chicago. 1893.) Cast iron, crushing stress: For plates, 15,000 lbs. per square inch; for lintels, brackets, or corbels, compression 13,500 lbs. per square inch, and tension 3000 lbs. per square inch. For girders, beams, corbels, brackets, and trusses, 16,000 lbs. per square inch for steel and 12,000 lbs. for iron. For plate girders : _. maximum bending moment in ft.-lbs. Flange area — CD. D = distance between centre of gravity of flanges in feet. c=r' j 13,500 for steel. 1 10,000 for iron. maximum shear „ ( 10,000 for steel, Web area = . C = -j 6 ,000 for iron. For rivets in single shear per square inch of rivet area : Steel. Iron. If shop-driven. 9000 lbs. 7500 lbs. If field-driven 7500 " 6000 " For timber girders : b = breadth of beam in inches. d = depth of beam in inches. _ cbd* I = length of beam in feet. " — fT ' (160 for long-leaf yellow pine, c = <120 for oak, ( 100 for white or Norway pine. Proportioning of Materials in the Memphis Bridge (Geo. S. Morison, Trans. A. S. C. E., 1893).— The entire superstructure of the Mem- phis bridge is of steel and it was all worked as steel, the rivet-holes being drilled in all principal members and punched and reamed in the lighter members. The tension members were proportioned on the basis of allowing the dead load to produce a strain of 20,000 lbs. per square inch, and the live load a strain of 10,000 lbs. per square inch. In the case of the central span, where the dead load was twice the live load, this corresponded to 15,000 lbs. total strain per square inch, this being the greatest tensile strain. The compression members were proportioned on a somewhat arbitrary basis. No distinction was made between live and dead loads. A maximum strain of 14,000 lbs. per square inch was allowed on the chords and other large compression members where the length did not exceed 16 times the least transverse dimension, this strain being reduced 750 lbs. for each addi- tional unit of length. In long compression members the maximum length was limited to 30 times the least transverse dimension, and the strains limited to 6,000 lbs. per square inch, this amount being increased by 200 lbs. for each unit by which the length is decreased. Wherever reversals of strains occur the member was proportioned to re- sist the sum of compression and tension on whichever basis (tension or compression) there would be the greatest strain per square inch ; and, in addition, the net section was proportioned to resist the maximum tension, and the gross section to resist the maximum compression. The floor beams and girders were calculated on the strain being limited to 10,000 lbs. per square inch in extreme fibres. Rivet-holes in cover-plates and flanges were deducted. 382 IRON AND STEEL. The rivets of steel in drilled or reamed holes were proportioned on the basis of a bearing strain of 15,000 lbs. per square inch and a shearing strain of 7500 lbs. per square inch, and special pains were taken to get the double shear in as many rivets as possible. This was the requirement for shop rivets. In the case of field rivets, the number was increased one-half. The pins were proportioned on the basis of a bearing strain of 18,000 lbs. per square inch and a bending strain of 20,000 lbs. per square inch in ex- treme fibre, the diameters of the pins being never made more than one inch less than the width of the largest eye-bar attaching to them. The weight on the rollers of the expansion joint on Pier II is 40,000 lbs. per linear foot of roller, or 3,333 lbs. per linear inch, the rollers being 15 ins. in diameter. As the sections of the superstructure were unusually heavy, and the strains from dead load greatly in excess of those from moving load, it was thought best to use a slightly higher steel than is now generally used for lighter structures, and to work this steel without punching, all holes being drilled. A somewhat softer steel was used in the floor-system and other lighter parts. The principal requirements which were to be obtained as the results of tests on samples cut from finished material were as follows: Max. Ultimate Strength, lbs. per sq. inch. High-grade steel. 78,500 Eye-bar steel 75,000 Medium steel — 72,500 Soft steel 63,000 Min. Ultimate Min. Elastic Strength, Limit, lbs, lbs. per per sq. in. sq. inch. 69,000 40,000 66,000 38,000 64.000 37,000 55,000 30,000 Min. per- centage of Elongation "1 inches . Min. Per- centage of Reduction at Fracture 18 20 22 40 44 50 TENACITY OF METALS AT VARIOUS TEMPERATURES. The British Admiralty made a series of experiments to ascertain what loss of strength and ductility takes place in gun-metal compositions when raised to high temperatures. It was found that all the varieties of gun-metal suffer a gradual but not serious loss of strength and ductility up to a certain temperature, at which, within a few degrees, a great change takes place, the strength falls to about one half the original, and the ductility is wholly gone. At temperatures above this point, up to 500, there is little, if any, further loss of strength ; the temperature at which this great change and loss of strength takes place, although uniform in the specimens cast from the same pot, varies about 100° in the same composition cast at different temperatures, or with some varying conditions in the foundry process. The temperature at which the change took place in No. 1 series was ascer- tained to be about 370°, and in that of No. 2, at a little over 250°. Whatever may be the cause of this important difference in the same composition, the fact stated may be taken as certain. Rolled Muntz metal and copper arc satisfactory up to 500°, and may be used as securing-bolts with safety. Wrought iron, Yorkshire and remanufactured, increase in strength up to 500°, but lose slightly in ductility up to 300°, where an increase begins and continues up to 500°, where it is still less than at the ordinary temperature of the atmosphere. The strength of Landore steel is not affected by temper- ature up to 500°, but its ductility is reduced more than one half. (Iron, Oct. 6, 1877.) Tensile Strength of Iron and Steel at High Tempera- tures.— James E. Howard's tests (Iron Age, April 10, J8 C J0), shows that the tensile strength of steel diminishes as the temperature increases from 0° until a minimum is reached between 200° and 300° F., the total decrease being about 4000 lbs. per square inch in the softer steels, and from 6000 to 8000 lbs. in steels of over 80,000 lbs. tensile strength. From this minimum point the strength increases up to a temperature of 400° to 650° F., the maximum being reached earlier in the harder steels, the increase amounting to from 10,000 to 20,000 lbs. per square inch above the minimum strength at from 200o TENACITY OF METALS AT VAKIOUS TEMPERATURES. 383 to 800°. From this maximum, the strength of all the steel decreases steadily at a rate approximating 10,000 lbs. decrease per 100° increase of tempera- ture. A strength of 20,000 lbs. per square inch is still shown by .10 C. steel at about 1000° F., and by .60 to 1.00 C. steel at about 1600° F. The strength of wrought iron increases with temperature from 0° up to a maximum at from 400 to 600° F., the increase being from 8000 to 10,000 lbs. per square inch, and then decreases steadily till a strength of only 6000 lbs. per square inch is shown at 1500° F. Cast iron appears to maintain its strength, with a tendency to increase, unt 1 900° is reached, beyond which temperature the strength gradually diminishes. Under the highest temperatures, 1500° to 1600° F., numerous cracks on the cylindrical surface of the specimen were developed prior to rupture. It is remarkable that cast iron, so much inferior in strength to the steels at atmospheric temperature, under the highest temperatures has nearly the same strength the high-temper steels then have. Strength of Iron and Steel Boiler-plate at High Tem- peratures. (Chas. Huston, Jour. F. I., 1877.) Average of Three Tests of Each. Temperature F. 68° 575° 925° Charcoal iron plate, tensile strength, lbs 55.366 63,080 65,343 " " " contr. of area % 26 23 21 Soft open-hearth steel, tensile strength, lbs 54,600 66,083 64,350 " contr. % 47 38 33 " Crucible steel, tensile strength, lbs 64,000 69,266 68,600 " " " contr. % 36 30 21 Strength of Wrought Iron and Steel at High Temper- atures. (Jour. F. L, cxii., 1881, p. 241.) Kollmann's experiments at Ober- hausen included tests of the tensile strength of iron and steel at tempera- tures ranging between 70° and 2000° F. Three kinds of metal were tested, viz., fibrous iron having an ultimate tensile strength of 52,464 lbs., an elastic strength of 38,280 lbs., and an elongation of 17.5$; fine-grained iron having for the same elements values of 56.892 lbs., 39,113 lbs., and 20$; and Bes- semer steel having values of 84,826 lbs., 55,029 lbs., and 14.5$. The mean ultimate tensile strength of each material expressed in per cent of that at ordinary atmospheric temperature is given in the following table, the fifth column of which exhibits, for purposes of comparison, the results of experi- ments carried on by a committee of the Franklin Institute in the years 1832-36. Fibrous Fine-grained Bessemer Franklin Temperature Wrought Iron, Steel, Institute Degrees F. Iron, p. c. per cent. per cent. per cent. 100.0 100.0 100.0 96.0 100 100.0 100.0 100.0 102.0 200 100.0 100.0 100.0 105.0 300 97.0 100.0 100.0 106.0 400 95.5 100.0 100.0 106.0 500 92.5 98.5 98.5 104.0 600 88.5 95.5 92.0 99.5 700 81.5 90.0 68.0 92.5 800 67.5 77.5 44.0 75.5 900 44.5 51.5 36.5 53.5 1000 26.0 36.0 31.0 "36.0 1100 20.0 30.5 26.5 1200 18.0 28.0 22.0 1300 16.5 23.0 18.0 1400 13.5 19.0 15.0 1500 10.0 15.5 12.0 1600 7.0 12.5 10.0 1700 5.5 10.5 8.5 1800 4.5 8.5 7.5 1900 3.5 7.0 6.5 2000 3.5 5.0 5.0 The Effect of Cold on the Strength of Iron and Steel.— The following conclusions were arrived at by Mr. Styffe in 1865 : (1) That the absolute strength of iron and steel is not diminished by cold, but that even at the lowest temperature which ever occurs in Sweden it is at least as great as at the ordinary temperature (about 60° F.). 384 IRON" AND STEEL. (2) That neither in steel nor in iron is the extensibility less in severe cold than at the ordinary temperature. (3) That the limit of elasticity in both steel and iron lies higher in severe cold. (4) That the modulus of elasticity in both steel and iron is increased on reduction of temperature, and diminished on elevation of temperature ; but that these variations never exceed 0.05 % for a change of temperature of 1.8° F., and therefore such variations, at least for ordinary purposes, are of no special importance. Mr. C. P. Sandberg made in 1867 a number of tests of iron rails at various temperatures by means of a falling weight, since he was of opinion that, although Mr. Styffe's conclusions were perfectly correct as regards tensile strength, they migbt not apply to the resistance of iron to impact at low temperatures. Mr. Sandberg convinced himself that " the breaking strain " of iron, such as was usually employed for rails, " as tested by sudden blows or shocks, is considerably influenced by cold ; such iron exhibiting at 10° F. only from one third to one fourth of the strength which it possesses at 84° F." Mr. J. J. Webster (Inst. C. E., 1880) gives reasons for doubting the accuracy of Mr. Sandberg's deductions, since the tests at the lower temperature were nearly all made with 21-ft. lengths of rail, while those at the higher temperatures were made with short lengths, the supports in every case being the same distance apart. W. H. Barlow (Proc. Inst. C. E.) made experiments on bars of wrought iron, cast iron, malleable cast iron, Bessemer steel, and tool steel. The bars were tested with tensile and transverse strains, and also by impact ; one half of them at a temperature of 50° F., and the other half at 5° F. The lower temperature was obtained by placing the bars in a freezing mixture, care being taken to keep the bars covered with it during the whole time of the experiments. The results of the experiments were summarized as follows : 1. When bars of wrought iron or steel were submitted to a tensile strain and broken, their strength was not affected by severe cold (5° F.), but their ductility was increased about \% in iron and 3% in steel. 2. When bars of cast iron were submitted to a transverse strain at a low temperature, their strength was diminished about 3% and their flexibility about 16%. 3. When bars of wrought iron, malleable cast iron, steel, and ordinary cast iron were subjected to impact at a temperature of 5° F., the force re- quired to break them, and the extent of their flexibility, were reduced as follows, viz.: Reduction of Force Reduction of Flexi- of Impact, per cent. bility, per cent. Wrought iron, about 3 18 Steel (best cast tool), about 3% 17 Malleable cast iron, about 4)4 I 5 Cast iron, about 21 not taken The experience of railways in Russia, Canada, and other countries where the winter is severe is that the breakages of rails and tires are far more numerous in the cold weather than in the summer. On this account a softer class of steel is employed in Russia for rails than is usual in more temperate climates. The evidence extant in relation to this matter leaves no doubt that the capability of wrought iron or steel to resist impact is reduced by cold. On the other hand, its static strength is not impaired by low temperatures. Effect of tow Temperatures on Strength of Railroad Axles. (Thos. Andrews, Proc. Inst. C. E., 1891.)— Axles 6 ft. 6 in. long between centres of journals, total length 7 ft. 3>£ in., diameter at middle 4J^ in., at wheel-sets b% in., journals 334 X 7 in. were tested by impact at temper- atures of 0° and 100° F. Between the blows each axle was half turned over, and was also replaced for 15 minutes in the water-bath. The mean force of concussion resulting from each impact was ascertained as follows : Let h = height of free fall in feet, w — weight of test ball, hiv = W — " energy," or work in foot-tons, x = extent of deflections between bearings, W hw then F (mean force) = — = — , DURABILITY OF IRON", CORROSION, ETC. 385 The results of these experiments show that whereas at a temperature of 0° F. a total average mean force of 179 tons was sufficient to cause the breaking of the axles, at a temperature of 100° F. a total average mean force of 428 tons was requisite to produce fracture. In other words, the re- sistance to concussion of the axles at a temperature of 0° F. was only about 42% of what it was at a temperature of 100° F. The average total deflection at a temperature of 0° F. was 6.48 in., as against 15.06 in. with the axles at 100° F. under the conditions stated; this represents an ultimate reduction of flexibility, under the test of impact, of about 57?S for the cold axles at 0° F., compared with the warm axles at 100° F. EXPANSION OF IRON AND STEEL BY HEAT. James E. Howard, engineer in charge of the U. S. testing- machine at Wa- fertown, Mass., gives the following results of tests made on bars 35 inches long (/row Age, April 10, 1890): Marks. Chemical composition. Coefficient of Expansion. Metal. C. Mn. Si. Feby difference. Per degree F. per unit of length. .0000067302 Steel la 2a 3a 4a 5a 6a 7a 8a 9a 10a .09 .20 .31 .37 .51 .11 .81 .89 .97 A i .57 .70 .58 .93 .58 .56 .57 .80 ".bk" .07 .08 .17 .19 .28 99.80 99.35 99.12 98.93 98.89 98.43 98.63 98.46 98.35 97.95 .0000067561 .0000066259 w .0000065149 lt .0000066597 .0000066202 11 .0000063891 It .0000064716 c< .0000062167 (C .0000062335 11 .0000061700 0000059261 .0000091286 DURABILITY OF IRON, CORROSION, ETC. Durability of Cast Iron.— Frederick Graff, in an article on the Philadelphia water-supply, says that the first cast-iron pipe used there was laid in 1820. These pipes were made of charcoal iron, and were in constant use for 53 years. They were uncoated, and the inside was well filled with tubercles. In salt water good cast iron, even uncoated, will last for a cen- tury at least; but it often becomes soft enough to be cut by a knife, as is shown in iron cannon taken up from the bottom of harbors after long sub- mersion. Close-grained, hard white metal lasts the longest in sea water.— Eng'g News, April 23. 1887, and March 26. 1892. Tests of Iron after Forty Years' Service.— A square link 12 inches broad, 1 inch thick and about 12 feet long was taken from the Kieff bridge, then 40 years old, and tested in comparison with a similar link which had been preserved in the stock-house since the bridge was built. The fol- lowing is the record of a mean of four longitudinal test -pieces, 1 X 1% X 8 inches, taken from each link (Stahl und Eisen, 1890): Old Link taken New Link from from Bridge. Store-house. Tensile strength per square inch, tons 21 .8 22.2 Elastic limit " " 11.1 11.9 Elongation, per cent 14.05 13.42 Contraction, per cent 17.35 18.75 Durability of Iron in Bridges. (G-. Lindenthal, Eng'g, May 2, 1884, p. 139.)— The Old Monongahela suspension bridge in Pittsburgh, built in 1845, was taken down in 1882. The wires of the cables were frequently strained to half of their ultimate strength, yet on testing them after 37 years' 386 IRO^ AND STEEL. use they showed a tensile strength of from 72,700 to 100,000 lbs. per square inch. The elastic limit was from 67,100 to 78,600 lbs. per square inch. Re- duction at point of fracture, 35$ to 75$. Their diameter was 0.13 inch. A new ordinary telegraph wire of same gauge tested for comparison showed: T. S., of'100,000 lbs.; E. L., 81,550 lbs.; reduction, 57%. Iron rods used as stays or suspenders showed: T. S , 43,770 u> 49,720 lbs. per square inch; E. L., 26,380 to 29,200. Mr. Lindenthal draws these conclusions from his tests: " The above tests indicate that iron highly strained for a long number of years, but still within the elastic limit, and exposed to slight vibration, will not deteriorate in quality. " That if subjected to only one kind of strain it will not change its texture, even if strained beyond its elastic limit, for many yer.rs. It will stretch and behave much as in a testing-machine during a long test. " That iron will change its texture only when exposed to alternate severe straining, as in bending in different directions. If the bending is slight but very rapid, as in violent vibrations, the effect is the same." Corrosion of Iron Bolts.— On bridges over the Thames in London, bolts exposed to the action of the atmosphere and rain-water v\ ere eaten away in 25 years from a diameter of % in. to % in., and from % in. diameter to 5/16 inch, Wire ropes exposed to drip in colliery shafts are very liable to corrosion. Corrosion of Iron and Steel.— Experiments made at the Riverside Iron Works, Wheeling, W. Va., on the comparative liability to rust of iron and soft Bessemer steel: A piece of iron plate and a similar piece of steel, both clean and bright, were placed in a mixture of yellow loam and sand, with which had been thoroughly incorporated some carbonate of soda, nitrate of soda, ammonium chloride, and chloride of magnesium. The earth as prepared was kept moist. At the end of 33 days the pieces of metal were taken out, cleaned, and weighed, when the iron was found to have lost 0.84$ of its weight and the steel 0.72$. The pieces were replaced and after 28 days weighed again, when the iron was found to have lost 2.06$ of its original weight and the steel 1.79$. (Eng'g, June 26, 1891.) Corrosive Agents in the Atmosphere.— The experiments of F. Crace Calvert (Chemical Neivs, March 3, 1871) show that carbonic acid, in the presence of moisture, is the agent which determines the oxidation of iron in the atmosphere. He subjected perfectly cleaned blades of iron and steel to the action of different gases for a period of four months, with results as follows : Dry oxygen, dry carbonic acid, a mixture of both gases, dry and damp oxygen and ammonia: no oxidation. Damp oxygen: in three experiments one blade only was slightly oxidized. Damp carbonic acid: slight appearance of a white precipitate upon the iron, found to be carbonate of iron. Damp carbonic acid and oxygen: oxidation very rapid. Iron immersed in water containing carbonic acid oxidized rapidly. Iron immersed in distilled water deprived of its gases by boiling rusted the iron in spots that were found to contain impurities. Galvanic action is a most active agent of corrosion. It takes place when two metals, one electro-negative to the other, are placed in contact and exposed to dampness. Sulphurous acid (the product of the combustion of the sulphur in coal) is an exceedingly active corrosive agent, especially when the exposed iron is coated with soot. This accounts for the rapid corrosion of iron in railway bridges exposed to the smoke from locomotives. (See account of experi- ments by the author on action of sulphurous acid in Jour. Frank. Inst. , June, 1875, p. 437.) An analysis of sooty iron rust from a railway bridge showed the presence of sulphurous, sulphuric, and carbonic acids, chlorine, and ammonia. Bloxam states that ammonia is formed from the nitrogen of the air during the process of rusting. Rustless Coatings for Iron and Steel.— Tinning, enamelling, lac- quering, galvanizing, electro-chemical painting, and other preservative methods are discussed in two important papers by M. P. Wood, in Trans. A. S. M. E., vols, xv and xvi. A Method of Producing an Inoxidizable Surface on iron and steel by means of electricity has been developed by M. A. de Meri- tens (Engineering.) The article to be protected is placed in a bath of ordi- nary or distilled water, at a temperature of from 158° to 176° F., and an electric current is sent through. The water is decomposed into its elements, DURABILITY OF IRON", CORROSION, ETC. 387 Oxygen and hydrogen, and the oxygen is deposited on the metal, while the hydrogen appears at the other pole, which may either be the tank in which the operation is conducted or a plate of carbon or metal. The current has only sufficient electromotive force to overcome the resistance of the circuit and to decompose the water; for if it be stronger than this, the oxygren com- bines with the iron to produce a pulverulent oxide, which has no adherence. If the conditions are as they should be, it is only a few minutes after the oxygen appears at the metal before the darkening of the surface shows that the gas has united with the iron to form the magnetic oxide Fe 3 4 , which it is well known will resist the action of the air and protect the metal beneath it. After the action has continued an hour or two the coating is sufficiently solid to resist the scratch-brush, and it will then take a brilliant polish. If a piece of thickly rusted iron be placed in the bath, its sesquioxide (Fe 2 3 ) is rapidly transformed into the magnetic oxide. This outer layer has no adhesion, but beneath it there will be found a coating which is actually a part of the metal itself. In the early experiments M. de Meritens employed pieces of steel only, but in wrought and cast iron he was not successful, for the coating came oif with the slightest friction. He then placed the iron at the negative pol of the apparatus, after it had been already applied to the positive pole. Here the oxide was reduced, and hydrogen was accumulated in the pores of the metal. The specimens were then returned to the anode, when it was found that the oxide appeared quite readily and was very solid. But the result was not quite perfect, and it w r as not until the bath was filled with distilled water, in place of that from the public supply, that a perfectly satisfactory result was attained. Manganese Plating of Iron as a Protection from Rust. —According to the Italian Progresso, articles of iron can be protected against rust by sinking them near the negative pole of an electric bath com- posed of 10 litres of water, 50 grammes of chloride of manganese, and 200 gramnies of nitrate of ammonium. Under the influence of the current the bath deposits on the articles a film of metallic manganese which prevents them from rusting. A Non-oxidizing Process of Annealing is described by H. P. Jones, in Eng'g Neivs, Jan. 2, 1892. The ordinary process of annealing, by means of which hard and brittle iron or steel is rendered soft and tough, consists in heating the metal to a good red-heat and then allowing it to cool gradually. While the metal is in a heated condition the surface becomes oxidized; and although for many classes of work this scale of oxide is of practical importance, yet in some cases it is very undesirable and even necessitates considerable expense in its removal. The new process uses a non-oxidizing gas, and is the invention of Mr. Horace K. Jones, of Hartford, Conn. The principal feature of this process consists in keeping the annealing-retort in communication with the gas holder or gas main during the entire process of heating and cooling, the j;as thus being allowed to expand back into the main, and being, therefore, kept at a practically constant pressure. The retorts used are made from wrought-iron tubes. The gas used is taken directly from the mains supplying the city with illuminating gas. It was noticed that if metal which had been blued or slightly oxidized was sub- jected to the annealing process it came out bright, the oxide being reduced by the action of the gas. Practical use has been made of this fact in deoxi- dizing metal. Comparative tests were made of specimens of metal annealed in illlumi- nating gas and of specimens annealed in nitrogen. The results of these tests were compared with the results of tests of specimens annealed in an open fire and cooled in ashes, and of specimens of the unannealed metal, and thus the relative efficiency of the gas process was determined. The specimens were made from steel wire .188 in. in diameter and were turned down to diameters of .156 and .150 in. Different lots of wire were tested in order to secure average results. The elongations were in each case referred to an original length of 1.15 ins. The difference in total per cent of elongation and in breaking load between the specimens annealed in nitrogen and those annealed in illuminating gas is very slight. The average results were as follows: 388 IRON AND STEEL. Gas used. No. Test Pieces . Breaking Load, lbs. per sq. in. Elongation. Total p. c. p. c. gained. A B C D E F G H I Nitrogen Illuminating Nitrogen Illuminating Nitrogen Illuminating Open fire Unannealed Unannealed 4 4 4 4 5 5 8 5 5 62,140 63,140 60.000 60,400 57,330 57.070 63,090 97,120 80,790 29.12 28.08 28.00 27.20 30.88 29.60 26.76 7.12 8.80 22.00 20.86 19.20 18.40 23.76 22.48 19.64 Painting Wood and Iron Structures. (E. H. Brown, Eng'rs Club of Phila., Engineering Neivs, April 20, 1893.)— A paint consists of two portions— the pigment and the vehicle or binder. The pigment is a solid substance which is more or less finely ground, so as to be capable (when mixed with the vehicle) of being spread out in a thin layer or coating over the surface to be painted. The vehicle or binder is the liquid in which the pigment is mixed or ground, which serves to spread the pigment over the surface to be painted, and which also holds it to that surface. For ordinary painting the most generally used vehicle is linseed oil. Linseed oil possesses the peculiar property of drying by uniting with the oxygen of the air to form a tough, leather-like compound called lindxin. For painting on wood, zinc white has valuable pigment properties, but these seem to be most fully developed when this pigment is used in con- junction with white lead, and then to the best advantage when the mixture is used as a final coat over an elastic undercoating of white lead. So far no other white base has been discovered which possesses at the same time the other properties which render white lead valuable, namely, covering and spreading capacit3 r . Of the inert pigments, lampblack is probably the" most valuable. Being almost pure carbon, it is practically unchangeable except by fire. It ha^ the peculiar property of absorbing great quantities of linseed oil, and hence of spreading over a large surface. French ochre, an earth pigment containing more or less of the hydrated oxide of iron, possesses the property of absorb- ing a large quantity of oil, and hence has considerable spreading capacity, and also holds very firmly to any wooden surface to which it may be applied. The various mineral and metallic paints are almost all natural or artificial iron oxides. While these are cheap and useful for painting rough wooden structures they are sometimes really quite dangerous for application to iron work, because, instead of preventing oxidation, they are apt to further it. Coal tar is much used as a paint for the roughest class of work, both wood and iron; in the latter case especially for cast-iron pipes, smoke-stacks, and work to be buried underground. It has the nature both of a resin and an oil. It has the disadvantage of becoming exceedingly brittle by the action of cold, and softening at 115° F. Asphalt permits of somewhat wider range of temper- ature, but otherwise exhibits the same peculiarities. These substances, while they last, are probably the most valuable of paints, especially under water; but they are unfortunate in their tendency to flow or crawl on the surface to which they are applied, finally leaving the upper portions almost or quite bare. This is the case even under ground. Red lead has long been regarded as the best possible preservative for clean dry iron. But in order to be most effective, the iron must be perfectly clean and free from any suspicions of rust, and absolutely dry. Red lead should be perfectly pure and of the best and most careful preparation. That from any well-known corroding house may be depended upon for purity, but not always for quality. It is simply a red oxide of lead. The best type is orange mineral, which is made by roasting white lead. On account of its expense this is not so frequently used as it would deserve. Red lead proper is made directly from the metal, which is first oxidized to the yellow litharge, and then to the red oxide. This, however, does not give as good a paiut as that made from the scrap, settlings, and tailings of the white lead works. As red lead saponifies very quickly with linseed oil, it must be used within a few days after being ground, and, moreover, it is rather difficult to work. CHEMICAL COMPOSITION AND PHYSICAL CHARACTER. 389 Hence there is great temptation to add some substance, such as whiting, to it in' order to make it work freer, as well as to cost less money for material. Before painting iron work it is essential that the iron itself should be ab- solutely free from rust. Rust has the peculiar property of spreading and extending from a centre, if there be the slightest chance to do so. Hence, a small amount of rust on the iron may grow under the surface of the paint, especially if it be true, as Dr. Dudley asserts, that linseed oii is permeable by air and moisture, and in time the paint will be flaked off by the rust un- derneath, thus gradually exposing the bare surface of the iron to the action of its destroying agent, oxygen in the presence of water. It is necessary to remove all the scale possible from wrought iron by means of stiff wire brushes, and then to remove the rust by a pickle of very dilute acid, which must afterward be thoroughly washed off before the paint is applied. The surface of the iron should be dry and at least moderately warmed before it is primed. The best method of painting a tin roof is to carefully remove all traces of oii or grease from the surface of the tin while it is yet bright with benzine; then to apply a coat of red lead and linseed oil, or the best quality of metallic paint, and to follow this with one or two coats of graphite paint. The graphite is almost unchangeable by atmospheric action, and is remarkably waterproof as well. Red Lead as a Preservative of Iron.— A. J. Whitney writes to Engineering News, August, 1891, that in 30 years' experience he has found red lead to be the best material for preserving iron under all circumstances. Quantity of Paint Required for a Given Surface. (M. P. Wood.)— Sq. ft. of surface -j- 200 = gallons of liquid paint for two coats; sq. ft. of surface h- 18 = lbs. of pure white lead for three coats. Qualities of Paints. — The Railroad and Engineering Journal, vols, liv and lv, 1890 and 1891, has a series of articles[on paint as applied to wooden structures, its chemical nature, application, adulteration, etc., by Dr. C. B. Dudley, chemist, and F. N. Pease, assistant chemist, of the Penna. R. R. They give the results of a long series of experiments on paint as applied to railway purposes. Graphite Paint. (M. P. Wood.)— Graphite, mixed with pure boiled linseed oil in which a small percentage of litharge, red lead, manganese, or other metallic salt has been added at the time of boiling to aid in the oxida- tion of the oil, forms a most effective paint for metallic surfaces, as well as for wood and fibrous substances. Wood surfaces protected by this paint, and exposed to the action of sea- water for a number of years, are found in a perfect state of preservation. STEEL. RELATION BETWREN THE CHEMICAL, COMPOSI- TION AND PHYSICAL CHARACTER OF STEEL. W. R. Webster (see Trans. A. I. M. E., vols, xxi and xxii, 1893-4) gives re- sults of several hundred analyses and tensile tests of basic Bessemer steel plates, and from a study of them draws conclusions as to the relation of chemical composition to strength, the chief of which are condensed as follows : The indications are that a pure iron, without carbon, phosphorus, man- ganese, silicon, or sulphur, if it could be obtained, would have a tensile strength of 34,750 lbs. per square inch, if tested in a %-inch plate. With this as a base, a table is constructed by adding the following hardening effects, as shown by increase of tensile strength, for the several elements named. Carbon, a constant effect of 800 lbs. for each 0.01$. Sulphur, " " 500 " " 0.01^. Phosphorus, the effect is higher in high-carbon than in low-carbon steels. With carbon hundreths % 9 10 11 12 13 14 15 16 17 Each .01^ P has an effect of lbs. 900 1000 1100 1200 1300 1400 1500 1500 1500 Manganese, the effect decreases as the per cent of manganese increases. ( .00 .15 .20 .25 .30 .35 .40 .45 .50 .55 Mn being per cent < to to to to to to to to to to ( .15 .20 .25 .30 .35 .40 .45 .50 .55 .65 Str'gth increases for .01 j* 240 240 220 200 180 160 140 120 100 100 lbs. Total incr. from Mn . . . 3600 4800 5900 6900 7800 8600 9'iOO 9900 10,400 1 1,400 390 STEEL. Silicon is so low in this steel that its hardening effect has not been con- sidered. With the above additions for carbon and phosphorus the following table has been constructed (abridged from the original by Mr. Webster). To the figures given the additions for sulphur and manganese should be made as above. Estimated Ultimate Strengths of Basic Bessemer Steel Plates. For Carbon, .06 to .24; Phosphorus, .00 to .10; Manganese and Sulphur, .00 in all cases. Carbon. .06 .08 .10 .12 44,950 .14 .16 .18 48,300' 49,900 .20 53,100 .24 Phos. .005 39,950 41,550 43,250 46,650 51,500 54,700 " .01 -10,350 41,950 43,750 45,550 47,350 49,050 50,650 52,250 53,850 55,450 " .02 41,150 42.750 44,750 46.750 48,750 50,550 52,150 53.750 55.350 56,950 " .03 41,950 43,550 45,750 47,950 50.150 52,050 53,650 55, -,'50 56,850 58.450 " .04 42.750 44,350 46,750 49,150 51,550 53.550 55,150 50,750 58,350 59,950 '• .05 43,550 45.150 47,750 50,350 52,950 55.050 56,650 58,250 59,850 61.450 " .06 44.350 45,950 48,750 51,550 54,350 56.550 58.150 59.750 61 350 62.950 " .0? 45,150 46,750 49,750 52,75C 55,750 58,050 59,650 61,250 62,850 64.450 " .08 45,950 47,550 50,750 53.950 57,150 59,550 61,150 62,750 64,350 65,950 " .09 46,750 48,350 51,750 55,150 58,550 61.050 62,650 64,250 65,850 67,450 " .10 47,550 49,150 52,750 56,35C 59,95( 62,550 64.150 65,750 67,350 68,950 .001 Phos = 80 lbs. 80 lbs. 100 lb 120 lb 140 lb 150 lb 150 lb 150 1b 150 1b 1501b In all rolled steel the quality depends on the size of the bloom or ingot from which it is rolled, the work put oh it, and the temperature at which it is finished, as well as the chemical composition. The above table is based on tests of plates % inch thick and under 70 inches wide; for other plates Mr. Webster gives the following corrections for thickness and width. They are made necessary only by the effect of thickness and width on the finishing temperature in ordinary practice. Steel is frequently spoiled by being finished at too high a temperature. Corrections for Size of Plates. Plates. Up to 70 ins. wide. Over 70 ins. wide. Inches thick. Lbs. Lbs. % and over. -2000 —1000 11/16 % 9/16 — 1750 — 750 — 1500 — 500 — 1250 — 250 — 1000 — — 500 ± 500 + 1000 + 3000 + 5000 Comparing the actual result of tests of 408 plates with the calculated results, Mr. Webster found the variation to range as in the table below. Summary of the Differences Between Calculated and Actual Results in 408 Tests of Plate Steel. In the first three columns the effects of sulphur were not considered; in the last three columns the effect of sulphur was estimated at 500 lbs. for each .01$ of S. "3 53S •6 1 "3 i i | A II a s si o ~ ^ «5g P w M P pq Per cent within 1000 lbs.. 23.4 32.1 28.4 24.6 27.0 26.0 28.4 2000 " .. 40.9 48.9 45.6 48.5 54.9 52.2 55.1 3000 " .. 62.5 71.3 67.6 67.8 73.0 70.8 74.7 4000 " .. 75.5 81.0 78.7 82.5 85.2 84.1 89.9 " 5000 " .. 89.5 91.1 90.4 93.0 92.8 92.9 94.9 STRENGTH OF BESSEMER AND OPEN-HEARTH STEELS. 391 The last figure in the table would indicate that if specifications were drawn calling for steel plates not to vary more than 5000 lbs. T. S. from a specified figure (equal to a total range of 10,000 lbs.), there would be a probability of the rejection of 5% of the blooms rolled, even if the whole lot was made from steel of identical chemical analysis. In 1000 heats only 2% of the heats failed to meet the requirements of the orders on which they were graded; the loss of plates was much less than 1%, as one plate was rolled from each heat and tested before rolling the remainder of the heat. R. A. Hadfield (Jour. Iron 6b Steel Inst., No. 1, 1894) gives the strength of very pure Swedish iron, remelted and tested as cast, 20.1 tons (45,024 lbs.) per sq. in.; remelted and forged, 21 tons (47.040 lbs.). The analvsis of the cast bar was : C, 0.08; Si, 0.04; S, 0.02 : P. 0.02; Mn, 0.01; Fe. 99.82. Effect of Oxygen upon Strength of Steel.— A. Lantz, of the Peine works, Germany, in a letter to Mr. Webster, says: "We have found ars and Sample Test-pieces of Same Steel Used in the Memphis Bridge. (Geo. S. Morison, Trans. A. S. C. E„ 1893.) Full-Sized Eyebars, Sample Bars fr om Same Melts, Sectio is 10" w de X 1 to 2 3/16" thick. about 1 in. area Reduc- Elongation. Elastic Max. Reduc- Elon- Elastic Max. tion of Limit, Load, tion, gation, Limit, Load, Area, p.c. Inches. p.c. lbs. per sq. in. p.c. p.c. lbs. per sq. in. 39.6 20.2 16.8 35.100 67,490 47.5 27.5 41,580 73,050 39.7 26.6 8.2 37,680 70,160 52.6 24.4 42,650 75,620 44.4 36.8 11.8 39,700 65,500 47.9 28.8 40,280 70,280 38.5 38.5 17.3 33,140 65,060 47.5 27.5 41,580 73,050 40.0 32.5 13.5 32,860 65,600 44.5 20.0 43,750 75,000 39.4 36.8 15.3 31,110 61.060 42.7 28.8 42,210 69,730 34.6 32.9 13.7 33,990 63,220 52.2 28.1 40,230 69,720 32.6 13.0 13.5 29,330 63,100 48.3 28.8 38,090 71,300 7.3 20 8 6.9 28,080 55,160 43.2 24.2 38,320 70,220 38.1 28.9 14.1 29,670 62.140 59.6 26.3 40,200 71,080 31.8 24.0 11.8 32,700 65,400 40.3 25.0 39,360 69,360 48.6 39.4 19.3 30,500 58,870 40.3 25.0 40,910 70,360 10.3 11.8 12.3 33,360 73,550 51.5 25.5 40,410 69,900 44.6 32.0 15.7 32,520 60,710 43.6 27.0 40,400 70,490 46.0 35.8 14.9 28,000 58,720 44.4 29.5 40,000 66,800 41.8 23.5 13.1 32,290 62,270 42.8 21.3 40,530 72,240 41.2 47.1 15.1 29,970 58,680 45.7 27.0 40,610 70,480 ..-sized eye-bars was about 6 the sample test-pieces. Tbe average st in., or about 12$ 1 ■ength of the f i 2ss than that of 394 STEEL. TREATMENT OF STRUCTURAL, STEEL. (James Christie, Trans. A. S. C. E., 1893.) Effect of Punching and Shearing.— There is no doubt that steel of higher tensile strength than is now accepted for structural purposes should not be punched or sheared, or that the softer material may contain elements prejudicial to its use however treated, but especially if punched. But extensive evidence is on record indicating that steel of good quality, in bars of moderate thickness and below or not much exceeding 80,000 lbs. tensile strength, is not any more, and frequently not as much, injured as wrought iron by the process of punching or shearing. The physical effects of punching and shearing as denoted by tensile test are for iron or steel: Reduction of ductility; elevation of tensile strength at elastic limit; reduc- tion of ultimate tensile strength. In very thin material the superficial disturbance described is less than in thick; in fact, a degree of thinness is reached where this disturbance prac- tically ceases. On the contrary, as thickness is increased the injury becomes more evident. The effects described do not invariably ensue; for unknown reasons there are sometimes marked deviations from what seems to be a general result. By thoroughly annealing sheared or punched steels the ductility is to a large extent restored and the exaggerated elastic limit reduced, the change being modified by the temperature of reheating and the method of cooling. It is probable that the best results combined with least expenditure can be obtained by punching all holes where vital strains are not transferred by the rivets; and by reaming for important joints where strains on riveted joints are vital, or wherever perforation may reduce sections to a minimum. The reaming should be sufficient to thoroughly remove the material dis- turbed by punching; to accomplish this it is best to enlarge punched holes at least % in diameter with the reamer. Riveting. — It is the current practice to perforate holes 1/16 in. larger than the rivet diameter. For work to be reamed it is also a usual require- ment to punch the holes from y% to 3/16 in. less than the finished diameter, the holes being reamed to the proper size after the various parts are assembled. It is also excellent practice to remove the sharp corner at both ends of the reamed holes, so that a fillet will be formed at the junction of the body and head of the finished rivets. The rivets of either iron or mild steel should be heated to a bright red or yellow heat and subjected to a pressure of not less than 50 tons per square inch of sectional area. For rivets of ordinary length this pressure has been found sufficient to completely fill the hole. If, however, tl e holes and the rivets are excep- tionally long, a greater pressure and a slower movement of the closing tool than is used for shorter rivets has been found advantageous in compelling the more sluggish flow of the metal throughout the longer hole. Welding.— No welding should be allowed on any steel that enters into structures. Upsetting.— Enlarged ends on tension bars for screw-threads, eyebars, etc., are formed by upsetting the material. With proper treatment and a sufficient increment of enlarged sectional area over the body of the bar the result is entirely satisfactory. The upsetting process should be performed so that the properly heated metal is compelled to flow without folding or lapping. Annealing.— The object of annealing structural steel is for the purpose of securing homogeneity of structure that is supposed to be impaired by un- equal heating, or by the manipulation necessarily attendant on certain pro- cesses. The objects to be annealed should be heated throughout to a uniform temperature and uniformly cooled. The physical effects of annealing, as indicated by tensile tests, depend on the grade of steel, or the amount of hardening elements associated with it; also on the temperature to which the steel is raised, and the method or rate of cooling the heated material. The physical effects of annealing medium-grade steel, as indicated by ten- sile test, are reported very differently by different observers, some claiming directly opposite results from others. It is evident, when all the attendant conditions are considered, that the obtained results must vary both in kind and degree. TREATMENT OF STRUCTURAL STEEL. 395 The temperatures employed will vary from 1000° to 1500° F.: possibly even a wider range is used. In some cases the heated steel is withdrawn at full temperature from the furnace and allowed to cool in the atmosphere: in others the mass is removed from the furnace, but covered under a muffle, to lessen the free radiation; or. again, the charge is retained in the furnace, and the whole mass cooled with the furnace, and more slowly than by either of the other methods. The best general results from annealing will probably be obtained by in- troducing the material into a uniformly-heated oven in which the tempera- ture is not so high as to cause a possibility of cracking by sudden and unequal changing of temperature, then gradually raising the temperature of the material until it is uniformly about 1?00° F., then withdrawing the material after the temperature is somewhat reduced and cooling under shelter of a muffle, sufficiently to prevent too free and unequal cooling on the one hand or excessively slow cooling on the other. G. G. Mehrtens, Trans. A. S. C. E. 1S93. says : " A good mild steel can be worked as readily as wrought iron in the shop or the field, and even bear still harder treatment. It was, however, often thought necessary to require preliminary annealing to remove the initial strains due to rolling. The an- nealing is undoubtedly of great advantage to all steel above 64,000 lbs. strength per square inch, but it is questionable whether it is necessary in softer steels. The distortions due to heating cause trouble in subsequent straightening, especially of thin plates. It cannot be denied, however, that annealing produces greater toughness. "In a general way all unannealed mild steel for a strength of 56,000 to 64.000 lbs. may be worked in the same way as wrought iron. Rough treat- ment or working at a blue heat must, however, be prohibited. Such treat- ment cannot be borne by wrought iron, although it does not suffer so-much as soft steel. Shearing is to be avoided, except to prepare rough plates, which should afterwards be smoothed by machine tools or files before using. Drifting is also to be avoided, because the edges of holes are thereby strained beyond the yield point. Reaming drilled holes is not necessary, particularly when sharp drills are used and neat work is done. A slight countersinking of the edges of drilled holes is all that is necessary. Work- ing the material while heated should be avoided as far as possible, and the engineer should bear this in mind when designing structures. Upsetting, cranking, and bending ought to be avoided, but when necessaiy the material should be annealed after completion: " The rivetiug of a mild-steel rivet should be finished as quickly as possible, before it cools to the dangerous heat. For this reason machine work is the best. There is a special advantage in machine work from the fact that the pressure can be retained upon the rivet until it has cooled sufficiently to prevent elongation and the rons^quenl loosening of the rivet." Punching and Drilling of Steel Plates. (Proc. Inst. M. E„ Aug. 1887. p. 3-'6.)— In Prof. Unwin's report the results of the greater num- ber of the experiments made on iron and steel plates lead to the general conclusion that, while thin plates, even of steel, do not suffer very much from punching, yet in those of ^ in. thickness and upwards the loss of te- nacity due to punching ranges from 10$ to 23? in iron plates and from 11$ to '13% in the case of mild steel. Mr. Parker found the loss of tenacity in steel plates to be as high as fully one third of the original strength of the plate. In drilled plates, on the contrary, there is no appreciable loss of strength. It is even possible to remove the bad effects of punching by subsequent reaming or annea'ing. "Working Steel at a Blue Heat.— Not only are wrought iron 'and steel much more brittle at a blue heat (i.e., the heat that would produce an ox- ide coating ranging from light straw to blue on bright steel, 430° to 600° F.), but while they are probably not seriously affected by simple exposure tob'ue- ness, even if prolonged, yet. if they be worked in this range of temperature they remain extremely brittle after cooling, and may indeed be more brittle than when at blueness; this last point, however, is not certain. (Howe, " Metallurgy of Steel," p. 534.) Tests by Prof. Krohn, for the German State Railways, show that working at blue heat has a decided influence on all materials tested, the injury done being greater on wrought iron and harder steel than on the softer steel. The fact that wrought iron is injured by working at a blue heat was reported by Stromeyer. (Engineering News. Jan. 9, 1892.) A practice among boiler-makers for guarding against failures due to work- ing at a blue heat consists in the cessation of work as soon as a plate which 396 had been red-hot becomes so cool that the mark produced by rubbing a hammer-handle or other piece of wood will not glow. A plate which is not hot enough to produce this effect, yet too hot to be touched by the hand, is most probablv blue-hot, and should under no circumstances be hammered or bent. (0. E. Stromever, Proc. Inst C. E. 1886.) Welding of Steel.— A. E. Hunt (A. I. M. E., 1892) says: I have never seen so-called " welded " pieces of steel pulled apart in a testing-machine or otherwise broken at the joint which have not shown a smooth cleavage- , plane, as it were, such as in iron would be condemned as an imperfect weld. My experience in this matter leads me to agree with the position taken by Mr. William Metcalf in his paper upon Steel in the Trans. A. S. C. E., vol. xvi., p. 301. Mr. Metcalf says, " I do not believe steel can be welded." INFLUENCE OF ANNEALING UPON MAGNETIC CAPACITY. Prof. D. E. Hughes (Eng'g, Feb. 8. 1884, p. 130) has invented a " Magnetic Balance," for testing the condition of iron and steel, which consists chiefly of a delicate magnetic needle suspended over a graduated circular index, and a magnet coil for magnetizing the bar to be tested. He finds that the fol- lowing laws hold with every variety of iron and steel : 1. The magnetic capacity is directly proportional to the softness, or mo- lecular freedom. 2. The resistance to a feeble external magnetizing force is directly as the hardness, or molecular rigidity. The magnetic balance shows that annealing not only produces softness in iron, .and consequent molecular freedom, but it entirely frees it from all strains previously introduced by drawing or hammering. Thus a bar of iron drawn or hammered has a peculiar structure, say a fibrous one, which gives a greater mechanical strength in one direction than another. This bar, if thoroughly annealed at high temperatures, becomes homogeneous in all directions, and has no longer even traces of its previous strains, provided that there has been no actual mechanical separation into a distinct series of fibres. Effect of Annealing upon tlie Magnetic Capacity of Different Wires; Tests by the Magnetic Balance. Description. Magnetic Capacity. Bright as sent. Annealed. Best Swedish charcoal iron, first variety. " " " " second " " third " Swedish Siemens-Martin iron Puddled iron, best best Bessemer steel, soft " " hard Crucible fine cast steel deg. on scale. 230 236 275 165 212 150 115 50 deg. on scale. 525 510 503 430 340 291 172 84 Crucible Fine Steel, Tempered. Bright-yellow heat, cooled completely in cold water. Yellow-red heat, cooled completely in cold water Bright yellow, let down in cold water to straw color. " " " " " " blue " ' ' cooled completely in oil " " let down in water to white Reheat, cooled completely in water "oil Annealed, " " "oil Magnetic Capacity. 51 58 66 SPECIFICATIONS FOR STEEL. 397 SPECIFICATIONS FOR STEEL. Structural Steel.— There has been a change during the ten years from 1880 to 1890, in the opinions of engineers, as to the requirements in specifica- tions for structural steel, in the direction of a preference for metal of low tensile strength and great ductility. The following specifications of differ- ent dates are given by A. E. Hunt and G. H. Clapp, Trans. A. I. M. E. 1890, xix, 926: Tension Members. 1879. 1881. 1882. 1885. 1887. 1888. Elastic limit 50,000 40@,45,000 40,000 40,000 40,000 38.000 Tensile strength 80,000 70@,80,000 70,000 70,000 67@,?5,000 63@70.000 Elongation in 8 in 12* 18* 18* 18* 20* 22* Reduction area 20* 30* 45* 42* 42* 45* Kind of steel O.H. O.H. or B. O.H. Not O.H. or B. O.H.or B. Compression Members : Elastic limit Same 50@55,000 50,000 50,000 Same as tension Tensile strength as 80@90,000 80,000 80,000 members. Elongation in 8 in ten- 12* 15* 15* " Reduction area sion. 20* 35* 35* " F. H. Lewis (Iron Age, Nov. 3, 1892) says: Regarding steel to be used under the same conditions as wrought iron, that is, to be punched without ream- ing, there seems to be a decided opinion (and a growing one) among engi- neers, that it is not safe to use steel in this way, when the ultimate tensile strength is above 65,000 lbs. The reason for this is, not so much because there is any marked change in the material of this grade, but because all steel, especially Bessemer steel, has a tendency to segregations of carbon and phosphorus, producing places in the metal which are harder than they normally should be. As long as the percentages of carbon and phosphorus are kept low, the effect of these segregations is inconsiderable; but when these percentages are increased, the existence of these hard spots in the metal becomes more marked, and it is therefore less adapted to the treat- ment to which wrought iron is subjected. There is a wide consensus of opinion that at an ultimate of 64,000 to 65,000 lbs. the percentages of carbon and phosphorus (which are the two harden- ing elements) reach a point where the steel has a tendency to become tender, and to crack when subjected to rough treatment. A grade of steel, therefore, running in ultimate strength from 54,000 to 62,000 lbs., or in some cases to 64,000 lbs., is now generally considered a proper material for this class of work. Millard Hunsicker, engineer of tests of Carnegie, Phipps & Co., writes as follows concerning grades of structural steel (Eng'g News, June 2, 1892): Grade of Steel.— Steel shall be of three grades— soft, medium, high. Soft Steel. — Specimens from finished material for test, cut to size speci- fied above, shall have an ultimate strength of from 54,000 to 62,000 lbs. per sq. in.; elastic limit one half the ultimate strength; minimum elongation of 26* in 8 in.; minimum reduction of area at fracture 50*. This grade of steel to bend cold 180° flat on itself, without sign of fracture on the outside of the bent portion. Medium Steel.— Specimens from finished material for test, cut to size specified above, shall have an ultimate strength of 60,000 to 68,000 lbs. per sq. in.; elastic limit one half the ultimate strength; minimum elongation 20* in 8 in.; minimum reduction of area at fracture, 40*. This grade of steel to bend cold 180° to a diameter equal to the thickness of the piece tested, without crack or flaw on the outside of the bent portion. High Steel.— Specimens from finished material for test, cut to size speci- fied above, shall have an ultimate strength of 66 000 to 74.000 lbs. per sq. in. ; elastic limit one half the ultimate strength; minimum elongation. 18* in 8 in. ; minimum reduction of area at fracture, 35*. This grade of steel to bend cold 180° to a diameter equal to three times the thickness of the test-piece, without crack or flaw on the outside of the bent portion. F. H. Lewis, Engineers' Club of Phila., 1891, gives specifications for struc- tural steel as follows: The phosphorus in acid open-hearth steel must be less than 0.10*, aud in all Bessemer or basic steel must be less than 0.08* The material will be tested in specimens of at least one half square inch section, cut from the finished material. Each melt of steel will be tested and each section rolled, and also widely differing gauges of the same section. 398 STEEL. Requirements. Soft Steel. Medium Steel. Elastic limit, lbs. per sq. in. , at least 32,000 35,000 Ultimate strength, lbs. per sq. in 54,000 to 62,000 60,000 to 70,000 Elongation in 8 in., at least 25$ 20$ Reduction of area, per cent, at least 45$ 40$ In soft steel for web-plates over 36 in. wide the elongation will be reduced to 20$ and the reduction of area to 40$. It must bend cold 180 degrees and close down on itself without cracking on the outside. %-inch holes pitched % inch from a roll-finished or machined edge and 2 inches between centres must not crack the metal; and %-inch holes pitched 1% inches between centres and V/% inches from the edge must not split the metal between the holes. Medium steel must bend 180 degrees on itself around a lj^-inch round bar. Full-sized eye-bars, when tested to destruction, must show an ultimate strength of at least 56,000 lbs., and stretch at least 10$ in a length of 10 feet. A. E. Hunt, in discussing Mr. Lewis's specifications, advises a requirement as to the character of the fracture of tensile tests being entirely silky in sections of less than 7 square inches, and in larger sections the test specimen not to contain over 25$ crystalline or granular fracture. He also advises the drifting test as a requirement of both soft and medium steel; the require- ment being worded about as follows: " Steel to be capable of having a hole, punched for a %" rivet, enlarged by blows of a sledge upon a drift-pin until the hole (which in the first case should be punched \y^" from the roll- finish or machined edge) is 114" diameter in the case of soft steel, and 1*4'' diameter in the case of medium steel, without fracture." This drifting test is an excellent requirement, not only as a matter of record, but as a meas- ure of the ductility of the steel. H. H. Campbell, Trans. A. I. M. E. 1893, says: In adhering to the safest course, engineers are continually calling for a metal with lower phosphorus. The limit has been 0.10$; it is now 0.08$: soon it will be 0.06$; it should be 0.04$. A. E. Hunt, Trans. A. I. M. E. 1892, says: Why should the tests for steel be so much more rigid than for iron destined for the same purpose ? Some of the reasons are as follows: Experience shows that the acceptable quali- ties of one melt of steel offer no absolute guarantee that the next melt to it, even though made of the same stock, will be equally satisfactory. Again, good wrought iron, in plates and angles, has a narrow range (from 25,000 to 27,000 lbs.) in elastic limit per square inch, and a tensile strength of from 46,000 to 52,000 lbs. per square inch; whereas for steel the range in elastic limit is from 27,000 to 80,000 lbs., and in tensile strength from 4S,000 to 120,000 lbs. per square inch, with corresponding variations in ductility. Moreover, steel is much more susceptible than wrought iron to widely vary- ing effects of treatment, by hardening, cold rolling, or overheating. It is now almost universally recognized that soft steel, if properly made and of good quality, is for many purposes a safe and satisfactory substitute for wrought iron, being capable of standing the same shop-treatment as wrought iron. But the conviction is equally general, that poor steel, or an unsuitable grade of steel, is a very dangerous substitute for wrought iron even under the same unit strains. For this reason it is advisable to make more rigid requirements in select- ing material which may range between the brittleness of glass and a duc- tilitv greater than that of wrought iron. Specifications for Steel for the World's Fair Buildings, Chicago, 1892.— No steel shall contain more than .08$ of phosphorus. From three separate ingots of each cast a round sample bar, not less than §4 in. in diameter, and having a length not less than twelve diameters be- tween jaws of testing machine, shall be furnished and tested by the manu- facturer. From these test-pieces alone the quality of the material in the steel works shall be determined as follows: All the test-bars must have a tensile strength of from 60.000 to 68,000 lbs. per square inch, an elastic limit of not less than half the tensile strength of the test-bar, an elongation of not less than 24$, and a reduction of area of not less than 40$ at the point of fracture. In determining the ductility, the elon- gation shall be measured after breaking on an original length of ten times th» shortest dimension of the test-piece Rivet steel shall have a tensile strength of from 52.000 to 58,000 lbs. per square inch, and an elastic limit, elongation, and reduction of area at the SPECIFICATIONS FOR STEEL. 399 point of fracture as stated above for test-bars, and be capable of bending double flat, without sign of fracture ou the convex surface of the bend. Boiler, Ship, and Tank Plates. W. F. Mattes (Iron Age, July 9, 1893.1 recommends that the different qualities of steel plates be classified as follows : " __■ Tensile test, longitudinal coupon Elongation in 8-in. longitu- dinal coupon, percent Bending test, longitudinal coupon Bending test, transverse coupon Phosphorus limit Sulphur limit Surface inspection Tank. i Limit, \ 75,000 0.15 Easy. Ship. S 55,000 1 to 65,000 20 Flat. - j Over 1 in, ( diam. 0.10 i Careful. Shell. | 55,000 I to 65,000 Flat (Over 3^ in. 1 diam. 0.06 0.065 Close. Fire-box. j 55,000 ( to 60,000 Flat. Flat. 0.045 0.05 Rigid. A steel-manufacturing firm in Pittsburgh advertises six different grades of steel as follows : Extra fire-box. Fire-box. Extra flange. Flange. Shell. Tiink. The probable average phosphorus content in these grades is, respectively: .02 .03 .04 0.6 0.8 .10. Different specifications for steel plates are the following (1889) : United States Navy.— Shell : Tensile strength, 58,000 to 67,000 lbs. per sq. in. ; elongation, 22$ in 8-in. transverse section, 25$ in 8-in. longitudiualseciion. Flange : Tensile strength, 50,000 to 58,000 lbs.; elongation. 26$ in 8 inches. Chemical requirements : P. not over .035$ ; S. not over .040$. Cold-bending test : Specimen to stand being bent flat on itself. Quenching test : Steel heated to cherry-red. plunged in water 82° F., and to be bent around curve \Vo, times thickness of the plate. British Admiralty.— Tensile strength, 58,240 to 67,200 lbs.; elongation in 8 in., 20$ ; same cold-bending and quenching tests as U. S. Navy. American Boiler -makers' 1 Association.— -Tensile strength, 55,000 to 65,000 lbs.; elongation in 8 in., 20$ for plates % in. thick and under ; 22$ for plates % in. to % in. ; 25$ for plates % in. and over. Cold-bending test : For plates ^ in. thick and under, specimen must bend back on itself without fracture ; for plates over J^ in. thick, specimen must withstand bending 180° around a mandril, 1% times the thickness of the plate. Chemical requirements : P. not over .040$ ; S. not over .030$. American Shipmasters'' Association. — Tensile strength, 62,000 to 72,000 lbs.; elongation, 16$ on pieces 9 in. long. Strips cut from plates, heated to a low red and cooled in water the tem- perature of which is 82° F., to undergo without crack or fracture being doubled over a curve the diameter of which does not exceed three times the thickness of the piece tested. Boiler Shell-plates, Front Tube-plate, and Butt-strips. (Penna. R. R., 1892.)— The metal desired is a homogeneous steel having a tensile strength of 60,000 lbs. per sq. in., and an elongation of 25$ in a section originally 8 in. long. These plates will not be accepted if the test- piece shows — 1. A tensile strength of less than 55,000 lbs. per sq. in. ; 2. An elongation in section originally 8 in. long less than 20$ ; 3. A tensile strength over 65.000 lbs. per sq. in. ; should, however, the elongatio» be 27$ or over, plates will not be rejected for high strength. Inside Fire-box Plates, including Back Tube-plate. (Penna. R. R. t 1892.)— The metal should show a tensile strength of 60,000 lbs. per sq. in., and an elongation of 28$ in a test section originally 8 in. long. Chemical Composition. Desired. Will be Rejected. Carbon 0.18 per cent. over 0.25, below 0.15 Phosphorus, not above 0.03 " over 0.04 Manganese, not above 0.40 " over 0.55 Silicon, not above 0.02 " over 0.04 Sulphur, not above 0.02 " over 05 Copper, not above 0.03 " over 0.05 400 STEEL. * These plates will not be accepted if the test-piece shows: 1. A tensile strength of less than 55,000 lbs. per sq. in. ; 2. An elongation in section originally 8 in. long, less than 2:2$ (20$ in plates 34 in ch thick) ; 3. A tensile strength over 65,000 lbs. per sq. in. (08,000 for plates 34 in. thick); should, however, the elongation be 30% or over, plates will not be rejected for high strength ; 4. Any single seam or cavity more than 34 in- long in either of the three fractures obtained on test for homogeneity, as described below. Homogeneity test : A portion of the test-piece is nicked with a chisel, or grooved on a machine, transversely about a sixteenth of an inch deep, in three places about 134 m « apart. The first groove should be made on one side, 134 m - from the square end of the piece; the second, 134 in. from it on the opposite side; and the third, 1J4 in. from the last, and on the opposite side from it. The test-piece is then put in a vise,' with the first groove about 34 ha, above the jaw, care being taken to hold it firmly. The projecting end of the test-piece is then broken off by means of a ham- mer, a number of light blows being used, and the bending being away from the groove. The piece is broken at the other two grooves in the same way. The object of this treatment is to open and render visible to the eye any seams due to failure to weld up, or to foreign interposed matter, or cavities due to gas bubbles in the ingot. After rupture, one side of each fracture is examined, a pocket lens being used if necessary, and the length of the seams and cavities is determined. The length of the longest seam or cavity determines the acceptance or rejection of the plate. Dr. C. B. Dudley, chemist of the Peuna. R. R. (Trans. A. I. M. E. 1892, vol. xx. p. 709), gives "as an example of the progressive improvement in specifi- cations the following : In the early days of steel boilers the specification in force called for steel of not less than 50,000 lbs. tensile strength and not less than 25$ elongation. Some metal was received having 75,000 lbs. tensile strength, and as the elongation was all right it was accepted ; but when those plates were being flanged in the boiler-shop they cracked and went to pieces. As a result, an upper limit of 65,000 lbs. tensile strength was established. Am. Ry. Master Mechanics'' Assn., 1894. — Same as Penna. R. R. Specifica- tions of 1892, including homogeneity test. Plate, Tank, and Sheet Steel. (Penna. R. R., 1888.*)— A test strip taken lengthwise of each plate, % in. thick and over, without annealing, should have a tensile strength of 60,000 lbs. per sq. in., and an elongation of 25$ in a section originally 2 in. long. Sheets will not be accepted if the tests show the tensile strength less than 55,000 lbs. or greater than 70,000 lbs. per sq. in., nor if the elongation falls below 20%. Steel Billets for Main and Parallel Rods. (Penna. R. R., 1884.) —One billet from each lot of 25 billets or smaller shipment of steel for main or parallel rods for locomotives will have a piece drawn from it under the hammer and a test-section will be turned down on this piece to % in. in diameter and 2 in. long. Such test-piece should show a tensile strength of 85,000 lbs. and an elongation of 15$. No lot will be acceptable if the test shows less than 80,000 lbs. tensile strength or 12% e'ongatinn in 2 in. ^Locomotive Spring Steel, (Penna. R. R., 1887.)— Bars which vary more than 0.01 in. in thickness, or more than 0.02 in. in width, from the size ordered, or which break where they are not nicked, or which, when properly nicked and held, fail to break square across where they are nicked, will be returned. The metal desired has the following composition: Carbon, 1.00$; manganese, 0.25$; phosphorus, not over 0.03$; silicon, not over 0.15$; sul- phur, not over 0.03$; copper, not over 0.03$. Shipments will not be accepted which show on analysis less than 0.90$ or over 1.10$ of carbon, or over 0.50$ of manganese, 0.05$ of phosphorus, 0.25$ of silicon, 0.05$ of sulphur, and 0.05$ of copper. Steel tor Locomotive Driving-axles. (Penna. R. R., 1883.)— Steel for driving-axles should have a tensile strength of 85,000 lbs. per sq. in. and an elongation of 15$ in section originally 2 in. long and % in. diameter, taken midway between centre and circumference of the axle. Axles will not be accepted if tensile strength is less than 80,000 lbs., nor if elongation is bplow 12$. Steel for Crank-pins. (Penna. R. R., 1886.)— Steel ingots for crank- * The Penna. R. R. specifications of the several dates given are still in force., July, 1894. SPECIFICATIONS FOR STEEL. 401 pins must be swaged as per drawings. For each lot of 50 ingots ordered, 51 must be furnished, from which one will be taken at random, and two pieces, with test sections % in. diameter and 2 in. long, will be cut from any part of it, provided that centre line of test-pieces falls \y% in. from centre line of in- got. Such test-pieces should have a tensile strength of 85,000 lbs. per sq. in. and an elongation of 15$. Ingots will not be accepted if the tensile strength is less than 80,000 lbs. nor if the elongation is below 12$. Dr. Chas. B. Dudley, Chemist of the P. R. R. (Trans. A. I. M. E. 1892), re- ferring to this specification, says : In testing a recent shipment, the piece from one side of the pin showed 88,000 lbs. strength and 22$ elongation, and the piece from the opposite side showed 106,000 lbs. strength and 14$ elonga- tion. Each piece was above the specified strength and ductility, but the lack of uniformity between the two sides of the pin was so marked that it was finally determined not to put the lot of 50 pins in use. To guard against trouble of this sort in future, the specifications are to be amended to require that the difference in ultimate strength of the two specimens shall not be more than 3000 lbs. Steel Car-axles. (Penna. R. R., 1891.)— For each 100 axles ordered 101 must be furnished, from which one will be taken at random, and subjected to tests prescribed. Axles for passenger cars and passenger locomotive and tender trucks must be made of steel and be rough turned throughout. Two test-pieces will be cut from an axle, and the test sections of % in. diameter by 2 in. long may fall at any part of the axle provided that the centre line of the test- section is 1 in. from the centre line of ihe axle. Such test-pieces should have a tensile strength of 80.000 lbs. per sq. in. and an elongation of 20$. Axles will not be accepted if the tensile strength is less than 75,000 lbs. or the elongation below 15$, nor if the fractures are irregular. Axles for freight cars and freight-locomotive tender trucks must be made of steel, and will be subjected to the following test, wdiich they must stand without fracture : Axles 4 in. diameter at centre — Five blows at 20 ft. of a 1640-lb. weight, striking midway between supports 3 ft. apart; axle to be turned over after each blow. Axles 4% in. diameter at centre— Five blows at 25 ft. of a 1640-lb. weight, striking midway between supports 3 ft. apart: axles to be turned over after each blow. Steel for Rails.— P. H. Dudley (Trans. A. S. C. E. 1893) recommends the following chemical composition for rails of the weights specified : Weights per yard 60, 65, and 70 lbs. 75 and 80 lbs. 100 lbs. Carbon 45 to .55$ .50 to .60$ .65 to .75$ For all weights: Manganese, .80$ to 1.00$; silicon, .10$ to .15$; phos- phorus, not over .06$; sulphur, not over .07$. Carbon by itself up to or over 1$ increases the hardness andtensile strength of the iron rapidly, and at the same time decreases the elongation. The amount of carbon in the early rails ranged from 0.25 to 0.5 of 1$. while in recent rails and very heavy sections it has been increased to 0.5, 0.6, and 0.75 of 1$. With good irons and suitable sections it can run from 0.55 to 0.75 of 1$. according to the section, and obtain fine-grain tough rails with low phosphorus. Manganese is a necessary ingredient in the first place to take up the oxide of iron formed in the bath of molten metal during the blow. It also is of great assistance to check red shortness of the ingots during the first passes in the blooming train. In the early rails 0.4 to 0.5 of 1$ was sufficient when the ingots were hammered or the reductions in the passes in the trains were very much lighter than to day. With the more rapid rolling of recent years the manganese is very often increased to 1.25$ to 1.5$. It makes the rails hard with a coarse crystallization and with a decided tendency to brittleness. Rails high in manganese seem to flow quite easily, especiallj- under severe service or the use of sand, and oxidize rapidly in tunnels. From 0.80 to 1.00$ seems to be all that is necessary for good rolling at, the present time. Steel Rivets. (H. C. Torrance, Amer. Boiler Mfrs. Assn., 1890.)— The Government requirements for the rivets used in boilers of the cruisers built in 1890 are: For longitudinal seams, 58,000 to 67,000 lbs. tensile strength; elongation, not less than 26$ in 8 in., and all others a tensile strength of 50,000 to 58,000 lbs., with an elongation of not less than 30$. They shall be capable of being flattened out cold under the hammer to a thickness of one half the diameter, and of being flattened oup hot to a thickness of one third 402 STEEL. the diameter without showing cracks or flaws. The steel must not contain more than .035 of l^of phosphorus, nor more than .04 of 1% of sulphur. A lot of 20 succesive tests of rivet steel of the low tensile strength quality and 12 tests of the higher tensile strength gave the following results: Low Steel. Higher. Tensile strength, lbs. per sq. in.. . 51,230 to 54,100 59,100 to 61,850 Elastic limit, lbs. per sq. in 31,050 to 33,190 32,080 to 33,070 Elongation in 8 in., per cent 30.5 to 35 25 28.5 to 31.75 Carbon, per cent 11 to .14 .16 to .18 Phosphorus 027 to .029 .03 Sulphur 033 to .035 .033 to .035 The safest steel rivets are those of the lowest tensile strength, since they are the least liable to become hardened and fracture by hammering, or to break from repeated coucussive and vibratory strains to which they are subjected in practice. For calculations of the strength of riveted joints the tensile strength may be taken as the average of the figures above given, or 52,665 lbs., and the shearing strength at 45,000 lbs. per sq. in. MISCELLANEOUS NOTES ON STEEL,. May Carbon be Burned. Out of Steel ?— Experiments made at the Laboratory of the Penna. Railroad Co. (Specifications for Springs, 1888) with the steel of spiral springs, show that the place from which the borings are taken for analysis has a very important influence on the amount of car- bon found. If the sample is a piece of the round bar, and the borings are taken from the end of this piece, the carbon is always higher than if the borings are taken from the side of the piece. It is common to find a differ- ence of 0.10$ between the centre and side of the bar-, and in some cases the difference is as high as 0.23$. Furthermore, experiments made with samples taken from the drawn out end of the bar show, usually, less carbon than samples taken from the round part of the bar, even though the borings may be taken out of the side in both cases. Apparently during the process of reducing the metal from the ingots to the round bar, with successive heatings, the carbon in the outside of the bar is burned out. " Recalescence " of Steel.— If we heat a bar of copper by a flame of constant strength, and note carefully the interval of lime occupied in passing from each degree to the next higher degree, we find that these in- tervals increase regularly, i.e., that the bar heats more and more slowly, as its temperature approaches that of the flame. If we substitute a bar of steel for one of copper, we find that these intervals increase regularly up to a certain point, when the rise of temperature is suddenly and in most cases greatly retarded or even completely arrested. After this the regular rise of temperature is resumed, though other like retardations ma3' recur as the temperature rises farther. So if we cool a bar of steel slowly the fall of temperature is greatly retarded when it reaches a certain point in dull red- ness. If the steel contains much carbon, and if certain favoring conditions be maintained, the temperature, after descending regularly, suddenly rises spontaneously very abruptly, remains stationary a while, and then rede- sceuds, This spontaneous reheating is known as " recalescence. 1 ' These retardations indicate that some change which absorbs or evolves heat occurs within the metal. A retardation while the temperature is rising points to a change which absorbs heat; a retardation during cooling points to some change which evolves heat. (Henry M. Howe, on " Heat Treatment of Steel," Trans. A. I. M. E., vol. xxii.) Effect of Nicking a Steel Bar.— The statement is sometimes made that, owing to the homogeneity of steel, a bar with a surface crack or nick in one of its edges is liable to fail by the gradual spreading of the nick, and thus break under a very much smaller load than a sound bar. With iron it is contended this does not occur, as this metal has a fibrous structure. Sir Benjamin Baker has, however, shown that this theory, at least so far as statical stress is concerned, is opposed to the facts, as he purposely made nicks in specimens of the mild steel used at the Forth Bridge, but found that the tensile strength of the whole was thus reduced by only about one torr per square inch of section. In an experiment by the Union Bridge Com- pany a full-sized steel counter-bar, with a screw-turned buckle connection, was tested under a heavy statical stress, and at the same time a weight weighing 1040 lbs. was allowed to drop on it from various heights. The bar was first broken by ordinary statical strain, and showed a breaking stress of MISCELLANEOUS NOTES ON" STEEL. 403 65,800 lbs. per square inch. The longer of the broken parts was then placed in the machine and put under the following loads, whilst a weight, as already- mentioned, was dropped on it- from various heights at a distance of five feet from the sleeve-nut of the turn-buckle, as shown below: Stress in pounds per sq. in 50,000 55,000 60,000 63,000 65,000 ft. in. ft. in. ft. in. ft. in. ft. in. Height of fall 21 26 30 40 50 The weight was then shifted so as to fall dirctly on the sleeve-nut, and the test proceeded as follows: Stress on specimen in lbs. per square inch 65,350 65,350 68,800 ft. ft. ft. Height of fall 3 6 6 It will be seen that under this trial the bar carried more than when origi- nally tested statically, showing that the nicking of the bar by screwing had not appreciably weakened its power of resisting shocks. — Eng , g Neics. Electric Conductivity of Steel.— Louis Campredon reports in Le Genie Civil the results of a series of experiments made to ascertain the rela- tions between electric resistance and chemical compositions of steel. The wires were No. 17, 3 mm. diameter. The results are given in the table below: Car- bon. 1 0.090 2 0.100 3 0.100 4 0.100 5 0.120 6 0.110 7 0.100 8 0.120 9 0.110 10 0.140 0.020 0.020 0.020 0.020 0.030 0.030 0.020 0.020 0.030 0.930 Sulphur. 0.050 0.050 0.060 0.050 0.070 0.060 0.070 0.070 0.060 0.060 Phos- phorus. 0.030 0.040 0.040 0.050 0.050 0.060 0.040 0.070 0.060 0.080 Manga- 0.210 0.240 0.260 0.310 0.330 0.350 0.400 0.400 0.490 0.540 0.410 0.450 0.480 0.530 0.600 0.610 0.630 0.680 0.750 0.850 Electric Resist - 127.7 133.0 137.5 140.3 142.7 144.5 149.0 150.3 156.0 173.0 An examination of these series of figures shows that the purer and softer steel the better is its electric conductivity, and, furthermore, that manga- nese is the element which most influences the conductivity. Specific Gravity of Soft Steel. (W. Kent, Trans. A. I. M. E., xiv. 585. »— Five specimens of boiler-plate of C. 0.14, P. 0.03 gave an average sp. gr. of 7.932, maximum variation 0.008. The pieces were first planed to re- move all possible scale indentations, then filed smooth, then cleaued in dilute sulphuric acid, and then boiled in distilled water, to remove all traces of air from the surface. The figures of specific gravity thus obtained by careful experiment on bright, smooth pieces of steel are, however, too high for use in determining the weights of rolled plates for commercial purposes. The actual average thickness of these plates is always a little less than is shown by the calipers, on account of the oxide of iron on the surface, and because the surface is not perfectly smooth and regular. A number of experiments on commercial plates, and comparison of other authorities, led to the figure 7.854 as the average specific gravity of open-hearth boiler-plate steel. This figure ;s easily remembered as being the same figure with change of position of the decimal point (.7854) which expresses the relation of the area of a circle to that of its circumscribed square. Taking the weight of a cubic foot of water at 62° F. as 62.36 lbs. (average of several authorities), this figure gives 489.775 lbs. as the weight of a cubic foot of steel, or the even figure, 490 lbs., may be taken as a convenient figure, and accurate within the limits of the error of observation. A common method of approximating the weight of iron plates is to con- sider them to weigh 40 lbs. per square foot one iuch thick. Taking this weight and adding 2% gives almost exactly the weight of steel boiler-plate given above (40 X 12 X 1.02 = 489.6 lbs. per cubic foot). Occasional Failnres of Bessemer Steel.— G. H. Clapp and A. E. Hunt, in their paper on "The Inspection of Materials of Construction in 404 STEEL. the United States " (Trans. A. I. M. E., vol. xix), say: Numerous instances could be cited to show the unreliability of Bessemer steel for structural pur- poses. One of the most marked, however, was the following: A 12-in. I-beam weighing 30 lbs. to the foot, 20 feet long, on being unloaded from a car broke in two about 6 feet from one end. The analyses and tensile tests made do not show any cause for the failure. The cold and quench bending tests of both the original %-in. round test- pieces, and of pieces cut from the finished material, gave satisfactory re- sults; the cold-bending tests closing down on themselves without sign of fracture. Numerous other cases of angles and plates that were so hard in places as to break off short in punching, or, what was worse, to break the punches, have come under our observation, and although makers of Bessemer steel claim that this is just as likely to occur in open-hearth as in Bessemer steel, we have as yet never seen an instance of failure of this kind in open-hearth steel having a composition such as C 0.25$, Mn 0.70$, P 0.80$. J. W. Wailes, in a paper read before the Chemical Section of the British Association for the Advancement of Science, in speaking of mysterious failures of steel, states that investigation shows that " these failures occur in steel of one class, viz., soft steel made by the Bessemer process.'" Segregation in Steel Ingots. (A. Pourcel, Trans. A. I. M. E. 1893.) — H. M. Howe, in his " Metallurgy of Steel,'' 1 gives a resume of observations, with the results of numerous analyses, bearing upon the phenomena of seg- regation. In 1881 Mr. Stubbs. of Manchester, showed the heterogeneous results of analyses made upon different parts of an ingot of large section. A test-piece taken 24 inches from the head of the ingot 7.5 feet in length gave by analysis very different results from those of a test-piece taken 30 inches from the bottom. C. Mn. Si. S. P. Top 0.92 0.535 0.043 0.161 0.261 Bottom 37 0.498 . 0.006 0.025 0.096 Windsor Richards says he had often observed in test-pieces taken from different points of one plate variations of 0.05$ of carbon. Segregation is specially pronounced in an ingot in its central portion, and around the space of the piping. It is most observable in large ingots, but in blocks of smaller weight and limited dimensions, subjected to the influence of solidification as rapid as casting within thick walls will permit, it may still be observed distinctly. An ingot of Martin steel, weighing about 1000 lbs., and having a height of 1.10 feet and a section of 10.24 inches square, gave the following: 1. Upper section: C. S. P. Mn. Border 0.330 0.040 0.033 0.420 Centre 0.530 0.077 0.057 0.430 2. Lower section: C. S. P. Mn. Border 0.280 0.029 0.016 0.390 Centre 0.290 0.030 0.038 0.390 3. Middle section: C. S. P. Mn. Border „ 0.320 0.025 0.025 0.100 Centre 0.320 0.048 0.048 0.4CH* Segregation is less marked in ingots of extra-soft metal cast in cast-iron moulds of considerable thickness. It is, however, still important, and ex- plains the difference often shown by the results of tests on pieces taken from different portions of a plate. Two samples, taken from the sound part of a flat ingot, one on the outside and the other in the centre, 7.9 inches from the upper edge, gave: • C. S. P. Mn. Centre 0.14 0.053 0.072 0.576 Exterior 0.11 0.036 0.027 0.610 Manganese is the element most uniformly disseminated in hard or soft steel. For cannon of large calibre, if we reject, in addition to the part cast in sand and called the masselotte (sinking-head), one third of the upper part of the ingot, we can obtain a tube practically homogeneous in composition, because the central part is naturally removed by the boring of the tube. With extra-soft steels, destined for ship- or boiler-plates, the solution for practically perfect homogeneity lies in the obtaining of a metal more closely deserving its name of extra-soft metal. STEEL CASTINGS. 405 The injurious consequences of segregation must be suppressed by reduc- ing, as far as possible, the elements subject to liquation. Earliest Uses of Steel for Structural Purposes. (G. G. Mehrtens, Trans. A. S. C. E. 1893).— The Pennsylvania Railroad Company first introduced Bessemer steel in America in locomotive boilers in the year 1863, but the steel was too hard and brittle for such use. The first plates made for steel boilers had a tenacity of 85,000 to 92,000 lbs. and an elongation of but 7$ to 10$. The results were not favorable, and the steel works were soon forced to offer a material of less tenacity and more ductility. The re- quirements were therefore reduced to a tenacity of 78,000 lbs. or less, and the elongation was increased to 15% or more. Even with this, between the years 1870 and 1S80, many explosions occurred and many careful examina- tions were made to determine their cause. It was found on examining the rivet-holes that there were incipient changes in the metal, many cracks around them, and points near them were corroded with rust, all caused by the shock, of tools in manufacturing. It was evident that the material was unsuitable, and that the treatment must be changed. In the beginning of 1878, Mr. Parker, chief engineer of the Lloyds, stated that there was then but one English steamer in possession of a steel boiler; a year later there were 120. In 1878 there were but five large English steamers built of steel, while in 1883 there were 116 building. The use of Bessemer steel in bridge- building was tried first on the Dutch State railways in 1863-61, then in Eng- land and Austria. In 1874 a bridge was built of Bessemer steel in Austria. The first use of cast steel for bridges was in America, for the St. Louis Arch Bridge and for the wire of the East River Bridge. These gave an impetus to the use of ingot metal, and before 1880 the Glasgow and Plattsmouth Bridges over the Missouri River were also built of ingot metal. Steel eye- bars were applied for the first time in the Glasgow Bridge. Since 1880 the introduction of mild steel in all kinds of engineering structures has steadily increased. STEEL CASTINGS. (E. S. Cramp, Engineering Congress, Dept. of Marine Eng'g, Chicago, 1893.) In 1891 American steel-founders had successfully produced a considerable variety of heavy and difficult castings, of which the following are the most noteworthy specimens: Bed-plates up to- 24,000 lbs.; stern-posts up to 54,000 lbs.; stems up to 21,000 lbs. ; hydraulic cylinders up to 11,000 lbs. ; shaft-struts up to 32,000 lbs. ; hawse-pipes up to 7500 lbs. ; stern-pipes up to 8000 lbs. ' The percentage of success in these classes of castings since 1890 has ranged from 65$ in the more difficult forms to 90$ in the simpler ones; the tensile strength has been from 62.000 to 78,000 lbs., elongation from 15$ to 25$. The best performance recorded is that of a guide, cast in January, 1893, which developed 84,000 lbs. tensile strength and 15.6$ elongation. The first steel castings of which anything is generally known were crossing-frogs made for the Philadelphia & Reading R. R. in July, 1867, by the William Butcher Steel Works, now the Midvale St<-el Co. The moulds were made of a mixture of ground fire-brick, black-lead crucible-pots ground fine, and fire-clay, and washed with a black-lead wash. The steel was melted in crucibles, and was about as hard as tool steel. The surface of these castings was very smooth, but the interior was very much honey- combed. This was before the days when the use of silicon was known for solidifying steel. The sponginess, which was almost universal, was a great obstacle to their general adoption.. The next step was to leave the ground pots out of the moulding mixture and to wash the mould with finely ground fire-brick. This was a great im- provement, especially in very heavy castings; but this mixture still clung so strongly to the casting that only comparatively simple shapes could be made with certainty. A mould made of such a mixture became almost as hard as fire-brick, and was such an obstacle to the proper shrinkage of castings, that, when at all complicated in shape, they had so great a tendency to crack as to make their successful manufacture almost impossible. By this time the use of silicon had been discovered, and the only obstacle in the way of making good castings was a suitable moulding mixture. This was ulti- mately found in mixtures having the various kinds of silica sand as the principal constituent. One of the most fertile sources of defects in castings is a bad design. Very intricate shapes can be cast successfully if they are so designed as to 406 STEEL. cool uniformly. Mr. Cramp says while he is not yet prepared to state that anything that can be cast successfully in iron can be cast in steel, indica- tions seem to point that way in all cases where it is possible to put on suit- able sinking-heads for feeding the casting. H. L. Gantt (Trans. A. S. M. E., xii. 710) says: Steel castings not only shrink much more than iron ones, but with less regularity. The amount of shrinkage varies with the composition and the heat of the metal; the hotter the metal the greater the shrinkage; and, as we get smoother castings from hot metal, it is better to make allowance for large shrinkage and pour the metal as hot as possible. Allow 3/10 or 14 in. per ft. in length for shrinkage, and J4 in. for finish on machined surfaces, except such as are cast '■up." Cope surfaces which are to be machined should, in large or hard castings, have an allowance of from % to % in. for finish, as a large mass of metal slowly rising in a mould is apt to become crusty on the sur- face, and such a crust is sure to be full of imperfections. On small, soft castings Y% in. on drag side and x A'vc\. on cope side will be sufficient. No core should have less than 34 in- fiuish on a side and very large ones should have as much as ^ in. on a side. Blow-holes can be entirely prevented in cast- ings by the addition of manganese and silicon in sufficient quantities; but both of these cause brittleness, and it is the object of the conscientious steel- maker to put no more manganese and silicon in his steel than is just suffi- cient to make it solid. The best results are arrived at when all portions of the castings are of a uniform thickness, or very nearly so. The following table will illustrate the effect of annealing on tensile strength and elongation of steel castings : Carbon. Unannealed. Annealed. Tensile Strength. Elongation. Tensile Strength. Elongation. .23$ .37 .53 68.738 85,540 90,121 22. 40$ 8.20 2.35 67.210 82,228 106.415 31.40$ 21.80 9.80 The proper annealing of large castings takes nearly a week. The proper steel for roll pinion*, hammer dies, etc., seems to be that con- taining about .60$ of carbon. Such castings, properly annealed, have woi;n well and seldom broken. Miscellaneous gearing should contain carbon .40$ to 60$, gears larger in diameter being softest. General machinery castings should, as a rule, contain less than .40$ of carbon, those exposed to great shocks containing as low at .20$ of carbon. Such castings will give a tensile strength of from 60,000 to 80,000 lbs. per sq. in. and at least 15$ extension in a 2 in. long specimen. Machinery and hull castings for war-vessels for the United States Navy, as well as carriages for naval guns, contain from .20$ to 30$ of carbon. The following is a partial list of castings in which steel seems to be rapidly taking the place of iron: Hydraulic cylinders, crossheadsand pistons for large engines, roughing rolls, rolling-mill spindles, coupling-boxes, roll pinions, gearing, hammer-heads and dies, riveter stakes, castings for ships, car-couplers, etc. For description of methods of manufacture of steel castings by the Besse- mer, open-hearth, and crucible processes, see paper by P. G. Salom, Trans. A. I. M. E. xiv, 118. Specifications for steel castings issued by the U. S. Navy Department, 1889 (abridged) : Steel for castings must be made by either the open-hearth or the crucible process, and must not show more than .06$ of phosphorus. All castings must be annealed, unless otherwise directed. The tensile strength of steel casiings shall be at least 60,000 lbs., with an elongation of at least 15$ in 8 in. for all castings for moving parts of the machinery, and at least 10$ in 8 in. for other castings. Bars 1 in. sq. shall be capable of bending cold, without fracture, through an angle of 90°, over a radius not greater than \% in. All castings must be sound, free from injurious roughness, sponginess, pitting, shrinkage, or other cracks, cavities, etc. Pennsylvania Railroad specifications, 1888: Steel castings should have a tensile strength of 70,000 lbs. per sq. in. and an elongation of 15$ in section originally 2 in. long. Steel castings will not be accepted if tensile strength MANGANESE, NICKEL, AND OTHER u ALLOY* 3 STEELS. 407 falls below 60,000 lbs., nor if the elongation is less than 12$, nor if cast- ings have blow-holes and shrinkage cracks. Castings weighing 80 lbs. or more must have cast with them a strip to be used as a test-piece. The di- mensions of this strip must be % in. sq. by 12 in. long. MANGANESE, NICKEL, AND OTHER "ALLOY" STEELS. manganese Steel. (H. M. Howe, Trans. A. S. M. E., vol. xii.)— Man- ganese steel is an alloy of iron and manganese, incidentally, and probably unavoidably, containing a considerable proportion of carbon. The effect of small proportions of manganese on the hardness, strength, and ductility of iron is probably slight. The point at which manganese begins to have a predominant effect is not known : it may be somewhere about 2.5$. As the proportion of manganese rises above 2.5$ the strength and ductility diminish, while the hardness increases. This effect reaches a maximum with somewhere about 6$ of manganese. When the proportion of this element rises beyond 6$ the strength and ductility both increase, while the hardness diminishes slightly, (he maximum of both strength and ductility being reached with about 14$ of manganese. With this proportion the metal is still so hard that it is very difficult to cut it with steel tools. As the proportion of manganese rises above 15$ the ductility falls off abruptly, the strength remaining nearly constant till the manganese passes 18$, when it in turn diminishes suddenly. Steel containing from 4$ to 6.5$ of manganese, even if it have but 0.37$ of carbon, is reported to be so extremely brittle that it can be powdered under a hand-hammer when cold ; yet it is ductile when hot. Manganese steel is very free from blow-holes ; it welds with great diffi- culty; its toughness is increased by quenching from a yellow heat ; its elec- tric resistance is enormous, and very constant with changing temperature ; it is low in thermal conductivity. Its remarkable combination of great hard- ness, which cannot be materially lessened by annealing, and great tensile strength, with astonishing toughness and ductility, at once creates and limits its usefulness. The fact that manganese steel cannot be softened, that it ever remains so hard that it can be machined only with great diffi- culty, sets up a barrier to its usefulness. The following comparative results of abrasion tests of manganese and other steel were reported by T. T. Morrell : Abrasion by Pressure Against a Revolving Hardened-Steel Shaft. Loss of weight of manganese steel 1.0 " blue-tempered hard tool steel 0.4 " annealed hard tool steel 7.5 " hardened Otis boiler-plate steel 7.0 " annealed " " " 14.0 Abrasion by an Emery-Wheel. Loss of weight of hard manganese-steel wheels 1.00 softer " " 1.19 " hardest carbon-steel Avheels 1.23 " soft " " 2.85 The hardness of manganese steel seems to be of an anomalous kind. The alloy is hard, but under some conditions not rigid. It is very hard in its resistance to abrasion ; it is not ahvays hard in its resistance to impact. Manganese steel forges readily at a yellow heat, though at a bright white heat it crumbles under the hammer. But it offers greater resistance to deformation, i.e., it is harder when hot, than carbon steel. The most important single use for manganese-steel is for the pins which hold the buckets of elevated dredgers. Here abrasion chiefly is to be resisted. Another important use is for the links of common chain-elevators. As a material for stamp-shoes, for horse-shoes, for the knuckles of an automatic car-coupler, manganese steel has not met expectations. Manganese steel has been regularly adopted for the blades of the Cyclone pulverizer. Some manganese-steel wheels are reported to have run over 300.000 miles each without turning, on a New England railroad. Nickel Steel.— The remarkable tensile strength and ductility of nickel steel, as shown by the test-bars and the behavior of nickel-steel armor- plate under shot tests, are witness of the valuable qualities conferred upon steel by the addition of a few per cent of nickel. 408 STEEL. The following tests were made on nickel steels by Mr. Maunsel White of the Bethlehem Iron Company (Eng. <& M. Jour., Sept. 16, 1893.) : „ & Tensile Elastic Specimen B ■ "&rt Str'gth, Limit, p. c. p. c. from — 5.2 5 B.S 03 lbs. per sq. in. lbs. per sq. in. ex. cont. Forged bars. * i .625 4 276,800 2.75 "Yof Special ~" 1 " 246,595 4.25 treatment. 03 1 " " 105,300 19.25 55.0 Annealed. EQ f .564 4 142,800 ' 74.000 13.0 28.2 % i n 143.200 74,000 12.32 27.6 M . 1^-in. round rolled bar.t 1 » " 117,600 64,000 17.0 46.0 ■= 1 ti " 119,200 65,000 16.66 42.1 s * • < ." 91,600 91,200 51,000 51,000 22.25 21.62 53.2 53.4 eo " " 85,200 86,000 53,000 48,000 21.82 21.25 49.5 47.4 rjH ".798 8 115,464 51,820 36.25 66.23 £ lj^in. sq. " 112,600 60,000 37.87 62.82 bar, rolled. $ " 102,010 39,180 41.37 69.59 Annealed. 1- " " 102,510 40,200 44.00 68.34 " \500 2 114,590 56,020 47.25 68.4 i 1 1-in. round " 115.610 59,080 45.25 62.3 - 1 bar, rolled. § - u " 105,240 45,170 49.65 72.8 Annealed. gl - " 106,780 45,170 55.50 63.6 * Forged from 6-in. ingot to % in. diam., with conical heads for holding. t Showing the effect of varying carbon. X Rolled down from 14-in. ingot to lJ4-in. square billet, and turned to size. § Rolled down from 14 in. ingot to 1-in. round, and turned to size. Nickel steel has shown itself to be possessed of some exceedingly valuable properties; these are, resistance to cracking, high elastic limit, and homo- geneity. Resistance to cracking, a property to which the name of non fissi- bility has been given, is shown more remarkably as the percentage of nickel increases. Bars of 27$ nickel illustrate this property. A lJ4-in. square bar was nicked J4 '"• deep and bent double on itself without further fracture than the splintering off, as it were, of the nicked portion. Sudden failure or rupture of this steel would be impossible ; it seems to possess the toughness of rawhide with the strength of steel. With this percentage of nickel the steel is practically non-corrodible and non-magnetic. The resistance to cracking shown by the lower percentages of nickel is best illustrated in the many trials of nickel-steel armor. The elastic limit rises in a very marked degree with the addition of about 3$ of nickel, the other physical properties of the steel remaining unchanged or perhaps slightly increased. In such places (shafts, axles, etc.) where failure is the result of the fatigue of the metal this higher elastic limit of nickel steel will tend to prolong in- definitely the life of the piece, and at the same time, through its superior toughness, offer greater resistance to the sudden strains of shock. Howe states that the hardness of nickel steel depends on the proportion of nickel and carbon jointly, nickel up to a certain percentage increasing the hardness, beyond this iessening it. Thus while steel with 2$ of nickel and 0.90$ of carbon cannot be machined, with less than 5$ nickel it can be worked cold readily, provided the proportion of carbon be low. As the proportion of nickel rises higher, cold-working becomes less easy. It forges easily whether it contain much or little nickel. The presence of manganese in nickel steel is most important, as it appears that without the aid of manganese in proper proportions, the conditions of treatment would not be successful. Tests of Nickel Steel.— Two heats of open-hearth steel were made by the Cleveland Rolling Mill Co., one ordinary steel made with 9000 lbs. each scrap and pig, and 165 lbs. ferro-manganese, the other the same with, the addition of 3$, or 540 lbs. of nickel. Tests of six plates rolled from each heat., 0.24 to 0.3 in. thick, gave results as follows : Ordinary steel, T. S. 52,500 to 56.500 ; E. L. 32,800 to 37,900 ; elong. 26 to 32$. Nickel steel, " 63,370 to 67,100 ; " 47,100 to 48,200 ; " 23J4 to 26$. MANGANESE, NICKEL, AND OTHER " ALLOY" STEELS. 409 The nickel steel averages 31$ higher in elastic limit, 20$ higher in ultimate tensile strength, with but slight reduction in ductility. {Eng. & M. Jour., Feb. 25, 1893.) Aluminum Steel.— R. A. Hadfield (Trans. A. I. M. E. 1890) says : Aluminum appears to be of service as an addition to baths of molten iron or steel unduly saturated with oxides, and this in properly regulated steel manufacture should not often occur. Speaking generally, its role appears to be similar to that of silicon, though acting more powerfully. The state- ment that aluminum lowers the melting-point of iron seems to have no foundation in fact. If any increase of heat or fluidity takes place by the addition of small amounts of aluminum, it may be due to evolution of heat, owing to oxidation of the aluminum, as the calorific value of this metal is very high— in fact, higher than silicon. According to Berthollet, the con- version of aluminum to A1 2 3 equals 7900 cal. ; silicon to Si0 2 is stated as 7800. The action of aluminum maybe classed along with that of silicon, sulphur, phosphorus, arsenic, and copper, as giving no increase of hardness to iron, in contradistinction to carbon, manganese, chromium, tungsten, and nickel. Therefore, whilst for some special purposes aluminum may be employed in the manufacture of iron, at any rate with our present knowledge of its properties, this use cannot be large, especially when taking into considera- tion the fact of its comparatively high price. Its special advantage seems to be that it combines in itself the advantages of both silicon and manganese; but so long as alloys containing these metals are so cheap and aluminum dear, its extensive use seems hardly probable. J. E. Stead, in discussion of Mr. Hadfield's paper, said: Every one of our trials has indicated that aluminum can kill the most fiery steel, providing, of course, that it is added in sufficient quantity to combine with all the oxy- gen which the steel contains. The metal will then be absolutely dead, and will pour like dead-melted silicon steel. If the aluminum is added as metal- lic aluminum, and not as a compound, and if the addition is made just be- fore the steel is cast, 1/10$ is ample to obtain perfect solidity in the steel. Chrome Steel. (F. L. Garrison, Jour. F. I., Sept. 1891.)— Chromium increases the hardness of iron, perhaps also the tensile strength and elastic limit, but it lessens its weldibility. Ferro chrome, according to Berthier, is made by strongly heating the mixed oxides of iron and chromium in brasqued crucibles, adding powdered charcoal if the oxide of chromium is in excess, and fluxes to scorify the earthy matter and prevent oxidation. Chromium does not appear to give steel the power of becoming harder when quenched or chilled. Howe states that chrome steels forge more readily than tungsten steels, and when not containing over 0.5 of chromium nearly as well as ordinary carbon steels of like percentage of carbon. On the whole the status of chrome steel is not satisfactory. There are other steel alloys coming into use, which are so much better, that it would seem to be only a question of time when it will drop entirely out of the race. Howe states that many experienced chemists have found no chromium, or but the merest traces, in chrome steel sold in the markets. J. W. Langley (Trans. A. S. C. E. 1892) says: Chromium, like manganese, is a true hardener of iron even in the absence of carbon. The addition of 1% or 2$ of chromium to a carbon steel will make a metal which gets exces- sively hard. Hitherto its principal employment has been in the production of chilled shot and shell. Powerful molecular stresses result during cooling, and the shells frequently break spontaneously months after they are made. Tungsten Steel— Mushet Steel. (J. B. Nau, Iron Age. Feb. 11, 1892.) — By incorporating simultaneously carbon and tungsten in iron, it is possi- ble to obtain a much harder steel than with carbon alone, without danger of an extraordinary brittleness in the cold metal or an increased difficulty in the working of the heated metal. When a special grade of hardness is required, it is frequently the custom to use a high tungsten steel, known in England as special steel. A specimen from Sheffield, used for chisels, contained 9.3$ of tungsten, 0.7$ of silver, and 0.6$ of carbon. This steel, though used with advantage in its untem- pered state to turn chilled rolls, was not brittle; nevertheless it was hard enough to scratch glass. A sample of Mushet's special steel contained 8.3$ of tungsten and 1.73$ of manganese. The hardness of tungsten steel cannot be increased by the or- dinary process of hardening. The only operation that it can be submitted to when cold is grinding. It has to be given its final shape through hammering at a red heat, and even 410 STEEL. then, when the percentage of tungsten is high, it has to be treated very carefully; and in order to avoid breaking it, not only is it necessary to reheat it several times while it is being hammered, but when the tool nas acquired the desired shape hammering must still be continued gently and with nu- merous blows until it becomes nearly cold. Then only can it be cooled en- tirely. Tungsten is not only emploj^ed to produce steel of an extraordinary hard- ness, but more especially to obtain a steel which, with a moderate hardness, allies great toughness, resistance, and ductility. Steel from Assailly, used for this purpose, contained carbon, 0.52%; silicon, 0.04$; tungsten, 0.3$; phosphorus, 0.04$; sulphur, 0.005$. Mechanical tests made by Styffe gave the following results : Breaking load per square inch of original area, pounds. . 172,424 Reduction of area, per cent 0.54 Average elongation after fracture, per cent 13 According to analyses made by the Due de Luynes of ten specimens of the celebrated Oriental damasked steel, eight contained tungsten, two of them in notable quantities (0.518$ to 1$), while in all of the samples analyzed nickel was discovered ranging from traces to nearly 4$. Stein & Schwartz of Philadelphia, in a circular say : It is stated that tungsten steel is suitable for the manufacture of steel magnets, since it re- tains its magnetism longer than ordinary steel. Mr. Kniesche has made tungsten up to 98$ fine a specialty. Dr. Heppe, of Leipsig, has written a number of articles in German publications on the subject. The following instructions are given concerning the use of tungsten: In order to produce cast iron possessing great hardness an addition of one half to one and one half of tungsten is all that is needed. For bar iron it must be carried up to 1$ to 2$, but should not exceed 2^$. For puddled steel the range is larger, but an addition beyond S}/ 2 % only increases the hardness, so that it is brought up to \y%% only for special tools, coinage dies, drills, etc. For tires 2^$ to 5$ have proved best, and for axles y% to 1^$. Cast steel to which tungsten has been added needs a higher temperature for tempering than ordinary steel, and should be hardened only between yellow, red, and white. Chisels made of tungsten steel should be drawn between cherry-red and blue, and stand well on iron and steel. Tempering is best done in a mixture of 5 parts of yellow rosin, 3 parts of tar, and 2 parts of tallow, and then the article is once more heated and then tempered as usual in water of about 15° C. Whitwortli Compressed Steel. (Proc. Inst. M. E.. May, 1887, p. 167 )— In this system a gradually increasing pressure up to 6 or 8 tons per square inch is applied to the fluid ingot, and within half an hour or less after the application of the pressure the column of fluid steel is shortened \y% inch per foot or one eighth of its length; the pressure is then kept on for several hours, the result being that the metal is compressed into a perfectly solid and homogeneous material, free from blow-holes. In large gun-ring ingots during cooling the carbon is driven to the centre, the centre containing 0.8 carbon and the outer ring 0.3. The centre is bored out until a test shows that the inside of the ring contains the same percent- age of carbon as the outside. Compressed steel is made by the Bethlehem Iron Co. and the Carnegie . Steel Co. for armor-plate and for gun and other heavy forgings. CRUCIBLE STEEL. Selection of Grades by the Eye, and Effect of Heat Treat- ment. (J. W. Langley, Amer. Chemist, November, 1876.)— In 1874, Miller, Metcalf & Parkin, of Pittsburgh, selected eight samples of steel which were believed to form a set of graded specimens, the order being based on the quantity of carbon which they were supposed to contain. They were num- bered from one to eight. On analysis, the quantity of carbon was found to follow the order of the numbers, while the other elements present— silicon, phosphorus, and sulphur— did not do so. The method of selection is described as follows : The steel is melted in black-lead crucibles capable of holding about eighty pounds; when thoroughly fluid it is poured into cast-iron moulds, and when cold the top of the ingot is broken off, exposing a freshly-fractured surface. The appearance presented is that of confused groups of crystals, all appear- ing to have started from the outside and to have met in the centre; this general form is common to all ingots of whatever composition, but to the trained eye, and only to one long and critically exercised, a minute but in- CRUCIBLE STEEL. 411 describable difference is perceived between varying samples of steel, and tbis difference is now known to be owing almost wholly to variations in the amount of combined carbon, as the following table will show. Twelve sam- ples selected by the eye alone, and analyses of drillings taken direct from the ingot before it had been heated or hammered, gave results as below: Ingot Nos. Iron by Diff. Carbon. Diff. of Carbon. Silicon. Phos. Sulph. 1 99.614 99.455 .302 .490 .019 .034 .047 .005 .018 2 .188 .016 3 99.363 .529 .039 .043 .047 .018 4 99.270 .649 .120 .039 .030 .012 5 99.119 .801 .152 .029 .035 .016 6 99.086 .841 .040 .039 .024 .010 7 99.044 .S67 .026 .057 .014 .018 8 99.040 .871 .004 .053 .024 .012 9 98.900 .955 .084 .059 .070 .016 10 98 861 1.005 .050 .088 .034 .012 11 98.752 1.058 .053 .120 .064 .006 12 98.834 1.079 .021 .039 .044 .004 Here the carbon is seen to increase in quantity in the order of the num- bers, while the other elements, with the exception of total iron, bear no rela- tion to the numbers on the samples. The mean difference of carbon is .071. In mild steels the discrimination is less perfect. The appearance of the fracture by which the above twelve selections were made can only be seen in the coid ingot before any operation, except the original one of casting, has been performed upon 'it. As soon as it is hammered, the structure changes in a remarkable manner, so that all trace of the primitive condition appears to be lost. Another method of rendering visible to the eye the molecular and chemi- cal changes which go on in steel is by the process of hardening or temper- ing. When the metal is heated and plunged into water it acquires an increase of hardness, but a loss of ductility. If the heat to which the steel has been raised just before plunging is too high, the metal acquires intense hardness, but it is so brittle as to be worthless; the fracture is of a bright, granular, or sandy character. In this state it is said to be burned, and it cannot again be restored to its former strength and ductility by annealing; it is ruined for all practical purposes, but in just this state' it again shows differences of structure corresponding with its content in carbon. The nature of these changes can be illustrated by plunging a bar highly heated at one end and cold at the other into water, and then breaking it off in pieces of equal length, when the fractures will be found to show appear- ances characteristic of the temperature to which the sample was raised. The specific gravity of steel is influenced not only by its chemical analy- sis, but by the heat to which it is subjected, as is shown by the following table (densities referred to 60° F.): Specific gravities of twelve samples of steel from the ingot; also of six hammered bars, each bar being overheated at one end and cold at the other, in this state plunged into water, and then broken into pieces of equal length. Ingot Bar: *Burned 1. . 2. . 3... 4... 1 7.855 2 7.836 3 7.841 7.818 7.814 7.823 7.826 7.831 7.844 4 7.829 7.791 7.811 7.830 7.849 7.806 7.824 5 7.838 6 7.824 7.789 7.784 7.780 7.808 7.812 7.829 7 1 8 7.819,7.818 . ... 7.752 7.755 ,7.758 7.773 17.790 . ... |7.825 9 7.813 10 7.807 7.744 7.749 7.755 7.789 7.812 7.826 11 7.803 12 7.805 7.690 7.741 7.769 7.798 5... 7.811 Cold 6... 7.825 * Order of samples from bar. 412 Effect of Heat on the Grain of Steel. (W. Metcalf,— Jeans on Steel, p. 642.) — A simple experiment will show the alteration produced in a high-carbon steel by different methods of hardening. If a bar of such steel be nicked at about 9 or 10 places, and about half an inch apart, a suitable specimen is obtained for the experiment. Place one end of the bar in a good fire, so that the first nicked piece is heated to whiteness, while the rest of the bar, being out of the fire, is heated up less and less as we approach the other end. As soon as the first piece is at a good white heat, which of course burns a high carbon steel, and the temperature of the rest of the bar gradually passes down to a very dull red, the metal should be taken out of the fire and suddenly plunged in cold water, in which it should be left till quite cold. It should then be taken out and carefully dried. An examina- tion with a file will show that the first piece has the greatest hardness, while the last piece is the softest, the intermediate pieces gradually passing from one condition to the other. On now breaking off the pieces at each nick it will be seen that very considerable and characteristic changes have been produced in the appearance of the metal. The first burnt piece is very open or crystalline in fracture; the succeeding pieces become closer and closer in the grain until one piece is found to possess that perfectly even grain and velvet-like appearance which is so much prized by experi- enced steel users. The first pieces also, which have been too much hard- ened, will probably be cracked; those at the other end will not be hardened through. Hence if it be desired to make the steel hard and strong, the temperature used must be high enough to harden the metal through, but not sufficient to open the grain. Changes in T 1 Him ate Strength and Elasticity due to Hammering, Annealing, and Tempering. (J. W. Langley, Trans. A. S. C. E. 1892.)— The following tabl« gives the result of tests made on some round steel bars, all from the same ingot, which were tested by tensile stresses, and also by bending till fracture took place: Treatment. si 85c Carbon. p p s eg 5 2 33 a ■a s eg ~ 53 a . W ■ a p ence of the intensities of the two ' L forces, Fig. 93, 418 MECHANICS. Thus the resultant of the two forces Q and P, Fig. 93, is equal to Q - P = R. Of any two parallel forces and their N resultant each is proportional to the dis- q< -p tance between the other two; thus in both / Figs. 92 and 93, P : Q : R : : SN : SM : MN. M-p — j %-p Couples.— If P and Q be equal and act / i in opposite directions, R — 0; that is, they / I have no resultant. Two such forces con- ~ @L ! >r stitute what is called a couple. * C The tendency of a couple is to produce Fig. 93. rotation; the measure of this tendency, called the moment of the couple, is the product of one of the forces by the distance between the two. Since a couple has no single resultant, no single force can balance a couple. To prevent the rotation of a body acted on by a couple the applica- tion of two other forces is required, forming a second couple. Thus in Fig. 94, Pand Q forming a couple, may be balanced by a second couple formed by R and S. The point of application of eitherP or £? may be a fixed pivot or axis. I P Moment of the couple PQ = P(c + b + a) = T moment of RS = Rb. Also, P + R = Q -f S. The forces R and S need not be parallel to P | < and Q, but if not, then their components parallel to PQ are to be taken Instead of the forces themselves. Equilibrium of Forces.— A system of forces applied at points of a solid body will be in equilibrium when they have no tendency to yS produce motion, either of translation or of rota- Fig. 94. tion. The conditions of equilibrium are : i. The algebraic sum of the compo- nents of the forces in the direction of any three rectangular axes must be separately equal to 0. 2. The algebraic sum of the moments of the forces, with respect to any three rectangular axes, must be separately equal to 0. If the forces lie in a plane : 1. The algebraic sum of the components of the forces, in the direction of any two rectangular axes, must be separately equal to 0. 2. The algebraic sum of the moments of the forces, with respect to any point in the plane, must be equal to 0. If a body is restrained by a fixed axis, as in case of a pulley, or wheel and axle, the forces will be in a equilibrium when the algebraic sum of the mo- ments of the forces with respect to the axis is equal to 0. CENTRE OF GRAVITY. The centre of gravity of a body, or of a system of bodies rigidly connected together, is that point about which, if suspended, all the parts will be in equilibrium, that is, there will be no tendency to rotation. It is the point through which passes the resultant of the efforts of gravitation on each of the elementary particles of a body. In bodies of equal heaviness through- out, the centre of gravity is the centre of magnitude. (The centre of magnitude of a figure is a point such that if the figure be divided into equal parts the distance of the centre of magnitude of the whole figure from any given plane is the mean of the distances of the centres . of magnitude of the several equal parts from that plane.) If a body be suspended at its centre of gravity, it will be in equilibrium in all positions. If it be suspended at a point out of its centre of gravity, it will swing into a position such that its centre of gravity is vertically beneath its point of suspension. To find the centre of gravity of any plane figure mechanically, suspend the figure by any point near its edge, and mark on it the direction of a plumb-line hung from that point ; then suspend it from some other point, and again mark the direction of tiie plumb-line in like manner. Then the centre of gravity of the surface will be at the point of intersection of the two marks of the plumb-line. The Centre of Gravity of Regular Figures, whether plane or solid, is the same as their geometrical centre ; for instance, a straight line, MOMENT OF INERTIA. 419 parallelogram, regular polygon, circle, circular ring, prism, cylinder, sphere, spheroid, middle frustums of spheroid, etc. Of a triangle : On a line drawn from any angle to the middle of the op- posite side, at a distance of one third of the line from the side; or at the intersection of such lines drawn from any two angles. Of a trapezium or trapezoid: Draw the two diagonals, dividing it into four triangles. Draw lines joining the centres of gravity of opposite pairs of triangles, and their intersection is the centre of gravity. Of a sector of a circle : On the radius which bisects the arc, ^ — from the centre, c being the chord, r the radius, and I the arc. Of a semicircle: On the middle radius, .4244/- from the centre. Of a quadrant : On the middle radius, .6002r from the centre. Of a segment of a circle ; c 3 -^- \2a from the centre, c = chord, a = area. Of a parabolic surface : In the axis, 3/5 of its length from the vertex. Of a semi-parabola (surface) : 3/5 length of the axis from the vertex, and % of the semi-base from the axis. Of a cone or pyramid : In the axis, Y± of its length from the base. Of a paraboloid : In the axis, % of its length from the vertex. Of a cylinder, or regidar prism ; In the middle point of the axis. Of a frustum of a cone or pyramid : Let a — length of a line drawn from the vertex of the cone when complete to the centre of gravity of the base, and a' that portion of it between the vertex and the top of the frustum; then distance of centre of gravity of the frustum from centre of gravity of its _ a 3a' 3 base-- - 4(a2 + aa / + a , 2) « For two bodies, fixed one at each end of a straight bar, the common centre of gravity is in the bar, at that point which divides the distance between ttieir respective centres of gravity in the inverse ratio of the weights. In this solution the weight of the bar is neglected. But it may be taken as a third body, and allowed for as in the following directions : For more than two bodies connected in one system: Find the common centre of gravity of two of them ; and find the common centre of these two jointly with a third body, and so on to the last body of the group. Another method, by the principle of moments : To find the centre of gravity of a system of bodies, or a body consisting of several parts, whose several centres are known. If the bodies are in a plane, refer their several centres to two rectangular co-ordinate axes. Multiply each weight by its distance from one of the axes, add the products, and divide the sum by the sum of the weights: the result is the distance of the centre of gravity from that axis. Do the same with regard to the other axis. If the bodies are not in a plane, refer them to three planes at right angles to each other, and determine the mean distance of the sum of the weights from each of the three planes. MOMENT OF INERTIA. The moment of inertia of the weight of a body with respect to an axis is the algebraic sum of the products obtained by multiplying the weight of each elementary particle by the square of its distance from the axis. If the moment of inertia with respect to any axis = 1, the weight of any element of the body = v, and its distance from the axis = r, we have / — 2(t«r 2 ). The moment of inertia varies, in the same body, according to the position of the axis. It is the least possible when the axis passes through the centre of gravity. To find the moment of inertia of a body, referred to a given axis, divide the body into small parts of regular figure. Multiply the weight of each part by the square of the distance of its centre of gravity froni the axis. The sum of the products is the moment of inertia. The value of the moment of inertia thus obtained will be more nearly exact, the smaller and more numerous the parts into which the body is divided. Moments op Inertia of Regular Solids.— Rod, or bar, of uniform thick- ness, with respect to an axis perpendicular to the length of the rod, 1=1 TF (lT+ d2 )' (1 > W— weight of rod, 21 = length, d = distance of centre of gravity from axis. Thin circular plate, axis in its I T TTr /r 2 . ,„\ own plane, f T = W (j + d * /! ( 2 > r — radius of plate. 420 MECHANICS. Circular plate,axis perpendicular / T TTr />" 2 , 7 „\ /AV to the plate, \ I = W \T + ^') (3) Circular ring, axis perpendicular I /> 2 + r' 2 V to its own plane, j * = ^ \ — g -f« 2 /' • • • • (4) j- and r' are the exterior and interior radii of the ring. Cylinder, axis perpendicular to) T TTr /r 2 , P , \ .... the axis of the cylinder, \ * = w \l[ + Y ^ '* ' ' ' ' ( ' r — radius of base, 21 — length of the cylinder. By making d — in any of the above formulse we find the moment of inertia for a parallel axis through the centre of gravity. The moment of inertia. 2ww* a ; numerically equals the weight of a body which, if concentrated at the distance unity from the axis of rotation, would require the same work to produce a given increase of angular velocity that the actual body requires. It bears the same relation to angular acceleration which weight does to linear acceleration (Rankine). The term moment of inertia is also used in regard to areas, as the cross-sections of beams under strain. In this case I = 2a?- 2 , in which a is any elementary area, and r its distance from the centre. (See Moment of Inertia, under Strength of Ma- terials, p. 247.) CENTRE AND RADIUS OF GYRATION. The centre of gyration, with reference to an axis, is a point at which, if the entire weight of a body be concentrated, its moment of inertia will re- main unchanged; or, in a revolving bodj% the point in which the whole weight of the body may be conceived to be concentrated, as if a pound of platinum were substituted for a pound of revolving feathers, the angular velocity and the accumulated work remaining: the same. The distance of this point from the axis is the radius of gyration. If W — the weight of a body, I = 2w 2 = its moment of inertia, and k = its radius of gyration, 1 = WW = 2wr 2 ; k = a/ 2^.. The moment of inertia = the weight x the square of the radius of gyration. To fiud the radius of gyration divide the body into a considerable number of equal small parts— the more numerous the more nearly exact is the re- sult, — then take the mean of all the squares of the distances of the parts from the axis of revolution, and find the square root of the mean square. Or, if the moment of inertia is known, divide it by the weight and extract the square root. For radius of gyration of an area, as a cross-section of a beam, divide the moment of inertia of the area by the area and extract the square root. The radius of gyration is the least possible when the axis passes through the centre of gravity. This minimum radius is called the principal radius of gyration. If we denote it by k and any other radius of gyration by k', we have for the five cases given under the head of moment of inertia above the following values : <1) lengt 1 h, aXiS PerPent ° \* = l^/h V= |/f+^. <2) Circular plate, axis ) in its plane, j (3) Circular plate, axis > perpen. to plane, f (4) Circular ring, axis \ perpen. to plane, r /,.« i A-2 _i_ r /2 /: (5) Cylinder, axis per- !'*._■. A 2 . P . ,, _ /r* pen. to length, f f I + j' i/ T " CENTRES OP OSCILLATION AND OF PERCUSSION. 421 Principal Radii of Gyration and Squares of Radii of Gyration. (For radii of gyration of sections of columns, see page 249.) Surface or Solid. Pavallelogram: | axis at its base height h J " mid-height Straight rod : ) • t ■, Rectangular prism: axes 2a, 2b, 2c, referred to axis 2a... Parallelopiped: length I, base 6, axis I at one end, at mid-breadth f Hollow square tube: out. side h, inn'r / length, referred to axis of cyl ) Hollow circ. cylinder, or flat ring:l 1, length; R, r., outer and inner I radii. Axis, 1, longitudinal axis; | 2, diam. at mid-length J Same: very thin, axis its diameter — " radius r; axis, longitud'l axis. . Circumf . of circle, axis its centre " •' " " " diam Sphere: radius r. axis its diam Spheroid : equatorial radius r, re- \ volving polar axis a — j Paraboloid : r = rad. of base, rev. i on axis f Ellipsoid: semi-axes a, b, c; re vol v- ( ing on axis 2a j Spherical shell: radii R, r, revolving (_ on its diam ) Same: very thin, radius r Solid cone: r — rad. of base, rev. on | axis j Square of R. Rad. of Gyration. jf Gyrat ion. .2886/i .5773? .2886/ .577 V& 2 + c 2 .289 V41* + & 2 .289 V/i 2 -f /t' 2 .408/i y& J4 f/i 2 +/i' 2 .289-V / / 2 4-3r 2 .7071r .7071 VR* +- r 2 59 i/ia + 3(JB a 4-r 9 ) .7071?- .6325r .6325r .5773r .4472 4/624 .63254 /R 5 - »;s 6325 f W~^ .8165r .5477r i^/i 2 l/12/i 2 1/12/ 2 (6 2 4- c 2 ) -4- : 4/2 4- fo2 (/I, 2 4- /l'2) h_ 12 /i 2 -*-6 CENTRES OF OSC1L.L.ATION AND OF PERCUSSION. Centre of Oscillation,,— If a body oscillate about a fixed horizontal axis, not passing through its centre of gravity, there is a point in the line drawn from the centre of gravity perpendicular to the axis whose motion is the same as it would be if the whole mass were collected at that point and allowed to vibrate as a pendulum about the fixed axis. This point is called the centre of oscillation. The Radius of Oscillation, or distance of the centre of oscillation from the point of suspension = the square of the radius of gyration h- dis- tance of the centre of gravity from the point of suspension or axis. The centres of oscillation and suspension are convertible. If a straight line, or uniform thin bar or cylinder, be suspended at one end, oscillating about it as an axis, the centre of oscillation is at % the length of 422 MECHANICS. the rod from the axis. If the point of suspension is at ^ the length from the end, the centre of oscillation is also at % the length from the axis, that is, it is at the other end. In both cases the oscillation will be performed in the same time. If the point of suspension is at the centre of gravity, the length of the equivalent simple pendulum is infinite, and therefore the time of vibration is infinite. For a sphere suspended by a cord, r= radius, h = distance of axis of motion from the centre of the sphere, h' — distance of centre of oscillation 2 ?- 2 from centre of the sphere, I = radius of oscillation = 7i + h' = h -\ — — • 5 h If the sphere vibrate about an axis tangent to its surface, It = r, and I = r + 2/5r. If h = lOr, I = lOr + ~ Lengths of the radius of oscillation of a few regular plane figures or thin plates, suspended by the vertex or uppermost point. 1st. Wheu the vibrations are flatwise, or perpendicular to the plane of the figure: In an isosceles triangle the radius of oscillation is equal to % of the height of the triangle. In a circle, % of the diameter. In a parabola, 5/7 of the height. 2d. When the vibrations are edgewise, or in the plane of the figure: In a circle the radius of oscillation is % of the diameter. In a rectangle suspended by one angle, % of the diagonal. In a parabola, suspended by the vertex, 5/7 of the height, plus y§ of the parameter. In a parabola, suspended by the middle of the base, 4/7 of the height plus i^ the parameter. Centre of Percussion.— The centre of percussion of a body oscillat- ing about a fixed axis is the point at which, if a blow is struck by the body, the percussive action is the same as if the whole mass of the body were con- centrated at the point. This point is identical with the centre of oscillation. THE PENDULUM. A body of any form suspended from a fixed axis about which it oscillates by the force of gravity is called a compound pendulum. The ideal body concentrated at the centre of oscillation, suspended from the centre of sus- pension by a string without weight, is called a simple pendulum. This equi- valent simple pendulum has the same weight as the given body, and also the same moment of inertia, referred to an axis passing through the point of suspension, and it oscillates in the same time. The ordinary pendulum of a given length vibrates in equal times when the angle of the vibrations does not exceed 4 or 5 degrees, that is, 2° or 2V£° each side of the vertical. This property of a pendulum is called its isochronism. The time of vibration of a pendulum varies directly as the square root of the length, and inversely as the square root of the acceleration due to grav- ity at the given latitude and elevation above the earth's surface. If T — the time of vibration, I — length of the simple pendulum, g — accel- eration = 32.16, T = t i / -; since n is constant, Tec . At a given loca- y y Vg tion g is constant and Tec Vl. If Z be constant, then for any location 1 . - 27 Tec — -. If Tbe constant, gT 2 = irH; I oc g; g -- Vg the force of gravity at any place may be determined if the length of the simple pendulum, vibrating seconds, at that place is known. At New York this length is 39.1017 inches = 3.2585 ft., whence g = 32.16 ft. At London the length is 39.1393 inches. At the equator 39.0152 or 39.0168 inches, according to different authorities. Time of vibration of a pendulum of a given length at New York : t : Y 39.K _ Vi_ 1017 6.253' t being in seconds and I in inches. Length of a pendulum having a given time of vibration, I — t 2 X 39.1017 inches. VELOCITY, ACCELERATION", FALLING BODIES. 423 The time of vibration of a pendulum may be varied by the addition of a weight at a point above the centre of suspension, which counteracts the lower weight, and lengthens the period of vibration. By varying the height of the upper weight the time is varied. To find the weight of the upper bob of a compound pendulum, vibrating seconds, when the weight of the lower bob, and the distances of the weights from the point of suspension are given: 19.1 + d) + d 2 ' W — the weight of the lower bob, w = the weight of the upper bob; D = the distance of the lower bob and d = the distance of the upper bob from the point of suspension, in inches. Thus, by means of a second bob, short pendulums may be constructed to vibrate as slowly as longer pendulums. By increasing w or d until the lower weight is entirely counterbalanced, the time of vibration may be made infinite. Conical Pendulum.— A weight suspended by a cord and revolving at a uniform speed in the circumference of a circular horizontal plane whose radius is r, the distance of the plane below the point of suspension be- ing h, is held in equilibrium by three forces — the tension in the cord, the cen- trifugal force, which tends to increase the radius r, and the force of gravity acting downward. If v = the velocity in feet per second, the centre of gravity of the weight, as it describes the circumference, g = 32.16, and r and h are taken in feet, the time in seconds of performing one revolution is PI = .8146/2. 4.7T 2 If t = 1 second, h - .8146 foot - 9.775 inches. The principle of the conical pendulum is used in the ordinary fly-ball governor for steam-engines. (See Governors.) CENTRIFUGAL. FORCE. A body revolving in a curved path of radius = E in feet exerts a force, called centrifugal force, F, upon the arm or cord which restrains it from moving in a straight line, or "flying off at a tangent." If W = weight of the body in pounds, N = number of revolutions per minute, V= linear velocity of the centre of gravity of the body, in feet per second, g = 32.16, then 2nRN „ Wv* W\P WWRN* WRN* niWAmvirvTm itJ V = -60-; F = ~W = 22AQR = 36000 = ^933~ = • 00034 1°^^ 2 **• l£n = number of revolutions per second, F = 1. 2276 WRn*. (For centrifugal force in fly-wheels, see Fly-wheels.) VELOCITY, ACCELERATION, FALLING BODIES. Velocity is the rate of motion, or the distance passed over by a body in a given time. If s = space in feet passed over in t seconds, and v = velocity in feet per second, if the velocity is uniform, v = -; s = vt; t = -. t v If the velocity varies uniformly, the mean velocity v = 1 j~ 2 , in which ^i is the velocity at the beginning and v? the velocity at the end of the time t. » = *Jp* 2 = a*; i? 2 — a* = 0; * = - 2 -. Combining (1) and (2), we have If Vl = 0, 8 = ^t. Retarded Motion.— If the body start with a velocity v x and come to rest, v 2 = 0; then s = ~t. In any case, if the change in velocity is v, v. i; 2 a JO '=«* S= W S =^ For a body starting from or ending at rest, we have the equations v + at2 o » v = at; s = -c; s = — ; v 1 — 2as. Falling Bodies.- In the case of falling bodies the acceleration due to gravity is 32.16 feet per second in one second, = g. Then if v = velocity acquired' at the end of t seconds, or final velocity, and h - height or space in feet passed over in the same time, . . _ 2h v - gt = 32.16* = V2gh - 8.02 \/h = -j ', 2 2g 64.32 2' ~g "32.16 ~y gr "4.01 ~ v ' w = space fallen through in the Tth second = g(T — y£). Value of g.— The value of g increases with the latitude, and decreases with the elevation. At the latitude of Philadelphia, 40°, its value is 32.16. At the sea-level, Everett gives g = 32.173 - .082 cos 2 lat. -.000003 height in feet. Values of \/2g, calculated by an equation given by C. S. Pierce, are given in a table in Smith's Hydraulics, from which we take the following : Latitude...^.. 0° 10° 20° 30° 40° 50° 60° Value of V2g.. &0112 8.0118 8.0137 8.0165 8.0199 8.0235 8.0269 The value of V2g decreases about .0004 for every 1000 feet increase in ele- vation above the sea-level. For all ordinary calculations for the United States, g is generally taken at 32.16, and V2g at 8.02. In England g = 32.2, V%g = 8.025. Practical limit- ing values of g for the United States, according to Pierce, are : Latitude 49° at sea-level g — 32.186 25° 10,000 feet above the sea g - 32.089 From the above formula for falling bodies we obtain the following : During the first second the body starting from a state of rest (resistance of the air neglected) falls g -=- 2 = 16.08 feet ; the acquired velocity is g — at 2 32.16 ft. per sec. ; the distance fallen in two seconds is h = -^ = 16.08 X 4 .= 64.32 ft. ; and the acquired velocity is v = gt = 64.32 ft. The acceleration, or increase of velocity in each second, is constant, aud is 32.16 ft. per sec. Solv- ing the equations for difterent times, we find for Seconds,* 1 2 3 4 5 6 Acceleration, g 32.16 X. 1 1 1 1 1 1 Velocity acquired at end of time, v 32.16x1 2 3 4 .5 6 Height of fall in each second, u... ... - 1 — X 1 3 5 7 9 11 Total height of fall, h ^^ X 1 4 9 16 25 36 / 2h VELOCITY, ACCELERATION, FALLIKG BODIES. 425 Fig. 95 represents graphically the velocity, space, etc., of a body falling for six seconds. The vertical line at the left is ■ the time in seconds, the horizontal lines represent one half the acquired velocities at the end of each second. The area of the small triangle at the top represents the height fallen through in the first second = y 2 g = 16.08 feet, and each of the other triangles is an equal space. The number of triangles between each pair of horizontal lines represents the height of fall in each second, and the number of triangles between any horizontal line and the top is the total height fallen during 16 the time. The figures under h, tt, and v adjoining the cut are to be multiplied by 16.08 to obtain the actual velocities and 25 heights for the given times. Angular and Linear Velocity of a Turning Body.— Let r = radius of a turning body in feet, n = number of revo- lutions per minute, v = linear velocity of a point on the circumference in feet per second, per minute. bO Angular velocity is a term used to denote the angle through which any radius of a body turns in a second, or the rate at which any point in it having a radius equal to unit3 r is moving, expressed in feet per second. The unit of angular velocity is the angle which at a distance = radius from the centre is subtended by an arc equal to the radius. This unit angle = ; — Fig. 95. : velocity in feet degrees = 2n X 57.3° = 360°, v 27m velocity, v = Ar, A = - — -^-. or the circumference. If A = angular Height Corresponding to a Given Acquired Velocity. >s >> j£ £» >, >> £> & & '5 £ £ ^3 M o &fi o be o to c bn _o b£ '53 *® "3 > ffi > K i> K > X > w > E feet feet. feet feet. feet feet. feet feet. feet feet. feet feet. p. sec. p. sec. p. sec. p. sec. p. sec. p. sec. .25 .0010 13 2.62 34 17.9 55 47.0 76 89.8 97 146 .50 .0039 14 3.04 35 19.0 56 48.8 77 92.2 98 149 .75 .0087 15 3.49 36 20.1 57 50.5 78 94.6 99 152 1.00 .016 16 3.98 37 21.3 58 52.3 79 97.0 100 155 1.25 .024 17 4.49 38 22.4 59 54.1 80 99.5 105 171 1.50 .035 18 5.03 39 23.6 60 56.0 81 102.0 110 188 1.75 .048 19 5.61 40 24.9 61 57.9 82 104.5 115 205 2 .062 20 6.22 41 26.1 62 59.8 83 107.1 120 224 2.5 .097 21 6.85 42 27.4 63 61.7 84 109.7 130 263 3 .140 22 7.52 43 28.7 64 63.7 85 112.3 140 304 3.5 .190 23 8.21 44 30.1 65 65.7 S6 115.0 150 350 4 .248 24 8.94 45 31.4 66 67.7 87 117.7 175 476 4.5 .314 25 9.71 46 32.9 67 69.8 88 120.4 200 622 5 .388 26 10.5 47 34.3 68 71 .9 89 123.2 300 1399 6 .559 27 11.3 48 35.8 69 74.0 90 125.9 400 2488 7 .761 28 12.2 49 37.3 70 76.2 91 128.7 500 3887 8 .994 29 13.1 50 38.9 71 78.4 92 131.6 600 5597 9 1.26 30 14.0 51 40.4 72 80.6 93 134.5 700 7618 10 1.55 31 14.9 52 42.0 73 82.9 94 137.4 800 9952 11 1.88 32 15.9 53 43.7 74 85.1 95 140.3 900 12593 12 2.24 33 16.9 54 45.3 75 87.5 96 143.3 1000 15547 426 MECHANICS. Falling Bodies Velocity Acquired by a Body Falling: a Given Height. s >> 5 £> *s J? 4* >> s i? A >> M 'o o .6X1 '5 o .SP 'o o •Sf o .y o tj) o w i> W 0) > w > M > B > w "3 > feet. feet feet. feet feet. feet feet. feet feet. feet feet. feet p. sec. p. sec. p. sec. p. sec. p. sec. p. sec. .005 .57 .39 5.01 1 20 8.79 5. 17.9 23. 38.5 72 68.1 .010 .80 .40 5.07 1 .22 8.87 .2 18.3 .5 38.9 73 68.5 .015 .98 .41 5.14 1.24 8.94 .4 18.7 24. 39.3 74 69.0 .020 1.13 .42 5.20 1.26 9.01 .6 19.0 .5 39.7 75 69.5 .025 1.27 .43 5.26 1.28 9.08 .8 19.3 25 40.1 76 69.9 .030 1.39 .44 5.32 1.30 9.15 6. 19.7 26 40.9 77 70.4 .035 1.50 .45 5.38 1.32 9.21 .2 20.0 27 41.7 78 70.9 .040 1.60 .46 5.44 1.34 9.29 .4 20.3 28 42.5 79 '71.3 .045 1.70 .47 5.50 1.36 9.36 .6 20.6 29 43.2 80 71.8 .050 1.79 .48 5.56 1.38 9.43 .8 20.9 30 43.9 81 72.2 .055 1.88 .49 5.61 1.40 9.49 7. 21.2 31 44.7 82 72.6 .060 1.97 .50 5.67 1.42 9.57 .2 21.5 32 45.4 83 73.1 .065 2.04 .51 5.73 1.44 9 62 .4 21.8 33 46.1 84 73.5 .070 2.12 .52 5.78 1.46 9.70 .6 22.1 34 46.8 85 74.0 .075 2.20 .53 5.84 1.48 9.77 .8 22.4 35 47.4 86 74.4 .080 2.27 .54 5.90 1.50 9.82 8. 22.7 36 48.1 87 74.8 .085 2.34 .55 5.95 1.52 9.90 .2 23.0 37 48.8 88 75.3 .090 2.41 .56 6.00 1.54 9.96 .4 23.3 38 49.4 89 75.7 .095 2.47 .57 6.06 1.56 10.0 .6 23.5 39 50.1 90 76.1 .100 2.54 .58 6.11 1.58 10.1 .8 23.8 40 50.7 91 76.5 .105 2.60 .59 6.16 1.60 10.2 9. 24.1 41 51.4 92 76.9 .110 2.66 .60 6.21 1.65 10.3 .2 24.3 42 52.0 93 77.4 .115 2.72 .62 6.32 1.70 10.5 .4 24.6 43 52.6 94 77.8 .120 2.78 .64 6.42 1.75 10.6 .6 24.8 44 53.2 95 78.2 .125 2.84 .66 6.52 1.80 10.8 .8 25.1 45 53.8 96 78.6 .130 2.89 .68 6.61 1.90 11.1 10. 25.4 46 54.4 97 79.0 .14 3.00 .70 6.71 2. 11.4 .5 26.0 47 55.0 98 79.4 .15 3.11 .72 6.81 2.1 11.7 11. 26.6 48 55.6 99 79.8 .16 3.21 !74 6.90 2.2 11.9 .5 27.2 49 56.1 100 80.2 .17 3.31 .76 6.99 2.3 12.2 12. 27.8 50 56.7 125 89.7 .18 3.40 .78 7.09 2.4 12.4 .5 28.4 51 57.3 150 98.3 .19 3.50 .80 7.18 2.5 12.6 13. 28.9 52 57.8 175 106 .20 3.59 .82 7.26 2.6 12.9 .5 29.5 53 58.4 200 114 .21 3.68 .84 7.35 2.7 13.2 14. 30.0 54 59.0 225 120 .22 3.76 .86 7.44 2.8 13.4 .5 30.5 55 59.5 250 126 .23 3.85 .88 7.53 2.9 13.7 15. 81.1 56 60.0 275 133 .24 3.93 .90 7.61 3. 13.9 .5 31.6 57 60.6 300 139 .25 4.01 .92 7.69 3.1 14.1 16. 32.1 58 61.1 350 150 .26 4.09 .94 7.78 3-2 14.3 .5 32.6 59 61.6 400 160 .27 4.17 .96 7.86 3.3 14.5 17. S3.1 60 62.1 450 170 .28 4.25 .98 7.94 3.4 14.8 .5 33.6 61 62.7 500 179 .29 4.32 1.00 8.02 3.5 15.0 18. 34.0 62 63.2 550 188 .30 4.39 1.02 8.10 3.6 15.2 .5 31.5 63 63.7 600 197 .31 4.47 1.04 8.18 37 15.4 19. 35.0 64 64.2 700 212 .32 4.54 1.06 8.26 3-8 15.6 .5 35.4 65 64.7 800 227 .33 4.61 1.08 8.34 3-9 15.8 20. 35.9 66 65.2 900 241 .34 4.68 1.10 8.41 4- 16.0 .5 36.3 67 65.7 1000 254 .35 4.74 1.12 8.49 .2 16.4 81. 36.8 68 66.1 2000 359 .36 4.81 1.14 8.57 .4 16.8 .5 37.2 69 66.6 3000 439 .37 4.88 1.16 8.64 .6 17.2 22. 37.6 70 67.1 4000 507 .38 4.94 1.18 8.72 .8 17.6 .5 38.1 71 67.6 5000 567 Parallelogram of Velocities.— The principle of the composition and resolution of forces may also be applied to velocities or to distances moved in given intervals of time. Referring to Fig. 88, page 416, if a body at O has a force applied to it which acting alone would give it a velocity represented by OQ per second, and at the same time it is acted on by VELOCITY, ACCELERATION, FALLING BODIES. 427 another force which acting alone would give it a velocity OP per second, the result of the two forces acting together for one second will carry it to R, OR being the diagonal of the parallelogram of OQ and OP, and the resultant velocity. If the two component velocities are uniform, the result- ant will be uniform and the line OR will be a straight line; but if either velocity is a varying one, the line will be a curve. Fig. 96 shows the resultant velocities, also the path traversed by a body acted on by two forces, one of which would carry it at; a uniform velocity over the intervals 1, 2, 3, B, and the other of which would carry it by an accelerated mo- tion over the intervals a, b, c, D in the same times. At the end of the respective inter- vals the body will be found at C,, C. 2 , C 3 , C, and the mean velocity during each interval is represented by the distances between these' points. Such a curved path is trav- ersed by a shot, the impelling force from the gun giving it a uniform velocity in the direction the gun is aimed, and gravity giv- ing it an accelerated velocity downward. The path of a projectile is a parabola. The distance it will travel is greatest when its initial direction is at an angle 45° above the horizontal. Mass— Force of Acceleration.-— The mass of a body, or the quantity of matter it contains, is a constant quantity, while the weight varies according to the variation in the force of gravity at different places. If g — the acceler- ation due to gravity, and w — weight, then the mass m — ~,io = mg. Weight here means the resultant of the force of gravity on the particles of a body, such as may be measured by a spriug-balance, or by the extension or deflection of a rod of metalloaded with the given weight. Force has been defined as that which causes, or tends to cause, or to destroy, motion. It may also be defined (Kennedy's Mechanics of Ma- chinery) as the cause of acceleration; and the unit of force as the force required to produce unit acceleration in a unit of free mass. Force equals the product of the mass by the acceleration, or f "'= ma. Also, if v = the velocity acquired in the time t, ft = mv; f — mv -*- t; the acceleration being uniform. The force required to produce an acceleration of g (that is, 32.16 ft. per sec.) in one second is / = mg = —g = w, or the weight of the body. Also, / = ma = m 2 1 , in which r 2 is the velocity at the end, and v x the W (Vn — V-,) W velocity at the beginning of the time t, and/ = mg = = —a; — = -; or, the force required to give any acceleration to a body is to the weight of the body as that acceleration is to the acceleration produced by gravity. (The weight iv is the weight where g is measured.) Example.— Tension in a cord lifting a weight. A weight of 100 lbs. is lifted vertically by a cord a distance of 80 feet in 4 seconds, the velocity uniformly increasing from to the end of the time. What tension must be maintained in the cord? Mean velocity = v = 20 ft. per sec; final velocity = u„ = 2v - 40; accele-ation a — ^ = — = 10. Force / — ma = — = -oTTr x t 4 g o4.1\j 10 = 31.1 lbs. This is the force required to produce the acceleration only; to it must be added the force required to lift the weight without accelera- tion, or 100 lbs., making a total of 131.1 lbs. The Resistance to Acceleration is the same as the force required to pro- Formulae for Accelerated Motion.— For cases of uniformly accelerated motion other than those of falling bodies, we have the formulae already given, / - - a, = H~* If tne body starts from rest, v x = 0, v 2 9 9 *■ 428 MECHANICS. = v, and/= — t, fgt -■ wv. We also have s = --. Transforming and sub- s' t -A stituting for g its value 32.16, we obtain / = wv 2 64.32s ~ wv 32.16* "" IVS 16708^ wv 2 _ 16.08/f 2 _ vt . ' 64.32/ " - wv - _i_ i/— 32.16/ ~ 4.01 \ f (^2 _ v 2\ - „ ) . (See also Work of Acceleration, under Work.) Motion on Inclined Planes.— The velocity acquired by a body descending an inclined plane by the force of gravity (friction neglected) is equal to that acquired by a body falling freely from the height of the plane. The times of descent down different inclined planes of the same height vary as the length of the planes. The rules for uniformly accelerated motion apply to inclined planes. If a is the angle of the plane with the horizontal, sin a = the ratio of the height to the length = - , and the constant accelerating force is g sin a. The final velocity at the end of t seconds is v — gt sin a. The distance passed over in t seconds is I = y% gt 2 sin a. The time of descent is I 4.01 Yh MOMENTUM, VIS-VIVA. Momentum, or quantity of motion in a body, is the product of the mass by the velocity at any instant = mv — — v. Since the moving force = product of mass by acceleration, / = ma; and if the velocity acquired in t seconds = v, or a = -, / = -r- ; ft = mv; that is, the product of a constant force into the time in which it acts equals numer- ically the momentum. Since ft = mv, if t — 1 second mv — /. whence momentum might be de- fined as numerically equivalent to the number of pounds of force that will stop a moving body in 1 second, or the number of pounds of force which acting during 1 second will give it the given velocity. Vis-viva, or living force, is a term used by early writers on Mechanics to denote the energy stored in a moving body. Some defined it as the pro- duct of the mass into the square of the velocity, mv 2 , — —v 2 others as one half of this quantity or y 2 mv 2 , or the same as what is now known as energy. The term is now practically obsolete, its place being taken by the word energy. WORK, ENERGY, POWER. Work is the overcoming of resistance through a. certain distance. It is measured by the product of the resistance into the space through which it is overcome. It is also measured by the product of the moving force into the distance through which the force acts in overcoming the resistance. Thus in lifting a body from the earth against the attraction of gravity, the resistance is the weight of the body, and the product of this weight into the height the body is lifted is the work done. The Unit of Work, in British measures, is the foot-pound, or the amount of work done in overcoming a pressure or weight equal to one pound through one foot of space. WORK, ENERGY, POWER. 429 The work performed by a piston in driving a fluid before it, or by a fluid in driving a piston before it, may be expressed in either of the following ways : Resistance X distance traversed = intensity of pressure x area x distance traversed ; =■ intensity of pressure X volume traversed. The work performed in lifting a body is the product of the weight of the body into the height through which its centre of gravity is lifted. If a machine lifts the centres of gravity of several bodies at once to heights either the same or different, the whole quantity of work performed in so doing is the sum of the several products of the weights and heights ; but that quantity can also be computed by multiplying the sum of all the weights into the height through which their common centre of gravity is lifted. (Rankine.) Power is the rate at which work is done, and is expressed by the quo- tient of the work divided by the time in which it is done, or by units of work per second, per minute, etc., as foot-pounds per second. The most common unit of power is the horse-jjoiver, established by James Watt as the power of a strong London draught-horse to do work during a short interval, and used by him to measure the power of his steam-engines. This unit is 33,000 foot- pounds per minute = 550 foot-pounds per second = 1,980,000 foot-pounds per hour. Expressions for Force, "Work, Power, etc. The fundamental conceptions in Dynamics are : Force, Time, Space, represented by the letters F, T, S. Velocity = space divided by time, V = — , if Fbe uniform. Work = product of force into space = JTS = W = FVT. (F uniform.) FS Power = rate of work = work divided by time = — - = P = product of force into velocity = FV. Power exerted for a certain time produces work; PT = FS = FVT = W. Effort is a name applied to a force which acts on a body in the direction of its motion. Resistance is that which is opposed to a moving force. It is equal and opposite force. Horse-power Hours, an expression for work measured as the product of a power into the time during which it acts = PT. Sometimes it is the summation of a variable power for a given time, or the average power multiplied by the time. Energy, or stored work, is the capacity for performing work. It is measured by the same unit as work, that is, in foot-pounds. It may be either -potential, as in the case of a body of water stored in a reservoir, capable of doing work by means of a water-wheel, or actual, sometimes called kinetic, which is the energy of a moving body. Potential energy is measured by the product of the weight of the stored body into the distance through which it is capable of acting, or by the product of the pressure it exerts into the distance through which that pressure is capable of acting. Potential energy may also exist as stored heat, or as stored chemical energy, as in fuel, gunpowder, etc., or as electrical energy, the measure of these energies being the amount of work that they are capable of performing. Actual energy of a moving body is the work which it is capable of performing against a retarding resistance before being brought to rest, and is equal to the work which must be done upon it to bring it from a state of rest to its actual velocity. The measure of actual energy is the product of the weight of the body into the height from which it must fall to acquire its actual velocity. If v = the velocitv in feet per second, according to the principle of falling bodies, v i h, the height due to the velocity = — , and if w = the weight, the energy = the mass into the square of the velocity = ^mv 2 . Since energy is the capacity for performing work, the units of work and energy are equivalent, or FS = l&nv 2 = — = ivh. Energy exerted = work done. 430 MECHANICS. The actual energy of a rotating body whose angular velocity is A and moment of inertia 2wr 2 = / is -■— , that is, the product of the moment of 2g inertia into the height due to the velocity, A, of a point whose distance from the axis of rotation is unity; or it is equal to — — , in which w is the weight of the bodv and v is the velocity of the centre of gyration. Work of Acceleration. -The work done in giving acceleration to a body is equal to the product of the force producing the acceleration, or of the resistance to acceleration, into the distance moved in a given time. This force, as already stated equals the product of the mass into the acceleration, or f = ma = ^-^ — - 1 . If the distance traversed in the time t = s, then 9 t W Vn — Vj work — fs = 7 s. g . t Example.— What work is required to move a body weighing 100 lbs. hori- zontally a distance of 80 ft. in 4 seconds, the velocity uniformly increasing, friction neglected ? Mean velocity v = 20 ft. per second; final velocity = Vn — 2v — 40; initial , . , . Vn - v, 40 An £ io 100 velocity Vi = 0; acceleration, a = -^— = — = 10; force = —a = ,— —^ x 10 = 31.1 lbs. ; distance 80 ft. ; work = fs = 31 .1x80 = 2488 foot-pounds." The energy stored in the body moving at the final velocity of 40 ft. per second is „ 1 w „ 100 X 402 y 2 mv-> = - - g v* = 2 x 3216 = 2488 foot-pounds, which equals the work of acceleration, IV Vn 10 Vn Vn, 1 io If a body of the weight W falls from a height H, the work of acceleration is simply WH, or the same as the work required to raise the body to the same height. Work of Accelerated Rotation.— Let A = angular velocity of a solid body rotating about an axis, that is, the velocity of a particle whose radius is unity. Then the velocity of a particle whose radius is r is v — Ar. If the angular velocity is accelerated from A] to^4 2 » the increase of the velocity of the particle is Vn — v 1 = r(A 1 - An), and the work of accelerating it is w v.;? - v^ _ wr* A^-AJ 9 X 2 ~ g 2 ' in which w is the weight of the particle. The work of acceleration of the whole body is ^l. \~ *9 - X 2wr 2 . The term 2?cr 2 is the moment of inertia of the bodv. " Force of the Blow " of a Steam Hammer or Other Fall- ing Weight.— The question is often asked: "With what force does a falling hammer strike?" The question cannot be answered directly, and it is based upon a misconception or ignorance of fundamental mechanical laws. The energy, or capacity of doing work, of a body raised to a given height and let fall cannot be expressed in pounds, simply, but only in foot- pounds, which is the product of the weight into the height through which it fails, or the product of its weight -t- 64.32 into the square of the velocity, in feet per second, which it acquires after falling through the given height. If F = weight of the body, M its mass, g the acceleration due to gravity, S the height of fall, and v the velocity at the end of the fall, the energy in the body just before striking, is FS = i^M-u 2 = Wv* -f- 2g = Wv* -*- 64.32, which is the general equation of energy of a moving body. Just as the energy of the body is a product of a force into a distance, so the work it does when it strikes is not the manifestation of a force, which can be ex- pressed simply in pounds, but it is the overcoming of a resistance through a certain distance, which is expressed as the product of the average resist- WORK, EtfERGY, POWER. 431 ance into the distance through which it is exerted. If a hammer weighing 100 lbs. falls 10 ft., its energy is 1000 foot-pounds. Before being brought to rest it must do 1000 foot-pounds of work against one or more resistances. These are of various kinds, such as that due to motion imparted to the body struck, penetration against friction, or against resistance to shearing or other deformation, and crushing and heating of both the falling body and the body struck. The distance through which these resisting forces act is gen- erally indeterminate, and therefore the average of the resisting forces, which themselves generally vary with the distance, is also indeterminate. Impact of Bodies.— If two inelastic bodies collide, they will move on together as one mass, with a common velocity. The momentum of the com- bined mass is equal to the sum of the momenta of the two bodies before im- pact. If m x and m a are the masses of the two bodies and v x and v 2 their re- spective velocities before impact, and v their common velocity after impact, (" l i + vi a )v = m^Vx X m 2 v 2 , _ ffli^i -f" "*2 V 2 * - 2 , or, the velocity ..„i + m 2 of two inelastic bodies after impact is equal to the algebraic sum of their momenta before impact, divided by the sum of their masses. If two inelastic bodies of equal momenta impinge directly upon one an- other from opposite directions they will be brought to rest. Impact of Inelastic Bodies Causes a JLoss of Energy, and this loss is equal to the sum of the energies due to the velocities lost and gained by the bodies, respectively. \bm x v x * + ^m 2 r 2 2 - %{m x -f m 2 )u 2 = V z m x {v x - v) 2 -f- i^m 2 (t; 2 - u) 2 . In which v x — vis the velocity lost by m 1 and v - v 2 the velocity gained by m 2 . Example— Let m, — 10, m 2 = 8, v x = 12, v 2 — 15. If the bodies collide they will come to rest, for v = = 0. The enei'gy loss is 3^10 X 144 + y 2 8 X 225 - ^ 18 X = ^10(12 - 0) 2 -f 3^8(15 - 0)« = 1620 ft. lbs. What becomes of the energy lost ? Ans. It is used doing internal work on the bodies themselves, changing their shape and heating them. For imperfectly elastic bodies, let e = the elasticity, that is, the ratio which the force of restitution, or the internal force tending to restore the shape of a body after it has been compressed, bears to the force of compres- sion; and let vi x and m 2 be the masses, v x and v 2 their velocities before im- pact, and v x 'v 2 ' their velocities after impact: then , _ niiVi -4- m 2 v 2 _ m 2 e{v x — v 2 ) _ 1 — Hij -f m 2 vi x + m 2 ' , m x v x + m 2 v 2 , m x e(v x — v 2 ) Vn' = p 1 . m 1 -\-m 2 m x -\-m 2 If the bodies are perfectly elastic, their relative velocities before and after impact are the same. That is : v x ' — v 2 ' = v 2 — v x . In the impact of bodies, the sum of "their momenta after impact is the same as the sum of their momenta before impact. m x v x ' -f m 2 v 2 ' — m x v x -f m 2 v 2 . For demonstration of these and other laws of impact, see Smith's Me- chanics; also, Weisbach's Mechanics. Energy of Recoil of Guns.— (Eng'g, Jan. 25, 1884, p. 72.) Let W — the weight of the gun and carriage; V = the maximum velocity of recoil; w = the weight of the projectile; v — the muzzle velocity of the projectile. Then, since the momentum of the gun and carriage is equal to the momen- tum of the projectile, we have WV = ivv, or V = wv -v- W. * The statement by Prof. W. D. Marks, in NystronTs Mechanics, 20th edi- tion, p. 454, that this formula is in error is itself erroneous. 432 MECHANICS. Taking the case of a 10-inch gun firing a 400-lb. projectile with a muzzle velocity of 1400 feet per second, the weight of the gun and carriage being 22 tons = 49,280 lbs., we find the velocity of recoil — Tr 1400X400 V= — g — =11 feet per second. Now the energy of a body in motion is WV 2 -f- 2g. 49 280 X ll 2 Therefore the energy of recoil = — | ■ = 92,593 foot-pounds. 400 X 1400 2 The energy of the projectile is = 12,173,913 foot-pounds. Conservation of Energy.— No form of energy can ever be pro- duced except by the expenditure of some other form, nor annihilated ex- cept by being reproduced in another form. Consequently the sum total of energy in the universe, like the sum total of matter, must always remain the same. (S. Newcomb.) Energy can never be destroyed or lost; it can be transformed, can be transferred from one body to another, but no matter what transformations are undergone, when the total effects of the exertion of a given amount of energy are summed up the result will be exactly equal to the amount originally expended from the source. This law is called the Conservation of Energy. (Cotterill and Slade.) A heavy body sustained at an elevated position has potential energy. When it falls, just before it reaches the earth's surface it has actual or kinetic energy, due to its velocity. When it strikes it may penetrate the earth a certain distance or may be crushed. In either case friction results by which the energy is converted into heat, which is gradually radiated into the earth or into the atmosphere, or both. Mechanical energy and heat are mutually convertible. Electric energy is also convertible into heat or mechanical energy, and either kind of energy may be converted into the other. Sources of Energy.— The principal sources of energy on the earth's surface are the muscular energy of men and animals, the energy of the wind, of flowing water, and of fuel. These sources derive their energy from the rays of the sun. Under the influence of the sun's rays vegetation grows and wood is formed. The wood may be used as fuel under a steam boiler, its carbon being burned to carbonic acid. Three tenths of its heat energy escapes in the chimney and by radiation, and seven tenths appears as potential energy in the steam. In the steam-engine, of this seven tenths six parts are dissipated in heating the condensing water and are wasted ; the remaining one tenth of the original heat energy of the wood is con- verted into mechanical work in the steam-engine, which may be used to drive machinery. This work is finally, by friction of various kinds, or pos- sibly after transformation into electric currents, transformed into heat, which is radiated into the atmosphere, increasing its temperature. Thus all the potential heat energy of the wood is, after various transformations, converted into heat, which, mingling with the store of heat in the atmos- phere, apparently is lost. But the carbonic acid generated by the combus- tion of the wood is, again, under the influence of the sun's rays, absorbed by vegetation, and more wood may thus be formed having potential energy equal to the original. Perpetual Motion.— The law of the conservation of energs% than which no law of mechanics is more firmly established, is an absolute barrier to all schemes for obtaining by mechanical means what is called " perpetual motion," or a machine which will do an amount of work greater than the equivalent of the energy, whether of heat, of chemical combination, of elec- tricity, or mechanical energy, that is put into it. Such a result would be the creation of an additional store of energy in the universe, which is not possible by any human agency. The Efficiency of a Machine is a fraction expressing the ratio of the useful work to the whole work performed, which is equal to the energy expended. The limit to the efficiency of a machine is unity, denoting the efficiency of a perfect machine in which no work is lost. The difference between the energy expended and the useful work done, or the loss, is usually expended either in overcoming friction or in doing work on bodies surrounding the machine from which no useful work is received. Thus in an engine propelling a vessel part of the energy exerted in the cylinder AHIMAL POWER. 433 does the useful work of giving motion to the vessel, and the remainder is spent in overcoming the friction of the machinery and in making currents and eddies in the surrounding water. ANIMAL, POWER. Work of a Man against Known Resistances. (Rankine.) Kind of Exertion. R, lbs. ft. per sec. T" 3600 (hours per day). RV, ft.-lbs. per sec. RVT, ft.-lbs. per day. Raising his own weight up stair or ladder Hauling up weights with rope, and lowering the rope un loaded Lifting weights by hand Carrying weights up-stairs and returning unloaded Shovelling up earth tc height of 5 ft. 3 in Wheeling earth in barrow up slope of 1 in 12, % horiz. veloc. 0.9 ft. per sec. and re- turning unloaded Pushing or pulling horizon- tally (capstan or oar) 8. Turning a crank or winch . , Working pump . . ). Hammering .... 143 40 44 143 6 132 26.5 12.5 18.0 20.0 13.2 15 0.75 0.55 0.13 1.3 2.0 5.0 2.5 14.4 2.5 72.5 30 24.2 18.5 7.8 53 62.5 2,088,000 648,000 522,720 280,800 356,400 1,526,400 2min. 10 1,296,000 1,188,000 480,000 Explanation.— R, resistance; V, effective velocity = distance through which R is overcome -h total time occupied, including the time of moving unloaded, if any; T", time of working, in seconds per day; T" -i- 3600, same' time, in hours per day; RV, effective power, in foot-pounds per second; RVT, daily work. Performance of a Man in Transporting Loads Horizontally. (Rankin e.) Kind of Exertion. 11. Walking unloaded, transport- ing his own weight 12. Wheeling load L in 2-whld. barrow, return unloaded. . 13. Ditto in 1-wh. barrow, ditto.. 14. Travelling with burden 15. Carrying burden, returning unloaded 16. Carrying burden, for 30 sec- onds only T LV, L, lbs. V, ft. -sec. 3600 lbs. (hours per con- veyed day). 1 foot. 140 5 10 700 224 1% 10 373 132 m 10 220 90 %H 7 225 140 Ws 6 233 (252 -h26 11.7 1474.2 1 o 23.1 LVT, lbs. con- veyed 1 foot. 25,200,000 428,000 920,000 670,000 5,032,800 Explanation.— L, load; V, effective velocity, computed as before; T'', time of working, in seconds per day; T" -i- 3600, same time in hours per day; LV, transport per second, in lbs. conveyed one foot; LVT, daily transport. 434 MECHANICS. In the first line only of each of the two tables above is the weight of the man taken into account in computing the work done. Clark says that the average net daily work of an ordinary laborer at a pump, a winch, or a crane may be taken at 3300 foot-pounds per minute, or one- tenth of a horse-power, for 8 hours a day; but for shorter periods from four to five times this rate may be exerted. Mr. Glynn says that a man may exert a force of 25 lbs. at the handle of a crane for short periods; but that for continuous work a force of 15 lbs. is all that should be assumed, moving through 220 feet per minute. Man-wheel.— Fig. 97 is a sketch of a very efficient man-power hoist- ing-machine which the author saw in Berne, Switzerland, in 1889. The face of the wheel was wide enough for three men to walk abreast, so that nine men could work in it at one time. Work of a Horse against a Known Resistance. (Rankine.) Kind of Exertion. R. V. T. 3600 RV. RVT. 1. Cantering and trotting, draw- ing a light railway carriage (thoroughbred) 2. Horse drawing cart or boat, walking (draught-horse) 3. Horse drawing a gin or mill, 1 min.22^ < mean 30^| ( max. 50 120 100 66 1 14% 3.6 3.0 6.5 4. 8 8 447^ 432 300 429 6,444,000 12,441,600 8,640,000 6,950,000 4. Ditto, trotting Explanation.— R, resistance, in lbs.; V, velocity, in feet per second; T" -*- 3600, hours work per day; RV, work per second; RVT, work per day. The average power of a draught-horse, as given in line 2 of the above table, being 432 foot-pounds per second, is 432/550 = 0.785 of the conventional value assigned by Watt to the ordinary unit of the rate of work of prime movers. It is the mean of several results of experiments, and may be considered the average of ordinary performance under favorable circumstances. Performance of a Horse in Transporting Loads Horizontally. (Rankine.) Kind of Exertion. L. V. T. LV. LVT. 5. Walking with cart, always 1500 750 1500 270 180 3.6 7.2 2.0 3.6 7.2 10 4^ 10 10 5400 5400 3000 972 1296 194,400,000 6. Trotting, ditto 7. Walking with cart, going load- ed, returning empty; V, 87,480,000 108,000,000 34,992,000 32,659,200 8. Carrying burden, walking 9. Ditto, trotting Explanation.— L, load in lbs.; V, velocity in feet per second; T-^3600, working hours per day; LV, transport per second; LVT, transport per day. This table has reference to conveyance on common roads only, and those evidently in bad order as respects the resistance to traction upon them. Horse Gin.— In this machine a horse works less advantageously than in drawing a carriage along a straight track. In order that the best ELEMENTS OF MACHINES. 435 possible results may be realized with a horse-gin, the diameter of the cir- cular track in which the horse walks should not be less than about forty feet. Oxen, Mules, Asses.— Authorities differ considerably as to the power of these animals. The following may he taken as an approximative com- parison between them and draught-horses (Rankine): Ox. — Load, the same as that of average draught-horse; best velocity and work, two thirds of horse. Mule. — Load, one half of that of average draught-horse; best velocity, the same with horse: work one half. Ass.— Load, one quarter that of average draught-horse; best velocity the same; work one quarter. Reduction of Draught of Horses by Increase of Grade of Roads. (Engineering Record, Prize Essays on Roads, 1892.)— Experi- ments on English roads by Gay frier & Parnell: Calling load that can be drawn on a level .100: On a rise of 1 in 100. 1 in 50. 1 in 40. 1 in 30. 1 in 26. 1 in 20. 1 in 10. A horse can draw only 90. 81. 72. 64. 54. 40. 25. The Resistance of Carriages on Roads is (according to Gen. Morin>given approximately by the following empirical formula: W R = ~[aA-b(u- 3.28)]. In this formula R = total resistance; r = radius of wheel in inches; W = gross load; u = velocity in feet per second; while a and b are constants, whose values are: For good broken-stone road, a — A to .55, b = .024 to .026; for paved roads, a = .27, b ~ .0684. Rankine states that on gravel the resistance is about double, and on sand five times, the resistance on good broken-stone roads. ELEMENTS OF MACHINES. The object of a machine is usually to transform the work or mechanical energy exerted at the point where the machine receives its motion into work at the point where the final resistance is overcome. The specific end may be to change the character or direction of mo- tion, as from circular to rectilinear, or vice versa, to change the velocity, or to overcome a great resistance by the application of a moderate force. In all cases the total energy exerted equals the total work done, the latter including the overcoming of all the f fictional Fig. 98. resistances of the machine as well as the use- ful work performed. No increase of power can be obtained from any machine, since this is impossible according to the law of conser- vation of energy. In a f rictionless machine the I B product of the force exerted at the driving- point into the velocity of the driving-point, or the distance it moves in a given interval of time, equals the product of the resistance into the distance through which the resist- ance is overcome in the same time. Fig. 99. The most simple machines, or elementary machines, are reducible to three classes, viz., the Lever, the Cord, and the Inclined Plane. The first class includes every machine con- sisting of a solid body capable of revolving g on an axis, as the Wheel and Axle. The second class includes every machine in which force is transmitted by means of flexi- ble threads, ropes, etc., as the Pulley. The third class includes every machine in jr IG jqo. which a hard surface inclined to the direc- tion of motion is introduced, as the Wedge and the Screw. A Lever is an inflexible rod capable of motion about a fixed point, called a fulcrum. The rod may be straight or bent at any angle, or curved. It is generally regarded, at first, as without weight, but its weight may be A C B p fj Ow Ow 436 MECHANICS. considered as another force applied in a vertical direction at its centre of gravity. The arms of a lever are the portions of it intercepted between the force, P, and fulcrum, C, and between the weight, W, and fulcrum. Levers are divided into three kinds or orders, according to the relative positions of the applied force, weight, and fulcrum. In a lever of the first order, the fulcrum lies between the points at which the force and weight act. (Fig. 98.) In a lever of the second order, the weight acts at a point between the fulcrum and the point of action of the force. (Fig. 99.) In a lever of the third order, the point of action of the force is between that of the weight and the fulcrum. (Fig. 100.) In all cases of levers the relation between the force exerted or the pull, P, and the weight lifted, or resistance overcome, W, is expressed by the equation P X AC = W X BC, in which AC is the lever-arm of P, and BC is the lever-arm of W, or moment of the force = the moment of the resist- ance. (See Moment.) In cases in which the direction of the force (or of the resistance) is not at right angles to the arm of the lever on which it acts, the " lever-arm" is the length of a perpendicular from the fulcrum to the line of direction of the force (or of the resistance). W : P : : A C : BC, or, the ratio of the resistance to the applied force is the inverse ratio of their lever-arms. Also, if Vw is the velocity of W, and Vp is the velocity of P, W : F : : Vp : Vw, and Px Vp = Wx Vw. If Sp is the distance through which the applied force acts, and Sw is the distance the weight is lifted or through which the resistance is overcome, W : P : : Sp : Sw; Wx Sw= PXSp, or the weight into the distance it is lifted equals the force into the distance through which it is exerted. These equations are general for all classes of machines as well as for levers, it being understood that friction, which in actual machines increases the resistance, is not at present considered. Tlie Bent Lever.— In the bent lever (see Fig. 91, page 416) the lever- arm of the weight m is cf instead of bf. The lever is in equilibrium when n X of = in x cf, but it is to be observed that the action of a bent lever may be very different from that of a straight lever. In the latter, so long as the force and the resistance act in lines parallel to each other, the ratio of the lever-arms remains constant, although the lever itself changes its inclina- tion with the horizontal. In the bent lever, however, this ratio changes: thus, in the cut, if the arm bf is depressed to a horizontal direction, the dis- tance cf lengthens while the horizontal projection of af shortens, the latter becoming zero when the direction of af becomes vertical. As the arm af approaches the vertical, the weight in which may be lifted with a given force s is very great, but the distance through which it may be lifted is very small. In all cases the ratio of the weight m to the weight n is the in- verse ratio of the horizontal projection of their respective lever-arms. Tlie Moving Strut (Fig. 101) is similar to the bent lever, except that one of the arms is missing, and that the force and the resistance to be overcome act at the same end of the single arm. The resistance in the case shown in the cut is not the weight W, but its resistance to being moved, R, which may be sim- ply that due to its friction on the horizontal plane, or some other op- posing force. When the angle be- tween the strut and the horizontal plane changes, the ratio of the resistance to the applied force changes. When the angle becomes very small, a moderate force will Fig. 101. overcome a very great resistance, which tends to become infinite as the angle approaches zero. If a = the angle, P X sin a — R x versin a. If a. = 5 degrees, sin a — 0.8716, versin a = .00381, R — 23 R, nearly. The stone-crusher (Fig. 102) shows a practical example of the use of two moving struts. The Toggle-joint is an elbow or knee-joint consisting of two bars so connected that they may be brought into a straight line and made to pro- duce great endwise pressure when a force is applied to bring them into this ELEMENTS OF MACHINES. 437 position. It is a case of two moving struts placed end to end, the moving force being applied at their point of junction, in a direction at right angles to the direction of the resistance, the other end of one of the struts resting against a fixed abutment, and that of the other against the body to be moved. If a — the angle each strut makes with the straight line joining the points about which their outer ends rotate, the ratio of the resistance to the applied force is R : P : : sin a : 2 versin a; 2R versin a = P sin a. The Fig. 102. Fig. 103. ratio varies when the angle varies, becoming infinite when the angle becomes zero. The toggle-joint is used where great resistances are to be overcome through very small distances, as in stone-crushers (Fig. 103). The Inclined. Plane, as a mechanical element, is supposed perfectly hard and smooth, unless friction be considered. It assists in sustaining a heavy body by its reaction. This reaction, however, being normal to the plane, cannot entirely counteract the weight of the body, which acts verti- cally downward Some other force must therefore be made to act upon the body, in order that it may be sustained. If the sustaining force act parallel to the plane (Fig. 104), the force is to the weight as the height of the plane is to its length, measured on the incline. If the force act parallel to the base of the plane, the power is to the weight as the height is to the base. If the force act at any other angle, let i = the angle of the plane with the horizon, and e = the angle of the direction of the applied force with the angle of the plane. P : W :: sin i : cos e; P X cos e Fig. 104. W sin i. Problems of the inclined plane may be solved by the parallelogram of forces thus : Let the weight Wbe kept at rest on the incline by the force P, acting in i he line bP, parallel to the plane. Draw the vertical line ba to represent the weight ; also bb' perpendicular to the plane, and complete the parallelo- gram b'c. Then the vertical weight ba is the resultant of bb', the measure of support given by the plan » to the weight, and be, the force of gravity tend- ing to draw the weight down the plane. The force required to maintain the weight in equilibrium is represented by this force be. Thus the force and the weight are in the ratio of be to ba. Since the triangle of forces abe is similar to the triangle of the incline ABC, the latter may be substituted for the former in determining the relative magnitude of the forces, and P : W : : be : ab : : BC : AB. The Wedge is a pair of inclined planes united by their bases. In the application of pressure to the head or butt end of the wedge, to cause it to penetrate a resisting body, the applied force is to the resistance as the thickness of the wedge is to its length. Let t be the thickness, I the length, IF the resistance, and P the applied force or pressure on the head of the wedge. Then, friction neglected, P : W : : t : I; P = — r— ; W — — . The Screw is an inclined plane wrapped around a cylinder in such a way that the height of the plane is parallel to the axis of the cylinder If the screw is formed upon the internal surface of a hollow cylinder, it is usually called a nut. When force is applied to raise a weight or overcome a resistance by means of a screw and nut, either the screw or the nut may 438 MECHANICS. be fixed, the other being movable. The force is generally applied at the end of a wrench or lever-arm, or at the circumference of a wheel. If r = radius of the wheel or lever-arm, andp = pitch of the screw, or distance between threads, that is, the height of the inclined plane for one revolution of the screw, P = the applied force, andW= the resistance overcome, then, neg- lecting resistance due to friction, 2nr X P = Wp ; W = 6.283Pr -f- p. The ratio of P to W is thus independent of the diameter of the screw. In actual screws, much of the power transmitted is lost through friction. The Cam is a revolv- ing inclined plane. It may be either an inclined plane wrapped around a cylin- der in such a way that the height of the plane is ra- dial to the cylinder, such as the ordinary lifting- cam, used in stamp-mills (Fig. 105), or it may be an inclined plane curved edgewise, and rotating in a plane parallel to it's base (Fig. 106). The relation of the weight to the applied force is calculated in the same manner as in the case of the screw. Fig. 105. Fig. 106. (\ A., Pulleys or Blocks.— P = force applied, or pull ; W = weight lifted or resistance. In the simple pulley A (Fig. 107) the point P on the pulling rope descends the same amount that the weight is lifted, therefore P = W. In B and C the point P moves twice as far as the weight is lifted, there- fore W — 2P. In B and C there is one movable block, and two plies of the rope engage with it. In D there are three sheaves in the movable block, each with two plies engaged, or six in all. Six plies of the rope are there- fore shortened by the same amount that the weight is lifted, and the point P moves six times as far as the weight, consequently W = 6P. In general, the ratio of W X.Q Pis equal to the number of plies of the rope that are shortened, and also is equal to the number of plies that eugage the lower block. If the lower block has 2 sheaves and the upper 3, the end of the rope is fastened to a hook in the top of the lower block, and then there are 5 plies shortened instead of 6, and W — 5P. If V = velocity of W. and v = velocity of P, then in all cases VW = vP, whatever the number of sheaves or their arrangement. If the hauling rope, at the pulling end. passes first around a sheave in the upper or stationary block, it makes no difference in what direction the rope is led from this block to the point at which the pull on the rope is applied ; but if it first passes around the movable block, it is necessary that the pull be exerted in a direction parallel to the line of action of the resistance, or a line joining the centres of the two blocks, in order to obtain the maximum effect. If the rope pulls on the lower block at an angle, the block will be pulled out of the line drawn between the weight and the upper block, and the effective pull will be less than the actual pull ELEMENTS OF MACHINES. 439 on the rope in the ratio of the cosine of the angle the pulling rope makes with the vertical, or line of action of the resistance, to unity. Differential Pulley. (Fig. 108.)— Two pulleys, i?aud C, of different radii, rotate as one piece about a fixed axis, A. An end- less chain, BDECLKH, passes over both pulleys. The rims of the pulleys are shaped so as to hold the chain and prevent it from slipping. One of the bights or loops in which the chain hangs, DE, passes under and supports the running block F. The other loop or bight, HKL, hangs freely, and is called the hauling part. It is evident that the velocity of the hauling part is equal to that of the pitch-circle of the pulley B. In order that the velocity-ratio may be exactly uniform, the radius of the sheave F should be an exact mean be- tween the radii of B and C. Consider that the point B of the cord BD moves through an arc whose length = AB, during the same time the point C or the cord CE will move downward a distance = AC. The length of the bight or loop BDEC will be shortened by AB — AC, which will cause the pulley F to be raised half of this amount. If P = the pulling force on the cord HK, and W the weight lifted at F, then P X AB = W X M(AB - AC). To calculatethe length of chain required for a differential pulley, take the following sum: Half the circumference of A + half the circumference of B -j- half the circumference of F -\- twice the greatest distance of F from A + the least length of loop HKL. The last quantity is fixed according to convenience. The Differential Windlass (Fig. 109) is identical in principle with the differential pulley, the difference in con- struction being that in the differential windlass the running block hangs in the bight of a rope whose two parts are wound round, and have their ends respec- tively made fast to two barrels of different radii, which rotate as one piece about the axis A. The dif- ferential windlass is little used in practice, because Hof the great length of rope which it requires. The Differential Screw (Fig. 110) is a com- pound screw of different pitches, in which the threads wind the same way. N t and. N 2 are the two nuts; SjSj, the longer-pitched thread; S 2 S 2 . the shorter-pitched thread: in the figure both these threads are left-handed. At each turn of the screw the nut N-2 advances relatively to iV 2 through a dis- tance equal to the difference of the pitch. The use of the differential screw is to combine the slowness of advance due to a fine pitch with the strength of thread which can be obtained by means of a coarse pitch only. A Wheel and Axle, or Windlass, resembles two pulleys on one axis, having different diameters. If a weight be lifted bj 7 means of a rope wound over the axle, the force being applied at the rim of the wheel, the action is like that of a lever of which the shorter arm is equal to the radius of the axle plus half the thick- ness of the rope, and the longer arm is equal to the radius of the wheel. A wheel and axle is therefore sometimes classed as a perpetual lever. If P = the applied force, D = W = the weight lifted, and d the diameter of the axle -J- the diameter of the rope, PD = Wd. Toothed-wheel Gearing is a combination of two or more wheels and axles (Fig. 111). If a series of wheels and pinions gear into each other, as in the cut, friction neglected, the weight lifted, or resistance over- come, is to the force applied inversely as the distances through which they act in a given time. If R, R x , R 2 be the radii of the successive wheels, measured to the pitch-line of the teeth, and r, r x , r 2 the radii of the cor- responding pinions, Pthe applied force, and W the weight lifted, Px V Fig. 109. Fig. 110. : diameter of the wheel, 440 MECHANICS. K X . R* X R 9 = W x r x r x X r 2 , or the applied force is to the weight as the product of the radii of the pinions is to the product of the radii of the wheels; or, as the product of the numbers expressing the teeth in each pinion is to the product of the numbers expressing the teeth in each Endless Screw, or Worm-gear. (Fig. Ii2.)-This gear is com- monly used to convert motion at high speed into motion at very slow* Fig. 111. 'Fig. 112. speed. When the handle P describes a complete circumference, the -pitch- line of the cog-wheel moves through a distance equal to the pitch of the screw, and the weight Wis lifted a distance equal to the pitch of the screw multiplied by the ratio of the diameter of the axle to the diameter of the pitch-circle of the wheel. The ratio of the applied force to the weight lifted is inversely as their velocities, friction not being considered; but the friction in the worm-gear is usually very great, amounting sometimes to three or four times the useful work done. If v = the distance through which the force Pacts in a given time, say 1 second, and V= distance the weight W is lifted in the same time, r = radius of the crank or wheel through which Pacts, t = pitch of the screw, and also of the teeth on the cog-wheel, d = diameter of the axle, and D = diameter of the pitch-line of the cog-wheel, v = — — - — XV; V=vXtd-r- 6.283rd. Pv = WV+ friction. STRESSES IN FRAMED STRUCTURES. Framed structures in general consist of one or more triangles, for the reason that the triangle is the one polygonal form whose shape cannot be changed without distorting one of its sides. Problems in stresses of simple framed structures may generally be solved either by the application of the triangle, paralellogram, or polygon of forces, by the principle of the lever, or by the method of moments. We shall give a few examples, referring the student to the works of Burr, Dubois, Johnson, and others for more elabo- rate treatment of the subject. 1. A Simple Crane. (.Figs. 113 and 114.) — ^4 is a fixed mast, B a brace or boom, T a tie, and P the load. Required the strains in B and T. The weight P, considered as acting at the end of the boom, is held in equilibrium by three forces: first, gravity acting downwards; second, the tension in T; and third, the thrust of B. Let the length of the line p represent the magnitude of the downward force exerted by the load, and draw a parallelogram with sides bt parallel, respectively, to B and T, such that pis the diagonal of the parallelogram. Then b and t are the components drawn to the same scale as p, p being the resultant. Then if the length p represents the load, t is the tension in the tie, and b is the compression in the brace. Or. more simply. T, B, and that portion of the mast included between them or A' may represent a triangle of forces, and the forces are proportional to the length of the sides of the triangle; that is, if the height of the triangle .4' = t he load, then B = the compression in the brace, and T = the tension in the T tie; or if P = the load in pounds, the tension in T = P x — , , and the com- STRESSES IK FRAMED STRUCTURES. 441 pression in B = P X B Also, if a = the angle the inclined member makes with the mast, the other member being horizontal, and the triangle being right-angled, then the length of the inclined member = height of the tri- angle x secant a, and the strain in the inclined member = P secant a. Also, the strain in the horizontal member = P tan a. The solution by the triangle or parallelogram of forces, and the equations Tension in 2' = P x T/A', and Compression ini? = P X B/A', hold true even if the triangle is not right-angled, as in Fig. 115; but the trigonometrical rela- Fig. 113. Fis. 114. Fig. 115. tions above given do not hold, except in the case of a right-angled triangle. It is evident that as A' decreases, the strain in both Tand B increases, tend- ing to become infinite as A' approaches zero. If the tie Tis not attached to the mast, but is extended to the ground, as shown in the dotted line, the tension in it remains the same. 2. A Guyed Crane or Derrick. (Fig. 116.)— The strain in B is, as before, PxB/A', A' being that portion of the vertical included between B and T, wherever Tmay be attached to A. If, however, the tie Tis attached to B beneath its extremity, there may be in addition a bending strain in B due to a tendency to turn about the point of attachment of Tas a fulcrum. The strain in T may be calculated by the principle of moments. The mo- ment of P is Pc, that is, its weight X its perpendicular distance from the point of rotation of B on the mast. The moment of the strain on T is the product of the strain into the perpendicular distance from the line of its «.__F— direction to the same point of rotation of B, or Td. The strain in T there- fore = Pc-i- d. As d decreases the strain on T increases, tending to infin- ity as d approaches zero. The strain on the guy-rope is also calculated by the method of moments. The moment of the load about the bottom of the mast O is, as before, Pc. If the guy is horizontal the strain in it is Fand its moment is Ff, and F — Pcs-f. If it is inclined, the moment is the strain G X the perpendicular distance of the line of its direction from O, or Gg, and G — Pc-i- g. The guy-rope having the least strain is the horizontal one F, and the strain 442 MECHANICS. in G = the strain in F X the se- cant of the angle between F and G. As G is made more nearly- vertical g decreases, and the strain increases, becoming infi- nite when g — 0. 3. Shea r-p oles with Guys. (Fig. 109.)— Resultant of strain in both masts = P X BD -i-BC. Resultant strain in both guys=Px AB-*-BC. The strain on each mast (or guy) will be half the above, multiplied by the se- cant of half the angle the masts Fig. 117.
^° n n ; ~_ a r> T> Fig. 119. Fig. 120. B, or one half of a uniformly dis buted load, then compression on AB = P (the floor-beam CD not being considered to have any resistance to a slight bending). Tension on AC or AD = *AP X AD + AB. Compression on CD = y>P X BD -f- AB. Queen-post Truss. (Fig. 123.)-If uniformly loaded, and the queen-posts di- vide the length into three equal bays, the load may be considered to be divided into three equal parts, two parts of which, P x andP 2 , are concentrated at the panel joiuts STRESSES IK FRAMED STRUCTURES. 443 and the remainder is equally divided between the abutments and supported by them directly. The two parts P x and P 2 only are considered to affect the members of the truss. Strain in the vertical ties BE and CF each equals P x or P 2 . Strain on AB and CD each = P t x CD + CF. Strain on the tie AE or EF ov ED = F X FD h- CF. Thrust on BC = tension on EF. For stability to resist heavy un- equal loads the queen-post truss should have diagonal braces from B to Pand from Cto E. Inverted Queen-post Truss. (Fig;. 123.) — Compression on EB and FC each = P. Tension on AB and CD each = P X AB + EB. Compression on AE or EF or FD = PX AE -^- EB. Tension on BC = compression on EF. For sta- bility to resist unequal loads, ties should be run from C to E and from Fig. 123. BtoF. Burr Truss of Five Panels. (Fig. 124. )— Four fifths of the load may be taken as concentrated at the points E, JKT, L and F, the other fifth being B G H C ®t <& © Fig. 124. supported directly by the two abutments. For the strains in BA and CD the truss may be considered as a queen-post truss, with the loads P x , P 2 concentrated at Pand the loads P 3 , P 4 concentrated at F. Then, compres- sive strain on AB = (P 1 + P 2 ) X AB^-BE. The strain on CD is the same if the loads and panel lengths are equal. The tensile strain on BE or CF = P l + P 2 . That portion of the truss between E and Pmay be considered as a smaller queen-post truss, supporting the loads P 2 , P 3 at K and L. The strain on EG or HF = P 2 X EG -f- GK. The diagonals GL and KH receive no strain unless the truss is unequally loaded. The verticals GK and HL each receive a tensile strain equal to P 2 or P 3 . For the strain in the horizontal members: BG and CH receive a thrust equal to the horizontal component of the thrust in AB or CD, — (Pj + P 2 ) X tan angle ABE, or (Pj + P 2 ) X AE-i-BE. GH receives this thrust and also, in addition, a thrust equal to the horizontal component of the thrust in EG or HF, or, in all, (P x -4- P 2 + P 3 )X AE-i-BE. The tension in AE or FD equals the thrust in BG or HC, and the tension in EK. KL. and LF equals the thrust in GH. Pratt or Whipple Truss. (Fig. 125.)— In this truss the diagonals are ties, and the verticals are struts or columns. Calculation by the method of distribution of strains: Consider first the load P x . The truss having six bays or panels, 5/6 of the load is transmitted to the abutment H, and 1/6 to the abutment O, on the principle of the lever. As the five sixths must be transmitted through J A and AH, write on these members the figure 5. The one sixth is transmitted successively through JC, CK, KD, DL, etc., passing alternately through a tie and a strut. Write on these members, up to the strut GO inclusive, the figure 1. Then consider the load P 2 , of which 4/6 goes to AH and 2/6 to GO. Write on KB, BJ, J A, and AH the figure 4, and on KD, DL, LE, etc., the figure 2. The load P a 444 MECHANICS. transmit 3/6 in each direction; write 3 on each of the members through which this stress passes, and so on for all the loads, when the figures on the several members will appear as on the cut. Adding them up, we have the following totals : j A J BH BK CJ CL DK DM EL EN FM FO GN 15 10 1 6 3 3 6 1 10 15 , AH BJ CK DL EM FN GO ! 15 10 7 6 7 10 15 Tension on diagonals -j Compression on verticals ■ Each of the figures in the first line is to be multiplied by 1/6PX. secant of angle HAJ, or 1/6P X A J-r- AH, to obtain the tension, and each figure in the lower line is to be multiplied by 1/6P to obtain the compression. The diag- onals HB and FO receive no strain. 6 6 6 5 6 P 3 P4 Fig. 125. It is common to build this truss with a diagonal strut at HB instead of the post HA and the diagonal AJ; in which case 5/6 of the load Pis carried through JB and the strut BH, which latter then receives a strain = 15/6P X secant of HBJ. The strains in the upper and lower horizontal members or chords increase from the ends to the centre, as shown in the case of the Burr truss. AB receives a thrust equal to the horizontal component of the tension in AJ, or 15/6PX tan AJB. BC receives the same thrust + the horizontal component of the tension in BK, and so on. The tension in the lower chord of each panel is the same as the thrust in the upper chord of the same panel. (For calcu- lation of the chord strains by the method of moments, see below.) The maximum thrust or tension is at the centre of the chords and is equal WL to — -, in which W is the total load supported by the truss, L is the length, and D the depth. This is the formula for maximum stress in the chords of a truss of any form whatever. The above calculation is based on the assumption that all the loads P lr P 2 , etc., are equal. If they are unequal the value of each has to be taken into account in distributing the strains. Thus the tension in AJ, with unequal loads, instead of being 15 X 1/6 P secant would be sec X (5/6?! + 4/6 P 2 + 3/6 P 3 + 2/6 P 4 4- 1/6 P 5 .) Each panel load, P x etc., includes its fraction of the weight of the truss. General Formula for Strains in Diagonals and Verticals. —Let n— total number of panels, x — number of any vertical considered from the nearest end, counting the end as 1, r = rolling load for each panel, P = total load for each panel, Strain on verticals (n— x)-\-(n— x) 2 — (x- 2n l)-Ha?-l) a TT . r(a;-l)+(a:-l) a + 2n For a uniformly distributed load, leave out the last term, [r(x-l)-\-\a>— l)?]-=r2M. Strain on principal diagonals = strain on verticals X secant 0, that is secant of the angle the diagonal makes with the vertical. Strain on the counterbraces : The strain on the counterbrace in the first panel is 0, if the load is uniform. On the 2d, 3d, 4th, etc., it is P secant X — , , — — — — , etc., P being the total load in one panel. STRESSES IN FRAMED STRUCTURES. 445 Strain in the Chords— Method of Moments. -Let the truss be uniformly loaded, the total load acting on it = W. Weight supported at each end, or reaction of the abutment = W/2. Length of the truss = L. Weight on a unit of length = W/L. Horizontal distance from the nearest abutment to the point (say Jl/in Fig. 125) in the chord where the strain is to be determined = x. Horizontal strain at that point (tension on the lower chord, compression in the upper) = H. Depth of the truss = D. By the method of moments we take the difference of the moments, about the point M, of the reaction of the abutment and of the load between and the abut- ments, and equate that difference with the moment of the resistance, or of the strain in the horizontal chord, considered with reference to a point in the opposite chord, about which the truss would turn if the first chord were severed at M. The moment of the reaction of the abutment is Wx/2. The moment of the load from the abutment to M is W/Lx X the distance of its centre of gravity from M, which is x/2, or moment = Wx 2 -h2L. Moment of the stress • 4.1. i, * riri Wx Wx* , TT Wy x*\ „ in the chord = HD = — — ~, whence H — — / x - — V If x — or L, WL~ H = 0. If x = L/2, H = -=-, which is the horizontal strain at the middle of the chords, as before given. The Howe Truss. (Fig. 126.)— In the Howe truss the diagonals are struts, and the verticals are ties. The calculation of strains may be made in the same method as described above for the Pratt truss. The Warren Girder. (Fig. 127.)— In the Warren girder, or triangular truss, there are no vertical struts, and the diagonals may transmit either Fig. 187. tension or compression. The strains in the diagonals may be calculated by the method of distribution of strains as in the case of the rectangular truss. On the principle of the lever, the load P t being 1/10 of the length of the span from the line of the nearest support a, transmits 9/10 of its weight to a and 1/10 to g. Write 9 on the right hand of the strut la. to represent the compression, and 1 on the right hand of 16, 2c, 3d. etc., to represent com- pression, and on the left hand of 62, c3, etc., to represent tension. The load P 2 transmits 7/10 of its weight to a and 3/10 to g. Write 7 on each member from 2 to a and 3 on each member from 2 to g, placing the figures representing compression on the right hand of the member, and those representing tension on the left. Proceed in the same manner with all the loads, then 446 MECHANICS. sum up the figures on each side of each diagonal, and write the difference of each sum beneath, and on the side of the greater sum, to show whether the difference represents tension or compression. The results are as follows: Compression, la, 25; 2b, 15; 3c, 5; 3d, 5; 4e, 15; 5a, 25. Tension, 16, 15; 2c, 5: 4d, 5; he. 15. Each of these figures is to be multiplied by 1/10 of one of the loads as P 1 , and by the secant of the angle the diagonals make with a vertical line. The strains in the horizontal chords may be determined by the method of moments as in the case of rectangular trusses. Roof-truss.— Solution by Method of Moments. — The calculation of strains in structures by the method of statical moments consists in taking a cross-section of the structure at a point where there are not more than three members (struts, braces, or chords). To find the strain in either one of these members take the moment about the intersection of the other two as an axis of rotation. The sum of the moments of these members must be if the structure is in equilibrium. But the moments of the two members that pass through the point of refer- ence or axis are both 0, hence one equation containing one unknown quan- tity can be found for each cross-section. A R / \ / P 3 S C y E^ i \ SJSP /r ^ ^^ X ! \ 20 / 1 ^»^*^ V \ / jj^ - ^^V Z i^, x /^ 25 ^" , ^*^ 12.5 ^S 15 ljj.5 \ / z 9 A \ * >- k D Fig. 128. In the truss shown in Fig. 128 take a cross-section at ts, and determine the strain in the three members cut by it, viz., CE, ED, and DF. Let X = force exerted in direction CE, Y = force exerted in direction DE, Z = force ex- erted in direction FD. For X take its moment about the intersection of F and Z at D = Xx. For Y take its moment about the intersection of Xand Z at A — Yy. For Z take its moment about the intersection of X and Y aUE — Zz. Let z = 15, x — 18.6, y - 38.4. AD = 50, CD = 20 ft. Let Pj, P 2 , P 3 , P 4 be equal loads, as shown, and 3}4 P the reaction of the abutment A. The sum of all the moments taken about D or A or E will be when the structure is at rest. Then - Xx + 3.5P X 50 - P 3 X 12.5 - P 2 X 25 - P, X 37.5 = 0. The +. signs are for moments in the direction of the hands of a watch or "clockwise " and — signs for the reverse direction or anti-clockwise. Since P= P, = Pz - P 3 , - 18.6X+ 175P - 75P = 0; - 18.6X = - 100P; X = 100P-^18.6 = 5.376P. - Yy + P 3 X 37.5 + P 2 X 25 f P, X 12.5 = 0; 38.4F = 75P; Y = 75P-=- 38.4 = 1 953P. -Zz + 3.5P X 37.5 - P, X 25 - P 2 X 12.5 - P 3 X = 0; lhZ = 93.75P; Z = 6. 25 P. In the same manner the forces exerted in the other members have been found as follows: EG = 6.73P; GJ = 8.07P; J A = 9.42P; JH = 1.35P; GF = 1.59P; AH= 8.75P; HF = 7.50P. The Fink Roof-truss. (Fig. 129.)— An analysis by Prof. P. H. Phil- brick (Van N. Mag., Aug. 1880) gives the following results: STRESSES IN FRAMED STRUCTURES. C 447 D Fig. 129. W = total load on roof; iV = No. of panels on both rafters; W/N = P = load at each joint 6, d, /, etc.; V = reaction at A = % W = y 2 NP = 4P; AD = S; AC=L; CD = D; *i> ^2> ^3 = tension on De, eg, gA, respectively; c x , c 2 , c 3 , c 4 = compression on Cb, fed, df, and/4. Strains in 1, or De = t x = 2PS -4- D; 2, " eg = t 2 = SPS -i- D; " gr4 = # 3 = 7/2PS-=-D; 4, " 4/ = c 4 = 7/2PL -f- D; 5, » fd = c 3 = 7/2PL/D -PD/L; " " dfe = c 3 = 7/2PL/D -2PD/L; 7, or &C = d =7/2 Pi/D - 3 PD/L: 8, " fee or/g = PS-r-L; 9, " de = 2PS + L; 10, " cd or dg = %PS -=- D; 11, " ec = PS^-D; 12, " cC - 3/2 PS -v- D. Example.— Given a Fink roof -truss of span 64 ft., depth 16 ft., with four panels on each side, as in the cut; total load 32 tons, or 4 tons each at the points/, d, b, C, etc. (and 2 tons each at A and B, which transmit no strain to the tr uss mem bers). Here W — 32 tons, P = 4 tons, S = 32 ft., D = 16 ft., L = YS" 2 + D 2 = 2.236 X D. L -*- D = 2.23( S + L= .8944 D 2 = 2.236 X D. _L -s- D = 2.236, D -*- £ = .4472, flf -s- D = 2, The strains on the numbered members then are as follows: 1, 2X4X2 2, 3X4X2 3, 7/2 X 4 X 2 =28 4, 7/2 X 4 X 2.236 = 31.3 5, 31.3-4 X .447 =29.52 6, 31.3- 8 X .447 =27.72 tons; 7, 3 1.3- 12 X .447 =25.94 tons 8, 4 X .8944 = 3.58 " 9, 8 X .8944 = 7.16 " 10, 2X2 =4 11, 4X2=8 12, 6 X 3 =12 448 HEAT. THERMOMETERS. The Fahrenheit thermometer is generally used in English-speaking coun- tries, and the Centigrade, or French thermometer, in countries that use the metric system. In many scientific treatises in English, however, the Centi- grade temperatures are also used, either with or without their Fahrenheit equivalents. The Reaumur thermometer is used in Russia, Sweden, Turkey, and Egypt. (Clark.) In the Fahrenheit thermometer the freezing-point of water is taken at 32°, and the boiling-point of water at mean atmospheric pressure at the sea- level, 14.7 lbs. per sq. in., is taken at 212°, the distance between these two points being divided into 180°. In the Centigrade and Reaumur thermometers the freezing-point is taken at 0°. The boiling-point is 100° in the Centigrade scale, and 80° in the Reaumur. 1 Fahrenheit degree — 5/9 deg. Centigrade = 4/9 deg. Reaumur. 1 Centigrade degree = 9/5 deg. Fahrenheit = 4/5 deg. Reaumur. 1 Reaumur degree = 9/4 cleg. Fahrenheit = 5/4 deg. Centigrade. Temperature Fahrenheit = 9/5 X temp. C. + 32° = 9/4 R. -f- 32°. Temperature Centigrade = 5/9 (temp. F. — 32°) = 5/4 R. Temperature Reaumur = 4/5 temp C. = 4/9 (F. — 32°). Mercurial Thermometer, (Rankine. S. E., p. 234.)— The rate of expansion of mercury with rise of temperature increases as the temperature becomes higher ; from which it follows, tnat if a thermometer showing the dilatation of mercury simply were made to agree with an air thermometer at 32° and 212°, the mercurial thermometer would show lower temperatures than the air thermometer between those standard points, and higher tem- peratures beyond them. For example, according to Regnanlt, when the air thermometer marked 350° C. (- 662° F.), the mercurial thermometer would mark 362.16° C. (= 683.89° F.), the error of the latter being in excess 12.16° C. (= 21.89° F.). Actual mercurial thermometers indicate intervals of temperature propor- tional to the difference between the expansion of mercury and that of glass. The inequalities in the rate of expansion of the glass (which are very different for different kinds of glass) correct, to a greater or less extent, the errors arising from the inequalities in the rate of expansion of the mercury. For practical purposes connected with heat engines, the mercurial ther- mometer made of common glass may be considered as sensibly coinciding with the air-thermometer at all temperatures not exceeding 500° F. PYROMETRY. Principles Used in Various Pyrometers.— Contraction of clay by heat, as in the Wedgwood pyrometer used by potters. Not accurate, as the contraction varies with the quality of the clay. Expansion of air, as in the air-thermometers, Wiborgh's pyrometer, Ueh- ling and Steinhart's pyrometer, etc. Specific heat of solids, as in the copper-ball, platinum-ball, and fire-clay pyrometers. Relative expansion of two metals or other substances, as copper and iron, as in Brown's and Bulkley's pyrometers, etc. Melting-points of metals, or other substances, as in approximate deter- minations of temperature by melting pieces of zinc, lead, etc. Measurement of strength of a thermo-electric current produced by heat- ing the junction of two metals, as in LeCliatelier"s pyrometer. Changes in electric resistance of platinum, as in the Siemens pyrometer. Time required to heat a weighed quantity of water enclosed in a vessel, as in the water pyrometer. Thermometer for Temperatures up to 800° F.— Mercury with compressed nitrogen in the tube above the mercury. Made by Queen & Co., Philadelphia. TEMPERATURES, CENTIGRADE AND 44Q FAHRENHEIT. — * c. F. C F. C. F. C. F. C. F. C. F. C. F. -40 -40. 26 78.8 92 197.6 158 316.4 224 435.2 290 554 950 1742 —39 -38.2 .'7 80.6 93 199.4 159 318.2 225 437. 572 960 1760 -38 -36.4 28 82.4 94 201.2 160 320. 220 438.8 310 590 970 1778 -37 —34 6 29 84.2 95 203. 161 321.8 227 440.6 320 608 980 1796 -36 -32.8 30 80. 90 204.8 162 323.6 228 442.4 330 626 990 1814 -35 -31. 31 87.8 97 206. (j 163 325.4 229 444.2 3l<) 644 1000 1832 -34 —29.2 32 89.6 98 208.4 164 327.2 230 446. 350 662 1010 1850 -33 -27.4 33 91.4 99 210.2 165 329. 231 447.8 300 680 1020 1868 -32 -25.6 34 93.2 100 212. 166 330.8 232 449.6 370 698 1030 1886 -31 -23.8 35 95. 101 213.8 167 332.6 233 451.4 380 716 1040 1904 -30 -22. 36 96.8 102 215.(j 168 334.4 234 453.2 734 1050 1922 -29 -20.2 37 98.6 103 217.4 169 336.2 235 455. 400 752 1060 1940 —28 -18.4 38 100.4 101 219.2 170 338. 230 456.8 410 770 1070 1958 -27 -16.6 39 102.2 105 221. 171 339.8 237 458.6 420 788 1080 1976 -26 -14.8 40 104. 100 -.22.8 172 341.6 238 460.4 430 806 1090 1994 -25 -13. 41 105.8 107 224.6 173 343.4 239 462 2 440 824 1100 2012 —24 -11.2 42 107.6 108 226.4 174 345.2 240 464. 450 842 1110 2030 -23 - 9.4 43 109.4 109 228.2 175 347. 241 465.8 460 860 1120 2048 -22 — 7.6 44 111.2 no 230. 176 348.8 242 467.6 470 878 1130 2066 -21 — 5.8 45 113. 111 231.8 177 350.6 243 469.4 480 896 1140 2084 -20 - 4. 46 114.8 112 233 6 178 352.4 244 471.2 490 914 1150 2102 -19 - 2.2 47 116.6 113 235.4 179 354.2 245 473. 500 932 1160 2120 -18 - 0.4 48 118.4 114 237.2 180 356. 240 474.8 510 950 1170 2138 -17 + 1.4 49 120.2 115 239. 181 357.8 247 476.6 968 1180 2156 -16 3.2 50 122. 110 240.8 182 359.6 248 478.4 986 1190 2174 —15 5. 51 123.8 117 242.6 183 361.4 249 480.2 540 1004 1200 2192 -14 6.8 52 125.6 118 244.4 184 363.2 250 482. 550 1022 1210 2210 -13 8.6 53 127.4 119 246.2 185 365. 251 483.8 1040 1220 V -12 10.4 54 129.2 120 248. 186 366.8 252 485.6 570 1058 1230 2246 -11 12.2 55 131. 121 249.8 187 368.6 253 487.4 1076 1240 2264 -10 14. 56 132.8 122 251.Q 188 370.4 254 489.2 1094 1250 - 9 15.8 57 134.6 123 253.4 189 372.2 255 491. 000 1112 1260 2300 - 8 17.6 58 136.4 124 255 . 2 190 374. 256 492.8 610 1130 1270 2318 7 19.4 59 138.2 125 257 '. 191 375.8 257 494.6 1148 1280 2336 - 6 21.2 GO 140. 120 258.3 192 377.6 496.4 1166 1290 - 5 23. 01 141.8 127 260.6 193 379.4 259 498.2 040 1184 1300 2372 - 4 24.8 62 143.6 128 262.4 194 381.2 260 500. 1202 1310 2390 - 3 26.6 03 145.4 129 264.2 195 383. 201 501.8 1220 1320 2408 - 2 28.4 04 147.2 130 266. 196 384.8 202 303.6 1238 1330 2426 — 1 30.2 05 149. 131 267.8 197 386.6 203 5D5rT4 1256 1340 2444 32. 00 150.8 132 269.6 198 388.4 204 507.2 1274 1350 2462 + 1 33.8 67 152.6 133 271.4 199 390.2 205 509. 700 1292 1360 2480 2 35.6 08 154.4 134 273.2 200 392. 260 510.8 710 1310 1370 2498 3 37.4 09 156 2 135 275. 201 393.8 267 512.6 132S 1380 2516 4 39.2 70 158 130 276.8 202 395.6 514.4 1346 1390 2534 5 41. 71 159 8 137 278.6 203 397.4 209 516.2 741 1364 1400 2552 6 42.8 72 101 6 li 280.4 204 399.2 270 518. 750 1382 1410 2570 7 44.6 73 103 4 139 282.2 205 401. 271 519.8 7 6i 1400 1420 2588 8 46.4 74 165 2 140 284. 206 402.8 521.6 770 1418 1430 2606 9 48.2 75 167 141 285.8 207 404.6 523.4 78C 1436 1440 2624 10 50. 70 168 8 142 287.6 20S 406.4 274 525.2 1454 1450 2642 11 51.8 77 170 6 143 289.4 209 408.2 275 527. 1472 1460 2660 12 53.6 78 172 4 144 291.2 210 410. 276 528.8 810 1490 1470 2678 13 55.4 79 174 2 145 293. 211 411.8 277 530.6 820 1508 1480 2696 14 57.2 80 170 146 294.8 212 413.6 278 532.4 1526 1490 2714 15 59. 81 177 8 147 296.6 213 415.4 279 534.2 810 1544 1500 16 60.8 82 179 6 14S 298.4 214 417.2 536. iS5i 1562 1510 2750 17 62.6 181 4 149 300.2 215 419. 281 537.8 800 1580 1520 2768 18 64.4 84 183 2 150 302. 216 420.8 ■:s2 539.6 1598 1530 2786 19 66.2 85 18.5 151 303.8 217 422.6 2S3 541.4 1616 1540 2804 20 68. 80 186 8 152 305.6 218 424.4 284 543.2 1034 1550 2822 21 69.8 87 188.6 153 307.4 219 426.2 285 545. 1052 1600 2912 22 71.6 190.4 154 309.2 220 428. 546.8 910 1670 1650 3002 23 73.4 192.2 155 311. 221 429.8 287 548.6 1688 1700 3092 24 75.2 90 194. 150 312.8 222 431.6 2SS 550.4 930 1706 1750 3182 25 77. 91 195.8 157 314.6 226 433.4 289 552.2 940 1724 1800 3272 AKf) fEMPERATURES, FAHRENHEIT AND ^ uv CENTIGRADE. F. C. F. C. F. C. F. c. F. c. F. C. F. c. -40 —40. 2G — 3.3 92 33.3 158 70. 224 106.7 290 143.3 3G0 182.2 —39 -39.4 27 — 2.8 93 33.9 159 70.6 107.2 143.9 370 187.8 —38 —38.9 28 — 2 2 94 34.4 160 71.1 107.8 2! 144.4 193.3 -37 -38.3 29 — 1.7 95 35. 161 71.7 108.31293 145. 198.9 -36 -37.8 30 — 1.1 96 35.6 162 72.2 . 108.9 145.6 400 204.4 -35 —37.2 31 — 0.6 97 36.1 163 72.8 . 109.4 295 146.1 41C 210. -34 -36.7 32 0. 98 36.7 164 73.3 110. 146.7 215.6 -33 -36.1 33 -f 0.6 99 37.2 165 73.9 110.6 297 147.2 221.1 -32 —35.6 34 1.1 100 37.8 166 74.4 111.1 29f 147.8 410 226.7 -31 -35. 35 1.7 101 38.3 167 75. 111.7 1 148.3 2322 —30 —34.4 36 2.2 102 38.9 75.6 234 112.2 300 148.9 . 237.8 —29 —33.9 37 2.8 103 39.4 169 76.1 112.8 301 149.4 470 243.3 —28 -33.3 38 3.3 104 40. 170 76.7 113.3 302 150. 480 248.9 —27 -32.8 39 3.9 105 40.6 171 77.2 113.9 150.6 490 254.4 —26 -32.2 40 4.4 106 41.1 172 77 8 !! 114.4 301 151.1 500 260. —25 -31.7 41 5. 107 41.7 173 78.3 239 115. 305 151.7 510 265.6 —24 -31.1 42 5.6 108 42.2 174 78.9 115.6 306 152.2 271.1 -23 -30.6 43 6.1 109 42.8 175 79.4 241 116.1 307 152.8 530 276.7 -22 —30. 44 6.7 110 43.3 176 80. 116.7 3'X 153.3 540 282.2 -21 —29.4 45 7.2 111 43.9 177 80.6 243 117.2 30! 153.9 550 287.8 -20 -28.9 46 7.8 112 44.4 178 81.1 244 117.8 310 154.4 5G0 293.3 -19 —28.3 47 8.3 113 45. 179 81.7 245 118.3 311 155. 570 298.9 -18 —27.8 48 8.9 114 45.6 180 82.2 24 ( 118.9 312 155.6 580 304.4 -17 —27.2 49 9.4 115 46.1 181 82.8 247 119.4 313 156.1 590 310. -16 —26.7 50 10. 116 46.7 182 83.3 120. 314 156.7 600 315.6 —15 —26.1 51 10.6 117 47.2 183 83.9 24'. 120.6 315 157.2 61.C 321.1 —14 —25.6 52 11.1 118 47.8 184 84.4 ;-.:-; 121.1 316 157.8 62C 326.7 —13 —25. 53 11.7 119 48.3 185 85. 251 121.7 317 158.3 G3C —12 —24.4 54 12.2 120 48.9 186 85.6 122.2 318 158.9 64C 337.8 -11 -23.9 55 12.8 121 49.4 187 86.1 2.>-: 122.8 319 159.4 650 ; —10 —23.3 56 13.3 122 50. 86.7 . 123.3 320 160. 660 348.9 - 9 —22.8 57 13.9 123 50.6 1S9 87.2 255 123.9 321 160.6 670 354.4 - 8 —22.2 58 14.4 124 51.1 190 87.8 256 124.4 322 161.1 680 360. — 7 —21.7 15. 125 51.7 191 88.3 25? 125. 323 161.7 691 365 6 — 6 —21.1 60 15.6 126 52.2 192 88.9 25.' 125.6 324 162.2 371.1 — 5 -20.6 61 16.1 127 52.8 193 89.4 259 126.1 325 162.8 710 376.7 - 4 —20. 62 16.7 128 53.3 194 90. 2fi! 126.7 326 163.3 720 382.2 — 3 —19.4 63 17.2 129 53.9 195 90.6 261 127.2 327 163.9 730 387.8 — 2 -18.9 64 17.8 130 54.4 196 91.1 127.8 328 164.4 740 393.3 — 1 —18 3 65 18.3 131 55. 197 91.7 128.3 329 165. 750 398.9 -17.8 66 18.9 132 55.6 198 92.2 264 128.9 330 165.6 760 404.4 + 1 —17.2 67 19.4 133 56.1 199 92.8 265 129.4 331 166.1 770 410. 2 -16.7 68 20. 134 56.7 0(5 93.3 130. 332 166.7 7S0 415.6 3 -16.1 69 20.6 135 57.2 201 93.9 267 130.6 333 167.2 790 421.1 4 —15.6 70 21.1 136 57.8 202 94.4 26! 131.1 334 167.8 SOO 426.7 5 —15. 71 21.7 137 58.3 203 95. 131.7 335 168.3 810 432.2 6 —14.4 72 22.2 138 58.9 204 95.6 270 132.2 336 168.9 437.8 7 -13.9 73 22.8 139 59.4 205 96.1 271 132.8 337 169.4 830 443.3 8 —13.3 74 23.3 140 60. 06 96.7 133.3 338 170. 840 448.9 9 —12.8 75 23.9 141 60.6 207 97.2 133.9 339 170.6 850 454.4 10 —12.2 76 24.4 142 61.1 97.8 274 134.4 340 171.1 860 460. 11 -11.7 77 25. 143 61.7 209 98.3 275 135. 341 171.7 870 465.6 12 —11.1 78 25.6 144 62.2 210 98.9 135.6 342 172.2 880 471.1 13 —10.6 79 26.1 145 62.8 211 99.4 136.1 343 172.8 890 476.7 14 -10. 80 26.7 146 63.3 100. 136.7 344 173.3 900 482.2 15 — 9.4 81 27.2 147 63.9 100.6 137.2 345 173.9 910 487.8 16 — 8.9 82 27.8 148 64.4 214 101.1 137.8 346 174.4 920 493.3 17 — 8.3 28.3 149 65. 215 101.7 138.3 347 175. 930 498.9 18 - 7.8 28.9 150 65.6 102.2 138.9 341 175.6 940 504.4 19 — 7.2 29.4 151 66.1 217 102.8 139.4 m 176.1 950 510. 20 — 6.7 86 30. 152 66.7 218 103.3 140. 350 176.7 960 515.6 21 — 6.1 87 30.6 153 67.2 103.9 140.6 351 177.2 970 521.1 22 - 5.6 31.1 154 67.8 104.4 141.1 177.8 80 526.7 23 — 5. 89 31.7 155 68.3 221 105. 141.7 353 178.3 990 532.2 24 — 4.4 00 32.2 156 68.9 222 105.6 142.2 354 178.9 1000 537.8 25 — 3.9 91 32.8 157 69.4 223 106.1 142.8 355 179.4 1010 543.3 PYROMETRY. 451 Platinum or Copper Ball Pyrometer.— A weighed piece of platinum, copper, or iron is allowed to remain in the furnace or heated chamber till it has attained the temperature of its surroundings. It is then suddenly taken out and dropped into a vessel containing water of a known weight and temperature. The water is stirred rapidly and its maximum temperature taken. Let W — weight of the water, w the weight of the ball, t = the original and T the final heat of the water, and S the specific heat of the metal ; then the temperature of fire may be found from the formula *." wS ^- For a fuller description, by J. C. Hoadley, see Trans. A. S. M. E., vi, 702. The mean specific heat of platinum above 32° is .03333 or l/30th that of water, and it increases with the temperature, the increase being about .000305 for each 100° F. For accuracy corrections are required for variations in the specific heat of the water and of the metal at different temperatures, for loss of heat by radiation from the metal during the transfer from the furnace to the water, and from the apparatus during the heating of the water; also for the heat- absorbing capacity of the vessel containing the water. Fire-clay or fire-brick may be used instead of the metal ball. lie Chatelier's Tliermo-electric Pyrometer. — For a very full description see paper by Joseph Struthers, School of Mines Quarterly, vol. xii, 1891; also, paper read by Prof. Roberts-Austen before the Iron and Steel Institute, May 7, 1891. The principle upon which this pyrometer is constructed is the measure- ment of a current of electricity produced by heating a couple composed of two wires, one platinum and the other platinum with \0% rhodium— the cur- rent produced being measured by a galvanometer. The composition of the gas which surrounds the couple has no influence on the indications. When temperatures above 2500° F. are to be studied, the wires must have an isolating support and must be of good length, so that all parts of a fur- nace can be reached. For a Siemens furnace, about llt£ feet is the general length. The wires are supported in an iron tube, J^ inch interior diameter and held in place by a cylinder of refractory clay having two holes bored through, in which the wires are placed. The shortness of time (five seconds) allows the tempera- ture to be taken without deteriorating the tube. Tests made by this pyrometer in measuring furnace temperatures under a great variety of conditions show that the readings of the scale uncorrected are always within 45° F. of the correct temperature, and in the majority of industrial measurements this is sufficiently accurate. Le Chatelier's py- rometer Is sold by Queen & Co., of Philadelphia. Graduation of I*e Chatelier's Pyrometer.— W. C. Roberts- Austen in his Researches on the Properties of Alloys, Proc. Inst. M. E. 1892, says : The electromotive force produced by heating the thermo-junction to any given temperature is measured by the movement of the spot of light on the scale graduated in millimetres. A formula for converting the divi- sions of the scale into thermometric degrees is given by M. Le Chatelier; but it is better to calibrate the scale by heating the thermo-junction to temper- atures which have been very carefully determined by the aid of the air- thermometer, and then to plot the curve from the data so obtained. Many fusion and boiling-points have been established by concurrent evidence of various kinds, and are now very generally accepted. The following table contains certain of these : Water boils. Lead melts. Mercury boils. Zinc melts. Sulphur boils. Aluminum melts. Selenium boils. The Temperatures Developed in Industrial Furnaces.— M. Le Chatelier states that by means of his pyrometer he has discovered that the temperatures which occur in melting steel and in other industrial operations have been hitherto overestimated. Deg. F. Deg. C. 212 100 618 326 676 358 779 415 , 838 448 ' 1157 625 1229 665 Deg. F. Deg. C. 1733 945 Silver melts. 1859 1015 Potassium sul- phate melts 1913 1045 Gold melts. 1929 1054 Copper melts. 2732 1500 Palladium melts. 3227 1775 Platinum melts. 452 HEAT. M. Le Cliatelier finds the melting heat of white cast iron 1135° (2075 6 F.), and that of gray cast iron 1220° (22J8° F.). Mild steel melts at 1475° (2687° F.), semi-mild at 1455° (2651° F.), and hard steel at 1410° (2570° F.). The furnace for hard porcelain at the end of the baking has a heat of 1370° (2498° F.). The heat of a normal incandescent lamp is 1800° (3272° F.), but it may be pushed to beyond 2100° (3812° F.). Prof. Roberts-Austen (Recent Advances in Pyrometry, Trans. A. I. M. E., Chicago Meeting, 1M?3) gives an excellent description of modern forms of pyrometers. The following are some of his temperature determinations. Gold-melting, Royal Mint. Degrees. Degrees. Centigrade. Fahr. Temperature of standard alloy, pouring into moulds. . . . 1180 2156 Temperature of standard alloy, pouring into moulds (on a previous occasion, by thermo-couple) 1147 2097 Annealing blanks for coinage, temperature of chamber.. 890 1634 Silver-melting, Royal Mint. Temperature of standard alloy, pouring into mould 980 1796 Ten-ton Open-hearth Furnace, Woolwich Arsenal. Temperature of steel. 0.3$ carbon, pouring into ladle 1645 2993 Temperature of steel, 0.3$ carbon, pouring into large mould 1580 2876 Reheating furnace, Woolwich Arsenal, temperature of interior 930 1706 Cupola furnace, temperature of No. 2, cast-iron pouring into ladle 1600 2912 The following determinations have been effected by M. Le Chatelier: Bessemer Process. Six- ton Converter. Degrees. Degrees. Centigrade Fahr. A. Bath of slag 1580 2876 B. Metal in ladle 1640 2984 C. Metal in ingot mould 1580 2876 D. Ingot in reheating furnace 1200 2192 E. Ingot under the hammer 1080 1976 Open-hearth Furnace (Siemens). Semi-Mild Steel. A. Fuel gas near gas generator 720 1328 B. Fuel gas entering into bottom of regenerator chamber 400 752 C. Fuel gas issuing from regenerator chamber 1200 2192 Air issuing from regenerator chamber 1000 1832 Chimney Gases. Furnace in perfect condition 300 590 Open-hearth Furnace. End of the melting of pig charge 1420 2588 Completion of conversion 1500 2732 Molten Steel. In the ladle— Commencement of casting 1580 2876 End of casting 1490 2714 Inthemoulds 1520 2768 For very mild (soft) steel the temperatures are higher by 50° C. Siemens Crucible or Pot Furnace. 1000° C, 2912° F. Rotary Puddling Furnace. Degrees C. Degrees F. Furnace ... 1340-1230 2444-224(1 Puddled ball— End of operation 1330 2420 PYltOMETllY. 453 Blast-furnace (6 ray -Bessemer Pig). Opening in face of tuyere 1930 3506 Molten metal — Commencement of fusion 1400 2552 End, or prior to tapping 1570 2858 Hoffman Red-brick Kiln. Burning temperatures 1100 2012 The Wihorgh Air-pyrometer. (E. Trotz, Trans. A. I. M. E. 1892.)— The inventor using the expansion-coefficient of air, as determined by Gay-Lussac, Dulon, Rudberg, and Regnaulfc, bases his construction on the following theory : If an air-volume, V, enclosed in a porcelain globe and connected through a capillary pipe with the outside air, be heated to the temperature T (which is to be'determined) and thereupon the connection be discontinued, aud there be then forced into the globe containing V another volume of air V of known temperature t, which was previously under atmospheric pressure H, the additional pressure /i, due to the addi- tion of the air-volume V to the air-volume V, can be measured by a ma- nometer. But this pressure is of course a function of the temperature 2'. Before the introduction of V, we have the two separate air-volumes, Fat the temperature T and V at the temperature f, both under the atmospheric pressure H. After the forcing in of V into the globe, we have, on the contrary, only the volume V ol the temperature T, but under the pressure H+h. The Wiborgh Air-pyrometer is adapted for use at blast-furnaces, smelt ing- works, hardening and tempering furnaces, etc., where determinations of temperature from 0° to 2400° F. are required. Seger's Fire-clay Pyrometer. (H. M. Howe, Eng. and Mining Jour., June 7, 1890.)— ProlVssor Seger uses a series of slender triangular fire-clay pyramids, about 3 inches high and % inch wide at the base, aud each a little less fusible than the next : these he calls "normai pyramids" ( u normal-kegel "). When the series is placed in a furnace whose temper- ature is gradually raised, one after another will bend over as its range of plasticity is reached ; and the temperature at which it has bent, or "wept," so far that its apex touches the hearth of the furnace or other level surface on which it is standing, is selected as a point on Seger's scale. These points may be accurately determined by some absolute method, or they may merely serve to give comparative results. Unfortunately, these pyramids afford no indications when the temperature is stationary or falling. Mesure and Nonel's Pyrometric Telescope. (Ibid.)— Mesure and Nouel's pyrometric telescope gives us an immediate determination of the temperature of incandescent bodies, and is therefore much better adapted to cases where a great number of observations are to be made, and at short intervals, than Seger's. Such cases arise in the careful heating of steel. The little telescope, carried in the pocket or hung from the neck, can be used by foreman or heater at any moment. It is based on the fact that a plate of quartz, cut at right angles to the axis, rotates the plane of polarization of polarized light to a degree nearly inversely proportional to the square of the length of the waves; and, further, on the fact that while a body at dull redness merely emits red light, as the temperature rises, the orauge, yellow, green, and blue waves successively appear. If, now, such a plate of quartz is placed between two Nicol prisms at right angles, "a ray of monochromatic light which passes the first, or polarizer, and is watched through the second, or analyzer, is not extin- guished as it was before interposing the quartz. Part of the light passes the analyzer, and, to again extinguish it, we must turn one of the Nicols a certain angle," depending on the length of the waves of light, and hence on the temperature of the incandescent object which emits this light. Hence the angle through which we must turn the analyzer to extinguish the light is a measure of the temperature of the object observed. The instrument is made by Ducietet, of Paris, in two sizes ; cost, $20 and $25. The Uehling and Steinbart Pyrometer. (For illustrated descrip- tion see Engineering, Aug. 24, 1894.)— The action of the pyrometer is based on a principle which involves the law of the flow of gas through minute apertures in the following manner : If a closed tube or chamber be supplied with a minute inlet and a minute outlet aperture and air be caused by a constant suction to flow in through one and out through the other of these apertures, the tension iu the chamber between the apertures will vary with 454 HEAT. the difference of temperature between the inflowing and outflowing air. If the inflowing air be made to vary with the temperature to be measured, and the outflowing air be kept at a certain constant temperature, then the tension in the space or chamber between the two apertures will be an exact measure of the temperature of the inflowing air, and hence of the tem- perature to be measured. In operation it is necessary that the air be sucked into it through the first minute aperture at the temperature, to be measure:!, through the second aperture at a lower but constant temperature, and that the suction be of a constant tension. The first aperture is therefore located in the end of a platinum tube in the bulb of a porcelain tube over which the hot blast sweeps, or inserted into the pipe or chamber containing the gas whose tem- perature is to be ascertained. The second aperture is located in a coupling, surrounded by boiling water, and the suction is obtained by an aspirator aud regulated by a column of water of constant height. The tension in the chamber between the apertures is indicated by a manometer. The Air-thermometer. (Prof. R. C. Carpenter, Eng'g News, Jan. 5, 1893.) — Air is a perfect thermometric substance, and if a given mass of air be considered, the product of its pressure and volume divided by iis absolute temperature is in every case constant. If the volume of air remain constant, the temperature will vary with the pressure; if the pressure remain constant the temperature will vary with the volume. As the former condition is more easily attained air-thermometers are usually constructed of constant volume, in which case the absolute temperature will vary with the pressure. If we denote pressure by p and p', the corresponding absolute temper- atures by T and T', we should have T p:p'::T:T' and T'=p'—.- * p The absolute temperature Tis to be considered in every case 460 higher than the thermometer-reading expressed in Fahrenheit degrees. From the form of the above equation, if the pressure be corresponding to a known absolute temperature, 2'can be found. The quotient is a constant which may be used in all determinations with the instrument. The pressure on the instrument can be expressed in inches of mercury, and is evidently the atmospheric pressure b as shown by a barometer, plus or minus an addi- tional amount h shown by a manometer attached to the air thermometer. That is, in general, p = b x h. The temperature of 32° F. is fixed as the point of melting ice, in which case 2' = 460 X 32 = 492° F. This temperature can be produced by sur- rounding the bulb in melting ice and leaving several minutes, so that the temperature of the confined air shall acquire that of the surrounding ice. When the air is at that temperature, note the reading of the attached manometer h, and that of a barometer; the sum will be the value of p cor- responding to the absolute temperature of 492° F. The constant of the instrument, K — 492 — p, once obtained, can be used in all future determina- tions. High Temperatures judged by Color.— The temperature of a body can be approximately judged by the experienced eye unaided, and M. Pouillet has constructed a table, which has been generally accepted,- giving the colors and their corresponding temperature as below: Deg. C. Deg. F. Deep orange heat. . . 1100 2021 Clear orange heat.. 1200 2192 White heat 1300 2372 Bright white heat.. 1400 2552 ) 1500 2732 Dazzling white heat > to to J 1600 2912 Deg. C. Deg. F Incipient red heat.. 525 977 Dull red heat 700 1292 Incipient cherry-red heat 800 1472 Cherry-red heat 900 1652 Clear cherry - red heat 1000 1832 The results obtained, however, are unsatisfactory, as much depends on the susceptibility of the retina of the observer to light as well as the degree of illumination under which the observation is made. QUANTITATIVE MEASUREMENT OF HEAT. 455 A bright bar of iron, slowly heated in contact with air, assumes the fol- lowing tints at annexed temperatures (Claudel): Cent. Fahr. Yellow at 225 437 Orange at 243 473 Red at 265 509 Violetat.. 277 531 Cent. Fahr. Indigo at 288 550 Blue at 293 559 Green at 332 630 ' ' Oxide-gray " 400 752 BOILING POINTS AT ATMOSPHERIC PRESSURE. 14.7 lbs. per square inch. Ether, sulphuric 100° F. Carbon bisulphide . 118 Ammonia 140 Chloroform 140 Bromine 145 Wood spirit . 150 Alcohol 173 Benzine 176 Water 212 Average sea- water 213.2° F. Saturated brine 226 Nitricacid 248 Oil of turpentine 315 Phosphorus 554 Sulphur 570 Sulphuric acid 590 Linseed oil 597 Mercury 676 The boiling points of liquids increase as the pressure increases. The boil- ing point of water at any given pressure is the same as the temperature of saturated steam of the same pressure. (See Steam.) MELTING-POINTS OF VARIOUS SUBSTANCES. The following figures are given by Clark (on the authority of Pouillet, Claudel, and Wilson), except those marked *, which are given by Prof. Rob- erts-Austen in his description of the Le Chatelier pyrometer. These latter are probably the most reliable figures. Sulphurous acid - 148° F. Alloy, 1 tin, 1 lead.. 370 to 466° F. Carbonic acid — 108 Mercury - 39 Bromine -f- 9.5 Turpentine 14 Hyponitric acid 16 Ice 32 Nitro-glycerine 45 Tallow 92 Phosphorus 112 Acetic acid. 113 Stearine 109 to 120 Spermaceti 120 Margaric acid 131 to 140 Potassium 136 to 144 Wax 142 to 154 Stearic acid 158 Sodium 194 to 208 Alloy, 3 lead, 2 tin, 5 bismuth 199 Iodine 225 Sulphur 239 Alloy, \y z tin, 1 lead 334 Tin 442 to 446 Cadmium 442 Bismuth 504 to 507 Lead 608 to 618* Zinc 680 to 779* Antimony 810 to 1150 Aluminum 1 157* Magnesium 1200 Calcium Full red heat. Bronze 1692 Silver 1733* to 1873 Potassium sulphate 1859* Gold 1913* to 2282 Copper 1929* to 1996 Cast iron, white... 1922 to 2075* gray 2012 to 2786 2228* Steel 2372 to 2532 " hard 2570*; mild, 2687* Wrought iron 2732 to 2912 Palladium 2732* Platinum 3227* For melting-point of fusible alloys, see Alloys. Cobalt, nickel, and manganese, fusible in highest heat of a forge. Tung- sten and chromium, not fusible in forge, but soften and agglomerate. Plati- num and iridium, fusible only before the oxyhydrogen blowpipe. QUANTITATIVE MEASUREMENT OF HEAT. Unit of Heat.— The British unit of heat, or British thermal unit (B. T. U.), is that quantity of heat which is required to raise the temperature of 1 lb. of pure water 1° Fahr., at or near 39°. 1 F., the temperature of maxi- mum density of water. The French thermal unit, or calorie, is that quantity of heat which is re- quired to raise the temperature of 1 kilogramme of pure water 1° Cent., at or about 4° C, which is equivalent to 39°. 1 F. 1 French calorie = 3.968 British thermal units; 1 B. T. U. = .252 calorie. The " pound calorie " is sometimes used by English writers; it is the quaes- 456 tity of heat required to raise the temperature of 1 lb. of water 1° C. 1 lb. calorie = 2.2046 B. T. U. = 5/9 calorie. The heat of combustion of carbon, to C0 2 , is said to be 8080 calories. This figure is used either for French calories or for pound calories, as it is the number of pounds of water that can be raised 1° O. by the complete combustion of 1 lb. of carbon, or the number of kilogrammes of water that can be raised 1° C. by the combustion of 1 kilo. of carbon; assumiug in each case that all the heat generated is transferred to the water. The Mechanical Equivalent of Heat is the number of foot- pounds of mechanical "energy equivalent to one British thermal unit, heat and mechanical energy being mutually convertible. Joule's experiments, 1843-50, gave the figure 772, which is known as Joule's equivalent. More re- cent experiments by Prof. Rowland (Proc. Am. Acad. Arts and Sciences,. 1880; see also Wood's Thermodynamics) give higher figures, and the most probable average is now considered to be 778. 1 heat-unit is equivalent to 778 ft.-lbs. of energy. 1 ft. lb. = 1/778 =.0012852 heat-units. 1 horse-power = 33,000 ft.-lbs. per minute = 2545 heat-uuits per hour = 42,416 4- per minute = .70694 per second. 1 lb. carbon burned to CO a = 14,544 heat-units. 1 lb. C. per H.P. per hour = 2545 -{-[14544 = 17i# efficiency (.174986). Heat of Combustion of Various Substances in Oxygen. Authority. Heat-units. Cent. Fahr. ( 34,462 62,032 \ 33,808 60,854 ( 34,342 61,816 28,732 51,717 ( 8,080 14,544 { 7,900 14,220 8,137 14,647 7,859 14,146 7,861 14,150 7,901 14,222 2,473 4,451 I 2,403 4,325 { 2,431 4,376 ( 2,3S5 4,293 5,607 10,093 ( 13,120 23,616 { 13,108 23,594 { 13,063 23,513 111,858 21,344 ■{ 11,942 21,496 (11,957 21,523 j 10,102 1 9,915 18,184 17,847 Hydrogen to liquid water at 0° C " to steam at 100° C Carbon (wood charcoal) to carbonic acid, CO a ; ordinary temperatures. Carbon, diamond to C0 2 . . " black diamond to C0 2 " graphite to C0 2 Carbon to carbonic oxide, CO Carbonic oxide to CO a , per unit of CO CO to C0 2 per unit of C = 2}£ X 2403 Marsh-gas, Methane, CH 4 to water and CO a defiant gas, Ethylene, C a H 4 to water and CO a . Benzole gas, C 6 H„ to water and CO^ Favre and Silbermaun. Andrews. Thomsen. Favre and Silbermann. Andrews. Berthelot. Favre and Silbermann. Andrews. Thomsen. Favre and Silbermann. Thomsen. Andrews. Favre and Silbermann. Andrews. Thomsen. Favre and Silbermann. In burning 1 pound of hydrogen with 8 pounds of oxygen to form 9 pounds of water, the units of heat evolved are b2,032 (Favre and S.); but if the resulting product is not cooled to the initial temperature of the gases, part of the heat is rendered latent in the steam. The total heat of 1 lb. of steam at 212° F. is 1146.1 heat-units above that of water at 32°, and 9 X 1146 1 = 10,315 heat-units, which deducted from 62,032 gives 51,717 as the heat evolved by the combustion of 1 lb. of hydrogen and 8 lbs. of oxygen at 32° F. to form steam at 212° F. By the decomposition of a chemical compound as much heat is absorbed or rendered latent as was evolved when the compound was formed. If 1 lb. of carbon is burned to C0 2 , generating 14,544 B.T.U., and the C0 2 thus formed is immediately reduced to CO in the presence of glowing carbon, by the reaction C0 2 + C = 2CO, the result is the same as if the 2 lbs. C had been burned directly to 2CO, generating 2 X 4451 = 8902 heat-units; consequently 14,544 — 8902 = 5642 heat-units have disappeared or become latent, and the SPECIFIC HEAT. 457 "unburning " of C0 2 to CO is thus a cooling operation. (For heats of com- bustion of various fuels, see Fuel.) SPECIFIC HEAT. Thermal Capacity.— The thermal capacity of a body is the quantity of heat required to raise its temperature one degree. The ratio of the heat required to raise the temperature of a given substance one degree to that required to raise the temperature of water one degree from the temperature of maximum density 39.1 is commonly called the specific heat of the sub- stance. Some writers object to the term as being an inaccurate use of the words " specific " and " heat." A more correct name would be " coefficient of thermal capacity." Determination of Specific Heat.— Method by Mixture.— The body whose specific beat is to be determined is raised to a known tempera- ture, and is then immersed in a mass of liquid of which the weight, specific heat, and temperature are known. When both the body and the liquid have attained the same temperature, this is carefully ascertained. Now the quantity of heat lost by the body is the same as the quantity of heat absorbed by the liquid. Let c, w, and t be the specific heat, weight, and temperature of the hot body, and c', w', and V of the liquid. Let T be the temperature the mix- ture assumes. Then, by the definition of specific heat, c X w X (t - T) = heat-units lost by the hot body, and c' X w' x (T - t') = heat-units gained by the cold liquid. If there is no heat lost by radiation or conduction, these must be equal, and av(t - T) = c'iv'(T- t') or c = C '™' l . T ~ *'\ Specific Heats of Various Substances. The specific heats of substances, as given by different authorities, show considerable lack of agreement, especially in the case of gases. The following tablesgive the mean specific heats of the substances named according to Regnaulr. (From Rontgen's Thermodynamics, p. 134.) These specific heats are average values, taken at temperatures which usually come under observation in technical application. The actual specific heats of all substances, in the solid or liquid state, increase slowly as the body expands or as the temperature rises. It is probable that the specific heat of a body when liquid is greater than when solid. For many bodies this has been verified by experiment. Solids. Steel (soft) 0.1165 Steel (hard) 1175 Zinc 0.0956 Brass 0.0939 Ice... 0.5040 Sulphur 0.2026 Charcoal 0.2410 Alumina 0.1970 Phosphorus 0.1887 Antimony 0.0508 Copper 0.0951 Gold 0.0324 Wrought iron 0.1138 Glass 0.1937 Cast iron 0. 1298 Lead 0.0314 Platinum . 0324 Silver 0.0570 Tin 0.0562 Water 1.0000 Lead (melted) 0.0402 Sulphur " 0.2340 Bismuth " 0.0308 Tin " 0.0637 Sulphuric acid 0.3350 Liquids. Mercury 0.0333 Alcohol (absolute) 0.7000 Fusel oil 0.5640 Benzine 0.4500 Ether 0.5034 458 HEAT. Gases. Constant Pressure Constant Volume. Air 0.23751 0.16847 Oxygen 0.21751 0.15507 Hydrogen 3.40900 2.41226 Nitrogen 0.24380 0.17273 Superheated steam 0.4805 0.346 - Carbonic acid 0.217 0.1535 Olefiant'Gas (CH a ) 0.404 0.173 Carbonic oxide 0.2479 0.1758 Ammonia 0.508 0.299 Ether....... 0.4797 0.3411 Alcohol 0.4534 0.3200 Acetic acid. , 0.4125 Chloroform.. 0.1567 In addition to the above, the [following are given by other authorities. (Selected from various sources.) Metals. Platinum at 32° F 0333 (increased .000305 for each 100° F.) Cadmium. .. . 0567 Copper, 32° to 212° F . .094 " 32° to 572° F 1013 Zinc 32°to212°F 0927 32° to 572° F 1015 Nickel ,. .1086 Aluminum, 0° F. to melting- point (A. E. Hunt) 0.2185 Other Solids. Wrought iron (Petit & Dulong). 32° to 212° 1098 " 32° to 39:.'° 115 " 32° to 572° 1218 32° to 662° 1255 Wrought iron (J. C. Hoadley, A. S. M. E., vi. 713), Wrought iron, 32° to 200° 1129 32° to 600° 1327 32° to 2000° 2619 Brickwork and masonry, about. .20 Marble 210 Chalk 215 Quicklime 217 Magnesian limestone 217 Silica 191 Corundum 198 Stones generally 2 to 22 Coal 20to241 Coke 203 Graphite 202 Sulphate of lime 197 Soda 231 Quartz .188 River sand 195 Woods. Pine (turpentine) 467 I Oak 570 Fir 650 | Pear 500 Alcohol, density .793 622 Sulphuric acid, density 1.87 335 1.30 661 Hydrochloric acid 600 Liquids. Olive oil 310 Benzine 393 Turpentine, density .872 472 Bromine 1.111 At Constant At Constant Pressure. Volume. Sulphurous acid 1553 .1246 Light carburetted hydrogen, marsh gas (CH 4 ). .5929 .4683 Blast-furnace gases 2277 Specific Heat of Salt Solution. (Schuller.) Per cent salt in solution 5 10 15 20 25 Specific heat 9306 .8909 .8606 .8490 .8073 Specific Heat of Air.— Regnault gives for the mean value Between — 30° C. and + 10° C 0.23771 0°C. " 100° C .0.23741 0°C. " 200°C 0.23751 Hanssen uses 0.1686 for the specific heat of air at constant volume. The value of this constant has never been found to any degree of accuracy by direct experiment. Prof. Wood gives 0.2375 -4- 1.406 = 0.1689. The ratio of EXPANSION BY HEAT. 459 the specific heat of a fixed gas at constant pressure to the sp. ht. at con- stant volume is given as follows by different writers (Eng'g, July 12, 1889): Regnault, 1.3953; Moll and Beck, 1.4085; Szathmari, 1.4027; J. Macfarlane Gray, 1.4. The first three are obtained from the velocity of sound in air. The fourth is derived from theory. Prof. Wood says: The value of'the ratio for air, as found in the days of La Place, was 1.41, and we have 0.2377 h- 1.41 = 0.1686, the value used by Clausius, Hanssen, and many others. But this ratio is not definitely known. Rankine in his later writings used 1.408, and Tait in a recent work gives 1.404, while some experiments gives less than 1.4 and others more than 1.41. Prof. Wood uses 1.406. Specific Heat of Gases.— Experiments by Mallard and Le Chatelier indicate a continuous increase in the specific heat at constant volume of steam, CO a , and even of the perfect gases, with rise of temperature. The variation is inappreciable at 100° C, but increases rapidly at the high tem- peratures of the gas-engine cylinder. (Robinson's Gas and Petroleum Engines.) Specific Heat and Latent Heat of Fusion of Iron and Steel. (H. H. Campbell, Trans. A. I. M. E., xix. 181.) Akerman. Troilius. Specific heat pig iron, to 1200° C 0.16 1200tol800°C 0.21 Oto 1500° C 0.18 1500tol800°C 0.20 Calculating by both sets of data we have : Akerman. Troilius. Heating from to 1800° C . . 318 330 calories per kilo. Hence probable value is about 325 calories per kilo. Specific heat, steel (probably high carbon) (Troilius) 1175 " soft iron " .1081 Hence probable value solid rail steel 1125 " " " melted rail steel 1275 Akerman. Troilius. Latent heat of fusion, pig iron, calories per kilo. .46 44 " gray pig 33 " *' " white pig 23 From which we may assume that the truth is about : Steel, 20 ; pig iron, 30. EXPANSION BY HEAT. In the centigrade scale the coefficient of expansion of air per degree is 0.003665 = 1/273; that is, the pressure being constant, the volume of a perfect gas increases 1/273 of its volume at 0° C. for every increase in temperature of PC. In Fahrenheit units it increases 1/491.2 = .002036 of its volume at 32° F. for every increase of 1° F. Expansion of Gases by Heat from 32° to 212° F. (Regnault.) Hydrogen Atmospheric air. . Nitrogen Carbonic oxide . . . Carbonic acid Sulphurous acid , Increase in Volume, Pressure Constant. Volume at 32° Fahr. = 1.0, for 0.3661 o.;: 70 0.3670 0.3669 0.3710 0.3903 0.002034 0.002039 0.002039 0.002038 0.002061 0.002168 Increase in Pressure, Volume Constant. Pressure at 32° Fahr. = 1.0, for 0.3667 0.3665 0.3668 0.3667 0.3688 0.3845 002037 002036 002039 002037 002039 002136 If the volume is kept constant, the pressure varies directly as the a temperature. 460 Lineal Expansion of Solids at Ordinary Temperatures. (British Board of Trade; from Clark ) For 1° Fahr. For 1° Cent. Coef- ficient of Expan- sion from 32° to 212° F. Accord- ing to Other Author- ities. Length = 1 Length=l Aluminum (cast) . . Antimony (crj-st.) Brass, cast " plate Brick Bronze (Copper, 17; Tin, 2^; Zinc 1) Bismuth Cement, Portland (mixed), pure ... . Concrete : cement, mortar, and pebbles Copper Ebonite Glass, English flint thermometer " hard Granite, gray, dry " red, dry Gold, pure Iridium, pure Iron, wrought " cast Lead Magnesium Marbles, various -j JJ om j from .00001234 .00000627 .00000957 .00001052 .00000306 .00000986 .00000975 .00000594 .00000795 .00000887 .00004278 .00000451 .00000499 .00000397 .00000438 .00000498 .00000786 .00000356 .00000648 .00000556 .00001571 Masonry, brick J Mercury (cubic expansion) Nickel Pewter Plaster, white Platinum Platinum, 85 per cent I Iridium, 15 " "{"•■•■ Porcelain , Quartz, parallel to major axis, t 0° to 40° C Quartz, perpendicular to major axis. t0° to 40° C Silver, pure Slate Steel, cast " tempered Stone (sandstone), dry .. Rauville Tin Wedgwood wan Wood, pine Zinc Zinc, 8 | Tin, If .00000308 .00000786 .00000256 .00000494 .00009984 .00000695 .00001129 .00000922 .00000479 .00000453 .00000200 .00000434 .00000788 .00001079 .00000577 .00000636 00000689 .00000652 .00000417 .00001163 .00000489 .00000276 .00001407 .00001496 .00002221 .00001129 .00001722 .00001894 .00000550 .00001774 .00001755 .00001070 .00001430 .00001596 .00007700 .00000812 .00000897 .00000714 .00000789 .00000897 .00001415 .00000641 .00001166 .00001001 .00002828 .00000554 .00001415 .00000460 .00000890 .00017971 .00001251 .00002033 .00001660 .00000863 .00000815 .00000360 .00000781 .00001419 .0000194: .00001038 .00001144 .00001240 .00001174 .00000750 .00002094 .00000881 .00000496 .00002532 .00002692 .002221 .001129 .001722 .001894 .000550 .001774 .001755 .001070 .001430 .001596 007700 .000812 .000897 .000714 .000789 .000897 .001415 .000641 .001166 .001001 000554 .001415 .000460 .000890 .017971 .001251 .002033 .001660 .000863 .000815 .000781 .001419 .001943 .001038 .001144 .001240 .001174 .00U750 .002094 .000881 .000496 .002532 Cubical expansion, or expansion of volume = linear expansion x c LATENT HEATS OF FUSION. 461 Absolute Temperature— Absolute Zero.— The absolute zero of a gas is a theoretical consequence of the law of expansion by heat, assuming that it is possible to continue the cooling of a perfect gas until its volume is diminished to nothing. If the volume of a perfect gas increases 1/273 of its volume at 0° C. for every increase of temperature of 1° C, and decreases 1/278 of its volume for every decrease of temperature of 1° C, then at - 273° C. the volume of the imaginary gas would be reduced to nothing. This point — 273° C. or 491.2° F. below the melting-point of ice on the air thermometer, or 492.66° F. be- low on a perfect gas thermometer = — 459.2° F. (or — 460.66°), is called the absolute zero; and absolute temperatures are temperatures measured, on either the Fahrenheit or centigrade scale, from this zero. The freezing point, 32° F., corresponds to 491.2° F. absolute. If p be the pressure and v the volume of a gas at the temperature of 32° F. = 491.2° on the absolute scale = T , and p the pressure, and v the volume of the same quantity of gas at any other absolute temperature T, then pv _ T _ t + 459.2 pv _ p v p^v ~ T ~ 491.2" ; ~T ~ ~1\ ' The value of p v -*- T for air is 53.37, and pv - 53.37T, calculated as fol- lows bv Prof. Wood: A cubic foot of dry air at 32° F. at theSsea-level weighs 0.080728 lb. The volume of one pound is v = ■ n o n ^ 9 ^ — 12.387 cubic feet. The pressure per square foot is 2116.2 lbs. p v _ 2116.2 X 12.387 _ 26214 _ ~T^ ~ 491.13 - 4903 ~ 5d ' d7 ' The figure 491.13 is the number of degrees that the absolute zero is below the melting-point of ice, by the air thermometer. On the absolute scale, whose divisions would be indicated by a perfect gas thermometer, the cal- culated value approximately is 492.66, which would make pv = 53.21 T. Prof. Thomson considers that - 273.1° C. = — 491.4° F., is the most probable value of the absolute zero. See Heat in Ency. Brit. Expansion of Liquids from 32° to 212° F.— Apparent ex- pansion in glass (Clark). Volume at 212°, volume at 32° being 1: Water 1.0466 Nitric acid 1.11 Water saturated with salt 1.05 Olive and linseed oils 1.08 Mercury.. 1 .0182 Turpentine and ether 1 .07 Alcohol!...; 1.11 Hydrochlor. and sulphuric acids 1.06 For water at various temperatures, see Water. For air at various temperatures, see Air. LATENT HEATS OF FUSION AND EVAPORATION. Latent Heat means a quantity of heat which has disappeared, having been employed to produce some change other than elevation of temperature. By exactly reversing that change, the quantity of heat which has dis- appeared is reproduced. Maxwell defines it as the quantity of heat which must be communicated to a body in a given state in order to convert it into another state without changing its temperature. Latent Heat of Fusion.— When a body passes from the solid to the liquid state, its temperature remains stationary, or nearly stationary, at a certain melting point during the whole operation of melting; and in order to make that operation go on, a quantity of heat must be transferred to the substance melted, being a certain amount for each unit of weight of the substance. This quantity is called the latent heat of fusion. When a body passes from the liquid to the solid state, its temperature remains stationary or nearly stationary during the whole operation of freez- ing; a quantity of heat equal to the latent heat of fusion is produced in the body and rejected into the atmosphere or other surrounding bodies. The following are examples in British thermal units per pound, as given by Rankine: qnhstanoes Melting Latent Heat Substances. Points. of Fusion. Ice (according to Person) 32 142.65 Spermaceti 56 148 Beeswax 140 175 Phosphorus 177 9.06 Sulphur... 405 16.86 Tin 426 500 462 HEAT. Prof. Wood considers 144 heat units as the most reliable value for the latent heat of fusion of ice. Box gives only 26.6 for tin. Clements gives 233 for cast iron. Latent Heat of Evaporation.— When a body passes from the solid or liquid to the gaseous state, its temperature during the operation remains stationary at a certain boiling point, depending on the pressure of the vapor produced ; and in order to make the evaporation go on, a quantity of heat must be transferred to the substance evaporated, whose amount for each unit of weight of the substance evaporated depends on the temperature. That heat does not raise the temperature of the substance, but disappears in causing it to assume the gaseous state, and it is called the latent heat of evaporation. When a body passes from the gaseous state to the liquid or solid state, its temperature remains stationary, during that operation, at the boiling-poiut corresponding to the pressure of the vapor: a quantity of heat equal to the latent heat of evaporation at that temperature is produced in the body; and in order that the operation of condensation may go on, that heat must be transferred from the body condensed to some other body. The following are examples of the latent heat of evaporation in British thermal units, of one pound of certain substances, when the pressure of the vapor is one atmosphere of 14.7 lbs. on the square inch: Q„v, danno Boiling-point under Latent Heat in bubstance. one atm Fahr British units. Water 212.0 965.7 (Regnault.) Alcohol , 172.2 364.3 (Andrews.) Ether 95.0 162.8 Bisulphide of carbon 114.8 156.0 The latent heat of evaporation of water at a series of boiling-points ex tending from a few degrees below its freezing-point up to about 375 degrees Fahrenheit has been determined experimentally by M. Regnault. The re- sults of those experiments are represented approximately by the formula, in British thermal units per pound, I nearly = 1091.7 - 0.7(£ - 32°) = 965.7- 0.7(£ - 212°). The Total Heat of Evaporation is the sum of the heat which disappears in evaporating one pound of a given substance at a given tem- perature (or latent heat of evaporation) and of the heat required to raise its temperature, before evaporation, from some fixed temperature up to the temperature of evaporation. The latter part of the total heat is called the sensible heat. In the case of water, the experiments of M„ Regnault show that the total heat of steam from the temperature of melting ice increases at a uniform rate as the temperature of evaporation rises. The following is the formula in British thermal units per pound: h = 1091.7 + 0.305(f - 32°). For the total heat, latent heat, etc., of steam at different pressures, see table of the Properties of Saturated Steam. For tables of total heat, latent heat, and other properties of steams of ether, alcohol, acetone, chloroform, chloride of carbon, and bisulphide of carbon, see Rontgen's Thermodynam- ics (Dubois's translation.) For ammonia and sulphur dioxide, see Wood's Thermodynamics; also, tables under Refrigerating Machinery, in this book. EVAPORATION AND DRYING. In evaporation, the formation of vapor takes place on the surface; in boil- ing, within the liquid : the former is a slow, the latter a quick, method of evaporation. If we bring an open vessel with water under the receiver of an air-pump and exhaust the air the water in the vessel will commence to boil, and if we keep up the vacuum the water will actually boil near its freezing-point. The formation of steam in this case is due to the heat which the water takes out of the surroundings. Steam formed under pressure has the same temperature as the liquid in which it was formed, provided the steam is kept under the same pressure. By properly coolirjg the rising steam from boiling water, as in the multiple- effect evaporating' systems, we can regulate the pressure so that the water b ils at low temperatures. EVAPORATION. 463 Evaporation of Water in Reservoirs.— Experiments at the Mount Hope Reservoir, Rochester, N. Y., in 1891, gave the following results: July. Aug. Sept. Oct. Meau temperature of air in shade 70.5 70.3 68.7 53.3 "' u " water in reservoir 68.2 70.2 66.1 54.4 " humidity of air, per cent 67.0 74.6 75.2 74.7 Evaporation in inches during month 5.59 4.93 4.05 3.23 Rainfall in inches during month 3.44 2.95 1.44 2.16 Evaporation of Water from Open Channels, (Flynn's Irrigation Canals and Flow of Water.)— Experiments from 1881 to 1885 in Tulare County, California, showed an evaporation from a pan in the river equal to an average depth of one eighth of an inch per day throughout the year. When the pan was in the air the average evaporation was less than 3/16 of an inch per day. The average for the month of August was 1/3 inch per day, and for March and April 1/12 of an inch per day. Experiments in Colorado show that evaporation ranges from .088 to .16 of an inch per day during the irrigating season. In Northern Italy the evaporation was from 1/12 to 1/9 inch per day, while in the south, under the influence of hot winds, it was from 1/6 to 1/5 inch per day. In the hot season in Northern India, with a decidedly hot wind blowing, the average evaporation was y% inch per day. The evaporation increases with the temperature of the water. Evaporation by the Multiple System.— A multiple effect is a series of evaporating vessels each having a steam chamber, so connected that the heat of the steam or vapor produced in the first vessel heats the second, the vapor or steam produced in the second heats the third, and so on. The vapor from the-last vessel is condensed in a condenser. Three vessels are generally used, in which case the apparatus is called a Triple Effect. In evaporating in a triple effect the vacuum is graduated so that the liquid is boiled at a constant and low temperature. Resistance to Roiling 1 .— Brine. (Rankine.)— The presence in a liquid of a substance dissolved in it (as salt in water) resists ebullition, and raises the temperature at which the liquid boils, under a given pressure; but unless the dissolved substance enters into the composition of the vapor, the relation between the temperature and pressure of saturation of the vapor remains unchanged. A resistance to ebullition is also offered by a vessel of a material which attracts the liquid (as when water bcils in a glass vessel), and the boiling take place by starts. To avoid the errors which causes of this kind produce in the measurement of boiling-points, it is advisable to place the thermometer, not in the liquid, but in the vapor, which shows the true boiling-point, freed from the disturbing effect of the attractive nature of the vessel. The boiling-point of saturated brine under one atmosphere is 226° Fahr., and that of weaker brine is higher than the boiling-point of pure water by 1.2° Fahr., for each 1/32 of salt that the water contains. Average sea-water contains 1/32; and the brine in marine boilers is not suf- fered to contain more than from 2/32 to 3/32. Methods of Evaporation Employed in the Manufacture of Salt. (F. E. Engelhardt, Chemist Onondaga Salt Springs; Report for 1889.)— 1. Solar heat— solar evaporation. 2. Direct fire, applied to the heat- ing surface of the vessels containing brine— kettle and pan methods. 3. The steam-grainer system— steam-pans, steam-kettles, etc. 4. Use of steam and a reduction of the atmospheric pressure over the boiling brine— vacuum system. When a saturated salt solution boils, it is immaterial whether it is done under ordinary atmospheric pressure at 2^8° F., or under four atmospheres with a temperature of 320° F., or in a vacuum under 1/10 atmosphere, the result will always be a fine-grained salt. The fuel consumption is stated to be as follows: By the kettle method, 40 to 45 bu. of salt evaporated per ton of fuel, anthracite dust burned on per- forated grates; evaporation, 5.53 lbs. of water per pound of coal.. By the pan method, 70 to 75 bu. per ton of fuel. By vacuum pans, single effect, 86 lbs. per ton of anthracite dust (2000 lbs.). With a double effect nearly double that amount can be produced. 464 Heat. Solubility of Common Salt in Pure Water. (Andrese.) Temp, of brine, F 100 parts water dissolve parts . . 100 parts brine contain salt — 52 50 5.63 35.69 5.27 26.30 56 104 140 176 3.03 36.32 37.06 38.00 3.49 26.64 27.04 27.54 According to Poggial, 100 parts of water dissolve at 229.66° F., 40.35 parts of salt, or in per cent of brine, 28.749. Gay Lussac found that at 229.72° F., 100 parts of pure water would dissolve 40.38 parts of salt, in per cent of brine, 28.764 parts. The solubility of salt at 229° F. is only 2.5$ greater than at 32°. Hence we cannot, as in the case of alum, separate the salt from the water by allowing a saturated solution at the boiling point to cool to a lower temperature. Solubility of Sulpbate of Lime in Pure "Water. (Marignac.) Temperature F. degrees. Parts water to dissolve I 1 part gypsum f Parts water to dissolve 1 ( part anhydrous CaS0 4 ) In salt brine sulphate of lime is much more soluble than in pure water. In the evaporation of salt brine the accumulation of sulphate of lime tends to stop the operation, and it must be removed from the pans to avoid waste of fuel. The average strength of brine in the New York salt districts in 1889 was 69.38 degrees of the salinometer. Strength of Salt Brines.— The following table is condensed from one given in U. S. Mineral Resources for 1888, on the authority of Dr. Englehardt. Relations between Salinometer Strength, Specific Gravity, Solid Contents, etc., of Brines of Different Strengths. 32 64.5 89.6 100.4 105.8 127.4 186.8 212 415 386 371 368 370 375 417 452 525 488 470 466 468 474 528 572 «H , ^ T3 CD 2 5b CD T3 U CD "S s o 1 W) a be o s H o a o . o c ^ g o.S IS o8«w '" CD . 'ob'S « £ . $$ CD O I 5 o u "* £ O of coal required t oduce a bushel o It, 1 lb. coal evapo ting 6 lbs. of water ** CO til cc cS O 03 &« Sg c$ ft ® 0^ S5S ^a P % O 6C-S fc cS 3 ££ 3 "efiE-i £-X3H W £ o >c3o > a . ffia Ht-ft ffiw W ^m Laurens Copper coils... 2 Copper coils. .292 .981 1.20 315 974 1120 Havrez.. Copper coil . . . .268 1.26 280 1200 Perkins. Iron coil .24 .22 215 208.2 j Steam pressure 1 = 100. ( Steam pressure \ = 10. Box Iron tube .... .235 .196 .206 230 207 210 Havrez.. Cast-iron boil- .077 .105 82 100 From the above it would appear that the efficiency of iron surfaces is less than that of copper coils, plate surfaces being far inferior. In all experiments made up to the present time, it appears that the tem- perature of the condensing water was allowed to rise, a mean between the initial and final temperatures being accepted as the effective temperature. But as water becomes warmer it circulates more rapidly, thereby causing the water surrounding the coil to become agitated and replaced bj cooler water, which allows more heat to be transmitted. CONDUCTION" AND CONVECTION OF HEAT. 473 Again, in accepting the mean temperature as that of the condensing me- dium, the assumption is made that the rate of condensation is in direct pro- portion to the temperature of the condensing water. In order to correct and avoid any error arising from these assumptions and approximations, experiments were undertaken, in which all the condi- tions were constant during each test. The pressure was maintained uniform throughout the coil, and provision was made for the free outflow of the condensed steam, in order to obtain at all times the full efficiency of the condensing surface. The condensing water was continually stirred to secure uniformity of temperature, which was regulated by means of a steam-pipe and a cold-water pipe entering the tank in which the coil was placed. The following is a condensed statement of the results Heat Transmitted per Square Foot of Cooling Surface, per Degree of Difference of Temperature. (British Thermal Units.) Temperature of Condens- ing Water. 1-in. Iron Pipe; Steam inside, 60 lbs. Gauge Pressure. 1^ in. Pipe; Steam inside, 10 lbs. Pressure. \y 2 in. Pipe; Steam inside, 10 lbs. Pressure. 1^ in. Pipe; Steam inside, 60 lbs. Pressure. 80 100 120 140 160 180 200 265 269 272 277 281 299 313 128 130 137 145 158 174 200 230 260 267 271 270 '239 247 276 306 349 419 The results indicate that the heat transmitted per degree of difference of temperature in general increases as the temperature of the condensing water is increased. The amount transmitted is much larger with the steam on the outside of the coil than with the steam inside the coil. This may be explained in part by the fact that the condensing water when inside the coil flows over the sur- face of conduction very rapidly, and is more efficient for cooling than when contained in a tank outside of the coil. This result is in accordance with that found by Mr. Thomas Craddock, which indicated that the rate of cooling by transmission of heat through metallic surfaces was almost wholly dependent on the rate of circulation of the cooling medium over the surface to be cooled. Transmission of Heat in Condenser Tubes. {Eng'g, Dec. 10, 1875, p. 449.).— In 1874 B. C. Nichol made experiments for determining the rate at which heat was transmitted through a condenser tube. The results went to show that the amount of heat transmitted through the walls of the tube per estimated degree of mean difference of temperature increased considerably with this difference. For example: Estimated mean difference of Vertical Tube. temperature between inside and outside of tube,, degrees Fahr. . Heat-units transmitted per hour per square foot of surface per degree of mean diff . of temp .... 422 531 561 These results seem to throw doubt upon Mr. Isherwood's statement that the rate of evaporation per degree of difference of temperature is the same for all temperatures. Mr. Thomas Craddock found that water was enormously more efficient than air for the abstraction of heat through metallic surfaces in the process of cooling. He proved that the rate of cooling by transmission of heat through metallic surfaces depends upon the rate of circulation of the cool- ing medium over the surface to be cooled. A tube filled with hot water, moved by rapid rotation at the rate of 59 ft. per second, through air, lost as much heat in one minute as it did in still air in 12 minutes. In water, at a velocity of 3 ft. per second, as much heat was abstracted in half a minute as was abstracted in one minute when it was at rest in the water. Mr. Crad- dock concluded, further, that the circulation of the cooling flujd became of Horizontal Tube 128 151.9 J52.9 111.6 146.2 150.4 610 737 474 greater importance as the difference of temperature on the two sides of the plate became less. (Clark, R. T. D.. p. 461.) Heat Transmission through Cast-iron Plates Pickled in Nitric Acid.— Experiments by R. C. Carpenter (Trans. A. S. M. E., xii 179) show a marked change in the conducting power of the plates (from steam to water), due to prolonged treatment with dilute nitric acid. The action of the nitric acid, by dissolving the free iron and not attacking the carbon, forms a protecting surface to the iron, which is largely com- posed of carbon. The following is a summary of results: Character of Plates, each plate 8.4 in by 5.4 in., exposed surface 27 sq. ft. Increase in Tempera- ture of 3.125 lbs. of Water each Minute. Proportionate Thermal Units Transmitted for each Degree of Difference of Temperature per Square Foot per Hour. Rela- tive Trans- mission of Heat. Cast iron— untreated skin on, but clean, free from rust Cast iron— nitric acid, 1% sol., 9 days . " ' 1% sol., 18 days. " " \% sol., 40 days. " " 5% sol., 9 days. . " ' " 5% sol., 40 days. Plate of pine wood, same dimensions as the plate of cast iron 13.90 11.5 9.7 9.6 io!e 113.2 97.7 80.08 77.8 87.0 1.! 100.0 86.3 70.7 68.5 1.6 "^ In the actual engine this efficiency can never be attained, for the difference between the quantity of heat received into the cylinder and that rejected into the condenser is not all converted into work, much of it being lost by radiation, leakage, etc. In the steam engine the phenomenon of cylinder condensation also tends to reduce the efficiency. PHYSICAL PROPERTIES OF GASES. (Additional matter on this subject will be found under Heat, Air, Gas, and Steam.) When a mass of gas is enclosed in a vessel it exerts a pressure against the vails. This pressure is uniform on every square inch of the surface of the vessel; also, at any point in the fluid mass the pressure is the same in every direction. In small vessels containining gases the increase of pressure due to weight may be neglected, since all gages are very light; but where liquids are con- cerned, the increase in pressure due to their weight must always be taken into account. Expansion of Gases, Marriotte's I*aw. — The volume of a gas diminishes in the same ratio as the pressure upon it is increased. This law is by experiment found to be very nearly true for all gases, and 3 known as Boyle's or Mariotte's law. If p = pressure at a volume v, and p x = pressure at a volume v x , p 1 v 1 = pv; Pi — — p; pv = a constant. The constant, C, varies with the temperature, everything else remaining the same. Air compressed by a pressure of seventy -five atmospheres has a volume about 2% less than that computed from Boyle's law, but this is the greatest divergence that is found below 160 atmospheres pressure. Law of Charles.— The volume of a perfect gas at a constant pressure is proportional to its absolute temperature. If v a be the volume of a gas at 32° F., and v x the volume at any other temperature, t t , then ft 1 + 459.2\ / :" t x .- 32°\ V - = V °V 491.2 )' V - = V + -49lX>»' or v x = [1 + 0.002036(^ - 32°)]tv If the pressure also change fromp top l5 „ _ v *o(*i+459.2\ 1 °pA 49L2~/- The Densities of Gases and Vapors are simply proportional to their atomic weights. Avogadro's Liaw.— Equal volumes of all gases, under the same con- ditions of temperature and pressure, contain the same number of mole- cules. To find the weight of a gas in pounds per cubic foot at 32° F., multiply half the molecular weight of the gas by .00559. Thus 1 cu. ft. marsh-gas, CH4, 12 4- 4 == -~— X .00559 = .0447 lb, 480 PHYSICAL PROPERTIES OF GASES. When a certain volume of hydrogen combines with one half its volume of oxygen, there is produced an amount of water vapor which will occupy the same volume as that which was occupied by the hydrogen gas when at the same temperature and pressure. Saturation-point of Vapors.— A. vapor that is not near the satura- tion-point behaves like a gas under changes of temperature and pressure; but if it is sufficiently compressed or cooled, it reaches a point where it be- gins to condense: it then no longer obeys the same laws as a gas, but its pressure cannot be increased by diminishing the size of the vessel containing it, but remains constant, except when the temperature is changed. The only gas that can prevent a liquid evaporating seems to be its own vapor. Dalton's Law of Gaseous Pressures.—- Every portion of a mass of gas inclosed in a vessel contributes to tiie pressure against the sides of the vessel the same amount that it would have exerted by itself had no other gas been present. Mixtures of Vapors and Gases.— The pressure exerted against the interior of a vessel by a given quantity of a perfect gas enclosed in it is the sum of the pressures which any number of parts into which such quan- tity might be divided would exert separately, if each were enclosed in a vessel of the same bulk alone, at the same temperature. Although this law is not exactly true for any actual gas, it is very nearly true for many. Thus if 0.080728 lb. of air at 32° F., being enclosed in a vessel of one cubic foot capacity, exerts a pressure of one atmosphere or 14.7 pounds, on each square inch of the interior of the vessel, then will each additional 0.080728 lb. of air which is enclosed, at 32°, in the same vessel, produce very nearly an addi- tional atmosphere of pressure. The same law is applicable to mixtures of gases of different kinds. For example, 0,12344 lb. of carbonic-acid gas, at 32°, being enclosed in a vessel of one cubic foot in capacity, exerts a pressure of one atmosphere; consequently, if 0.080728 lb. of air and 0.12344 lb. of carbonic acid, mixed, be enclosed at the temperature of 32°, in a vessel of one cubic foot of capacity, the mixture will exert a pressure of two atmos- pheres. As a second example: Let 0.080728 lb. of air, at 212°, be enclosed in a vessel of one cubic foot; it will exert a pressure of 212 + 459.2 . MC , oc . , ,. n n — 1 .366 atmospheres. 32 + 409.2 Let 0.03797 lb. of steam, at 212°, be enclosed in a vessel of one cubic foot ; it will exert a pressure of one atmosphere. Consequently, if 0.080728 lb. of air and 0.03797 lb. of steam be mixed and enclosed together, at 212°, in a vessel of one cubic foot, the mixture will exert a pressure of 2.366 atmospheres. It is a common but erroneous practice, in elementary books on physics, to de- scribe this law as constituting a difference between mixed and homogeneous gases; whereas it is obvious that for mixed and homogeneous gases the law of pressure is exactly the same, viz., that the pressure of the whole of a gaseous mass is the sum of the pressures of all its parts This is one of the laws of mixture of gases and vapors. A second law is that the presence of a foreign gaseous substance in con- tact with the surface of a solid or liquid does not affect the density of the vapor of that solid or liquid unless there is a tendency to chemical com- bination between the two substances, in which case the density of the vapor is slightly increased. (Rankine, S. E., p. 239.) Flow of Gases.— By the principle of the conservation of energy, it may be shown that the velocity with which a gas under pressure will escape into a vacuum is inversely proportional to the square root of its density; that is, oxygen, which is sixteen times as heavy as hydrogen, would, under exactly the same circumstances, escape through an opening only one fourth as fast as the latter gas. Absorption of Gases foy Liquids.— Many gases are readily ab- sorbed by water. Other liquids also possess this power in a greater or less degree. Water will for example, absorb its own volume of carbonic-acid gas, 430 times its volume of ammonia, 2^g times its volume of chlorine, and only about 1/20 of its volume of oxygen. The weight of gas that is absorbed by a given volume of liquid is propor- tional to the pressure. But as the volume of a mass of gas is less as the pressure is greater, the volume which a given amount of liquid can absorb at a certain temperature will be constant, whatever the presstire. Water, for example, can absorb its own volume of carbonic-acid gas at atmospheric pressure; it will also dissolve its own volume if the pressure is twice as great, but in that case the gas will be twice as dense, and consequently twice the weight of gas is dissolved, PRESSURE OF THE ATMOSPHERE. 481 AIR. Properties of Air.— Air is a mechanical mixture of the gases oxygen and nitrogen; 21 parts O and 79 parts N by volume, 23 parts O and 77 parts N by weight. The weight of pure air at 32° F. and a barometric pressure of 29.92 inches of mercury, or 14.6963 lbs. per sq. in., or 2116.3 lbs. per sq. ft., is .080728 lbs. The volume of 1 lb. is 12.387 cubic feet. At any other temperature and barometric pressure its weight in lbs. per cubic foot is W~ '' , where B = height of the barometer, T= temperature Fahr., and 1.3253 = weight in lbs. of 459.2 c. ft. of air at 0° F. and one inch barometric pressure. Air expands 1/491.2 of its volume for every increase of 1° F., and its volume varies inversely as the pressure. Volume, Density, and Pressure of Air at Various Temperatures. (D. K. Clark.) Volume at Atmos. Pressure at Constant Pressure. Density, lbs. per Cubic Foot at Atmos. Pressure. Volume. Fahr. Cubic Feet Compara- Lbs. per Sq. In. Compara- in 1 lb. tive Vol. tive Pres. 11.583 .881 .086331 12.96 .881 32 12.387 .943 .080728 13.86 .943 40 12.586 .958 .079439 14.08 .958 50 12.840 .977 .077884 14.36 .977 62 13.141 1.000 .076097 14.70 1.000 70 13.342 1.015 .074950 14.92 1.015 80 13.593 1 .034 .073565 15.21 1.034 90 13.845 1.054 .072230 15.49 1.054 100 14.096 1.073 .070942 15.77 1.073 110 14.344 1.092 .069721 16.05 1.092 120 14.592 1.111 .06^500 16.33 1.111 130 14.846 1.130 .067361 16.61 1.130 140 15.100 1.149 .066221 16.89 1.149 150 15.351 1.168 .065155 17.19 1.168 160 15.603 1.187 .064088 17.50 1.187 170 15.854 1.206 .063089 17.76 1.206 180 16.106 1.226 .062090 18.02 1.226 200 16.606 1.264 .060210 18.58 1.264 210 16.860 1.283 .059313 18.86 1.283 212 16.910 1.287 .059135 18.92" 1.287 The Air-manometer consists of a long vertical glass tube, closed at theuppir end, open at the lower end, containing air, provided with a scale, and immersed, along with a thermometer, in a transparent liquid, such as water or oil, contained in a strong cylinder of glass, which communicates with the vessel in which the pressure is to be ascertained. The scale shows the volume occupied by the air in the tube. Let v be that volume, at the temperature of 32° Fahrenheit, and mean pressure of the atmosphere, p ; let v x be the volume of the air at the tem- perature t, and under the absolute pressure to be measured p, ; then Pi = " _ (t + 459.2° ) Po v Pressure of the Atmosphere at Different Altitudes. At the sea-level the pressure of the air is 14.7 pounds per square inch; at J4 of a mile above the sea-level it is 14.02 pounds; at % mile, 13.33; at $£ mile, 12,66; at 1 mile, 12.02; at V& mile, 11.42; at \\i mile, 10.88; and at 2 482 miles, 9.80 pounds per square inch. For a rough approximation we may- assume that the pressure decreases Y» pound per square inch for every 1000 feet of ascent. It is calculated that at a height of about 3V£ miles above the sea-level the weight of a cubic foot of air is only one half what it is at the surface of the earth, at seven miles only one fourth, at fourteen miles only one sixteenth, at twenty-one miles only on9 sixty-fourth, and at a height of over forty- ' five miles it becomes so attenuated as to have no appreciable weight. The pressure of the atmosphere increases with the depth of shafts, equal to about one inch rise in the barometer for each 900 feet increase in depth: this may be taken as a rough-and-ready rule for ascertaining the depth of shafts. Pressure of tlie Atmosphere per Square Inch and per Square Foot at Various Readings of the Barometer. Rule.— Barometer in inches x .4908 = pressure per square inch; pressure per square inch x 144 = pressure per square foot. Pressure Pressure Pressure Pressure per Sq. In. per Sq. Ft. per Sq. In. per Sq. Ft. in. lbs. lbs.* in. lbs. lbs.* 28.00 13.74 1978 29.75 14.60 2102 28.25 13.86 1995 30.00 14.72 2119 28.50 13.98 2013 30.25 14.84 2136 28.75 14.11 2031 30.50 14.96 2154 29.00 14.23 2049 30.75 15.09 2172 29.25 14.35 2066 31.00 15.21 2190 29.50 14.47 2083 * Decimals omitted. For lower pressures see table of the Properties of Steam. Barometric Readings corresponding with Different Altitudes, in French and English Measures. Read- Reading Reading Reading Alti- tude. ing of Barom- Altitude. of Barom- Alti- tude. of Barom- Altitude. of Barom- eter. eter. eter. eter. meters. mm. feet. inches. meters. mm. feet. inches. 762 0. 30. 1147 660 3763.2 25.98 21 760 68.9 29.92 1269 650 4163.3 25.59 127 750 416.7 29.52 1393 640 4568.3 25.19 234 740 767.7 29.13 1519 630 4983.1 24.80 342 730 1122.1 28.74 1647 620 5403.2 24.41 453 720 1486.2 28.35 1777 610 5830.2 24.01 564 710 1850.4 27.95 1909 600 6243. 23.62 678 700 2224.5 27.55 2043 590 6702.9 23.22 793 690 2599.7 27.16 2180 580 7152.4 22.83 909 680 2962.1 26.77 2318 570 7605.1 22.44 1027 670 3369.5 26.38 2460 560 8071. 22.04 Levelling by the Barometer and hy Boiling Water. (Trautwine.) — Many circumstances combine to render the results of this kind of levelling unreliable where great accuracy is required. It is difficult to read off from an aneroid (the kind of barometer usually employed for engineering purposes) to within from two to five or six feet, depending on its size. The moisture or dryness of the air affects the results; also winds, the vicinity of mountains, and the daily atmospheric tides, which cause incessant and irregular fluctuations in the barometer. A barometer hang- ing quietly in a room will often vary 1/4 of an inch within a few hours, cor- responding to a difference of elevation of nearly 100 feet. No formula can possibly be devised that shall embrace these sources of error. MOISTURE IN THE ATMOSPHERE. 483 To Find the Difference in Altitude of Two Places.— Take from the table the altitudes opposite to tue two boiling temperatures, or to the two barometer readings. Subtract the one opposite the lower reading from that opposite the upper reading. The remainder will be the required height, as a rough approximation. To correct this, add together the two thermometer readings, and divide the sum by 2, for their mean. From table of corrections for temperature, take out the number under this mean. Multiply the approximate height just found by this number. At 70° F. pure water will boil at 1° less of temperature for an average of about 550 feet of elevation above sea-level, up to a height of 1/2 a mile. At the height of 1 mile, 1° of boiling temperature will correspond to about 560 feet of elevation. In the table the mean of the temperatures at the two stations is assumed to be 32°F., at which no correction for temperature is necessary in using the table. ^ftd £ I>1- ^•s ^ a . s" if^h 2.2 $%i$ 2.9 eg 5"Sl« 03"" '**~ (i«M « •- PQ < & M < & pq S M < OQ 184° 16.79 15,221 196 21.71 8,481 208 27.73 2,063 185 17.16 14,649 197 22.17 7,932 208.5 28.00 1,809 186 17.54 14,075 198 22.64 7,381 209 28.29 1,539 187 17.93 13,498 199 23.11 6,843 209.5° 28.56 1,290 188 18.32 12,934 200 23.59 6,304 210 28.85 1,025 189 18.72 12,367 201 24.08 5,764 210.5 29.15 754 190 19.13 11,799 202 24.58 5,225 211 29.42 512 191 19.54 11,243 203 25.08 4,697 211.5 29.71 255 192 19.96 10,685 204 25.59 4,169 212 30.00 S.L.= 193 20.39 10,127 205 26.11 3,642 212.5 30.30 -261 194 20.82 9,579 206 26.64 3,115 213 30.59 -511 195 21.26 9,031 207 27.18 2,589 Corrections for Temperature. Mean temp. F. in shade. ! 10° I 20° I Multiply by .933 |.954|.975|. M 40° I 50° > 1.016|l.036|l.058|l. 70° 180° .079|1. 100|1. 121 1.143 I 100«| IWoisture in the Atmosphere.— Atmospheric air always contains a small quantity of carbonic-acid gas and a varying quantity of aqueous vapor. Pure mountain air contains about 3 to 4 parts of carbonic acid in 10,000. A properly ventilated room should contain not more than six parts in 10,000. The degree of saturation or relative humidity of the air is determined by the use of the dry and wet bulb thermometer. The degree of saturation for a number of different readings of the thermometer is given in the following table : Indications op the Hygrometer (Dry and Wet Bulb), from Mr. Glaisher's Observations at Greenwich. Difference of Temperature or Degrees of Cold in the Wet- bulb Thermometer. Temperature of the Air, Fahrenheit. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1516 17 18 19 20 21 22 23 24 Degrees of Humidity, Saturation being 100. 32° 87 75 42° 7K 72 m 6(1 54 49 44 40 36 33 30 27 52° W St* 74 69 64 59 54 50 46 42 39 36 33 30 27 25 62° 94 88 82 77 72 67 62 58 54 50 47 44 41 38 35 32 30 28 26 21 72° 04 S9 84 79 74 65 61 57 54 51 48 45 42 39 36 34 32 30 28 26 24 23 22 82° 90 85 72 64 60 57 54 51 48 45 42 40 3 y 35 33 31 29 27 25 92° 95 90 85 81 " 73 71 66 62 59 5t; 53 50 47 45 43 41 38 36 34 32 30 28 26 484 air. Weights of Air, Vapor of Water, and Saturated Mixtures of Air and Vapor at Different Temperatures, under the Ordinary Atmospheric Pressure of 29.921 inches of Mercury. r 3 H III Mixtures of Air Saturated with Vapor. Weight of Cubic Foot of the © §3 Elastic Force of the Air in Mixture Mixture of Air and Vapor. Weight of 3."S cS t« 3 %% Vapor ~s3 J5 33 a P.? ©'IS £° of Airand Vapor, Weight of the Air, lbs. Weight of the Total W'ghtof mixed with lib. of Air, pounds. 11 '33 tw 979 •«» ■*» -™ ™ ' " Fliegner's Equations for Flow of Air from a Reservoir through an Orifice. (Peabody's Thermodynamics, p. 135.) For Pi > 2pa, G = 0.530 F J^ ; Vt 1 Pl >2pa, Q = l.mF^/ pa( ^- pa H G = flow of air through the orifice in lbs. per sec, F — area of orifice in square inches, p 1 = pressure in reservoir in lbs. per sq. in., pa — pressure of atmosphere, T x — absolute temperature, Fahrenheit, of air in reservoir. Clark (Rules, Tables, and Data, p. 891) gives, for the velocity of flow of air through an orifice due to small differences of pressure, ,= 0j /fxrr, 2 x( 1 + l^)x^, or, simplified, !Ci/(l+.00203U- 32^ /p in which V— velocity in feet per second; 2g — 64.4; h — height of the column of water in inches, measuring the difference of pressure; / = the tempera- ture Fahr. ; and p = barometric pressure in inches of mercury. 773.2 is the volume of air at 32° under a pressure of 29.92 inches of mercury when that of an equal weight of water is taken as 1. For 62° F., the formula becomes V = 363C A/ -, and if p = 29.92 inchesF = 66.35C Vh The coefficient of efflux C, according to Weisbach, is: For conoidal mouthpiece, of form of the contracted vein, with pressures of from .23 to 1.1 atmospheres C — .97 to .99 Circular orifices in thin plates C = .56 to .79 Short cylindrical mouthpieces C — .81 to .84 The same rounded at the inner end C = .92 to .93 Conical converging mouthpieces C — .90 to .99 Flow of Air in Pipes.— Hawksley (Proc. Inst. C. E.. xxxiii, 55) states that his formula for flow of water in pipes v = 48 a/ — =r- may also be employed for flow of air. In this case H. — height in feet of a column of air required to produce the pressure causing the flow, or the loss of head 486 for a given flow; v = velocity in feet per second, D = diameter in feet, L - length in feet. If the head is expressed in inches of water, h, the air being taken at 62° F , its weight per cubic foot at atmospheric pressure — .0761 lb. Then H = 076 ^" x 12 - 68.37i. If d = diameter in inches, D = — and the formula becomes v = 114.5 - / hd ■ »hich h — inclies of. water column, d = dia Lv* Lv* The quantity in cubic feet per second is "144 T L ' a Y .39/,' ' l --^w- -- .7851 - The horse-power required to drive air through a pipe is the volume Q in cubic feet per second multiplied by the pressure in pounds per square f.ot and divided by 550. Pressure in pounds per square foot = P — inches -f water column x 5.196, whence horse-power = = QP _ Qh Q 3 L 41.3d 5 " If the head or pressure causing the flow is expressed in pounds per square inch = p, then h = 27.71p, and the above formulae become 363,300p Q=c y £ ' P - 363,300d' iTi/ifE. v - Q* L ■ d - //ZM QlUp = .2&18Qp = .02421^. Volume of Air Transmitted in Cubic Feet per Minute in Pipes of Various Diameters. Formula Q = ^d>v x 60. 144 >>* Actual Diameter of Pipe in Inches. £ OS 1 2 3 4 5 6 8 10 12 16 20 24 frft 1 .327 1.31 2.95 5.24 8.18 11.78 20.94 32.73 47.12 83.77 130.9 188.5 2 .655 2.62 5.89 10.47 16.36 23.56 41 89 65.45 94.25 167.5 261.8 377 3 .982 3.93 8.84 15.7 24.5 35.3 62.8 98.2 141.4 251.3 392.7 565.5 4 1.31 5,24 11.78 20.9 32.7 47.1 83.8 131 188 335 523 754 5 1.64 6.54 14.7 26.2 41 59 104 163 235 419 654 942 6 1.96 7.85 17.7 31.4 49.1 70.7 125 196 283 502 785 1131 7 2.29 9.16 20.6 36.6 57.2 82.4 146 229 330 586 916 1319 8 2.62 10 5 23.5 41.9 65.4 94 167 262 377 670 1047 1508 9 2.95 11.78 26.5 47 73 106 188 294 424 754 1178 1C96 10 3.27 13.1 29.4 52 82 118 209 327 471 838 1309 1885 12 3.93 15.7 35.3 63 98 141 251 393 565 1005 1571 2262 15 4.91 19.6 44.2 78 122 177 314 491 707 1256 1963 2827 18 5.89 23.5 53 94 147 212 377 589 848 1508 2356 3393 20 6.54 26.2 59 105 164 235 419 654 942 1675 2618 3770 24 7.85 31.4 71 125 196 283 502 785 1131 2010 3141 4524 25 8.18 32.7 73 131 204 294 523 818 1178 2094 3272 4712 28 9.16 36.6 82 146 229 330 586 916 1319 2346 3665 5278 30 9.8 39.3 88 157 245 353 628 982 1414 2513 3927 5655 FLOW OF AIR IN" PIPES. 487 In Hawksley's formula and its derivatives the numerical coefficients are constant. It is scarcely possible, however, that they can be accurate except within a limited range of conditions. In the case of water it is found that the coefficient of friction, on which the loss of head depends, varies with the length and diameter of the pipe, and with the velocity, as well as with the condition of the interior surface. In the case of air and other gases we have, in addition, the decrease in density and consequent increase in volume and in velocity due to the progressive loss of head from one end of the pipe to the other. Clark states that according to the experiments of D'Aubisson and those of a Sardinian commission on the resistance of air through long conduits or pipes, the diminution of pressure is very nearly directly as the length, and as the square of the velocity and inversely as the diameter. The resistance is not varied by the density. If these statements are correct, then the formulae h = — — and h = ¥-—■ cd c'd a and their derivatives are correct in form, and they may be used when the numerical coefficients c and c' are obtained by experiment. If we take the forms of the above formulas as correct, and let C be a vari- able coefficient, depending upon the length, diameter, and condition of sur- face of the pipe, and possibly also upon the velocity, the temperature and the density, to be determined by future experiments, then for h = head in inches of water, d = diameter in inches, L = length in feet, v — velocity in ; feet per second, and Q = quantity in cubic feet per second: •-y$ Lv* m '' CVi ' .005454C • loss of 33683(g 2 L CVi ' For difference or loss of pressure p in pounds per square inch, h = 27.71p Vh = 5.264 \'p\ /m3Q*L. _ 1213Q2L d -\/ (For other formulae for flow of air, see Mine Ventilation.) Loss of Pressure in Ounces per Square Inch.— B. F. Sturte- vant Company uses the following formulae : Lv* m _ \ / 250Q0dp 1 . _ Lifl . Pl ~ 25000d ' V ~ L '■ 25000?!.' in which p t = loss of pressure in ounces per square inch, v = velocity of air in feet per second, and L — length of pipe in feet. If p is taken in pounds per square inch, these formulae reduce to ._ nnno .Lv2 .00158 Vdp , .00000251^2 p = .0000020—— ; v = =-^ — — ; d = ■ . ^ d L p trom tne common rormuia (weisoacns;, p — f which /= .0001608. The following table is condensed from one given in the catalogue of B. F. Sturtevant Company. Loss of pressure in pipes 100 feet long, in ounces per square inch. For any other length, the loss is proportional to the length, 488 AIR. '-5.9 >>53 '5 -O Diameter of Pipe in Inches. 1 2 3 4 5 6 7 8 9 10 11 12 13 v Loss of Pressure in Ounces. 600 .400 .200 .133 .100 .080 .067 .057 .050 .044 040 036 033 1200 1.600 .800 .533 .400 .320 .267 .229 .200 .178 .160 .145 133 1800 3.60C 1.800 1.200 .900 .720 .600 .514 .450 400 360 327 .300 2400 6.400 3.200 2.133 1.600 1.280 1.067 .914 .800 .711 .640 582 .533 3000 10. 5. 3.333 2.5 2. 1.667 1 421) 1.250 1 111 1 000 .909 833 3600 14.4 7.2 4.8 3.6 2.88 2.4 2.057 1.8 1 6 1.44 1.309 1 200 4200 9.8 6.553 4.9 3.92 3.267 2.8 2.45 2 178 1.96 1 782 1 633 4800 12.8 8.533 6.4 5.12 4.267 3.657 3.2 2.844 2.56 2 327 2 133 6000 20. 13.333 10.0 8.0 6.667 5.714 5.0 4.444 4.0 3.636 Diameter of Pipe in Inches. 14 16 18 20 22 24 28 32 36 40 44 48 Loss of Pressure in Ounces. 600 .029 .026 .022 .020 .018 .017 .014 .012 .011 .010 .009 .008 1200 .114 .100 .089 .080 .073 .067 .057 .050 .044 .040 .036 .033 1800 .257 .225 .200 .180 .164 .156 .129 .112 .100 .090 082 .075 2400 .457 .400 .356 .320 .291 .267 .239 .200 .178 .160 145 .133 3600 1.029 .900 .800 .720 .655 .600 .514 .450 .400 .360 .327 .300 4200 1.400 1.225 1.089 .980 .891 .817 .700 .612 .544 .490 .445 408 4800 1.829 1.600 1.422 1.280 1.164 1.067 .914 .800 .711 640 582 .533 6000 2.857 2.500 2.222 2.000 1.818 1.667 1.429 1.250 1.111 1.000 .909 .833 Effect of Bends in Pipes. (Norwalk Iron Works Co.) Radius of elbow, in diameter of pipe =53 2 1J^ 1J4 1 % M Equivalent lgths. of straight pipe, diams 7.85 8.24 9.03 10.36 12.7217.51 35.09 121.2 Compressed-air Transmission. (Frank Richards, Am. Mach., March 8, 1894 )— The volume of free air transmitted may be assumed to be directly as the number of atmospheres to which the air is compressed. Thus, if the air transmitted be at 75 pounds gauge-pressure, or six atmos- pheres, the volunle of free air will be six times the amount given in the table (page 486). It is generally considered that for economical transmission the velocity in main pipes should not exceed 20 feet per second. In the smaller distributing pipes the velocity should be decidedly less than this. The loss of power in the transmission of compressed air in general is not a serious one, or at all to be compared with the losses of power in the opera- tion of compression and in the re-expansion or final application of the air. The formulas for loss by friction are all unsatisfactory. The statements of observed facts in this line are in a more or less chaotic state, and self- evidently unreliable. A statement of the friction of air flowing through a pipe involves at least all the following factors: Unit of time, volume of air, pressure of air, diam- eter of pipe, length of pipe, and the difference of pressure at the ends of the pipe or the head required to maintain the flow. Neither of these factors can be allowed its independent and absolute value, but is subject to modifi- cations in deference to its associates. The flow of air being assumed to be uniform at the entrance to the pipe, the volume and flow are not uniform after that. The air is constantly losing some of its pressure and its volume is constantly increasing. The velocity of flow is therefore also somewhat accelerated continually. This also modifies the use of the length of the pipe as a constant factor. Then, besides the fluctuating values of these factors, there is the condition of the pipe itself. The actual diameter of the pipe, especially in the smaller sizes, is different from the nominal diameter. The pipe may be straight, or it may be crooked and have numerous elbows. Mr. Richards considers one elbow as equivalent to a length of pipe, FLOW OF COMPRESSED AIR IN" PIPES. 489 Head or Additional Pressure in pounds per sq. in. required to deliver Air at ¥5 Pounds Gau$;e-pressure through Pipes of Various Sizes and Lengths. (Frank Richards.) Length in feet. ;- = 50 100 150 200 100 150 200 250 300 100 200 400 500 100 200 400 500 250 500 50 100 300 500 1,000 Loss of pre .245 .49 .981! 1.962 3.925 7.85 8.829 17.66 lbs. p. 1*4" Pipe. 1.000 1,250 1,500 Length in feet. .64 1. 1.44 of pre 24 54 1.5 2.16 400 1,000 ssure, lbs. p. .4 .8 .9 1.8 1.6 3.2 2.5 5. 3.6 7.2 2,000 sq. in. 1.6 3.6 6.4 10. 14.4 25 .056 .112 50 .224 .449 100 .897 1.79 150 2.02 3.94 200 3.59 7.18 .336 1.35 .017 .034 .103 .171 .068 .137 .411 .685 .274 .548 1.64 2.74 .616 1.23 3.69 6.16 1.09 2.19 6.57 10.96 .34 1.37 5. 500 1,000 2,000 4,000 500 .11 .22 .44 .88 1,000 .44 .881 1.76 3.52 1,500 .99 1.98 3.96 7.92 2,000 1.76 3.52 7.04 14.08 2,500 2.75 5.5 11. 5,000 1.1 4.4 9.9 .019 .038 .114 .19 .076 .152 .457 .761 .171 .343 1.03 1.71 .304 .609 1.83 3 04 .476 .952 2.86 4.76 .685 1.37 4.11 6.85 6.09 9.53 13, 1,000 1,500 2,000 2,500 3,000 1,000 2,000 4,000 5,000 1.77 .354 .708 1.42 .799 1.599 3.2 3.99 1.417 2.83 5.67 7.09 2.22 4.44 8.89 11.1 3.18 6.37 12.7 15.9 10,000 3.54 7.99 14.17 200 300 500 1,000 2,000 .087 .13 .217 .434 .87 .347 .521 .868 1.74 3.47 .781 1.17 1.95 3.91 7.81 1.39 2.08 3.47 6.94 13.89 2.17 3.25 5.42 10.85 21.7 2,000 2,500 3,000 4.000 5,000 2,000 4,000 8,000 10,000 .598 .935 1.25 2.39 3.74 1.19 1.87 2.49 4.79 7.48 2.39 3.74 4.99 9.58 14.97 2.99 4.68 6.24 11.97 18.71 4.48 7.02 .0333 .05 .0833 .166 .133 .2 .333 .666 .3 .45 .75 1.5 .533 .8 1.33 2.66 .833 1.25 2.08 4.16 .33 1.33 3 5.33 2,500 .286 5,000 1.14 7,500 2.57 10,000 4.57 .57 2.29 5.15 9.14 1.14 4.57 10.29 1.43 5.71 12.86 2.15 8.56 .0832 .125 .208 .416 .332 .499 .832 1.66 .748 1.12 1.87 3.75 1 328 1.99 3.33 6.66 2.08 3.12 5.2 10.4 3.32 7.49 2,500 5,000 7,500 10,000 2,000 4,000 .22 .88 1.98 3.52 8,000 10,000 .55 4*95 8.81 .11 .44 .99 1.76 .44 1.76 3.96 7.05 20,000 1.101 4.4 9.91 17 6 Although Mr. Richards does not give any formula with this table, an nspection of it shows that for any given diameter the loss of head is 490 AIK. taken to vary directly as the length and as the square of the quantity delivered, but for a given quantity and length the loss of head appears to vary inversely as some higher power of the diameter than the fifth, ap- proximately the 5.5 power; or, in other words, that the coefficient of fric- /pd* tion is variable. If we take the formula of the form Q' - -V 1 ■ ~~,V~J~v an d solve for c'-- ■ y d 5 p - cubic feet of free air per minute, we find values of the coefficient as follows: For diameter, inches Value of c' — 4 552 10 664 12 676 The following table is condensed from one given by F. A. Halsey in the catalogue of the Rand Drill Co.: -2 0) Cubic feet of free air compressed to a gauge-pressure of 60 lbs. and passing through the pipe each minute. 50 100 200 400 600 800 1000 1500 2000 3000 4000 -)000 s'g Loss of pressure in lbs. per square inch for each 1000 ft. ^ of straight pipe. 1 10.40 1M 2.63 1W 1.22 4.89 2 .35 1.41 5.64 2U .14 .57 2.30 9.20 3 .20 .78 3.14 7.05 4 .20 .80 .26 1.81 .59 .23 3.22 1.04 .41 5.02 1.63 .64 3.66 1.46 6.50 2.59 5.81 10.30 5 6 8 .10 .16 .37 .65 1.47 2.61 4.08 10 .12 .21 .47 .19 .84 .34 .16 1 30 19 ^3 14 .24 This table appears to follow more closely than does Richards' table the law of the formula p = vf /2 , 5 , but the coefficients differ considerably from those of Richards. Solving for C", we obtain— For diameter, inches . . 2 4 5 6 8 10 12 14 Value of C 471 442 443 448 437 436 435 431 Comparing some of the losses of pressure in the two tables, we find- Length, feet 1000 1000 1000 5000 5000 5000 Quantity, cu. ft 1000 1000 1000 4000 4000 4000 Diameter, inches 4 5 6 8 10 12 Loss, Richards 3.2 .881 .354 7.48 2.29 .88 " Halsey 5.02 1.63 _ .64 13.05 4.20 170 The two tables are not calculated for the same amount of compression, but the difference is not sufficient to account for the difference in the coeffi- cients. If we multiply the coefficients derived from Halsey's table by 5/4, the ratio of the pressures 75 and 60 lbs., they become for a 2-inch pipe 589, and for a 12-inch pipe 531, against Richards's figures of 453 and 676 for the same pipes. To compare Richards's figures for loss of pressure with Hal- sey's, the former should be multiplied by 25/16. In the absence of experi- mental data no opinion can be formed as to which table is the more accurate, but either one is probably of sufficient accuracy for practical purposes. MEASUREMENT OF VELOCITY OF AIR. 491 Mr. Richards, in Am. Mack., Dec. 27, 1894, publishes a new formula, viz.: ' 10,000d 5 a ; "= z=5V=500Va X \/ 1 and from the values of a given by Mr. Richards we find values of c' as follows: . For diameter, nominal, inches = Value of c' 2 4 6 8 10 12 ! 374 458 500 530 548 561 Measurement of the Velocity of Air in Pipes by an Ane- mometer.— Tests were made by B. Donkin, Jr. (Inst. Civil Engrs. 1892), to compare the velocity of air in pipes from 8 in. to 24 in. diam., as shown by an anemometer 2% in. diam. with the true velocity as measured by the time of descent of a gas-holder holding 1622 cubic feet. A table of the results with discussion is given in Eng'g News, Dec. 22, 1892. In pipes from 8 in. to 20 in. diam. with air velocities of from 140 to 690 feet per minute the anemome- ter showed errors varying from 14.5$ fast to 10$ slow. With a 24-inch pipe and a velocity of 73 ft. per minute, the anemometer gave from 44 to 63 feet, or from 13.6 to 39.6$ slow. The practical conclusion drawn from these ex- periments is that anemometers for the measurement of velocities of air in pipes of these diameters should be used with great caution. The percentage of error is not constant, and varies considerably with the diameter of the pipes and the speeds of air. The use of a baffle, consisting of a perforated plate, which tended to equalize the velocity in the centre and at the sides in some cases diminished the error. 492 The impossibility of measuring the true quantity of air by an anemometer held stationary in one position is shown by the following figures, given by Wm. Daniel (Proc. Inst. M. E., 1875), of the velocities of air found at different points in the cross-sections of two different airways in a mine. Differences of Anemometer Readings in Airways. 8 ft. square. 5 X 8 f t. 1712 1795 1859 1329 1622 1685 1782 1091 1477 1262 1344 1524 1049 1356 1293 1333 1170 948 1209 1288 1104 1177 1134 1049 1106 Average 1469. Average 1132. Equation of Pipes.— It is frequently desired to know what number of pipes of a given size are equal in carrying capacity to one pipe of a larger size. At the same velocity of flow the volume delivered by ttvo pipes of different sizes is proportional to the squares of their diameters; thus, one 4-inch pipe will deliver the same volume as four 2-inch pipes. With the same head, however, the velocity is less in the smaller pipe, and the volume de- livered varies about as the square root of the fifth power (i.e., as the 2.5 power). The following table has been calculated on this basis. The figures opposite the intersection of any two sizes is the number of the smaller-sized pipes required to equal one of the larger. Thus, one 4-inch pipe is equal to 5.7 2-inch pipes. 5 S3 5'" 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 24 2 3 5.7 15.fi 1 2.8 1 4 32 5.7 2.1 1 5 55,1) 9.9 3.6 1.7 1 6 SS.2 15.6 5 7 2.8 1.6 1 7 130 22.9 8 3 4.1 2.3 1.5 1 8 181 32 11. ^ 5.7 3.2 2.1 1.4 1 9 243 43. 15.6 7.6 4.3 2.8 1.9 1.3 1 10 ma 55 9 20.3 9 9 5,7 3.6 2.4 1.7 1 3 1 11 401 70 9 25 . 7 12.5 7 2 4.6 3.1 2,2 1.7 1.3 12 m KS 2 32 ir - si 5.7 3,8 2.8 2 1 1.6 1 13 609 108 39.1 19 10.9 7.1 4.7 3.4 2.5 1.9 1 2 14 733 130 47 22.'.) 18.1 h.;j 5.7 4.1 3.0 2.3 1.5 1 15 787 154 55. 9 27.2 15.6 9.9 6.7 4.8 3.6 2 8 1.7 1.2 16 181 65.7 32 18.: J 11.7 7.9 5.7 4.2 3.2 2.1 1.4 1 17 •211 76 4 37 2 21 3 13.5 9.2 6 , 6 4,9 3,8 2,4 1.6 1 2 18 -'43 88.2 43 15.6 10 6 7.6 5.7 4.3 2.8 1.9 1.3 1 19 27S 101 49 1 28 . 1 17.8 12.1 8.7 6.5 5 3.2 2.1 1.5 1.1 20 316 115 55 9 32 20,3 13 8 9.9 7.4 5.7 3.6 2.4 1.7 1.3 1 22 101 146 70.!) 40 . 6 25.7 17.5 12 5 9.3 7' 2 4.6 3 1 2.2 1.7 1.3 24 m 181 8S.2 50.5 32 21.8 15.6 11.6 8.9 5.7 3 8 2.8 2.1 1.6 1 26 .ion 221 108 61.7 39.1 26.6 19. 14.2 10.9 7.1 4.7 3.4 2.5 1.9 1.2 28 T33 266 130 7-4.2 47 32 22.9 17 1 13.1 8.3 5.7 4,1 3 2.3 1.5 30 T87 316 154 88. 2 55 9 38 27.2 20.3 15.6 9 9 6.7 4.8 3.6 2.8 1.7 36 (9!) 243 130 88 2 60 43 32 24.6 15.6 10. (i 7.6 5.7 4.3 2.8 42 733 357 205 130 88. -J 63.2 47 36.2 19 15,6 11.2 8.3 6.4 4.1 48 499 286 181 123 88.2 62.7 50.5 32 21.8 15.6 11.6 8.9 5.7 54 670 383, 243 165 118 88.2 67.8 43 29 2 20 9 15.6 12 7.6 60 787 499 316 215 154 115 38 27.2 15.6 9.9 493 Loss of Pressure in Compressed Air Pipe-main, at St. Gothard Tunnel. (E. Stockalper.) s i s per second air.orequi- volume at pheric pres- ud 32° F. o'cH '3 3 II «"3 .S c O u Observed Pressures. 1 o .S 6 ts-S 1 Loss of "Value of c' in for- mula •r OS ® S a o =* 5^5 -oaS « Ch £& "fl Pressure. P = V £ ~£iiS£ e3 Oi> 5 &CQ, 0^3 Q'L .i: o o >tf w "So el ««H £ S c' 2 d 6 > > § is: g Oh h lbs. per No in. cu.ft. cu.ft. den. lbs. feet. at. at. sq.in. % . ( r sr J- 33.056] 6.534 .00650 2.609 19.32 5.60 5.24 5.292 0.4 610 l \ 5,91 7.063 .00603 2 669 37.14 5.24 5.00 3.528 4.6 515 H ;,s? [■ 28.002 -1 5.509 .00514 1.776 16.30 4.35 4.13 3.234 5.1 519 5 91 5.863 .00482 1.776 4.13 H 7 ST j- 18.364] 5.262 .00449 1.483 15.58 3.84 3.65 2.793 5.0 466 5.91 5.580 .00423 1.483 29.34 3.65 3.54 1.617 3.0 422 The length of the pipe 7.87 in diameter was 15,092 ft., and of the smaller pipe 1712.6 ft. The mean temperature of the air in the large pipe was 70° F. and in the small pipe 80° F. WIND. Force of tlie Wind.— Smeaton in 1759 published a table of the velocity and pressure of wind, as follows: Velocity and Force of Wind, in Pounds per Square Inch. 16 u . 55 v. 33 "2 2*~ p x p, fa 1.47 0.005 2.93 0.020 4.4 0.044 5.87 0.079 7.33 0.123 8.8 0.177 10.25 0.241 11.75 0.315 13.2 0.400 14.67 0.49-2 17.6 0.708 20.5 0.964 22.00 1.107 23.45 1.25 Common Appelte tion of the Force of Wind. Hardly percepti- ble. - Just perceptible. Gentle pleasant wind. Pleasant brisk i,. 53-d a« -r 3 Png fa 2 ero owp. fa 18 26.4 1.55 20 29.34 1.968 25 36.67 3.075 30 44.01 4.429 35 51.34 6.027 40 58.68 7.873 45 66.01 9.963 50 73.35 12.30 55 80.7 14.9 60 88.02 17.71 66 95.4 20.85 70 102.5 24.1 75 110. 27.7 80 117.36 31.49 100 146.67 49.2 Common Appella- tion of the Force of Wind. >Very brisk. y High wind. 1 j- Very high storm. Hurricane. Immense hurri- The pressures pnr square foot in the above table correspond to the formula P = 0.005F ra , in which V is the velocity in miles per hour. Eng^g News. Feb. 9, 1893, says that the formula was never well established, and has floated chiefly on Smeaton's name and for lack of a better. It was put forward only for surfaces for use in windmill practice. The trend of modern evidence is that it is approximately correct only for such surfaces, and that for large solid bodies it often gives greatly too large results. Observations by others are thus compared with Smeaton's formula: Old Smeaton formula P = .005FS As determined by Prof. Martin P = .004 V* " Whipple and Dines P=.0029F 2 494 air. At 60 miles per hour these formulas give for the pressure per square foot, 18, 14.4 and 10.44 lbs., respectively, the pressure varying by all of them as the square of the velocity. Lieut. Crosby's experiments (Eng'g, June 13, 1890), claiming to prove that P = fV instead of P = fV 2 , are discredited. A. R. Wolff (The Windmill as a Prime Mover, p. 9) gives as the theoretical pressure per sq. ft. of surface, P = —£-, in which d = density of air in pounds per cu. ft. = ' - — ~^ — - ; p being the barometric pressure per square foot at any level, and temperature of 32° F., t any absolute temperature, Q = volume of air carried along per square foot in one second, v — velocity of the wind in feet per sec, g = 32.16. Since Q = v cu. ft. per sec, P= — . Multiplying this by a coefficient 0.93 found by experiment, and substituting ^ - ii ' ' i e ^ t, 1,4 • -d 0.017431 Xp . the above value of d, he obtains P = — - ■, and when p - 2116.5 lbs. per sq ft. or average atmospheric pressure at the sea-level, 36 8929 > = - — — -^ — , an expression in which the pressure is shown to vary v 2 with the temperature; and he gives a table showing the relation between velocity and pressure for temperatures from 0° to 100° F., and velocities from 1 to 80 miles per hour. For a temperature of 45° F. the pressures agree with those in Smeaton's table, for 0° F. they are about 10 per cent greater, and for 100° 10 per cent less. Prof. H. Allen Hazen, Eng^g News, July 5, 1890, says that experiments with whirling arms, by exposing plates to direct wind, and on locomotives with velocities running up to 40 miles per hour, have invariably shown the resistance to vary with V 2 . In the formula P = .005SF 2 , in which P — pressure in pounds, S — surface in square feet, V= velocity in miles per hour, the doubtful question is that regarding the accuracy of the first two factors in the second member of this equation. The first factor has been variously determined from .003 to .005 [it has been determined as low as .0014.— Ed. Eng'g News]. The second factor has been found in some experiments with very short whirling arms and low velocities to vary with the perimeter of the plate, but this entirely disappears with longer arms or straight line motion, and the only question now to be determined is the value of the coefficient. Per- haps some of the best experiments for determining this value were tried in France in 1886 by carrying flat boards on trains. The resulting formula in this case was, for 44.5 miles per hour, p = .00535SF 2 . Mr. Crosby's whirling experiments were made with an arm 5.5 ft. long. It is certain that most serious effects from centrifugal action would be set up by using such a short arm, and nothing satisfactory can be learned with arms less than 20 or 30 ft. long at velocities above 5 miles per hour. Prof. Kernot, of Melbourne {Engineering Record, Feb. 20, 1894), states that experiments at the Forth Bridge showed that the average pressure on sur- faces as large as railway carriages, houses, or bridges never exceeded two thirds of that upon small surfaces of one or two square feet, such as have been used at observatories, and also that an inertia effect, which is frequently overlooked, may cause some forms of anemometer to give false results enormously exceeding the correct indication. Experiments of Mr. O. T. Crosby showed that the pressure varied directly as the velocity, whereas all the early investigators, from the time of Smeaton onwards, made it vary as the square of the veloi ity. Experiments made by Prof . Kernot at speeds varying from 2 to 15 miles per hour agreed with the earlier authorities, and tended to negative Crosby's results. The pressure upon one. side of a cube, or of a block proportioned like an ordinary carriage, was found to be .9 of that upon a thin plate of the same area. The same result was obtained for a square tower. A square pyramid, whose height was three times its base, experienced .8 of the pressure upon a thin plate equal to one of its sides, but if an angle was turned to the wind the pressure was iucreased by fully 20%. A bridge consisting of two plate-girders connected by a deck at the top was found to experience .9 of the pressure on a thin plate equal in size to one girder, when the distance between the girders was equal to their depth, and this was increased by one fifth when the distance between the girders was WINDMILLS. 495 double the depth. A lattice-work in which the area of the openings was 55% of the whole area experienced a pressure of 80% of that upon a plate of the same area. The pressure upon cylinders and cones was proved to be equal to half that upou the diametral planes, and that upon an octagonal prism to be 20^ greater than upon the circumscribing cylinder. A sphere was sub- ject to a pressure of .36 of that upon a thin circular plate of equal diameter. A hemispherical cup gave the same result as the sphere; when its convexity was turned to the wind the pressure was 1.15 of that on a flat plate of equal diameter. When a plane surface parallel to the direction of the wind was brought nearly into contact with a cylinder or sphere, the pressure on the latter bodies was augmented by about 20$, owing to the lateral escape of the air being checked. Thus it is possible for the security of a tower or chimney to be impaired by the erection of a building nearly 'touching it on one side. Pressures of Wind. Registered in Stornis.— Mr. Frizell has examined the published records of Greenwich Observatory from 1849 to 1869, and reports that the highest pressure of wind he finds recorded is 41 lbs. per sq. ft., and there are numerous instances in which it was between 30 and 40 lbs. per sq. ft. Prof. Henry says that on Mount Washington, N. H., a ve- locity of 150 miles per hour has been observed, and at New York City 60 miles an hour, and that the highest winds observed in 1870 were of 72 and 63 miles per hour, respectively. Lieut. Dun woody, U. S. A., says, in substance, that the New England coast is exposed to storms which produce a pressure of 50 lbs. per sq. ft. Engi- neering Neivs, Aug. 20, 1880. WINDMILLS. Power and Efficiency of Windmills.— Rankine, S. E., p. 215. gives the following: Let Q = volume of air which acts on the sail, or part of a sail, in cubic feet per second, v = velocity of the wind in feet per second, s = sectional area of the cylinder, or annular cjdinder of wind, through which the sail, or part of the sail, sweeps in one revolution, c = a coefficient to be found by experience; then Q = cvs. Rankine, from experi- mental data given by Smeaton, and taking c to include an allowance for friction, gives for a wheel with four sails, proportioned in the best manner, c = 0.75. Let A = weather angle of the sail at any distance from the axis, i.e., the angle the portion of the sail considered makes with its plane of revolution. This angle gradually diminishes from the inner end of the sail to the tip; u = the velocity of the same portion of the sail, and E = the effi- ciency. The efficiency is the ratio of the useful work performed to whole energy of the stream of wind acting on the surface s of the wheel, which Dsv 3 energy is -= — , D being the weight of a cubic foot of air. Rankine's formula for efficiency is E =^f = C \^ sia2A ~ g ct-^^+/)-/)h in which c = 0.75 and / is a coefficient of friction found from Smeaton's data = 0.016. Rankine gives the following from Smeaton's data: A = weather-angle =7° 13° 19° V -f- v = ratio of speed of greatest effi- ciency, for a given weather- angle, to that of the wind =2.63 1.86 1.41 E = efficiency =0.24 0.29 0.31 Rankine gives the following as the best values for the angle of weather at different distances from the axis: Distance in sixths of total radius. .. 1 2 3 4 5 6 Weather angle ... 18° 19° 18° 16° 12^° 7° But Wolff (p. 125) shows that Smeaton did not term these the best angles, but simply says they " answer as well as any,' 1 '' possibly any that were in ex- istence in'his time. Wolff says that they "cannot in the nature of things be the most desirable angles." Mathematical considerations, he says, con- clusively show that the angle of impulse depends on the relative velocity of each point of the sail and the wind, the angle growing larger as the ratio be- comes greater. Smeaton's angles do not fulfil this condition. Wolff devel- 496 AIR. ops a theoretical formula for the best angle of weather, and from it calculates a table for different relative velocities of the blades (at a distance of one seventh of the total length from the centre of the shaft) and the wind, from which the following is condensed: Ratio of the Speed of Blade at 1/7 of Radius to Velocity of Wind. Distance from the axis of the wheel in sevenths of radius. Best angles of weather. 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 42° 9' 39° 21' 36° 39' 34° 6' 31° 43' 29° 31' 40 44 36 39 32 53 29 31 26 34 24 39 21 34 6 29 31 25 40 22 30 19 54 37 59 36 43 26 34 22 30 19 20 16 51 36 39 29 31 24 19 54 16 51 14 32 35 21 27 30 21 48 17 46 14 52 12 44 34 6 25 40 19 54 16 13 17 11 19 32 53 24 18 16 14 32 11 59 10 10 31 43 22 30. 16 51 13 17 10 54 9 13 27° 30' 21 48 17 46 14 52 12 44 11 6 58 The effective power of a windmill, as Smeaton ascertained by experiment, varies as s, the sectional area of the acting stream of wind; that is, for simi- lar wheels, as the squares of the radii. The value 0.75, assigned to the multiplier c in the formula Q = cvs, is founded on the fact, ascertained by Smeaton, that the effective power of a windmill with sails of the best form, and about 15^ ft. radius, with a breeze of 13 ft. per second, is about 1 horse-power. In the computations founded on that fact, the mean angle of weather is made — 13°. The efficiency of this wheel, according to the formula and table given, is 0.29, at its best speed, when the tips of the sails move at a velocity of 2.6 times that of the wind. Merivale (Notes and Formulas for Mining Students), using Smeaton's co- efficient of efficiency, 0.29, gives the following: U = units of work in foot lbs. per sec. ; W = weight, in pounds, of the cylinder of wind passing the sails each second, the diameter of the cylinder being equal to the diameter of the sails ; V = velocity of wind in feet per second; H.P. = effective horse-power; TT-KE1. °- 29 wv * 64 ' ~ 64 X 550' A. R. Wolff, in an article in the American Engineer, gives the following (see also his treatise on Windmills): Let c = velocity of wind in feet per second; n = number of revolutions of the windmill per minute; 6<>i &n b 2 , b x be the breadth of the sail or blade at distances l , l x , 7 2 , Z 3 , and I, respectively, from the axis of the shaft; l — distance from axis of shaft to beginning of sail or blade proper; I — distance from axis of shaft to extremity of sail proper; v oi v ii v 2* v 3> v x — tne ve l°city of the sail in feet per second at dis- tances l . l l% l 2 , Z, respectively, from the axis of the shaft; a , «i, a 2 , a 3 , a x = the angles of impulse for maximum effect at dis- tances l , l t , l 2 . l s , I respectively from the axis of the shaft; a = the angle of impulse when the sails or blocks are plane surfaces, so that there is but one angle to be considered; N = number of sails or blades of windmill; K = .93. d — density of wind (weight of a cubic foot of air at average tempera- ture and barometric pressure where mill is erected); W '= weight of wind-wheel in pounds; / = coefficient of friction of shaft and bearings; D = diameter of bearing of windmill in feet. WINDMILLS. 4£>? The effective horse-power of a windmill with plane sails will equal (l-l )Kc*dN J . . v £— X mean of ^ (sin a cos a)b cos a v x .. \ f\V X .05236?il> v x (sin a cos a) b x cos a J— - r^c . The effective horse-power of a windmill of shape of sail for maximum effect equals N(l - l )Kdc 3 „ .12 sin 2 a -l. 2 sin* a x - 1 , n " X mean off . . ° 6 , r-^ b x . . . 22Q0g \ sin 2 a °' sin 2 a x 1 2 sin 2 a x - 1 ■ : b. sin 2 a x fW X .05236w£> 550 The mean value of quantities in brackets is to be found according to Simpson's rule. Dividing I into 7 parts, finding the angles and breadths corresponding to these divisions by substituting them in quantities within brackets will be found satisfactory. Comparison of these formulae with the only fairly reliable experiments in windmills (Coulomb's) showed a close agreement of results. Approximate formulae of simpler form for windmills of present construc- tion can be based upon the above, substituting actual average values for a, c, d, aud e. but since improvement in the present angles is possible, it is better to give the formulae in their general and accurate form. Wolff gives the following table based on the practice of an American manufacturer. Since its preparation, he says, over 1500 windmills have been sold oh its guaranty (1885), and in all cases the results obtained did not vary sufficiently from those presented to cause any complaint. The actual re- sults obtained are in close agreement with those obtained by theoretical analysis of the impulse of wind upon windmill blades. Capacity of the Windmill. a> § oi> a P^ §5° 3 ii £^ Gallons of Water raised per Minute to "3 hM o O 3-u &^3 I he 25 50 75 100 150 200 |o|| > feet. feet. feet. feet. feet. feet. 3 3 £ < wheel 8^ ft. 16 70 to 75 6.162 3.016 0.04 8 10 " 16 60 to 65 19.179 9.563 6.638 4.750 0.12 8 12 " 16 55 to 60 33.941 17.952 11.851 8.485 5.680 0.21 8 14 " 16 50 to 55 45.139 22.569 15.304 11.246 7.807 4 998 0.28 8 16 " 16 45 to 50 64 600 31.654 19.542 16.150 9.771 8.075 0.41 8 18 " 16 40 to 45 97.682 52.165 32.513 24.421 17.485 12.211 0.61 8 20 " 16 35 to 40 124.950 63.750 40.800 31.248 19.284 15.938 0.78 8 25 " 16 30 to 35 212.381 106.964 71.604 49.725 37.349 26.741 1.34 8 These windmills are made in regular sizes, as high as sixty feet diameter of wheel; but the experience with the larger class of mills is too limited to enable the presentation of precise data as to their performance. If the wind can be relied upon in exceptional localities to average a higher velocity for eight hours a day than that stated in the above table, tbe per- formance or horse-power of the mill will be increased, and can be obtained by multiplying the figures in the table by the ratio of the cube of the higher average velocity of wind to the cube of the velocity above recorded. He also gives the following table showing the economy of the windmill. All the items of expense, including both interest and repairs, are reduced to the hour by dividing the costs per annum by 365 x 8 = 2920; the interest, 493 AIR. etc., for the twenty-four hours being charged to the eight hours of actual work. By multiplying the figures in the 5th column by 584, the first cost of the windmill, in dollars, is obtained. Economy of the Windmill. -3 to Expense of Actual Useful Power I=>f o =•£ Developed, in cents, per hour. i a Designation D O umbei * Day s Qua ised. or Interest on First Cost (Firs Cost, including Cost of Wind- mill, Pump, an Tower. 5% per annum). T3 — & 5 C+J 5 ffi cd of Mill. o ^ o w 3 || _> CO '3 o verage N Hours pe which thi will be ra or Repair Deprecia of First C annum). a < o «i aO O O < fc fe Ik h H s 8J^ ft. wheel 370 0.04 8 0.25 0.25 0.06 0.04 0.60 15.0 10 " " 1151 0.12 8 0.30 0.30 0.06 0.04 0.70 5.8 12 " 2036 0.21 8 0.36 0.36 06 0.04 3.9 14 " 2708 0.28 8 0.75 0.75 0.06 0.07 1.63 5.8 16 " 3876 0.41 8 1.15 1.15 06 0.07 2.43 5.9 18 " 5861 061 8 1.35 1.35 (1 Of, l) 07 4.6 20 " 7497 0.79 8 1.70 1.70 0.06 4.5 25 " 12743 1.34 8 2.05 2.05 i Id 4.26 3.2 Lieut. I. N. Lewis (Eng'g Mag., Dec. 1894) gives a table of results of ex- periments with wooden wheels, from which the following is taken : Diameter of wheel, Feet. Velocity of Wind, miles per hour. Actual Useful Horse-power developed. 16 20 25 V* m w 3 IV2 4y* 1 214 5^ The wheels were tested by driving a differentially wound dynamo. The " useful horse-power " was measured by a voltmeter and ammeter, allow- ing 500 watts per horse-power. .Details of the experiments, including the means used for obtaining the velocity of the wind, are not given. The re- sults are so far in excess of the capacity claimed by responsible manufactu- rers that they should not be given credence until established by further experiments. ' A recent article on windmills in the Iron Age contains the following: Ac- cording to observations of the United States Signal Service, the average velocity of the wind within the range of its record is 9 miles per hour for the year along the North Atlantic border and Northwestern States, 10 miles on the plains of the West, and 6 miles in the Gulf States. The horse-powers of windmills of the best construction are proportional to the squares of their diameters and inversely as their velocities; for ex- ample, a 10-ft. mill in a 16-mile breeze will develop 0.15 horse-power at 65 revolutions per minute; and with the same breeze A 20-ft. mill, 40 revolutions, 1 horse-power. A 25-ft. mill, 35 revolutions, \% horse-power. A 30-ft. mill, 28 revolutions, 3^| horse-power. A 40-ft. mill, 22 revolutions, 7J4 horse-power. A 50-ft. mill, 18 revolutions, 12 horse-power. The increase in power from increase in velocity of the wind is equal to the square of its proportional velocity; as for example, the 25-ft. mill rated COMPRESSED AIR. 499 above for a 16-niile wind will, with a 32-mile wind, have its horse-power in- creased to 4 X 1% = 7 horse-power, a 40-ft. mill in a 32-mile wind will run up to 30 horse-power, and a 50-ft. mill to 48 horse-power, with a small de d notion for increased friction of air on the wheel and the machinery. The modern mill of medium and large size will run and produce work in a 4-mile breeze, becoming very efficient in an 8 to 16-mile breeze, and increase its power with safety to the running-gear up to a gale of 45 miles per hour. Prof. Thurston, in an article on modern uses of the windmill, Engineer- hut Magazine, Feb. 1893, says : The best mills cost from about $600 for the 10-ft. wheel of % horse-power to $1200 for the 25-ft. wheel of l^j horse-power or less. In the estimates a working-day of 8 hours is assumed ; but the ma- chine, when used for pumping, its most common application, may actually do its work 24 hours a day for clays, weeks, and even months together, whenever the wind is "stiff" enough to turn it. It costs, for work done in situations in which its irregularity of action is no objection, only one half or one third as much as steam, hot-air, and gas engines of similar power. At Faversbam, it is said, a 15-horse-power mill raises 2,000,000 gallons a month from a depth of 100 ft., saving 10 tons of coal a month, which would other- wise be expended in doing the work by steam. Electric storage and lighting from the power of a windmill has been tested on a large scale for several years bj r Charles F. Brush, at Cleveland, Ohio. In 1887 he erected on the grounds of his-dwelling a windmill 56 ft. in diam- eter, that operates with ordinary wind a dynamo at 500 revolutions per minute, with an output of 12,000 amperes— 16 electric horse-power— charging a storage system that gives a constant lighting capacity of 100 16 to 20 candle-power lamps. The current from the dynamo is automatically regu- lated to commence charging at 330 revolutions and 70 volts, and cutting the circuit at 75 volts. Thus, by its 21 hours' work, the storage system of 408 cells in 12 parallel series, each cell having a capacity of 100 ampere hours, is kept in constaut readiness for all the requirements of the establishment, it being fitted up with 350 incandescent lamps, about 100 being in use each evening. The plant runs at a mere nominal expense for oil. repairs, and at- tention. (For a fuller description of this plant, and of a more recent one at Marblehead Neck, Mass., see Lieut. Lewis's paper in Engineering Magazine, Dec. 1894, p. 475.) COMPRESSED AIR. Heating of Air by Compression.— Kimball, in his treatise on Physi- cal Properties of Gases, says: When air is compressed, all the work which is done in the compression Is converted into heat, and shows itself in the rise in temperature of the compressed gas. As the gas becomes hotter it is com- pressed with more difficulty; so in practice many devices are employed to carry off the heat as fast as it is developed, and keep the temperature down. But it is not possible in any way to totally remove this difficulty. But, it may be objected, if all the work done in compression is converted into heat, and if this heat is got rid of as soon as possible, then the work may be virtually thrown away, and the compressed air can have no more energy than it had before compression. It s true that the compressed gas has no more energy than the gas had before compression, if its temperature is no higher, but the advantage of the compression lies in bringing its energy into more avail- able form. The total energy of the compressed and uncompressed gas is the same at the same temperature, but the available energy is much greater in the former. The rise in temperature due to compression is so great that if a mass of air at 32° F. is compressed to one fourth its original volume, its temperature will be raised 376° F., if no heat is allowed to escape. When the compressed air is used in driving a rock-drill, or any other piece of machinery, it gives tip energy equal in amount to the work it does, and its temperature is accordingly greatly reduced. Causes of IiOss of Energy in Use of Compressed Air. (Zahner, on Transmission of Power by Compressed Air.)— 1. The compression of air always develops heat, and as the compressed air always cools down to the temperature of the surrounding atmosphere before it is used, the me- chanical equivalent of this dissipated heat is work lost. 2. The heat of compression increases the volume of the air, and hence it is necessary to carry the air to a higher pressure in the compressor in order that we may finally have a given volume of air at a given pressure, and at the temperature of the surrounding atmosphere. The work spent in effect- ing this excess of pressure is work lost. 500 3. The great cold which results when air expands against a resistance forbids expansive working, which is equivalent to saying, forbids the reali- zation of a high degree of efficiency in the use of compressed air. 4. Friction of the air in the pipes', leakage, dead spaces, the resistance of- fered by the valves, insufficiency of valve-area, inferior workmanship, and slovenly attendance, are all more or less serious causes of loss of power. The first cause of loss of work, namely, the heat developed by compres- sion, is entirely unavoidable. The whole of the mechanical energy which the compressor-piston spends upon the air is converted into heat. This heat is dissipated by conduction and radiation, and its mechanical equivalent is work lost. The compressed air, having again reached thermal equilibrium with the surrounding atmosphere, expands and does work in virtue of its intrinsic energy. The intrinsic energy of a fluid is the energy which it is capable of exert- ing against a piston in changing from a given state as to temperature and volume, to a total privation of heat and indefinite expansion. Volumes, Mean Pressures per Stroke, Temperatures, etc., in the Operation of Air-compression from 1 Atmosphere and 60° Fahr. (F. Richards, Am. Alack., March 30, 1893.) 5 ■r. © 1 o a < £3 © > o o * > a H 1 < > o3 > S 3 H 1 3 4 5 6 2 3 4 5 6 7 1 1 1 60° 80! 6. 442|. 1552 .267 27.38 36.64 432 1 1.068 .9363 .95 .96 .975 71 851 6.782 .1474 2566 28.16 37.94 447 2 1.136 .8803 .91 1.87 1.91 80.4 90; 7.1221.1404 .248 28.89 39.18 459 3 1.204 .8305 .876 2.72 2.8 88.9 95! 7.462!. 134 100 1 7.802 .1281 .24 29.57 40.4 472 4 1.272 .7861 .84 3.53 3.67 98 .232 30.21 41.6 485 5 1.34 .7462 .81 4.3 4 5 106 105! 8.142J.1228 .2254 30.81 42.78 496 10 1.68 .5952 .69 7.62 8.27 145 110! 8. 483!. 1178 .2189 31.39 43.91 507 15 2.02 .495 .606 10.33 11.51 178 lisl 8 823' 1133 .2129 31.98 44.98 518 20 2.36 .4237 .543 12.62 14.4 207 120! 9.163 .1091 .2073 32.54 46.04 529 25 2.7 .3703 .494 14.59 17.01 234 125 1 9.503 .1052 .2020 33.07 47.06 540 3ii 3.04 3289 .4538 16.34 19.4 252 130 9.843 .1015 .1969 33.57 48.1 550 35 3.381 .2957 .42 17.92 21.6 281 135 10.183 .0981 .1922 34.05 49.1 560 40 3.721 .2687 .393 19.32 23.66 302 140 30.523 .095 .1878 34.57 50.02 570 45 4.061 .2462 37 20.57 25.59 321 145 10.864 .0921 .1837 35.09 51. 580 no 4.401 .2272 .35 21.69 27.39 339 150 11.204 .0892 .1796 35.48 51.89 589 55 4.741 .2109 .331 22.76 29.11 357 160 11.88 .0841 .1722 36.29 53.65 607 CO 5.081 .1968 .3141 23.78 30.75 375 170 12.56 .0796 .1657 37.2 55.39 624 65 5.423 .1844 .301 24.75 32.32 389 180 13.24 .0755 .1595 37.96 57.01 640 70 5.762 .1735 .288 25.67 33.83 405 190 13.92 .0718 .154 38.68 58.57 657 75 6.102 .1639 .276 26.55 35.27 420 200 14.6 .0685 .149 39.42 60.14 672 Column 3 gives the volume of air after compression to the given pressure and after it is cooled to its initial temperature. After compression air loses its heat very rapidly, and this column may be taken to represent the volume of air after compression available for the purpose for which the air has been compressed. Column 4 gives the volume of air more nearly as the compressor has to deal with it. In any compressor the air will lose some of its heat during compression. The slower the compressor runs the cooler the air and the smaller the volume. Column 5 gives the mean effective resistance to be overcome by the air- cylinder piston in the stroke of compression, supposing the air to remain constantly at its initial temperature. Of course it will not so remain, but this column is the ideal to be kept in view in economical air-compression. COMPRESSED AIR. 501 Column 6 gives the mean effective resistance to be overcome by the pis- ton, supposing that there is no cooling of the air. The actual mean effec- tive pressure will be somewhat less than as given in this column; but for computing the actual power required for operating air-compressor cylinders the figures in this column may be taken and a certain percentage added — say 10 per cent— and the resuk will represent very closely the power required by the compressor. The mean pressures given being for compression from one atmosphere upward, they will not be correct for computations in compound compression or for any other initial pressure. Loss Due to Excess of Pressure caused by Heating In the Compression-cylinder.— If the air during compression weie kept at a constant temperature, the compression-curve of an indicator-dia- gram taken from the cylinder would be an isothermal curve, and would fol- low the law of Boyle and Marriotte, pv = a constant, or pjV, = PoV Q , or Pi =Po — i Po ana " v o being the pressure and volume at the beginning of compression, a,ndp 1 v 1 the pressure and volume at the end, or at any inter- mediate point. But as the air is heated during compression the pressure increases faster than the volume decreases, causing the work required for any given pressure to be increased. If none of the heat were abstracted by radiation or by injection of water, the curve of the diagram would be an adiabatic curve, with the equation p x = p (^— ) ' Cooling the air dur- ing compression, or compressing it in two cylinders, called compounding, and cooling- the air as it passes from one cylinder to the other, reduces the exponent, of this equation, and reduces the quantity of work necessary to effect a given compression. F. T. Gause (Am. Much., Oct. 20, 1892), describ- ing the operations of thePopp air-compressors in Paris, says : The greatest saving realized in compressing in a single cylinder was 33 per cent of that theoretically possible. In cards taken from the 2000 H.P. compound com- pressor at Quai De La Gare. Paris, the saving realized is 85 per cent of the theoretical amount. Of this amount only 8 per cent is due to cooling dur- ing compression, so that the increase of economy in the compound com- pressor is mainly due to cooling the air between the two stages of compres- sion. A compression-curve with exponent 1.25 is the best result that was obtained for compression in a single cylinder and cooling with a very fine spray. The curve with exponent 1.15 is that which must be realized in a single cylinder to equal the present economy of the compound compressor at Quai De La Gare. Horse-power required to compress one cubic foot of Free Air per minute to a given Pressure with no cooling of the air during the compression; also the horse-power required, sup- posing the air to be maintained at constant temperature during the compression. (Richards.) Gauge- Air not Air constant 70 60 50 40 cooled. .22183 .20896 .19521 .17989 .164 .14607 .12433 .10346 .076808 .044108 .024007 Temperature, .14578 .13954 .13251 .12606 .11558 .10565 .093667 .079219 .061188 Horse - power required to deliver one cubic foot of Air per minute at a given Pressure with no cooling of the air during the compiessiou; also the horse-power required, supposing the air to be maintained at constaut temperature during the compres- sion. (Richards.) Gauge- pressure. 100 90 80 70 60 50 40 30 Air not cooled. 1.7317 1.4883 1.25779 1.03683 .83344 .64291 .46271 .31456 .181279 .074106 .032172 In computing the above table an allowance of 10 per cent for friction of the compressor. Air constant Temperature. 1.13801 .99387 .8538 .72651 .58729 .465 .34859 .24086 .14441 .06069 .027938 has been made 502 TaMe for Adiabatlc Compression or Expansion of Air. (Proc. Inst. M.E., Jan. 1881, p. 123.) Absolute Pressure. Absolute Temperature. "Volume. Ratio of Ratio of Ratio of Ratio of Ratio of Ratio of Greater Less to Greater Less to Greater Less to to Less. Greater. to Less. Greater. to Less. Greater. (Expan- (Compres- (Expan- (Compres- (Compres- (Expan- sion.) sion.) sion.) sion.) sion.) sion.) 1.8 .833 1.054 .948 1.138 .879 1.4 .714 1.102 .907 1.270 .788 1.6 .625 1.146 .873 1.396 .716 1.8 .556 1.186 .843 1.518 .659 2.0 .500 1.222 .818 1.636 .611 2.2 .454 1.257 .796 ■ 1.750 .571 2.4 .417 1.289 .776 1.862 .537 2.6 .385 1.319 .758 1.971 .507 2.8 .357 1.348 .742 2.077 .481 3.0 .333 1.375 .727 2.182 .458 3.2 .312 1.401 .714 2.284 .438 3.4 .294 1.426 .701 2.384 .419 3.6 .278 1.450 .690 2.483 .403 3.8 .263 1.473 .679 2.580 .388 4.0 .250 1.495 .669 2.676 .374 4.2 .238 1.516 .660 2.770 .361 4.4 .227 1.537 .651 2.863 .349 4.6 .217 1.557 .642 2.955 .338' 4.8 .208 1.576 .635 3.046 .328 5,0 .200 1.595 .627 3.135 .319 6.0 .167 1.681 .595 3.569 .280 7.0 .143 1.758 .569 3.981 .251 8.0 .125 1.828 .547 4.377 .228 9.0 .111 1.891 .529 4.759 .210 10.0 .100 1.950 .513 5.129 .195 Mean Effective Pressures for tlie Compression Part only of the Stroke when compressing and delivering Air from one Atmosphere to given Gauge-pressure in a Sin- gle Cylinder. (b\ Richards, Am. Much., Dec. 14, 1893.) Gauge- Adiabatic Isothermal Gauge- Adiabatic Isothermal pressure. Compression Compression . pressure. Compression. Compression. 1 .44 .43 45 13.95 12.62 2 .96 .95 50 15.05 13.48 3 1.41 1.4 55 15.98 14.3 4 1.86 1.84 60 16.89 15.05 5 2.26 2.22 65 17.88 15.76 10 4.26 4.14 70 18.74 16.43 15 5.99 5.77 75 19.54 17.09 20 7.58 7.2 80 20.5 17.7 25 9.05 8.49 85 31.22 18.3 30 10.39 9.66 90 22. 18.87 35 11.59 10.72 95 22.77 19.4 40 12.8 11.7 100 23.43 19.92 The mean effective pressure for compression only is always lower than the mean effective pressure for the whole work COMPRESSED AIR. 503 Mean and Terminal Pressures of Compressed Air used Expansively for Gauge-pressures from 60 to 100 lbs. (Frank Richards, Am. Much., April 13, 1893.) Initial Pres- 60. 70. 80. 90. 100. sure. ojb ,2 13 CD a , £ % © <6 | ^ £ .2 i. 8 a , £ "3 g 6 3 £ 4J o a3 u S as L 3 •9^3 § - P o 3 ©'=3 '{I ft H ft ft $ ft £3f H ft ft 39 04 £ ft ft H ft .25 23.6 JO. 65 28.74 12.07 33.89 13.49 U.91 44.19 1.33 .30 28.9 13.77 34.75 .6 40.61 1 2.44 46.46 4.27 53.32 6.11 Vb 32.13 .96 38.41 3.09 44.69 5.22 50.98 7.35 57.26 9.48 .35 33.66 2.33 40.15 4.38 46.64, 6.66 53.13 8.95 59.62 11.23 y% 35.85 3.85 42.63 6.36 49.411 7.88 56.2 11.39 62.98 13.89 .40 37.93 5.64 44.99 8.39 52.05 11.14 59.11 13.88 66.16 16.64 .45 41.75 10.71 49.31 12.61 56.9 1 15.86 64 45 19.11 72.02 22.36 .50 45.14 13.26 53.16 17. 61.18 20.81 69.19 24.56 77.21 2S.33 .60 50.75 21.53 59.51 26.4 68.28 31.27 77.05 36.14 85.82 41.01 y& 51.92 23.69 60.84 28.85 69.76 34.01 78.69 39.16 87.61 44.32 Vb 53.67 27.94 62.83 33.03 71.99 38.68 81.14 44.33 90.32 49.97 .70 54.93 30.39 64 25 36.44 73.57| 42.49 82.9 48.54 92.22 54.59 .75 56.52 35.01 66.05 41.68 75.59 1 48.35 85.12 55.02 94.66 61.69 .80 57.79 39.78 67.5 47.08 77.2 54.38 86.91 61.69 96.61 68.99 Vs 59.15 47.14 69.03 55.43 78.92, 63.81 88.81 72. 98.7 80.28 .90 59.46 49.65 69.38 58.27 79 31 1 66.89 89.24 75.52 99.17 87.82 The pressures in the table are all gauge-pressures except those in italics, which are absolute pressures (above a vacuum). Straight-line Air-compressors, Ingersoll-Sergeant Rock-drill Co. Diameter Steam- cylinder, inches. Diameter of Air- cylinder, inches. Length of Stroke, inches. No. of Revolu- tions per minute. Piston Speed in feet per minute. Cubic Feet Free Air per minute (Theo- retical). Horse- power of Boiler required. 4 m 10 175 291 28 6 5 b}4 10 175 291 42 8 6 12 160 320 66 10 7 m 12 160 320 91 12 8 &A 12 160 320 117 15 9 9H 12 160 320 148 20 10 IO14 14 155 361 207 30 12 12^4 14J4 14 155 361 295 40 14 18 120 360 398 55 16 16J4 18 120 360 518 70 18 1814 24 94 376 683 100 20 20^ 24 94 376 840 130 22 22J4 30 75 375 1011 155 24 24^4 30 75 375 1202 t.'00 The same sizes are made to be driven by belt or gearing. Compressors at High Altitudes.— Cubic feet of compressed air delivered by air-compressors at high altitudes, expressed as a percentage of the air delivered at the sea-level. Altitude above Sea- | level, feet. j 1000 2000 3000 4000 5000 60C0 7000 8000 9000 10000 Air delivered, per cent. . 100 97 94 91 89 86 84 81 78 70 74 504 AIR. Standard Air-compressors driven by Steam. (Norwalk Iron Works Co.) In the following list the large air-cylinder gives the capacity of the ma- chine. For actual capacity, allowance of 10 per cent may be made for contingencies. The small piston only encounters the pressure of the final compression. 3 . js lis teg u . fee's sssZ HOt«C c *> a oQ£c oretical pacity, bic feet r minute, ee Air. & "ft E 00 2 3 Kft ft ft 63 ft 'ft -. .SCO $§.£ .2co o E S o c Ha £3 £ ft level. feet. feet. feet. .2 « i © Sh o O ft '5 S3 ft •L J O O ft '3 eS ft , 3 V >. t c - ft >> '3 03 ft . 3 o a ft J Q A ti o w O w O w o W 12 12 7 10 190 298 35 2N0 34 24-1 32 214 30 16 16 9U 14 150 558 70 524 68 462 64 405 60 20 20 13W 18 120 872 110 819 107 72-,' 100 634 94 22 24 13U 20 110 1160 145 lO'.IO 140 9(H) 132 843 124 26 30 17^ 24 90 1659 215 1500 207 195 1200 184 The delivery and power of the compressors decrease as the height in- creases. As the capacity decreases in a greater ratio than the power necessary to compress, it follows that operations at a high altitude are more expensive than at sea-level. At 10,000 feet this extra expense amounts to over 20 per cent. COMPRESSED AIR. 505 Rand Drill Co.'s Air-coiiipressors. ■ ) and ■{ B i A f and I B Geared C 1 u Theo S3 S of Air- eyliuders •£ 2 in 5,'- inches. O S3 > pel Free. 10x16 |d*.: 100 100 145.44 290.88 14x22] I'- 85 85 333.20 666.40 16^x30 1|; 75 75 556.83 1113.66 18x30-1 D "" 75 75 662.68 1325.36 20x48] »;;; 50 50 872.66 1745.32 28x4s]S;;; 40 1368.34 40 - 32x48] 1;;; 40 40 1787.22 3574.44 32 x 60 ] |- • • 35 35 1954.77 3909.55 36 x 60 ] |- • 30 30 2120.61 4241.22 8x12 120 83.78 10x14 110 139.95 12x16 100 209.44 14x22 95 372.40 16x24 90 502.66 17^x24 .... 90 601.29 20x30... . 80 872.67 Theoretical Volume of Air delivered in cubic feet per minute, at Sea-level. Compressed to a Gauge-pressure of- 10 20 40 60 80 100 lbs. lbs. lbs. lbs. lbs. lbs. 86.56 61.61 39.08 28.62 22.57 18.64 173.12 123.23 78.17 57.24 45.15 37.28 198.31 141.10 89.51 65.54 51.93 42.67 396.61 282.20 179.01 131.07 103.86 85.31 331.39 235.89 149.64 109.57 86.43 71.36 662.79 471.79 299.28 219.15 172.86 142.72 394.39 280.73 178.08 130.40 102.86 84.92 788.78 561.46 356.17 260.81 205.72 169.84 519.36 369.69 234.51 171.72 135.46 111.84 1038.72 739.38 469.03 270.92 223.68 814.36 579.67 367.72 212.40 175.36 1628.71 1159.34 735.45 538.54 424.80 350.73 1063.65 757.12 480.29 351.70 277.42 229.05 2127.30 1514.24 960.58 703.40 554.85 458.10 1163.37 828.10 525.32 384.67 303.43 250.52 : 1656.20 1050.63 769.34 606.86 501.05 1262.07 898.35 572.07 417.72 329.16 272.82 2524.14 1796.70 1144.14 835.44 658.32 545.64 49.86 35.49 22; 51 16.49 13.00 10.74 83.27 59 29 37.62 27.50 21.72 17.94 124.65 88.73 56.28 41.22 32.51 26.66 221.64 157.70 100.04 73.25 58.04 47.69 299.15 212.94 135.08 98.92 78.03 64.42 357.85 254.95 161.60 118.33 93.33 77.06 519.36 369.69 234.52 171.73 135.46 111.84 *S, Single; D, Duplex. Practical Results with Compressed Air.— Compressed-air System at the Chapiu Mines, Iron Mountain, Mich. — These mines are three miles from the falls which supply the power. There are four turbines at the falls, one of 1000 horse-power and three of 900 horse-power each. The pressure is 60 pounds at 60° Fahr. Each turbine runs a pair of compressors. The pipe to the mines is 24 inches in diameter. The power is applied at the mines to Corliss engines, running pumps, hoists, etc., and direct to rock- drills. A test made in 1888 gave 1430.27 horse-power at the compressors, and 390.17 horse-power as the sum of the horse-power of the engines at the mines. Therefore, only 27$ of the power generated was recovered at the mines. This includes the loss due to leakage and the loss of energy in heat, but not the friction in the engines or compressors. (F. A. Pocock, Trans. A. I. M. E., 1890.) W. L. Saunders (Jour. F. I. 1892) says: " There is not a properly designed compressed-air installation in operation to-day that loses over h% by trans- mission alone. The question is altogether one of the size of pipe; and if the pipe is large enough, the friction loss is a small item. The largest com- pressed-air power plant in America is that at the Chapin Mines in Michigan, where power is generated at Quinnesec Falls, and transmitted three miles. This is is not an economical plant, but the loss of pressure as shown by the gauge is only 2 lbs., and this is the loss which may be laid strictly to trans- mission. " The loss of power in common practice, where compressed air is used to drive machinery in mines and tunnels, is about 70$. I refer to cases where common American air-compressors are used, and where the air is trans- mitted far enough to lose its heat .of compression and is exhausted without 506 AIR. reheating. In the best practice, with the best air-compressors, and without reheating, the loss is about 60*. " These losses may be reduced to a point as low as 20% by combining the best systems of reheating with the best air-compressors. 11 Prof. Kennedy says compressed air transmission system is now being carried on, on a "large commercial scale, in such a fashion that a small motor four miles away from the central station can indicate in round numbers 10 horse-power, for 20 horse power at the station itself, allowing for the value of the coke used in heating the air. The limit to successful reheating lies in the fact that air-engines cannot work to advantage at temperatures over 350°. The efficiency of the common system of reheating is shown by the re- sults obtained with the Popp system in Paris. Air is admitted to the re- heater at about 83°, and passes to the engine at about 315°, thus being in- creased in volume about 42*. The air used in Paris is about 11 cubic feet of free air per minute per horse-power. The ordinary practice in America with cold air is from 15 to 25 cubic feet per minute per horse-power. When the Paris engines were worked without reheating the air consumption was increased to about 15 cubic feet per horse-power per minute. The amount of fuel consumed during reheating is trifling. Efficiency of Compressed-air Engines.— The efficiency of an air-engine, that is, the percentage which the power given out by the air-en- gine bears to that required to compress the air in the compressor, depends on the loss by friction in the pipes, valves, etc., as well as in the engine itself. This question is treated at length in the catalogue of the Norwalk Iron Works Co., from which the following is condensed. As the friction increases the most economical pressure increases. In fact, for any given friction in a pipe, the pressure at the compressor must not be carried below a certain limit. The following table gives the lowest pressures which should be used at the compressor with varying amounts of friction in the pipe: Friction, lbs 2.9 5.8 8.8 11.7 14.7 17.6 20.5 23.5 26.4 29.4 Lbs. at Compressor... 20.5 29.4 38.2 47. 52.8 61.7 70.5 76.4 82.3 88.2 Efficiency* 70.9 64.5 60.6 57.9 55.7 54.0 52.5 51.3 50.2 49.2 An increase of pressure will decrease the bulk of air passing the pipe and its velocity. This will decrease the loss by friction, but we subject ourselves to a new loss, i.e. the diminishing efficiencies of increasing pressures. Yet as each cubic foot of air is at a higher pressure and therefore carries more power, we will not need as many cubic feet as before, for the same work. With so many sources of gain or loss, the question of selecting the proper pressure is not to be decided hastily. The losses are, first, friction of the compressor. This will amount ordinarily to 15 or 20 per cent, and cannot probably be reduced below 10 per cent. Second, the loss occasioned by pumping the air of the engine-room, rather than the air drawn from a cooler place. This loss varies with the season and amounts from 3 to 10 per cent. This can all be saved. The third loss, or series of losses, arises in the compressing cylinder, viz., insufficient supply, difficult discharge, defective cooling arrangements, poor lubrication, etc. The fourth loss is found in the pipe. This loss varies with the situation, and is subject to somewhat complex influences. The fifth loss is chargeable to fall of temperature in the cylinder of the air-engine. Losses arising from leaks are often serious. Air should be drawn from outside the engine-room, and from as cool a place as possible. The gain amounts to one per cent for every five degrees that the air is taken in lower than the temperature of the engine-room. The inlet conduit should have an area at least 50* of the area of the air- piston, and should be made of wood, brick, or other non-conductor of heat. Discharge of a compressor having an intake capacity of 1000 cubic feet per minute, and volumes of the discharge reduced to cubic feet at atmos- pheric pressure and at temperature of 62 degrees Fahrenheit: Temperature of Intake, F 0° 32° 62° 75 c 80° 90° 100° 1 10° Relative volume discharged, cubic ft.. . 1135 1060 1000 975 966 949 932 916 Requirements of Rock-drills Driven by Compressed Air. (Norwalk Iron Works Co.)— The speed of the drill, the pressure of air, and the nature of the rock affect the consumption of power of rock- drills. A three-inch drill using air at 30 lbs. pressure made 300 blows per minute and consumed the equivalent of 64 cubic, feet of free air per minute, The COMPRESSED AIR. 507 same drill, with air of 58 lbs. pressure, made 450 blows per minute and con- sumed 160 cubic f> j et of free air per minute. At Hell Gate different machines doing the same work used from 80 to 150 cubic feet free air per minute. An average consumption may be taken generally from 80 to 100 cubic feet per minute, according to the nature of the work. The Popp Compressed-air System in Paris.— A most exten- sive system of distribution of power by means of compressed air is that of M. Popp, in Paris. One of the central stations is laid out for 24,000 horse- power. For a very complete description of the system, see Engineering, Feb. 15, June 7, 21. and 28, 1889, and March 13 and 29, April 10, and May 1, 1891. Also Proc. Inst. M. E., July, 1889. A condensed description will be found in Modern Mechanism, p. 12. Utilization of Compressed Air in Small Motors.— In the earliest stages of the Popp system in Paris it was recognized that no good results could be obtained if the air were allowed to expand direct into the motor; not only did the formation of ice due to the expansion of the air rapidly accumulate and choke the exhaust, but the percentage of useful work obtained, compared with that put into the air at the central station, was so small as to render commercial results hopeless. After a number of experiments M. Popp adopted a simple form of cast- iron stove lined with fire-clay, heated either by a gas jet or by a small coke fire. This apparatus answered the desired purpose until some better ar- rangement was perfected, and the type was accordingly adopted through- out the whole system. The economy resulting from the use of an improved form was very marked, as will be seen from the following table. Efficiency of Air-heating Stoves. © o w be a 3 P. is 3 c3 O < Temperature of Air in Oven. Value of Heat Absorbed per Hour. Nature of Stove. .2 -a O Eh Per Square Foot of Heat i n g Surface. £0 Cast-iron box j stoves j Wrought-i r o n coiled tubes.. sq. ft. 14 14 46.3 cub.ft. 20,342 11,054 38,428 45 45 41 215 361 347 cal. 17,900 17,200 39,200 cal. 1278 1228 830 cal. 2032 2058 2545 The results given in this table were obtained from a large number of trials. From these trials it was found that more than 70# of the total num- ber of calories in the fuel employed was absorbed by the air and trans- formed into useful work. Whether gas or coal be employed as the fuel, the amount required is so small as to be scarcely worth consideration; accord- ing to the experiments carried out it does not exceed 0.2 lb. per horse-power per hour, but it is scarcely to be expected that in regular prac- tice this quantity is not largely exceeded. The efficiency of fuel consumed in this way is at least six times greater than when utilized in a boiler and steam-engine. According to Prof. Riedler, from 15$ to 20$ above the power at the central station can be obtained by means at the disposal of the power users, and it has been shown by experiment that by heating the air to 480° F. an in- creased efficiency of 30% can be obtained. A large number of motors in use among the subscribers to the Compressed Air Company of Paris are rotary engines developing 1 horse-power and less, and these in the early times of the industry were very extravagant in their consumption. Small rotary engines, working cold air without expan- sion, used as high as 2330 cu. ft. of air per brake horse-power per hour, and with heated air 16$4 cu. ft. Working expansively, a 1 horse- power rotary engine used 1469 cu. ft. of cold air, or 960 cu. ft. of heated air, and a 2- horse-power rotary engine 1059 cu. ft. of cold air. or 847 cu. ft. of air, heated to about 50° C. The efficiency of this type of rotary motors, with air heated to 50° C , may now be assumed at 43$. With such an efficiency the use of small motors in II. III. IV. 69.7 85 71 356 388 46 68 77 350 310 243 477 376 316 791 900 1148 508 air. many industries becomes possible, while in cases where it is necessary to have a constant supply of cold air economy ceases to be a matter of the first importance. The following table shows the results of tests of a small rotary engine used for driving sewing-machines, and indicating about a tenth of a horse-power: Trials of a Small Rotary Riedinger Engine. Numbers of trials I. II. Initial air-pressure, lbs. per sq. in 86 71.8 Initial temperature, deg. Fahr 54° 338° Ft.-lbs. per sec, measured on the brake 51.63 34.07 Revolutions per minute 384 300 Consumption of air per 1 horse-power per hour 1377 988 The following table shows the results obtained with a one-half horse- power variable expansive Riedinger rotary engine. These trials represent the best practice that has been obtained up to the present time (1890). The volumes of air were in all cases taken at atmospheric pressure: Trials of a 5-Horse-power Riedinger Rotary Engine. Numbers of trials I. Initial pressure of air, lbs. per sq. in 54 " temperature of air, deg. Fahr 338 Final " " " " " .. 77 Revolutions per minute 335 Ft.-lbs. per second, measured on brake . . 271 Consumption of air per horse-power per hour 883 Trials made with an old single-cylinder 80-horse-power Farcot steam-en- gine, indicating 72 horse-power, gave a consumption of air per brake horse- power as low as 465 cu. ft. per hour. The temperature of admission was 320° F., and of exhaust 95° F. Prof. Elliott gives the following as typical results of efficiency for various systems of compressors and air-motors : Simple compressor and simple motor, efficiency S9A% Compound compressor and simple motor, " 44.9 " " " compound motor, efficiency 50.7 Triple compressor and triple motor, " 55.3 The efficiency is the ratio of the indicated horse-power in the motor cylin- ders to the indicated horse-power in the steam-cylinders of the compressor. The pressure assumed is 6 atmospheres absolute, and the losses are equal to those found in Paris over a distance of 4 miles. Summary of Efficiencies of Compressed-air Transmission at Paris, between the Central Station at St. Fargeau and a 10-horse-power Motor Working with Pressure Re- duced to 4>o Atmospheres. (The figures below correspond to mean results of two experiments cold and two heated.) 1 indicated horse-power at central station gives 0.845 indicated horse-power in compressors, and corresponds to the compression of 348 cubic feet of air per hour from atmospheric pressure to 6 atmospheres absolute. (The weight of this air is about 25 pound s.} 0.845 indicated horse-power in compressors delivers as much air as will do 0.52 indicated horse-power in adiabatic expansion after it has fallen in tem- perature to the normal temperature of the mains. The fall of pressure in mains between central station and Paris (say 5 kilo- metres) reduces the possibility of work from 0.52 to 0.51 indicated horse- power. The further fall of pressure through the reducing valve to 4J^ atmospheres (absolute) reduces the possibility of work from 0.51 to 0.50. Incomplete expansion, wire-drawing, and other such causes reduce the actual indicated horse-power of the motor from 0.50 to 0.39. By heating the air before it enters the motor to about 320° F., the actual indicated horse-power at the motor is. however, increased to 0.54. The ratio of gain by heating the air is, therefore, r 1 ^ = 1.38. COMPRESSED AIR. 500 In this process additional heat is supplied by the combustion of about 0.39 pounds of coke per indicated horse-power per hour, and if this be taken into account, the real indicated efficiency of the whole process becomes 0.47 instead of 0.54. Working with cold air the work spent in driving the motor itself reduces the available horse-power from 0.39 to 0.26. Working with heated air the work spent in driving the motor itself reduces the available horse-power from 0.54 to 0.44. A summary of the efficiencies is as follows : Efficiency of main engines 0.845. Efficiency of compressors 0.52 -^ 0.845 = 0.61. Efficiency of transmission through mains 0.51 -v- 0.52 = 0.98. Efficiency of reducing valve 0.50-i- 0.51 = 0.98. The combined efficiency of the mains and reducing valve between 5 and 4)4 atmospheres is thus 0.98 X 0.9S = 0.96. If the reduction had been to 4, 3J^, or 3 atmospheres, the corresponding efficiencies would have been 0.93, 0.89, and 0.85 respectively. Indicated efficiency of motor 0.39 -h- 0.50 = 0.78. Indicated efficiency of whole process with cold air 0.39. Apparent indi- cated efficiency of whole process with heated air 0.54. Real indicated efficiency of whole process with heated air 0.47. Mechanical efficiency of motor, cold, 0.67. Mechanical efficiency of motor, hot, 0.81. Most of the compressed air in Paris is used for driving motors, but the work done by these is of the most varied kind. A list of motors driven from St. Fargeau station shows 225 installations, nearly all motors working at from % horse-power to 50 horse-power, and the great majority of them more than two miles away from the station. The new station at Quai de la Gare is much larger than the one at St. Fargeau. Experiments on the Riedler air-compressors at Paris, made in December, 1891, to determine the ratio between the indicated work done by the air-pistons and the indicated work in the steam-cylinders, showed a ratio of 0.8997. The compressors are driven by four triple-expansion Corliss engines of 2000 horse-power each. Shops Operated by Compressed Air.— The Iron Age, March 2, 1893, describes the shops of the Wuerpel Switch and Signal Co. .East St. Louis, the machine tools of which are operated by compressed air, each of the larger tools having its own air engine, and the smaller tools being belted from shafting driven by an air engine. Power is supplied by a compound compressor rated at 55 horse-power. The air engines are of the Kriebel make, rated from 2 to 8 horse-power. Pneumatic Postal Transmission.— A paper by A Falkenau, Eng'rs Club of Philadelphia, April 1894, entitled the "First United States Pneumatic Postal System, 1 ' gives a description of the system used in London and Paris, and that recently introduced in Philadelphia between the main post-office and a substation. In London the tubes are 214 and 3 inch lead pipes laid in cast-iron pipes for protection. The carriers used in 2^-inch tubes are but 1*4 inches diameter, the remaining space being taken up by packing. Carriers are despatched singly. First, vacuum alone was used; later, vacuum and compressed air. The tubes used in the Continental cities in Europe are wrought iron, the Paris tubes being 2^ inches diameter. There the carriers are despatched in trains of six to ten, propelled by a' piston. In Philadelphia the size of tube adopted is 6J-6 inches, the tubes being of cast iron bored to size. The lengths of the outgoing and return tubes are 2928 feet each. The pressure at the main station is 7 lbs., at the substation 4 lbs., and at the end of the return pipe atmospheric pressure. The compressor has two air-cylinders 18 X 24 in. Each carrier holds about 200 letters, but 100 to 150 are taken as an average. Eight carriers may be despatched in a minute-, giving a delivery of 48,000 to 72,000 letters per hour. The time required in transmission is about 57 seconds. The Mekarski Compressed-air Tramway at Berne, Switzerland. (Eng'g Neivs, April 20, 1893.)— -The Mekarski system has been introduced in Berne, witzerland, on a line about two miles long, with grades of 0.25$? to 3.7$ and 5.2$. A special feature of the Mekarski system is the heating of the air, to maintain it at a constant temperature, by passing it through superheated water at 330° F. The air thus becomes saturated with steam, which subsequently partly condenses, its latent heat being absorbed by the expanding air. The pressure in the car reservoirs is 440 lbs. per sq. in. The engine is constructed like an ordinary steam tramway locomotive. 510 AIR. and drives two coupled axles, the wheel-base being 5.2 ft. It has a pair of outside horizontal cylinders, 5.1 x 8.6 in.; four coupled wheels, 27.5 in. diameter. The total weight of the ear including compressed air is 7.25 tons, and with 30 passengers, including the driver and conductor, about 9.5 tons. The authorized speed is about 7 miles per hour. Taking the resistance due to the grooved rails and to curves under unfavorable conditions at 30 lbs. per ton of car weight, the engine has to overcome on the steepest grade, b%, a total resistance of about 0.63 ton, and has to develop 25 H.P. At the maximum authorized working pressure in cylinders of 176 lbs. persq. in. the motors can develop a tractive force of 0.64 ton. This maximum is, there- fore, just sufficient to take the car up the h.2% grade, while on the flatter sections of the line the working pressure does not exceed 73 to 147 lbs. per sq. in. Sand has to be frequently used to increase the adhesion on the 2% to 5% grades. Between the two car frames are suspended ten horizontal compressed-air storage-cylinders, varying in length according to the available space, but of uniform inside diameter of 17.7 in., composed of riveted 0.27-in. sheet iron, and tested up to 588 lbs. per sq. in. These cylinders have a collective capacity of 64.25 cu. ft., which, according to Mr. Mekarski's estimate, should have been sufficient for a double trip, 3% miles. The trial trips, however, showed this estimate to be inadequate, and two further small storage-cylinders had therefore to be added of 5.3 cu. ft. capacity each, bringing the total cubic contents of the 12 storage-cylinders per car up to 75 cu. ft., divided into two groups, the working and the reserve battery, the former of 49 cu. ft. the latter of 26 cu. ft. capacity. From the results of six official trips, the pressure and the mean consump- tion of air during a double journey per motor car are as follows: Working, Reserve, Storage-cylinders. ^ P er ^fjf Pressure of air on starting 440 440 Pressure of air at end of up journey 176 260 Pressure of air at end of down journey 103 176 Lbs. Consumption of air at end of up journey 92 Consumption of air during down journey 31 This has been fully confirmed by the working experience of 1891, when the consumption of air per motor car and double journey was as follows: Minimum, 103 lbs „. . 28 lbs. per car-mile. Maximum, 154 lbs 42 " " " Mean, 123 lbs 35 " The principal advantages of the compressed-air system for urban and suburban tramway traffic as worked at Berne consist in the smooth and. noiseless motion ; in the absence of smoke, steam, or heat, of overhead or underground conductors, of the more or less grinding motion of most electric cars, and of the jerky motion to which underground cable traction is subject. On all these grounds the system has vindicated its claims as being preferable to any other so far known system of mechanical traction for street tramways. Its disadvantages, on i he other hand, consist in the extremely delicate adjustment of the different parts of the system, in the comparatively small supply of air carried by one motor car, which necessi- tates the car returning to the depot for refilling after a run of only four miles or 40 minutes, although on the Nogent and Paris lines the cars, which are, moreover, larger, and carry outside passengers on the top, run seven miles, and the loading pressure is 517 lbs. per sq. in. as against oidy 440 lbs. at Berne. Longer distances in the same direction would involve either more power- ful motors, a larger number of storage-cylinders, and consequently heavier cars, or loading stations every four or seven miles; and in this respect the system is manifestly inferior to electric traction, which easily admits of a line of 10 to 15 miles in length being continuously fed from one central station without the loss of time and expense caused by reloading. The cost of -working the Berne line is compared in the annexed table with some other tramways worked under similar conditions by horse and mechanical traction for the year 1891. As is seen, both in the case of com- pressed air and of electric traction, the cost of working is considerably FANS AND BLOWERS. 511 increased where steam at a high cost of fuel has to be used instead of hydraulic power. Given the latter, the cost of working by air is about the same as that by steam-locomotives or steam-cars; but over both of these last-named, compressed-air offers, at equal cost and for such short lines with constant traffic, certain advantages: Constr. Opera- 1891. Length of Line, Motive Power, and equip't tion, miles. per mile. p. car mi Geneva, city 8.68 Horse $60,800 19.4 cts Zurich, city 5.58 Horse 39,700 11.6 Geneva, suburban 40.30 Steam locomotive. 32,000 13.2 Mulhouse, city 18.00 Steam locomotive.22,400 17.8 Montreux, suburban 6.82 Hydro-electric . . 20,800 10.4 Florence, suburban 4 . 96 Steam -electric .... 32,000 20.0 Tours, suburban 6.20 Steam cars 19,200 17.2 Nogent (Paris), suburban 7.44 Steam-compr. air.46,100 25.6 Berne, city . . 1.86 Hydro-compr. air.48,950 17.8 For description of the Mekarski system as used at Nantes, France, see paper by Prof. D. S. Jacobus, Trans. A. I. M. E. xix. 553. Compressed Air for Working Underground Pumps in Mines.— Eng'g Record, May 19, 1894, describes an installation of com- pressors for working a number of pumps in the Nottingham No. 15 Mine, Plymouth, Pa., which is claimed to be the largest in America. The com- pressors develop above 2300 H.P., and the piping, horizontal and vertical, is 6000 feet in length. About 25,000 gallons of water per hour are raised. FANS AND BLOWERS. Centrifugal Fans.— The ordinary centrifugal fan consists of a num- ber of blades fixed to arms, revolving on a shaft at high speed. The width of the blade is parallel to the axis of the shaft. Most engineers 1 reference books quote the experiments of VV. Buckle, Proc. Inst. M.E., 1847, as still standard. Mr. Buckle's conclusions are given below, together with data of more recent experiments. Experiments were made as to the proper size of the inlet openings and on the proper proportions to be given to the vane. The inlet openings in the sides of the fan-chest were contracted from 17^ in., the original diameter, to 12 and 6 in. diam., when the following results were obtained: First, that the power expended with the opening contracted to 12 in. diam. was as 2)4, to 1 compared with the opening of 17J^ in. diam. ; the velocity of the fan being nearly the same, as also the quantity and density of air delivered. Second, that the power expended with the opening contracted to 6 in. diam. was as 2% to 1 compared with the opening of 17J^ in. diam.; the velocity of the fan being nearly the same, and also the area of the efflux pipe, but the density of the air decreased one fourth. These experiments show that the inlet openings must be made of sufficient size, that the air may have a free and uninterrupted action in its passage to the blades of the fan; for if we impede this action we do so at the expense of power. With a vane 14 in. long, the tips of which revolve at the rate of 236.8 ft. per second, air is condensed to 9.4 ounces per square inch above the pres- sure of the atmosphere, with a power of 9.6 H. P. ; but a vane 8 inches long, the diameter at the tips being the same, and having, therefore, the same velocity, condenses air to 6 ounces per square inch only, and takes 12 H. P. Thus the density of the latter is little better than six tenths of the former, while the power absorbed is nearly 1.25 to 1. Although the velocity of the tips of the vanes is the same in each case, the velocities of the heels of the respective blades are very different, for, while the tips of the blades in each case move at the same rate, the velocity of the heel ofthe 14-inch is in the ratio of 1 to 1.67 to the velocity of the heel of the 8-inch blade. The longer blades approaching nearer the centre, strikes the air with less velo- city, and allows it to enter on the blade with greater freedom, and with considerably less force than the shorter one. The inference is, that the short blade must take more power at the same time that it accumulates a less quanta of air. These experiments lead to the conclusion that the length of the vane demands as great a consideration as the proper diameter of the inlet opening. If there were no other object in view, it 512 AIR. would be useless to make the vanes of the fan of a greater width than the inlet opening can freely supply. On the proportion of the length and width of the vane and the diameter of the inlet opening rest the three most im- portant points, viz., quantity and density of air, and expenditure of power. In the 11-inch blade the tip has a velocity 2.6 times greater than the heel; and, by the laws of centrifugal force, the air will have a density 2.6 times greater at the tip of the blade than that at the heel. The air cannot enter on the heel with a density higher than that of the atmosphere; but in its passage along the vane it becomes compressed in proportion to its centrifugal force. The greater the length of the vane, the greater will be the difference of the centrifugal force between the heel and the tip of the blade; consequently the greater the density of the air. Reasoning from these experiments, Mr. Buckle recommends for easy ref- erence the following proportions for the construction of the fan: 1. Let. the width of the vanes be one fourth of the diameter; 2. Let the diameter of the inlet openings in the sides of the fan-chest be one half the diameter of the fan; 3. Let the length of the vanes be one fourth of the diameter of the fan. In adopting this mode of construction, the area of the inlet openings in the sides of the fan-chest will be the same as the circumference of the heel of the blade, multiplied by its width; or the same area as the space described by the heel of the blade. Best Proportions of Fans. (Buckle.) Pressure from 3 ounces to 6 ounces per square inch; or 5.2 inches to 10.1 inches of Water. Diameter Vanes. Diameter of Inlet Open- ings. Diameter of Fan. Vanes. Diameter of Inlet Open- ings. Width. Length. Width. Length. ft. ins. 3 3 6 4 ft. ins. 9 10 J* 1 ft. ins. 9 10% 1 ft. ins. 1 6 1 9 2 ft. ins. 4 6 5 6 ft. ins. / 6 ft. ins. i m 1 3 1 6 ft. ins. 2 3 2 6 3 Pressure from 6 ounces to 9 ounces per square inch, and upwards, or 10.4 inches to 15.6 inches of Water. 3 7 1 1 4 6 10% 1 4% 1 9 3 6 8)4 1 Wz 1 3 5 1 o 1 6 2 4 o 9% i sy 2 1 6 6 1 2 1 10 2 4 The dimensions of the above tables are not laid down as prescribed limits, but as approximations obtained from the best results in practice. Experiments were also made with reference to the admission of air into the transit or outlet pipe. By a slide the width of the opening into this pipe was varied from 12 to 4 inches. The object of this was to proportion the opening to the quantity of air required, and thereby to lessen the power necessary to drive the fan. It was found that the less this opening is made, provided we produce sufficient blast, the less noise will proceed from the fan; and by making the tops of this opening level with the tips of the vane, the column of air has little or no reaction on the vanes. The number of blades may be 4 or 6. The case is made of the form of an arithmetical spiral, widening the space between the case and the revolv- ing blades, circumferentially, from the origin to the opening for discharge. The following rules deduced from experiments are given in Spretson's treatise on Casting and Founding: The fan-case should be an arithmetical spiral to the extent of the depth of the blade at least. The diameter of the tips of the blades should be about double the diameter of the hole in the centre; the width to be about two thirds of the radius of the tips of the blades, The velocity of the tips of the blades should be rather FANS AND BLOWERS. 513 more than the velocity due to the air at the pressure required, say one eighth more velocity. In some cases, two fans mounted on one shaft would be more useful than one wide one, as in such an arrangement twice the aiva of inlet opening is obtained as compared with a single wide fan. Such an arrangement may be adopted where occasionally half the full quantity of air is required, as one of them may be put out of gear, thus saving power. Pressure due to Velocity of the Fan-blades.— " By increas- ing the number of revolutions of the fan the head or pressure is increased, the law being that the total head produced is equal (in centrifugal fans) to twice the height due to the velocity of the extremities of the blades, or v i H = — approximatelyin practice" (W. P. Trowbridge, Trans. A. S. M. E., vii. 536.) This law is analogous to that of the pressure of a jet striking a plane surface. T. Hawksley, Proc. Inst. M. E., 1882, vol. lxix.. says: "The pressure of a fluid striking a plane surface perpendicularly and then escap- ing at right angles to its original path is that due to twice the height h due the velocity." (For discussion of this question, showing that it is an error to take the pressure as equal to a column of air of the height h = -u 2 -j- 2g, see Wolff on Windmills, p. 17.) Buckle says: " From the experiments it further appears that the velocity of the tips of the fan is equal to nine tenths of the velocity a body would acquire in falling the height of a homogeneous column of air equivalent to the density." D. K. Clark (R. T. & D., p. 924), paraphrasing Buckle, appar ently, says: " It further appears that the pressure generated at the circum ference is one ninth greater than that which is due to the actual circumfer- ential velocity of the fan." The two statements, however, are not in harmony, for if v = 0.9 V2gH, H = n a ? _ = 1.234 ~ and not 14 ~. 0.81 x 2(? 2g 9 2g If we take the pressure as that equal to a head or column of air of twice the height due the velocity, as is correctly stated by Trowbridge, the para- doxical statements of Buckle and Clark— which would indicate that the actual pressure is greater than the theoretical— are explained, and the v 2 , — , formula becomes H- .617— and v - 1.273 VgH = 0.9 \ 2gH, in which H is the head of a column producing the pressure, which is equal to twice the theoretical head due the velocity of a falling body (orh = —\, multiplied by the coefficient .617. The difference between 1 and this coefficient ex- presses the loss of pressure due to friction, to the fact that the inner por- tions of the blade have a smaller velocity than the outer edge, and probably to other causes. The coefficient 1.273 means that the tip of the blade must be given a velocity 1.273 times that theoretically required to produce the head H. To convert the head H expressed in feet to pressure in lbs. per sq. in. multiply it by the weight of a cubic foot of air at the pressure and tempera- ture of the air expelled from the fan (about .08 lb. usually) and divide by 144. Multiply this by 16 to obtain pressure in ounces per sq. in. or by 2.035 to obtain inches of mercury, or by 27.71 to obtain pressure in inches of water column. Taking .08 as the weight of a cubic foot of air, p lbs. per sq. in. = ,00001066u 2 ; v = 310 \ |T nearly; Pi ounces per sq. in. = .0001706-u 2 ; v = 80 \/px " p 2 inches of mercury = .00002169u 2 ; v — 220 \ p~ 2 " p 3 inches of water = .0002954i> 2 ; v= 60 4-^ " in which v = velocity of tips of blades in feet per second. Testing the above formula by the experiment of Buckle with the vane 14 inches long, quoted above, we have p = .00001066w 2 = 9.56 oz. The ex- periment grave 9.4 oz. Testing it by the experiment of H. I. Snell, given below, in which the circumferential speed was about 150 ft. per second, we obtain 3.85 ounces, while the experiment gave from 2.38 to 3.50 ounces, according to the amount of opening for discharge. The numerical coefficients of the above formulae are all based on Buckle's statement that the velocity of the tips of the fan js equal to nine tenths of the velocity a body would acquire in falling the 514 height of a homogeneous column of air equivalent to the pressure. Should other experiments show a different law, the coefficients can be corrected accordingly. It is probable that they will vary to some extent with differ- ent proportions of fans and different speeds. Taking the formula y — 80 Vpi, we have for different pressures in ounces per square inch the following velocities of the tips of the blades in feet per second: p x = ounces per square inch.. v = feet per second 2 3 4 5 113 139 160 179 1 10 12 I 212 226 253 14 299 A rule in App. Cyc. Mech, article " Blowers," gives the following velocities of circumference for different densities of blast in ounces: 3, 170; 4, 180; 5, 195; 6, 205; 7, 215. The same article gives the following tables, the first of which shows that the density of blast is not constant for a given velocity, but depends on the ratio of area of nozzle to area of blades: Velocity of circumference, feet per second. Area of nozzle -=- area of blades Density of blast, oz. per square inch 150 150 150 170 200 200 220 2 i y 2 ya % V6 Vs 12 3 4 4 6 6 Quantity of Air of a Given Density Delivered by a Fan. Total area of nozzles in square feet X velocity in feet per minute corre- sponding to density (see table) = air delivered in cubic feet per minute. S Velocity, feet perTq Ce m. ^ minute ' 1 5000 2 7000 3 8600 4 10,000 JESTS' Velocity, feet .er?qin. Per m ' n - 5 11,000 6 12,250 7 13,200 8 14,150 Density, ounces per sq. in. 9 10 11 12 Velocity, feet per minute. 15,000 15,800 16,500 17,300 Experiments with Blowers. (Henry I. Snell, Trans. A. S. M. E. ix. 51.)— The following tables give velocities of air discharging through an aperture of any size under the given pressures into the atmosphere. The volume discharged can be obtained by multiplying the area of discharge opening by the velocity, and this product by the coefficient of contraction: .65 for a thin plate and .93 when the orifice is a conical tube with a conver- gence of about 3.5 degrees, as determined by the experiments of Weisbach. The tables are calculated for a barometrical pressure of 14.69 lbs. (= 235 oz.), and for a temperature of 50° Fahr., from the formula V "== \/2gh. Allowances have been made for the effect of the compression of the air, but none for the heating effect due to the compression. At a temperature of 50 degrees, a cubic foot of air weighs .078 lbs., and calling g — 32.1602, the above formula may be reduced to V x = 60 1/31.5812 X (235 - P) X P, where Vi = velocity in feet per minute. P — pressure above atmosphere, or the pressure shown by gauge, in oz. per square inch. Pressure per sq. in. in inches of water. Corre- sponding Pressure in oz. per sq. inch. Velocity due the Pressure in feet per minute. Pressure per sq. in. in inches of water. Corre- sponding Pressure in oz. per sq. inch. Velocity due the Pressure in feet per minute. 1/32 .01817 696.78 % .36340 3118.38 1/16 .03634 987.66 Va .43608 3416.64 H .07268 1393.75 Vs .50870 3690.62 3/16 .10902 1707.00 1 .58140 3946.17 Va .14536 1971.30 m .7267 4362.62 5/16 .18170 2204.16 Wz .8721 4836.06 % .21804 2414.70 Wa 1.0174 5224.98 Xt .29072 2788.74 2 1.1628 5587.58 FANS AND BLOWERS. 515 Press- Velocity Press- Velocity Press- Velocity Velocity ure due the ure due the ure due the Pressure due the in oz. . Pressure in oz. Pressure in oz. Pressure in oz. Pressure per sq. in ft. per persq. in ft. per per sq. in ft. per persq. in. in ft. per inch. minute. inch. minute. inch. minute. minute. .25 2,582 2.25 7,787 5.50 12,259 11.00 17,534 .50 3,658 2.50 8,213 6.00 12,817 12.00 18,350 .75 4,482 2.75 8,618 6.50 13,354 13.00 19,138 1.00 5,178 3.00 9,006 7.00 13.873 14.00 19,901 1.25 5,792 3.50 9,739 7.50 14,374 15.00 20.641 1.50 6,349 4.00 10,421 8.00 14,861 16.00 21,360 1.75 6,861 4.50 11,065 9.00 15,795 2.00 7,338 5.00 11,676 10.00 16,684 Pressure in ounces Velocity in feet per square inch. per minute. .04 .05 516.90 722.64 895.26 1033.86 1155.90 Pressure in ounces Velocity in feet per per square inch. minute. 12C6.24 1367.76 1550.70 1635.00 Experiments on a Fan with. Varying Discharge-opening Revolutions nearly constant. 1485 1465 1468 1500 1426 OB'S •go -O "S u ^ < O > w ,S % «a 3 Sa heore min. disch IH.P Press CD cS c SE » o EH H 1048 1048 .337 1048 .496 1048 .66 1048 .709 1078 .718 1126 .70 1222 .635 1222 .646 1544 .536 The fan wheel was 23 inches in diameter, %% inches wide at its periphery, and had an inlet of 12)^ inches in diameter on either side, which was partially obstructed by the pulleys, which were 5 9/16 inches in diameter. It had eight blades, each of an area of 45.49 square inches. The discharge of air was through a conical tin tube with sides tapered at an angle of 3^ degrees. The actual area of opening was 7% greater than given in the tables, to compensate for the vena contracta. In the last experiment, 89.5 sq. in. represents the actual area of the mouth of the blower less a deduction for a narrow strip of wood placed across it for the purpose of holding the pressure-gauge. In calculating the volume of air discharged in the last experiment the value of vena contractu is taken at .80. 516 AIR. Experiments were undertaken for the purpose of showing the results ob- tained by running the same fan at different speeds with the discharge-open- ing the same throughout the series. The discharge-pipe was a conical tube 8% inches inside diameter at the end, having an area of 56.74, which is 7% larger than 53 sq. inches ; therefore 53 square inches, equal to .368 square feet, is called the area of discharge, as that is the practical area by which the volume of air is computed. Experiments on a Fan with Constant Discharge-open- ing and Varying Speed.— The first four columns are given by Mr. Snell, the others are calculated by the author. a i > o a o a w 1 o S CO 2! o u CO o ft o EH ft Is & ol.e, fcl II Is 5 ^1 ! Ill 111 lit it. ill o o CO s a Z ft c '5 5F ti Ph > W > > O > Eh H 600 .50 1336 .25 60.2 56.6 85.1 3,630 .182 73 800 .88 1787 .70 80.3 75.0 85.6 4,856 .429 61 1000 1.38 2245 1.35 100.4 94. 85.4 6,100 .845 6?, 1200 2.00 2712 2.20 120.4 113. 85.1 7,370 1.479 67 1400 2.75 3177 3.45 140.5 133. 84.8 8,633 2.283 66 1600 3.80 3670 5.10 160.6 156. 82.4 9,973 3.803 74 1800 4.80 4172 8.00 180.6 175. 82.4 11,337 5.462 68 2000 5.95 4674 11.40 200.7 195. 85.6 12,701 7.586 67 Mr. Snell has not found any practical difference between the efficiencies of blowers with curved blades and those with straight radial ones. From these experiments, says Mr. Snell, it appears that we may expect to receive back 65$ to 75$ of the power expended, and no more. The great amount of power often used to run a fan is not due to the fan itself, but to the method of selecting, erecting, and piping it. (For opinions on the relative merits of fans and positive rotary blowers, see discussion of Mr. Snell 's paper, Trans. A. S. M. E, ix. 66, etc.) Comparative Efficiency of Fans and Positive Blowers.— (H. M. Howe, Trans. A. I. M. E., x. 482.)— Experiments with fans and positive (Baker; blowers working at moderately low pressures, under 20 ounces, show that they work more efficiently at a given pressure when delivering large volumes {i.e., when working nearly up to their maximum capacity) than when delivering comparatively small volumes. Therefore, when great vari- ations in the quantity and pressure of blast required are liable to arise, the highest efficiency would be obtained by having a number of blowers, always driving them up to their full capacity, and regulating the amount of blast by altering the number of blowers at work, instead of having one or two very large blowers and regulating the amount of blast by the speed of the blowers. There appears to be little difference between the efficiency of fans and of Baker blowers when each works under favorable conditions as regards quantity of work, and when each is in good order. For a given speed of fan, any diminution in the size of the blast-orifice de- creases the consumption of power and at the same time raises the pressure of the blast ; but it increases the consumption of power per unit of orifice for a given pressure of blast. When the orifice has been reduced to the normal size for any given fan, further diminishing it causes but slight elevation of the blast pressure: and, when the orifice becomes com- paratively small, further diminishing it causes no sensible elevation of the blast pressure, which remains practically constant, even when the orifice is entirely closed. Many of the failures of fans have been due to too low speed, to too small pulleys, to improper fastening of belts, or to the belts being too nearly ver- tical; in brief, to bad mechanical arrangement, rather than to inherent de- fects in the principles of the machine. FANS AND BLOWERS. 51? If several fans are used, it is probably essential to high efficiency to pro- vide a separate blast pipe for each (at least if the fans are of different size or speed), while any number of positive blowers may deliver into the same pipe without lowering their efficiency. Capacity of Fans and Blowers. The following tables show the guaranteed air-supply and air-removal of leading forms of blowers and exhaust fans. The figures given are often exceeded in practice, especially when the blowers and fans are driven at higher speeds than stated. The ratings, particularly of the blowers, are below those generally given in catalogues, but it was the desire to present only conservative and assured practice. (A. R. Wolff on Ventilation.) Quantity of Air SUPPLIED to Buildings by Blowers op Various Sizes. Capacity cu. ft. Capacity Diam- Ordinary Horse- Diam- Ordinary Horse- cu. ft. eter of Number power against a Pressure of 1 ounce eter of Number power against a Pressure of 1 ounce Wheel of Revs. to Drive Wheel of Revs. to Drive in feet. per mm. Blower. in feet. per mm. Blower. per sq. in per sq. in. 4 350 6. 10,635 9 175 29 56,800 5 325 9.4 17,000 10 160 35.5 70,340 6 275 13.5 29,618 12 130 49.5 102,000 7 230 18.4 42,700 11 110 66 139,000 8 200 24 46,000 15 100 77 160,000 If the resistance exceeds the pressure of one ounce per square inch, of above table, the capacity of the blower will be correspondingly decreased, or power increased, and allowance for this must be made when the distrib- uting ducts are small, of excessive length, and contain many contractions and bends. Quantity of Air moved by an Approved Form of Exhaust Fan, the fan discharging directly from room into the atmosphere. Diam- eter of Wheel in feet. Ordinary Number of Revs, per min. Horse- power to Drive Fan. Capacity in cu. ft. per min. Diam- eter of Wheel in feet. Ordinary Number of Revs, per min. Horse- power to Drive Fan. Capacity in cu. ft. per min. 2.0 2.5 3.0 3.5 600 550 500 500 0.50 0.75 1.00 2.50 5,000 8,000 12,000 20,000 4.0 5.0 6.0 7.0 475 350 300 250 3.50 4.50 7.00 9.00 28,000 35,000 50,000 80,000 The capacity of exhaust fans here stated, and the horse-power to drive them, are for free exhaust from room into atmosphere. The capacity de- creases and the horse-power increases materially as the resistance, resulting from lengths, smallness and bends of ducts, enters as a factor. The differ- ence in pressures in the two tables is the main cause of variation in the re- spective records. The fan referred to in the second table could not be used with as high a resistance as one ounce per square inch, the rated resistance of the blowers. 518 CENTRIFUGAL FANS. Pressures, Velocities, Vol nine Required, etc. (B. F. of Air, Horse-Power Sturtevant Co.) ^ _ ~ >> .5 --3 « C bjC . 3 s a COO'" ill. elocity in feet per minute of Air (at 50 °F.) escaping into open air through any shaped hole from any pipe or reservoir in which the Air is compressed. ubic feet of Air per minute (at 50° F.), which may be discharged through a proper shaped mouth- piece, the diameter of which must be 1.362 inches, the area being 1 07 square inches. && o P.-I.8 111 I !ubic feet of Air per minute that may be discharged with one H. P., no allowance heing made for friction in the blast-machine (whatever power that friction amounts to must be added). It makes no difference how the Air is discharged, provided the pres- sure is steady, the same as given in the first column. Number of mouth-pieces de- scribed in column 3, required to discharge one H. P. of wind, no allowance being made for fric- tion in the blast-machine. Ph t> O < # +- 1 2 3 4 5 6 Ya 2584.80 17.944 0.001224 14662.76 817.00 g 3657.60 25.400 0.003463 7333.70 288.70 4482.00 31.124 0.005659 4889.11 157.08 t 5175.00 35.93 0.0098 3666.62 102.05 2 7338.24 50.96 0.0278 1833.00 35.970 3 9006 42 62.54 0.0512 1222.30 19.540 4 10421.58 72.37 0.0789 916.27 12.660 5 11676.00 81.08 0.1106 733.39 9.015 6 12817.08 89.01 0.1456 611.10 6.867 7 13872.72 96.34 0.1839 523.81 5.440 8 14861.16 103.20 0.2251 458.43 4.440 9 15795.06 109.69 0.2692 407.42 3.715 10 16683.51 115.86 0.3160 366.69 3.165 11 17533.50 121.76 0.3652 333.40 2.738 12 18350 34 127.43 0.4170 305.56 2.398 13 19138.26 132.90 0.4712 282.05 2.136 . 14 19900.68 138.20 0.5277 261.91 1.895 *15 20640.48 143.34 0.5864 244.44 1.705 16 21360.00 148.33 0.6473 229.17 1.545 17 22060.80 153.26 0.7103 215.77 1.408 18 22745.40 157.96 0.7754 203.71 1.290 19 23415.00 162.60 0.8426 192.98 1.187 20 24070.80 167.16 0.9118 183.33 1.097 * Always give the wind a good wide opening into the furnace or forge ; see by this table how much more wind can be discharged with one H. P. at low pressure than at high. This table shows the great advantage of large tuyeres, large pipes, large blower, and slow speed when the nature of the w r ork will admit. t Number of forges driven with 1.2 H. P. with Sturtevant blower. CENTRIFUGAL FANS. 519 Engines, Fans, and Steam-coils combined for the Blower System of Heating. (Buffalo Forge Co.) a H Is P* H > o . ■*8 ° 13 1 i?© © $ '1 fl S & O C v ©^ ? eg.S o 8-2 ^ •si® 3*3 8'5b m © fgls 5oB 03 ©■'Wco 4x3 52 450 8,740 1,200 49 x 38 3.1 1,000 12 4x4 60 425 11,000 1,525 51 x 45 4 1,200 15 5x4 70 390 15,280 1,700 52 x 50 4.5 1,600 20 5x5 80 360 19,900 2,200 52 x 56 6 2,000 25 6x5 90 330 25,900 2,450 59 x 74 7.2 2,500 30 6x6 100 290 32,500 2,700 62 x 84 9.1 3,000 35 7x6 110 260 39,300 3,200 69 x 94 11 3,500 42 7x7 120 235 49,161 3,900 79 x 104 13.5 4,000 48 8x7 130 210 57,720 4,500 83 x 111 15 4,500 54 8x8 150 180 81,120 5.300 87 x 133 20 5,000 62 10 x 9 170 165 101,250 6,000 92 x 148 22 6,000 72 The Sturtevant Steel Pressure- blower, applied to Cupola Furnaces. .2* «H . id u Power Saved by Reducing H ® — © o 0) E d 2 ©*! © eeq 5~' the Speed and Pressure of Blast. o o c "2 2* S =o &c© o2 d © a XII .« Ph d • ml 0h N © Oh W £ s s° & 3 © O ft 5~ 5* a w a 1 22 1200 *4 324 4135 0.5 2 26 1900 5.7 507 3756 6 1 3445 5 o.s 8100 4 0.6 3 30 2880 8 768 3250 7 1.8 3000 6 1.5 5 1.1 4 35 4130 10.7 1102 3100 8 3 2900 7 2 5 6 2. 5 40 6178 14.2 1646 2900 10 5.5 2560 8 4 7 3.3 6 46 8900 18.7 2375 2820 12 9.7 2550 10 7 4 8 5.3 7 53 12500 24.3 3353 2600 14 16. 23S0 12 12.7 2 inn 10 9.4 8 60 1656C 32 4416 2270 14 22. 2100 12 16.7 10 12.7 9 72 2380C 43 6364 2100 16 35. 1960 14 128.4 12 22.4 10 84 33300 60 8880 1815 16 48. 1700 14 39.6 12 31.7 *One square inch of blast is sufficient for one forge-fire, or 90 square inches area of cupola furnaces. The speed given is regulated so as to give the pressure of blast stated in ounces per square inch. The term " square inches of blast " refers to the area of a proper shaped mouth-piece discharging blast into the open air. The melting capacity per hour in pounds of iron is made up from an average of tests on a few of the best cupolas found, and is reliable in cases where the cupolas are well constructed and driven with the greatest force of blast given in the table. For tables of the steel pressure -blower as applied to forge-fires, and for sizes, etc., of other patterns of blowers and exhausters, see catalogue of B. F. Sturtevant Co. (For other data concerning Cupolas, see Foundry Practice.) Diameter of Blast-pipes for Pressure-blowers for Cupola Furnaces and Forges. (B. F. Sturtevant Co.) The following table has been constructed on this basis, namely : Allowing a loss of pressure of y% oz. in the process of transmission through any length of pipe of any size as a standard, the increased friction due to lengthening the pipe has been compensated for by an enlargement of the pipe sufficient 520 AIR. to keep the loss still at % oz. The quantities of air in the left-hand column of each division indicate the capacity of the given blower when working under pressures of 4, 8, 12, and 16 ozs. Thus a No. 6 Blower will force 2678 cubic ft. of air, at 8 oz. pressure, through 50 ft. of 12J4-in. pipe, with a loss of % oz pressure. If it is desired to force the air 300 ft. without an increased loss by friction, the pipe must be enlarged to 17*4 hi. diameter. Blower No. 1. Blower No. 515 635 740 Lengths of Blast-pipe in Feet. 50 100 150 200 300 S15 Diameter in inches. 734 vm S%. I 9 9 9% 1872 2678 Lengths of Blast-pipe in Feet. 50 100 150 200 300 Diameter in inches. 10% 12% 13M 13% 15 12V, 14 15*6 16 17M 1314 15% 16% 1% 18% 14% 16*6 17y * 18% 20*6 Blower No. 2. Blower No. 7. 7*4 I 8*4 734 m m 9 I 9% 1 10*4 9% 1 10% 1 11 10% 11 UH 2592 12 1334 15 15% 17*4 18*4 3708 13% 15% 45V2 1^/8 17% 18% 18% 19% 5328 16 20 21*4 17% 1934 Blower No. 3. Blower No. 8. 720 7*4 8*4 9 9% 1030 834 9i/o 1U46 11 •1270 9*6 1034 11*4 11% 1480 9% 11 12 M% 1014 1134 13*6 3312 13*4 15% 16% 17% 4738 1514 Vi% 19% 20% 5842 16% 19% 2034 22 6808 l'<% 20*4 22% «% 18% Blower No. 4. Blower No. 9. 1008 8*4 9% 10% 10*4 10% 1442 W», 11% 12i/ 3 1778 103/ 8 11% 12% 13% 2072 11 12% 1334 14% 4320 14% 17 183/6 19% 6180 17 19Vo 21 V A 22% 7620 18% 21% 23% 243^ 8880 19% ^% 24% 26 21% 24% 26% Blower No. 5. Blower No. 10. 1440 2060 2540 9% 10% 11% 1334 12% 11 12% 14*6 11% 13% 14% 15% 1234 14% 15% 16% 15% 5760 16% 19 20% 21% 8240 18% 2134 2334 25% 25% 10160 20% 2514 273/ 8 11840 22% 27% 29% 27*4 29% 31% CENTRIFUGAL FANS. 521 Centrifugal Ventilators for Mines.— Of different appliances for ventilating mines various forms of centrifugal machines having proved their efficiency have now i lmost completely replaced all others. Most if not all of the machines in use in this country are of this class, being either open- periphery fans, or closed, with chimney and spiral casing, of a more or less modified Guibal type. The theory of such machines has been demonstrated by M.v. Daniel Murgue in " Theories and Practices of Centrifugal Ventilating Machines, " translated by A. L Stevenson, and is discussed in a paper by R. Van A. Norris, Trans. A. I. M. E. xx. 637. From this paper the following for- mulae are taken: Let a = area in sq. ft. of an orifice in a thin plate, of such area that its re- sistance to the passage of a given quantity of air equals the resistance of the mine; o = orifice in a thin plate of such area that its resistance to the pas- sage of a given quantity of air equals that of the machine; Q — quantity of air passing in cubic feet per minute ; V= velocity of air passing through a. in feet per second; F = velocity of air passing through o in feet per second; h = head in feet air -column to produce velocity V; h = head in feet air-column to produce velocity V . ; V2gh; Q = 0.65a Y2gh; a = v = equivalent orifice of mine; 0.65 Y2gh or, reducing to water-gauge in inches and quantity in thousands of feet per minute, Q = 0.65oF ; V = Y2gh - Q = 0.65o V2gh ; equivalent orifice of machine. The theoretical depression which can be produced by any centrifugal ven- tilator is double that due to its tangential speed. The»formula in which Tis the tangential speed, Fthe velocity of exit of the air from the space between the blades, and H the depression measured in feet of air- column, is an expression for the theoretical depression which can be pro- duced by an uncovered ventilator; this reaches a maximum when the air leaves the blades without speed, that is, V= 0, and H = T 2 -t- 2g. Hence the theoretical depression which can be produced by any uncovered ventilator is equal to the height due to its tangential speed, and one half- that which can be produced by a covered ventilator with expanding chimney. So long as the condition of the mine remains constant: The volume produced by any ventilator varies directly as the speed of rotation. The depression produced by any ventilator varies as the square of the speed of rotation. For the same tangential speed with decreased resistance the quantity of air increases and the depression diminishes. The following table shows a few results, selected from Mr. Norris's paper, giving the range of efficiency which may be expected under different cir cumstances. Details of these and other fans, with diagrams of the results are given in the paper. 522 ALU. Experiments on Mine-ventilating Fans. .2 - OS "3 . & ft £,05 <1 V 2 3 2^ .3 fa Ms © 0)« *.£& 0) s a U % o . ft.il o S o 0) a '5b . a I >>fa §■« .2 c Out > 3-d ft Q 53 f o W c3 he"- to 1-1 O • ° $ v •2 8? ft O s3 S P5 U 5? . § ^ a PL a k a 8 >> § as 25,797 32,575 41,929 47,756 For 0.65 2.29 4.42 7.41 series 1.682 440 534 612 1.257 1.186 1.146 1.749 1.262 1.287 1.139 1.851 3.523 1.843 1.677 11.140 5.4 2.4 3.97 4. .9553 1.062 .9358 340 20,372 26,660 31,649 36,543 For 0.76 1.99 3.86 6.47 series .7110 453 536 627 1.332 1.183 1.167 1.761 1.308 1.187 1.155 1.794 2.618 1.940 1.676 8.513 3.55 3.86 3.59 3.63 .6063 .5205 .4802 340 430 534 570 9,983 13,017 17,018 18,649 For 1.12 3.17 6.07 8.46 series 0.28 0.47 0.75 0.87 'i!265' 1.242 1.068 1.676 'l'.304 1.307 1.096 1.704 "2". 837 1.915 1.394 7.554 '3!93' 2.25 3.63 3.24 i!65' 1.74 1.60 1.81 .3939 .3046 .3319 .3027 330 8,399 10,071 11,157 For 1.31 3.27 6.00 series 0.26 0.45 0.75 '6*31 3.66 5.35 siofr 4.96 3.72 .2631 437 516 1.324 1.181 1.563 1.199 1.108 1.329 3.142 1.457 4.580 .2188 .2202 Nature of the Experiments.— First Series: Drawing air through 30 ft. of 48-in. diam. pipe on inlet side of the fan. Second Series: Forcing air through 30 ft. of 48-in. diam. pipe on outlet side of the fan. Third Series: Drawing air through 30 ft. of 48-in. pipe on inlet side of the fan— the pipe being obstructed by a diaphragm of cheese-cloth. Fourth Series: Forcing air through 30 ft. of 48-in. pipe on outlet side of fan —the pipe being obstructed by a diaphragm of cheese cloth. Mr. Babcock says concerning these experiments : The first four experi- ments are evidently the subject of some error, because the efficiency is such as to prove on an average that the fan was a source of power sufficient to overcome all losses and help drive the engine besides. The second series is less questionable, but still the efficiency in the first two experiments is larger than might be expected. In the third and fourth series the resistance of the cheese-cloth in the pipe reduces the efficiency largely, as would be expected. In this case the value has been calculated from the height equivalent to tne water-pressure, rather than the actual velocity of the air. This record of experiments made with the disk fan shows that this kind of fan is not adapted for use where there is any material resistance to the flow of the air. In the centrifugal fan the power used is nearly proportioned to the amount of air moved under a given head, while in this fan the power re- quired for the same number of revolutions of the fan increases very mate- rially with the resistance, notwithstanding the quantity of air moved is at the same time considerably reduced. In fact, from the inspection of the third and fourth series of tests, it would appear that the power required is very nearly the same for a given pressure, whether more or less air be in motion. It would seem that the main advantage, if any, of the disk fan over the cen- trifugal fan for slight resistances consists in the fact that the delivery is the full area of the disk, while with centrifugal fans intended to move the same quantity of air the opening is much smaller, DISK FAHS. 525 It will be seen by columns 8 and 9 of the table that the power used in- creased much more rapidly than the cube of the velocity, as in centrifugal fans. The different experiments do not agree with each other, but a general average may be assumed as about the cube root of the eleventh power. Cubic Feet of Air removed by Exhaust Disk-wheel per minute, (Buffalo Forge Co.) Number of Revo- lutions of Wheel per minute, Diameter of Wheel. !4 Inch. 30 Inch. 36 Inch. 42 Inch. 48 Inch. 54 Inch. ]60 Inch. 72 Inch. Amount of Air in cubic feet per minute. 100.. 150.. 200.. 250.. 300.. 350.. 400.. 450. . 500 . 550. 600. 650. . 700. 1,307 1,684 2,014 2,375 2,770 3,197 3,656 4,148 4.671 5.221 3,338 1,042 3.594 1,541 3,550 3.621 7.755 B,950 23,420 4,245 6,059 6,405 9,154 8,686 12,410 11,098 15,822 13,641 19,408 16,315 23,147 19,119 27,048 22,055 31,112 25,127 35,338 28,325 39,727 31,518 44,277 34,310 48,992 36,940 53,-858 : 8,387 12,822 17,457 22.292 ^7.:«7 32.565 37,997 43,632 49,467 55,152 60,401 14,936 22,926 31,267 39.956 48,996 58,386 67,985 76,900 Efficiency ©t Disk Fans. — Prof. A. B. W. Kennedy {Industries, Jan. 17, 1890) made a series of tests on two disk fans, 2 and 3 ft. diameter, known as the Verity Silent Air-propeller. The principal results and conclusions are condensed below. In each case the efficiency of the fan, that is, the quantity of air delivered per effective horse-power, increases very rapidly as the speed diminishes, so that lower speeds are much more economical than higher ones. On the other hand, as the quantity of air delivered per revolution is very nearly constant, the actual useful work done by the fan increases almost directly with its speed. Comparing the large and small fans with about the same air delivery, the former (running at a much lower speed, of course) is much the more economical. Comparing the two fans running at the same speed, however, the smaller fan is very much the more economical. The delivery of air per revolution of fan is very nearly directly proportional to the area of the fan's diameter. The air delivered per minute by the 3-ft. fan is nearly 12.522 cubic feet (R being the number of revolutions made by the fan per minute). For the 2-ft. fan the quantity is 5.7R cubic feet. For either of these or any other similar fans of which the area is A square feet, the delivery will be about 1.8AR cubic feet. Of course any change in the pitch of the blades might entirely change these figures. The net H.P. taken up is not far from proportional to the square of the number of revolutions above 100 per minute. Thus for the 3-ft. fan the net (R - 100)2 n m n ti n ti „ , „„ . (R— 100)2 H.P. , while for the 2-ft. fan the net H.P. is 200,000 ' "*""" l "' """ ~ *"■ """ """ """ "■■"■ " 10 1,000,000 ' The denominators of these two fractions are very nearly proportional in- versely to the square of the fan areas or the fourth power of the fan diam- eters. The net H.P. required to drive a fan of diameter D feet or area A square feet, at a speed of R revolutions per minute, will therefore be ap- . . . DKR - 100)2 A i( B _ 10 0)2. proximately -^^^ or 'j^^. The 2-ft. fan was noiseless at all speeds. The 3-ft. fan was also noiseless up to over 450 revolutions per minute. 526 Speed of fan, revolutions per minute. Net H.P. to drive fan and belt. Cubic feet of air per minute Mean velocity of air in 3-ft. flue, feet per minute Mean velocity of air in flue, same diameter as fan Cu.f t.of air per min.per effective H.P. Motion given to air per rev. of fan, ft. Pubic feet of a ir per rev, of fan Propeller, 2 ft. diam. 750 0.42 1.7' 5.58 676 0.32 3,830 543 1,220 11,970 1.81 5.66 577 0.227 3,410 15.000 1.88 5.90 Propeller, 3 ft. diam. 576 1.02 7,400 1,046 7,250 l."~ 12. 459 0.575 10,070 1.7f 12.6 373 0.324 4,470 13,800 1.70 12.0 POSITIVE ROTARY BLOWEKS. (P. H. & F. M. Roots.) Size number Cubic feet per revolution Revolutions per minute, Smith fires " Furnishes blast for Smith fires Revolutions per minute for J ' cupola, melting iron ) Size of cupola, inches, side lining 5 200 350 300 275 2 6 10 to to to 4 8 14 275 225 24 30 200 4 13 150 ■:!: Will melt iron per hour, tons- 375 325 300 275 18 24 30 36 to to to to 24 30 36 42 1^ 2V 2 3 4% to 175 47 to 67 170 to 250 42 42 100 to 150 70 to 100 150 to 200 50 135 137 to 175 72 to to to 2 3 4^ m 5^ 8 60 2-55's 12}4 17% to 7 12 16% 22% Horse-power required 1 2 3^ 5^ 8 llj^ 17% 27 40 The amount of iron melted is based on 30.000 cubic feet of air per ton of iron. The horse-power is for maximum speed and a pressure of % pound, ordinary cupola pressure. (See also Foundry Practice.) BLOWING-ENGINES. Blast-furnace Blowing-engines of the Variable Puppet- valve Cut-off Type. (Philada. Engineering Wo ks.) Diameter Diameter Shop Revolu- Displace- Maximum of of Weights. tions, Blast-pres- Steam - Blowing- approxi- ordinary sure for Reg- cylinder. cylinder. mate. speed. ordinary speed. ular Work. in. in. in. pounds. cubic feet. lbs. persq.in. 28 66 36 80,000 60 8,550 10 28 66 48 90,000 50 9,500 10 32 72 48 106.000 50 11,308 12 36 72 48 130.000 50 11,308 15 36 84 48 140,000 50 15,392 11 36 84 60 165,000 40 15,392 11 42 84 48 165,000 50 15,392 15 42 84 60 190,000 40 15,392 15 42 90 48 170,000 50 17,700 13 42 90 60 195,000 40 17,700 13 48 96 48 220,000 50 20.000 15 48 96 60 280.000 40 20,000 15 The blowing-engines of the country are usually very wasteful of steam. by reason of wire-drawing valve-gear, and especially of slow piston-speed. The latter is perhaps the greatest and the least recognized of all steam- engine defects. Almost any expense to increase the economy of blowing- engines is warranted. (A. L. Holley, Trans. A. I. M. E., vol. iv. p. 81.) STEAM-JET BLOWER, AND EXHAUSTER. 527 The calculations of power, capacity, etc., of blowing-engines are the same as those for air-compressors. They are built without any provision for cooling the air during compression. About 400 feet per minute is the usual piston-speed for recent forms of engines, but with positive air-valves, which have been introduced to some extent, .this speed may be increased. The efficiency of the engine, that is, the ratio of the I.H.P. of the air cylinder to that of the steam cylinder, is usually taken at 90 per cent, the losses by friction, leakage, etc., being taken at 10 per cent. STEAM-JET BLOWER AND EXHAUSTER. A blower and exhauster is made by L. Schutte & Co., Philadelphia, on the principle of the steam-jet ejector. The following is a table of capacities: Size No. Quantity of Air per hour in cubic feet. Diameter of Pipes in inches. Size No. Quantity of Air per hour in cubic feet. Diameter of Pipes in inches. Steam. Air. Steam. Air. 000 00 1 2 3 4 1,000 2,000 4,000 6,000 12.000 18,000 24,000 M M 2 V 2 V2 4 5 6 8 9 10 30,000 36,000 42,000 48,000 54,000 60,000 2^ ft 3 3^ 5 6 6 7 S The admissible vacuum and counter pressure, for which the apparatus is constructed, is up to a rarefaction of 20 inches of mercury, and a counter- pressure up to one sixth of the steam-pressure. The table of capacities is based on a steam- pressure of about 60 lbs., and a counter-pressure of about 8 lbs. With an increase of steam-pressure or decrease of counter-pressure the capacity will largely increase. Another steam-jet blower is used for boiler-firing, ventilation, and similar purposes where a low counter-pressure or rarefaction meets the require- ments. The volumes as given in the following table of capacities are under the supposition of a steam-pressure of 45 lbs. and a counter-pressure of, say, 2 inches of water : Cubic Diameter Diameter in Cubic Diam. Diameter in Size No. feet of of inches of— Size No. feet of of inches of— Aii- delivered Steam- pipe in Air de- livered Steam - pipe in per hour. inches. Inlet Disch. per hour inches. Tnlet. Disch. 00 6,000 % 4 3 4 250,000 1 17 14 12,000 Mi 5 4 6 500,000 iy s 24 20 1 30,000 y* 8 6 8 1,000,000 32 27 2 60,000 M 11 8 10 2,000,000 2 42 36 3 125,000 l 14 10 The Steam-jet as a Means for Ventilation.— Between 1810 and 1850 the steam-jet was employed to a considerable extent for veutilat- ing English collieries, and in 1852 a committee of the House of Commons reported that it was the most powerful and at the same time the cheapest meihod for the ventilation of mines ; but experiments made shortly after- wards proved that this opinion was erroneous, and that furnace ventilation was less than half as expensive, and in consequence the jet was soon aban- doned as a permanent method of ventilation. For an account of these experiments see Colliery Engineer, Feb. 1890. The jet, however, is sometimes advantageously used as a substitute, for instance, in the case of a fan standing for repairs, or after an explosion, when the furnace may not be kept going, or in the case of the fan having been rendered useless. 528 HEAT1H0 AND VtfNTILATtOH. HEATING AND VENTILATION. Ventilation. (A. R. Wolff, Stevens Indicator, April, 1890.)— The pop- ular impression that the impure air falls to the bottom of a crowded room is erroneous. There is a constant mingling of the fresh air admitted with the impure air due to the law of diffusion of gases, to difference of temper- ature, etc. The process of ventilation is one of dilution of the impure ?ir by the fresh, and a room is properly ventilated in the opinion of the hygien- ists when the dilution is such that the carbonic acid in the air does not ex- ceed from 6 to 8 parts by volume in 10,000. Pure country air contains about 4 parts C0 2 in 10,000. and badly-ventilated quarters as high as 80 parts. An ordinary man exhales 0.6 of a cubic foot of C0 2 per hour. New York gas gives out 0.75 of a cubic foot of C0 2 for each cubic foot of gas burnt. An ordinary lamp gives out 1 cu. ft. of C0 2 per hour. An ordinary candle gives out 0.3 cu. ft. per hour. One ordinary gaslight equals in Vitiating effect about 5J^ men, an ordinary lamp 1% men, and an ordinary candle ^ man. To determine the quantity of air to be supplied to the inmates of an un- lighted room, to dilute the air to a desired standard of purity, we can estab- lish equations as follows: Let v = cubic feet of fresh air to be supplied per hour; r = cubic feet of C0 2 in each 10,000 cu. ft. of the entering air: R = cubic feet of C0 2 which each 10,000 cu. ft. of the air in the room may contain for proper health conditions; n = number of persons in the room; .6 = cubic feet of C0 2 exhaled by one man per hour. Then ■ -J- .6?i equals cubic feet of C0 2 communicated to the room dur- ing one hour. This value divided by v and multiplied by 10,000 gives the proportion of C0 2 in 10,000 parts of the air in the room, and this should equal B, the stan- dard of purity desired. Therefore or the quantity of air to be supplied per person is 3000 cubic feet per hour. If the original air in the room is of the purity of external air, and the cubic contents of the room is equal to 100 cu. ft. per inmate, only 3000 - 100 = 2900 cu. ft. of fresh air from without will have to be supplied the first hour to keep the air within the standard purity of 6 parts of C0 2 in 10,000. If the cubic contents of the room equals 200 cu. ft. per inmate, only 3000 — 200 = 2800 cu. ft. will have to be supplied the first hour to keep the air within the standard purity, and so on. Again, if we only desire to maintain a standard of purity of 8 parts of carbonic acid in 10,000, equation (1) gives as the required air-supply per hour v = -£ — -n = 1500n, or 1500 cu. ft. of fresh air per inmate per hour. Cubic feet of air containing 4 parts of carbonic acid in 10,000 necessary per person per hour to keep the air in room at the composition of 6 7 8 9 10 15 20 j parte of rarbonic acid in 3000 2000 1500 1200 1000 545 375 cubic feet. If the original air in the room is of purity of external atmosphere (4 parts of carbonic acid in 10,000), the amount of air to be supplied the first hour, for given cubic spaces per inmate, to have given standards of purity not exceeded at the end of the hour is obtained from the following table : 10 ' 000 Llo^oo- + -H ] ftPfl= S9222L. . . . . (1) . . . (2) v E — r and B at 6, v - „ r n - 3000?i, b — 4 VENTILATION. 529 Cubic Feet of Space in Room Proportion of Carbonic Acid in 10,000 Parts of the Air be Exceeded at End of Hour. not to 6 7 8 9 10 15 20 Individual. Cubic Fe 3t of Air, of Composition 4 Parts of Carbonic Acid in 10,000, to be Supplied the First Hour. 100 200 300 400 500 600 700 2900 2800 2700 2600 2500 2400 2300 2200 2100 2000 1500 1000 500 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 500 None 1400 1300 1200 1100 1000 900 800 700 600 500 None 1100 1000 900 800 700 600 500 400 300 200 None 900 800 700 600 500 400 300 200 100 None 445 345 245 145 45 None 275 175 75 None 800 900 1000 1500 2000 2500 It is exceptional that systematic ventilation supplies the 3000 cubic feet per inmate per hour, which adequate health considerations demand. Large auditoriums in which the cubic space per individual is great, and in which the atmosphere is thoroughly fresh before the rooms are occupied, and the occupancy is of two or three hours' duration, the systematic air-supply may be reduced, and 2000 to 2500 cubic feet per inmate per hour is a satisfactory allowance. Hospitals where, on account of unhealthy excretions of various kinds, the air-dilution must be largest, an air-supply of from 4000 to 6000 cubic feet per inmate per hour should be provided, and this is actually secured in some hospitals. A report dated March 15, 1882, by a commission appointed to examine the public schools of the District of Columbia, says : " In each class-room not less than 15 square feet of floor-space should be allotted to each pupil. In each class-room the window-space should not be less than one fourth the floor-space, and the distance of desk most remote from the window should not be more than one and a half times the height of the top of the window from the floor. The height of the class-room should never exceed 14 feet. The provisions for ventilation should be such as to provide for each person in a class-room not less than 30 cubic feet of fresh air per minute (1800 per hour), which amount must be introduced and thoroughly distributed without creatiug unpleasant draughts, or causing any two parts of the room to differ in temperature more than 2° Falir., or the maximum temperature to exceed 70° Fahr." When the air enters at or near the floor, it is desirable that the velocity of inlet should not exceed 2 feet per second, which means larger sizes of register openings and flues than are usually obtainable, and much higher velocities of inlet than two feet per second are the rule in practice. The velocity of current into vent-flues can safely be as high as 6 or even 10 feet per second, without being disagreeably perceptible. The entrance of fresh air into a room is co-incident with, or dependent on, the removal of an equal amount of air from the room. The ordinary means of removal is the vertical vent-duct, rising to the top of the buildingr. Some- times reliance for the production of the current in this vent-duct is placed solely on the difference of temperature of the air in the room and that of the external atmosphere: sometimes a steam coil is placed within the flue near its bottom to heat the air within the duct; sometimes steam pipes (risers and returns) run up the duct performing the same functions; or steam jets within the flue, or exhaust fans, driven by steam or electric power, act directly as exhausters; sometimes the heating of the air in the flue is ac- complished by gas-jets. The draft of such a duct is caused by the difference of weight of ths 530 HEATING AND VENTILATION. heated air in the duct, and a column of equal height and cross-sectional area of weight of the external air. Let d — density, or weight in pounds, of a cubic foot of the external air. Let d 1 = density, or weight in pounds, of a cubic foot of the heated air within the duct. Let h = vertical height, in feet, of the vent-duct. h(d — d,) = the pressure, in pounds per square foot, with which the air is forced into and out of the vent-duct. This pressure can be expressed in height of a column of the air of density within the vent-duct, and evidently the height of such column of equal ... ,, h(d — d t ) /Q , presssure would be_, — - (o) Or, if t = absolute temperature of external air, and t x = absolute temper- ature of the air in vent-duct in the form, then the pressure equals h(t 1 - t), (4) The theoretical velocity, in feet per second, with which the air would travels through the vent-duct under this pressure is .y^pi'^-V'^ <»> The actual velocity will be considerably less than this, on account of loss due to friction. This friction will vary with the form and cross-sectional area of the vent duct and its connections, and with the degree of smooth- ness of its interior surface. On this account, as well as to prevent leakage of air through crevices in the wall, tin lining of vent-flues is desirable. The loss by friction may be estimated at approximately 50$, and so we find for the actual velocity of the air as it flows through the vent-duct : v = -a/ 2gh- (*i - t) or, approximately, v= A a/ h- [t t - t) (6) If V= velocity of air in vent-duct, in feet per minute, and the external air be at 32° Fahr., since the absolute temperature on Fahrenheit scale equals thermometric temperature plus 459.4, = 240^-^! (7) from which has been computed the following table : Quantity of Air, in Cubic Feet, Discharged per Minute through a Ventilating J>uct, of which the Cross-sec- tional Area is One Square Foot (the External Tempera- ture of Air being 32° Fahr.). Excess of Temperature of Air in Vent-duct above that of Height of External Air. feet. 5° 10° 15° 20° 25° 30° 50° 100° 150° 10 77 108 133 153 171 188 242 342 419 15 94 133 162 188 210 230 297 419 514 20 108 153 188 217 242 265 342 484 593 25 121 171 210 242 271 297 383 541 663 30 133 188 230 265 297 325 419 593 726 35 143 203 248 286 320 351 453 640 784 40 153 217 265 306 342 375 484 656 838 45 162 230 282 325 363 398 514 476 889 50 1 171 242 297 342 383 419 541 278 937 Multiplying the figures in above table by 60 gives the cubic feet of air dis- charged per hour per square foot of cross-section of vent-duct. Knowing MINE-VENTILATION". 531 the cross-sectional area of vent-ducts we can find'the total discharge; or for a desired air-removal, we can proportion the cross-sectional area of vent-ducts required. Artificial Cooling of Air lor Ventilation. (Engineering News, July 7, 1892.) — A pound of coal used to make steam for a fairly effi- cient refrigerating-machine can produce an .actual cooling effect equal to that produced by the melting of 16 to 46 lbs. of ice, the amount varying with the conditions of working. Or, 855 heat-units per lb. of coal converted into work in the refrigerating plant (at the rate of 3 lbs. coal per horse- power hour) will abstract 2275 to 6545 heat-units of heat from the refriger- ated body. If we allow 2000 cu. ft. of fresh air per hour per person as suffi- cient for fair ventilation, with the air at an initial temperature of 80° F., its weight per cubic foot will be .0736 lb.; hence the hourly supply per person will weigh 2000 x .0736 lb. = 147.2 lbs. To cool this 10°, the specific heat of air being 0.238. will require the abstraction of 147.2 X 0.238 X 10 - 350 heat- units per person per hour. Taking the figures given for the refrigerating effect per pound of coal as above stated, and the required abstraction of 350 heat-units per person per hour to have a satisfactory cooling effect, the refrigeration obtained from a pound of coal will produce this cooling effect for 2275 -h- 350 = 6J^ hours with the least efficient working, or 6545 -=- 350 = 18.7 hours with the most efficient working. With ice at $5 per ton, Mr. Wolff computes the cost of cooling with ice at about $5 per hour per thousand persons, and concludes that this is too expensive for any generaluse. With mechanical refrigeration, however, if we assume 10 hours' cooling per person per pound of coal as a fair practical service in regular w : ork, we have an expense of only 15 cts. per thousand persons per hour, coal being estimated at S3 per short ton. This is for fuel alone, and the various items of oil. attendance, interest, and depreciation on the plant, etc., must be considered in making up the actual total cost of mechanical refrigeration. Mine-ventilation— Friction of Air in Underground Pas- sages.— In veuiilaiin^ a mine or other underground passage the resistance to be overcome is, according to most writers on the subject, proportional to the extent of the f rictional surface exposed ; that is, to the product lo of the length of the gangway by its perimeter, to the density of the air in circula- tion, to the square of its average speed, v, and lastly to a coefficient k, whose numerical value varies according to the nature of the sides of the gangway and the irregularities of its course. The formula for the loss of head, neglecting the variation in density as unimportant, is p = , in which p — loss of pressure in pounds per square foot, s = square feet of»rubbing-surface exposed to the air, v the velocity of the air in feet per minute, a the area of the passage in square feet, and k the coefficient of friction. W. Fairley, in Colliery Engineer, Oct. and Nov. 1893, gives the following formulas for all the quantities involved, using the same notation as the above, with these additions : /*, = horse-power of ven- tilation; I = length of air-channel; o = perimeter of air-channel; q = quan- tity of air circulating in cubic feet per minute; u — units of work, in foot- pounds, applied to circulate the air: w — water-gauge in inches. Then, _ lesv 2 _ ksv^q _ ksv 3 v — pv 5 2qiv > 2 sv 3 sv 2 ■+■ a sv 2 -*- a pa ~ kv 2 o ' _ pa ~'kvH' v* _ u _ 5 n w _ I / 2l\ ks — fcs« 3 _ _M_ T ~q " \f ks) « 3 a>v 532 HEATING AND VENTILATION. 7. pa = ksv* = [ i/~\ ks = !?; pa3 = fcsg*. a u ksv 3 /pa / u p p Y ks y ks 9 s = P?L = _ — _ 3P_ _ 2^a _ Zo fcv 2 ~~ kv 3 kv 3 kv 3 10. w = qp = vpa = SV q = fcsv' = o.2qw = 33,0007i. pa a y fcs y fcs y fcs fcs fcs fcs . . p fcsv 3 14 ' W = -572 = ^a To find the quantity of air with a given horse-power and efficiency (e) of engine: h X 33,000 X e q = i — • The value of fc, the coefficient of friction, as stated, varies according to the nature of the sides of the gangway. Widely divergent values have been given by different authorities (see Colliery Engineer, Nov. 1893), the most generally accepted one until recently being probably that of J. J. Atkinson, .0000000217, which is the pressure per square foot in decimals of a pound for each square foot of rubbing-surface and a velocity of one foot per minute. Mr. Fairley, in his " Theory and Practice of Ventilating Coal-mines, ,, gives a value less than half of Atkinson's, or .00000001 ; and recent experiments by D. Murgue show that even this value is high under most conditions. Murgue's results are given in his paper on Experimental Investigations in the Loss of Head of Air-currents in Underground Workings, Tnans. A. I. M. E., 1893. vol. xxiii. 63. His coefficients are given in the following table, as determined in twelve experiments: Coefficient of Loss of Head by Friction. French. British. f Straight, normal section 00092 .000,000,00486 Rock. J Straight, normal section 00094 .000,000,00497 gangways. | Straight, large section 00104 .000,000,00549 [Straight, normal section 00122 .000,000,00645 f Straight, normal section 00030 .000,000,00158 Brick-lined | Straight, normal section .00036 . 000,000,00190 arched -( Continuous curve, normal section 00062 .000,000,00328 gangways. | Sinuous, intermediate section 00051 .000,000,00269 (.Sinuous, small section 00055 .000,000,00291 rr- k *a ( Straight, normal section 00168 .000,000,00888 nmoerea J straight, normal section 00144 .000,000.00761 gangways. ( slightly sinuous, small section 00238 .000,000,01257 The French coefficients which are given by Murgue represent the height of water-gauge in millimetres for each square metre of rubbing-surface and a velocity of one metre per second. To convert them to the British measure of pounds per square foot for each square foot of rubbing-surface and a velocity of one foot per minute they have been multiplied by the factor of conversion, .000005283. For a velocity of 1000 feet per minute, since the loss of head varies as v a , move the decimal point in the coefficients six places to the right. FANS AND HEATED CHIMNEYS FOR VENTILATION. 533 Equivalent Orifice.— The head absorbed by the working-chambers of a mine cannot be computed a priori, because the openings, cross-pas- sages, irregular-shaped gob-piles, and daily changes in the size and shape of the chambers present much too complicated a network for accurate analysis. In order to overcome this difficulty Murgue proposed in 1872 the method of equivalent orifice. This method consists in substituting for the mine to be considered the equivalent thin-lipped orifice, requiring' the same height of head for the discharge of an equal volume of air. The area of this orifice is obtained when the head and the discharge are known, by means of the following formulae, as given by Fairley: Let Q = quantity of air in thousands of cubic feet per minute; to — inches of water-gauge; A — area in square feet of equivalent orifice. Then A = ±KQ = __L ; * _ AX_VK. _ / Qy Vio 2.7 Via y_ 0.37 ' w ~ °" 3C9 X \a ' ' Motive Column or the Head of Air Due to Differences of Temperature, etc. (Fairley.) Letii = motive column in feet; T = temperature of upcast; / = weight of one cubic foot of the flowing air; t = temperature of downcast; D = depth of downcast. Then To find diameter of a round airway to pass the same amount of air as a square airway the length and power remaining the same: Let D = diameter of round airway, A = area of s quare airway; O = peri- b / ~A 3 X 3.1416 meter of square airway. Then D 3 — 4/ .7S54 3 x O ' If two fans are employed to ventilate a mine, each of which when worked separately produces a certain quantity, which may be indicated by A and B then the quantity of air that will pass wLien the two fans are worked together will be a/A 3 -f B 3 . (For mine-ventilating fans, see page 521.) Relative Efficiency of Fans and Heated Chimneys for Ventilation.— W. P. Trowbridge, Trans. A. S. M. E. vii. 531, gives a theo- retical solution of the relative amounts of heat expended to remove a given volume of impure air by a fan and by a chimney. Assuming the total effi- ciency of a fan to be only 1/25, which is made up of an efficiency of 1/10 for the engine, 5/10 for the fan itself, and 8/10 for efficiency as regards friction, the fan requires an expenditure of heat to drive it of only 1/38 of the amount that would be required to produce the same ventilation by a chimney 100 ft. high. For a chimney 500 ft. high the fan will be 7.6 times more efficient. In all cases of moderate ventilation of rooms or buildings where the air is heated before it enters the rooms, and spontaneous ventilation is pro- duced by the passage of this heated air upwards through vertical flues, no special heat is required for ventilation; and if such ventilation be suffi- cient, the process is faultless as far as cost is concerned. This is a condition of things which may be realized in most dwelling-houses, and in many halls, schoolrooms, and public buildings, provided inlet and outlet fines of ample cross-section be provided, and the heated air be properly distributed. If a more active ventilation be demanded, but such as requires the small- est amount of power, the cost of this power may outweigh the advantages of the fan. There are many cases in which steam-pipes in the base of a chimney, requiring no care or attention, may be preferable to mechanical ventilation, on the ground of cost, and trouble of attendance, repairs, etc. * Murgue gives A — _ -, and Norris A = — — rdr. See page 521 , ante. 534 HEATING AND VENTILATION. The following' figures are given by Atkinson (Coll. Engr., 1889), showing the minimum depth at which a furnace would be equal to a ventilating- machine, assuming that the sources of loss are the same in each case, i.e., thar, the loss of fuel in a furnace from the cooling in the upcast is equivalent to the power expended in overcoming the friction in the machine, and also assuming that the ventilating-machine utilizes 60% of the engine-power. The coal consumption of the engine per I.H.P. is taken at 8 lbs. per hour: Average temperature in upcast 100° F. 150° F. 200° F: Minimum depth for equal economy... 960 yards. 1040 yards. 1130 yards. Heating and Ventilating of Large Buildings. (A. R. Wolff, Jour. Frank. Inst., 1893.)— The transmission of heat from the interior to the exterior of a room or building, through the walls, ceilings, windows, etc., is calculated as follows : S = amount of transmitting surface in square feet; t = temperature F. inside, / = temperature outside; K = a coefficient representing, for various materials composing buildings, the loss by transmission per square foot of surface in British ther- mal units per hour, for each degree of difference of temperature on the two sides of the material ; Q — total heat transmission — SK (t - t ). This quantity of heat is also the amount that must be conveyed to the room in order to make good the loss by transmission, but it does not cover the addiiional heat to be conveyed on account of the change of air for pur- poses of ventilation. The coefficients K given below are those prescribed by law by the German Government in the design of the heating plants of its public buildings, and generally used in Germany for all buildings. They have been converted into American units by Mr. Wolff, and he finds that they agree well with good American practice: Value of K for Each Square Foot of Brick Wall. Th brick e wall f [ 4 " 8 " 12 " 16 " 20 " 24 " 28 " 32 " 36 " 40 " '£=0.68 0.46 0.32 0.26 0.23 0.20 0.174 0.15 0.129 0.115 1 sq. ft., wooden-beam construction, ) as flooring, K — 0.083 planked over or ceiled, j as ceiling-, £"=0.104 1 sq. ft., fireproof construction, i as flooring, K = 0.1 -J4 floored over, j as ceiling, K— 0.145 1 sq. ft., single window K = 0.776 1 sq. ft., single skylight £=1.118 1 sq. ft. , double window K = 0.518 1 sq. ft., double skylight K = 0.621 1 sq. ft., door K = 0.414 These coefficients are to be increased respectively as follows: 10% when the exposure is a northerly one, and winds are to be counted on as important factors; 10% when the building is heated during the daytime only, and the location of the building is not an exposed one; 30% when the building is heated during the daytime only, and the location of the building is exposed; 50% when the building is heated during the winter months intermittently, with long intervals (say days or weeks) of non-heating. The value of the radiating-surface is about as follows: Ordinary bronzed cast-iron radiating-surfaces, in American radiators (of Bundy or similar type), located in rooms, give out about 250 heat-units per hour for each square foot of surface, with ordinary steam-pressure, say 3 to S lbs. per sq. in., and about 0.6 this amount with ordinary hot-water heating. Non-painted radiating-surfaces, of the ordinary "indirect" type (Climax or pin surfaces), give out about 400 heat-units per hour for each square foot of heating-surface, with ordinary steam-pressure, say 3 to 5 lbs. per sq. in.; and about 0.6 this amount with ordinary hot-water heating. A person gives out about 400 heat-units per hour; an ordinary gas-burner, about 4800 heat-units per hour; an incandescent electric (16 candle-power) lij-rht, about 1600 heat-units per hour. The following example is given by Mr. Wolff to show the application of the formula and coefficients: Lecture-room 40 X 60 ft., 20 ft. high, 48,000 cubic feet, to be heated to 69° F.; exposures as follows: North wall, 60 x 20 ft., with four windows, each 14 x 4 feet, outside temperature 0° F. Room beyond west wall and HEATING AND VENTILATING OF LARGE BUILDINGS. 535 room overhead heated to 69°, except a double skylight in ceiling, 14 X 24 ft., exposed to the outside temperature of 0°. Store-room beyond east wall at 30°. Door X 12 ft. in wall. Corridor beyond south wall heated to 59°. Two doors, 6 X 12, in wall. Cellar below, temperature 36°. The following table shows the calculation of heat transmission: fa © 1^ Kind of Transmitting Surface. o5.S • Ho Calculation of Area of Transmitting Surface. Is w ° 7 k "3 . SB Eh 69° 69 33 33 Outside wall ... Four windows (single) Inside wall (store-room) 36" 36" 24" 36" 63 X 22 - 448 4x 8X 14 42X22- 72 6X12 45X22- 72 6X12 17X22- 72 6X 12 32 X 42 - 336 14X24 62 X 42 938 448 852 72 918 72 302 72 1,008 336 2,604 9 72 4 19 2 5 1 5 10 43 4 8,442 32,256 3,408 1,368 1,836 360 302 360 10,080 14,448 10,416 10 10 10 10 69 69 Inside wall (corridor) Door Inside wall (corridor) Door Roof 33 Supplementary allowance, r " " i Exposed location and intern Total thermal units orth outside wall, 1C lorth outside windov littent day or night % 83,276 844 ps, 10% jse, 30$ . . . . 3,226 87,346 26.204 113.550 If we assume that the lecture- room must be heated to 69 degrees Fahr. in the daytime when unoccupied, so as to be at this temperature when first persons arrive, there will be required, ventilation not being considered, and bronzed direct low-pressure steam-radiators being the heating media, about 113,550 -h 250 = 455 sq. ft. of radiating-surface. (This gives a ratio of about 105 cu. ft. of contents of room for each sq. ft. of heating-surface.) If we assume that there are 160 persons in the lecture-room, and we pro- vide 2500 cubic feet of fresh air per person per hour, we will supply 160 X 2500 = 400,000 cubic feet of air per hour (i.e., ' - over eight changes of contents of room per hour). To heat this air from 0° Fahr. to 69° Fahr. will require 400,000 X 0.0189 X 69 = 521,640 thermal units per hour (0.0189 being the product of a weight of a cubic foot by the specific heat of air). Accordingly there must be provided 521,640 h-400 = 1304 sq. ft. of indirect surface, to heat the air required for ventilation, in zero weather. If the room were to be warmed entirely indi- rectly, that is, by the air supplied to room (including the heat to be'conveyed to cover loss by transmission through walls, etc.), there would have to be conveyed to the fresh-air supply 521,640 + 113,550 = 635,190 heat-units. This would imply the provision of an amount of indirect heating-surface of the " Climax " type of 635,190 -*- 400 = 1589 sq. ft., and the fresh air entering the room would have to be at a temperature of about 84° Fahr., viz., 69° = The above calculations do not, however, take into account that 160 per- sons in the lecture-room give out 160 X 400 = 64.000 thermal units per hour; and that, say, 50 electric lights give out 50 x 1600 — 80,000 thermal units per hour; or, say, 50 gaslights. 50 X 4800 = 240,000 thermal units per hour. The presence of 160 people and the gas-lighting would diminish considerably the amount of heat required. Practically, it appears that the heat generated by the presence of 160 people, 64,000 heat-units, and by 50 electric lights, 80,000 heat-units, a total of 144,000 heat-units, more than covers the amount of heat transmitted through walls, etc. Moreover, that if the 50 gaslights give out 240,000 thermal units per hour, the air supplied for ventilation must enter considerably below 69° Fahr., or the room will be heated to an unbearably high temperature. If 400,000 cubic feet of fresh air per hour 536 HEATIHG AHt) VEHTILATIOK. are supplied, and 240,000 thermal units per hour generated by the gas must be abstracted, it means that the air must, under these conditions, enter 400 000 V 0189 = about 3 ~° less tnan 84°, or at about 52° Fahr. Further- more, the additional vitiation due to gaslighting would necessitate a rmich larger supply of fresh air than when the vitiation of the atmosphere by the people alone is considered, one gaslight vitiating the air as much as five men. Various Rules for Computing Radiating-surface.— The following rules are compiled from various sources. They are more in the nature of "rule-of -thumb " rules than those given by Mr. Wolff, quoted above, but they may be useful for comparison. Divide the cubic feet of space of the room to be heated, the square feet of wall surface, and the square feet of the glass surface by the figures given under these headings in the following table, and add the quotients together; the result will be the square feet of radiating-surface required. (F. Schumann.) Space, Wall and Glass Surface which One Square Foot of Radiating- surface will Heat. 2 It k§ as a m 1 3 5 o '2 3 O .2 O D ft* - m 190 210 225 Exposure of Rooms. 0) All Sides. Northwest. : Southeast. c3 .q O < Wall Surface, sq. ft. Glass Surface, sq. ft. Wall Surface, sq. ft. Glass Surface, sq. ft. Wall Surface, sq. ft. Glass Surface, sq. ft. Once per hour. 13.8 15.0 16.5 7 7.7 8.5 15.87 17.25 18.97 8.05 8.85 9.77 16.56 18.00 19.80 8.4 9.24 10.20 Twice per hour. 1 3 5 75 82 90 11.1 12.1 13.0 5.7 6.2 6.7 12.76 13.91 14.52. 6.55 7.13 7.60 13.22 14.52 15.60 6.84 7.44 8.04 Emission of Heat-units per square foot per Hour from Cast-iron Pipes or Radiators. Temp, of Air in Room, 70° F. (F. Schumann.) Mean Temperature of By Contact. By Radi- ation. By Radiation and Contact. Heated Pipe, Radia- tor, etc. Air quiet. Air moving. Air quiet. Air moving. Hot water 140° 55.51 65.45 75.68 86.18 96.93 107.90 119.13 130.49 142.20 153.95 165.90 178.00 189.90 202.70 215.30 228.55 240.85 92.52 109.18 126.13 143.30 161.55 179.83 198.55 217.48 237.00 256.58 279.83 296.63 31.6.50 337.83 358.85 380.91 401.41 59.63 69.69 80.19 91.12 102.15 114.45 127.00 139.96 155.27 169.56 184.58 200.18 214.36 233.42 251.21 267.73 279.12 115.14 135.11 155 87 177.30 199.43 222.35 246.13 270.49 297.47 323.51 350.48 378.18 404.26 436.12 466.51 496.28 519.97 152 15 " 150° 44 160° " 170° 178.87 206.32 234.42 180° " 190° 44 200° 14 44 or steam.. 210° Steam 220° 230° 264.05 294.28 325.55 357.48 392.27 426.14 •' 240° 250° 464.41 496.81 " 260° 270° 44 ...280° 530.86 571.25 610.06 290° 300° 648.64 680.53 INDIRECT HEATING-SURFACE. 537 Radiating-surface required for Different Kinds of Buildings. (From practice of the Dubuque Steam Supply Co., External Air 0° F. Chas. A. Smith.) Cubic ft. of Room heated by 1 sq. ft. of Surface. Direct Indirect System. System. Dwellings 50 40 Stores, wholesale 125 100 retail 100 80 Cubic ft. of Room heated by 1 sq. ft. of Surface. Direct Indirect System. System. Banks, offices, drug-stores 70 60 Large hotels 125 100 Churches 200 150 The Nason Mfg. Co.'s catalogue gives the following: One square foot of surface will heat from 40 to 100 cu. ft. of space to 75° in — 10° latitudes. This range is intended to meet conditions of exposed or corner rooms of buildings, and those less so, as intermediate ones of a block. As a general rule, 1 sq. ft. of surface will heat 70 cu. ft. of air in outer or front rooms and 100 cu. ft. in inner rooms. In large stores in cities with buildings on each side, 1 to 100 is ample. Approximate Proportions of Radiating-surfaces. One square foot radiating-surface will heat: Indwellings, In hall, stores, In churches, large schoolrooms, lofts, factories, auditoriums, offices, etc. etc. etc. By direct radiation. . . 60 to 80 ft. 75 to 100 ft. 150 to 200 ft. By indirect radiation . 40 to 50 " 50 to 70 •« 100 to 140 " Isolated buildings exposed to prevailing north or west winds should have a generous addition made to the heating-surface on their exposed sides. The following rule is given in the catalogue of the Babcock & Wilcox Co., and is also recommended by the Nason Mfg. Co.: Radiating surface may be calculated by the rule: Add together the square feet of glass in the windows, the number of cubic feet of air required to be changed per minute, and one twentieth the surface of external wall and roof; multiply this sum by the difference between the required temperature of the room and that of the external air at its lowest point, and divide the product by the difference in temperature between the steam in the pipes and the required temperature of the room. The quotient is the required rad latin g-surf ace in square feet. Overhead Steam-pipes. (A. R. Wolff, Stevens Indicator, 1887.)— When the overhead system of steam-heating is employed, in which system direct radiating-pipes, usually 1*4 in. in diam., are placed in rows overhead, suspended upon horizontal racks, the pipes running horizontally, and side by side, around the whole interior of the building, from 2 to 3 ft. from the walls, and from 2 to 4 ft. from the ceiling, the amount of 1J4 in. pipe re- quired, according to Mr. C. J. H. Woodbury, for heating mills (for which use this system is deservedly much in vogue), is about 1 ft. in length for every 90 cu. ft. of space. Of course a great range of difference exists, due to the special character of the operating machinery in the mill, both in re- spect to the amount of air circulated by the machinery, and also the aid to warming the room by the friction of the journals. Indirect Heating-surface.— J. H. Kinealy, in Heating and Ven- tilation, May 15, 1894, gives the following formula, deduced from results of experiments by C. B. Richards, W. J. Baldwin, J. H. Mills, and others, upon indirect heaters of various kinds, supplied with varying amounts of air per hour per square foot of surface: N= cubic feet of air, reduced to 70° F., supplied to the heater per square foot of heating-surface per hour; T = temperature of the steam or water in the heater; Tx = temperature of the air when it enters the heater; T 2 = temperature of the air when it leaves the heater. As the formula is based upon an average of experiments made upon all sorts of indirect heaters, the results obtained by the use of the equation may in some cages be slightly too small and in others slightly too large, 538 HEATING AND VENTILATION. although the error will in no case be great. No single formula ought to be expected to apply equally well to all dispositions of heating-surface in in- direct heaters, as the efficiency of such heater can be varied between such wide limits by the construction and arrangement of the surface. In indirect heating, the efficiency of the radiating-surface will increase, and the temperature of the air will diminish, when the quantity of the air caused to pass through the coil increases. Thus 1 sq. ft. radiating-surface, with steam at 212°, has been found to heat 100 cu. ft. of air per hour from zero to 150°, or 30J cu. ft. from zero to 100° in the same time. The best re- sults are attained by using indirect radiation to supply the necessary venti- lation, and direct radiation for the balance of the heat. (Steam.) ■■ In indirect steam-heating the least flue area should be 1 to 1*4 sq- in. to every square foot of heating-surface, provided there are no long horizon- tal reaches in the duct, with little rise. The register should have twice the area of the duct to allow for the fretwork. For hot water heating from 25% to '30% more heating-surface and flue area should be given than for low- pressure steam. (Engineering Record, May 26, 1894.) Boiler Heating-surface Required. (A. R. Wolff, Stevens Indi- cator, 1887.) — When the direct system is used to heat buildings in which the street floor is a store, and the upper floors are devoted to sales and stock- rooms and to light manufacturing, and in which the fronts are of stone or iron, and the sides and the rear of building of brick— a safe rule to follow is to supply 1 sq. ft. of boiler heating-surface for each 700 cu. ft., and 1 sq. ft. of radiating-surface for each 100 cu. ft. of contents of building. For heating mills, shops, and factories, 1 sq. ft. of boiler heating-surface should be supplied for each 475 cu. ft. of conteuts of building; and the same allowance should also be made for heating exposed wooden dwellings. For heating foundries and wooden shops, 1 sq. ft. of boiler heating-surface should be provided for each 400 cu. ft. of contents; and for structures in which glass enters very largely in the construction— such as conservatories, exhibition buildings, and the like— 1 sq. ft. of boiler heating-surface should be provided for each 275 cu. ft. of contents of building. When the indirect system is employed, the radiator-surface and the boiler capacity to be provided will each have to be, on an average, about 25$ more than where direct radiation is used. This percentage also marks approxi- mately the increased fuel consumption in the indirect system. Steam (Babcock& Wilcox Co.) has the following: 1 sq. ft. of boiler-surface will supply from 7 to 10 sq. ft. of radiating-surface, depending upon the size of boiler and the efficiency of its surface, as well as that of the radiating- surface. Small boilers for house use should be much larger proportionately than large plants. Each horse-power of boiler will supply from 240 to 360 ft. of 1-in. steam pipe, or 80 to 120 sq. ft. of radiating surface. Cubic feet of space has little to do with amount of steam or surface required, but is a convenient factor for rough calculations. Under ordinary conditions 1 horse-power will heat, approximately, in — Brick dwellings, in blocks, as in cities 15,000 to 20,000 cu. ft. " stores " " 10,000 " 15,000 " " dwellings, exposed all round 10,000 " 15,000 " mills, shops, factories, etc 7,000 " 10,000 " Wooden dwellings, exposed 7,000 " 10,000 " Foundries and wooden shops 6,000 " 10,000 " Exhibition buildings, largely glass, etc 4,000 " 15,000 " Proportion of Grate-surface to Radiator-surface. (J. R. Willett, Heating and Ventilation, Feb. 1894.) R sq lf f t° r ~ SUI f ' } 10 ° 200 400 600 80 ° 100 ° 1200 140 ° 160 ° 1800 2000 G 'sq te hi UrfaCe ' f 120 208 362 501 630 754 8T2 986 110 ° 1210 131 ° Steam-consumption in Car-heating. C, M. & St. Paul Railway Tests. (Engineering, June 27, 1890, p. 764.) Water of Condensation Outside Temperature. Inside Temperature. per Car per Hour. 40 70 70 lbs. 30 70 85 JO 70 100 REGISTERS AND COLD-AIR DUCTS. 539 Internal Diameters of Steam Supply-mains, with Total Resistance equal to 2 inches of Water-column.* Steam, Pressure 10 lbs. per square inch above atm., Temperature 239° F. Formula, d ^ 0.5374 a/ ^-; where d = internal diameter in inches; g = s .2 cub c feet of steam pei minute per 100 sq. ft, of radiat ng-su face ; l — length of ma ns in feet; h = 159.3 feet head of steam to produce flow. Inter ial Diameters in inches for Lengths of Mains from 1 ft. to 600 ft. 1ft. 10 ft 20 ft. 40 ft. 60 ft. 80 ft. 100 ft. 200 ft. 300 ft. 400 ft. 600 ft. sq.ft. inch. inch. inch. inch. inch. inch. inch. inch. inch. inch. inch. 1 0.075 119 0.136 0.157 0.170 0.180 0.189 0.216 0.234 0.248 0.270 10 0.19 0.30 0.34 0.39 0.43 0.45 0.47 0.54 0.59 0.62 0.68 20 0.25 0.39 0.45 0.52 0.56 60 0.62 0.72 0.78 0.82 0.89 40 0.33 0.52 0.60 0.69 0.74 0.79 0.82 0.95 1.03 1.09 1.18 60 0.39 0.61 0.71 0.81 0.87 0.93 0.97 1.11 1.21 1.28 1.39 HO 0.43 0.68 0.79 0.90 0.98 1.04 1.09 1.25 1.35 1.43 1.55 100 0.47 0.75 0.86 0.99 1.07 1.14 1.19 1.36 1.48 1.57 1.70 200 62 0.99 1.14 1.30 1.41 1.50 1.57 1.80 1.95 2.07 2.24 300 0.73 1.16 1.34 1.53 1.66 1.76 1.84 2 12 2.30 2.43 2.64 400 0.82 1.30 1.50 1.72 1.86 1.98 2.07 2.37 2.57 2.73 2.96 500 0.90 1.43 1.64 1.88 2.04 2.16 2.26 2.60 2.81 2.98 3.23 600 0.97 1.53 1.76 2.03 2.20 2.33 2.43 2.79 3.03 3.21 3.48 800 1.09 1.72 1.98 2.27 2 46 2.61 2.73 3.13 3.40 3.60 3.90 1,000 1.19 1.88 2.16 2.48 2.69 2.85 2.98 3.43 3.71 3.94 4.27 1,200 1.28 2.04 2.33 2.67 2.90 3.07 3.21 3.68 4.00 4.23 4.59 1,400 1.36 2.15 2.47 2.84 3.08 3.26 3.41 3.92 4.25 4.50 4.83 1,600 1.43 2.27 2.61 3.00 3.25 3.44 3.60 4.13 4.49 4.75 5.15 1,800 1.50 2.38 2.74 3.14 3.41 3.61 3.78 4.34 4.70 4.98 5.40 2,000 1.57 2.48 2.85 3.28 3.55 3.76 3.93 4.52 4.90 5.19 5.63 3,000 1.84 2.92 3.36 3.85 4.18 4.43 4 63 5.32 5.77 6.11 6.63 4,000 2.07 3.28 3.76 4.32 4.69 4.96 5.19 5.96 6.47 6.85 7.44 * From Robert Briggs's paper on American Practice of Warming Buildings by Steam (Proc. Inst. C. E., 1882, vol. lxxi). For other resistances and pressures above atmosphere multiply by the respective factors below : Water col . C in. 12 in. 24 in. I Press, ab. atm. lbs. 3 lbs. 30 lbs. 60 lbs. Multiply by 0.8027 0.6988 0.6084 | Multiply by 1.023 1.015 0.973 0.948 Registers and Cold-air Ducts for Indirect Steam Heating. —The Locomotive gives the following table of openings for registers and cold-air ducts, which has been found to give satisfactory results. The cold- air boxes should have \y% sq. in. area for each square foot of radiator suface, and never less than % the sectional area of the hot air ducts. The hot air ducts should have 2 sq. in. of sectional area to each square foot of radiator surface on the first floor, and from 1^ to 2 inches on the second floor. Heating Surface in Stacks. Cold-air Supply, First Floor. Size Register. Supply, 2d Floor. inches inches inches 30 square feet 45 square inches = 5 by 9 9 by 12 4 by 10 40 •' 60 " = 6 by 10 10 by 14 4 by 14 50 " 75 ,l " = 8 by 10 10 by 14 5 by 15 60 " 90 " " = 9 by 10 12 by 15 6 by 15 70 108 " " = 9 by 12 12 by 19 6 by 18 80 " 120 " " = 10 by 12 12 by 22 8 by 15 90 " 135 " " = 11 by 12 14 by 24 9 by 15 100 " 150 ■■ " =12 by 12 16 by 20 12 by 12 The sizes in the table approximate to the rules given, and it will be found that they will allow an easy flow of air and a full distribution throughout the room to be heated. 540 HEATING AND VENTILATION. Physical Properties of Steam and Condensed Water, under Conditions of Ordinary Practice in Warming by Steam. (Brig^s.) A ( Steam-pressure j above atm. . . } per square inch | total lbs. lbs. 14.7 3 17.7 10 24.7 30 44.7 60 74.7 F, Fahr. Fahr. Fahr. >■ units Fahr. 212° 60° 152° 456 965° 222° 60 162° 486 958° 239° 60° 179° 537 946° 274° 60° 214° 642 921° 307° € D E F Temperature of air Difference = B — C 1 Heat given out per minute per ■I 100 sq. ft. of radiatiug-sur- I face = D X 3 Latent heat of steam 60* 247° 741 898° G H J Volume of 1 lb. weight of steam Weight of 1 cubic foot of steam ( Volume Q of steam per minute < to give out E units ( =EXG-F. cu. ft. lb. tcu.ft. 26.4 0.0380 12.48 22.1 0.0452 11.21 16.2 0.0618 9.20 9.24 0.1082 6.44 5.70 0.1752 4.70 K L M ( Weight of 1 cubic foot of con- < densed water at tempera- [ ture B, I Volume of condensed water to < return to boiler per minute j =JXH-K, i Head of steam equivalent to ■< 12 inches water-column I =K-4-H. t lbs. tcu.ft. V feet 59.64 0.0079 1569 59.51 0.0085 1317 59.05 0.0096 955.5 58.07 0.0120 536.7 57.03 0.0144 325.5 N P B S Steam-supply Mains. fHead h of steam, equivalent J to assumed 2 inches water- 1 column for producing steam L flow Q, = M -f- 6, j Internal diameter d of tube* \ for flow Q when I = 1 foot, Do. do. when I = 100 feet, Ratios of values of d. }■ feet J >■ inch inch ratio 261.5 0.484 1.217 1.023 219.5 0.481 1.207 1.015 159.3 0.474 1.190 1.000 89.45 0.461 1.158 0.973 54.25 0.449 1.128 0.948 T U V W Water-Return Mains. ( Head h assumed at J^-inch < water-column for producing [ full-bore water-flow Q, j Internal diameter d of tube* j for flow Q when I = 1 foot, Do. do. when I = 100 feet, Ratios of values of d V foot y inch inch ratio 0.0417 0.147 0.368 0.926 0.0417 151 0.379 0.952 0.0417 0.158 0.398 1.000 0.0417 0.173 0.434 1.092 0.0417 0.186 0.468 1.176 * P, P, U, V are each determined from the formula d = 0.5374 m Size of Steam Pipes for Steam Heating. (See also Flow of Steam in Pipes.)— Sizes of vertical main pipes. Direct radiation. (J. R. Willett, Heating and Ventilation, Feb., 1894.) Diameter of pipe, inches. 1 1J4 1}4 2 2^ 3 3^ 4 5 6 Sq. ft. of radiator surface 40 70 110 220 360 560 810 1110 2000 3000 A horizontal branch pipe for a given extent of radiator surface should be one size larger than a vertical pipe for the same surface. No return from a main should be more than two sizes smaller than the feed at its commence- ment (or than its largest dimension). A. R. Wolff (Stevens Indicator, 1887) says: For determining the cross- sectional area of pipes (in square inches) for steam mains and returns it will be ample to allow a constant of .375 sq. in. for each hundred square HEATING A GREENHOUSE BY STEAM. 541 feet of heating-surface in coils and radiators, when exhaust steam is used, .19 sq. in. when live steam is used, and .09 sq. in. for the return. If the cross- sectional areas thus obtained are each mulitplied by 1.273, and the square root extracted from each product, the respective figures obtained will represent the proper diameters in inches of the several steam-pipes referred to. Steam, by the Babcock & Wilcox Co., says : Where the condensed water is returned" to the boiler, or where low pressure of steam is used, the diame- ter of mains leading from the boiler to the radiating-surface should be equal in inches to one tenth the square root of the radiating-surface, mains included, in square feet. Thus a 1-inch pipe will supply 100 square feet of surface, itself included. Return-pipes should be at least % inch in diame- ter, and never less than one half the diameter of the main— longer returns requiring larger pipe. A thorough drainage of steam-pipes will effectually prevent all cracking and pounding noises therein. The Nason Mfg. Co. gives the following : Radiating-surface in square Size of Steam- Size of Return- feet to be supplied. pipes. pipes. 125 1M 1 125to200 \\i V/ A 200 to 500 2 \% 500 to 1000 2)4 2 1000tol500 * 3 2}4 1500to2500 33^ 3 When mains and surfaces are very much above the boiler the pipes need not be as large as given above; under very favorable circumstances and conditions a 4-inch pipe may supply from 2000 to 2500 sq. ft. of surface, a 6- inch pipe for 5000 sq. ft., and a 10-inch pipe for 15,000 to 20,000 sq. ft., if the distance of run from boiler is not too great. Less than lj^-inch pipe should not be used horizontally in a main unless for a single radiator connection. The return sizes named are large enough in ordinary pipe-work, though when horizontal pipes with many fittings are used they should be of the same diameter as the steam-pipes. Generally, when condensation is returned to the boiler by gravity, the diameter of mains in inches should equal one tenth of the square root of the radiating-surfaces in square feet; thus a 1-inch pipe will supply 100 sq. ft. of surface, or with 900 sq. ft. the supply-pipe should be V900 = 30 -s- 10 = 3" diameter. Heating a Greenhouse by Steam.— Wm. J. Baldwin answers a question in the American Macliinist as below: With five pounds steam- pressure, how many square feet or inches of heating-surface is necessary to heat 100 square feet of glass on the roof, ends, and sides of a greenhouse in order to maintain a night heat of 55° to 65°, while the thermometer out- side ranges at from 15° to 20° below zero ; also, what boiler-surface is neces- sary ? What is the best way to set pipes in a greenhouse — hang them or lay them down ? Which is the best for the purpose to use— 2" pipe or 134" pipe? Ans.— Reliable authorities agree that 1.25 to 1.50 cubic feet of air in an enclosed space will be cooled per minute per sq. ft. of glass as many degrees as the internal temperature of the house exceeds that of the air outside. Between -f- 65° and — 20° there will be a difference of 85°, or, say, one cubic foot of air cooled 127.5° F. for each sq ft. of glass for the most extreme condition mentioned. Multiply this by the number of square feet of glass and by 60, and we have the number of cubic feet of air cooled 1° per hour within the building or house. Divide the number thus found by 48, and it gives the units of heat required, approximately. Divide again by 953, and it will give the number of pounds of steam that must be condensed from a pressure and temperature of five pounds above atmosphere to water at the same temperature in an hour to maintain the heat. Each square foot of surface of pipe will condense from 34 to nearly y% lb. of steam per hour, according as the coils are exposed or well or poorly arranged, for which an average of \£ lb. may be taken. According to this, it will require 3 sq. ft. of pipe surface per lb. of steam to be condensed. Proportion the heating- surface of the boiler to have about one fifth the actual radiating-surface, if you wish to keep steam over night, and proportion the grate to burn not more than six pounds of coal per sq. ft. of grate per hour. With very slow combustion, such as takes place in base-burning boilers, the grate might be proportioned for four to five pounds of coal per hour. It is cheaper to make coils of 1J4" pipe than of 2", and there is nothing to be gained by using 2" pipe unless the coii^ are very long. The pipes in a greenhouse should be 542 HEATING AND VENTILATION. under or in front of the benches, with every chance for a good circulation of air. " Header" coils are better than "return-bend 11 coils for this purpose. Mr. Baldwin's rule may be given the following form : Let H = heat-units transferred per hour, T ~ temperature inside the greenhouse, t — tempera- ture outside, 8= sq. ft. of glass surface; then H = 1.5S(!T- t) X 60 -=- 48 = 1.8755(7' - t). Mr. Wolff's coefficient K for single skylights would give H= 1.1l8S(r- t). Heating a Greenhouse by Hot Water.— W. M. Mackay, of the Richardson & Boynton Co., in a lecture before the Master Plumbers' Asso- ciation, N. Y., 1889, says : I find that while greenhouses were formerly heated by 4-inch and 3-inch cast-iron pipe, on account of the large body of water which they contained, and the supposition that they gave better satis- faction and a more even temperature, florists of long experience who have tried 4-inch and 3-inch cast-iron pipe, and also 2 inch wrought-iron pipe for a number of years in heating their greenhouses by hot water, and who have also tried steam-heat, tell me that they get better satisfaction, greater economy, and are able to maintain a more even temperature with 2- inch wrought-iron pipe and hot water than by any other system they have used. They attribute this result principally to the fact that this size pipe contains less water and on this account tUe beat can be raised and lowered quicker than by any other arrangement of pipes, and a more uniform tem- perature maintained than by steam or any other system. HOT- WATER HEATING. (Nason Mfg. Co.) There are two distinct forms or modifications of hot- water apparatus, de- pending upon the temperature of the water. In the first or open-tank system the water is never above 212° tempera- ture, and rarely above 200°. This method always gives satisfaction where the surface is sufficiently liberal, but in making it so its cost is considerably greater than that for a steam-heating apparatus. In the second method, sometimes called (erroneously) high-pressure hot- water heating, or the closed-system apparatus, the tank is closed. If it is provided with a safety-valve set at 10 lbs. it is practically as safe as the open- tank system. Law of Velocity of Flow.— The motive power of the circulation in a hot -water apparatus is the difference between the specific gravities of the ascending and the descending pipes. This effective pressure is very small, and is equal to about one grain for each foot in height for each de- gree difference between the pipes; thus, with a height of 12" in " up " pipe, and a difference between the temperatures of the up and down pipes of 8°, the difference in their specific gravities is equal to 8.16 grains on each square inch of the section of return-pipe, and the velocity of the circulation is pro- portioned to these differences in temperature and height. To Calculate Velocity of Flow.— Thus, with a height of ascend- ing pipe equal to 10' and a difference in temperatures of the flow and return pipes of 8°, the difference in their specific gravities will equal 81.6 grains, or -h 7000 = .01166 lbs., or X 2.31 (feet of water in one pound ) = . 0269 ft., and by the law of falling bodies the velocity will be equal to 8 V.0-J69 = 1.312 ft. per second, or X 60 = 78.7 ft. per minute. In this calculation the effect of fric- tion is entirely omitted. Considerable deduction must be made on this account. Even in apparatus where length of pipe is not great, and with pipes of larger areas and with few bends or angles, a large deduction for friction must be made from the theoretical velocity, while in large and complex apparatus with small head, the velocity is so much reduced by friction that sometimes as much as from 50$ to 90$ must be deducted to ob- tain the true rate of circulation^ Main flow-pipes from the heater, from which branches may be taken, are to be preferred to the practice of taking off nearly as many pipes from the heater as there are radiators to supply. It is not necessary that the main flow and return pipes should equal in capacity that of all their branches. The hottest water will seek the highest level, while gravity will cause an even distribution of the heated water if the surface is properly proportioned. It is good practice to reduce the size of the vertical mains as they ascend, say at the rate of one size for each floor. As with steam, so with hot water, the ninoa ™yst De unconfined to allow HOT- WATER HEATING. 543 for expansion of the pipes consequent on having their temperatures in- creased. An expansion tank is required to keep the apparatus filled with water, which latter expands 1/24 of its bulk on being heated from 40° to 212°, and the cistern must have capacity to hold certainly this increased bulk. It is recommended that the supply cistern be placed on level with or above the highest pipes of the apparatus, in order to receive the air which collects in the mains and radiators, and capable of holding at least 1/20 of the water in the entire apparatus. Approximate Proportions of Radiatiiig-surfaces to Cubic Capacities of Space to be Heated. One Square Foot of Ra- In Dwellings, In Halls, Stores, In Churches, diating-surface will School-rooms, Lofts, Facto - Large Audito- heat with— Offices, etc. , ries, etc. riums, etc. High temperature di- ) rect hot- water radi- V 50 to 70 cu. ft. 65 to 90 cu. ft. 130 to 180 cu. ft. ation ) Low temperature di- ) rect hot-water radi- > 30 to 50 " " 35 to 65 " " 70 to 130 " " High temperature in- 1 direct hot- water ra- V 30 to 60 " " 35 to 75 " " 70 to 150 " " diation ) Low temperature in- ) direct hot-water ra- >■ 20 to 40 " " 25 to 50 " " 50 to 100 " " diation J Diameter of Main and Branch Pipes and square feet of coil surface they will supply, in a low-pressure hot-water apparatus (212°) for direct or indirect radiation, when coils are at different altitudes for direct radiation or in the lower story for indirect radiation: 04 oa lS a o.2 Ej£> .a * Direct Radiation. Height of Coil above Bottom of Boiler, ■§■■2 . a MA Q 10 20 30 40 sq. ft. 'sq.ft. 50 60 70 80 90 100 sq. ft. sq. ft. sq. ft. sq. ft. sq. ft. sq. ft. sq.ft. sq. ft. sq. ft. % 49 50 52 53 55 57 59 61 63 65 68 1 87 89 92 95 98 101 103 108 112 116 121 4\i 136 140 144 149 153 158 161 169 175 182 189 w<& 196 202 209 214 222 228 235 243 252 261 271 349 359 370 380 393 405 413 433 449 465 483 m 546 561 577 595 613 633 643 678 701 727 755 3 785 807 835 856 888 912 941 974 1009 1046 1086 3^ 1069 1099 1132 1166 1202 1241 1283 1327 1374 1425 1480 4 1395 1436 1478 1520 1571 1621 1654 1733 1795 1861 1933 *H 1767 1817 1871 1927 1988 2052 2120 2193 2272 2356 2445 5 2185 2244 2309 2376 2454 2531 2574 2713 2805 2907 3019 6 3140 3228 3341 3424 3552 3648 3763 3897 4036 4184 4344 7 4276 4396 4528 4664 4808 4964 5132 5308 5496 5700 5920 8 5580 5744 5912 60S0 6284 6484 6616 6932 7180 7444 7735 9 7068 7268 7484 7708 7952 8208 8482 8774 9088 9424 9780 10 8740 8976 9236 9516 9816 10124 10296 10852 11220 11628 12076 11 10559 10860 11180 11519 11879 12262 12666 13108 13576 14078 14620 12 12560 12912 13364 13696 14208 14592 15052 15588 16144 16736 17376 13 14748 15169 15615 16090 |16591 17126 17697 18307 18961 19633 20420 14 17104 17584 18109 18656 ! 19232 19856 20528 21232 21984 22800 23680 15 19634 20195 20789 21419 22089 22801 23561 24373 25244 26179 27168 16 22320 22978 23643 24320 25136 25936 2G464 27728 28720 29776 30928 544 HEATING AND VENTILATION. The best forms of hot- water- heating boilers are proportioned about as follows: 1 sq. ft. of grate-surface to about 40 sq. ft. of boiler-surface. 1 " " boiler- " " 5 " " radiating-surface. 1 " " grate- " " 200 " " Rules for Hot-water Heating.— J. L. Saunders (Heating and Ventilation, Dec. 15, 1894) gives the following : Allow 1 sq. ft. of radiating surface for every 3 ft. of glass surface, and 1 sq. ft. for every 30 sq. ft. of wall surface, also 1 sq. ft. for the following numbers of cubic feet of space in the several cases mentioned. In dwelling-houses: Libraries and dining-rooms, first floor. . 35 to 40 cu. ft. Reception halls, first floor 40 to 50 " " Stairhalls, " " 40 to 55 '• " Chambers above, " " 50 to 65 " " Libraries, sewing-rooms, nurseries, etc., above first floor 45 to 55 " " Bath-rooms 30 to 40 " " Public-schoolrooms 60 to 85 " " Offices 50 to 65 " " Factories and stores 65 to 90 " " Assembly halls and churches 90 to 150 " " To find the necessary amount of indirect radiation required to heat a room: Find the required amount of direct radiation according to the foregoing method and add 50$. This if wrought-iron pipe coil surface is used ; if cast- iron pin indirect -stack surface is used it is advisable to add from 70$ to 80$. Sizes of hot-air flues, coLt-air ducts, and registers for indirect work. — Hot-air flues, first floor: Make the net internal area of the flue equal to % sq. in. to every square foot of radiating surface in the indirect stack. Hot- air flues, second floor: Make the net internal area of the flue equal to % sq. in. to every square foot of radiating surface in the indirect stack. Cold-air ducts, first floor : Make the net internal area of the duct equal to % sq. in. to every square foot of radiating surface in the indirect stack. Cold air ducts, second floor : Make the net internal area of the duct equal to H> sq. in. to every square foot of radiating surface in the indirect stack. Hot-air registers should have their net area equal in full to the area of the hot-air flues. Multiply the length by the w r idth of the register in inches ; % of the product is the net area of register. Arrangement of Mains for Hot-water Heating. (W. M. Mackay, Lecture before Master Plumbers' Assoc, N. Y., 1889 )— There are two different systems of mains in general use, either of which, if properly placed, will give good satisfaction. One is the taking of a single large-flow main from the heater to supply all the radiators on the several floors, with a corresponding return main of the same size. The other is the taking of a number of 2-inch wrought-iron mains from the heater, with the same num- ber of return mains of the same size, branching off to the several radiators or coils with l^-inch or 1-inch pipe, according to the size of the radiator or coil. A 2-inch main will supply three 1 14-inch or four 1-inch branches, and these branches should be taken from the top of the horizontal main with a nipple and elbow, except in special cases where it is found necessary to retard the flow of water to the near radiator, for the purpose of assisting the circu- lation in the far radiator ; in this case the branch is taken from the side of the horizontal main. The flow and return mains are usually run side by side, suspended from the basement ceiling, and should have a gradual ascent from the heater to the radiators of at least 1 inch in 10 feet. It is customary, and an advantage where 2-inch mains are used, to reduce the size of the main at every point where a branch is taken off. The single or large main system is best adapted for large buildings ; but there is a limit as to size of main which it is not wise to go beyond— gener- ally 6- inch, except in special cases. The proper area of cold- air pipe necessary for 100 square feet of indirect radiation in hot-water heating is 75 square inches, while the hot-air pipe should have at least 100 square inches of area. There should be a damper in the cold-air pipe for the purpose of controlling the amount of air admitted to the radiator, depending on the severity of the weather. BLOWER SYSTEM OF HEATING AND VENTILATING. 545 THE BLOWER SYSTEM OF HEATING AND VENTILATING. The sj^stem provides for the use of a fan or blower which takes its supply of fresh air from the outside of the building to be heated, forces it over steam coils, located either centrally or divided up into a number of indepen- dent groups, and then into the several ducts or flues leading to the various rooms. The movement of the warmed air is positive, and "the delivery of the air to the various points of supply is certain and entirely independent of atmospheric conditions. For engines, fans, and steam-coils used with the blower system, see page 519. Experiments with Radiators of 60 sq. ft. of Surface. (Mech. News, Dec, 1893.) — After having determined the volume and tem- perature of the warm air passing through the flues and radiators from natural causes, a fan was applied to each flue, forcing in air, and new sets of measurements were made. The results showed that more than t\\ o and one- third times as much air was warmed with the fans in use, and the falling off in the temperature of this greatly increased air-volume was only about 12. 6$. The condensation of steam in the radiators with the forced-air circulation also was only 66%$ greater than with natural air draught. One of the several sets of test figures obtained is as follows : ^ ■ Natural Forced- Draught air in Flue. Circulation. Cubic feet of air per minute 457.5 1227 Condensation of steam per minute in ounces 11.7 19.6 Steam pressure in radiator, pounds 9 9 Temperature of air after leaving radiator 142° 124° " " " before passing through radiator. 61° 61° Amount of radiating surface in square feet 60 60 Size of flue in both cases 12 x 18 inches. There was probably an error in the determination of the volume of air in these tests, as appears from the following calculation. (W. K.) Assume that 1 lb. of steam in condensing from 9 lbs. pressure and cooling to the tem- perature at which the water may have been discharged from the radiator gave up 1000 heat-units, or 62.5 h. u. per ounce; that the air weighed .076 lb. per cubic foot, and that its specific heat is .238. We have Natural Forced Draught. Draught. Heat given up by steam, ounces x 62 5 — 731 1225 H.U. Heat received by air, cu. ft. x. 076 xdiff. of tern. x. 238= 673 1399 '.' Or, in the case of forced draught the air received \\% more heat than the steam gave out, which is impossible. Taking the heat given up by the steam as the correct measure of the work done by the radiator, the temperature of the steam at 237°, and the average temperature of the air in the case of natural draught at 102° and in the other case at 93°, we have for the tem- perature difference in the two cases 135° and 144° respectively; dividing these into the heat- units we find that each square foot of radiating surface transmitted 5.4 heat-units per hour per degree of difference of temperature, in the case of natural draught, and 8.5 heat-units in the case of forced draught. In the Women's Homoeopathic Hospital in Philadelphia, 2000 feet of one-inch pipe heats 250,000 cubic feet of space, ventilating as well; this equals one square foot of pipe surface for about 350 cubic feet of space, or less than 3 square feet for 1000 cubic feet. The fan is located in a sepa- rate building about 100 feet from the hospital, and the air, after being heated to about 135°, is conveyed through an underground brick duct with a loss of only five or six degrees in cold weather. (H. I. Snell, Trans. A. S. M. E ,ix. 106. Heating a Building to 7O F. Inside when the Outside Temperature is Zero.— It is customary in some contracts for heating to guarantee that the apparatus will heat the interior of the building to 70° in zero weather. As it may not be practicable to obtain zero weather for the purpose of a test, it may be difficult to prove the performance of the guarantee. E. E. Macgovern, in Engineering Record, Feb. 3, 1894, gives a calculation tending to show that a test may be made in weather of a higher temperature than zero, if the heat of the interior is raised above 70°. The higher the temperature of the rooms the lower is the efficiency of the radi- ating-surface, since the efficiency depends upon the difference between the 546 HEATING AND VENTILATION. temperature inside of the radiator and the temperature of the room. He concludes that a heating apparatus sufficient to heat a given building to 70° in zero weather with a given pressure of steam will be found to heat the same building, steam-pressure constant, to 110° at 60°, 95° at 50°, 82° at 40°, and 74° at 32°, outside temperature. The accuracy of these figures, however has not been tested by experiment. The following solution of the question is proposed by the author. It gives results quite different from those of Mr. Macgovern, but, like them, lacks ex- perimental confirmation. Let S = sq. ft. of surface of the steam or hot-water radiator; W = sq. ft. of surface of exposed walls, windows, etc.; Ts = temp, of the steam or hot water, T x = temp, of inside of building or room, T = temp, of outside of building or room; a — heat-units transmitted per sq. ft. of surface of radiator per hour per degree of difference of temperature; b — average heat-units transmitted per sq. ft. of walls per hour, per degree of difference of temperature, including allowance for ventilation. It is assumed that within the range of temperatures considered Newton's law of cooling holds good, viz., that it is proportional to the difference of temperature between the two sides of the radiating-surface. bW Then aS(Ts - T x ) = bW{T x - T )._ Let - - = C ; then T x = CKTj. - T ) ; T x = aS Ts + CT . Ts - T x 1 _j_ C ' ~ Ti If T t = 70, and T = 0, C = Ts ~ 7 ° . LetTs = 140°, 213.5°, 308°; Then C = 1, 2.05, 3.4. From these we derive the following: Temperature of Outside Temperatures, T . Steam or Hot - 20° - 10° 0° 10° 20° Water, Ts. Inside Temperatures, T,. 140° 213.5 54.5 62.3 70 75 80 85 90 70 76.7 83.4 90.2 96.9 70 77.7 85.5 93.2 100.9 If eating by Electricity.— If the electric currents are generated oy a dynamo driven by a steam-engine, electric heating will prove very expen- sive, since the steam-engine wastes in the exhaust-steam and by radiation about 90$ of the heat-units supplied to it. In direct steam-heating, with a good boiler and properly covered supply-pipes, we can utilize about 60$ of the total heat value of the fuel. One pound of coal, with a heating value of 13,000 heat-units, would supply to the radiators about 13,000 X -CO = 7800 heat-units. In electric heating, suppose we have a first-class condensing- engine developing 1 H.P. for every 2 lbs. of coal burned per hour. This would be equivalent to 1,980,000 ft.-lbs. -h 778 = 2545 heat-units, or 1272 heat-units for 1 lb. of coal. The friction of the engine and of the dynamo and the loss by electric leakage, and by heat radiation from the conducting wires, might reduce the heat-units delivered as electric current to the elec- tric radiator, and these converted into heat to 50% of this, or only 636 heat- units, or less than one twelfth of that delivered to the steam -radiators in direct steam -heating. Electric heating, therefore, will prove uneconomical unless the electric current is derived from water or wind power, which would otherwise be wasted. (See Electrical Engineering.) WEIGHT OF WATEB. 547 WATER. Expansion of Water.— The following table gives the relative vol- umes of water at different temperatures, compared with its volume at 4° C. according to Kopp, as corrected by Porter. Cent. Fahr. Volume . Cent. Fahr. Volume. Cent, Fahr. Volume. 40 39.1° 1.00000 35° 95° 1.00586 70° 158° 1.02241 5 41 1.00001 40 104 1.00767 75 167 1.02548 10 50 1.00025 45 113 1.00967 80 176 1.02872 15 59 1.00083 50 122 1.01186 85 185 1.03213 20 68 1 .00171 55 131 1.01423 90 194 1.03570 25 77 1.00286 60 140 1.01678 95 203 1.03943 30 86 1.00425 65 149 1.01951 100 212 1.04332 Weight of 1 cu. ft. at c ft. at 212° F. U° F. = 62.4245 lb. -- 1.04332 = E S3, weight of 1 cu. "Weight of Water at Different Temperatures.— The weight of water at maximum density, 39.1°, is generally taken at the figure given by Rankine, 62.425 lbs. per cubic foot. Some authorities give as low as 62.379. The figure 62.5 commonly given is approximate. The highest authoritative figure is 62.425. At 62° F. the figures range from 62.291 to 62.360. The figure 62.355 is generally accepted as the most accurate. At 32° F. figures given by different writers range from 62.379 to 62.418. Clark gives the latter figure, and Hamilton Smith, Jr., (from Rosetti,) gives 62.416. Weight of Water at Temperatures above 212° F.— Porter (Richards' "Steam-engine Indicator,' 1 p. 52) says that nothing is known about the expansion of water above 212°. Applying formulae derived from experiments made at temperatures below 212°, however, the weight and volume above 212° may be calculated, but in the absence of experimental data we are not certain that the formulae hold good at higher temperatures. Thurston, in his " Engine and Boiler Trials," gives a table from which we take the following (neglecting the third decimal place given by him) : £■§ S-S J§.2 •S.2 2 fe KtbD K,Q eg ,• e-Fb'r o3 _• o.gfbi ^-3 sh 43 3 to- ^r 3 43 3 4»£s-o §5-o ■s &s *£% '53 a<2 e=^ £ S.S £5.3 •s&S S S-S '33 £ H £ H £ H F H £ H 212 59.71 280 57.90 350 55.52 420 52.86 490 50.03 220 59.64 290 57.59 360 55.16 430 52.47 500 49.61 230 59.37 300 57.26 370 54.79 440 52.07 510 49.20 240 59.10 310 56.93 380 54.41 450 51.66 520 48.78 250 58.81 320 56.58 390 54.03 460 51.26 530 48.36 260 58 52 330 56.24 400 53.64 470 50.85 540 47.94 270 58.21 310 55.88 410 53.26 480 50.44 550 47.52 Box on Heat gives the following : Temperature F Lbs. per cubic foot. . 212° 59.82 250° 58.85 00° 350° r.42 55.94 400° 450° 500° 600° 54.34 52.70 51.02 47.64 te At 212° figures given by different writers (see Trans. A. S. M. E., xiii. 409) h'ange from 59.56 to 59.845, averaging about 59.77. 548 WATER. Weight of Water per Cubic Foot, from 32° to 212° F., and heal- units per pouud, reckoned above 32° F.: The following table, made by in- terpolating the table given by Clark as calculated from Rankine's formula, with corrections for apparent errors, was published by the author in 1884, Trans. A. S. M. E., vi. 90. (For heat units above 212° see Steam Tables.) If & ftg'ti If cc «S • II S « ■J* P&JD bo 5s o '3 3 2 & §•£ tub bjc£ o '3 '3 1-8 ■-Do «3 JJ23 33p.fi 55 Mean Specific Heat between 32° F. and the given Temp. Cent. Fahr. Cent. Fahr. 0° 32" 0.000 1.0000 120° 218° 217.449 1.0177 1.0067 10 50 18.004 1.0005 1.0002 130 266 235.791 1.0204 1.0076 20 68 36.018 1.0012 1.0005 140 284 254.187 1.0232 1.0087 30 86 54.047 1.0020 1.0009 150 302 272.628 1.0262 1.0097 40 104 72.090 1.0030 1.0013 160 320 291.132 1.0294 1.0109 50 122 90.157 1.0042 1.0017 170 3~" 309.690 1.0328 1.0121 60 140 108.247 1.0056 1.0023 180 35d 328.320 1.0364 1.0133 70 158 126.378 1.0072 1.0030 190 374 347.004 1.0401 1.0146 SO 176 144.508 1.0089 1.0035 200 392 365.760 1.0440 1.0160 90 194 162.686 1.0109 1.0042 210 410 384.588 1.0481 1.0174 100 212 180.900 1.0130 1.0050 220 428 403.-18!- 1.0524 1.0189 110 230 199.152 1.0153. 1.0058 230 446 422.47S- 1.0568 1.0204 Compressibility of Water.— Water is very slightly compressible. Its compressibility is from .000040 to .000051 for one atmosphere, decreasing with increase of temperature. For each foot of pressure distilled water will be diminished in volume .0000015 to .0000013. Water is so incompressible that even at a depth of a mile a cubic foot of water will weigh only about half a pound more than at the surface. THE IMPURITIES OF WATER. (A. E. Hunt and G. H. Clapp, Trans. A. I. M. E. xvii. : 3.) Commercial analyses are made to determine concerning a given water: (1) its applicability for making "3am; (2) its hardness, or the facility with which it will " form a lather" necessary for washing; or (3) its adaptation to other manufacturing purposes. At the Buffalo meeting of the Chemical Section of the A. A. A. S. it was de- cided to report all water analyses in parts per thousand, hundred-thousand, and million. To convert grains per imperial (British) gallons into parts per 100,000, di- vide by 0.7. To convert parts per 100,000 into grains per U. S. gallon, mul- tiply by 7/12 or .583. The most common commercial analysis of water is made to determine its fitness for making steam. Water containing more than 5 parts per 100,000 of free sulphuric or nitric acid is liable to cause serious corrosion, not only of the metal of the boiler itself, but of the pipes, cylinders, pistons, and valves with which the steam comes in contact. The total residue in water used for making steam causes the interior lin- ings of boilers to become coated, and often produces a dangerous hard scale, which prevents the cooling action of the water from protecting the metal against burning. Lime and magnesia bicarbonates in water lose their excess of carbonic acid on boiling, and often, especially when the water contains sulphuric acid, produce, with the other solid residues constantly bein^ formed by the evaporation, a very hard and insoluble scale. A larger amount than 100 parts per 100,000 of total solid residue will ordinarily cause troublesome scale, and should condemn the water for use in steam-boilers, unless a better supply can be obtained. The following is a tabulated form of the causes of trouble with water for steam purposes, and the proposed remedies, given by Prof. L. M. Norton. Causes of Incrustation. 1. Deposition of suspended matter. 2. Deposition of deposed salts from concentration. 3. Deposition of carbonates of lime and magnesia by boiling off carbonic acid, which holds them in solution, 552 4. Deposition of sulphates of lime, because sulphate of lime is but slightly soluble in cold water, less soluble in hot water, insoluble above 270° F. 5. Deposition of magnesia, because magnesium salts decompose at high temperature. 6 Deposition of lime soap, iron soap, etc., formed by saponification of grease. Means for Preventing Incrustation. 1. Filtration. 2. Blowing off. 3. Use of internal collecting apparatus or devices for directing the cir- culation. 4. Heating feed-water. 5. Chemical or other treatment of water in boiler. 6. Introduction of zinc into boiler. 7. Chemical treatment of water outside of boiler. Troublesome Substance. Sediment, mud, clay, etc. Readily soluble salts. Bicarbonates of lime, magnesia, } iron. j Sulphate of lime. Chloride and sulphate of magne- ) sium. f Carbonate of soda in large ) amounts. f Acid (in mine waters). Dissolved carbonic acid and | oxygen. j Grease (from condensed water). Organic matter (sewage). Organic matter. Tabular View. Trouble. Incrustation. Priming. Corrosion. Priming. Corrosion. Remedy or Palliation. Filtration ; blowing off. Blowing off. Heating feed. Addition of caustic soda, lime, or magnesia, etc. j Addition of carb. soda, I barium chloride, etc. j Addition of carbonate of } soda, etc. j Addition of barium chlo- j ride, etc. Alkali. I Heating feed. Addition of •< caustic soda, slacked ( lime, etc. Slacked lime and filtering. Carbonate of soda. Substitute mineral oil. Precipitate with alum or ferric chloride and filter. Ditto. The mineral matters causing the most troublesome boiler-scales are bicar- bonates and sulphates of lime and magnesia, oxides of iron and alumina, and silica. The analyses of some of the most common and troublesome boiler-scales are given in the following table : Analyses of Boiler-scale. (Chandler.) Sul- phate of Lime. Mag- nesia. Silica. Per- oxide of Iron. Water. Car- bonate of Lime. N.Y.C &H.R.Ry.,No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 " " No. 7 " No. 8 " " No. 9 No. 10 74.07 71.37 62.86' 53.05 46.83 30.80 4.95 0.88 4.81 30.07 9.19 "18.95' "3ili7 2.61 2.84 0.65 1.76 2.60 4.79 5.32 7.75 2.07 0.65 2.92 8.24 0.08 1.14 14.78 " 0.92 1.28 12.62 G §S = o - i J ^ o o 2 © 73 o — 5.5 S3 o o. o 3 o ?* 5 O 3 S-c - i. fcH cc U 1-1 O < o no 151 130 80 32 30 25 38 89 •21 70 82 50 119 1.90 95 161 94 01 41 39 48 120 33 81 1.04 68 890 3150 310 •210 219 28 890 590 990 21 38 210 1.90 42 7S0 38 75 70 90 38 23 30 •21 10 640 30 80 13.10 Spring 1 36 a u Allegheny R., near Oil-works Many substances have been added with the idea of causing chemical action which will prevent boiler-scale. As a general rule, these do more harm than good, for a boiler is one of the worst possible places in which to carry on chemical reaction, where it nearly always causes more or less corrosion of the metal, and is liable to cause dangerous explosions. In cases where water containing large amounts of total solid residue is necessarily used, a heavy petroleum oil, free from tar or wax, which is not acted upon by acids or alkalies, not having sufficient wax in it to cause saponification, and which has a vaporizing-point at nearly 600° F., will give the best results in preventing boiler-scale. Its action is to form a thin greasy film over the boiler linings, protecting them largely from the action of acids in the water and greasing the sediment which is formed, thus pre- venting the formation of scale and keeping the solid residue from the evaporation of the water in such a plastic suspended condition that it can be easily ejected from the boiler by the process of " blowing off. 1 ' If the water is not blown off sufficiently often, this sediment forms into a " putty" that will necessitate cleaning the boilers. Any boiler using bad water should be blown off every twelve hours. Hardness of Water.— The hardness of water, or its opposite quality, indicated by the ease with which it will form a lather with soap, depends almost altogether upon the presence of compounds of lime and magnesia. Almost all soaps consist, chemically, of oleate, stearate, and palmitate, of an alkaline base, usually soda and potash. The more lime and magnesia in a sample of water, the more soap a given volume of the water will decompose, so as to give insoluble oleate, palmitate, and stearate of lime and magnesia, and consequently the more soap must be added to a gallon of water in order that the necessary quantity of soap may remain in solution to form the lather. The relative hardness of samples of water is generally expressed in terms of the number of standard soap-measures consumed by a gallon of water in yielding a permanent lather. The standard soap-measure is the quantity required to precipitate one grain of carbonate of lime. It is commonly reckoned that one gallon of pure distilled water takes one soap-measure to produce a lather. Therefore one is deducted from the total number of soap-measures found to be necessary to use to produce a lather in a gallon of water, in reporting the number of soap-measures, or " degrees " of hardness of the water sample. In actually making tests for hardness, the " miniature gallon," or seventy cubic centimetres, is used rather than the inconvenient larger amount. The standard measure is made by completely dissolving ten grammes of pure castile soap (containing 60 per- cent olive-oil) in a litre of weak alcohol (of about 35 per cent alcohol). This yields a solution containing exactly sufficient soap in one cubic centimeter of the solution to precipitate one milligramme of carbonate of lime, or, in other words, the standard soap solution is reduced to terms of the " minia- ture gallon" of water taken. If a water charged with a bicarbonate of lime, magnesia, or iron is boiled, 554 WATER. it will, on the excess of the carbonic acid being expelled, deposit a, consid- erable quantity of the lime, magnesia, or iron, and consequently the water will be softer. The hardness of the water after this deposit of lime, after long boiling, is called the permanent hardness and the difference between it and the total hardness is called temporary hardness. Lime salts in water react immediately on soap-solutions, precipitating the oleate, palmitate, or stearate of lime at once. Magnesia salts, on the con- trary, require some considerable time for reaction. They are, however, more powerful hardeners ; one equivalent of magnesia salts consuming as much soap as one and one-half equivalents of lime. The presence of soda and potash salts softens rather than hardens water. Each grain of carbonate of lime per gallon of water causes an increased expenditure for soap of about 2 ounces per 100 gallons of water. {Eng^g. Neivs, Jan. 31, 1885.) Purifying Feed-water for Steam-boilers.— To effect the purification of water before and after being fed into a boiler, a device man- ufactured by the Albany Steam Trap Company, Albany, N. Y. removes the impurities by the process of a continuous circulation of the water from the boiler, through the filter and back into the boiler, The scale forming impurities that are held in suspension are thus brought in contact with and "arrested" by the filtering agent contained in the filter while under pressure, and at a temperature limited only by that contained in the boiler. It is sometimes desirable, in the removal of the sulphates and carbonates from the feed-water, to heat the water up to nearly the same temperature as it is in the boiler, and then to filter the same before feeding it into the boiler. The operation in a general way is : The water is first forced into the usual exhaust -heater by the feed-pump, and there it is heated by the ex- haust from the engine, say to 200°, and at this temperature it enters the re- heater. The reheater consists of a vertical, cylindrical shell containing a series of water pans or shelves, and so arranged that as the water enters it it delivered into the top pan, and then overflows into the second, and so on down the series to the bottom, and during its transit deposits the scale- forming material. The circulating-pump takes the water from the bottom of the reheater and forces it through the filter on its way into the boiler. Mr. W. B. Coggswell, of the Solvay Process Co.'s Soda Works in Syracuse, N. Y., thus describes the system of purification of boiler feed-water in use at these works (Trans. A. S. M. E., xiii. 255): For purifying, we use a weak soda liquor, containing about 12 to 15 grams Na 2 Co 3 per litre. Say 1^ to 2 M 3 (or 397 to 530 gals.) of this liquor is run into the precipitating tank. Hot water about 60° C. is then turned in, and the reaction of the precipitation goes on while the tank is filling, which re- quires about 15 minutes. When the tank is full the water is filtered through the Hyatt (4), 5 feet diameter, and the Jewell (1), 10 feet diameter, filters in 30 minutes. Forty tanks treated per 24 hours. Charge of water purified at once 35 M 3 , 9,275 gallons. Soda in purifying reagent 15 kgs. Na 2 C0 3 . Soda used per 1,000 gallons 3.5 lbs. A sample is taken from each boiler every other day and tested for deg. Baume, soda and salt. If the deg. B is more than 2, that boiler is blown to reduce it below 2 deg. B. The following are some analyses given by Mr. Coggswell : Lake Water, grams per litre. Mud from Hyatt Filter. Scale from Boiler- tube. Scale found in Pump. .261 .186 .091 .015 .087 .63 3.70 51.24 10.9 Calcium carbonate Magnesium carbonate 63.37 1.11 19.76 25.21 87. Salt, NaCl .14 2.29 1.10 Silica 15.17 3.75 .8 Iron and aluminum oxide. . . 1.2 Total 1.270 87.10 99.74 99.9 FLOW OF WATER. 555 Softening Hard Water for Locomotive Use.— A water-soft- ening plant in operation at Fossil, in Western Wyoming, on the Union Pa- cific Railway, is described in Encfg News, June 9. 1892. It is the invention of Arthur Pennell, of Kansas City. The general plan adopted is to first dis- solve the chemicals in aclo-ed tank, and then connect this to the supply main so that its contents will be forced into the main tank, the supply-pipe being so arranged that thorough mixture ot the solution with the water is ob- tained. A waste-pipe from the bottom of the tank is opened from time to time to draw off the precipitate. The pipe leading to the tender is arranged to draw the water from near the surface. A water-tank 24 feet in diameter and 16 feet high will contain about 46,600 gallons of water. About three hours should be allowed for this amount of water to pass through the tank to insure thorough precipitation, giving a permissible consumption of about 15,000 gallons per hour. Should more than this be required, auxiliary settling-tanks should be provided. The chemicals added to precipitate the scale-forming impurities are so- dium carbonate and quicklime, varying in proportions according to the rela- tive proportions of sulphates and carbonates in the water to be treated. Sufficient sodium carbonate is added to produce just enough sodium sulphate to combine with the remaining lime and magnesia sulphate and produce glauberite or its corresponding magnesia salt, thereby to get rid of the sodium sulphate, which produces foaming, if allowed to accumulate. HYDRAULICS-FLOW OP WATER. Formulae for Discharge of Water though Orifices and Weirs.— For rectangular or circular orifices, with the head measured from centre of the orifice to the surface of the still water in the feeding reservoir. Q= CVtoJHX a (1) For weirs with no allowance for increased head due to velocity of approach: Q=C%+2gHxLH . . . (2) For rectangular and circular or other shaped vertical or inclined orifices; formula based on the proposition that each successive horizontal layer of water passing through the orifice has a velocity due to its respective head: Q = cL% \2y X ( \Hb 3 - VHt*) (3) For rectangular vertical weirs: Q = c%\2gHxLh (4) Q — quantity of water discharged in cubic feet per second; C = approxi- mate coefficient for formulas (1) and (2) ; c — correct coefficient for (3) and (4). Values of the coefficients c and Care given below. g = 32.16; V2g = 8.02; H — head in feet measured from centre of orifice to level of still water; Hb = head measured from bottom of orifice; Ht = head measured from top of orifice; h = H, corrected for velocity of ap- 4 Va? proach, Va, — H-\-- - — ; a = area in square feet; L = length in feet. Flow of Water from Orifices.— The theoretical velocity of water flowing from an orifice is the same as the velocity of a falling body which has fallen from a height equal to the head of water, — \'2gH. The actual velocity at the smaller section of the vena contractu is substantially the same as the theoretical, but the velocity at the plane of the orifice is C \'2gH, in which the coefficient C has the nearly constant value of .62. ^The smallest diameter of the vena contractu is therefore about .79 of that of the orifice. If C be the approximate coefficient = .62, and c the correct coeffi- 556 HYDBATTLICS. cient, the ratio - varies with different ratios of the head to the diameter c H 10. 1. For vertical rectangular orifices of ratio of head to width W : .6 .8 .1 1.5 2. 3. 4. 5. 8. 9657 .9823 .9890 .9953 .9974 .9988 .9993 .9996 .999* c For H -¥- D or H -h W over 8, C — c, practically. o Weisbacb gives the following values of c for circular orifices in a thin wall. H — measured head from centre of orifice. D ft H ft. .066 .33 .82 2.0 3.0 45. 340. .033 .066 .10 .13 .711 .665 .637 .629 .622 .614 .628 .621 .614 .607 .641 .632 .600 For an orifice of D = . effective head in feet, 5 ft. and a well-rounded mouthpiece, H being the H = . : .959 l.C 11.5 56 .994 338 .994 Hamilton Smith, Jr., found that for great heads, 312 ft. to 336 ft., with con- verging mouthpieces, c has a value of about one. and for small circular orifices in thin plates, with full contraction, c — about .60. Some of Mr. Smith's experimental values of c for orifices in thin plates discharging into air are as follows. All dimensions in feet. Circular, in steel, D — .«*>. f^Z .739 .6495 2.43 .6298 3.19 .6264 Circular, in brass, D = .050,{ ^ = .185 .6525 .536 .6265 1.74 .6113 2.73 .6070 3.57 4.63 .6060 .6051 Circular, in brass, D = ■m\ H c z .129 .6337 .457 .6155 .900 .6096 1.73 .6042 2.05 3.18 .6038 .6025 Circular, in iron, D = •™'\ H c = .80 .6061 1.81 .6041 2.81 .6033 4.68 .6026 Square, in brass, .05 x ^m .313 .6410 .877 .6238 1.79 .6157 2.81 .6127 3.70 4.63 .6113 .6097 Square, in brass, .10 X •io, K = .181 .6292 .939 .6139 1.71 ..6084 2.75 .6076 3.74 4.59 .6060 .6065 Rectangular, in brass, j H = .261 .917 1.82 2.83 3.75 4.70 Z= .300, W = .050.. . ...1 c = .6476 .6280 .6203 .6180 .6176 .6168 For the rectangular orifice, L, the length, is horizontal. Mr. Smith, as the result of the collation of much experimental data of others as well as his own, gives tables of the value of c for vertical orifices, with full contraction, with a free discharge into the air, with the inner face of the plate, in which the orifice is pierced, plane, and with sharp inner corners, so that the escaping vein only touches these inner edges. These tables are abridged below. The coefficient c is to be used in the formulas (3) and (4) above. For formulas (1) and (2) use the coefficient found from the C values of the ratios — above. HYDRAULIC FORMULAE. 557 Values of Coefficient e for Vertical Orifices with Sharp Edges, Full Contraction, and Free Discharge into Air. (.Hamilton Smith, Jr.) ^6 1.0 3.0 6.0 10. 20. 100. (?) Square Orifices. Length of the Side of the Square, in feet. .03 .04 .643 .05 .637 .07 .628 .10 .621 .12 .616 .15 .20 .40 .60 .80 ,611 .645 .636 .630 .623 .617 .613 .610 .605 .601 .598 .596 .636 .628 .622 .618 .613 .610 .608 .605 603 .601 .600 .622 .616 .612 .609 .607 .606 .606 , 605 .605 .604 ,603 .616 .612 .609 .607 .605 .605 .605 .604 .604 .603 .602 .611 608 .606 .605 .604 .604 .603 .603 .603 .602 602 .605 .604 .603 .602 .602 .602 602 602 601 601 .601 .598 .598 .598 .598 .598 .598 .598 .598 598 .598 .598 Circular Orifices. Diameters, in feet. 10. 20. 50.(?) 100.(?) .637 .624 .617 .610 .605! .604' .601 .598; .5951 .592 07 .10 .12 .15 .20 .40 .60 .80 62^ .618 .612 .606 CIS .613 .609 .605 .601 .596 .593 .590 612 .608 .605 .603 .600 .598 .595 .593 ho; .604 .601 .600 .599 .599 .597 .596 603 .602 .600 .599 .599 .598 .597 .597 .60(1 .599 .599 .598 .598 .597 .596 599 .598 .598 .597 .597 .597 .596 .596 597 .596 .596 .596 596 .596 .596 .595 594 .594 .594 .594 .594 .594 .594 .593 592 ■ 502 .592 .592 .592 .592 .592 .592 .591 .595 .596 .596 .595 .594 .593 .592 HTDRAFLIC FORMULJE.-FLOW OF WATER IN OPEN AND CLOSED CHANNELS. Flow of Water in Pipes.— The quantity of water discharged through a pipe depends on the •'head;' 1 that is, the vertical distance be- tween the level surface of still water in the chamber at the entrance end of the pipe and the level of the centre of the discharge end of the pipe ; also upon the length of the pipe, upon the character of its interior surface as to smoothness, and upon the number and sharpness of the bends: but it is independent of the position of the pipe, as horizontal, or inclined upwards or downwards. The head, instead of being an actual distance between levels, may be caused by pressure, as by a pump, in which case the head is calculated as a vertical distance corresponding to the pressure 1 lb. per sq. in. = 2.309 ft. head, or 1 ft. head = .433 lb. per sq. in. The total head operating to cause flow is divided into three parts: 1. The velocity-head, which is the height through which a body must fall in vacuo to acquire the velocity with which the water flows into the pipe = v 2 h- 2g, in which v is the velocity in ft. per sec. and 2g = 64.32; 2. the entry-head, that required to overcome the resistance to entrance to the pipe. With sharp- edged entrance the entry-head = about V^ the velocity-head; with smooth rounded entrance the entry-head is inappreciable; 3. the friction-head, due to the frictional resistance to fl >w within the pipe. In ordinary cases of pipes of considerable length the sum of the entry and velocity heads required scarcely exceeds 1 foot. In the case of Ions: pipes with low heads the sum of the velocity and entry heads is generally so small that it may be neglected. General Formula for Flow of Water in Pipes or Co nduits. Mean velocity in ft. per sec. = c \ n ieau hydraulic ra dius X slope „ „ . „ ,, /diameter Do. for pipes running full = ca/ x slope, in which c is a coefficient determined by experiment. (See pages 559-564.) 558 HYDRAULICS. m , ■' ' :.. ,. £.rea of wet cross-section The mean hydraulic radius = : . wet perimeter. In pipes running full, or exactly half full, and in semicircular open chan- nels running full it is equal to 34 diameter. The slope = the head (or pressure expressed as a head, in feet) -=- length of pipe measured in a straight line from end to end. In open channels the slope is the actual slope of the surface, or its fall per unit of length, or the sine of the angle of the slope with the horizon. If r = mean hydraulic radius, s = slope = head -h lengthy = velocity in feet per second (all dimensions in feet), v = c \'r \/s = c )/rs. Quantity of Water Discharged. -If Q = discharge in cubic feet per second and a — area of channel, Q = av — uc Vrs< a V'r is approximately proportional to the discharge. It is a maximum at 308°, corresponding to 19/20 of the diameter, and the flow of a conduit 19/20 full is about 5 per cent greater than that of one completely filled. Table giving Fall in Feet per Mile, the Distance on Slope corresponding to a Fall of 1 Ft., and also_the Values of s and V* for Use in the Formula v — c \ rs. s = H-^-L— sine of angle of slope = fall of water-surface (H), in any dis- tance (L), divided by that distance. Fall in Slope, Sine of Fall in Slope, Sine of Feet 1 Foot Slope, Vs. Feet 1 Foot Slope, Vs. per Mi. in s. per Mi. in s. 0.25 21120 .0000473 .006881 17 310.6 .0032197 .056742 .30 17600 .0000568 .007538 18 293.3 .0034091 .058S88 .40 13200 .0000758 .008704 19 277.9 .0035985 .059988 .50 10560 .0000947 .009731 20 264 .0037879 .061546 .60 8800 .0001136 .010660 22 240 .0041667 .064549 .702 7520 .0001330 .011532 24 220 .0045455 .067419 .805 6560 .0001524 .012347 26 203.1 .0049242 .070173 .904 5840 .0001712 .013085 28 188.6 .0053030 .072822 1. 5280 .0001894 .013762 30 176 .0056818 .075378 1.25 4224 .0002367 .015386 35.20 150 .0066667 .081650 1.5 3520 .0002841 .016854 40 132 .0075758 .087039 1.75 3017 .0003314 .018205 44 120 .0083333 .091287 2. 2640 .0003788 .019463 48 110 .0090909 .095346 2.25 2347 .0004261 .020641 52.8 100 .010 .1 2.5 2112 .0004735 .021760 60 88 .0113636 .1066 2.75 1920 .0005208 .022822 66 80 .0125 .111803 3. 1760 .0005682 .023837 70.4 75 .0133333 .115470 3.25 1625 .0006154 .024807 80 66 .0151515 .123091 3.5 1508 .0006631 .025751 88 60 .0166667 .1291 3.75 1408 .0007102 .026650 96 55 .0181818 .134839 4 1320 .0007576 .027524 105.6 50 .02 .141421 5 1056 .0009470 .030773 120 44 .0227273 .150756 6 880 .0011364 .03371 132 40 .025 .158114 7 754.3 .0013257 .036416 160 33 .0303030 .174077 8 660 .0015152 .038925 220 24 .0416667 .204124 9 586.6 .0017044 .041286 264 20 .05 .223607 10 528 .0018939 .043519 330 16 .0625 .25 11 443.6 .0020833 .045643 440 12 .0833333 .288675 12 440 .0022727 .047673 528 10 .1 .316228 13 406.1 .0024621 .04962 660 8 .125 .353553 14 377.1 .0026515 .051493 880 6 .1666667 .408248 15 352 .0028409 .0533 1056 5 .2 .447214 16 330 .0030303 .055048 1320 4 !25 .5 HYDRAULIC FORMULAE. 559 Values of yr for Circular Pipes, Sewers, and Conduits of different Diameters. r = mean hydraulic depth = : — = 14 diam. for circular pipes run- ■ 7 * perimeter * * ning full or exactly half full. Diam. . \ r Y Di Mil., \'r Diam., Yr Diam., in Feet. ft. in. in Feet. i't. in. in Feet. ft. in. in Feet. ft, 111. ¥s .088 2 .707 4 6 1.061 9 1.500 y* .102 2 1 .722 4 7 1.070 9 3 1.521 % .125 2 2 .736 4 8 1.0S0 9 6 1.541 .144 2 3 .750 4 9 1.089 9 9 1.561 m .161 2 4 .764 4 10 1.099 10 1.581 V4 .177 2 5 .777 4 11 1.109 10 3 1.601 m •191 2 6 .790 5 1.118 10 6 1.620 2 .204 2 7 .804 5 1 1.127 10 9 1.639 %\& .228 2 8 .817 5 2 1.137 11 1.658 3 .251 2 9 .829 5 3 1.146 11 3 1.677 4 .290 2 10 .842 5 4 1.155 11 6 1.696 5 .323 2 11 .854 5 5 1.164 11 9 1.714 6 .354 3 .866 5 6 1.173 12 1.732 7 .382 3 1 .878 5 7 1.181 12 3 1 .750 8 .408 3 2 .890 5 8 1.190 12 6 1.768 9 .433 3 3 .901 5 9 1.199 12 9 1.785 10 .456 3 4 .913 5 10 1.208 13 1.083 11 .479 3 5 .924 5 11 1.216 13 3 1.820 1 .500 3 6 .935 6 1.225 13 6 1.837 1 1 .520 3 7 .946 6 3 1.250 14 1.871 1 2 .540 3 8 .957 6 6 1.275 14 6 1.904 1 3 .559 3 9 .968 6 9 1.299 15 1.936 1 4 .577 3 10 .979 7 1.323 15 6 1.968 1 5 .595 3 11 .990 7 3 1.346 16 2. 1 6 .612 4 1. 7 6 1.369 16 6 2.031 1 7 .629 4 1 1.010 7 9 1.392 17 2.061 1 8 .646 4 2 1.021 S 1.414 17 6 2.091 1 9 .661 4 3 1.031 8 3 1.436 18 2.121 1 10 .677 4 4 1.041 8 6 1.458 19 2.180 1 11 .692 4 5 1.051 8 9 1.479 20 2.236 Values of the Coefficient c. (Chiefly condensed from P. J. Flynn on Flow of Water.)— Almost all the old hydraulic formulae for finding 'the mean velocity in open and closed channels have constant coefficients, and are therefore correct for only a small range of channels. They have often been found to give incorrect results with disastrous effects. Ganguillet and Kut- ter thoroughly investigated the American, French, and other experiments, and they gave as the result of their labors the formula now generally known as Kutter's formula. There are so many varying conditions affecting the flow of water, that all hydraulic formulas are only approximations to the correct result. When the surface-slope measurement is good, Kutter's formula will give results seldom exceeding 7^4% error, provided the rugosity coefficient of the formula is known for the site. For small open channels D'Arcy's and Bazin's formulas, and for cast-iron pipes D'Arcy's formulas, are generally accepted as being approximately correct. Kutter's Formula for measures in feet is f I 1.811 + 4U .00281 i'+O X Vrs, s y " Yr i which v — mean velocity in feet per second ; r • hydraulic mean 560 HYDRAULICS. depth in feet = area of cross-section in square feet divided by wetted perim- eter in lineal feet ; s = fall of water-surface (h) in any distance (I) divided by that distance, = -, == sine of slope ; n — the coefficient of rugosity, de- pending on the nature of the lining or surface of the channel. If we let the first term of the right-hand_side of the equation equal c, we have Chezy's formula, v = c Yrs = c X Vr X Vs- Values ofn in Kutter's Formula.— The accuracy of Kutter's for- mula depends, in a great measure, on the proper selection of the coefficient of roughness n. Experience is required in order to give the right value to this coefficient, and to this end great assistance can be obtained, in making this selection, by consulting and comparing the results obtained from ex- periments on the flow of water already made in different channels. In some cases it would be well to provide for the contingency of future deterioration of channel, by selecting a high value of n, as, for instance, where a dense growth of weeds is likely to occur in small channels, and also where channels are likely not to be kept in a state of good repair. The following table, giving the value of n for different materials, is com- piled from Kutter, Jackson, and Hering, and this value of n applies also in each instance, to the surfaces of other materials equally rough. Value of n in Kutter's Formula for Different Channels. n — .009, well-planed timber, in perfect order and alignment ; otherwise, perhaps .01 would be suitable. n = .010, plaster in pure cement ; planed timber ; glazed, coated, or en- amelled stoneware and iron pipes ; glazed surfaces of every sort in perfect order. n = .011, plaster in cement with one third sand, in good condition ; also for iron, cement, and terra-cotta pipes, well joined, and in best order. n = .012, unplaned timber, when perfectly continuous on the inside ; flumes. n — .013, ashlar and well-laid brickwork ; ordinary metal ; earthen and stoneware pipe in good condition, but not new ; cement and terra-cotta pipe not well jointed nor in perfect order , plaster and planed wood in imperfect or inferior condition ; and, generally, the materials mentioned with n — .010, when in imperfect or inferior condition. n — .015, second class or rough-faced brickwork ; well-dressed stonework ; foul and slightly tuberculated iron ; cement and terra-cotta pipes, with im- perfect joints and in bad order ; and canvas lining on wooden frames. n = .017, brickwork, ashlar, and stoneware in an inferior condition ; tu- berculated iron pipes ; rubble in cement or plaster in good order ; fine gravel, well rammed, y% to % inch diameter ; and, generally, the materials men- tioned with n = .013 when in bad order and condition. n — .020, rubble in cement in an inferior condition ; coarse rubble, rough set in a normal condition ; coarse rubble set dry ; ruined brickwork and masonry ; coarse gravel well rammed, from 1 to l£g inch diameter ; canals with beds and banks of very firm, regular gravel, carefully trimmed and rammed in defective places ; rough rubble with bed partially covered with silt and mud ; rectangular wooden troughs, with battens on the inside two inches apart ; trimmed earth in perfect order. it = .0225, canals in earth above the average in order and regimen. n — .025. canals and rivers in earth of tolerably uniform cross-section ; slope and direction, in moderately good order and regimen, and free from stones and weeds. n — .0^75, canals and rivers in earth below the average in order and regi- men. n = .030, canals and rivers in earth in rather bad order and regimen, hav- ing stones and weeds occasionally, and obstructed by detritus. n = .035, suitable for rivers and canals with earthen beds in bad order and regimen, and having stones and weeds in great quantities. n — .05, torrents encumbered with detritus. Kutter's formula has the advantage of being easily adapted to a change in the surface of the pipe exposed to the flow of water, by a change in the value of n. For cast-iron pipes it is usual to use n = .013 to provide for the future deterioration of the surface. Reducing Kutter's formula to the form v = cX Vr X Vs, and taking n, the coefficient of roughness in the formula = .011, .012, and .013, and s = .001, we have the following values of the coefficient c for different diameters cf conduit. HYDRAULIC FORMULAE. 561. Values of c in Formula v - c x \ r x \ s for Metal Pipes and Moderately Smooth Conduits Generally. By Kutter's Formula, (s = .001 or greater.) Diameter. n = .011 n = .012 ft = .013 Diameter. n = .011 ft = .012 ft = .013 ft. in. c = c = c = ft. c = c = c = 1 2 4 47.1 61.5 77.4 7 8 9 152.7 155.4 157.7 139.2 141.9 144.1 127.9 130.4 132.7 6 87.4 77.5 69.5 10 159.7 146 134.5 1 105.7 94.6 85.3 11 161.5 147.8 136.2 1 6 116.1 104.3 94.4 12 163 149.3 137.7 2 123.6 111.3 101.1 14 165.8 152 140.4 3 133.6 120.8 110.1 16 168 154.2 142.1 4 140.4 127.4 116.5 18 169.9 156.1 144.4 5 145.4 132.3 121.1 20 171.6 157.7 146 6 149.4 136.1 124.8 For circular pipes the hydraulic mean depth r equals *4 of tlie diameter. According to Kutter's formula the value of c, the coefficient of discharge, is the same for all slopes greater than 1 in 1000; that is, within these limits c is constant. We further find that up to a slope of 1 in 2640 the value of c is, for all practical purposes, constant, and even up to a slope of 1 in 5000 the difference in the value of c is very little. This is exemplified in the following : Value of c for Different Values of \'r and s in Kutter's Formula, with nj= .013. -- c V r X Ys Slopes. Vr 1 in 1000 1 in 2500 1 in 3333.3 1 in 5000 1 in 10,000 .6 1 2 93.6 116.5 142.6 91.5 115.2 142.8 90.4 114.4 143.0 88.4 113.2 143.1 83.3 109.7 143.8 The reliability of the values of the coefficient of Kutter's formula for pipes of less than 6 in. diameter is considered doubtful. (See note under table on page 564.) Values of c for Earthen Channels, by Kutter's Formula, for Use in Formula v — c Vrs. Coefficient of Roughness, Coefficient of Roughness, ft = .0225. ft = .035. Vr in feet. Yr in feet. 0.4 1.0 1.8 2.5 4.0 0.4 1.0 1.8 2.5 4.0 Slope, 1 in c c c c c c c c c c 1000 35.7 62.5 80.3 89.2 99.9 19.7 37.6 51.6 59.3 69.2 1250 35 5 62.3 80.3 89.3 100.2 19.6 37.6 51.6 59.4 69.4 1667 35.2 62.1 80.3 89.5 100 6 19.4 37.4 51.6 59.5 £9.8 2500 34.6 61.7 80.3 89.8 101.4 19.1 37.1 51.6 59.7 70.4 3333 34. 61.2 80.3 90.1 102.2 18.8 36.9 51.6 59.9 71.0 5000 33. 60.5 80.3 90.7 103.7 18.3 36.4 51.6 60.4 72.2 7500 31.6 59.4 80.3 91.5 106.0 17.6 35.8 51.6 60.9 73.9 10000 30.5 58.5 80.3 92.3 107.9 17.1 35.3 51.6 60.5 75.4 15840 28.5 56.7 80.2 93.9 112.2 16.2 34.3 51.6 62.5 78.6 20000 27.4 55.7 80.2 J 94.8 115.0 15.6 33.8 51.5 63.1 80.6 562 HYDRAULICS. Mr. Molesworth, in the 22d edition of his " Pocket-book of Engineering Formulae," gives a modification of Kutter's formula as follows : For flow in cast-iron pipes, v = c )/rs, in which 'l + ^+^l in which d — diameter of the pipe in feet. (This formula was given incorrectly in Molesworth's 21st edition.) Molesworth's Formula.- v = Vkrs, in which the values of k are as follows : Nature of Channel. Values of k for Velocities. Less than 4 ft. per sec. More than 4 ft. per sec. 8800 7200 6400 5300 8500 Earth 6800 5900 Rough, with bowlders 4700 In very large channels, rivers, etc., the description of the channel affects the result so slightly that it may be practically neglected, and k assumed = from 8500 to 9000. Flynn's Formula.— Mr. Flynn obtains the following expression of the value of Kutter's coefficient for a slope of .001 and a value of n = .013 : 183.72 - (44.41 5 The following table shows the close agreement of the values of c obtained from Kutter's, Molesworth's, and Flynn's formulas : Diameter. Slope. Kutter. Molesworth. Flynn. 6 inches lin 40 71.50 71.48 69.5 6 inches 1 in 1000 69.50 69.79 69.5 4 feet 1 in 400 117. 117. 116.5 4 feet 1 in 1000 116.5 116.55 116.5 8 feet 1 in 700 130.5 130.68 130.5 8 feet 1 in 2600 129.8 129.93 130.5 Mr. Flynn gives another simplified form of Kutter's formula for use with different values of n as follows : A 1+ H x i-))" In the following table the value of Kis given for the several values of n : n K n K n K n K n K .009 .010 .011 245.63 225.51 209.05 .012 .013 .014 195.33 183.72 137.77 .015 .016 .017 165.14 157.6 150.94 .018 .019 .020 145.03 139.73 184.96 .021 .022 .0225 130.65 126.73 124.9 If in the application of Mr. Flynn's formula given above within the limits of n as given in the table, we substitute for n, K, and Vr their values, we have a simplified form of Kutter's formula. HYDRAULIC FORMULA. 563 For instance, when n = .011, and d = 3 feet, we have >9.05 Vrs. / -01l\ ' Bazin's Formulae : For very even surfaces, fine plastered sides and bed, planed planks, etc., v =|/l -*- .0000045(l0.16 -f~) X Vn For even surfaces such as cut-stone, brickwork, unplaned planking, mortar, etc. : v = i/l -*- .000013(4.354 + i) x Vrs. For slightly uneven surfaces, such as rubble masonry : v = i/l -*- .00006(l.219 + i) x VrsI For uneven surfaces, such as earth : v = i/l -4- .00035(0.2438 -f -) X Vrs. A modification of Bazin's formula, known as D'Arcy's Bazin's : - • / 1 000"s V ~ ? \ .08534r -f 0.35* For small channels of less than 20 feet bed Bazin's formula for earthen channels in good order gives very fair results, but Kutter's formula is super- seding it in almost all countries where its accuracy has been investigated. The last table on p. 561 shows the value of c, in Kutter's formula, for a wide range of channels in earth, that will cover anything likely to occur in the ordinary practice of an engineer. D'Arcy's Formula for clean iron pipes under pressure is _ ( rs ) V -! ™™,.,™ , .00000162 1 j .00007726 -\ f Flynn's modification of D'Arcy's formula is / 155256 \^ .j— in which d = diameter in feet. D'Arcy's formula, as given by J. B. Francis, C.E., for old cast-iron pipe lined with deposit and under pressure, is v = ( U4d* s \y 2 V.0082(12d + 1 V . . • Flynn's modification of D'Arcy's formula for old cast-iron pipe is / 70243.9 \ A/ — 564 HYDRAULICS. For Pipes Less _than 5 inches in Diameter, coefficients (c) in the formula v — c Vrs, from the formula of D'Arcy, Kutter, and Fanning. Diam. in inches. D'Arcy, for Clean Pipes. Kutter, for n = .011 s = .001 Fanning, for Clean Iron Pipes Diam. in inches D'Arcy, for Clean Pipes. Kutter, for n = .011 s = .001 Fanning, for Clean Iron Pipes. % H m 59.4 65.? 74.5 80.4 84.8 88.1 32. 36.1 42.6 47.4 51.9 55.4 80.4 88. ¥ 4 5 90.7 92.9 96.1 98.5 101.7 103.8 58.8 61.5 66. 70.1 77.4 82.9 92.5 94.8 96.6 103.4 Mr. Flynn, in giving the above table, says that the facts show that the co- efficients diminish from a diameter of 5 inches to smaller diameters, and it is a safer plan to adopt coefficients varying with the diameter than a con- stant coefficient. No opinion is advanced as to what coefficients should be used with Kutter's formula for small diameters. The facts are simply stated, giving the results of well-known authors. Older Formulae,— The following are a few of the many formulae for flow of water in pipes given by earlier writers. As they have constant coef- ficients, they are not considered as reliable as the newer formulas. Prony, v = 97 Vrs - .08; Eytelwein, v = 50 a/- .^ or v = 108 Vrs - 0.13 ; y I -\- 50a — 3 /— Neville, v = 140 Vrs - 11 yrs. Hawksley, Wd 54cT : head of water in feet; I = r — mean hydraulic depth, In these formulae d = diameter in feet; h = length of pipe in feet; s = sine of slope = y; == area -v- wet perimeter = — for circular pipe. Mr. Santo Crimp (Eug'g, August 4, 1893) states that observations on flow in brick sewers show that the actual discharge is 33% greater than that cal- culated by Eytelwein's formula. He thinks Kutter's formula not superior to D'Arcy's for brick sewers, the usual coefficient of roughness in the former, viz., .013, being too low for large sewers and far too small in the case of small sewers. D'Arcy's formula for brickwork is v = ^rs ; m = a(\ + — ) ; a = .0037285; B = .229663. m \ r ' VELOCITY OF WATER IN OPEN CHANNELS. Irrigation Canals.— The minimum mean velocity required to prevent the deposit of silt or the growth of aquatic plants is in Northern India taken at \y» feet per second. It is stated that in America a higher velocity is required for this purpose, and it varies from 2 to 3J^ feet per second. The maximum allowable velocity will vary with the nature of the soil of the bed. A sandy bed will be disturbed if the velocity exceeds 3 feet per second. Good loam with not too much sand will bear a velocity of 4 feet per second. The Cavour Canal in Italy, over a gravel bed, has a velocity of about 5 per second. ( Flynn 's "Irrigation Canals.") Mean Surface and Bottom Velocities.— According to the for- mula of Bazin, = Umax — 25.4 Vt'i -- vb + 10.87 Vrs. VELOCITY OF WATER IN OPEN CHANNELS. 565 .-•. vb ~ v — 10.87 \ ,r rs, in which v = mean velocity in feet per second, vmak - maximum surface velocity in feet per second, vb = bottom velocity in feet per second, r = hydraulic mean depth in feet = area of cross-section in square feet divided by wetted perimeter in feet, s = sine of slope. The least velocity, or that of the particles in contact with the bed, is almost as much less than the mean velocity as the greatest velocity is greater than the mean. Rankine states that in ordinary cases the velocities may be taken as bear- ing to each other nearly the proportions of 3, 4, and 5. In very slow cur- rents they are nearly as 2, 3, and 4. Safe Bottom and Mean Velocities.— Ganguillet & Kutter give the following table of safe bottom and mean velocity in channels, calculated from the formula v = vb -\- 10.87 \'rs' Material of Channel. Soft brown earth Soft loam Sand Gravel Pebbles Broken stone, flint Conglomerate, soft slate. Stratified rock Hard rock Safe Bottom Veloc Mean Velocity v, ity vb, in feet in feet per second. per second. 0.249 0.328 0.499 0.656 1.000 1.312 1.998 2.625 2.999 3.938 4.003 5.579 4.988 6.564 6.006 8.204 10.009 13.127 Ganguillet & Kutter state that they are unable for want of observations to judge how far these figures are trustworthy. They consider them to be rather disproportionately small than too large, and therefore recommend them more confidently. Water flowing at a high velocity and carrying large quanties of silt is very destructive to channels, even when constructed of the best masonry. Resistance of Soils to Erosion by Water.— W. A. Burr, Eng'g Nexus, Feb. 8, 1894, gives a diagram showing the resistance of various soils to erosion by flowing water. Experiments show that a velocity greater than 1.1 feet per. second will erode sand, while pure clay will stand a velocity of 7.35 feet per second. The greater the proportion of clay carried by any soil, the higher the per- missible velocity. Mr. Burr states that experiments have shown that the line describing the power of soils to resist erosion is parabolic. From his dia- gram the following figures are selected representing different classes of soils: Pure sand resists erosion by flow of 1.1 feet per second. Sandy soil, 15$ clay 1.2 " " Sandy loam, 40$ clay 1.8 " " Loamy soil, 65$ clay 3.0 " " Clay loam, 85$ clay 4.8 " " Agricultural clay, 95$ clay 6.2 " " Clay 7.35 " Abrading and Transporting Power of Water.— Prof. J. LeConte, in his "Elements of Geology," states : The erosive power of water, or its power of overcoming cohesion, varies as the square of the velocity of the current. The transporting power of a current varies as the sixth power of the ve- locity. * * * If the velocity therefore be increased ten times, the transport- ing power is increased 1,000,000 times. A current running three feet per second, or about two miles per hour, will bear fragments of stone of the size of a hen's egg, or about three ounces weight. A current of ten miles an hour will bear fragments of one and a half tons, and a torrent of twenty miles an hour will carry fragments of 100 tons. The transporting power of water must not be confounded with its erosive power. The resistance to be overcome in the one case is weight, in the other, cohesion ; the latter varies as the square : the former as the sixth power of the velocity. In many cases of removal of slightly cohering material, the resistance is a 566 HYDRAULICS. mixture of these two resistances, and the power of removing material will vary at some rate between i> 2 and v*. Baldwin Latham has found that in order to prevent deposits of sewage silt in small sewers or drains, such as those from 6 inches to 9 inches diameter, a mean velocity of not less than 3 feet per second should be produced. Sewers from 12 to 24. inches diameter should have a velocity of not less than 2% feet per second, and in sewers of larger dimensions in no case should the velocity be less than 2 feet per second. The specific gravity of the materials has a marked effect upon the mean velocities necessary to move them. T. E. Blackwell found that coal of a sp. gr. of 1.26 was moved by a current of from 1.25 to 1.50 ft. per second, while stones of a sp. gr. of 2 32 to 3.00 required a velocity of 2.5 to 2.75 ft. per second. Chailly gives the following formula for finding the velocity required to move rounded stones or shingle : v = 5.67 Vag, in which v = velocity of water in feet per second, a = average diameter in feet of the body to be moved, g = its specific gravity. Geo. Y. Wisner, Eng'g News, Jan 10, 1895, doubts the general accuracy of statements made by many authorities concerning the rate of flow of a cur- rent and the size of particles which different velocities will move. He says: The scouring action of any river, for any given rate of current, must be an inverse function of the depth. The fact that some engineer has found that a given velocity of current on some stream of unknown depth will move sand or gravel has no bearing whatever on what may be expected of cur- rents of the same velocity in streams of greater depths. In channels 3 to 5 ft. deep a mean velocity of 3 to 5 ft. per second may produce rapid scouring, while in depths of 18 ft. and upwards current velocities of 6 to 8 ft. per second often have no effect whatever on the channel bed. Grade of Sewers.— The following empirical formula is given in Bau- meister's " Cleaning and Sewerage of Cities," for the minimum grade for a sewer of clear diameter equal to d inches, and either circular or oval in section : Minimum grade, in per cent, = , . As the lowest limit of grades which can be flushed, 0.1 to 0.2 per cent may be assumed for sewers which are sometimes dry, while 0.3 per cent is allow- able for the trunk sewers in large cities. The sewers should run dry as rarely as possible. Relation of Diameter of Pipe to Quantity Discharged.— In many cases which arise iu practice the information sought is the diame- ter necessary to supply a given quautity of water under a given head. The diameter is commonly taken to vary as the two-fifth power of the dis- charge. This is almost certainly too large. Hagen's formula, with Prof. f O \ 387 Unwin's coefficients, give d = el j^ V , where c = .239 when d and Q \Q)> are in feet and cubic feet per second. Mr. Thrupp has proposed a formula which makes d vary as the .383 power of the discharge, and the formula of M. Vallot, a French engineer, makes d vary as the .375 power of the discharge. (Engineering.) FliOW OF WATER-EXPERIMENTS AND TABLES. The Flow of Water through New Cast-iron Pipe was recently measured by S. Bent Russell, of the St. Louis, Mo., Water-works. The pipe was 12 inches in diameter, 1631 feet long, and laid on a uniform grade from end to end. Under an average total head of 3.36 feet the flow was 43,200 cubic feet in seven hours; under an average head of 3.37 feet the flow was the same; under an average total head of 3.41 feet the flow was 46,700 cubic feet in 8 hours and 35 minutes. Making allowance for loss of head due to entrance and to curves, it was found that the value of c in the formula v = c \/rs was from 88 to 93 (Eng'g Record, April 14, 1894. Flow of Water in a 20-inch Pipe 75,000 Feet Long.— A comparison of experimental data with calculations by different formulae is FLOW OF WATER — EXPERIMENTS AttD TABLES. 567 given by Chas. B. Brush, Trans. A. S. C. E., 1888. The pipe experimented with was that supplying the city of Hoboken, N. J. Results Obtained by the Hackensack Water Company, from 1882-1887, in Pumping Through a 20-in. Cast-iron Main 75,000 Feet Long. Pressure in lbs. per sq. in. at pumping-station: 95 100 105 110 115 120 125 130 Total effective head in feet : 55 66 77 89 100 112 123 135 Discharge in U. S. gallons in 24 hours, 1 = 1000 : 2,848 3,165 3,354 3,566 3,804 I Actuai velocity in main in feet per second : 2.00 2.24 2.36 2.52 2.68 4,116 4,255 2 92 3.00 Cost of coal consumed in delivering each million gals, at given velocities : $8.40 $8.15 $8.00 $8.10 $8.30 $8.60 $9.00 $9.60 Theoretical discharge by D'Arcy's formula : 2,743 3,004 3,244 3,488 3,699 3,915 4,102 4,297 Velocities in Smooth Cast-iron Water-pipes from 1 Foot to 9 Feet in Diameter, on Hydraulic Grades of 0.5 Foot to 8 Feet^per Mile; with Corresponding Values of c in V- c \rs. (D. M. Greene, in Eng'g News, Feb. 24, 1894.) 0) S !«■- hydraulic Grade; Feet per Mile = h. sr„ ■sil h = 0.5 1.0 1.5 2.0 3.0 4.0 D. r. s = 0.0000947 0.0001894 0.0002841 0.0003788 0.0005682 0.0007576 1. 0.25] V= 0.4542 0.6673 0.8356 0.9803 1.2277 1.4402 c= 92.7 97.0 99.1 100.7 103.0 104.7 2. 0.5 -j V= 0.7359 1.0793 1.3516 1.5856 1.9857 2.3294 c= 106.6 110.9 113.4 115.2 117.9 119.7 3. 0.75J V= 0.9733 1.4298 1.7906 2.1017 2.6306 3.0860 c= 115.5 119.9 122.6 124.4 127.5 129.5 4. 1.0 F= 1.1883 1.7456 2.1861 2.5645 3.2116 3.7676 c = 122.1 126.8 129.7 131.8 134.7 136.9 5. 1.25 -j V= 1.3872 2.0379 2.5521 2.9939 3.7493 4.39S3 c = 127.5 132.4 135.5 137.6 140.7 142.9 6. '•M V= 1.5742 2.3126 2.8961 3.3975 4.2548 4.9913 c = 132.1 137.8 140.3 142.6 145.8 148.1 7. 1.75^ V= 1.7518 2.5736 3.2230 3.7809 4.7350 5.5546 c = 135.9 141.4 146.0 146,8 150.2 152.5 8. 2.0 -j V= 1.9218 2.8234 3.5358 4.1479 5.1945 6.0936 c = 139.7 145.1 148.4 150.7 154.1 156.5 9. 2.25J F = 2.0854 3.0638 3.8368 4.5010 5.6368 6.6125 c = 142.9 148.4 151.7 154.2 157.6 160.1 The velocities in this table have been calculated by Mr. Greene's modifi- cation of the Chezy formula, which modification is found to give results which differ by from 1.29 to — 2.65 per cent (average 0.9 per centj from very carefully measured flows in pipes from 16 to 48 inches in diameter, on grades from 1.68 feet to 10.296 feet per mile, and in which the velocities ranged from 1.577 to 6.195 feet per second. The only assumption made is that the modified formula for V gives correct results in conduits from 4 feet to 9 feet in diameter, as it is known to do in conduits less than 4 feet in diameter. Other articles on Flow of Water in long- tubes are to be found in Eng'g News as follows : G. B. Pearsons, Sept, 23, 1876; E. Sherman Gould, Feb. 16, 23, March 9, 16, and 23, 1889; J. L. Fitzgerald, Sept. 6 and 13, 1890; Jas. Duane, Jan. 2, 1892; J. T. Fanning, July 14, 1892; A. N. Talbot, Aug. 11, 1892. 568 HYDRAULICS. Flow of Water in Circular Pipes, Sewers, etc., Flowing Full. Based on Kutter's Formula, with n = .013. Discbarge in cubic feet per second. Slope, or Head Divided by Length of Pipe. Diam- eter. lin40 Iin70 1 in 100 1 in 200 1 in 300 1 in 400 1 in 500 1 in 600 5 in. .456 .344 .288 .204 .166 .144 .137 .118 6 " .762 .576 .482 .341 .278 .241 .230 .197 7 " 1.17 .889 .744 .526 .430 .372 .355 .304 8 " 1.70 1.29 1.08 .765 .624 .54 .516 .441 9 " 2.37 1.79 1.50 1.06 .868 .75 .717 .613 Slope .... 1 in 60 1 in 80 1 in 100 1 in 200 1 in 300 1 in 400 1 in 500 1 in 600 10 in. 2.59 2.24 2.01 1.42 1.16 1.00 .90 .82 11 " 3.39 2 94 2.63 1.86 1.52 1.31 1.17 1.07 12 " 4.32 3.74 3.35 2.37 1.93 1.67 1.5 1.37 13 " 5.38 4.66 4.16 2.95 2.40 2.08 1.86 1.70 14 " 6.60 5.72 5.15 3.62 2.95 2.57 2.29 2.09 Slope... 1 in 100 1 in 200 1 in 300 1 in 400 1 in 500 1 in 600 1 in 700 1 in 800 15 in. 6.18 4 37 3.57 3.09 2.77 2.52 2.34 2.19 16 " 7.38 5.22 4.26 3.69 3.30 3.01 2.79 2.61 18 " 10.21 7.22 5.89 5.10 4.56 4.17 3.86 3.61 20 " 13.65 9.65 7.88 6.82 6.10 5.57 5.16 4.83 22 " 17.71 12.52 10.22 8.85 7.92 7.23 6.69 6.26 Slope 1 in 200 1 in 400 1 in 600 1 in 800 1 in 1000 1 in 1250 1 in 1500 1 in 1800 2 ft. 15.88 11.23 9.17 7.94 7.10 6.35 5 80 5.29 2fr.2iu. 19.73 13.96 11.39 9.87 8.82 7.89 7.20 6.58 2 " 4 " 24.15 17.07 13 94 12.07 10.80 9 66 8.82 8.05 2 " 6 " 29.08 20.56 16.79 14.54 13.00 11.63 10.62 9.69 2 " 8 " 34.71 24.54 20.04 17 .35 15.52 13.88 12.67 11.57 Slope ... I in 500 1 in 750 1 in 1000 1 in 1250 1 in 1500 1 in 1750 1 in 2000 1 in 2500 2 ft. 10 in. 25.84 21.10 18.27 16.34 14.92 13.81 12.92 11.55 3 " 30.14 24.61 21.31 19.06 17.40 16.11 15.07 13.48 3 " 2 in. 34.90 28.50 24.68 22.07 20.15 18.66 17.45 15.61 3 " 4 " 40.08 32 72 28 34 25.35 23.14 21.42 20.04 17.93 3 " 6 " 45.66 37.28 32.28 28.87 26.36 24.40 22.83 20.41 Slope 1 in 500 1 in 750 1 in 1000 1 in 1250 1 in 1500 1 in 1750 1 in 2000 1 in 2500 3 ft. 8 in. 51.74 42.52 36.59 32.72 29.87 27.66 25.87 23.14 3 " 10 " 58.36 47.65 41.27 36 91 33.69 31.20 29.18 26.10 4 " 65.47 53.46 46.30 41.41 37.80 34.50 32.74 29.28 4 " 6 in. 89.75 73.28 63.47 56.76 51.82 47.97 44.88 40.14 5 " 118.9 97.09 84.08 75.21 68.65 63.56 59.46 53.18 Slope . . 1 in 750 1 in 1000 1 in 1500 1 in 2000 1 in 2500 1 in 3000 1 in 3500 1 in 4000 5fr.6in. 125.2 108.4 88.54 76.67 68.58 62.60 57.96 54.21 6 " 157.8 136.7 111 6 96.66 86.45 78.92 73.07 68 35 6 " 6 " 195.0 168.8 137.9 119.4 106.8 97.49 90.26 84.43 7 " 237.7 205.9 168.1 145.6 130.2 118.8 110.00 102.9 285.3 247.1 201.7 174.7 156.3 142.6 132.1 123.5 Slope.... 1 in 1500 1 in 2000 1 in 2500 1 in 3000 1 in 3500 1 in 4000 1 in 4500 1 in 5000 8 ft. 239.4 207.3 195.4 169.3 156.7 146.6 138.2 131.1 8 " 6 in. 281.1 243.5 217.8 198.8 184.0 172.2 162.3 154.0 9 " 327.0 283.1 253.3 231.2 214.0 200.2 188.7 179.1 9 " 6 " 376.9 326.4 291.9 266.5 246.7 230.8 217.6 206.4 10 " 431.4 373.6 334.1 305.0 282.4 264.2 249.1 236.3 For U. S. gallons multiply the figures in the table by 7.4805. For a given diameter the quantity of flow varies as the square root of the sine of the slope. From this principle the flow for other slopes than those FLOW OF WATER IK CTRCULAK PIPES, ETC. 569 {riven in the table may be found. Thus, what is the flow for a pipe 8 feet diameter, slope 1 in 125 ? From the table take Q = 207.3 for slope 1 in 2000. The given slope 1 in 125 is to 1 in 2000 as 16 to 1, and the square root of this ratio is 4 to 1. Therefore the flow required is 207.3 x 4 = 829.2 cu. ft. Circular Pipes, Conduits, etc., Flowing Full. "Values of the factor ac \ r in tbe formula Q = ac \'r X \'s correspond- ing to different values of the coefficient of roughness, n. (Based on Kutter's formula.) § Value of acVr. 5 ft. in. n = .010. n = .011. n = .012. n = .013. n = .015. n = .017. 6 6.906 6.0627 5.3800 4.8216 3.9604 3 329 9 21.25 18.742 16.708 15.029 12.421 10.50 1 46.93 41.487 37.149 33.497 27.803 23 60 1 3 86.05 76.347 68.44 61.867 51.600 43.93 1 6 141.2 125.60 112.79 102.14 85.496 72.99 1 9 214.1 190.79 171.66 155.68 130.58 111.8 2 307.6 274.50 247.33 224.63 188.77 164 2 3 421.9 377.07 340.10 309.23 260.47 223.9 2 6 559.6 500.78 452.07 411.27 347.28 299.3 2 9 722.4 647.18 584.90 532.76 451.23 388.8 3 911.8 817.50 739.59 674.09 570.90 493.3 3 3 1128.9 1013.1 917.41 836.69 709.56 613.9 3 6 1374.7 1234.4 1118.6 1021.1 866.91 750.8 3 9 1652.1 1484.2 1345.9 1229.7 1045 906 4 1962.8 1764.3 1600.9 1463.9 1245.3 1080.7 4 6 2682.1 2413.3 2193 2007 1711.4 1487.3 5 3543 3191.8 2903.6 2659 2272.7 1977 5 6 4557.8 4111.9 3742.7 3429 2934.8 2557.2 6 5731.5 5176.3 4713.9 4322 3702.3 3232.5 6 6 7075.2 6394.9 5825.9 5339 4588.3 4010 7 8595.1 7774 3 7087 6510 5591.6 4893 7 6 10296 9318! 8 8501.8 7814 6717 5884.2 8 12196 11044 1008C 9272 7978.3 6995.3 8 6 14298 12954 11832 10889 9377.9 8226.3 9 16604 15049 13751 12663 10917 9580.7 9 6 19118 17338 15847 14597 12594 11061 10 21858 19834 18134 16709 14426 12678 10 6 24823 22534 20612 18996 16412 14434 11 28020 25444 23285 21464 18555 16333 11 6 31482 28593 26179 24139 20879 18395 12 35156 31937 29254 26981 23352 20584 12 6 39104 35529 32558 30041 26012 22938 13 43307 39358 36077 33301 28850 25451 13 6 47751 43412 39802 36752 31860 28117 14 52491 47739 43773 40432 35073 30965 14 6 57496 52308 47969 44322 38454 33975 15 62748 57103 52382 48413 42040 37147 16 74191 67557 62008 57343 49823 44073 17 86769 79050 72594 67140 58387 51669 18 100617 91711 84247 77932 67839 60067 19 115769 105570 96991 89759 78201 69301 20 132133 120570 110905 102559 89423 79259 Flow of Water in Circular Pipes, Conduits, etc., Flowing under Pressure. Based on D'Arcy's formulae for the flow of water through cast-iron pipes. With comparison of results obtained by Kutter's formula, with n = .013. (Condensed from Flynn on Water Power.) Values of a, and also the values of the factors c Vr and ac fr for use in the formulae Q = av; v — c V'r X V s , an d Q = ac Vr X V$« 570 HYDRAULICS. Q = discharge in cubic feet per second, a = area in square feet, v = veloc- ity in feet per second, r — mean hydraulic depth, J4 diam. for pipes running full, s = sine of slope. (For values of 4/s see page 558.) Clean Cast-iron Old Cast- iron Pipes Size Ol .ripe. Pipes. Value of ac \/r by Lined with Deposit. d= diam. a = area For For Dis- Kutter's Formula For For in square feet. Velocity, charge, when Velocity, Discharge, ft. in. c Vr- ac Vr- n = .013. cYr- ac |/r. % .00077 5.251 .00403 3.532 .00272 Vq .00136 6.702 .00914 4.507 .00613 H .00307 9.309 .02855 6.261 .01922 1 .00545 11.61 .06334 7.811 .04257 VA .00852 13.68 .11659 9.255 .07885 .01227 15.58 .19115 10.48 .12855 m .01670 17.32 .28936 11.65 .19462 2 .02182 18.96 .41357 12.75 .27824 W/2 .0341 21.94 .74786 14.76 .50321 .0491 24.63 1.2089 16.56 .81333 4 .0873 29.37 2.5630 19.75 1.7246 5 .136 33.54 4.5610 22.56 3.0681 6 .196 37.28 7.3068 4.822 25.07 4.9147 7 .267 40.65 10 852 27.34 7.2995 8 .349 43.75 15.270 29.43 10.271 9 .442 46.73 20.652 15.03 31.42 13.891 10 .545 49.45 26.952 33.26 18.129 11 .660 52.16 34.428 35.09 23.158 .785 54.65 42.918 33.50 36.75 28.867 1 2 1.000 59.34 63.435 39 91 42 668 1 4 1.396 63.67 88.886 42.83 59.788 1 6 1.767 67.75 119.72 102.14 45.57 80.531 1 8 2.182 71.71 156.46 48.34 105.25 1 10 2.640 75.32 198.83 50.658 133.74 2 3.142 78.80 247.57 224.63 52.961 166.41 2 2 3.687 82.15 302.90 55.258 203.74 2 4 4.276 85.39 365.14 57.436 245.60 2 6 4.909 88.39 433.92 411.37 59.455 291.87 2 8 5.585 91.51 511.10 61.55 343.8 2 10 6.305 94.40 595.17 63.49 400.3 3 7.068 97.17 686.76 674.09 65.35 461.9 3 2 7.875 99.93 786.94 67.21 529.3 3 4 8.726 102.6 895.7 69 602 3 6 9.621 105.1 1011.2 1021.1 70.70 680.2 3 8 10 559 107.6 1136.5 72.40 764.5 3 10 11.541 110.2 1271.4 74.10 855.2 4 12.566 112.6 1414.7 1463.9 75.73 951.6 4 3 14.186 116.1 1647.6 78.12 1108.2 4 6 15.904 1196 1901.9 2007 80.43 1279.2 4 9 17.721 1:22.8 2176.1 82.20 1456.8 5 19.6:5 126.1 2476.4 • 2659 84.83 1665.7 5 3 21.648 129.3 2799.7 86.99 1883.2 5 6 23.758 132.4 3146.3 3429 89.07 2116.2 5 9 25.967 135.4 3516 91.08 2365 6 2S.274 138.4 3912.8 4322 93.08 2631.7 6 6 33.183 144.1 4782.1 5339 96.93 3216.4 7 38.485 149.6 5757.5 6510 100.61 3872.5 7 6 44.179 154 9 6841.6 7814 104.11 4601.9 8 50.266 160 8043 9272 107.61 5409.9 8 6 56.745 165 9364.7 10889 111 6299.1 9 63.617 169.8 10804 12663 114.2 7267.3 9 6 70.882 174.5 12370 14597 117.4 8320.6 10 78.540 179.1 14066 16709 120.4 9460.9 FLOW OP WATER IK CIRCULAR PIPES, ETC. 571 Clean Cast-iron Old Cast-iron Pipet Pipes. Value of ac Vr by Lined with Deposit. d= diam. a = area For For Dis- Kutter's For For HI square feet. Velocity, ci] arge, when Velocity, Discharge, ft. in. c Vr. ac Vr- n = .013 c Vr. ac Vr. 10 6 86.590 183.6 15893 18996 123.4 10690 11 95.033 187.9 17855 21464 126.3 12010 11 6 103.869 192.2 19966 24139 129.3 13429 12 113.098 196.3 22204 26981 132 14935 12 6 122 719 200.4 24598 30041 134.8 16545 13 132.733 204.4 27134 33301 137.5 18252 13 6 143.139 208.3 29818 36752 140.1 20056 14 153.938 212.2 32664 40432 142.7 21971 14 6 165.130 216.0 35660 44322 145.2 23986 15 176.715 219.6 38807 48413 147.7 26103 15 6 188.692 223.3 42125 52753 150.1 28335 16 201.062 226.9 45621 57343 152.6 30686 16 6 213.825 230 4 49273 62132 155 33144 1? 226.981 233.9 53082 67140 157.3 35704 17 6 240.529 237.3 57074 72409 159.6 38389 18 254.470 240.7 61249 77932 161.9 41199 19 283.529 247.4 70154 89759 166.4 47186 20 314.159 253.8 79736 102559 170.7 53633 Flow of Water in Circular Pipes from % inch to 12 inches Diameter. Based on D'Arcy's formula for clean cast-ii on pipes. Q = ac Vr Vs' Value of Dia. Slope, or Head Divided by Length of Pipe. ac Vr- linlO. lin 20. 1 in 40. 1 in 60. 1 in 80. 1 in 100. 1 in 150. lin 200. Quan tity in cubic feet p er sec ond. .00403 % .00127 .00090 .00064 .00052 .00045 .00040 .00033 .00028 .00914 X, .00289 .00204 .00145 .00118 .00102 .00091 .00075 .00065 .02855 34 .00903 .00638 .00451 .00369 .00319 .00286 .00233 .00202 .06334 1 .02003 .01416 .01001 .00818 .00708 .00633 .00517 .00448 .11659 114 03687 .02607 .01843 .01505 .01303 .01166 .00952 .00824 .19115 \X4> .06044 .04274 .03022 .02468 .02137 .01912 .01561 .01352 .28936 W A .09140 .06470 .04575 .03736 .03235 .02894 .02363 .02046 .41357 2 .13077 .09247 .06539 .05339 .04624 .04136 .03377 .02927 .74786 2Vo .23647 .16722 .11824 .09655 .08361 .07479 .06106 .05288 1.2089 3 .38225 .27031 .19113 .15607 .13515 .12089 .09871 .08548 2.5630 4 .81042 .57309 .40521 .33088 .28654 .25630 .20927 .18123 4.5610 5 1.4422 1.0198 .72109 .58882 .50992 .45610 .37241 .32251 7.3068 6 2.3104 1.6338 1 . 1552 .94331 .81690 .73068 .59660 .51666 10.852 7 3.4314 2.4265 1.7157 1.4110 1.2132 1 .0852 .88607 .76734 15.270 8 4.8284 3.4143 2.4141 1.9713 1.7072 1.5270 1.2468 1.0797 20.652 9 6.5302 4.6178 3.2651 2.6662 2.3089 2.0652 1.6862 1.4603 26.952 10 8.5222 6.0265 4.2611 3.4795 3.0132 2.6952 2.2006 1.9058 34.428 11 10.886 7.6981 5.4431 4.4447 3.8491 3.4428 2.8110 2 4344 42.918 12 13.571 9.5965 6.7853 5.5407 4.7982 4.2918 3.5043 3.0347 Value of \ s = .3162 .2236 .1581 .1291 .1118 .1 .08165 .07071 572 HYDRAULICS. Slope, or Head Divided by Length of Pipe. Value of Dia. in. ac Vr. 1 in 1 in lin 1 in lin lin lin 1 in 250. 300. 350. 400. 450. 500. 550. 600. .00403 % .00025 .00023 .00022 .00020 .00019 .00018 .00017 .00016 .00914 v« .00058 .00053 .00049 .00046 .00043 .00041 .00039 .00037 .02855 Z A .00181 .00165 .00153 .00143 .00317* .00134 .00128 .00122 .00117 .06334 i .00400 .00366 .00339 .00298 .00283 .00270 .00259 .11659 1M .00737 .00673 .00623 .00583 .00549 .00521 .00497 .00476 .19115 .01209 .01104 .01022 .00956 .00901 .00855 .00815 .00780 .28936 .01830 .01671 .01547 .01447 .01363 .01294 .01234 .01181 .41357 2 .02615 .02388 .02211 .02068 .01948 .01849 .01763 .01688 .74786 21/, .0473C .04318 .03997 .03739 .03523 .03344 .03189 .03053 1.2089 8 .07645 .06980 .06462 .06045 .05695 .05406 .05155 .04935 2.5630 4 .16208 .14799 .13699 .12815 .12074 .11461 .10929 .10463 4.5610 5 .28843 .26335 .24379 .22805 .21487 .20397 .19448 .19620 7.3068 6 .46208 .42189 .39055 .36534 .34422 .32676 .31156 .29830 10.852 7 .68628 .62660 .58005 .54260 .51124 .48530 .46273 .44303 15.270 8 .96567 .88158 .81617 .76350 .71936 .68286 .65111 .62340 20.652 9 1.3060 1.1924 1.1038 1.0326 .97292 .92356 .88060 .84310 26.952 10 1.7044 1.5562 1.4405 1.3476 1.2697 1 2053 1 . 1492 1.1003 34.428 11 2.1772 1.9878 1.8402 1.7214 1.6219 1.5396 1.4680 1.4055 42.918 12 2.7141 2.4781 2.2940 2.1459 2.0219 1.9193 1.8300 1.7521 Value of V s = .06324 .05774 .05345 .05 .04711 .04472 .04264 .04082 For U. S. gals, per sec, multiply the figures in the table by 7.4805 " " " " min., ',' " " 448.83 " " " " horn, *'■ " " 26929.8 " " " " 24hi^., " " " 646315. For any other slope the flow is proportional to the square i-oot of the slope ; thus, flow in slope of 1 in 100 is double that in slope of 1 in 400. Flow of Water in Pipes from % Inch to 12 Inches Diameter for a Uniform Velocity of 100 Ft. per Min. Diameter Area Flow in Cubic Flow in U. S Flow in U. S. in in Feet per Gallons per Gallons per Inches. Square Feet. Minute. Minute. Hour. % .00077 0.077 .57 34 M. .00136 0.136 1.02 61 u .00307 0.307 2.30 138 1 .00545 0.545 4.08 245 M .00852 0.852 6.38 383 M .01227 1.227 9.18 551 m .01670 1.670 12.50 750 2 .02182 2.182 16.32 979 ®4 .0341 3.41 25.50 1,530 3 .0491 4.91 36.72 2,203 4 .0873 8.73 65.28 3,917 5 .136 13.6 102.00 6,120 6 .196 19.6 146.88 8,813 7 .267 26.7 199.92 11,995 8 .349 34.9 261.12 15,667 9 .442 44.2 330.48 19,829 10 .545 54.5 408.00 24,480 11 .660 66.0 493.68 29,621 12 .785 78.5 587.52 35,251 Given the diameter of a pipe, to find the quantity in gallons it will deliver, the velocity of flow being 100 ft. per minute. Square the diameter in inches and multiply by 4.08. LOSS OF HEAD. 573 If Q f = quantity in gallons per minute and d — diameter in inches, then g , = d* X .7854 X 100 X 7.4805 = ^^ V For any other velocity, V, in feet per minute, Q' = 4.08d a — - .0408d a F\ Given diameter of pipe in inches and velocity in feet per second, to find discharge in cubic feet and in gallons per minute. _ d a X -7854 X v X 60 _ 33-05^ cub i c feet per minute. * 144 = .32725 x 7,4805 or 2.448d 2 v U. S. gallons per minute. To find the capacity of a pipe or cylinder in gallons, multiply the square of the diameter in inches by the length in inches and by .0034. Or multiply the square of the diameter in inches by the length in feet and by .0408. LOSS OF HEAD. The loss of head due to friction when water, steam, air, or gas of any kind flows through a straight tube is represented by the formula AlV* , /64.4 hd --flT^ whence V =j/ — — J d 2g' in which I = the length and d = the diameter of the tube, both in feet; v = velocity in feet per second, and / is a coefficient to be determined by experi- ment. According to Weisbach, / = .00644, in which case |/^ = 50, and , = S0j/f which is one of the older formulae for flow of water (Downing's). Prof. Un- win says that the value of / is possibly too small for tubes of small bore, and he would put/ = .006 to .01 for 4-inch tubes, and/ = .0084 to .012 for 2- inch tubes. Another formula by Weisbach is Rankine gives 0+4)- From the general equation for velocity of flow of water v = c Yr Vs, = for round pipes c,i/ — if -, we have v" 2 = c 2 -- - and h = — — , in which c is the coefficient c of D'Arcy's, Bazin's, Kutter's, or other formula, as found by experiment. Since this coefficient varies with the condition of the inner surface of the tube, as well as with the velocity, it is to be expected that values of the loss of head given by different writers will vary as much as those of quantity of flow. Twotables for loss of head per 100 ft. in length in pipes of different diameters with different velocities are given below. The first is given by Clark, based on Ellis' and Howland's experiments; the second is from the Pelton Water-wheel Co.'s catalogue, authority not stated. The loss of head as given in these two tables for any given diameter and velocity differs considerably. Either table should be used with caution and the re- sults compared with the quantity of flow for the given diameter and head as given in the tables of flow based on Kutter's and D'Arcy's formulae. 574 HYDRAULICS. Relative Loss of Head by Friction for each 100 Feet Length of Clean Cast-iron Pipe. (Based on Ellis and Howland's experiments.) Velocity Diameter of Pipes in Inches. in Feet per S | 4 5 | 6 | 7 | 8 9 1 10 n\ 14 Second. Loss of Head in Feet, per 100 Feet Long Feet Feet Feet Feet Feet Feet Feet Feet Feet Feet Feet of of of of or of of of of of Head Head Head Head Head Head Head Head Head Head 2 .97 .55 .41 .32 .27 .23 .19 .18 .15 .12 2.5 1.49 .92 .64 .50 .43 .36 .30 .27 .23 .19 3 1.9 1.2 .82 .72 .61 .51 .44 .39 .33 .27 3.5 2.6 1.6 1.2 1.0 .7 .71 .61 .52 .45 .37 4 3.3 2.2 1.7 1.3 .9 .92 .79 .69 .59 .49 4.5 1.6 1.2 1.2 1.01 .87 .75 .61 5 1.2 1.1 .90 .76 5.5 .92 6 15 18 21 24 27 30 33 36 42 48 2 .11 .095 .075 .065 .055 .052 .049 .047 .036 .030 2.5 .17 .147 .117 .109 .088 .085 .076 .067 .056 .046 3 .25 .21 .17 .15 .13 .12 .108 .10 .081 .067 3.5 .34 .29 .23 .20 .18 .16 .15 .14 .111 .092 4 .44 .36 .31 .27 .23 .22 .20 .17 .14 .116 4.5 .50 .46 .39 .34 .30 .28 .25 .22 .18 .15 5 .70 .58 .48 .41 .37 .34 .30 .27 .22 .18 5.5 .84 .70 .59 .50 .44 .39 .36 .32 .27 .22 6 .59 .53 .49 43 .4 .32 .27 Loss of Head in Pipe by Friction.— Loss of head by friction in each 100 IVet in length of different diameters of pipe when discharging the following quantities of water per minute (Pelton Water-wheel Co.) : ~ Inside Diameter of Pipe in Inches. 43 1 2 3 4 5 | 6 _, C -v3 ft -3 o3 ft u ft .■=, ° W •g w . -g W "g a -g w 4) ® fa"3 K © b™ *o % ei_, O 1) r ®.2 o w fe o a « a fa o a fa fa g 3 fa t> a _o ©•■« ■sa 8-fl 3i o-S ?§ tie c 3 § %n 5» As •la > J o iJ o hJ O hJ o o hJ O V h Q h Q h 5.89 ft head or pressure is what would be indicated by a pressure-gauge attached to the pipe near the wheel. Headings of gauge should be taken while the water is flowing from the nozzle. To reduce heads in feet to pressure in pounds multiply by .433. To reduce pounds pressure to feet multiply by 2.309. Cox's Formula,- Weisbach's formula for loss of head caused by the friction of water in pipes is as follows : Friction-head = /o.0144 + ~^\ hll, \ VV I 5.367d where L — length of pipe in feet; V = velocity of the water in feet per second ; d = diameter of pipe in inches. William Cox {Amer. Mach., Dec. 28, 1893) gives a simpler formula which gives almost identical results : H = friction-head in feet = — — (1) Hd _ 4V* A-5V-2 L " 1200" ( } 576 HYDRAULICS. He gives a table by means of which the value of obtained when V is known, and vice versa. 4F 2 -f 5V - 2 [F2 + 5F-2 . Values op - 1200 V 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .00583 .00695 .00813 .00938 .01070 .01208 .01353 .01505 .01663 .01828 2 .02000 .02178 .02363 .02555 .02753 .02958 .03170 .03388 .03613 .03845 3 .04083 .04328 .04580 .04838 .05103 .05375 .05653 .05938 .06230 .06528 4 .06833 .07145 .07463 .07788 .08120 .08458 .08803 .09155 .09513 .09878 5 .10250 .10628 .11013 .11405 .11803 .12208 .12620 .13038 .13463 .13895 6 .14333 .14778 .15230 .15688 .16153 .16625 .17103 .17588 .18080 . 18578 7 .19083 .19595 .20113 .20638 .21170 .21708 .22253 .22805 .22363 .23928 8 .24500 .25078 .25663 .26255 .26853 .27458 .28070 .28688 .29313 .29945 9 .30583 .31228 .318S0 .32538 .33203 .33875 .34553 .35238 .35930 .36628 10 .37333 .38045 .38763 .39488 .40220 .40958 .41703 .42455 .43213 .43978 11 .44750 .45528 .46313 .47105 .47903 .48708 .49520 .50338 .51163 .51995 12 .52833 .53678 .54530 .55388 .56253 .57125 .58003 .58888 .59780 .60678 13 .61583 .62495 .63413 .64338 .65270 .66208 .67153 .68105 .69063 .70028 14 .71000 .71978 .72963 .73955 .74953 .75958 .76970 .7798S .79013 .80045 15 .81083 .82128 .83180 .84238 .85303 .86375 .87453 .88538 .89630 .90728 16 .91833 .92945 .94063 .95188 .96320 .97458 .98603 .99755 1.00913 1.02078 17 1.03-250 1 .04428 1.05613 1.06805 1.08003 1.09208 1 . 10420 1.11638 1.12863 1.14095 18 1.15333 1.16578 1.17830 1.19088 1.20353 1.21625 1.22903 1.24188 1.25480 1.26778 19 1.28083 1.29395 1.30713 1.32038 1.33370 1.34708 1.36053 1.37405 1.38763 1.40128 20 1.41500 1.42878 1.44263 1.45655 1.47053 1.48458 1.49870 1.51288 1.52713 1.54145 21 1.55583 1.57028 1.58480 1.59938 1.61403 1.62875 1.64353 1.65838 1.67330 1.68828 The use of the formula and table is illustrated as follows: Given a pipe 5 inches diameter and 1000 feet long, with 49 feet head, what will the discharge be? If the velocity Fis known in feet per second, the discharge is 0.32725d 2 F cubic foot per minute. By equation 2 we have 4F2 + 5F-2 _Hd L _ 49x5 . 1200 L 1000 whence, by table, V = real velocity = 8 feet per second. The discharge in cubic feet per minute, if V is velocity in feet per second and d diameter in inches, is 0.32725d 2 F, whence, discharge = 0.32725 x 25 X 8 = 65.45 cubic feet per minute. The velocity due the head, if there were no friction, is 8.025 4/ff = 56.175 feet per second, and the discharge at that velocity would be 0.32725 x 56.175 x 8 = 460 cubic feet per minute. Suppose it is required to deliver this amount, 460 cubic feet, at a velocity of 2 feet per second, what diameter of pipe will be required and what will be the loss of head by friction? d = diameter J VX 0.32725 \\ 2 X 0.32725 = 4/703" = 26.5 inches. H-. -- 0.75 foot, Having now the diameter, the velocity, and the discharge, the friction-head is calculated by equation 1 and use of the table; thus, L 4F 2 -f5F-2 _ 1000 20 " d 1200 26.5 ' 26.5 = thus leaving 49 — 0.75 = say 48 feet effective head applicable to power-pro- ducing: purposes. Problems of the loss of head may be solved rapidly by means of Cox's Pipe Computer, a mechanical device on the principle of the slide-rule, for sale by Keuffel & Esser, New York. LOSS OF HEAD. 57^ Frictional Heads at Given Rates of Discharge in Clean Cast-iron Pipes for Each 1000 Feet of Length. (Condensed from Ellis and HowlancTs Hydraulic Tables.) 4-inch 6-inch 8-inch 10-inch 12- nch 14-inch Pipe. Pipe. Pipe. Pipe. Pipe. Pipe. o y-g jj a 6 +j c 6- ^j +j a ,n jj c 6 ^; = MS — o> 3 - "5 03 S3 «8 o - £73 O - ST3 o - •£T3 '^ 5 - o<2 o .. & 5ft j££ E-S g£ &•* z* 3>H ■gjy >£ 25 .64 1.28 .59 2.01 .28 .57 .11 .32 .16 .32 .04 .10 .10 .20 .02 .04 .07 .14 .01 .02 50 10 .01 100 2.55 7.36 1.18 1.08 .64 .29 .41 .11 .28 .05 .21 .03 150 3.83 16.05 1.70 2.28 .96 .60 .61 .22 43 .10 .31 .05 200 5.11 28.09 2.27 3.92 1.28 1.01 .82 .36 .57 .16 ,42 .08 250 6.37 43.47 2.84 6.00 1.60 1.52 1.02 .54 .71 .24 52 .12 300 7.66 62.20 3.40 8.52 1.91 2.13 1.23 .75 .85 32 .63 .16 350 8.94 84.26 3.97 .11.48 2.23 2.85 1.43 .99 .99 .43 73 .21 400 10.21 109.68 4.54 14.89 2.55 3.68 1.63 1.27 1.13 .54 S3 .27 500 12.77 170.53 5.67 23.01 3.19 5.64 2.04 1.93 1.42 .81 1.04 .40 600 15.32 244.76 6.81 32.89 3.83 8.03 2.45 2.72 1.70 1.14 1 25 .55 700 17.87 332.36 7.94 44.54 4.47 10.83 2.86 3.66 1.98 1.52 1 46 .73 800 9.08 10.21 11.35 13.61 15.88 18.15 20.42 22.69 57.95 73.12 90.05 129.20 175.38 228 62 288.90 356.22 5.09 5.74 6.38 7.66 8.94 10.21 11.47 12.77 15.96 14.05 17.68 21.74 31.10 42.13 54.84 69.22 85.27 132.70 3.27 3.68 4.08 4.90 5.72 6.53 7.35 8.17 10.21 4.73 5.93 7.28 10.38 14.02 18.22 22.96 2.27 2.55 2.84 3.40 3.97 4.54 5 11 1.96 2.45 3.00 4.26 5.74 7.44 9.36 11.50 17.82 1.67 1 88 2.08 2.50 2 91 3.33 3.75 4.17 5 21 .94 000 1 17 1000 1 43 1200 2 02 1400 2 72 1600 3 51 1800 4 41 2000 28.25'5.67 43.87,7.09 5 41 2500 8.35 3000 12.25 62.92,8.51 25.51 6.25 11.93 4000 1.... 8.34 21.00 16-inch 18-i nch 20-inch 24-inch 30- inch 36-inch 03 Pipe. Pi je. Pipe. Pipe. P pe. Pipe. Si? S ^ a 6 ^j a 6 4J ^j ^j c 6 ^ 3 MS i<2 o .. •43^3 £3 c-8 is 1! .2 - - y o3 •r v i. - o - £5 o"l •.Co y <=3 •r a> 500 .80 .22 .63 .13 .51 .08 .35 .04 .23 .01 16 .01 1000 1.60 .76 1.26 .44 1.02 .27 .71 .12 45 .04 82 .02 1500 2.39 1.6c 1.89 .93 1.53 .56 1.06 .24 68 .OK .47 .04 2000 3 19 2.82 2.52 1.6C 2.04 .96 1.42 .41 .91 .15 63 .06 2500 3.99 4.34 3.15 2 45 2.55 1.47 1.77 .62 1.13 .22 .79 09 3000 4.79 6. IS 3.78 3.48 3.06 2.09 2. IS .87 1.36 .3C .95 .13 3500 5.59 8.37 4.41 4.7C 3.57 2.81 2.48 1.16 1.59 .4C 1.10 .17 4000 6.38 10.87 5.04 6.09 4.08 3.64 2.84 1.50 1.82 .52 1.26 .22 4500 7.18 13. 7C 5.67 7.67 4.59 4.58 3.19 1.88 2,04 .64 1.42 .27 5000 7.98 16.85 6 30 9.43 5.11 5.62 3.55 2.31 2.27 .78 1 . 58 33 6000 7.57 13.49 6.13 7.15 8.03 10.86 4.26 4.96 5.67 6.38 3.28 4.43 5.75 7 25 2.72 3.18 3.63 4.08 4 . 54 5.44 t; 3i j 1.11 1.49 1.93 2.43 2.98 4.25 5.75 1.89 2 21 2.52 2.84 3.15 3.78 4 41 46 7000 .62 8000 80 9000 1 00 10000 1.23 12000 1.74 14000 .2.35 16000 5.05 3.04 18000 5 68 3 83 20000 4.71 578 HYDRAULICS. Effect of Bends and Curves in Pipes.— Weisbach's rule for bends : Loss of head in feet = .131 + 1.847i--\ 2 X -^~ X -r|U in which r - radius of curvature of axis of pipe, v - the central angle, or angle subtended = internal radius of pipe in feet, R - — velocity in feet per second, and a - by the bend. Hamilton Smith, Jr., in his work on Hydraulics, says: The experimental data at hand are entirely insufficient' to permit a satisfactory analysis of this quite complicated subject; in fact, about the only experiments of value are those made by Bossut and Dubuat with small pipes. Curves.— If the pipe has easy curves, say with radius not less than 5 diameters of the pipe, the flow will not be materially diminished, provided the tops of all curves are kept below the hydraulic grade-line and provision be made for escape of air from the tops of all curves. (Trautwine.) Hydraulic Grade-line.— In a straight tube of uniform diameter throughout, running full and discharging freely into the air, the hydraulic grade-line is a straight line drawn from the discharge end to a point imme- diately over the entry end of the pipe and at a depth below the surface equal to the entry and velocity heads. (Trautwine.) In a pipe leading from a reservoir, no part of its length should be above the hydraulic grade-line. Flow of Water in House-service Pipes. Mr. E. Kuichling, C.E., furnished the following table to the Thomson Meter Co.: „ Discharge, or Quantity capable of being delivered, in '3 .£ a .2 Cubic Feet per Minute, from the Pipe, under the conditions specified in the first column. Condition of Discharge. £-c£ Nominal Diameters of Iron or Lead Service-pipe in 3 2 °3 III Hi Inches. y z % H | I IX 2 33.34 3 4 6 Through 35 30 1.10 1.92 3.01 6.13 16.58 88.16 173.85 444.63 40 1.27 2.22 3.48 7.08 19.14 38.50 101.80 200.75 513.42 50 1.42 2.48 3.89 7.92 21.40 43.04 113.82 224.44 574.02 60 1.56 2.71 4.26 8.67 23.44 47.15 124.68 245.87 628.81 back 75 1.74 3.03 4.77 9.70 26.21 52.71 139.39 274.89 703.03 100 2.01 3.50 5.50 11.20 30.27 60.87 160.96 317.41 811.79 130 2.29 3.99 6.28 12.77 34.51 69.40 183.52361.91 925.58 Through 100 feet of 30 0.66 1.16 1.84 3.78 10.40 21.30 58.19118.13 317.23 40 0.77 1.34 2.12 4.36 12.01 24.59 67. 191136.41 366.30 50 0.86 1.50 2.37 4.88 13.43 27.50 75.13152.51 409.54 60 0.94 1.65 2.60 5.34 14.71 30.12 82. 30! 167. 06 448.63 back 75 1.05 1.84 2.91 5.97 16.45 33.68 92.01 186.78 501 .58 100 1.22 2.13 3.36 6.90 18.99 38.89 106.24 215.68 579.18 130 1.39 2.42 3.83 7.86 21.66 44.34 121.14 245.91 660.36 Through 30 0.55 0.96 1.52 3.11 8.57 17.55 47.90 97.17 260.56 100 feet O! 40 0.66 1.15 1.81 8 72 10.24 20.95 57.20116.01 311.09 service- 50 0.75 1.31 2.06 4.24 11.67 23.87 65. 18 132. 2C 354.49 pipe and 60 0.83 1.45 2.29 4.70 12.94 26.48 72.28 146 61 393.13 15 feet 75 0.94 1.64 2.59 5.32 14.64 29.96 81. 79, 165. 9( 444.85 vertical 100 1.10 1.92 3.02 6.21 17. 1C 35. 0C 95.55! 193. 82 519.72 rise. 130 1.26 2.20 3.48 7.14 19.66 40.23 109.82 222.75 597.31 Through 30 0.44 0.77 1.22 2.50 6.80 14.11 38.63 78.54 211.54 100 feet o: 40 0.55 0.97 1.53 3.15 8.68 17. 7E 48.68 98.98 266.59 service- 50 0.65 1.14 1.79 3.69 10.16 20.82 56.98 115.87:312.08 pipe, and 60 0.73 1.28 2.02 4.15 11.45 23.47 64.22:130.59 351.73 30 feet 75 0.84 1.47 2.32 4.77 13 15 26.95 73. 76 1149.99 403.98 vertical 100 1.00 1.74 2.75 5.65 15.58 31.93 87.38 177.67 478.55 rise. 130 1.15 2.02 3.19 6.55 18.07 37.02 101.33 ^06.04 554.96 FIRE-STREAMS. 579 In this table it is assumed that the pipe is straight and smooth inside; that the friction of the main and meter are disregarded; that the inlet from the main is of ordinary character, sharp, not flaring or rounded, and that the outlet is the full diameter of pipe. The deliveries given will be increased if, first, the pipe between the meter and the main is of larger diameter than the outlet; second, if the main is tapped, say for 1-inch pipe, but is enlarged from the tap to 1J4 or 1}4 inch; or, third, if pipe on the outlet is larger than that on the inlet side of the meter. The exact details of the conditions given are rarely met in practice; consequently the quantities of the table may be expected to be decreased, because the pipe is liable to be throttled at the joints, additional bends may interpose, or stop-cocks may be used, or the back-pressure may be increased. Air-bound. Pipes.— A pipe is said to be air-bcfcnd when, in conse- quence of air being entrapped at the hign points of vertical curves in the line, water will not flow out of the pipe, although the supply is higher than the outlet. The remedy is to provide cocks or valves at the high points, through which the air may be discharged. The valve may be made auto- matic by means of a float. Vertical Jets. (Molesworth.)— H = head of water, h = height of jet, d = diameter of jet, K — coefficient, varying with ratio of diameter of jet to head; then h = KH. IfiJ=dX300 600 1000 1500 1800 2800 3500 4500, K= .96 .9 .85 .8 .7 .6 .5 .25 Water Delivered through Meters. (Thomson Meter Co.).— The best modern practice limits the velocity in water-pipes to 10 lineal feet per second. Assume this as a basis of delivery, and we find, for the several sizes of pipes usually metered, the following approximate results: Nominal diameter of pipe in inches: % % H 1 1^2 3 4 6 Quantity delivered, in cubic feet per minute, due to said velocity: 0.46 1.28 1.85 3.28 7.36 13.1 29.5 52.4 117.9 Prices Charged for Water in Different Cities (National Meter Co.;: Average minimum price for 1000 gallons in 163 places 9.4 cents. " maximum " " " " " " " 28 " Extremes, 2J^ cents to 100 " FIRE-STREAMS. Discharge from Nozzles at Different Pressures. (J. T. Fanning, Am. Water-works Ass'n, 1892, Eng'g Neivs, July 14, 1892.) Nozzle diam., in. Height of stream, ft. Pressure at Play- pipe, lbs. Horizon- tal Pro- jection of Streams, ft. Gallons per minute. Gallons per 24 hours. Friction per 100 ft. Hose, lbs. Friction per 100 ft. Hose, Net Head, ft. 1 70 46.5 59.5 303 292,298 10.75 24.77 1 80 59.0 67.0 230 331,200 13.00 31.10 1. 90 79.0 76.6 267 384,500 17.70 40.78 1 100 130.0 88.0 311 447.900 22.50 54.14 m 70 44.5 61.3 249 358,520 15.50 35.71 80 55.5 69.5 281 404,700 19.40 44.70 1^4 90 72.0 78.5 324 466,600 25.40 58.52 ty* 100 103.0 89.0 376 541,500 33.80 77.88 va 70 43.0 66.0 306 440,613 22.75 52.42 m 80 53.5 72.4 343 493.900 28.40 65.43 m 90 68.5 81.0 388 558,800 35.90 82.71 m 100 93.0 92.0 460 662,500 57.75 86.98 Ws 70 41.5 77.0 368 530,149 32.50 74.88 1% 1% 80 51.5 74.4 410 590,500 40.00 92.16 90 65.5 82.6 468 674,000 51.40 118.43 m 100 88.0 92.0 540 777,700 72.00 165.89 580 HYDRAULICS. Friction Losses in Hose.— In the above table the volumes of water discharged per jet were for stated pressures at the play-pipe. In providing for this pressure due allowance is to be made for friction losses in each hose, according to the streams of greatest discharge which are to be used. The loss of pressure or its equivalent loss of head (h) in the hose may be found by the formula h = vK4m)^-^. In this formula, as ordinarily used, for friction per 100 ft. of 2^-in. hose there are the following constants : 2)4, in. diameter of hose d = .20833 ft.; length of hose I = 100 ft., and 2g = 64.4. The variables are : v = velocity in feet per second; h^ loss of head in feet per 100 ft. of hose; m = a coeffi- cient found by experiment ; the velocity v is found from the given dis- charges of the jets through the given diameter of hose. Head and Pressure Losses by Friction in 100 -ft. Lengths of Rubber-lined Smooth 2^-in. Hose. Discharge Velocity Coefficient, Head Lost, Pressure Gallons per per minute, per second, m. ft. Lost, lbs. 24 hours. gallons. ft. per sq. in. 200 13.072 .00450 22.89 9.93 288,000 250 16.388 .00446 35.55 15.43 360,000 300 18.858 .00442 46.80 20.31 432,000 347 21.677 .00439 61.53 26.70 499,680 350 22.873 .00439 68.48 29.73 504,000 400 26.144 .00436 88.83 38.55 576,000 450 29.408 .00434 111.80 48.52 648,000 500 32.675 .00432 137.50 59.67 720,000 520 33.982 .00431 148.40 64.40 748,800 These frictions are for given volumes of flow in the hose and the veloci- ties respectively due to those volumes, and are independent of size of nozzle. The changes in nozzle do not affect the friction in the hose if there is no change in velocity of flow, but a larger nozzle with equal pressure at the nozzle augments the discharge and velocity of flow, and thus materially increases the friction loss in the hose. Loss of Pressure (p) and Head (/*) in Rubber-lined Smooth 2Hj-in. Hose may be found approximately by the formulae p = jl and h = ^ , in which p = pressure lost by friction, in pounds per square inch; I = length of hose in feet; q — gallons of water discharged per minute: d = diam. of the hose in inches, 2J^j in.; h = friction- head in feet. The coefficient of d b would be decreased for rougher hose. The loss of pressure and head for a lj^-in. stream with power to reach a height of 80 ft. is, in each 100 ft. of 2^-in. hose, approximately 20 lbs., or 45 ft. net, or, say, including friction in the hydrant, ^ ft. loss of head for each foot of hose. If we change the nozzles to 1J4 or 1% in. diameter, then for the same 80 ft. height of stream we inci'ease the friction losses on the hose to approxi- mately % ft. and 1 ft. head, respectively, for each foot-length of hose. These computations show the great difficulty of maintaining a high stream through large nozzles unless the hose is very short, especially for a gravity or direct-pressure system. This single 1^-in. stream requires approximately 56 lbs pressure, equiva- lent to 129 ft. head, at the play-pipe, and 45 to 50 ft. head for each 100 ft. length of smooth 2i^-in. hose, so that for 100, 200, and 300 ft. of hose we must have available heads at the hydrant or fire-engine of 1C6, 156, and 206 ft., respectively. If we substitute 1 J^-in. nozzles for same height of stream we must have available heads at the hydrants or engine of 185, 255 and 325 ft., respectively, or we must increase the diameter of a portion at least of the long hose and save friction-loss of head. Rated Capacities of Steam Fire-engines, which is perhaps one third greater than their ordinary rate of work at fires, are substantially as follows : 3d size, 550 gals, per min., or 792,000 gals, per 24 hours. 2d " 700 " " 1,008,000 1st " 900 " " 1,296,000 1 ext., 1,100 " " 1,584,000 THE SIPHON". 581 Pressures required at Nozzle and at Pump, with Quantity and Pressure of Water Necessary to throw Water Various Distances through Different-sized Nozzles- using 234-inch Rubber Hose and Smooth Nozzles. (From Experiments of Ellis & Leshure, Farming's " Water Supply.") Size of Nozzles. 1% Inch. Pressure at nozzle, lbs. per sq. in * Pressure at pump or hydrant with 100 ft. 2)4 inch rubber hose Gallons per minute. . Horizontal distance thrown, feet Vertical distance thrown, feet 135 310 193 Size of Nozzles. 1J4 Inch. 1% Inch. Pressure at nozzle, lbs. per sq. in * Pressure at pump or hydrant with 100 feet 2^}-inch rubber hose Gallons per minute Horizontal distance thrown, feet Vertical distance thrown, feet 115 14-2 1G4 80 100 144 180 413 462 200 224 146l 169 * For greater length of 2^-inch hose the increased friction can be ob- tained by noting the differences between the above given "pressure at nozzle 1 ' and "pressure at pump or hydrant with 100 feet of hose." For instance, if it requires at hydrant or pump eight pounds more pressure than it does at nozzle to overcome the friction when pumping through 100 feet of 2^-inch hose (using 1-inch nozzle, with 40-pound pressure at said nozzle) then it requires 16-pounds pressure to overcome the friction in forcing through 200 feet of same size hose. Decrease of Flow due to Increase of Length of Hose. {J. R. Freeman's Experiments, Trans. A. S. C. E. 1889.)— If the static pres- sure is 80 lbs. and the hydrant-pipes of such size that the pressure at the hy- drant is 70 lbs., the hose 2)4 in. nominal diam., and the nozzle \% in. diam., the height of effective fire-stream obtainable and the quantity in gallons per minute will be : Best Rubber- Linen Hose. lined Hose. Height, Gals. Height, Gals, feet. per min. feet, per min. 73 261 81 282 42 184 61 229 27 146 46 192 With 50 ft. of 2^-in. hose . . " 250 " " " " ., .4 50Q « U K «. _ With 500 ft. of smoothest and best rubber-lined hose, if diameter be exactly 2)4 in., effective height of stream will be 39 ft. (177 gals.); if diameter be 14 m « larger, effective height of stream will be 46 ft. (192 gals.) THE SIPHON. The Siphon is a bent tube of unequal branches,' open at both ends, and is used to convey a liquid from a higher to a lower level, over an intermedi- ate point higher than either. Its parallel branches being in a vertical plane and plunged into two bodies of liquid whose upper surfaces are at different levels, the fluid will stand at the same level both within and without each branch of the tube when a vent or small opening is made at the bend. If the air be withdrawn from the siphon through this vent, the water will rise in the branches by the atmospheric pressure without, and when the two columns unite and the vent is closed, the liquid will flow from the upper reservoir as long as the end of the shorter branch of the siphon is below the surface of the liquid in the reservoir. If the water was free from air the height of the bend above the supply level might be as great as 33 feet. 582 HYDRAULICS. If A = area of cross-section of the tube in square feet, H= the difference in level between the two reservoirs in feet, D the density of the liquid in pounds per cubic foot, then ADH measures the intensity of the force which causes the movement of the fluid, and V= V'tyH = 8.02 VH is the theoretical velocity, in feet per second, which is reduced by the loss of head for entry and friction, as in other cases of flow of liquids through pipes. In the case of the difference of level being greater than 33 feet, however, the velocity of the water in the shorter leg is limited to that due to a height of 33 feet, or that due to the difference between the atmospheric pressure at the entrance and the vacuum at the bend. Leicester Allen {Am. Mack., Nov. 2, 1893) says: The supply of liquid to a siphon must be greater than the flow which would take place from the dis- charge end of the pipe, provided the pipe were filled with the liquid, the supply end stopped, and the discharge end opened when the discharge end is left free, unregulated, and unsubnierged. To illustrate this principle, let us suppose the extreme case of a siphon having a calibre of 1 foot, in which the difference of level, or between the point of supply and discharge, is 4 inches. Let us further suppose this siphon to be at the sea-level, and its highest point above the level of the supply to be 27 feet. Also suppose the discharge end of this siphon to he un- regulated, unsubnierged. It would be inoperative because the water in the longer leg would not be held solid by the pressure of the atmosphere against it, and it would therefore break up and run out faster than it could be re- placed at the inflow end under an effective head of only 4 inches. Long Siphons.— Prof. Joseph Torrey, in the Amer. Machinist, describes a long siphon which was a partial failure. The length of the pipe was 1792 feet. The pipe was 3 inches diameter, and rose at one point 9 feet above the initial level. The final level was 20 feet below the initial level. No automatic air valve was provided. The highest point in the siphon was about one third the total distance from the pond and nearest the pond. At this point a pump was placed, whose mission was to fill the pipe when necessary. This siphon would flow for about two hours and then cease, owing to accumulation of air in the pipe. When in full operation it discharged 43J^ g-allons per minute. The theoretical discharge from such a sized pipe witli the specified head is 55^ grallons per minute. Siphon on the Water-supply of Mount Vernon, N. Y. (Enifa News, May 4, 1893.)— A 12-inch siphon, 925 feet lone:, with a maximum lift of 22.12 feet and a 45° change in alignment, was put in use in 1892 by the New York City Suburban Water Co., which supplies Mount Vernon, N. Y. At its summit the siphon crosses a supply main, which is tapped to charge the siphon. The air-chamber at the siphon is 12 inches by 16 feet long. A J^-inch tap and cock at the top of the chamber provide an outlet for the collected air. It was found that the siphon with air-chamber as desc.ibed would run until 125 cubic feet of air had gathered, and that this took place only half as soon with a 14-foot lift as with the full lift of 22.12 feet. The siphon will operate about 12 hours without being recharged, but more water can be gotten over by charging every six hours. It can be kept running 23 hours out of 24 with only one man in attendance. With the siphon as described above it is necessary to close the valves at each end of the siphon to recharge it. It has been found by weir measurements that the discharge of the siphon before air accumulates at the summit is practically the same as through a straight pipe. MEASUREMENT OF FLOWING WATER. Piezometer.— If a vertical or oblique tube be inserted into a pipe con- taining water under pressure, the water will rise in the former, and the ver- tical height to which it rises will be the head producing the pressure at the point where the tube is attached. Such a tube is called a piezometer or pressure measure. If the water in the piezometer falls below its proper level it shows that the pressure in the main pipe has been reduced by an obstruction between the piezometer and the reservoir. If the water rises above its proper level, it indicates that the pressure there has been in- creased by an obstruction beyond the piezometer. If we imagine a pipe full of water to be provided with a number of pie- zometers, then a line joining the tops of the columns of water in them is the hydraulic grade-line. MEASUREMENT OF PLOWING WATER. 583 Pitot Tube Gauge.— The Pitot tube is used for measuring the veloc- ity of fluids in motion. It has been used with great success in measuring the flow of natural gas. (S. W. Robinson, Report Ohio Geol. Survey, 1S90.) (See also Van No strand's Mag., vol. xxxv.) It is simply a tube so bent that a short leg extends into the current of fluid flowing from a tube, with the plane of the entering orifice opposed at right angles to the direction of the current. The pressure caused by the impact of the current is transmitted through the tube to a pressure' gauge of any kind, such as a column of water or of mercury, or a Bourdon spring-gauge. From the pressure thus indicated and the known density and temperature of the flowing gas is ob- tained the head corresponding to the pressure, and from this the velocity. In a modification of the Pitot tube described by Prof. Robinson, there are two tubes inserted into the pipe conveying the gas, one of which has the plane of the orifice at right angles to the current, to receive the static pres- sure plus the pressure due to impact; the other has the plane of its orifice parallel to the current, so as to receive the static pressure only. These tubes are connected to the legs of a U tube partly filled with mercury, which then registers the difference in pressure in the two tubes, from which the velocity may be calculated. Comparative tests of Pitot tubes with gas- meters, for measurement of the flow of natural gas, have shown an agree- ment within Sfo. The Venturi Meter, invented by Clemens Herschel, and described in a pamphlet issued by the Builders' Iron Foundry of Providence, R. I., is named from Venturi, who first called attention, in 1796, to the relation be- tween the velocities and pressures of fluids when flowing through converging and diverging tubes. It consists of two parts— the tube, through which the water flows, and the recorder, which registers the quantity of water that passes through the tube. The tube takes the shape of two truncated cones joined in their smallest diameters by a short throat-piece. At the up-stream end and at the throat there are air-chambers, at which points the pressuees are taken. The action of the tube is based on that property which causes the small section of a gently expanding frustum of a cone to receive, without material resultant loss of head, as much water at the smallest diameter as is dis- charged at the large end, and on that further property which causes the pressure of the water flowing through the throat to be less, by virtue of its greater velocity, than the pressure at the up-stream end of the tube, each pressure being at the same time a function of the velocity at that point and of the hydrostatic pressure which would obtain were the water motionless within the pipe. The recorder is connected with the tube by pressure-pipes which lead to it from the chambers surrounding the up-stream end and the throat of the tube. It may be placed in any convenient position within 1000 feet of the tube. It is operated by a weight and clockwork. The difference of pressure or head at the entrance and at the throat of the meter is balanced in the recorder by the difference of level hi two columns of mercury in cylindrical receivers, one within the other. The inner carries a float, the position of which is indicative of the quantity of water flowing through the tube. By its rise and fall the float varies the time of contact between an integrating drum and the counters by which the successive readings are registered. There is no limit to the sizes of the meters nor the quantity of water that may be measured. Meters with 21-inch, 36-inch, 48-inch, and even 20-foot tubes can be readily made. Measurement by Venturi Tubes. (Trans. A. S. C. E., Nov., 1887, and Jan., 1888.) — Mr. Herschel recommends the use of a Venturi tube, in- serted in the force-main of the pumping engine, for determining the quantity of water discharged. Such a tube applied to a 24-inch main has a total length of about 20 feet. At a distance of 4 feet from the end nearest the engine the inside diameter of the tube is contracted to a throat having a diameter of about 8 inches. A pressure-gauge is attached to each of two chambers, the one surrounding and communicating with the entrance or main pipe, the other with the throat. According to experiments made upon two tubes of this kind, one 4 in. in diameter at the throat and 12 in. at the en- trance, and the other about 36 in. in diameter at the throat and 9 feet at its entrance, the quantity of water which passes through the tube is very nearly the theoretical discharge through an opening having an area equal' to that of the throat, and a velocity which is that due to the difference in head Miown 584 HYDRAULICS. by the two gauges. Mr. Herschel states that the coefficient for these two widely-varying sizes of tubes and for a wide range of velocity through the pipe, was found to be within two per cent, either way, of 98%. In other words, the quantity of water flowing through the tube per second is ex- pressed within two per cent by the formula W — 0.98 X A X V%gh, in which A is the area of the throat of the tube, h the head, in feet, correspond- ing to the difference in the pressure of the water entering the tube and that found at the throat, and q = 32.16. measurement of Discharge of Pumping-engines toy Means of Nozzles. (Trans. A. S. M. E., xiii, 557). — The measurement of water by computation from its discharge through orifices, or through the nozzles of fire-hose, furnishes a means of determining the quantity of water delivered by apumping-engine which can be applied without much difficulty. John R. Freeman, Trans. A. S. C. E., Nov., 1889, describes a series of experi- ments covering a wide range of pressures and sizes, and the results showed that the coefficient of discharge for a smooth nozzle of ordinary good form was within one half of one per cent, either way, of 0.977 ; the diameter of the nozzle being accurately calipered, and the pressures being determined by means of an accurate gauge attached to a suitable piezometer at the base of the play-pipe. In order to use this method for determining the quantity of water dis- charged by a pumping-engine, it would be necessary to provide a pressure- box, to which the water would be conducted, and attach to the box as many nozzles as would be required to carry off the water. According to Mr. Freeman's estimate, four 134-inch nozzles, thus connected, with a pressure of 80 lbs. per square inch, would discharge the full capacity of a two-and a- half-million engine. He also suggests the use of a portable apparatus with a single opening for discharge, consisting essentially of a Siamese nozzle, so-called, the water being carried to it by three or more lines of fire-hose. To insure reliability for these measurements, it is necessary that the shut- off valve in the force-main, or the several shut-off valves, should be tight, so that all the water discharged by the engine may pass through the nozzles. Flow through Rectangular Orifices. (Approximate. Seep. 556.) Cubic Feet of Water Discharged per Minute through an Orifice One Inch Square, under any Head of Water from 3 to 72 Inches. For any other orifice multiply by its area in square inches. Formula, Q' — .624 Vh"X a. Q' — cu. ft. per min. ; a = area in sq. in. -u'd +=-ci +3 "a CO CD -u'd +si3 -M-d CO CD CD CD CD CD CD CD 0) CD CD feJC • CD &J0 . <^ too • CD 6J0 • CD &C . /-'-=- |! Mli< \\' Lik Fig. 130. OTiners' Incli Measurements. (Pelton Water Wheel Co.) The cut, Fig. 130, shows the form of measuring-box ordinarily used, and the following table gives the discharge in cubic feet per minute of a miner's inch of water, as measured under the various heads and different lengths and heights of apertures used in California. Length Openin gs 2 Inches High. Openi ngs 4 Inches High. of Opening Head to Head to Head to Head to Head to Head to in Centre, Centre, Centre, Centre, Centre, Centre, inches. 5 inches. 6 inches. 7 inches. 5 inches. 6 inches. 7 inches. Cu. ft. Cu. ft. Cu. ft. Cu. ft. Cu. ft. Cu. ft. 4 1.348 1.473 1.589 1.320 1.450 1.570 6 1.355 1.480 1.596 1.336 1.470 1.595 8 1.359 1.484 1.600 1.344 1.481 1.608 10 1.361 1.485 1.602 1.349 1.487 1.615 12 1.363 1.487 1.604 1.352 1.491 1.620 14 1.364 1.488 1.604 1.354 1.494 1.623 16 1.365 1.489 1.605 1.356 1.496 1.626 18 1.365 1-489 1.606 1.357 1.498 1.628 20 1.365 1.490 1.606 1.359 1.499 1.630 22 1.366 1.490 1.607 1.359 1.500 1.631 24 1.366 1.490 1.607 1.360 1.501 1.632 26 1.366 1.490 1.607 1.361 1.502 1.633 28 1.367 1.491 1.607 1.361 1.503 1.634 30 1.367 1.491 1.608 1.362 1.503 1.635 40 1.367 1.492 1.608 1.363 1.505 1.637 50 1.368 1.493 1.609 1.364 1.507 1.639 60 1.368 1.493 1.609 1.365 1.508 1.640 70 1.368 1.493 1.609 1.365 1.508 1.641 80 1.368 1.493 1.609 1.366 1.509 1.641 90 1.369 1.493 1.610 1.366 1.509 1.641 100 1.369 1.494 1.610 1.366 1.509 1.642 Note.— The apertures from which the above measurements were obtained 586 HYDRAULICS. were through material \y± inches thick, and the lower edge 2 inches above the bottom of the measuring-box, thus giving full contraction. Flow of Water Over Weirs. Weir Dam Measurement. (Pelton Water Wheel Co.)— Place a board or plank in the stream, as shown Fig. 131. in the sketch, at some point where a pond will form above. The length of the notch in the dam should be from two to four times its depth for small quantities and longer for large quantities. The edges of the notch should be bevelled toward the intake side, as shown. The overfall below the notch should not be less than twice its depth, that is, 12 inches if the notch is 6 inches deep, and so on. In the pond, about 6 ft. above the dam, drive a stake, and then obstruct the water until it rises precisely to the bottom of the notch and mark the stake at this level. Then complete the dam so as to cause all the water to flow through the notch, and, after time for the water to settle, mark the stake again for this new level. If preferred the stake can be driven with its top precisely level with the bottom of the notch and the depth of the water be measured with a rule after the water is flowing free, but the marks are pre- ferable in most cases. The stake can then be withdrawn ; and the distance between the marks is the theoretical depth of flow corresponding to the quantities in the table. Francis's Formulae for Weirs. As given by As modified by Francis. Smith. Weirs with both end contractions I n — ^ qq7/,§ onofn h. ~\]$ suppressed j v ' v "*" 7 ' W s 6 u S ressed 116 ^ contraction j- Q = 3.33(J - .lh)h% 3.29Z^ Weirs with full contraction Q = 3.33(Z - .2h)h? 3.29^ - —Jh* The greatest variation of the Francis formulas from the values of c given by Smith amounts to 3*/£#. The modified Francis formulas, says Smith, will give results sufficiently exact, when great accuracy is not required, within the limits ©f h, from .5 ft. to 2 ft., I being not less than 3 h. MEASUREMENT OF FLOWING WATER. 587 Q — discharge in cubic feet per second, I = length of weir in feet, /(, =effec- tive head in feet, measured from the level of the crest to the level of still water above the weir. If Q' -. discharge in cubic feet per minute, and V and h' are taken in inches, the first of the above formulas reduces to Q' = 0.41'h' 71 . From this formula the following table is calculated. The values are sufficiently accu- rate for ordinary computations of water-power for weirs without end con- traction, that is, for a weir the full width of the channel of approach, and are approximate also for weirs with end contraction when I - at least 10/i, but about 6$ in excess of the truth when I = 4h. Weir Table. Giving Cubic Feet op Water per Minute that will Flow over a Weir one inch wide and from % to 20% inches deep. For other widths multiply by the width in inches. in. cu. ft. .00 1 .40 2 1.13 3 2.07 4 3.20 5 4.47 6 5.87 7 7.40 8 9.05 9 10.80 10 12.64 11 14.59 12 16.62 13 18.74 14 20.95 15 23.23 16 25.60 17 28.03 18 30.54 19 33.12 20 35.77 ^i". 14 in. % in. ^in. Y&in. 34 in. %m. cu. ft. cu. ft. cu. ft. cu. ft. cu. ft. cu. ft. cu. ft. .01 .05 .09 .14 .19 .26 .32 .47 .55 .64 .73 .82 .92 1.02 1.23 1.35 1.46 1.58 1.70 1.82 1.95 2.21 2.34 2.48 2.61 2.76 2.90 3.05 3.35 3.50 3.66 3.81 3.97 4.14 4.30 4.64 4.81 4.98 5.15 5.33 5.51 5.69 6.06 6.25 6.44 6.62 6.82 7.01 7.21 7.60 7.80 8.01 8.21 8.42 8.63 8.83 9.26 9.47 9.69 9.91 10.13 10.35 10.57 11.02 11.25 11.48 11.71 11.94 12.17 12.41 12.88 13.12 13.36 13.60 13.85 14.09 14.34 14.84 15.09 15 34 15.59 15.85 16.11 16.36 16.88 17.15 17.41 17.67 17.94 18.21 18.47 19.01 19.29 19.56 19.84 20.11 20 39 20.67 21.23 21.51 21.80 22.08 22.37 22.65 22.94 23.52 23.82 24.11 24.40 24.70 25.00 25.30 25.90 26.20 26.50 26.80 27.11 27.42 27.72 28.34 28.65 28.97 29.28 29.59 29.91 30.22 30.86 31.18 31.50 31.82 32.15 32.47 32.80 33.45 33.78 34.11 34.44 34.77 35.10 35.44 36.11 36.45 36.78 37.12 37.46 37.80 38.15 For more accurate computations, the coefficients of flow of Hamilton Smith, Jr., or of Bazin should be used. In Smith's hydraulics will be found a collection of results of experiments on orifices and weirs of various shapes made by many different authorities, together with a discussion of their several formulae. (See also Trautwine's Pocket Book.) Bazln'S Experiments. - M. Bazin (Annates des Pouts et Cliaussees, Oct., 1888, translated by Marichal and Trautwine, Proc. Engrs. Club of Phila., Jan , 1890), made an extensive series of experiments with a sharp-crested weir without lateral contraction, the air being admitted freely behind the falling sheet, and found values of m varying from 0.42 to 0.50, with varia- tions of the length of the weir from 19% to 78% in., of the height of the crest above the bottom of the channel from 0.79 to 2.46 ft., and of the head from 1.97 to 23.62 in. From these experiments he deduces the following formula : Q =[0.425 + ^(jrf^) 2 ]^ V^H, in which P is the height in feet of the crest of the weir above the bottom of the channel of approach, L the length of the weir, H the head, both in feet, and Q the discharge in cu. ft. per sec. This formula, says M. Bazin, is en- tirely practical where errors of 2% to 3% are admissible. The following table is condensed from M. Bazin's paper : 588 WATER-POWER. Values of the Coefficient m in the Formula Q = mLH V'igH, for a Sharp-crested Weir without Lateral Contraction ; the Air being Admitted Freely Behind the Falling Sheet. Height of Crest of Weir Above Bed of Channel. Head, H. I Feet ...0.66 0.98 1.31 1.64 1.97 2.62 3.28 4.92 6 56 oo Inches 7.87 11.81 15,75 19.69 23.62 31.50 39.38 59.07 I 78.76 GO Ft. Tn. m m m m VI m m | m m m .164 1.97 0.458 0.453 0.451 0.450 0.449 0.449 0.449 0.448 0.448 0.4481 .230 2.76 0.455 0.448 0.445 0.443 0.442 0.441 0.440 0.440 0.439 0.4391 .295 3.54 0.457 0.447 0.442 0.440 0.438 0.436 0.436 0.435 0.434 0.4340 .394 4.72 0.462 0.448 0.442 0.438 0.436 0.433 0.432 0.43C 430 0.4291 .525 6.30 0.471 0.453 0.444 0.438 0.435 0.431 0.429! 0.427 0.426 0.4246 .656 7.87 0.480 0.459 0.447 0.440 0.436 0.431 0.428 0.425 0.423 0.4215 .787 9.45 0.488 0.465 0.452 0.444 0.438 0.432 0.428,0.424 422 0.4194 .919 11.02 0.496 0.472 0.457 0.448 0.441 0.433 0.429 0.424 0.422 0.4181 1.050 12 60 478 0.462 0.452 0.444 436 0.430 0.424 0.421 0.4168 1.181 14 17 483 0.467 0.456 0.448 438 0.432 0.424 0.421 0.4156 1.312 15.75 0.489 0.472 0.459 0.451 0.440 0.433 0.421 0.4144 1.444 17.3V 494 0.476 463 0.454 0.442 0.435 0.425 0.421 0.4134 1.575 18.90 30.47 0.480 0.483 0.467 470 0.457 0.460 0.444 0.446 0.436 0.425 0.438 0.426 0.421 0.421 0.4122 1.706 0.4112 1.837 ' 0.487 0.490 0.473 0.476 0.463 0.466 0.448 0.451 0.4390.427 0.441 0.427 0.421 0-421 0.4101 1.969 23.62 0.4092 i r A comparison of the results of this formula with those of experiments, says M. Bazin, justifies us in believing that, except in the unusual case of a very low weir (which should always be avoided), the preceding table will give the coefficient m in all cases within 1%; provided, however, that the ar- rangements of the standard weir are exactly reproduced. It is especially important that the admission of the air behind the falling sheet be perfectly assured. If this condition is not complied with, m may vary within much wider limits. The type adopted gives the least possible variation in tho coefficient. WATER-POWER. Power of a Fall of Water— Efficiency.— The gross power of a fall of water is the product of the weight of water discharged in a unit of time into the total head, i.e., the difference of vertical elevation of the upper surface of the water at the points where the fall in question begins and ends. The term " head " used in connection with water-wheels is the difference in height from the surface of the water in the wheel-pit to the surface in the pen-stock when the wheel is running. If Q = cubic feet of water discharged per second, D = weight of a cubic foot of water = 62.36 lbs. at 60° F., H = total head in feet; then DQH = gross pow r er in foot-pounds per second, arid DQH -*- 550 =.ll%4 QH = gross horse-power. If Q' if. taken in cubic feet per minute, H> P. = ■ ' = oo,U00 'H. A water-wheel or motor of any kind cannot utilize the whole of the head H, since there are losses of head at both the entrance to and the exit from i he wheel. There are also losses of energy due to friction of the water in its passage through the wheel, t'he ratio of the power developed by the wheel to the gross power of the fall is the efficiency of the wheel. For 75% efficiency, net horse-power = .00142Q' H = -^r . MILL-POWER. 589 A head of water can be made use of in one or other of the following ways viz. : 1st. By its weight, as in the water-balance and overshot-wheel. 2d. By its pressure, as in turbines and in the hydraulic engine, hydraulic press, crane, etc. 3d. By its impulse, as in the undershot- wheel, and in the Pelton wheel. 4th. By a combination of the above. Horse-power of a Running Stream.— The gross horse power is, H. P. = QH X 62.30 -*- 550 = MMQH, in which Q is the discharge in cubic feet per second actually impinging on the float or bucket, and H — theoret- ic v 2 ical head due to the velocity of the stream = — = rr- - , in which v is the 2g 64.4 velocity in feet per second. If Q' be taken in cubic feet per minute, H.P. = .00189Q'if. Thus, if the floats of an undershot-wheel driven by a current alone be 5 feet x 1 foot, and the velocity of stream = 210 ft. per minute, or 3% ft. per sec, of which the theoretical head is .19 ft., Q = 5 sq. ft. X 210 = 1050 cu. ft. per minute ; H = .19 ft. ; H. P. = 1050 X .19 X. 00189 = .377 H. P. The wheels would realize only about .4 of this power, on account of friction and slip, or .151 H. P., or about .03 H.P. per square foot of float, which is equivalent to 33 sq. ft. of float per H. P. Current Motors. — A current motor could only utilize the whole power of a running stream if it could take all the velocity out of the water, so that it would leave the floats or buckets with no velocity at all ; or in other words, it would require the backing up of the whole volume of the stream until the actual head was equivalent to the theoretical head due to the velocity of the stream. As but a small fraction of the velocity of the stream can be taken up by a current motor, its efficiency is very small. Current motors may be used' to obtain small amounts of power from large streams, but for large powers they are not practicable. Horse-power of Water Flowing in a "Tube.— The head due to i> 2 f the velocity is — ; the head due to the pressure is - ; the head eiue to actual 2g w height above the datum plane is h feet. The total head is the sum of these = v 9 f — 4- h -f — , in feet, in which v = velocity in feet per second,/ = pressure 2g w in lbs. per sq. ft., to — weight of 1 cu. ft. of water = 62.36 lbs. If p = pres- sure in lbs. per sq. in., — = 2.309jp. In hydraulic transmission the velocity and the height above datum are usually small compared with the pressure- head. The work or energy of a given quantity of water underpressure = its volume in cubic feet X its pressure in lbs. per sq. ft.; or if Q = quantity in cubic feet per second, and p = pressure in lbs. per square inch, W = U4 P Q, and the H. P. = ^^ = .2618pQ. Maximum Efficiency of a Long Conduit.— A. L. Adams and R. G. Uenimel {Euy'y News, May 4, 1893), show by mathematical analysis that the conditions for securing the maximum amount of power through a long conduit of fixed diameter, without regard to the economy of water, is that the draught from the pipe should be such that the frictional loss in the pipe will be equal to one third of the entire static head. Mill-Power. — A "mill-power " is a unit used to rate a water-power for the purpose of renting it. The value of the unit is different in different localities. The following are examples (from Emerson): Holyoke, Mass. — Each mill-power at the respective falls is declared to be the right during 16 hours in a day to draw 38 cu. ft. of water per second at the upper fall when the head there is 20 feet, or a quantity proportionate to the height at the falls. This is equal to 86.2 horse-power as a maximum. Lowell, M ass. —The right to draw during 15 hours in the day so much water as shall give a power equal to 25 cu. ft. a second at the great fall, when the fall there is 30 feet. Equal to 85 H. P. maximum. Lawrence, Mass. — The right to draw during 16 hours in a day so much water as shall give a horse-power equal to 30 cu. ft. per second when the head is 25 feet. Equal to 85 H. P. maximum. Minneapolis, Minn.— 30 cu. ft. of water per second with head of 22 feet. Equal to 74.8 H. P. Manchester, N. H.— Divide 725 by the number of feet of fall minus 1, and 590 WATER-POWER. the quotient will be the number of cubic feet per second in that fall. For 20 feet fall this equals 38.1 cu. ft., equal to 86.4 H. P. maximum. Cohoes, N. Y.— " Mill-power " equivalent to the power given by 6 cu. ft. per second, when the fall is 20 feet. Equal to 13.6 H. P.,«maximum. Passaic, N. J.— Mill-power: The right to draw 8J^ cu. ft. of water per sec, fall of 22 feet, equal to 21.2 horse-power. Maximum rental $700 per year for each mill-power = $33.00 per H. P. The horse-power maximum above given is that due theoretically to the weight of water and the height of the fall, assuming the water-wheel to have perfect efficiency. It should be multiplied by the efficiency of the wheel, say 75# for good turbines, to obtain the H. P. delivered by the wheel. Value of a Water-power.— In estimating the value of a water- power, especially where such value is used as testimony for a plaintiff whose water-power has been diminished or confiscated, it is a common custom for the person making such estimate to say that the value is represented by a sum of money which, when put at interest, would maintain a steam-plant of the same power in the same place. Mr. Charles T. Main (Trans. A. S. M. E. xiii. 140) points out that this sys- tem of estimating is erroneous; that the value of a power depends upon a great number of conditions, such as location, quantity of water, fall or head, uniformity of flow, conditions which fix the expense of dams, canals, founda- tions of buildings, freight charges for fuel, raw materials and finished prod- uct, etc. He gives an estimate of relative cost of steam and water-power for a 500 H. P. plant from which the following is condensed: The amount of heat required per H. P. varies with different kinds of busi- ness, but in an average plain cotton-mill, the steam required for heating and slashing is equivalent to about 25$ of isteam exhausted from the high- pressure cylinaer of a compound engine of the power required to run that mill, the steam to be taken from the receiver. The coal consumption per H. P. per hour for a compound engine is taken at 1% lbs. per hour, when no steam is taken from the receiver for heating purposes. The gross consumption when 25$ is taken from the receiver is about 2.06 lbs. 75% of the steam is used as in a compound engine at 1.75 lbs. = 1.31 lbs. 25% " " " " high-pressure " 3.00 lbs. = .75 " 2.06 " The running expenses per H. P. per year are as follows : 2.06 lbs. coal per hour = 21.115 lbs. for 10J4 hours or one day = 6503.42 lbs. for 308 days, which, at $3.00 per long ton = $8 71 Attendance of boilers, one man @ $2.00, and one man @ $1.25 = 2 00 " engine, ■« '■ " $3.50. 2 16 Oil, waste, and supplies. 80 The cost of such a steam-plant in New England and vicinity of 500 H. P. is about $65 per H. P. Taking the fixed expenses as 4% on engine, 5% on boilers, and 2% on other portions, repairs at 2%, in- terest at 5%, taxes at 1 \^% on % cost, an insurance at y%% on exposed portion, the total average per cent is about 12J^#, or $65 X -12^ = 8 13 Gross cost of power and low-pressure steam per H. P. $21 80 Comparing this with water-power, Mr. Main says : "At Lawrence the cost of dam and canals was about $650,000, or $65 per H. P. The cost per H. P. of wheel-plant from canal to river is about $45 per H. P. of plant, or about $65 per H. P. used, the additional $20 being caused by making the plant large enough to compensate for fluctuation of power due to rise and fall <>f river. The total cost per H. P. of developed plant is then about $130 per H. P. Placing the depreciation on the whole plant at 2%, repairs at 1%, interest at 5%, taxes and insurance at 1 %, or a total o£ 9%, gives: Fixed expenses per H. P. $130 X .09 = $11 70 Running " " " (Estimated) 2 00 $13 70 " To this has to be added the amount of steam required for heating pur- poses, said to be about 25$ of the total amount used, but in winter months the consumption is at least 37^$. It is therefore necessary to have a boiler plant of about S7},4% of the size of the one considered with" the steam-plant. TURBINE WHEELS. 591 costing about $20 X .375 = $7.50 per H. P. of total power used. The ex- pense of running this boiler-plant is, per H. P. of the the total plant per year: Fixed expenses 12^$ on $7.50 $0.94 Coal 3.26 Labor 1 .23 Total $5.43 Making a total cost per year for water-power with the auxiliary boiler plant $13.70 + $5.43 = $19.13 which deducted from" $21.80 make a difference in favor of water-power of $2.67, or for 10,000 H. P. a saving of $26,700 per year. " It is fair to say," says Mr. Main," that the value of this constant power is a sum of money which when put at interest will produce the saving ; or if 6% is a fair interest to receive on money thus invested the value would be $26,700-- .06 = $445,000." Mr. Main makes the following general statements as to the value of a water-power : "The value of an undeveloped variable power is usually noth- ing if its variation is great, unless it is to be supplemented by a steam-plant. It is of value then only when the cost per horse-power for the double-plant is less than the cost of steam-power under the same conditions as mentioned for a permanent power, and its valuecan be represented in the same man- ner as the value of a permanent power has been represented. " The value of a developed power is as follows: If the power can be run cheaper than steam, the value is that of the power, plus the cost of plant, less depreciation. If it cannot be run as cheaply as steam, considering its cost, etc., the value of the power itself is nothing, but the value of the plant is such as could be paid for it new, which would bring the total cost of run- ning down to the cost of steam-power, less depreciation." Mr. Samuel Webber, Iron Age, Feb. and March, 1893, writes a series of articles showing the development of American turbine wheels, and inci- dentally criticises the statements of Mr. Main and others who have made comparisons of costs of steam and of water-power unfavorable to the latter. Hesays : ' ' They have based their calculations on the cost of steam, on large compound engines of 1000 or more H. P. and 120 pounds pressure of steam in their boilers, and by careful 10-hour trials succeeded in figuring down steam to a cost of about $20 per H. P., ignoring the well-known fact that its average cost in practical use, except near the coal mines, is from $40 to $50. In many instances dams, canals, and modern turbines can be all completed for a cost of $100 per H. P. ; and the interest on that, and the cost of attend- ance and oil, will bring water-power up to but about $10 or $12 per annum; and with a man competent to attend the dynamo in attendance, it can probably be safely estimated at not over $15 per H. P." TURBINE WHEELS. Proportions of Turbines.— Prof. De Volson Wood discusses at length the theory of turbines in his paper on Hydraulic Reaction Motors, Trans. A. S. M. E. xiv. 266. His principal deductions which have an imme- diate bearing upon practice are condensed in the following : Notation. Q = volume of water passing through the wheel per second, h l = head in the supply chamber above the entrance to the buckets, /< 2 = head in the tail-race above the exit from the buckets, Z) — fall in passing through the buckets. H — h x + z-i — /i 2 , the effective head, l*. x = coefficient of resistance along the guides, ju., = coefficient of resistance along the buckets, r x = radius of the initial rim, r 2 = radius of the terminal rim, V = velocity of the water issuing from supply chamber, v x = initial velocity of the water in the bucket in reference to the bucket, v 2 = terminal velocity in the bucket, co = angular velocity of the wheel, a = terminal angle between the guide and initial rim = CAB, Fig. 132, Vi = angle between the initial element of bucket and initial rim = BAD. V 2 = OFI, the angle between the terminal rim and terminal element of the bucket. a == eb, Fig. 133 = the arc subtending one gate opening, 592 WATEft-POWEft. <*! = the arc subtending one bucket at entrance. (In practice a t is larger than «,) « 2 = gh, the arc subtending one bucket at exit, K = £>/, normal section of passage, it being assumed that the passages and buckets are very narrow, k l = bd, initial normal section of bucket, fc 2 = gi, terminal normal section, to?! = velocity of initial rim, wr 2 = velocity of terminal rim, = HFI, angle between the terminal rim and actual direction of the water at exit, Y — depth of K. y, of a,, and y 2 of K^ then K = Ya sin a; K x = y x a x sin y t ; 1T 2 == # 2 a 2 sin y 2 . Fig. 132. Fig. 133. Three simple systems are recognized, i\ < ?- 2 ,{called outward flow; r x > r 3 , called inward flow; r t = r 2 , called parallel flow. The first and second may be combined with the third, making a mixed system. Value of v? (the quitting angle).— The efficiency is increased as y 9 de- creases, and is greatest for y 2 = 0. Hence, theoretically, the terminal ele- ment of the bucket should be tangent to the quitting rim for best efficiency. This, however, for the discharge of a finite quantity of water, would require an infinite depth of bucket. In practice, therefore, this angle must have a finite value. The larger the diameter of the terminal rim the smaller may be this angle for a given depth of wheel and given quantity of water discharged. In practice y 2 is from 10° to 20°. In a wheel in which all the elements except y 2 are fixed, the velocity of the wheel for best effect must increase as the quitting angle of the bucket decreases. Values of a -{- y } must be less than 180°, but the best relation cannot be determined by analysis. However, since the water should be deflected from its course as much as possible from its entering to its leaving the wheel, the angle a for this reason should be as small as practicable. In practice, a cannot be zero, and is made from 20° to 30°. The value ?j = 1.4r a makes the width of the crown for internal flow about the same as for i\ =r 3 \/% for outward flow, being approximately 0.3 of the external radius. Values of v-! and /u. 2 . — The frictional resistances depend upon the construc- tion of the wheel as to smoothness of the surfaces, sharpness of the angles, TUKBLNfi WHEELS. 593 regularity of the curved parts, and also upon the speed it is run. These values cannot be definitely assigned beforehand, but Weisbach gives for good conditions /u^ = /u. 2 = 0.05 to 0.10. They are not necessarily equal, and /u-i may be from 0.05 to 0.075, and /u. 2 from 0.06 to 0.10 or even larger. Values of y t must be less than 180° — a. To be on the safe side, Vi may be 20 or 30 degrees less than 180° -2a, giving Vl = 180° - 2a - 25 (say) = 155 - 2a. Then if a = 30°, y 1 = 95°. Some designers make y± 90°; others more, and still others less, than that amount. Weisbach suggests that it be less, so that the bucket will be shorter and friction less. This reasoning appears to be correct for the inflow wheel, but not for the outflow wheel. In the Tre- mont turbines, described in the Lowell Hydraulic Experiments, this angle is 90°, the angle a 20°, and y 2 10°, which proportions insured a positive pressure in the wheel. Fourneyron made y x = 90°, and a from 30° to 33°, which values made the initial pressure in the wheel near zero. Form of Bucket —The form of the bucket cannot be determined analytic- ally. From the initial and terminal directions and the volume of the water flowing through the wheel, the area of the normal sections may be found. The normal section of the buckets will be : V Vi v 7 The depths of those sections will be : K k x _ « 2 sin "y 3 The changes of curvature and section must be gradual, and the general form regular, so that eddies and whirls shall not be formed. For the same reason the wheel must be run with the correct velocity to secure the best effect. In practice the buckets are made of two or three arcs of circles, mutually tangential. The Value of <■>.— So far as analysis indicates, the wheel may run at any speed; but in order that the stream shall flow smoothly from the supply chamber into the bucket, the velocity V should be properly regulated. If ix j = nz = 0.10, r 2 -^ n = 1.40, a = 25°, ?i = 90°, v 2 .= 12°, the velocity of the initial rim for outward flow will be f or m aximum efficiency 0.614 of the velocity due to the head, or wj-j = 0.614 V2 gH. The velocity due to the head would be V2yH = 1.414 VgH. For an inflow wheel for the case in w hich r^ = 2?- a 2 , and the other dimen sions as given above, wrj = 0.682 S/2gH. The highest efficiency of the Tremont turbine, found experimentally, was 0.79375, and the corresponding velocity, 0.62645 of that due to the head, and for all velocities above and below this value the efficiency was less. In the Tremont wheel a = 20° instead of 25°, and y ? = 10° instead of 12°. These would make the theoretical efficiency and velocity of the wheel some- what, greater. Experiment showed that the velocit}' might be considerably larger or smaller than this amount without much diminution of the efficiency. It was found that if the velocity of the initial (or interior) rim was not less than 44$ nor more than 75$ of that due to the fall, the efficiency was 75$ or more. This wheel was allowed to run freely without any brake except its own friction, and the velocity of the initial rim was observed to be 1.335 V2gH, half of which is 0.6675 V^gH, which is not far from the velocity giving maximum effect; that is to say, when the gate is fully raised the coeffi- cient of effect is a maximum when the wheel is moving with about half its maximum velocity. Number of Buckets.— Successful wheels have been made in which the dis- tance between the buckets was as small as 0.75 of an inch, and others as much as 2.75 inches. Turbines at the Centennial Exposition had buckets from i\4, inches to 9 inches from centre to centre. If too large they will not work properly. Neither should they be too deep. Horizontal partitions are sometimes introduced. These secure more efficient working in case the gates are only partly opened. The form and number of buckets for com- mercial purposes are chiefly the result of experience. 594 WATER-POWEtfc. Ratio of Radii.— Theory does not limit the dimensions of the wheel. In practice, for outward flow, r 2 ■*- r 2 is from 1.25 to 1.50; for inward flow, r 2 -s- r x is from 0.66 to 0.80. It appears that the inflow-wheel has a higher efficiency than the outward- flow wheel. The inflow-wheel also runs somewhat slower for best effect. The centrifugal force in the outward-flow wheel tends to force the water outward faster than it would otherwise flow ; while in the inward-flow wheel it has the contrary effect, acting as it does in opposition to the velocity in the buckets. It also appears that the efficiency of the outward-flow wheel increases slightly as the width of the crown is less and the velocity for maximum efficiency is slower ; while for the inflow-wheel the efficiency slightly in- creases for increased width of crown, and the velocity of the outer rim at the same time also increases. Efficiency. — The exact value of the efficiency for a particular wheel must be found by experiment. It seems hardly possible for the effective efficiency to equal, much less exceed, 86$, and all claims of 90 or more per cent for these motors should be discarded as improbable. A turbine yielding from 75$ to 80$ is extremely good. Experiments with higher efficiencies have been reported. The celebrated Tremont turbine gave 79*4$ without the " diffuser," which might have added some 2$. A Jonval turbine (parallel flow) was reported as yielding 0.75 to 0.90, but Morin suggested corrections reducing it to 0.63 to 0.71. Weisbach gives the results of many experiments, in which the effi- ciency ranged from 50$ to 84$. Numerous experiments give E = 0.60 to 0.65. The efficiency, considering only the energy imparted to the wheel, will ex- ceed by several per cent the efficiency of the wheel, for the latter will in- clude the friction of the support and leakage at the joint between the sluice and wheel, which are not included in the former ; also as a plant the resist- ances and losses in the supply-chamber are to be still further deducted. The Crowns.— The crowns may be plane annular disks, or conical, or curved. If the partitions forming the buckets be so thin that they may be discarded, the law of radial flow will be determined bv the form of the crowns. If the crowns be plane, the radial flow (or radial component) will diminish, for the outward flow-wheel, as the distance from the axis increases —the buckets being full — for the angular space will be greater. Prof. Wood deduces from the formulae in his paper the tables on page 595. It appears from[these tables: 1. That the terminal angle, a, has frequently been made too large in practice for the best efficiency. 2. That the terminal angle, a, of the guide should be for the inflow less than 10° for the wheels here considered, but when the initial angle of the bucket is 90°, and the terminal angle of the guide is 5° 28', the gain of effi- ciency is not 2$ greater than when the latter is 25°. 3. That the initial angle of the bucket should exceed 90° for best effect for outflow-wheels. 4. That with the initial angle between 60° and 120° for best effect on inflow wheels the efficiency varies scarcely 1$. 5. In the outflow-wheel, column (9) shows that for the outflow for best effect the direction of the quitting water in reference to the earth should be nearly radial (from 76° to 97°), but for the inflow wheel the water is thrown forward in quitting. This shows that the velocity of the rim should some- what exceed the relative final velocity backward in the bucket, as shown in columns (4) and (5). 6. In these tables the velocities given are in terms of Vtyh, and the co- efficients of this expression will be the part of the head which would produce that velocity if the water issued freely. There is only one case, column (5), where the coefficient exceeds unity, and the excess is so small it may be dis- carded; and it may be said that in a properly proportioned turbine with the conditions here given none of the velocities will equal that due to the head in the supply-chamber when running at best effect. 7. The inflow turbine presents the best conditions for construction for producing a given effect, the only apparent disadvantage being an increased first cost due to an increased depth, or an increased diameter for producing a given amount of work. The larger efficiency should, however, more than neutralize the increased first cost. TURBINE WHEELS. 595 8 N (D ^ O > !D !• 00 O O O O r4 4" l&cc.- &3 ^ m fcq ■S'lSS^&Ioi j5 g 55 £J ~ o © o © © © d o H°.S 1 tM , U O O « ^jj© £ a 's c ce « as L- i- 00 © 54°^ •£ _ .2 © t- CO In ^h £ "3 ^.m^ „ ft c 3 o 5 ti CO h ^ o w §S S S gi?.. |H||&3|H!|aj £ a £ ft ft ft ft o Q-o I oo © o © © ££ 6 a5 c a ItCJ jtCJ itcj ltd [? § ft ft ft ft >• g l«bi m « w>> .£l=j ? S3 o ■* to to in NOOn .S o M « «■* © >> o d d © tf.t 1 i- i |tr3|Hj|H;|tq P'Q ft ft ft ft > M 0* 1 0? 0* 1 0* ^^ « v-w> o © © oS °- c3 >> t-I d © d ti fe \ imituitnifti & ft & & l« «l«l7i ~tf II «. WW 1 I o co o in d odd > ~ H s ifccj Itcj Ifec? itq s ft ft ft ft t§\ l f l-l-^ ss £ CO [i S OS © OS 0O i- l- "a3 d d d d t> ieucy. 2 go co 66 os d d d d » 1 *2 Sic - o o o d «f *"* co os itEJ |tX3 IB33 |S=CS ft ft ft ft 1 0* 1 0* 1 OJ 1 a* rt NM0O © © © © 53 a > ft ft ft ft !0* la* la* I'M OS 00 OO -rf t, CO CO CO d odd N © o c; oo d d d d £ d d o o o os a* in 596 WATER-POWER. Tests of Turbines.— Emerson says that in testing turbines it is a rare thing to find two of the same size which can be made to do their best at the same speed. The best speed of one of the leading wheels is invariably wide from the tabled rate. It was found that a 54-in. Leffel wheel under 12 ft. head gave much better results at 78 revolutions per minute than at 90. Overshot wheels have been known to give 75$ efficiency, but the average performance is not over 60$. A fair average for a good turbine wheel may be taken at 75$. In tests of 18 wheels made at the Philadelphia Water-works in 1859 and 1860, one wheel gave less than 50$ efficiency, two between 50$ and 60$, six between 60$ and 70$, seven between 71$ and 77$, two 82^. and one 87.77$. (Emerson.) Tests of Turbine Wheels at the Centennial Exhibition, 1876. (From a paper by R. H. Thurston on The Systematic Testing of Turbine Wheels in the United States, Trans. A. S. M. E., viii. 359.)— In 1876 the judges at the International Exhibition conducted a series of trials of turbines. Many of the wheels offered for tests were found to be more or less defective in fitting and workmanship. The following is a statement of the results of all turbines entered which gave an efficiency of over 75$. Seven other wheels were tested, giving results between 65$ and 75$. Maker's Name, or Name the Wheel is Known By. 3 CO * 5 ^ O B m ® Ph 87.68 83.79 83.30 82.13 81.21 78.70 79.59 77.57 77.43 76.94 76.16 75.70 75.15 £ 03 go U Ph 3.2 OQ « . a P o°5 Ph 3 w -^ . Ph 3.2? B P 3*5 ^NOD-B Ph J c8 Ph 75.35 3.22 OQ ce — BO S, Ph 86.20 82.41 70.79 Geyelin (single) Thos. Tait 71.66 Yi.bV 81.24 70.40 55.90 68.60 79.92 66.35 51.03 67.23 55.00 Tyler Wheel Geyelin (duplex) 69.59 74.25 73.88 74.89 62.75 69.92 70.87 62.06 71.74 York Manufacturing Co. . . W. F. Mosser & Co 67.08 71.90 67.57 70.52 66.04 The limits of error of the tests, says Prof. Thurston, were very uncertain; they are undoubtedly considerable as compared with the later work done in the permanent flume at Holy oke— possibly as much as 4$ or 5$. Experiments with "draught-tubes," or "suction-tubes," which were actually " diff users " in their effect, so far as Prof. Thurston has analyzed them, indicate the loss by friction which should be anticipated in such cases, this loss decreasing as the tube increased in size, and increasing as its diameter approached that of the wheel— the miuimum diameter tried. It was sometimes found very difficult to free the tube from air completely, and next to impossible, during the interval, to control the speed with the brake. Several trials were often necessary before the power due to the full head could be obtained. The loss of power by gearing and by belting was variable with the proportions and arrangement of the gears and pulleys, length of belt, etc.. but averaged not far from 30$ for a single pair of bevel- gears, uncut and dry, but smooth for such gearing, and but 10$ for the same gears, well lubricated, after they had been a short time in operation. The amount of power transmitted was, however, small, and these figures are probably much higher than those representing ordinary practice. Intro- ducing a second pair— spur-gears— the best figures were but little changed, although the difference between the case in which the larger gear was the driver, and the case in which the small wheel was the driver, was perceiv- able, and was in favor of the former arrangement. A single straight belt gave a loss of but 2$ or 3$, a crossed belt 6$ to 8$, when transmitting 14 TURBINE WHEELS. 597 horsepower with maximum tightness and transmitting power. A " quarter turn ■" wasted about 10;? as a maximum, and a "quarter twist" about 5%. Dimensions of Turbines.— For dimensions, power, etc., of stand- ard makes of turbines consult the catalogues of different manufacturers. The wheels of different makers vary greatly in their proportions for any given capacity. The Pelton Water-wheel.— Mr. Ross E. Browne (Eng'g News, Feb. 20, 1892) thus outlines the principles upon which this water-wheel is constructed : The function of a water-wheel, operated by a jet of water escaping from a nozzle, is to convert the energy of the jet, due to its velocity, into useful work In order to utilize this energy fully the wheel-bucket, after catching the jet, must bring it to rest before discharging it, without inducing turbu- lence or agitation of the particles. This cannot be fully effected, and unavoidable difficulties necessitate the loss of a portion of the energy. The principal losses occur as follows: First, in sharp or angular diversion of the jet in entering, or in its course through the bucket, causing impact, or the conversion of a portion of the energy into heat instead of useful work. Second, in the so-called frictional resistance offered to the motion of the water by the wetted surfaces of the buckets, causing also the conversion of a portion of the energy into heat instead of useful work, Third, in the velocity of the water, as it leaves the bucket, representing energy which has not been converted into work. Hence, in seeking a high efficiency: 1. The bucket-surface at theentrance should be approximately parallel to the relative course of the jet, and the bucket should be curved in such a manner as to avoid sharp angular de- flection of the stream. If, for example, a jet strikes a surface at an angle and is sharply deflected, a portion of the water is backed, the smoothness of the stream is disturbed, and there results considerable loss by impact and other- wise. The entrance and deflection in the Pelton bucket are such as to avoid these losses in the main. (See Fig. 136.) 2. The number of buckets should be small, and the path of the jet in the bucket short; in other words, the total wetted surface should be small, as the loss by friction will be proportional to this. 3. The discharge end of the bucket should be as nearly tangential to the wheel periphery as compatible with the clearance of the bucket which follows; and great differences of velocity in the parts of the escaping water should be avoided. In order to bring the water to rest at the discharge end of the bucket, it is shown, mathematically, that the velocity of the bucket should be one half the velocity of the jet. A bucket, such as shown in Fig. 135, will cause the heaping of more or less dead or turbulent water at the point indicated by dark shading. This dead water is subsequently thrown from the wheel with considerable velocity, and represents a large loss of energy. The introduction of the wedge in the Pelton bucket (see Fig. 134) is an efficient means of avoiding this loss. A wheel of the form of the Pelton conforms closely in construction to each of these requirements. In a test made by the proprietors of the Idaho mine, Fig 136 near Grass Valley, Cal., the dimensions and results were as follows : Main supply-pipe, 22 in. diameter, 6900 ft. long, with a head of 386^ feet above centre of nozzle. The loss by friction in the pipe was 1.8 ft., reducing the effective head to 384.7 ft. The Pelton wheel used in the test was 6 ft. in diameter and the nozzle was 1.89 in. diameter. The work done was measured by a Prony brake, and the mean of 13 tests showed a useful effect of 87.3#. The Pelton wheel is also used as a motor for small powers. A test by M. E. Cooley of a 12-inch wheel, with a %-inch nozzle, under 100 lbs. pressure, gave 1.9 horse-power. The theoretical discharge was .0935 cubic feet per second, and the theoretical horse-power 2.45; the efficiency being 80 per cent. Two other styles of water-motor tested at the same time each gave efficiencies of 55 per cento Fig. 134. Fig. 135. 598 WATER-POWER. Pelton Water-wheel Tables. (Abridged.) The smaller figures uuder those denoting the various heads give the spouting velocity of the water in feet per minute. The cubic-feet measure- ment is also based on the flow per minute. Head in ft. Size of Wheels. 6 in No.l .05 1.67 684 12 in. No. 2 18 in. No. 3 18 in. No. 4 24 in. No. 5 3 ft. 4 ft. 5 ft. 6 ft. 20 2151.97 Horse-power. Cubic feet.... Revolutions.. .12 3.91 342 .20 6.62 228 .37 11.72 228 .66 20.83 171 1 50 46.93 114 2.64 83.32 85 4.18 130.36 70 6.00 187.72 57 30 2635.62 Horse-power. Cubic feet .... Revolutions.. .10 2.05 837 .23 4.79 418 .38 8.11 279 .69 14.36 279 1.22 25.51 209 2.76 57.44 139 4.88 102.04 104 7.69 159.66 83 11.04 229.76 69 40 3043.39 Horse-power. Cubic feet. . . . Revolutions.. .15 2.37 969 .35 5.53 484 .59 9.37 323 1.06 16.59 323 1.89 29.46 242 4.24 66.36 161 7.58 107.84 121 11.85 184.36 96 16.96 265.44 80 50 3402.61 Horse-power. Cubic feet.... Revolutions.. .2! - 1083 .49 6.18 541 .84 10.47 361 1.49 18.54 361 2.65 32.93 270 5.98 74.17 180 10.60 131.72 135 16.63 206.13 108 23.93 296.70 90 60 Horse-power. Cubic feet Revolutions.. Horse-power. Cubic feet Revolutions.. .28 11 85 .35 3.13 1281 .65 6.77 592 .82 7.31 640 1.10 11.47 395 1.39 12.39 427 1.96 20.31 395 2.47 21.94 427 3.48 36.08 296 7.84 81.25 197 13.94 144.32 148 21.77 225.80 118 31.36 325.00 98 70 4026.00 4.39 38.97 320 9.88 87.76 213 17.58 155.88 160 27.51 243.89 130 39.52 351.04 106 80 4303.99 Horse-power. Cubic feet Revolutions.. .43 3.35 1308 1.00 7.82 684 1.70 13 25 456 3.01 23.46 456 5.36 41.66 342 12.04 93 84 228 21.44 166.64 171 33.54 260.73 137 48.16 375.36 114 90 4565.04 Horse-power. Cubic feet Revolutions. . .51 3.55 1152 1.20 8.29 726 2.03 14.05 481 3.60 24.88 484 6.39 44.19 363 14.40 99.52 242 25.59 176.75 181 40.04 276.55 145 57.60 398.08 121 100 4812.00 Horse-power. Cubic feet Revolutions.. .60 3.74 1530 1.40 8.74 765 2.32 14.81 510 4.21 26.22 510 7.49 46.58 382 16.84 104.88 255 29.93 186.32 191 46.85 291.51 152 67.36 419.52 127 120 5271.30 Horse-power. Cubic feet Revolutions.. Horse-power. Cubic feet. .. Revolutions.. .79 4.10 1677 .09 4 . 43 1812 1 84 9.57 838 2.33 10.31 906 3.12 16.21 559 5.54 28.72 559 9.85 51.02 419 22.18 114.91 279 39.41 204.10 209 61.66 319.33 167 88.75 459.64 139 140 5693.65 3.94 17.53 604 4.82 18.74 646 6.99 31.03 604 12.41 55.11 453 27.96 124.12 302 49.64 220.44 226 77.71 344.92 181 111.85 496.48 151 160 6086 74 Horse-power. Cubic feet Revolutions.. 1.22 4.73 1938 2.84 1.1.05 969 8.54 33.17 646 15.17 58.92 484 34.16 132.68 323 60.68 235.68 242 94.94 368.73 193 136.65 530.75 161 180 6455.97 Horse power. Cubic feet.. . Revolutions. . 1.45 3.39 11.72 1024 5.75 19.87 683 10.19 35.18 683 18.10 62.49 513 40.77 140.74 342 72.41 249.97 256 113.30 391.10 206 163.08 562.96 171 200 6805.17 Horse-power. Cubic feet Revolutions.. 1.70 2100 3.97 12.36 1080 6.74 20.94 720 11.93 37.08 720 21.20 65.87 540 47.75 148.35 360 84.81 263.49 270 132.70 412 25 216 191.00 593.40 180 250 7608.44 Horse power. Cubic feet. . . . Revolutions. . 5.56 13.82 1209 9.42 23.42 800 16.68 41.46 806 29.63 73.64 605 66.74 165.86 403 118.54 291.59 302 185.4? 460.91 241 266.96 663.45 202 POWER OF OCEAN WAVES. 59§ Pelton Water-wlieel Tables.— Continued. Head in ft. Size of Wheels. in. No.l 12 in. No. 2 18 in. No. 3 18 in. No. 4 24 in. No. 5 3 ft. 4 ft. 1 5 ft. 6 ft. 300 8334.62 Horse-pow'r Cubic feet... Revolutions 3.13 6.48 2G52 7.31 15.13 1326 12.38 25.66 884 21.93 45.42 884 38. 95 1 87.73 80.67 181.69 663 j 442 155.83 322.71 331 243.82 504.91 265 350.94 726.76 221 350 9002.43 Horse-pow'r Cubic feet... Revolutions 3.94 7.00 9.21 16.35 1432 15.61 27.71 955 27.64 49.06 955 49.09 110.56 87.14 196.25 716 477 196.38 348.57 358 307.25 545.36 285 442.27 785.00 238 400 9624.00 Horse-pow'r Cubic feet... Revolutions 4.82 7.40 11.25 17.48 1531 19.0 29.63 1021 33.77 52.45 1021 59.98 93.16 765 135.08 209.80 510 239.94 372.64 382 375.40 583.02 306 540.35 839.20 255 450 10207.79 Horse-pow'r Cubic feet... Revolutions 5.75 7.94 3-219 13.43 18.54 1624 22.76 31.42 1083 40.29 55.63 1083 71.57 98.81 812 161.19 222.52 541 188.80 234.56 571 286.31 395.24 406 447.95 618.38 324 644.78 890.11 270 500 10759.96 Horse-pow'r Cubic feet... Revolutions 6.74 8.37 3420 15.73 19.54 1713 26.66 33.12 1142 47.20 58.64 1142 83.83 104.15 856 335.34 416.62 428 524.66 651.83 342 755.20 938.25 285 600 Horse-pow'r 62.04 64.24 1251 110.19 114.09 938 248.16 256.95 625 440.77 456.38 469 689.63 714.05 375 992.65 1027.80 11786.94 Revolutions — '- 312 650 Horse-pow'r 69.95 66.86 1302 124.25 118.75 976 279.82 267.44 651 497.01 475.02 488 777.62 743.21 390 1119.29 1069.77 12268.24 Revolutions 325 700 Horse-pow'r 78.18 69.38 1351 138.86 123.23 1013 675 555.46 492.95 506 869.06 771.26 405 1250.92 1110.16 12731.34 Revolutions 337 750 13178.19 Horse-pow'r Cubic feet... Revolutions 86.70 71.82 1399 154.00 127.56 1049 346.83 - 699 616.03 510.25 524 963 . 82 798.33 419 1387.34 1149.13 319 800 13610.40 Horse-pow'r Cubic feet... Revolutions 95.52 74.17 1444 169.66 131.74 1083 382.09 296.70 722 678.66 526.99 542 1061.81 824.51 433 1528.36 1186.81 361 900 Horse-pow'r Cubic feet... Revolutions 113.98 78.67 1532 202.45 139.74 1149 455.94 314.70 766 809.82 558.96 574 1267.02 874.53 459 1823.76 14436.00 11111 1258.81 383 1000 Horse-pow'i 133 50 237.12 147.30 1210 534.01 80? 948.48 589.19 605 1483.9? 921 . S3 2136 04 82.93 1615 132(i.91 15216.89 Revolutions 484' 403 THE POWER OF OCEAN WAVES. Albert W. Stahl, U. S. N. (Trans. A. S. M. E., xiii. 438), gives the following formulae and table, based upon a theoretical discussion of wave motion: The total energy of one whole wave-length of a wave iJfeet high, L feet long, and one foot in breadth, the length being the distance between succes- sive crests, and the height the vertical distance between the crest and the trough, is E= 8LH* (l - 4.935 ^) foot-pounds. The time required for each wave to travel through a distance equal to its own length is P = a/ r— ^~ seconds, and the number of waves passing any 600 WATER-POWER. 60 given point in one minute is N of an indefinite series of such waves, expressed in horse-power per foot of breadth, is 04/ —7— • Hence the total energy T~L(\ By substituting various values for H -f- L, within the limits of such values actually occurring in nature, we obtain the following table of Total Energy of Deep-sea Waves in Terms op Horse-power per Foot op Breadth. Ratio of Length of Waves to Length of Waves in Feet. Height of Waves. 25 50 75 100 150 200 [300 400 50 .04 .23 .64 1.31 3.62 7.43 20.46 42.01 40 .06 .36 1.00 2.05 5.65 11.59 31.95 65.58 30 .12 .64 1.77 3.64 10.02 20.57 56.70 116.38 20 .25 1.44 3.96 8.13 21 79 45.98 120.70 260.08 15 .42 2.83 6.97 14.31 39.43 80.94 223.06 457.89 10 .98 5.53 15.24 31.29 86.22 177.00 487.75 1001.25 5 3.30 18.68 51 48 105.68 291.20 597.78 1647.31 3381.60 The figures are correct for trochoidal deep-sea waves only, but they give a close approximation for any nearly regular series of waves in deep water and a fair approximation for waves in shallow water. The question of the practical utilization of the energy which exists in ocean waves divides itself into several parts : fe 1. The various motions of the water which may be utilized for power purposes. 2. The wave motor proper. That is, the portion of the apparatus in direct contact with the water, and receiving and transmitting the energy thereof ; together with the mechanism for transmitting this energy to the machinery for utilizing the same. C. Regulating devices, for obtaining a uniform motion from the irregular and more or less spasmodic action of the waves, as well as for adjusting the apparatus to the state of the tide and condition of the sea. 4. Storage arrangements for insuring a continuous and uniform output of power during a calm, or when the waves are comparatively small. The motions that may be utilized for power purposes are the following: 1. Vertical rise and fall of particles at and near the surface. 2. Horizontal to-and-fro motion of particles at and near the surface. 3. Varying slope of surface of wave. 4. Impetus of waves rolling up the beach in the form of breakers. 5. Motion of distorted verticals. All of these motions, except the last one mentioned, have at various times been proposed to be utilized for power purposes; and the last is proposed to be used in apparatus described by Mr. Stahl. The motion of distorted verticals is thus defined: A set of particles, origi- nally in the same vertical straight line when the water is at rest, does not remain in a vertical line during the passage of the wave; so that the line connecting a set of such particles, while vertical and straight in still water, becomes distorted, as well as displaced, during the passage of the wave, its upper portion moving farther and more rapidly than its lower portion. Mr. StahPs paper contains illustrations of several wave-motors designed upon various principles. His conclusions as to their practicability is as fol- lows: " Possibly none of the methods described in this paper may ever prove commercially successful; indeed the problem may not be susceptible of a financially successful solution. My own investigations, however, so far as I have yet been able to carry them, incline me to the belief that wave-power can and will be utilized on a paying basis." Continuous Utilization of Tidal Power. (P. Decoeur, Proc. Inst. C. E. 1890.)— In connection with the training-walls to be constructed in PUMPS AND PUMPING ENGINES. 601 the estuary of the Seine, it is proposed to construct large basins, by means of which the power available from the rise and fall of the tide could be util- ized. The method proposed is to have two basins separated by a bank rising above high water, within which turbines would be placed. The upper basin would be in communication with the sea during the higher one third of the tidal range, rising, and the lower basin during the lower one third of the tidal range, falling. If H be the range in feet, the level in the upper basin would never fall below %H measured from low water, and the level in the lower basin would never rise above X /&H. The available head varies between 0.53Hand 0.80H, the mean value being %H. If S square feet be the area of the lower basin, and the above conditions are fulfilled, a quantity l/'dSH cu. ft. of water is delivered through the turbines in the space of 9*4 hours. The mean flow is, therefore, SH -f- 99,900 cu. ft. per sec , and, the mean fall being ?£H, the available gross horse-power is about l/306"i? 2 , where S' is measured in acres. This might be increased by about one third if a variation of level in the basins amounting to %H were permitted. But to reach this end the number of turbines would have to be doubled, the mean head being reduced to J^i?, and it would be more difficult to transmit a constant power from the turbines. The turbine proposed is of an improved model designed to utilize a large flow with a moderate diameter. One has been designed to produce 300 horse-power, with a minimum head of 5 ft. 3 in. at a speed of 15 revolutions per minute, the vanes having 13 ft. internal diameter. The speed would be maintained constant by regulating sluices. PUMPS AND PUMPING ENGINES. Theoretical Capacity of a Pump.— Let Q' = cu. ft. per min.; G' = Amer. gals, per min. = 7.4805(5'; d = diam. of pump in inches; I — stroke in inches; N = number of single strokes per min. 77- d 2 IN Capacity in cu. ft. per min. = Q' = - ■ — — - . -- - .0004545iVd 2 Z; ,7 NdW Capacity in gals, per min. G' = - . —— = .OOMNdH; Capacity in gals, per hour = .204iVd 2 Z. Diameter required for ^^^ — Aaq./Q^ _ ir- -.c . / & given capacity per min. \ 1/ jVZ 1/ jyi ' If v — piston speed in feet per min., d = 13.54 j/ jL == 4.95a / . If the piston speed is 100 feet per min.: Nl = 1200, and d = 1.354 V~Q' = .495 V~G'\ G' = 4.08cZ 2 per min. The actual capacity will be from 60% to 95% of the theoretical, according to the tightness of the piston, valves, suction-pipe, etc. Theoretical Horse-power required to raise "Water to a given Height.— Horse-power == Volume in cu. ft. per min. X pressure per sq. ft. _ Weight x height of lift 33,000 ~ " 33,000 ' Q' == cu. ft. per min.; G' — gals, per min.; W — wt. in lbs.; P = pressure in lbs. per sq. ft.; p — pressure in lbs. per sq. in.; H = height of lift in ft.: W- 62.36Q', P= U4p,p = .433iJ, H = 2.309p, G' = 7.4805Q'. Q'P _ 33,000 Q'H X 144 X .433 Q'H G'H _ 33,000 ~ 529.2 ~ 3958.7' WH 33,000 ~ Q' X 62.36 X 2.309p Q'p G'p 33,000 ~ 229.2 ~ 1714.5' For the actual horse-power require/1 an allowance must be made for the friction, slips, etc., of engine, pump, valves, and passages, 602 WATER-POWER. Depth of Suction.— Theoretically a perfect pump will draw water from a height of nearly 34 feet, or the height corresponding to a perfect vacuum (14.7 lbs. X 2.309 = 33.95 feet); but since a perfect vacuum cannot be obtained, on account of valve-leakage, air contained in the water, and the vapor of the water itself, the actual height is generally less than 30 feet. When the water is warm the height to which it can be lifted by suction de- creases, on account of the increased pressure of the vapor. In pumping hot water, therefore, the water must flow into the pump by gravity. The fol- lowing table shows the theoretical maximum depth of suction for different temperatures, leakage not considered: Temp. F. Absolute Pressure ofVapor, lbs. per sq. in. Vacuum in Inches of Mercury. Max. Depth of Suction, feet. Temp. F. Absolute Pressure ot Vapor, lbs. per sq. in. Vacuum in Inches of Mercury. Max. Depth of Suction, feet. 101.4 126.2 144.7 153.3 162.5 170.3 177.0 1 2 3 4 5 6 7 27.88 25.85 23.81 21.77 19.74 17.70 15.66 31.6 29.3 27.0 24.7 22.4 20.1 17.8 183.0 188.4 193.2 197.6 201.9 205.8 209.6 8 9 10 11 12 13 14 13.63 11.59 9.55 7.51 5.48 3.44 1.40 15.5 13.2 10.9 8.5 6.2 3.9 1.6 Amount of Water raised by a Single-acting Lift-pump. —It is common to estimate that the quantity of water raised b}' a single-acting bucket-valve pump per minute is equal to the number of strokes in one direction per minute, multiplied by the volume traversed by the piston in a single stroke, on the theory that the water rises in the pump only when the piston or bucket ascends; but the fact is that the column of water does not cease flowing when the bucket descends, but flows on con- tinuously through the valve in the bucket, so that the discharge of the pump, if it is operated at a high speed, may amount to nearly double that calculated from the displacement multiplied by the number of single strokes in one direction. Proportioning the Steam-cylinder of a Direct-acting Pump.— Let A — area of steam-cylinder; a — area of pump-cylinder; D = diameter of steam-cylinder; d = diameter of pump-cylinder; P = steam-pressure, lbs. per sq. in. ; p = resistance per sq. in. on pumps; H= head = 2.309p; p = ASSH: _, „ . . .., work done in pump-cylinder E — efficiency of the pump = ■ r—j , ^. „ — t^-t- • work done by the steam-cyhnder •^ — T7TD a}) _ EP' EAP p y EP' \ p EA EAP H = 2.B09EP — ; If E = 75*, H = 1 .732P - A - JL - - 433g . a~ EP ~ EP ' E is commonly taken at 0.7 to 0.8 for ordinary direct-acting pumps. For the highest class of pumping-engines it may amount to 0.9. The steam- pressure Pis the mean effective pressure, according to the indicator-dia- gram; the water-pressure p is the mean total pressure acting on the pump plunger or piston, including the suction, as could be shown by an indicator- diagram of the water-cylinder. The pressure on the pump-piston is fre- quently much greater than that due to the height of the lift, on account of the friction of the valves and passages, which increases rapidly with velocity of flow. Speed of Water through Pipes and Pump-passages.— The speed of the water is commonly from 100 to 200 feet per minute. If 200 feet per minute is exceeded, the loss from friction may be considerable. The diameter of pipe required is 4. _ / gallons per minute \ velocity in feet per minute' For a velocity of 200 feet per minute, diameter =,35 x ^'gallons per roiq. PUMPS. 003 Sizes of Direct-acting Pumps.— The two following tables are se- lected from catalogues of manufacturers, as representing the two common types of direct-acting pump, viz., the single-cylinder and the duplex. Both types are now made by most of the leading manufacturers. The Deane Direct-acting Pump. Standard Sizes for Ordinary Service. a | = 55 - £ 6 6 o3 fl a ft M o p : S Capacity per Minute ■B "5) B en ■S . S a a? Ift o ho O Z ■- a CD a5 at G lven 32 > 03 03 O 03 cS ft 5 -a oft .2 ft a w ■- 1. i ■- °.a a ft O "5 ft 03 o Speed. Stks. Gals. os u cc« w o 0) 5 o Q 5 *3 O OQ S H w w 33 w 4 3i o 5 .14 1 to 300 130 18 33 9k k Va 2 Ik 4 4 ~ 5 .27 1 to 300 130 35 33 9k k % 2 ik 5 4 7 .39 1 to 300 125 49 45k 15 H 3 2k m 5 7 .51 1 to 275 125 64 45k 15 % 3 2k 5Va m 7 .72 1 to 275 125 90 45k 15 H 3 2k 7 10 1.64 1 to 250 110 180 58 17 l Ik 5 4 7k 7L> 10 1.91 1 to 250 110 210 58 17 l Ik 5 4 10 8 14 14 24 15.99 1 to 150 50 800 112 34 2 2k 12 10 14 16 16 13.92 1 to 175 80 1114 84 34 2 2k 2k 12 10 14 16 24 20.88 1 to 150 50 1044 112 38 2 12 10 16 14 18 12.00 1 to 175 70 840 89 27 2 2k 8 8 16 14 24 15.99 1 to 150 50 800 109 34 2 2k 12 10 16 16 16 13.92 1 to 175 80 1114 85 34 2 2k 12 10 16 1Q 24 20.88 1 to 150 50 1044 115 34 2 2k 12 10 16 18 24 26.43 1 to 125 50 1322 115 40 2 2k 14 12 18 16 24 20.88 1 to 125 50 1044 118 38 3 3k 12 10 18 18 24 26.43 1 to 125 50 1322 118 40 3 3k 14 12 18 20 24 32.64 1 to 125 50 1632 118 40 3 3k 16 14 20 18 24 26.43 1 to 125 50 1322 118 40 3 3k 14 12 20 20 24 32.64 1 to 125 50 1632 118 40 3 3k 16 14 20 22 24 39.50 1 to 125 50 1975 120 40 3 3k 18 14 Efficiency of Small Direct-acting Pumps.— Chas. E. Emery, in Reports of Judges of Philadelphia Exhibition, 1876, Group xx., says : "Ex- periments made with steam-pumps at the American Institute Exhibition of 1867 showed that average sized steam-pumps do not, on the average, utilize more than 50 per cent of the indicated power in the steam-cylinders, the re- mainder being absorbed in the friction of the engine, but more particularly in the passage of the water through the pump. Again, all ordinary steam- pumps for miscellaneous uses require that the steam -cylinder shall have three to four times the area of the water-cylinder to give sufficient power 604 WATER-POWER. when the steam is accidentally low; hence as such pumps usually work against the atmospheric pressure, the net or effective pressure forms a small percentage of the total pressure, which, with the large extent of radiating surface exposed and the total absence of expansion, makes the expenditure of steam ve\y large. One pump tested required 120 pounds weight of steam per indicated horse-power per hour, and it is believed that the cost will rarely fall below 60 pounds ; and as only 50 per cent of the in- dicated power is utilized, it may be safely stated that ordinary steam-pumps rarely require less than 120 pounds of steam per hour for each horse-power utilized in raising water, equivalent to a duty of only 15,000,000 foot-pounds per 100 pounds of coal. With larger steam-pumps, particularly when they are proportioned for the work to be done, the duty will be materially in- creased. " The Worthlngton Duplex Pump. Standard Sizes for Ordinary Service. ki =1 . Sizes of Pipes for S to-s 8*0 g = a 2 1* ft ft S 0> ft 'ft | © ft "ft a a 'ft 5 I 03 g c3 tx c ID fao -S'3'fe c3^^ .2 ss oj -f 3 £ 3 s J 5 ft q CO ft 3 2 3 .04 100 to 250 8 to 20 W/a 1J4 l *H Wa 4 .10 100 to 200 20 to 40 4 Vz i J i 2 i« 514 Wz 5 .20 100 to 200 40 to 80 5 2^ m 6 4 6 .33 100 to 150 70 to 100 5% 1 3 2 W% Q& G .42 100 to 150 85 to 125 6% m 2 4 3 TA 5 G .51 100 to 150 100 to 150 7 n/ 2 2 4 3 w* 4^ 10 .69 75 to 125 100 to 170 6% 2 4 3 9 5'/4 10 .93 75 to 125 135 to 230 7^ ■i ^% 4 3 10 6 10 1.22 75 to 125 180 to 300 8fc& 2 5 4 10 7 10 1.66 75 to 135 245 to 410 9% 2 2^ 6 5 12 7 10 1.66 75 to 125 245 to 410 9% 2J^ 3 6 5 14 7 10 1.66 75 to 125 215 to 410 9% 3 6 5 12 Wz 10 2.45 75 to 125 365 to 610 12 3 6 5 14 m 10 2.45 75 to 125 365 to 610 12 3 6 5 16 8^2 10 2.45 75 to 125 365 to 610 12 3 6 5 18)4 32 10 2.45 75 to 125 365 to 610 12 3 o% 6 5 20 10 2.45 75 to 125 365 to 610 12 4 5 6 5 12 1014 10 3.57 75 to 125 530 to 890 1414 •2 l/o 3 8 7 14 1014 10 3.57 75 to 125 530 to 890 1414 3 8 7 16 1014 10& 10 3.57 75 to 125 530 to 890 14J4 3 8 7 18V6 10 3.57 75 to 125 530 to 890 14J4 3 31 -a 8 7 20 io# 10 3.57 75 to 125 530 to 890 14J4 4 5 8 7 14 12 10 4.89 75 to 125 730 to 1220 17 3 10 8 16 12 10 4.89 75 to 125 730 to 1220 17 3 10 8 18^ 12 10 4.89 75 to 125 730 to 1220 17 3 3^ 10 8 20 12 10 4.89 75 to 125 730 to 1220 17 4 5 10 8 \&y 2 14 10 6.66 75 to 125 990 to 1660 19% 3 12 10 20 14 10 6.66 75 to 122 990 to 1660 1 5 12 10 17 10 15 5.10 50 to 100 510 to 1020 14 3 10 8 20 12 15 7.34 50 to 100 730 to 1460 17 4 5 12 10 20 15 15 11.47 50 to 100 1145 to 2290 21 25 15 15 11.47 50 to 100 1145 to 2290 21 PUMPS. 605 Speed of Piston.— A piston speed of 100 feet per minute is commonly assumed as correct in practice, but for short-stroke pumps this gives too high a speed of rotation, requiring too frequent a reversal of the valves.' For long stroke pumps, 3 feet and upward, this speed may be considerably exceeded, if valves and passages are of ample area. Number of Strokes required to Attain a Piston Speed from 50 to 125 Feet per Minute for Pumps having Strokes from 3 to 18 Inclies in Length. s! . Length of Stroke in Inches. 9 & &5 p. Number of Strokes per Minute. 50 200 150 120 100 86 75 60 50 40 33 55 220 165 132 110 94 82.5 66 55 44 37 60 240 180 144 120 103 90 72 60 48 40 65 260 195 156 130 111 97.5 78 65 52 43 70 280 210 168 140 120 105 84 70 56 47 75 300 225 180 150 128 112.5 90 75 60 50 80 320 240 192 160 137 120 96 80 64 53 85 340 255 204 170 146 127.5 102 85 68 57 90 360 270 216 180 154 135 108 90 72 60 95 380 285 228 190 163 142.5 114 95 76 63 100 400 300 240 200 171 150 120 100 80 67 105 420 315 252 210 180 157.5 126 105 84 70 110 440 330 264 220 188 165 132 110 88 73 115 460 345 276 230 197 172.5 138 115 92 77 120 480 360 288 ,240 206 180 144 120 96 80 125 500 375 300 250 214 187.5 150 125 100 83 Piston Speed of Pumping-engines. (John Birkinbine, Trans. A. I. M. E., v. 459.)— In dealing with such a ponderous and unyielding sub- stance as water there are many difficulties to overcome in making a pump work with a high piston speed. The attainment of moderately high speed is, however, easily accomplished. Well-proportioned pumping-engines of large capacity, provided with ample water-ways and properly constructed valves, are operated successfully against heavy pressures at a speed of 250 ft. per minute, without "thug.'" concussion, or injury to the apparatus, and there is no doubt that the speed can be still further increased. Speed of Water through Valves.— If areas through valves and water passages are sufficient to give a velocity of 250 ft. per min. or less, they are ample. The water should be carefully guided and not too abruptly deflected. (F. W. Dean. Eng. News, Aug. 10, 1893.) Boiler-feed Pumps.— Practice has shown that 100 ft. of piston speed per minute is the limit, if excessive wear and tear is to be avoided. The velocity of water through the suction-pipe must not exceed 200 ft. per minute, else the resistance of the suction is too great. The approximate size of suction-pipe, where the length does not exceed 25 ft. and there are not more than two elbows, may be found as follows : 7/10 of the diameter of the cylinder multiplied by 1/100 of the piston speed in feet. For duplex pumps of small size, a pipe one size larger is usually employed. The velocity of flow in the discharge-pipe should not exceed 500 ft. per minute. The volume of discharge and length of pipe vary so greatly in different installations that where the water is to be forced more than 50 ft. the size of discharge-pipe should be calculated for the particular conditions, allowing no greater velocity than 500 ft. per minute. The size of discharge-pipe is calculated in single-cylinder pumps from 250 to 400 ft. per minute. Greater velocity is permitted in the larger pipes. , In determining the proper size of pump for a steam-boiler, allowances must be made for a supply of water sufficient to cover all the demands of engines, steam-heating, etc., up to the capacity of generator, and should not be calculated simply according to the requirements of the engine. In prac- tice engines use all the way from 12 up to 50, or more, pounds of steam per H.P. per hour when being worked up to capacity. When an engine is over- loaded or underloaded more water per H.P. will be required than when operating at its rated capacity. The average run of horizontal tubular 606 WATER-POWER. boilers will evaporate from 2 to 3 lbs. of water per sq. ft. of heating-surface per hour, but may be driven up to 6 lbs. if the grate-surface is too large or the draught too great for economical working. Pump- Valves.— A. F. Nagle (Trans. A. S. M. E., x. 521) gives a number of designs with dimensions of double-beat or Cornish valves used in large pumping-engines, with a discussion of the theory of their proportions. The following is a summary of the proportions of the valves described. Summary of Valve Proportions. Location of Engine. Providence high-ser- vice engine Providence Cornish- engine St. Louis Water Wks, Milwaukee " " Chicago " " wood seats , Chicago Water Wks 1 lb. reduced to .66 lb. 1.41 1.31 1.16 .96 ®|5 $ a r CO «D »- 3 o* 73 < sw J£t3 Hi h »-i a 3 a ce o a o* 3 «8 CO c«5 £^.S 16% 377 lbs. 12 680 67 250 88 120 75 151 85 140 94 132 75 151 Good Some noise Some noise at high speed. Noisy Mr. Nagle says : There is one feature in which the Cornish valves are necessarily defective, namely, the lift must always be quite large, unless great power is sacrificed to reduce it. It is undeniable that a small lift is prefer- able to a great one, and hence it naturally leads to the substitution of numerous small valves for one or several large ones. To what extreme re- duction of size this view might safely lead must be left to the judgment of the engineer for the particular case in hand, but certainly, theoretically, we must adopt small valves. Mr. Corliss at one time carried the theory so far as to make them only 1% inches in diameter, but from 3 to 4 inches is the more common practice now. A small valve presents proportionately a larger surface of discharge with the same lift than a larger valve, so that whatever the total area of valve-seat opening, its full contents can be dis- charged with less lift through numerous small valves than with one large one. Henry R. Worthington was the first to use numerous small rubber valves in preference to the larger metal valves. These valves work well under all the conditions of a city pumping-engine. A volute spring is generally used to limit the rise of the valve. In theLeavitt high-duty sewerage-engine at Boston (Am. Machinist. May 31, 1884), the valves are of rubber, %-inch thick, the opening in valve-seat being 13^ x 4^ inches. The valves have iron face and back-plates, and form their own hinges. CENTRIFUGAL PUMPS. Relation of Height of Iiift to Velocity.— The height of lift depepds only on the tangential velocity of the circumference, every tangen- tial velocity giving a constant height of lift— sometimes termed "head "— whether the pump is small or large. The quantity of water discharged is in proportion to the area of the discharging orifices at the circumference, or in proportion to the square of the diameter, when the breadth is kept the same. R. H. Buel (App. Cyc. Mech., ii, 606) gives the following: Let Q represent the quantity of water, in cubic feet, to be pumped per minute, h the height of suction in feet, h' the height of discharge in feet, and d the diameter of suction-pipe, equal to the diameter of discharge-pipe, in CENTRIFUGAL PUMPS. f V'2g 607 , g being the accel- feet; then, accordingto Fink, d = .-„- f \2g (h + h') eration due to gravity. If the suction takes place on one side of the wheel, the inside diameter of the wheel is equal to 1 .2d, and the outside to 2 Ad. If the suction takes place at both sides of the wheel, the inside diameter of the wheel is equal to 0.85d, and the outside to 1.7d. Then the suction-pipe will have two branches, the area of each equal to half the area of d. The suction-pipe should be as short as possible, to prevent air from entering the pump. The tangenti al velocity of the outer edge of wheel for the delivery Q is equal to 1.25 \'2y{h -\-h') feet per second. The arms are six in number, constructed as follows : Divide the central angle of 60°, which incloses the outer edges of the two arms, into any num- ber of equal parts by dividing the radii, and divide the breadth of the wheel in the same manner by drawing concentric circles. The intersections of the several radii with the corresponding circles give points of the arm. In experiments with AppokTs pump, a velocity of circumference of 500 ft. per min. raised the water 1 ft. high, and maintained it at that level without discharging any; and double the velocity raised the water to four times the height, as the centrifugal force was proportionate to the square of the velocity; consequently, 500 ft. per min. raised the water 1 ft. without discharge. 1000 " lv " " •' 4 " 2000 " " " " " 16 " 4000 " " " " " 64 " The greatest height to which the water had been raised without discharge, in the experiments with the 1-ft. pump, was 67.7 ft., with a velocity of 4153 ft. per min., being rather less than the calculated height, owing probably to leakage with the greater pressure. A velocity of 1128 ft. per min. raised the water 5}^ ft. without any discharge, and the maximum effect from the j^ower employed in raising to the same height 5J^ ft. was obtained at the "* velocity of 1678 ft. per min., giving a discharge of 1400 gals, per min. from the 1-ft. pump. The additional velocity required to effect a discharge of 1400 gals, per min., through a 1-ft. pump working at a dead level without any height of lift, is 550 ft. per min. Consequently, adding this number in each case to the velocity given above, at which no discharge takes place, the fol- lowing velocities are obtained for the maximum effect to be produced in each case : 1050 ft. per min., velocity for 1 ft. height of lift. 1550 " '' " " 4 " 2550 " " " " 16 " " " 4550 " " " " 64 " " " Or, in general terms, the velocity in feet per minute for the circumference of the pum p to be driven, to rai se the water to a certain height, is equal to 550 -f 500 Vheight of lift in feet. Lawrence Centrifugal Pumps, Class B— For Lifts from 15 to 35 ft. Size of Pipes. Economical Total Horse-power Capacity, Capacity, per Ft. Lift, Dis- charge. in gallons in gallons for smaller Suction. per min. per min. quantity. No. iy % 2 in. l^in. 20 to 50 150 .024 " 2 2V, 2 60 to 80 300 .035 " 3 w% 3 80 to 160 650 .055 " 4 V/z 4 160 to 350 1,250 .075 " 5 6 5 330 to 600 1,850 .175 " 6 6 6 500 to 900 2,600 .22 " 8 8 8 1,100 to 2,000 4,750 .45 " 10 10 10 1,600 to 3,000 7,500 .62 " 12 12 12 2,000 to 3,000 10,000 1.00 " 14 14 14 3,000 to 5,000 14,000 1.25 " 15 15 15 3,500 to 7,000 16,000 1.40 " 18 18 18 6.000 to 11,000 22,000 2.40 608 WATER-POWER. Table of Diameters and Width of Pulley s. Width of Belts, and Number of Revolutions per Minute Necessary to raise Minimum Quantity of Water to Different Heights with Different Sizes of Pumps of Class B. u So «M Height in Feet and Revolutions per dJ ■S 3 o Minute. o a 33 6 8 10 12 16 20 25 30 35 6 § Ins. Ins. Ins. Ins. v& 5 5 3 40 465 515 560 605 6S0 745 820 885 945 1M 2 5 5 4 60 425 475 515 560 625 750 810 870 2 3 Wz 7 6 80 390 435 475 510 575 . ! 750 800 3 4 m 7 7 160 365 405 445 475 535 590 645 700 745 4 5 12 11 8 330 355 !9l 415 470 520 570 610 750 5 6 14 11 9 500 285 315 345 370 415 460 500 540 6 8 16 12 10 1100 215 240 260 280 310 340 375 410 435 8 10 18 12 10 1600 170 190 210 225 250 275 300 350 10 12 22 14 12 2000 150 165 185 195 265 285 310 12 14 24 14 13 3000 135 150 165 175 195 215 240 295 275 14 15 28 15 14 3500 125 145 155 165 190 210 230 360 15 18 28 16 14 6000 110 120 130 135 160 175 190 255 220 18 Efficiencies of Centrifugal and Reciprocating Pumps.— W. O. Webber (Trans. A. S. M. E., vii. 598) gives diagrams showing the relative efficiencies of centrifugal and reciprocating pumps, from which the following figures are taken for the different lifts stated : Lift, feet: 2 5 10 15 20 25 30 35 40 50 60 80 100 120 160 200 240 280 Efficiency reciprocating pump: 30 .45 .55 .61 .66 .68 .71 .75 .77 .82 .85 .87 .90 .89 .88 .85 Efficiency centrifugal pump: .50 .56 .64 .68 .69 .68 .66 .62 .58 .50 .40 The term efficiency here used indicates the value of W. H. P. -=- 1. H. P., or horse-power of the water raised divided by the indicated horse-power of the steam-engiue,and does not therefore show the full efficiency of the pump, but that of the combined pump and engine. It is, however, a very simple way of showing the relative values of different kinds of pumping-engines having their motive power forming a part of the plant. The highest value of this term, given by Mr. Webber, is .9164 for a lift of 170 ft., and 3615 gals, per min. This was obtained in a test of the Leavitt pumping engine at Lawrence, Mass., July 24, 1879. With reciprocating pumps, for higher lifts than 170 ft., the curve of effi ciencies falls, and from 200 to 300 ft. lift the average value seemo about .84. Below 170 ft. the curve also falls reversely and slowly, until at about 90 ft. its descent becomes more rapid, and at 35 ft. .727 appears the best recorded performance. There are not any very satisfactory records below this lift, but some figures are given for the yearly coal consumption and total number of gallons pumped by engines in Holland under a 16-ft. lift, from which an efficiency of .44 has been deduced. With centrifugal pumps, the lift at which the maximum efficiency is ob- tained is approximately 17 ft. At lifts from 12 to 18 ft. some makers of large experience claim now to obtain from 65$ to 70$ of useful effect, but .613 appears to be the best done at a public test under 14.7 ft. head. The drainage-pumps constructed some years ago for the Haarlem Lake were designed to lift 70 tons per min. 15 ft., and they weighed about 150 tons. Centrifugal pumps for the same work weigh only 5 tons. The weight of a centrifugal pump and engine to lift 10,000 gals, per min. 35 ft. high is 6 tons. The pumps placed by Gwynne at the Ferrara Marshes, Northern Italy, in 1865, are, it is believed, capable of handling more water than other set of pumping-engines in existence. The work performed by these pumps is the lifting of 2000 tons per min.— over 600.000,000 gals, per 24 hours— on a mean lift of about 10 ft. (maximum of 12.5 ft.). (See Engineering, 1876.) The efficiency of centrifugal pumps seems to increase as the size of pump DUTY TEIALS OF PUMPIKG-ENGINES. 609 increases, approximately as follows: A 2" pump (this designation meaning always the size of discharge-outlet in inches of diameter), giving an effi- ciency of 38#, a 3" pump 45#, and a 4" pump 52#, a 5" pump 60$, and a 6" pump 64$ efficiency. Tests of Centrifugal Pumps. W. O. Webber, Trans. A. S. M. E., ix. 237. Berlin. Schwartz- kopff. Size Diam. discharge . " suction ... " disk Rev. per minute. Galls, per minute Height in feet.. . . Water H.P Dynam'eter H.P. Efficiency An- drews. An- drews. An- drews. Heald & Sisco. Heald & Sisco. Heald & Sisco. No. 9. No. 9. No. 9. No. 10. No. 10. No. 10. $Vs" 9^" $H" 10" 10" 10" 9H" 9M" Q%" 12" 12" 12" 26" 26" 26" 30.5" 30.5" 30.5" 191.9 195.5 200.5 188.3 202.7 213.7 1513.12 2023.82 2499.33 1673.37 2044.9 2371.67 12.25 12.62 13.08 12.33 12.58 13.0 4.69 6.47 8.28 5.22 6.51 7.81 10.09 12.2 14.38 8.11 10.74 14.02 46.52 53.0 57.57 64.5 60.74 55.72 No. 9. Wi" 10.3" 20.5" 500 1944.8 16.46 "ii Vanes of Centrifugal Pumps.— For forms of pump vanes, see paper by W. O. Webber, Trans. A. S. M. E., ix. 228, and discussion thereon by Profs. Thurston, Wood, and others. The Centrifugal Pump used, as a Suction Dredge.— The Andrews centrifugal pump was used by Gen. Gillmore, U. S. A., in 1871, in deepening tbe channel over the bar at the mouth of tbe St. John's River, Florida. The pump was a No. 9, with suction and discharge pipes each 9 inches diam. It was driven at 300 revolutions per minute by belt from an engine developing 26 useful horse-power. Although 200 revolutions of the pump disk per minute will easily raise 3000 gallons of clear water 12 ft. high, through a straight vertical 9-inch pipe, 300 revolutions were required to raise 2500 gallons of sand and water 11 ft. high, through two inclined suction-pipes having two turns each, dis- charged through a pipe having one turn. The proportion of sand that can be pumped depends greatly upon its specific gravity and fineness. The calcareous and argillaceous sands flow more freely than the silicious, and fine sands are less liable to choke the pipe than those that are coarse. When working at high speed, 50$ to 55$ of sand can be raised through a straight vertical pipe, giving for every 10 cubic yards of material discharged 5 to 5% cubic yards of compact sand. With the appliances used on the St. John's bar, the proportion of sand seldom exceeded 45$, generally ranging from 30$ to 35$ when working under the most favorable conditions. In pumping 2500 gallons, or 12.6 cubic yards of sand and water per minute, there would therefore be obtained from 3.7 to 4.3 cubic yards of sand. Dur- ing the early stages of the work, before the teeth under the drag had been properly arranged to aid the flow of sand into the pipes, the yield was con- siderably below this average. (From catalogue of Jos. Edwards & Co., Mfrs. of the Andrews Pump, New York.) DUTY TRIALS OF PUMPING-JENGINES. A committee of the A. S. M. E. (Trans., xii. 530) reported in 1891 on a standard method of conducting duty trials. Instead of the old unit of duty of foot-pounds of work per 100 lbs. of coal used, the committee recom- mend a new unit, foot-pounds of work per million heat-units furnished by the boiler. The variations in quality of coal make the old standard unfit as a basis of duty ratings. The new unit is the precise equivalent of 100 lbs. of coal in cases where each pound of coal imparts 10,000 heat-units to the water in the boiler, or where the evaporation is 10,000 -e- 965.7 = 10 355 lbs. of water from and at 212° per pound of fuel. This evaporative result is readily obtained from all grades of Cumberland bituminous coal, used in horizontal return tubular boilers, and, in many cases, from the best grades of anthra- cite coal, 610 WATER-POWER. The committee also recommend that the work done be determined by plunger displacement, after making a test for leakage, instead of by meas- urement of flow by weirs or other apparatus, but advise the use of such apparatus when practicable for obtaining additional data. The following extracts are taken from the report. When important tests are to be made the complete report should be consulted. The necessary data having been obtained, the duty of an engine, and other quantities relating to its performance, may be computed by the use of the following formulae: „ _ , Foot-pounds of work done . nnn .... 1. Duty = rf — — — X 1,000,000 Total number of heat-units consumed = A(P± P +£XLXN x 1)000)0()0 (foot . pounds) , C X 144 3. Percentage of leakage = ~ - - x 100 (per cent). ■A. X -Li X iv 3. Capacity = number of gallons of water discharged in 24 hours A X L X NX 7.4805 X 24 AxLx NX 1.24675 D X 144 4. Percentage of total frictions, rw. - ^p±p+ - (gallons). D X 60 X 33,000 ~ i-~~ I.H.P. = L 1 - isXM.E.P.Xi,XiyJ X 10 ° (per CeQt); or, in the usual case, where the length of the stroke and number of strokes of the plunger are the same as that of the steam-piston, this last formula becomes: tA(P + v-V- s) "1 1 - / x MEF ' X 10 ° ( ' per cent )* In these formulae the letters refer to the following quantities: A = Area, in square inches, of pump plunger or piston, corrected for area of piston rod or rods; P = Pressure, in pounds per square inch, indicated by the gauge on the force main; p — Pressure, in pounds per square inch, corresponding to indication of the vacuum-gauge on suction -main (or pressure -gauge, if the suction- pipe is under a head). The indication of the vacuum-gauge, in inches of mercury, may be converted into pounds by dividing it by 2.035; s = Pressure, in pounds per square inch, corresponding to distance be- tween the centres of the two gauges. The computation for this pressure is made by multiplying the distance, expressed in feet, by the weight of one cubic foot of water at the temperature of the pump-well, and dividing the product by 144; L = Average length of stroke of pump-plunger, in feet; N = Total number of single strokes of pump-plunger made during the trial; As = Area of steam-cylinder, in square inches, corrected for area of piston- rod. The quantity As X M.E.P., in an engine having more than one cylinder, is the sum of the various quantities relating to the respec- tive cylinders; Ls = Average length of stroke of steam -piston, in feet; JVs = Total number of single strokes of steam-piston during trial; M-E.P. = Average mean effective pressure, in pounds per square inch, measured from the indicator-diagrams taken from the steam-cylin- der; I.H.P. = Indicated horse-power developed by the steam-cylinder; C = Total number of cubic feet of water which leaked by the pump-plunger during the trial, estimated from the results of the leakage test; D = Duration of trial in hours; DUTY TUtALS OF PUMPING-ENGI^ES. 611 H— Total number of heat-units (B. T. U.) consumed by engine = weight of water supplied to boiler by main feed-pump x total beat of steam of boiler pressure reckoned from temperature of main feed-water -f weight of water supplied by jacket-pump X total heat of steam of • boiler-pressure reckoned from temperature of jacket-water -f- weight of any other water supplied X total heat of steam reckoned from its temperature of supply. The total heat of the steam is corrected for the moisture or superheat which the steam may contain. No allow- ance is made for water added to the feed water, which is derived from auy source, except the engine or some accessory of the engine. Heat added to the water by the use of a flue -heater at the boiler is not to be deducted. Should heat be abstracted from the flue by means of a steam reheater connected with the intermediate re- ceiver of the engine, this heat must be included in the total quantity supplied by the boiler. Leakage Test of Pump.— The leakage of an inside plunger (the only type which requires testing) is most satisfactorily determined by mak- ing the test with the cylinder-head removed. A wide board or plank may be temporarily bolted to the lower part of the end of the cylinder, so as to hold back the water in the manner of a dam, and an opening made in the temporary head thus provided for the reception of an overflow-pipe. The plunger is blocked at some intermediate point in the stroke (or, if this posi- tion is not practicable, at the end of the stroke), and the water from the force main is admitted at full pressure behind it. The leakage escapes through the overflow-pipe, and it is collected in barrels and measured. The test should be made, if possible, with the plunger in various positions. In the case of a pump so planned that it is difficult to remove the cylinder- head, it may be desirable to take the leakage from one of the openings which are provided for the inspection of the suction-valves, the head being allowed to remain in place. It is assumed that there is a practical absence of valve leakage. Exami- nation for such leakage should be made, and if it occurs, and it is found to be due to disordered valves, it should be remedied before making the plunger test. Leakage of the discharge valves will be shown by water passing down into the empty cylinder at either end when they are under pressure. Leak- age of the suction-valves will be shown by the disappearance of water which covers them. If valve leakage is found which cannot be remedied the quantity of water thus lost should also be tested. One method is to measure the amount of water required to maintain a certain pressure in the pump cylinder when this is introduced through appe temporarily erected, no water being al- lowed to enter through the discharge valves of the pump. Table of Data and Results.— In order that uniformity may be se- cured, it is suggested that the data and results, worked out in accordance with the standard method, be tabulated in the manner indicated in the fol- lowing scheme : DUTY TRIAL OF ENGINE. DIMENSIONS. 1. Number of steam-cylinders 2. Diameter of steam-cylinders ins. 3. Diameter of piston -rods of steam-cylinders ins. 4. Nominal stroke of steam-pistons , .... ft. 5. Number of water-plungers '. 6. Diameter of plungers ins. 7. Diameter of piston-rods of water-cylinders ins. 8. Nominal stroke of plungers ft. 9. Net area of steam-pistons „ sq. ins. 10. Net area of plungers sq. ins. 11. Average length of stroke of steam-pistons during trial ft. 12. Average length of stroke of plungers during trial ft. (Give also complete description of plant.) TEMPERATURES. 13. Temperature of water in pump-well degs. 14. Temperature of water supplied to boiler by main feed-pump. . degs. 15. Temperature of water supplied to boiler from various other sources degs. 612 WATER-POWER. FEED-WATER. 16. Weight of water supplied to boiler by main feed-pump lbs. 17. Weight of water supplied to boiler from various other sources, lbs. 18. Total weight of feed-water supplied from all sources . lbs. PRESSURES. 19. Boiler pressure indicated by gauge lbs. 20. Pressure indicated by gauge on force main lbs. 21. Vacuum indicated by gauge on suction main ins. 22. Pressure corresponding to vacuum given in preceding line lbs. 23. Vertical distance between the centres of the two gauges ins. 24. Pressure equivalent to distance between the two gauges lbs. MISCELLANEOUS DATA. 25. Duration of trial hrs. 26. Total number of single strokes during trial 27. Percentage of moisture in steam supplied to engine, or number of degrees of superheating % or deg„ 28. Total leakage of pump during trial, determined from results of leakage test lbs. 29. Mean effective pressure, measured from diagrams taken from steam-cylinders M.E.P. PRINCIPAL RESULTS. 30. Duty , ft. lbs. 31. Percentage of leakage % 32. Capacity gals. 33. Percentage of total friction % ADDITIONAL RESULTS. 34. Number of double strokes of steam-piston per minute 35. Indicated horse-power developed by the various steam-cylinders I.H.P. 36. Feed- water consumed by the plant per hour lbs. 37. Feed-water consumed by the plant per indicated horse-power per hour, corrected for moisture in steam lbs. 38. Number of heat units consumed per indicated horse-power per hour B.T.U. 39. Number of heat units consumed per indicated horse-power per minute B.T.U. 40. Steam accounted for by indicator at cut-off and release in the various steam-cylinders lbs. 41. Proportion which steam accounted for by indicator bears to the feed-water consumption 42. Number of double strokes of pump per minute 43. Mean effective pressure, measured from pump diagrams ...... M.E.P. 44. Indicated horse-power exerted in pump-cylinders I.H.P. 45. Work done (or duty) per 100 lbs. of coal ft. lbs. SAMPLE DIAGRAM TAKEN FROM STEAM-CYLINDERS. (Also, if possible, full measurement of the diagrams, embracing pressures at the initial point, cut off, release, and compression ; also back pressure, and the proportions of the stroke completed at the various points noted.) SAMPLE DIAGRAM TAKEN FROM PUMP-CYLINDERS. These are not necessary to the main object, but it is desirable to give them. DATA AND RESULTS OF BOILER TEST. (In accordance with the scheme recommended by the Boiler-test Com- mittee of the Society.) VACUUM PUMPS-AIR-LIFT PUMP. Tlie Pulsometer.-In the pulsometer the water is raised by suction into the pump-chamber by the condensation of steam within it, and is then forced into the delivery-pipe by the pressure of a new quantity of steam on the surface of the water. Two chambers are used which work alternately, one raising while the other is discharging. Test of a Pulsometer.— A test of a pulsometer is described by De Volson Wood in Trans. A. S. M. E. xiii. It had a 3^-inch suction-pipe, stood 40 ii high, and weighed 695 lbs. The steam-pipe was 1 inch in diameter. A throttle was placed about 2 feet VACUUM PUMPS— AIR-LIFT PUMP. 612 from the pump, and pressure gauges placed on both sides of the throttle, and a mercury well and thermometer placed beyond the throttle. The wire drawing due to throttling caused superheating. The pounds of steam used were computed from the increase of the tern perature of the water in passing through the pump. Pounds of steam x loss of heat = lbs. of water sucked in x increase of temp. The loss of heat in a pound of steam is the total heat in a pound of satu- rated steam as found from "steam tables " for the given pressure, plus the heat of superheating, minus the temperature of the discharged water ; or Pounds of steam = lbs. water X increase of temp. H - 0.48* - T. The results for the four tests are given in the following table : Data and Results. Strokes per minute Steam press. in pipe before throttPg ■ Steam press, in pipe after throttPg Steam temp, after throttling,deg.F. Steam am'nt of superheat'g.deg.F. Steam used asdet'd from temp., lbs. Water pumped, lbs. Water temp. before entering pump, Water temp., rise of Water head by gauge on lift, ft ... . Water head by gauge on suction. . . Water head by gauge, total (H) Water head by measure, total (h) Coeff . of friction of plant (h) -=- (H) Efficiency of pulsometer Effic. of plant exclusive of boiler. . . Effic. of plant if that of boiler be 0.7 Duty,if 1 lb.evaporates 10 lbs.w^ater Number of Test. 71 114 19 270.4 3.1 1617 404,786 75.15 4.47 29.90 12.26 42.16 32.8 0.777 0.012 0.0)93 0.1065 3.4 931 186.362 80.6 5.5 54.05 12.26 66.31 57.80 0.877 0.0155 0.0136 0.0095 10.5li,4Ul)j 1:1391,000 43.8 309.0 17.4 1518 228,425 76.3 7.49 54.05 19.67 66^6 0.911 0.0126 0.0115 0.0080 11,059,000 26.1 270.1 1.4 1019.9 248.053 70.25 4.55 49.57 41.60 0.839 0.0138 0.0116 0.0081 12,036,300 Of the tw r o tests having the highest lift (54.05 ft.), that was more efficient which had the smaller suction (12.26 ft.), and this was also the most efficient of the four tests. But, on the other hand, the other two tests having the same lift (,29.9 ft.), that was the more efficient which had the greater suction (19.67), so that no law in this regard was established. The pressures used, 19, 30, 43.8, 26.1, follow the order of magnitude of the total heads, but are not proportional thereto. No attempt was made to determine what press- ure would give the best efficiency for any particular head. The pressure used was intrusted to a practical runner, and he judged that when the pump was running regularly and well, the pressure then existing was the proper one. It is peculiar that, in the first test, a pressure of 19 lbs. of steam should pro- duce a greater number of strokes and pump over 50$ more water than 26.1 lbs., the lift being the same, as in the fourth experiment. Chas. E. Emery in discussion of Prof. Wood's paper says, referring to tests made by himself and others at the Centennial Exhibition in 1876 (see Report of Judges, Group xx.), says that a vacuum-pump tested by him in 1871 gave a duty of 4.7 millions; one tested by J. F. Flagg, at the Cincinnati Exposition in 1875, gave a maximum duty of 3.25 millions. Several vacuum and small steam-pumps, compared later on the same basis, were reported to have given duties of 10 to 11 millions, the steam-pumps doing no better than the vacuum-pumps. Injectors, when used for lifting water not re- quired to be heated, have an efficiency of 2 to 5 millions; vacuum-pumps vary generally between 3 and 10; small steam-pumps between 8 and 15 ; larger steam-pumps, between 15 and 30, and pumping-engiues between 30 and 140 millions. A very high record of test of a pulsometer is given in Enc/'g. Nov. 24, 1893, p. 639, viz. : Height of suction 11.27 ft. ; total height of lift, 102.6 ft. ; hori- zontal length of delivery-pipe, 118 ft. ; quantity delivered per hour, 26,188 British gallons. Weight of steam used per H. P. per hour, 92.76 lbs. ; work 614 WATER-POWER. done per pound of steam 21,345 foot-pounds, equal to a duty of 21,345,000 foot-pounds per 100 lbs. of coal, if 10 lbs of steam were generated per pound of coal. The Jet-pump.— This machine works by means of the tendency of a stream or jet of fluid to drive or carry contiguous particles of fluid along with it. The water-jet pump, in its present form, was invented by Prof. James Thomson, and first described in 1852. In some experiments on a small scale as to the efficiency of the jet-pump, the greatest efficiency was found to take place when the depth from which the water was drawn by the suction-pipe was about nine tenths of the height from which the water fell to form the jet ; the flow up the suction-pipe being in that case about one fifth of that of the jet, and the efficiency, consequently, 9/10 X 1/5 = 0.18. This is but a low efficiency; but it is probable that it may be increased by improvements in proportions of the machine. (Rankine, S. E.) Tlte Injector when used as a pump has a very low efficiency. (See Injectors, under Steam-boilers.) Air-lift Pump.— The air-lift pump consists of a vertical water-pipe with its lower end submerged in a well, and a smaller pipe delivering air into it at the bottom. The rising column in the pipe consists of air mingled with water, the air being in bubbles of various sizes, and is therefore lighter than a column of water of the same height; consequently the water in the pipe is raised above the level of the surrounding water. This method of raising water was proposed as early as 1797, by Loeseher, of Freiberg, and was mentioned by Collon in lectures in Paris in 1876, but its first practical application probably was by Werner Siemens in Berlin in 1885. Dr. J. G. Pohle experimented on the principle in California in 1886, and U. S. patents on apparatus involving it were granted to Pohle and Hill in the same year. A paper describing tests of the air-lift pump made by Randall, Browne and Behr was read before the Technical Society of the Pacific Coast in Feb. 1890. The diameter of the pump-column was 3 in., of the air-pipe 0.9 in., and of the air-discharge nozzle % in. The air-pipe had four sharp bends and a length of 35 ft. plus the depth of submersion. The water was pumped from a closed pipe-well (55 ft. deep and 10 in. in diameter). The efficiency of the pump was based on the least work theo- retically required to compress the air and deliver it to the receiver. If the efficiency of the compressor be taken at 70$, the efficiency of the pump and compressor together would be 70$ of the efficiency found for the pump alone. For a given submersion (70 and lift (II), the ratio of the two being kept within reasonable limits, (H) being not much greater than (h), the efficiency was greatest when the pressure in the receiver did not greatly exceed the head due to the submersion. The smaller the ratio H-t-h, the higher was the efficiency. The pump, as erected, showed the following efficiencies : For H+h= 0.5 1.0 1.5 2.0 Efficiency = 50$ 40$ 30$ 25$ The fact that there are absolutely no moving parts makes the pump especially fitted for handling dirty or gritty water, sewage, mine water, and acid or alkali solutions in chemical or metallurgical works. In Newark, N. J., pumps of this type are at work having a total capacity of 1,000,000 gallons daily, lifting water from three 8-in. artesian wells. The Newark Chemical Works use an air-lift pump to raise sulphuric acid of 1.72° gravity. The Colorado Central Consolidated Mining Co., in one of its mines at Georgetown, Colo., lifts water in one case 250 ft., using a series of lifts. For a full account of the theory of the pump, and details of the tests above referred to, see Eng^g News, June 8, 1893. THE HYDRAULIC RAM. Efficiency.— The hydraulic ram is used where a considerable flow of water with a moderate fall is available, to raise a small portion of that flow to a height exceeding that of the fall. The following are rules given by Eytelwein as the results of his experiments (from Rankine): Let Q be the whole supply of water in cubic feet per second, of which q is lifted to the height h above the pond, and Q — q runs to waste at the depth H below the pond; L, the length of the supply -pipe, from the pond to the waste-clack ; D, its diameter in feet; then D - 4/(1.63Q); L = H+h + fiX 2 feet; Volume of air vessel = volume of feed pipe; THE HYDRAULIC RAM. 615 Efficiency, qh -™i/h (Q-q)H- 1 -*- ( 1 + Tqtt) nearly, when — does not exceed 12. D'Aubisson gives Clark, using five sixths of the values given by D , Aubisson , s formula, gives: Ratio of lift to fall. ... 4 6 8 10 12 14 16 18 20 22 24 26 Efficiency per cent 72 61 52 44 37 31 25 19 14 9 4 Prof. R. C. Carpenter (Eng'g Mechanics, 1894) reports the results of four tests of a ram constructed by Runisey & Co., Seneca Falls. The ram was fitted for pipe connection for l^-inch supply and i^-inch discharge. The supply-pipe used was 1}^ inches in diameter, about 50 feet long, with 3 elbows, so that it was equivalent to about - 65 feet of straight pipe, so far as resist- ance is concerned. Each run'was made with a different stroke for the waste or clack-valve, the supply and delivery head being constant; the object of the experiment was to find that stroke of clack-valve which would give the highest efficiency. Length of stroke, per cent Number of strokes per minute Supply head, feet of water Delivery head, feet of water. . Total water pumped, pounds. . Total water supplied, pounds.. Efficiency, per cent , 100 80 60 52 56 61 5.67 5.77 5.58 19.75 19.75 19.75 .297 296 301 1615 1567 1518 64.9 66 74.9 66 5.65 19.75 297.5 1455.5 70 The efficiency, 74.9, the highest realized, was obtained when the clack-valve travelled a distance equal to 60$ of its full stroke, the full travel being 15/16 of one inch. Quantity of Water Delivered, by the Hydraulic Ram, (Chadwick Lead Works.)— From 80 to 100 feet conveyance, one seventh of supply from spring can be discharged at an elevation five times as high as the fall to supply the ram; or, one fourteenth can be raised and discharged say ten times as high as the fall applied. Water can be conveyed by a ram 3000 feet, and elevated 200 feet. The drive-pipe is from 25 to 50 feet long. The following table gives the capacity of several sizes of rams, the dimen- sions of the pipes to be used, and the size of the spring or brook to which they are adapted: Quantity of Water Furnished per Min. by the Spring or Brook to which the Ram is Adapted. Caliber of Pipes. Weight of Pipe (Lead), if Wrought Iron, then of Ordinary Weight. Size of Ram. 03 > & 5 Drive-pipe for head or fall not over 10 ft. Discharge- pipe for not over 50 ft. rise. Discharge- pipe for over 50 ft. and not ex- ceeding 100 ft. in height. No. 2 " 3 " 4 " 5 " 6 " 7 "10 Gals, per min. U to 2 iy 2 " 4 3 " 7 6 " 14 12 " 25 20 " 40 25 " 75 inch. % 1 2 inch. Vs y* y% .* 2 per foot. 2 lbs. 3 " 5 " 8 " 13 " 13 " 21 " per foot. 10 ozs. 12 " 12 " lib. 4 " 2 " 3 " per foot, lib. 1 " 4 ozs. 1 " 4 ozs. 2 " 3 " 4 " 8 " 616 WATER-POWER. HYDRAULIC-PRESSURE TRANSMISSION, Water under high pressure (700 to 2000 lbs. per square inch and upwards) affords a very satisfactory method of transmitting power to a distance, especially for the movement of heavy loads at small velocities, as by cranes and elevators. The system consists usually of one or more pumps capable of developing the required pressure; accumulators, which are vertical cylin- ders with heavily-weighted plungers passing through stuffing-boxes in the upper end, by which a quantity of water may be accumulated at the pres- sure to which the plunger is weighted ; the distributing-pipes; and the presses, cranes, or other machinery to oe operated. The earliest important use of hydraulic pressure probably was in the Bramah hydraulic press, patented in 1796. Sir W. G. Armstrong in 1846 was one of the pioneers in the adaptation of the hydraulic system to cranes. The use of the accumulator by Armstrong led to the extended use of hydraulic machinery. Recent developments and applications of the system are largely due to Ralph Tweddell, of London, and Sir Joseph Whitworth. Sir Henry Bessemer, in his patent of May 13, 1856, No. 1292, first suggested the use of hydraulic pressure for compressing steel ingots while in the fluid state. The Gross Amount of Energy of the water under pressure stored in the accumulator, measured in foot-pounds, is its volume in cubic feet X its pressure in pounds per square foot. The horse-power of a given quantity l44r>Q steadily flowing is H.P. = ^ = .26l8pQ, in which Q is the quantity flowing in cubic feet per second andp the pressure in pounds per square inch. The loss of energy due to velocity of flow in the pipe is calculated as fol- lows (R. Gr. Blaine, Eng'g, May 22 and June 5, 1891): According to D'Arcy, every pound of water loses -=— times its kinetic energy, orenergy due to its velocity in passing along a straight pipe L feet in length and D feet diameter, where A is a variable coefficient. For clean cast-iron pipes it may be taken as A = .005 ( 1 -f- — - ), or for diameter in inches — d. d= l£ 1 2 3 45 6 7 8 9 10 12 A = .015 .01 .0075 .00667 .00625 .006 .00583 .00571 .00563 .00556 .0055 .00542 The loss of energy per minute is 60 x 62.36Q X -jr- .5-, andthehorse- «. /. • «. • • ur -6363AL(H.P.)3 . ... ? . power wasted in the pipe is W = 3 -=-g — — , m which A varies with the diameter as above, p = pressure at entrance in pounds per square inch. Values of .6363A for different diameters of pipe in inches are: d= Y 2 1 2 3 4 5 6 7 8 9 10 12 .00954 .00636 .00477 .00424 .00398 .00382 .00371 .00363 .00358 .00353 .00350 .00345 Efficiency of Hydraulic Apparatus.- The useful effect of a direct hydraulic plunger or ram is usually taken at 93$. The following is given as the efficiency of a ram with chain-and-pulley multiplying gear properly proportioned and well lubricated: Multiplying.... 2 to 1 4 to 1 6 to 1 8 to 1 lOtol 12 to 1 14 to 1 16 to 1 Efficiency*.... 80 76 72 67 63 59 54 50 With large sheaves, small steel pins, and wire rope for multiplying gear the efficiency has been found as high as 66* for a multiplication of 20 to 1. Henry Adams gives the following formula for effective pressure in cranes and hoists: P — accumulator pressure in pounds per square inch; m — ratio of multiplying power; E = effective pressure in pounds per square inch, including all allowances for friction ; E = P(.84 - .02m). J. E. Tuit (Eng^g, June 15, 1888) describes some experiments on the fric- tion of hydraulic jacks from 3J4 to 13^-inch diameter, fitted with cupped leather packings. The friction loss varied from 5.6* to 18.8$ according to the condition of the leather, the distribution of the load on the ram, etc. The friction increased considerably with eccentric loads. With hemp pack- ing a plunger, 14 inch diameter, showed a friction loss of from 11.4* to 3.4$, the load being central, and from 15.0* to 7.6* with eccentric load, the per- centage of loss decreasing in both cases with increase of lo&cl. HYDRAULIC-PRESSURE TRANSMISSION". 017 Thickness of Hydraulic Cylinders.- -From a table used by Sir W. W. Armstrong we take the following, tor cast-iron cylinders, for an in- terior pressure ot 10U0 lbs. per square inch: Diam. of cylinder, iuches.. 2 4 6 8 10 12 16 20 24 Thickness,' inches 0.832 1.146 1.552 1.875 2.222 2.578 3.19 3.69 4.11 For any other pressure multiply by the ratio of that pressure to 1000. These figures correspond nearly to the formula t — 0.175cZ -f 0.4S, in which t = thickness and d — diameter in inches, up to 16 inches diameter, but for 20 inches diameter the addition 0.48 is reduced to 0.19 and at 24 inches it disappears. For formulae for thick cylinders see page 287, ante. Cast iron should not be used for pressures exceeding 2000 lbs. per square inch. For higher pressures steel castings or forged steel should be used. For working pressures of 750 lbs. per square inch the test pressure should be 2500 lbs. per square inch, and for 1500 lbs. the test pressure should not be less than 3500 lbs. Speed, of Moisting by Hydraulic Pressure.— The maximum allowable speed for warehouse cranes is 6 feet per second; for platform cranes 4 feet per second; for passenger and wagon hoists, heavy loads, 2 feet per second. The maximum speed under any circumstances should never exceed 10 feet per second. The Speed of Water Through Valves should never be greater than 100 feet per second. Speed of Water Through Pipes.— Experiments on water at 1600 lbs. pressure per square inch flowing into a flanging-machine ram, 20-inch diameter, through a J^-inch pipe contracted at one point to ^-incli, gave a velocity of 114 feet per second in the pipe, and 456 feet at the reduced sec- tion. Through a J^-inch pipe reduced to ^s-inch at one point the velocity was 213 feet per second in the pipe and 381 feet at the reduced section In a J^-inch pipe without contraction the velocity was 355 feet per second. For many of the above notes the author is indebted to Mr. John Piatt, consulting engineer, of New York. High-pressure Hydraulic Presses in Iron-works are de- scribed by R. M. Daeleu, of Germany, in Trans. A. I. M. E. 1892. The fol- lowing distinct arrangements used in different systems of high-pressure hydraulic work are discussed and illustrated: 1. Steam-pump, with fly-wheel and accumulator. 2. Steam pump, without fly-wheel and with accumulator. 3. Steam-pump, without fly-wheel and without accumulator. In these three systems the valve-motion of the working press is operated in the high-pressure column. This is avoided in the following: 4. Single-acting steam-intensifier without accumulator. 5. Steam-pump with fly-wheel, without accumulator and with pipe-circuit. 6. Steam-pump with fly-wheel, without accumulator and without pipe- circuit. The disadvantages of accumulators are thus stated: The weighted plungers which formerly served in most cases as accumulators, cause violent shocks in the pipe-line when changes take place in the movement of the water, so that in many places, in order to avoid bursting from this cause, the pipes are made exclusively of forged and bored steel. The seats and cones of the metallic valves are cut by the water (at high speed), and in such cases only the most careful maintenance can prevent great losses of power. Hydraulic Power in London.— The general principle involved is pumping water into mains laid in the streets, from which service-pipes are carried into the houses to work lifts or three-cylinder motors when rotatory power is required. In some cases a small Pelton wheel has been tried, working under a pressure of over 700 lbs. on the square inch. Over 55 miles of hydraulic mains are at present laid (1892). The reservoir of power consists of capacious accumulators, loaded to a pressure of 800 lbs. per square inch, thus producing the same effect as if large supply-tanks were placed at 1700 feet above the street-level. The water is taken from the Thames or from wells, and all sediment is removed therefrom by filtration before it reaches the main engine-pumps. There are over 1750 machines at work, and the supply is about 6,500,000 gallons per week. It is essential that the water used should be clean. The storage-tank ex- tends over the whole boiler-house and coal-store. The tank is divided, and a certain amount of mud is deposited here. It then passes through the sur- face condenser of the engines, and it is turned into a set of filters, eight in number. The body of the filter is a cast-iron cylinder, containing a layer of 618 WATER-POWER* granular filtering material resting upon a false bottom; under this is the dis- tributing arrangement, affording passage for the air, and under this the real bottom of the tank. The dirty water is supplied to the filters from an over- head tank. After passing through the filters the clean effluent is pumped into the clean-water tank, from which the pumping-engines derive their supply. The cleaning of the filters, which is done at intervals of 24 hours, is effected so thoroughly in situ that the filtering material never requires to be removed. The engine-house contains six sets of triple-expansion engines. The cylinders are 15-inch, 22-inch, 36 inch X 24-inch. Each cylinder drives a single plunger-pump with a 5-inch ram, secured directly to the cross-head, the connecting-rod being double to clear the pump. The boiler-pressure is 150 lbs. on the square inch. Each pump will deliver 300 gallons of water per minute under a pressure of 800 lbs. to the square inch, the engines making about 61 revolutions per minute. This is a high velocity, considering the heavy pressure; but the valves work silently and without perceptible shock. The consumption of steam is 14.1 pounds per horse per hour. The water delivered from the main pumps passes into the accumulators. The rams are 20 inches in diameter, and have a stroke of 23 feet. They are each loaded with 110 tons of slag, contained in a wrought-iron cylindrical box suspended from a cross-head on the top of the ram. One of the accumulators is loaded a little more heavily than the other, so that they rise and fall successively; the more heavily loaded actuates a stop- valve on the main steam-pipe. If the engines suppty more water than is wanted, the lighter of the two rams first rises as far as it can go; the other then ascends, and when it has nearly reached the top, shuts off steam and checks the supply of water automatically. The mains in the public streets are so constructed and laid as to be per- fectly trustworthy and free from leakage. Every pipe and valve used throughout the system is tested to 2500 lbs. per square inch before being placed on the ground and again tested to a reduced pressure in the trenches to insure the perfect tightness of the joints. The jointing material used is gutta-percha. The average rate obtained by the company is about 3 shillings per thou- sand gallons. The principal use of the power is for intermittent work in cases where direct pressure can be employed, as, for instance, passenger elevators, cranes, presses, warehouse hoists, etc. An important use of the hydraulic power is its application to the extin- guishing of fire by means of Greathead's injector hydrant. By the use of these hydrants a continuous fire-engine is available. Hydraulic Riveting-machines.— Hydraulic riveting was intro- duced in England by Mr. R. H. Tweddell. Fixed riveters were first used about 1868. Portable riveting-machines were introduced in 1872. The riveting of the large steel plates in the Forth Bridge was done by small portable machines working with a pressure of 1000 lbs. per square inch. In exceptional cases 3 tons per inch was used. (Proc. Inst. M. E., May, 1889.) An application of hydraulic pressure invented by Andrew Higginson, of Liverpool, dispenses with the necessity of accumulators. It consists of a three-throw pump driven by shafting or worked by steam, and depends partially upon the work accumulated in a heavy fly-wheel. The water in its passage from the pumps and back to them is in constant circulation at a very feeble pressure, requiring a minimum of power to preserve the tube of water ready for action at the desired moment, when by the use of a tap the current is stopped from going back to the pumps, and is thrown upon the piston of the tool to be set in motion. The water is now confined, and the driving-belt or steam-engine, supplemented by the momentum of the heavy fly-wheel, is employed in closing up the rivet, or bending or forging the ob- ject subjected to its operation. Hydraulic Forging.— In the production of heavy forgings from cast ingots of mild steel it is essential that the mags of metal should be operated on as equally as possible throughout its entire thickness. When employing a steam-hammer for this purpose it has been found that the ex- ternal surface of the ingot absorbs a large proportion of the sudden impact of the blow, and that a comparatively small effect only is produced on the central portions of the ingot, owing to the resistance offered by the inertia of the mass to the rapid motion of the falling hammer— a disadvantage that is entirely overcome by the slow, though powerful, compression of the hydraulic forging-press, which appears destined to supersede the steam- hammer for the production of massive steel forgings. HYDRAULIC-PRESSURE TRANSMISSION. 619 It) the Allen forging-press the force-pump and the large or main cylinder of the press are in direct and constant communication. There are no inter- mediate valves of any kind, nor has the pump any clack-valves, but it simply forces its cylinder full of water direct into the cylinder of the press, and receives the same water, as it were, back again on the return stroke. Thus, when both cylinders and the pipe connecting them are full, the large ram of the press rises and falls simultaneously with each stroke of the pump, keeping up a continuous oscillating motion, the ram, of course, travelling the shorter distance, owing to the larger capacity of the press cylinder. (Journal Iron and Steel Institute, 1891. See also illustrated article in "' Modern Mechanism/' page 668.) For a very complete illustrated account of the development of the hy- draulic forging-press, see a paper by R. H. Tweddell in Proc. Inst. C. E., vol. cxvii. 1893-4. Hydraulic Forging-press.— A 2000-ton forging-press erected at the Couillet forges in Belguim is described in Eng. and M. Jour., Nov. 25, 1893. The press is composed essentially of two parts— the press itself and the compressor. The compressor is formed of a vertical steam-cylinder and a hydraulic cylinder. The piston-rod of the former forms the piston of the latter. The hydraulic piston discharges the water into the press proper. The distribution is made by a cylindrical balanced valve; as soon as the pressure is released the steam-piston falls automatically under the action of gravity. During its descent the steam passes to the other face of the piston to reheat Ihe cylinder, and finally escapes from the upper end. When steam enters under the piston of the compressor-cylinder the pis- ton rises, and its rod forces the water into the press proper. The pressure thus exerted on the piston of the latter is transmitted through a cross-head to the forging which is upon the anvil. To raise the cross-head two small single-acting steam-cylinders are used, their piston-rods being connected to the cross-head ; steam acts only on the pistons of these cylinders from below. The admission of steam to the cylinders, which stand on top of the press frame, is regulated by the same lever which directs the motions of the com- pressor. The movement given to the dies is sufficient for all the ordinary purposes of forging. A speed of 30 blows per minute has been attained. A double press on the same system, having two compressors and giving a maximum pressure of 6000 tons, has been erected in the Krupp works, at Essen. The Aiken Intensifies {Iron Age, Aug. 1890.)— The object of the machine is to increase the pressure obtained by the ordinary accumulator which is necessary to operate powerful hydraulic machines requiring very high pressures, without increasing the pressure carried in the accumulator and the general hydraulic system. The Aiken Intensifier consists of one outer stationary cylinder and one inner cylinder which moves in the outer cylinder and on a fixed or stationary hollow plunger. When operated in connection with the hydraulic bloom- shear the method of working is as follows: The inner cylinder having been filled with water and connected through the hollow plunger with thehydrau- lic cylinder of the shear, water at the ordinary accumulator-pressure is ad- mitted into the outer cylinder, which being four times the sectional area of the plunger gives a pressure in the inner cylinder and shear cylinder con- nected therewith of four times the accumulator-pressure— that is, if the ac- cumulator-pressure is 500 lbs. per square inch the pressure in the intensifier will be 2000 lbs. per square inch. Hydraulic Engine driving an Air-compressor and a Forging-hammer. {Iron Age, May 12, 1892.)— The great hammer in Terni, near Rome, is one of the largest in existence. Its falling weight amounts to 100 tons, and the foundation belonging to it consists of a block of cast iron of 1000 tons. The stroke is 16 feet \% inches; the diameter of the cylinder 6 feet S}4 inches: diameter of piston-rod 13% inches; total height of the hammer, 62 feet 4 inches. The power to work the hammer, as well as the two cranes of 100 and 150 tons respectively, and other auxiliary appli- ances belonging to it, is furnished by four air-compressors coupled together and driven directly by water-pressure engines, by means of which the air is compressed to 73.5 pounds per square inch. The cylinders of the water- pressure engines, which are provided with a bronze lining, have a 13%-inch bore. The stroke is 47% inches, with a pressure of water on the piston amounting to 264.6 pounds per square inch. The compressors are bored out to 31^2 inches diameter, and have 47%-inch stroke. Each of the four cylin- ders requires a power equal to 280 horse-power. The compressed air is de- 620 FUEL. livered into huge reservoirs, where a uniform pressure is kept up by means of a suitable water-column. The Hydraulic Forging Plant at Bethlehem, Pa., is de- scribed in a paper by R. W. Davenport, read before the Society of Naval Engineers and Marine Architects, 1893. It includes two hydraulic forging- presses complete, with engines and pumps, one of 1500 and one of 4500 tons capacity, together with two Whitworth hydraulic travelling forging-cranes and other necessary appliances for each press ; and a complete fluid-compres- sion plant, including a press of 7000 tons capacity and a 125 ton hydraulic travelling crane for serving it (the upper and lower heads of this press weighing respectively about 135 and 120 tons). A new forging-press has been designed by Mr. John Fritz, for the Bethle- hem Works, of 14,000 tons capacity, to be run by engines and pumps of 15,000 horse-power. The plane is served by four open-hearth steel furnaces of a united capacity of 120 tons of steel per heat. Some References on Hydraulic Transmission.— Reuleaux's "Constructor;" "Hydraulic Motors, Turbines, and Pressure-engines, 1 ' G. Bodmer, London, 1889; Robinson's "Hydraulic Power and Hydraulic Ma- chinery," London, 1888; Colyer's " Hydraulic Steam, and Hand- power Lift- ing and Pressing Machinery," London, 1881. See also Engineering (London), Aug. 1, 1884, p. 99; March 13, 1885, p. 262; May 22 and June 5, 1891, pp. 612, 665; Feb. 19, 1892, p. 25; Feb. 10, 1893, p. 170. FUEL. Theory of Combustion of Solid Fuel. (From Rankine, some- what altered.) — The ingredients of every kind of fuel commonly used may be thus classed: (1) Fixed or free carbon, which is left in the form of char- coal or coke after the volatile ingredients of the fuel have been distilled away. These ingredients burn either wholly in the solid state (C to C0 2 ), or part in the solid state and part in the gaseous state (CO + O = C0 2 ), the lat- ter part being first dissolved by previously formed carbonic acid by the re- action C0 2 + C = 2CO. Carbonic oxide, CO, is produced when the supply of air to the fire is insufficient. (2) Hydrocarbons, such as olefiant gas, pitch, tar, naphtha, etc., all of which must pass into the gaseous state before being burned. If mixed on their first issuing from amongst the burning carbon with a large quantity of hot air, these inflammable gases are completely burned with a transparent blue flame, producing carbonic acid and steam. When mixed with cold air they are apt to be chilled and pass off uuburned. When raised to a red heat, or thereabouts, before being mixed with a sufficient quantity of air for perfect combustion, they disengage carbon in fine pow- der, and pass to the condition partly of marsh gas, and partly of free hydro- gen; and the higher the temperature, the greater is the proportion of carbon thus disengaged. If the disengaged carbon is cooled below the temperature of ignition be- fore coming in contact with oxygen, it constitutes, while floating in the gas, smoke, and when deposited on' solid bodies, soot. But if the disengaged carbon is maintained at the temperature of ignition, and supplied with oxygen sufficient for its combustion, it burns while float- ing in the inflammable gas. and forms red, yellow, or white flame. The flame from fuel is the larger the more slowly its combustion is effected. The flame itself is apt to be chilled by radiation, as into the heating surface of a steam-boiler, so that the combustion is not completed, and part of the gas and smoke pass off unburned. (3) Oxygeu or hydrogen either actually forming water, or existing in combination with the other constituents.in the proportions which form water. Such quantities of oxygen and hydrogen are to left be out of account in deter- mining the heat generated by the combustion. If the quantity of water actually or virtually present in each pound of fuel is so great as to make its latent heat of evaporation worth considering, that heat is to be deducted from the total heat of combustion of the fuel. (4) Nitrogen, either free or in combination with othar constituents. This substance is simply inert. (5) Sulphuret of iron, which exists in coal and is detrimental, as tending to cause spontaneous combustion. (6) Other mineral compounds of various kinds, which are also inert, and form the ash left after complete combustion of the fuel, and also the clinker j material produced by fusion of the ash, which tends to choke the FUEL. 621 Total Heat of Combustion of Fuels. (Rankine.)— The follow- ing table shows the total heat of combustion with oxygen of one pound of each of the substances named in it, in British thermal units, and also in lbs. of water evaporated from 212 p . It also shows the weight of oxygen re- quired to combine with each pound of the combustible and the weight of air necessary in order to supply that oxygen. The quantities of heat are given on the authority of MM. Favre and Silbermann. Hydrogen gas Carbon imperfectly burned so as to make carbonic oxide Carbon perfectly burned so as to make carbonic acid defiant gas, 1 lb Various liquid hydrocarbons, 1 lb. Carbonic oxide, as much as is made by the imperfect combustion of 1 lb of carbon, viz., 2^ lbs. . . Lbs. Oxy- gen per lb. Com- bustible. Lb. Air (about). VA 3 3/7 12 15 3/7 Total Brit- ish Heat- units. Evapora- tive Power from 212° F., lbs. 4,400 14,500 21,344 from 21,700 to 19,000 10,000 64.2 4.55 15.0 22.1 from 22££ to 20 10.45 The imperfect combustion of carbon, making carbonic oxide, produces iess than one third of the heat which is yielded by the complete combustion. The total heat of combustion of any compound of hydrogen and carbon is nearly the sum of the quantities of heat which the constituents would pro- duce separately by their combustion. (Marsh-gas is an exception.) In computing the total heat of combustion of compounds containing oxy- gen as well as hydrogen and carbon, the following principle is to be observed: When hydrogen and oxygen exist in a compound in - the proper proportion to form water (that is, by weight one part of hydrogen to eight of oxygen), these constituents have no effect on the total heat of combus- tion. If hydrogen exists in a greater proportion, only the surplus of hydro- gen above'that which is required by the oxygen is to be taken into account. The following is a general formula (Dulong's) for the total heat of combus- tion of any compound of carbon, hydrogen, and oxygen : Let C, H, and O be the fractions of one pound of the compound, which consists respectively of carbon, hydrogen, and oxygen, the remainder being nitrogen, ash, and other impurities. Let h be the total heat of combustion of one pound of the compound in British thermal units. Then h = 14,500 i C + 4.28(if - ~ ) j • I The following table shows the composition of those compounds which are of importance, either as furnishing oxygen for combustion, as entering into the composition, or as being produced by the combustion of fuel : O S3 •§ I'll flip. >-~ r m 3 111 PI Air Water Ammonia Carbonic oxide Carbonic acid defiant gas Marsh-gas or fire-damp. . Sulphurous acid Sulphuretted hydrogen. . Sulphuret of carbon H 2 NH, CO co 2 CHo CH 4 SOo SH 2 s 2 c N77 H2 H3 C12 C12 C12 C 12 S 32 S32 S64 + 23 + 16 + N14 + 16 + 32 + H2 + H4 + 32 + H2 + C12 N 79 + O 21 H2 +0 H3 fN C + O C + 02 C +H2 C +H4 622 Since each lb. of C requires 2%$ lbs. of O to burn it to C0 2 , and air contains 23$ of O, by weight, 2% -^0.23 or 11.6 lbs. of air are required to burn 1 lb. of C. Analyses of Gases of Combustion. — The following are selected from a large number o£ analyses of gases from locomotive boilers, to show the range of composition under different circumstances (P. H. Dudley, Trans. A. I. M. E., iv. 250) : Test. co 2 CO o N 1 2 3 4 5 6 8 9 13.8 11.5 8.5 2.3 5.7 8.4 12 3.4 6 2.5 i'.k 1 2.5 6. 8. 17.2 14.7 8.4 4.4 16.8 13.5 81.6 82.5 83. 80.5 79.6 82. 82.6 76.8 81.5 No smoke visible. Old fire, escaping gas white, engine working hard. Fresh fire, much black gas, " " '.' Old fire, damper closed, engine standing still. " " smoke white, engine working hard. New fire, engine not working hard. Smoke black, engine not working hard. " dark, blower on, engine standing still. " white, engine working hard. In analyses on the Cleveland and Pittsburgh road, in every instance when the smoke was the blackest, there was found the greatest percentage of uuconsumed ox} r gen in the product, showing that something besides the mere presence of oxygen is required to effect the combustion of the volatile carbon of fuels. J. C. Hoadley (Trans. A. S. M. E., vi. 749) found as the mean of a great number of analyses of flue gases from a boiler using anthracite coal: C0 2 , 13.10; CO, 0.30; O, 11.94; N, 74.66. The loss of heat due to burning C to CO instead of to C0 2 was 2.13$. The surplus oxygen averaged 113.3$ of the O required for the C of the fuel, the average for different weeks ranging from 88.6$ to 137$. Analyses made to determine the CO produced by excessively rapid firing gave results from 2.54$ to 4.81$ CO and 5.12$ to 8.01$ C0 2 ; the ratio of C in the CO to total carbon burned being from 43.80$ to 48.55$, and the number of pounds of air supplied to the furnace per pound of coal being from 33 2 to 19.3 lbs. The loss due to burning C to CO was from 27.84$ to 30.86$ of the fnll power of the coal. Temperature of the Fire. (Rankine, S. E., p. 283.)— By temper- ature of the fire is meant the temperature of the products of combustion at the instant that the combustion is complete. The elevation of that temper- ature above the temperature at which the air and the fuel are supplied to the furnace may be computed by dividing the total heat of combustion of one lb. of fuel by the weight and by the mean specific heat of the whole products of combustion, and of the air employed for their dilution under constant pressure. The specific heat under constant pressure of these prod- ucts is about as follows : Carbonic-acid gas, 0.217 ; steam, 0.475 ; nitrogen (probably), 0.245 ; air, 0.238; ashes, probably about 0.200. Using: these data, the following results are obtained for pure carbon and for olefiant gas burned, respectively, first, in just sufficient air, theoretically, for their combustion, and, second, when, an equal amount of air is supplied in addition for dilution. Fuel. Products undiluted. Products diluted. Carbon. Olefiant Gas. Carbon. Olefiant Gas. Total heat of combustion, per lb. . . Wt. of products of combustion, lbs. . Their mean specific heat Specific heat x weight Elevation of temperature, F 14,500 13 0.237 3.08 4580° 21,300 •16.43 0.257 4.22 5050° 14,500 0?238 5.94 2440° 21,300 31.86 0.248 7.9 2710° [The above calculations are made on the assumption that the specific heats of the gases are constant, but they probably increase with the in- crease of temperature (see Specific Heat), in which case the temperatures would be less than those above given. The temperature would be further CLASSIFICATION OF FUEL. 623 reduced by the heat rendered latent by the conversion into steam of any water present in the fuel.] Rise of Temperature in Combustion of Gases. (Evg'g, March 12 and April 2, 1886.)— It is found that the. temperatures obtained by experiment fall short of those obtained by calculation. Three theo- ries have been given to account for this : 1. The cooling effect of the sides of the containing vessel; 2. The retardation of the evolution of heat caused by dissociation; 3. The increase of the specific heat of the gases at very high temperatures. The calculated temperatures are obtainable only on the condition that the gases shall combine instantaneously and simulta- neously throughout their whole mass. This condition is practically impos- sible in experiments. The gases formed at the beginning of an explosion dilute the remaining combustible {gases and tend to retard or check the combustion of the remainder. CLASSIFICATION OF SOLID FUELS. Gruner classifies solid fuels as follows (2£ug'<; and M'g Jour., July, 1874) : ._.-., Ratio — Proportion of Coke or fcameofFuel. " Charcoal yielded by or O + N* . the Dry p ure FueL H Pure cellulose 8 0.28 @, 0.30 Wood (cellulose and encasing matter) 7 .30 @ .35 Peat and fossil fuel 6 @, 5 .35 @, .40 Lignite, t or brown coal 5 .40 @ .50 Bituminous coals 4 @ 1 .50 @, .90 Anthracite 1 @ 0.75 .90 @, .92 The bituminous coals he divides into five classes as below: Name of Type. Elementary Composition. Ratio — O+N*. 01 — Propor- tion of Coke yielded by Dis- tilla- tion. Nature C. H. O. Appear- ance of Coke. 1. Long flaming dry { coal, \ 2. Long flaming fat ) or coking coals, >- or gas coals, ) 3. Caking fat coals, ) or blacksmiths' \ coals, ) 4. Short flaming fat ) or caking coals. > coking coals, ) 5. Lean or anthra- } citic coals, j 75@80 80@85 84@89 88@91 90@,93 5.5@4.5 5.8@5 5@4.5 5.5@4.5 4.5@4 19.5@15 14.2@10 11 @5.5 6.5@5.5 5.5@3 4@3 3@2 2@1 1 1 0.50@,.60 .60©.68 .68®. 74 .74@.82 .82@.90 j Pulveru- 1 lent. \ Melted, •< but ( friable, f Melted; | some- ■{ what com- t pact, f Melted; J very j com- [ pact. \ Pulveru- 1 lent. * The nitrogen rarely exceeds 1 per cent of the weight of the fuel. tNot including bituminous lignites, which resemble petroleums. . Rankine gives the following: The extreme differences in the chemical composition and properties of different kinds of coal are very great. The proportion of free carbon ranges from 30 to 93 per cent; that of hydrocar- bons of various kinds from 5 to 58 per eent; that of water, or oxygen and hydrogen in the proportions which form water, from an inappreciably small quantity to 27 per cent; that of ash, from IJ^ to 26 per cent. The numerous varieties of coal may be divided into principal classes as follows: 1, anthracite coal; 2, semi-bituminous coal; 3, bituminous coal; 4, long flaming or cannel coal; 5, lignite or brown coal. 624 - FUEL. Diminution of H and O in Series from Wood to Anthracite. (Groves and Thorp's Chemical Technology, vol. i., Fuels, p. 58.) Substance. Carbon. Hydrogen. Oxygen. Woody fibre 52.65 5.25 42.10 Peat from Vulcaire 59.57 5.96 34.47 Lignite from Cologne 66.04 5.27 28.69 Earthy brown coal 73.18 5.88 21.14 Coal from Belestat, secondary 75.06 5.84 19.10 Coal from Rive de GHer 89.29 5.05 5.66 Anthracite, Mayenne, transition formation 91.58 3.96 4.46 Progressive Change from Wood to Graphite. (J. S. Newberry in Johnson's Cyclopedia.) Wood. Loss. I£ L„ss.J^, Loss. ^" Loss. Grajh- Carbon 49.1 18.65 30.45 12.35 18.10 3.57 14.53 1.42 13.11 Hydrogen... 6.3 3.25 3.05 1.85 1.20 0.93 0.27 0.14 0.13 Oxygen 44.6 24.40 20.20 18.13 2.07 1.32 0.65 0.65 0.00 100.0 46.30 53.70 32.33 21.37 5.82 15.45 2.21 13.24 Classification of Coals, as Anthracite, Bituminous, etc.— Prof. Persifer Frazer (Trans. A. I. M. E., vi, 430) proposes a classifica- tion of coals according to their " fuel ratio," that is, the ratio the fixed car- bon bears to the volatile hydrocarbon. In arranging coals under this classification, the accidental impurities, such as sulphur, earthy matter, and moisture, are disregarded, and the fuel con- stituents alone are considered. Carbon Fixed Volatile Ratio. Carbon. Hydrocarbon. I. Hard dry anthracite. 100 to 12 100. to 92.31$ 0. to 7.69$ II. Semi -anthracite 12 to 8 92.31to88.89 7.69 to 11.11 IH. Semi-bituminous. ... 8 to 5 88.89 to 83.33 11.11 to 16.67 IV. Bituminous 5 to 83.33 to 0. 16.67tol00 It appears to the author that the above classification does not draw the line at the proper point between the semi-bituminous and the bituminous coals, viz., at a ratio of C -*- V.H.C. = 5, or fixed carbon 83.33$, volatile hy- drocarbon 16.67$, since it would throw many of the steam coals of Clearfield and Somerset counties, Penn., and the Cumberland, Md., and Pocahontas, Va., coals, which are practically of one class, and properly rated as semi-bituminous coals, into the bituminous class. The dividing: line be- tween the semi -anthracite and semi-bituminous coals, C ■*■ V.H.C. = 9, would place several coals known as semi-anthracite in the semi-bituminous class. The following is proposed by the author as a better classification : Carbon Ratio. Fixed Carbon. Vol. H.C. I. Hard dry anthracite.. 100 to 12 100 to 92.31$ to 7.69$ II. Semi-anthracite 12 to 7 92.31to87.5 7.69to 12.5 III. Semi-bituminous 7 to 3 87.5 to 75 12.5 to 25 IV. Bituminous 3. to 75 to 25 to 100 Rhode Island Graphitic Anthracite.— A peculiar graphite is found at Cranstdn, near Providence, R. I. It resembles both graphite and anthracite coal, and has about the following composition (A. E. Hunt, Trans. A. I. M. E., xvii., 678): Graphitic carbon, 78$; volatile matter, 2.60$; silica, 15.06$; phosphorus, .045$. It burns with extreme difficulty. ANALYSES OF COALS. Composition of Pennsylvania Anthracites. (Trans. A. I. M. E., xiv., 706.)— Samples weighing 100 to 200 lbs. were collected from lots of 100 to 200 tons as shipped to market, and reduced by proper methods to laboratory samples. Thirty-three samples were analyzed by McCreath, giv- ing results as follows. They show the mean character of the coal of the more important coal-beds in the Northern field in the vicinity of Wilkesbarre, in the Eastern Middle (Lehigh) field in the vicinity of Hazleton, in the Western ANALYSES OF COALS. 625 Middle field in the vicinity of Shenandoah, and in the Southern field between Mauch Chunk and Tamaqua. Name of Bed. o . 3§ ^6 < % CO *2 c °3 q O Wharton. . . E. Middle 3.71 3.08 86.40 6.22 .58 3.44 28.07 Mammoth.. E. Middle 4.12 3.08 86.38 5.92 .49 3.45 27.99 Primrose . . W. Middle 3.54 3.72 81.59 10.65 .50 4.36 21.93 Mammoth . W. Middle 3.16 3.72 81.14 11.08 .90 4.38 21.83 Primrose F Southern 3.01 4.13 87.98 4.38 .50 4.48 21.32 BuckMtn.. W. Middle 3.04 3.95 82.66 9.88 .41) 4.56 20.93 Seven Foot W. Middle 3.41 3.98 80.87 11.23 .51 4.69 20.32 Mammoth . Southern 3.09 4.28 83.81 8.18 .64 4.85 19.62 Mammoth . Northern 3.42 4.38 83.27 8.20 .73 5.00 19.00 B. Coal Bed Loyalsock 1.30 8.10 83.34 6.23 1.03 8.86 10.29 The above analyses were made of coals of all sizes (mixed). When coal is screened into sizes for shipment the purity of the different sizes as regards ash varies greatly. Samples from one mine gave results as follows: Screened Analyses. Name of Through Over Fixed Coal. inches. inches. Carbon. Ash. Egg 2.5 1.75 88.49 5.66 Stove 1.75 1.25 83.67 10.17 Chestnut 1.25 .75 80.72 12.67 Buckwheat. . .50 .50 ) 05 76.92 14.6 16.62 Sulphur. 0.24 1.04 Sulphur. 0.91 0.59 Bernice Basin, Pa., Coals. Water. Vol. H.C. Fixed C. Ash. Bernice Basin, Pullivan andj * 6 '\f 8 ? 52 3 £ 7 Lycoming Cos.; range of 8. . j j m g 5g g9 39 QM This coal is on the dividing-line between the anthracites and semi-anthra- cites, and is similar to the coal of the Lykens Valley district. More recent analyses (Trans. A. I. M. E., xiv. 721) give : Water. Vol. H.C. Fixed Carb. Ash. Working seam 65 9.40 83.69 5.34 60 ft. below seam .... 3.67 15.42 71.34 8.97 The first is a semi-anthracite, the second a semi-bituminous. . Space Occupied by Anthracite Coal. (J. C. I. W., vol. hi.)— The cubic contents of 2240 lbs. of hard Lehigh coal is a little over 36 feet ; an average Schuylkill W. A., 37 to 38 feet ; Shamokin, 38 to 39 feet; Lor berry, nearly 41. According to measurements made with Wilkesbarre anthracite coal from the Wyoming Valley, it requires 32.2 cu. ft. of lump, 33.9 cu. ft. broken, 34.5 cu. ft. egg, 34.8 cu. ft. of stove, 35.7 cu. ft, of chestnut, and 36.7 cu. ft. of pea, to make one ton of coal of 2240 lbs.; while it requires 28.8 cu. ft. of lump, 30.3 cu. ft. of broken, 30.8 cu. ft. of egg, 31.1 cu. ft. of stove, 31.9 cu. ft. of chestnut, and 32.8 cu. ft. of pea, to make one ton of 2000 lbs. Composition of Anthracite and Semi-bituminous Coals. (Trans. A. I. M. E., vi. 430.)— Hard dry anthracites, 16 analyses by Rogers, show a range from 94.10 to 82.47 fixed carbon, 1.40 to 9.53 volatile matter, and 4.50 to 8.00 ash, water, and impurities. Of the fuel constituents alone, the fixed carbon ranges from 98.53 to 89.63, and the volatile matter from 1.47 to 10.37, the corresponding carbon ratios, or C -*- Vol. H.C. being from 67.02 to 8.64. Semi-anthracites. — 12 analyses by Rogers show a range of from 90.23 to 74.55 fixed carbon, 7.07 to 13.75 volatile matter, and 2.20 to 12.10 water, ash, and impurities. Excluding the ash, etc., the range of fixed carbon is 92.75 to 84.42, and the volatile combustible 7.27 to 15.58, the corresponding carbon ratio being from 12.75^0 5.41. 626 FUEL. Semi-bituminous Coals. — 10 analyses of Penna. and Maryland coals give fixed carbon 68.41 to 84.80, volatile matter 11.2 to 17.28, and ash, water, and impurities 4 to 13.99. The percentage of the fuel constituents is fixed carbon 79.84 to 88.80, volatile combustible 11.20 to 20.16, and the carbon ratio 11.41 to American Semi-bituminous and Bituminous Coals. (Selected chiefly from various papers in Trans. A. I. M. E.) Moist- ure. Vol. Hydro- arbon Fixed Carbon Sul- phur. Penna. Semi-bituminous : Broad Top, extremes of 5 Somerset Co., extremes of 5 Blair Co., average of 5 Cambria Co. , average of 7, ) lower bed, B. j Cambria Co. , 1 , | upper bed, C. \ Cambria Co., South Fork, 1 Centre Co., 1 Clearfield Co., average of 9, I upper bed, C. f " " Clearfield Co., average of 8, j lower bed, D. j — Clearfield Co., range of 17 anal. . Bituminous : Jefferson Co., average of 26 Clarion Co., average of 7 Armstrong Co., 1 Connellsville Coal Coke from Conn'ville (Standard) Youghiogheny Coal Pittsburgh, Ocean Mine jl.27 (1.89 1.07 0.74 1.14 0.60 0.70 0.81 0.41 to 1.94 1.21 1.97 .49 1.03 13.84 17.38 14.33 18.51 26.72 21.21 15.51 22.60 23.94 21.10 20.09 to 25.19 32.53 38.60 42.55 30.10 0.01 78.46 76.14 74.08 66.69 60.99 54.15 49.69 59.61 87.46 59.05 57.33 6.00 4.81 10.62 9.45 5.84 5.40 2.65 to 7.65 3.76 4.10 4.58 8.23 11.32 2.61 2.20 1.98 2.69 1.42 0.42 0.43 1.00 1.19 2.00 The percentage of volatile matter in the Kittaning lower bed B and the Freeport lower bed D increases with great uniformity from east to west; thus: Volatile Matter. Fixed Carbon. Clearfield Co, bed D 20.09 to 25.19 68.73 to 74.76 " B 22.56 to 26.13 64.37 to 69.63 ClarionCo., "B 35.70 to 42.55 47.51 to 55.44 " D 37.15 to 40.80 51.39 to 56.36 Connellsville Coal and Coke. (Trans. A. I. M. E., xiii. 332.) — The Connellsville coal-field, in the southwestern part of Pennsylvania, is a strip about 3 miles wide and 60 miles in length. The mine workings are confined to the Pittsburgh seam, which here has its best development as to size, and its quality best adapted to coke-making. It generally affords from 7 to 8 feet of coal. The following analyses by T. T. Morrell show about its range of composi- tion : Moisture. Vol. Mat. Fixed C. Ash. Sulphur. Phosph's. Herold Mine .... 1.26 28.83 60.79 8.44 .67 .013 Kintz Mine 0.79 31.91 56.46 9.52 1.32 .02 In comparing the composition of coals across the Appalachian field, in the western section of Pennsylvania, it will be noted that the Connellsville variety occupies a peculiar position between the rather dry semi-bituminous coals eastward of it and the fat bituminous coals flanking it on the west. Beneath the Connellsville or Pittsburgh coal-bed occurs an interval of from 400 to 600 feet of "barren measures," separating it from the lower productive coal measures of Western Pennsylvania. The following tables ANALYSES OE COALS. 627 show the great similarity in composition in the coals of these upper and lower coal-measures in the same geographical belt or basin. Analyses froni the Upper Coal-measures (Penna.) in a Westward Order. Localities. Moisture. Vol. Mat. Fixed Carb. Ash. Sulphur. Anthracite 1.35 3.45 89.06 5.81 0.30 Cumberland, Md 0.89 15.52 74.28 9.29 0.71 Salisbury, Pa 1.66 22.35 68.77 5.96 1.24 Connellsville, Pa 31.38 60.30 7.24 1.09 Green sbnrg, Pa 1.02 33.50 61.34 3.28 0.86 Irwin's, Pa 1.41 37.66 54.44 5.86 0.64 Analyses from tlie Lower Coal-measures in a Westward Order. Localities. Moisture. Vol. Mat. Fixed Carb. Ash. Sulphur. Anthracite 1.35 3.45 89.06 5.81 0.30 Broad Top 0.77 18.18 73.34 6.69 1.02 Bennington 1.40 27.23 61.84 6.93 2.60 Johnstown 1.18 16.54 74.46 5.96 1.86 Blairsville 0.92 24.36 62.22 7.69 4.92 Armstrong Co 0.96 38.20 52.03 5.14 3.66 Pennsylvania and Ohio Bituminous Coals. Variation in Character of Coals of the same Beds in different Dis- tricts.— From 50 analyses in the reports of the Pennsylvania Geological Survey, the following are selected. They are divided into different groups, and the extreme analysis in each group is given, ash and other impurities being neglected, and the percentage in 100 of combustible matter being alone considered. Waynesburg coal-bed, upper bench. . . . Jefferson township, Greene Co Hopewell township, Washington Co.. Waynesburg coal-bed, lower bench Morgan township, Greene Co Pleasant Valley, Washington Co Sewickley coal-bed. Whitely Creek. Greene Co Gray's Bank Creek, Greene Co Pittsburgh coal-bed: Upper bench, Washington Co Lower bench, " " Main bench, Greene Co Frick & Co., Washington Co., average Lower bench, Greene Co Somerset Co., semi-bituminous (showing decrease of vol. mat. to the eastward) Beaver Co., Pa Diehl's Bank, Georgetown Bryan's Bank, Georgetown Ohio. Pittsburgh coal-bed in Ohio: Jefferson Co., Ohio Belmont Co., Ohio Harrison Co., Ohio Pomeroy Co., Ohio "No. of Fixed Vol. Analyses Carbon H. C. 5 59.72 40.28 53.22 46.78 9 60.69 39.31 54.31 45.69 3 64.39 35.61 60.35 39.65 j 60.87 39.13 1 59.11 40.89 5 j 63.54 36.46 j 50.97 49.03 3 j 61.80 38.20 j 54.33 45.67 66.44 33.56 1 57.83 42.17 \ 8 (79.73 | 75.47 20.27 24.53 ' 40.68 59.32 62.57 37.43 61.45 38.55 j 63.46 36.54 )66.14 33.86 I 63.46 36.54 (64.93 35.07 ( 60.92 39.08 "(62.33 37.67 Carbon Ratio. 1.48 1.13 1.54 1.19 1.80 1.52 1.74 1.04 1.61 1.19 1.98 1.37 0.68 1.66 1.59 1.73 1.95 1.73 1.85 1.55 1.65 628 Analyses of Southern and Western Coals. Moisture. Vol. Mat. Fixed C. Ash. Sul- phur. Ohio. j 5.00 1 7.40 j 95 | 1.23 j from 0.67 | to 2.46 1.48 j 0.40 1 1.79 ( 1 .57 "I 1.56 ] from 1 to .. ( 0.52 | 0.62 J from 0.76 I to 0.94 j 0.34 1 1.35 (from 0.80 | to 2.01 (from 1.26 | to 1.32 j from 3.60 1 to 7.06 j from 1 to .... j from 70 1 to 1.83 1.75 2.74 94 1.60 1.30 1.20 3.01 .12 1.59 2.00 1.78 32.80 29.20 19.13 15.47 27.28 38.60 32.24 18.60 23.96 9.64 14.26 21.33 30.50 26.06 23.90 18.48 17.57 18.19 29.59 25.35 31.44 36.27 35.15 39.44 30.60 38.70 40.201 63.301 32.33 41.29 26.62 26.50 23.72 29.30 21.80 23.05 42.76 26.11 38.33 32.90 30.60 53.15 60.45 72.70 73.51 46.70 67.83 58.89 71.00 59.98 79.93 81.61 54.97 70.80 63.75 74.20 75.22 75.89 79.40 69.00 70.67 54.80 63.50 60.85 52.48 58.80 53.70 59.80 coke 33.70 coke 46.61 61.66 60.11 67.08 63.94 61.00 74.20 60.50 48.30 71.64 54.64 53.08 66.58 9.05 2.95 6.40 9.09 2.00 15.76 7.72 10.00 14.28 8.86 2.24 3.35 22.60 10.06 1.38 5.68 1.11 4.92 1.07 2.10 1.73 8.25 1.23 5.52 3.40 6.50 8.81 4.80 16.94 1.11 11.52 3.68 11.40 7.80 2.70 15.16 3.21 2.03 5.45 11.34 1.09 0.44 Maryland. 0.93 0.78 Virginia. South of James River, 23 anal- yses, range 0.70 0.58 2.89 1.45 0.23 North of James River, eastern outcrop, Carbonite or Natural Coke Western outcrop, 11 analyses, range Pocahontas Flat-top* (Castner & Curran's Circular) West Virginia (New River.) Quinnimont,t 3 analyses 0.52 0.28 0.23 0.30 Virginia and Kentucky. Big Stone Gap Field, % 9 anal- yses, range Kentucky. Pulaski Co., 3 analyses, range Muhlenberg Co., 4 analyses, range Kentucky Cannel Coals, § 5 an- alyses, range Tennessee. Scott Co., Range of several. 1.. Roane Co., Rockwood Hamilton Co., Melville 0.08 0.56 1.72 0.40 1.00 0.79 3.16 0.96 1.32 3.37 0.77 1.49 91 1 19 Sewanee Co., Tracy City Kelly Co., Whiteside Georgia. 84 Alabama. Warren Field: Jefferson Co., Birmingham.. " Black Creek.. Tuscaloosa Co Cahaba Field, 1 Helena Vein . Bibb Co ("Coke Vein.... 2.72 .10 1.33 .68 .04 * Analyses of Pocahontas Coal by John Pattinson, F.C.S., C. H. O. N. S. Ash. Water. Coke. J°{/ Lumps... 86.51 4.44 4.95 0.66 0.61 1.54 1.29 78.8 21.2* Small ... 83.13 4.29 5.33 0.66 0.56 4.63 1.40 79.8 20.2 Calorific value, by Thomson's Calorimeter: Lumps = 15.4 lbs. of water evaporated from and at 212°; small = 14.7 lbs. t These coals are coked in beehive ovens, and yield from 63$ to 64$ of coke. JThis field covers about 120 square miles in Virginia, and about 30 square miles in Kentucky. § The principal use of the cannel coals is for enriching illuminating-gas. || Volatile matter including moisture. 1 Single analyses from Morgan, Rhea, Anderson, and Roane counties fall within this range. ANALYSES OF COALS. G29 Alabama Coals. (W. B. Phillips, Eng. & M. J., June 3, 1 Location. Proximate. Ultimate. Name of Seam. ill c o is d o 3 o bJ3 O O ■a O t3 o i s < o © si |i r=> o o q E-1 Per cent of Yield. a 6 < it "S CD I 3 s ?? ^ fU h. m. lb. lb. lb. lb. lb. 1 67 00 12,420 99 385 7,518 7,903 00.80 3 10 60.53 63.63 35.57 •2 08 00 11,090 90 359 6,580 6,939 00.81 3.24 59.33 62.57 36.62 3 45 00 9,120 77 272 5,418 5,690 00.84 2.98 59.41 62.39 36 77 4 45 00 9,020 41,650 74 349 5,334 24.850 5,683 00.82 3.87 59.13 63.00 36.18 340 1365 26,215 00.82 3.28 59.66 62.94 36.24 These results show, in a general average, that Connellsville coal carefully coked in a modern beehive oven will yield 66.17$ of marketable coke, 2.30'^ of small coke or braize, and 0.82$ of ash. 638 FUEL. The total average loss in volatile matter expelled from the coal in coking amounts to 30.71$. The modern beehive coke oven is 12 feet in diameter and 7 feet high at crown of dome. It is used in making 48 and 12 hour coke. In making these tests the coal was weighed as it was charged into the oven; the resultant marketable coke, small coke or braize and ashes weighed drv as they were drawn from the oven. Coal Washing.— In making coke from coals that are high in ash and sulphur, it is advisable to crush and wash the coal before coking it. A coal- washing plant at Brookwood, Ala., has a capacity of 50 tons per hour. The average percentage of ash in the coal during ten days' run varied from 14$ to 21$, in the washed coal from 4 8$ to 8.1$. and in the coke from 6.1$ to 10.5$. During three months the average reduction of ash was 60.9$. (Eng. and Mining Jour., March 25, 1893.) Recovery of By-products in Coke Manufacture.— In Ger- many considerable progress has been made in the recovery of by-products. The Hoffman-Otto oven has been most largely used, its principal feature being that it is connected with regenerators. In 1884 40 ovens on this system were running, and in 1892 the number had increased to 1209. A Hoffman-Otto oven in Westphalia takes a charge of 6*4 tons of dry coal and converts it into coke in 48 hours. The product of an oven annually is 1025 tons in the Ruhr district, 1170 tons in Silesia, and 960 tons in the Saar dis- trict. The yield from dry coal is 75$ (o 77$ of coke, 2.5$ to 3$ of tar, and 1.1$ to 1.2$ of sulphate of ammonia in the Ruhr district; 65$ to 70$ of coke, 4$ to 4.5$ of tar, and l$to 1.25$ of sulphate of ammonia in the Upper Silesia region and 68$ to 72$ of coke, 4$ to 4.3$ of tar and 1.8$ to 1.9$ of sulphate of ammonia in the Saar district. A group of 60 Hoffman ovens, therefore, yields annually the following: Poke Tar Sulphate District. <**£ Tar Ammonia , tons. Ruhr 51,300 1860 780 Upper Silesia 48,000 3000 840 Saar 40,500 2400 492 An oven which has been introduced lately into Germany in connection with the recovery of by-products is the Semet-Solvay, which works hotter than the Hoffman -Otto, and for this reason 73$ to 77$ of gas coal can be mixed with 23$ to 27$ of coal low in volatile matter, and yet yield a good coke. Mixtures of this kind yield a larger percentage of coke, but, oh the other hand, the amount of gas is lessened, and therefore the yield of tar and ammonia is not so great. In the manufacture of coke from soft coal in retort ovens, particularly in those constructed so as to save the by-products formed in the coking oper- ations, the coke has the disadvantage of being more porous, softer, with more easily crushed" cell-walls than when the same coal is coked in the ordinary beehive-oven. References: F. W. Luerman, Verein Deutscher Eisenhuettenleute 1891, Iron Age, March 31, 1892 ; Amer: Mfr., April 28, 1893. An excellent series of articles on the manufacture of coke, by John Fulton, of Johnstown, Pa., is published in the Colliery Engineer, beginning in January, 1893. Making Hard Coke.— J. J. Fronheiser and C. S Price, of the Cam- bria Iron Co., Johnstown. Pa., have made an improvement in coke manu- facture by which coke of any degree of hardness may be turned out. It is accomplished by first grinding the coal to a coarse powder and mixing it with a hydrate of lime (air or water slacked caustic lime) before it is charged into the coke-ovens. The caustic lime or other fluxing material used is mechanically combined with the coke, filling up its cell-walls. It has been found that about 5$ by weight of caustic lime mixed with the fine coal gives the best results. However, a larger quantity of lime can be added to coals containing more than 5$ to 7$ of ash (Amer. Mfr.) Generation of Steam from the "Waste Meat and Gases 6f Coke-ovens. (Erskine Ramsey, Amer. Mfr., Feb. 16, 1894.)— The gases from a number of adjoining ovens of the beehive type are led into a long horizontal flue, and thence to a combustion-chamber under a battery of boilers. Two plants are in satisfactory operation at Tracy City, Tenn., and two at Pratt Mines. Ala. A Bushel of Coal.— The weight of a bushel of coal in Indiana is 70 lbs., in Penna. 76 lbs.; in Ala., Colo., Ga., 111., Ohio, Tenn., and W. Va. it is 80 lbs. A Bushel of Coke is almost uniformly 40 lbs., but in exceptional WOOD AS FUEL. 639 cases, when the coke is very light, 38, 36, and 33 lbs. are regarded as a bushel. In others, from 42 to 50 lbs are given as the weight of a bushel ; in this case the coke would be quite heavy. Products of the Distillation of Coal.— S. P. Sadler's Handbook of Industrial Organic Chemistry gives a diagram showing over 50 chemical products that are derived from distillation of coal. The first derivatives are coal-gas, gas-liquor, coal-tar, and coke. From the gas-liquor are derived ammonia and sulphate, chloride and carbonate of ammonia. The coal-tar is split up into oils lighter than water or crude naphtha, oils heavier than water — otherwise dead oil or tar. commonly called creosote, — and pitch. From the two former are derived a variety of chemical products. From the coal-tar there comes an almost endless chain of known combina- tions. The greatest industry based upon their use is the manufacture of dyes, and the enormous extent to which this has grown can be judged from the fact that there are over GOO different coal-tar colors in use, and many more which as yet are too expensive for this purpose. Many medicinal prepara- tions come from the series, pitch for paving purposes, and chemicals for - the photographer, the rubber manufacturers and tanners, as well as for preserving timber and cloths. The composition of the hydrocarbons in a soft coal is uncertain and quite complex; but the ultimate analysis of the average coal shows that it ap- proaches quite nearly to the composition of CH 4 (marsh-gas). (W. H. Blauvelt, Trans. A. I. M. E., xx. 625.) WOOD AS FUEL. Wood, when newly felled, contains a proportion of moisture which varies very much in different kinds and in different specimens, ranging between 30$ and 50$, and being on an average about 40%. After 8 or 12 months' ordi- nary drying in the air the proportion of moisture is from 20 to 25$. This degree of dryness, or almost perfect dryness if required, can be produced by a few days' drying in an oven supplied with air at about 240° F. When coal or coke is used as the fuel for that oven. 1 lb. of fuel suffices to expel about 3 lbs. of moisture from the wood. This is the result of experiments on a large scale by Mr. J. R. Napier. If air- dried wood were used as fuel for the oven, from 2 to 2% lbs. of wood would 'probably be required to produce the same effect. The specific gravity of different kinds of wood ranges from 0.3 to 1.2. Perfectly dry wood'contains about 50$ of carbon, the remainder consisting almost entirely of oxygen and hydrogen in the proportions which form water. The coniferous family contain a small quantity of turpentine, which is a hydrocarbon. The proportion of ash in w r ood is from 1$ to 5$. The total heat of combustion of all kinds of wood, when dry, is almost ex- actly the same, and is that due to the 50$ of carbon. The above is from Rankine; but according to the table by S. P. Sharpless in Jour. 0. 1. W., iv. 36, the ash varies from 0.03$ to 1.20$ in American woods, and the fuel value, instead of being the same for all woods, ranges from 3667 (for white oak) to 5546 calories (for long-leaf pine) = 6600 to 9883 British thermal units for dry wood, the fuel value of 0.50 lbs. carbon being 7272 B. T. TJ. Heating Value of Wood.— The following table is given in several books of reference, authority and quality of coal referred to not stated. The weight of one cord of different woods (thoroughly air-dried) is about as follows : Hickory or hard maple 4500 lbs. equal to 1800 lbs. coal. (Others give 2000.) White oak 3850 " " 1540 " " ( " 1715.) Beech, red and black oak.. 3250 " " 1300 " " ( " 1450.) Poplar, chestnut, and elm.. 2350 " " 940 " " ( " 1050.) The average pine 2000 " " 800 " " ( ' 925.) Referring to the figures in the last column, it is said : From the above it is safe to assume that 2*4 lbs. of dry wood are equal to 1 lb. average quality of soft coal and that the full value of the same weight of different woods is very nearly the same — that is, a pound of hickory is worth no more for fuel than a pound of pine, assuming both to be dry. It is important that the wood be dry, as each 10$ of water or moisture in wood will detract about 12$ from its value as fuel. Taking an average wood of the analysis C 51$, H 6.5$, O 42.0$, ash 0.5%, perfectly dry, its fuel value per pound, according to Dulong's formula, Y — 640 [l4,500 C + 62,000 (H -^)], is 8170 British thermal units. If the wood, as ordinarily dried in air, contains 25$ of moisture, then the heating value of a pouud of such wood is three quarters of 8170 — 6127 heat-units, less the heat required to heat and evaporate the J4 lb. of water from the atmospheric temperature, and to heat the steam made from this water to the tempera- ture of the chimney gases, say 150 heat-units per pound to heat the water to 212°, 966 units to evaporate it at that temperature, and 100 heat-units to raise the temperature of the steam to 420° F., or 1 216 in all = 304 for 14 lb., which subtracted from the 6127, leaves 5821 heat-units as the net fuel value of the wood per pound, or about 0.4 that of a pound of carbon. Composition of Wood. (Analysis of Woods, by M. Eugene Chevandier.) Composition. Carbon. Hydrogen. Oxygen. Nitrogen. Ash. 49.36$ 49.64 50.20 49.37 49.96 6.01$ 5.92 6.20 6.21 5.96 42.69$ 41.16 41.62 41.60 39.56 0.91$ 1.29 1.15 0.96 0.96 1.06$ Oak .. 1.97 Birch 0.81 1.86 Willow.......... 3.37 Average 49.70$ 6.06$ 41.30$ 1.05$ 1.80$ The following table, prepared by M. Violette, shows the proportion of water expelled from wood at gradually increasing temperatures: Temperature. Water Expelled from 100 Parts of Wood. Oak. Ash. Elm. Walnut. 257° Fahr 15.26 17.93 32.13 35.80 44.31 14.78 16.19 21.22 27.51 33.38 15.32 17.02 36.94? 33.38 40.56 15.55 302° Fahr 347° Fahr 17.43 21.00 41.77? 437° Fahr 36.56 The wood operated upon had been kept in store during two years. When wood which has been strongly dried by means of artificial heat is left ex- posed to the atmosphere, it reabsorbs about as much water as it contains in its air-dried state. A cord of ivood = 4 X 4 X 8 = 128 cu. ft. About 56$ solid wood and 44$ interstitial spaces. (Marcus Bull, Phila.. 1829. J. C. I. W., vol. i. p. 293.) B. E. Fernow gives the per cent of solid wood in a cord as determined offi- cially in Prussia (J. C. I. W., vol. iii. p. 20): Timber cords, 74.07$ = 80 cu. ft. per cord; Firewood cords (over 6" diam.), 69.44$ = 75 cu. ft. per cord; "Billet" cords (over 3" diam.j, 55.55$ = 60 cu. ft. per cord; " Brush " woods less than 3" diam., 18.52$; Roots, 37.00$. CHARCOAL. Charcoal is made by evaporating the volatile constituents of wood and peat, either by a partial combustion of a conical heap of the material to be charred, covered with a layer of earth, or by the combustion of a separate portion of fuel in a furnace, in which are placed retorts containing the ma- terial to be charged. According to Peelet, 100 parts by weight of wood when charred in a heap yield from 17 to 22 parts by weight" of charcoal, and when charred in a retort from 28 to 30 parts. This has reference to the ordinary condition of the wood used in charcoal- making, in which 25 parts in 100 consist of moisture. Of the remaining 75 parts the carbon amounts to one half, or 37^$ of the gross weight of the wood, Hence it appears that on an average nearly half of the carbon in the CHARCOAL. 641 wood is lost during the partial combustion in a heap, and about one quarter during the distillation in a retort. To char 100 parts by weight of wood in a retort, 12^> parts of wood must be burned in the furnace. Hence in this process the whole expenditure of wood to produce from 28 to 30 parts of charcoal is 112^ parts; so tbat if the weight of charcoal obtained is compared with the whole weight of wood expended, its amount is from 25$ to 27%; and the proportion lost is on an average U% s- 37J^ = 0.3, nearly. According to Peclet, good wood charcoal contains about 0.07 of its weight of ash. The proportion of ash in peat charcoal is very variable, and is es- timated on an average at about 0.18. (Rankine.) Much infoi'mation concerning charcoal may be found in the Jom*nal of the Charcoal-iron Workers' Assn., vols. i. to vi. From this source the following notes have been taken: Yield of Charcoal from a Cord of Wood.— From 45 to 50 bushels to the cord in the kiln, and from 30 to 35 in the meiler. Prof. Egle*- ton in Trans. A. I. M. E., viii. 395, says the yield from kilns in the Lake Champlain region is often from 50 to 60 bushels for hard wood and 50 for soft wood; the average is about 50 bushels. The apparent yield per cord depends largely upon whether the cord is a full cord of 128 cu. ft. or not. In a four months' test of a kiln at Goodrich, Tenn., Dr. H. M. Pierce found results as follows: Dimensions of kiln— inside diameter of base, 28 ft. 8 in.; diam. at spring of arch, 26 ft. 8 in. ; height of walls, 8 ft. ; rise of arch, 5 ft.; capacity, 30 cords. Highest yield of charcoal per cord of wood (measured) 59.27 bushels, lowest 50.14 bushels, average 53.65 bushels. No. of charges 12, length of each turn or period from one charging to another 11 days. (J. C. I. W., vol. vi. p. 26.) Results from Different Methods of Charcoal-making. Character of Wood used. Yield. |8o I- O *°° 63.4 54.2 66.7 62.0 59.5 43.9 45.0 35 22 Coaling Methods. - Z - -j c ■_ l f "z > ~ •z%£ Odelstjerna's experiments Mathieu's retorts, fuel ex- cluded Mathieu's retorts, fuel in- Birch dried at 230 F ( Air dry, av. good yel- ) < low pine weighing > { abt. 28 lbs. percu. ft. ) j Good dry fir and pine, ) ) mixed. j j Poor wood, mixed fir | j and pine j I Fir and white-pine < wood, mixed. Av. 25 j- ( lhs. per cu. ft. \ Av. good yellow pine < weighing abt. 25 lbs. > ( per cu. ft. j 77.0 65.8 81.0 70.0 72.2 52.5 54.7 42.9 35.9 28.3 24.2 27.7 25 8 24 7 IS 3 22.0 17.1 15.7 15.7 Swedish ovens, av. results Swedish ovens, av. results Swedish meiiers excep- 13.3 13.3 13.3 Swedish meiiers. av. results American kilns, av. results American meiiers, av. re- sults 13.3 17 5 17.5 Consumption of Charcoal in Blast-furnaces per Ton of Pig Iron ; average consumption according to census of 1880, 1.14 tons charcoal per ton of pig. The consumption at the best furnaces is much below this average. As low as S53 ton, is recorded of the Morgan furnace; Bay furnace, 0.858; Elk Rapids. 0.884. (1892.) Absorption of Water and of Oases by Charcoal. -Svedlius, in his hand-book for charcoal-burners, prepared for the Swedish Govern- ment, says: Fresh charcoal, also reheated charcoal, contains scarcely any water but when cooled it absorbs it very rapidly, so that after twenty-four hours, it may contain 4% to 8% of water. After the lapse of a few weeks the moisture of charcoal may not increase perceptibly, and may be estimated at 10$ to 15#, or an average of 12$. A thoroughly charred piece of charcoal ought, then, to contain about 84 parts carbon, 12 parts water, 3 parts ash, and 1 part hydrogen. 642 FUEL. M. Saussure, operating with blocks of fine boxwood charcoal, freshly- burnt, found that by simply placing such blocks in contact with certain gases they absorbed them in the following proportion: Volumes. Carbonic oxide 9.42 Oxygen 9.25 Nitrogen 6.50 Carburetted hydrogen ...... 5.00 Hydrogen 1.75 Volumes. Ammonia 90.00 Hydrochloric-acid gas 85.00 Sulphurous acid 65.00 Sulphuretted hydrogen 55.00 Nitrous oxide (laughing-gas). . 40.00 Carbonic acid . . 35.00 It is this enormous absorptive power that renders of so much value a comparatively slight sprinkling of charcoal over dead animal matter, as a preventive of the escape of odors arising from decomposition. In a box or case containing one cubic foot of charcoal may be stored without mechanical compression a little over nine cubic feet of oxygen, representing a mechanical pressure of one hundred and twenty -six pounds to the square inch. From the store thus preserved the oxygen can be drawn by a small hand-pump. Composition of Charcoal Produced at Various Tempera- tures. (By M. Violette.) Temperature of Car bonization. Cent. 150° 200 250 300 350 432 392 482 592 662 810 1873 Composition of the Solid Product. Hydro- gen. Per cent. 6.12 3.99 4.81 4.25 4.14 4.96 2.30 Per cent. 46.29 43.98 28.97 21.96 18.44 15.24 14.15 Per cent. 0.08 0.23 0.63 0.57 0.61 1.61 1.60 Per cent. 47.51 39.88 32.98 24.61 22.42 15.40 15.30 The wood experimented on was that of black alder, or alder buckthorn, which furnishes a charcoal suitable for gunpowder. It was previously dried at 150 deg. C. = 302 deg. F. MISCELLANEOUS SOLID FUELS. Dust Fuel— Dust Explosions. —Dust when mixed in air burns with such extreme rapidity as in some cases to cause explosions. Explosions of flour-mills have been attributed to ignition of the dust in confined passages. Experiments in England in 1876 on the effect of coal-dust in carrying flame in mines showed that in a dusty passage the flame from a blown-out shot may travel 50 yards. Prof. F. A. Abel (Trans. A. I. M. E , xiii. 260) says that coal- dust in mines much promotes and extends explosions, and that it may read- ily be brought into operation as a fiercely burning agent which will carry flame rapidly as far as its mixture with air extends, and will operate as an explosive agent though the medium of a very small proportion of fire-damp in the air of the mine. The explosive violence of the combustion of dust is largely due to the instantaneous heating and consequent expansion of the air. (See also paper on " Coal Dust as an Explosive Agent," by Dr. R. W. Raymond, Trans. A. I. M. E. 1894.) Experiments made in Germany in 1893. show that pulverized fuel may be burned without smoke, and with high economy. The fuel, instead of being introduced into the fire-box in the ordinary manner, is first reduced to a powder by pulverizers of any con- struction. In the place of the ordinary boiler fire-box there is a combustion chamber in the form of a closed furnace lined with fire-brick and provided with an air-injector similar in construction to those used in oil-burning fur- naces. The nozzle throws a constant stream of the fuel into the chamber. This nozzle is so located that it scatters the powder throughout the whole MISCELLANEOUS SOLID FUELS. 643 space of the fire-box. When this powder is once ignited, and it is very readily done by first raising the lining to a high temperature by an open fire, the combustion continues in an intense and regular manner under the action of the current of air which carries it in. (Mfrs. Record, April, 1893.) Powered fuel was used in the Crompton rotary puddhng-furnace at Woolwich Arsenal. England, in 1873. (Jour. I. & S. I., i. 1873, p. 91.) Peat or Turf, as usually dried in the air, contains from 25% to 30$ of water, which must be allowed for in estimating its heat of combustion. This water having been evaporated, the analysis of M. Regnault gives, in 100 parts of perfectly dry peat of the best quality: C 58$, H 6%, O 31%, Ash 5%. In some examples of peat the quantity of ash is greater, amounting to 7% and sometimes to 11%. The specific gravitj 7 of peat in its ordinary state is about 0.4 or 0.5. It can be compressed by machinery to a much greater density. (Rankine.) Clark (Steam-engine, i. 61) gives as the average composition of dried Irish peat: C 59j6, H 6%, O 30, N 1.25%, Ash 4. Applying Dulong's formula to this analysis, we obtain for the heating value of perfectly dry peat 10,260 heat-units per pound, and for air-dried peat con- taining 25% of moisture, after making allowance for evaporating the water, 7391 heat-units per pounds. Sawdust as Fuel.— The heating power of sawdust is naturally the same per pound as that of the wood from which it is derived, but if allowed to get wet it is more like spent tan (which see below). The conditions neces- sary for burning sawdust are that plenty of room should be given it in the furnace, and sufficient air supplied on the surface of the mass. The same applies to shavings, refuse lumber, etc. Sawdust is frequently burned in saw-mills, etc., by being blown into the furnace by a fan-blast. Horse-manure has been successfully used as fuel by the Cable Rail- way Co. of Chicago. It was mixed with soft coal and burned in an ordinary urnace provided with a fire-brick arch. "Wet Tan Bark as Fuel.— Tan, or oak bark, after having been used in the processes of tanning, is burned as fuel. The spent tan consists of the fibrous portion of the bark. According to M. Peclet, five parts of oak bark produce four parts of dry tan; and the heating power of perfectly dry tan, containing 15% of ash, is 6100 English units; whilst that of tan in an ordinary state of dryness, containing 30% of water, is only 4284 English units. The weight of water evaporated from and at 212° by one pound of tan, equiva- lent to these heating powers, is, for perfectly dry tan, 5.46 lbs., for tan with 30% moisture, 3.84 lbs. Experiments by Prof. R.H. Thurston (Jour. Frank. Inst., 1874) gave with the Crockett furnace, the wet tan containing 59% of water, an evaporation from and at 212° F. of 4.24 lbs. of water per pound of the wet tan, and with the Thompson furnace an evaporation of 3.19 lbs. per pound of wet tan containing 55% of water. The Thompson furnace con- sisted of six fire-brick ovens, each 9 feet x 4 feet 4 inches, containing 234 square feet of grate in all, for three boilers with a total heating surface of 2000 square feet, a ratio of heating to grate surface of 9 to 1. The tan was fed through holes in the top. The Crockett furnace was an ordinary fire- brick furnace, 6x4 feet, built in front of the boiler, instead of under it, the ratio of heating surface to grate being 14.6 to 1. According to Prof. Thurs- ton the conditions of success in burning wet fuel are the surrounding of the mass so completely with heated surfaces and with burning fuel that it may be rapidly dried, and then so arranging the apparatus that thorough com- bustion may be secured, and that the rapidity of combustion be precisely equal to and never exceed the rapidity of desiccation. Where this rapidity of combustion is exceeded the dry portion is consumed completely, leaving an uncovered mass of fuel which refuses to take fire. Straw as Fuel. (Eng'g Mechanics, Feb., 1893, p. 55.)— Experiments in Russia showed that winter-wheat straw, dried at 230° F., had the following composition: C, 46.1; H, 5.6; N, 0.42; O, 43.7; Ash, 4.1. Heating value in British thermal units: dry straw, 6290; with 6% water, 5770; with 10% water, 5448. With straws of other grains the heating value of dry straw ranged from 5590 for buckwheat to 6750 for flax. Clark (S. E., vol. 1, p. 62) gives the mean composition of wheat and barley straw as C, 36; H. 5; O, 38; O, 0.50; Ash, 4.75; water, 15.75, the two straws varying less than 1%. The heating value of straw of this composition, accord- ing to Dulong's formula, and deducting the heat lost in evaporating the water, is 5155 heat units. Clark erroneously gives it as 8144 heat units. Bagasse as Fuel in Sugar Manufacture.- -Bagasse is the name given to refuse sugar-cane, after the juice has been extracted. Prof. L. A. 644 FUEL. Becuel, in a paper read before the Louisiana Sugar Chemists' Association, in 1892, says; " With tropical cane containing 12.5$ woody fibre, a juice contain- ing 16.13$ solids, and 83.37$ water, bagasse of, say, 66$ and 72$ mill extrac- tion would have the following percentage composition: Woody Combustible Wotm , Fibre. Salts. water. 66$ bagasse 37 10 53 72% bagasse 45 9 46 "Assuming that the woody fibre contains 51$ carbon, the sugar and other combustible matters an average of 42.1$, and that 12,906 units of heat are generated for every pound of carbon consumed, the 66$ bagasse is capable of generating 297,834 heat units as against 345,200, or a difference of 47,366 units in favor of the 72$ bagasse. "Assuming the temperature of the waste gases to be 450° F., that of the surrounding atmosphere and water in the bagasse at 86° F., and the quan- tity of air necessary for the combustion of one pound of carbon at 24 lbs., the lost heat will be as follows: In the waste gases, heating air from 86° to 450° F., and in vaporizing the moisture, etc., the 66$ bagasse will require 112,546 heat units, and 116,150 for the 72$ bagasse. " Subtracting these quantities from the above, we find that the 66$ bagasse will produce 185,288 available heat units, or nearly 38% less than the 72$ bagasse, which gives 299,050 units. Accordingly, one ton of cane of 2000 lbs. at 66$ mill extraction will produce 680 lbs. bagasse, equal to 125,995,840 avail- able heat units, while the same cane at 72$ extraction will produce 560 lbs. bagasse, equal to 167.468,000 units. "A similar calculation for the case of Louisiana cane containing 10$ woody fibre, and 16$ total solids in the juice, assuming 75$ mill extraction, shows that bagasse from one ton of cane contains 157,395,640 heat units, from which 56,146,500 have to be deducted. " This would make such bagasse worth on an average nearly 92 lbs. coal per ton of cane ground. Under fairly good conditions, 1 lb. coal will evap- orate 7% l° s - water, while the best boiler plants evaporate 10 lbs. Therefore, the bagasse from 1 ton of cane at 75$ mill extraction should evaporate from 689 lbs. to 919 lbs. of water. The juice extracted from such cane would un- der these conditions contain 1260 lbs. of water. If we assume that the water added during the process of manufacture is 10$ (by weight) of the juice made, the total water handled is 1410 lbs. From the juice represented in this case, the commercial massecuite would be about 15$ of the weight of the original mill juice, or say 225 lbs. Said mill juice 1500 lbs., plus 10$, equals 1650 lbs. liquor handled; and 1650 lbs., minus 225 lbs., equals 1425 lbs., the quantity of water to be evaporated during the process of manufacture. To effect a 7^-lb. evaporation requires 190 lbs. of coal, and 1421^ lbs. for a 10- lb. evaporation. " To reduce 1650 lbs. of juice to syrup of, say, 27° Baume. requires the evap- oration of 1770 lbs. of water, leaving 480 lbs. of syrup: If this work be ac- complished in the open air, it will require about 156 lbs. of coal at 7\i lbs. boiler evaporation, and 117 at 10 lbs. evaporation. " With a double effect the fuel required would be from 59 to 78 lbs., and with a triple effect, from 36 to 52 lbs. " To reduce the above 480 lbs. of syrup to the consistency of commercial massecuite means the further evaporation of 255 lbs. of water, requiring the expenditure of 34 lbs. coal at 7J^ lbs. boiler evaporation, and 25^ lbs. with a 10-lb. evaporation. Hence, to manufacture one ton of cane into sugar and molasses, it will take from 145 to 190 lbs. additional coal to do the work by the open evaporator process; from 85 to 112 lbs. with a double effect, and only 7J^ lbs. evaporation in the boilers, while with 10 lbs. boiler evaporation the bagasse alone is capable of furnishing 8% more heat than is actually re- quired to do the work. With triple-effect evaporation depending on the ex- cellence of the boiler plant, the 1425 lbs. of w T ater to be evaporated from the juice will require between 62 and 86 lbs. of coal. These values show that from 6 to 30 lbs. of coal can be spared from the value of the bagasse to run engines, grind cane, etc. "It accordingly appears." says Prof. Becuel, "that with the best boiler plants, those taking up all the available heat generated, by using this heat economically the bagasse can be made to supply all the fuel required by our sugar-houses." PETROLEUM. 645 PETROLEUM. Products of the Distillation of Crude Petroleum. Crude American petroleum of sp. gr. 0.800 may be split up by fractional distillation as follows (Robinson's Gas and Petroleum Engines): Temp, of ' Distillation Fahr. Distillate. Percent- ages. Specific Gravity. Flashing Point. Deg. F. 113° Rhigolene. |_ traces. 1.5 10. 2.5 2. .590 to .625 .636 to .657 .680 to .700 .714 to .718 .725 to .737 113 to 140° 140 to 158° Chymogene. ) Gasolene (petroleum spirit)... Benzine, naphtha C, benzolene. ( Benzine, naphtha B ^ " " A 158 to 248° 248° to 14 32 347° 338° and 1 upwards, j 482° Kerosene (lamp-oil) Lubricating oil Paraffine wax Residue and Loss 50. 15. 2. 16. .802 to .820 .850 to .915 100 to 122 230 Lima Petroleum, produced at Lima, Ohio, is of a dark green c very fluid, and marks 48° Baume at 15° C. (sp. gr., 0.792). The distillation in fifty parts, each part representing 2% by volume, the following results : Per Sp. Per Sp. Per Sp. Per Sp. Per Sp. Per Sp. cent. Gr. cent. Gr. cent. Gr. cent. Gr. cent. Gr. cent. Gr. 2 680 18 0.720 34 0.764 50 0.802 68 0.820 88 0.815 4 683 20 .728 36 .768 52! 70 .825 90 .815 6 685 22 .730 38 .772 to^ .806 72 .830 s 8 690 24 .735 40 .778 58^ 73 .830 92) 10 694 26 .740 42 .782 60 .800 76 .810 to V 3 12 698 28 .742 44 .788 62 .804 78 .820 100 ) s 14 700 30 .746 46 .792 64 .808 82 .818 a; 16 706 32 .760 48 .800 66 .812 86 .816 03 RETURNS. 16 per cent naphtha, 70° Baume. 6 per cent paraffine oil. 68 " burning oil. 10 " residuum. The distillation started at 23° O, this being due to the large amount of naphtha present, and when 60$ was reached, at a temperature of 310° C, the hydrocarbons remaining in the retort were dissociated, then gases escaped, lighter distillates were obtained, and, as usual in such cases, the temperature decreased from 310° O. down gradually to 200° C, until 75$ of oil was obtained, and from this point the temperature remained constant until the end of the distillation. Therefore these hydrocarbons in statu moriendi absorbed much heat. {Jour. Am. Chem. Soc.) Value of Petroleum as Fuel.— Thos. Urquhart, of Russia (Proc. Inst. M. E., Jan. 1889), gives the following table of the theoretical evapora- tive power of petroleum in comparison with that of coal, as determined by Messrs. Favre & Silbermann: Fuel. Specific Gravity at 32° F., Water = 1.000. Chem. Com p. Heating- power, British Thermal Units. Theoret. Evap., lbs. C. H. O. lb. Fuel, from and at 212° F. Penna. heavy crude oil Caucasian light crude oil. . " heavy " " .. S. G. 0.886 0.884 0.938 0.928 1.380 p. c. 84.9 86.3 86.6 87.1 80.0 P o C „ 13.7 13.6 12.3 11.7 5.0 p. c. 1.4 0.1 1.1 1.2 8.0 Units. 20,736 22,027 20,138 ' 19,832 14,112 lbs. 21.48 22.79 20.85 20.53 Good English Coal, Mean of 98 Samples 14.61 646 FUEL. In experiments on Russian railways with petroleum as fuel Mr. Urquhart obtained an actual efficiency equal to 82$ of the theoretical heating-value The petroleum is fed to the furnace by means of a spray-injector driven by steam. An induced current of air is cariied in around the injector-nozzle, and additional air is supplied at the bottom of the furnace. Oil vs. Coal as Fuel. (Iron Age, Nov. 2, 1893.)— Test by the Twin City Rapid Transit Company of Minneapolis and St. Paul. This test showed that with the ordinary Lima oil weighing 6 6/10 pounds per gallon, and costing 2J4 cents per gallon, and coal that gave an evaporation of 7^£ lbs. of water per pound of coal, the two fuels were equally economical when the price of coal was $3 85 per ton of 2000 lbs. With the same coal at $2.00 per ton, the coal was 37$ more economical, and with the coal at $4.85 per ton, the coal was 20$ more expensive than the oil. These results include the difference in the cost of handling the coal, ashes, and oil. In 1892 there were reported to the Engineers 1 Club of Philadelphia some comparative figures, from tests undertaken to ascertain the relative value of coal, petroleum, and gas. Lbs. Water, from and at 212° F. 1 lb. anthracite coal evaporated 9 . 70 1 lb. bituminous coal 10.14 1 lb. free oil, 36° gravity 16 48 1 cubic foot gas, 20 C. P 1.28 The gas used was that obtained in the distillation of petroleum, having about the same fuel-value as natural or coal-gas of equal candle-power. Taking the efficiency of bituminous coal as a basis, the calorific energy of petroleum is more than 60$ greater than that of coal; whereas, theoretically, petroleum exceeds coal only about 45$— the one containing 14,500 heat-units, and the other 21,000. Crude Petroleum vs. Indiana Block Coal for Steam- raising at tne South Chicago Steel Works. (E. C. Potter, Trans. A. I. M. E., xvii, 807.)— With coal, 14 tubular boilers 16 ft. X 5 ft. re- quired 25 men to operate them ; with fuel oil, 6 men were required, a saving of 19 men at $2 per day, or $38 per day. For one week's work 2731 barrels of oil were used, against 848 tons of coal required for the same work, showing 322 barrels of oil to be equivalent to 1 ton of coal. With oil at 60 cents per barrel and coal at $2.15 per ton, the rel- ative cost of oil to coal is as $1.93 to $2.15. No evaporation tests were made. Petroleum as a Metallurgical Fuel.— C. E. Felton (Trans. A. I. M. E., xvii, 809) reports a series of trials with oil as fuel in steel-heating and open-hearth steel-furnaces, and in raising steam with results as follows: 1. In a run of six weeks the consumption of oil, partly refiued (the paraffine and some of the naphtha being removed), in heating 14-inch ingots in Siemens furnaces was about 6^ gallons per ton of blooms. 2. In melting in a 30-ton open-hearth furnace 48 gallons of oil were used per ton of ingots. 3. In a six weeks' trial with Lima oil from 47 to 54 gallons of oil were required per ton of ingots. 4. In a six months' trial with Siemens heating-furnaces the consumption of Lima oil was 6 gallons per ton of ingots. Under the most favorable circumstances, charging hot ingots and running full capacity, 4^ to 5 gallons per ton were required. 5. In raising steam in two 100-H.P. tubular boilers, the feed-water being supplied at 160° F., the average evap- oration was about 12 pounds of water per pound of oil, the best 12 hours' work being 16 pounds. In all of the trials the oil was vaporized in the Archer producer, an apparat- us for mixing the oil and superheated steam, and heating the mixture to a high temperature. From 0.5 lb. to 0.75 lb. of pea-coal was used per gallon of oil in the producer itself. FUEL. GAS. The following notes are extracted from a paper by W. J. Taylor on " The Energy of Fuel " (Trans. A. I. M. E., xviii. 205): Carbon Gas.— In the old Siemens producer, practically, all the heat of primary combustion— that is, the burning of solid carbon to carbon monox- ide, or about 30$ of the total carbon energy— was lost, as little or no steam was used in the producer, and nearly all the sensible heat of the gas was dissipated in its passage from the producer to the furnace, which was usu- ally placed at a considerable distance. Modern practice has improved on this plan, by introducing steam with the FUEL GAS. 647 air blown into the producer, and by utilizing the sensible heat of the gas in the combustion-furnace. It ought to be possible to oxidize one out of every four lbs. of carbon with oxygen derived from water- vapor. The thermic reactions in this operation are as follows: Heat-units. 4 lbs. C burned to CO (3 lbs. gasified with air and 1 lb. with water) develop 17,600 1.5 lbs. of water (which furnish 1.33 lbs. of oxygen to combine with 1 lb. of carbon) absorb by dissociation 10,333 The gas, consisting of 9.333 lbs. CO, 0.167 lb. H, and 13.39 lbs. N, heated 600°, absorbs 3,748 Leaving for radiation and loss 3,519 ?7,600 The steam which is blown into a producer with the air is almost all con- densed into finely-divided water before entering the fuel, and consequently is considered as water in these calculations. The 1.5 lbs. of water liberates .167 lb. of hydrogen, which is delivered to the gas, and yields in combustion the same heat that it absorbs in the pro- ducer by dissociation. According to this calculation, therefore, 60$ of the heat of primary combustion is theoretically recovered by the dissociation of steam, and, even if all the sensible heat of the gas be counted, with radia- tion and other minor items, as loss, yet the gas must carry 4 X 14,500 — (37'48 + 3519) = 50,733 heat-units, or 87$ of the calorific energy of the carbon. This estimate shows a loss in conversion of 13$, without crediting the gas with its sensible heat, or charging it with the heat required for generating the necessary steam, or taking into account the loss due to oxidizing some of the carbon to C0 2 . In good producer-practice the proportion of C0 2 in the gas represents from 4$ to 7$ of the C burned to C0 2 , but the extra heat of this combustion should be largely recovered in the dissociation of mora water-vapor, and therefore does not represent as much loss as it would indi- cate. As a conveyer of energy, this gas has the advantage of carrying 4.46 lbs. less nitrogen than would be present if the fourth pound of coal had been gasified with air; and in practical working the use of steam reduces the amount of clinkering in the producer. Anthracite Gas.— In anthracite coal there is a volatile combustible varying in quantity from 1.5$ to over 7$. The amount of energy derived from the coal is shown in the following theoretical gasification made with coal of assumed composition: Carbon, 85$; vol. HC, 5$; ash, 10$; 80 lbs. car- bon assumed to be burned to CO; 5 lbs. carbon burned to C0 2 ; three fourths of the necessary oxygen derived from air, and one fourth from water. , Products. , Process. Pounds. Cubic Feet. Anal, by Vol. 80 lbs. C burned to CO 186.66 2529.24 33.4 5 lbs. C burned to C0 2 18.33 157.64 2.0 5 lbs. vol. HC (distilled) ... , 5.00 116.60 1.6 120 lbs. oxygen are required, of which 30 lbs. from H 2 liberate H 3.75 712.50 9.4 90 lbs. from air are associatied with N 301 .05 4064.17 53.6 514.79 7580.15 100.0 Energy in the above gas obtained from 100 lbs. anthracite: 186.66 lbs. CO 807,304 heat-units. 5.00 " CH 4 117,500 3.75 " H 232,500 1,157,304 Total energy in gas per lb 2,248 " " " 100 lbs. of coal.. 1,349,500 " Efficiency of the conversion 86$. The sum of CO and H exceeds the results obtained in practice. The sen- sible heat of the gas will probably account for this discrepancy, and, there- fore, it is safe to assume the possibility of delivering at least 82$ of the energy of the anthracite. Bituminous Gas.— A theoretical gasification of 100 lbs. of coal, con- taining 55$ of carbon and 32$ of volatile combustible (which is above the average of Pittsburgh coal), is made in the following table. It is assumed that 50 lbs, of C are burned to CO and 5 lbs. to C0 2 ; one fourth of the O is 648 FUEL. derived from steam and three fourths from air; the heat value of the volatile combustible is taken at 20,000 heat-units to the pound. In comput- ing' volumetric proportions all the volatile hydrocarbons, fixed as well as condensing, are classed as marsh-gas, since it is only by some such tenta- tive assumption that even an approximate idea of the volumetric composi- tion can be formed. The energy, however, is calculated from weight: , Products. — , Process. Pounds. Cubic Feet. Anal, by Vol. 50 lbs. C burned to CO 116.66 1580.7 27.8 5 lbs. C burned to C0 2 18.33 157.6 2.7 32 lbs. vol. HC (distilled) 32.00 746.2 13.2 80 lbs. O are required, of which 20 lbs., derived from H 2 0, liberate H 2.5 475.0 8.3 60 lbs. O, derived from air, are asso- ciated with N 200.70 2709.4 47.8 370.19 5668.9 99.8 Energy in 116.66 lbs. CO 504,554 heat-units. " 32. 00 lbs. vol. HC... 640,000 " 2.50 lbs. H 155,000 1 299 554 Energy in coal 1,437,'500 Per cent of energy delivered in gas Heat-units in 1 lb. of gas , Water-gas.— Water-gas is made in an intermittent process, by blowing up the fuel-bed of the producer to a high state of incandescence (and in some cases utilizing the resulting gas, which is a lean producer-gas), then shutting off the air and forcing steam through the fuel, which dissociates the water into its elements of oxygen and hydrogen, the former combining with the carbon of the coal, and the latter being liberated. This gas can never play a very important part in the industrial field, owing to the large loss of energy entailed in its production, yet there are places and special purposes where it is desirable, even at a great excess in cost per unit of heat over producer-gas; for instance, in small high-temperature fur- naces, where much regeneration is impracticable, or where the " blow-up " gas can be used for other purposes instead of being wasted. The reactions and energy required in the production of 1000 feet of water- gas, composed, theoretically, of equal volumes of CO and H, are as follows: 500 cubic feet of H weigh 2.635 lbs. 500 cubic feet of CO weigh 36.89 " Total weight of 1000 cubic feet 39.525 lbs. Now, as CO is composed of 12 parts C to 16 of O, the weight of C in 36.89 lbs. is 15.81 lbs. and of O 21.08 lbs. When this.oxygen is derived from water it liberates, as above, 2.635 lbs. of hydrogen. The heat developed and ab- sorbed in these reactions (roughly, as we will not take into account the en- ergy required to elevate the coal from the temperature of the atmosphere to say 1800°) is as follows: Heat-units. 2.635 lbs. H absorb in dissociation from water 2.635 X 62,000.. = 163,370 15.81 lbs. C burned to CO develops 15.81 X 4400 = 69,564 Excess of heat- absorption over heat-development = 93,806 If this excess could be made up from C burnt to C0 2 without loss by radi- ation, we would only have to burn an additional 4.83 lbs. C to supply this heat, and we could then make 1000 feet of water-gas from 20.64 lbs. of car- bon (equal 24 lbs. of 85% coal). This would be the perfection of gas-making, as the gas would contain really the same energy as the coal; but instead, we require in practice more than double this amount of coal, and do not deliver more than 50% of the energy of the fuel in the gas, because the supporting heat is obtained in an indirect way and with imperfect combustion. Besides this, it is not often that the sum of the CO and H exceed 90%, the balance be- ing CO a and N. But water-gas should be made with much less loss of en- ergy by burning the "blow-up" (producer) gas in brick regenerators, the stored -up heat of which can be returned to the producer by the air used in blowing-up. The following table shows what may be considered average volumetric FUEL GAS. 649 analyses, and the weight and energy of 1000 cubic feet, of the four types of gases used for heating and illuminating purposes: Natural Gas. Coal- gas. Water- gas. Producer-gas. CO 0.50 2.18 92.6 0.31 0.26 3.61 0.34 6.0 46.0 40.0 4.0 0.5 1.5 0.5 1.5 32.0 735,000 45.0 45.0 2.0 "4.0 ' 2.0 0.5 1.5 45.6 322,000 Anthra. 27.0 12.0 1.2 '"2.5 57.0 0.3 Bitu. 27.0 H 12.0 CH 4 2.5 0.4 co 2 N 2.5 56.2 0.3 3(45.6 1,100,000 65.6 137,455 65.9 Heat units in 1000 cubic feet 156,917 Natural Gas in Ohio and. Indiana. (Eng. and M. J., April 81, 1894.) Ohio. Indiana. Description. Fos- toria. Findlay St. Mary's. Muncie. Ander- son. Koko- 1110. Mar- ion. Hydrogen 1.89 92.84 .20 .55 .20 .35 3.82 .15 1.64 93.35 .35 .41 .25 .39 3.41 .20 1.94 93.85 .20 .44 .23 .35 2.98 .21 2.35 92.67 .25 .45 .25 .35 3.53 .15 1.86 93.07 .47 .73 .26 .42 3.02 .15 1.42 94.16 .30 .55 .29 .30 2.80 .18 1.20 93.57 Olefiant gas Carbon monoxide.. Carbon dioxide Oxygen Nitrogen Hydrogen sulphide .15 .60 .30 .55 3.42 .20 Approximately 30,000 cubic feet of gas have the heating power of one ton of coal. Producer-gas from One Ton of Coal. (W. H. Blauvelt, Trans. A. I. M. E., xviii. 614.) Analysis by Vol. Per Cent. Cubic Feet. Lbs. Equal to — CO H CH 4 25.3 9.2 3.1 0.8 3.4 58.2 33,213.84 12,077.76 4,069.68 1,050.24 4,463.52 76,404.96 2451.20 63.56 174.66 77.78 519.02 5659.63 1050.51 lbs. C+ 1400.7 lbs. O. 63.56 " H. 174.66 " CH 4 . N (by difference . 77.78 " C 2 H 4 . 141.54 " C + 377.44 lbs. O. 7350.17 " Air. 100.0 131,280.00 8945.85 Calculated upon this basis, the 131,280 ft. of gas from the ton of coal con- tained 20,311,162 B.T.U., or 155 B.T.U. per cubic ft., or 2270 B.T.U. per lb. The composition of the coal from which this gas was made was as follows: Water. 1.26$; volatile matter, 36.22$; fixed carbon, 57.98$; sulphur, 0.70$; ash, 3.78$. One ton contains 1159.6 lbs. carbon and 724.4 lbs. volatile com- bustible, the energy of which is 31,302,200 B.T.U. Hence, in the processes of gasification and purification there was a loss of 35.2$ of the energy of the coal. The composition of the hydrocarbons in a soft coal is uncertain and quite complex; but the ultimate analysis of the average coal shows that it ap- proaches quite nearly to the composition of CH 4 (marsh-gas). Mr. Blauvelt emphasizes the folio wing. points as highly important in soft- coal producer-practice: 650 £UeL. First. That a large percentage of the energy of the coal is lost when the gas is made in the ordinary low producer and cooled to the temperature of the air before being used. To prevent these sources of loss, the producer should be placed so as to lose as little as possible of the sensible heat of the gas, and prevent condensation of the hydrocarbon vapors. A high fuel-bed should be carried, keeping the producer cool on top, thereby preventing the breaking-down of the hydrocarbons and the deposit of soot, as well as keep- ing the carbonic acid low. Second. That a producer should be blown with as much steam mixed with the air as will maintain incandescence. This reduces the percentage of nitrogen and increases the hydrogen, thereby greatly enriching the gas. The temperature of the producer is kept down, diminishing the loss of heat by radiation through the walls, and in a large measure preventing clinkers. The Combustion of Producer-gas. (H. H. Campbell, Trans. A. I. M. E., xix, 128.)— The combustion of the components of ordinary pro- ducer-gas may be represented by the following formulae: C 2 H 4 + 60 = 2C0 2 + 2H 2 ! 2H + O = H 2 ; CH 4 + 40 = C0 2 + 2H 2 ; CO -f O = C0 2 . Average Composition by Volume op Producer-gas: A, made with Open Grates, no Steam in Blast; B, Open Grates, Steam-jet in Blast. 10 Samples op Each. C0 2 . O. C 2 H 4 . CO. H. CH 4 . N. Amin 3.6 0.4 0.2 20.0 5.3 3.0 58.7 A max 5.6 0.4 0.4 24.8 8.5 5.2 64.4 A average... 4.84 0.4 0.34 22.1 6.8 3.74 61.78 B min 4.6 0.4 .0.2 20.8 6.9 2.2 57.2 B max 6.0 0.8 0.4 24.0 9.8 3.4 62.0 B average... 5.3 0.54 0.36 22.74 8.37 2.56 60.13 The coal used contained carbon 82#, hydrogen 4.7%. The following are analyses of products of combustion : C0 2 . O. CO. CH 4 . H. N. Minimum 15.2 0.2 trace, trace. trace. 80.1 Maximum 17.2 1.6 2.0 0.6 2.0 83.6 Average 16.3 0.8 0.4 0.1 0.2 82.2 Use of Steam in Producers and in Boiler-furnaces. (R. W. Raymond, Trans. A. I. M. E., xx. 635.)— No possible use of steam can cause a gain of heat. If steam be introduced into a bed of incandescent carbon it is decomposed into hydrogen and oxygen. The heat absorbed by the reduction of one pound of steam to hydrogen is much greater in amount than the heat generated by the union of the oxygen thus set free with carbon, forming either carbonic oxide or carbonic acid. Consequently, the effect of steam alone upon a bed of incandescent fuel is to chill it. In every water-gas apparatus, designed to produce by means of the decomposition of steam a fuel -gas relatively free from nitro- gen, the loss of heat in the producer must be compensated by some reheat- ing device. This loss may be recovered if the hydrogen of the steam is subsequently , burned, to form steam again. Such a combustion of the hydrogen is con- t templated, in the case of fuel-gas, as secured in the subsequent use of that ' gas. Assuming the oxidation of H to be complete, the use of steam will '. cause neither gain nor loss of heat, but a simple transference, the heat f absorbed by steam decomposition being restored by hydrogen combustion. . In practice, it may be doubted whether this restoration is ever complete. < But it is certain that an excess of steam would defeat the reaction alto- gether, and that there must be a certain proportion of steam, which per- mits the realization of important advantages, without too great a net loss in heat. The advantage to be secured (in boiler furnaces using small sizes of anthracite) consists principally in the transfer of heat from the lower side of the fire, where it is not wanted, to the upper side, where it is wanted. The decomposition of the steam below cools the fuel and the grate-bars, whereas a blast of air alone would produce, at that point, intense combus- tion (forming at first C0 2 ), to the injury of the grate, the fusion of part of the fuel, etc. The proportion of steam most economical is not easily determined. The temperature of the steam itself, the nature of the fuel mixture, and the use or non-use of auxiliary air- supply, introduced into the gases above or ILLUMINATING-GAS. 651 beyond the fire -bed, are factors affecting the problem. (See paper by R. J. Foster on the Use of the McClave Grate and Argand Steam Blower, etc., in Trans. A. I. M. E., xx. 625.) Gas-fuel for Small Furnaces. E. P. Reichhelm (Am. Mach., Jan. 10,1895) discusses the use of gaseous fuel for forge fires, for drop - forging, in annealing-ovens and furnaces for melting brass and copper, for case-hardening, muffle-furnaces, and kilns. Under ordinary conditions, in such furnaces he estimates that the loss by draught, radiation, and the heating of space not occupied by work is, with coal, 80$, with petroleum 70$, and with gas above the grade of producer-gas 25%. He gives the following table of comparative cost of fuels, as used in these furnaces : Kind of Gas. o o 3 o No. of Heat- units in Fur- naces after Deducting 25$ Loss. bCr-T < Cost of 1,000,- 000 Heat- units Ob- tained in Fur- naces. Natural gas Coal-gas, 20 candle-power Carburetted water-gas 1,000,000 675,000 646,000 690,000 313,000 377,000 185,00( 150,00f 306,36. r 750,000 506,250 484,500 517,500 234,750 282,750 138,750 112,500 229,774 $1.25 1.00 .90 .40 .45 .20 .15 .15 $2.46 2.06 1.73 Water-gas from coke. Water-gas from bituminous coal Water-gas and producer-gas mixed. . . 1.70 1.59 1.44 i.33 Naphtha-gas, fuel 2% gals, per 1000 ft. .65 Coal, $4 per ton, per 1 ,000,000 heat-units Crude petroleum, 3 cts. per gal., per 1,C utilizec 00,000 he .73 at-units. .73 Mr. Reichhelm gives the following figures from practice in melting brass with coal and with naphtha converted into gas: 1800 lbs. of metal require 1080 lbs. of coal, at $4.65 per ton, equal to $2.51, or, say, 15 cents per 100 lbs. Mr, T.'s report : 2500 lbs. of metal require 47 gals, of naphtha, at 6 cents per gal., equal to $2.82, or, say, 11J4 cents per 100 lbs. ILLUMINATING-GAS. Coal-gas is made by distilling bituminous coal in retorts. The retort is usually a long horizontal semi-cylindrical or a shaped chamber, holding from 160 to 300 lbs. of coal. The retorts are set in " benches " of from 3 to 9, heated by one fire, which is generally of coke. The vapors distilled from the coal are converted into a fixed gas by passing through the retort, which is heated almost to whiteness. The gas passes out of the retort through an " ascension-pipe " into a long horizontal pipe called the hydraulic main, where it deposits a portion of the tar it contains: thence it goes into a condenser, a series of iron tubes surrounded by cold water, where it is freed f rom condensable vapors, as ammonia-water, then into a washer, where it is exposed to jets of water, and into a scrubber, a large chamber partially filled with trays made of wood or iron, containing coke, fragments of brick or paving-stones, which are wet with a spray of water. By the washer and scrubber the gas is freed from the last portion of tar and ammonia and from some of the sulphur compounds. The gas is then finally purified from sulphur compounds by passing it through lime or oxide of iron. The gas is drawn from the hy- draulic main and forced through the washer, scrubber, etc., by an exhauster or gas pump. The kind of coal used is generally caking bituminous, but as usually this coal is deficient in gases of high illuminating power, there is added to it a portion of cannel coal or other enricher. The following table, abridged from one in Johnson's Cyclopedia, shows the analysis, candle power, etc., of some gas-coals and enrichers: 652 ILLUMINATING-GAS. Gas-coals, etc. i - ,a e . > Coke per -2 ©JS^ ton of 2240 o ^o^ &M lbs. S sjg -CO o > M £ < |o-S 6" lbs. bush. 36.76 51.93 7.07 36.00 58.00 6.00 10,642 16.62 1544 40 37.50 56.90 5.60 18.81 1480 36 40.00 53.30 6.70 10,765 20.41 1540 36 43.00 40.00 17.00 9,800 34.98 1320 32 46.00 41.00 13.00 13,200 42.79 1380 32 53.50 44.50 2.00 15,000 28.70 1056 44 73 OS m Pittsburgh, Pa Westmoreland, Pa Sterling, O Despard, W. Va... Darlington, O Petonia, W. Va — Grahamite, W. Va. 6420 3993 2494 2806 4510 The products of the distillation of 100 lbs. of average gas-coal are about as follows. They vary according to the quality of coal and the temperature of distillation. Coke, 64 to 65 lbs.; tar, 6.5 to 7.5 lbs.; ammonia liquor, 10 to 12 lbs.; puri- fied gas, 15 to 12 lbs.; impurities and loss, 4.5$ to 3.5$. The composition of the gas by volume ranges about as follows: Hydro- gen, 38$ to 48$; carbonic oxide, 2$ to 14$; marsh-gas (Methane, CH 4 ), 43$ to 31$; heavy hydrocarbons (C«H 2 «, ethylene, propylene, benzole vapor, etc.), 7.5$ to 4.5$; nitrogen, 1$ to 3$. In the burning of the gas the nitrogen is inert; the hydrogen and carbonic oxide give heat but no light. The luminosity of the flame is due to the de- composition by heat of the heavy hydrocarbons into lighter hydrocarbons and carbon, the latter being separated in a state of extreme subdivision. By the heat of the flame this separated carbon is heated to intense white- ness, and the illuminating effect of the flame is due to the light of incandes- cence of the particles of carbon. The attainment of the highest degree of luminosity of the flame depends upon the proper adjustment of the proportion of the heavy hydrocarbons (with due regard to their individual character) to the nature of the diluent mixed therewith. Investigations of Percy F. Frankland show that mixtures of ethylene and hydrogen cease to have any luminous effect when the proportion of ethy- lene does not exceed 10$ of the whole. Mixtures of ethylene and carbonic oxide cease to have any luminous effect when the proportion of the former does not exceed 20$, while all mixtures of ethylene and marsh-gas have more or less luminous effect. The luminosity of a mixture of 10$ ethylene and 90$ marsh-gas being equal to about 18 candles, and that of one of 20$ ethylene and 80$ marsh-gas about 25 candles. The illuminating effect of marsh-gas alone, when burned in an argand burner, is by no means inconsiderable. For further description, see the Treatises on Gas by King. Richards, and Huemes; also AppletoiTs Cyc. Mech., vol. i. p. 900. "Water-gas.— Water-gas is obtained by passing. steam through a bed of coal, coke, or charcoal heated to redness or beyond. The steam is decom- posed, its hydrogen being liberated and its oxygen burning the carbon of the fuel, producing carbonic-oxide eas. The chemical reaction is. C + H 2 = CO + 2H, or 2C + 2H 2 = C -4- C0 2 + 4H, followed by a splitting up of the C0 2 , making 2CO + 4H. By weight the normal gas CO + 2H is com- posed of C + O + H = 28 parts CO and 2 parts H, or 93.33$ CO and 6.67$ H; 12 + 16 + 2 by volume it is composed of equal parts of carbonic oxide and hydrogen. Water-gas produced as above described has great heating-power, but no illuminating-power. It may, however, be used for lighting by causing it to heat to whiteness some solid substance, as is done in the Welsbach incan- descent light. An illuminating-gas is made from water-gas by adding to it hydrocarbon gases or vapors, which are usually obtained from petroleum or some of its products. A history of the development of modern illuminating water-gas processes, together with a description of.the most recent forms of apparatus, is given by Alex. C. Humphreys, in a paper on " Water-gas in the United States," read before the Mechanical Section of the British Association for Advancement of Science, in 1S89. After describing many earlier patents, he states that success in the manufacture of water-gas may be said to date ANALYSES OF WATER-GAS AND COAL-GAS COMPARED. 653 from 1874, when the process of T. S. C. Lowe was introduced. All the later most successful processes are the modifications of Lowe's, the essential features of which were " an apparatus consisting of a generator and super- heater internally fired; the superheater being heated by the secondary combustion from the generator, the heat so stored up in the loose brick of the superheater being used, in the second part of the process, in the fixing or rendering permanent of the hydrocarbon gases; the second part of the process consisting in the passing of steam through the generator fire, and the admission of oil or hydrocarbon at some point between the fire of the generator and the loose filling of the superheater.' 1 The water-gas process thus has two periods: first the '" blow," during which air is blown through the bed coal in the generator, and the partially burned gaseous products are completely burned in the superheater, giving up a great portion of their heat to the fire-brick work contained in it, and then pass out to a chimney; second, the "run" during which the air blast is stopped, the opening to the chimney closed, and steam is blown through the incandescent bed of fuel. The resulting water-gas passing into the car- buretting chamber in the base of the superheater is there charged with hy- drocarbon vapors, or spray (such as naphtha and other distillates or crude oil) and passes through the superheater, where the hydrocarbon vapors be- come converted into fixed illuminating gases. From the superheater the combined gases are passed, as in the coal-gas process, through washers, scrubbers, etc., to the gas-holder. In this case, however, there is no am- monia to be removed. The specific gravity of water-gas increases with the increase of the heavy hydrocarbons which give it illuminating power. The following figures, taken from different authorities, are given by F. H. Shelton in a paper on Water- gas, read before the Ohio Gas Light Association, in 1894: Candle-power ... 19.5 20. 22.5 24. 25.4 26.3 28.3 29.6 .30 to 31.9 Sp.gr. (Air = 1).. .571 .630 .589 .60 to .67 .64 .602 .70 .65 .65 to .71 Analyses of Water-gas and Coal-gas Compared, The following analyses are taken from a report of Dr. Gideon E. Moore on the Granger Water-gas, 1885: Composition by Volume. Composition by Weight. Water-gas. Coal-gas. Heidel- berg. Water-gas. Coal- Wor- cester. Lake. Wor- cester. Lake. gas. 2.64 0.14 0.06 11.29 0.00 1.53 28.26 18.88 37.20 3.85 0.30 0.01 12.80 0.00 2.63 23.58 20.95 35.88 2.15 3.01 0.65 2.55 1.21 1.33 8.88 34.02 46.20 0.04402 0.00365 0.00114 0.18759 0.06175 0.00753 0.00018 0.20454 04559 Carbonic acid 0.09992 01569 05389 Propylene Benzole vapor Carbonic oxide.. . Marsh-gas Hydrogen 03834 0.07077 0.46934 0.17928 0.04421 0.11700 0.37664 0.19133 0.04103 0.07825 0.18758 0.41087 0.06987 100.00 100.00 100.00 1.00000 1.00000 1.00000 0.5S25 0.5915 0.6057 0.6018 0.4580 B. T. U. from 1 cu. 650.1 597.0 5311. 2°F. 688.7 646.6 5281. 1°F. 642.0 577.0 5202. 9°F. ft. : Water liquid. Av. candle-power. 22.06 26.31 The heating values (B. T. U.) of the gases are calculated from the analysis by weight, by using the multipliers given below (computed from results of 654 ILLUMINATING-GAS. J. Thomsen), and multiplying the result by the weight of 1 cu. ft. of the gas at 62° F., and atmospheric pressure. The flame temperatures (theoretical) are calculated on the assumption of complete combustion of the gases in air, without excess of air. The candle-power was determined by photometric tests, using a pressure of l^-in. water-column, a candle consumption of 120 grains of spermaceti per hour, and a meter rate of 5 cu. ft. per hour, the result being corrected for a temperature of 62° F. and a barometric pressure of 30 in. It appears that the candle-power may be regulated at the pleasure of the person in charge of the apparatus, the range of candle-power being from 20 to 29 candles, according to the manipulation employed. Calorific Equivalents of Constituents of Illuminating- gas. Heat-units from 1 lb. Heat-units from 1 lb. Water Water - Water Water Liquid. Vapor. Liquid. Vapor. Ethylene 21,524.4 20,134.8 Carbonic oxide. . 4,395.6 4,395.6 Propylene 21,222.0 19,834.2 Marsh- gas 24,021.0 21,592.8 Benzole vapor.... 18,954.0 17,847.0 Hydrogen 61,524.0 51,804.0 Efficiency of a Water-gas Plant.— The practical efficiency of an illuminating water-gas setting is discussed in a paper by A. G. Glasgow (Proc. Am. Gaslight Assn., 1890), from which the following is abridged : The results refer to 1000 cu. ft. of unpurified carburetted gas, reduced to 60° F. The total anthracite charged per 1000 cu. ft. of gas was 33.4 lbs., ash and unconsumed coal removed 9.9 lbs., leaving total combustible consumed 23.5 lbs., which is taken to have a fuel-value of 14500 B. T. U. per pound, or a total of 340,750 heat- units. Composi- tion by Volume. Weight per 100 cu. ft. Composi- tion by Weight. Specific Heat. fC0 2 + H a S.. i C n H 2n CO I. Carburetteu J CH 4 Water-gas. J H I N 3.8 14.6 28.0 17.0 35.6 1.0 .465842 1.139968 2.1868 .75854 .1991464 .078596 4.8288924 .09647 .23607 .45285 .15710 .04124 .01627 1.00000 .02088 .08720 .11226 .09314 .14041 .00397 1 100.0 .45786 fC0 2 3.5 43.4 51.8 1.3 .429065 3.389540 .289821 .102175 .1019 .8051 .0688 .0242 .02205 1 CO .19958 .23424 .00591 1 100.0 4.210601 1.0000 .46178 f co a 17.4 3.2 79.4 2.133066 .2856096 6.2405224 .2464 .0329 .7207 .05342 .00718 escaping from -J N .17585 100.0 8.6591980 1.0000 .23645 fCO a 9.7 17.8 72.5 1.189123 1.390180 5.698210 .1436 .1680 .6884 .031075 .041647 IV. Generator J ^ .167970 blast- gases. ] 1 100.0 8.277513 1.0000 .240692 The heat energy absorbed by the apparatus is 23.5 X 14,500 = 340,750 heat- units = A. Its disposition is as follows : E, the energy of the CO produced; C, the energy absorbed in the decomposition of the steam ; D, the difference between the sensible heat of the escaping illuminating- gases and that of the entering oil; E, the heat carried off by the escaping blast products; F, the heat lost by radiation from the shells; EFFICIENCY OF A WATER-GAS PLANT. 655 G, the heat carried away from the shells by convection (air-currents); H, the heat rendered latent in the gasification of the oil; J, the sensible heat in the ash and unconsumed coal recovered from the generator. The heat equation is A = B+C+D+E+ F+ G + H+I; A being known. A comparison of the CO in Tables I and II show that-r^- , or 64.5£ of the volume of carburetted gas is pure water-gas, distributed thus : CO a , 2.3£; CO, 28.0^; H, 33.4^; N, 0.8^; = 64.5^. 1 lb. of CO at 60° F. = 13.531 cu. ft. CO per 1000 cu. ft. of gas = 280 -h 13.531 = 20.694 lbs. Energy of the CO = 20.694 x 4395.6 = 91,043 heat-units, = B. 1 lb. of H at 60° F. = 189.2 cu. ft. H per M of gas = 334-^-189.2 = 1.7653 lbs. Energy of the H per lb. (according to Thomsen, considering the steam generated by its combustion to be condensed to water at 75° F.) = 61,524 B. T. U. In Mr. Glasgow's ex- periments the steam entered the generator at 331° F. ; the heat required to raise the product of combustion of 1 lb. of H, viz., 8.9S lbs. H 2 0, from water at 75° to steam at 331° must therefore be deducted from Thomsen's figure, or 61,524 - (8.98 X 1140.2) = 51,285 B. T. U. per lb. of H. Energy of the H, then, is 1.7653 X 51,285 = 90,533 heat-units, = C. The heat lost due to the sensible beat in the illuminating-gases, their temperature being 1450° F., and that of the entering oil 235° F., is 48.29 (weight) X .45786 sp. heat X 1215 (rise of tem- perature) = 26,864 heat-units = D. (The specific heat of the entering oil is approximately that of the issuing gas.) The heat carried off in 1000 cu. ft. of the escaping blast products is 86.592 (weight) X .23645 (sp. heat) X 1474° (rise of temp.) = 30,180 heat-units: the temperature of the escaping blast gases being 1550° F., and that of the entering air 76° F. But the amount of the blast gases, by registra- tion of an anemometer, checked by a calculation from the analyses of the blast gases, was 2457 cubic feet for every 1000 cubic feet" of carburetted gas made. Hence the heat carried off per M. of carburetted gas is 30,180 x 2.457 = 74,152 heat-units = E. Experiments made by a radiometer covering four square feet of the shell of the apparatus gave figures for the amount of heat lost by radiation = 12,454 heat-units = F, and by convection = 15,696 heat-units = G. The heat rendered latent by the gasefication of the oil was found by taking the difference between all the heat fed into the carburetter and super- heater and the total heat dissipated therefrom to be 12,841 heat-units = H. The sensible heat in the ash and unconsumed coal is 9.9 lbs. X 1500° x .25 (sp. ht.) = 3712 heat-units — I. The sum of all the items B+ C + D + E+F-\- G-\- H+I= 327,295 heat- units, which substracted from the heat energy of the combustible consumed, 340,750 heat-units, leaves 13,455 heat-units, or 4 percent, unaccounted for. Of the total heat energy of the coal consumed, or 340,750 heat-units, the energy wasted is the sum of items D, E, F, G, and I, amounting to 132,878 heat-units, or 39 per cent; the remainder, or 207,872 heat-units, or 61 per cent, being utilized. The efficiency of the apparatus as a heat machine is therefore 61 per cent. Five gallons, or 35 lbs. of crude petroleum were fed into the carburetter per 1000 cu. ft. of gas made; deducting 5 lbs. of tar recovered, leaves 30 lbs. X 20,000 = 600,000 heat-units as the net heating value of the petroleum used. Adding this to the heating value of the coal, 340,750 B. T. U., gives 940,750 heat-units, of which there is found as heat energy in the carburetted gas, as in the table below, 764,050 heat units, or 81 per cent, which is the commer- cial efficiency of the apparatus, i.e., the ratio of the energy contained in the finished product to the total energy of the coal and oil consumed. The heating power per M. of the uncarburetted gas is C0 2 35.0 CO 434.0 X .078100 X 4395.6 = 148991 H 518.0 X .005594 X 61524.0 = 178277 N 13.0 The heating power per M. cu. ft. of the carburetted gas is C0 2 38.0 C 3 H 6 * 146.0 X .117220 X 21222.0 = 363200 CO 280.0 X .078100 X 4395.6 = 96120 CH 4 170.0 X .044620 X 24021.0 = 182210 H 356.0 X 005594 X 61524.0 = 122520 N 10.0 1000.0 764050 1000.0 327268 * The heating value of the illuminants C n H 2 » is assumed to equal that of C 3 H 6 . 656 ILLUMINATING-GAS. The candle-power of the gas is 31, or 6.2 candle-power per gallon of oil used. The calculated specific gravity is .6355, air being 1. For description of the operation of a modern carburetted water-gas plant, see paper by J. Stelfox, Eng'g, July 20, 1894, p. 89. Space required for a Water-gas Plant.— Mr. Shelton, taking 15 modern plants of the form requiring the most floor-space, figures the average floor-space required per 1000 cubic feet of daily capacity as follows: Water-gas Plants of Capacity Require an Area of Floor-space for in 24 hours of each 1000 cu. ft. of about 100,000 cubic feet .4 square feet. 200,000 " " 3.5 " 400,000 " " 2.75" 600,000 " " 2 to 2.5 sq.ft. 7 to 10 million cubic feet 1.25 to 1.5 sq. ft. These figures include scrubbing and condensing rooms, but not boiler and engine rooms. In coal-gas plants of the most modern and compact forms one with 16 benches of 9 retorts each, with a capacity of 1,500,000 cubic feet per 24 hours, will require 4.8 sq. ft. of space per 1000 cu. ft. of gas, and one of 6 benches of 6 retorts each, with 300,000 cu. ft. capacity per 24 hours wdl re- quire 6 sq. ft. of space per 1000 cu. ft. The storage-room required for the gas-making materials is: for coal-gas, 1 cubic foot of room for every 232 cubic feet of gas made; for water-gas made from coke, 1 cubic foot of room for every 373 cu. ft. of gas made; and for water-gas made from anthracite, 1 cu. ft. of room for every 645 cu. ft. of gas made. The comparison is still more in favor of water-gas if the case is considered of a water-gas plant added as an auxiliary to an existing coal-gas plant; for, instead of requiring further space for storage of coke, part of that already required for storage of coke produced and not at once sold can be cut off, by reason of the water-gas plant creating a constant demand for more or less of the coke so produced. Mr. Shelton gives a calculation showing that a water-gas of .625 sp. gr. would require gas-mains eight per cent greater in diameter than the same -quantity coal-gas of .425 sp. gr. if the same pressure is maintained at the holder. The same quantity may be carried in pipes of the same diameter if the pressure is increased in proportion to the specific gravity. With the same pressure the increase of candle-power about balances the decrease of flow. With five feet of coal-gas, giving, say, eighteen candle-power, 1 cubic foot equals 3.6 candle-power; with water-gas of 23 candle-power, 1 cubic foot equals 4.6 candle-power, and 4 cubic feet gives 18.1 candle-power, or more than is given by 5 cubic feet of coal-gas. Water-gas may be made from oven-coke or gas-house coke as well as from anthracite coal. A water- gas plant may be conveniently run in connection with a coal-gas plant, the surplus retort coke of the latter being used as the fuel of the former. In coal-gas making it is impracticable to enrich the gas to over twenty candle-power without causing too great a tendency to smoke, but water-gas of as high as thirty candle-power is quite common. A mixture of coal-gas and water-gas of a higher C.P. than 20 can be advantageously distributed. Fuel-value of. Illuminating-gas.— E. G. Love (School of Mines Qtly, January, 1892) describes F. W. Hartley's calorimeter for determining the calorific power of gases, and gives results obtained in tests of the car- buretted water-gas made by the municipal branch of the Consolidated Co. of New York. The tests were made from time to time during the past two years, and the figures give the heat-units per cubic foot at 60° F. and 30 inches pressure: 715. 692, 725, 732, 691, 738, 735, 703, 734, 730, 731, 727. Average, 721 heat units. Similar tests of mixtures of coal- and water-gases made by other branches of the same company give 694, 715, 684, 692, 727, 665, 695, and 686 heat-units per foot, or an average of 694.7. The average of all these tests was 710.5 heat-units, and this we may fairly take as representing the calorific power of the illuminating gas of New York. One thousand feet of this gas, costing $1.25, would therefore yield 710,500 heat-units, which would be equivalent to 568,400 heat-units for $1.00. The common coal-gas of London, with an illuminating power of 16 to 17 candles, has a calorific power of about 668 units per foot, and costs from 60 to 70 cents per thousand. The product obtained by decomposing steam by incandescent carbon, as effected in the Motay process, consists of about 40$ of CO, and a little over W% of H. FLOW OF GAS IK PIPES. 657 This mixture would have a heating-power of about 300 units per cubic foot, and if sold at 50 cents per 1000 cubic feet would furnish 600,000 units for $1.00, as compared with 568,400 units for $1.00 from illuminating gas at $1 .25 per 1000 cubic feet. This illuminating-gas if sold at $1.15 per thousand would there- fore be a more economical heating agent than the fuel-gas mentioned, at 50 cents per thousand, and be much more advantageous than the latter, in that one main, service, and meter could be used to furnish gas for both lighting and heating. A large number of fuel-gases tested by Mr. Love gave from 184 to 470 heat- units per foot, with an average of 309 units. Taking the cost of heat from illuminating-gas at the lowest figure given by Mr. Love, viz., $1.00 for 600,000 heat-units, it is a very expensive fuel, equal to coal at $40 per ton of 2000 lbs., the coal having a calorific power of only 12,000 heat-units per pound, or about 83$ of that of pure carbon : 600,000 : (12,000 X 2000) :: $1 : $40. FLOW OF GAS IN PIPES. The rate of flow of gases of different densities, the diameter of pipes re- quired, etc., are given in King's Treatise on Coal Gas, vol. ii. 374, as follows: If d = diameter of pipe in inches, . Q = quantity of gas in cu. ft. pe: hour, I — length of pipe in yards, h = pressure in inches of water, s = specific gravity of gas, air be ingl, = »/ Q*sl Y (1350) 2 /i ' ft - _ Q* sl f (1350)W_'_ Molesworth gives Q = 1000 ju — si / d 6 h J. P. Gill, Am. Gas-light Jour. 1894, gives Q = 1291 A/ - + d) This formula is said to be based on experimental data, and to make allow- ance for obstructions by tar, water, and other bodies tending to check the flow of gas through the pipe. A set of tables in Appleton's Cyc. Mech. for flow of gas in 2, 6, and 12 in. pipes is calculated on the supposition that the quantity delivered varies as the square of the diameter instead of as d 2 x Vd, or Yd 6 . These tables give a flow in large pipes much less than that calculated by the formulae above given, as is shown by the following example. Length of pipe 100 yds., specific gravity of gas .042, pressure 1-in. water-column. 2 -in. Pipe. 6-in. Pipe. 12-in. Pipe. Q=12S0a/^-j- 1178 18,368 103,912 , - nn /d*h ) = 1000|/--, Q = 1291 A/ , " • 1116 16,327 93,845 y s(l -J- a) Table in App. Cyc 1290 11,657 46,628 An experiment made by Mr. Clegg, in London, with a 4-in. pipe, 6 miles long, pressure 3 in. of water, specific gravity of gas .398, gave a discharge into the atmosphere of 852 cu. ft. per hour, after a correction of 33 cu. ft. was made for leakage. Substituting this value, 852 cu. ft., for Q in the formula Q = C Vd 6 h ■+- si, we find C, the coefficient, = 997, which corresponds nearly with the formula given by Molesworth. 658 ILLUMINATING-GAS. Services -for Lamps. (Molesworth.) Lamps. 4.. 6.. 10.. Ft. from Main. . . . . 40 .... 40 . . . . 50 .... 100 Require Pipe-bore. Lamps. 15.... 20.... 25 ... . Ft. from Main. 150 180 Require Pipe-bore. 1 in. 1*4 in. l^in. 1% in. (In cold climates no service less than % in. should be used.) Maximum Supply of Gas through Pipes in en. ft. per Hour, Specific Gravity oeing taken at .45, calculated from the Formula Q = 1000 \/tVh -=- si. (Molesworth.) Length of Pipe = 10 Yards. Pressure by the Water-gauge in Inches. of Pipe in .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 % 13 18 22 26 29 31 34 36 38 41 14 26 37 46 53 59 64 70 74 79 83 H 73 103 126 145 162 187 192 205 218 230 1 149 211 25S 298 333 365 394 422 447 471 VA 260 368 451 521 582 638 689 737 781 823 •m 411 581 711 821 918 1006 1082 1162 1232 1299 2 843 1192 1460 1686 1886 2066 2231 2385 2530 2667 Length of Pipe = 100 Yards. Pressure by the "Water-gauge in Inches. .1 R .2 12 .3 14 .4 .5 .75 1.0 1.25 29 1.5 2 2.5 % 17 19 23 26 32 36 42 ¥4. 23 32 42 46 51 63 73 81 89 103 115 1 47 67 82 94 105 129 149 167 183 211 236 m 82 116 143 165 184 225 260 291 319 368 412 m 130 184 225 260 290 356 411 459 503 581 649 267 377 462 533 596 730 843 943. 1033 1193 1333 2y 2 466 659 K07 932 1042 1276 1473 1647 1804 2083 2329 3 735 1039 1270 1470 1643 2012 2323 2598 2846 3286 3674 3^ 152H 1S71 2161 2416 2958 3416 3820 4184 4831 5402 4 3017 3373 4131 4770 5333 5842 6746 7542 Length of Pipe = 1000 Yards. Pressure by the Water-gauge in Inches. .5 .75 1.0 1.5 2.0 2.5 3.0 1 33 41 47 58 67 75 82 m 92 113 130 159 184 205 226 2 189 231 267 327 377 422 462 2H 329 403 466 571 659 737 807 3 520 636 735 900 1039 1162 1273 4 1067 1306 1508 1847 2133 2385 2613 5 1863 2282 2635 3227 3727 4167 4564 6 2939 3600 4157 5091 5879 6573 7200 659 Length of Pipe = 5000 Yards. Diameter Pressure by the Water-gauge in Inches. of Pipe in Inches. 1.0 1.5 2.0 2.5 3.0 2 119 146 169 189 207 3 329 402 465 520 569 4 675 826 955 1067 1168 5 1179 1443 1667 1863 2041 6 1859 2277 2629 2939 3220 7 2733 3347 3865 4321 4734 8 3816 4674 5397 6034 6610 9 5123 6274 7245 8100 8873 10 6667 8165 9428 10541 11547 12 10516 12880 14872 16628 18215 Mr. A. C. Humphreys says his experience goes to show that these tables give too small a flow, but it is difficult to accurately check the tables, on ac- count of the extra friction introduced by rough pipes, bends, etc. For bends, one rule is to allow 1/42 of an inch pressure for each right-angle bend. Where there is apt to be trouble from frost it is well to use no service of less diameter than % in., no matter how short it may be. In extremely cold climates this is now often increased to 1 in., even for a single lamp. The best practice in the U. S. now condemns any service less than % in. STEAM. The Temperature of Steam in contact with water depends upon the pressure under which it is generated. At the ordinary atmospheric pressure (14.7 lbs. per sq. in.) its temperature is 212° F. As the pressure is increased, as by the steam being generated in a closed vessel, its tempera- ture, and that of the water in its presence, increases. Saturated Steam is steam of the temperature due to its pressure- not superheated. Superheated Steam is steam heated to a temperature above that due to its pressure. I>ry Steam is steam which contains no moisture. It may be either saturated or superheated. "Wet Steam is steam containing intermingled moisture, mist, or spray. It has the same temperature as dry saturated steam of the same pressure. Water introduced into the presence of superheated steam will flash into vapor until the temperature of the steam is reduced to that due its pres- sure. Water in the presence of saturated steam has the same temperature as the steam. Should cold water be introduced, lowering the temperature of the whole mass, some of the steam will be condensed, reducing the press- ure and temperature of the remainder, until an equilibrium is established. Temperature and Pressure of Saturated Steam.— The re- lation between the temperature and the pressure of steam, according to Regnault's experiments, is expressed by the formula (Buchanan's, as given by Clark) t = _ .)— The formulae constructed by Regnault are strictly empirical, and were based entirely on his experiments. They are therefore not valid beyond the range of temper- atures and pressures observed Mr. Gray has made a most elaborate calculation, based not on experiments but on fundamental principles of thermodynamics, from which he deduces formulas for the pressure and total heat of steam, and presents tables calcu- 662 lated therefrom which show substantial agreement with Regnault's figures. He gives the following examples of steam-pressures calculated for tempera- tures beyond the range of Regnault's experiments. Temperature. Pounds per sq. in. Tempe rature. Pounds per C. Fahr. C. Fahr sq. in. 230 446 406.9 340 644 2156.2 240 464 488.9 360 680 2742.5 250 482 579.9 380 716 3448.1 260 500 691.6 400 752 4300.2 280 536 940.0 415 779 5017.1 300 572 1261.8 427 800.6 5659.9 320 608 1661.9 These pressures are higher than those obtained by Regnault's formula, which gives for 415° C. only 4067.1 lbs. per square inch. Table of the Properties of Saturated Steam.— In the table of properties of saturated steam on the following pages the figures for tem- perature, total heat, and latent heat are taken, up to 210 lbs. absolute pres- sure, from the tables in Porter's Steam-engine Indicator, which tables have been widely accepted as standard by American engineers. The figures for total heat, given in the original as from 0° F., have been changed to heat above 32° F. The figures for weight per cubic foot and for cubic feet per pound have been taken from Dwelshauvers-Dery's table, Trans. A. S. M. E., vol. xi., as being probably more accurate than those of Porter. The figures for relative volume are from Buel's table, in Dubois's translation of Weis- bach, vol. ii. They agree quite closely with the relative volumes calculated from weights as given by Dery. From 211 to 219 lbs. the figures for temper- ature, total heat, and latent heat are from Dery's table; and from 220 to 1000 lbs. all the figures are from Buel's table. The figures have not been carried out to as many decimal places as they are in most of the tables given by the different authorities: but any figure beyond the fourth significant figure is unnecessary in practice, and beyond the limit of error of the observations and of the formulae from which the figures were derived. Weight of 1 Cubic Foot of Steam in Decimals of a Pound. Comparison of Different Authorities. _c Weight of 1 cubic foot a a a m £2& Weight of 1 cubic foot SP* according to— according to— £2^ JD Por- ter. Clark Buel. Dery. Pea- body. Por- ter. Clark Buel. Dery. Pea- body 1 .0030 .003 .00303 .00299 .00299 120 .27428 .2738 .2735 .2724 .2695 14.7 .03797 .0380 .03793 .0376 140 .31386 .3162 .3163 .3147 .3113 20 .0511 .0507 .0507 .0507 .0502 160 .35209 .3590 .3589 .3567 .3530 40 .0994 .0974 .0972 .0972 .0964 180 .38895 .4009 .4012 .3983 .3945 60 .1457 .1425 .1424 .1422 .1409 200 .42496 .4431 .4433 .4400 .4359 80 .19015 .1865 .1866 .1862 .1843 220 .4842 .4852 .4772 100 .23302 .2307 .2303 .2296 .2271 240 .5248 .5270 .5186 There are considerable differences between the figures of weight and vol- ume of steam as given by different authorities. Porter's figures are based on the experiments of Fairbairn and Tate. The figures given by the other authorities are derived from theoretical formula? which are believed to give more reliable results than the experiments. The figures for temperature, total heat, and latent heat as given by different authorities show a practical agreement, all being derived from Regnault's experiments. See Peabody's Tables of Saturated Steam; also Jacobus, Trans. A. S. M. E., vol. xii., 593. STEAM. 663 Properties of Saturated Steam. oTi , Total Heat oi , fi'S 2> ®-5 II above 32° F. "el j.-£ 2 Z^ 3.S o=S In the In the S CD . SOU o aj = PUS Water h Steam H 1^1 < £ units. units. j tf > 29.74 .089 32 1091.7 1091.7 208080 3333.3 .00030 29.67 .122 40 8. 1094.1 1086.1 154330 2472.2 .00040 29.56 .176 50 18. 1097.2 1079.2 107630 1724.1 .00058 29.40 .254 60 28.01 1100.2 1072.2 76370 1223.4 .00082 29.19 .359 70 38.02 1103.3 1065.3 54660 875.61 .00115 28.90 .502 80 48.04 1106.3 1058.3 39690 635.80 .00158 28.51 .692 90 58.06 1109.4 1051.3 29290 469.20 .00213 28.00 .943 100 68.08 1112.4 1044.4 21830 349.70 .00286 27.88 1 102.1 70.09 1113.1 1043.0 20623 334.23 .00299 25.85 2 126.3 94.44 1120.5 1026.0 10730 173.23 .00577 23.83 3 141.6 109.9 1125.1 1015.3 7325 117.98 .00848 21.78 4 153.1 121.4 1128.6 1007.2 5588 89.80 .01112 19.74 5 162.3 130.7 1131.4 1000.7 4530 72.50 .01373 17.70 6 170.1 138.6 1133.8 995.2 3816 61.10 .01631 15.67 7 176.9 145.4 1135.9 990.5 3302 53.00 .01887 13.63 8 182.9 151.5 1137.7 986.2 2912 46.60 .02140 11.60 9 188.3 156.9 1139.4 982.4 2607 41.82 .02391 9.56 10 ' 11 193.2 161.9 1140.9 979.0 2361 37.80 .02641 7.52 197.8 166.5 1142.3 975.8 2159 34.61 .02889 5.49 12 202.0 170.7 1143.5 972.8 1990 31.90 .03136 3.45 13 205.9 174.7 1144.7 970.0 1846 29.58 .03381 1.41 14 209.6 178.4 1145.9 967.4 1721 27.59 .03625 Gauge Pressure lbs. per sq. in. 14.7 212 180.9 1146.6 965.7 1646 26.36 .03794 0.304 15 213.0 181.9 1146.9 965.0 1614 25.87 .03868 1.3 16 216.3 185.3 1147.9 962.7 1519 24.33 .04110 2.3 17 219.4 188.4 1148.9 960.5 1434 22.98 .04352 3.3 18 222.4 191.4 1149 8 958.3 1359 21.78 .04592 4.3 19 225.2 194.3 1150.6 956.3 1292 20.70 .04831 5.3 20 227.9 197.0 1151.5 954.4 1231 19.72 .05070 6.3 21 230.5 199.7 1152.2 952.6 1176 18.84 .05308 7.3 22 233.0 202.2 1153.0 950.8 1126 18.03 .05545 8.3 23 235.4 204.7 .7 949.1 1080 17.30 .05782 9.3 24 237.8 207.0 1154.5 947.4 1038 16.62 .06018 10.3 25 240.0 209.3 1155.1 945.8 998.4 15.99 .06253 11.3 26 ' 242.2 211.5 .8 944.3 962.3 15.42 .06487 12.3 27 244.3 213.7 1156.4 942.8 928.8 14.88 .06721 13.3 28 246.3 215.7 1157.1 941.3 897.6 14.38 .06955 14.3 29 248.3 217.8 .7 939.9 868.5 13.91 .07188 15.3 30 250.2 219.7 1158.3 938.9 841.3 13.48 .07420 16.3 31 252.1 221.6 .8 937.2 815 8 13.07 .07652 17.3 32 254.0 223.5 1159.4 935.9 791.8 12.68 .07884 18.3 33 255.7 225.3 .9 934.6 769.2 12.32 .08115 19.3 34 257.5 227.1 1160.5 933.4 748.0 11.98 .08346 20.3 35 259.2 228.8 1161.0 932.2 727.9 11.66 .08576 21.3 36 260.8 230.5 1161.5 931.0 708.8 11.36 .08806 22.3 37 262.5 232.1 1162.0 929.8 690.8 11.07 .09035 664 STEAM. Properties of Saturated Steam. of = ' CO Total Heat above 32° F. ^ . CD ; £ 5 Z-O P- 1 -.2 CO S -us m . o In the In the o « *2 £ U CD A* Water h Steam H a if > .OS •J3 — 00 TO ©+-> s— too . §>:2 w ~ §* Heat- Heat- tgllffl 1> «s o s '<££ o :§ H units. units. J ti > £ 23.3 38 264.0 233.8 1162.5 928.7 673.7 10.79 .09264 20 39 265.6 235.4 .9 927.6 657.5 10.53 .09493 25.3 40 267.1 236.9 1163.4 926.5 642.0 10.28 .09721 26.3 41 268.6 238.5 .9 925.4 627.3 10.05 .09949 27.3 42 270.1 240.0 1164.3 924.4 613.3 9.83 .1018 28.3 43 271.5 241.4 .7 923.3 599.9 9.61 .1040 29.3 44 272.9 242.9 1165.2 922.3 587.0 9.41 .1063 30.3 45 274.3 244.3 .6 921.3 574.7 9.21 .1086 31.3 46 275.7 245.7 1166.0 920.4 563 9.02 .1108 32.3 47 277.0 247.0 .4 919.4 551.7 8.84 .1131 33.3 48 278.3 248.4 .8 918.5 540.9 8.67 .1153 34.3 49 279.6 249.7 1167.2 917.5 530.5 8.50 .1176 35.3 50 280.9 251.0 .6 916.6 520.5 8.34 .1198 36.3 51 282.1 252.2 1168.0 915.7 510.9 8.19 .1221 37.3 52 283.3 253.5 .4 914.9 501.7 8.04 .1243 38.3 53 284.5 254.7 7 914.0 492.8 7.90 .1266 39.3 54 285.7 256.0 1169.1 913.1 484.2 7.76 .1288 40.3 55 286.9 257.2 .4 912.3 475.9 7.63 .1311 41.3 56 288.1 258.3 .8 911.5 467.9 7.50 .1333 42.3 57 289.1 259.5 1170.1 910.6 460.2 7.38 .1355 43.3 58 290.3 260.7 .5 909.8 452.7 7.26 .1377 44.3 59 291.4 261.8 .8 909.0 445.5 7.14 .1400 45.3 60 292.5 262.9 1171.2 908.2 438.5 7.03 .1422 46.3 61 293.6 264.0 .5 907.5 431.7 6.92 .1444 47.3 62 294.7 265.1 .8 906.7 425.2 6.82 .1466 48.3 63 295.7 266.2 1172.1 905.9 418.8 6.72 .1488 49.3 64 296.8 267.2 .4 905.2 412.6 6.62 .1511 50.3 65 297.8 268.3 .8 904.5 406.6 • 6.53 .1533 51.3 66 298.8 269.3 1173.1 903 7 400.8 6.43 .1555 52.3 67 299.8 270.4 .4 903.0 395.2 6.34 .1577 53.3 68 300.8 271.4 .7 902.3 389.8 6.25 .1599 54.3 69 301.8 272.4 1174.0 901.6 384.5 6.17 .1621 55.3 70 302.7 273.4 .3 900.9 379.3 6.09 .1643 56.3 71 303.7 274.4 .6 900.2 374.3 6.01 .1665 57.3 72 304.6 275.3 .8 899.5 369.4 5.93 .1687 58.3 73 305.6 276.3 1175.1 898.9 364.6 5.85 .1709 59.3 74 306.5 277.2 .4 898.2 360.0 5.78 .1731 60.3 75 307.4 278.2 .7 897.5 355.5 5.71 .1753 61.3 76 308.3 279.1 1176.0 896.9 351.1 5.63 .1775 62.3 77 309.2 280.0 .2 896.2 346.8 5.57 .1797 63.3 78 310.1 280.9 .5 895.6 342.6 5.50 .1819 64.3 79 310.9 281.8 1176.8 895.0 338.5 5.43 .1840 65.3 80 311.8 282.7 1177.0 894.3 334.5 5.37 .1862 66.3 81 312.7 283.6 .3 893.7 330.6 5.31 .1884 67.3 82 313.5 284.5 .6 893.1 326.8 5.25 .1906 68.3 83 314.4 285.3 .8 892.5 323.1 5.18 .1928 69.3 84 315.2 286.2 1178.1 891.9 319.5 5.13 .1950 70.3 85 316.0 287.0 .3 891.3 315.9 5.07 .1971 665 Properties of Saturated Steam. €.5 is a a CD ^ Total above Heat 32° F. i4 . W | ? ■gnw -e 3 ~ ® .O 3h © a us In the Water h Heat- In the Steam H Heat- "53 «S o < H units. units. J P5 - >'" £ 71.3 86 316.8 287.9 1178.6 890.7 312.5 5 02 .1993 72.3 87 317.7 288.7 .8 890.1 309.1 4.96 .2015 73.3 88 318.5 289.5 1179.1 889.5 305.8 4.91 .2036 74.3 89 319.3 290.4 .3 888.9 302.5 4.86 .2058 75.3 90 320.0 291.2 .6 888.4 299.4 4.81 .2080 76.3 91 320.8 292.0 .8 887.8 296.3 4.76 .2102 77.3 92 321.6 292.8 1180.0 887.2 293.2 4.71 .2123 78.3 93 322.4 293.6 .3 886.7 290.2 4.66 .2145 79.3 94 323.1 294.4 .5 886.1 287.3 4.62 .2166 80.3 95 323.9 295.1 .7 885.6 284.5 4.57 .2188 81.3 96 324.6 295.9 1181.0 885.0 281.7 4.53 .2210 82.3 97 325.4 296.7 .2 884.5 279.0 4.48 .2231 83.3 98 326.1 297.4 .4 884.0 276.3 4.44 .2253 84.3 99 3:26.8 298.2 .6 883.4 273.7 4.40 .2274 85.3 100 327.6 298.9 .8 882.9 271.1 4.36 .2296 86.3 101 3-28.3 299 7 1182.1 882.4 268.5 4.32 .2317 87.3 102 329.0 300.4 .3 881.9 266.0 4.28 .2339 88.3 103 329.7 301.1 .5 881.4 263.6 4.24 .2360 89.3 104 330.4 301.9 .7 880.8 261.2 4.20 .2382 90.3 105 331.1 302.6 .9 880.3 258.9 4.16 .2403 91.3 106 331.8 303.3 1183.1 879.8 256.6 4.12 .2425 92.3 107 332.5 304.0 .4 879.3 254.3 4.09 .2446 93.3 108 333.2 304.7 .6 878.8 252.1 4.05 .2467 94.3 109 333.9 305.4 .8 878.3 249.9 4.02 .2489 95.3 110 334.5 306.1 1184.0 877.9 247.8 3.98 .2510 96.3 111 335.2 306.8 .2 877.4 245.7 3.95 .2531 97.3 112 335.9 307.5 .4 876.9 243.6 3.92 .2553 98.3 113 336.5 308.2 .6 876.4 241.6 3.88 .2574 99.3 114 337.2 308.8 .8 875.9 239.6 3.85 .2596 100.3 115 337.8 309.5 1185.0 875.5 237.6 3.82 .2617 101.3 116 338.5 310.2 .2 875.0 235.7 3.79 .2638 102.3 117 339.1 310.8 '.4 874.5 233.8 3.76 .2660 103.3 . 118 339.7 311.5 .6 874.1 231.9 3 73 .2681 104.3 119 340.4 312.1 .8 873.6 230.1 3>0 .2703 105.3 120 341.0 312.8 .9 873.2 228.3 3.67 .2724 106.3 121 341.6 313.4 1186.1 872.7 226.5 3.64 .2745 107.3 122 342.2 314.1 .3 872.3 224.7 3.62 .2766 108.3 123 342.9 314.7 .5 871.8 223.0 3.59 .2788 109.3 124 343.5 315.3 .7 871.4 221.3 3.56 .2809 110.3 125 344.1 316 .9 870.9 219.6 3.53 .2830 111.3 126 344.7 316.6 1187.1 870.5 218.0 3.51 .2851 112.3 127 345.3 317.2 .3 870.0 216.4 3.48 .2872 113.3 128 345.9 317.8 .4 869.6 214.8 3.46 .2894 114.3 129 346.5 318.4 .6 869.2 213.2 3.43 .2915 115.3 130 347.1 319.1 .8 868.7 211.6 3.41 .2936 116.3 131 347.6 319.7 1188.0 868.3 210.1 3.38 .2957 117.3 132 348.2 320.3 .2 867.9 208.6 3.36 .2978 118.3 133 34S.8 320.8 .3 867.5 207.1 3.33 .3000 119,3 134 349.4 321.5 .5 867.0 205.7 3.31 .3021 666 Properties of Saturated Steam. Total Heat 11 £*S 3 — fc ..S ©j§ Q Is "a above 32° F. O02 &s to a" © 02 In the In the !| £~ £<« II S°fe V o "3 fl '§£ 1 st Pi r ^<£ 169.3 184 374.6 347.7 1196.2 848.5 152.4 2.46 .4066 170.3 185 375.1 348.1 .3 848.2 151.6 2.45 .4087 171.3 186 375.5 348.6 .5 847.9 150.8 2.43 .4108 172.3 187 375.9 349.1 .6 847.6 150.0 2.42 .4129 173.3 188 376.4 349.5 .7 847.2 149.2 2.41 .4150 174.3 189 376.9 350.0 .9 846.9 148.5 2.40 .4170 175.3 190 377.3 350.4 1197.0 846.6 147.8 2.39 .4191 176.3 191 377.7 350.9 .1 846.3 147.0 2.37 .4212 177.3 192 378.2 351.3 .3 845.9 146.3 2.36 .4233 178.3 193 378.6 351.8 .4 845.6 145.6 2.35 .4254 179.3 194 379.0 352.2 .5 845.3 144.9 2.34 .4275 180.3 195 379.5 352.7 7 845.0 144.2 2.33 .4296 181.3 196 380.0 353.1 !8 844.7 143.5 2.32 .4317 182.3 197 380.3 353.6 .9 844.4 142.8 2.31 .4337 183.3 198 380.7 354.0 1198.1 844.1 142.1 2.29 .4358 184.3 199 381.2 354.4 .2 843.7 141.4 2.28 .4379 185.3 200 381.6 354.9 .3 843.4 140.8 2.27 .4400 186.3 201 382.0 355.3 .4 843.1 140.1 2.26 .4420 187.3 202 382.4 355.8 .6 842.8 139.5 2.25 .4441 188.3 203 382.8 356.2 .7 842.5 138.8 2.24 .4462 189.3 204 383.2 356.6 .8 842.2 138.1 2.23 .4482 190.3 205 383.7 357.1 1199.0 841.9 137.5 2.22 .4503 191.3 206 384.1 357.5 .1 841.6 136.9 2.21 .4523 192.3 207 384.5 357.9 .2 841.3 136.3 2.20 .4544 193.3 208 384.9 358.3 .3 841 135.7 2.19 .4564 194.3 209 385.3 358.8 .5 840.7 135.1 2.18 .4585 195.3 210 385.7 359.2 .6 840.4 134.5 2.17 .4605 196.3 211 386.1 359.6 .7 840.1 133.9 2.16 .4626 197.3 212 386.5 360.0 .8 839.8 133.3 2.15 .4646 198.3 213 386.9 360.4 .9 839.5 132.7 2.14 .4667 199.3 214 387.3 360.9 1200.1 839.2 132.1 2.13 .4687 200.3 215 387.7 361.3 .2 838.9 131.5 2.12 .4707 201.3 216 388.1 361.7 .3 838.6 130.9 2.12 .4728 202.3 217 388.5 362.1 .4 838.3 130.3 2.11 .4748 203.3 218 388.9 362.5 .6 838.1 129.7 2.10 .4768 204.3 219 389.3 362.9 .7 837.8 129.2 2.09 .4788 205.3 220 389.7 362.2* 1200.8 838.6* 128.7 2.06 .4852" 215.3 230 393.6 366.2 1202.0 835.8 123.3 1.98 .5061 225.3 240 397.3 370.0 1203.1 833.1 118.5 1.90 .5270 235.3 250 400.9 373.8 1204.2 830.5 114.0 1.83 .5478 245.3 260 404.4 377.4 1205.3 827.9 109.8 1.76 .5686 255.3 270 407.8 380.9 1206.3 825.4 105.9 1.70 .5894 265.3 280 411.0 384.3 1207.3 823.0 102.3 1.64 .6101 275.3 290 414.2 387.7 1208.3 820.6 99.0 1.585 .6308 285.3 300 417.4 390.9 1209.2 818.3 95.8 1.535 .6515 335.3 350 432.0 406.3 1213.7 807.5 82.7 1.325 .7545 * The discrepancies at 205.3 lbs. gauge are due to the change from Dery's to Buel's figures. 668 STEAM. Properties of Saturated Steam. a 1 & .8 3'v In the Water In the Steam 33 co !>o £- 385.3 400 444.9 419.8 1217.7 797.9 72.8 1.167 .8572 435.3 450 456.6 432.2 1221.3 , 789.1 65.1 1.042 .9595 485.3 500 467.4 443.5 1224.5 781.0 58.8 .942 1.062 535.3 550 477.5 454.1 1227.6 773.5 53.6 .859 1.164 585.3 600 486.9 464.2 1230.5 766.3 49.3 .7,90 1.266 635.3 650 495.7 473.6 1233.2 759.6 45.6 .731 1.368 685.3 700 504.1 482.4 1235.7 753.3 42.4 .680 1.470 735.3 750 512.1 490.9 1238.0 747.2 39.6 .636 1.572 785.3 800 519.6 498.9 1240.3 741 .4 37.1 .597 1.674 835.3 850 526.8 506.7 1242.5 735.8 34.9 .563 1.776 885.3 900 533.7 514.0 1244.7 730.6 33.0 .532 1.878 935.3 950 540.3 521.3 1246.7 725.4 31.4 .505 1.980 985.3 1000 546.8 528.3 1248.7 720.3 30.0 .480 2.082 FLOW OF STEAM. Flow of Steam through a Nozzle. (From Clark on the Steam- engine.)— The flow of steam ot a greater pressure into an atmosphere of a less pressure increases as the difference of pressure is increased, until the external pressure becomes only 58$ of the absolute pressure in the boiler. The flow of steam is neither increased nor diminished by the fall of the ex- ternal pressure below 58$, or about 4/7ths of the inside pressure, even to the extent of a perfect vacuum. In flowing through a nozzle of the best form, the steam expands to the external pressure, and to the volume due to this pressure, so long as it is not less than 58$ of the internal pressure. For an external pressure of 58$, and for lower percentages, the ratio of expansion is 1 to 1.624. The following table is selected from Mr. Brownlee's data exem- plifying the rates of discharge under a constant internal pressure, into various external pressures: Outflow of Steam ; from a Given Initial Pressure into Various Lower Pressures. Absolute initial pressure in boiler, 75 lbs. per sq. in. Absolute External Ratio of Velocity of Actual Discharge Boiler per square inch. Pressure per square inch. Expansion in Nozzle. Outflow at Constant Density. Velocity of Outflow Expanded. inch of Orifice per minute. lbs. lbs. ratio. feet per sec. feet p. sec. lbs. 75 74 1.012 227.5 230 16.68 75 72 1.037 386.7 401 28.35 75 70 1.063 490 521 35.93 75 65 1.136 660 749 48.38 75 61.62 1.198 736 • 876 53.97 75 60 1.219 765 933 56.12 75 50 1.434 873 1252 64 75 45 1.575 890 1401 65.24 75 ( 43.46 J 1 58 p. cent f 1.624 890.6 1446.5 65.3 75 15 1.624 890.6 1446.5 65.3 75 1.624 890.6 1446.5 65.3 FLOW OF STEAM, When steam of varying initial pressures is discharged into the atmos- phere—the atmospheric pressure being not more than 58$ of the initial pressure— the velocity of outflow at constant deusity, that is, supposing the initial density to be maintained, is given by the formula V — 3.5953 y/i. V — the velocity of outflow in feet per minute, as for steam of the initial density ; h = the height in feet of a column of steam of the given absolute initial pressure of uniform density, the weight of which is equal to the pres- sure on the unit of base. The lowest initial pressure to which the formula applies, when the steam is discharged into the atmosphere at 14.7 lbs. per square inch, is (14.7 X 100/58 =) 25.37 lbs. per square inch. Examples of the application of the formula are given in the table below. From the contents of this table it appears that the velocity of outflow into the atmosphere, of steam above 25 lbs. per square inch absolute pressure, or 10 lbs. effective, increases very slowly with the pressure, obviously be- cause the density, and the weight to be moved, increase with the pressure. An average of 900 feet per second may, for approximate calculations, be taken for the velocity of outflow as for constant density, that is, taking the volume of the steam at the initial volume. Outflow of Steam into the Atmosphere.— External pressure per square inch 14.7 lbs. absolute. Ratio of expansion in nozzle, 1.624. "3 u . ?%>> >> « h . a o £■ e*> o . © ©55 co O £t3 Hi cso ft tToW 8.1 S ©.5 ft &£© © 8 to . --co g S a 2 oil '3 fe a o of? 1** m o < Q w < > ft W lbs. feet p. sec. feet per sec. lbs. H.P. lbs. feet p. sec. feet per sec. lbs. H.P. 25.37 863 1401 22.81 45.6 90 895 1454 77.94 155.9 30 867 1408 26.84 53.7 100 898 1459 86.34 172.7 40 874 1419 35.18 70.4 115 902 1466 98.76 197.5 50 880 1429 44.06 88.1 135 906 1472 115.61 231.2 60 885 1437 52.59 105.2 155 910 1478 132.21 264.4 70 889 1444 61.07 122.1 165 912 1481 140.46 280.9 75 891 1447 65.30 130.6 215 919 1493 181.58 363.2 Napier's Approximate Rule.— Flow in pounds per second = ab- solute pressure x area in square inches -f- 70. This rule gives results which closely correspond with those in the above table, as shown below. Abs. press., lbs. p. sq. in. 25.37 40 60 75 100 135 165 215 Discharge per mm., by table, lbs 22.81 35.18 52.59 65.30 86.34 115.61 140.46 181.58 By Napier's rule 21.74 34.23 51.43 64.29.85.71 115.71 141.43 184.29 Prof. Peabody, in Trans. A. S. M. E., xi, 187, reports a series of experi- ments on flow of steam through tubes J4 inch in diameter, and J4, J^, and \y& inch long, with rounded entrances, in which the results agreed closely with Napier's formula, the greatest difference being an excess of the experimental over the calculated result of 3.2$. An equation derived from the theory of thermodynamics is given by Prof. Peabody, but it does not agree with the experimental results as well as Napier's rule, the excess of the actual flow being 6.6$. Flow ot Steam in Pipes.— A formula commonly used for velocity of flow of steam in pipes is the same as Downing's for the flow of water in smooth cast-iron pipes, viz., V = 50 i/f a ■ in which V = velocity in feet per second, L — length and D = diameter of pipe in feet, H = height in feet of a column of steam, of the pressure of the steam at the entrance, 670 STEAM. which would produce a pressure equal to the difference of pressures at the two ends of the pipe. (For derivation of the coefficient 50, see Briggs on "Wanning Buildings by Steam," Proc. Inst. C. E. 1882.) If Q = quantity in cubic feet per minute, d = diameter in inches, L and H being in feet, the formula reduces to <2 = 4.7233|/^, #=.0448^, d = .5374^^. (These formulae are applicable to. air and other gases as well as steam.) If Pi = pressure in pounds per square inch of the steam (or gas) at the en- trance to the pipe, p 2 = the pressure at the exit, then 144(pj — p 2 ) — differ- ence in pressure per square foot. Let w = density or weight per cubic foot of steam at the pressure p t , then the height of column equivalent to the difference in pressures = H= Ui ^~P*\ and Q = 60 X .7854 X 5oW ***<£■' ~ P * )D . to y %vL If W = weight of steam flowing in pounds per minute =. Qw, and d is taken in inches, L being in feet, S|/ M( P'2 Pa,)d. -t-A. -H D. -*- A. -^A. 300 0.194 0.0375 0.223 0.0500 0.03 400 0.224 0.0500 0.258 0.0667 04 500 0.250 0.0625 0.288 0.0833 0.05 600 0.274 0.0750 0.316 0.1000 0.06 700 0.296 0.0875 0.341 0.1167 0.07 800 0.316 0.1000 0.365 0.1333 0.08 900 0.335 0.1125 0.387 0.1500 0.09 1000 0.353 0.1250 0.400 0.1667 0.10 STEAM PIPES. Bursting-tests of Copper Steam-pipes. (From Report of Chief Engineer Melville, U. S. N., for 1892.)— Some tests were made at the New York Navy Yard which show the unreliability of brazed seams in cop- per pipes. Each pipe was 8 in. diameter inside and 3 ft. 1% in . long. Both ends were closed by ribbed heads and the pipe was subjected to a hot- water pressure, the temperature being maintained constant at 371° F- Three STEAM-PIPES. 675 of the pipes were made of No. 4 sheet copper (" Stubbs " gauge) and the fourth was made of No. 3 sheet. The following were the results, in lbs. per sq. in., of bursting-pressure: Pipe number Actual bursting-strength. Calculated " " Difference 1 2 3 4 4' 835 785 950 1225 1275 336 1336 1569 1568 1568 501 551 619 343 293 The theoretical bursting-pressure of the pipes was calculated by using the figures obtained in the tests for the strength of copper sheet with a brazed joint at 350° F. Pipes 1 and 2 are considered as having been annealed. The tests of specimens cut from the ruptured pipes show the injurious action of heat upon copper sheets; and that, while a white heat does not change the character of the metal, a heat of only slightly greater degree causes it to lose the fibrous nature that it has acquired in rolling, and a serious reduction in its tensile strength and ductility results. All the brazing was done by expert workmen, and their failure to make a pipe-joint without burning the metal at some point makes it probable that, with copper of this or greater thickness, it is seldom accomplished. That it is possible to make a joint without thus injuring the metal was proven in the cases of many of the specimens, both of those cut from the pipes and those made separately, which broke with a fibrous fracture. Rule for Thickness of Copper Steam-pipes. (U. S. Super- vising Inspectors of Steam Vessels.)— Multiply the working steam-pressure in lbs. per sq. in. allowed the boiler by the diameter of the pipe in inches, then divide the product by the constant whole number 8000, and add .0625 to the quotient; the sum will give the thickness of material required. Example.— Let 175 lbs. = working steam-pressure per sq. in. allowed the boiler, 5 in. = diameter of the pipe; then — ^r— — \- .0625 = .1718 -f- inch, thickness required. Reinforcing Steam-pipes. (Eng., Aug. 11, 1893.)— In the Italian Navy copper pipes above 8 in. diam. are reinforced by wrapping them with a close spiral of copper or Delta-metal wire. Two or three independent spirals are used for safety in case one wire breaks. They are wound at a tension of about lVg tons per sq. in. Wire-wound Steam-pipes.— The system instituted by the British Admiralty of winding all steam -pipes over 8 in. in diameter with 3/16-in. copper wire, thereby about doubling the bursting-pressure, has within re- cent years been adopted on many merchant steamers using high-pressure steam, says the London Engineer. The results of some of the Admiralty tests showed that a wire pipe stood just about the pressure it ought to have stood when unwired, had the copper not been injured in the brazing. Riveted Steel Steam-pipes have recently been used for high pressures. See paper on A Method of Manufacture of Large Steam-pipes, by Chas. H. Manning, Trans. A. S. M. E., vol. xv. Valves in Steam-pipes.— Should a globe-valve on a steam-pipe have the steam -pressure on top or underneath the valve is a disputed question. With the steam-pressure on top, the stuffing-box around the valve-stem can- not be repacked without shutting off steam from the whole line of pipe; on the other hand, if the steam-pressure is on the bottom of the valve it all has to be sustained by the screw-thread on the valve-stem, and there is danger of stripping the thread. A correspondent of the American Machinist, 1892, says that it is a very uncommon thing in the ordinary globe-valve to have the thread give out, but by water-hammer and merciless screwing the seat will be crushed down quite -frequently. Therefore with plants where only one boiler is used he advises placing the valve with the boiler-pressure underneath it. On plants where several boilers are connected to one main steam-pipe he would re- verse the position of the valve, then when one of the valves needs repacking the valve can be closed and the pressure in the boiler whose pipe it controls can be reduced to atmospheric by lifting the safety-valve. The repacking can then be done without interfering with the operation of the other boilers of the plant. He proposes also the following other rules for locating valves: Place valves with the stems horizontal to avoid the formation of a water-pocket. Never put the junction-valve close to the boiler if the main pipe is above the boiler, but put it on the highest point of the junction-pipe. If the other 676 STEAM. plan is followed, the pipe fills with water whenever this boiler is stopped and the others are running, and breakage of the pipe may cause serious re- sults. Never let a junction-pipe run into the bottom of the main pipe, but into the side or top. Always use an angle-valve where convenient, as there is more room in them. Never use a gate valve under high pressure unless a by-pass is used with it. Never open a blow-off valve on a boiler a little and then shut it; it is sure to catch the sediment and ruin the valve; throw it well open before closing. Never use a globe-valve on an indicator-pipe. For water, always use gate or angle valves or stop-cocks to obtain a clear pas- sage. Buy if possible valves with renewable disks. Lastly, never let a man go inside a boiler to work, especially if he is to hammer on it, unless you break the joint between the boiler and the valve and put a plate of steel between the flanges. Flanges for Steam-nozzles and Steam-pipe, used with the Gill Water-tube Boiler, Phila., 1892. Sizeofpipe 3 4 5 6 7 8 9 Outside diameter of flange, inches.. 9 10 11 12 13 14 15 Pitch-circle for bolts, diam., " .. 7 8 9 10 11 12 13 Outside diam. of gaskets, " .. $% 6% 7% 8)4 9% 10J^ llj^ Inside diam. of gaskets, " .. 3^| 4^| 5^ in. long under the head. All bolts to have square heads and hexagon nuts. The "Steam Lioop" is a system of piping by which water of con- densation in steam-pipes is automatically returned to the boiler. In its simplest form it consists of three pipes, which are called the riser, the hori- zontal, and the drop-leg. When the steam-loop is used for returning to the boiler the water of condensation and entrainment from the steam-pipe through which the steam flows to the cylinder of an engine, the riser is gen- erally attached to a separator; this riser empties at a suitable height into the horizontal, and from thence the water of condensation is led into the drop-leg, which is connected to the boiler, into which the water of condensa- tion is fed as soon as the hydrostatic pressure in drop-leg in connection with the steam-pressure in the pipes is sufficient to overcome the boiler-pressure. The action of the device depends on the following principles: Difference of pressure may be balanced by a water-column: vapors or liquids tend to flow to the point of lowest pressure; rate of flow depends on difference of pres- sure and mass; decrease of static pressure in a steam-pipe or chamber is proportional to rate of condensation; in a steam-current water will be car- ried or swept along rapidly by friction. (Illustrated in Modern Mechanism, p. 807.) Lioss from an Uncovered Steam-pipe. (Bjorling on Pumping- engines.) — The amount of loss by condensation in a steam-pipe carried down a deep mine-shaft has been ascertained by actuai practice at the Clay Cross Colliery, near Chesterfield, where there is a pipe 7\4 in. internal diam.. 1 100 ft. long. The loss of steam by condensation was ascertained by direct measurement of the water deposited in a receiver, and was found to be equivalent to about 1 lb. of coal per I.H.P. per hour for every 100 ft. of steam-pipe; but there is no doubt that if the pipes had been in ihe upcast shaft, and well covered with a good non-conducting material, the loss would have been less. (For Steam-pipe Coverings, see p. 469, ante.) THE HORSE-POWER OF A STEAM-BOILER. 677 THE STEAM-BOILER. The Horse-power of a Steam-boiler.-— The term horsepower has two meanings in engineering : First, an absolute unit or measure of the rate of work, that is, of the work done in a certain definite period of time, by a source of energy, as a steam-boiler, a waterfall, a current of air or water, or by a prime mover, as a steam-engine, a water-wheel, or a wind- mill. The value of this unit, whenever it can be expressed in foot-pounds of energ}^, as in the case of steam-engines, water-wheels, and waterfalls, is 33,000 foot-pounds per minute. In the case of boilers, where the work done, the conversion of water into steam, cannot be expressed in foot-pounds of available energy, the usual value given to the term horse-power is the evap- oration of 30 .'bs. of water of a temperature of 100° F. into steam at 70 lbs. pressure above the atmosphere. Both of these units are arbitrary; the first, 33,000 foot-pounds per minute, first adopted by James Watt, being considered equivalent to the power exerted by a good London draught-horse, and the 30 lbs. of water evaporated per hour being considered to be the steam re- quirement per indicated horse-power of an average engine. The second definition of the term horse-power is an approximate measure of the size, capacity, value, or " rating " of a boiler, engine, water-wheel, or other source or conveyer of energy, by which measure it may be described, bought and sold, advertised, etc. No definite value can be given to this measure, which varies largely with local custom or individual opinion of makers and users of machinery. The nearest approach to uniformity which can be arrived at in the term "horse power," used in this sense, is to say that a boiler, engine, water-wheel, or other machine, " rated 1 ' at a certain horse-power, should be capable of steadily developing that horse-power for a long period of time under ordinary conditions of use and practice, leaving to local custom, to the judgment of the buyer and seller, to written contracts of purchase and sale, or to legal decisions upon such contracts, the interpre- tation of what is meant by the term "ordinary conditions of use and practice.' 1 (Trans. A. S. M. E., vol. vii. p. 226.) The committee of the A. S. M. E. on Trials of Steam-boilers in 1S84 (Trans., vol. vi. p. 265) discussed the question of the horse-power of boilers as follows: The Committee of Judges of the Centennial Exhibition, to whom the trials of competing boilers at that exhibition were intrusted, met with this same problem,. and finally agreed to solve it, at least so far as the work of that committee was concerned, by the adoption of the unit, 30 lbs. of water evap- orated into dry steam per hour from feed-water at 100° F., and under a pressure of 70 lbs. per square inch above the atmosphere, these conditions being considered by them to represent fairly average practice. The quan- tity of heat demanded to evaporate a pound of water under these conditions is 1110.2 British thermal units, or 1.1496 units of evaporation. The unit of power proposed is thus equivalent to the development of 33,305 heat units per hour, or 34 488 units of evaporation. . . . Your committee, after due consideration, has determined to accept the Centennial Standard, the first above mentioned, and to recommend that in all standard trials the commercial horse-power be taken as an evaporation of 30 lbs. of water per hour from a feed-water temperature of 100° F. into steam at 70 lbs. gauge pressure, which shall be considered to be equal to 34}^ units of evaporation, that is, to 34^ lbs. of water evaporated from a feed- water temperature of 212° F. into steam at the same temperature. This standard is equal to 33.305 thermal units per hour. It is the opinion of this committee that a boiler rated at any stated number of horse-powers should be capable of developing that power with easy firing, moderate draught, and ordinary fuel, while exhibiting good economy ; and further, that the boiler should be capable of developing at least one third more than its rated power to meet emergencies at times when maximum economy is not the most important object to be attained. Unit ot Evaporation. — It is the custom to reduce results of boiler- tests to the common standard of weight of water evaporated by the unit weight of the combustible portion of the fuel, the evaporation being consid- ered to have taken place at mean atmospheric pressure, and at the temper- ature due that pressure, the feed-water being also assumed to have been supplied at that temperature. This is, in technical language, said to be the equivalent evaporation from and at the boiling-point at atmospheric pres- sure, or "from and at 212° F." This unit of evaporation, or one pound of 678 THE STEAM-BOILER, water evaporated from and at 212°, is equivalent to 965.7 British thermal units. Measures for Comparing the Buty of Boilers.— The meas- ure of the efficieuey of a boiler is the number of pounds of water evaporated per pound of combustible, the evaporation being reduced to the standard of "from and at 212°;'" that is, the equivalent evaporation from feed-water at a temperature of 212° F. into steam at the same temperature. The measure of the capacity of a boiler is the amount of "boiler horse- power " developed, a horse-power being defined as the evaporation of 30 lbs. of water per hour from 100° F. into steam at 70 lbs. pressure, or 34J/£ lbs. per hour from and at 212°. The measure of relative rapidity of steaming of boilers is the number of pounds of water evaporated per hour per square foot of water-heating sur- face. The measure of relative rapidity of combustion of fuel in boiler-furnaces is the number of pounds of coal burned per hour per square foot of grate- surface. STEAM-BOILER PROPORTIONS. Proportions of Grate and Heating Surface required for a given Horse-power. — The term horse-power here means capacity to evaporate 30 lbs. of water from 100° F., temperature of feed-water, to steam of 70 lbs., gauge-pressure = 34.5 lbs. from and at 212° F. Average proportions for maximum economy for land boilers fired with good anthracite coal: Heating surface per horse-power 1 1.5 sq. ft. Grate " " " 1/3 " Ratio of heating to grate surface 34.5 " Water evap'd from and at 212° per sq. ft H.S. per hour 3 lbs. Combustible burned per H. P. per hour 3 " Coal with 1/6 refuse, lbs. per H.P. per hour 3.6 " Combustible burned per sq. ft. grate per hour 9 " Coal with 1/6 refuse, lbs. per sq.ft. grate pe 1- hour 10.8 " Water evap'd from and at 212° per lb. combustible. .. 11.5 " " " " " " coal (1/6 refuse) 9.6 >l The rate of evaporation is most conveniently expressed in pounds evapo- rated from and at 212° per sq. ft. of water-heating surface per hour, and the rate of combustion in pounds of coal per sq. ft. of grate-surface per hour. Heating-surface.— For maximum economy with any kind of fuel a boiler should be proportioned so that at least one square foot of heating- surface should be given for every 3 lbs. of water to be evaporated from and at 212° F. per hour. Still more liberal proportions are required if a portion of the heating-surface has its efficiency reduced by: 1. Tendency of the heated gases to short-circuit, that is, to select passages of least resistance and flow through them with high velocity, to the neglect of other passages. 2. Deposition of soot from smoky fuel. 3. Incrustation. If the heating-sur- faces are clean, and the heated gases pass over it uniformly, little if any increase in economy can be obtained by increasing the heating-surface be- yond the proportion of 1 sq. ft. to every 3 lbs. of water to be evaporated, and with all conditions favorable but little decrease of economy will take place if the proportion is 1 sq. ft. to every 4 lbs. evaporated; but in order to pro- vide for driving of the boiler beyond its rated capacity, and for possible decrease of efficiency due to the causes above named, it is better to adopt 1 sq. ft. to 3 lbs. evaporation per hour as the minimum standard proportion. Where economy may be sacrificed to capacity, as where fuel is very cheap, it is customary to proportion the heating-surface much less liberally. The following table shows approximately the relative results that may be ex- pected with different rates of evaporation, with anthracite coal. Lbs. water evapor 1 d from and at 212° per sq. ft. heating-surface per hour: 2 2.5 3 3.5 4 5 6 7 8 9 10 Sq. ft. heating-surface required per horse-power: 17.3 13.8 11.5 9.8 8.6 6.8 5.8 4.9 4.3 3.8 3.5 Ratio of heating to grate surface if 1/3 sq. ft. of G. S. is required per H.P.: 52 41.4 34.5 29.4 25.8 20.4 17.4 13.7 12.9 11.4 10.5 Probable relative economy: 100 100 100 95 90 85 80 75 70 65 60 Probable temperature of chimney gases, degrees F.: 450 450 450 518 585 652 720 787 855 922 990 STEAM-BOILER PROPORTIONS. 679 The relative economy will vary not only with the amount of heating-sur- face per horse-power, but with the efficiency of that heating-surface as regards its capacity for transfer of heat from the heated gases to the water, which will depend on its freedom from soot and. incrustation, and upon the circulation of the water and the heated gases. With bituminous coal the efficiency will largely depend upon the thorough- ness with which the combustion is effected in the furnace. The efficiency with any kind of fuel will greatly depend upon the amount of air supplied to the furnace in excess of that required to support com- bustion. With strong draught and thin fires this excess may be very great, causing a serious loss of economy. Measurement of Heating-surface. —Authorities are not agreed as to the methods of measuring the heating-surface of steam-boilers. The usual rule is to consider as heating-surface all the surfaces that are sur- rounded by water on one side and by flame or heated gases on the other, but there is a difference of opinion as to' whether tubular heating-surface should be figured from the inside or from the outside diameter. Some writers say, measure the heating-surface always on the smaller side— the fire side of the tube in a horizontal return tubular boiler and the water side in a water-tube boiler. Others would deduct from the heating-surface thus measured an allowance for portions supposed to be ineffective on account of being cov- ered by dust, or being out of the direct current of the gases. For the sake of uniformity, however, it would appear to be the best method to consider all surfaces as heating-surfaces which transmit heat from the flame or gases to the water, making no allowance for different degrees of effectiveness; also, to use the external instead of the internal diameter of tubes, for greater convenience in calculation, the external diameter of boiler-tubes usually being made in even inches or half inches. There would seem to be no good reason for considering the smaller surface in a tube as the heating-surface, for the transmission of heat through plates that are ribbed or corrugated on one side does not appear to be proportional to the smaller surface, but rather to the larger. Thus the Serve ribbed tube trans- mits more heat to the water per foot of length than a plain tube of same external diameter, and a ribbed steam-radiator radiates more heat than a plain radiator having the same internal or smaller surface. Rule for finding the heating-surface of vertical tubular boilers : Multiply the circumference of the fire-box (in inches) by its height above the grate ; multiply the combined circumference of all the tubes by their length, and to these two products add the area of the lower tube-sheet ; from this sum subtract the area of all the tubes, and divide by 144 : the quotient is the number of square feet of heating-surface. Rule for finding the heating-surface of horizontal tubular boiler« : Multi- ply two thirds of the circumference of the shell (in inches) by its length ; multiply the combined length of the tubes by their combined circumference, to the sum of these products add two thirds of the area of both tube-sheets; from this sum subtract the combined area of all the tubes, and divide the remainder by 144: the result is the number of square feet of heatiug-surface. Rule for finding the square feet of heating -surface in tubes : Multiply the number of tubes by the diameter of a tube in inches, by its length in feet, and by .2618. Horse-power, Builder's Rating. Heating-surface per Horse-power.— If is a general practice among builders to furnish about 12 square feet of heating-surface per horse-power, but as the practice is not uniform, bids and contracts should always specify the amount of heating- surface to be furnished. Not less than one third square foot of grate-surface should be furnished per horse-power. Engineering Neivs, July 5, 1894, gives the following rough-and-ready rule for finding approximately the commercial horse-power of tubular or water- tube boilers : Number of tubes X their length in feet X their nominal diameter in inches -s- 50 = nLd -*- 50. The number of square feet of surface in the tubes is = -5-0.5, an( * tne horse-power at 12 square feet of surface of tubes per horse-power, not counting the shell, = nLd -s- 45.8. If 15 square feet of surface of tubes be taken, it is nLd -4- 57.3. Making allowance for the heating-surface in the shell will reduce the divisor to about 50. Horse-power of Marine and Locomotive Boilers.— The term horse-power is not generally used in connection with boilers in marine practice, or with locomotives. The boilers are designed to suit the engines, and are rated by extent of grate and heating-surface only, 680 THE STEAM-BOILER. Grate-surface.— The amount of grate-surface required per horse power, and the proper ratio of heating-surface to grate-surface are ex- tremely variable, depending chiefly upon the character of the coal and upon the rate of draught. With good coal, low in ash, approximately equal results may be obtained with large grate-surface and light draught and with small grate-surface and strong draught, the total amount of coal burned per hour being the same in both cases. With good bituminous coal, like Pittsburgh, low in ash, the best results apparently are obtained with strong draught and high rates of combustion, provided the grate-surfaces are cut down so that the total coal burned per hour is not too great for the capacity of the heating-surface to absorb the heat produced. With coals high in ash, especially if the ash is easily fusible, tending to choke the grates, large grate-surface and a slow rate of combustion are required, unless means, such as shaking grates, are provided to get rid of the ash as fast as it is made. The amount of grate-surface required per horse-power under various con- ditions may be estimated from the following table : Lbs. Water from and at 212° per lb., Coal. Pounds of Coal burned per square foot 6ffi2 rA t-> '-• 52 5** of Grate per hour. 8 | 10 | 12 | 15 | 20 | 25 | 30 35 | 40 J 10 3.45 Sq. Ft. Grate per H. P. Good coal .48 35 28 .23 .17 14 11 10 09 and boiler, 1 9 3.83 .48 38 . : J .2 . 25 19 .15 13 .11 ,10 Fair coal or boiler, ( 8.61 1 8 4. 4.31 .50 54 .40 .48 .33 .36 .26 .29 .20 .22 .16 .17 .13 .14 .12 .13 .10 .11 I 7 4.93 . 62 .49 .41 .88 .24 .20 .17 .14 .12 Poor coal or boiler, I 6.9 1 6 5. 5.75 .63 .72 .50 .58 .42 .48 .34 .38 .25 .29 .20 .23 .17 .19 .15 17 .13 .14 1 5 6.9 .86 .69 58 .46 .35 .28 .23 .22 .17 Lignite and poor boiler, [ 3.45 10. 1.25 1.00 .83 .67 .50 .40 .33 .29 .25 In designing a boiler for a given set of conditions, the grate-surface should be made as liberal as possible, say sufficient for a rate of combustion of 10 lbs. per square foot of grate for anthracite, and 15 lbs. per square foot for bituminous coal, and in practice a portion Of the grate-surface may be bricked over if it is found that the draught, fuel, or other conditions render it advisable. Proportions of Areas of Flues and other Gas-passages. — Rules are usually given making the area of gas-passages bear a certain ratio to the area of the grate-surface; thus a common rule for horizontal tubular boilers is to make the area over the bridge wall 1/7 of the grate- surface, the flue area 1/8, and the chimney area 1/9. For average conditions with anthracite coal and moderate draught, say a rate of combustion of 12 lbs. coal per square foot of grate per hour, and a ratio of heating to grate surface of 30 to 1, this rule is as good as any, but it is evi- dent that if the draught were increased so as to cause a rate of combustion of 24 lbs., requiring the grate-surface to be cut down to a ratio of 60 to 1, the areas of gas-passages should not be reduced much, because the grate-sur- face is reduced. The coal burned being the same under the changed condi- tions, and there being no reason why the gases should travel at a higher velocity, the actual areas of the passages should remain as before, but the ratio of the area to the grate-surface would in that case be doubled. Mr. Barrus states that the highest efficiency with anthracite coal is obtained when the tube area is 1/9 to 1/10 of the grate-surface, aud with bituminous coal when it is 1/6 to 1/7, for the conditions of medium rates of combustion, such as 10 to 12 lbs. per square foot of grate per hour, and 12 .square feet of heating surface allowed to the horse-power. The tube area should be made large enough not to choke the draught, and so lessen the capacity of the boiler; if made too large the gases are apt to select the passages of least resistance and escape from them at a high velocity and high temperature. This condition is very commonly found in horizontal tubular boilers where PERFORMANCE OF BOILERS. 681 the gases go chiefly through the upper rows of tubes; sometimes also in vertical tubular boilers, where the gases are apt to pass most rapidly through the tubes nearest to the centre. Air-passages through Grate-bars.— The usual practice is, air- opening = 30$ to 50$ of area of the grate ; the larger the better, to avoid stoppage of the air-supply by clinker; but with coal free from clinker much smaller air-space may be used without detriment. See paper by F. A. Scheffler, Trans. A. S. M. E., vol. xv. p. 503. PERFORMANCE OF ISOIL.ERS. Clark (Steam-engine, vol. i. p. 327) gives the following formulas for the relation of coal and water consumed in steam-boilers per square foot of grate-area per hour, and the ratio of the heating-surface to the area of the fire-grate. Water taken as evaporated from and at 212° F. Stationary boilers to = M22r* + 9.56c Marine boilers w = .016r a + 10.25c Portable-engine boilers ru = .OOSr 2 -j- 8.6c Locomotive boilers icoal-burning) iv — .009r a -j- 9.7c Locomotive boilers (coke-burning) to = ,0178r 2 -j- 7.94c In which w = weight of water in pounds per square foot of grate per hour; c = pounds of fuel per square foot of grate per hour; r = ratio of heating to grate surface. There are minimum rates of consumption of fuel below which these formulas are not applicable. The limit varies for each kind of boiler, and it varies with the surface-ratio. It is imposed by the fact that the maximum evaporative power of fuel is a fixed quantity, and is naturally at that point where the reduction of the rate of combustion for a given ratio procures the absorption into the boiler of the whole of the proportion of the heat which is available for evaporation. In the combustion of good coal the limit of evaporative efficiency may be taken as measured by 12}^ lbs. of water from: and at 212° F. ; and in that of good coke by 12 lbs. of water from and at- 212° F. Based on these formulas Clark gives the following table : Evaporative Performance of Steam-boilers for increasing Rates of Combustion and different Surface-ratios. For best coal; surface-ratio 30. Kind of Boiler. Stationary. Marine. Portable. Locomotive. Water from and at 212° F. per hour. Per sq. ft. of grate Per lb. of coal Per sq. ft. of grate Per lb. of coal Per sq. ft. of grate Per lb. of coal Per sq. ft. of grate Per lb. of coal.. . Fuel per Square Foot of Grate per hour, lbs. 62.5* 12.5 62.5* 12.5 50 10 57 11.4 10 lbs. 116 11.56 117 11. 6J 93 9.3 15 20 30 40 50 10.89 16f 11.25 136 9.01 154 10.26 lbs. 211 10.56 219 10.95 179 8.95 202 10.10 lbs. 307 10.23 lbs. 402 10.06 424 10.61 351 8.77 lbs. 498 10.54 437 8.74 493 9.86 Surface-ratio 50. 5 10 15 20 30 40 50 lbs. lbs. lbs. lbs. lbs. lbs lbs, Stationary. Per sq. ft. of grate 62.5* 125* 187.5* 247 342 438 534 " Per lb. of coal 12.5 12.5 12.5 12.33 11.41 10.95 10.67 Marine. Per sq. ft. of grate 62.5* 125* 187.5* 245 348 450 552 " Per lb. of coal 12.5 12.5 12.5 12.25 11.58 11.25 11.05 Portable. Per sq. ft. of grate 62.5* 106 149 192 278 364 450 " Per lb. of coal 12.5 10.6 9.93 9.6 9.27 9.10 9.00 Locomotive. Per sq. ft. of grate 62.5* 120 168 217 314 411 508 " Per lb. of coal 12.5 11.95 11. zO 10.85 10 45 10.26 10.15 * These quantities fall below the scope of the formulas for the water, as explained in the text. 682 THE STEAM-BOlLim Surface ratio 75. 30 40 50 60 75 90 100 Locomotive. Pei* sq. ft. of grate. Per lb. of coal lbs. 342 11.39 lbs. 439 10.97 lbs. 536 10.71 lbs. 633 10.65 lbs. 778 10.37 lbs. 927 10.26 lbs. 102C 10.20 General Conditions which secure Economy of Steam- boilers.— In general, the highest results are produced where the tempera- ture of the escaping gases is the least. An examination of this question is made by Mr. G. H. Barrus in his book on " Boiler Tests, 11 by selecting those tests made by him, six in number, in which the temperature exceeds the average, that is, 375° F., and comparing with five tests in which the temper- ature is less than 375°. The boilers are all of the common horizontal type, and all use anthracite coal of either egg or broken size. The average flue temperatures in the two series v/as 444° and 343° respectively, and the dif- ference was 101°. The average evaporations are 10.40 lbs. and 11.02 lbs. re- spectively, and the lowest result corresponds to the case of the highest flue temperature. In these tests it appears, therefore, that a reduction of 101° in the temperature of the waste gases secured an increase in the evaporation of 6%. This result corresponds quite closely to the effect of lowering the temperature of the gases by means of a flue-heater where a reduction of 107° was attended by an increase of 7% in the evaporation per pound of coal. A similar comparison was made on horizontal tubular boilers using Cum- berland coal. The average flue temperature in four tests is 450° and the average evaporation is 11.34 lbs. Six boilers have temperatures below 415°, the average of which is 383°, and these give an average evaporation of 11.75 lbs. With 67° less temperature of the escaping gases the evaporation is higher by about 4%. The wasteful effect of a high flue temperature is exhibited by other boilers than those of the horizontal tubular class. This source of waste was shown to be the main cause of the low economy produced in those vertical boilers which are deficient in heating-surface. Relation between the Heating-surface and Grate-surface to obtain the Highest Efficiency. — A comparison of thi*ee tests of horizontal tubular boilers with anthracite coal, the ratio of heating- surf ace to grate-surf ace being 364 to 1, with three other tests of similar boilers, in which the ratio was 48 to 1, showed practically no difference in the results. The evidence shows that a ratio of 36 to 1 provides a sufficient quantity of heating-surface to secure the full efficiency of anthracite coal where the rate of combustion is not more than 12 lbs. per sq. ft. of grate per hour. In tests with bituminous coal an increase in the ratio from 36.8 to 42.8 se- cured a small improvement in the evaporation per pound of coal, and a high temperature of the escaping gases indicated that a still further increase would be beneficial. Among the high results produced on common horizon- tal tubular boilers using bituminous coal, the highest occurs where the ratio is 53.1 to 1. This boiler gave an evaporation of 12.47 lbs. A double-deck boiler furnishes another example of high performance, an evaporation of 12.42 lbs. having been obtained with bituminous coal, and in this. case the ratio is 65 to 1. These examples indicate that a much larger amount of heating-surface is required for obtaining the full efficiency of bituminous coal than for boilers using anthracite coal. The temperature of the escap- ing gases in the same 1 oiler is invariably higher when bituminous coal is used than when anthracite coal is used. The deposit of soot on the surfaces when bituminous coal is used interferes with the full efficiency of the sur- face, and an increased area is demanded as an offset to the loss which this deposit occasions. It would seem, then, that if a ratio of 36 to 1 is sufficient for anthracite coal, from 45 to 50 should be provided when bituminous coal is burned, especially in cases where the rate of combustion is above 10 or 12 lbs. per sq. ft. of grate per hour. The number of tubes controls the ratio between the area of grate-surface and area of tube opening. A certain minimum amount of tube-opening is required for efficient work. The best results obtained with anthracite coal in the common hoiizontal boiler are in cases where the ratio of area of grate-surface to area of tube- opening is larger than 9 lo 1. The conclusion is drawn that the highest effi- ciency with anthracite coal is obtained when the tube-opening is from 1/9 to 1/10 of the grate-surface. PERFORMANCE OF BOILERS. 683 When bituminous coal is burned the requirements appear to be different. The effect of a large tube opening does not seem to make the extra tubes inefficient when bituminous coal is used. The highest result on any boiler of the horizontal tubular class, fired with bituminous coal, was obtained where the tube-opening was the largest. This gave an evaporation of 12.47 lbs., the ratio of grate-surface to tube-opening being 5.4 to 1. The next highest re- sult was 12.42 lbs., the ratio being 5.2 tol. Three high results, averaging 12.01 lbs., were obtained when the average ratio was 7.1 to 1. Without going to extremes, the ratio to be desired when bituminous coal is used is that which gives a tube-opening having an area of from 1/6 to 1/7 of the grate- surface. This applies to medium rates oi ; combustion of, say, 10 to 12 lbs. per sq. ft. of grate per hour, 12 sq. ft. of water-heating surface being allowed per horse-power. A comparison of results obtained from different types of boilers leads to the general conclusion that the economy with which different types of boilers operate depends much more upon their proportions and the condi- tions under which they work, than upon their type ; and, moreover, that when these proportions are suitably carried out, and when the conditions are favorable, the various types of boilers give substantially the same eco- nomic result. Efficiency of a Steam-boiler.— The efficiency of a boiler is the percentage of the total heat generated by the combustion of the fuel which is utilized in heating the water and in raising steam. With anthracite coal the heating-value of the combustible portion is very nearly 14,500 B. T. U. per lb., equal'to an evaporation from and at 212° of 14,500-^966 = 15 lbs. of water. A boiler which when tested with anthracite coal shows an evaporation of 12 lbs. of water per lb. of combustible, has an efficiency of 12 -=- 15 = 80$, a figure which is approximated, but scarcely ever quite reached, in the best practice. With bituminous coal it is necessary to have a determination of its heating-power made by a coal calorimeter before the efficiency of the boiler using it can be determined, but a close estimate may be made from the chemical analysis of the coal. (See Coal.) The difference between the efficiency obtained by test and 100$ is the sum of the numerous wastes of heat, the chief of which is the necessary loss due to the temperature of the chimney-gases. If we have an analysis and a calorimetric determination of the heating-power of the coal (properly sam- pled), and an average analysis of the chimney-gases, the amounts of the several loses may be determined with approximate accuracy by the method described below. Data given : 1. Analysis of the Coal. 2. Analysis of the Dry Chimney- Cumberland Semi-bituminous. gases, by Weight. Carbon 80.55 Hydrogen 4.50 C0 2 = 13.6 Oxygen 2.70 CO = .2 Nitrogen 1.08 O = 11.2 Moisture 2.92 N = 75.0 Ash 8.25 o. N. 9.89 .11 11.20 75.00 100.00 100.0 Heating-value of the coal by Dulong's formula, 14,243 heat-units. The gases being collected over water, the moisture in them is n&t deter- mined. 3. Ash and refuse as determined by boiler-test, 10.25, or 2$ more than that found by analysis, the difference representing carbon in the ashes obtained in the boiler-test. 4. Temperature of external atmosphere, 60° F. 5. Relative humidity of air, 60$, corresponding (see air tables) to .007 lb. of vapor in each lb. of air. 6. Temperature of chimney-gases, 560° F. Calculated results : The carbon in the chimnej^-gases being 3.8$ of their weight, the total weight of dry gases per lb. of carbon burned is 100 -h 3.8 = 26.32 lbs. Since the carbon burned is 80.55 — 2 = 78.55$ of the weight of the coal, the weight of the dry gases per lb. of coal is 26.32 X 78.55 -h 100 = 20.67 lbs. Each pound of coal furnishes to the dry chimney-gases .7855 lb. C, .0108N, and (g.70 - ^-) -s- 100 = .0214 lb, O; a total of .8177, say .82 lb. This sub- 684 THE STEAM-BOILER. tracted from 20.6? lbs. leaves 19.85 lbs. as the quantity of dry air (not includ- ing moisture) which enters the furnace per pound of coal, not counting the air required to burn the available hydrogen, that is, the hydrogen minus one eighth of the oxygen chemically combined in the coal. Each lb. of coal burned contained .045 lb. H, which requires .045 X 8 = .36 lb. O for its com- bustion. Of this, .027 lb. is furnished by the coal itself, leaving .333 lb. 10. come from the air. The quantity of air needed to supply this oxygen (air containing 23$ by weight of oxygen) is .333 -h .23 — 1.45 lb., which added to the 19.85 lbs. already found gives 21.30 lbs. as the quantity of dry air sup- plied to the furnace per lb. of coal burned. The air carried in as vapor is .0071 lb. for each lb. of dry air, or 21.3 X .0071 = 0.15 lb. for each lb. of coal. Each lb. of coal contained .029 lb. of mois- ture, which was evaporated and carried into the chimney-gases. The .045 lb. of H per lb. of coal when burned formed .045 x9= .405 lb. of H 2 0. From the analysis of the chimney-gas it appears that .09 -s- 3.80 — 2.37$ of the carbon in the coal was burned to CO instead of to C0 2 . We now have the data for calculating the various loses of heat as follows, for each pound of coal burned : Heat- units. Per cent of Heat-value of the Coal. 21.3 lbs. dry air X (560° - 60°) X sp. heat .238 = 2534.7 17.80 .15 lb. vapor in air X (560° - 60°) x sp. heat .48 = 36.0 0.25 .029 lb. moisture in coal heated from 60° to 212° — 4.4 0.03 evaporated from and at 212°; .029 X 966 = 28.0 0.20 steam (heated from 212° to 560°) X 348 X .48 = 4.8 0.03 .405 lb. H 2 from H in coal X (560° - 60°) x .48 = 97.2 0.68 .0237 lb. C burned to CO ; loss by incomplete com- bustion, .0237 X (14544 - 4451) = 239.2 1.68 .02 lb. coal lost in ashes ; .02 x 14544 = 290.9 2.04 Radiation and unaccounted for, by difference = 712.1 5.00 3,947.3 27.71 Utilized in making steam, equivalent evaporation 10.66 lbs. from and at 212° per lb. of coal = 10,295.7 72.29 14,243.0 100.00 The heat lost by radiation from the boiier and furnace is not easily deter- mined directly, especially if the boiler is enclosed in brickwork, or is pro- tected by non-conducting covering. It is customary to estimate the heat lost by radiation by difference, that is, to charge radiation with all the heat lost which is not otherwise accounted for. One method of determining the loss by radiation is to block off a portion of the grate-surface and build a small fire on the remainder, and drive this fire with just enough draught to keep up the steam-pressure and supply the heat lost by radiation without allowing any steam to be discharged, weigh- ing the coal consumed for this purpose during a test of several hours' dura- tion. Estimates of radiation by difference are apt to be greatly in error, as in this difference are accumulated all the errors of the analyses of the coal and of the gases. An average value of the heat lost by radiation from a boiler set in brickwork is about 4 per cent. When several boilers are in a battery and enclosed in a boiler-house the loss by radiation maybe very much less, since much of the heat radiated from the boiler is returned to it in the air supplied to the furnace, which is taken from the boiler-room. An important source of error in making a "heat balance" such as the one above given, especially when highly bituminous coal is used, may be due to the non-combustion of part of the hydrocarbon gases distilled from the coal immediately after firing, when the temperature of the furnace may be reduced below the point of ignition of the gases. Each pound of hydro- gen which escapes burning is equivalent to a loss of heat in the furnace of 62,500 heat-units. In analyzing the chimney gases by the usual method the percentages of the constituent gases are obtained by volume instead of by weight. To reduce percentages by volume to percentages by weight, multiply the per- centage by volume of each gas by its specific gravity as compared with air, and divide each product by the sum of the products. TESTS OF STEAM-BOILER. 685 The pounds of air required to burn a pound of carbon may be obtained directly from the analysis by volume by the following formula: Lbs. of air required to burn I _ t j 3(CQ 2 -f O) + CO < + 23- one pound of carbon j 3) C0 2 + CO j In which O, CO.^, and CO are the per cents, by volume, of the several con- stituents of the flue gases. Lbs. of air per pound | _ j Lbs. of air per pound \ v \ Per cent of carbon of coal f ) of carbon \ ) in coal. To reduce to volume at temperature of 32° F. make use of the formula V = 12.387 X lbs. of air per pound of coal. TESTS OF STEAM-BOILERS. Boiler-tests at the Centennial Exhibition, Philadel- phia, 18 76. -(See Reports and Awards Group XX, International Exhibi- tion, Phila., 1876; also, Clark on the Steam-engine, vol. i, page 253.) Competitive tests were made of fourteen boilers, using good anthracite coal, one boiler, the Galloway, being tested with both anthracite and semi- bituminous coal. Two tests were made with each boiler : one called the capacity trial, to determine the economy and capacity at a rapid rate of driving; and the other called the economy trial, to determine the economy when driven at a rate supposed to be near that of maximum economy and rated capacilv. The following table gives the principal results obtained in the economy "trial, together with the capacity and economy figures of the capacity trial for comparison Economy Tests. Capacity Tests. co - ^ 6 o ^ §J=CO J* s "g'FI oS O Name of Boiler. .5 u v i "~ - ■°ta -a < a o z* 1- ft£ aS J. V & & a g co o © o ft © o ft © 41 & 5 s-j * = «2% H o ft o o ^^^ m o Ph £f- ^ ww EH S X M K £ lbs. p.Ct lbs. lbs. deg * deg H.P. H.P. lbs. 84.6 64.3 9.1 12 10 4 •> OS, 12.094 11.988 393 415 41.4 119.8 57.8 148.6 68.4 10.441 Firmenich 10.4 1.68 11.064 • J ,ll fi 6.8 12.1 11.31.87 11.1 2.42 11.923 11.906 333 411 l"3 9.4 47.0 99.8 69.3 125.0 11.163 Smith. 45 8 11.925 Bahcock & Wilcox 37.7 10.0 11.02.43 11.822 296 2.7 135.6 186.6 10.330 Galloway 23.7 9.(5 ll.l!3.63 11 583 303 1.4 103.3 133.8 11.216 Do. semi-bit. coal 23.7 7.9 8.8 3.20 12.125 825 0.3 90.9 125.1 11.609 Andrews 15.6 8.0 10.32.32 11.039 420 71.7 42.6 58.7 9.745 Harrison 27.3 12.4 8.5 2.75 10.930 517 0.9 82.4 108.4 9.889 Wiegand 30. 7 12.3 9.5 3.30 10.834 524 20.5 147.5 162.8 9.145 Anderson 17.5 9.7 9.3 2.64 10.618 417 15.7 98.0 132.8 9.568 Kellv 20.9 33.5 10.8 9.3 9.0 3.82 11.41.38 10.312 10.041 '430 5.6 4.2 81.0 72.1 99.9 108.0 8.397 Exeter 9.974 14.0 19.0 8.C 8.6 11.0 4.44 9.9 3.43 10.021 9.613 871 572 5.2 2.1 51.7 45.7 67.8 67.2 9.865 Rogers & Black . . . 9.429 Averages 2.77 11.123 85.0 110.8 10.251 The comparison of the economy and capacity trials shows that an average increase in capacity of 30 per cent was attended by a decrease in economy of 8 per cent, but the relation of economy to rate of driving varied greatly in the different boilers. In the Kelly boiler an increase in capacity of 22 per cent was attended by a decrease in economy of over 18 per cent, while the Smith boiler with an increase of 2k- per cent in capacity showed a slight increase in economy. 686 THE STEAM-BOILER. One of the most important lessons gained from the above tests is that there is no necessary relation between the type of a boiler and economy. Of the five boilers that gave the best results, the total range of variation be- tween the highest and lowest of the five being only 2.3%, thi'ee were water- tube boilers, one was a horizontal tubular boiler, and the fifth was a com- bination of the two types. The next boiler on the list, tbe Galloway, was an internally fired boiler, all of the others being externally fired. The following is a brief description of the principal constructive features of the fourteen boilers: !,--,. j 4-in. water-tubes, incliued 20° to horizontal; reversed Root 1 draught. 3 in. water-tubes, nearly vertical; reversed draught. Cylindrical shell, multitubular flue. ( Cylindrical shell, multitubular flue — water-tubes in ) side flues. j 33^-in. water-tubes, inclined 15° to horizontal; re- I versed draught. Galloway Cylindrical shell, furnace-tubes and water-tubes. Andrews Square fire-box and double return multitubular flues. tto,...^™ j 8 slabs of cast-iron spheres, 8 in. in diameter; re- nai f lson f versed draught. wTi a „ a „A j 4-in. water tubes, vertical, with internal circulating wiegana 1 tubes. Anderson 3-in. flue-tubes, nearly horizontal; return circulation. K ,j ) 3-in. water-tubes, slightly inclined; each divided by y j internal diaphragm to promote circulation. Exeter 27 hollow rectangular cast-iron slabs. Pierce Rotating horizontal cylinder, with flue-tubes. Rogers & Black Vertical cylindrical boiler, with external water-tubes. Tests of Tubulous Boilers.— The following tables are given by S. H. Leonard, Asst. Engr. U. 8. N., in Jour. Am. Soc. Naval Engrs. 1890. The tests were made at different times by boards of U. S. Naval Engineers, ex- cept the test of the locomotive-torpedo boiler, which was made in England. Firmenich Lowe Smith Babcock & Wilcox. Type. Herreshoff Towne Ward Scotch Locom'tive torpedo, Ward Thorny- croft, (U. S.S.Cush- ing.) ■33 o O 25. 1 4.3 24.1 7 A 15.1 j 24 I 38 Evaporation from and at 212° F. 17.1 •,'0.05 23.8 10 Weights, lbs. E40,6 S 42,7 E 2,9 S 3,0 E 1,3 S 1,6 E 1,6 S 1,9 E18,9 S 30,0 S 34,9 E26,5 S 30.4 E 20.1 S -J4.C . £ a - m* CX(t -2)XB Lloyd's : P = — . t = thickness of plate in sixteenths ; B and D as before ; C = a constant depending on the kind of joint. When longitudinal seams have double butt-straps, C = 20. When longi- tudinal seams have double butt-straps of unequal width, only covering on one side the reduced section of plate at the outer line of rivets, C — 19.5. When the longitudinal seams are lap-jointed, C = 18.5. U. S. Statutes.— Using same notation as for Board of Trade, t X 2 X T P — -— — for single-riveting ; add 20$ for double-riveting ; D X o where 7 is the lowest T. S. stamped on any plate. Mr. Foley criticises the rule of the United States Statutes as follows : The rule ignores the riveting, except that it distinguishes between single and double, giving the latter 20$ advantage; the circumferential riveting or class of seam is altogether ignored. The rule takes no account of workman- ship or method adopted of constructing the joints. The factor, one sixth, simply covers the actual nominal factor of safety as well as the loss of strength at the joint, no matter what its percentage ; we may therefore dismiss it as unsatisfactory. C(t + 1 )2 Rules for Flat Plates.— Board of Trade ; P= ^ _ ^ . P — working- pressure in lbs. per square inch; S = surface supported in square inches; t = thickness in sixteenths of an inch; C = a constant as per following table: C = 125 for plates not exposed to heat or flame, the stays fitted with nuts and washers, the latter at least three times the diameter of the stay and % the thickness of the plate; C = 187.5 for the same condition, but the washers % the pitch of stays in diameter, and thickness not less than plate; C — 200 for the same condition, but doubling plates in place of washers, the width of which is % the pitch and thickness the same as the plate; C = 112.5 for the same condition, but the stays with nuts only; C = 75 when exposed to impact of heat or flame and steam in contact with the plates, and the stays fitted with nuts and washers three times the diameter of the slay and % the plate's thickness; 702 THE STEAM-BOILER. C — 67.5 for the same condition, but stays fitted with nuts only; C — 100 when exposed to heat or flame, and water in contact with the plates, and stays screwed into the plates and fitted with nuts; C = 66 for the same condition, but stays with riveted heads. C X U. S. Statutes.— U sing same notation as for Board of Trade. P — — > where p = greatest pitch in inches, P and t as above; C — 112 for plates 7/16" thick and under, fitted with screw stay-bolts and nuts, or plain bolt fitted with single nut and socket, or riveted head and socket; C — 120 for plates above 7/16", under the same conditions; C — 140 for flat surfaces where the stays are fitted with nuts inside and outside; C = 200 for flat surfaces under the same condition, but with the addi- tion of a washer riveted to the plate at least J^ plate's thick- ness, and of a diameter equal to 2/5 pitch. N.B.— Plates fitted with double angle-irons and riveted to plate, with leaf at leasts the thickness of plate and depth at least J4 of pitch, would be allowed the same pressure as determined by formula fur plate with washer riveted on. N.B.— No brace or stay-bolt used in marine boilers to have a greater pitch than 10^" on fire-boxes and back connections. Certain experiments were carried out by the Board of Trade which showed that the resistance to bulging does not vary as the square of the plate's thickness. There seems also good reason to believe that it is not inversely as the square of the greatest pitch. Bearing in mind, says Mr. Foley, that mathematicians have signally failed to give us true theoretical foundations for calculating the resistance of bodies subject to the simplest forms of stresses, we therefore cannot expect much from their assistance in the matter of flat plates. The Board of Trade rules for flat surfaces, being based on actual experi- ment, are especially worthy of respect; sound judgment appears also to have been used in framing them. Furnace Formulae.— Board of Trade.— Long Furnaces.— P = -=— ■, -, but not where L is shorter than (1 1.5* — 1), at which length the rule for short furnaces comes into play. P = working-pressure in pounds per square inch; t — thickness in inches; D = outside diameter in inches; L — length of furnace in feet up to 10 ft.; C — a constant, as per following table, for drilled holes : C — 99,000 for welded or butt-jointed with single straps, double- riveted; C = 88,000 for butts with single straps, single-riveted; C = 99,000 for butts with double straps, single-riveted. Provided always that the pressure so found does not exceed that given by the following formulas, which apply also to short furnaces : _ OX*/. tXl2\ . ... '•"/. P — o w r, ( 5 — <^~7 j. ) when with Adamson rings. o X D\ 67.5 X */ C— 8,800 for plain furnaces; C = 14,000 for Fox; minimum thickness 5/16", greatest %"; plain part not to exceed 6" in length; C = 13,500 for Morison; minimum thickness 5/16", greatest %''; plain part not to exceed 6" in length; C = 14,000 for Purves-Brown ; limits of thickness 7/16" and %" ; plain part 9" in length; C — 8,800 for Adamson rings; radius of flange next fire 1%" '• U. S. Statutes.— Long Furnaces.— Same notation. 89 600 X * 2 P = — '— — , but L not to exceed 8 ft. IXD N.B.— If rings of wrought iron are fitted and riveted on properly arour.d and to the flue in such a manner that the tensile stress on the rivets shall STRENGTH OF STEAM-BOILERS. 703 not exceed 6000 lbs. per sq. in., the distance between the rings shall be taken as the length of the flue in the formulas. Short Furnaces, Plain and Patent.— P, as before, when not 8 ft. 89.600 X < 2 l ° ng = LXD > „ txo . P — — - — when C = 1-1,000 for Fox corrugations where D = mean diameter; C — 1-1,000 for Purves-Brown where D = diameter of flue; C ~ 5677 for plain flues over 16" diameter and less than 40", when not over 3 ft. lengths. Mr. Foley comments on the rules for long furnac s as follows: The Board of Trade general formula, where the length is a factor, has a very limited range indeed, viz., 10 ft. as the extreme length, and 135 thicknesses — 12", Cx t 2 as the short limit. The original formula, P — j -, is that of Sir W. Fairbairn, and was, I believe, never intended by him to apply to short fur- naces. On the very face of it, it is apparent, on the other hand, that if it is true for moderately long furnaces, it cannot be so for very long ones. We are therefore driven to the conclusion that any formula which includes simple L as a factor must be founded on a wrong basis. With Mr. Traill's form of the formula, namely, substituting (L + 1) for L, the results appear sufficiently satisfactory for practical purposes, and in- deed, as far as can be judged, tally with the results obtained from experi- ment as nearly as could be expected. The experiments to which I refer were six in number, and of great variety of length to diameter; the actual factors of safety ranged from 4.4 to 6.2, the mean being 4.78, or practically 5. It seems tome, therefore, that, within the limits prescribed, the Board of Trade formula may be accepted as suitable for our requirements. The United States Statutes give Fairbairn's rule pure and simple, except that the extreme limit of length to which it applies is fixed at 8 feet. As far as can be seen, no limit for the shortest length is prescribed, but the rules to me are by no means clear, flues and furnaces being mixed or not well distinguished. Material for Stays.— The qualities of material prescribed are as follows: Board of Trade.— The tensile strength to lie between the limits of 27 and 32 tons per square inch, and to have an elongation of not less than 20$ in 10". Steel stays which have been welded or worked in the fire should not be used. Lloyd's.— 26 to 30 ton steel, with elongation not less than 20% in 8". U. S. Statutes.— The only condition is that the reduction of area must not be less than 40% if the test bar is over %" diameter. Loads allowed on Stays.— Board of Trade.— 9000 lbs. per square inch is allowed on the net section, provided the tensile strength ranges from 27 to 32 tons. Steel stays are not to be welded or worked in the fire. Lloyd's.— For screwed and other stays, not exceeding 1J^" diameter effec- tive, 8000 lbs. per square inch is allowed; for stays above 1J^", 9000 lbs. No stays are to be welded. U. S. Statutes.— Braces and stays shall not be subjected to a greater stress than 6000 lbs. per square inch. [Raukine, S. E., p. 459, says: " The iron of the stays ought not to be ex- posed to a greater working tension than 3000 lbs. on the square inch, in order to provide against their being weakened by corrosion. This amounts to making the factor of safety for the working pressure about 20." It is evident, however, that an allowance in the factor of safety for corrosion may reasonably be decreased with increase of diameter. W. K.] C X d 2 X t I Girders.— Board of Trade. P == — rr- -, P = working pres- ( *y — P)ls X j-i sure in lbs. per sq. in.; W — width of flame-box in inches; L = length of girder in inches; p — pitch of bolts in inches; D = distance between girders from centre to centre in inches; d = depth of girder in inches; t = thick- ness of sum of same in inches; C — a constant = 6600 for 1 bolt, 9900 for 2 or 3 bolts, and 11,220 for 4 bolts. Lloyd's.— The same formula and constants, except that C = 11,000 for 4 or 5 bolt's, 11,550 for 6 or 7, and 11,880 for 8 or more. L 7 . S. Statutes.— The matter appears to he left to the designers, 704 THE STEAM-BOILER. t(D- Tube-riates.— Board of Trade. P - horizontal distance between centres of tubes in inches; d — inside diameter of ordinary tubes; t = thickness of tube-plate in inches; W — extreme width of combustion-box in inches from front tube-plate to back of fire- box, or distance between combustion-box tube plates when the boiler is double-ended and the box common to both ends. The crushing stress on tube-plates caused by the pressure on the flame- box top is to be limited to 10,000 lbs. per square inch. Material for Tubes.— Mr. Foley proposes the following: If iron, the ,' quality to be such as to give at least 22 tons per square inch as the minimum tensile strength, with an elongation of not less than 15$ in 8". If steel, the elongation to be not less than 26$ in 8" for the material before being rolled into strips; and after tempering, the test bar to stand completely closing together. Provided the steel welds well, there does not seem to be any ob- ject in providing tensile limits. The ends should be annealed after manufacture, and stay-tube ends should be annealed before screwing. Holding-power of Boiler-tubes.— Experiments made in Wash- ington Navy Yard show that wit h 2}/ 2 in . brass tubes in no case was the holding- power less, roughly speaking, than 6000 lbs., while the average was upwards of 20,000 lbs. It was further shown that with these tubes nuts were super- fluous, quite as good results being obtained with tubes simply expanded into the tube-plate and fitted with a ferrule. When nuts were fitted it was shown that they drew off without injuring the threads. In Messrs. Yarrow's experiments on iron and steel tubes of 2" to 2%" diameter the first 5 tubes gave way on an average of 23,740 lbs., which would appear to be about % the ultimate strength of the tubes themselves. In all these cases the hole through the tube-plate was parallel with a sharp edge to it, and a ferrule was driven into the tube. Tests of the next 5 tubes were made under the same conditions as the first 5, with the exception that in this case the ferrule was omitted, the tubes be- ing simply expanded into the plates. The mean pull required was 15,270 lbs., or considerably less than half the ultimate strength of the tubes. Effect of beading the tubes, the holes through the plate being parallel and ferrules omitted. The mean of the first 3, which are tubes of the same kind, gives 26,876 lbs. as their holding-power, under these conditions, as com- pared with 23,740 lbs. for the tubes fitted with ferrules only. This high figure is, however, mainly due to an exceptional case where'the holding- power is greater than the average strength of the tubes themselves. It is disadvantageous to cone the hole through the tube-plate unless its sharp edge is removed, as the results are much worse than those obtained, with parallel holes, the mean pull being but 16.031 lbs., the experiments be- ing made with tubes expanded and ferruled but not beaded over. In experiments on tubes expanded into tapered holes, beaded over and fitted with ferrules, the net result is that the holding-power is, for the size experimented on, about % of the tensile strength of the tube, the mean pull being 28,797 lbs. With tubes expanded into tapered holes and simply beaded over, better results were obtained than with ferrules; in these cases, however, the sharp edge of the hole was rounded off, which appears in general to have a good effect. In one particular the experiments are incomplete, as it is impossible to reproduce on a machine the racking the tubes get by the expansion of a boiler as it is heated up and cooled down again, and it is quite possible, therefore, that the fastening giving the best results on the testing-machine may not prove so efficient in practice. N.B.— It should be noted that the experiments were all made under the cold condition, so that reference should be made with caution, the circum- stances in practice being very different, especially when there is scale on the tube-plates, or when the tube-plates are thick and subject to intense heat. Iron versus Steel Boiler-tubes. (Foley.) — Mr. Blechynden prefers iron tubes to those of steel, but how far he would go in attributing the leaky-tube defect to the use of steel tubes we are not aware. It appears, however, that the results of his experiments would warrant him in going a considerable distance in this direction. The test consisted of heating and cooling two tubes, one of wrought iron and the other of steel. Both tubes were 2^4 in. in diameter and .16 in. thickness of metal. The tubes were Steel. Iron. 55.495 in. 55.495 in. 0.52 " 0.48 " .0000067 .0000062 .00? in. .003 in. .031 in. .004 in. .017 in. .006 in. .055 in. .013 in. STRENGTH OF STEAM-BOILERS. 705 put in the same furnace, made red-hot, and then dipped in water. The length was gauged at a temperature of 46° F. This operation was twice repeated, with results as follows : Steel. Original length 55.495 in. Heated to 186° F. ; increase Coefficient of expansion per degree F Heated red-hot and dipped in water; decrease Second heating and cooling, decrease Third heating and cooling, decrease Total contraction Mr. A. C. Kirk writes : That overheating of tube ends is the cause of the leakage of the tubes in boilers is proved by the fact that the ferrules at present used by the Admiralty prevent it. These act by shielding the tube ends from the action of the flame, and consequently reducing evaporation, and so allowing free access of the water to keep them cool. Although many causes contribute, there seems no doubt that thick tube- plates must bear a share of causing the mischief. Rules for Construction of Boilers in Merchant Vessels in tne United States. (Extracts from General Rules and Regulations of the Board of Supervising Inspectors of Steam-vessels (as amended 1893 and 1894).) Tensile Strength of Plate. (Section 3.)— To ascertain the tensile strength and other qualities of iron plate there shall be taken from each sheet to be used in shell or other parts of boiler which are subject to tensile strain a test piece prepared in form according to the following diagram, viz.: 10 inches in length, 2 inches in width, cut out in the centre in the manner indicated. To ascertain the tensile strength and other qualities of steel plate, there small be taken from each sheet to be used in shell or other parts of boiler which are subject to tensile strain, a test- piece prepared in form according to the following diagram, the length "T~| fs. /[ of straight part in centre varying as called for by different thickness of material, as follows: The straight portion shall be in length at least eight times the width multiplied by the thickness of said pa' t, and have a reduction of area as called for by the present rules of the Boaio.. and an elongation of at least 2".#. The stra'ght part shall be of a width of 1 inch. This rule to take effect on and after July 1, 1894. Provided, that where contracts for boilers for ocean-going steamers re- quire a test of material in compliance with the British Board of Trade, British Lloyd's, or Bureau Veritas rules for testing, the inspectors shall make the tests in compliance with the following rules: Steel plates shall in all cases to have an ultimate elongation not less than 20# in a length of 8 inches. It is to be capable of being bent to a curve of which the inner radius is not greater than one and a half times the thickness of the plates after having been heated uniformly to a low cherry-red, and quenched in water of 82° F. [Prior to 1894 the shape of test-piece for steel was the same as that for iron, viz., the grooved shape. This shape has been condemned by authorities on strength of materials for over twenty years. It always gives results which are too high, the error sometimes amounting to 25 per cent. See pages 242, 243, ante; also. Strength of Materials, W. Kent. Van N. Science Series No. 41, and Beardslee on Wrought-iron and Chain Cables.] Ductility. (Section 6.)— To ascertain the ductility and other lawful qualities, iron of 45.000 lbs. tensile strength shall show a contraction of area of 15 per cent, and each additional 1000 lbs. tensile strength shall show 1 per cent additional contraction of area, up to and including 55,000 tensile strength. Iron of 55,000 tensile strength and upwards, showing 25 per cent reduction of. area, shall be deemed to have the lawful ductility. All steel plate of % inch thickness and under shall show a contraction of area of not less than 50 per cent. Steel plate over y % inch in thickness, up to % inch in -6-inches- "-*T??4*j ° 'rp r^rJP* 1 6-iirches— jy xl 706 THE STEAM-BOILER. thickness, shall show a reduction of not less than 45 per cent. All steel plate over % inch thickness shall show a reduction of not less than 40 per cent. Bumped Heads of Boilers. (Section 17 as amended 1894.) — Pressure Allowed on Bumped Heads.— Multiply the thickness of the plate by one sixth of the tensile strength, and divide by six tenths of the radius to which head is bumped, which will give the pressure per square inch of steam allowed. Pressure Allowable for Concaved Heads of Boilers.— Multiply the pressure per square inch allowable for bumped heads attached to boilers or drums convexly, by the constant .6, and the product will give the pressure per square inch allowable in concaved heads. Tlie pressure on unstayed flat-heads on steam-drums or shells of boilers, when flanged and made of wrought iron or steel or of cast steel, shall be determined by the following rule: The thickness of plate in inches multiplied by one sixth of its tensile strength in pounds, which product divided by the area of the head in square inches multiplied by .09 will give pressure per square inch allowed. The material used in the construction of flat-heads when tensile strength has not been officially determined shall be deemed to have a tensile strength of 45,000 lbs. Table of Pressures allowable on Steam-boilers made of Riveted Iron or Steel Plates. (Abstract from a table published in Rules and Regulations of the U. S, Board of Supervising Inspectors of Steam-vessels.) Plates 14 i ncn thick. For other thicknesses, multiply by the ratio of the thickness to J4 inch. 50,000 Tensile 55,000 Tensile 60,000 Tensile 65,000 Tensile 70,000 Tensile O .£ Strength. Strength. Strength. Strength. Strength. |.s £ ■ "^ s , "^ s a ai . "3 33 ■ "3 3 •a 3 ts 9 •ag 3 ^•g 3 -ig ■So on GO I'' 3 <§ ©^ «j.2 36 115.74 138.88 127.31 152.77 138.88 166.65 150.46 180.55 162.03 191.43 38 109.64 131.56 120.61 144.73 131.57 157.88 142.54 171.04 153.5 184.20 40 104.16 124.99 114.58 137.49 125 150 135.41 162.49 145.83 174.99 42 99.2 119.04 109.12 130.94 119.04 142 81 128.96 154.75 138.88 166.65 44 94.69 113.62 104.16 124.99 113.63 136.35 123.1 147.72 132.56 159.07 46 90.57 108.68 99.63 119.55 108.69 130.42 117.75 141.3 126.8 153.16 48 86.8 104.16 95.48 114.57 104.16 124.99 112.84 135.4 121.52 145.82 54 77.16 92.59 84.87 101.84 92.59 111.10 100.3 120.36 108.02 129.62 60 69.44 83.32 76.38 91.65 83.33 99.99 90.27 108.32 97.22 116.66 66 63.13 75.75 69.44 83.32 75.75 90.90 8.2.07 98.48 88.37 106.04 72 57.87 69.44 63.65 76.38 69 44 83.32 75.22 90.26 81.01 97.21 78 53.41 64.09 58.76 70.5 64.4 76.92 69.44 83.32 74.78 89.73 84 49.6 59.52 54.56 65.47 59.52 71.42 64.48 77.37 69.44 83.32 90 46.29 55.44 50.92 61.1 55.55 66.66 60.18 72.21 64.81 77.77 96 43.4 52.08 47.74 57.28 52.08 62.49 56.42 67.67 60.76 72.91 The figures under the columns headed "pressure" are for single-riveted boilers. Those under the columns headed " 20$ Additional 1 ' are for double- riveted. U. S. Rule for Allowable Pressures. The pressure of any dimension of boilers not found in the table annexed to these rules must be ascertained by the following rule: Multiply one sixth of the lowest tensile strength found stamped on any plate in the cylindrical shell by the thickness (expressed in inches or parts of an inch) of the thinnest plate in the same cylindrical shell, and divide by the radius or half diameter (also expressed in inches), and the sum will be the pressure allowable per square inch of surface for single-riveting, to which add twenty per centum for double-riveting. The author desires to express his condemnation of the above rule, and of the tables derived from it, as giving too low a factor of safety. (See also criticism by Mr. Foley, page 701, ante.) STRENGTH OF STEAM-BOILERS. 70? If Pb = bursting-pressure, t = thickness, T — tensile strength, c = coef- ficient of strength of riveted joint, that is, ratio of strength of the joint to that of the solid plate, d = diameter, Pb = —p, or if c be taken for double- 1 4tT riveting at 0.7, then Pb — - L -r--. ■\/QtT 4tT By the U. S. rule the allowable pressure Pa = -jy-r X 1.20 = - 1 — - ; whence Pb = Z.bPa\ that is, the factor of safety is only 3.5, provided the "tensile strength found stamped in the plate 11 is the real tensile strength of the material. But in the case of iron plates, since the stamped T.S. is obtained from a grooved specimen, it may be greatly in excess of the real T.S., which would make the factor of safety still lower. According to the table, a boiler 40 in. diam., % in - thick, made of iron stamped 60,000 T.S., would be licensed to carry 150 lbs. pressure if double-riveted. If the real T.S. is only 50,000 lbs. the calculated bursting-strength would be p= 2JTC = L xa»mxMXM = 43M lbs _ and the factor of safety only 437.5 -=- 150 = 2.91 ! The author's formula for safe working-pressure of extern ally -fired boilers with longitudinal seams double-riveted, is P-— - — ; t = -—-;P = gauge- pressure in lbs. per sq. in. ; t = thickness and d — diam. in inches. This is derived from the formula P = -j— , taking c at 0.7 and / = 5 for steel of 50,000 lbs. T.S., or 6 for 60,000 lbs. T.S.; the factor of safety being increased* in the ratio of the T.S., since with the higher T.S. there is greater danger of cracking at the rivet-holes from the effect of punching and rivet- ing and of expansion and contraction caused by variations of temperature. For external shells of internally -fired boilers, these shells not being exposed to the fire, with rivet-holes drilled or reamed after punching, a lower factor of safety and steel of a higher T.S. may be allowable. If the T.S. is 60,000, a working pressure P = would give a factor of safety of 5.25. The following table gives safe working pressures for different diameters of shell and thicknesses of plate calculated from the author's formula. Safe Working Pressures in Cylindrical Shells of Boilers, Tanks, Pipes, etc., in Pounds per Square Inch. Longitudinal seams double'-riveted. (Calculated from formula P = 14,000 x thickness -*- diameter.) Diameter in Inches 50 1 — 1 24 36.5 30 29.2 36 38 40 42 44 46 48 50 52 l 24.3 23.0 21.9 20.8 19.9 19.0 18.2 17.5 16 8 2 72.9 58.3 48.6 46.1 43.8 41.7 39.8 38.0 36.5 25.0 33.7 3 109.4 87.5 72.9 69.1 65.6 62.5 59.7 57.1 54.7 52.5 50.5 4 145.8 116.7 97.2 92.1 87.5 83.3 79.5 76.1 72.9 70.0 67.3 5 182.3 145.8 121.5 115.1 109.4 104.2 99.4 95.1 91.1 87.5 84.1 6 218.7 175.0 145.8 138.2 131.3 125.0 119.3 114.1 109.4 105.0 101.0 7 255.2 204.1 170.1 161.2 153.1 145.9 139.2 133.2 127.6 122.5 117.8 8 291.7 233.3 194.4 184.2 175.0 166.7 159.1 152.2 145.8 140.0 134.6 9 328.1 262.5 218.8 207.2 196.9 187.5 179.0 171.2 164.1 157.5 151.4 10 364.6 291.7 243.1 230.3 218.8 208.3 198.9 190.2 182.3 175.0 168.3 11 401.0 320.8 267.4 253.3 240.6 229.2 218.7 209.2 200.5 192.5 185.1 12 437.5 350.0 291.7 276.3 262.5 250.0 238.6 228.3 218.7 210.0 201.9 13 473.9 379.2 316.0 299.3 284.4 270.9 258.5 247.3 337.0 227.5 218.8 14 410.4 408.3 340.3 322.4 306.3 291.7 278.4 266.3 255.2 245.0 235.6 15 546.9 437.5 364.6 345.4 328.1 312.5 298 3 285.3 273.4 266.5 252.4 16 583.3 466.7 388.9 368.4 350.0 333.3 318.2 304.4 291.7 280.0 269.2 708 THE STEAM-BOILER. Diameter in Inches. Eh.2 e8 54 60 66 72 78 84 10.4 90 96 102 108 114 7 I' 120 1 16.2 14.6 13.3 12.2 11.2 9.7 9.1 8.6 8 1 7 3 2 32.4 29.2 26.5 24.3 22.4 20.8 19.4 18.2 17.2 16.2 15 4 14 fi 3 48.6 43.7 39.8 36.5 33.7 31 .3 29.2 27 3 25 7 24 3 23 21.9 4 64.8 58.3 53.0 48.6 44.9 41.7 38.9 36 5 34 3 32.4 3(1 7 29.2 5 81.0 72.9 66.3 60.8 56.1 52.1 48.6 45.6 42 9 40 5 aft 4 36.5 6 97.2 87.5 79.5 72.9 67.3 62.5 58.3 54.7 51 5 48 6 46 1 43.8 7 113.4 102.1 92.8 85.1 78.5 72.9 C8.1 63.8 60.0 56 7 S3 , 7 51.0 8 129.6 116.7 106.1 97.2 89.7 83.3 77.8 72.9 68 6 64 8 61.4 58.3 9 145.8 131.2 119.3 109.4 101.0 93.8 87.5 82.0 77 2 72.9 69.1 65.6 10 162.0 145.8 132.6 121.5 112.2 104.2 97.2 91 1 85 8 81,0 76 8 72.9 11 178.2 160.4 145.8 133.7 123.4 114.6 106 9 100 3 94.4 89 1 84.4 80.2 12 194.4 175.0 159.1 145.8 134.6 125.0 116.7 109 4 102.9 97 2 92 1 87 5 13 210.7 189.6 172.4 158.0 145.8 135.4 126.4 118 5 111 5 105 3 99 8 94.8 14 226. 9 204.2 185.6 170.1 157.1 145.8 136.1 127 6 120 1 113 4 107 5 102.1 15 243.1 218.7 198.9 182.3 168.3 156.3 145.8 136.7 128 7 121 5 115.1 109 4 16 259.3 233.3 212.1 194.4 179.5 166.7 155.6 145.8 137.3 129.6 122.8 116.7 Rules governing Inspection of Boilers in Philadelphia. In estimating the strength of the longitudinal seams in the cylindrical shells of boilers the inspector shall apply two formulae, A and B : j Pitch of rivets — diameter of holes punched to receive the rivets ' ' pitch of rivets percentage of strength of the sheet at the seam. B, ( Area of hole filled by rivet x No. of rows of rivets in seam x shear- < ipg strength of rivet pitch of rivets X thickness of sheet x tensile strength of sheet percentage of strength of the rivets in the seam. Take the lowest of the percentages as found by formulae A and B and apply that percentage as the " strength of the seam " in the following formula C, which determines the strength of the longitudinal seams: ( Thickness of sheet in parts of inch X strength of seam as obtained p -< by formula A or B X ultimate strength of iron stamped on plates _ ' internal radius of boiler in inches X 5 as a factor of safety safe working pressure. Table of Proportions and Safe Working Pressures with Formula A AND C, @ 50,000 LBS., T.S. Diameter of rivet Diameter of rivet-hole. Pitch of rivets Strength of seam, %.. .. Thickness of plate. . . . 11/16" 2" .656 Y4," 11/16 2 1/16 13/16 2 3/16 .60 7/16 15/16 .58 Diameter of boiler, in. Safe Working Pressure with Longitudinal £ Single-riveted. 24 137 165 193 220 242 30 109 132 154 176 194 32 102 124 144 165 182 34 96 117 136 155 171 36 91 110 129 147 161 38 86 104 122 139 153 40 82 99 116 132 145 44 74 91 105 120 132 48 68 83 96 110 121 54 60 73 86 98 107 60 55 66 77 88 97 STRENGTH OF STEAM-BOILERS. 709 Diameter of rivet %" 11/16 % 13/16 % Diameter of rivet-hole. . . 11/16" u 13/16 % 15/16 3" .76 .75 m .74 3^o .73 Strength of seam, % Thickness of plate ji" 5/16 % 7/16 X Safe Working Pressure with Longitudinal Seams, Double- riveted. 24 160 198 235 269 305 30 127 158 188 215 243 32 119 148 176 202 228 34 112 140 166 190 215 36 106 132 156 179 203 . 38 101 125 148 170 192 40 96 119 141 161 183 44 87 108 128 147 166 48 79 99 118 135 152 54 70 88 104 120 135 60 64 79 94 108 122 Flues and. Tubes for Steam-boilers.— (From Rules of U. S- Supervising Inspectors. Steam-pressures per square inch allowable on riveted and lap-welded flues made in sections. Extract from table in Rules of U. S. Supervising Inspectors.) T = least thickness of material allowable, D = greatest diameter in inches, P — allowable pressure. For thickness greater than T with same diameter P is increased in the ratio of the thickness. D=in. 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 T=h\. .18 .20.21 .21 .22 .22 .23 .24 .25 .26 .27.28.29 30.31 .32 .33 P = lbs. 189 184 179 174 172 158 152 147 143 139 136 134 131129 126 125 122 D = in. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 T = in. .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .50 P= lbs. 121 120 119 117 116 115 115 114 112 112 110 110 109 109 108 108 107 For diameters not over 10 inches the greatest length of section allowable is 5 feet; for diameters 10 to 23 inches, 3 feet; for diameters 23 to 40 inches, 30 inches. If lengths of sections are greater than these lengths, the allowable pressure is reduced proportionately. TheU. S. rule for corrugated flues, as amended in 1894, is as follows: Rule II, Section 14. The strength of all corrugated flues, when used for furnaces or steam chimneys (corrugation not less than \% inches deep and notexceed- ing 8 inches from centres of corrugation), and provided that the plain parts at the ends do not exceed 6 inches in length, and the plates are not less than 5/16 inch thick, when new, corrugated, and practically true circles, to be calculated from the following formula: 14,000 XT— pressure. T — thickness, in inches; D — mean diameter in inches. Ribbed Flues. — The same formula is given for ribbed flues, with rib projections not less than 1% inches deep and not more than 9 inches apart. Flat Stayed Surfaces in Steam-boilers.— Rule II., Section 6, of the rules of the U. S. Supervising Inspectors provides as follows: No braces or stays hereafter employed in the construction of boilers shall be allowed a greater strain than 6000 lbs. per square inch of section. Clark, in his treatise on the Steam-engine, also in his Pocket-book, gives the following formula: p = 40?fo -=- d, in which p is the internal pressure in pounds per square inch that will strain the plates to their elastic limit, t is the thickness of the plate in inches, d is the distance between two rows of stay-bolts in the clear, aud s is the tensile stress in the plate in tons of 2240 lbs. per square inch, at the elastic limit. Substituting values of s for iron, steel, and copper, 12, 14, and 8 tons respectively, we have the following : 710 THE STEAM-BOILER. FORMULAE FOR ULTIMATE ELASTIC STRENGTH OF FLAT STAYED SURFACES. Iron. Steel. Copper. p = 5000-, d . pXd 5000 5000* ~ p p= 5700 j . p X d 1 ~ 5700 d _ 5700 t P p = 3300-* PXd ~ 3300 3300f Thickness of plate Pitch of bolts P For Diameter of the Stay-bolts, Clark gives d' = .0024i, ' PP'p in which d' = diameter of screwed bolt at bottom of thread, P = longitudi- nal and P' transverse pitch of stay-bolts between centres, p = internal pressure in lbs. per sq. in. that will strain the plate to its elastic limit, s = elastic strength of the stay-bolts in lbs. per sq. j n . Taking s = 12, 14, and 8 tons, respectively for iron, steel, and copper, we have For iron, d' = .00069 ^PP'^'or if P = P, d' = .00069P Vp; For steel, d' = .00064 VPP'p , " " d' = .00064 P Vp; For copper, d' = .00084 VPP'p, " " d' = .00084P Vp. In using these formulas a large factor of safety should be taken to allow for i-eduction of size by corrosion. Thurston's Manual of Steam-boilers, p. 144, recommends that the factor be as large as 15 or 20. The Hartford Steam Boiler Insp. & Ins. Co. recommends not less than 10. Strength of Stays.— A. F. Yarrow (Engr., March 20, 1891) gives the following results of experiments to ascertain the strength of water-space stays : Description. Length between Plates. Diameter of Stay over Threads. Ulti- mate Stress. Hollow stays screwed into j plates and hole expanded ( Solid stays screwed into j plates and riveted over. ) 1 in. (hole 7/16 in. and 5/16 in. lin.(hole 9/16 in. and 7/16 in. lbs. 25,457 20,992 22,008 22,070 The above are taken as a fair average of numerous tests. Stay-bolts in Curved Surfaces, as In Water-legs of Verti- cal Boilers.— The rules of the U. S. Supervising Inspectors provide as follows: All vertical boiler-furnaces constructed of wrought iron or steel plates, and having a diameter of over 42 in. or a height of over 40 in. shall be stayed with bolts as provided by § 6 of Rule II, for flat surfaces; and the thickness of material required for the shells of such furnaces shall be de- termined by the distance between the centres of the stay-bolts in the fur- nace and not in the shell of the boiler; and the steam-pressure allowable shall be determined by the distance from centre of stay-bolts in the furnace and the diameter of such stay-bolts at the bottom of the thread. The Hartford Steam-boiler Insp. & Ins. Co. approves the above rule {The Locomotive, March, 1892) as far as it states that curved surfaces are to be computed the same as flat ones, but prefers Clark's formulae for flat stayed surfaces to the rules of the U. S. Supervising Inspectors. Fusible-plugs.— Fusible-plugs should be put in that portion of the heating-surface which first becomes exposed from lack of water. The rules of the U. S. Supervising Inspectors specify Banca tin for the purpose. Its melting-point is about 445° F. The rule says: All steamers shall have inserted in their boilers plugs of Banca tin, at least \i> in. in diameter at the smallest end of the internal opening, in the following manner, to wit: Cylinder- boilers with flues shall have one plug inserted in one flue of each boiler; and also one plug inserted in the shell of each boiler from the inside, immediately before the fire line and not less than 4 ft. from the forward end of the boiler. All fire-box boilers shall have one plug inserted in the crown of the back connection, or in the highest fire-surface of the boiler. IMPROVED METHODS OF FEEDING COAL. 711 All upright tubular boilers used for marine purposes shall have a fusible plug inserted in one of the tubes at a point at least 2 in. below the lower gauge-cock, and said plug may be placed in the upper head sheet when deemed advisable by theHocal inspectors. Steam-domes.— Steam domes or drums were formerly almost univer- sally used on horizontal boilers, but their use is now generally discontinued, as they are considered a useless appendage to a steam-boiler, and unless properly designed and constructed are an element of weakness. Height of Furnace.— Recent practice in the United States makes the height of furnace much greater than it was formerly. With large sizes of anthracite there is no serious objection to having the furnace as low as 12 to 18 in., measured from the surface of the grate to the nearest portion of the heating surface of the boiler, but with coal containing much volatile matter and moisture a much greater distance is desirable. With very vola- tile coals the distance may be as great as 4 or 5 ft. Rankine (S. E., p. 457) says: The clear height of the " crown " or roof of the furnace above the grate- bars is seldom less than about 18 in., and often considerably more. In the fire-boxes of locomotives it is on an average about 4 ft. The height of 18 in. is suitable where the crown of the furnace is a brick arch. Where the crown of the furnace, on the other hand, forms part of the heating-surface of the- boiler, a greater height is desirable in every case in which it can be obtained; for the temperature of the boiler-plates, being much lower than that of the flame, tends to check the combustion of the inflammable gases which rise from the fuel. Asa general principle a high furnace is favorable to complete combustion. IMPROVED METHODS OF FEEDING COAIi, Mechanical Stokers. (William R. Roney, Trans. A. S. M. E., vol. xii.)— Mechanical stokers have been used in England to a limited extent since 1785. In that year one was patented by James Watt. It was a simple device to push the coal, after it was coked at the front end of the grate, back towards the bridge. It was worked intermittently by levers, and was designed primarily to prevent smoke from bituminous coal. (See D. K. Clark's Treatise on the Steam-engine.) After the year 1840 many styles of mechanical stokers w r ere patented in England, but nearly all were variations and modifications of the two forms of stokers patented by John Jukes in 1841, and by E. Henderson in 1843. The Jukes stoker consisted of longitudinal fire-bars, connected by links, so as to form an endless chain, similar to the familiar treadmill horse-power. The small coal was delivered from a hopper on the front of the boiler, on to the grate, which slowly moving from front to rear, gradually advanced the fuel into the furnace and discharged the ash and clinker at the back. The Henderson stoker consists primarily of two horizontal fans revolving on vertical spindles, which scatter the coal over the fire. Numerous faults in mechanical construction and in operation have limited the use of these and other mechanical stokers. The first American stoker was the Murphy stoker, brought out in 1878. It consists of tw r o coal maga- zines placed in the side walls of the boiler furnace, and extending back from the boiler front 6 or 7 feet. In the bottom of these magazines are rectangu- lar iron boxes, which are moved from side to side by means of a rack and pinion, and serve to push the coal upon the grates, which incline at an angle of about 35° from the inner edge of the coal magazines, forming a V-shaped receptacle for the burning coal. The grates are composed of narrow parallel bars, so arranged that each alternate bar lifts about an inch at the lower end, while at the bottom of the V, and filling the space between the ends of the grate-bars, is placed a cast-iron toothed bar, arranged to be turned by a crank. The purpose of this bar is to grind the clinker coming in contact with it. Over this V-shaped receptacle is sprung a fire-brick arch. In the Roney mechanical stoker the fuel to be burned is dumped into a hopper on the boiler front. Set in the lower part of the hopper is a "pusher" to which is attached the u feed-plate " forming the bottom of the hopper. The " pusher, 1 ' by a vibratory motion, carrying with it the "feed-plate," gradually forces the fuel over the " dead-plate " and on the grate. The grate-bars, in their normal condition form a series of steps, to the top step of which coal is fed from the " dead-plate." Each bar rests in a concave seat in the bearer, and is capable of a rocking motion through an adjustable angle. All the grate-bars are coupled together by a "rocker- bar." A vari- able bagk-and-forth motion being given to the «' rocker-bar," through a corj- 712 ! THE STEAM-BOILEK. \ necting-rod, the grate-bars rock in unison, now forming a series of steps, and now approximating to an inclined plane, with the grates partly over- lapping, like shingles on a roof. When the grate-bars rock forward the fire will tend to work down in a body. But before the coal can move too far the bars rock back to the stepped position, checking the downward motion, breaking up the cake over the whole surface, and admitting a free volume of air through the fire. The rocking motion is slow, being from 7 to 10, strokes per minute, according to the kind of coal. This alternate starting j and checking motion is continuous, and finally lands the cinder and ash on ' the dumping-grate below. Mr. Roney gives the following record of six tests to determine the com- parative economy of the Roney mechanical stoker and hand-firing on return tubular boilers. 60 inches X 20 feet, burning Cumberland coal with natural draught. Rating of boiler at 12.5 square feet, 105 H. P. Three tests, hand-firing. Three tests, Stoker. E 3Sora^ r aS" C !bs r ^ >«•« >«■« »■«» "•» *» !»« H.P. developed above rating, % 5.8 13.5 68 54.6 66.7 84.3 Results of comparative tests like the above should be used with caution in drawing generalizations. It by no means follows from these results that a stoker will always show such comparative excellence, for in this case the results of hand-firing are much below what may be obtained under favor- able circumstances from hand-firing with good Cumberland coal. Tlie Mawley Down-draught Furnace.— A foot or more above the ordinary grate there is carried a second grate composed of a series of water-tubes, opening at both ends into steel drums or headers, through which water is circulated. The coal is fed on this second grate, and as it is par- tially consumed falls through it upon the lower grate, where the combustion is completed in the ordinary manner. The draught through the coal on the upper grate is downward through the coal and the grate. The volatile gases are therefore carried down through the bed of coal, where they are thor- oughly heated, and are burned in the space beneath, where they meet the excess of hot air drawn through the fire on the lower grate. In tests in Chicago, from 30 to 45 lbs. of coal were burned per square foot of grate upon this system, with good economical results. (See catalogue of the Hawley Down Draught Furnace Co., Chicago, 1894.) Under-feed. Stokers.— Results similar to those that may be obtained with downward draught are obtained by feeding the coal at the bottom of the bed. pushing upward the coal already on the bed which has had its volatile matter distilled from it. The volatile matter of the freshly fired coal then has to pass through a body of ignited coke. (See circular of the Jones Un- der-feed Stoker, Fraser & Chalmers, Chicago, 1894.) SMOKE PREVENTION. A committee of experts was appointed in St. Louis in 1891 to report on the smoke problem. A summary of its report is given in the Iron Age of April 7, 1892. It describes the different means that have been tried to prevent smoke, such as gas-fuel, steam-jets, fire-brick arches and checker-work, hollow walls for preheating air, coking arches or chambers, double combus- tion furnaces, and automatic stokers. All of these means have been more or less effective in diminishing smoke, their effectiveness depending largely upon the skill with which they are operated ; but none is entirely satisfac- tory. Fuel-gas is objectionable chiefly on account of its expense. The average quality of fuel-gas made from a trial run of several car-loads of Illinois coal, in a well-designed fuel-gas plant, showed a calorific value of 243,391 heat-units per 1000 cubic feet. This is equivalent to 5052.8 heat-units per lb. of coal, whereas by direct calorimeter test an average sample of the coal gave 11,172 heat-units. One lb. of the coal showed a theoretical evap- oration of 11.56 lbs. water, while the gas from 1 lb. showed a theoretical evaporation of 5.23 lbs. 48.17 lbs. of coal were required to furnish 1000 cubic feet of the gas. In 39 tests the smoke-preventing furnaces showed only 74% of the capacity of the common furnaces, reduced the work of the boilers 28$, and required about 2% more fuel to do the same work. In one case with steam-jets the fuel consumption was increased 12% for the same work. Prof. O. H. Landreth, in a report to the State Board of Health of Tennes- see {Engineering News, June 8, 1S93), writes as follows on the subject of smoke prevention; SMOKE PREVENTION. 713 As pertaius to steam-boilers, the object must be attained by one or more of the following: agencies : 1. Proper design and setting of the boiler-plant. This implies proper grate area, sufficient draught, the necessary air-space between grate-oars and through furnace, and ample combustion-room under boilers. 2. That system of firing that is best adapted to each particular furnace to secure the perfect combustion of bituminous coal. This may be either: (a) ••coke-firing," or charging all coal into the front of the furnace until par- tially coked, then pushing back and spreading; or (b) "alternate side-fir- ing"; or (c) "spreading," by which the coal is spread over the whole grate area in thin, uniform layers at each charging. 3. The admission of air through the furnace-door, bridge- wall, or side walls. 4. Steam-jets and other artificial means for thoroughly mixingjthe air and combustible gases. 5. Preventing the cooling of the furnace and boilers by the inrush of cold air when the furnace-doors are opened for charging coal and handling the fire. 6. Establishing a gradation of the several steps of combustion so that the coal may be charged, dried, and warmed at the coolest part of the furnace, and then moved by successive steps to the hottest place, where the final combustion of the coked coal is completed, and compelling the distilled combustible gases to pass through this hottest part of the fire. 7. Preventing the cooling by radiation of the unburned combustible gases until perfect mixing and combustion have been accomplished. 8. Varying the supply of air to suit the periodic variation in demand. 9. The substitution of a continuous uniform feeding of coal instead of intermittent charging. 10. Down-draught burning or causing the air to enter above the grate and pass down through the coal, carrying the distilled products down to the high temperature plane at the bottom of the fire. The number of smoke-prevention devices which have been invented is legion. A brief classification is : (a) Mechanical stokers. They effect a material saving in the labor of firing, and are efficient smoke-preventers when not pushed above their capacity, and when the coal does not cake badly. They are rarely suscepti- ble to the sudden changes in the rate of firing frequently demanded in service. (b) Air-flues in side walls, bridge-wall, and grate-bars, through which air when passing is heated. The results are always beneficial, but the flues aie difficult to keep clean and in order. (c) Coking arches, or spaces in front of the furnace arched over, in which the fresh coal is coked, both to prevent cooling of the distilled gases, and to force them to pass through the hottest part of the furnace just beyond the arch. The results are good for normal conditions, but ineffective when the fires are forced. The arches also are easily burned out and injured by working the fire. (d) Dead-plates, or a portion of the grate next the furnace-doors, reserved for warming and coking the coal before it is spread over the grate. These give good results when the furnace is not forced above its normal capacity. This embodies the method of "coke-firing" mentioned before. (e) Down-draught furnaces, or furnaces in which the air is supplied to the coal above the grate, and the products of combustion are taken away from beneath the grate, thus causing a downward draught through the coal, carry- ing the distilled gases down to the highly heated incandescent coal at the bottom of the layer of coal on the grate. This is the most perfect manner of producing combustion, and is absolutely smokeless. (/) Steam- jets to draw air in or inject air into the furnace above the grate, and also to mix the air and the combustible gases together. A very efficient smoke-preventer, but one liable to be wasteful of fuel by inducing too rapid a draught. (g) Baffle-plates placed in the furnace above the fire to aid in mixing the combustible gases with the air. (h) Double furnaces, of which there are two different styles; the first of which places the second grate below the first grate; the coal is coked on the first grate, during which process the distilled gases are made to pass over the second grate, where they are ignited and burned ; the coke from the first grate is dropped onto the second grate: a very efficient and economical smoke-preventer, but rather complicated to construct and maintain. In the second form the products of combustion from the first furnace pass through ^14 THE STEAM-BOiLER\ the grate and fire of the second, each fttrnace being charged with fresh fuel when needed, the latter generally with a smokeless coal or coke : an irra- tional and unpromising method. Mr. C. F. White, Consulting Engineer to the Chicago Society for the Pre- vention of Smoke, writes under date of May 4, 1893 : The experience had in Chicago has shown plainly that it is perfectly easy to equip steam-boilers with furnaces which shall burn ordinary soft coal iu such a manner that the making of smoke dense enough to obstruct the vision shall be confined to one or two intervals of perhaps a couple of minutes' duration in the ordinary day of 10 hours. Gas-fired Steam-boilers,— Converting coal into gas in a separate producer, before burning it under the steam-boiler, is an ideal method of smoke-prevention, but its expense has hitherto prevented its general intro- duction. A series of articles on the subject, illustrating a great number of devices, by F. J. Rowan, is published in the Colliery Engineer, 1889-90. See also Clark on the Steam-engine. FORCED COMBUSTION IN STEAM-BOILERS. For the purpose of increasing the amount of steam that can be generated by a boiler of a given size, forced draught is of great importance. It is universally used in the locomotive, the draught being obtained by a steam- jet in the smoke-stack. It is now largely used in ocean steamers, especially in ships of war, and to a small extent in stationary boilers. Economy of fuel is generally not attained by its use, its advantages being confined to the securing of increased capacity from a boiler of a given bulk, weight, or cost. The subject of forced draught is well treated in a paper by James Howden, entitled, "Forced Combustion in Steam-boilers" (Section G, Engineering Congress at Chicago, in 1893), from which we abstract the following: Edwin A. Stevens at Bordentown, N. J., in 1827, in the steamer "North America," fitted the boilers with closed ash-pits, into which the air of com- bustion was forced by a fan. In 1828 Ericsson fitted in a similar manner the steamer "Victory," commanded by Sir John Ross. Messrs. E. A. and R. L. Stevens continued the use of forced draught for a considerable period, during which they tried three different modes of using the fan for promoting combustion: 1, blowing direct into a closed ash-pit; 2, exhausting the base of the funnel by the suction of the fan; 3, forcing air into an air-tight boiler-room or stoke-hold. Each of these three methods was attended with serious difficulties. In the use of the closed ash-pit the blast-pressure would frequently force the gases of combustion, in the shape of a serrated flame, from thie joint around the furnace doors in so great a quantity as to affect both the effi- ciency and health of the firemen. The chief defect of the second plan was the great size of the fan required to produce the necessary exhaustion. The size of fan required grows in a rapidly increasing ratio as the combustion increases, both on account of the greater air-supply and the higher exit temperature enlarging the volume of the waste gases. The third method, that of forcing cold air by the fan into an air-tight boiler-room— the present closed stoke-hold system— though it overcame the difficulties in working belonging to the two forms first tried, has serious defects of its own, as it cannot be worked, even with modern high-class boiler-construction, much, if at all, above the power of a good chimney draught, in most boilers, without damaging them. In 1875 John I. Thorn ycroft & Co., of London, began the construction of torpedo-boats with boilers of the locomotive type, in which a high rate of combustion was attained by means of the air-tight boiler-room, into which air was forced by -means of a fan. In 1882 H.B.M. ships "Satellite" and "Conqueror" were fitted with this system, the former being a small ship of 1500 I.H.P., and the latter an iron- clad of 4500 I.H.P. On the trials with forced draught, which lasted from two to three hours each, the highest rates of combustion gave 16.9 I.H.P. per square foot of fire-grate in the " Satellite," and 13.41 I.H.P. in the " Con- queror." None of the short trials at these rates of combustion were made without injury to the seams and tubes of the boilers, but the system was adopted, and it has been contiuued in the British Navy to this day (1893). In Mr. Howden's opinion no advantage arising from increased combustion over natural-draught rates is derived from using forced draught in a closed ash-pit sufficient to compensate the disadvantages arising from difficulties FUEL ECONOMIZERS. 715 in working, there being either excessive smoke from bituminous coal or reduced evaporative economy. In 1880 Mr. Howden designed an arrangement intended to overcome the defects of both the closed ash-pit and closed stokehold systems. An air-tight reservoir or chamber is placed on the front end of the boiler and surrounding the furnaces. This reservoir, which projects from 8 to 10 inches from the end of the boiler, receives the air under pressure, which is passed by the valves into the ash-pits and over the fires in proportions suited to 'the kind of fuel used and the rate of combustion required. The air nsed above the fires is admitted to a space between the outer and inner furnace-doors, the inner having perforations and an air-distributing box through which the air passes under pressure. By means of the balance of air-pressure above and below the fires all tendency for the fire to blow out at the furnace-door is removed. By regulating the admission of the air by the valves above and below the fires, the highest rate of combustion possible by the air-pressure used can be effected, and in same manner the rate of combustion can be reduced to far below that of natural draught, while complete and economical combus- tion at all rates is secured. A feature of the system is the combination of the heating of the air of combustion by the waste gases with the controlled and regulated admission of air to the furnaces. This arrangement is effected most conveniently by passing the hot fire- gases after they leave the boiler through stacks of vertical tubes enclosed in the uptake, their lower ends being immediately above the smoke-box doors. Installations on Howden's system have hitherto been arranged for a rate of combustion to give at full sea-power an average of from 18 to 22 I.H.P. per square foot of fire-grate with fire-bars from 5' 0" to 5' 6" in length. It is believed that with suitable arrangement of proportions even SO I.H.P. per square foot can be obtained. For an account of recent uses of exhaust-fans for increasing draught, see paper by W. R. Roney, Trans. A. S. M. E., vol. xv. FUEL ECONOMIZERS. Green's Fuel Economizer.— Clark gives the following average re- sults of comparative trials of three boilers at Wigan used with and without economizers : Without With Economizers. Economizers. Coal per square foot of grate per hour 21.6 21.4 Water at 100° evaporated per hour 73 . 55 79 . 32 Water at 212° per pound of coal 9.60 10.56 Showing that in burning equal quantities of coal per hour the rapidity of evaporation is increased 9.3$ and the efficiency of evaporation 10$ by the addition of the economizer. The average temperatures of the gases and of the feed-water before and after passing the economizer were as follows: With 6-f t. grate. With 4-ft. grate. Before. After. Before. After. Average temperature of gases 649 340 501 312 Average temperature of feed- water. 47 157 41 137 - Taking averages of the two grates, to raise the temperature of the feed- water 100° the gases were cooled down 250°. Performance of a Green Economizer with a Smoky Coal. —The action of Green's Economizer was tested by M. W. Grosseteste for a period of three weeks. The apparatus consists of four ranges of vertical pipes, Q}4 feet high, 3% inches in diameter outside, nine pipes in each range, connected at top and bottom by horizontal pipes. The water enters all the tubes from below, and leaves them from above. The system of pipes is en- veloped in a brick casing, into which the gaseous products of combustion are introduced from above, and which they leave from below. The pipes are cleared of soot externally by automatic scrapers. The capacity for water is 24 cubic feet, and the total external heating-surface is 290 square feet. The apparatus is placed in connection with a boiler having 355 square feet of surface. This apparatus had been at work for seven weeks continuously without having been cleaned, and had accumulated a J^-inch coating of soot and 716 THE STEAM-BOILER. ash, when its performance, in the same condition, was observed for one week. During the second week it was cleaned twice every day; but during the third week, after having been cleaned on Monday morning, it was worked continuously without further cleaning. A smoke-making coal was used. The consumption was maintained sensibly constant from day to day. Green's Economizer.— Results of Experiments on its Efficiency as AFFECTED BY THE STATE OF THE SURFACE. (W, GrOSSeteSte.) Temperature of Feed-, water. Temperature of Gas- eous Products. Time (February and March). Enter- ing Feed- heater. Leav- ing Feed- heater. Differ- ence. Enter- ing Feed- heater Leav- ing Feed- heater. Differ- ence. 1st Week 2d Week Fahr. 73.5° 77.0 73.4 73.4 79.0 80.6 80.6 79.0 Fahr. 161.5° 230 196.0 181.4 178.0 170.6 169.0 172.4 Fahr. 88.0° 153.0 122.6 108.0 99.0 90.0 88.4 93.4 Fahr. 849° 882 831 871 952 889 901 Fahr. 261° 297 284 309 329 338 351 Fahr. 588° 585 3d Week— Monday Tuesday Wednesday Thursday Friday Saturday 547 562 623 551 550 Coal consumed per hour Water evaporated from 32° F. per hour Water per pound of coal 1st Week. 2d Week. 3d Week. . 214 lbs. 216 lbs. 213 lbs. . 1424 1525 1428 . 6.65 7.06 6.70 It is apparent that there is a great advantage in cleaning the pipes daily —the elevation of temperature having been increased by it from 88° to 153°. In the third week, without cleaning, the elevation of temperature relapsed in three days to the level of the first week; even on the first day it was quickly reduced by as much as half the extent of relapse. By cleaning the pipes daily an increased elevation of temperature of 65° F., was obtained, whilst a gain of 6% was effected in the evaporative efficiency. INCRUSTATION AND CORROSION. Incrustation and Scale.— Incrustation (as distinguished from mere sediments due to dirty water, which are easily blown out, or gathered up, by means of sediment-collectors) is due to the presence of salts in the feed-water (carbonates and sulphates of lime and magnesia for the most part), which are precipitated when the water is heated, and form hard de- posits upon the boiler-plates. (See Impurities in Water, p. 551, ante.) Where the quantity of these salts is not very large (12 grains per gallon, say) scale preventives may be found effective. The chemical preventives either form with the salts other salts soluble in hot water; or precipitate them in the form of soft mud, which does not adhere to the plates, and can be washed out from time to time. The selection of the chemical must de- pend upon the composition of the water, and it should be introduced regu- larly with the feed. Examples.— Sulphate-of -lime scale prevented by carbonate of soda: The sulphate of soda produced is soluble in water; and the carbonate of lime falls down in grains, does not adhere to the plates, and may therefore be blown out or gathered into sediment- collectors. The chemical reaction is: Sulphate of lime H CaSOj Carbonate of soda = Sulphate of soda -f- Carbonate of lime NA 9 COa NA 2 S0 4 CaC0 3 Sodium phosphate will decompose the sulphates of lime and magnesia: Sulphate of lime -4- Sodium phosphate = Calcium phos. + Sulphate of soda. CaS0 4 Na 2 HP0 4 CaHP0 4 Na 2 S0 4 Sul. of magnesia 4- Sodium phosphate = Phosphate of magnesia + Sul. of soda. MgS0 4 Na 2 HP0 4 MgHP0 4 Na 2 S0 4 INCRUSTATION" AND CORROSION. 717 Where the quantity of salts is large, scale preventives are not of much use. Some other source of supply must be sought, or the bad water purified before it is allowed to enter the boilers. The damage done to boilers by un- suitable water is enormous. Pure water may be obtained by collecting rain, or condensing steam by means of surface condensers. The water thus obtained should be mixed with a little bad water, or treated with a little alkali, as undiluted, pure water corrodes iron ; or, after each periodic cleaning, the bad may be used for a day or two to put a skin upon the plates. Carbonate of lime and magnesia may be precipitated either by heating the water or by mixing milk of lime (Porter Clark process) with it, the water being then filtered. Corrosion may be produced by the use of pure water, or by the presence of acids in the water, caused perhaps in the engine-cylinder by the action of high- pressure steam upon the grease, resulting in the production of fatty acids. Acid water may be neutralized by the addition of lime. Amount of Sediment which may collect in a 100-H.P. steam-boiler, evaporating 3000 lbs. of water per hour, the water containing different amounts of impurity in solution, provided that no water is blown off: Grains of solid impurities per gallon: 5 10 20 30 40 50 60 70 80 90 100 Equivalent parts per 100,000: 8.57 17.14 34.28 51.42 68.56 85.71 102.85 120 137.1 154.3 171.4 Sediment deposited in 1 hour, pounds: 2.57 5.14 10.28 15.42 20.56 25.71 30.85 36 41.1 46.3 51.4 In one day of 10 hours, pounds: 25.7 51.4 102.8 154.2 205.6 257.1 308.5 360 411 463 514 In one week of 6 days, pounds: 154.3 3u8.5 617.0 925.5 1234 1543 1851 2160 2468 2776 3085 If a 100-H.P. boiler has 1200 sq. ft. heating-surface, one week's running without blowing off, wish water containing 100 grains of solid matter per gallon in solution, would make a scale nearly 1/5 in. thick, if evenly depos- ited all over the heating-surface, assuming the scale to have a sp. gr. of 2.5 = 156 lbs. percu. ft.; 1/5 x 1200 X 156 X 1/12 - 3120 lbs. fSoiler-scale Compounds.— The Bavarian Steam-boiler Inspection Assn. in 1885 reported as follows: Generally the unusual substances in water can be retained in soluble form or precipitated as mud by adding caustic soda or lime. This is especially desirable when the boilers have small interior spaces. It is necessary to have a chemical analysis of the water in order to fully determine the kind and quantity of the preparation to be used for the above purpose. All secret compounds for removing boiler-scale should be avoided. (A list of 27 such compounds manufactured and sold by German firms is then given which have been analyzed by the association.) Such secret preparations are either nonsensical or fraudulent, or contain either one of the two substances recommended by the association for re- moving scale, generally soda, which is colored to conceal its presence, and sometimes adulterated with useless or even injurious matter. These additions as well as giving the compound some strange, fanciful name, are meant simply to deceive the boiler owner and conceal from him the fact that he is buying colored soda or similar substances, for which he is paying an exorbitant price. The Chicago, Milwaukee & St. P. R. R. uses for the prevention of scale in locomotive-boilers an alkaline compound consisting of 3750 gals, of water, 2G00 lbs. of 70£ caustic soda, and 1600 lbs. of 58$ soda-ash. Between Milwau- kee and Madison the water-supply contains from 1 to 4]4 lbs. of incrusting solids per 1000 gals., principally calcium carbonate and sulphate and mag- nesium sulphate. The amount of compound necessary to prevent the in- crustation is V& to 7 pints per 1000 gals, of water. This is really only one fourth of the quantity needed for chemical combination, but the action of the compound is regenerative. The soda-ash (sodium carbonate) extracts carbonic acid from the carbonates of lime and magnesia and precipitates them in a granular form. The bicarbonate of soda thus formed, however, loses its carbonic acid by the heat, and is again changed to the active car- bonate form. Theoretically this action might continue indefinitely; but on 718 THE STEAM-BOILER. account of the loss by blowing off and the presence of other impurities in the water, it is found that the soda-ash will precipitate only about four times the theoretical quantity. Scaling is entirely prevented. One engine made 122,000 miles, and inspection of the boiler showed that it was as clean as when new. This compound precipitates the impurities in a granular form, and careful attention must be paid to washing out the precipitate. The practice is to change the water every 600 miles and wash out the boiler every 1200 miles, using the blow-off cocks also whenever there is any indica- tion of foaming, winch seems to be caused by the precipitate in the water, but not by the alkali itself. (Eng'g News, Dec. 5, 1891.) Kerosene and otner Petroleum Oils ; Foaming.— Kerosene has recently been highly recommended- as a scale preventive. See paper by L. F. Lyne (Trans. A. S. M. E., ix. 247). The Am. Mach., May 22, 1890,. says: Kerosene used in moderate quantities will not make the boiler foam;, it is recommended and used for loosening the scale and for preventing the formation of scale. Neither will a small quantity of common oil always cause foaming; it is sometimes injected into small vertical boilers to pre- vent priming, and is supposed to have the same effect on the disturbed sur- face of the water that oil has when poured on the rough sea. Yet oil in boilers will not have the same effect, and give the desired results in all cases. The presence of oil in combination with other impurities increases the tendency of many boilers to foam,as the oil with the impurities impedes the free escape of steam from the water surface. The use of common oil not only tends to cause foaming, but is dangerous otherwise. The grease appears to combine with the impurities of the water, and when the boiler is at rest this com- pound sinks to the plates and clings to them in a loose, spongy mass, pre- venting the water from coming in contact with the plates, and thereby pro- ducing overheating, which may lead to an explosion. Foaming may also be caused by forcing the fire, or by taking the steam from a point over the furnace or where the ebullition is violent; the greasy and dirty state of new boilers is another good cause for foaming. Kerosene should be used at first in small quantities, the effect carefully noted, and the quantity increased if necessary for obtaining the desired results. R. 0. Carpenter (Trans. A. S. M. E., vol. xi.) says: The boilers of the State Agricultural College at Lansing, Mich., were badly incrusted with a hard scale. It was fully three eighths of an inch thick in many places. The first application of the oil was made while the boilers were being but little used, by inserting a gallon of oil, filling with water, heating to the boiling-point and allowing the water to stand in the boiler two or three weeks before removal. By this method fully one half the scale was removed during the warm season and before the boilers were needed for heavy firing. The oil was then added in small quantities when the boiler was in actual use. For boilers 4 ft. in diam. and 12 ft. long the best results were obtained by the use of 2 qts. for each boiler per week, and for each boiler 5 ft. in diam. 3 qts. per week. The water used in the boilers has the following analysis: CaC0 3 (carbonate calcium) 20G parts in 1,000,000. Mg0O 3 (carbonate magnesium) 78 " " " 1< 2 C0 3 (carbonate iron) 22 " " " Traces of sulphates and chlorides of potash and soda. Total solid parts, 325 to 1,000,000. Tannate of Soda Compound.— T. T. Parker writes to Am. Mach.: Should you find kerosene not doing any good, try this recipe: 50 lbs. sal-soda, 35 lbs. japonica; put the ingredients in a 50-gal. barrel, fill half full of Avater, and run a steam hose into it until it dissolves and boils. Remove the hose, fill up with water, and allow to settle. Use one quart per day of ten hours for a40-H.P. boiler, and, if possible, introduce it as you do cylinder- oil to your engine. Barr recommends tannate of soda as a remedy for scale com- posed of sulphate and carbonate of lime. As the japonica yields the tannic acid, I think the resultant equivalent to the tannate of soda. Petroleum Oils heavier than kerosene have been used with good re- sults. Crude oil should never be used. The more volatile oils it contains make explosive gases, and its tarry constituents are apt to form a spongy incrustation. Removal of Hard Scale.— When boilers are coated with a hard scale difficult to remove the addition of J4 lb. caustic soda per horse-power, and steaming for some hours, according to the thickness of" the scale, just before cleaning, will greatly facilitate that operation, rendering the scale INCRUSTATIOK AKD CORROSION. 719 soft and loose. This should be done, if possible, when the boilers are not otherwise in use. {Steam.) Corrosion in Marine Boilers. (Proc. Inst. M. E., Aug. 1884).— The investigations of the Committee on Boilers served to show that the internal corrosion of boilers is greatly due to the combined action of air and sea- water when under steam, and when not under steam to the combined action of air and moisture upon the unprotected surfaces of the metal. There are other deleterious influences at work, such as the corrosive action of fatty- acids, the galvanic action of copper and brass, and the inequalities of tem- perature; these latter, however, are considered to be of minor importance. Of the several methods recommended for protecting the internal surfaces of boilers, the three found most effectual are: First, the formation of a thin layer of hard scale, deposited by working the boiler with sea-water; second, the coating of the surfaces with a thin wash of Portland cement, partially wherever there are signs of decay; third, the use of zinc slabs suspended in the water and steam spaces. As to general treatment for the preservation of boilers in store or when laid up in the reserve, either of the two following methods is adopted, as may be found most suitable in particular cases. First, the boilers are dried as much as possible by airing-stoves, after which 2 to 3 cwt. of quick- lime, according to the size of the boiler, is placed on suitable trays at the bottom of the boiler and on the tubes. The boiler is then closed and made as air-tight as possible. Periodical inspection is made every six months, when if the lime be found slacked it is renewed. Second, the other method is to fill the boilers up with sea or fresh water, having added soda to it in the proportion of 1 lb. of soda to every 100 or 120 lbs. of water. The sufficiency of the saturation can be tested by introducing a piece of clean new iron and leaving it in the boiler for ten or twelve hours; if it shows signs of rusting, more soda should be added. It is essential that the boilers be entirely filled, to the complete exclusion of air. Great care is taken to prevent sudden changes of temperature in boilers. Directions are given that steam shall not be raised rapidly, and that care shall be taken to prevent a rush of cold air through the tubes by too sud- denly opening the smoke-box doors. The practice of emptying boilers by blowing out is also prohibited, except in cases of extreme urgency. As a rule the water is allowed to remain until it becomes cool before the boilers are emptied. Mineral oil has for many years been exclusively used for internal lubrica- tion of engines, with the view of avoiding the effects of fatty acid, as this oil does not readily decompose and possesses no acid properties. Of all the preservative methods adopted in the British service, the use of zinc properly distributed and fixed has been found the most effectual in saving the iron and steel surfaces from corrosion, and also in neutralizing by its own deterioration the hurtful influences met with in water as ordina- rily supplied to boilers. The zinc slabs now used in the navy boilers are 12 in. long, 6 in. wide, and y% inch thick; this size being found convenient for general application. The amount of zinc used in new boilers at present is one slab of the above size for every 20 I.H.P., or about one square foot of zinc surface to two square feet of grate surface. Rolled zinc is found the most suitable for the purpose. To make the zinc properly efficient as a protector especial care must betaken to insure perfect metallic contact between the slabs and the stays or plates to which they are attached. The slabs should be placed in such positions that all the surfaces in the boiler shall be protected. Each slab should be periodically examined to see that its connection remains perfect, and to renew any that may have decayed ; this examination is usually made at intervals not exceeding three months. Under ordinary circumstances of working these zinc slabs may be expected to last in fit condition from sixty to ninety days, immersed in hot sea-water; but in new boilers they at first decay more rapidly. The slabs are generally secured by means of iron straps 2 in. wide and % inch thick, and long enough to reach the nearest stay, to which the strap is firmly attached by screw-bolts. To promote the proper care of boilers when not in use the following order has been issued to the French Navy by the Government: On board all ships in the reserve, as well as those which are laid up, the boilers will be com- pletely filled with fresh water. In the case of large boilers with large tubes there will be added to the water a certain amounts of milk of lime, or a solution of soda may be used instead. In the case of tubulous boilers with small tubes milk of lime or soda may be added, but the solution will not be no THE STEAM-BOILER. so strong as in the case of the larger tube, so as to avoid any danger of contracting the effective area by deposit from the solution ; but the strength of the solution will be just sufficient to neutralize any acidity of the water. {Iron Age, Nov. 2, 1893.) Use of Zinc— Zinc is often used in boilers to prevent the corrosive action of water on the metal. The action appears to be an electrical one, the iron being one pole of the battery and the zinc being the other. The hydrogen goes to the iron shell and escapes as a gas into the steam. The oxygen goes to the zinc. On account of this action it is generally believed that zinc will always prevent corrosion, and that it cannot be harmful to the boiler or tank. Some experiences go to disprove this belief, and in numerous cases zinc has not only been of no use, but has even been harmful. In one case a tubular boiler had beeu troubled with a deposit of scale consisting chiefly of or- ganic matter and lime, and zinc was tried as a preventive. The beneficial action of the zinc was so obvious that its continued use was advised, with frequent opening of the boiler and cleaning out of detached scale until all the old scale should be removed and the boiler become clean. Eight or ten months later the water supply was changed, it being now obtained from another stream supposed to be free from lime and to contain only organic matter. Two or three months after its introduction the tubes and shell were found to be coated with an obstinate adhesive scale, and composed of zinc oxide and the organic matter or sediment of the water used. The deposit had become so heavy in places as to cause overheating and bulging of the plates over the fire. {The Locomotive.) Effect of Deposit on Flues. (Rankine.) — An external crust of a carbonaceous kind is often deposited from the flame and smoke of the fur- naces in the flues and tubes, and if allowed to accumulate seriously impairs the economy of fuel. It is removed from time to time by means of scrapers and wire brushes. The accumulation of this crust is the probable cause of the fact that in some steamships the consumption of coal per indicated horse-power per hour goes on gradually increasing until it reaches one and a half times its original amount, and sometimes more. Dangerous Steam-boilers discovered by Inspection.— The Hartford Steam-boiler Inspection and Insurance Co. reports that its inspectors during 1893 examined 163,328 boilers, inspected 66,698 boilers, both internally and externally, subjected 7861 to hydrostatic pressure, and found 597 unsafe for further use. The whole number of defects reported was 122,893, of which 12,390 were considered dangerous. A summary is given below. (The Locomotive, Feb. 1894.) Summary, by Defects, for the Year 18! Nature of Defects. Whole Dan- No. geroos Nature of Defects. Whole Dan- No. gerous. Leakage around tubes. . . 21 ,211 2,90S Leakage at seams 5,424 Water-gauges defective. 3,670 Blow outs defective 1,620 Deficiency of water 204 Safety-valves overloaded 723 Safety-valves defective.. 942 Pressure-gauges def'tive 5,953 Boilers without pressure- gauges 115 Unclassified defects 755 482 Deposit of sediment. . Incrustation and scale. . .18,369 Internal grooving 1,249 Internal corrosion 6,252 External corrosion 8,600 Deftive braces and stays 1,966 Settings defective 3,094 Furnaces out of shape. . . 4,575 Fractured plates 3,532 Burned plates 2,762 Blistered plates 3,331 Defective rivets 17,415 Defective heads 1,357 The above-named company publishes annually a classified list of boiler- explosions, compiled chiefly from newspaper reports, showing that from 200 to 300 explosions take place in the United States every year, killing from 200 to 300 persons, and injuring from 300 to 450. The lists are not pretended to be complete, and may include only a fraction of the actual number of explosions. Steam-boilers as Magazines of Explosive Energy.— Prof. R. H. Thurston (Trans. A. 8. M. E., vol. vi.), in a paper with the above title, presents calculations showing the stored energy in the hot water and s'eam of various boilers. Concerning the plain tubular boiler of the form and dimensions adopted as a standard by the Hartford Steam-boiler Total. . .122,893 12,3 SAFETY-VALVES. 721 Insurance Co., he says: It is 60 inches in diameter, containing 66 3-inch tubes, and is 15 feet long. It has 850 feet of heating and 30 feet of grate surface; is rated at 60 horse-power, but isoftener driven up to 75; weighs 9500 pounds, and contains nearly its own weight of water, but only 21 pounds of steam when under a pressure of 75 pounds per square inch, which is below its safe allowance. It stores 52,000,000 foot-pounds of en- ergy, of which but 4 per cent is in the steam, and this is enough to drive the' boiler just about one mile into the air, with an initial velocity of nearly 600 feet per second. SAFETY-VALVES. Calculation of Weight, etc., for Lever Safety-valves. Let W = weight of ball at end of lever, in pounds; to — weight of lever itself, in pounds; V = weight of valve and spindle, in pounds; L = distance between fulcrum and centre of ball, in inches; I — " " " " " " valve, in inches; g — " " " " " " gravity of lever, in in. ; A = area of valve, in square inches; ,P — pressure of steam, in lbs. per sq. in., at which valve will open. Then PA x I = W x L + w X g + V x l; whence P = WL -f tog + VI . Al ' w ^ PAl-wg- Vl m T PAl - wg - VI L = w • Example.— Diameter of valve, 4"; distance from fulcrum to centre of ball, 36"; to centre of valve, 4"; to centre of gravity of lever, \5%"\ weight of valve and spindle, 3 lbs.; weight of lever, 7 lbs. ; required the weight of ball to make the bio wing-off pressure 80 lbs. per sq. in.; area of 4" valve = 12.566 sq. in. Then w _ PAl - wg - VI _ 80 X 12.566 X 4 - 7 X 15^2 - 4 X 4 _ 10g Jbg L 36 The following rules governing the proportions of lever- valves are given by the U. S. Supervisors. The distance from the fulcrum to the valve-stem must in no case be less than the diameter of the valve-opening; the length of the lever must not be more than ten times the distance from the fulcrum to the valve-stem; the width of the bearings of the fulcrum must not be less than three quarters of an inch; the length of the fulcrum-link must not be less than four inches; the lever and fulcrum-link must be made of wrought iron or steel, and the knife-edged fulcrum points and the beatings for these points must be made of steel and hardened; the valve must be guided by its spindle, both above and below the ground seat and above the lever, through supports either made of composition (gun-metal) or bushed with it; and the spindle must fit loosely in the bearings or supports. Rules for Area of Safety-valves. (Rule of U. S. Supervising Inspectors of Steam-vessels (as amended 1891).) Lever safety-valves to be attached to marine boilers shall have an area of not less than 1 sq. in. to 2 sq. ft. of the grate surface in the boiler, and the seats of all such safety-valves shall have an angle of inclination of 45° to the centre line of their axes. Spring-loaded safety-valves shall be required to have an area of not less than 1 sq. in. to 3 sq. ft. of grate surface of the boiler, except as hereinafter otherwise provided for water-tube or coil and sectional boilers, and each spring-loaded valve shall be supplied with a lever that will raise the valve from its seat a distance of not less than that equal to one eighth the diam- eter of the valve-opening, and the seats of all such safety-valves shall have an angle of inclination to the centre line of their axes of 45°. All spring- loaded safety-valves for water-tube or coil and sectional boilers required to 722 THE STEAM-BOILER. carry a steam -pressure exceeding 175 lbs. per square inch shall be required to have an area of not less than 1 sq. in. to 6 sq. ft. of the grate surface of the boiler. Nothing herein shall be construed so as to prohibit rhe use of two safety-valves on one water- tube or coil and sectional boiler, provided the combined area of such valves is equal to that required by rule for one such valve. Rule In Philadelphia Ordinances : Bureau of Steam- engine and Boiler Inspection.— Every boiler when fired sepa- rately, and every set or series of boilers when placed over one fire, shall have attached thereto, without the interposition of any other valve, two or more safety-valves, the aggregate area of which shall have such relations to the area of the grate and the pressure within the boiler as is expressed in schedule A. Schedule A. — Least aggregate area of safety-valve (being the least sec- tional area for the discharge of steam) to be placed upon all stationary boil- ers with natural or chimney draught [see note a]. a - 22 - hG P+8.62' in which A is area of combined safety-valves in inches; G is area of grate in square feet; P is pressure of steam in pounds per square inch to be carried in the boiler above the atmosphere. The following table gives the results of the formula for one square foot of grate, as applied to boilers used at different pressures: Pressures per square inch: 10 20 30 40 50 60 70 80 90 110 120 Area corresponding to one square foot of grate: 1.21 0.79 0.58 0.46 0.38 0.33 0.29 0.25 0.23 0.21 0.19 0.17 [Note a.] Where boilers have a forced or artificial draught, the inspector must estimate the area of grate at the rate of one square foot of grate-sur- face for each 10 lbs. of fuel burned on the average per hour. Comparison of Various Bules for Area of Lever Safety- valves. (From an article by the author in American- Machinist, May z4, 1894, with some alterations and additions.) — Assume the case of a boiler rated at 100 horse-power; 40 sq. ft. grate; 1200 sq. ft. heating-surface; using 400 lbs. of coal per hour, or 10 lbs. per sq. ft. of grate per hour, and evapora- ting 3600 lbs. of water, or 3 lbs. per sq. ft. of heating-surface per hour; steam-pressure by gauge, 100 lbs. What size of safety-valve, of the lever type, should be required ? A compilation of various rules for finding the area of the safety-vale disk, from The Locomotive of July, 1892, is given in abridged form below, to- gether with the area calculated by each rule for the above example. Disk Area in sq. in. U. S. Supervisors, heating-surface in sq. f t. -h 25 * 48 English Board of Trade, grate-surface in sq. ft. -s- 2 20 Molesworth, four fifths of grate-surface in sq. f t 32 Thurston, 4 times coal burned per hour X (gauge pressure + 10) 14.5 „, 1 (5 X heating-surface) Thurston,- — . 273 2 gauge pressure -4- 10 Rankine, .006 X water evaporated per hour 21 .6 Committee of U. S. Supervisors, .005 X water evaporated per hour 18 Suppose that, other data remaining the same, the draught were increased- so as to burn 13^, lbs. coal per square foot of grate per hour, and the grate- surface cut down to 30 sq. ft. to correspond, making the coal burned per hour 400 lbs., and the water evaporated 3600 lbs., the same as before; then the English Board of Trade rule and Molesworth's rule would give an area of disk of only 15 and 24 sq. in., respectively, showing the absurdity of mak- ing the area of grate the basis of the calculation of disk area. Another rule by Prof. Thurston is given in American Machinist, Dec. 1877, viz.: Disk area - ^ max. wt. of water evap. per hour gauge pressure + 10 This gives for the example considered 16.4 sq. in. * The edition of 1893 of the Rules of the Supervisors does not contain this rule, but gives the rule grate-surface -5- 2. SAFETY-VALVES. 723 One rule by Rankine is 1/150 to 1/180 of the number of pounds of water evaporated per hour, equals for the above case 27 to 20 sq. in. A communi- tion in Power, July, 1890. gives two other rules: 1st. 1 sq. in. disk area for 3 sq. ft. grate, which would give 13.3 sq. in. 2d. % sq. in. disk area for 1 sq. ft. grate, which would give 30 sq. in.; but if the grate-surface were reduced to 30 sq. ft. on account of increased draught, these rules would make the disk area only 10 and 22.5 sq. in., respectively. The Philadelphia rule for 100 lbs. gauge pressure gives a disk area of 0.21 sq. in. for each sq. ft. of grate area, which would give an area of 8.4 sq. in. for 40 sq. ft. grate, and only 6.3 sq. in. if the grate is reduced to 30 sq. ft. According to the rule this aggregate area would have to be divided between two valves. But if the boiler was driven by forced draught, then the in- spector " must estimate the area of grate at 1 sq. ft. for each 16 lbs. of fuel burned per hour." Under this condition the actual grate-surface might be cut down to 400 -=- 16 = 25 sq. ft., and by the rule the combined area of the two safety-valves would be only 25 X 0.21 = .25 sq. in. Nystrom's Pocket-book, edition of 1891, gives % sq. in. for 1 sq. ft. grate; also quoting from Weisbach, vol. ii, 1/3000 of the heating-surface^ This in the case considered is 1200/3000 = .4 sq. ft. or 57.6 sq. in. We thus have rules w r hich give for the area of safety-valve of the same 100- horse-power boiler results ranging all the way from 5.25 to 57.6 sq. in. All of the rules above quoted give the area of the disk of the valve as the thing to be ascertained, and it is this area which is supposed to bear some direct ratio to the grate-surface, to the heating-surface, to the water evap- orated, etc. It is difficult to see why this area has been considered even approximately proportional to these quantities., for with small lifts the area, of actual opening bears a direct ratio, not to tne area of disk, but to the circumference. Thus for various diameters of valve : Diameter 1 2 3 4'' " 6 7 Area 785 3.14 7.07 12.57 19.64 28.27 38.48 Circumference 3.14 6.28 9.42 12.57 15.71 18.85 21.99 Ciicum. X lift of 0.1 in 31 .63 .94 1.26 1.57 1.89 2.20 Ratio to area 4 .2 .13 .1 .08 .067 .057 The apertures, therefore, are therefore directly proportional to the diam- eter or to the circumference, but their relation to the area is a varying one. If the lift = J4 diameter, then the opening would be equal to the area of the disk, for circumference X f4 diameter = area, but such a lift is far beyond the actual lift of an ordinary safety-valve. A correct rule for size of safety-valves should make the product of the diameter and the lift proportional" to the weight of steam to be discharged. A " logical " method for calculating the size of safety-valve is given in The Locomotive, July, 1892, based on the assumption that the actual opening should be sufficient to discharge all the steam generated by the boiler. Napier's rule for flow of steam is taken, viz., flow through aperture of one sq. in. in lbs. per second = absolute pressure -^- 70, or in lbs. per hour — 51.43 X absolute pressure. If the angle of the seat is 45°, as specified in the rules of the U. S. Super- visors, the area of opening in sq. in. = circumference of the disk X the lift X .71, .71 beiug the cosine of 45°; or diameter of disk X lift X 2.23. A. G. Brown in his book on The Indicator and -its Practical Working (London, 1894) gives the following as the lift of the ordinary lever safetj : - valve for 100 lbs. gauge-pressure: Diam. of valve.. 2 2^ 3 3}& 4 4% 5 6 inches. Rise of valve 0583 .0523 .0507 .049~2 .0478 .0462 .0446 .0430 inch. The lift decreases with increase of steam -pressure; thus for a 4-inch valve: Abs. pressure, lbs. 45 65 85 105 115 135 155 175 195 215 Gauge-press., lbs.. 30 50 70 90 100 120 140 160 180 200 Rise, inch 1034 .0775 .0620 .0517 .0478 .0413 .0365 .0327 .0296 .0270 The effective area of opening Mr. Brown takes at 70$£ of the rise multiolied by the circumference. An approximate formula corresponding to Mr. Brown's figures for diam- eters between 2J^ and 6 in. and gauge-pressures between 70 and 200 lbs. is Lift = (.0603 - 0031d) x -* — , in which d = diam. of valve in in, abs. pressure 724 THE STEAM-BOILER. If we combine this formula with the formulae Flow in lbs. per hour = area of opening in sq. in. x 51.43X abs. pressure, and Area = diameter of valve X lift X 2.23, we obtain the following, which the author suggests as probably a more correct formula for the discharging capacity of the ordinary lever safety-valve than either of those above given. Flow 'in lbs. per hour = d(.0603 - .0031cZ) X 115 X 2.23 X 51.43 = d(7% — Aid). From which we obtain : Diameter, inches ... . 1 1*4 2 2\i 3 3^ 4 5 6 7 Flow, lbs. per hour.. 754 1100 1426 1733 2016 2282 2524 2950 3294 3556 Horse-power 25 37 47 58 67 76 84 98 110 119 the horse-power being taken as an evaporation of 30 lbs. of water per hour. If we solve the example, above given, of the boiler evaporating 3600 lbs. of water per hour by this table, we find it requires one 7-inch valve, or a 2)4- and a 3-inch valve combined. The 7-inch valve has an area of 38.5 sq. in., and the two smaller valves taken together have an area of only 12 sq. in.; another evidence of the absurdity of considering the area of disk as the factor which determined the capacity of the valve. It is customary in practice not to use safety-valves of greater diameter than 4 in. If a greater diameter is called for by the rule that is adopted, then two or more valves are used instead of one. Spring-loaded Safety-valves.— Instead of weights, springs are sometimes employed to hold down safety-valves. The calculations are similar to those for lever safety-valves, the tension of the spring correspond- ing to a given rise being first found by experiment (see Springs, page 347). The rules of the U. S. Supervisors allow an area of 1 sq. in. of the valve to 3 sq. ft. of grate, in the case of spring-loaded valves, except in water-tube, coil, or sectional boilers, in which 1 sq. in. to 6 sq. ft. of grate is allowed. Spriug-loaded safety-valves are usually of the reactionary or "pop " type, in which the escape of the steam is opposed by a lip above the valve-seat, against which the escaping steam reacts, causing the valve to lift higher than the ordinary valve. A. G. Brown gives the following for the rise, effective area, and quantity of steam discharged per hour by valves of the "pop " or Richardson type. The effective is taken at only 50% of the actual area due to the rise, on account of the obstruction which the lip of the valve offers to the escape of steam. Dia. value, in 1 Wo 2 2Yo, 3 3% 4 4Vo, 5 6 Lift, inches. .125 .150 .175 .200 .225 .250 .275 .300 .325 .375 Area, sq. in. .196 .354 .550 .785 1.061 1.375 1.728 2.121 2.553 3.535 Gauge-pres., Steam discharged per hour, lbs. 30 lbs. 474 856 1330 1897 2563 3325 4178 5128 6173 8578 50 669 1209 1878 2680 3620 4695 5901 7242 8718 12070 70 861 1556 2417 3450 4660 6144 7596 9324 11220 15535 90 1050 1897 2947 4207 5680 7370 9260 11365 13685 18945 100 1144 2065 3208 4580 6185 8322 10080 12375 14895 20625 120 1332 2405 3736 5332 7202 9342 11735 14410 173-10 24015 140 1516 2738 4254 6070 8200 10635 13365 16405 19745 27340 160 1696 3064 4760 6794 9175 11900 14955 18355 22095 30595 180 1883 3400 5283 7540 10180 13250 16595 20370 24520 33950 200 2062 3724 5786 8258 11150 14165 18175 22310 37185 If we take 30 lbs. of steam per hour, at 100 lbs. gauge-pressure = 1 H.P., we have from the above table: Diameter, inches... 1 1]4 2 2% 3 3^ 4 4J^ 5 6 Horse-power 38 69 107 153 206 277 336 412 496 687 A safety-valve should be capable of discharging a much greater quantity of steam than that corresponding to the rated horse-power of a boiler, since a. boiler having ample grate surface and strong draught may generate more than double the quantity of steam its rating calls for. The Consolidated Safety-valve Co.'s circular gives the following rated capacity of its nickel-seat " pop " safety-valves: Size, in .... 1 1J4 Vy 2 2 2\& 3 3}^ 4 4)4 5 5^£ Boiler t from 8 10 20 35 60 75 100 125 150 175 200 H.P. 1 to 10 15 30 50 75 100 125 150 175 200 275 The figures in the lower line from 2 inch to 5 inch, inclusive, correspond to the formula H.P. — 50(diameter — 1 inch). THE INJECTOR. 725 Lets W J L 778 Then THE INJECTOR. Equation of the Injector. be the number of pounds of steam used ; the number of pounds of water lifted and forced into the boiler; the height in feet of a column of water, equivalent to the absolute pressure in the boiler; the height in feet the water is lifted to the injector; the temperature of the water before it enters the injector; the temperature of the water after leaving the injector; the total heat above 32° F. in one pound of steam in the boiler, in heat-units; the lost work in friction and the equivalent lost work due to radia- tion and lost heat; the mechanical equivalent of heat. 8[H - (t 2 - 32°)] = W(t. z - t,) + (W+ S)h -f- irft + Z, An equivalent formula, neglecting Wh -f Las small, is = [w 2 ■*i) + W+S 144T 1 d ' * '778J.H" - (t 2 - 32°)' _ W[(t 2 - tjd + .1851p] H-{t 2 - 32°)cZ - .185123' in which d = weight of 1 cu. ft. of water at temperature £ 2 ; p — absolute pressure of steam, lbs. per sq. in. The rule for finding the proper sectional area for the narrowest part of the- nozzles is given as follows by Rankine, S. E. p. 477: . , cubic feet per hour gross feed-water Area in square inches = • 800 ^/pressure in atmospheres An important condition which must be fulfilled in order that the injector will work is that the supply of water must be sufficient to condense the steam. As the temperature of the supply or feed -water is higher, the amount of water required for condensing purposes will be greater. The table below gives the calculated value of the maximum ratio of water to the steam, and the values obtained on actual trial, also the highest admis- sible temperature of the feed-water as shown by theory and the highest actually found hy trial with several injectors. Maximum Ratio Water to Steam. Gauge- pres- sure, pounds per sq. in. Maximum Temperature of Feed -Water. Gauge- pres- Calculated from Theory. Actual Expe- riment. Theoretical. Experrtal Results. pounds per 111 H.| T3 H. P. M. sq. in. H. P. M. S. 10 36.5 25.6 20.9 17.87 16.2 14.7 13.7 12.9 12.1 11.5 30.9 2^2.5 19.0 15.8 13.3 11.2 12.3 11.4 19^9 17.2 15.0 14.0 11.2 11.7 11.2 21 ^5 19.0 15.86 13.3 12.6 12.9 10 20 30 40 50 60 70 80 90 100 120 150 135° 140* 141* 141* 120° 113 lis 118 130° 125 123 123 122 132° 20 30 40 50 60 70 80 90 142° 132 126 120 114 109 105 99 95 87 77 173° 162 156 150 143 139 134 129 125 117 107 134 134 132 131 130 130 131 132* 100 132* 134* 121* * Temperature of delivery above 212°. Waste-valve closed. H, Hancock inspirator; P, Park injector; M, Metropolitan injector; S, lers 1876 injector. Sel- 72b THE STEAM-BOILER. Efficiency of the Injector.— Experiments at Cornell University, described by Prof. R. C. Carpenter, in Cassier , s Magazine, Feb. 1892, show- that the injector, when considered merely as a pump, has an exceedingly low efficiency, the duty ranging from 161^000 to 2,752.000 under different cir- cumstances of steam and delivery pressure. Small direct-acting pumps, such as are used for feeding boilers, show a duty of from 4 to 8 million lbs , and the best pumping-engines from 100 to 140 million. When used for feeding- water into a boiler, however, the injector has a thermal efficiency of 100$, less the trifling loss due to radiation, since all the heat re- jected passes into the water which is carried into the boiler. The loss of work in the injector due to friction reappears as heat which is carried into the boiler, and the heat which is converted into useful work in the injector appears in the boiler as stored-up energy. Although the injector thus has a perfect efficiency as a boiler-feeder, it is nevertheless not the most economical means for feeding a boiler, since it can draw only cold or moderately warm water, while a pump can feed water which has been heated by exhaust steam which would otherwise be wasted. Performance of Injectors.— In Am. Mach., April 13, 1893, are a number of letters from different manufacturers of injectors in reply to the question: "What is the best performance of the injector in l-aising or lifting water to any height ?" Some of the replies are tabulated below. W. Sellers & Co.— 25.51 lbs. water delivered to boiler per lb. of steam; tem- perature of water, 64°; steam pressure, 65 lbs. Schaeffer & Budenberg— 1 gal. water delivered to boile- for 0.4 to 0.8 lb. steam. Injector will lift by suction water of 140° F. 136° to 133° 122° to 180° 113° to 107° If boiler pressure is. 30 to 60 lbs. 60 to 90 lbs. 90 to 120 lbs. 120 to 150 lbs. If the water is not over 80° F., the injector will force against a pressure 75 lbs. higher than that of the steam. Hancock Inspirator Co.: Lift in feet 22 22 22 11 Boiler pressure, absolute, lbs 75.8 54.1 95.5 . 75.4 Temperature of suction 34.9° 35.4° 47.3° 53.2° Temperature of delivery 134° 117.4° 173.7° 131.1 Water fed per lb. of steam, lbs... 11.02 13.67 8.18 13.3 The theory of the injector is discussed in Wood's. Peabody's, and Ront- gen's treatises on Thermodynamics. See also "Theory and Practice of the Injector,"" by Strickland L. Kueass, New York, 1895. Boiler-feeding Pumps.- Since the direct-acting pump, commonly used for feeding boilers, has a very low efficiency, or less than one tenth that of a good engine, it is generally better to use a pump driven by belt from the main engine or driving shaft. The mechanical work needed to feed a boiler may be estimated as follows: If the combination of boiler and en- gine is such that half a cubic foot, say 32 lbs. of water, is needed per horse- power, and the boiler-pressure is 100 lbs. per sq. in., then the work of feed- ing the quantity of water is 100 lbs. X 144 sq. in. x J4 ft. -lbs. per hour = 120 ft.-lbs. per min. = 120/33.000 = .0036 H.P., or less than 4/10 of H of the power exerted by the engine. If a direct-acting pump, which discharges its exhaust steam into the atmosphere, is used for feeding, and it has only 1/10 the efficiency of the main engine, then the steam used by the pump will be equal to nearly 4% of that generated by the boiler. The following table by Prof. D. S. Jacobus gives the relative efficiency of steam and power pumps and injector, with and without heater, as used upon a boiler with 80 lbs. gauge-pressure, the pump having a duty of 10,000,000 ft.-lbs. per 100 lbs. of coal when no heater is used ; the injector heating the water from 60° to 150° F. Direct-acting pump feeding water at 60°, without a heater 1 .000 Injector feeding water at 150°, without a heater 985 Injector feeding water through a heater in which it is heated from 150° to 200° 938 Direct-acting pump feeding water through a heater, in which it is heated from 60° to 200° 879 Geared pump, run from the engine, feeding water through a heater, in which it is heated from 60° to 200° .868 PEED-WATER HEATERS. m FEED-WATER HEATERS. Percentage of Saving for Each Degree of Increase in Tem- perature of Feed-water Heated by Waste Steam. Pressure of Steam in Boiler lbs. per sq . iu. above Initial Temp. of Atmosphere. Initial Temp. Feed. 20 40 60 80 100 120 140 160 180 200 32° .0872 .0861 .0855 .0851 .0847 .0844 .0841 .0839 .0837 .0835 .0833 32 40 .0878 .0867 .0861 .0856 .0853 .0850 .0847 .0845 .0813 .0841 .0839 40 50 .0886 .0875 .0868 .0864 .0860 .08571.0854 .0852 .0850 .0848 .0846 50 60 .0894 .0883 .0876 .0872 .0867 .0864 |.0862 .0859 .0850 .0855 .osr,3 60 70 .0902 .0890 .0884 .0879 .0875 .08721.0869 .0867 .0801 .0802 .0800 70 80 .0910 .0898 .0891 .0887 .oss:-; .0879 .0877 .0874 . 0872 .0870 .0808 80 90 .0919 .0907 .0900 .0895 .osss .08871.0884 .0883 .0879 .087; .0875 90 100 .0927 .0915 .0908 .0903 .0899 .08951.089.2 0890 .0887 .0885 .OS 83 100 110 .0936 .0923 0916 .0911 .0907 .0903; .0900 .0898 .0895 .0^93 .0891 110 120 .0945 .0932 .0925 .0919 0915 .09111.0908 .0906 .0903 .0901 .0899 120 130 .0954 .0941 .0934 .0928 .0924 .09201.0917 .0914 .0912 .0909 .0907 130 140 .0963 .0950 .0943 .0937 .0932 .0929 .0925 .0923 .0920 .0918 .0916 140 150 .0973 .0959 .0951 .0946 .0941 .0937 .0934 .0931 .11929 .0926 .0921 150 . 160- .0982 .0968 .0961 .0955 .0950 .0946 .0943 .0940 .0937 .0935 .0933 160 170 .0992 .0978 .0970 .0964 .0959 .0955 .0952 .0949 .0946 .0944 .0941 170 180 .1002 .0988 .0981 .0973 .0969 .0965 .0961 .0958 0955 .0953 .0951 180 190 .1012 .0998 .0989 .0983 .0978 .0974 .0971 .0968 .0964 .0962 0960 190 200 .1022 .1008 .0999 .09'.):; .0988 .0984 .0980 .0977 .0974 .0972 .0969 200 210 .1033 .1018 .1009 .1003 .0998 .0994 .0990 .0987 .0984 .0981 .0979 210 220 .1029 .1019 .1013 .1008 .1001 .1000 .0997 .0994 .0991 .0989 220 230 .1039 .1031 .1024 .1018 .1012 .1010 .1007 .1003 .1001 .0999 230 240 .1050 .1041 .1034 .1029 .1021 .1020 .1017 .1014 .1011 .1009 210 250 .1062 .1052 .1045 .1040 .1035 .1031 .1027 .1025 .1022 .1019 250 An approximate rule for the conditions of ordinary practice is a saving of 1$ is made by each increase of 11° in the temperature of the feed-water. This corresponds to .0909$ per degree. The calculation of saving is made as follows: Boiler-pressure, 100 lbs. gauge; total heat in steam above 32° = 1185 B.T.U. Feed-water, original temperature 60°, final temperature 209° F. Increase in heat-units, 150. Heat-units above 32° in feed -water of original temperature = 28. Heat- units in steam above that in cold feed-water, 1185 - 28 = 1157. Saving by the feed- water heater = 150/1157 = 12.96$. The same result is obtained by the use of the table. Increase in temperature 150° X tabular figure .0864 = 12.96$. Let total heat of 1 lb. of steam at the boiler-pressure = H\ total heat of 1 lb. of feed-water before entering the heater = h 1 , and after pass- ing through the heater = ft 2 ! t^ en the saving made by the heater is -| - 1 . Strains Caused by Cold Feed-water.— A calculation is made in The Locomotive of March, 1893, of the possible strains caused in the sec- tion of the shell of a boiler by cooling it by the injection of cold feed-water. Assuming the plate to be cooled 200° F., and the coefficient of expansion of steel to be .0000067 per degree, a strip 10 in. long would contract .013 in., if it were free to contract. To resist this contraction, assuming that the strip is firmly held at the ends and that the modulus of elasticity is 29,000,000, would require a force of 37,700 lbs. per sq. in. Of course this amount of strain can- not actually take place, since the strip is not firmly held at the ends, but is allowed to contract to some extent by the elasticity of the surrounding metal. But, says The Locomotive, we may feel pretty confident that in the case considered a longitudinal strain of somewhere in the neighborhood of 8000 or 10,000 lbs. per sq. in. may be produced by the feed-water striking directly upon the plates; and this, in addition to the normal strain pro- duced by the steam-pressure, is quite enough to tax the girth-seams beyond their elastic limit, if the feed-pipe discharges anywhere near them. Hence it is not surprising that the girth-seams develop leaks and cracks in 99 cases out of every 100 in which the feed discharges directly upon the fire- sheets. m THE STEAM-BOILEK. STEAM SEPARATORS, If moist steam flowing at a high velocity in a pipe has its direction sud- denly changed, the particles of water are by their momentum projected in Cheir original direction against the bend in the pipe or wall of the chamber in which the change of direction takes place. By making proper provision for drawing off the water thus separated the steam may be dried to a greater or less extent. For long steam-pipes a large drum should be provided near the engine for trapping the water condensed in the pipe. A drum 3 feet diameter, 15 feet high, has given good results in separating the water of condensation of a steam-pipe 10 inches diameter and 800 feet long. Efficiency of Steam Separators.— Prof. R. C. Carpenter, in 1S91, made a series of tests of six steam separators, furnishing them with steam containing different percentages of moisture, and testing the quality of steam before entering and after passing the separator. A condensed table of the prin cipal results is given below. o| Test with Steam of about 10$ of Moisture. Tests with Varying Moisture. 1* Quality of Steam before. Quality of Steam after. Efficiency per cent. Quality of Steam before. Quality of Steam after. Av'ge Effi- ciency. B A D C E F 87.0$ 90.1 89.6 90.6 88.4 88.9 98.8$ 98.0 95.8 93.7 90.2 93.1 90.8 80.0 59.6 33.0 15.5 28.8 66.1 to 97.5$ 51.9 " 98 72.2 " 96.1 67.1 " 96.8 68.6 " 98.1 70.4 " 97.7 97.8 to 99$ 97.9 " 99.1 95.5 " 98.2 93.7 " 98.4 79.3 " 98.5 84.1 " 97.9 87.6 76.4 71.7 63.4 36.9 28.4 Conclusions from the tests were: 1. That no relation existed between the eolume of the several separators and their efficiency. 2. No marked decrease in pressure was shown by any of the separators, the most being 1.7 lbs. in E. 3. Although changed direction, reduced velocity, and perhaps centrifugal force are necessary for good separation, still some means must be provided to lead the water out of the current of the steam. The high efficiency obtained from B and A was largely due to this feature, [n B the interior surfaces are corrugated and thus catch the water thrown out of the steam and readily lead it to the bottom. In A, as soon as the water falls or is precipitated from the steam, it comes in contact with the perforated diaphragm through which it runs into the space below, where it is not subjected to the action of the steam. In D, the next in efficiency, this is accomplished by means of a >-shaped diaphragm which throws the water back into the corners out of the current of steam. DETERMINATION OF THE MOISTURE IN STEAM- STEAM CALORIMETERS. In all boiler tests it is important to ascertain the quality of the steam, i.e., 1st, whether the steam is "saturated" or contains the quantity of heat due to the pressure according to standard experiments; 2d, whether the quantitv of heat is deficient, so that the steam is wet; and 3d. whether the heat is in excess and the steam superheated. The best method of ascer- taining the quality of the steam is undoubtedly that employed by a com- mittee which tested the boilers at the American Institute Exhibition of 1871-2, of which Prof. Thurston was chairman, i.e., condensing all the water evaporated by the boiler by means of a surface condenser, weighing the condensing water, and taking its temperature as it enters and as it leaves the condenser; but this plan cannot always be adopted. A substitute for this method is the barrel calorimeter, which with careful operation and fairly accurate instruments may generally be relied on to give results within Iwo per cent of accuracy (that is, a sample of steam which gives the apparent result of 2$ of moisture may contain anywhere be tween and 4$). This calorimeter is described as follows: A sample of the steam is taken by inserting a perforated J^-inch pipe into and through the main pipe near the boiler, and led by a hose, thoroughly felted, to a barrel, holding preferably 400 lbs. of water, which is set upon a platform scale and DETERMINATION OF THE MOISTURE IN STEAM. 729 provided with a cock or valve for allowing the water to flow to waste, and with a small propeller for stirring the water. To operate the calorimeter the barrel is filled with water, the weight and temperature ascertained, steam blown through the hose outside the barrel until the pipe is thoroughly warmed, when the hose is suddenly thrust into the water, and the propeller operated until the temperature of the water is increased to the desired point, say about 110° usually. The hose is then withdrawn quickly, the temperature noted, and the weight again taken. An error of 1/10 of a pound in weighing the condensed steam, or an error of y 2 degree in the temperature, will cause an error of over \% in the calcu- lated percentage of moisture. See Trans. A. S. M. E., vi. 293. When all the steam generated is not condensed, the method of making the connection for the purpose of taking out a sample is of the utmost impor- tance. Unless great care be exercised, the results will frequently show that the steam is superheated when the boiler has no superheating surface. The samples should be taken from the main steam-pipe, but not from the bottom, as this would take all the water draining to that point. The calculation of the percentage of moisture is made as below: Q = =4— [-Uh - h) ~ (T - >h)l H - T I- iv J Q = quality of the steam, dry saturated steam being unity. H = total heat of 1 lb. of steam at the observed pressure. T "== " " " " " water at the temperature of steam of the ob- served pressure. h = " " " " " condensing water, original. /ij = " " " " " " " final. W — weight of condensing w r ater, corrected for water-equivalent of the apparatus. w = weight of the steam condensed. Percentage of moisture = 1 — Q. If Q is greater than unity, the steam is superheated, and the degrees of superheating = 2.0833 (H - T) (Q - 1). Coil Calorimeters.— Instead of the open barrel in which the steam is condensed, a coil acting as a surface-condenser may be used, which is placed in the barrel, the water in coil and barrel being weighed separately. For description of an apparatus of this kind designed by the author, which he lias found to give results w r ith a probable error not exceeding ^ per cent of moisture, see Trans. A. S. M. E., vi. 294. This calorimeter may be used continuously, if desired, instead of intermittently. In this case a continu- ous flow of condensing water into and out of the barrel must be established, and the temperature of inflow and outflow and of the condensed steam read at short intervals of time. Throttling Calorimeter.— For percentages of moisture not ex- ceeding 3 per cent the throttling calorimeter is most useful and convenient and remarkably accurate. In this instrument the steam which reaches it in a J^-inch pipe is throttled by an orifice 1/16 inch diameter, opening into a chamber which has an outlet to the atmosphere. The steam in this cham- ber has its pressure reduced nearly or quite to the pressure of the atmos- phere, but the total heat in the steam before throttling causes the steam in the chamber to be superheated more or less according to whether the steam before throttling was dry or contained moisture. The only observa- tions required are those of the temperature and pressure of the steam on each side of the orifice. The author's formula for reducing the observations of the throttling calorimeter is as follows (Experiments on Throttling Calorimeters, Am. Mach., Aug. 4, 1892) : w = 100 X j , in which iv — percent- age of moisture in the steam ; H = total heat, and L = latent heat of steam in the main pipe; h = total heat due the pressure in the discharge side of the calorimeter, — 1146 6 at atmospheric pressure: K— specific heat of su- perheated steam; T = temperature of the throttled and superheated steam in the calorimeter; t = temperature due the pressure in the calorimeter, = 212° at atmospheric pressure. Taking K a,t 0.48 and the pressure in the discharge side of the calorimeter as atmospheric pressure, the formula becomes « = ioox H "" 46 - fl - £ °- 48(r - 218B \ From this formula the fojlowipg table is calculated ; 730 THE STEAM-BOILER. Moisture in Steam— Determinations by Throttling Calorimeter. £ Gauge-pressures. & tuO°? © = 0.0S07 — , where t is the absolute temperature at 32° F., = 493., t x the absolute tem- perature of the chimney gases and t 2 that of the external air. Substituting these values the formula for force of draught becomes 41 - 41 -) = /,(^-^Y y v t 2 t x y .192/1 / 39.79 ^ t 2 .5 ^ • ® a Temperature of the Ex ternal Air— Barometer, 14.7 lbs per sq. in. 1*1 E-i O 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 200 .453 .419 -.384 .353 .321 .292 .263 .234 .209 .182 .157 220 .488 .453 .419 .388 .355 .326 .298 .269 .244 .217 .192 240 .520 .488 .451 .421 .388 .359 .330 .301 .276 .250 .225 260 .555 .528 .484 .453 .420 .392 .363 .331 .309 .282 .257 280 .584 .549 .515 .482 .451 .422 .394 .365 .310 .313 .288 300 .611 .576 .541 .511 .478 .449 .420 .392 .367 .340 .315 320 .637 .603 .568 .538 .505 .476 .447 .419 .394 .367 .342 340 .662 .638 .593 .563 .530 .501 .472 .443 .419 .392 .367 360 .687 .653 .618 .588 .555 .526 .497 .468 .444 .117 .392 380 .710 .676 .641 .611 .578 .549 .520 .492 .467 .440 .415 400 .732 .697 .662 .632 .598 .570 .541 .513 .488 .461 .436 420 .753 .718 .684 .653 .620 .591 .563 .534 .509 .482 .457 440 .774 .739 .705 .674 .641 .612 .584 .555 .530 .503 .478 460 .793 .758 ,724 .694 .660 .632 .603 .574 .549 .522 .497 480 .810 .776 .741 .710 .678 .649 .620 .£91 .566 .540 .515 500 .829 .791 .760 .730 .697 .669 .689 .610 .586 .559 .534 To find the maximum intensity of draught for any given chimney, the heated column being 600° F., and the external air 60°, multiply the height above grate in feet by .0073, and the product is the draught in inches of water. Height of Water Column Due to Unbalanced Pressure in Chimney 100 Feet High. {The Locomotive, 1884.) * Much confusion to students of the theory of chimneys has resulted from their understanding the words maximum draught to mean maximum inten- sity or pressure of draught, as measui'ed by a draught-gauge. It here means maximum quantity oj- weight of gases passed up the chimney. The maxi- mum intensity is found only with maximum temperature, but after the temperature reaches about 622° F. the density of the gas decreases more rapidly than its velocity increases, so that the weight is a maximum about 622° F., as shown by Rankine. — W. K. CHIMKEYS. 738 For any other height of chimney than 100 ft. the height of water column is found by simple proportion, the height of water column being directly proportioned to the height of chimney. The calculations have been made for a chimney 100 ft. high, with various temperatures outside and inside of the flue, and on the supposition that the temperature of the chimney is uniform from top to bottom. This is the basis on which all calculations respecting the draught-power of chimneys have been made by Rankine and other writers, but it is very far from the truth in most cases. The difference will be shown by comparing the read- ing of the draught-gauge with the table given. In one case a chimney 122 ft. high showed a temperature at the base of 320°, and at the top of 230°. Box, in his " Treatise on Heat," gives the following table : Draught Powers op Chimneys, etc., with the Internal Air at 552°, and the External Air at 02°, and with the Damper nearly Closed. , . S"" 1 fD in feet pe • second. S'=53 in feet per second. cs Z £ (SI'S Pi £PPs£ £*3 eg Z ? a Is p-i Cold Air Hot Air Cold Aii- Hot Air Entering. at Exit. MS Entering. at Exit. 10 .073 17.8 35.6 80 .585 50.6 101.2 20 .146 25.3 50.6 90 .657 53.7 107.4 30 .219 31.0 62.0 100 .730 56,5 113.0 40 .292 35.7 71.4 120 .876 62.0 124.0 50 .365 40.0 80.0 150 1.095 69.3 138.6 60 .438 43.8 87.6 175 1.277 74.3 149 6 70 .511 47.3 94.6 200 1.460 80.0 160.0 Rate of Combustion Due to Height of Chimney. — Trowbridge's "Heat and Heat Engines 1 ' gives tne following table showing the heights of chimney for producing certain rates of combustion per sq. ft. of section of the chimney. It may be approximately true for anthracite in moderate and large sizes, but greater heights than are given in the table are needed to secure the given rates of combustion with small sizes of anthracite, and for bituminous coal smaller heights will suffice if the coal is reasonably free from ash— 5% or less. Lbs. of Coal Lbs. of Coal Lbs. of Coal Burned per Lbs. of Coal Burned per Burned per Sq. Ft. of Burned per Sq. Ft. of Heights Hour per Grate, the Heights Hour per Sq. Ft. Grate, the in Sq. Ft. Ratio of in Ratio of feet. of Section Grate to Sec- feet. of Section Grate to Sec- of tion of of tion of Chimney. Chimney be- ing 8 to 1. Chimney. Chimney be- ing 8 to 1. 20 60 7.5 70 126 15.8 25 68 8.5 75 131 16.4 30 76 9.5 80 135 16.9 35 84 10.5 85 139 17.4 40 93 11.6 90 144 18.0 45 99 12.4 95 148 18.5 50 105 13.1 100 152 19 55 111 13.8 105 156 19.5 60 116 14.5 110 160 20 65 121 15.1 Thurston's rule for rate of combustion effected by a given fieight of chim- ney (Trans. A. S. M. E., xi. 991) is: Subtract 1 from twice the square root of the height, and the result is the rate of combustion in pounds per square foot of grate per hour, for anthracite. Or rate ~ 2 yTi — \, in which h is the height in feet. This rule gives the following: _ h = .50 60 70 80 90 100 110 125 150 175 2G0 2 tyJi- 1 = 13.14 14.49 15.73 16.89 17.97 19 19.97 21.36 23.49 25.45 27.28 The results agree closely with Trowbridge's table given above. In prac- m CHIMNEYS. tice the high rates of combustion for high chimneys given by the formula are not generally obtained, for the reason that with high chimneys there are usually long horizontal flues, serving many boilers, and the friction and the interference of currents from the several boilers are apt to cause the inten- sity of draught in the branch flues leading to each boiler to be much less than that at the base of the chimney. The draught of each boiler is also usually restricted by a damper and by bends in the gas-passages. In a bat- tery of several boilers connected to a chimney 150 ft. high, the author found a draught of %-inch water-column at the boiler nearest the chimney, and only 14-inch at the boiler farthest away. The first boiler was wasting fuel from too high temperature of the chimney-gases, 900°, having too large a grate-surface for the draught, and the last boiler was working below its rated capacity and with poor economy, on account of insufficient draught. The effect of changing the length of the flue leading into a chimney 60 ft. high and 2 ft. 9 in. square is given in the following table, from Box on " Heat" : Length of Flue in feet. Horse-power. Length of Flue in feet. Horse-power. 50 100 200 400 600 107.6 100.0 85.3 70.8 62.5 800 1,000 1,500 2,000 3,000 56.1 51.4 43.3 38.2 31.7 The temperature of the gases in this chimney was assumed to be 552° F., and that of the atmosphere 62°. High Chimneys not Necessary.— Chimneys above 150 ft. in height are very costly, and their increased cost is rarely justified by increased ef- ficiency. In recent practice it has become somewhat common to build two or more smaller chimneys instead of one large one. A notable example is the Speckels Sugar Refinery in Philadelphia, where five separate chimneys are used for one boiler-plant of 7500 H.P. The five chimneys are said to have cost several thousand dollars less than a single chimney of their combined capacity would have cost. Very tall chimneys have been characterized by one writer as " monuments to the folly of their builders. 1 '' Heights of Chimney required for Different Fuels.— The minimum height necessary varies with the fuel, wood requiring the least, then good bituminous coal, and fine sizes of anthracite the greatest. It also varies with the character of the boiler — the smaller and more circuitous the gas-passages the higher the stack required; also with the number of boilers, a single boiler requiring less height than several that discharge into a horizontal flue. No general rule can be given. SIZE OF CHIMNEYS. The formula given below, and the table calculated therefrom for chimneys up to 96 in. diameter and 200 ft. high, were first published by the author in 1884 (Trans. A. S. M. E. vi , 81). They have met with much approval since that date by engineers who have used them, and have been frequently published in boiler-makers' catalogues and elsewhere. The table is now extended to cover chimneys up to 12 ft. diameter and 300 ft. high. The sizes corresponding to the given commercial horse-powers are believed to be ample for all cases in which the draught areas through the boiler-flues and connections are sufficient, say not less than 20$ greater than the area of the chimney, and in which the draught between the boilers and chimney is not checked by long horizontal passages and right-angled bends. Note that the figures in the table correspond to a coal consumption of 5 lbs. of coal per horse-power per hour. This liberal allowance is made to cover the contingencies of poor coal being used, and of the boilers being driven beyond their rated capacity. In large plants, with economical boilers and engines, good fuel and other favorable conditions, which will reduce the maximum rate of coal consumption at any one time to less than 5 lbs. per H. P. per hour, the figures in the table may be multiplied by the ratio of 5 to the maximum expected coal consumption per H.P. per hour. Thus, with conditions which make the maximum coal consumption only 2.5 lbs. per hour, the chimney 300 ft. high X 12 ft. diameter should be sufficient for 6155 X 2 = 12,310 horse-power. The formula is based on the following data : SIZE OF CHIMNEYS. 735 I'.Ji (OCOOCO COiOtOOO - - _■ z 77 .<• "i -^ :-t^coc© 00 00 t^ t- CO -*" iH 38S 333§ s?l 736 CHIMNEYS. 1. The draught power of the chimney varies as the square root of the height. 2. The retarding of the ascending gases by friction may be considered as equivalent to a diminution of the area of the chimney, or to a lining of the chimney by a layer of gas which has no velocity. The thickness of this lining is assumed to be 2 inches for all chimneys, or the diminution of area equal to the perimeter x 2 inches (neglecting the overlapping of the corners of the lining). Let D = diameter in feet, A = area, and E — effective area in square feet. For square chimneys, E — D? — — = A — ■= VA. For round chimeys, E = ^(l> 2 - ~) = A - 0.591 ]/A. For simplifying calculations, the coefficient of YA may be taken as 0.6 for both square and round chimneys, and the formula becomes E = A - 0.6 YA. 3. The power varies directly as this effective area E. 4. A chimney should be proportioned so as to be capable of giving sufficient draught to cause the boiler to develop much more than its rated power, in case of emergencies, or to cause the combustion of 5 lbs. of fuel per rated horse-power of boiler per hour. 5. The power of the chimney varying directly as the effective area, E, and as the square root of the height, H, the formula for horse-power of boiler for a given size of chimney will take the form H.P. = CE YH, in which C is a constant, the average value of which, obtained by plotting the results obtained from numerous examples in practice, the author finds to be 3.33. The formula for horse-power then is H.P. = 3.33.EJ \/H, or H.P. = 3.33(4 - .6 YA) Y~H. If the horse-power of boiler is given, to find the size of chimney, the height being assumed, For round chimneys, diameter of chimney = diam. of E -\- 4". For square chimneys, side of chimney = * ' E + 4". If effective area E is taken in square feet, the diameter in inches_is d — 13.54 YE 4-4", and the side of a square chimney in inches is s — 12 YE-\- 4". (nou p \ 2 — ^ 'J . In proportioning chimneys the height is generally first assumed, with due consideration to the heights of surrounding buildings or hills near to the proposed chimney, the length of horizontal flues, the character of coal to be used, etc., and then the diameter required for the assumed height and horse-power is calculated by the formula or taken from the table. The Protection of Tall Chimney-shafts from Lightning. — C. Molyneux and J. M. Wood (Industries, March 28, 1890) recommend for tall chimneys the use of a coronal or hea\y band at the top of the chimney, with copper points 1 ft. in height at intervals of 2 ft. throughout the circum- ference. The points should be gilded to prevent oxidation. The most ap- proved form of conductor is a copper tape about % in. by % in. thick, weighing 6 ozs. per ft. If iron is used it should weigh not less than 2J4 lbs. per ft. There must be no insulation, and the copper tape should be fastened to the chimney with holdfasts of the same material, to prevent voltaic action. An allowance for expansion and contraction should be made, say 1 in. in 40 ft. Slight bends in the tape, not too abrupt, answer the purpose. For an earth terminal a plate of metal at least 3 ft. sq. and 1/16 in. thick should be buried as deep as possible iu a damp spot. The plate should be of the same metal as the conductor, to which it should be soldered. The best earth terminal is water, and when a deep well or other large body of water is at hand, the conductor should be carried down into it. Pught-angled beucla in the conductor should be avoided. No bend in it should be over 30*, SIZE OF CHIMNEYS. 737 Some Tall Brick Chimneys. 1. Hallsbriickner Hiitte, Sax. 2. Townsend's. Glasgow 3. Tennant's, Glasgow 4. Dobson & Barlow, Bolton. Eng 5. Fall River Iron Co., Boston 6. Clark Thread Co., Newark, N.J 7. Merrimac Mills, Low'l,Mass 8. Washington Mills, Law- rence, Mass . 9. Amoskeag Mills, Manches- ter, N. H 10. Narragansett E. L. Co., Providence, R. I 11. Lower Pacific Mills, Law- rence, Mass . ... 12. Passaic Print Works, Pas saic, N.J 13. Edison Sta,B'kIyn,Two e'ch a 5 S D a Outside Diameter. Capacity by the Author's Formula. be 6 M o H. P. Pounds Coal per hour. 460 454 435 15.7' 33' 32 40 16' 13^221 9,795 66,105 48,975 367^ 350 13' 2" 11 33'10" 30 21 8,245 5,558 41,225 27, 790 335 282'9" 11 12 28' 6" 14 5,435 5,980 27,175 29,900 250 10 3,839 19,195 250 10 3,839 19,195 238 14 7,515 37,575 214 8 2,248 11,240 200 150 9 50" x 120" each 2,771 1,541 13,855 7,705 Notes on the Above Chimneys. — 1. This chimney is situated near Freiberg, on the right bank of the Mulde, at an elevation of 219 feet above that of the foundry works, so that its total height above the sea will be 711% feet. The works are situated on the bank of the river, and the furnace- gases are conveyed across the river to the chimney on a bridge, through a pipe 3227 feet in length. It is built throughout of brick, and will cost about $40,000.— Mfr. and Bldr. 2. Owing to the fact that it was struck by lightning, and somewhat damaged, as a precautionary measure a copper extension subsequently was added to it, making its entire height 488 feet. 1, 2, 3, and 4 were built of these great heights to remove deleterious gases from the neighborhood, as well as for draught for boilers. 5. The structure rests on a solid granite foundation, 55 X 30 feet, and 16 feet deep. In its construction there were used 1,700,000 bricks, 2000 tons of stone, 2000 barrels of mortar, 1000 loads of sand, 1000 barrels of Portland cement, and the estimated cost is $40,000. It is arranged for two flues, 9 feet 6 inches by 6 feet, connecting with 40 boilers, which are to be run in connection with four triple-expansion engines of 1350 horse-power each. 6. It has a uniform batter of 2.85 inches to every 10 feet. Designed for 21 boilers of 200 H. P. each. It is surmounted by a cast-iron cop- ing which weighs six tons, and is composed of thirty-two sections, which are bolted together by inside flanges, so as to present a smooth exterior. The foundation is in concrete, composed of crushed lime- stone 6 parts, sand 3 parts, and Portland cement 1 part. It is 40 feet square and 5 feet deep. Two qualities of brick were used; the outer portions were of the first quality North River, and the backing up was of good quality New Jersey brick. Every twenty feet in vertical measurement an iron ring, 4 inches wide and % to y% inch thick, placed edgewise, was built into the walls about 8 inches from the outer circle. As the chimney starts from the base it is double. The outer wall is 5 feet 2 inches in thick- ness, and inside of this is a second wall 20 inches thick and spaced off about 20 inches from main wall. From the interior surface of the main wall eight buttresses are carried, nearly touching this inner or main fluB wall in order to keep it in line should it tend to sag. The interior wall, starting with the thickness described, is gradually reduced until a height of about 90 feet is reached, when it js diminished to 8 inches. At 165 feet it ceases, 738 CHIMNEYS. and the rest of the chimney is without lining. The total weight of the chim- ney and foundation is 5000 tons. It w r as completed in .September, 1888. 7. Connected to 12 boilers, with 1200 square feet of grate-surface. Draught- Degauge 1 9/16 inches. 8. Connected to 8 boilers, 6' 8" diameter X 18 feet. Grate-surface 448 square feet. 9. Connected to 64 Manning vertical boilers, total grate surface 1810 sq. ft. Designed to burn 18,000 lbs. anthracite per hour. 10. Designed for 12,000 H. P. of engines; (compound condensing). 11. Grate-surface 431 square feet; H.P. of boilers (Galloway) about 2500. 13. Eight boilers (water-tube) each 450 H.P. ; 12 engines, each 300 H.P. Plant designed for 36,000 incandescent lights. For the first 60 feet the exterior wall is 28 inches thick, then 24 inches for 20 feet, 20 inches for 30 feet, 16 inches for 20 feet, and 12 inches for 20 feet. The interior wall is 9 inches thick of fire-brick for 50 feet, and then 8 inches thick of red brick for the next 30 feet. Illustrated in Iron Age, January 2, 1890. A number of the above chimneys are illustrated in Power, Dec, 1890. Chimney at Knoxville, Tenn., illustrated in Eng'g Neivs, Nov. 2, 1893. 6 feet diameter, 120 feet high, double wall: Exterior wall, height 20 feet, 30 feet, 30 feet. 40 feet; " " thickness 211^ in., 17in., 13 in,, 8*^ in. ; Interior wall, height 35 ft., 35 ft., 29 ft., 21ft.; " " thickness 13J^ in., 8}^ in., 4 in., 0. Exterior diameter, 15' 6" at bottom ; batter, 7/16 inch in 12 inches from bot- tom to 8 feet from top. Interior diameter of inside wall, 6 feet uniform to top of interior wall. Space between walls, 16 inches at bottom, diminishing to at top of interior wall. The interior wall is of red brick except a lining of 4 inches of fire-brick for 20 feet from bottom. Stability of Chimneys.— Chimneys must be designed to resist the maximum force of the wind in the locality in which they are built, (see Weak Chimneys, below). A general rule for diameter of base, of brick chimneys, approved by many years of practice in England and the United States, is to make the diameter of the base one tenth of the height. If the chimney is square or rectangular, make the diameter of the inscribed circle of the base one tenth of the height. The " batter " or taper of a chimney should be from 1/16 to 14 inch to the foot on each side. The brickwork should be one brick (8 or 9 inches) thick for the first 25 feet from the top, in- creasing }4 brick (4 or 4}^ inches) for each 25 feet from the top downwards. If the inside diameter exceed 5 feet, the top length should be 1J^ bricks; and if under 3 feet, it may be \y% brick for ten feet. (From The Locomotive, 1884 and 1886.) For chimneys of four feet in diam- eter and one hundred feet high, and upwards, the best form is circular, with a straight batter on the outside. A circular chimney of this size, in addition to being cheaper than any other form, is lighter, stronger, and looks much better and more shapely. Chimneys of any considerable height are not built up of uniform thickness from top to bottom, nor with a uniformly varying thickness of wall, but the wall, heaviest of course at the base, is reduced by a series of steps. Where practicable the load on a chimney foundation should not exceed two tons per square foot in compact sand, gravel, or loam. Where a solid rock- bottom is available for foundation, the load may be greatly increased. If the rock is sloping, all unsound portions should be removed, and the face dressed to a series of horizontal steps, so that there shall be no tendency to slide after the structure is finished. All boiler-chimneys of any considerable size should consist of an outer stack of sufficient strength to give stability to the structure, and an inner stack or core independent of the outer one. This core is by many engineers extended up to a height of but 50 or 60 feet from the base of the chimney, but the better practice is to run it up the whole height of the chimney; it may be stopped off, say, a couple feet below the top, and the outer shell con- tracted to the area of the core, but the better way is to run it up to about 8 or 12 inches of the top and not contract the outer shell. But under no cir- cumstances should the core at its upper end be built into or connected with the outer stack. This has been done in several instances by bricklayers, and the result has been the expansion of the inner core wiiich lifted the top of the outer stack squarely up and crpcked the brickwork. For a height of 100 feet we would make the outer shell in three steps, the first 20 feet high, 16 inches thick, the second 30 feet high, 12 inches thick, the SIZE OF CHIMNEYS. 739 third 50 feet high and 8 inches thick. These are the minimum thicknesses admissible for chimneys of this height, and the batter should be not less than 1 in 36 to give stability. The core should also be built in three steps each of which maybe about one third the height of the chimney, the lowest 12 inches, the middle 8 inches, and the upper step 4 inches thick. This will insure a good sound core. The top of a chimney may be protected by a cast-iron eap; or perhaps a cheaper and equally good plan is to lay the ornamental part in some good cement, and plaster the top with the same material. Weak Chimneys.— James B. Francis, in a report to the Lawrence Mfg. Co. in 1873 (Emfg News, Aug. 28, 1880), gives some calculations con- cerning the probable effects of wind on that company's chimney as then constructed. Its outer shell is octagonal. The inner shell is cylindrical, with an air-space between it and the outer shell; the two shells not being bonded together, except at the openings at the base, but with projections in the brickwork, at intervals of about 20 ft. in height, to afford lateral sup- port by contact of the two shells. The principal dimensions of the chimney are as follows : Height above the surface of the ground 211 ft. Diameter of the inscribed circle of the octagon near the ground . 15 " Diameter of the inscribed circle of the octagon near the top 10 ft. 1% in. Thickness of the outer shell near the base, 6 bricks, or 23J^ in. Thickness of the outer shell near the top, 3 bricks, or 11^ " Thickness of the inner shell near the base, 4 bricks, or .15 " Thickness of the inner shell near the top, 1 brick, or 3% " One tenth of the height for the diameter of the base is the rule commonly adopted. The diameter of the inscribed circle of the base of the Lawrence Manufacturing Company's chimney being 15 ft., it is evidently much less than is usual in a chimney of that height. Soon after the chimney was built, and before the mortar had hardened, it ■ was found that the top had swayed over about 29 in. toward the east. This was evidently due to a strong westerly wind which occurred at that time. It was soon brought back to the perpendicular by sawing into some of the joints, and other means. The stability of the chimney to resist the force of the wind depends mainly on the weight of its outer shell, and the width of its base. The cohesion of the mortar may add considerably to its strength; but it is too uncertain to be relied upon. The inner shell will add a little to the stability, but it may be cracked by the heat, and its beneficial effect, if any, is too uncertain to be taken into account. The effect of the joint action of the vertical pressure due to the weight of the chimney, and the horizontal pressure due to the force of the wind is to shift the centre of pressure at the base of the chimney, from the axis to- ward one side, the extent of the shifting depending on the relative magni- tude of the two forces. If the centre of pressure it brought too near the side of the chimne}^, it will crush the brickwork on that side,, and the chim- ney will fall. A line drawn through the centre of pressure, perpendicular to the direction of the wind, must leave an area of brickwork between it and the side of the chimney, sufficient to support half the weight of the chim- ney; the other half of the weight being supported by the brickwork on the windward side of the line. Different experimenters on the strength of brickwork give very different results. Kirkaldy found the weights which caused several kinds of bricks, laid in hydraulic lime mortar and in Roman and Portland cements, to fail slightly, to vary from 19 to 60 tons (of 2000 lbs.) per sq. ft. If we take in this case 25 tons per sq. ft., as the weight that would cause it to begin to fail, we shall not err greatly. To support half the weight of the outer shell of the chimney, or 322 tons, at this rate, requires an area of 12.88 sq. ft. of brick- work. From these data and the drawings of the chimney, Mr. Francis cal- culates that the area of 12.88 sq. ft. is contained in a portion of the chimney extending 2.428 ft. from one of its octagonal sides, and that the limit to which the centre of pressure may be shifted is therefore 5.072 ft. from the axis. If shifted beyond this, he says, on the assumption of the strength of the brickwork, it will crush and the chimney will fall. Calculating that the wind-pressure can affect only the upper 141 ft. of the chimney, the lower 70 ft. being protected by buildings, he calculates that a wind-pressure of 44.02 lbs. per sq. ft. would blow the chimney down. Rankine, in a paper printed in the transactions of the Institution of Engi- 740 CHIMNEYS. neers, in Scotland, for 1867-68, says: " It had previously been ascertained by observation of the success and failure of actual chimneys, and especially of those which respectively stood and fell during the violent storms of 1856, that, in order that a round chimney may be sufficiently stable, its weight should be such that a pressure of wind, of about 55 lbs. per sq. ft. of a plane surface, directly facing the wind, or 27^ lbs. per sq. ft. of the plane projec- tion of a cylindrical surface, . . . shall not cause the resultant pressure at any bed-joint to deviate from the axis of the chimney by more than one quarter of the outside diameter at that joint," According to Eankine's rule, the Lawrence Mfg. Co.'s chimney is adapted to a maximum pressure of wind on a plane acting on the whole height of 18.80 lbs. per sq. ft., or of a pressure of 21.70 lbs. per sq. ft. acting on the uppermost 141 ft. of the chimney. Steel Chimneys are largely coming into use, especially for tall chim- neys of iron-works, from 150 to 300 feet in height. The advantages claimed are: greater strength and safety; smaller space required; smaller cost, by 30 to 50 per cent, as compared with brick chimneys; avoidance of infiltra- tion of air and consequent checking of the draught, common in brick chim- neys. They are usually made cylindrical in shape, with a wide curved flare for 10 to 25 feet at the bottom. A heavy cast-iron base-plate is provided, to which the chimney is riveted, and the plate is secured to a massive founda- tion by holding-down bolts. No guys are used. F. W. Gordon, of the Phila. Engineering Works, gives the following method of calculating their resist- ance to wind pressure (Poirer, Oct. 1893) : In tests by Sir William Fairbairn we find four experiments to determine the strength of thin hollow tubes. In the table will be found their elements, with their breaking strain. These tubes were placed upon hollow blocks, and the weights suspended at the centre from a block fitted to the inside of the tube. Clear Span, ft. in. Thick- ness Iron, Outside Diame- ter, in. Sectional Area, Breaking Weight, lbs. Breaking W't, lbs, by Clarke's Formula, Constant 1.2. I. II. III. IV. 15 7% 23 5 23 5 .037 .113 .0631 .119 12.4 17.68 18.18 6.74 2,704 11,440 6,400 14,240 9,184 7,302 13,910 Edwin Clarke has formulated a rule from experiments conducted by him during his investigations into the use of iron ana steel for hollow tube bridges, which is as follows : Area of material in sq.in. X Mean depth in in. X Constant Clear span in feet. Center break- | ing load,in tons. j When the constant used is 1.2, the calculation for the tubes experimented upon by Mr. Fairbairn are given in the last column of the table. D. K. Clark's "Rules, Tables, and Data," page 513, gives a rule for hollow tubes as follows : W= 3A4D 2 TS -=-i. W= breaking weight in pounds in centre; D— extreme diameter in inches; T= thickness in inches; L = length be- tween supports in inches; S — ultimate tensile strength in pounds per sq. in. Taking S, the strength of a square inch of a riveted joint, at 35,000 lbs. per. sq. in., this rule figures as follows for the different examples experi- mented upon by Mr. Fairbairn : I, 2870; II, 10,190; III, 7700; IV, 15,320. . This shows a close approximation to the breaking weight obtained by experiments and that derived from Edwin Clarke's and D. K. Clark's rides. We therefore assume that this system of calculation is practically correct, and that it is eminently safe when a large factor of safety is provided, and from the fact that a chimney may be standing for many years without receiving anything like the strain taken as the basis of the calculation, viz., fifty pounds per square foot. Wind pressure at fifty pounds per square foot may be assumed to be travelling in a horizontal direction, and be of the same velocity from the top to the bottom of the stack. This is the extreme assumption. If, however, the chimney is round, its effective area would be only half of its diameter plane. We assume that the entire force may be concentrated in the centre of the height of the section of the chimney under consideration. SIZE OE CHIMNEYS. 741 Taking as an example a 125-foot iron chimney at Poughkeepsie, N. Y., the average diameter of which is 90 inches, the effective surface in square feet upon which the force of the wind may play will therefore be 7% times 125 divided by 2, which multiplied by 50 gives a total wind force of 23,43? pounds. The resistance of the chimney to breaking across the top of the foundation would be 3-14 X 168 2 (that is, diameter of base) X .25 x 35,000 -*- (750 X 4) = 258,486, or 10.6 times the entire force of the wind. We multiply the half height above the joint in inches, 750, by 4, because the chimney is considered a fixed beam with a load suspended on one end. In calculating its strength half way up, we have a beam of the same character. It is a fixed beam at a line half way up the chimney, where it is 90 inches in diam- eter and .187 inch thick. Taking the diametrical section above this line, and the force as concentrated in the centre of it, or half way up from the point under consideration, its breaking strength is: 3.14 X 90 2 x .187 X 35,000 -5- (381 X 4) = 109,220; and the force of the wind tc tear it apart through its cross-section, 7J4 x 62^ x 50-5- 2 = 11,352, or a little more than one tenth of the strength of the stack. The Babcock & Wilcox Co/s book "Steam 1 ' illustrates a steel chimney at the works of the Maryland Steel Co., Sparrow's Point, Md. It is 225 ft. in height above the base, with internal brick lining: 13' 9" uniform inside diameter. The shell is 25 ft. diam. at the base, tapering in a curve to 1? ft. 25 ft. above the base, thence tapering almost imperceptibly to 14' 8" at the top. The upper 40 feet is of J4 _mcn plates, the next four sections of 40 ft. each are respectively 9/32, 5/16, 11/32, and % inch. Sizes of Foundations for Steel Chimneys. (Selected from circular of Phila. Engineering Works.) Half-Lined Chimneys. Diameter, clear, feet 3 Height, feet 100 Least diameter foundation.. 15'9" Least depth foundation 6' Height, feet Least diameter foundation Least depth foundation "Weight of Sheet-iron Smoke-stacks per Foot. (Porter Mfg. Co.) 4 5 6 7 91 11 100 150 150 150 150' 150 16'4" 20'4' 21'10" 22'7" 23'8" 24'H' 6' 9' 8' 9' 10' 10' 125 200 200 250 275 300 18'5" 23'8" 25' 29'8" 33'6" 36' 7' 10' 10' 12' 12' 14' Diam., Thick- Weight Diam., Thick- Weight Diam. Thick- Weight inches. w. g: per ft. inches. W. G. per ft. inches. W. G. per ft. 10 No. 16 7.20 26 No. 16 17.50 20 No. 14 18.33 12 " 8.66 28 " 18.75 22 " 20.00 14 " 9.58 30 " 20.00 24 " 21.66 16 <( 11.68 10 No. 14 9.40 26 " 23.33 20 •' 13.75 12 " 11.11 28 " 25.00 22 " 15.00 14 " 13.69 30 " 26.66 24 " 16.25 16 " 15.00 Sheet-iron Chimneys. (Columbus Machine Co.) Diameter Chimney, inches. Length Chimney, feet. Thick- ness Iron, B. W. G. Weight, lbs. Diameter Chimney, inches. Length Chimney, feet. Thick- ness Iron, B. W. G Weight, lbs. 10 20 No. 16 160 30 40 No. 15 960 15 20 ". 16 240 32 40 " 15 1,020 20 20 " 16 320 34 40 " 14 1,170 22 20 " 16 350» 36 40 Y 14 1,240 24 40 " 16 760 38 40 " 12 1,800 26 40 " 16 826 40 40 " 12 1,890 28 40 •' 15 900 742 THE STEAM-ENGINE. THE STEAM-ENGINE. Expansion of Steam. Isothermal and Adlabatic— Accord- ing to Mariotte's law, the volume of a perfect gas, the temperature being kept constant, varies inversely as its pressure, or p > 1 pv = a constant. The curve constructed from this formula is called the isothermal curve, or curve of equal temperatures, and is a common or rectangular hyperbola. The relation of the pressure and volume of saturated steam, as deduced from Regnault's experiments, and as given in Steam tables, is approxi- mately, according to Rankine (S. E., p. 403), for pressures not exceeding 1^0 lbs., p oc — , orp qc v~ ib, or pv ~\i — pv~ ' = a constant. Zeuner has 1516 found that the exponent 1.0646 gives a closer approximation. When steam expands in a closed cylinder, as in an engine, according to Rankine (S. E., p. 385), the approximate law of the expansion is p <*- — — , or pccv~ 5, ovpv~ = a constant. The curve constructed from this for- mula is called the adiabatic curve, or curve of no transmission of heat. Peabody (Therm., p. 112) says : " It is probable that this equation was obtained by comparing the expansion lines on a large number of indicator- diagrams. . . . There does not appear to be any good reason for using an exponential equation in this connection, . . . and the action of a lagged steam- engine cylinder is far from being adiabatic. . . . For general purposes the hyperbola is the best curve for comparison with the expansion curve of an indicator-card. . . ." Wolff and Denton, Trans. A. S. M. E., ii. 175, say : " From a number of cards examined from a variety of steam-engines in cur- rent use, we find that the actual expansion line varies between the 10/9 adiabatic curve and the Mariotte curve. 1 ' Prof. Thurston (A. S. M. E , ii. 203), says he doubts if the exponent ever becomes the same in any two engines, or even in the same engines at dif- ferent times of the day and under varying conditions of the clay. Expansion of Steam according* to Mariotte's Law and to the Adiabatic Law, (Trans. A. S. M. E., ii. 156.)— Mariotte's law: pv- Pm 1 = Pi^i ; values calculated from formula — = — (1 + hyp log R), in which R — Vj-t-Vi, Pi = absolute initial pressure, Pm — absolute mean pressure, Vj = initial volume of steam incylinder at pressure p x , v 2 = final volume of steam at final pressure. Adiabatic law: pv$ = PiV 1 x $\ values calculated from formula-— = \0R ~ 1 - 9R ~ s - Pi Ratio of Mean Ratio of Mean Ratio of Mean to Initial Ratio to Initial Ratio to Initial Ratio of Pressure. of Pressure. of Pressure. Expan- Expan- sion R. Expan- sion R. Mar. Adiab. Mar. Adiab. sion R. Mar. Adiab. 1.00 1.000 1.000 3.7 .624 .600 6. .465 .438 1.25 .978 .976 3.8 .614 .590 6.25 .453 .425 1.50 .937 .931 3.9 .605 .580 6.5 .442 .413 1.75 .891 .881 4. .597 .571 6.75 .431 .403 2. .847 .834 4.1 .588 .562 7 .421 .393 2.2 .813 .798 4.2 .580 .554 7~25 .411 .383 2.4 .781 .765 4.3 .572 .546 7.5 .402 .374 2.5 .766 .748 4.4 .564 .538 7.75 .393 .365 2.6 .752 .733 4.5 .556 .530 8. .385 .357 2.8 .725 .704 4.6 .549 .523 8.25 .377 .349 3. .700 .678 4.7 .542 .516 8.5 .369 .342 3.1 .688 .666 4.8 .535 .509 8.75 .362 .335 3.2 .676 .654 4.9 .528 .502 9. .355 .328 3.3 .665 .642 5.05 .562 .495 9.25 .349 .321 3.4 .654 .630 5 2 .506 .479 9.5 .342 .315 3.5 .644 .620 5.5 .492 .464 9.75 .336 .309 3.6 .634 .610 5.75 .478 .450 10. .330 .303 MEAN" AND TERMINAL ABSOLUTE PRESSURES. 743 Mean Pressure of Expanded Steam.— For calculations of engines it is generally assumed that steam expands according to Mariotte's law, the curve of the expansion line being a hyperbola. The mean pressure, measured above vacuum, is then obtained from the formula Pm = p 1 + hyp log R in which Pm is the absolute mean pressure, p, the absolute initial pressure taken as uniform up to the point of cut-off, and R the ratio of expansion. If I — length of stroke to the cut-off, L = total stroke, L Pm = - Pjl+Pilhyiplog- and if R = Pm = p 1 + hyp log R Mean and Terminal Abso lute Pressures.— Mariotte's Law. — The values in the following table are based on Mariotte's law, except those in the last column, which give the mean pressure of superheated steam, which, according to Rankine, expands in a cylinder according to the law p V -c~ —^-—l B Area of ABCD = Pl (l + c)(l + hyp log - B = p b (L-x); C = pcc(l -f byp log ^±^) = p b (x + c)(l + hyp log £±£) D = (p, - pc)c = p x c - p h (x + c). D = (Pi - pc)c = p x c - p b (x + c), Area of A = ABCD -(B + C-fD) L + c l + c x -\- c = pS + c)(l + hyp log 4^f) [p b (L - x) 4- jo b (z + c)(l + hyp log ^~^J + Pic - p 5 (x + c)J = Pi(i+c)(l+hyplog^±^) ■Pb [>- ») + (* + c) hyp log x-\- c~ Mean effective pressure = - Example.— Let L = 1, Z = 0.25, « = 0.25, c = 0.1, p x = 1 Area A = 60(.25 4 .l)(l +hyp log -H-). - 2 [(I - .25) + .35 hyp log ■ >lbs., p & = 2 lbs. f] = 21(1 + 1.145) -2[.75- = 45.045 - 2.377 - 6 = 'c 5 X 1.253] - 6 68 = mean effective pressure. The actual indicator-diagram generally shows a mean pressure consider- ably less than that due to the initial pressure and the rate of expansion. The causes of loss of pressure are: 1. Friction in the stop-valves and steam- pipes. 2. Friction or wire-drawing of the steam during admission and cut- off, due chiefly to defective valve-gear and contracted steam-passages. 3. Liquefaction during expansion. 4. Exhausting before the engine has completed its stroke. 5. Compression due to early closure of exhaust. 6. Friction in the exhaust-ports, passages, and pipes. Re-evaporation during expansion of the steam condensed during admis- sion, and valve-leakage after cut off, tend to elevate the expansion line of the diagram and increase the mean pressure. If the theoretical mean pressure be calculated from the initial pressure a-nd the rate of expansion on the supposition that the expansion curve fol- EXPANSION OF STEAM, 745 lows Mariotte's law, pv — a constant, and the necessary corrections are made for clearance and compression, the expected mean pressure in practice may be found by multiplying the calculated results by the factor in the following table, according to Seaton. Particulars of Engine. Factor. Expansive engine, special valve-gear, or with a separate cut-off valve, cylinder jacketed 0.94 Expansive engine having large ports, etc., and good or- dinary valves, cylinders jacketed 0.9 to 0.92 Expansive engines with the ordinary valves and gear as in general practice, and unjacketed 0.8 to 0.85 Compound engines, with expansion valve to h.p. cylin- der; cylinders jacketed, and with large ports, etc 0.9 to 0.92 Compound engines, with ordinary slide-valves, cylinders jacketed, and good ports, etc 0.8 to 0.85 Compound engines as in general practice in the merchant service, with early cut-off in both cylinders, without jackets and expansion-valves 0.7 to 0.8 Fast-running engines of the type and design usually fitted in war-ships 0.6 to 0.8 If no correction be made for clearance and compression, and the engine is in accordance with general modern practice, the theoretical mean pres* sure may be multiplied by 0.96, and the product by the proper factor in the table, to obtain the expected mean pressure. Given the Initial Pressure and the Average Pressure, to Find the Ratio of Expansion and the Period of Admis- sion. P = initial absolute pressure in lbs. per sq. in. ; p = average total pressure during stroke in lbs. per sq. in.; L = length of stroke in inches; I — period of admission measured from beginning of stroke; c = clearance in inches; R = actual ratio of expansion = , T" (1) n _ P(l + hyp log R) p ~ R To find average pressure p, taking account of clearance, n _ P(Z + c) + P{1 + c) hyp lug R-Pc p ~ L ' {4) whence pL + Pc = P(l + c)(l + hyp log R) ; —L + c Given p and P, to find R and I (by trial and error). — There being two un- known quantities R and I, assume one of them, viz., the period of admission I, substitute it in equation (3) and solve for R. Substitute this value of R in the formula (1), or I = — ^ c, obtained from formula (1), and find I. If the result is greated than the assumed value of /, then the assumed value of the period of admission is too long; if less, the assumed value is too short. Assume a new value of /, substitute it in formula (3) as before, and continue by this method of trial and error till the required values of R and I are obtained. Example.— P = 70, p = 42.78, L = 60", c = 3'-, to find I. Assume I = 21 in hyp log B = -j^ - 1 = -4^-3 1 = 1.653 - 1 = .653; hyp log R = .653, whence R = 1.92, HQ THE STEAM-ENGINE. which is greater than the assumed value, 21 inches. Now assume I — 15 inches : hyp log R = - 1 = 1.204, whence £ = 3.5; 15 -f- d Z = — ^ c = -— — 3 = 18 — 3 = 15 inches, the value assumed. K 6.0 Therefore R = 3.5, and I — 15 inches. Period of Admission Required for a Given Actual Ratio of Expansion: I = — =- c, in inches . (4) In percentage of stroke, I = — ^- — '— — p. ct. clearance. . (5) P L+c ~ R' Pressure at any other' a Point of the Expansion.— Let L x = length of stroke up to the given point. Pressure at the given point = (7) L x -f c WORE OF STEAM IN A SINGLE CYLINDER. To facilitate calculations of steam expanded in cylinders the table on the next page is abridged from Clark on the Steam-engine. The actual ratios of expansion, column 1, range from 1.0 to 8.0, for which the hyperbolic logarithms are given in column 2. The 3d column contains the periods of admission i*elative to the actual ratios of expansion, as percentages of the stroke, calculated by formula (5) above. The 4th column gives the values of the mean pressures relative to the initial pressures, the latter being taken as 1, calculated by formula (2). In the calculation of columns 3 and 4, clear- ance is taken into account, and its amount is assumed at 7% of the stroke. The final pressures, in the 5th column, are such as would be arrived at by the continued expansion of the whole of the steam to the end of the stroke, the initial pressure being equal to 1. They are the reciprocals of the ratios of expansion, column 1. The 6th column contains the relative total per- formances of equal weights of steam worked with the several actual ratios of expansion; the total performance, when steam is admitted for the whole of the stroke, without expansion, being equal to 1. They are obtained by dividing the figures in column 4 by those in column 5. The pressui-es have been calculated on the supposition that the pressure of steam, during its admission into the cylinder, is uniform up to the point of cutting off, and that the expansion is continued regularly to the end of the stroke. The relative performances have been calculated without any allow- ance for the effect of compressive action. The calculations have been made for periods of admission ranging from 100$, or the whole of the stroke, to 6.4$, or 1/16 of the stroke. And though, nominally, the expansion is 16 times in the last instance, it is actually only 8 times, as given in the first column. The great difference between the nominal and the actual ratios of expansion is caused by the clearance, which is equal to 7% of the stroke, and causes the nominal volume of steam admitted, namely, 6.4$, to be augmented to 6.4 -J- 7 = 13.4$ of the stroke, or, say, double, for expansion. When the steam is cut off at 1/9, the actual expansion is only 6 times; when cut off at 1/5, the expansion is 4 times; when cut off at %, the expansion is 2% times; and to effect an actual expan- sion to twice the initial volume, the steam is cut off at 46J^$ of the stroke, not at half-stroke. WORK OF STEAM IN A SINGLE CYLINDER. 747 Expansive Working; of Steam— Actual Ratios of Expan- sion, with the Relative Periods ot Admission, Press- ures, and Performance. Steam-pressure 100 lbs. absolute. Clearance atjeach end of the cylinder 7% of the stroke. (Single Cylinder.) 1 2 3 4 5 6 7 8 9 ctual Ratio of Ex- pansion, or No. of Volumes to which the Initial Volume is Expanded. vperbolic Loga- rithm of Actual Ratio of Expan- sion. ±riod of Admis- sion or Cut-off, 7% Clearance. veras-e Total Press- 11 1 1 •-, CD ^ 1 a & . eg o Sa Ratio of Total Per- formance of Equal Weights of Steam. (Col. 4.-*- Col 5.) ® Cfi" = >>£ uantity of Steam Consumed per H.P. of Actual Work done per houi et Capacity of Cyl- inder per lb. of 100 lbs. Steam ad- mitted in 1 stroke. Cubic feet. < W cu £ 1 .0000 100 1 000 1.000 1.000 58,273 34.0 4.05 1.1 .0953 90.3 996 .909 1.096 63,850 31.0 4.45 1.18 .1698 83.3 986 .847 1.164 67,836 29.2 4.78 1.23 .2070 80 980 .813 1.206 70,246 28.2 4.98 1.3 .2624 75.3 969 .769 1.261 73,513 26.9 5.26 1.39 .3293 70 953 .719 1.325 77,242 25.6 5.63 1.45 .3716 66.8 942 .690 1.365 79,555 24.9 5.87 1 54 .4317 62.5 925 .649 1.425 83,055 23.8 6.23 1.6 .4700 59.9 913 .625 1.461 85,125 23.3 6.47 1.75 .5595 54.1 883 .571 1.546 90,115 22.0 7.08 1.88 .6314 50 860 .532 1.616 94,200 21.0 7.61 2 .3931 46.5 836 .5 1.672 97,432 20.3 8.09 2.28 .8241 40 787 .439 1.793 104,466 19.0 9.23 2.4 .8755 37.6 766 .417 1.837 107,050 18.5 9.71 2.65 .9745 33.3 726 .377 1.925 112,220 17.7 10.72 2.9 1.065 29.9 692 .345 2.006 116,88? 16.9 11.74 3.2 1.163 26.4 652 .313 2.083 121,386 16.3 12.95 3.35 1.209 25 637 .298 2.129 124,066 16.0 13.56 3.6 1.281 22.7 608 .278 2.187 127,450 15.5 14.57 3.8 1.335 21.2 589 .263 2.240 130,533 15.2 15.38 4 1.386 19.7 569 .250 2.278 132.770 14.9 16.19 4.2 1.435 18.5 551 .238 2.315 134.900 14.7 17.00 4.5 1.504 16.8 526 .222 2.370 138,130 14.34 18.21 4.8 1.569 15.3 503 .208 2.418 140,920 14.05 19.43 5 1.609 14.4 488 .200 2.440 142,180 13.92 20.23 5.2 1.649 13.6 476 .193 2.466 143,720 13.78 21.04 5.5 1.705 12.5 457 .182 2.511 146.325 13.53 22.25 5.8 1.758 11.4 438 .172 2.547 148,390 13.34 23.47 5.9 1.775 11.1 432 .169 2.556 148,940 13.29 23.87 6.2 1.825 10.3 419 .161 2.585 150,630 13.14 25.09 6.3 1.841 10 413 .159 2.597 151,370 13.08 25.49 6.6 1.887 9.2 398 .152 2.619 152,595 12.98 26.71 7 1.946 8.3 381 .143 2.664 155,200 12.75 28.33 7.3 ',1.988 7.7 369 .137 2.693 156,960 12.61 29.54 7.6 2.028 7.1 357 .132 2.711 157,975 12.53 30.76 7.8 2.054 6.7 348 .128 2.719 158,414 12.50 31.57 8 2.079 6.4 342 .125 2.736 159,433 11.83 32.38 Assumptions op the Table.— That the initial pressure is uniform; that the expansion is complete to the end of the stroke; that the pressure in ex- pansion varies inversely as the volume; that there is no back-pressure of exhaust or of compression, and that clearance is 7% of the stroke at each end of the cylinder. No allowance has been made for loss of steam by cyl- inder-condensation or leakage. Volume of 1 lb. of steam of 100 lbs. pressure per sq. in., or 14,400 lbs. per sq, ft 4.33 cu. ft. Product of initial pressure and volume 62,352 f t.-lbs. 748 THE STEAM-EKGIKE. Though a uniform clearance of 7% at each end of the stroke has been assumed as an average proportion for the purpose of compiling the table, the clearance of cylinders with ordinary slides varies considerably— say from 5% to 10$. (With Corliss engines it is sometimes as low as 2%.) With the clearance, 7%, that has been assumed, the table gives approximate re- sults sufficient for most practical purposes, and more trustworthy than re- sults deduced by calculations based on simple tables of hyperbolic loga- rithms, where clearance is neglected. Weight of steam of 100 lbs. total initial pressure admitted for one stroke, per cubic foot of net capacity of the cylinder, in decimals of a pound = reciprocal of figures in column 9. Total actual work done by steam of 100 lbs. total initial pressure in one stroke per cubic foot of net capacity of cylinder, in foot-pounds = figures in column 7 -j- figures in column 9. Rule 1: To find the net capacity of cylinder for a given weight of steam admitted for one stroke, and a given actual ratio of expansion. (Column 9 of table.)— Multiply the volume of 1 lb. of steam of the given pressure by the given weight in pounds, and by the actual ratio of expansion. Multiply the product by 100, and divide by 100 plus the percentage of clearance. The quotient is the net capacity of the cylinder. Rule 2: To find the net capacity of cylinder for the performance of a given amount of total actual work in one stroke, with a given initial press- ure and actual ratio of expansion — Divide the given work by the total actual work done by 1 lb. of steam of the same pressure, and with the same actual ratio of expansion; the quotient is the weight of steam necessary to do the given work, for which the net capacity is found by Rule 1 preceding. Note.— 1. Conversely, the weight of steam admitted per cubic foot of net capacity for one stroke is the reciprocal of the cylinder-capacity per pound of steam, as obtained by Rule 1. 2. The total actual work done per cubic foot of net capacity for one stroke is the reciprocal of the cylinder-capacity per foot-pound of work done, as obtained by Rule 2. 3. The total actual work done per square inch of piston per foot of the stroke is 1 /144th part of the work done per cubic foot. 4. The l esistance of back pressure of exhaust and of compression are to be added to the net work required to be done, to find the total actual work. Appendix to above Table— Multipliers for Net Cylinder-capacity, and Total Actual Work done. (For steam of other pressures than 100 lbs. per square inch.) Multipliers. Total Pres- sures per square inch. Multipliers. Total Pres- sures per square inch. For Col. 7. Total Work by 1 lb. of Steam. For Col. 9. Capacity of Cylinder. For Col. 7. Total Work by 1 lb. of Steam. For Col. 9. Capacity of Cylinder. lbs. 65 70 75 80 85 90 95 .975 .981 .986 .988 .991 .995 .998 1.50 1.40 1.31 1.24 1.17 1.11 1.05 lbs. 100 110 120 130 140 150 160 1.000 1.009 1.011 1.015 1.022 1.025 1.031 1.00 .917 .843 .781 .730 .683 .644 The figures in the second column of this table are derived by multiplying the total pressure per square foot of any given steam by the volume in cubic feet of 1 lb. of such steam, and dividing the product by 62,352, which is the product in foot-pounds for steam of 100 lbs. pressure. The quotient is the multiplier for the given pressure. The figures in the third column are the quotients of the figures in the second column diyided by the ratio of the pressure of the given steam to 100 lbs. Measures tor Comparing the Duty of Engines.— Capacity is measured in horsepowers, expressed by the initials, H.P.: 1 H.P. = 33.000 ft. -lbs. per minute, = 550 ft.-lbs. per second, = 1,980,000 ft.-lbs. per hour. WOKE OF STEAM IN A SINGLE CYLINDER. 749 1 ft. -lb. = a pressure of 1 lb. exerted through a space of 1 ft. Economy is measured, 1, in pounds of coal per horse-power per hour; 2, in pounds of steam per horse-power per hour. The second of these measures is the more accurate and scientific, since the engine u»3es steam and not coal, and it is indepndent of the economy of the boiler. In gas-engine tests the common measure is the number of cubic feet of gas (measured at atmospheric pressure) per horse-power, but as all gas is not of the same quality, it is necessary for comparison of tests to give the analysis of the gas. When the gas for one engine is made in one gas-pro- ducer, then the number of pounds of coal used in the producer per hour per horse-power of the engine is the proper measure of economy. Economy, or duty of an engine, is also measured in the number of foot- pounds of work done per pound of fuel. As 1 horse-power is equal to 1,980,- 000 ft. -lbs. of work in an hour, a duty of 1 lb. of coal per H.P. per hour would be equal to 1,980,000 ft. -lbs. per lb. of fuel; 2 lbs. per H.P. per hour equals 990,000 ft. -lbs. per lb. of fuel, etc. The duty of pumping-engines is commonly expressed by the number of foot-pounds of work done per 100 lbs. of coal. When the duty of a pumping-engine is thus given, the equivalent number of pounds of fuel consumed per horse-power per hour is found by dividing 198 by the number of millions of foot-pounds of duty. Thus a pumping- engine giving a duty of 99 millions is equivalent to 198/99 = 2 lbs. of fuel per horse-power per hour. Efficiency Measured in Thermal Units per Minute.— Some writers express the efficiency of an engine in terms of the number of thermal units used by the engine per minute for each indicated horse-power, instead of by the number of pounds of steam used per hour. The heat chargeable to an engine per pound of steam is the difference be- tween the total heat in a pound of steam at the boiler-pressure and that in a pound of the feed-water entering the boiler. In the case of condensing engines, suppose we have a temperature in the hot-well of 101° F., corre- sponding to a vacuum of 28 in. of mercury, or an absolute pressure of 1 lb. per sq. in. above a perfect vacuum : we may feed the water into the boiler at that temperature. In the case of a non-condensing-engine, by using a por- tion of the exhaust steam in a good feed-water heater, at a pressure a trifle above the atmosphere (due to the resistance of the exhaust passages through the heater), we may obtain feed-water at 212°. One pound of steam used by the engine then would be equivalent to thermal units as follows : Pressure of steam by gauge: 50 75 100 125 150 175 200 Total heat in steam above 32° : 1172.8 1179.6 1185.0 1189.5 1193.5 1197.0 1200.2 Subtracting 69.1 and 180.9 heat-units, respectively, the heat above 32° in feed -water of 101° and 212° F., we have- Heat given by boiler: Feed at 101° 1103.7 1110.5 1115. a 1120.4 1124.4 1127.9 1131.1 Feed at 212° 991.9 998.7 1004.1 1008.6 1012.6 1016.1 1019.3 Thermal units per minute used by an engine for each pound of steam used per indicated horse-power per hour: Feedatl0l° 18.40 18.51 18.60 18.67 la. 74 IS. 80 18.85 Feed at 212° 16.53 16.65 16.74 16.81 16.88 16.94 16.99 Examples. — A triple-expansion engine, condensing, with steam at 1751bs., gauge and vacuum 28 in., uses 13 lbs. of water per I.H.P. per hour, and a high-speed non-condensing engine, with steam at 100 lbs. gauge, uses 30 lbs. How many thermal units per minute does each consume ? Ans— 13 X 18.80 = 244.4, and 30 X 16.74 = 502.2 thermal units per minute. A perfect engine converting ail the heat -energy of the steam into work would require 33,000 ft. -lbs. -f- 778 = 42.4164 thermal units per minute per indicated horse-power. This figure, 42.4164, therefore, divided by the num- ber of thermal units per minute per I.H.P. consumed by an engine, gives its efficiency as compared with an ideally perfect engine. In the examples above, 42.4164 divided by 244.4 and by 502.2 gives 17.35^ and 8Ao% efficiency, respectively. Total Work Done by One Pound of Steam Expanded in a Single Cylinder. (Column 7 of table.)— If 1 pound of water be con- verted into steam of atmospheric pressure = 2116.8 lbs. per sq. ft., it occu- pies a volume equal to 26.36 cu. ft. The work done is equal to 2116.8 lbs. 750 THE STEAM-EKGINE. X 26.36 ft. = 55,788 ft. -lbs. The heat equivalent of this work is (55,788 -f- 778 =) 71.7 units. This is the work of 1 lb. of steam of one atmosphere acting on a piston without expansion. The gross work thus done on a piston by 1 lb. of steam generated at total pressures varying from 15 lbs. to 100 lbs. per sq. in. varies in round numbers from 56,000 to 62,000 ft.-lbs., equivalent to from 72 to 80 units of heat. This work of 1 lb. of steam without expansion is reduced by clearance according to the proportion it bears to the net capacity of the cylinder. If the clearance be 7$ of the stroke, the work of a given weight of steam with- out expansion, admitted for the whole of the stroke, is reduced in the ratio of 107 to 100. Having determined by this ratio the quantity of work of 1 lb. of steam with- out expansion, as reduced by clearance, the work of the same weight of steam for various ratios of expansion may be found by multiplying it by the relative performance of equal weights of steam, given in the 6th column of the table. Quantity of Steam Consumed per Horse-power of Total Work, per Hour. (Column 8 of table.)— The measure of a horse-power is the performance of 33,000 ft.-lbs. per minute, or 1,980,000 ft.-lbs. per hour. This work, divided by the w r ork of 1 lb of steam, gives the weight of steam required per horse-power per hour. For example, the total actual work done in the cylinder by 1 lb. of 100 lbs. steam, without expansion and with 1% of clearance, is 58,273 ft.-lbs. ; and ' 5g g ' 73 = 34 lbs. of steam, is the weight of steam consumed for the total work done in the cylinder per horse-power per hour. For any shorter period of admission with expansion the weight of steam per horse-pow'er is less, as the total work of 1 lb. of steam is more, and may be found by dividing 1,980,000 ft.-lbs. by the respective total work done; or by dividing 34 lbs. by the ratio of performance, column 6 in the table. Real Ratios of Expansion with Clearances from Q to K%. ' g S Points of Cut-off. 5 a> .10 10 01 9.111 0125 9 0150 8.826 .0175 8.659 .02 8.5 .0225 8.346 .0250 8.2 .0275 8.088 .03 7.933 03-25 7.792 0350 7.666 .0375 7.545 .04 7.428 .0425 7.315 .0450 7.206 .0475 7.102 .05 7 0525 6.901 .0550 6.806 .0575 6.714 06 6.625 .0625 0.538 .065C 6.454 .0675 6.373 .07 6.294 7.481 7.363 7.25 7.133 7.034 6.932 6.833 6.738 6.645 6.555 6.468 6.390 5.605 5.545 4.677 4. 4.595 4.555 4.516 4.41' 4.440 4.404 4.484 4.333 4.130 4.106 4.076 4.017 4.045 3 .25 .30 3.333 3.258 3.24 3.222 3.204 .333 4 3.884 3.875 3.830 3.803 3 2.944 2.930 2.916 2.902 3.777 3.752 3.727 3.702 3.187 3.170 3.153 3.137 2.889 2.876 2.863 2.850 3.678 3.654 3.631 3.608 3.121 3.105 3.089 3.074 2.837 2.824 2.812 2.800 3.58 3.564 3.542 3.521 3.058 3.043 3 028 3.014 2.788 2.776 2.764 2.752 3.5 3.478 3.459 3.439 3 2.986 2.971 2.957 2.741 2.730 2.719 2. 70S 3.418 3 407 3.380 3.362 2.944 2.931 2.917 2.904 2.697 2.686 2.675 2.665 3.342 2.892 2.655 2.667 2.623 2.612 2.602 2.574 2.562 2.552 2 543 2.533 2.524 2.515 2.506 2.49' 2.470 2.461 2.453 2.445 2.428 2.420 2.412 2.5 2.463 2.454 2.445 2.428 2.420 2.411 2.403 2.395 2.: 2.379 2.371 50 .60 .625 2 1.983 1.975 1 970 1.1 1.952 1.947 1.943 1.938 1.934 1.930 2.363 1.925 2.355 1.921 2.348 1.917 2.340 1.913 1.1 1.904 1.900 2.325 2.318 2.311 2.304 2 2.290 2.283 2.276 1.884 1.881 1.43 1.42 1.42 1.42 1.42 1.42 1.41 1.41 41 1.41 1.41 1.41 1.41 1.40 1.40 1.40 1.40 1.40 1.40 1 39 1.39 1.39 1.39 WORK OF STEAM IN A SINGLE CYLINDER. 751 Relative Efficiency of 1 lb. of Steam with and without Clearance; back pressure and compression not considered. Mean total pressure = p = Jg + + Pff + e^hyp. tog, g - ft LetP=l; I, = 100; Z = 25; c = 7. 107 _ 32 ~ ' _ 32-f 32X l.i *~ 100 ~ 100 If the clearance be added to the stroke, so that clearance becomes zero, the same quantity of steam being used, admission I being then = I -f c = 32, and stroke L + c = 107. : .707. 107 107 That is, if the clearance be reduced to 0, the amount of the clearance 7 being added to both the admission and the stroke, the same quantity of steam will do more work than when the clearance is 7 in the ratio 707 : 637, or 11$ more. Back Pressure Considered.— If back pressure = .10 of P, this amount has to be subtracted from p andp x giving p = .537, p x = .607, the work of a given quantity of steam used without clearance being greater than when clearance is 7 per cent in the ratio of 607 : 537, or 13% more. Effect of Compression. —By early closure of the exhaust, so that a portion of the exhaust-steam is compressed into the clearance-space, much of the loss due to clearance may be avoided. If expansion is continued down to the back pressure, if the back pressure is uniform throughout the exhaust-stroke, and if compression begins at such point that the exhaust- steam remaining in the cylinder is compressed to the initial pressure at the end of the back stroke, then the work of compression of the exhaust-steam equals the work done during expansion by the clearance-steam. The clear- ance-space being filled by the exhaust-steam thus compressed, no new steam is required to fill the clearance-space for the next forward stroke, and the work and efficiency of the steam used in the cylinder are just the same as if there were no clearance and no compression. When, however, there is a drop in pressure from the final pressure of the expansion, or the terminal pressure, to the exhaust or back pressure (the usual case), the work of com- pression to the initial pressure is greater than the work done by the expan- sion of the clearance-steam, so that a loss of efficiency results. In this case a greater efficiency can be attained by inclosing for compression a less quantity of steam than that needed to fill the clearance-space with steam of the initial pressure. (See Clark, S. E., p. 399, et seq.; also F. H. Ball, Trans. A. S. M. E., xiv. 1067.) It is shown by Clark that a somewhat greater effi- ciency is thus attained whether or not the pressure of the steam be carried down by expansion to the back exhaust-pressure. As a result of calcula- tions to determine the most efficient periods of compression for various percentages of back pressure, and for various periods of admission, he gives the table on the next page : Clearance in Low- and High-speed Engines. (Harris Tabor, Am. Mach., April 17, 1891.) — The consrruction of the high-speed engine is such, with its i - elatively short stroke, that the clearance must be much larger than in the releasing-valve type. The short-stroke engine is, of necessity, an engine with large clearance, which is aggravated when a variable compression is a feature. Conversely, the releasing-valve gear is, from necessity, an engine of slow rotative speed, where great power is obtainable from long stroke, and small clearance is a feature in its construc- tion. In one case the clearance will vary from 8% to 12% of the piston-dis- placement, and in the other from 2% to 3%. In the case of an engino with a clearance equalling 10% of the piston-displacement the waste room becomes enormous when considered in connection with an early cut-off. The system of compounding reduces the waste due to clearance in proportion as the steam is expanded to a lower pressure. The farther expansion is carried through a train of cylinders the greater will be the reduction of waste due to clear- ance. This is shown from the fact that the high-speed engine, expanding 752 THE STEAM-ENGINE. steam much less than the Corliss, will show a greater gain when changed from simple to compound than its rival under similar conditions. Compression of Steam in the Cylinder. Best Periods of Compression; Clearance 7 per cent. Total Back Pressure, in percentages of the total initial pressure. Cut-off in Percent- ages of % 5 10 15 20 25 30 35 the Stroke. Periods of Compression, in parts of the stroke. 10$ 65$ 57$ 44$ 32$ 15 58 52 40 29 23$ 20 52 47 37 27 22 25 47 42 34 26 21 17$ 30 42 39 32 25 20 16 14$ 12$ 35 39 35 29 23 19 15 13 11 40 36 32 27 21 18 14 13 11 45 33 30 25 20 17 14 12 10 50 30 27 23 18 16 13 12 10 55 27 24 21 17 15 13 11 9 60 24 22 19 15 14 12 11 9 65 22 20 17 15 14 12 10 8 70 19 17 16 14 14 12 10 8 75 17 16 14 13 12 11 9 8 Notes to Table.— 1. For periods of admission, or percentages of back pressure, other than those given, the periods of compression may be readily found by interpolation. 2. For any other clearance, the values of the tabulated periods of com- pression are to be altered in the ratio of 7 to the given percentage of clearance. Cylinder-condensation may have considerable effect upon the best point of compression, but it has not yet (1893) been determined by experiment. (Trans. A. S. M. E.. xiv. 1078.) Cylinder- condensation.— Rankine, S. E., p. 421, says : Conduction of heat to and from the metal of the cylinder, or to and from liquid water contained in the cylinder, has the effect of lowering the pressure at the be- ginning and raising it at the end of the stroke, the lowering effect being on the whole greater than the raising effect. In some experiments the quantity of steam wasted through alternate liquefaction and evaporation in the cylinder has been found to be greater than the quantity wnich performed the work. Percentage of Loss l>y Cylinder-condensation, taken at Cut-off. (From circular of the Ashcroft Mfg. Co. on the Tabor Indicator, 1889.) 13 §J2 ! l! / ^f\ A R N c clearance. The clearance may also be found from the expansion-line by constructing a rectangle efhg, and drawing a diagonal gf to intersect the line XO. This will give the point O, and by erecting a perpendicular to XO we obtain a clearance-line OY. Both these methods for finding the clearance require that the expansion and compression curves be hyperbolas. Prof. Carpenter (Power, Sept. 1893) says that with good diagrams the methods are usually very accurate, and give results which check substantially. The Buckeye Engine Co., however, say that, as the results obtained are seldom correct, being sometimes too little, but more frequently too much, and as the indications from the two curves seldom agree, the operation lias little practical value, though when a clearly defined and apparently undis- torted compression curve exists of sufficient extent to admi,t of the applica- tion of the process, it may be relied on to give much more correct results than the expansion curve. To draw the Hyperholic Curve on the Indicator-dia- gram.— Select any point /in the actual curve, and from this point draw a M B line perpendicular to the line JB, meet- ing the latter in the point J. The line JB may be the line of boiler-pressure, but this is not material ; it may be drawn at any convenient height near the top of diagram and parallel to the atmospheric line. From J draw a diagonal to K, the latter^ point being the intersection of ( the vacuum and clearance lines; from / _ draw IL parallel with the atmospheric line. From L, the point of intersection - of the diagonal JK and the horizontal line IL, draw the vertical line LM. The point M is the theoretical point of cut-off, and LM the cut-off line. Fix upon any number of points 1, 2, 3, etc., on the line JB, and from these points draw diagonals to K. From the intersection of these diagonals with LM draw horizontal lines, and from 1, 2, 3, etc., vertical lines. Where these lines meet will be points in the hyperbolic curve. Pendulum Indicator Rig.— Power (Feb. 1893) gives a graphical representation of the errors in indicator-diagrams, caused by the use of in- Ftg. 140. 760 THE STEAM-ENGIKE. correct form of the pendulum rigging. It is shown that the " brumbo " pulley on the pendulum, to which the cord is attached, does not genet- ally give as good a reduction as a simple piu attachment. When the end of the pendulum is slotted, working in a pin on the crosshead, the error is apt to be considerable at both ends of the card. With a vertical slot in a plate fixed to the crosshead, and a pin on the pendulum working in this slot, the reduction is perfect, when the cord is attached to a pin on the pen- dulum, a slight error being introduced if the brumbo pulley is used. With the connection between the pendulum and the crosshead made by means of a horizontal link, the reduction is nearly perfect, if the construction is such that the connecting link vibrates equally above and Fig. 141. below the horizontal, and the cord is attached by a pin. If the link is horizontal at mid-stroke a serious error is introduced, which is magnified if a brumbo pulley also is used. The adjoining figures show the two forms recommended. Theoretical Water-consumption calculated from the Indicator-card.— The following method is given by Prof. Carpenter (Poiver, Sept. 18y3) : p = mean effective pressure, I = length of stroke in feet, a = area of piston in square inches, a -s- 144 = area in square feet, c = percentage of clearance to the stroke, b = percentage of stroke at point where water rate is to be computed, n — number of strokes per minute, 60w = number per hour, w = weight of a cubic foot of steam having a pres- sure as shown by the diagram corresponding to that at the point where water rate is required, io' — that corresponding to pressure at end of com- pression. Number of cubic feet per stroke = l(^ ~T )-rrj> Corresponding weight of steam per stroke in lbs. — l{^ "t" Jrrrr lea „ . . L ' , ^ . , Icaio' Weight of steam m clearance = - . Total weight of | _ rf b+c ^wa _ Icaw' _ la r -, steam per stroke) ~ l V ioo / 144 14,400 ~ 14,400|_ ^ ' J' Total weight of steam { _ Wnla r , ~\ from diagram per hour, ~ i^iOO L "*" ' " ,CW J' The indicated horse-power is p I a n -i- 33,000. Hence the steara-consump* tion per indicated horse-power is lM00L (b + c)w - CW/ J 137.50,- L , n -p-Ta~^ = -^~\} b + ° )W ~ CW [ 33,000 Changing the formula to a rule, we have: To find the water rate from the indicator diagram at any point in the stroke. Rule.— To the percentage of the entire stroke which has been completed by the piston at the point under consideration add the percentage of clear- ance. Multiply this result by the weight of a cubic foot of steam, having a pressure of that at the required point. Subtract from this the product of percentage of clearance multiplied by weight of a cubic foot of steam hav- ing a pressure equal to that at the end of the compression. Multiply this result by 137.50 divided by the mean effective pressure.* Note.— This method only applies to points in the expansion curve or be- tween cut-off and release. * For compound or triple-expansion engines read: divided by the equiva- lent mean effective pressure, on the supposition that all work is done in one cylinder. COMPOUND ENGINES. 761 The beneficial effect of compression in reducing the water-consumption of an engine is clearly shown by the formula. If the compression is carried to such a point that it produces a pressure equal to that at the point under consideration, the weight of steam per cubic foot is equal, and w = w'. In this case the effect of clearance entirely disappears, and the formula becomes — —(bw). P In case of no compression, 10' becomes zero, and the water-rate = Prof. Denton (Trans. A. S. M. E., xiv. 1363) gives the following table of theoretical water-consumption for a perfect Mariotte expansion with steam at 150 lbs. above atmosphere, and 2 lbs. absolute back pressure : Ratio of Expansion, r. M.E.P., lbs. per sq. in. Lbs. of Water per hour per horse-power, W. 10 15 20 25 30 35 52.4 38.7 30.9 25.9 22.2 19.5 9.68 8.74 8.20 7.84 7.63 7.45 The difference between the theoretical water -consumption found by the formula and the actual consumption as found by test represents " water not accounted for by the indicator, 1 ' due to cylinder condensation, leakage through ports, radiation, etc. Leakage of Steam.— Leakage of steam, except in rare instances, has so little effect upon the lines of the diagram that it can scarcely be detected. The only satisfactory way to determine the tightness of an engine is to take it when not in motion, apply a full boiler-pressure to the valve, placed in a closed position, and to the piston as well, which is blocked for the purpose at some point away from the end of the stroke, and see by the eye whether leakage occurs. The indicator-cocks provide means for bringing into view steam which leaks through the steam-valves, and in most cases that which leaks by the piston, and an opening made in the exhaust-pipe or observa- tions at the atmospheric escape-pipe, are generally sufficient to determine the fact with regard to the exhaust-valves. The steam accounted for by the indicator should be computed for both the cut-off and the release points of the diagram. If the expansion -line de- parts much from the hyperbolic curve a very different result is shown at one point from that shown at the other. In such cases the extent of the loss occasioned by cylinder condensation and leakage is indicated in a much more truthful manner at the cut-off than at the release. (Tabor Indicator Circular.) COMPOUND ENGINES. Compound, Triple- and Quadruple-expansion Engines. — A compound engine is one having two or more cylinders, and in which the steam after doing work in the first or high-pressure cylinder completes its expansion in the other cylinder or cylinders. The term "compound" is commonly restricted, however, to engines in which the expansion takes place in two stages only— high and low pressure, the terms triple-expansion and quadruple-expansion engines being used when the expansion takes place respectively in three and four stages. The number of cylinders may be greater than the number of stages of expansion, for constructive reasons; thus in the compound or two-stage expansion engine the low-pressure stage may be effected in two cylinders so as to obtain the advantages of nearly equal sizes of cylinders and of three cranks at angles of 120°. In triple- expansion engines there are frequently two low-pressure cylinders, one of them being placed tandem with the high-pressure, and the other with the intermediate cylinder, as in mill engines with two cranks at 90°. In the triple-expansion engines of the steamers Campania and Lucania, 762 THE STEAM-ENGINE* with three cranks at 120°, there are five cylinders, two high, one intermedi- ate, and two low, the high-pressure cylinders being tandem with the low. Advantages of Compounding.— The advantages secured by divid- ing the expansion into two or more stages are twofold: 1. Reduction of wastes of steam by cylinder-condensation, clearance, and leakage; 2. Dividing the pressures on the cranks, shafts, etc., in large engines so as to avoid excessive pressures and consequent friction. The diminished loss by cylinder-conden- sation is effected by decreasing the range of temperature of the metal sur- faces of the cylinders, or the difference of temperature of the steam at; admission and exhaust. When high-pressure steam is admitted into a single- cylinder engine a large portion is condensed by the comparatively cold, metal surfaces; at the end of the stroke and during the exhaust the w:ater is re-evaporated, but the steam so formed escapes into the atmosphere or into the condenser, doing no work; while if it is taken into a second cylinder, as in a compound engine, it does work. The steam lost in the first cylinder by leakage and clearance also does work in the second cylinder. Also, if there is a second cylinder, the temperature of the steam exhausted from the first cylinder is higher than if there is only one cylinder, and the metal surfaces therefore are not cooled to the same degree. The difference In temperatures and in pressures corresponding to the work of steam of 150 lbs. gauge-pressure expanded 20 times, in one, two, and three cylinders, is shown in the following table, by W. H. Weightman, Am. Mach., July 28, Single Cyl- inder. Compound Cylinders. Triple-expansion Cylinders. Diameter of cylinders, in. . Area ratios Expansions Initial steam - pressures- absolute— pounds Mean pressures, pounds. . Mean effective pressures, pounds Steam temperatures into cylinders Steam temperatures out of the cylinders Difference in temperatures Horse-power developed. . . Speed of piston ... Total initial pressures on pistons, pounds 165 32.1 184°. 2 181.8 455.218 33 1 5 165 86.11 53.11 366° 259°. 9 106.1 399 290 112,900 19.68 15.68 184°. 2 175.7 403 290 28 1 2.714 165 121.44 293°. 5 72.5 46 2.70 2.714 61 4.746 2.714 60.8 44.75 22.4 16.49 22.35 12.49 293°. 5 234°. 1 234°. 1 59.4 268 184°. 2 49.9 264 817 53,773 64 Woolf " and Receiver Types of Compound Engines.- The compound steam-engine, consisting of two cylinders, is reducible to two forms, 1, in which the steam from the h.p. cylinder is exhausted direct into the 1. p. cylinder, as in the Woolf engine; and 2, in which the steam from the h. p. cylinder is exhausted into an intermediate reservoir, whence the steam is supplied to, and expanded in, the 1. p. cylinder, as in the " receiver- engine.' 1 If the steam be cut off in the first cylinder before the end of the stroke, the total ratio of expansion is the product of the ratio of expansion in the first cylinder, into the ratio of the volume of the second to that of the first cylinder: that is, the product of the two ratios of expansion. Thus, let the areas of the first and second cylinders be as 1 to 3J4 the strokes being equal, and let the steam be cut off in the first at'J^ stroke ; then Expansion in the 1st cylinder „ 1 to 2 " "2d " ■. lto3^| Total or combined expansion, the product of the two ratios. . . 1 to 7 Woolf Engine, without Clearance— Ideal Diagrams. - The diagrams of pressure of an ideal Woolf engine are shown in Fig. 142, as I they would be described by the indicator, according to the arrows. In these j diagrams pq is the atmospheric line, mn the vacuum line, pc| the admission COMPOUKD ENGINES. 763 line, dg the hyperbolic curve of expansion in the first cylinder, and gh the con- secutive expansion-line of back pressure for the return -stroke of the first piston, and of positive pressure for the steam- stroke of the second piston. At the point h, at the end of the stroke of the second piston, the steam is exhausted into the condenser, and the pressure falls to the level of perfect vacuum, mn. The diagram of the second cylinder, below gh, is characterized by the absence of any specific period of admission ; the whole of the steam-line gh being expan- sional, generated by the expansion of the initial body of steam contained in the first cylinder into the second. When the return-stroke is completed, the whole of the steam transferred from the first is shut into the second cylin- w 142 WnnTW w wrTwl . t^at der Thft final nvossnm pnrl vnlnn p nf * 1G - l?^.— WOOLF ENGINE— IDEAL the steam in the second cylinder are the same as if the whole of the initial steam had been admitted at once into the second cylinder, and then expanded to the end of the stroke in the manner of a single-cylinder engine. The net work of the steam is also the same, according to both distributions. Receiver-engine, without Clearance— Ideal Diagrams.— In the ideal receiver-engine the pistons of the two cylinders are con- nected to cranks at right angles to each other on the same shaft. The receiver takes the steam exhausted from the first cylinder and supplies it to the second, in which the steam is cut off and then expanded to the end of the stroke. On the assumption that the initial pressure in the second cylin- der is equal to the final pressure in the first, and of course equal to the pres- sure in the receiver, the volume cut off in the second cylinder must be equal to the volume of the first cylinder, for the second cylinder must admit as much steam at each stroke as is discharged from the first cylinder. In Fig. 143 cd is the line of admission and hg the exhaust-line for the first < i 3 7 f r __ i ■* V *^ 2 ! -60 lbs d - \ -40 —-n--> < x A-h-> -2C i y / h _ P k L 0. 7 <1 Fig. 143.— Receiver-engine, Ideal Indicator-diagrams. Fig. 144.— Receiver Engine, Ideal Diagrams reduced and combined. cylinder; and dg is the expansion-curve and pq the atmospheric line. In the region below the exhaust-line of the first cylinder, between it and the line of perfect vacuum, ol, the diagram of the second cylinder is formed; hi, the second line of admission, coincides with the exhaust-line hg of the first cylinder, showing in the ideal diagram no intermediate fall of pressure, and ik is the expansion-curve. The arrows indicate the order in which the dia- grams are formed. In the action of the receiver-engine, the expansive working of the steam, though clearly divided into two consecutive stages, is, as in the Woolf engine, essentially continuous from the point of cut-off in the first cylinder to the end of the stroke of the second cylinder, where it is delivered to the condenser; and the first and second diagrams may be placed together and 764 THE STEAM-EXGIHE. combined to form a continuous diagram. For this purpose take the second diagram as the basis of the combined diagram, namely, hiklo, Fig. 144. The period of admission, hi, is one third of the stroke, and as the ratios of the cylinders are as 1 to 3, hi is also the proportional length of the first diagram as applied to the second. Produce oh upwards, and set off oc equal to the total height of the first diagram above the vacuum-line; and, upon the shortened base hi, and the height he, complete the first diagram with the steam-line cd, and the expansion-line di. It is shown by Clark (S. E., p. 432, et seq.) in a series of arithmetical cal- culations, that the receiver-engine is an elastic system of compound engine, in which considerable latitude is afforded for adapting the pressure in the receiver to the demands of the second cylinder, without considerably dimin- ishing the effective work of the engine. In the Woolf engine, on the contrary, it is of much importance that the intermediate volume of space between the first and second cylinders, which is the cause of an interme- diate fall of pressure, should be reduced to the lowest practicable amount. Supposing that there is no loss of steam in passing through the engine, by cooling and condensation, it is obvious that whatever steam passes through the first cylinder must also find its way through the second cylin- der. By varying, therefore, in the receiver-engine, the period of admission in the second cylinder, and thus also the volume of steam admitted for each stroke, the steam will be measured into it at a higher pressure and of a less bulk, or at a lower pressure and of a greater bulk; the pressure and density naturally adjusting themselves to the volume that the steam from the re- ceiver is permitted to occupy in the second cylinder. With a sufficiently restricted admission, the pressure in the receiver may be maintained at the pressure of the steam as exhausted from the first cylinder. On the con- trary, with a wider admission, the pressure in the receiver may fall or "drop" to three fourths or even one half of the pressure of the exhaust- steam from the first cylinder. (For a more complete discussion of the action of steam in the Woolf and receiver engines, see Clark on the Steam-engine.) Combined Diagrams of Compound Engines,— The only way of making a correct combined diagram from the indicator-diagrams of the several cylinders in a compound engine is to set off all the diagrams on the same horizontal scale of volumes, adding the clearances to the cylinder ca- Fig. 145. pacifies proper. When this is attended to, the successive diagrams fall ex- actly into their right places relatively to one another, and would compare properly with any theoretical expansion-curve. (Prof. A. B. W. Kennedy, Proc. Inst. M. E., Oct. 1886.) COMPOUND ENGINES. ?65 This method of combining diagrams is commonly adopted, but there are objections to its accuracy, since the whole quantity of steam consumed in the first cylinder at the end of the stroke is not carried forward to the second, but a part of it is retained in the first cylinder for compression. For a method of combining diagrams in which compression is taken account of, see discussions by Thomas Mudd and others, in Proc. Inst. M. E., Feb., 1887, p. 48. The usual method of combining diagrams is also criticised by Frank H. Ball as inaccurate and misleading (Am. Mach., April 12, 1891; Trans. A. S. M. E., xiv. 1405, and xv. 403). Figure 145 shows a combined diagram of a quadruple-expansion engine, drawn according to the usual method, that is, the diagrams are first reduced in length to relative scales that correspond wiih the relative piston-displace- ment of the three cylinders. Then the diagrams are placed at such distances from the clearance-line of the proposed combined diagram as to correctly represent the clearance in each cylinder. Calculated Expansions and Pressures in Two-cylinder Compound Engines. (James Tribe, Am. Mach., Sept. 8i Oct. 1891.) Two-cylinder Compound Non-condensing. Back pressure ^3 lb. above atmosphere. Initial gauge pressure Initial absolute pressure Total expansion . Exp a n s i o n s in each cylinder.. Hyp. log:, plus 1. Forward \ High, pressures ) Low. . Back j High, pressures I Low Mean effective pressures , Ratio-c y 1 i n d e 1 areas a J High. ^"JLow. 100 110 120 130 140 150 115 7.39 125 7.84 135 8.41 145 9 155 9.61 165 10.24 2.7 1.993 84.8 31.3 42.5 15.5 2.8 2.029 90.5 32.3 44.6 15.5 2.9 2.064 96 33.1 46.5 15.5 3 2.008 101.4 33.7 48.3 15.5 3.10 2.131 106.5 34.3 50 15.5 3.2 2.163 111.5 31.8 51.5 15.5 42.3 15.8 45.9 16.8 49.5 17.6 53.1 18.2 56.5 18.8 60 19.3 2.67 2.73 2.81 2.91 3 3.11 lv5 10.89 3.3 2.193 116.3 35.2 53 15.5 63.3 19.7 185 11.56 3.4 2.223 120.9 3.6 51.4 15.5 66.5 20 1 2.238 123.2 35.7 55 15.5 68.2 20.2 Two-cylinder Compound Condensing. Back pressure, 6.5 lbs. above vacuum . Initial gauge-pressures Initial absolute pressures. . Probable per cent of loss. . Total expansions Exps. in each cylinder. . .. Hyp. log. plus 1 Mean forward j High pressures 1 Low Mean back j High pressures ( Low Mean effective pressures Terminal pressures (High., | Low . . High. Low. [j 1 cnauyc: Initial pressure in 1. p. cyl. Ratio of cylinder areas. . 105 2.6 15.7 3.96 2.376 62.9 15.2; 5 4.3 10.95 26.5 25.3 3.32 100 115 2.9 17 4.13 2.418 67.3 15.55 27.8 4.3 19.5 11.25 27.8 6.45 !6.6 3.51 110 125 3.3 18.5 4.3 2.458 71.4 15.9 29 4.3 42.4 11.6 19.0 6.45 27.8 3.66 120 135 3.6 20 4.47 2.49; 75.4 16.2 30.2 4.3 45.2 11.9 30.2 6.5 21.5 4.64 2.534 79.3 16.5 31.4 4.3 47.9 12.2 31.4 6.55 ;o.2 140 155 4.0 22.7 4.77 2.562 83.2 16.75 32.4 4.3 50.8 12.45 31.4 4.08 150 165 4.3 24.2 4.92 2.593 87 17.05 33.5 4.3 53.5 12.75 33 5 6.6 32.4 4.19 The probable percentage of loss, line 3, is thus explained : There is always a loss of heat due to condensation, and which increases with the pressure of steam. The exact percentage cannot be predetermined, as it depends largely upon the quality of the non-conducting covering used on the cylin- der, receiver, and pipes, etc . but will probably be abonr as shown. Proportions of Cylinders in Compound Engines.— Authori- ties differ as to the proportions by volume of the high and low pressure cylinders v and V. Thus Grashof gives V-+- v = 85 frl Krabak, 0.90 4/rJ 766 THE STEAM-EHGIHE. Werner, |r; and Rankine,|/r2, r being the ratio of expansion. Busley makes the ratio dependent on the boiler-pressure thus: Lbs. per sq . in 60 90 105 120 V-i-v = 3 4 4.5 5 (See Season's Manual, p. 95, etc.. for analytical method; Sennett, p. 496, etc. ; Clark's Steam-engine, p. 445, etc; Clark, Rules, Tables, Data, p. 849, etc.) Mr. J. McFarlane Gray states that he finds the mean effective pressure in the compound engine reduced to the low-pressure cylinder to be approxi- mately the square root of 6 times the hoiler-pressure. Approximate Horse-power of a Modern Compound Marine-engine. (Seaton.)— The following rule will give approximately the horse-power developed by a compound engine made in accordance with „ . . , ,„„ D* X VpX RX S modern marine practice. Estimated H.P. = ~-r . D = diameter of l.p. cylinder; p — boiler-pressure by gauge; B = revs, per min.; S = stroke of piston in feet. Ratio of Cylinder Capacity in Compound Marine En- gines. (Seaton.)— The low-pressure cylinder is the measure of the power of a compound engine, for so long as the initial steam-pressure and rate of expansion are the same, it signifies very little, so far as total power only is concerned, whether the ratio between the low and high-pressure cylinders is 3 or 4; but as the power developed should be nearly equally divided be- tween the two cylinders, in order to get a good and steady working engine, there is a necessity for exercising a considerable amount of discretion in fixing on the ratio. In choosing a particular ratio the objects are to divide the power evenly and to avoid as much as possible "drop " and high initial strain. If increased economy is to be obtained by increased boiler-pressures, the rate of expansion should vary with the initial pressure, so that the pressure at which the steam enters the condenser should remain constant. In this case, with the ratio of cylinders constant, the cut-off in the high-pressure cylinder will vary inversely as the initial pressure. Let R be the ratio of the cylinders; r, the rate of expansion; p t the initial pressure: then cut-off in high-pressure cylinder = R -r- r; r varies withpj, so that the terminal pressure p n is constant, and consequently r — Pt -*- J>n; therefore, cut-off in high-pressure cylinder — R X pn -s-Pi- Ratios of Cylinders as Found in Marine Practice.— The rate of expansion may be taken at one tenth of the boiler-pressure (or about one twelfth the absolute pressure), to work economically at full speed. Therefore, when the diameter of the low-pressure cylinder does not exceed 100 inches, and the boiler-pressure 70 lbs., the ratio of the low-pressure to the high-pressure cylinder should be 3.5; for a boiler-pressure of 80 lbs., 3.75; for 90 lbs., 4.0; for 100 lbs., 4.5. If these proportions are adhered to, there will be no need of an expansion-valve to either cylinder. If, however, to avoid " drop,' 1 the ratio be reduced, an expansion-valve should be fitted to the high-pressure cylinder. Where economy of steam is not of first importance, but rather a, large power, the ratio of cylinder capacities may with advantage be decreased, so that with a boiler-pressure of 100 lbs. it may be 3.75 to 4. In tandem engines there is no necessity to divide the work equally. The ratio is generally 4, but when the steam-pressure exceeds 90 lbs. absolute 4.5 is better, and for 100 lbs. 5.0. When the power requires that the 1. p. cylinder shall be more than 100 in. diameter, it should be divided in two cylinders. In this case the ratio of the combined capacity of the two 1. p. cylinders to that of the h. p. may be 3.0 for 85 lbs. absolute, 3.4 for 95 lbs., 3.7 for 105 lbs., and 4.0 for 115 lbs. Receiver Space in Compound Engines should be from 1 to 1.5 times the capacity of the high-pressure cy Under, when the cranks are at an angle of from 90° to 120°. When the cranks are at 180° or nearly this, the space may be very much reduced. In the case of triple-compound en- gines, with cranks at 120°, and the intermediate cylinder leading the high- pressure, a very small receiver will do. The pressure in the receiver should never exceed half the boiler-pressure. (Seaton.) COMPOUND ENGINES. 767 Formula for Calculating the Expansion and the Work of Steam in Compound Engines. (Condensed from Clark on the " Steam-engine.") a = area of the first cylinder in square inches; a' — area of the second cylinder in square inches; r = ratio of the capacity of the second cylinder to that of the first; L = length of stroke in feet, supposed to be the same for both cylinders; I = period of admission to the first cylinder in feet, excluding clearance; c = clearance at each end of the cylinders, in parts of the stroke, in feet; L' = length of the stroke plus the clearance, in feet; I' = period of admission plus the clearance, in feet; s = length of a given part of the stroke of the second cylinder, in feet; P = total initial pressure in the first cylinder, in lbs. per square inch, sup- posed to be uniform during admission; P' = total pressure at the end of the given part of the stroke s; p — average total pressure for the whole stroke; B = nominal ratio of expansion in the first cylinder, or L -j- Z; B' = actual ratio of expansion in the first cylinder, or L' -*- V; B" = actual combined ratio of expansion, in the first and second cylinders together; n = ratio of the final pressure in the first cylinder to any intermediate fall of pressure between the first and second cylinders; N = ratio of the volume of the intermediate space in the Woolf engine, reckoned up to, and including the clearance of, the second piston, to the capacity of the first cylinder plus its clearance. The value of N is correctly expressed by the actual ratio of the volumes as stated, on the assumption that the intermediate space is a vacuum when it receives the exhaust-steam from the first cylinder. In point of fact, there is a residuum of unexhausted steam in the interme- diate space, at low r pressure, and the value of N is thereby prac- tically reduced below the ratio here stated. N = — 1. n — 1 to = whole net work in one stroke, in foot-pounds. Ratio of expansion in the second cylinder: In the Woolf engine, -^ y . Total actual ratio of expansion = product of the ratios of the thrse con- secutive expansions, in the first cylinder, in the intermediate space, and in the second cylinder, In the Woolf engine, B' (r — -f- N j . ' I' Combined ratio of expansion behind the pistons = — — — rR' = B". n Work done in the two cylinders for one stroke, with a given cut-off and a given combined actual ratio of expansion: Woolf engine, iv = 11 72 Ratios of cylinder areas = ^=^ = 1 to 3.189. In this calculation no account is taken of clearance, nor of drop between cylinders, nor of area of piston-rod. It also assumes that the diagrams in both cylinders are the full theoretical diagrams, with hyperbolic expansion curves, with no allowance for rounding of the corners. Calculation of Diameters of Cylinders of a 500 H.P. Compound Non-con- densing Engine. — Assuming initial pressure 170 lbs. above atmosphere, back pressure 15.5 lbs., absolute piston-speed 600 feet per minute. Total Expansions -185 h- 15.5 = 11.9. Expansions in each cylinder = |/H-9 = 3.45; hyp log = 1.238. Terminal pressure h. p. cyl. = 185 -h 3.45 = 53.6 lbs. Mean total pressure, " " = 53.6 X (1 + 1.238) = 120.0. Back pressure h. p. cyl. = terminal pressure 53.6 lbs. Mean effective pressure = 120 - 53.6 = 66.4 lbs. Terminal pressure 1. p. cyl. = 53.6 -s- 3.45 = 15.5 lbs. Mean total pressure " " = 15.5 X 2.238 = 34.7 lbs. Mean effective pressure 1. p. cyl. = 34.7 — 15.5 ^ 19.2 lbs. 19 2 Ratio of areas of cylinders = x—, = 1 to 3.46. bo. A Area of 1. p. cyl. = 33000 X H.P. 30.2" piston-speed X M.E.P." 600 X 19.2 Area of h. p. cyl., 716 -=- 3.46 = 207 sq. in. = 16.2 in. diameter, TRIPLE-EXPANSION ENGINES. 769 TRIPLE-EXPANSION ENGINES. Proportions of Cylinders.— H. H. Suplee, Mechanics, Nov. 1887, ives the following method of proportioning cylinders of triple-expansion engines: As in the case of compound engines the diameter of the low-pressure cylinder is first determined, being made large enough to furnish the entire power required at the mean pressure due to the initial pressure and expan- ; sion ratio given; and then this cylinder is only given pressure enough to per- form one third of the work, and the other cylinders are proportioned so as to i divide the other two thirds between them. Let us suppose that an initial pressure of 150 lbs. is used and that 900 H.P. is to be developed at a piston-speed of 800 ft. per min., and that an expan- sion ratio of 16 is to be reached with an absolute back pressure of 2 lbs. The theoretical M.E.P. with an absolute initial pressure of 150 X 14.7 = 164.7 lbs. initial at 16 expansions is fq+hypio g i6) = 1M7 sjrm lb lb less 2 lbs. back pressure, = 38.83 - 2 = 36.83. In practice only about 0.7 of this pressure is actually attained, so that S.83 x 0.7 = 25.781 lbs. is the M.E.P. upon which the engine is to be pro- portioned. To obtain 900 H.P. we must have 33,000 X 900 = 29,700,000 foot-pounds, and this divided by the mean pressure (25.78) and by the speed in feet (800) will give 33000 X 900 800-X-25T8 =1440Sq - ln ' for the area of the 1. p. cylinder, which is about equivalent to 43 in. diam. Now as one third of the work is to be done in the 1. p. cylinder, the M.E.P. in it will be 25.78 -h 3 = 8.59 lbs. The cut-off in the high-pressure cylinder is generally arranged to cut off at 0.6 of the stroke, and so the ratio of the h. p. to the 1. p. cylinder is equal to 16 X 0.6 = 9.6, and the h. p. cylinder will be 1440 -=- 6 = 150 sq. in. area, or about 14 in. diameter, and the M.E.P. in the h. p. cylinder is equal to 9.6 X 8.59 = 82.46 lbs. If the intermediate cylinder is made a mean size between the other two, its size would be determined by dividing the area of the 1. p. cylinder by the square root of the ratio between the low and the high; but in practice this is found to give a result too large to equalize the stresses, so that instead the area of the 1. p. cylinder is found by dividing the area of the 1. p. piston by 1.1 times the square root of the ratio of 1. p. to h. p. cylinder, which in this case is 1440 -=- (1.1 V9.6) = 422.5 sq. in., or a little more than 23 in. diam. To put the above into the form of rules, we have Area of low-pressure piston Area h. p. cyl. = — Area intermediate cjl Cutoff in h. p. cyl. X rate of expansion. Area of low-pressure p ; ston 1.1 X Vratio of 1. p. to h. p. cyl. The choice of expansion ratio is governed by the initial pressure, and is generally chosen so that the terminal pressure in the 1. p. cylinder shall be about 10 lbs. absolute. Annular Ring Method.— Jay M. Whitham, Trans. A. S. M. E., x. 577, give,s the following method of ascertaining the diameter of pistons of triple expansion engines: Lay down a theoretical indicator-diagram of a simple engine for the par- ticular expansion desired. By trial find (with the polar planimeter or other- wise) the position of horizontal lines, parallel to the back-pressure line, such that the three areas into which they divide the diagram, representing low, intermediate, and high pressure diagrams, marked respectively A, B, and C, are equal. Find the mean ordinate of each area: that of " C ,1 will be the mean un- balanced pressure on the small piston; that of " B " will be the mean unbal- anced pressure on the area remaining after subtracting the area Of the small piston from that of the intermediate; and that of the area "A " will denote 770 THE STEAM-ENGINE. the mean unbalanced pressure on a square inch of the annular ring of the large piston obtained by subtracting tne intermediate from the large piston We thus see that the mean ordinates of the two lower cards act on annular rings. Let H = area of small piston in square inches; 1 = " " intermediate piston in square inches; L = " " large piston in square inches; Ph — mean unbalanced pressure per square inch from card "C"; Pi = " " " " B"; Pi = " " " A"; S = piston-speed in feet per minute; (I.H.P.) = indicated horse-power of engine. Then for equal work in each cylinder we have: Area of small piston H = 33,000 X 5S-- _=_ {ph x s)\ . • (1 Area of annular ring of j _ , intermediate cylinder j — ' 1,000 x t^ -*■ ( Pi X S); Area of interme- { diate piston j I = H 4- 33,000 X - Area of annular ring of large piston = 33,000 x Area of large piston = L = I + 33,000 x — ~ >- (Pi X S) I.H.P. ^ 3 : -*-(PlX8); (p X 8); (3) This method is illustrated by the following example: Given I.H.P. = 3000, piston-speed S = 900 ft. permin., ratio of expansion 10, initial steam=pres- sure at cylinder 127 lbs. absolute, and back-pressure in large cylinder 4 lbs. absolute. Find cylinder diameters for equal work in each. The mean ordinate of " C ,1 is found to be ph = 37.414 lbs. per sq. in. 44 "B" " " " pi = 15.782 " " " " " " "A" " " " pi = 11.730 " " " Then by (1), (2), and (3) we have: H = 33,000 x —^ ■+■ 37.414 x 900 = 980 sq. in., diam. 35%"; I = £ 3 + 33,000 X 3000 15.782 X 900 = I sq. in., diam. 65" - 11.730 X y00 = 6432 sq. in., diam. 90^ Mr. Whitham recommends the following cylinder ratios when the piston- speed is from 750 to 1000 ft. per min., the terminal pressure in the large cylinder being about 10 lbs. absolute. Cylinder Ratios recommended for Triple-expansion Engines. Boiler-pressure (Gauge). Small. Intermediate. Large. 130 140 150 160 1 1 1 1 2.25 2.40 2.55 2.70 5.00 5.85 6.90 7.25 170 and upwards— quadruple-expansion engine to be used. He gives the following ratios from examination of a number of actual engines : No. of Engines Steam-boiler Averaged. 3 11 Pressure. 130 135 140 145 150 160 h.p. Cylinder Ratios, int. 2.10 2.07 2.40 2.35 2.54 4.88 5.00 5.84 TRIPLE-EXPANSION ENGINES. m A Common Rule for Proportioning the Cylinders of mul- tiple-expansion engines is: for two-cylinder compound engines, the cylinder ratio is the square root of the number of expansions, and for triple-expansion engines the ratios of the high to the intermediate and of the intermediate to the low are each equal to the cube root of the number of expansions, the ratio of the high to the low being the product of the two ratios, that is, the square of the cube root of the number of expansions. Applying this rule to the pressures above given, assuming a terminal pressure (absolute) of 10 lbs. and 8 lbs. respectively, we have, for triple-expansion engines: Boiler- Terminal Pressure, 10 lbs. (Absolute). No. of Ex- pansions. Cylinder Ratios, areas. 130 140 150 160 13 14 15 16 1 to 2.35 to 5.53 1 to 2.41 to 5.81 1 to 2.47 to 6.08 1 to 2.52 to 6.35 Terminal Pressure, 8 lbs. No. of Ex- pansions. Cylinder Ratios, areas. 16J4 20 1 to 2.53 to 6.42 1 to 2.60 to 6.74 1 to 2.66 to 7.06 1 to 2.71 to 7.37 The ratio of the diameters is the square root of the ratios of the areas, and the ratio of the diameters of the first and third cylinders is the same as the ratio of the areas of first and second. Seaton, in his Marine Engineering, says: When the pressure of steam em- ployed exceeds 115 lbs. absolute, it is advisable to employ three cylinders, through each of which the steam expands in turn. The ratio of the low- pressure to high- pressure cylinder in this system should be 5, when the steam-pressure is 125 lbs. absolute; when 135 lbs. absolute, 5.4; when 145 lbs. absolute, 5.8; when 155 lbs. absolute, 6.2; when 165 lbs. absolute, 6.6. The ratio of low-pressure to intermediate cylinder should be about one half that between low-pressure and high- pressure, as given above. That is, if the ratio of 1. p. to h. p. is 6, that of 1. p. to int. should be about 3, and conse- quently that of int. to h. p. about 2. In practice the ratio of int. to h. p. is nearly 2.25, so that the diameter of the int. cylinder is 1.5 that of the h. p. The introduction of the triple-compound engine has admitted of ships being propelled at higher rates of speed than formerly obtained without exceeding the consumption of fuel of similar ships fitted with ordinary compound engiues; in such cases the higher power to obtain the speed has been devel- oped by decreasing the rate of expansion, the low-pressure cylinder being only 6 times the capacity of the high-pressure, with a working pressure of 170 lbs. absolute. It is now a very general practice to make the diameter of the low pressure cylinder equal to the sum of the diameters of the h. p. and int. cylinders; hence, Diameter of int. cylinder = 1.5 diameter of h. p. cylinder; Diameter of 1. p. cylinder = 2.5 diameter of h. p. cylinder. In this case the ratio of 1. p. to h. p. is 6.25; the ratio of int. to h. p. is 2.25; and ratio of 1. p. to int. is 2.78. Ratios of Cylinders for Different Classes of Engines. (Proc. Inst. M. E., Feb. 1887, p. 36.)— As to the best ratios for the cylinders iu a triple engine there seems to be great difference of opinion. Considera- ble latitude, however, is due to the requirements of the case, inasmuch as it would not be expecied that the same ratio would be suitable for an eco- nomical land engaie. where the space occupied and the weight were of minor importance, as in a war-ship, where the conditions were reversed. In the land engine, for example, a theoretical terminal pressure of about 7 lbs. above absolute vacuum would probably be aimed at, which would give a ratio of capacity of high pressure to low pressure of 1 to 8J^ or 1 to 9; whilst in a war-ship a terminal pressure would be required of 12 to 13 lbs. which would need a ratio of capacity of 1 to 5; yet in both these instances the cylinders were correctly proportioned and suitable to the requirements of the case. It is obviously unwise, therefore, to introduce any hard-and- fast rule. Types of Three-stage Expansion Engines.— 1. Three cranks at 120 deg. 2. Two cranks with 1st and 2d cylinders tandem. 3. Two cranks with 1st and 3d cylinders tandem. The most common type is the first, with cylinders arranged in the sequence high, intermediate, low. Tel THE STEAM-ENGIKE. Sequence of Cranks.- Mr. Wyllie (Proc. lust. M. E., 1887) favors the sequence high, low, intermediate, while Mr. Mudd favors high, intermediate, low. The former sequence, high, low, intermediate, gave an approximately horizontal exhaust-line, and thus minimizes the range of temperature and the initial load; the latter sequence, high, intermediate, low, increased the range and also the load. Mr. Morrison, in discussing the question of sequence of cranks, presented a diagram showing that with the cranks arranged in the sequence high, low, intermediate, the mean compression into the receiver was 19)^ per cent of the stroke; with the sequence high, intermediate, low r , it was 5? percent. In the former case the compression was just what was required to keep the receiver-pressure practically uniform; in the latter case the compression caused a variation in the receiver-pressure to the extent sometimes of 22^ lbs. Velocity of Steam through Passages in Compound Engines. (l J roc. Inst. M. E., Feb. 1887.)— In the SS. Para, taking the area of the cylinder multiplied by the piston-speed in feet per second and dividing by the area of the port the velocity of the initial steam through the high-pressure cylinder port would be about 100 feet per second; the ex- haust w r ould be about 90. In the intermediate cylinder the initial steam had a velocity of about 180, and the exhaust of 120. In the low-pressure cylinder the initial steam entered through the port with a velocity of 250, and in the exhaust-port the velocity was about 140 feet per second. QUADRUPLE-EXPANSION ENGINES. H. H. Suplee (Trans. A. S. M. E., x. 583) states that a study of 14 different quadruple-expansion engines, nearly all intended to be operated at a pres- sure of 180 lbs. per sq. in., gave average cylinder ratios of 1 to 2, to 3.78, to 7.70, or nearly in the proportions 1, 2, 4, 8. If we take the ratio of areas of any two adjoining cylinders as the fourth root of the number of expansions, the ratio of the 1st to the 4th will be the cube of the fourth root. On this basis the ratios of areas for different pres- sures and rates of expansion will be as follows : Gauge- Absolute Terminal Ratio of Ratios of Areas pressures. Pressures. Pressures. Expansion. of Cylinders. \\l 14.6 1 : 1.95 :3/81 : 7.43 160 175 17.5 1 : 2.05: 4.18: 8.55 i 8 21.9 1 : 2.16:4.68: 10.12 18 16.2 1 :2.01 : 4.02: 8.07 180 195 ho 19.5 1 : 2.10: 4.42: 9.28 1 8 24.4 1 : 2.22:4.94: 10.98 112 17.9 1 : 2.06: 4.23: 8.70 200 215 ho 21.5 1:2.15:4.64: 9.98 I 8 26.9 1 : 2.28: 5.19: 11.81 112 19.6 1:2.10:4.43: 9.31 220 235 ho 23.5 1 : 2.20: 4.85: 10.67 I 8 29.4 1 : 2.33: 5.42: 12.62 Seaton says: When the pressure of steam employed exceeds 190 lbs. abso- lute, four cylinders should be employed, with the steam expanding through each successively; and the ratio of 1. p. to h. p. should be at least 7.5, and if economy of fuel is of prime consideration it should be 8; then the ratio of first intermediate to h. p. should be 1.8, that of second intermediate to first int. 2, and that of L p. to second int. 2.2. In a paper read before the North East Coast Institution of Engineers and Shipbuilders, 1890, William Russell Cummins advocates the use of a four- cylinder engine with four ctanks as being more suitable for high speeds than the three-cylinder three-crank engine. The cylinder ratios, he claims, should be designed so as to obtain equal initial loads in each cylinder. The ratios determined for the triple engine are 1, 2.01. 6.54, and for the quadru- ple 1, 2.08, 4.46, 10.47. He advocates long stroke, high piston- speed, 100 rev- olutions per minute, and 250 lbs. boiler-pressure, unjacketed cylinders, and separate steam and exhaust valves. QUADRUPLE-EXPANSION ENGINES. 773 Diameters of Cylinders of Recent Triple-expansion Engines, Chiefly Marine. Compiled from several sources, 1890-1893. Diam. in inches: H — high pressure, J = intermediate, L = low pressure H I L H I L H I 36 L H I L 3 5 8 16 25.6 41 22 (40 J 40 36 58 94 43/f 7.5 13 16J4 237^ 38.5 38 61.5 100 5 6.5 8 10.5 12 16.5 16.5 24.5 j31 131 23 38 38 61 60 28 1 28 if 56 86 7 9 12.5 17 27 44 24 37 56 39 61 97 7.1 1-1.8 18.9 17 26.5 42 25 40 64 40 59 88 7.5 12 19 17 28 45 26 42 69 40 67 106 8 11.5 16 18 27 40 26 42.5 70 40 66 100 9 14.5 22.5 18 29 48 28 44 72 41 66 101 9.8 15.7 25.6 18 305. 51 293/ s 44 70 41 3^ 67 10634 10 16 25 18.7 29.5 43.3 29.5 48 78 42 59 92 11 lij 24 18^ 23.6 35.4 30 48 77 43 66 9-2 11 18 25 19.7 29.6 47.3 32 46 70 43 68 110 11 18 30 20 30 45 32 51 82 43% 67 106^4 11.5 18 28.5 20 32.5 136 32 54 82 45 71 113 11.5 17.5 30.5 33 58 88 32.5J 32.5 j 68 J 85.7 12 19.2 30.7 20 33 52 33.9 55.1 84.6 185.7 13 22 33.5 21 32 48 34 54 85 47 r-~ J 81.5 14 22.4 36 21 36 51 34 50 90 to ] 81.5 14.5 24 39 21.7 33.5 49.2 34.5 51 85 37| 79 j 98 15 24 39 21.9 34 57 34.5 57 92 37 ( )98 15 24.5 38 22 34 51 "Where the figures are bracketed there are two cylinders of a kind. Two 28" = one 39.6", two 31" = one 43.8", two 32.5" = one 46.0 ', two 36" = one 50.9", two 37" = one 52.3", two 40" = one 56.6", two 81.5" = one 115", two 85.7" = one 121", two 98" = one 140". The average ratio of diameters of cylinders of all the engines in the above table is nearly 1 to 1.60 to 2.56 and the ratio of areas nearly 1 to 2.56 to 6.55. The Progress in Steam-engines between 1876 and 1893 is shown in the following comparison of the Corliss engine at the Centennial Exhibi- tion in 1876 and the Allis-Corliss quadruple-expansion engine at the Chicago Exhibition. j Quadruple- ) 1876. . Simple 2 40 in. 72 in. 120 in. 30 ft. 30 ft. 76 in. 24 in. 136,000 lbs. 125,440 lbs. 60 36 2000 H.P. 1400 H.P. 3000 H.P. 2500 H.P. 650,000 lbs. 1,360,588 lbs. The Crank-shaft body or wheel-seat of the All is engine has a diameter of 21 inches, journals 19 inches, and crank bearings 18 inches, with a total length of 18 feet. The crank-disks are of cast iron and are 8 feet in diam- eter. The crank-pins are 9 inches in diameter by 9 inches long. A Double-tandem Triple-expansion Engine, built by Watts, Campbell & Co., Newark, N. J., is described in Am. Alack., April 26, 1894. It is two three-cylinder tandem engines coupled to one shaft, cranks at 90°, cylinders 21, 32 and 48 by 60 in. stroke, 65 revolutions per minute, rated H.P. 2000; fly-wheel 28 feet diameter, 12 ft. face, weight 174,000 lbs; main shaft 22 in. diameter at the swell; main journals 19 X 38 in.; crank-pins 9^ X 10 in.; distance between centre lines of two engines 24 ft. \% in.; Corliss valves, with separate eccentrics for the exhaust-valves of the l.p. cylinder. j expansion. Cylinders, number " diameter 24, 40, " stroke Fly-wheel, diameter " width of face. . " weight Revolutions per minute. . . Capacity, economical " maximum Total weight 774 THE STEAM-ENGINE. o -mm© ■ooicmocco •jfrao -uoog umiu -IXT3M -j-h-I ©©0©©©tfI©© < > fi c3 ji © © d © fi ciJi o © £ © u o c3 © a © fig- a £ to ail orrespondi ual Result in Practic ing Slight 3 O a CD-° P-l uj- 8§ 3 spondi Result Practic Slight a 's 53 1* 11 3? © Ct 01 6 3 s So, 03 03 ©£ gig" S-3 bo fi §~ O O fi § O o f 60 8.70 37.26 40.95 r 60 14.42 18.22 20.00 1 70 12.39 30.99 33.68 1 70 16.96 17.96 19.69 1-. 3^:1. 4^:1. 3^:1. 4*6 : 1. VA • 1. 4^:1. Pk Ph SO 90 130 150 80 90 130 150 80 90 130 150 W w J lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. bs. lbs. lbs. 5% 63^ 12 10 370 7 15 19 32 23 31 35 40 44 55 04 79 7W 12 318 9 19 24 40 29 39 45 59 56 70 81 101 r^ 9 ; 14 277 14 28 30 60 43 58 07 87 83 104 121 159 9 0U 19 16 246 1S 37 47 78 57 70 87 114 109 158 196 10W 22 U 18 222 26 53 68 112 81 109 125 104 156 195 220 281 12 3U 25 20 185 32 65 84 139 100 135 154 202 192 241 279 346 lWo 24 158 43 88 112 186 135 LSI 200 271 258 323 374 464 16 28 138 57 118 151 180 242 277' 303 346 433 502 023 18 i 38 32 120 74 1 52 194 321 232 312 357 408 446 558 047 803 20 £ 43 31 112 94 194 249 412 297 400 457 001 572 715 1030 -Jf., • 52 42 93 13S 285 3!i5 003 430 587 070 WHO 838 048 1215 1508 28J^ t 3 60 48 80 ISO 374 477 789 570 707 877 1151 1096 370 1589 1973 Mean eff ec. press.. .lbs 3.3 6.8 8.7 14.4 10.4 14.0 16 21 20 25 29 36 Ratio of expansion — 13*£ ism I 10J4 13M 6% m Cyl. condensation, %... 14 14 16 | 16 12 12 13 13 10 10 11 11 Ter. press, (about). lbs. 7.3 7.7 7.9, 9 9 2 10.4 10.5 12 14 5.5 14.0 17.8 Loss from expanding 1 below atmosphere, % 34 15 17 3 5 St per I.H.P. p. hr.lbs 55 42 47 I 29 27 . 7 2 a? Wz • l. 4 : 1. %:1. 4 : 1. 3*6:1. 4 : 1. P-' P-i 80 110 115 125 80 110 115 125 80 110 115 125 X a lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 6 H* 12 10 370 44 59 53 62 55 70 68 75 70 97 95 106 (U-o ?i.> 13* 12 318 56 76 67 78 70 90 87 95 90 123 120 134 ^1 9 14 277 83 112 100 116 104 133 129 141 133 183 1 79 200 9i « 10* 19 lfi 246 109 117 131 152 136 174 169 185 174 239 234 261 11 1-2 IS 222 156 210 187 218 195 25d 242 265 250 343 335 374 1SU 2a 20 185 192 260 231 269 241 308 298 327 308 423 414 462 14 24 158 25S 348 310 361 323 413 400 439 413 568 555 619 17 28 138 840 467 415 484 433 554 536 588 554 761 744 830 19 38 35 120 446 602 535 624 558 714 691 758 714 981 059 1070 21 43 34 112 572 772 686 801 715 915 887 972 915 1258 1373 26 55 42 93 83S 1131 1174 1048 1341 1299 1341 1844 1801 2012 30 33 60 48 80 1096 1480 1316 1534 1370 1757 1699 1757 2411 2356 2632 Mean effec. press Ratio of Expansi C3'l. condensatioi ..lbs. on... , %. . . 20 27 24 28 25 32 31 34 32 44 43 48 13* mi 10 12J4 6% m 18 I 18 20 | 20 15 1 15 18 1 18 12 1 12 14 I 14 St. per I. H.P. p. h r.lbs. 17.3116.6 16.6|15.2 17.0|16.4 16.3|15.8 17.5|17.0 16.8|l6.0 The original table contains figures for 95 lbs., cylinder ratio 3 120 lbs., ratio 4 to 1. Triple-expansion Engines, Non-condensing.— Receiver only Jacketed. Horse-power Horse-power Horse-power i. CD when Cutting when Cutting when Cutting Cylinders, i off at 42 per off at 50 per off at 67 per ■j cent of Stroke cent of Stroke cent of Stroke ■- .2 =,: in First Cylin- in First Cylin- in First Cylin- 1 der. der. der. H.P. LP. L. P. 12 180 lbs. 200 lbs. 180 lbs. 200 lbs. 180 lbs. 200 lbs 43/f 7* 370 55 64 70 84 95 108 5* 8* 13V, 12 318 70 81 90 106 120 137 6* 10* 16* 14 27, 104 121 133 158 179 204 7* 12 19 246 136 158 174 207 234 267 9 14* 22* 18 222 195 226 250 296 335 382 10 16 25 20 185 241 279 308 366 414 471 11* 18 28* 24 158 323 374 413 490 555 632 13 22 33* 28 138 433 502 554 657 744 848 15 24*, 38 32 120 558 647 714 847 959 1093 17 27 43 34 112 715 829 915 1089 1230 1401 20 33 52 42 93 1048 1215 1341 1592 1801 2053 23* 38 60 48 80 1370 1589 1754 2082 2356 2685 Mean effective press., lbs. 25 29 32 38 43 49 16 13 10 Per cent cyl. condens — 14 12 10 Steam p. I. H.P. p.hr., lbs. 20.76 1 19.36 19.25 | 17.00 17.89 i 17.20 Lbs. c aal at S lb. evt ip. lbs. 2.59 2.39 2.40 2.12 2.23 2.15 780 THE STEAM-ENGINE. Triple-expansion Engines— Condensing— Steam- Jacketed. Diameter Cylinders, inches. 4M JO ny 2 13 15 17 20 2sy 2 Horse-power when Cut- ting off at J4 Stroke in First Cylin- der. Mean effec. press., lbs. No. of expansions Percent cyl. condens. St.p.I.H.P.p. hr.,lbs. Coal at 8 lb. evap., lbs. Horse-power when Cut- ting off at H Stroke in First Cylin der. 120 140 160 lbs. lbs. ;lbs 125 154 1 206 245 27? 329 35? 424 458 543 670 796 877 1041 19 19 14.7 13.9 13.3 1.8 1.73 1 Horse-power when Cut- ting off at Yz Stroke in First Cylin- der. „ 140 160 lbs. lbs. lbs. 120 140 160 lbs. lbs. lbs. 16 | 16 16 | 12 12 14.3 13.98 13.2 14.3 13.6 13.0 1.78 1.74 1.65 1.78 1.70 1.62 Horse-power when Cut- ting off at y^ Stroke in First Cylin- der. 120 140 160 lbs. lb?, lbs. 110 140 208 272 390 481 645 865 1115 1430 15.7 14. 1.96 1.8 1.7' Type of Engine to be used where Exhaust-steam is needed for Heating.— In many factories more or less of the steam exhausted from the engines is utilized for boiling, drying, heating, etc. Where all the exhaust-steam is so used the question of economical use of steam in the engine itself is eliminated, and the high-pressure simple engine is entirely suitable. Where only part of the exhaust-steam is used, and the quantity so used varies at different times, the question of adopting a simple, a condensing, or a compound engine becomes more complex. This problem is treated by C. T. Main in Trans. A. S. M. E., vol. x. p. 48. He shows that the ratios of the volumes of the cylinders in compound engines should vary according to the amount of exhaust-steam that can be used for heating. A case is given in which three different pressures of steam are required or could be used, as in a worsted dye-house: the high or boiler pressure for the engine, an intermediate pressure for crabbing, and low-pressure for boiling, drying, etc. If it did not make too much complication of parts in the engine, the boiler-pressure might be used in the high-pressure cylinder, exhausting into a receiver from which steam could be taken for running small engines and crabbing, the steam remaining in the receiver passing into the intermediate cylinder and expanded there to from 5 to 10 lbs. above the atmosphere and exhausted into a second receiver. From this receiver is drawn the low-pressure steam needed for drying, boiling, warming mills,- etc., the steam remaining in receiver passing into the condensing cylinder. Comparison of the Economy of Compound and Single- cylinder Corliss Condensing Engines, each expanding about Sixteen Times. (D. S. Jacobus, Trans. A. S. M. E., xii. 943.) The engines used in obtaining comparative results are located at Stations I. and II. of the Paw tucket Water Co. The tests show that the compound engine is about 30# more economical than the single-cylinder engine. The dimensions of the two engines are as follows: Single 20" X 48"; compound 15" and 30^" x 30". The steam used per horse-power per hour was: single 20.35 lbs., compound 13.73 lbs. Both of the engines are steam-jacketed, practically on the barrels only, with steam at full boiler-pressure, viz. single 106.3 lbs., compound 127.5 lbs. PERFORMANCES OF STEAM-ENGINES. 781 The steam-pressure in the case of the compound engine is 127 lbs., or 21 lbs. higher than for the single engine. If the steam-pressure be raised this amount in the case of the single engine, and the indicator-cards be increased accordingly, the consumption for the single-C3 T linder engine would be 19.97 lbs. per hour per horse-power. Two-cylinder vs. Three-cylinder Compound Engine.— A Wheelock triple-expansion engine, built for the Merrick Thread Co., Holyoke, Mass., is constructed so that the intermediate cylinder maybe cut out of the circuit and the high-pressure and low-pressure cylinders run as a two-cylinder compound, using the same conditions of initial steam-pressure and load. The diameters of the cylinders are 12, 16, and 24^§ inches, the stroke of the first two being 36 in. and that of the low r -piessure cylinder 48 i. The results of a test reported by S. M. Green and G. I. Rockwood, Trans. . S. M. E., vol. xiii. 647, are as follows: In lbs. of dry steam used per I.H.P. per hour, 12 and 24|§ in. cylinders only used, two tests 13.06 and 12.76 lbs., average 12.91. All three cylinders used, two tests 12.67 and 12.90 lbs., average 12.79. The difference is only 1%, and would indicate that more than two cylin- ders are unnecessary in a compound engine, but it is pointed out by Prof. Jacobus, that the conditions of the test were especially favorable for the two-cylinder engine, and not relatively so favorable for the three cylinders. The steam-pressure was 142 lbs. and the number of expansions about 25. (See also discussion on the Rockwood type of engine, Trans. A. S. M. E., vol. cvi.) Effect of Water contained in Steam on the Efficiency of the Steam-engine. (From a lecture by Walter C. Kerr, before the Franklin Institute, 1891.) — Standard writers make little mention of the effect of entrained moisture on the expansive properties of steam, but by common consent rather than any demonstration they seem to agree that moisture produces an ill effect simply to the percentage amount of its presence. That is, 5% moisture will increase the water rate of an engine 5%. Experiments reported in 1893 by R. C. Carpenter and L. S. Marks, Trans. A. S. M. E., xv., in which water in varying quantity w r as introduced into the steam-pipe, causing the quality of the steam to range from 99$ to 58$ dry, showed that throughout the range of qualities used the consumption of dry steam per indicated horse-power per hour remains practically constant, and indicated that the water was an inert quantity, doing neither good nor harm. It appears thac the extra work done by the heat of the entrained water during expansion is sensibly equal to the extra, negative work which it does during exhaust and compression, that the heat carried in by the entrained water performs no useful function, and that a fair measure of the economy of an engine is the consumption of dry and saturated steam. Relative Commercial Economy of Best Modern Types of Compound and Triple-expansion Engines. (J. E. Denton, American Machinist, Dec. 17, 1891.) — The following table and deductions show the relative commercial economy of the compound and triple type for the best stationary practice in steam plants of 500 indicated horse-power. The table is based on the tests of Prof. Schroter, of Munich, of engines built at Augsburg, and those of Geo. H. Barrus on the best plants of America, and of detailed estimates of cost obtained from several first-class builders. Trip motion, or Corliss engines of [^SS l 136 14 "° the twin-compound-receiver con- J T £; ^'^ in ^ ^'f^ ie " t ; Trip motion, or Corliss engines of [ Lbs. water per hour per | .„ 5g - g0 the triple-expansion four-cylin- j H. P., by measurement, f der-receiver condeusing type, ex- \ Lbs. coal per hour per 1 panding 2^ times. Boiler pressure, ! H. P., assuming 8.5 lbs. > 1.48 1.50 150 lbs. i. actual evaporation. ) The figures in the first column represent the best recorded performance (1891), and tnose in the second column the probable reliable performance. Increased cost of triple -expansion plant per horse-power, including boilers, chimney, heaters, foundations, piping and erection $4.50 The following table shows the total annual cost of operation, with coal at $4.00 per ton, the plant running 300 days in the year, for 10 hours and for 24 hours per day : 782 THE STEAM-ENGINE. 10 24 Per H.P. $9.90 9.00 0.90 Per H.P. $28.50 Expense for coal. Triple plant Annual saving of triple plant in fuel 25.92 2.60 Annual interest at 5% on $4.50 Annual depreciation at 5% on $4.50 Annual extra cost of oil, 1 gallon per 24-hour day, at $0.50, or 15$ of extra fuel cost Annual extra cost of repairs at 3% on $4.50 per $0.23 0.23 0.15 0.06 $0.23 0.23 0.36 0.14 $0.67 $0.96 Annual saving per H.P $0.23 $1.64 The saving between the compound and triple types is much less than that involved in the step from the single-expansion condensing to the compound engine. The increased cost per horse-power of the triple plant over the compound is due almost entirely to the extra cost of the triple engine and its foundations, the boilers costing the same or slightly more, owing to their extra strength. In the case of the single versus the compound, however, about one third of the increased cost of the compound engine is offset by the less cost of the latter's boilers. Taking the total cost of the plants at $33.50, $36.50 and $41 per horse- power respectively, the figures in the table imply that the total annual sav- ing is as follows for coal at $4 per ton: 1. A compound 500 horse-power plant costs $18,250, and saves about $1630 for 10 hours 1 service, and $4885 for 24 hours 1 service, per year over a single plant costing $16,750. That is, the compound saves its extra cost in 10-hour service in about one year, or in 24-hour service in four months. 2. A triple 500 horse-power plant costs $20,500, and saves about $114 per year in 10-hour service, or $826 in 24-hour service, over a compound plant, thereby saving its extra cost in 10-hour service in about 19% years, or in 24- hour service in about 2% years. Triple - expansion Pumping-engine at Milwaukee- Highest Economy on Record, 1893. (See paper on "Maximum Contemporary Economy of the Steam-engine," by R. H. Thurston, Trans. A. S. M. E., xv. 313.)— Cylinders 28, 48 and 74 in. by 60 in. stroke; ratios of volumes 1 to 3 to 7; total number of expansions 19.55; clearances, h.p. 1.4$; int. 1.5%; 1. p. 0.77$; volume of receivers: 1st, 101.3 cu. ft.; 2d, 181 cu. ft.; steam-pressure gauge during test, average 121.5 lbs.; vacuum 13.84 lbs. absolute; revolutions 20.3 per minute; indicated horse-power, h.p. 175.4, int. 109.6, 1. p. 228.9; total, 573.9; total friction, horse-power 52.91 = 9.22$; dry steam per I. H.P. per hour 11.678; B.T.U. per I. H.P. per min. 217.6; duty in foot-pounds per 100 lbs. of coal, 143,306,000; per million B.T.U., 137,656,000. Steam per I. H.P. per hour, from diagram, at cut-off 9.35 9.12 8.37 " release.. . 10.1 10.0 8.92 Steam accounted for by indicator at cut-off, per cent. . . 87.1 85.0 78.2 " " " " " release, " ... 94.0 93.2 83.2 Per cent of total steam used by jackets 9.25 Highest Economy of the Two - cylinder Compound Pumping-engines,- Repeated tests of the Pawtucket-Corliss engine, 15 and '60% by 30 in. stroke, gave a water consumption of 13.69 to 14.16 lbs. per I. H.P. per hour. Steam -pressure 123 lbs.; revolutions per min. 48 lbs.; expansions about 16. Cylinders jacketed. The lowest water rate was with jackets in use; both jackets supplied with steam of boiler pressure. The average saving due to jackets was only about 2^ per cent. (Trans. A. S. M. E., xi. 328 and 1038; xiii. 176.) This record was beaten in 1894 by a Leavitt pumping-engine at Louisville, Ky. (Trans. A. S. M. E. xvi.) Cylinders 27.21 and 54.13 in. diam. by 10 ft. stroke; revolutions per min. 18.57; piston speed 371.5 ft. ; expansions 20.4; steam -pressure, gauge, 140 lbs. Cylinders and receiver jacketed. Steam PERFORMANCES OF STEAM-ENGINES. 783 used per I.H.P. per hour, 12.223 lbs. Duty per million B.T.U. = 138,126,000 ft.-lbs. Test of a Triple-expansion Funiping-engine with and without Jackets, at Laketon, Ind., by Prof. J. E. Denton (Trans. A. S. M. E., xiv. 1340).— Cylinders 24, 34 and 54 in. by 36 in. stroke; 28 revs, per min. ; H.P. developed about 320; boiler-pressure 150 lbs. Tests made on eight different days with different sets of conditions in jackets. At 150 lbs. boiler- pressure, and about 20 expansions, with any pressure above 43 lbs. in all of the jackets and reheaters, or with no pressure in the high jacket, the per- formance was as follows: With 2.5$ of moisture in the steam entering the engine, the jackets used 16$ of the total feed-water. About 20$ of the latter was condensed during admission to the high cylinder, and about 13.85 lbs. of feed-water was consumed per hour per indicated horse-power. With no jackets or reheaters in action the feed-water consumption was 14.99 lbs., or 8.3% more than with jackets and reheaters. The consumption of lubricating oil was two thirds of a gallon of machine oil and one and three quarter gal- lons of cylinder oil per 24 hours. The friction of the engine in eight tests on different days varied from 5.1% to 8.7%. If we regard the measurements of indicated horse-power and water as liable to an error of one per cent, which is probably a minimum allowance for the most careful determinations, the steam economy is the same for the following conditions: (a) Any pressure from 43 to 131 in the intermediate and low jackets and receivers. (b) Any pressure from to 151 in the jacket of high cylinder. (c) Any cut-off from 21$ to 23$ in high cylinder, from 39$ to 43$ in inter- mediate cylinder, from 40$ to 53$ in low cylinder. Water Consumption of Three Types of Sulzer Engines. (B. Donkin, Jr., Eng'g, Jan. 15, 1892, p. 77.) Summary and Averages of Twenty-one Published Experiments op the Sulzer Type of Steam-engine. All Horizontal Condensing and Steam- jacketed. From 1872 to 1891. Single j Cyl. 1 Com. j pound. ( Triple. . -j lbs. 72 to 95 84 to 104 104 to 156 ft. per min 272 to 444 to 607 157 to 400 133 to 524 198 to 615 Steam Consump tion, pounds per I.H.P. per hour, includingSteam- pipe water and Jacket Water. lbs. j 18.7 to 19 8 ( Mean ]9.4 j 13.35 to 16.0 1 Mean 14.44 j 11.85 to 12.86 j Mean 12.36 Steam Consump- tion, pounds per I.H.P. per hour, exclud'g Steam- pipe water, but includingjacket Water. lbs. 17.9 to 19.2 Mean 18.95 13.4 to 15.5 Mean 14.3 11.7 to 12.7 Mean 12.18 I 5 exp. J 1872-78 I 10 exp. f 1888-91 I 6 exp. ) 1888-89 Triple-expansion Corliss engine at Narragansett E. L. Co., Providence, R. I., built by E. P. Allis Co. Cylinder 14, 25 and 33 in. by 48 in. stroke tested at 99 revs, per min.; 125 lbs. steam-pressure; steam per I.H.P. per hour 12.94 lbs. ; H.P. 516. A full account of this engine, with records of tests is given by J. T. Henthorn, in Trans. A. S. M. E., xii. 643. Buckeye-cross compound engine, tested at Chicago Exposition, by Geo. H. Barrus (Evg'g Record. Feb. 17, 1894). Cylinder 14 and 28 by 24 in. stroke; tested at 165 r. p. m. ; 120 lbs. steam-pressure. I.H.P. in four tests condens- ing and one non-condensing 295 224 123 277 267 Steam per horse-power per hour 16.07 15.71 17.22 16.07 23.24 Relative Economy of Compound Non-condensing En- gines under Variable IiOads.— F. M. Rites, in a paper on the Steam Distribution in a Form of Single-acting Engine (Trans. A. S. M. E. xiii. 537), discusses an engine designed to meet the following problem : Given an 784 THE STEAM-EHGIKE. extreme range of conditions as to load or steam -pressure, either or both, to fluctuate together or apart, violently or with easy gradations, to construct an engine whose economical performance should be as good as though the engine were specially designed for a momentary condition— the adjustment to be complete and automatic. In the ordinary non-condensing compound engine with light loads the high -pressure cylinder is frequently forced to supply all the power and in addition drag along with it the low-pressure piston, whose cylinder indicates negative work. Mr. Rites shows the peculiar value of a receiver of predetermined volume which acts as a clear- ance chamber for compression in the high-pressure cylinder. The Westiug- house compound single-acting engine is designed upon this principle. The following results of tests of one of these engines rated at 175 H.P. for most economical load are given : Watek Rates under Varying Loads, lbs. per H.P. per hour. Horse-power 210 170 140 115 100 80 50 Non-condensing 22.6 21.9 22.2 22.2 22.4 24.6 28.8 Condensing 18.4 18.1 18.2 18.2 18.3 18.3 20.4 Efficiency of Non-condensing Compound Engines. (W. Lee Church, Am. Mach., Nov. 19, 1891.)— The compound engine, non-con- densing, at its best performance will exhaust from the low-pressure ejdin- der at a pressure 2 to 6 pounds above atmosphere. Such an engine will be limited in its economy to a very short range of power, for the reason that its valve-motion will not permit of any great increase beyond its rated power, and any material decrease below its rated power at once brings the expansion curve in the low-pressure cylinder below atmosphere. In other words, decrease of load tells upon the compound engine somewhat sooner, and much more severely, than upon the non-compound engine. The loss commences the moment the expansion line crosses a line parallel to the atmospheric line, and at a distance above it representing the mean effective pressure necessary to carry the f fictional load of the engine. When expan- sion falls to this point the low-pressure cylinder becomes an air-pump over more or less of its stroke, the power to drive which must come from the high pressure cylinder alone. Under the light loads common in many industries the low-pressure cylinder is thus a positive resistance for the greater portion of its stroke. A careful study of this problem revealed the functions of a fixed intermediate clearance, always in communication with the high-pressure cylinder, and having a volume bearing the same ratio to that of the high-pressure cylinder that the high-pressure cylinder bears to the low-pressure. Diagrams were laid out on this principle and justified until the best theoretical results were obtained. The designs were then laid down on these lines, and the subsequent performance of the engines, of which some 600 have been built, have fully confirmed the judgment of the designers. The effect of this constant clearance is to supply sufficient steam to the low-pressure cylinder under light loads to hold its expansion curve up to atmosphere, and at the same time leave a sufficient clearance volume in the high -pressure cylinder to permit of governing the engine on its compression under light loads. Economy of Engines under Varying Loads. (From Prof. W. C. Unwin's lecture before the Society of Arts, London, 1892.)— The gen- eral result of numerous trials with large engines was that with a constant load an indicated horse-power should be obtained with a consumption of 1^2 pounds of coal per indicated horse-power for a condensing engine, and \% pounds for a non-condensing engine, figures which correspond to about 1% pounds to ,'.% pounds of coal per effective horse-power. It was much more difficult to ascertain the consumption of coal in ordinary everyday work, but such facts as were known showed it was more than on trial. In electric-lighting stations the engines work under a very fluctuating load, and the results are far more unfavorable. An excellent Willans non- condensing engine, which on full-load trials worked with under 2 pounds per effective horse-power hour, in the ordinary daily working of the station used 7^a pounds per effective horse-power hour in 1886, which was reduced to 4.3 pounds in 1890 and 3.8 pounds in 1891. Probably in very few cases were the engines at electric-light stations working under a consumption of 4i^> pounds per effective horse-power hour. In the case of small isolated motors working with a fluctuating load, still more extravagant results were obtained. PERFORMANCES OF STEAM-ENGINES. 785 Engines in Electric Central Stations. Year 1886. 1890. 1892. Coal used per hour per effective H.P 8.4 5.6 4.9 " " " " indicated " 6.5 4.35 3.8 At electric-lighting stations the load factor, viz., the ratio of the average load to the maximum, is extremely small, and the engines worked under very unfavorable conditions, which largely accounted for the excessive fuel consumption at these stations. In steam-engines the fuel consumption has generally been reckoned on the indicated horse-power. At full-power trials this was satisfactory enough, as the internal friction is then usually a small fraction of the total. Experiment has, however, shown that the internal friction is nearly con- stant, and hence, when the engine is lightly loaded, its mechanical efficiency is greatly reduced. At full load small engines have a mechanical efficiency of 0.8 to 0.85, and large engines might reach at least 0.9, but if the internal friction remained constant this efficiency would be much reduced at low powers. Thus, if an engine working at 100 indicated horse power had an effi- ciency ot 0.85, then when the indicated horse-power fell to 50 the effective horse-power would be 35 horse-power and the efficiency only 0.7. Similarly, at 25 horse-power the effective horse-power would be 10 and the efficiency 0.4 Experiments on a Corliss engine at Creusot gave the following results : Effective power at full load 1.0 0.75 0.50 0.25 0.125 Condensing, mechanical efficiency 0.82 0.79 0.74 0.63 0.48 Non condensing, " " 0.86 0.83 0.78 0.67 0.52 At light loads the economy of gas and liquid fuel engines fell off even more rapidly than in steam-engines. The engine friction was large and nearly constant, and in some cases the combustion was also less perfect at light loads. At the Dresden Central Station the gas-engines were kept working at nearly their full power by the use of storage-batteries. The results of some experiments are given below : Brake load, per Gas-engine, cu. ft. Petroleum Eng., Petroleum Eng., cent of full of Gas per Brake Lbs. of Oil per Lbs. of Oil per Power. H.P. per hour. B.H.P. per hr. B.H.P. per hr. 100 22.2 0.96 0.88 75 23.8 1.11 0.99 59 28.0 1.44 1.20 20 40.8 2.38 1.82 12}^ 66.3 4.25 3.07 Steam Consumption of Engines of Various Sizes.— W. C. Unwin (Cassier's Magazine, 1894) gives a table showing results of 49 tests of engines of different types. In non-condensing simple engines, the steam consumption ranged from 65 lbs. per hour in a 5-horse-power engine to 22 lbs. in a 134-H.P. Harris-Corliss engine. In non-condensing compound en- gines, the only type tested was the Willans, which ranged from 27 lbs. in a 10 H.P. slow-speed engine, 122 ft. per minute, with steam-pressure of 84 lbs. to 19.2 lbs. in a 40-H.P. engine, 401 ft. per minute, with steam-pressure 165 lbs. A Willans triple-expansion non-condensing engine, 39 H.P., 172 lbs. pressure, and 400 ft. piston speed per minute, gave a consumption of 18.5 lbs. In condensing engines, nine tests of simple engines gave results ranging only from 18.4 to 22 lbs., and, leaving out a beam pumping-engine running at slow speed (240 ft. per minute) and low steam -pressure (45 lbs.), the range is only from 18.4 to 19.8 lbs. In compound-condensing engines over 100 H.P., in 13 tests the range is from 13.9 to 20 lbs. In three triple- expansion engines the figures are 11.7, 12 2, and 12.45 lbs., the lowest being a Sulzer engine of 360 H.P. /In marine compound engines, the Fusiyama and Colchester, tested by Prof. Kennedy, gave steam consumption of 21.2 and 21.7 lbs.; and the Meteor and Tartar triple-expansion engines gave 15.0 and 19.8 lbs. Taking the most favorable results which can be regarded as not excep- , tional, it appears that in test trials, with constant and full load, the expen- diture of steam and coal is about as follows: Per Indicated Horse- Per. Effective Horse- power Hour. power Hour. Kind of Engine. > * > , ' * Coal, Steam, . Coal, Steam, lbs. lbs. lbs. lbs. Non-condensing 1.80 16.5 2.00 18.0 Condensing 1.50 13.5 1.75 15.8 186 THE STEAM-EKGIKE. These may be regarded as minimum values, rarely surpassed by the most efficient machinery, and only reached with very good machinery in the favorable conditions of a test trial. Small Engines and Engines with Fluctuating Loads are usually very wasteful of fuel. The following figures, illustrating their low economy, are given by Prof. Unwin, Cassier's Magazine, 1894. Coal Consumption per Indicated Horse-power in Small Engines. In Workshops in Birmingham, Eng. Probable I.H.P. at full load... 12 45 60 45 75 60 60 Average I.H.P. during obser- vation 2.96 7.37 8.2 8.6 23.64 19.08 20.08 Coal per I.H.P. per hour dur- ing observation, lbs 36.0 21.25 22.61 18.13 11.68 9.53 8.50 It is largely to replace such engines as the above that power will be dis- tributed from central stations. Steam Consumption in Small Engines. Tests at Royal Agricultural Society's show at Plymouth, Eng. Engineer- ing, June 27, 1890. Rated H.P. Com- pound or Simple. Diam. of Cylinders. Stroke, ins. Max. Steam- pressure. Per Brake H. P., per hour. %4-i h.p. l.p. Coal. Water. £&o 5 3 2 simple compound simple 3 "6 10 6 1Yz 75 110 75 12.12 4.82 11.77 78.1 lbs. 42.03 '« 89.9 " 6.1 lb. 8.72" 7.61" Steam-consumption of Engines at Various Speeds. (Profs. Denton and Jacobus, Trans. A. S. M. E., x. 722)— 17 X 30 in. engine, non-condensing, fixed cut-off, Meyer valve. Steam-consumption, lbs. per I.H.P. per Hour. Figures taken from plotted diagram of results. Revs, per min 8 12 16 20 24 32 40 48 56 72 88 Ys cut-off, lbs 39 35 32 30 29.3 29 28.7 28.5 28.3 28 27.7 M " " 39 34 31 29.5 29 28.4 28 27.5 27.1 26.3 25.6 Y% " " 39 36 34 33 32 30.8 29.8 29.2 28.8 28.7 .... Steam-consumption of Same Engine; Fixed Speed, 60 Revs, per Min. Varying cut-off compared with throttling-engine for same horse-power and boiler-pressures: Cut-off, fraction of stroke 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 Boiler-pressure, 90 lbs... 29 27.5 27 27 27.2 27.8 28.5 60 lbs.. 39 34.2 32.2 31.5 31.4 31.6 32.234.136.5 39 Throttling-engine, % Cut-off, for Corresponding Horse-powers. Boiler-pressure, 90 lbs... 42 37 33.8 31.5 29.8 - 601bs 50.1 49 46.8 44.6 41 Some of the principal conclusions from this series of tests are as follows : 1. There is a distinct gain in economy of steam as the speed increases for Yz, %, and Ya cut-off at 90 lbs. pressure. The loss in economy for about £4 cut-off is at the rate of 1/12 lb. of water per H.P. for each decrease of a revolution per minute from 86 to 26 revolutions, and at the rate of % lb. of water below 26 revolutions. Also, at all speeds the J4 cut-off is more eco- nomical than either the ]^or^ cut-off. 2. At 90 lbs. boiler-pressure and above Y cut-off, to produce a given H.P. requires about 20$ less steam than to cut off at % stroke and regulate by the throttle. 3. For the same conditions with 60 lbs. boiler-pressure, to obtain, by throttling, the same mean effective pressure at % cut-off that is obtained by PERFORMANCES OF STEAM-ENGINES. 787 cutting off about ^, requires about 30£ more steam than for the latter condition. High l»iston-speed in Engines. (Proc. Inst. M. E., July, 1883, p. 3-21).— The torpedo boat is an excellent example of the advance towards high speeds, and shows what can be accomplished by studying lightness and strength in combination. In running at 22% knots an hour, an engine with cylinders of 16 in. stroke will make 480 revolutions per minute, which gives 1280 ft. per minute for piston-speed; and it is remarked that engines running at that high rate work much more smoothly than at lower speeds, and that the difficulty of lubrication diminishes as the speed increases. A High-speed Corliss Engine.— A Corliss engine, 20x42 in., has been running a wire-rod mill at the Trenton Iron Co.'s works since 1877, at 160 revolutions or 1120 ft. piston -speed per minute (Trans. A. S. M. E., ii. 72). A piston-speed of 1200 ft. per min. has been realized In locomotive pmctice. The Limitation of Engine-speed. (Chas. T. Porter, in a paper on the Limitation of Engine-speed, Trans. A. S. M. E., xiv. 806.)— The practical limitation to high rotative speed in stationary reciprocating steam- engines is not found in the danger of heating or of excessive wear, nor, as is generally believed, in the centrifugal force of the fly-wheel, nor in the tendency to knock in the centres, nor in vibration. He gives two objections to very high speeds: First, that " engines ought not to be run as fast as they can be ;" second, the large amount of waste room in the port, which is required for proper steam distribution. In the important respect of economy of steam, the high-speed engine has thus far proved a failure. Large gain was looked for from high speed, because the loss by condensa- tion on a given surface would be divided into a greater weight of steam, but this expectaiion has not been realized. For this unsatisfactory result we have to lay the blame chiefly on the excessive amount of waste room. The ordinary method of expressing the amount of waste room in the percentage added by it to the total piston displacement, is a misleading one. It should be expressed as the percentage which it adds to the length of steam admis- sion. For example, if the steam is cut off at 1/5 of the stroke, 8% added by the waste room to the total piston displacement means 40$ added to the volume of steam admitted. Engines of four, five and six feet stroke may properly be run at from 700 to 800 ft. of piston travel per minute, but for ordinary sizes, says Mr. Porter, COO ft. per minute should be the limit. Influence of the Steam-jacket.— Tests of numerous engines with and without steam-jackets show an exceeding diversity of results, ranging all the way from 30$ saving down to zero, or even in some cases showing an actual loss. The opinions of engineers at this date (1894) is also as diverse as the results, but there is a tendency towards a general belief that the jacket is not as valuable an appendage to an engine as was formerly supposed. An ex- tensive resume of facts and opinions on the steam-jacket is given by Prof. Thurston, in Trans. A. S. M. E., xiv. 462. See also Trans. A. S. M. E., xiv. 873 and 1340; xiii. 176: xii. 426 and 1340; and Jour. F. I., April, 1891, p. 276. The following are a few statements selected from these papers. The results of tests reported by the research committee on steam-jackets appointed by the British Institution of Mechanical Engineers in 1886, indi- cate an increased efficiency due to the use of the steam-jacket of from \% to over 30$, according to varying circumstances. Sennett asserts that ''it has been abundantly proved that steam- jackets are not only advisable but absolutely necessary, in order that high rates of expansion may be efficiently carried out and the greatest possible economy of heat attained." Isherwood finds the gain by its use. under the conditions of ordinary practice, as a general average, to be about 2% on small and 8% or 9% on large engines, varying through intermediate values with intermediate sizes, it being understood that the jacket has an effective circulation, and that both heads and sides are jacketed. Professor Unwin considers that " in all cases and on all cylinders the jacket is useful: provided, of course, ordinary, not superheated, steam is used; but the advantages may diminish to an amount not worth the interest on extra cost." Professor Cotterill says: Experience shows that a steam-jacket is advan- tageous, but the amount to be gained will vary according to circumstances. In many cases it may be that the advantage is small. Great caution is necessary in drawing conclusions from any special set of experiments on the influence of jacketing. "7SS THE STEAM-ENGINE. Mr. E. D. Leavitt has expressed the opinion that, in his practice, steam- jackets produce an increase of efficiency of from 15$ to 20$. In the Pawtucket pumping engine, 15 and 2,0% x 30 in., 50 revs, per min., steam-pressure 125 lbs. gauge, cut-off J4 in h.p. and % in l.p. cylinder, the barrels only jacketed, the saving by the jackets was from 1$ to 4$. The superintendent of the Holly Mfg. Co. (compound pumping-engines) says: "In regard to the benefits derived from steam-jackets on our steam- cylinders, I am somewhat of a skeptic. From data taken on our own en- gines and tests made I am yet to be convinced that there is any practical value in the steam-jacket." . . . " You might practically say that there is no difference." Professor Schroter from his work on the triple-expansion engines at Augs- burg, and from the results of his tests of the jacket efficiency on a small engine of the Sulzer type in his own laboratory, concludes: (1) The value of the jacket may vary within very wide limits, or even become nega- tive. (2) The shorter the cut-off the greater the gain by the use of a jacket. (3) The use of higher pressure in the jacket than in the cylinder produces an advantage. The greater this difference the better. (4) The high-pressure cylinder may be left unjacketed without great loss, but the others should always be jacketed. The test of the Laketon triple-expansion pumping-engine showed a gain of 8.3$ by the use of the jackets, but Prof. Denton points out (Trans. A. .S M. E., xiv. 1412) that all but 1.9$ of the gain was ascribable to the greater range of expansion used with the jackets. Test of a Compound Condensing Engine with and with- out Jackets at different Loads. (R. C. Carpenter, Trans. A. S. M. E,xiv. 428.)— Cylinders 9 and 1(5 in.Xl4 in. stroke; 112 lbs. boiler-pressure; rated capacity 100 H.P. ; 265 revs, per min. Vacuum, 23 in. From the results of several tests curves are plotted, from which the following principal figures are taken. Indicated H.P 30 40 50 60 70 80 90 100 110 120 125 Steam per I. H.P. per hour: With jackets, lbs 22.6 21.4 20.3 19.6 19 18.7 18.6 18.9 19.5 20.4 21.0 Without jackets, lbs 22. 20.5 19.6 19.2 19.1 19.3 20.1 .... Saving by jacket, p. c 10.9 7.3 4.6 3.1 1.0-1.0-1.5 .... This table gives a clue to the great variation in the apparent saving due to the steam-jacket as reported by different experimenters. With this par- ticular engine it appears that when running at its most economical rate of 100 H.P., without jackets, very little saving is made by use of the jackets. When running light the jacket makes a considerable saving, but when over- loaded it is a detriment. At the load which corresponds to the most economical rate, with no steam in jackets, or 100 H.P., the use of the jacket makes a saving of only 1$; but at a load of 60 H.P. the saving by use of the jacket is about 11$, and the shape of the curve indicates that the relative advantage of the jacket would be still greater at lighter loads than 60 H.P. Counterbalancing Engines.— Prof. Unwin gives the formula for counterbalancing vertical engines: W x = W 2 -; (1) in which W x denotes the weight of the balance weight and p the radius to its centre of gravity, W 2 the weight of the crank-pin and half the weight of the connecting-rod, and r the length of the crank. For horizontal engines: W 1 =%(W 2 + W 3 )~ to %(W* + W 3 )-, in which W 3 denotes the weight of the piston, piston-rod, cross-head, and the other half of the weight of the connecting-rod. The American Machinist, commenting on these formulae, says: For hori- zontal engines formula (2) is often used; formula (1) will give a counter- balance too light for vertical engines. We should use formula (2) for computing the counterbalance for both horizontal and vertical engines, excepting locomotives, in which the counterbalance should be heavier. PERFORMANCES OF STEAM-ENGINES. 789 Preventing Vibrations of Engines.— Many suggestions have been made for remedying the vibration and noise attendant on the working of the big engines which are employed to run dynamos. A plan which has given great satisfaction is to build hair-felt into the foundations of the engine. An electric company has had a DO-horse-power engine removed from its foundations, which were then taken up to the depth of 4 feet. A layer of felt 5 inches thick was then placed on the foundations and run up 2 feet on all sides, and on the top of this the brickwork was built up. — Safety Valve. Steam-engine Foundations Embedded in Air.— In the sugar- refinery of Claus bpreckels, at Philadelphia, Fa., the engines are distributed practically all over the buildings, a large proportion of them being on upper floors. Some are bolted to iron beams or girders, and are consequently innocent of all foundation. Some of these engines ran noiselessly and sat is- factorily, while others produced more or less vibration and rattle. To cor- rect the latter the engineers suspended foundations from the bottoms of the engines, so that, in looking at them from the lower floors, they were literally hanging in the air.— Iron Age. Mar. 13. 1890. Cost of Coal for Steam-power.— The following table shows the amount and the cost of coal per day and per year for various horse-powers, from 1 to 1000, based on the assumption of 4 lbs. of coal being used per hour pei 1 horse-power. It is useful, among other things, in estimating the saving that may be made in fuel by substituting more economical boilers and engines for those already in use. Thus with coal at $3.00 per ton, a saving of $9000 per year in fuel may be made by replacing a steam plant of 1000 H.P., requiring 4 lbs. of coal per hour per horse-power, with one requiring only 2 lbs. Coal Consumption , at 4 lbs. per H.P. per hour ; 10 hours a $1.50. $2.00. $3.00. $4.00. day ; 300 days in x Yea 1 Lbs. Long Tons. Short Per Per Per Per a 1 Tons. Short Ton. Short Ton. Short Ton. Short Ton. Cost in Cost in Cost in Cost in Dollars. Dollars. Dollars. Dollars. Per Day. Per Day. Per Year. Per Day. Per Year Per Per Per Per Per Per Per Per .0179 .02 6 Day. .03 Year 9 Day. .04 Year. Day. .06 Year. Day. .08 Year l 40 5.357 12 18 24 10 400 .1786 53.57 .20 60 .30 90 .40 120 .60 180 .80 240 25 1,000 .4464 133.92 .50 150 .75 225 1.00 300 1.50 450 2.00 600 50 2,000 .8928 267.85 1.00 300 1.50 450 2.00 600 3.00 900 4.00 1,200 75 3,000 1.3393 401.78 1.50 450 2.'25 675 3.00 900 4.50 1,350 6.00 1,800 100 4,000 1.7857 535.71 2.00 600 3.00 900 4.00 1,200 6.00 1,800 8.00 2,400 150 6,000 2.6785 803.56 3.00 900 4.50 1,350 6.0C 1,800 9.00 2,700 12.00 3.000 200 8,000 3.5714 1,071.42 4.00 1,200 6.00 1,800 8.00 2,400 12.00 3,600 16.00 1.800 250 10,000 4.4642 1,339.27 5.00 1,500 7.50 2.250 10.00 3,000 15.00 4,500 20.00 6,000 300 1-2.000 5 3571 1,607.13 6.00 1,800 9.00 2,700 12.00 3,600 18.00 5,400 24.00 7,200 350 14,000 6.2500 7.00 2,100 10.50 8.150 14.00 4,200 21.00 6,200 28.00 8,400 400 16,000 7.1428 8.00 2,400 12.00 3,000 16.00 4,800 24.00 7,200 9,600 450 18,000 8.0356 9.00 2,700 13.50 4,050 18 00 5,400 8,100 36.00 io.xoo 500 20,000 8.9285 10.00 3,000 15.00 4,500 20.00 6,000 30.00 9,000 40.00 12,000 600 24,000 10.7142 3,214.26 12.00 3,600 18.00 5,400 24.00 7,200 36.00 10,800 48.00 14,400 700 28,000 12.4999 3,749.97 14.00 4,200 21.00 6.300 28.(11) 8,400 42.00 11,600 56.00 16,800 800 32.000 14.2856 4,285.68 16.00 4,800 24.00 7,200 32.00 9,601 48.00 12,400 64.00 900 36,000 16.0713 4,821.39 18.00 5,400 27.00 8,100 36.00 |o,80( 54.00 14,200 72.00 1.000 40,000 17.8570 5,357.10 20.00 6,000 30.00 40.00 12,001 18,000 80.00 24,000 Storing Steam Heat.— There is no satisfactory method for equalizing the load on the engines and boilers in electric-light stations. Storage-batteries have been used, but they are expensive in first cost, repairs, and attention. Mr. Halpin, of London, proposes to store heat during the day in specially constructed reservoirs. As the water in the boilers is raised to 250 lbs. pres- sure, it is conducted to cylindrical reservoirs resembling English horizontal boilers, and stored there for use when wanted. In this way a comparatively small boiler-plant can be used for healing the water to 250 lbs. pressure all through the twenty-four hours of the day, and the stored water may be drawn on at any time, according to the magnitude of the demand. The 790 THE STEAM-ENGINE. steam-engines are to be worked by the steam generated by the release of pressure from this water, and the valves are to be arranged in such a way that the steam shall work at 130 lbs. pressure. A reservoir 8 ft. in diameter and 30 ft. long, containing 84,000 lbs. of heated water at 250 lbs. pressure, would supply 5250 lbs. of steam at 130 lbs. pressure. As the steam consump- tion of a condensing electric -light engine is about 18 lbs. per horse-power hour, such a reservoir would supply 286 effective horse-power hours. In 1878, in France, this method of storing steam was used on a tramway. M. Francq, the engineer, designed a smokeless locomotive to work by steam- power supplied by a reservoir containing 400 gallons of water at 220 lbs. pressure. The reservoir was charged with steam from a stationary boiler at one end of the tramway. Cost of Steam-power. (Chas. T. Main, A. S. M. E., x. 48.)— Estimated costs in .New England in 1S88, per horse-power, tyised on engines of 1000 H. P. Compound Engine. 1. Cost engine and piping, complete $2.5.00 2. Engine-house 8.00 3. Engine foundations 7.00 4. Total engine plant 40.00 5. Depreciation, 4% on total cost 1 .60 6. Repairs, 2% " " " 0.80 7. Interest, 5$ " " " 2.00 8. Taxation, 1.5$ on % cost 0.45 9. Insurance on engine and house 0.165 10. Total of lines 5, 6, 7, 8, 9 5.015 11. Cost boilers, feed-pumps, etc.. 9.33 12. Boiler-house , 2.92 13. Chimney and flues 6.11 14. Total boiler-plant 18.; 15. Depreciation, 5$ on total cost 16. Repairs, 2% " " '• 17. Interest, 5$ " " " .... 18. Taxation, 1.5$ on % cost 19. Insurance, 0.5$ on total cost 20. Total of lines 15 to 19 21. Coal used per I.H.P. per hour, lbs 22. Cost of coal per I.H.P. per day of 10J4 cts. hours at $5.00 per ton of 2240 lbs 4.00 23. Attendance of engine per day 0.60 24. " " boilers " " 0.53 25. Oil, waste, and supplies, per clay 0.25 26. Total daily expense 5.38 27. Yearly running expense, 308 days, per I.H.P $16,570 28. Total yearly expense, lines 10, 20. and 27. . 24.087 29. Total yearly expense per I.H.P. for power if 50$ of exhaust-steam is used for heat- ing 12.597 30. Total if all ex. -steam is used for heating. . . 8.624 Non-con- 20.00 $17.50 7.50 7.50 5.50 4.50 53.00 29.50 7l8 0.59 1 .475 0.332 0.125 3.702 16.00 5.00 8.00 Condens- 1.32 0.66 1.65 0.371 0.138 4.139 18.36 24.80 29.00 918 1.240 1.450 .367 .496 .580 .918 1.240 1.450 .207 .279 .326 .092 .124 .145 2.502 3.379 3.951 1.75 2.50 3.00 cts. 5.72 0.40 0.75 0.22 7.09 $21,837 29.355 14.907 7.916 0.35 0.90 0.20 $25,595 33.248 16.663 7.700 When exhaust- steam or a part of the receiver-steam is used for heating, or if part of the steam in a condensing engine is diverted from the condenser, and used for other purposes than power, the value of such steam should ROTARY STEAM-EKGIKES. ?91 be deducted from the cost of the total amount of steam generated, in order to arrive at the cost properly chargeable to power. The figures in lines 29 and 30 are based on an assumption made by Mr. Main of losses of heat amounting to 25$ between the boiler and the exhaust-pipe, an allowance which is probably too large. ROTARY STEAM-ENGINES. Steam Turbines.— The steam turbine is a small turbine wheel which runs with steam as the ordinary turbine does with water. (For description of the Parsons and the Dow steam turbines see Modern Mechanism, p. 298, etc.) The Parsons turbine is a series of parallel-flow turbines mounted side by side on a shaft; the Dow turbine is a series of radial outward-flow tur- bines, placed like a series of concentric rings in a single plane, a stationary guide-ring being between each pair of movable rings. The speeds of the steam turbines enormously exceed those of any form of engine with recip- rocating piston, or even of the so-called rotary engines. The three- and four- cyliuder engines of the Brotherhood type, in which the several cylinders are usually grouped radially about a common crank and shaft, often exceed 1000 revolutions per minute, and have been driven, experimentally, above 2000; but the steam turbine of Parsons makes 10.000 and even 20,000 revolu- tions, and the Dow turbine is reputed to have attained 25,000. (See Trans. A. S. M. E., vol. x. p. 680, and xii. p. 888; Trans. Assoc, of Eng'g Societies, vol. viii. p. 583; Eng'g, Jan. 13, 1888, and Jan. 8, 1892; Eng'g Neios, Feb. 27, 1892.) A Dow turbine, exhibited in 1889, weighed 68 lbs., and developed 10 HP., with a consumption of 47 lbs. of steam per H.P. per hour, the steam pressure being 70 lbs. The Dow turbine is used to spin the fly-wheel of the Howell torpedo. The dimensions of the wheel are 13.8 in. diam., 6.5 in. width, radius of gyration 5.57 in. The energy stored in it at 10,000 revs, per min. is 500,000 ft.-lbs. The De Laval Steam Turbine, shown at the Chicago exhibition, 1893, is a reaction wheel somewhat similar to the Pelton water-wheel. The steam jet is directed by a nozzle against the plane of the turbine at quite a small angle and tangentially against the circumference of the medium periphery of the blades. The angle of the blades is the same at the side of admission and discharge. The width of the blade is constant along the entire thickness of the turbine. The steam is expanded to the pressure of the surroundings before arriv- ing at the blades. This expansion takes place in the nozzle, and is caused simply by making its sides diverging. As the steam passes through this channel its specific volume is increased in a greater proportion than the cross section of the channel, and for this reason its velocity is increased, and also its momentum, till the end of the expansion at the last sectional area of the nozzle. The greater the expansion in the nozzle the greater its velocity at this point. A pressure of 75 lbs. and expansion to an absolute pressure of one atmosphere give a final velocity of about 2625 ft. per second. Expansion is carried further in this steam turbine than in ordinary steam- engines. This is on account of the steam expanding completely during its work to the pressure of the surroundings. For obtaiuing the greatest possible effect the admission to the blades must be free from blows and the velocity of discharge as low as possible. These conditions would require iu the steam turbine an enormous velocity of periphery — as high as 1300 to 1650 ft. per second. The centrifugal force, nevertheless, puts a limit to the use of very high velocities. In the 5 horse- power turbine the velocity of periphery is 574 ft. per second, and the num- ber of revolutions 30,000 per minute. However carefully the turbine may be manufactured it is impossible, on account of unevenness of the material, to get its centre of gravity to corre- spond exactly to its geometrical axle of revolution; and however small this difference may be, it becomes very noticeable at such high velocities. De Laval has succeeded in solving the problem by providing the turbine with a flexible shaft. This yielding shaft allows the turbine at the high rate of speed to adjust itself and revolve around its true centre of gravity, the centre line of the shaft meanwhile describing "a surface of revolution. In the gearing-box the speed is reduced from 30,000 revolutions to 3000 by means of a driver on the turbine shafts, which sets in motion a cog- wheel of 10 times its own diameter. These gearings are provided with spiral cogs placed at an angle of about 45°. The shaft of the larger cog-wheel, running at a speed of 3000 revolutions, is provided at its outer end with a pulley for the further transmission of the power. 792 THE STEAM-ENGINE. Rotary Steam-engines, other thaD steam turbines, have been invented by the thousands, but not oue has attained a commercial success. The possible advantages, such as saving of space, to be gained by a rotary engine are overbalanced by its waste of steam. The Tower Spherical Engine, one of the most recent forms of rotary-engine, is described in Proc. lust. M. E., 1885, also in Modern Mechanism, p. 296. DIMENSIONS OF PARTS OF ENGINES. The treatment of this subject by the leading authorities on the steam-en- gine is very unsatisfactory, being a confused mass of rules and formulae based partly upon theory and partly upon practice. The practice of builders shows an exceeding diversity of opinion as to correct dimensions. The treatment given below is chiefly the result of a study of the works of Rankine, Seaton, Unwin, Thurston, Marks, and Whitham, and is largely a condensa- tion of a series of articles by the author published in the American Ma- chinist, in 1894, with many alterations and much additional matter. In or- der to make a comparison of many of the formula? they have been applied to the assumed cases of six engines of different sizes, and in some cases this comparison has led to the construction of new formulae. Cylinder. (Whitham.)— Length of bore = stroke -f- breadth of piston- ring — % to 14 ' n ? length between heads = stroke + thickness of piston -f sum of clearances at both ends; thickness of piston = breadth of ring -j- thickness of flange on one side to carry the ring — + thickness of follower- plate. Thickness of flange or follower — % to \4> in. % in. 1 in. For cylinder of diameter 8 to 10 in. 36 in. 60 to 100 in. Clearance of Piston. (Seaton.)— The clearance allowed varies with the size of tbe engine from J^ to % in. for roughness of castings and \/\§ to "4 bn. for each working joint. Naval and other very fast-running engines lave a larger allowance. In a vertical direct-acting engine the parts which wear so as to bring the piston nearer the bottom are three, viz., the shaft journals, the crank-pin brasses, and oiston-rod gndgeon-brasses. Thickness of Cylinder. (Thurston.) — For engines of the older types and under moderate steam-pressures, some builders have for many years restricted the stress to about 2550 lbs. per sq. in. t = apiD + fc (1) is a common proportion; t, D, and b being thickness, diam., and a constant added quantity varying from to % in., all in inches; p l is the initial unbal- anced steam-pressure per sq. in. In this expression b is made larger for horizontal than for vertical cylinders, as, for example, in large engines 0.5 in the one case and 0.2 in the other, the one requiring re-boring more than the other. The constant a is from 0.0004 to 0.0005; the first value for verti- cal cylinders, or short strokes; the secon-d for horizontal engines, or for long strokes. Thickness of Cylinder and its Connections for Marine Engines. (Seaton).— D = the diam. ot the cylinder in inches: p = load on the safety-valves in lbs. per sq. in.; /, a constant multiplier = thickness of barrel 4- .25 in. Thickness of metal of cylinder barrel or liner, not to be less than p x D ■+■ 3000 when of cast iron.* (2) Thickness of cylinder-barrel = - ■ -f- 0.6 in (3) " liner = 1.1 X/. (4) Thickness of liner when of steel p x D -i- 6000 + 0.5 " metal of steam-ports =0.6 x /. " " valve-box sides = 0.65 X/. * "When made of exceedingly good material, at least twice melted, the thickness may be 0.8 of that given by the above rules. ha DIMENSIONS OF PARTS OF ENGINES. 793 Thickness of metal of valve-box covers = 0.7 X f- " " cylinder bottom = 1.1 x /, if single thickness. " = 0.65 X /, if double " " " covers =1.0 x /, if single " " " " " =0.6 x/, if double " *' cylinder flange =1.4 X /. " " cover-flange =1.3 X /. " " valve-box" =1.0 x /. " " door-flange =0.9 X /. " " face over ports = 1.2 X /. " " " " =1.0 X /. when there is a false-face. " " false-face =0.8 X /, when cast iron. " " " =0.6 X /, when steel or bronze. Whitham gives the following from different authorities: VanBuren:-H = °- 0001 ^+ - 15 ^; & I t = 0.03 VDp (6) ™ d: 1 = <-TO m Weisbach: t = 0.8 + 0.00033pD (8) Seaton : t = 0.5 -f 0,00Q4pD (9) Harwell • }t = 0.0004pD+M (vertical); (10) nasweii . ( £ = .0005pD + y 8 (horizontal) (11) Whitham recommends (6) where provision is made for the reboring, and where ample strength and rigidity are secured, for horizontal or vertical cylinders of large or small diameter; (9) for large cylinders using steam under 100 lbs. gauge -pressure, and t = 0.003D Vp for small cylinders (12) Marks gives t = 0.00028pD (13) This is a smaller value than is given by the other formulae quoted; but Marks says that it is not advisable to make a steam-cylinder less than 0.75 in. thick under any circumstances. The following table gives the calculated thickness of cylinders of engines of 10, 30. and 50 in. diam., assuming p the maximum unbalanced pressure on the piston = 100 lbs. per sq. in. As the same engines will be used for calcu- lation of other dimensions, other particulars concerning them are here given for reference. Dimensions, etc., of Engines. Engine No Indicated horse -power I.H.P. Diam. of cyl., in D Stroke, feet. L Revs, per min Piston speed, ft. per min S Area of piston, sq. in a Mean effective pressure . . M.E.P. Max. total unbalanced press P Max. total per sq. in p 50 10 7854 100 .... 5 4 .... ... 65 90 .... 650 700 706.86 1963.5 32.3 30 70,686 196,350 100 100 794 THE STEAM-ENGINE. Thickness of Cylinder by Formula. (1) .OOOipD 4- 0.5, short stroke., (1) .OOOopD + 0.5, long stroke . . . (2) ,00033pZ> (3) .0002pZ> + 0.6 . _ (5) .0001pZH-.15 \/D , (6) .03 4/Dp .. (0+2.5) (T) 1 — Z '. p 1900 (8) .00033pZ> + 8 (9) .0004pD-f 05 ( 10) .0004pD + Vh (vertical) . (11) .OOOopD + V 8 (horizontal) (12) .003D Vp (small engines). (13) .00038pD. 90 00 33 80 1.70 2.00 .99 1.40 .57 1.18 .95 1.64 .66 1.71 .13 .90 .53 .63 1.79 1.70 1.33 1.63 .30(?) .28(?) ".84 2.50 3.00 1 67 l.fiO 1.56 2.12 2.45 2.50 2.13 Average of first eleven , 1.48 The average corresponds nearly to the formula t = .00037Z)p -4- 0.4 in. A convenient approximation is t = .0004Z)p ~f- 0.3 in., which gives for Diameters 10 20 30 40 50 60 in. Thicknesses 70 1.10 1.50 .1.90 2.30 2.70 in. The last formula corresponds to a tensile strength of cast iron of 12,500 lbs., with a factor of safety of 10 and an allowance of 0.3 in. for reboring. Cylinder-heads.— Thurston says: Cylinder-heads may be given a thickness, ac l he edges and in the flanges, exceeding somewhat that of the cylinder. An excess of not less than 25$ is usual It may be thinner in the middle. Where made, as is usual in large engines, of two disks with inter- mediate radiating, connecting ribs or webs, that section which is safe against shearing is probably ample. An examination of the designs of experienced builders, by Professor Thurston, gave D being the diameter of that circle in which the thickness is taken. Thurston also gives t = .005D Vp _+ °- 25 ( 2 ) Marks gives t = 0.003Z) Vp $) He also says a good practical rule for pressures under 100 lbs. per sq. in. is to make the thickness of the cylinder-heads 1J4 times that of the walls; and applying this factor to his formula for thickness of walls, or .00028pZ>, we have ... t = .00035pD (4) Whitham quotes from Seaton, . _pD + 500 (1) 2000 -, which is equal to .0005pZ> + .25 inch. (5) Seaton's formula for cylinder bottoms, quoted above, is t = 1.1/, in which / = .0002pZ) + -85 inch, or t = .00022pD + .93. . (6) Applying the above formulas to the engines of 10, 30, and 50 inches diame- ter, with maximum unbalanced steam-pressure of 100 lbs. per sq. in., we Cylinder diameter, inches =10 30 50 (1) t = .00033Dp_-H .25 (2) t .= .005D Vp + .25 (3) t = .003Z) Vp (4) t - .00035 Dp (5) t = .0005 Dp + .25 (6) t = .0002 J 'Dp + .93 Average of 6 . . , .53 1.25 1.82 .75 1.75 2.75 .30 .35 .75 1.15 .90 1.05 1.75 1.59 1.50 1.75 2.75 2.03 ,65 1.38 2.10 DIMENSIONS OF PARTS OF ENGINES. 795 The average is expressed by the formula t = .00036Z)p -4- .31 inch. Meyer's " Modern Locomotive Construction, 1 ' p. 24, gives for locomotive cylinder-heads for pressures up to 120 lbs. : For diameters, in 19 to 22 16 to 18 14 to 15 11 to 13 9 to 10 Thickness, in 134 • * 1 V& % Taking the pressure at 120 lbs. per sq. in., the thicknesses 1*4 in- and % in. for cylinders 22 and 10 in. diam., respectively, correspond to the formula t = ,00035Dp-f .33 inch. Web-stiffened Cylinder-covers.— Seaton objects to webs for stiffening cast-iron cylinder-covers as a source of danger. The strain on the web is one of tension, and if there should be a nick or defect in the outer edge of the web the sudden application of strain is apt to start a crack. He recommends that high-pressure cylinders over 24 in. and low- pressure cylinders over 40 in. diam. should have their covers cast hollow, with two thicknesses of metal. The depth of the cover at the middle should be about J4 the diam. of the piston for pressures of 80 lbs. and upwards, and that of the low-pressure cylinder-cover of a compound engiue equal to that of the high-pressure cylinder. Another rule is to make the depth at the middle not less than 1.3 times the diameter of the piston-rod. In the British Navy the cylinder-covers are made of steel castings, % to 1J4 in. thick, generally cast without webs, stiffness being obtained by their form, which is often a series of corrugations. Cylinder-head Bolts.— Diameter of bolt-circle for cylinder-head = diameter of cylinder -f 2 x thickness of cylinder + 2 X diameter of bolts. The bolts should not be more than 6 inches' aoart (Whitham).. Marks gives for number of bolts b = '' ' , = area of a single bolt, p = boiler-pressure in lbs. per sq. in.; 5000 lbs. is taken as the safe strain per sq. in. on the nominal area of the bolt. Seaton says: Cylinder-cover studs and bolts, when made of steel, should be of such a size that the strain in them does not exceed 5000 lbs. per sq. in. When of less than % inch diameter it should not exceed 4500 lbs. per sq. in. When of iron the strain should be 20$ less. Thurston says : Cylinder flanges are made a little thicker than the cylin- dei\ and usually of equal thickness with the flanges of the heads. Cylinder- bolts should be so closely spaced as not to allow springing of the flanges and leakage, say, 4 to 5 times the thickness of the flanges. Their diameter should be proportioned for a maximum stress of not over 4000 to 5000 lbs. per square inch. If D = diameter of cylinder, p = maximum steam -pressure, 6 = number of bolts, s — size or diameter of each bolt, and 5000 lbs. be allowed per sq. in. of nominal area of the bolt, .7854D 2 p = 39276«2; whence 6s 2 = .00021^; , 2 ^£: „ = .01414i/|. Diameter of cylinder, inches Diameter of bolt-circle, approx . . Circumference of circle, approx. Minimum No. of bolts, circ. -*- 6. . Diam. of bolts, s = .01414D 10 30 13 35 40.8 110 7 18 Vi- The diameter of bolt for the 10 inch cylinder is 0.54- in. by the formula, but % inch is as small as should be taken, on account of possible overstrain by the wrench in screwing: up the nut. The Piston. Details of Construction of Ordinary Pis- tons. (Seaton.)— Let D be the diameter of the piston in inches, p the effec- tive pressure per square inch on it, x a constant multiplier, foundas follows: X = £xVp4-l. 796 THE STEAM-ENGINE. The thickness of front of piston near the boss =0.2 X % " " " " rim = 0.17 X x. " back " =0.18 X x. " boss around the rod =0.3 X x. " flange inside packing-ring = 0.23 X x. at edge = 0.25 x x. " packing-ring = 0.15 x x. " junk-ring at edge — 0.23 X x. " " inside packing-ring =0.21 X x. " "at bolt-holes = 0.35 X x. "' metal around piston edge = 0.25 X x The breadth of packing-ring = 0.63 X x. " depth of piston at centre =1.4 x x. " lap of junk-ring on the piston = 0.45 X x. " space between piston body and packing-ring =0.3 x x. " diameter of junk-ringbolts =0.1 X x -f 0.25 in. " pitch " " " = 10 diameters. " number of webs in the piston = (D + 20) h- 12. " thickness " " " = 0.18 X x. A. Marks gives the approximate rule: Thickness of piston-head= \/ld, in which I = length of stroke, and d — diameter of cylinder in inches. Whit- ham says in a horizontal engine the rings support the piston, or at least a part of it, under ordinary conditions. The pressure due to the weight of the piston upon an area equal to 0.7 the diameter of the cylinder X breadth of ring-face should never exceed 200 lbs. per sq. in. He also gives a formula much used in this country: Breadth of ring-face = 0.15 X diam- eter of cylinder. For our engines we have diameter = 10 30 50 Thickness of piston -head. Marks, VlO; long stroke 3.31 5.48 7.00 Marks, % ' ; short stroke 3.94 6.51 8.32 Seatou, depth at centre = 1.4a? . 4.30 9.80 15.40 Seaton, breadth of ring = .63.T 1.89 4.41 6.93 Whitham, breadth of ring = .151) 1.50 4.50 7.50 Diameter of Piston Packing- rings. — These are generally turned, before they are cut, about *4 inch diameter larger than the cylinder, for cylinders up to 20 inches diameter, and then enough is cut out of the ring to spring them to the diameter of the cylinder. For larger cylinders the rings are turned proportionately larger. Seaton recommends an excess of \% of the diameter of the cylinder. Cross-section of the Rings.— The thickness is commonly made l/30th of the diam. of cyl. -j- J^inch, and the width = thickness + % inch. For an eccentric ring the mean thickness may be the same as for a ring of uniform thickness, and the minimum thickness = % the maximum. A circular issued by J. H. Dunbar, manufacturer of packing -rings, Youngstown, O., says: Unless otherwise ordered, the thickness of rings will be made equal to .03 x their diameter. This thickness has been found to be satisfactory in practice. It admits of the ring being made about 3/16" to the foot larger than the cylinder, and has. when new, a tension of about two pounds per inch of circumference, which is ample to prevent leakage if the surface of the ring and cylinder are smooth. As regards the width of rings, authorities " scatter " from very narrow to verv wide, the latter being fully ten times the former. For instance, Unwin gives W= d .014 -|- .08. Whitham's formula is W = d .15. In both for- mula? IF is the width of the ring in inches, and d the diameter of the cylinder in inches. Unwin's formula makes the width of a 20" ring W = 20 X .014 -J- .08 = .36", while Whitham's is 20 X .15 = 3" for the same diameter of ring. There is much less difference in the practice of engine-builders in thi< respect, but there is still room for a standard width ot ring. It is believed that for cylinders over 16" diameter %" is a popular and practical width, and Vz" for evlinders of that size and under. Fit of Piston-rod into Piston. (Seaton.)— The most convenient and reliable practice is to turn the piston-rod end with a shoulder of 1/16 inch for small engines, and % inch for large ones, make the taper 3 in. to DIMENSIONS OF PARTS OF ENGINES. 797 the foot until the section of the rod is three fourths of that of the body, then i urn the remaining part parallel; the rod should then fit into the piston so as to leave y% incli between it and the shoulder for large pistons, and 1/16 in. for small. The shoulder prevents the rod from splitting the piston, and allows of the rod being turned true after long wear without encroaching on the taper. The piston is secured to the rod by a nut, and the size of the rod should be such that the strain on the section at the bottom of the thread does not exceed 5500 lbs. per sq. in. for iron, 7000 lbs. for steel. The depth of this nut need not exceed the diameter which would be found by allowing these strains. The nut shouid be locked to prevent its working loose. Diameter of Piston-rods.— Unwin gives d" = bD Vp, (1 in which D is the cylinder diameter in inches, p is the maximum unbalanced pressure in lbs. per sq. in., and the constant b — 0.0167 for iron, and b = 0.0144 for steel. Thurston, from an examination of a considerable number of rods in use, gives (L in feet, D and d in inches), in which a — 10,000 and upward in the various types of engines, the marine screw engines or ordinary fast engines on shore giving the lowest values, while "low-speed engines" being less liable to accident from shock give a = 15,000, often. Connections of the piston-rod to the piston and to thecrosshead should have a factor of safety of at least 8 or 10. Marks gives d" = 0.0179D j/p, for iron; for steel d" = 0.01 05Z> Yp; . . (3) and d" ^ 0.03901 \ 'DH*p, for iron; for steel d" = 0.03525 \'DH^p, ^4) in which I is the length of stroke, all dimensions in inches. Deduce the diameter of piston-rod by (3), and if this diameter is less than 1/12/, then use (4). ~ , . -^. . „ . , , Diameter of cylinder ,_ Seaton gives: Diameter of piston-rod = ^ — j/p. The following are the values of F: Naval engines, direct-acting F — 60 " " return connuecting-rod, 2 rods F = 80 Mercantile ordinary stroke, direct-acting F = 50 long " " " F= 48 " very long " " F — 45 " medium stroke, oscillating F — 45 Note.— Long and very long, as compared with the stroke usual for the power of engine or size of cylinder. In considering an expansive engine p. the effective pressure should be taken as the absolute working pressure, or 15 lbs. above that to which the boiler safety-valve is loaded; for a compound engine the value of p for the high-pressure piston should be taken as the absolute pressure, less 15 lbs., or the same as the load on the safety-valve; for the medium-pressure the load may be taken as that due to half the absolute boiler-pressure; and for the low-pressure cylinder the pressure to which the escape-valve is loaded -J- 15 lbs., or the maximum absolute pressure, which can be got in the re- ceiver, or about 25 lbs. It is an advantage to make all the rods of a com- pound engine alike, and this is now the rule. Applying the above formulae to the engines of 10, 30, and 50 in. diameter, both short and long stroke, we have: THE STEAM-EKGINE. Diameter of Piston-rods. Diameter of Cylinder, inches. . Thurston 'A Stroke, inches Unwin, iron, .0167Z) Yp.. Unwin, steel, .0144 D Yp /D*pL? D 10,000 + 80 Thurston, same with a = 15,000 Marks, iron, .0179Z> Vp Marks, iron, .03901 fW 2 p ... Marks, steel, .01052) Yp Marks, steel, .035-25 V D*l*p Seaton, naval engines, — y'p . . . (L in feet), 1.79 1.35 (1.05) 1.22 1.67 D Seaton, land engine, — y p Average of four for iron. 1.G7 1.44 5.37 (3.15) 3.34 5.01 2.22 1.82 5.01 4.32 48 8.35 7.20 5.10 3.88 5.37 5.13 6.04 (5.25) 8.95 8.54 The figures in brackets opposite Marks' third formula would be rejected since they are less than y$ of the stroke, and the figures derived by his fourth formula would be taken instead. The figure 1.79 opposite his first formula would be rejected for the engine of 24-inch stroke. An empirical formula which gives results approximating the above aver- ages is d" = .013 VDlp- The calculated results from this formula, for the six engines, are, respec- tively, 1.42, 1.88, 3.90, 5.61, 6.37, 9.01. Piston-rod. Guides.— The thrust on the guide, when the connecting- rod is at its maximum angle with the line of the piston-rod, is found from the formula: Thrust = total load on piston X tangent of maximum angle of connecting-rod — p tan 0. This angle is the angle whose tangent = half stroke of piston -*- length of connecting-rod. Ratio of length of connecting-rod to stroke 2 2)4 3 Maximum angle of connecting-rod with line of piston-rod 14° 29' 11° 19' 9° 36' Tangent of the angle 25 .20 .1667 Secant of the angle 1.0308 1.0198 1.0138 Seaton says: The area of the guide-block or slipper surface on which the thrust is taken should in no case be less than will admit of a pressure of 400 lbs. on the square inch; and for good working those surfaces which take the thrust when going ahead should be sufficiently large to prevent the maxi- mum pressure exceeding 100 lbs. per sq. in. When the surfaces are kept well lubricated this allowance may be exceeded. Thurston says: The rubbing surfaces of guides are so proportioned that if Vbe their relative velocity in feet per minute, and p be ihe intensity of pressure on the guide in lbs. per sq. in., pV < 60,000 and pV > 40,000. The lower is the safer limit; but for marine and stationary engines it is allowable to take p = 60,000 -s- V. According to Rankine, for locomotives, p = where p is the pressure in lbs. per sq. in. and F"the velocity of rubbing in feet per minute. This includes the sum of all pressures forcing the two rubbing surfaces together. Some British builders of portable engines restrict the pressure between the guides and cross-heads to less than 40, sometimes 35 lbs. per square inch. For a mean velocity of 600 feet per minute, Prof. Thurston's formulas give, p < 100, p > 66.7; Rankine's gives p = 72.2 lbs. per sq. in. DIMENSIONS OF PARTS OF ENGINES. 799 Whitham gives, A = area of slides in square inches = — = — — ===z, p Vn* - 1 p Vn* - 1 in which P = total unbalanced pressure, p^ = pressure per square inch on piston, d = diameter of cylinder, p = pressure allowable per square inch on slides, and n = length of connecting-rod -f- length of crank. This is equivalent to the formula, A = P tan -f- 2> . For n = 5, p l = 100 and p = 80, A = .2004d 2 . For the three engines 10, 30 and 50 in. diam., this would give for area of slides, A = 20, 180 and 500 sq. in., respectively. Whitham says: The normal pressure on the slide may be as high as 500 lbs. per sq. in., but this is when there is good lubrication and freedom from dust. Station- ary and marine engines are usually designed to carry 100 lbs. per sq. in., and the area in this case is reduced from 50$ to 60$ by grooves. In locomo- tive engines the pressure ranges from 40 to 50 lbs. per sq. in. of slide, on ac- count of the inaccessibility of the slide, dirt, cinder, etc. There is perfect agreement among the authorities as to the formula for area of the slides, A = P tan -*- p ; but the value given to p , the allow- able pressure per square inch, ranges all the way from 35 lbs. to 500 lbs. The Connecting-rod. Ratio of length of connecting-rod to length of stroke.— Experience has led generally to the ratio of 2 or 2% to 1, the latter giving a long and easy-working rod, the former a rather short, but yet a manageable one (Thurston). Whitham gives the ratio of from 2 to 4J^, and Marks from 2 to 4. Dimensions of the Connecting-rod. — The calculation of the diameter of a connecting-rod on a theoretical basis, considering it as a strut subject to both compressive and bending stresses, and also to stress due to its inertia, in high-speed engines, is quite complicated. See Whitham, Steam-engine Design, p. 217; Thurston, Manual of S. E., p. 100. Empirical formulas are as follows: For circular rods, largest at the middle, D — diam. of cylinder, I ~ length of connecting-rod in inches, p = maximum steam-pressure per sq. in. (1) Whitham, diam. at middle, d" - 0.0272 V Dl Vp. (2) Whitham, diam. at necks, d" = 1.0 to 1.1 x diam. of piston-rod. (3) Sennett, diam. at middle, d" = — Vv- (4) Sennett, diam. at necks, d" — — Vp- 60 (5) Marks, diam., d" — 0.0179Z) Vp. if d iam. is greater than 1/24 length. (6) Marks, diam., d" = 0.02758 \/ Dl Vp if diam. found by (5) is less than 1/24 length. (7) Thurston, diam. at middle, d" = a yD L Vp + C, D in inches, L in feet, a = 0.15 and C = % inch for fast engines, a — 0.08 and C — % inch for moderate speed. (8) Seaton says: The rod may be considered as a strut free at both ends, and, calculating its diameter accordingly, diameter at middle = — — ^-^-r > 4o.5 where R = the total load on piston P multiplied by the secant of the maxi- mum angle of obliquity of the connecting-rod. For wrought iron and mild steel a is taken at 1/3000. The following are the values of r in practice: Naval engines— Direct-acting r = 9 toll; " " Return connecting-rod r = 10 to 13, old; " " " " r = 8 to 9, modern; " " Trunk r= 11.5 to 13. Mercantile " Direct-acting, ordinary r — 12. " " " long stroke r = 13 to 16. (9) The following empirical formula is given by Seaton as agreeing closely with good modern practice: Diameter of conne cting-rod at middle — X'lK-±- 4, I = length of rod in inches, and K= 0.03 V effective loacTon piston in pounds, 800 THE STEAM-ENGINE. The diam. at the ends may be 0.875 of the diam. at the middle. Seaton's empirical formula when translated into terms of D andp is the same as the second one by Marks, viz., d" = 0.02758 V Dl Vp~- Whitham's (1) is also practically the same. (10) Taking Seaton's more complex formula, with length of connecting- rod = 2.5 x length of stroke^and r = 12 and 16, respectively, it reduces to: Diam. at middle = .02294 VP and .02411 VP for short and long stroke en- gines, respectively. Applying the above formulas to the engines of our list, we have Diameter of Connecting-rods. Diameter of Cylinder, inches. . Stroke, inches Length of connecting-i-od I (3) d" = ;| Vp = .0182Z) |/p.... (5) d" = .0179D Vp (6) d" = .02758^ Dl Vp (7) d" = OAbVDL Vp + %, (7) d" = 0.08 V DL Vp + U (9) d" = .03 VF. (10) d" = .02294 VP; .02411 VP. . 1.82 1.79 2.67 2.03 Average . 2.24 2. 5.46 5.37 2.54 2.67 2.14 13.29 10.16 6.38 6.27 10.52 10 240 9.09 13.29 10.68 Formulae 5 and 6 (Marks), and also formula 10 (Seaton), give the larger diameters for the long-stroke engine; formulas? give the larger diameters for the short-stroke engines. The average figures show but little difference in diameter between long- and short-stroke engines; this is what might be expected, for while the connecting-rod, considered simply as a column, would require an increase of diameter for an increase of length, the load remaining the same, yet in an engine generally the shorter the connecting- rod the greater the number of revolutions, and consequently the greater the strains due to inertia. The influences tending to increase the diameter therefore tend to balance each other, and to render the diameter to some extent independent of the length. The average figures correspond nearly to the simple formula d" = .021 D Vp. The diameters of rod for the three diameters of engine by this formula are, respectively, 2.10, 6.30, and 10.50 in. Since the total pressure on the piston P — .7854D 2 p, the formula is equiva lent to d' = .0287 VP. Connecting-rod. Ends.— For a connecting-rod end of the marine type, where the end is secured with two bolts, each bolt should be propor- tioned for a safe tensile strength equal to two thirds the maximum pull or thrust in the connecting-rod. The cap is to be proportioned as a beam loaded with the maximum pull of the connecting-rod, and supported at both ends. The calculation should be made for rigidity as well as strength, allowing a maximum deflection of 1/100 inch. For a strap-and-key connecting-rod end the strap is designed for tensile strength, considering that two thirds of the pull on the conuecting- rod may come on one arm. At the point where the metal is slotted for the key and gib, the straps must be thickened to make the cross-section equal to that of the remainder of the strap. Between the end of the strap and the slot the strap is liable to fail in double shear, and sufficient metal must be provided at the end to prevent such failure. The breadth of the key is generally one fourth of the width of the strap, and the length, parallel to the strapj should be such that the cross-section will have a shearing strength equal to the tensile strength of the section of the strap. The taper of the key is generally about % inch to the foot. DIMENSIONS OF PARTS OF ENGINES. 801 Tapered Connecting-rods.— In modem high-speed engines it is customary to make the connecting-rods of rectangular instead of circular section, the sides being parallel, and the depth increasing regularly from the crosshead end to the crank-pin end. According to Grashof, the bending action on the rod due to its inertia is greatest at 6/10 the length from the crosshead end, and, according to this theory, that is the point at which the section should be greatest, although in practice the section is made greatest at the crank-pin end. Professor Thurston furnishes the author with the following rule for tapered connecting-rod of rectangular section : Take the section as computed by the formula d" = O.lV DL Vp + 3/4 for a circular section, and for a rod 4/3 the actual length, placing the computed section at 2/3 the length from the small end, and carrying the taper straight through this fixed section to the large end. This brings the computed section at the surge point and makes it heavier than the rod for which a tapered form is not required. Taking the above formula, multiplying L by 4/3, and changing it to I in inches, it becomes d = 1/30 V Dl Vp -|- 3/4". Taking a rectangular section of the same area as the round section whose diameter is d, and making the depth of the section h = twice the thickness t, we have .7854d 2 = lit = 2i 2 , whence t = .627d = .0209 V Dl Vp + .47", which is the formula for the thick- ness or distance between the parallel sides of the rod. Making the depth at the crosshead end = IM, and at 2/3 the length = 2t, the equivalent depth at the crank end is 2.25t. Applying the formula to the short-stroke engines of our examples, we have Diameter of cylinder, inches Stroke, inches Length of connecting-r od . Thickness, t = .0209 VdI Vp -f .47 =.. Depth at crosshead end, \M — Depth at crank end, 2%t 10 12 30 30 30 75 1 61 3.60 2.42 3.62 5.41 8.11 48 120 5.59 8.39 12.58 The thicknesses t, found by the formula t = .0209 V Dl Vp + .47, agree closely with the more simple formula t = MD Vp -f- .60", the thicknesses calculated by this formula being respectively 1.6, 3.6, and 5.6 inches. The Crank-pin.— A crank-pin should be designed (1) to avoid heating, (2) for strength, (3) for rigidity. The heating of a crank-pin depends on the pressure on its rubbing-surface, and on the coefficient of friction, which latter varies greatly according to the effectiveness of the lubrication. It also depends upon the facility with which the heat produced may be carried away: thus it appears that locomotive crank-pins may be prevented to some degree from overheating by the cooling action of the air through which they pass at a high speed. Marks gives 1 = .0000247 fpNW = 1.038/^^^ (1) Whitham gives I = 0.9075/ - J ^ P, - ) , (2) in which I = length of crank-pin journal in inches, f — coefficient of friction, which may be taken at .03 to .05 for perfect lubrication, and .08 to .10 for im- perfect; p = mean pressure in the cylinder in pounds per square inch; D = diameter of cylinder in inches; N = number of single strokes per minute; I.H.P. = indicated horse-power; L = length of stroke in feet. These formulEe are independent of the diameter of the pin, and Marks states as a general law, within reasonable limits as to pressure and speed of rubbing, the longer a bearing is made, for a given pressure and number of revolutions, the cooler it will work; and its diameter Las no effect upon its heating. Both of the above formulae are deduced empirically from dimensions of crank-pins of existing marine engines. Marks says that about one-fourth the length required for crank-pins of propeller engines will serve for the pins Of side-wheel engines, and one tenth for locomotive engines, making the 802 THE STEAM-ENGINE. formula for locomotive crank-pins I = .00000247/piVD 2 , or if p = 150, / = .08, and N= 600, / = .013D 2 . Whitham recommends for pressure per square inch of projected area, for naval engines 500 pounds, for merchant engines 400 pounds, for paddle-wheel engines 800 to 900 pounds. Thurston says the pressure should, in the steam engine, never exceed 500 or 600 pounds per square inch for wrought-iron pins, or about twice that figure for steel. He gives the formula for length of a steel pin, in inches, 1 = PR+- 600.000, (3) in which P and R are the mean total load on the pin in pounds, and the number of revolutions per minute. For locomotives, the divisor may be taken as 500,000. Where iron is used this figure should be reduced to 300,000 and 250,000 for the two cases taken. Pins so proportioned, if well made and well lubricated, may always be depended upon to run cool; if not well formed, perfectly cylindrical, well finished, and kept well oiled, no crank-pin can be relied upon. It is assumed above that good bronze or white-metal bearings are used. Thurston also says : The size of crank-pins required to prevent heating of the journals may be determined with a fair degree of precision hy either of the formulae given below : l = l t^T ^nkine,1865); (4) ^-60^J (ThUrStOI1 ' 1862); (5) < = 3l|o¥o^ anB ^ en < 1866) (6) The first two formulas give what are considered by their authors fair work- ing proportions, and the last gives minimum length for iron pins. (V — velocity of rubbing-surface in feet per minute.) Formula (1) was obtained by observing locomotive practice in which great liability exists of annoyance by dust, and great risk occurs from inaccessi- bility while running, and (2) by observation of crank-pins of naval screw- engines. The first formula is therefore not well suited for marine practice. Steel can usually be worked at nearly double the pressure admissible with iron running at similar speed. Since the length of the crank- pin will be directly as the power expended upon it and inversely as the pressure, we may take it as l=a^,. . . (7) in which a is a constant, and L the stroke of piston, in feet. The values of the constant, as obtained by Mr. Skeel, are about as follows: a — 0.04 where water can be constantly used; a = 0.045 where water is not generally used; a = 0.05 where water is seldom used; a - 0.06 where water is never needed. Unwin gives ; IH.P. rR I = a (8 r ' in which r = crank radius in inches, a = 0.3 to a = 0.4 for iron and for marine engines, and a = 0.066 to a — 0.1 for the case of the best steel and for loco- motive work, where it is often necessary to shorten up outside pins as much as possible. J. B. Stanwood (Eng^g, June 12, 1891), in a table of dimensions of parts of American Corliss engines from 10 to 30 inches diameter of cylinder, gives sizes of crank-pins which approximate closely to the formula 1= .275D" + .5 in.; d = .25D" (9) By calculating lengths of iron crank-pins for the engines 10. 30, and 50 inches diameter, long and short stroke, by the several formulae above given, it is found that there is a great difference in the results, so that one formula in certain cases gives a length three times as great as another. Nos. (4). (5), and (6) give lengths much greater than the others. Marks (1), Whitham (2), Thurston (7), I = .06 I.H.P. -h L, and Unwin (8), I = 0.4 I.H.P. -*- r, give re- sults which agree more closely. DIMENSIONS OF PARTS OF ENGINES. 803 The calculated lengths of iron crank pins for the several cases by formulae (1), (2), (7), and (8) are as follows: Length of Cranlc-pins. Diameter of cylinder D Stroke I. (ft.) Revolutions per minute R 250 125 130 65 90 45 Horse-power I.H.P. 50 50 450 450 1,250 1,250 Maximum pressure lbs. 7,854 7,854 70,686 70,686 196,350 196,350 Mean pressure per cent of max 42 42 32.3 32.3 30 30 Mean pressure P. 3,299 3,299 22,832 22,832 58,905 58, 905 Length of crank-pin (1) Whitham, I = .9075 X .05 I.H.P. -s- L. (2) Marks, I = 1.038 X .05 I.H.P. -s- L. (7) Thurston, I = .06 I.H.P. +L (8)Unwin, I = .4 I.H.P. -s- r 3.33 1.67 12.0 6.0 20.83 10.42 (8) " I = .3 I.H.P. -r- r 2.50 1.25 9.0 4.5 15.62 7.81 Average (8) Unwin, best steel, I = .ilJ^J 83 .42 3.0 1.5 5.21 2.61 r (3) Thurston, steel, I = -**L 1.37 .69 4.95 2.47 8.84 4.42 o00,000 The calculated lengths for the long-stroke engines are too low to prevent excessive pressures. See " Pressures on the Crank-pins," below. The Strength of the Crank-pin is determined substantially as is' that of the crank. In overhung cranks the load is usually assumed as carried at its extremity, and, equating its moment with that of the resist- ance of the pin, 10 10 30 30 50 1 2 W» 5 4 250 125 130 65 90 50 50 450 450 1,250 7,854 7.854 70,686 70,686 196,350 42 42 32.3 32.3 30 3,299 3,299 22,832 22,832 58,905 2.18 1.09 8.17 4.08 14.18 2.59 1.30 9.34 4.67 16.22 3.00 1.50 10.80 5.40 18.75 3.33 1.67 12.0 6.0 20.83 2.50 1.25 9.0 4.5 15.62 2.72 1.36 9.86 4.93 17.12 .83 .42 3.0 1.5 5.21 1.37 .69 4.95 2.47 8.84 J4PI = l/32trrd*, and d -{/^ in which d = diameter of pin in inches, P = maximum load on the piston, t = the maximum allowable stress on a square inch of the metal. For iron it may be taken at 9000 lbs. For steel the diameters found by this formula may be reduced 10%. (Thurston.) Unwin gives the same formula in another form, viz. : 4/x^W* the last form to be used when the ratio of length to diameter is assumed. For wrought iron, t = 6000 to 9000 lbs. per sq. in., 0- For steel, t - 9000 to 13,000 lbs. per sq. in,, Whitham gives d = 0.0827 \/Pl = 2.1058, . 1/?= sq. in,, 0291 to .0238. for strength, and d = 0.405 yPl 3 for rigidity, and recommends that the diameter be calculated by both formulae, and the largest result taken. The first is the same as Unwinds formula, with t taken at 9000 lbs. per sq. in. The second is based upon an erroneous assumption. 804 THE STEAM-EtfGItfE. Marks, calculating the diameter for rigidity, gives d = O.OffifypPD 2 = 0.' . 4 /(H.P.)/ 3 p = maximum steam-pressure in pounds per square inch, D — diameter of cylinder in inches, L — length of stroke in feet, N= number of single strokes per minute. He says there is no need of an investigation of the strength of a crank-pin, as the condition of rigidity gives a great excess of strength. Marks's formula is based upon the assumption that the whole load may be concentrated at the outer end, and cause a deflection of .01 inch at that point. It is serviceable, he says, for steel and for wrought iron alike. Using the average lengths of the crank-pins already found, we have the following for our six engines : Diameter of Crank-pins. Diameter of cylinder. . . Stroke, ft Length of crank-pin. . . TT . , ' 3/5.IPZ Unwin, d = a/ — j- ... Marks, d = .066 typPD*. 10 1 2.72 10 2 1.36 30 9.86 30 5 4.93 50 4 17.12 2.29 1.82 7.34 5.82 12.40 1.39 .85 6.44 3.78 12.41 50 8 8.56 9.84 7.39 Pressures on the Crank-pins.— If we take the mean pressure upon the crank-pin = mean pressure on piston, neglecting the effect of the vary- ing angle of the connecting-rod, we have the following, using the average lengths already found, and the diameters according to Unwin and Marks: Engine No 1 2 3 4 5 6 10 1 3,299 6.23 3.78 530 873 10 2 3,299 236 1.16 1,398 2,845 30 2% 72 .4 63.5 315 360 30 5 22,832 28.7 18.6 796 1,228 50 4 58,905 212.3 212.5 277 277 50 8 Mean pressure on pin, pounds 58,905 84.2 63.3 Pressure per square inch, Unwin " " " " Marks 700 930 The results show that the application of the formulae for length and diam- eter of crank-pins give quite low pressures per square inch of projected area for the short-stroke high-speed engines of the larger sizes, but too high pressures for all the other engines. It is therefore evident that after calcu- lating the dimensions of a crank-pin according to the formulae given that the results should be modified, if necessary, to bring the pressure per square inch down to a reasonable figure. In order to bring the pressures down to 500 pounds per square inch, we divide the mean pressures by 500 to obtain the projected area, or product of length by diameter. Making I = 1.5d for engines Nos. 1, 2, 4 and 6, the revised table for the six engines is as follows : Engine, No". 1 2 Length of crank- pin, inches 3.15 3.15 Diameter of crank-pin 2.10 2.10 .34 4 5 6 3.37 17.12 13.30 5.58 12.40 8.87 Crosshead-pin or Wrist-pin.— Whitham says the bearing surface for the wrist-pin is found by the formula for crank-pin design. Seaton says the diameter at the middle must, of course, be sufficient to withstand the bending action, and general^ from this cause ample surface is provided for good working; but in any case the area, calculated by multiplying the diam- eter of the journal by its length, should be such that the pressure does not exceed 1200 lbs. per'sq. in., taking the maximum load on the piston as the total pressure on it. For small engines with the gudgeon shrunk into the jaws of the connect- DIMENSIONS OF PARTS OF ENGINES. 805 ing-rod, and working in brasses fitted into a recess in the piston-rod end and secured by a wrought- iron cap and two bolts, Seaton gives: Diameter of. gudgeon = 1.25 X diam. of piston-rod. Length of gudgeon = 1.4 x diam. of piston-rod. If the pressure on the section, as calculated by multiplying length by diameter, exceeds 1200 lbs. per sq. in., this length should be increased. J. B. Stanwood, in his "Ready Reference 1 ' book, gives for length of crosshead-pin 0.25 to 0.3 diam. of piston, and diam. = 0.18 to 0-.2 diam. of piston. Since he gives for diam. of piston-rod 0.14 to 0.17 diam. of piston, his dimensions for diameter and length of crosshead-pin are about 1.25 and 1.8 diam. of piston-rod respectively. Taking the maximum allowable press- ure at 1200 lbs. per sq. in. and making the length of the crosshead-pin = 4/3 of its diameter, we have d = l/P-^40, I = \/~P -s- 30, in which P = max- imum total load on piston in lbs., d = diam. and Z = length of pin in inches. For the engines of our example we have: Diameter of piston, inches 10 30 50 Maximum load on piston, lbs. 7854 70,686 196,350 Diameter of crosshead-pin, inches 2.22 6.65 11.08 Length of crosshead-pin, inches 2.96 *8.86 14.77 Stanwood's rule gives diameter, inches 1.8to2 5.4 to 6 9.0 to 10 Stan wood's rule gives length, inches 2.5 to 3 7.5 to 9 12.5 to 15 Stan wood's largest dimensions give pressure per sq. in., lbs 1309 1329 1309 Which pressures arc greater than the maximum allowed by Seaton. The Crank-arm.— The crank-arm is to be treated as a lever, so that if a is the thickness in direction paral.el to the shaft-axis and b its breadth at a section x inches from the crank-pin centre, then, bending moment M at that section = Px, P being the thrust of the connecting-rod, and / the safe strain per square inch, „ fab* . aXb* T 6T , P*=- W - and -g- = j, or a = ^^ f ; b w%- If a crank-arm were constructed so that b varied as Vx (as given by the above rule) it would be of such a curved form as to be inconvenient to man- ufacture, and consequently it is customary in practice to find the maxi- mum value of b and draw tangent lines to the curve at the points ; these lines are generally, for the same reason, tangential to the boss of the crank- arm at the shaft. The shearing strain is the same throughout the crank-arm; and, conse- quently, is large compared with the bending strain close to the crank-pin ; and so it is not sufficient to provide there only for bending strains. The section at this point should be such that, in addition to what is given by the calculation from the bending moment, there is an extra square inch for every 8000 lbs. of thrust on the connecting-rod (Seaton). The length of the boss h into which the shaft is fitted is faoin 0.75 to 1.0 of the diameter of the shaft D, and its thickness e must be calculated from the twisting strain PL. (L = length of crank.) For different values of length of boss h, the following values of thickness of boss e are given by Seaton: When h = D, then e = 0.35 D; if steel, 0.3. h = 0.9 D, then e = 0.38 D, if steel, 0.32. h = 0.8 D, then e = 0.40 D, if steel, 0.33. h = 0.7 D. then e = 0.41 D, if steel, 0.34. The crank-eye or boss into which the pin is fitted should bear the same relation to the pin that the boss does to the shaft. The diameter of the shaft-end onto which the crank is fitted should be 1.1 X diameter of shaft. Thurston says: The empirical proportions adopted by builders will com- monly be found to fall well within the calculated safe margin. These pro- portions are, from the practice of successful designers, about as follows : For the wrought-iron crank, the hub is 1.75 to 1.8 times the least diameter of that part of the shaft carrying full load; the eye is 2.0 to 2.25 the diame- ter of the inserted portion of the pin, and their depths are, for the hub, 1.0 to 1.2 the diameter of shaft, and for the eye, 1.25 to 1.5 the diameter of pin. 806 THE STEAM-EKGIXE. The web is made 0.7 to 0.75 the width of adjacent hub or eye, and is given a depth of 0.5 to 0.6 that of adjacent hub or eye. For the cast-iron crank the hub and eye are a little larger, ranging in diameter respectively from 1.8 to 2 and from 2 to 2.2 times the diameters of shaft and pin. The flanges are made at either end of nearly the full depth of hub or eye. Cast-iron has, however, fallen very generally into disuse. The crank-shaft is usually enlarged at the seat of the crank to about 1.1 its diameter at the journal. The size should be nicely adjusted to allow for the shrinkage or forcing on of the crank. A difference of diameter of one fifth of l$,"wiil usually suffice ; and a common rule of practice gives an allowance of but one half of this, or .001. The formulas given by different writers for crank-arms practically agree, since they all consider the crank as a beam loaded at one end and fixed at the other. The relation of breadth to thickness may vary according to the taste of the designer. Calculated dimensions for our six'engines are as fol lows : ©line u si oiis of Crank-arms, Diam. of cylinder, ins.. . Stroke S, ins * Max. pressure on pin P, (approx.) lbs Diam. crank-pin d 7T.H.P. Diam. shaft, (a = 4.69, 5.09 and 5.22). Length of boss, .8D Thickness of boss, AD. Diam. of boss, 1.8D Length crank-pin eye,. 8d Thickness of crank-pin eye, Ad Max. mom. Tat distance }4S - y%D from centre of pin, inch-lbs Thickness of crank-arm a = .75D Greatest breadth, : y 9001 6r 9000a Min.mom. T at distance d from centre of pin^l'd Least breadth, 10 12 10 24 30 30 30 60 50 48 7854 2.10 7854 2.10 70,686 7.34 70,686 5.58 196,350 12.40 1-2.74 2.19 1.10 4.93 1.76 3.46 7.70 9.70 12.55 2.77 1.39 6.23 1.76 6.16 3.08 13.86 5.87 7.76 3.88 17.46 4.48 10.04 5.02 22.59 9.92 ,88 .88 2.94 2.23 4.46 37, 149 80,661 788,149 1,848,439 3,479,322 2.05 2.60 5.78 7.28 9.41 3.48 4.55 9.54 13.0 15.7 16,493 16,493 528,835 394,428 2,434,740 2.32 2.06 7.81 6.01 13.13 196,350 8.87 28.47 7.10 7,871,671 11.87 1,741,6! The Shaft.— Twisting Resistance. for torsion, we have: T= — d 3 S - -From the g enera l formula .19635d 3 S, whence d = 1/ ^—, in which T = torsional moment in inch-pounds, d = diameter in inches, and S = the shearing resistance of the material in pounds per square inch. If a constant force P were applied to the crank-pin tangentially to its path, the work done per minute would be 13,000 XLH.P., in which L = length of cank in inches, and R = revs, per min., and the mean twisting moment T = ' x 63,025. Therefore 3 /s.ijr y ?I.H.P. DIMENSIONS OF PARTS OF ENGINES. 807 This may take the form 3 /l.H.P. X F, or d - c in which .Fand a are factors that depend on the strength of the material and on the factor of safety. Taking S at 45,000 pounds per square inch for wrought iron, and at 60,000 for steel, we have, for simple twisting by a uni- form tangential force, Factor of safety = 5 Iron F= 35.7 Steel F= 26.8 6 8 10 42.8 57.1 71.4 32.1 42.8 53.5 6 8 3.5 3.85 3.18 3.5 10 4.15 Unwin, taking for safe working strength of wrought iron 9000 lbs., steel 13.500 lbs., and cast iron 4500 lbs., gives a = 3.294 for wrought iron, 2.877 for steel, and 4.15 for cast iron. Thurston, for crank-axles of wrought iron, gives a = 4.15 or more. Seaton says: For wrought iron, /, the safe strain per square inch, should not exceed 9000 lbs., and when the shafts are more than 10 inches diameter, 8000 lbs. Steel, when made from the ingot and of good materials, will ad- mit of a stress of 12,000 lbs. for small shafts, and 10,000 lbs. for those above 10 inches diameter. The difference in the allowance between large and small shafts is to com- pensate for the defective material observable in the heart of large shafting, owing to the hammering failing to affect it. a / i h p The formula d = ai/ ' assumes the tangential force to be uniform and that it is the only acting force. For engines, in which the tangential force varies with the angle between the crank and the connecting-rod, and with the variation in steam-pressure in the cylinder, and also is influenced by the inertia of the reciprocating parts, and in which also the shaft may be subjected to bending as well as torsion, the factor a must be increased, to provide for the maximum tangential force aud for bending. Seaton gives the following table showing the relation between the maxi- mum and mean twisting moments of engines working under various condi- tions, the momentum of the moving parts being neglected, which is allow- able: Description of Engine. Steam Cut-off Max. Twist Divided by Mean Twist. Mome't Cube Root of the Ratio. Single-crank expansive Two-cylindev expansive, cranks at 90° — Three-cylinder compound, cranks 120°. . . " '■' 1. p. cranks ] opposite one another, andh.p. midway j 0.7 0.8 h.p.0.5, l.p. 0.6 2.625 2.125 1.835 1.698 1.616 1.415 1.298 1.256 1.270 1.329 1.357 1.40 1.26 1.38 1.29 1.22 1.20 1.17 1.12 1.08 1.10 1.11 1.12 Seaton also gives the following rules for ordinary practice for ordinary two-cylinder marine engines: Diameter of the tunnel-shafts = a/ ' • XF, or a. ;/ I.H P. 808 THE STEAM-ENGINE. Compound engines, cranks at right angles: Boiler pressure 70 lbs., rate of expansion 6 to 7, F = 70, a = 4.12. Boiler pressure 80 lbs., rate of expansion 7 to 8, F = 72, a — 4.16. Boiler pressure 90 lbs., rate of expansion 8 to 9, F = 75, a — 4.22. Triple compound, three cranks at 120 degrees: Boiler pressure 150 lbs., rate of expansion 10 to 12, F = 62, a = 3.96. Boiler pressure 160 lbs., rate of expansion 11 to 13, F = 64, a = 4. Boiler pressure 170 lbs., rate of expansion 12 to 15, F — 67, a — 4.06. Expansive engines, cranks at right angles, and the rate of expansion 5, boiler-pressure 60 lbs., F — 90, a — 4.48. Single-crank compound engines, pressure 80 lbs., F — 96, a — 4.58. For the engines we are considering it will be a very liberal allowance for ratio of maximum to mean twisting moment if we take it as equal to the ratio of the maximum to the mean pressure on the piston. The factor a, then, in the formula for diameter of the shaft will be multiplied by the cube root of this ratio, or a/— =1.34, i/^- = 1.45, and a/~ = 1.49 for the y 42 ' |/ 32.3 y .30 10, 30, and 50-in. engines, respectively. Taking a = 3.5, which corresponds to a shearing strength of 60,000 and a factor of safety of 8 for steel, or to 45,000 and a factor of 6 for iron, we have for the new coefficient cij in the 3 /l H P formula d x = a x A/ ■' ' * , the values 4.69, 5.08, and 5.22, from which we obtain the diameters of shafts of the six engines as follows: Engine No 1 2 3 4 5 6 Diana.- of cyl 10 10 30 30 50 50 Horse-power, I.H.P 50 50 450 450 1250 1250 Revs, per min., R 250 125 130 65 90 45 Diam. of shaft d = a 1 i/ I J5l!:.... 2.74 3.46 7.67 9.70 12.55 15.82 y r These diameters are calculated for twisting only. When the shaft is also subjected to bending strain the calculation must be modified as below : Resistance to Bending.— The strength of a circular-section shaft to resist bending is one half of that to resist twisting. If B is the bending moment in inch-lbs., and d the diameter of the shaft in inches, B= -^ x/; and d = i/y X 10.2; f is the safe strain per square inch of the material of which the shaft is composed, and its value may be taken as given above for twisting (Seaton). Equivalent Twisting Moment.— When a shaft is subject to both twisting and bending simultaneously, the combined strain on any sec- tion of it may be measured by calculating what is called the equivalent twisting moment; that is, the two strains are so combined as to be treated as a twisting strain only of the same magnitude and the size of shaft cal- culated accordingly. Rankine gave the following solution of the combined action of the two strains. If T = the twisting moment, and B — the bending mome nt on a se ction of a shaft, then the equivalent twisting moment T± = B-\- YB 2 -f- T 2 . Seaton says: Crank-shafts are subject always to twisting, bending, and shearing strains; the latter are so small compared with the former that they are usually neglected directly, but allowed for indirectly by means of the factor /. The two principal strains vary throughout the revolution, and the maxi- mum equivalent twisting moment can only be obtained accurately by a series of calculations of bending and twisting moments taken at fixed inter- vals, and from them constructing a curve of strains. Considering the engines of our examples to have overhung cranks, the maximum bending moment resulting from the thrust of the connecting-rod on the crank-pin will take place when the engine is passing its centres (neglecting the effect of the inertia of the reciprocating parts), and it will be the product of the total pressure on the piston by the distance between DIMENSIONS OP PARTS OP ENGINES. 809 two parallel lines passing through the centres of the crank-pin and of the shaft bearing, at right angles to their axes; which distance is equal to y& length of crank-pin bearing -f length of hub -f-i^ length of shaft bearing -f any clearance that may be allowed between the crank and the two bearings. For our six engines we may take this distance as equal to Yo, length of crank-pin -f- thickness of crank-arm-)- 1.5 X the diameter of the shaft as already found by the calculation for twisting. The calculation of diameter is then as below: Engine No. 1 2 3 4 5 Diain. of cyl., in. . 10 10 30 30 50 Horse-power 50 50 450 450 1250 Revs, per min.. .. 250 125 130 65 90 Max. press, on pis,P 7,854 7,854 70,686 70,686 196,350 Leverage,* L in 6.32 7.94 22.20 26.00 36.80 Bd.mo.PL^Pin.-lb 49,637 62,361 1,569,222 1,837,836 7,225,680 Twist, morn. T. 47,124 94,248 1,060,290 2,120,580 4,712,400 Equiv. Twist, mom. T!=B+ VB^+T 2 (approx.) 118.000 175.000 3,463.000 4.647.000 15,840,000 1250 45 196,350 42.25 8,295,788 9,424,800 * Leverage = distance between centres of crank-pin and shaft bearing = Having already found the diameters, on the assumption that the shafts were subjected to a twisting moment T only, we may find the diameter for resisting combined bending and twisting by multiplying the diameters already found by the cube roots of the ratio Ti -5- T, or Giving corrected diameters d 1 =. 1.40 1.27 1.46 1.34 1.64 1.36 4.39 11.35 12.99 20.58 21.52 By plotting these results, using the diameters of the cylinders for abscissas and diameters of the shafts for ordinates, we find that for the long-stroke engines the results lie almost in a straight line expressed by the formula, diameter of shaft = .43 X diameter of cylinder; for the short -stroke engines the line is slightly curved, but does not diverge far from a straight line whose equation is, diameter of shaft = .4 diameter of cylinder. Using these two formulas, the diameters of the shafts will be 4.0, 4.3, 12.0, 12.9, 20.0, 21.5. J. B. Stanwood, in Engineering, June 12, 1891, gives dimensions of shafts of Corliss engines in American practice for cylinders 10 to 30 in. diameter. The diameters range from 4 15/16 to 14 15/ 16, following precisely the equation, diameter of shaft = % diameter of cylinder - 1/16 inch. Fly-wtieel Shafts.— Thus far we have considered the shaft as resist- ing the force of torsion and the bending moment produced by the pressure on the crank-pin. In the case of fly-wheel engines the shaft on the opposite side of the bearing from the crank pin has to be designed with reference to the bending moment caused by the weight of the fly wheel, the weight of the shaft itself, and the strain of the belt. For engines in which there is an outboard bearing, the weight of fly-wheel and shaft being supported by two bearings, the point of the shaft at which the bending moment is a maximum may be taken as the point midway between the two bearings or at the middle of the fly-wheel hub, and the amount of the moment is the product of the weight supported by one of the bearings into the distance from the centre of that bearing to the middle point of the shaft. The shaft is thus to be treated as a beam supported at the ends and loaded in the middle. In the case of an overhung fly-wheel, the shaft having only one bearing, the point of maximum moment should be taken as the middle of the bearing, and its amount is very nearly the product of half the weight of the fly-wheel and the shaft into the distance from the middle of its hub from the middle of the bearing. The bending moment should be calculated and combined with the twisting moment as above shown, to obtain the equivalent twisting moment, and the diameter necessary at the point of maximum moment calculated therefrom. In the case of our six engines we assume that the weights of the fly- wheels, together with the shaft, are double the weight of fly-wheel rim obtained from the formula] W— 785,400 -^r^r' (given under Fly-wheels); 810 THE STEAM-ENGINE. that the shaft is supported by an outboard bearing, the distance between the two bearings being 2%, 5, and 10 feet for the 10-in., 30-in., and 50-in. engines, respectively. The diameters of the fly-wheels are taken such that their rim velocity will be a little less than 6000 feet per minute. Engine No 1 2 3 4 5 6 Diam. of cyl., inches 10 10 30 30 50 50 Diam. of fly-wheel, ft 7.5 15 14.5 29 21 42 Revs, per min 250 125 130 65 90 45 Half wt.fly-wh'l and shaft,lb. ■ 268 536 5,963 11,936 26,470 52,940 Lever arm for max.mom.,in. 15 15 30 30 60 60 Max. bending moment, in.-lb. 4020 8040 179,040 358,080 1,588,200 3,176,409 As these are very much less than the bending moments calculated from the pressures on the crank-pin, the diameters already found are sufficient for the diameter of the shaft at the fly-wheel hub. In the case of engines with heavy band fly-wheels and with long fly-wheel shafts it is of the utmost importance to calculate the diameter of the shaft with reference to the bending moment due to the weight of the fly-wheel and the shaft. B. H. Coffey (Power, October, 1892) gives the formula for combined bend- ing and twisting resistance, T t = .196d3£, in which T x = B + j/^ + r 2 ; T being the maximum, not the mean twisting moment; and finds empirical working values for .196S as below. He says: Four points should be consid- ered in determining this value: First, the nature of the material; second, the manner of applying the loads, with shock or otherwise; third, the ratio of the bending moment to the torsional moment— the bending moment in a revolving shaft produces reversed strains in the material, which tend to rup- ture it; fourth, the size of the section. Inch for inch, large sections are weaker than small ones. He puts the dividing line between large and small sections at 10 in. diameter, and gives the following safe values of S X .196 for steel, wrought iron, and cast iron, for these conditions. Value of S X .196. Ratio. Heavy Shafts with Shock. Light shafts with Shock. Heavy Shafts No Shock. Light Shafts No Shock. Bto T. Steel . Wro't Iron. Cast Iron. Steel. Wro't Iron. Cast Iron. Steel. Wro't Iron. Cast Iron. 3 to 10 or less 3 to 5 or less 1 to 1 or less B greater than T. . 1045 941 855 784 880 785 715 655 440 393 358 328 1566 1410 1281 1176 1320 1179 1074 984 660 589 537 492 2090 1882 1710 1568 1760 1570 1430 1310 880 785 715 655 Mr. Coffey gives as an example of improper dimensions the fly-wheel shaft of a 1500 H.P. engine at Willimantic, Conn., which broke while the en- gine was running at 425 H.P. The shaft was 17 ft. 5 in. long between centres of bearings, 18 in. diam. for 8 ft. in the middle, and 15 in. diam. for the re- mainder, including the bearings. It broke at the base of the fillet connect- ing the two large diameters, or 56% in. from the centre of the bearing. He calculates the mean torsional moment to be 446,654 inch-pounds, and the maximum at twice the mean; and the total weight on one bearing at 87,530 lbs., which, multiplied by 56^ in., gives 4,945,445 in. -lbs. bending moment at the fillet. Applying the formula 7\ = B + VB* + T 2 , gives for equivalent twisting moment 9,971,045 in.-lbs. Substituting this value in the formula T x = .196, Sd 3 gives for S the shearing strain 15,070 lbs. per sq. in., or if the metal had a shearing strength of 45,000 lbs., a factor of safety of only 3. Mr. Coffey considers that 6000 lbs. is all that should be allowed for S under these circumstances. This would give d = 20.35 in. If we take from Mr. Coffey's table a value of .1965 = 1100, we obtain d 3 = 9000 nearly, or d — 20.8 in., instead of 15 in., the actual diameter. Length of Shaft-bearings.— There is as great a difference of opinion among writers, and as great a variation in practice concerning length of journal-bearings, as there is concerning crank-pins. The length of a DIMENSIONS OF PARTS OF ENGINES. 811 journal being determined from considerations of its heating, the observa- tions concerning heating of crank-pins apply also to shaft-bearings, and the formulae for >ength of crank-pins to avoid heating may also be used, using for the total load upon the bearing the resultant of all the pressures brought upon it, by the pressure on the crank, by the weight of the fly-wheel, and by the pull of the belt. After determining this pressure, however, we must resort to empirical values for the so-called constants of the formulae, really variables, which depend on the power of the bearing to carry away heat, and upon the quantity of heat generated, which latter depends on the pres- sure, on the number of square feet of rubbing surface passed over in a minute, and upon the coefficient of friction. This coefficient is an exceed- ingly variable quantity, ranging from .01 or less with perfectly polished journals, having end-play, and lubricated by a pad or oil-bath, to .10 or more with ordinary oil-cup lubrication. For shafts resisting torsion only, Marks gives for length of bearing I = .0000247 fpND 2 , in which /is the coefficient of friction, p the mean pressure in pounds per square inch on the piston, JVthe number of single strokes per minute, and D the diameter of the piston. For shafts under the combined stress due to pressure on the crank-pin, weight of fly-wheel, etc., he gives the following: Let Q = reaction at bearing due to weight, S = stress due steam pressure on piston , and R x — the resultant force ; for horizontal engines, Ri = YQ 2 -f- S 2 , for vertical engines R x = Q + S, when the pressure on the crank is in the same direction as the pressure of the shaft on its bearings, and R t — Q - S when the steam pressure tends to lift the shaft from its bearings. Using empirical values for the work of friction per square inch of projected area, taken from dimensions of crank-pins in marine vessels, he finds the formula for length of shaft-journals I — .0000325/RiiV, and recommends that to cover the. defects of workmanship, neglect of oiling, and the introduction of dust, / be taken at .16 or even greater. He says that 500 lbs. per sq. in. of projected area may be allowed for steel or wrought- iron shafts in brass bearings with good results if a less pressure is not attain- able without inconvenience. Marks says that the use of empirical rules that do not take account of the number of turns per minute has resulted in bear- ings much too long for slow- speed engines and too short for high-speed engines. Whitham gives the same formula, with the coefficient .00002575. Thurston says that the maximum allowable mean intensity of pressure PV may be, for all cases, computed by his formula for journals, I = — , or b(J,u(J0ct P(V + 20) by Rankine's, I — -, in which Pis the mean total pressure in pounds, 44,o00u Fthe velocity of rubbing surface in feet per minute, and d the diameter of the shaft in inches. It must be borne in mind, he says, that the friction work on I he main bearing next the crank is the sum of that due the action of the piston on the pin, and that due that portion of the weight of wheel and shaft and of pull of the belt which is carried there. The outboard bearing carries practically only the latter two parts of the total. The crank-shaft journals will be made longer on one side, and perhaps shorter on the other, than that of the crank-pin, in proportion to the work falling upon each, i.e., to their respective products of mean total pressure, speed of rubbing sur- faces, and coefficients of friction. Unwin says: Journals running at 150 revolutions per minute are often only one diameter long. Fan shafts running 150 revolutions per minute have journals six or eight diameters long. The ordinary empirical mode of pro- portioning the length of journals is to make the length proportional to the diameter, and to make the ratio of length to diameter increase with the speed. For wrought-iron journals: Revs, per min. = 50 100 150 200 250 500 1000 -i = .004R + 1. Length -f- diam. = 1.2 1.4 1.6 1.8 2.0 3.0 5.0. Cast-iron journals may have I -*- d — 9/10, and steel journals I -h d = 3J4, of the above values. 0.4 H. P. Unwin gives the following, calculated from the formula I = ; , in which r is the crank radius in inches, and H.P. the horse-power transmitted to the crank- pin. 812 THE STEAM-ENGINE. Theoretical Journal Length in Inches. Load on Revolutions of Journal per minute. in pounds. 50 100 200 300 500 1000 1,000 .2 .4 .8 1.2 2. 4. 2,000 .4 8 1.6 2.4 4. 8. 4,000 .8 1 6 3.2 4.8 8. 16. 5,000 1.0 2 4. 6. 10. 20. 10,000 2. 4 8. 12. 20. 40. 15,000 3. 6 12. 18. 30. 20,000 4. 8 16. 24. 40. 30,000 6. 12 24. 36. 40,000 8. 16 32. 50,000 10. 20 40. Applying these different formluae to our six engines, we have: Engine No.. 1 2 3 4 5 10 50 250 3,299 268 10 50 125 3,299 536 30 450 130 23,185 5,968 30 450 65 23,185 11,936 50 1,250 90 58,905 26,470 3,310 3.84 3,335 4.39 23,924 11.35 26,194 12.99 64,580 20.58 5.38 2.71 20.87 11.07 37.78 4.27 2.15 16.53 8.77 29.95 3.61 1.82 14.00 7.43 25.36 5.22 2.78 21.70 10.85 35.16 7.68 6.59 17.25 16.36 27.99 3.33 1.60 12.00 6.00 20.83 4.92 2.99 17.05 10.00 29.54 Diam. cyl Horse -power Revs, per min Mean pressure on crank-pin = S... Half wc. of fly-wheel and shaft = ^ Resultant press, on bearing Diam. of shaft journal Length of shaft journal: Marks, I = .0000325/S 1 2V(/=.10) Whitham, I = .0000515/i?,B(/= 10). PV Thurston, Z Rankine, I Unwin, " 60,000d" ' " ' _P(F+20) 44,800d "■* I = (.004B + l)d TT . , 0.4 H.P. Unwin, 1= Average 4. 92 50 1,250 45 58,905 52,940 23.17 18.35 15.55 22.47 25.39 10.42 19.22 If we divide the mean resultant pressure on the bearing by the projected area, that is, by the product of the diameter and length of the journal, using the greatest and smallest length out of the seven lengths for each journal given above, we obtain the pressure per square inch upon the bearing, as follows: Engine No . 1 3 4 5 6 176 336 151 353 97 123 83 145 124 202 106 191 155 175 Pressure per sq. in., shortest journal Longest journal .... Average journal Journal of length = diam Many of the formulae give for the long-stroke engines a length of journal less than the diameter, but such short journals are rarely used in practice. The last line in the above table has been calculated on the supposition that DIMENSIONS OF PARTS OF ENGINES. 813 the journals of the long-stroke engines are made of a length equal to the diameter. In the dimensions of Corliss engines given hy J. B. Stan wood (Eng., June 12, 1891), the lengths of the journals for engines of diam. of cyl. 10 to 20 in. are the same as the diam. of the cylinder, and a little more than twice the diam. of the journal. For engines above 20 in. diam. of cyl. the ratio of length to diam. is decreased so that an engine of 30 in. diam. has a journal 26 in. long, its diameter being 14£f in. These lengths of journal are greater than those given by anj^ of the formulae above quoted. There thus appears to be a hopeless confusion in the various formulae for length of shaft journals, but this is no more than is to be expected from the variation in the coefficient of friction, and in the heat-conducting power of journals in actual use, the coefficient varying from .10 (or even .16 as given by Marks) down to .01, according to the condition of the bearing surfaces PV and the efficiency of lubrication. Thurston's formula, I =rrrrr,-, reduces to oU,UUUa the form I - .000004363PP, in which P = mean total load on journal, and R = revolutions per minute. This is of the same form as Marks' and Whitham's formulae, in which, if /the coefficient of friction be taken at .10, the coefficients of PR are, respectively, .0000065 and .00000515. Taking the mean of these three formulae, we have I - .0000053PP, if / = .10 or I = .000053/PP for any other value of/. The author believes this to be as safe a formula as any for length of journals, with the limitation that if it brings a result of length of journal less than the diameter, then the length should be made equal to the diameter. Whenever with / = .10 it gives a length which is inconvenient or impossible of construction on account of limited space, then provision should be made to reduce the value of the coefficient of friction below .10 by means of forced lubrication, end play, etc.. and to carry away the heat, as by water-cooled journal-boxes. The value of P should be taken as the resultant of the mean pressure on the crank, and the load brought on the bearing by the weight of the s haft, fl y-wheel, etc., as calculated by the formula already given, viz., R x = YQ' 2 + S 2 for horizontal engines, and P x = Q 4- S for vertical engines. For our six engines the formula I — .0000053PP gives, with the limitation for the long-stroke engines that the length shall not be less than the diam- eter, the following: Engine No 1 Length of journal 4.39 Pressure per square inoh on journal. . 196 Crank - shafts with Centre-crank and Double-crank Arms,— In centre-crank engines, one of the crank-arms, and its adjoining journal, called the after journal, usually transmit the power of the engine to the work to be done, and the journal resists both twisting and bending moments, while the other journal is subjected to bending moment only. For the after crank-journal the diameter should be calculated the same as for an overhung crank, using: the formula for combined bending and twist- ing moment, Tj = B + VB 2 -f T 2 , in which T x is the equivalent twisting moment, B the bending moment, and T the twisting moment. This value 3/5 17 of T, is to be used in the formula diameter —a/ — The bending mo- y 8 ment is taken as the maximum load on piston multiplied by one fourth of the length of the crank-shaft between middle points of the two journal bearings, if the centre crank is midway between the bearings, or by one half the distance measured parallel to the shaft from the middle of the crank-pin to the middle of the after bearing. This supposes the crank- shaft to be a beam loaded at its middle and supported at the ends, but Whitham would make the bending moment only one half of this, consider- ing the shaft to be a beam secured or fixed at the ends, with a point of con- traflexure one fourth of the length from the end. The first supposition is the safer, but since the bending moment will in any case be much less than the twisting moment, the resulting diameter will be but little greater than if Whitham's supposition is used. For the forward journal, which is sub- 3/jo 2B jected to bending moment only, diameter of shaft = a/ — ! , in which B 3 4 5 6 16.48 12.99 30.80 21.52 128 155 102 171 814 THE STEAM-ENGINE. is the maximum bending moment and S the safe shearing strength of the metal per square inch. For our six engines, assuming them to be centre-crank engines, and con- sidering the crank-shaft to be a beam supported at the ends and loaded in the middle, and assuming lengths between centres of shaft bearings as given below, we have: Engine No Length of shaft, assumed, inches, L Max. press, on crank-pin, P Max. bending moment. B = 14PL, inch-lbs Twisting moment, T Equiv. twisting moment, B + VB* + T* Diameter of after journal. d = l/^ V 8000 Diam. of forward journal, d^// 1 ^ V 8000 20 7,854 39,270 47,124 101,000 3. 49,637 94, "" 156,000 4.60 848,232 1,060,290 2,208,000 11.15 60 70,61 1.060,290 2,120,580 3,430,000 13.00 76 196,350 729,750 712,400 ,740,000 18.25 196,350 4,712.400 9,424,800 15,240,000 21.20 The lengths of the journals would be calculated in the same manner as in the case of overhung cranks, by the formula I = .000053/Pi?, in which P is the resultant of the mean pressure due to pressure of steam on the piston, and the load of the fly-wheel, shaft, etc., on each of the two bearings. Unless the pressures are equally divided between the two bearings, the calculated lengths of the two will be different; but it is usually customary to make them both of the same length, and in no case to make the length less than the diameter. The diameters also are usually made alike for the two journals, using the largest diameter found by calculation. The crank-pin for a centre crank should be of the same length as for an overhung crank, since the length is determined from considerations of heating, and not of strength. The diameter also will usually be the same, since it is made great enough to make the pressure per square inch on the projected area (product of length by diameter) small enough to allow of free lubrication, and the diameter so calculated will be greater than is re- quired for strength. C rant- shaft with Two Cranks coupled at 90°. — If the whole power of the engine is transmitted through the after journal of the after crank-shaft, the greatest twisting moment is equal to 1.414 times the maximum twisting moment due to the pressure on one of the crank-pins. If T — the maximum twisting moment produced by the steam-pressure on one of the pistons, then Tj the maximum twisting moment on the after part of the crank-shaft, and on the line-shaft, produced when each crank makes an angle of 45° with the centre line of the engine, is 1.41421 Substituting this value in the formula for diameter to resist simple torsion, viz., d = , we have d = f 5.1 Xl.4142 7 or d = 1 . 932 n- , in which T is the maximum twisting moment produced by one of the pistons, d = diam- eter in inches, and S = safe working shearing strength of the material. For the forward journal of the after crank, and the after journal of the forward crank, the torsional moment is that due to the pressure of steam on the forward piston only, and for the forward journal of the forward crank, if none of the power of the engine is transmitted through it, the torsional moment is zero, and its diameter is to be calculated for bending moment only. For Combined Torsion and Flexure.— Let B x = bending mo- ment on either j nirual of the forward crank due to maximum pressure on DIMENSIONS OF I'ARTS OF ENGINES. 815 forward piston, i? 2 = bending moment on either journal of the after crank due to maximum pressure on after piston, 1\ = maximum twisting momeno on after journal of forward crank, and T 2 — maximum twisting moment on after journal of after crank due to pressure on the after piston. Then equivalent twisting moment on after journal of forward crank = B } + VBJ -f 2\ 2 . _____ On forward journal of after crank = _ a + Vb 3 * + TJ. On after journal of after crank = £ 2 + VBf + {T x + T,) 2 . These values of equivalent twisting moment are to be used in the formula for diameter of journals d = A/ - — . For the forward journal of the 3 /l0 2B forward crank-shaft d = 4/ — : — - 1 . V s It is customary to make the two journals of the forward crank of one diameter, viz., that calculated for the after journal. For a Three-cylinder Engine with cranks at 120°, the greatest twisting moment on the after part of the shaft, if the maximum pressures on the three pistons are equal, is equal to twice the maximum pressure on any one piston, and it takes place when two of the cranks make angles of 30° with the centre line, the third crank being at right angles to it. (For de- monstration, see WhithanVs " Steam-engine Design, 11 p. 252.) For combined torsion and flexure the same method as above given for two crank engines is adopted for the first two cranks; and for the third, or after crank, if all the power of the three cylinders is transmitted through it, we have the equivalent twisting momenton the forward journal = B 3 -\- | / 5 3 2 -|-(T' 1 +T 2 ) 2 , and on the after journal = B 3 + \ B^ + (T x -f 2 T 2 + r 3 ) 2 , B 3 and T s being respectively the bending and twisting moments due to the pressure on the third piston. Crank - shafts for Triple-expansion marine Engines, according to an article in The Engineer, April 25, 1890, should be made larger than the formulae would call for, in order to provide for the stresses due to the racing of the propeller in a sea-way, which can scarcely be cal- culated. A kind of unwritten law has sprung up for fixing the size of a crank-shaft, according to which the diameter of the shaft is made about 0.45Z) , where D is the diameter of the high-pressure cylinder. This is for solid shafts. When the speeds are high, as in war-ships, and the stroke short, the formula becomes 4D, even for hollow shafts. The Valve-stem or Valve-rod.— The valve-rod should be designed to move the valve under the most unfavorable conditions, which are when the stem acts by thrusting, as a long column, when the valve is unbalanced (a balanced valve may become unbalanced by the joint leaking) and when it is imperfectly lubricated. The load on the valve is the product of the ar Q a into the greatest unbalanced pressure upon it per square inch, and the co- efficient of friction may be as high as 20%. The product of this coefficient and the load is the force necessary to move the valve, w T hich equals the maximum thrust on the valve-rod. From this force the diameter of the valve-rod may be calculated by Hodgkinson's formula for columns. An •V: Ibp empirical formula given by Sea ton is: Diam. of rod = d I = length and 6 = breadth of valve, in inches; p = maximum absolute pressure on the valve in lbs. per sq in., and Fa coefficient whose values are, for iron: long rod 10,000, short 12,000; for steel: long rod 12,000, short 14,500. Whitham gives the short empirical rule: Diam. of valve-rod — 1/30 diam. of cyl. = % diam of piston-rod. Size of Slot-link. (Seaton.)— Let D be the diam. of the valve-rod »Vi5< then Diameter of block-pin when overhung = D. " " " secured at both ends = 0.75 x D. " eccentric-rod pins = 0.7 x Z). suspension-rod pins = 0.55 x D. " " '• pin when overhung = 0.75 x D. 816 THE STEAM-ENGINE. Breadth of link = 0.8 to 9 x D. length of block = 1.8 to 1.6 x D. Thickness of bars of link at middle = 0.7 x D. If a single suspension rod of round section, its diameter = 0.7 X D. If two suspension rods of round section, their diameter = 0.55 x D. Size of I>ouble-bar liinlts.— When the distance between centres of eccentric pins = 6 to 8 times throw of eccentrics (throw = eccentricity = half- travel of valve at full gear) D as before : Depth of bars = 1.25 X D -f- % in. Thickness of bars =0.5 x D + Vi ^ Length of sliding-block = 2.5 to 3 X D. Diameter of eccentric-rod pins = 0.8 x D + \i in. " centre of sliding-block = 1.3 x D. When the distance between eccentric -rod pins = 5 to 5J/£ times throw of eccentrics: Depth of bars = 1.25 x D + Ja in. Thickness of bars =0.5 X D -j- J4 in. Length of sliding-block = 2.5 to 3 x D. Diameter of eccentric-rod pins = 0.75 X D. Diameter of eccentric bolts (top end) at bottom of thread = 0.42 X D when of iron, and 0.38 x D when of steel. The Eccentric.— Diam. of eccentric-sheave = 2.4 X throw of eccentric + 1.2 X diam. of shaft. D as before Breadth of the sheave at the shaft = 1.15 X D + 0.65 inch Breadth of the sheave at the strap = D 4- 0.6 inch. Thickness of metal around the shaft = 0.7 x D + 0.5 inch. Thickness of metal at circumference = 0.6 x D 4- 0.4 inch. Breadth of key # = 0.7 X D+ 0.5 inch. Thickness of key = 0.25 x D -f 0.5 inch. Diameter of bolts connecting parts of strap = 0.6x^+0.1 inch. Thickness of Eccentric-strap. When of bronze or malleable cast iron: Thickness of eccentric-strap at the middle = 0.4 x D + 0.6 inch. " '•' " " " sides = 0.3 X D + 0.5 inch. When of wrought iron or cast steel: Thickness of eccentric-strap at the middle = 0.4 X D + 0.5 inch. " " " sides = 0.27 X D + 0.4 inch. The Eccentric-rod.— The diameter of the eccentric-rod in the body and at the eccentric end may be calculated in the same way as that of the connecting-rod, the length being taken from centre of strap to centre of pin. Diameter at the link end = 0.8D -f 0.2 inch. This is for wrought-iron; no reduction in size should be made for steel. Eccentric-rods are often made of rectangular section. Reversing-gear should be so designed as to have more than sufficient strength to withstand the strain .of both the valves and their gear at the same time under the most unfavorable circumstances; it will then have the stiffness requisite for good working. Assuming the work done in reversing the link-motion, W, to be only that due to overcoming the friction of the valves themselves through their whole travel, then, if T be the travel of valves in inches; for a compound engine T/ l x bxp \ T ( V X& 1 Xp' \ 12V 5 / ■+" 12\ 5 /' l l , b 1 andp 1 being length, breadth and maximum steam-pressure on valve of the second cylinder; and for an expansive engine |(Lx|x,) ; or | (IX6XP) . 12 To provide for the friction of link-motion, eccentrics and other gear, and for abnormal conditions of the same, take the work at one and a half times the above amount. FLY-WHEELS. 817 To find the strain at any part of the gear having motion when reversing, divide the work so found by the space moved through by that part in feet; the quotient is the strain in pounds; and the size may be found from the ordinary rules of construction for any of the parts of the gear. (Seaton.) Engine-frames or Red-plates.— No definite rules for the design of engine-frames have been given by authors of works on the steam-engine. The proportions are left to the designer who uses " rule of thumb," or copies from existing engines. F. A. Halsey {Am. Mach., Feb. 14, 1895) has made a comparison of proportions of the frames of horizontal Corliss engines of several builders. The method of comparison is to compute from the measurements the number of square inches in the smallest cross-sec- tion of the frame, that is, immediately behind the pillow-block, also to compute the total maximum pressure upon the piston, and to divide the latter quantity by the former. The result gives the number of pounds pressure upon the piston allowed for each square inch of metal in the frame. He finds that the number of pounds per square inch of smallest section of frame ranges from 217 for a 10 X 30-in. engine up to 575 for a 28 X 48-inch. A 30 X 60-inch engine shows 350 lbs., and a 32-iuch engine which has been running for many years shows 667 lbs. Generally the strains increase with the size of the engine^ and more cross-section of metal is allowed with relatively long strokes than with short ones. From the above Mr. Halsey formulates the general rule that in engines of moderate speed, and having strokes up to one and one-half times the diameter of the cylinder, the load per square inch of smallest section should be for a 10-inch engine 300 pounds, which figure should be increased for larger bores up to 500 pounds for a 30- inch cylinder of same relative stroke. For high speeds or for longer strokes the load per square inch should be reduced. FL.Y-WHEEL.S. The function of a fly-wheel is to store up and to restore the periodical fluc- tuations of energy given to or taken from an engine or machine, and thus to keep approximately constant the velocity of rotation. Rankine calls the AE quantity — =- the coefficient of fluctuation of speed or of unsteadiness, in which E is the mean actual energy, and AS the excess of energy received or of work performed, above the mean, during a given interval. The ratio of the periodical excess or deficiency of energy AS to the whole energy exerted in one period or revolution General Morin found to be from 1/6 to J4 for single-cylinder engines using expansion; the shorter the cut-off the higher the value. For a pair of engines with cranks coupled at 90° the value of the ratio is about J4, and for three engines with cranks at 120°, 1/12 of its value for single cylinder engines. For tools working at intervals, such as punch- ing, slotting and plate-cutting machines, coining-presses, etc., AS is nearly equal to the whole work performed at each operation. AE A fly-wheel reduces the coefficient jr— to a certain fixed amount, being 4Eq about 1/32 for ordinary machinery, and 1/50 or 1/60 for machinery for fine purposes. If m be the reciprocal of the intended value of the coefficient of fluctua- tion of speed, AE the fluctuation or energy, /the moment of inertia of the fly-wheel alone, and a its mean angular velocity, I = — — *. As the rim of a fly-wheel is usually heavy in comparison with the arms, /may be taken to equal Wr*, in which W — weight of rim in pounds, and r the radius of the wheel; then W = — —-r- = — ''— — , if v be the velocity of the rim in feet per second. The usual mean radius of the fly-wheel in steam-engines is from three to five times the length of the crank. The ordinary values of the prod- uct m g, the unit of time being the second, lie between 1000 and 2000 feet. (Abridged from Rankine, S E., p. 62.) Thurston gives for engines with automatic valve-gear W = 250,000 'ig , in which A = area of piston in square inches, S = stroke in feet, p = mean steam pressure in lbs. per sq. in., R = revolutions per minute, D — out- side diameter of wheel in feet. Thurston also gives for ordinary forms of 818 THE STEAM-EtfGitfE. non- condensing engine with a ratio of expansion between 3. and 5, W =. j^r%, in which a ranges from 10,000,000 to 15,000,000, averaging 12,000,000. For gas-engines, in which the charge is fired with every revolution, the Amer- ican Machinist gives this latter formula, with a doubled, or 24,000,000. Presumably, if the charge is fired every other revolution, a should be again doubled. Rankine C" Useful Rules and Tables," p. 247) gives W = 475,000 ^SL^ , in which Fis the variation of speed per cent, of the mean speed. Thurston's first rule above given corresponds with this if we take Fat 1.9 per cent. Hartnell (Proc. Inst., M. E. 1882. 427) says : The value of V, or the variation permissible in portable engines, should not exceed 3 per cent, with an ordinary load, and 4 per cent when heavily loaded. In fixed engines, for ordinary purposes, V = 2^ to 3 per cent. For good governing or special purposes, such as cotton -spinning, the variation should not exceed \% to 2 per cent. F. M. Rites (Trans. A. S. M. E., xiv. 100) develops a new formula for weight C X I H P C of rim, viz., W — pi"?™" - " - "' an ^ wei g Qt of rim per horse-power = pj™, in which C varies from 10,000,000,000 to 20,000,000,000; also using the latter value Mv 2 W 3 14 2 Z) 2 i? 2 of C, he obtains for the energy of the fly-wheel — — = ^j— nnnn — = £ t>4 .4 ooOO CxH.P.(3 .14) 2 P 2 i Z 2 _ 850,000 H.P. B 3 D 2 X 64.4 X 3600 ~ R The limit of variation of speed with such a weight of wheel from excess of power per fraction of revolution is less than .0023. The value of the constant C given by Mr. Rites was derived from practice of the Westinghouse single-acting engines used for electric-lighting. For double-acting engines in ordinary service a value of C = 5,000,000,000 would probably be ample. From these formulae it appears that the weight of the fly-wheel for a given horse-power should vary inversely with the cube of the revolutions and the square of the diameter. J. B. Stanwood {Eng'g, June 12, 1891) says: Whenever 480 feet is the lowest piston-speed probable for an engine of a certain size, the fly-wheel weight for that speed approximates closely to the formula r/ 2 <} W = weight in pounds, d = diameter of cylinder in inches, s = stroke in inches, D = diameter of wheel in feet, R = revolutions per minute, corre- sponding to 480 feet piston -speed. In a Ready Reference Book published by Mr. Stanwood, Cincinnati, 1892, he gives the same formula, with coefficients as follows: For slide-valve en- gines, ordinary duty, 350,000; same, electric-lighting, 700,000; for automatic high-speed engines, 1,000,000; for Corliss engines, ordinary duty 700,000, electric-fighting 1,000,000. Thurston's formula above given, W = ^pjr 2 , with a = 12,000,000, when re- d 2 s duced to terms of d and s in inches, becomes W = 785,400 ^,, ' .. . 33,000 in which P = mean effective pressure. Taking this at 40 lbs., we obtain W = 5,000,000,000^^. If mean effective pressure = 30 lbs., then W = 6,666,000,000^^. Emil Theiss {Am. Much., Sept. 7 and 14, 1893) gives the following values or d, the coefficient of steadiness, which is the reciprocal of what Rankine calls the coefficient of fluctuation : FLY-WHEELS. 819 For engines operating — Hammering and crushing machinery d = 5 Pumping and shearing machinery d = 20 to 30 Weaving and paper-making machinery d = 40 Milling machinery d = 50 Spinning machinery d = 50 to 100 Ordinary driving-engines (mounted on bed-plate), belt transmission d = 35 Gear-wheel transmission d = 50 Mr. Theiss's formula for weight of fly-wheel in pounds is W= i X 2 ' — —'■> vhere d is the coefficient of steadiness, V the mean velocity of the fly- wheel rim in feet per second, n the number of revolutions per minute, i = a coefficient obtained by graphical solution, the values of which for dif- ferent conditions are given in the following table. In the lines under "cut- off," p means " compression to initial pressure," and " no compression ": Values of i. Single-cylinder Non-condensing Engines. 5 73 3 3 Cut-off, 1/6. Cut-off, y A . Cut-off, %. Cut-off, X. * So. Comp. P Comp. P Comp. P O Comp. P 200 400 600 272.690 240,810 194,670 158,200 218.580 187,430 145,400 108,690 242.010 208,200 168.590 162,070 209,170 179,460 136,460 135,260 220,760 188,510 165,210 201,920 170.040 146,610 193,340 174,630 182,840 167,860 800 Single-cylinder Condensing Engines. a £.5 Cut-off, ^. Cut-off, 1/6. Cut-off, y A . Cut-off, %. Cut-off, 14 Comp- P O Comp P O Oomp. P O 167,140 133,080 Comp. P O Comp P O 200 400 265.560 194,550 148.780 176,560 117,870 234, 16C 174,38( 173.660 118,350 204,210 164,720 189,600 174,630 161.830 151,680 172,69 D 156,990 600 140,090 TWO- CYLINDER ENGINES, CRANKS AT 90°. Cut-off, 1/6. Cut-off, y A . Cut-off, %. Cut-off, fc. 03 0> . Comp. P Comp. V O Comp. P Comp. P O 200 400 600 800 71,080 70,160 70,040 70,040 ! Mean (60,140 J 59,420 57,000 57,480 60,140 \ Mean ( 54,340 J 49,272 49.150 49,220 \ Mean f 50,000 37,920 35,500 [ Mean j 36,950 Three-cylinder Engines, Cranks at 120°. Cut-off, 1/6. Cut-off, J4. Cut-off, y B . Cut-off, jjg; s|| Comp. P O Comp. P O Comp. V O Comp. P O 200 800 33.810 30,190 32,240 31,570 33,810 35,140 35,500 33,810 34,540 36,470 33.450 32,850 35,260 33,810 32,370 32,370 As a mean value of'/ for these engines we may use 33,810. 820 THE STEAM-ENGINE. Centrifugal Force in Fly-wheels. — Let W = weight of rim in pounds; R — mean radius of rim in feet; r — revolutions per minute, g — 32.16: v — velocity of rim in feet per second = 2wRr -=- 60. Centrifugal force of whole rim = F= -^r- = ," - = .000341 WiJr*. b gR 3600g The resultant, acting at right angles to a diameter of half of this force, tends to disrupt one half of the wheel from the other half, and is resisted by the section of the rim at each end of the diameter. The resultant of half the 2 radial forces taken at right angles to the diameter is 1 h- y$n = - of the sum of these forces; hence the total force F is to be divided by 2 x 2 X 1.5108 = 6.2832 to obtain the tensile strain on the cross-section of the rim,. or, total strain on the cross-section = S = .00005427 WRr*. The weight W, of a rim of cast iron 1 inch square in section is 2nR X 3.125 = 19.635.R pounds, whence strain per square inch of sectional area of rim = Si = .0010656.K 2 j 2 = .0002664D 2 r 2 _ .0000270 V, in which D = diameter of wheel in feet, and V is velocity of rim in feet per minute. Si = .0972v 2 , if v is taken in feet per second. For wrought iron S x = .0011366i?2,-2 _ .0002842Z> 2 v 2 = .0000288F 2 . For steel S x = .001l593i? 2 r 2 = .0002901 Z) 2 ? 2 = .0000294 F 2 . For wood S x = .0000888E 2 r 2 = .0000222.D 2 ?- 2 = .00000225F 2 . The specific gravity of the wood being taken at 0.6 = 37.5 lbs. per cu. ft., or 1/12 the weight of cast iron. Example.— Required the strain per square inch in the rim of a cast-iron wheel 30 ft. diameter, 60 revolutions per minute. Answer. 15 2 X 60 2 X .0010656 = 863.1 lbs. Required the strain per square inch in a cast-iron wheel-rim running a mile a minute. Answer. .000027 X 5280 2 = 752.7 lbs. In cast-iron fly-wheel rims, on account of their thickness, there is difficulty in securing soundness, and a tensile strength of 10,000 lbs. per sq. in. is as much as can be assumed with safety. Using a factor of safety of 10 gives a maximum allowable strain in the rim of 1000 lbs. per sq. in., which corre- sponds to a rim velocity of 6085 ft. per minute. For any given material, as cast iron, the strength to resist centrifugal force depends only on the velocity of the rim, and not upon its bulk or weight. Chas. E. Emery (Cass. Mag., 1892) says: By calculation half the strength of the arms is available to strengthen the rim, or a trifle more if the fly- wheel centres are relatively large. The arms, however, are subject to trans- verse strains, from belts and from changes of speed, and there is, moreover, no certainty that the arms and rim will be adjusted so as to pull exactly together in resisting disruption, so the plan of considering the rim by itself and making it strong enough to resist disruption by centrifugal force within safe limits, as is assumed in the calculations above, is the safer way. It does not appear that fly-wheels of customary construction should be unsafe at the comparatively low speeds now in common use if proper materials are used in construction. The cause of rupture of fly-wheels that have failed is usually either the " running away " of the engine, such as may be caused by the breaking or slackness of a governor-belt, or incorrect design or defective materials of the fly-wheel. Chas. T. Porter (Trans. A. S. M. E., xiv. 808) states that no case of the bursting of a fly-wheel with a solid rim in a high-speed engine is known. He attributes the bursting of wheels built in segments to insufficient strength of the flanges and bolts by which the segments are held together. (See also Thurston, " Manual of the Steam-engine.' 1 Part II, page 413, etc.) Arms of Fly-wheels and Pulleys. — Professor Torrey (Am. Mack , July 30, 1891) gives the following formula for arms of elliptical cross- section of cast-iron wheels : W = load in pounds acting on one arm; .9 = strain on belt in pounds per inch of width, taken at 56 for single and 112 for double belts; v = width of belt in inches; n — number of arms; L = length of arm in feet; b = breadth of arm at hub; d = depth of arm at hub, both in inches : W = n ' WL b = T^Tr. ■ The breadth of the arm is its least dimension = minor axis of dOa 2 the ellipse, and the depth the major axis. This formula is based on a factor of safety of 10- FLY-WHEELS. 821 In using the formula, first assume some depth for the arm, and calculate fthe required breadth to go with it. If it gives too round an arm, assume the breadth a little greater, and repeat the calculation. A second trial will ialmost always give a good section. The size of the arms at the hub having been calculated, they may be somewhat reduced at the rim end. The actual amouut cannot be calculated, as there are too many unknown quantities. However, the depth and breadth can be reduced about one third at the rim without danger, and this ill give a well-shaped arm. Pulleys are often cast in halves, and bolted together. When this is done the greatest care should be taken to provide sufficient metal in the bolts. This is apt to be the very weakest point in such pulleys. The combined area of the bolts at each joint should be about 28/100 the cross-section of the pul ley at that point. (Torrey.) Unwin gives d = O.GSST'i/ ~^r~ for single belts ; 3/BD /BD IT for double belts; D being the diameter of the pulley, and B the breadth of the rim, both in inches. These formulae are based on an elliptical section of arm in which b = OAd or d = 2.5b on a width of belt = 4/5 the width of the pulley rim, a maximum driving force transmitted by the belt of 56 lbs. per inch of width for a single belt and 112 lbs. for a double belt, and a safe working stress of cast iron of 2250 lbs. per square inch. If in Torrey 's formula we make b = OAd, it reduces to <-tf WL s/WL 187. 5 ; d= V 12 • Example.— Given a pulley 10 feet diameter; 8 arms, each 4 feet long; face, 6 inches wide; belt, 30 inches: required the breath and depth of the arm at the hub. According to Unwin, 3 /BD s /36 X 120 d = 0.6337 4/ — = 0.633 j/ g = 5 - 16 for sin S le belt » b = 2 - 06 ! 3/36 «j/ ~ X 120 -g — ■ = 6.50 for double belt, 6 = 2.60. WL According to Torrey, if we take the formula b ■= -— and assume d = 5 and 6.5 inches, respectively, for single and double belts, we obtain b = 1.08 and 1.33, respectively, or practically oidy one half of the breadth according to Unwin. and, since transverse strength is proportional to breadth, an arm only one half as strong. Torrey 's formula is said to be based on a factor of safety of 10, but this factor can be only apparent and not real, since the assumption that the strain on each area is equal to the strain on the belt divided by the number of arms, is, to say the least, inaccurate. It would be more nearly correct to say that the strain of the belt is divided among half the number of arms. Unwin makes the same assumption in developing his formula, but says it is only in a rough sense true, and that a large factor of safety must be allowed. He therefore takes the low figure of 2250 lbs. per square inch for the safe working strength of cast iron. Unwin says that his equations agree well with practice. Diameters off Fly-wheels for Various Speeds.— If 6000 feet per minute be the maximum velocity of rim allowable, then 6000 = nRD, in which B = revolutions per minute, and D — diameter of wheel in feet, ^ 6000 1910 whence D = — — = -— -. nR K 822 THE STEAM-ENGINE. Maximum Diameter of Fly-wheel Allowable for Different Numbers of Revolutions. Assuming Maxi mum Speed of Assuming Maximum Speed Revolutions 5000 feet per minute. . of 6000 feet per minute. per minute. Circum. ft. Diam. ft. Circum. ft. Diam. ft. 40 125 39.8 150. 47.7 50 100 31.8 120. 38.2 60 83.3 26.5 100. 31.8 70 71.4 22.7 85.72 27.3 80 62.5 19.9 75.00 23.9 90 55.5 17.7 66.66 21.2 100 50. 15.9 60.00 19.1 120 41.67 13.3 50.00 15.9 140 35.71 11.4 42.86 13.6 160 31.25 9.9 37.5 11.9 180 27.77 8.8 33.33 10.6 200 25.00 8.0 30.00 9.6 220 22.73 7.2 27.27 8.7 240 20.83 6.6 25.00 8.0 260 19.23 6.1 23.08 7.3 280 17.86 5.7 21.43 6.8 300 16.66 5.3 20.00 6.4 350 14.29 4.5 17.14 5.5 400 12.5 4.0 15.00 4.8 450 11.11 3.5 13.33 4.2 500 10.00 3.2 12.00 3.8 Strains in the Rims of Fly-band Wheels Produced by Centrifugal Force. (James B. Stanwood, Trans. A. S. M. E., xiv. 251.) —Mr. Stanwood mentions one case of a fly-band wheel where the periphery velocity on a 17' 9" wheel is over 7500 ft. per minute. In band saw-mills the blade of the saw is operated successfully over wheels 8 and 9 ft. in diameter, at a periphery velocity of 9000 to 10,000 ft. per minute. These wheels are of cast iron throughout, of heavy thickness, with a large number of arms. In shingle-machines and chipping-machines where cast-iron disks from 2 to 5 ft. in diameter are employed, with knives inserted radially, the speed is frequently 10,000 to 11,000 ft. per minute at the periphery. If the rim of a fly-wheel alone be considered, the tensile strain in pounds F 2 per square inch of the rim section is T — —— — nearly, in which V = velocity in feet per second; but this strain is modified by the resistance of the arms, which prevent the uniform circumferential expansion of the rim, and induce a bending as well as a tensile strain. Mr. Stanwood discusses the strains in band-wheels due to transverse bending of a section of the rim between a pair of arms. When the arms are few in number, and of large cross-section, the ring will be strained transversely to a greater degree than with a greater number of lighter arms. To illustrate the necessary rim thicknesses for various rim velocities, pulley diameters, number of arms, etc., the following table is given, based upon the formula * = - .475d V p-a 10/ in which t = thickness of rim in inches, d = diameter of pulley in inches, JV= number of arms, V= velocity of rim in feet per second, and F — the greatest strain in pounds per square inch to which any fibre is subjected. The value of F is taken at 6000 lbs. per sq. in. ELY-WHEELS. 823 Thickness of Rims in Solid Wheel s. Diameter of Pulley in inches. Velocity of Rim in feet per second. Velocity of Rim iii feet per minute. No. of Arms. Thickness in inches. 24 24 48 108 108 50 88 88 184 184 3,000 5,280 5,280 11,040 11,040 6 6 6 16 36 2/10 15/32 15/16 If the limit of rim velocity for all wheels be assumed to be 88 ft. per sec- ond, equal to 1 mile per minute, F — 6000 lbs., the formula becomes t-'^L -0 7-*-. .67iV 2 N 2 When wheels are made in halves or in sections, the bending strain may be such as to make t greater than that given above. Thus, when the joint comes half way between the arms, the bending action is similar to a beam supported simply at the ends, uniformly loaded, and t is 50$ greater. Then the formula becomes .713d Vp 10/ or for a fixed maximum rim velocity of 88 ft. per second and F = 6000 lbs., t = ' ' . In segmental wheels it is preferable to have the joints opposite the arms. Wheels in halves, if very thin rims are to be employed, should have double arms along the line of separation, Attention should be given to the proportions of large receiving and tight- ening pulleys. The thickness of rim for a 48-in. wheel (shown in table) with a rim velocity of 88 ft. per second, is 15/16 in. Many wrecks have been caused by the failure of receiving or tightening pulleys whose rims have been too thin. Fly-wheels calculated for a given coefficient of steadiness are fre- quently lighter than the minimum safe weight. This is true especially of large wheels. A rough guide to the minimum weight of wheels can be de- duced from our formulae. The arms, hub, lugs, etc., usually form from one quarter to one third the entire weight of the wheel. If b represents the fact of a wheel in inches, the weight of the rim (considered as a simple annular ring) will be w = .82dtb lbs. If the limit of speed is 88 ft. per second, then for solid wheels t = 7d -e- N 2 . For sectional wheels (joint between arms) t = 1.05d h-F. Weight of rim for solid wheels, w =. .57d 2 b -*- N 2 in pounds. Weight of rim in sectional wheels with joints between arms, w = .8Qd 2 b -*- JV 2 in pounds. Total weight of wheel: for solid wheel, W = .76d 2 6 -s- N 2 to .86<2 2 6 -T- N 2 , in pounds. For segmental wheels with joint between arms, W = 1.05d 2 b -h N 2 to 1.3d 2 b -f- N 2 , in pounds. (This subject is further discussed by Mr. Stanwood, in vol. xv., and by Prof. Gaetano Lanza, in vol. xvi.. Trans. A. S. M. E.) A Wooden Rim Flywheel,, built in 1891 for a pair of Corliss en- gines at the Amoskeag Mfg. Co.'s mill, Manchester, N. H., is described by C H. Manning in Trans. A. S. M. E., xiii. 618. It is 30 ft. diam. and 108 in. face. The rim is 12 inches thick, and is built up of 44 courses of ash plank, 2, 3, and 4 inches thick, reduced about % inch in dressing, set edgewise, so as to break joints, and glued and bolted together. There are two hubs and two sets of arms, 12 in each, all of cast iron. The weights are as follows: Weight (calculated) of ash rim 31 ,855 lbs. " of 24 arms (foundry 45,0'20) 40,349 " " 2hubs( " 35,030) 31,394^ " Counter- weights in 6 arms 664 " Total, excluding bolts and screws 104,262± " The wheel was tested at 76 revs, per min., being a surface speed of nearly 7200 feet per minute. 824 THE STEAM -ENGlM. ]\Ir. Manning discusses the relative safety of cast iron and of wooden wheels as follows: As for safety, the speeds being the same in both cases, i he hoop tension in the rim per unit of cross-section would be directly as the weight per cubic unit; and its capacity to stand the strain directly as the tensile strength per square unit; therefore the tensile strengths divided by the weights will give relative values of different materials. Cast iron weighing 450 lbs. per cubic foot and with a tensile strength of 1,440,000 lbs. per square foot would give a value of 1,440,000 -=- 450= 3200, whilst ash, of which the rim was made, weghing 34 lbs. per cubic foot, and with 1,152,000 lbs. tensile strength per square foot, gives a result 1,152,000 -=- 34 = 33,882, and 33,882-^-3200 = 10.58, or the wood-rimmed pulley is ten times safer than the cast-iron when the castings are good. This would allow the wood- rimmed pulley to increase its speed to 4/10.58 =3.25 times that of a sound cast-iron one with equal safety. "Wooden Fly-wneel of tlie Willimantic Linen Co. (Illus- trated in Power, March, 1893.) — Rim 28 ft. diam., 110 in. face. The rim is carried upon three sets of arms, one under the centre of each belt, with 12 arms in each set. The material of the rim is ordinary whitewood, % in. in thickness, cut into segments not exceeding 4 feet in length, and either 5 or 8 inches in width. These were assembled by building a complete circle 13 inches in width, first with the 8 inch inside and the 5-inch outside, and then beside it another cir- cle with the widths reversed, so as to break joints. Each piece as it was added was brushed over with glue and nailed with three-inch wire nails to the pieces already in position. The nails pass through three and into the fourth thickness. At the end of each arm four 14-inch bolts secure the rim, the ends being covered by wooden plugs glued and driven into the face of the wheel. Wire-wound Fly-wlieels for Extreme Speeds. (Eng'gNews, August 2, 1890.)— The power required to produce the Mannesmann tubes is very large, varying from 2000 to 10,000 H.P., according to the dimensions of the tube. Since this power is only needed for a short time (it takes only 30 to 45 seconds to convert a bar 10 to 12 ft. long and 4 in. in diameter into a tube), and then some time elapses before the next bar is ready, an engine of 1200 HP. provided with a large fly-wheel for storing the energy will supply power enough for one set of rolls. These fly-wheels are so large and run at such great speeds that the ordinary method of constructing them cannot be followed. A wheel at the Mannesmann Works, made in Komotau, Hungary, in the usual manner, broke at a tangential velocity of 125 ft. per second. The fly-wheels designed to hold at more than double this speed consist of a cast-iron hub to which two steel disks, 20 ft. in diameter, are bolted; around the circumference of the wheel thus formed 70 tons of No. 5 wire are wound under a tension of 50 lbs. In the Mannesmann Works at Landore, Wales, such a wheel makes 240 revolutions a minute, corresponding to a tangential velocity of 15,080 ft. or 2.85 miles per minute. THE SLIDE-VALVE. Definitions.*— Travel = total distance moved by the valve. Throw of the Eccentric = eccentricity of the eccentric = distance from the centre of the shaft to the centre of the eccentric disk = J^ the travel of the valve. (Some writers use the term " throw " to mean the whole travel of the valve.) Lap of the valve, also called outside lap or steam-lap = distance the outer or steam edge of the valve extends beyond or laps over the steam edge of the port when the valve is in its central position. Inside lap, or exhaust-lap ■— distance the inner or exhaust edge of the valve extends beyond or laps over the exhaust edge of the port when the valve is in its central position. The inside lap is sometimes made zero, or even negative, in which latter case the distance between the edge of the valve and the edge of the port is sometimes called exhaust clearance, or inside clearance. Lead of the valve = the distance the steam-port is opened when the engine is on its centre and the piston is at the beginning of the stroke. Lead-angle = the angle between the position of the crank when the valve begins to be opened and its position when the piston is at the beginning of the stroke. The valve is said to have lead when the steam-port opens before the piston THE SLIDE-VALYE. 825 begins its stroke. If the piston begins its stroke before the admission of steam begins the valve is said to have negative lead, and its amount is the lap of the edge of the valve over the edge of the port at the instant when "ie piston stroke begins. Lap-angle = the angle through which the eccentric must be rotated to cause the steam edge to travel from its central position the distance of the lap. Angular advance of the eccentric = lap-angle -\- lead angle. Linear advance = lap -f- lea : — — ; r = X6- : — —- . sin \4>a ' '* sin J^|8 Ratio of Lap and of Port-opening to Valve-travel .—The table on page 831, giving the ratio of lap to travel of valve and ratio of travel to port opening, is abridged from one given by Buel in Weisbach-Dubois, vol. ii. It is calculated from the above formulae. Intermediate values may be found by the formulae, or with sufficient accuracy by interpolation from the figures in the table. By the table on page 830 the crank-angle may be found, that is, the angle between its position when the engine is on the centre and its position at cut-off, release, or compression, when these are known in fractions of the stroke. To illustrate the use of the tables the following example is given by Buel: width of port = 2.2 in. ; width of port opening = width of port + 0.3 in.; over overtravel = 2.5 in.; length of connecting-rod = 2% times stroke; cut-off, .75 of stroke; release, .95 of stroke; lead-angle, 10°. From the first table we find crank-angle = 114.6; 830 THE STEAM-ENGIJSTE. )etweeti ; = 3.72, add lead-angle, making 121.6.° From the second table, for angle beti admission and cut-off, 125°, we have ratio of travel to port-opening = or for 124.6° — 3.74, which, multiplied by port-opening 2.5, gives 9.45 in travel. The ratio of lap to travel, by the table, is .2324, or 9.45 X .2324 = 2.2 in. lap. For exhaust-lap we have, for release at .95, crank-angle = 151.3; add lead-angle 10° = 161.3°. From the second table, by interpolation, ratio of lap to travel = .0811, and .0811 X 9.45 — 0.77 in., the exhaust-lap. Lap-angle = y> (180° — lead- angle — crank- angle at cut-off); = y z (180° - 10 - 114.6; = 27.7°. Angular advance = lap-angle X lead-angle = 27.7 -f- 10 = 37.7°. Exhaust lap-angle = crank-angle at release -j- lap-angle -f lead-angle — 180°; =-. 151.3 -f 27.7 -f 10 - 180° = 9°. Crank-angle at com- 1 pression measured >- = 180° — lap -angle — lead-angle — exhaust lap-angle; on return stroke ) = 180 -27.7-10-9= 133.3° ; corresponding, by table, to a piston position of .81 of the return stroke; or Crank-angle at compression = 180° — (angle at release - angle at cut-off) ■+ lead-angle; = 180 - (151.3 - 114.6)+10 = 133.3°. The positions determined above for cut-off and release are for the forward stroke of the piston. On the return stroke the cut-off will take place at the same angle, 114.6°, corresponding by table to 66.6$ of the return stroke, instead of 75$. By a slight adjustment of the angular advance and the length of the eccentric rod the cut-off can be equalized. The width of the bridge should be at least 2.5 -j- 0.25 — 2.2 = 0.55 in. Crank Angles for Connecting-rods of Different Length, Forward and Return Strokes. t? s § Ratio of Length of Connecting-rod to Length of Stroke. c2S 2 ®/% 3 3^ 4 5 Infi- nite. For. ^02 s o O For. Ret. For. Ret. For. Ret. For. Ret. For. Ret. For. Ret. or Ret. .01 10.3 13.2 10.5 12.8 10.6 12 6 10.7 12.4 10.8 12.3 10.9 12.1 11.5 .02 14.6 18.7 14.9 18.1 15.1 17.8 15.2 17.5 15.3 17.4 15.5 17.1 16.3 .03 17.9 22.9 18.2 22.2 18.5 21.8 18.7 21.5 18.8 21.3 19.0 21.0 19.9 .04 20.7 26.5 21.1 25.7 21.4 25.2 21.6 24.9 21.8 24.6 22.0 24.3 23.1 .05 23.2 29.6 23.6 28.7 24.0 28.2 24.2 27.8 24.4 27.5 24.7 27.2 25.8 .10 33.1 41.9 33.8 40.8 34.3 40.1 34 6 39.6 34.9 39.2 35.2 38.7 36.9 .15 41 51.5 41.9 50.2 42.4 49.3 42.9 48.7 43.2 48.3 43.6 47.7 45.6 .20 48 59.6 48.9 58.2 49.6 57.3 50.1 56.6 50.4 56.2 50.9 55.5 53.1 .25 54.3 66.9 55.4 65.4 56.1 64.4 56.6 63.7 57.0 63 3 57.6 62.6 60.0 .30 60.3 73.5 61.5 72.0 62.2 71.0 62.8 70.3 63.3 69.8 63.9 69.1 66.4 .35 66.1 79 8 67.3 78.3 68.1 77.3 68.8 76.6 69 2 76.1 69.9 75.3 72.5 .40 71.7 85.8 73.0 84.3 73.9 83.3 74.5 82.6 75.0 82.0 75.7 81.3 78.5 .45 77.2 91.5 78.6 90.1 79.6 89.1 80.2 88.4 80.7 87.9 81.4 87.1 84.3 .50 82.8 97.2 84.3 95.7 85.2 94.8 85.9 94.1 86.4 93.6 87.1 92.9 90.0 .55 88.5 102.8 89.9 101.4 90.9 100.4 91.6 99.8 92.1 99.3 92.9 98. C 95.7 .60 94.2 108.3 95.7 107.0 96.7 106.1 97.4 105.5 98.0 105.0 98.7 104.3 101.5 .65 100.2 113.9 101.7 112.7 102.7 111.9 103.4 111.2 103.9 110.8 104.7 110.1 107.5 .70 106.5 119.7 108.0 118.5 109.0 117.8 109.7 117.2 110.2 116.7 110.9 116.1 113.6 .75 113.1 125.7 114.6 124.6 115.6 123.9 116.3 123.4 116.7 123.0 117.4 122.4 120.0 .80 120.4 132 121.8 131.1 122.7 130.4 123.4 129.9 123.8 129.6 124.5 129.1 126.9 .85 128.5 139 129.8 138.1 130.7 137.6 131.3 137.1 131.7 136.8 132.3 136.4 134.4 .90 138.1 146.9 139.2 146.2 139.9 145.7 140.4 145.4 140.8 145.1 141.3 144.8 143.1 .95 150.4 156.8 151.3 156.4 151.8 156.0 152.2 155.8 152.5 155.6 152.8 155.3 154.2 .96 153.5 159.3 154.3 158.9 154.8 158.6 155.1 158.4 155.4 158.2 155.7 158.0 156.9 .97 157.1 162.1 157.8 161.8 158.2 161.5 158.5 161.3 158.7 161.2 159.0 161.0 160.1 .98 161.3 165.41161.9 165.1 162.2 164.9 162.5 164.8 162.6 164.7 162.9 164.5 163.7 .99 166.8 169.7 167.2 169.5 167.4 169.4167.6 169.3 167.7 109.21167.9 169.1 168.5 1.00 ISO 180 1180 180 180 180 Il80 180 180 180 Il80 180 180 THE SLIDE-VALVE. 831 Relative Motions of Cross-head and Crank.— If L = length of connecting-rod, R — length of crank, 6 = angle of crank with centre line of engine, D — displacement of cross-head from the beginning of its stroke, = R(l - cos 9) = L - VL* - R* sin 2 6. Lap and Travel of Valve. a Otl £ W V,<* A >s o eg *3 >o .2« ?§ 1 = 3° 53 > 2 «, ? > £§• 0-30-0 a c O S2Sl B _ ^ a ~ -On P. 03° s °* ® a cs 1 * _ ® esS O £ a -2 a) a 03 h£ 1*2$ a ^ ^5 £^303 . c3 t-3 H 5 ngle be of Cra Admisfc or Rel pressio •goJ "3-2.2 c o ° be «-2S SB © a o t-J 2 1^ m 1 Vs % % ^ ¥s in. in. % % % % % % % % % 12 Va 88 90 93 95 96 97 98 98 99 99 10 A 82 87 89 92 95 96 97 98 98 99 8 Va 72 78 84 88 92 94 95 96 98 98 6 M 50 62 71 79 86 89 91 94 96 97 Wz A 43 56 68 77 85 88 91 94 96 97 5 ** 32 47 61 72 82 86 89 92 95 97 4^ H 14 35 51 66 78 83 87 90 94 96 4 H 17 39 57 72 78 83 88 92 95 3 % % 20 44 23 63 50 71 61 79 71 84 79 90 86 94 91 2^ 2 y 8 27 43 57 70 80 88 33 52 70 81 832 THE STEAM-ENGINE. Periods of Admission, or Points of Cut-off, for given I Travels and Laps of Slide-valves. Constant lead, 5/16. Travel. Lap. Inches. % % % % 1 V/a m Wa VA Wa m m 2 19 39 47 17 55 61 65 68 74 76 34 42 50 55 59 63 67 2% 2\i m 14 30 38 45 49 56 13 27 36 43 2% 12 26 m 3 78 80 81 70 73 74 59 62 65 47 50 55 32 38 44 11 23 30 &/a 10 m 83 76 68 59 48 34 22 3% 84 78 71 62 51 40 29 9 sy 2 85 80 73 64 53 45 34 20 m 86 81 75 66 57 49 38 26 9 m 3% 87 82 76 68 60 52 42 32 19 87 83 78 70 63 55 46 36 25 4 88 84 79 72 66 58 49 40 29 4J4 89 86 81 76 70 63 4^ 90 87 83 79 73 67 61 54 45 92 89 85 81 76 70 65 58 51 5 93 90 87 83 78 73 67 62 56 5^ 94 92 89 86 82 78 73 68 6 95 93 91 88 85 82 78 74 69 Diagram for Port-opening, Cut-off, and Lap.— The diagram on the opposite page was published in Power, Aug., 1893. It shows at a glance the relations existing between the outside lap, steam port-opening, and cut-off in slide valve engines. In order to use the diagram to find the lap, having given the cut-off and maximum port-opening, follow the ordinate representing the latter, taken on the horizontal scale, until it meets the oblique line representing the given cut-off. Then read off this height on the vertical lap scale. Thus, with a port-opening of 1J4 i Q ch and a cut-off of .50, the intersection of the two lines occurs on the horizontal 3. The required lap is therefore 3 in. If the cut off and lap are given, follow the horizontal representing the latter until it meets the oblique line representing the cut-off. Then vertically below this read the corresponding port-opening on the horizontal scale. If the lap and port-opening are given, the resulting cut-off may be ascer- tained by finding the point of intersection of the ordinate representing the port-opening with the horizontal representing the lap. The oblique line passing through the point of intersection will give the cut-off. If it is desired to take lead into account, multiply the lead in inches by the numbers in the following table corresponding to the cut-off, and deduct the result from the lap as obtained from the diagram: Cut-off. Multiplier. Cut-off. Multiplier. .20 4.717 .60 1.358 .25 3.731 .625 1.288 .30 3.048 .65 1.222 .33 2.717 .70 1.103 .375 2.381 .75 1.000 .40 2.171 .80 0.904 .45 1.930 .85 0.815 .50 1.706 .875 0.772 .55 1.515 .90 0.731 THE SLIDE-VALVE. 833 Cut-off .20 .25 .30 .35 .375.40 .45 .50 .55 .60 JJ utt l-T-J (-UL 44-44 tti - i -t 7 / 444441 t - i t V Z 1141414 -1 -t 7--//.' ttttt't -t 1 /Z/ t h t-u £ t-t L-tf 4-044-4 i 1 4-T7- . -444-tttl-f --T-tZ-t -* jJitt rri -7 74-f-L-Z i lfitrh t rJ-fz z 1 tttutt 4 4 tZ-A 7 lAJLlT'L 1 -44W -f -, : tiita-TT. -tit V ? - Uihftir '-■/-■A -7 -7 s aiiLttl-/ ff 7 7 1 ^±tuP-4-t7-i -t z -/ r zivttttlJA § ARutuott/ / 7 >/ l Jwi'lii _af / ~7 / ° MttrLZtZv -^ 7 ±* -O^ Mtniizrz 7 ^ y ^ - ^ * *s mil 1 1 / /// / / iii i / / /// / / / ' / ^^ ^^^ Willi / // / s /' ' y^ ^^ \lll// ////// y ' ' ^ \a^ w\\y /j//// /' s^' ' ^^ Will//// / / \c^^ J/f 1 /Wj '/ ' / y^l^^ wS0^^^^ wM%^^^ 12 3 4 5 Maximum Steam Port opening in Inches. DIAGRAM FOR SLIDE VALVES. 834 THE STEAM-ENGINE. Piston-valve.— The piston-valve is a modified form of the slide valve. The lap, lead, etc., are calculated in the same manner as for the common slide-valve. The diameter of valve and amount of port-opening are calcu- lated on the basis that the most contracted portion of the steam-passage between the valve and the cylinder should have an area such that the velocity of steam through it will not exceed 6000 ft. per minute. The area of the opening around the circumference of the valve should be about double the area of the steam-passage, since that portion of the opening that is opposite from the steam-passage is of little effect. Setting the Valves of an Engine. — The principles discussed above are applicable not only to the designing of valves, but also to adjust- ment of valves that have been improperly set; but the final adjustment of the eccentric and of the length of the rod depend upon the amount of lost motion, temperature, etc., and can be effected ouly after trial. After the valve has been set as accurately as possible when cold, the lead and lap for the forward and return strokes being equalized, indicator diagrams should be taken and the length of the eccentric-rod adjusted, if necessary, to cor-' rect slight irregularities. To Put an Engine on its Centre.— Place the engine in a posi- tion where the piston will have nearly completed its outward stroke, and opposite some point on the cross-head, such as a corner, make a mark upon .the guide. Against the rim of the pulley or crank-disk place a pointer and mark a line with it on the pulley. Then turn the engine over the centre until the cross-head is again in the same position on its inward stroke. This will bring theci»ank as much below the centre as it was above it before. With the pointer in the same position as before make a second mark on the pulley- rim. Divide the distance between the marks in two and mark the middle point. Turn the engine until the pointer is opposite this middle point, and it will then be on its centre. To avoid the error that may arise from the looseness of crank-pin and wrist-pin bearings, the engine should be turned a little above the centre and then be brought up to it, so that the crank pin will press against the same brass that it does when the first two marks are made. Iiink-motion.— Link-motions, of which the Stephenson link is the most commonly used, are designed for two purposes: first, for reversing the motion of the engine, and second, for varying the point of cut-off by varying the travel of the valve. The Stephenson link-motion is a combination of two eccentrics, called the forward and back eccentric, with a link connect- ing the extremeties of the eccentic-rods; so that by varying the position of the link the valve-rod may be put in direct connection with either eccentric, or may be given a movement controlled in part by one and in part bjvthe other eccentric. When the link is moved by the reversing lever into a posi- tion such that the block to which the valve-rod is attached is at either end of the link, the valve receives its maximum travel, and when the link is in mid-gear the travel is the least and cut-off takes place early in the stroke. In the ordinary shifting-link with open rods, that is, not crossed, the lead of the valve increases as the link is moved from full to mid-gear, that is, as the period of steam admission is shortened. The variation of lead is equa- lized for the front and back strokes by curving the link to the radius of the eccentric-rods concavely to the axles. With crossed eccentric-rods the lead decreases as the link is moved from full to mid-gear. In a valve-motion with stationary link the lead is constant. (For illustration see Clark's Steam- engine, vol. ii. p. 22.) The linear advance of each eccentric is equal to that of the valve in full gear, that is, to lap -|- lead of the valve, when the eccentric-rods are attached to the link in such position as to cause the half- travel of the valve to equal the eccentricity of the eccentric. The angle between the two eccentric radii, that is, between lines drawn from the centre of the eccentric disks to the centre of the shaft equals 180° less twice the angular advance. Buel, in Appleton's Cyclopedia of Mechanics, vol. ii. p. 316, discusses the Stephenson link as follows: " The Stephenson link does not give a perfectly correct distribution of steam; the lead varies for different points of cut-off. The period of admission and the beginning of exhaust are not alike for both ends of the cylinder, and the forward motion varies from the backward. " The correctness of the distribution of steam by Stephenson's link-motion depends upon conditions which, as much as the circumstances will permit, ought to be fulfilled, namely: 1. The link should be curved in the arc of a circle whose radius is equal to the length of the eccentric- rod. 2, The THE SLIDE-VALVE. 830 eccentric-rods ought to be long ; the longer they are in proportion to the eccentricity the more symmetrical will the travel of the valve be on both sides of the centre of motion. 3. The link ought to be short. Each of its points describes a curve in a vertical plane, whose ordinatesgrow larger the farther the considered point is from the centre of the link; and as the hori- zontal motion only is transmitted to the valve, vertical oscillation will cause irregularities. 4. The link-hanger ought to be long. The longer it is the nearer will be the arc in which the link swings to a straight line, and thus the less its vertical oscillation. If the link is suspended in its centre, the curves that are described by points equidistant on both sides from the centre are not alike, and hence results the variation between the forward and back- ward gear. If the link is suspended at its lower end, its lower half will have less vertical oscillation and the upper half more. 5. The centre from which the link-hanger swings changes its position as the link is lowered or raised, and also causes irregularities. To reduce them to the smallest amount the arm of the lifting-shaft should be made as long as the eccentric-rod, and the centre of the lifting-shaft should be placed at the height corresponding to the central position of the centre on which the link- hanger swings." All these conditions can never be fulfilled in practice, and the variations in the lead and the period of admission can be somewhat regulated in an artificial way, but for one gear only. This is accomplished by giving differ- ent lead to the two eccentrics, which difference will be smaller the longer the eccentric-rods are and the shorter the link, and by suspending *he link not exactly on its centre line but at a certain distance from it, giving what is called " the offset." For application of the Zeuner diagram to link-motion, see Holmes on the Steam-engine, p. 290. See also Clark's Railway Machinery (1855), Clark's Steam-engine and Zeuner's and Auchincloss's Treatises on Slide-valve Gears. The following rules are given by the American Machinist for laying out a link for an upright slide-valve engine. By the term radius of link is meant the radius of the link-arc ab, Fig. 150, drawn through the centre of the slot; (cf ^y ~~ -Q -^ka^ rrg^ Fig. 150. this radius is generally made equal to the distance from the centre of shaft to centre of the link-block pin P when the latter stands midway of its travel. The distance between the centres of the eccentric-rod pins e x e 2 should not be less than 2J^ times, and, when space will permit, three times the throw of the eccentric. "By the throw we mean twice the eccentricity of the eccentric. The slot link is generally suspended from the end next to the forward eccen- The following table gives the ratio of capacity of cylinder or cylinders to that of the air-pump; in the case of the compound engine, the low-pressure cylinder capacity only is taken. Description of Pump. Description of Engine. Ratio. Single-acting vertical Double-acting horizontal. . Jet-condensing, expansion 1^ to 2 — Surface " " 1)4 to 2.... Jet " " 3 to 5.... Surface " " 3 to 5 — Surface " compound Jet " expansion 1J^ to 2 Surface " " l^to2.... Jet " " 3 to 5 ... Surface " " 3 to 5 ... Surface " compound 6 to 8 8 to 10 10 to 12 12 to 15 15 to 18 10 to 13 13 to 16 16 to 19 19 to 24 24 to 28 The Area tlirougli Valve-seats and past the valves should not be less than will admit the full quantity of water for condensation at a velocity not exceeding 400 ft. per minute. In practice the area is generally in excess of this. Area through foot- valves = D 2 X S-*- 1000 square inches. Area through head-valves = D 2 X Sj- 800 square inches. Diameter of discharge-pipe = D x VS -s- 35 inches. D — diam. of air-pump in inches, S = its speed in ft. per min. James Tribe (Am. Much., Oct. 8, 1891) gives the following rule for air- CONDENSERS, AIR-PUMPS, ETC. 843 pumps used with jet-condensers: Volume of single-acting- air-pump driven by main engine = volume of low-pressure cylinder in cubic feet, multiplied by 3.5 and divided by the number of cubic feet contained in one pound of exhaust-steam of the given density. For a double-acting air-pump the same rule will apply, but the volume of steam for each stroke of the pump will be but one half. Should the pump be driven independently of the engine, then the relative speed must be considered. Volume of jet-con- denser = volume of air-pump X 4. Area of injection valve = vol. of air- pump in cubic inches -^ 520. Circulating-pump. — Let Q be the quantity of cooling water in cubic fe*t, n the number of strokes per minute, and S the length of stroke in feet. Capacity of circulating-pump = Q -=- n cubic feet. 3.55i/— y n : The following table gives the ratio of capacity of steam-cylinder or cylin- ders to that of the circulating-pump : Description of Pump. Description of Engine. Ratio. Single-acting. Expansive 1^ to 2 times. 13 to 16 3 to 5 " 20 to 25 " " Compound. 25 to 30 Double " Expansive 1]4 to 2 times. 25 to 30 3 to 5 " 36 to 46 " " Compound. 46 to 56 The ctear area through the valve-seats and past the valves should be such that the mean velocity of flow does not exceed 450 feet per minute. The flow through the pipes should not exceed 500 ft. per min. in small pipes and 600 in large pipes. For Centrifugal Circulating -pumps, the velocity of flow in the inlet and outlet pipes should not exceed 400 ft. permiu. The diameter of the fan- wheel is from 2^ to 3 times the diam. of the pipe, and the speed at its periphery 450 to 500 ft. per min. If W = quantity of water per minute, in American gallons, d = diameter of pipes in inches, R — revolutions of wheel per min., - , Jiam. of fan-wheel = not less than -!=—. Breadth of blade at 16.44 R --■/: w tip = -r^-r. Diam. of cylinder for driving the fan = about 2.8 V'diain. of pipe, and its stroke = 0.28 X diam. of fan. Feed-pumps for Marine Engines.— With surface-condensing engines the amount of water to be fed by the pump is the amount condensed from the main engine plus what may be needed to supply auxiliary engines and to supply leakage and waste. Since an accident may happen to the surface-condenser, requiring the use of jet-condensation, the pumps of engines fitted with surface-condensers must be sufficiently large to do duty under such circumstances. With jet-condensers and boilers using salt water the dense salt water in the boiler must be blown off at intervals to keep the density so low that deposits of salt will not be formed. Sea-water contains about 1/32 of its weight of solid matter in solution. The boiler of a surface- condensing engine may be worked with safety when the quantity of salt is four times that in sea-water. If Q — net quantity of feed-water required in a given time to make up for what is used as steam, n = number of times the saltness of the water in the boiler is to that of "sea- water, then the gross feed- water = — Z~jQ- * n orc * er to De capable of filling the boiler rapidly each feed-pump is made of a capacity equal to twice the gross feed-water. Two feed-pumps should be supplied, so that one may be kept in reserve to be used while the other is out of repair. If Q be the quantity of net feed- water in cubic feet, I the length of stroke of feed-pump in feet, and n the num- ber of strokes per minute, Diameter of each feed-pump plunger in inches /550 xQ 844 THE STEAM-ENGINE. W If Wbe the net feed- water in pounds, /g yy j] Diameter of each feed-pump plunger in inches = A/ — — — — y n X l An Evaporative Surface Condenser built at the Virginia Agri cultural College is described by James H. Fitts (Trans. A. S. M. E., xiv. 690). It consists of two rectangular end chambers connected by a series of hori- zontal rows of tubes, each row of tubes immersed in a pan of water. Through the spaces between the surface of the water in each pan and the bottom of the pan above air is drawn by means of an exhaust-fan. At the top of one of the end-chambers is an inlet for steam, and a horizontal dia- phragm about midway causes the steam to traverse the upper half of the tubes and back through the lower. An outlet at the bottom leads to the air- pump. The condenser, exclusive of connection to the exhaust fan, occupies a floor space of 5' 4}4" x 1' 9%", and 4' 1%" high. There are 27 rows of tubes, 8 in some and 7 in others; 210 tubes" in all. The tubes are of brass, No. 20 B.W G., %" external diameter and 4' 9J4" in length. The cooling sur- face (internal) is 176.5 sq. ft. There are 27 cooling pans, each 4' 9J4" X 1' 9«%", and 1 7/16" deep. These pans have galvanized iron bottoms which slide into horizontal grooves 34" wide and J4" deep, planed into the tube-sheets. The total evaporating surface is 234.8 sq. ft. Water is fed to every third pan through small cocks, and overflow-pipes feed the rest. A wood casing con- nects one side with a 30" Buffalo Forge Co.'s disk- wheel. This wheel is belted to a 3" x 4" vertical engine The air-pump is 5%" diameter with a 6" stroke, is vertical and single-acting. The action of this condenser is as follows: The passage of air over the water surfaces removes the vapor as it rises and thus hastens evaporation. The heat necessary to produce evaporation is obtained from the steam in the tubes, causing the steam to condense. It was designed to condense 800 lbs. steam per hour and give a vacuum of 22 in., with a terminal pressure in the cylinder of 20 lbs. absolute. Results of tests show that the cooling-water required is practically equal in amount to the steam used by the engine. And since consumption of steam is reduced by the application of a condenser, its use will actually reduce the total quantity of water required. From a curve showing the rate* of evapora- tion per square foot of surface in still air, and also one show ng the rate when a current of air of about 2300 ft. per min. velocity is passed over its surface, the following approximate figures are taken: Temp. F. Evaporation, lbs. per sq. ft. per hour. Temp. F. Evaporation, lbs. per sq. ft. per hour. Still Air. Current. Still Air. Current. 100° 110 120 130 0.2 0.25 0.4 0.6 1.1 1.6 2.5 3.5 140° 150 160 170 0.8 1.1 1.5 2.0 5.0 6.7 9.5 fhe Continuous Use of Condensing-water is described in a series of articles in Poiver, Aug.-Dec, 1892. It finds its application in situa- tions where water for condensing purposes is expensive or difficult to obtain. In San Francisco J. C. H. Stut cools the water after it has left the bot- well by means of a system of pans upon the roof. These pans are shallow troughs of galvanized iron arranged in tiers, on a slight incline, so that the water flows back and forth for 1500 o* ?000 ft., cooling by evaporation and radiation as it flows. The pans are about 5 ft. in width, and the water as it flows has a depth of about half an inch, the temperature being reduced from about 140° to 90°. The water from the hot-well is pumped up to the highest point of the cooling system and allowed to flow as above described, discharg- ing finally into the main tank or reservoir, whence it again flows to the con- denser as required. As the water in the reservoir lowers from evaporation, an auxiliary feed from the city mains to the condenser is operated, thereby keeping the amount of water in circulation practically constant. An accu- mulation of oil from the engines, with dust from the surrounding streets, makes a cleaning necessary about once in six weeks or two months. It is found by comparative trials, running condensing and non condensing, that CONDENSERS, AIR-PUMPS, ETC. 845 about 50$ less water is taken from the city mains when the whole apparatus is in use than when the engine is run non-condensing. 22 to 23 in. of vacuum are maintained. A better vacuum is obtained on a warm day with a brisk breeze blowing than on a cold day with but a slight movement of the air. In another plant the water from the hot- well is sprayed from a number of fountains, and also from a pipe extending around its border, into a large pond, the exposure cooling it sufficiently for the obtaining of a good vacuum by its continuous use . In the system patented by Messrs. See, of Lille, France, the water is dis- charged from a pipe laid in the form of a rectangle and elevated above a pond through a series of special nozzles, by which it is projected into a fine spray. On coming into contact with the air in this state of extreme divi- sion the water is cooled 40° to 50°, with a loss by evaporation of only one tenth of its mass, and produces an excellent vacuum. A 3000-H.P. cooler upon this system has been erected at Lannoy, one of 2500 H.P. at Madrid, and one of 1200 H.P. at Liege, as well as others at Roubaix and Tourcoing. The system could be used upon a roof if ground space were limited. In an arrangement adopted by the Worthington Pump Co. for supplying water to condensers attached to vacuum pans, the injection-water is taken from a tank, and after having passed through the condenser is discharged in a heated condition to the top of a cooling tower, where it is scattered by means of distributing-pipes. The water falling from top to bottom of the tower is lowered in temperature by the cooling effect of the atmosphere and the absorption of heat caused by a portion of the water being vaporized, and is led to the tank to be again started, on its circuit. In the evaporative condenser of T. Ledward & Co. of Brockley, London, the water trickles over the pipes of the large condenser or radiator, and by evaporation carries away the heat necessary to be abstracted to condense the steam inside. The condensing pipes are fitted with corrugations mounted with circular ribs, whereby the radiating or cooling j-urface is largely increased. The pipes, which are cast in sections about 76 in. long by Zy 2 ha. bore, have a cooling surface of 26 sq. ft., which is found sufficient under favorable conditions to permit of the condensation of 20 to 30 lbs. of steam per hour when producing a vacuum of 13 lbs. per sq. in. In a condenser of this type at Rixdorf , near Berlin, a vacuum ranging from 24 to 26 in. of mercury was constantly maintained during the hottest weather of August. The initial temperature of the cooling-water used in the appara- tus under notice ranged from 80° to 85° F., and the temperature in the sun, to which the condenser was exposed, varied each day from 100° to 115° F. During the experiments it was found that it was possible to run one engine under a load of 100 horse-power and maintain the full vacuum without the use of any cooling water at all on the pipes, radiation afforded by the pipes alone sufficing to condense the steam for this power. In Klein's condensing water-cooler, the hot water coming from the con- denser enters at the top of a w T ooden structure about twenty feet in height, and is conveyed into a series of parallel narrow metal tanks. The water overflowing from these tanks is spread as a thin film over a series of wooden partitions suspended vertically about 3^ inches apart within the tower. The upper set of partitions, corresponding to the number of metal tanks, reaches half-way down the tower. From there down to the well is sus- pended a second set of partitions placed at right angles to the first set. This impedes the rapidity of the downflow of the water, and also thoroughly mixes the water, thus affording a better cooling. A fan-blower at the base of the tower drives a strong current of air with a velocity of about twenty feet per second against the thin film of water running down over the partitions. It is estimated that for an effectual cooling two thousand times more air than water must be forced through the apparatus. With such a velocity the air absorbs about two per cent of aqueous vapor. The action of the strong air-current is twofold: first, it absorbs heat from the hot water by being itself warmed by radiation; and, secondly, it increases the evapora- tion, which process absorbs a great amount of heat. These two cooling effects are different during the different seasons of the year. During the winter months the direct cooling effect of the cold air is greater, while during summer the heat absorption by evaporation is the more important factor. Taking all the year round, the effect remains very much the same. The evaporation is never so great that the deficiency of water would not be supplied by the additional amount of water resulting from the condensed steam, whiie in very cold winter months it may be necessary to occasionally rid the cistern of surplus water. It was found that the vacuum obtained by 846 THE STEAM-ENGINE. this eontinual use of the same condensing-water varied during the year between 27.5 and 28.7 inches. The great saving of space is evident from the fact that only the five-hundredth part of the floor-space is required as if cooling tanks or ponds were used. For a 100-horse-power engine the floor-space required is about four square yards by a height of twenty feet. For one horse-power 3.6 square yards cooling-surface is necessary. The vertical suspension of the partitions is very essential. With a ventilator 50 inches in diameter and a tower 6 by 7 feet and 20 feet high, 10,500 gallons of water per hour were cooled from 104° F. to 68° F. The following record was made at Mannheim, Germany: Vacuum in condenser, 28.1 inches; tem- perature of condensing-water entering at top of tower, 104° to 108° F.; temperature of water leaving the cooler. 6(5.2° to 71.6° F. The engine was of the Sulzer compound type, of 120 horse-power. The amount of power necessary for the arrangement amounts to about three per cent of the total horse-power of the engine for the ventilator, and from one and one half to three per cent for the lifting of the water to the top of the' cooler, the total being four and one half to six per cent. A novel form of condenser has been used with considerable success in Germany and other parts of the Continent. The exhanst-steam from the engine passes through a series of brass pipes immersed in water, to which it gives up its heat. Between each section of tubes a number of galvanized disks are caused to rotate. These disks are cooled by a current of air supplied by a fan and pass down into the water, cooling it by abstract- ing the heat given out by the exhaust-steam and carrying it up where it is driven off by the air-current. The disks serve also to agitate the water and thus aid it in abstracting the heat from the steam. With 85 per cent vacuum the temperature of the cooling water was about 130° F., and a consumption of water for condensing is guaranteed to be less than a pound for each pound of steam condensed. For an engine 40 in. X 50 in., 70 revo- lutions per minute, 90 lbs. pressure, there is about 1150 sq. ft. of condensing- surface. Another condenser, 1600 sq. ft. of condensing-surface, is used for three engines, 32 in. x 48 in., 27 in. X 40 in., and 30 in. X 40 in., respectively. — The Steamship. The Increase of Power that may be obtained by adding a condenser giving a vacuum of 26 inches of mercury to a non-condensing engine may be approximated by considering it to be equivalent to a net gain of 12 pounds mean effective pressure per square inch of piston area. If A — area of piston in square inches, S = piston-speed in ft. per minute, then = — — - = H.P. made available by the vacuum. If the vacuum = 13.2 lbs. per sq. in. = 27.9 in. of mercury, then H.P. = AS -+- 2500. The saving of steam for a given horse-power will be represented approxi- mately by the shortening of the cut-off when the engine is run with the condenser. Clearance should be included in the calculation. To the mean effective pressure non-condensing, with a given actual cut-off, clearance considered, add 3 lbs. to obtain the approximate mean total pressure, con- densing. From tables of expansion of steam find what actual cut-off will give this mean total pressure. The difference between this and the original actual cut-off, divided by the latter and by 100, will give the percentage of saving. The following diagram (from catalogue of H. R. Worthington) shows the percentage of power that may be gained by attaching a condenser to a non- condensing engine, assuming that the vacuum is 12 lbs. per sq. in. The mean effective pressures are those of a non-condensing engine exhausting at atmospheric pressure, clearance and compression not considered. The left-hand vertical column of figures are the initial steam-pressures (above the atmosphere), and the upper horizontal column the several points of cut-off that represent the point of the stroke at which the steam is shut off and admission ceases; directly under this column is a similar one of the mean effective pressures. To determine the mean effective pressure produced by 90 pounds steam, cut-off at one quarter, find 90 in the initial- pressure column, and follow the line to the rig..t until it intersects the oblique line that corresponds to the 34 cut-off. Now read the mean effective pressure from the figures directly above, which in this case is 49 pounds. By glancing down and reading on the lower scale the figure that corresponds with this point of intersection the percentage of gain in power will be seen to be between 25 and 30 per cent of the power of the engine when running non-condensing. GAS, PETROLEUM, AND HOT-AIR ENGINES. 847 (' Point of Cut-off VJfefoflafe f/9 Vs f/s 'M ' -t/3 'fa j j j 7 Mean j Effect j\fe Pressure Per Cenl or Power gamed by Vacuum Fig. 151. Evaporators and Distillers are used with marine engines for the purpose of providing fresh water for the boilers or for drinking purposes. Weirds Evaporator consists of a small horizontal boiler, contrived so as to be easily taken to pieces and cleaned. The water in it is evaporated by the steam from the main boilers passing through a set of tubes placed in its bottom. The steam generated in this boiler is admitted to the low- pressure valve-box, so that there is no loss of energy, and the water con- densed in it is returned to the main boilers. In Weir's Feed-heater the feed-water before entering the boiler is heated up very nearly to boiling-point by means of the waste water and steam from the low-pressure valve-box of a compound engine. GAS, PETROLEUM, AND HOT-AIR ENGINES. Gas-engines. — For theory of the gas-engine, see paper by Dugald Clerk, Proc. Inst. C. E. 1882, vol. lxix.; and Van Nostrand's Science Series, No. 62. See also Wood's Thermodynamics. For construction of gas-engines, see Robinson's Gas and Petroleum Engines; articles by Albert' Spies in Cassier's Magazine, 1893; also Appleton's Cyc. of Mechanics, and Modern Mechanism. In the ordinary type of single-cylinder gas-engine (for example the Otto) known as a four-cycle engine one ignition of gas takes place in one end of the cylinder every two revolutions of the fly-wheel, or every two double strokes. The following sequence of operations takes place during four con- secutive strokes: (a) inspiration during an entire stroke; (b) compression during the second (return) stroke; (c) ignition at the dead-point, and expan- sion during the third stroke; (d) expulsion of the burnt gas during the fourth (return) stroke. Beau de Rochas in 1862 laid down the law that there are 848 GAS, PETROLEUM, AHD HOT-AIR ENGIHES. four conditions necessary to realize the best results from the elastic force of gas: (1) The cylinders should have the greatest capacity with the smallest circumferential surface; (2) the speed should be as high as possible; (3) the cut-off should be as early as possible; (4) the initial pressure should be as high as possible. In modern engines it is customary for ignition to take place, not at the dead point, as proposed by Beau de Rochas, but somewhat later, when the piston has already made part of its forward stroke. At first sight it might be supposed that this would entail a loss of power, but experi- ence shows that though the area of the diagram is diminished, the power registered by the friction-brake is greater. Starting is also made easier by this method of working. (The Simplex Engine, Proc. Inst. M. E. 1889.) In the Otto engine the mixture of gas and air is compressed to about 3 atmospheres. When explosion takes place the temperature suddenly rises to somewhere about 2900° F. (Robinson.) The two great sources of waste in gas-engines are: 1. The high tempera- ture of the rejected products of combustion ; 2. Loss of heat through the cylinder walls to the water-jacket. As the temperature of the water-jacket is increased the efficiency of the engine becomes higher. With ordinary coal-gas the consumption maybe taken at 20 cu. ft. per hour per I.H.P., or 24 cu. ft. per brake H.P. The consumption will vary with the quality of the gas. When burning Dowson producer-gas the consump- tion of anthracite (Welsh) coal is about 1.3 lbs. per I.H.P. perhourfor ordinary working. With large twin engines, 100 H.P., the consumption is reduced to about 1.1 lb. The mechanical efficiency or B. H.P. -+- I.H.P. in ordinary engines is about 85%; the friction loss is less in larger engines. Efficiency of the Gas-engine. (Thurston on Heat as a Form of Energy.) Heat transferred into useful work 17# " to the jacket-water 52 " lost in the exhaust-gas 16 " " by conduction and radiation 15 - 83# This represents fairly the distribution of heat in the best forms of gas- engine. The consumption of gas in the best engines ranges from a mini- mum of 18 to 20 cu. ft. per I.H.P. per hour to a maximum exceeding in the smaller engines 25 cu. ft. or 30 cu. ft. In small engines the consumption per brake horse-power is one third greater than these figures. The report of a test of a 170-H.P. Crossley (Otto) gas-engine in England, 1892, using producer-gas, shows a consumption of but ,85 lb. of coal per H.P. hour, or an absolute combined efficiency of 21.3$ for the engine and pro- ducer. The efficiency of the engine alone is in the neighborhood of 25$. The Taylor gas-producer is used in connection with the Otto gas-engine at the works of Schleicher, Schumm & Co., of Philadelphia. The only loss is due to radiation through the walls of the producer and a small amount of heat carried off in the water from the scrubber. Experiments on a 100-H.P. engine show a consumption of 97/100 lb. of carbon per I.H.P. per hour. This result is superior to any ever obtained on a steam-engine. (Iron Age, 1893.) Tests of the Simplex Gas-engine. (Proc. Inst. M. E. 1889.)— Cylinder 7% X 15% in., speed 160 revs, per min. Trials were made with town gas of a heating value of 607 heat-units per cubic foot, and with Dowson gas, rich in CO, of about 150 heat-units per cubic foot. Town Gas. Dowson Gas. T 2. 3. 7. 2. 3? Effective H.P 6.70 8.67 9.28 7.12 3.61 5.26 Gas per H.P. per hour, cu. ft.. 21.55 20.12 20.73 88.03 114.85 97.88 Water per H.P. per hour, lbs. 54.7 44.4 43.8 58.3 Temp, water entering, F 51° 51° 51° 48° " effluent 135° 144° 172° 144° The gas volume is reduced to 32° F. and 30 in barometer. A 50-H.P. engine working 35 to 40 effective H.P. with Dowson generator consumed 51 lbs. English anthracite per hour, equal to 1.48 to 1.3 lbs. per effective H.P. A 16- H.P. engine working 12 H.P. used 19.4 en. ft. of gas per effeciive H.P. A 320-H.F". Gas-engine.— The flour-mills of M. Leblanc, at Pantin, France, have been provided with a 320-horse-power fuel-gas engine of the Simplex type. With coal-gas the machine gives 450 horse-power. There is one cylinder, 34.8 in. diaui. ; the piston-stroke is 40 in.; and the speed 100 revs. GAS-ENG1XES. 849 per min. Special arrangements have been devised in order to keep the different parts of the machine at appropriate temperatures. The coal used is 0.81 J lb. per indicated or 1 .03 lb. per brake horse-power. The water used is 8% gallons per brake horse-power per hour. Test ofam Otto Gas-engine. (Jour. F. /., Feb. 1890, p. 115.)— En- gine 7 11. P. nominal; working capacity of cylinder .2594 cu. ft.; clearance space .179(3 cu. ft. Per cent Heat-units. of Heat received. Transferred into work 22.84 Taken by jacket- water 49 . 94 " " exhaust 27.22 Temperature of gas supplied . . 62.2 '_' " " exhaust... 774.3 " " enteringwater 50.4 " " exit water 89.2 Pressure of gas, in. of water.. 3.06 Revolution per min., av'ge 161.6 Explosions missed per min., average 6.8 Mean effective pressure, lbs. per sq. in 59. Horse -power, indicated 4.94 Work per explosion, foot- pounds 2204. Explosions per minute 74. Gas used perl.H.P. per hour, cu. ft 23.4 Composition of the gas: C0 2 ... C 3 H 4 .. O CO.... CH 4 . H .... N... . r olume. By Weight. 0.50# 1.923# 4.32 10.520 1.00 2.797 5.33 15.419 27.18 38.042 51.57 9.021 9.06 22.273 Temperatures and Pressures developed in a Gas-engine. (Clerk on the Gas-engine.)— Mixtures of air and Oldham coal-gas. Temper- ature before explosion, 17° C. Mixture. Max. Press Gas. Air. lbs. per sq. in 1 vol. 14 vols. 40. 1 " 13 " 51.5 1 " 12 " 60. 1 •* 11 " 61. 1 " 9 " 78. 1 " 7 " 87. 1 " 6 " 90. 91. Temp, of Explo- sion calculated from observed Pressure. 806° C. 1033 1202 1220 1557 1733 1792 1812 1595 Theoretical Temp, of Explo- sion if all Heat were evolved. 1786° C. 1912 Test of the Clerk Gas-engine. (Proc Inst. C. E. 1882, vol. lxix.)— Cylinder 6 X 12 in., 150 revs, per min.; mean available pressure 70.1 lbs., 9 I.H.P.; maximum pressure, 220 lbs. per sq. in. above atmosphere; pressure before ignition, 41 lbs. above atm. ; temperature before compression 60° F., after compression, 313° F.; temperature after ignition calculated from pressure, 2800° F. ; gas required per I.H.P. per hour, 22 cu. ft. Combustion of the Gas in the Otto Engine.— John Imray, in discussion of Mr. Clerk's paper on Theory of the Gas-engine, says: The change which Mr. Otto introduced, and which rendered the engine a success, was that, instead of burning in the cylinder an explosive mixture of gas and air, he burned it in company with, and arranged in a certain way in respect of, a large volume of incombustible gas which was heated by it, and which diminished the speed of combustion. W. R. Bousfield, in the same discus- sion, says: In the Otto engine the charge varied from a charge which was an explosive mixture at the point of ignition to a charge which was merely an inert fluid near the piston. When ignition took place there was n explo- sion close to the point of ignition that was gradually communicated through- out the mass of the cylinder. As the ignition got farther away from the primary point of ignition the rate of transmission became slower, and if the engine were not worked too fast the ignition should gradually catch up to the piston during its travel, all the combustible gas being thus consumed. This theory of slow combustion is, however, disputed by Mr. Clerk, who holds that the whole quantity of combustible gas is ignited in an instant. Use of Carburetted Air in Gas-engines.— Air passed over 850 GAS, PETROLEUM, AtfD HOT-AIR E^GItfES. gasoline or volatile petroleum spirit of low sp. gr., 0.65 to 0.70, liberates some of the gasoline, and the air thus saturated with vapor is equal in heat- ing or lighting power to ordinary coal-gas. It may therefore be used as a fuel for gas-engines. Since the vapor is given off at ordinary temperatures gasoline is very explosive and dangerous, and should be kept in an under- ground tank out of doors. A defect in the use of carburetted air for gas- engines is that the more volatile products are given off first, leaving an oily residue which is often useless. Some of the substances in the oil that are taken up by the air are apt to form troublesome deposits and incrustations when burned in the engine cylinder. The Otto Gasoline-engine. (Eng'g Neivs, May 4, 1893.)— It is claimed that where but a small gasoline-engine is used and the gasoline bought at retail the liquid fuel will be on a par with a steam-engine using 6 lbs. of coal per horse -power per hour, and coal at $3.50 per ton, and will besides save all the handling of the solid fuel and ashes, as well as the at- tendance for the boilers. As very few small steam-engines consume less than 6 lbs. of coal per hour, this is an exceptional showing for economy. At 8 cts. per gallon for gasoline and 1/10 gal. required per H.P. per hour, the cost per H.P. per hour will be 0.8 cent. The Priestman Petroleum-engine. (Jour. Frank. Inst., Feb. 1893 )— The following is a description of the operation of the engine: Any ordinary high -test (usually 150° test) oil is forced under air-pressure to an atomizer, where the oil is met by a current of air and broken up into atoms and sprayed into a mixer, where it is mixed with the proper proportion of supplementary air and sufficiently heated by the exhaust from the cylinder passing around this chamber. The mixture is then drawn by suction into the cylinder, where it is compressed by the piston and ignited by an electric spark, a governor controlling the supply of oil and air proportionately to the work performed. The burnt products are discharged through an ex- haust-valve which is actuated by a cam. Part of the air supports the com- bustion of the oil, and the heat generated by the combustion of the oil expands the air that remains and the products resulting from the explosion, and thus develops its power from air that it takes in while running. In other words, the engine exerts its power by inhaling air, heating that air, and expelling the products of combustion when done with. In the largest engines only the 1/250 part of a pint of oil is used at any one time, and in the smallest sizes the fuel is prepared in correct quantities varying from 1/7000 of a pint upward, according to whether the engine is running on light or full duty. The cycle of operations is the same as that of the Otto gas- engine. Trials of a 5-H.P. Priestman Petroleum-engine. (Prof. W. C. Unvvin, Proc. Inst. C. E. 1892.)— Cylinder, 8% X 12 in., making normally 200 revs, per min. Two oils were used, Russian and American. The more important results were given in the following table: Trial V. Full Power. Trial I. Full Power. Trial IV. Full Power. Trial II. Half Power. Trial III. Light. Oil used | Brake H.P Day- light. 7.722 9.3G9 0.824 0.842 0.694 33.4 151.4 35.0 35.4 Russo- lene. 6.765 7.408 0.91 0.946 0.864 31.7 134.3 27.6 23.7 Russo- lene. 6.882 8.332 0.876 0.988 0.816 43.2 128.5 26.0 25.5 Russo- lene. 3.62 4.70 0.769 1.381 1.063 21.7 48.5 14.8 15.6 Russo- lene. I.H.P Mechanical efficiency. . . Oil used per brake H.P. 0.889 Oil used per brake H.P. hour, lb Lb. of air per lb. of oil. . Mean explosion pressure, 5.734 10.1 9.6 Mean compression pres- sure, lbs. per sq. in .. Mean terminal pressure, lbs. per sq in 6.0 To compare the fuel consumption with that of a steam-engine, 1 lb. of oil might be taken as equivalent to 1*4 lbs. of coal. Then the consumption EFFICIENCY OF LOCOMOTIVES. 851 in the oil-engine was equivalent, in Trials I., IV., and V., to 1.18 lbs., 1.23 lbs., and 1.02 lbs. of coal per brake horse-power per hour. From Trial IV. the following values of the expenditure of heat were obtained: Per cent. Useful work at brake 13.31 Engine friction 2.81 Heat shown on indicator-diagram. 16.12 Rejected in jacket-water 47.54 " in exhaust -gases 26.72 Radiation and unaccounted for 9.61 Total 99.99 Naphtha-engines are in use to some extent in small yachts and launches. The naphtha is vaporized in a boiler, and the vapor is used ex- pansively in the engine-cylinder, as steam is used; it is then condensed and returned to the boiler. A portion of the naphtha vapor is used for fuel un- der the boiler. According to the circular of the builders, the Gas Engine and Power Co. of New York, a 2-H.P. engine requires from 3 to 4 quarts of naphtha per hour, and a 4-H.P. engine from 4 to 6 quarts. The chief advan- tages of the naphtha-engine and boiler for launches are the saving of weight and the quickness of operation. A 2-H.P. engine weighs 200 lbs., a 4-H.P. 300 lbs. It takes only about two minutes to get under headway. (Modern Mechanism, p. 270.) Mot-air (or Caloric) Engines.— Hot-air engines are used to some extent, but their bulk is enormous compared with their effective power. For an account of the largest hot-air engine ever built (a total failure) see Church's Life of Ericsson. For theoretical investigator, see Rankine's Steam-engine and Rontgen's Thermodynamics. For description of con- structions, see Appletoifs Cyc. of Mechanics and Modern Mechanism, and Babcock on Substitutes for Steam, Trans. A. S. M. E., vii., p. 693. Test of a Hot-air Engine (Robinson).— A vertical double-cylinder (Caloric Engine (Jo.'s) 12 nominal H.P. engine gave 20.19 I.H.P. in the work- ing cylinder and 11.38 I.H.P. in the pump, leaving 8.81 net I.H.P.; while the effective brake H.P. was 5.9, giving a mechanical efficiency of 67^. Con- sumption of coke, 3.7 lbs. per brake H.P. per hour. Mean pressure on pistons 15.37 lbs. per square inch, and in pumps 15.9 lbs., the area of working cylinders being twice that of the pumps. The hot air supplied was about 1160° F. and that rejected at end of stroke about 890° F. The b st result of Stirling's he?t-engine was 2.7 lbs. per brake H.P. per hour. Bailey's hot-air engine, 2 H.P. nominal, gave 4.2 I.H.P., 2.6 B.H.P.; mechanical efficiency 62^; estimated temperature at highest pressure 1500° F., and at atmospheric pressure 700° F. Highest pressure, 14 lbs. per square inch above atmosphere. Consumption of fuel, 7 lbs. per hour per brake H.P., and of cooling water, 30 lbs. LOCOMOTIVES. Efficiency of Iiocomotives and Resistance of Trains. (George R. Henderson, Proc. Engrs. Club of Phila. 1886.)— The efficiency of locomotives can be divided into two principal parts : the first depending upon the size of the cylinders and wheels, the valve-gear, boiler and steam- passages, of which the tractive power is a function; and the second upon the speed, grade, curvature, and friction, which combine to produce the resistance. The tractive pow r er may be determined as follows : Let P = tractive power; p = average effective pressure in cylinder; £? = stroke of piston ; d = diameter of cylinders; D — diameter of driving-wheels. Then 47rrf2 p g _ tftpS 4nD '" D ' 852 LOCOMOTIVES. The average effective pressure can be obtained from an indicator-dia- gram, or by calculation, when the initial pressure and ratio of expansion are known, together with the other properties of the valve-motion. The sub- joined table from " Auchincloss " gives the proportion of mean effective pressure to boiler-pressure above atmosphere for various proportions of cut-off. Stroke, Cut off at— M.E.P. (Boiler- pres. = 1). Stroke, Cut off at— (M.E.P. Boiler- pres. = 1). Stroke, Cut off at— M.E.P. (Boiler- pres. = 1). A25 = H .15 .175 '.25 = H .3 .15 .2 .24 .28 .32 .4 .46 .333 = y s .375 = % .4 .45 .55 .5 = % .55 .57 .62 .67 .72 .625 = % .666 = %j •875 = % .79 .82 .85 .89 .93 .98 These values were deduced from experiments with an English locomotive by Mr. Gooch. As diagrams vary so much from different causes, this table will only fairly represent practical cases. It is evident that the cut-off must be such that the boiler will be capable of supplying sufficient steam at the given speed. , In the following calculations it is assumed that the adhesion of the engine is at least equal to the tractive power, which is generally the case — if the engine be well designed— except when starting, or running at a very low rate of speed, with a small expansive ratio. When running faster, economy, and also the size of the boiler, necessitate a higher ratio of expansion, thus reducing the tractive power below the adhesion. If the adhesion be less than the tractive power, substitute it for the latter in the following for- mulae. The resistances can be computed in the following manner, first consider- ing the train: There is a resistance due to friction of the journals, pressure of wind, etc., which increases with the speed. Most of the experiments made with a view of determining the resistance of trains have been with European rolling-stock and on European railways. The few trials that have been made here seem to prove that with American systems this resistance is less. The following table gives the resistance at different speeds, assumed for American practice : Speed in miles per hour : s = 5 10 15 20 25 30 35 40 45 50 55 60 Resistance in pounds per ton of 2240 lbs. : y = 3.1 3.4 4. 4.8 5.8 7.1 8.6 10.2 12.1 14.3 16.8 19.2 Coefficient of resistance in terms of load : I = .0015 .0017 .0020 .0024 .0029 .0035 .0043 .0051 .0060 .0071 .0084 .0096 0+£> I = .0015 The resistance due to curvature is about .5 lb. per ton per degree of cur- vature, or the coefficient = .00025c, where c = the curvature in degrees. The effect of grades may be determined by the theory of the inclined plane. Consider a load Lona grade of m feet per mile. The component of the weight L acting in the line of traction, or parallel to the track, is L sin d = ^ = -00019 Lot. 52b0 To combine these coefficients in one equation representing the resistance of the train : Let L = weight of train, exclusive of engine, in pounds; R = resistance of train, in pounds. s, c } and m, as above. Then B = L [.0015(l + J^)+ .00025c ± .00019m], INERTIA AND RESISTANCES OF RAILROAD TRAINS. 853 the ± sign meaning that this coefficient is positive for ascending and nega- tive for descending grades. To find a grade upon which a train would descend by itself, take the last coefficient minus and make R — U, whence " i = r - 9 ( 1 + bfo) + 1 - 3c - As locomotives usually have a long rigid wheel-base, the coefficient for curvature had better be doubled. The resistance due to the friction of the working parts will be considered as being proportional to the tractive power, so that the effective tractive power will be represented by itP, the resistance being (1 - it) P. Combining all these values, there results the equation between the trac- tive power and the weight of the train and engine: uP- W (.0005c ± .00019m) = Ll+ .00025c ± .00019m, IF being weight of engiue aud tender, and u being probably about .8. Transforming, we have _ uF - TF(. 0005c ± ,00019m) Z+ .00025c ± .00019m ' and L(l + .00025c ± .00019m) + IF(.0005c ± .00019m) These deductions, says Mr. Henderson, agree well with railroad practice. The figures given above for resistances are very much less than those given by the old formulge (which were certainly wrong), but even Mr. Hen- derson's figures for high speed are too high, according to a diagram given by D. L. Barnes in Eng'g M'tg., June, 1894, from which the following figures are derived: Speed, miles per hour 50 60 70 80 90 100 Resistance, pounds per gross ton . . 12 12.4 13.5 15 17 20 Eng'g News, March 8, 1894, gives a formula which for high speeds gives figures for resistance between those of Mr. Barnes and Mr. Henderson. See' tests reported in Eng'g News of June 9, 1892. The formula is, resistance in pounds per ton = J4 velocity in miles per hour -\- 2. This gives for Speed 5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 Resistance.. 3J4 4.5 0% 7 8}/ 4 9.5 10^12 13^14.5 17 19.5 22 24.5 27 For tables showing that the resistance varies with the area exposed to the resistance and friction of the air per ton of load, see Dashiell, Trans. A. S. M. E., vol. xiii. p. 371. Inertia and Resistances of Railroad Trains at Increasing Speeds.— A series of tables and diagrams is given in R. R. Gaz., Oct. 31, 15*90, to show the resistances due to inertia in starting trains and accelerat- " ig their speeds. The mechanical principles and formulae from which these data were cal- culated are a.s follows: iS = speed in miles per hour to be acquired at the end of a mile. S -5- 2 = average speed in miles per hour during the first mile run. V = velocity in feet per second at the end of a mile; then F-h 2 = aver- age velocity in feet per second during the first mile run. 5280 -i- V/2 = time in seconds required to run first mile = 10560 -h V. F-r (10560 ■+- V) - V 2 -4- 10560 = .0000947 F 2 = Constant gain in velocity or acceleration in feet per second necessary to the acquirement of a velocity V at the end of a mile. g — acceleration due to the force of gravity, i.e., 32.2 feet per second. The forces required to accelerate a given mass in a given time to different velocities are in proportion to those velocities. The weight of a body is the measure of the force which accelerates it in the case of gravity, and as we are considering 1 lb., or the unit of weight, as the mass to be accelerated, we have g: (F 2 -7- 10560) : : 1 is to the force required to accelerate 1 lb. to the velocity Fat the end of a mile run, or, what is the same, to accelerate it at the rate of F 2 -r- 10560 feet per second. From this the pull on the drawbar— it is the same as the force just men- tioned, and is properly termed the inertia— in pounds per pound of train weight is F 2 -r- (10560 X 32.2), which equals .00000294F 2 . 854 LOCOMOTIVES. This last formula also gives the grade in per cent which will give a resist- ance equal to the inertia due to acceleration. The grade in feet per mile is .00000294 F 2 X 5280 = . 01553 F 2 . The resistance offered in pounds per ton is 2000 times as much as per pound, or .00588F 2 . When the adhesion of locomotive drivers is 600 lbs. per ton of weight thereon— this is about the maximum— then the tons on drivers necessary to overcome the inertia of each ton of total train load are .00588F 2 h- 600 = . 0000098 F 2 . In this determination of resistances no account has been taken of the rotative energy of the wheels. Efficiency of the Mechanism of a Locomotive. — Druitt Halpin (Proc. Inst. M. E., January, 1889,) writes as follov\s, concerning the tractive efficiency of locomotives; With simple two-cylinder engines, hav- ing four wheels coupled, experiments have been made by the late locomo- tive superintendent of the Eastern Railway of Fiance, M. Regray, with the greatest possible care and with the best apparatus, and the result arrived at was that out of 100 I.H.P in the cylinders 43 H.P. only was available on the draw-bar. The loss of 57% was rather a high price to pay for the efficiency of the engine. How much of that loss was due to coupling-rods no one could yet say; but a considerable amount of it must be due to the rods, be- j cause it was known that large engines with a single pair of driving-wheels not coupled were doing their work more economically, while advanced loco- motive engineers who had not yet gone in for compounding were at any rate going back to the single pair of driving-wheels. Moreover, that astonishing loss of b'i% had been confirmed independently on the Pennsylvania Railroad, trials made with an engine having 18*4 X 24-in. cylinders and 6 ft. in. wheels four-coupled; by taking indicator diagrams up to 65 miles an hour, which were professed to be taken correctly, the power on the draw-bar was found to be only 42$ of that in the cylinders, or only \% less than in the French experiments. The Size of Locomotive Cylinders is usually taken to be such that the engine will just overcome the adhesion of its wheels to the rails un- der favorable circumstances. The adhesion of the wheel is about one third the weight when the rail is clean and sanded, but is usually assumed at 0.25. (Thurston.) A committee of the American Association of Master Mechanics, after studying the performance reports of the best engines, proposes the follow- SdCcI^P"^ iug formula for weight on driving-w r heels: W = — in which the mean pressure in the cylinder is taken at 0.85 of the boiler-pressure at starting, C is a numerical coefficient of adhesion, d the diameter of cylinder in inches, D that of the drivers in inches, P the pressure in the boiler in pounds per square inch. S the stroke of piston in inches. C is taken as 0.25 for passenger engines, 0.24 for freight, and 0.22 for " switching " engines. The common builder's rule for determining the size of cylinders for the locomotive is the following, in which we accept Mr. Forney's assumption that the steam-pressure at the engine may be taken as nine tenths that in the boiler: The tractive force is, approximately, F — - — -= where C is the circumference of tires of driving-wheels, S — the stroke in inches, Pi = the initial unbalanced steam-pressure in the cylinder in pounds per square inch, and A = the area of one cylinder in square inches. If D — 9p v d 2 S diameter of driving wheel and d = diameter of cylinder, F = Taking the adhesion at one fourth the weight W, 0.9p 1 X A X iS _ O.Vp^S , C D ' whence the area of each piston i 0.25CIF . /0.25DW / 0.25£>H y o.9p 1 s 0.9 X 4 X piiS' y o.^s ' The above formulae give the maximum tractive force; for the mean trac- tive force substitute for pi in the formulae the mean effective pressure. BOILERS, GRATE-SURFACE, SMOKE-STACKS, ETC. 855 Von Borries's rule for the diameter of the low-pressure cylinder of a com- pound locomotive is d 2 = — — , ph where d — diameter of l.p. cylinder in inches; D = diameter of driviiig-wheel in inches; p = mean effective pressure per sq. in., after deducting internal machine friction; h = stroke of piston in inches; Z = tractive force required, usually 0.14 to 0.16 of the adhesion. The value of p depends on the relative volume of the two cylindei-s, and from indicator experiments may be taken as follows: nioco ^f -Pnonno Ratio of Cylinder p in percentage p for Boiler-press Ulass oi engine. Volumes. of Boiler-pressure. ureofl761bs. Large-tender eng's 1 : 2 or 1 : 2.05 42 74 Tank-engines l:2orl:2.2 40 71 The Size oi Locomotive Boilers. (Forney's Catechism of the Locomotive.)— They should be proportioned to the amount of adhesive weight and to the speed at which the locomotive is intended to work. Thus a locomotive with a great deal of weight on the driving-wheels could pull a heavier load, would have a greater cylinder capacity than one with little ad- hesive weight, would consume more steam, and therefore should have a larger boiler. The weight, and dimensions of locomotive boilers are in nearly all cases determined by the limits of weight and space to which they are necessarily confined. It may be stated generally that within these limits a locomotive boiler cannot be made too large. In other words, boilers for locomotives should always be made as large as is possible under the conditions that de - termine the weight and dimensions of the locomotives. Wootten's Locomotive. (Clark's Steam-engine ; see also Jour. Frank. Inst. 1891, and Modern Mechanism, p. 485.) — J. E. Wootten designed and constructed a locomotive boiler for the combustion of anthracite and lignite, though specially for the utilization as fuel of the waste produced in the mining and preparation of anthracite. The special feature of the engine is the fire-box, which is made of great length and breadth, extending clear over the wheels, giving a grate-area of from 64 to 85 sq. ft. The draught diffused over these large areas is so gentle as not to lift the fine panicles of the fuel. A number of express-engines having this type of boiler are engaged on the fast trains between Philadelphia and Jersey City. The fire-box shell is 8 ft. 8 in. wide and 10 ft. 5 in. long ; the fire-box is 8x9^ ft., making 76 sq. ft. of grate-area. The grate is composed of bars and water-tubes alternately. The regular types of cast-iron shaking grates are also used. The height of the fire-box is only 2 ft. 5 in. above the grate. The grate is terminated by a bridge of fire-brick, beyond which a combustion-chamber, 27 in. long, leads to the flue-tubes, about 184 in number, 1% in. diam. The cylinders are 21 in. diam., with a stroke of 22 inches. The driving-wheels, four-coupled, are 5 ft. 8 in. diam. The engine weighs 44 tons, of which 29 tons are on driv- ing wheels. The heating-surface of the fire-box is 135 sq. ft., that of the flue-tubes is 982 sq. ft.; together, 1117 sq. ft., or 14.7 times the grate-area. Hauling 15 passenger-cars, weighing with passengers 360 tons, at an average speed of 42 miles per hour, over ruling gradients of 1 in 89, the engine con- sumes 62 lbs. of fuel per mile, or 34*4 lbs. per sq. ft. of grate per hour. Qualities Essential for a Free-steaming Locomotive. (From a paper by A. E. Mitchell, read before the N. Y. Railroad Club; Eng'g Neivs, Jan. 24, 1891.)— Square feet of boiler-heating surface for bitu- minous coal should not be less than 4 times the square of the diameter in inches of a cylinder 1 inch larger than the cylinder to be used. One tenth of this should be in the fire-bOx. On anthracite locomotives more heating- surface is required in the fire-box, on account of the larger grate-area required, but the heating-surface of the flues should not be materially decreased. Grate-surface, Smoke-stacks, and Exhaust-nozzles for Locomotives. (Am. Mac//,., Jan. 8, 1891.)— For grate-surface for anthra- cite coal: Multiply the displacement in cubic feet of one piston during a stroke by 8.5; the product will be the area of the grate in square feet. For bituminous coal : Multiply the displacement in feet of one piston during a stroke by 6^; the product will be the grate-area in square feet for engines with cylinders 12 in. in diameter and upwards. For engines with 856 LOCOMOTIVES. smaller cylinders the ratio of grate-area to piston-displacement should be 7J^ to 1, or even more, if the design of the engine will admit this proportion. The grate-areas in the following table have been found by the foregoing rules, and agree very closely with the average practice : Smoke-stacks.— The internal area of the smallest cross-section of the stack should be 1/1? of the area of the grate in soft-coal-burning engines. A. E. Mitchell, Supt. of Motive Power of the N. Y. L. E. & W. R. R., says that recent practice varies from this rule. Some roads use the same size of , stack, 13J4 in. diam. at throat, for all engines up to 20 in. diam. of cylinder. The area of the orifices in the exhaust-nozzles depends on the quantity and quality of the coal burnt, size of cylinder, construction of stack, and the condition of the outer atmosphere. It is therefore impossible to give rules for computing the exact diameter of the orifices. All that can be done is to give a rule by which an approximate diameter can be found. The exact diameter can only be found by trial. Our experience leads us to believe that the area of each orifice in a double exhaust-nozzle should be equal to 1/400 part of the grate-surface, and for single nozzles 1/200 of the grate-surface. These ratios have been used in finding the diameters of the nozzles given in the following table. The same sizes are often used for either hard or soft coal-burners. Double Single Nozzles. Nozzles. Size of for Anthra- cite Coal, in sq. in. for Bitumin- ous Coal, in sq. in. Diameter Cylinders, in inches. of Stacks, in inches. Diam. of Orifices, in Diam. of Orifices, in inches. inches. 12 X 20 1591 1217 &A 2 2 13/16 13 X 20 1873 1432 io*2 2^ 3 14 X 20 2179 1666 HJ4 2 5/16 314 15 X 22 2742 2097 12^ 2 9/16 3 11/16 16 X 24 3415 2611 14 2% 4 1/16 17 X 24 3856 2948 15 3 1/16 4 5/16 18 X 24 4321 3304 15% 3^ 4% 19 X 24 4810 3678 16£ 3 7/16 4 13/16 20 X 24 5337 4081 1-T& Ws 5 1/16 Exhaust-nozzles in Locomotive Boilers.— A committee of the Am. Ry. Master Mechanics' Assn. in 1890 reported that they had, after two years of experiment and research, come to the conclusion that, owing to the great diversity in the relative proportions of cylinders and boilers, together with the difference in the quality of fuel, any rule which does not recognize each and all of these factors would be worthless. The committee was unable to devise any plan to determine the size of the exhaust-nozzle in proportion to any other part of the engine or boiler, and believes that the best practice is for each user of locomotives to adopt a nozzle tliat will make steam freely and fill the other desired conditions, best determined by an intelligent use of the indicator and a check on the fuel account. The conditions desirable are : That it must create draught enough on the fire to make steam, and at the same time impose the least possible amount of work on the pistons in the shape of back pressure. It should be large enough to produce a nearly uniform blast without lifting or tearing the fire, and he economical in its use of fuel. Fire-brick Arches in Locomotive Fire-boxes.— A com- mittee of the Am. Ry. Master Mechanics' Assn. in 1S90 reported strongly in favor of the use of brick arches in locomotive fire-boxes. They say : It is the unanimous opinion of all who use bituminous coal and brick arch, that it is most efficient in consuming the various gases composing black smoke, and by impeding and delaying their passage through the tubes, and ming- ling: and subjecting them to the heat of the furnace, greatly lessens the volume ejected, and intensifies combustion, and does not in the least check but rather augments draught, with the consequent saving of fuel and in- creased steaming capacity that might be expected from such results. This in particular when used in connection with extension front. Size, Weight, Tractive Power, etc., of Different Sizes of Locomotives. (J. G. A. Meyer, Modern Locomotive Construction, Am. SIZE, WEIGHT, TRACTIVE POWER, ETC. 857 Mach.< Aug. 8, 1885.)— The tractive power should not be more or less than the adhesion. In column 3 of each table the adhesion is given, and since the adhesion and tractive power are expressed by the same number of pounds, these figures are obtained by finding the tractive power of each engine, for this purpose always using the small diameter of driving-wheels given in column 2. The weight on drivers is shown in column 4, which is obtained by multiplying the adhesion by 5 for all classes of engines. Column 5 gives the weights on the trucks, and these are based upon observations. Thus, the weight on the truck for an eight-wheeled engine is about one half of that placed on the drivers. For Mogul engines we multiply the total weight on drivers by the decimal 2, and the product will be the weight on the truck. For ten-wheeled engines the total weight on the drivers, multiplied by the decimal .32, will he equal to the weight on the truck. And lastly, for consolidation engines, the total weight on drivers multi- plied by the decimal .16, will determine the weight on the truck. In column 6 the total weight of each engine is given, which is obtained by adding the weight on the drivers to the weight on the truck. Dividing the adhesion given in column 1 by 7% will give the number of tons of 2000 lbs. that the engine is capable of hauling on a straight and level track, column 7. The weight of engines given in these tables will be found to agree gen- erally with the actual weights of locomotives recently built, although it must not be expected that these weights will agree in every case with the actual weights, because the different builders do not build the engines alike. The actual weight on trucks for eight-wheeled or ten-wheeled engines will not differ much from those given in the tables, because these weights depend greatly on the difference between the total and rigid wheel-base, and these ai-e not often changed by the different builders. The proportion between the rigid and total wheel-base is generally the same. The rule for finding the tractive power is : ( Square of dia. of | | piston in inches j ( Mean effect, steam | I press, per sq. in. j i stroke | i in feet j Diameter of wheel in feet. tractive power. Eight wheeled Locomotives. Ten- 5VHEELED Engines. so el t« iT & a m i-sl q| o ri ft o 3 1 §lil i I > ft o C ft C o p H C lilt p 11 ■d 1 1 H £ O II ft ? "0 < 1 3| 1 a S 4 '» 6 7 1 2 3 4 5 6 7 in. in. lbs. lbs. lbs. lbs. in. in lbs lbs. lbs. lbs. 10x20 45-51 4000 20000 10000 30000 . 533 12X18 39-43 5981 29907 9570 39477 797 11x22 45-51 5324 2C620 13310 39930 709 13X18 41-45 6677 33387 II 44070 890 48-54 5940 29700 14850 44550 792 U -20 43-47 t 8205 41023 13127 54150 1093 49-57 6828 34140 17070 51210 910 15 ,-22 45-50 1 9900 49500 65340 1320 55-61 7697 384S5 19242 57727 1026 48-54 11520 57600 76032 1536 15X24 55-66 8836 44180 22090 66270 1178 17X24 51-56 12240 ' 80784 1632 16X24 58-66 9533 47665 71497 1271 51-56 13722 68611 219o5 90566 1829 17X24 60-66 10404 52020 26010 78030 1387 19X24 54-60 14440 72200 2310+ 95304 1925 18X24 61-66 11472 57360 28680 86640 1529 1 Mogul Engines. Consolidation Engines. in. in. lbs. lbs. lbs. lbs. in. in. lbs. lbs. lbs. lbs. 11x16 35-40 4978 24891 4978 663 14x16 36-38 1 7840 39200 ' 6272 45472 1045 12 > 18 36-41 6480 32400 6480 38880 864 15xlS 30-38 10125 50625 8100 58725 1350 13X18 37-42 7399 36997 7399 44396 986 20X24 48-50 18000 90000 14 400 104400 2400 14X20 39-43 9046 45230 9046 54270 1206 22x24 50 52 i 20909 :::; 121271 2787 15. 22 42-47 10607 53035 looo; 03,1 12 1414 16X24 45-51 12288 61440 12288 73738 1638 17x24 49 54 127 39 63697 12739 76436- 1698 18X24 51-50 13722 68611 13722 82333 1829 19X24 54 00 14440 72200 14440 86640 1925 858 LOCOMOTIVES. Leading American Types of Locomotive for Freight and Passenger Service. 1. The eight-wheel or "American" passenger type, having four coupled driving-wheels and a four-wheeled truck in front. 2. The •' ten- wheel " type, for mixed traffic, having six coupled drivers and a leading four-wheel truck. 3. The " Mogul " freight type, having six coupled driving-wheels and a pony or two-wheel truck in front. 4. The " Consolidation " type, for heavy freight service, having eight coupled driving-wheels and a ponj' truck in front. Besides these there is a great variety of types for special conditions of service, as four-wheel and six- wheel switching-engines, without trucks; the Forney type used on elevated railroads, with four coupled wheels under the engine and a four-wheeled rear truck carrying the water-tank and fuel; locomotives for local and suburban service with four coupled driving-wheels, with a two-wheel truck front and rear, or a two-wheel truck front and a four-wheel truck rear, etc. "Decapod 1 ' engines for heavy freight service have ten coupled driving-wheels and a two-wheel truck in front. Steam-distribution for High-speed Locomotive©. (C. H. Quereau, Eng'g News, March 8, 1894.) Balanced Valves.— Mr. Philip "Wallis, in 1886, when Engineer of Tests for the C, B. & Q. R. R., reported that while 6 H.P. was required to work un- balanced valves at 40 miles per hour, for the balanced valves 2.2 H.P. only was necessary. Effect of Speed on Average Cylinder-pressure.— Assume that a locomotive has a train in motion, the reverse lever is placed in the running notch, and the track is level ; by what is the maximum speed limited ? The resistance of the train and the load increase, and the power of the locomotive de- creases with increasing speed till the resistance and power are equal, when the speed becomes uniform. The power of the engine depends on the average pressure in the cylinders. Even though the cut-off and boiler- pressure remain the same, this pressure decreases as the speed increases; because of the higher piston-speed and more rapid valve-travel the steam has a shorter time in which to enter the cylinders at the higher speed. The following table, from indicator-cards taken from a locomotive at varying speeds, shows the decrease of average pressure with increasing speed: Miles per hour 46 51 51 53 54 57 60 66 Speed, revolutions 224 248 248 258 263 277 292 321 Average pressure per sq. in.: Actual 51.5 44.0 47.3 43.0 41.3 42.5 37.3 36.3 Circulated 46.5 46.5 44.7 43.8 41.6 39.5 35.9 The "average pressure calculated" was figured on the assumption that the mean effective pressure would decrease in the same ratio that the speed increased. The main difference lies in the higher steam-line at the lower speeds, and consequent higher expansion-line, showing that more steam entered tne cylinder. The back pressure and compression-lines agree quite closely for all the cards, though they are slightly better for the slower speeds. That the difference is not greater may safely be attributed to the large exhaust-ports, passages, and exhaust tip, which is 5 in. diameter. These are matters of great importance for high speeds. Boiler-pressure.— The increase of train resistance with increased speed is not as the square of the velocity, as is commonly supposed. It is more likely that it increases as the speed after about 20 miles an hour is reached. As- suming that the latter is true, and that an average of 50 lbs. per square inch is the greatest that can be realized in the cylinders of a given engine at 40 miles an hour, and that this pressure furnishes just sufficient power to keep the train at this speed, it follows that, to increase the speed to 50 miles, the mean effective pressure must be increased in the same proportion. To in- crease the capacity for speed of any locomotive its power must be increased, and at least by as much as the speed is to be increased. One way to accom- plish this is to increase the boiler-pressure. That this is generally realized, is shown by the increase in boiler-pressure in the last ten years. For twenty- three single-expansion locomotives described in the railway journals this year the steam-pressures are as follows; 3, 160 lbs.; 4, 165 lbs ■ 2, 170 lbs, ; 13, 180 lbs.; 1, 190 lbs, SOME LARGE AMERICAN LOCOMOTIVES, 1893. 859 Valve-travel. — An increased average cylinder-pressure may also be obtained by increasing the valve-travel without raising the boiler-pressure, and better results will be obtained by increasing both. The longer travel gives a higher steam-pressure in the cylinders, a later exhaust-opening, later exhaust-closure, and a larger exhaust-opening— all necessary for high speeds and economy. I believe that a 20-in. port and 6^-in. (or even 7-in.) travel could be successfully used for high-speed engines, and that frequently by so doing the cylinders could be economically reduced and the counter- balance lightened. Or, better still, the diameter of the drivers increased, securing lighter counterbalance and better steam-distribution. Size of Drivers.— Economy will increase with increasing diameter of drivers, provided the work at average speed does not necessitate a cut-off longer than one fourth the stroke. The piston-speed of a locomotive with 62-in. drivers at 55 miles per hour is the same as that of one with 68-in. drivers at 61 miles per hour. Steam-ports.— The length of steam-ports ranges from 15 in. to 23 in., and lias considerable influence on the power, speed, and economy of the loco- motive. In cards from similar engines the steam-line of the card from the engine with 23-in. ports is considerably nearer boiler-pressure than that of the card from the engine with 17J4-in. ports. That the higher steam-line is due to the greater length of steam-port there is little room for doubt. The 23-in. port produced 531 H.P. in an 18^-m. cylinder at a cost of 23.5 lbs. of indicated water per I. H.P. per hour. The lTJ^-in. port, 424 H.P., at the rate of 22 9 lbs. of water, in a 19-in. cylinder. Allen Valves. — There is considerable difference of opinion as to the advan- tage of the Allen pnrted-valve (See Eng. Netvs, July 6, 1893.) Speed of Railway Trains.— In 1834 the average speed of trains on the Liverpool and Manchester Railway was twenty miles an hour; in 1838 it was twenty-five miles an hour. But by 1840 tnere'were engines on the Great Western Railway capable of running fifty miles an hour with a train, and eighty miles an hour without. A speed of 86 miles per hour was made in England with the T. W. Worsdell compound locomotive. The total weight of the engine, tender, and train was 695,000 lbs.; indicator-cards were taken showing 1068.6 H.P. on the level. At a speed of 75 miles per hour on a level, and the same train, the indicator-cards showed 1040 H.P. developed. (Trans. A. S. M. E., vol. xiii., 363.), The limitation to the increase of speed of heavy locomotives seems at present to be the difficulty of counterbalancing the reciprocating parts. The unbalanced vertical component of the reciprocating parts causes the pres- sure of the driver on the rail to vary with every revolution. Whenever the speed is high, it is of considerable magnitude, and its change in direction is so rapid that the resulting effect upon the rail is not inappropriately called a "hammer blow. 1 ' Heavy rails have been kinked, and bridges have been shaken to their fall under the action of heavily balanced drivers revolving at high speeds. The means by which the evil is to be overcome has not yet been made clear. See paper by W. F. M. Goss, Trans. A. S. M. E.. vol. xvi. Engine No. 999 of the New York Central Railroad ran a mile in 32 seconds, equal to 112 miles per hour, May 11, 1893. Speed in miles | _ circum. of driving-wheels in in. x no. of rev, per min. x 60 per hour f - 63,360 = diam, of driving-wheels in in. X no. of rev. per min. X .003 (approximate, giving result 8/10 of 1 per cent too great). DIMENSIONS OF SOME LARGE AMERICAN LOCOMOTIVES, 1893. The four locomotives described below were exhibited at the Chicago Exposition in 1893. The dimensions are from Engineering News, June, 1893. The first, or Decapod engine, has ten-coupled driving-wheels. It is one of the heaviest and most powerful engines ever built for freight service. The Philadelphia & Reading engine is a new type for passenger service, with four- coupled drivers. The Rhode Island engin-e has six drivers, with a 4-wheel leading truck and a 2-wheel trailing truck. These three engines have all compound cylinders. The fourth is a simple engine, of the standard Ameri- can 8-wheel type, 4 driving-wheels, and a 4-wheel truck in front. This engine holds the world's record for speed (1893) for short distances, having run a mile in 32 seconds. 860 LOCOMOTIVES. Baldwin. N. Y., L. E. & W. R. R. Decapod Freight. Baldwin. Phila. & Read. R. R Express Passenger. Rhode Id. Loeomoti 1 e Works. Heavy Express. N. Y. C. & H. R. R. Empire State Express, No. 999. Running-gear : Driving-wheels, diam Truck " " Journals, driving-axles... " truck- " ... " tender- " , .. Wheel-base : Driving Total engine " tender " engine and tender. . . Wt. in working-order: On drivers On truck-wheels Engine, total. Tender " Engine and tender, loaded Cylinders : h.p. (2) lp. (2) Distance centre to centre. Piston-rod, diam — Connecting-rod, length... Steam-ports Exhaust-ports Slide-valves, out. lap, h.p. " " out. lap, l.p.. " " in. lap, h.p... " " in. lap, l.p. . . " " max. travel . " " lead, h.p lead, l.p 9 x 10 in. 5 xlO " 4^x 9 " 8 " 4 " 170,000 lbs. 29,500 " 192,500 " 117,500 " 310,000 '.« 16x28 in. 27x28 " 7 ft. 5" 4 in. 9' 8 7/16'" 28^ x 2 in. 28^x8 " %in. Boiler— Type Diam. of barrel inside. . Thickness of barrel-plates Height from rail to centre line Length of smoke-box Working steam-pressure.. Firebox— type Length inside Width " Depth at front Thickness of side plates . . " " back plate. . . Thickness of crown-sheet. " " tube " Grate-area Stay-bolts, diam., \% in. Tubes— iron Pitch Diam., outside Length betw'n tube-plates Heating-surface : Tubes, exterior Fire-box Miscellaneous : Exhaust-nozzle, diam Sniokestack,smarst diam. " height from rail to top 1/16 in. 5/16 " Straight 5 ft. 2}4 in. 8 ft. in. 5 " 7% " 180 lbs. Wootten 10' 11 9/16' 8 ft. 2y 8 in 4 " 6 " 5/16 in. 5/16 '• Vs " M " 89.6 sq. ft. pitch,4*4 in 354 234 in. 2 " 11 ft. 11 in 2,208.8 ft. 234.3 " 5 in. 1 ft. 6 " 15 " 6^ " 6 ft. 6 in. 4 " " 8V£xl2in. 6'^xl0 " 4J/ 2 x 8 " 6 ft. 10 in. 23 " 4 " 16 " " 47 " 3 " 82,700 lbs. 47,000 " 129,700 " 80,573 " 210,273 " 13x24 in. 22x24 " 7 ft. 4}^ in. 3^ in. 8 ft, 0^ in. 24x1^ in. 24x4J^ " %in. . %" (neg.) y H in. None 5 in. Straight 4 ft. 8J4 in. % i". 180 lbs. Wootten 9 ft. 6 in. 8 " oy 8 " 3 " 234 " 5/16 in. 5/16 " 5/16 * 76.8 sq. ft. '324*'" 2 1/16 in. 11/2 in. 10 ft. in. 1,262 sq. ft, 173 " " 5^ in. 1 ft. 6 in. 14 ft. 0M in. x- 834 in. 5^x10 " 41/4 x 8 " 15 " " 50 ',' 6% " 88,500 lbs. 54,500 " 143,000 " 75,000 " 218,000 " one 21 x26 one 31 x 26 7 ft. 1 iu. 3'^ in. 10 ft. 3)4 in. 11^x20 and xl2^in. 6^x10 " 4% x 8 " 8 ft. 6 in. 23 " 11 " 15 ft. 2}4 " 47 " m " 84,000 lbs. 40,000 " 121,000 " 80,000 " 204,000 " 19x24 in. 8 ft. l^in. 1^x18 in. 234x18 " lin. 1/10 in. "5^'in." Wagon top 5 ft. 2 in. ft. 11 in. '.* 1 " 200 lbs. Radial stay 10 ft. in. 2 '• 9% » 6 " 1034 " 5/16 in. 28 sq. ft. 4 in. 272 12 ft. 8% in Wagon top 4 ft. 9 in. 9/16 in. 7 ft. 11^ in. 1 " 8 r 190 lbs. Buchanan 9 ft. 15% in. 3 " 1% " 6 " iy 4 " 5/16 in. 5/16 il 1 ft. 3 in. 15 »* 2 " 12 ft. in. 1,697 sq.ft. 3^ in. 1 ft. 3^ in. DIMENSIONS OF AMERICAN LOCOMOTIVES- 861 •noysgq -py joj Q\qv J -inMy'-uiSi^v 01 J9Avod-.opin -\£q jo op^a T^TjicococO'rrcoo: B -J ".O ?? ' fMHt-WNOOiOOlOii. - > ?tina)o >30 i- co o> to t~ o» *-< ooooooooooooooooooooooooooooo S9qnx joimaiq | •in puB -5 j 'twqnj, * © -^r© © '«N(Dooxooqcxjj«oicq to iom i-Tso so '. too* i-' >o oo'co'co'© ic'co'cd'o x' i~ C- © ~> "( ?J CO TC ".> CC ~! i- Tf O? if! T? ?? ?» ?> - = o o ioc oTcd'o'c WWOfWWO SSgggSSS » — ©c f)«nn«« -00(-fOO OOOTPOJO oi 'spgqM-Sni -AUQ JO "111131(1 2^2 COOSOO-tOiOC jiioj^ jo -ojsr I jo)«Tfoion •S.I9AUQ JO -OR J< S^S^fQ^Z^Q. ^5tf5:| da^a- OutHO^O?0?SpHO^OO!^fflOW ft ^ Ph Kh" C g S 862 LOCOMOTIVES. Dimensions of Some American Locomotives.— The table on page 861 is condensed from one given by D. L. Barnes, in his paper on " Distinctive Features and Advantages of American Locomotive Practice,' 1 Trans. A.S.C.E., 1893. The formula from which column marked "Ratio of cylinder-power to weight available for adhesion" is calculated as follows: 2 X cylinder area X boiler-pressure x stroke Weight on drivers X diameter of driving-wheel' (Ratio of cylinder-power of compound engines cannot be compared with that of the single-expansion engines.) Where the boiler-pressure could not be determined from the description of the locomotives, as given by the builders and operators of the locomotives, it has been assumed to be 160 lbs. per sq. in. above the atmosphere. For compound locomotives the figures in the last column of ratios are based on the capacity of the low-pressure cylinders only, the volume of the high-pressure being omitted. This has been done for the purpose of com- parison, and because there is no accurate simple way of comparing the cylinder-power of single-expansion and compound locomotives. Dimensions of Standard Locomotives on tlie N. Y. C. & II. It. R. and Penna. It. It., 1882 and 1893. C. H. Quereau, Eng'g News March 8, 189 4. N. Y. C. & H. R. R. Pennsylvania R. R. Through Passenger. Through Freight. Through Passenger. Through Freight. 1882. 1893. 1882. 1893. 1882. 1893. 1882. 1893. Grate surface, sq. ft " Heating surface, sq. ft.. 17.87 1353 50 70 150 17X24 hy A 1/16 % 15« Am. 27.3 1821 58 78,86 180 19X24 1/16 1 18 1U Am. 17.87 1353 50 64 150 17X24 5M 1/16 % 1/162 15V6 m Am. 29.8 1763 58 67 160 19X26 1/16 % 3/32Z 18 Mog. 17.6 1057 50 62 125 17X24 5 1/16 % 16 m Am. 33.2 1583 57 78 175 I8ix2± V&cl Am. 23. 1260 54 50 125 -.20X24 5 Va U 1/221 16 Cons. 31.5 1498 60 Driver, diam., in Steam- pressure, lbs. .. Cylin., diam. and stroke. Valve-travel, ins . . . Lead at full gear, ins 50 140 20X24 5 1/16 1/32Z 16 1% Cons. Inside lap or clearance. . Sream-ports, length width Type of engine Two-cylinder Compound. Single-expansion. Revolu- tions. Speed, miles per hour. Water per I.H.P. per hour. Revolu- tions. Miles per Hour. Water. 100 to 150 150 " 200 200 " 250 250 " 275 21 to 31 31 " 41 41 " 51 51 " 56 18.33 lbs. 18.9 " 19.7 " 21.4 " 151 219 253 307 321 31 45 52 63 66 21.70 20.91 20.52 20.23 20.01 Indicated Water Consumption of Single and Compound Locomotive Engines at Varying Speeds. C . H. Quereau, Eng'g News, March 8, 1894. It appears that the compound engine is the more economical at low speeds, the economy decreasing as the speed increases, and that the single engine increases in economy with increase of speed within ordinary limits, becom- ing more economical than the compound at speeds of more than 50 miles per hour. The C, B. & Q. two-cylinder compound, which was about 30$ less eco- nomical than simple engines of the same class when tested in passenger service, has since been shown to be 15$ more economical in freight service ADVANTAGES OF COMPOUNDING, 863 than the best single-expansion engine, and 29#~more economical than the average record of 40 simple engines of the some class on the same division. Indicator-tests of a Locomotive at High Speed. (Locomo- tive Eng'g, June, 189:1)— Cards were taken by Mr. Angus Sinclair on the locomotive drawing the Empire State Express. Results of Indicator-diagrams. Card No. 3VS. Miles per hour. I.H.P. Card No 160 37.1 648.3 V 260 60.8 728 8 190 44 551 9 250 58 891 10 260 60 960 11 Revs. -'- LH.P. Miles, per hour. 304 70.5 977 296 68.6 972 300 69.6 1,045 304 70.5 1,059 340 78.9 1,120 12 310 71.9 1,026 The locomotive was of the eight-wheel type, built by the Schenectady Locomotive Works, with 19 X 24 in. cylinders, 78-in. drivers, and a large boiler and fire-box. Details of important dimensions are as follows : Heating-surface of fire-box, 150.8 sq. ft.: of tubes, 1670.7 sq. ft.; of boiler, 1821.5 sq ft. Grate area, 27.3 sq. ft. Fire-box: length, 8 ft.; width, 3 ft 4% in. Tubes, 268; outside diameter, 2 in. Ports: steam, 18 X % in.; exhaust, 18 X 2% in. Valve-travel, 5^ in. Outside lap, 1 in.; inside lap, 1/64 in. Journals: driving-axle, 8l£ X 10^2 in.; truck-axle, 6 x 10 in. The train consisted of four coaches, weighing, with estimated load, 340,000 lbs. The locomotive and tender weighed in working order 200,000 lbs,, making the total weight of the train about 270 tons. During the time that the engine was first lifting the train into speed diagram No. 1 was taken. It shows a mean cylinder-pressure of 59 lbs. According to this, the power exerted on the rails to move the train is 6553 lbs., or 24 lbs. per ton. The speed is 37 miles an hour. When a speed of nearly 60 miles an hour was reached the average cylinder-pressure is 40.7 lbs., representing a total traction force of 4520 lbs., without making deductions for internal friction. If we deduct 10^ for friction, it leaves 15 lbs. per ton to keep the train going at the speed named. Cards 6, 7, and 8 represent the work of keeping the train running 70 miles an hour. They were taken three miles apart, when the speed was almost uniform. The average cylinder-pressure for the three cards is 47.6 lbs. Deducting \0% again for friction, this leaves 17.6 lbs. per ton as the pow r er exerted in keeping the train up to a velocity of 70 miles. Throughout the trip 7 lbs. of water were evaporated per lb. of coal. The work of pulling the train from New York to Albany was done on a coal con- sumption of about 3% lbs. per H.P. per hour. The highest power recorded was at the rate of 1120 H.P. Locomotive-testing Apparatus at the Iiaboratory of Purdue University. (W. F. M. Goss, Trans. A. S. M. E., vol. xiv. 826 )— The locomotive is mounted with its drivers upon supporting wheels which are carried by shafts turning in fixed bearings, thus allowing the engine to be run without changing its position as a whole. Load is supplied by four friction-brakes fitted to the supporting shafts and offering resistance to the turning of the supporting wheels. Traction is measured by a dynamometer attached to the draw-bar. The boiler is fired in the usual way, and an exhaust-blower above the engine, but not in pipe connection with it, carries off all that may be given out at the stack. A Standard Method of Conducting Locomotive-tests is given in a report by a Committee of the A. S. M. E. in vol. xiv. of the Transactions, page 1312. '"Waste of Fuel in Locomotives.— In American practice economy of fuel is necessarily sacrificed to obtain greater economy due to heavy train-loads. D. L. Barnes, in Eng. Mag., June, 1894, gives a diagram showing the reduction of efficiency of boilers due to high rates of combustion, from which the following figures are taken: Lbs. of coal per sq. ft. of grate per hour 12 40 80 120 160 200 Per cent efficiency of boiler 80 75 67 59 51 43 A rate of 12 lbs. is given as representing stationary-boiler practice, 40 lbs. is English locomotive practice, 120 lbs. average American, and 200 lbs. max- imum American, locomotive practice. Advantages of Compounding.— Report of a Committee of the American Railway Master Mechanics' Association on Compound Locomotives (Am. Mach., July 3, 1890) gives the following summary of the advantages gained by compounding: (a) It has achieved a saving in the fuel burnt averaging 18$ at reasonable boiler pressures, with encouraging possibilities 864 LOCOMOTIVES. of further improvement in pressure and in fuel and water economy. (6) It has lessened the amount of water (dead weight) to be hauled, so that (c) the tender and its load are materially reduced in weight, (d) It has increased the possibilities of speed far beyond 60 miles per hour, without unduly straining the motion, frames, axles, or axle-boxes of the engine, (e) It has increased the haulage-power at full speed, or, in other words, has increased the continuous H.P. developed, per given weight of engine and boiler. (/) In some classes has increased the starting-power, (g) It has materially lessened the slide-valve friction per H.P. developed, (h) It has equalized or distrib- uted the turning force on the crank-pin, over a longer portion of its path, which, of course, tends to lengthen the repair life of the engine, (i) In the two-cylinder type it has decreased the oil consumption, and has even done so in the Woolf four-cylinder engine. (,;') Its smoother and steadier draught on the fire is favorable to the combustion of all kinds of soft coal; and the sparks thrown being smaller and less in number, it lessens the risk to prop- erty from destruction by fire, (k) These advantages and economies are gained without having to improve the man handling the engine, less being left to his discretion (or careless indifference) than in the simple engine. (I) Valve-motion, of every locomotive type, can be used in its best working and most effective position, (m) A wider elasticity in locomotive design is per- mitted; as, if desired, side-rods can be dispensed with, or articulated engines of 100 tons weight, with independent trucks, used for sharp curves on moun- tain service, as suggested by Mallet and Brunner. Of 2? compound locomotives in use on the Phila. and Reading Railroad (in 1892), 12 are in use on heavy mountain grades, and are designed to be the equivalent of 22 X 24 in. simple consolidations; 10 are in somewhat lighter service and correspond to 20 x 24 in. consolidations; 5 are in fast passenger service. The monthly coal record shows: Class of Engine. No. «£"£-' Mountain locomotives 12 25fctoS0% Heavy freight service 10 12% to 11% Fast passenger 5 % to 11$ (Report of Com. A. R. M. M. Assn. 1892.) For a description of the various types of compound locomotive, with discussion of their relative merits, see paper by A. Von Borries, of Germany, The Development of the Compound Locomotive, Trans. A. S. M. E. 1893, vol. xiv., p. 1172. Counterbalancing Iioconiotives.— The following rules, adopted by different locomotive- builders, are quoted in a paper by Prof. Lanza (trans. A. S. M. E., x. 302): A. " For the main drivers, place opposite the crank-pin a weight equal to one half the weight of the back end of the connecting-rod plus one half the weight of the front end of the connecting-rod, piston, piston-rod, and cross- head. For balancing the coupled wheels, place a weight opposite the crank- pin equal to one half the parallel rod plus one half of the weights of the front end of the main-rod, piston, piston-rod, and cross-head. The centres of gravity of the above weights must be at the same distance from the axles as the crank-pin." B. The rule given by D. K. Clark : " Find the separate revolving weights of crank-piu boss, coupling-rods, and connecting-rods for each wheel, also the reciprocating weight of the piston and appendages, and one half the connecting-rod, divide the reciprocating weight equally between each wheei and add the part so allotted to the revolving weight on each wheel: the sums thus obtained are the weights to be placed opposite the crank-pin, and at the same distance from the axis. To find the counterweight to be used when the distance of its centre of gravity is known, multiply the above weight by the length of the crank in inches and divide by the given dis- tance. 11 This rule differs from the preceding in that the same weight is placed in each wheel. C. " W= ^ — — , in which S = one half the stroke, G = distance from centre of wheel to centre of gravity in counterbalance, w -■ weight at crank-pin to be balanced, W = weight in counterbalance, / = coefficient of friction so called, — 5 in. ordinary practice. The reciprocating weight is found by adding together the weights of the piston, piston-rod, cross-head, and one half of the main rod. The revolving weight for the main wheel is found by adding together the weights of the crank-pin hub, crank-pin, one PETROLEUM-BURNING LOCOMOTIVES. 865 half of the main rod, and one half of each parallel-rod connecting to this wheel; to this add the reciprocating weight divided by the number of wheels. The revolving weight for the remainder of the wheels is found in the same manner as for the main wheel, except one half of the main rod is not added. The weight of the crank-pin hub and the counterbalance does not include the weight of the spokes, but of the metal inclosing them. This calculation is based for one cylinder and its corresponding wheels. 1 ' D. "Ascertain as nearly as possible the weights of crank-pin, additional weight of wheel boss for the same, add side rod, and main connections, piston-rod and head, with cross-head on one side: the sum of these multi- plied by the distance in inches of the centre of the crank-pin from the centre of the wheel, and divided by the distance from the centre of the wheel to the common centre of gravity of the counterweights, is taken for the total counterweight for that side of the locomotive which is to be divided among the wheels on that side." E. " Balance the wheels of the locomotive with a weight equal to the weights of crank-pin, crank-pin hub, main and parallel rods, brasses, etc., plus two thirds of the weight of the reciprocating parts (cross-head, piston and rod and packing)'" F. '• Balance the weights of the revolving parts which are attached to each wheel with exactness, and divide equally two thirds of the weights of the reciprocating parts between all the wheels. One half of the main rod is computed as reciprocating, and the other as revolving weight.' 1 See also articles on Counterbalancing Locomotives, in R. R. & Eng. Jour., March and April, 1890, and a paper by W. F. M. Goss, in Trans. A. S. M. E., vol xvi. Maximum Safe Load for Steel Tires on Steel Rails. (A. S. M. E., vii., p. 786.)— Mr. Chanute's experiments led to the deduction that 12,000 lbs. should be the limit of load for any one driving-wheel. Mr. Angus Sinclair objects to Mr. Chanute's figure of 12,000 lbs., and says that a locomotive tire which has a light load on it is more injurious to the rail than one which has a heavy load. In English practice 8 and 10 tons are safely used. Mr. Obeiiin Smith has used steel castings for cam-rollers 4 in. diam. and 3 in. face, which stood well under loads of from 10,000 to 20,000 lbs. Mr. C. Shaler Smith proposed a formula for the rolls of a pivot- bridge which may be reduced to the form : Load = 1760 x face X Vdiam., all in lbs. and inches. See dimensions of some large American locomotives on pages 860 and 861. On the " Decapod " the load on each driving-wheel is 17,000 lbs., and on "No. 999," 21.000 lbs. Narrow-gauge Railways in Manufacturing Works.— A tramway of 18 inches gauge, several miles in length, is in the works of tb,e Lancashire and Yorkshire Railway. Curves of 13 feet radius are used. The locomotives used have the following dimensions (Proc. Inst. M. E.. July, 1888): The cylinders were 5 in. diameter with 6 in. stroke, and 2 ft. 3*4 in. centre to centre. The wheels were 16J4 in- diameter, the wheel-base 2 ft. 9 in. ; the frame 7 ft. 4J4 in. long, and the extreme width of the engine 3 feet. The boiler, of steel, 2 ft. 3 in. outside diameter and 2 ft. long between tube plates, containing 55 tubes of 1% in. outside diameter; the fire-box, of iron and cylindrical, 2 ft. 3 in. long and 17 in. inside diameter. The heating- surface 10*42 sq. ft. in the fire-box and 36.12 in the tubes, total 46.54 sq. ft.; the grate-area, 1.78 sq. ft.; capacity of tank, 26J/£ gallons; working-pressure, 170 lbs. per sq. in.; tractive power, say, 1412 lbs., or 9.22 lbs. per lb. of effec- tive pressure per sq. in. on the piston. Weight, when empty, 2.80 tons; When full and in working order, 3.19 tons. For description of a system of narrow-gauge railways for manufactories, see circular of the C. W. Hunt Co., New York. Light Locomotives.— For dimensions of light ocomotives used for. mining, etc., and for much valuable information concerning them, see cata- logue of H K. Porter & Co., Pittsburgh. Petroleum-burning Locomotives. (From Clark's Steam-en- gine.)— The combustion of petroleum refuse in locomotives has been success f idly practised by Mr. Thos. Urquhart. on the Grazi and Tsaritsin Railway, Southeast Russia. Since November, 1884, the whole stock of 143 locomotives under his superintendence has been fired with petroleum refuse. The oil is injected from a nozzle through a tubular opening in the back of the fire-box, by means of a jet of steam, with an induced current of air. A brickwork cavity or "regenerative or accumulative combustion-cham- ber" is formed in the fire-box, into which the combined current breaks as 866 LOCOMOTIVES. spray against the rugged brickwork slope. In this arrangement the brick- work is maintained at a white heat, and combustion is complete and smoke- less. The form, mass, and dimensions of the brickwork are the most im- portant elements in such a combination. Compressed air was tried instead of steam for injection, but no appreciable reduction in consumption of fuel was noticed. The heating-power of petroleum refuse is given as 19,832 heat-units, equivalent to the evaporation of 20.53 lbs. of water from and at 212° F., or to 17.1 lbs. at 8^ atmospheres, or 125 lbs. per sq. in., effective pressure. The highest evaporative duty was 14 lbs. of water under 8J^ atmospheres per lb. of the fuel, or nearly 82$ efficiency. There is no probability of any extensive use of petroleum as fuel for loco- motives in the United States, on account of the unlimited supply of coal and the comparatively limited supply of petroleum. Fireless Ijocomotive.— The principle of the Francq locomotive is that it depends for the supply of steam on its spontaneous generation from a body of heated water in a reservoir. As steam is generated and drawn off the pressure falls; but by providing a sufficiently large volume of water heated to a high temperature, at a pressure correspondingly high, a margin of surplus pressure may be secured, and means may thus be provided for supplying the required quantity of steam for the trip. The fireless locomotive designed for the service of the Metropolitan Rail- way of Paris has a cylindrical reservoir having segmental ends, about 5 ft. 7 in. in diameter, 26J4 ft. in length, with a capacity of about 620 cubic feet. Four fifths of the capacity is occupied by water, which is heated by the aid of a powerful jet of steam supplied from stationary boilers. The water is heated until equilibrium is established between the boilers and the reser- voir. The temperature is raised to about 390° F., corresponding to 225 lbs. per sq. in. The steam from the reservoir is passed through a reducing- valve, by which the steam is reduced to the required pressure. It is then passed through a tubular superheater situated within the receiver at the upper part, and thence through the ordinary regulator to the cylinders. The exhaust-steam is expanded to a low pressure, in order to obviate noise of escape. In certain cases the exhaust-steam is condensed in closed vessels, which are only in part filled with water. In the upper free space a pipe is placed, into which the steam is exhausted. Within this pipe another pipe is fixed, perforated, from which cold water is projected into the sur- rounding steam, so as to effect the condensation as completely as may be. The heated water falls on an inclined plane, and flows off without mixing with the cold water. The condensing water is circulated by means of a centrifugal pump driven by a small three -cylinder engine. In working off the steam from a pressure of 225 lbs. to 67 lbs., 530 cubic feet of water at 390° F.as sufficient for the traction of the trains, for working the circulating-pump for the condensers, for the brakes, and for electric- lighting of the train. At the stations the locomotive takes from 2200 to 3300 lbs. of steam — nearly the same as the weight of steam consumed during the run between two consecutive charging stations. There is 210 cubic feet of condensing water. Taking the initial temperature at 60° F., the tempera- ture rises to about 180° F. after the longest runs underground. The locomotive has ten wheels, on a base 24 ft. long, of which six are coupled, 4]4 ft- in diameter. The extreme wheels are on radial axles. The cylinders are 23^j in. in diameter, with a stroke of 23^ in. The engine weighs, in working order, 53 tons, of which 36 tons are on the coupled wheels. The speed varies from 15 miles to 25 miles per hour. The trains weigh about 140 tons. Compressed-air Iiocomotives.— For an account of the Mekarski system of compressed-air locomotives see page 509, ante. SHAFTIKG. 867 SHAFTING. (See also Torsional Strength; also Shafts op Steam-engines.) For diameters of shafts to resist torsional strains only, Molesworth gives 3/pf d = A/ — , in which d = diameter in inches, P = twisting force in pounds applied at the end of a lever-arm whose length is I in inches, K = a coeffi- cient whose values are, for cast iron 1500, wrought iron 1700, cast steel 3200, gun-bronze 460, brass 425, copper 380, tin 220, lead 170. The value given for cast steel probably applies only to high-carbon steel. Thurston gives: For head shafts well supported against springing: For line shafting, m hangers 8 ft. apart: For transmission sim- ply, no pulleys: H.P. = d*R . 125' d a R 3/125 H.P. -V-R-' for iron; -*, for cold-rolled iron. H. P . = *^ = ;/?^P-, for iron; 90 y R , 3 /55 H.P for cold-rolled iron. ?:R = i | tf= |/^, for)ron; -; d =f 3 -^. for cold-rolled iron. H.P. = horse-power transmitted, d = diameter of shaft in inches, R = rev- olutions per minute. .-,. , 3 /100 H.P. J. B. Francis gives for turned-iron shafting d = ju — — — . Jones and Laughlins give the same formulae as Prof. Thurston, with the following exceptions: For line shafting, hangers 8 ft. apart: c oW -.-onedi,-on,H.P. = ^, 2.7 3.6 4.5 5.6 6.7 7.9 9.0 10 11 12 13 m 4.3 5.6 7.1 8.9 10.6 12.4 14.2 16 18 19 21 2 6.4 8.5 10.7 13 16 19 21 24 26 29 32 2^ 9 12 15 19 23 26 30 34 38 42 46 2^2 12 17 21 26 31 36 41 47 52 57 62 2% 16 22 27 35 41 48 55 62 70 76 82 3 21 29 36 45 54 63 72 81 90 98 108 3M 27 36 45 57 68 80 91 103 114 126 136 3J^ 34 45 57 71 86 100 114 129 142 157 172 3% 42 56 70 87 105 123 140 158 174 193 210 4 51 69 85 106 128 149 170 192 212 244 256 4^ 73 97 121 151 182 212 243 273 302 333 364 HORSE-POWER AT DIFFERENT SPEEDS. As Second Movers or Line-shafting, Bearings 8 ft. apart. Formula : H.P. = d s R -f- £0. s £ Number of Revolutions per Minute. cS O Jj 5 is^:«+ Jos ^g^ ^ cilia c*c-§^ M Oi COCO CO C4C^O?COff o« w ^ C4 t u = number of arms, D = diameter of pulley, S = thickness of belt. / = hiekness of rim at edge, T = thickness in middle, B = width of rim, /3 = viclrh of belt, h = breadth of arm at hub, h 1 = breadth of arm at rim, e = hickness of arm at hub e x = thickness of arm at rim, c = amount of crown- ng; dimensions in inches. Unwin. Reuleaux. 5 = widthof rim..., 9/8 ((8 f 0.4) ,,,9/8/3 to 5/4/3 t = thickness at edge of rim . T= " " middle of i-in: m . 75" + . 005D -[ (tl j/^ ^y^ ' ) 2t + c .." For single , / BD belts = .6337 j/ ^ YaT B D_ For double \ / BD 4 4 20?i \ = breadth of arm at hub.. \ For double . / - / belts = .798y li= " " " " rim %h O.SJi : thickness of arm at hub. 0.4ft. 0.5/i " " " rim 0.47i x 0.5/i x number of arms, for a} „ , BD ,,/► D \ single set, V'"; 3 + T 5 V A 5X 2b) t i™~fi, nf v,„k i not ^ ess than 2.5S, \ Bfor sin. -arm pulleys. L = length of hub -j is of fcen % ^ j-^ u ^^^.^ ,. M— thickness of metal in hub h to %h crowning of pulley 1/245 The number of arms is really arbitrary, and may be altered if necessary. (Unwin.) Pulleys with two or three sets of arms may be considered as two or three separate pulleys combined in one, except that the proportions of the arms should be 0.8 or 0.7 time that of single-arm pulleys. (Reuleaux.) Example.— Dimensions of a pulley 60" diam., 16" face, for double belt y^' thick. Solution by n h h x e e^ t T L M c Unwin 9 3.79 2.53 1.52 1.01 .65 1.97 10.7 3.8 .67 Reuleaux .... 4 5.0 4.0 2.5 2.0 1.25 16 5 The following proportions are given in an article in the Amer. Machinist, authority not stated: h = .0625Z) + .5 in., h ± = .041) + 3125 in., e = .025D -j- .2 in., e x = .016D -f- 125 in. These give for the above example: h — 4.25 in., /i t = 2.71 in., e = 1.7 in., 'i = 1.09 in. The section of the arms in all cases is taken as elliptical. The following solution for breadth of arm is proposed by the author: Assume a belt pull of 45 lbs. per inch of width of a single belt, that the whole strain is taken in equal proportions on one half of the arms, and that the arm is a beam loaded at one end and fixed at the other. We have the formula for a beam of elliptical section fP = .0982 — j— , in which P = the load, R = the modulus of rupture of the cast iron, b = breadth, d = depth, and I = length of the beam, and/ = factor of safety. Assume a modulus of rupture of 36.000 lbs., a factor of safety of 10, and an additional allow- ance for safety in taking / = i^ the diameter of the pulley instead of y%D :ss the radius of the hub. Take d — h, the breadth of the arm at the hub, and b = e = 0.4/i, the ^ ■ , TtT „ , ^„ „„ 45B nnn B 3535 X 0.4/i 3 thickness. We then have fP = 10 x ^ = 900- = — tk , whence n^-2 n y%D 3 /900BD " 3 /BD ,_. , . ,. „ ^ h — 4/ n ^r- = -633 i/ — , which is practically the same as the value y 6o6d)l y n reached by Unwin from a different set of assumptions. 8U PULLEYS. Convexity of Pulleys.— Authorities differ. Morin gives a rise equal to 1/10 of the face; Molesworth, 1/24; others from y H to 1/96. Seott A. Smith says the crown should not be over y 8 inch for a 24-inch face. Pulleys for shifting belts should be " straight, 11 that is, without crowning. CONE OR STEP PULLEYS. To find the diameters for the several steps of a pair of cone-pulleys: 1. Crossed Belts. — Let D and d be the diameters of two pulleys con- nected by a crossed belt, L = the distance between their centres, and /3 = the angle either half of the belt makes with a line joining the centres of the ■n- •*■« pulleys : then total length of belt z /3 = angle whose sine is - 2L i/--(^). The length of the belt is constant when D + d is constant; that is, in a pair of step- pulleys the belt tension will be uniform when the sum of the diameters of each opposite pair of steps is constant. Crossed belts are seldom used for cone-pulleys, on account of the friction between the rubbing parts of the belt. To design a pair of tapering speed-cones, so that the belt may fit equally tight in all positions : When the belt is crossed, use a pair of equal and similar cones tapering opposite ways. 2. Open Belts. — When the belt is uncrossed, use a pair of equal and similar conoids tapering opposite ways, and bulging in the middle, accord- ing to the following formula: Let L denote the distance between the axes of the conoids; R the radius of the larger end of each; r the radius of the smaller end; then the radius in the middle, r„, is found as follows: B-\-r , (R - r) i 6.28£ (Rankine.) If D = the diameter of equal steps of a pair of cone-pulleys, D and d — the diameters of unequal opposite steps, and L = distance between the D±d . (D - d^ axes, D = -^- + j^qZ- If a series of differences of radii of the steps, R — r, be assumed, then . R + r _ (R-r)* 2 ° 6.28L for each pair of steps - -, and the radii of each may be computed from their half sum and half difference, as follows : l + r R- r . . _ R-\-r _ R-r 2 + 2 ' ? ~ 2 2 * R-- A. J. Frith (Trans. A. S. M. E., x. 298) shows the following application of Rankine's method: If we had a set of cones to design, the extreme diame- ters of which, including thickness of belt, were 40" and 10", and the ratio desired 4, 3, 2, and 1, we would make a table as follows, L being 100": Trial Sum of D + d. Ratio. Trial Diameters. Values of (D - d)2 12.56Z. Amount to be Added. Corrected Values. D d D d 50 50 50 50 4 3 2 1 40 37.5 33.333 25 10 12.5 16.666 25 .7165 .4975 .2212 .0000 .0000 .2190 .4953 .7165 40 37.2190 33.8286 25.7165 10 12.2190 17.8286 25.7165 The above formulae are approximate, and they do not give satisfactory results when the difference of diameters of opposite steps is large and when the axes of the pulleys are near together, giving a large belt-angle. The following more accurate solution of the problem is given by C. A. Smith (Trans. A. S. M. E., x. 269) (Fig 152): Lay off the centre distance Cor EF, and draw the circles D x and d t equal to the first pair of pulleys, which are always previously determined by known conditions. Draw HI tangent to the circles Dj and d x . From B, midway between E and F, erect the perpendicular BG, making the length COKE Oil STEP PULLEYS. 875 BG = .314C. With G as a centre, draw a circle tangent to HI. Generally this circle will be outside of the belt-line, as in the cut, but when is short and the first pulleys D x and d x are large, it will fall on the inside of the belt- line. The belt-line of any other pair of pulleys must be tangent to the cir- cle G; hence any line, as JK or LM, drawn tangent to the circle G, will give G I /' f L \ 9? I J > i r"~' =~. \ s si / ^^-<__J4 / _X— ^S^> = 2 5*^ >\)i / i -K= \L---r^\ r^y 7^--^ p\\ K [^ff E J 1 ' i / f f F ^~^^Vi-M \ V ^ J J i B 1 i \ Jd z J J \ <*K_ y i \ / DjN,. / -~" Fig. 152. the diameters D 2 , d 2 or D 3 , d 3 of the pulleys drawn tangent to these lines from the centres E and F. The above method is to be used when the belt-angle A does not exceed 18°. When it is between 18° and 30° a slight modification is made. In that case, in addition to the point G, locate another point in on the line BG .298 C nbove B. Draw a tangent line to the circle G, making an angle of 18? to the Hue of centres EF, and from the point m draw an arc tangent to this tan- gent line. All belt-lines with angles greater than 18° are tangent to this arc. The following is the summary of Mr. Smith's mathematical method: A M angle in degrees between the centre line and the belt of any pair of pulleys; a = .314 for belt-angles less than 18°, and .298 for angles between 18° and 30°; B° ==. an angle depending on the velocity ratio; C = the centre distance of the two puileys; D, d &r diameters of the larger and smaller of the pair of pulleys; E° — an angle depending on B u ; L = .the length of the belt when drawn tight around the pulleys; r — D -i- d, or the velocity ratio (larger divided by smaller). 2C (3) Sin E° = sin 5° (cos A - r + 1 4- d\ 4aC (4) A = B° — E° when sin E° is positive ; = B° 4- E° when sin E° is negative ; (6) D = rd; (7) L = 2Ccos A -f .01745d[180 + (?■ - 1)(90 + ^)]. Equation (1) is used only once for any pair of cones to obtain the constant cos A, by the aid of tables of sines and cosines, for use in equation (3). 876 BELTING. Theory of Belts and Bands.— A pulley is driven by a belt by means of the friction between the surfaces in contact. Let 1\ be the tension on the driving side of the belt, '1\ the tension on the loose side; then S, — r l\, — T 2 , is the total friction between the band and the pulley, which is equal to the tractive or driving force. Let/ = the coefficient of friction, the ratio of the length of the arc of contact to the length of the radius, a — the angle of the arc of contact in degrees, e = the base of the Naperian logarithms = 2.71828, m = the modulus of the common logarithms = 0.434295. The following formulae are derived by calculus (Rankine's Mach'y & Millwork, p. 351 ; Carpenter's Exper. Eng'g, p. 173): T 2 ~ *>*; T* = T, e f0 ; t, - T, T, = r, - — - 1 e/» = 7\(i - e~f e ). T t - r 2 = 2\(1 - - e -/») = 3*i(l- - io- /0m ) = tjjx- 1Q -. 00758/a, T, = 10 .00758/a. Ti = r2Xlo .00758/a. 1C Ti 2 7 2 .00758/a • If the arc of contact between the band and the pulley expressed in turns and fractions of a turn = n, = 2n-?i; ef 9 = io 2 - 7288 /"; that is, e? 9 is the natural number corresponding to the common logarithm 2.7288/n. The value of the coefficient of friction / depends on the state and material of the rubbing surfaces. For leather belts on iron pulleys, Morin found f = .56 when dry, .36 when wet. .23 when greasy, and .15 when oily. In calcu- lating the proper mean tension for a belt, the smallest value, / — .15, is to be taken if there is a probability of the belt becoming wet with oil. The experiments of Henry R. Towne and Robert Briggs, however (Jour. Frank. Inst., 1868). show that such a state of lubrication is not of ordinary occur- rence; and that in designing machinery we may in most cases safely take f = 0.42. Reuleanx takes/ = 0.25. The following table shows the values of the coefficient 2.7288/, by which n is multiplied in the last equation, corre- sponding to different values of f; also the corresponding values of various ratios among the forces, when the arc of contact is half a circumference : f= 0.15 2.7588/ = 0.41 0.25 0.68 0.42 0.15 0.56 1.53 Lete = it and n = %, then T, -s- T« = 1 . 603 ^ ■+■&= 2.66 2\ + Ti+-2S = 2.16 2.188 1 84 1.34 3.758 1.86 0.86 5.821 1.21 0.71 In ordinary practice it is usual to assume T a = 2S; Ti = 2S; T, -f T 7 -j- 2S = 1.5. This corresponds to/ = 0.22 nearly. For a wire rope on cast iron / maybe taken as 0.15 nearly: and if the groove of the pulley is bottomed with gutta percha, 0.^5. (Kankine ) Centrifugal Tension of Belts.— When a belt or band runs at a high velocity, centrifugal force produces a tension in addition to that exist- ing when the belt is at rest or moving at a low velocity. This centrifugal tension diminishes the effective driving force. Rankine says : If an endless band, of any figure whatsoever, runs at a given speed, the centrifugal force produces a uniform tension at each cross- section of the band, equal to the weight of a piece of the band whose length is twice the height from which a heavy body must fall, in order to acquire the velocity of the band. (See Cooper on Belting, p. 101.) If To = centrifugal tension; V — velocity in feet per second ; q = acceleration due to gravity = 32.2; W = weight of a piece of the belt 1 ft. long and 1 sq. in. sectional area,— Leather weighing 56 lbs. per cubic foot gives W = 56 -+- 144 = .388. WV* .388F2 ntnTro 2101 30 lbs., then wd X rpm. BELTING PRACTICE. 87? Belting Practice. Handy Formulae for Belting. — Since in the practical application of the above formulae the value of the coefficient of friction must be assumed, its actual value varying within wide limits (\b% to 135$), and since the values of T^ and T 2 aiso are fixed arbitrarily, it is cus- tomary in practice to substitute for these theoretical formulae more simple empir.cal formulae and rules, some of which are given below. Let d = diam. of pulley in inches; rrd = circumference; V — velocity of belt in ft. per second; v = vel. in ft. per minute; a — angle of the arc of contact; L - length of arc of contact in feet = ndn ■+- (12 x 360) ; F — tractive force per square inch of sectional area of belt; w — width in inches; t — thickness; S = tractive force per inch of width = F-i- t ; rpm. = revs, per minute; rps. = revs, per second = rpm. -5- 60. v = -— X rpm. ; = .2618d X rpm. Horse-power, H.P. = §£ = ^ = ^F = Mm*** X rp m . If F = working tension per square inch = 275 lbs., and t = 7/32 inch, S = 60 lbs. nearly, then H.P. = j™ = .mvw = .00047Qwd x rpm. I wd X rpm " 550 If F = 180 lbs. per square inch, and t = 1/6 inch, S = H.P. =^jk= -055Vw = .000838wd X rpm. = '"" 2™*"' • • ( 2 ) If the working strain is 60 lbs. per inch of width, a belt 1 inch wide travel- ling 550 ft. per minute will transmit 1 horse-power. If the working strain is 30 lbs. per inch of width, a belt 1 inch wide, travelling 1100 ft. per minute, will transmit 1 horse-power. Numerous rules are given by different writers on belting which vary between these extremes. A rule commonly used is : 1 inch wide travelling 1000 ft. per min. = I.H.P. H.P.=^ = .06^=.000882»dXrpm.==^ |jp. . . (3) This corresponds to a working strain of 33 lbs. per inch of width. ^ Many writers give as safe practice for single belts in good condition a working tension of 45 lbs. per inch of width. This gives H.P. =S" = .0818FW = .000357wd X rpm. = wd *Jl pm . . (4) loo «K 219 .250 .312 .333 .375 .437 .500 10 .51 .59 .63 .73 .84 1.05 1.18 15 1.69 1.94 2.42 2.58 2.91 3.39 3.87 15 .75 .88 1 .00 1.16 1.32 1 . 66 1.77 20 2.24 2.57 3.21 3.42 3 .. 85 4.49 5.13 20 1.00 1.17 T32 1.54 1.75 2.19 25 2.79 3.98 4.25 4.78 5.57 6.37 25 I 23 1 43 1.61 1.88 2.16 30 3.31 3.79 4 74 5.05 5 67 6.62 7.58 30 1 47 1.72 1.93 3.22 3.44 35 3.82 4 37 5.46 5.83 6 56 7 65 8,75 35 1 m 1 97 3.70 3.94 40 4.33 4.95 6.19 6.60 7.42 8.66 9 90 40 1 90 2.22 4.15 4.44 45 4.85 5,49 6.86 7.32 8.43 9.70 10.98 45 2. OH 2.45 2.75 3.67 4.58 50 6.01 7.51 8.02 9 02 10.52112.03 50 2.27 2.65 >s 5.30 55 5.68 6.50 8.12 8.66 9.74 11.36!l3.00 55 2 41 2.84 3.19 5.32 60 6.:;6 8.70 9.28 10 43 12. 17(13.91 60 2 58 3.01 3.38 4.51 5.64 65 7.37 9.22 9.83 11 06 12.9014.75 65 2.71 3.16 3.55 4.14 4.74 6.32 70 7.75 9 69 10.33 11 62 13.56jl5.50 70 2.8! 3.27 4.91 6.14 75 7.09 8.11 10.13 10.84 12,16 14. 18116. 21 75 3.37 3.79 4.42 5.05 6.31 6.73 80 7.36 8.41 10 51 11.21 12 61 14.71|16.81 80 4.50 5.15 6.44 85 7.58 10 82 11,55 13 00 15.1617.32 85 3.47 6.50 90 7.74 8.85 11.06 11 80 13 27 15. 48117.69 90 2.97 3.47 3.90 4.55 5.20 6.50 100 7.96 9.10 11.37 12.13 13.65 15.92:18.20 The H.P. becomes a maximum The H.P. becomes a maximum at at 87.41 ft. per sec, = 5245 ft. p. min. 105.4 ft. per sec. = 6324 ft. per min. In the above table the angle of sub tension, a, is taken at 1 Should it be I 90°|100°|110 |120°| Multiply above values by | .65 | .70 I .75 j .79 | '1140° 1.87 160° 1 170° 1 180° 1 200° .94 | .97 | 1 ll.05 A. F. Nagle's Formula (Trans. A. S. M. E., vol. ii., 1881, p. 91. Tables published in 1882.) H.,=c^(^r , ) ; c = i - io - - 00758/a ; a = degrees of belt conta st; f — coefficient of friction; w = width in inches; 550 = thickness in inches; - velocity in feet per second: = T x — 1\ = stress upon belt per square inch. WIDTH OF BELT FOR A GIVEX HORSE-POWER. 879 Taking & at 275 lbs. per sq. in. for laced belts and 400 lbs. per sq in. for lapped and riveted belts, the formula becomes H.P. = H.P. = CVtw{M - .0000218F2) for laced belts; CFtiv(.727 - .0000218F 2 ) for riveted belts. Values of C = 1 - 10 -- 00758 /a (Nagi,e.) i!| Degrees af contact = a. * II "8 S 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° 200° .15 .210 .230 .250 .270 .288 .307 .325 .342 .359 .376 .408 .20 .270 .295 .319 .342 .364 .386 .408 .428 .448 .467 .503 .25 .325 .354 .381 .407 .432 .457 .480 .503 .524 .544 .582 .30 .376 .408 .438 .467 .494 .520 .544 .567 .590 .610 .649 .35 .423 .457 .489 .520 .548 .575 .600 .624 .646 .667 .705 .40 .467 .502 .536 .567 .597 .624 .649 .673 .695 .715 .753 .45 .507 .544 .579 .610 .640 .667 .692 .715 .737 .757 .792 .55 .578 .617 .652 .684 .713 .739 .763 .785 .805 .822 .853 .60 .610 .649 .684 .715 .744 .769 .792 .813 .832 .848 .877 1.00 .792 .825 .853 .877 .897 .913 .927 .937 .947 .956 .969 The following table gives a comparison of the formulae already given for the case of a belt one inch wide, with arc of contact 180°. Horse-power of a Belt One Inch wide, Arc of Contact 180°. Comparison op Different Formula. .2 6 la |*2 Form. 1 H.P. = wv "550* Form. 2 H.P. = wv iioo' Form. 3 H.P. = wv 1000 ' Form. 4 H.P. = wv w Form. 5 dbl.belt H.P. = wv "513" Nagle's Form. 7/32"single belt o a Laced. Riveted 10 20 30 40 50 60 70 80 90 100 110 120 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200 50 100 150 200 250 300 350 400 450 500 550 600 1.09 2.18 3.27 4.36 5.45 6.55 7.63 8.73 9.82 10.91 .35 1.09 1.64 2.18 2.73 3.27 3.82 4.36 4.91 5.45 .60 1.20 1.80 2.40 3.00 3.60 4.20 4.80 5.40 6.00 .82 1.64 2.46 3.27 4.09 4.91 5.73 6.55 7.37 8.18 1.17 2.34 3.51 4.68 5.85 7.02 8.19 9.36 10.53 11.70 .73 1.54 2.25 2.90 3.48 3.95 4.29 4.50 4.55 4.41 4.05 3.49 1.14 2.24 3.31 4.33 5.26 6.09 6.78 7.36 7.74 7.96 7.97 7.75 Width of Belt for a Given Horse-power.— The width of belt required for any given horse-power may be obtained by transposing the for- mulae for horse-power so as to give the value of w. Thus: 550H.P. 9.17 H.P. 2101 H.P. 275H.P. From formula (1), w - From formula (2), w - From formula (3), io - From formula (4), w = From formula (5),* w — * For double belts. 1100 H.P. 18.33 H.P. 1000 H.P. 16.67 H.P. 733 H.P. 12.22 H.P. 8.56 H.P. V d X rpm. _ 4202 H.P. ~ d X rpm. _ 3820 HP. ~ d X rpm. _ 2800 H.P . ~ d X rpm. 1960 H.P. L x rpm. ' _ 530 H.P. ~ LX rpm." 500 H.P. L X rpm.' 360 H.P. L x rpm.' 257 H.P. d x rpm. L x rpm.' 880 BELTIKG. Many authorities use formula (1) for double belts and formula (2) or (3i for ' single belts. To obtain the width by Nagle's formula, w = rVN a _ oi'op^V or divide the given horse-power by the figure in the table corresponding to the given \ thickness of belt and velocity in feet per second. The formula to be used in any particular case is largely a matter of judgment. A single belt proportioned according to formula (1), if tightly stretched, and if the surface is in good condition, will transmit the horse-power calculated by the formula, but one so proportioned is objec- tionable, first, because it requires so great an initial tension that it is apt to stretch, slip, and require frequent restretching and re lacing; and second, because this tension will cause an undue pressure on the pulley- shaft, and therefore an undue loss of power by friction. To avoid these difficulties, formula (2), (3), or (4,) or Mr. Nagle's table, should be used; the latter espe- cially in cases in which the velocity exceeds 4000 ft. per min. Taylor's Mules for Belting. -F. W. Taylor (Trans. A. S. M. E., xv. 204) describes a nine years' experiment on belting in a machine-shop, giving results of tests of 42 belts running night and day. Some of these belts were run on cone pulleys and others on shifting, or fast-and-loose, pul- leys. The average net working load on the shifting belts wasonly 4/10 of that of the cone belts. The shifting belts varied in dimensions from 39 ft. 7 in. long, 3.5 in. wide, .25 in. thick, to 51 ft. 5 in. long, 6.5 in. wide, .37 in. thick. The cone belts varied in dimensions from 24 ft. 7 in. long, 2 in. wide, .25 in. thick, to 31 ft. 10 in. long, 4 in. wide, .37 in. thick. Belt-clamps were used having spring-balances between the two pairs of clamps, so that the exact tension to which the belt was subjected was accurately weighed when the belt was first put on, and each time it was tightened. The tension under which each belt was spliced was carefully figured so as to place it under an initial strain — while the belt was at rest immediately after tightening — of 71 lbs. per inch of width of double belts. This is equiv- alent, in the case of Oak tanned and fulled belts, to 192 lbs. per sq. in. section; Oak tanned, not fulled belts, to 229 " " " " " Semi-raw-hide belts, to 253 " " " " " Raw-hide belts, to 284 " " " " " From the nine years' experiment Mr. Taylor draws a number of conclu- sions, some of which are given in an abridged form below. In using belting so as to obtain the greatest economy and the most satis- factory results, the following rules should be observed: Other Types of Leather Belts and 6- to 7-ply Rubber Belts. A double belt, having an arc of contact of 180°, will give an effective pull on the face of a pulley per inch of width of belt of Or, a different form of same rule: The number of sq. ft. of double Belt passing around a pulley per minute required to transmit one horse power is Or : The number of lineal feet of double- belting 1 in. wide passing around a pulley per minute required to transmit one horse- power is Or : A double belt 6 in. wide, running 4000 to 5000 ft. per min., will transmit 90 sq. ft. 1100 ft. 25 H.P. The terms "initial tension," " effective pull," etc., are thus explained by Mr. Taylor : When pulleys upon which belts are tightened are at rest, both strands of the belt (the upper and lower) are under the same stress per in. of width. By " tension," " initial tension," or " tension while at rest," we Taylor's rules for belting. 881 fcaean the stress per in. of width, or sq, in. of section, to which one of the strands of the belt is tightened, when at rest. After the belts are in motion and transmitting power, the stress on the slack side, or strand, of the belt becomes less, while that on the tight side — or the side which does the pull- ing—becomes greater than when the belt was at rest. By the term " total load " we mean the total stress per in. of width, or sq. in. of section, on the tight side of belt while in motion. The difference between the stress on the tight side of the belt and its slack side, while in motion, represents the effective force or pull which is trans- mitted from one pulley to another. By the terms "working load, 1 ' " net working load, 1 ' or "effective pull, 11 we mean the difference in the tension of the tight and slack sides of the belt per in. of width, or sq. in. section, while in motion, or the net effective force that is transmitted from one pul- ley to another per in. of width or sq. in. of section. The discovery of Messrs. Lewis and Bancroft (Trans. A. S. M. E., vii. 749) that the "sum of the tension on both sides of the belt does not remain constant, 11 upsets all previous theoretical belting formulae. The belt speed for maximum economy should be from 4000 to 4500 ft. per minute. The best distance from centre to centre of shafts is from 20 to 25 ft. Idler pulleys work most satisfactorily when located on the slack side of the belt about one quarter way from the driving-pulley. Belts are more durable and work more satisfactorily made narrow and thick, rather than wide and thin. It is safe and advisable to use: a double belt on a pulley 12 in. diameter or larger; a triple belt on a pulley 20 in. diameter or larger; a quadruple belt on a pulley 30 in. diameter or larger. As belts increase in width they should also be made thicker. The ends of the belt should be fastenej together by splicing and cement- ing, instead of lacing, wiring, or using hooks or clamps of any kind. A V-splice should be used on triple and quadruple belts and when idlers are used. Stepped splice, coated with rubber and vulcanized in place, is best for rubber belts. For double belting the rule works well of making the splice for all belts up to 10 in. wide, 10 in. long; from 10 in. to 18 in. wide the splice should be the same width as the belt, 18 in. being the greatest length of splice required for double belting. Belts should be cleaned and greased every five to six months. Double leather belts will last well when repeatedly tightened under a strain (when at rest) of 71 lbs. per in. of width, or 240 lbs per sq. in. section. They will not maintain this tension for any length of time, however. Belt-clamps having spring- balances between the two pairs of clamps should be used for weighing the tension of the belt accurately each time it is tightened. The stretch, durability, cost of maintenance, etc., of belts proportioned (A) according to the ordinary rules of a total load ot 111 lbs. per inch of width corresponding to an effective pull of 65 lbs. per inch of width, and (B) according to a more economical rule of a total load of 54 lbs., corresponding to an effective pull of 26 lbs. per inch of width, are found to be as follows: When it is impracticable to accurately weigh the tension of a belt in tight- ening it, it is safe to shorten a double belt one half inch for every 10 ft. of length for (A) and one inch for every 10 ft. for (B), if it requires tightening. Double leather belts, when treated with great care and run night and day at moderate speed, should last for 7 years (A); 18 years (B). The cost of all labor and materials used in the maintenance and repairs of double belts, added to the cost of renewals as they give out, through a term of years, will amount on an average per year to 37% of the original cost of the belts (A); 14$ or less (B). In figuring the total expense of beltiug, and the manufacturing cost chargeable to this account, by far the largest item is the time lost on the machines while belts are being relaced and repaired. The. total stretch of leather belting exceeds %% of the original length. The stretch during the first six months of the life of belts is 36$ of their entire stretch (A); \5% (B). A double belt will stretch 47/100 of \% of its length before requiring to be tightened (A); 81/100 of \% (B). The most important consideration in making up tables and rules for the use and care of belting is how to secure the minimum of interruptions to manufacture from this source. 882 BELTIKG. The average double belt (A), when running night and day in a machine- shop, will cause at least 26 interruptions to manufacture during its life, or 5 interruptions per year, but with (B) interruptions to manufacture will not average oftener for each belt than one in sixteen months. The oak-tanned and fulled belts showed themselves to be superior in all respects except the coefficient of friction to either the oak-tanned not fulled, the semi-raw-hide, or raw-hide with tanned face. Belts of any width can be successfully shifted backward and forward on tight and loose pulleys. Belts running between 5000 and COOO ft. per min. and driving 300 H.P. are now being daily shifted on tight and loose pulleys, to throw lines of shafting in and out of use. The best form of belt-shifter for wide belts is a pair of rollers twice the width of belt, either of which can be pressed onto the flat surface of the belt on its slack side close to the driven pulley, the axis of the roller making an angle of 75° with the centre line of the belt. Remarks on Mr. Taylor's Rules. (Trans. A. S. M. E., xv., 242.) —The most notable feature in Mr. Taylor's paper is the great difference be- tween his rules for proper proportioning of belts and those given by earlier writers. A very commonly used rule is, one horse-power may be transmitted by a single belt 1 in. wide running x ft. per min., substituting for x various values, according to the ideas of different engineers, ranging usually from 550 to 1100. The practical mechanic of the old school is apt to swear by the figure 600 as being thoroughly reliable, while the modern engineer is more apt to use the figure 1000. Mr. Taylor, however, instead of using a figure from 550 to 1100 for a single belt, uses 950 to 1100 for double belts. If we assume that a double belt is twice as strong, or will carry twice as much power, as a single belt, then he uses a figure at least twice as large as that used in modern practice, and would make the cost of belting for a given shop twice as large as if the belting were proportioned according to the most liberal of the customary rules. This great difference is to some extent explained by the fact that the problem which Mr. Taylor undertakes to solve is quite a different one from that which is solved by the ordinary rules with their variations. The prob- lem of the latter generally is, " How wide a belt must be used, or how nar- row a belt may be used, to transmit a given horse-power ?" Mr. Taylor's problem is: "How wide a belt must be used so that a given horse-power may be transmitted with the minimum cost for belt repairs, the longest life to the belt, and the smallest loss and inconvenience from stopping the machine while the belt is being tightened or repaired ?" The difference between the old practical mechanic's rule of a l-in.-wide single belt, 600 ft. per min., transmits one horse-power, and the rule com- monly used by engineers, in which 1000 is substituted for 600, is due to the belief of the engineers, not that a horse-power could not be transmitted by the belt proportioned by the older rule, but that such a proportion involved undue strain from overtightening to prevent slipping, which strain entailed too much journal friction, necessitated frequent tightening, and decreased the length of the life of the belt. Mr. Taylor's rule substituting 1100 ft. per min. and doubling the belt is a further step, and a long one, in the same direction. Whether it will be taken in any case by engineers will depend upon whether they appreciate the ex- tent of the losses due to slippage of belts slackened by use under overstrain, and the loss of time in tightening and repairing belts, to such a degree as to induce them to allow the first cost of the belts to be doubled in order to avoid these losses. It should be noted that Mr. Taylor's experiments were made on rather narrow belts, used for transmitting power from shafting to machinery, and his conclusions may not be applicable to heavy and wide belts, such as engine fly-wheel belts. MISCELLANEOUS NOTES ON BELTING. Formulae are useful for proportioning belts and pulleys, but they furnish no means of estimating how much power a particular belt may be trans- mitting at any given time, any more than the size of the engine is a measure of the load it is actually drawing, or the known strength of a horse is a measure of the load on the wagon. The only reliable means of determining the power actually transmitted is some form of dynamometer. (See Trans. A. S. M. E., vol. x'ii. p. 707.) MISCELLANEOUS NOTES ON BELTING. 883 If we increase the thickness, the power transmitted ought to increase in proportion; and for double belts we should have half the width required for a siugle belt under the same conditions. With large pulleys and moderate velocities of belt it is probable that this holds good. With small pulleys, however, when a double belt is used, there is not such perfect contact between the pulley-face and the belt, due to the rigidity of the latter, and more work is necessary to bend the belt-fibres than when a thinner and more pliable belt is used. The centrifugal force teuding to throw the belt from the pulley also increases with the thickness, and for these reasons the width of a double belt required to transmit a given horse-power when used with small pulleys is generally assumed not less than seven tenths the width of a single belt to transmit the same power. (Flather on " Dynamom- eters and Measurement of Power. 1 ') F. W. Taylor, however, finds that great pliability is objectionable, and favors thick belts even for small pulleys: The power consumed in bending the belt around the pulley he considers inappreciable. According to Ban- kine's formula for centrifugal tension, this tension is proportional to the sectional area of the belt, and hence it does not increase with increase of thickness when the width is decreased in the same proportion, the sectional area remaining constant. Scott A. Smith (Trans. A. S. M. E., x. 765) says: The best belts are made from all oak-tanned leather, and curried with the use of cod oil and tallow, all to be of superior quality. Such belts have continued in use thirty to forty years when used as simple driving-belts, driving a proper amount of power, and having had suitable care. The flesh side should not be run to the pulley-face, for the reason that the wear from contact with the pulley should come on the grain side, as that surface of the belt is much weaker in its tensile strength than the flesh side; also as the grain is hard it is more enduring for the wear of attrition; further, if the grain is actually worn off, then the belt may not suffer in its integrity from a ready tendency of the hard grain side to crack. The most intimate contact of a belt with a pulley comes, first, in the smoothness of a pulley -face, including freedom from ridges and hollows left by turning-tools; second, in the smoothness of the surface and evenness in the texture or body of a belt; third, in having the crown of the driving and re ceiving pulleys exactly alike, — as nearly so as is practicable in a commercial sense; fourth, in having the crown of pulleys not over \Q' for a 24" face, that is to say, that the pulley is not to be over J4" larger in diameter in its centre; fifth, in having the crown other than two planes meeting at the centre; sixth, the use of any material on or in a belt, in addition to those necessarily used in the currying process, to keep them pliable or increase their tractive quality, should wholly depend upon the exigencies arising in the use of belts: non-use is safer than over-use; seventh, with reference to the lacing of belts, it seems to be a good practice to cut the ends to a convex shape by using a former, so that there may be a nearly uniform stress on the lacing through the centre as compared with the edges. For a belt 10" wide, the centre of each end should recede 1/10". Lacing of Belts.— In punching a belt for lacing, use an oval punch, the longer diameter of the punch being parallel with the sides of the belt. Punch two rows of holes in each end, placed zigzag. In a 3-in. belt there should be four holes in each end— tv, o in each row. In a 6-inch belt, seven holes— four in the row nearest the end. A 10-inch belt should have nine holes. The edge of the holes should not come nearer than % of an inch from the sides, nor % of an inch from the ends of the belt. The second row should be at least 1% inches from the end. On wide belts these distances should be even a little greater. Begin to lace in the centre of the belt and take care to keep the ends exactly in line, and to lace both sides with equal tightness. The lacing should not be crossed on the side of the belt that runs next the pulley. In taking up belts, observe the same rules as putting on new ones. Setting a Belt on Quarter-twist.— A belt must run squarely on to the pulley. To connect with a belt two horizontal shafts at right angles with each other, say an engine-shaft near the floor with a line attached to the ceiling, will require a quarter-turn. First, ascertain the central point on the face of each pulley at the extremity of the horizontal diameter where the belt will leave the pulley, and then set that point on the driven pulley plumb over the corresponding point on the driver. This will cause the belt to run squarely on to each pulley, and it will leave at an angle greater or Jess, according to the size of the pulleys and their distance from each other. 884 BELTING. In quarter-twist belts, in order that the belt may remain on the pulleys, ■ the central plane on each pulley must pass through the point of delivery of I the other pulley. This arrangement does nor, admit of reversed motion. To find the Length of Belt required for two given \ Pulleys. — When the length cannot be measured directly by a tape-line, ! the following approximate rule may be used : Add the diameter of the two pulleys together, divide the sum by 2, and multiply the quotient by 3J4, and add the product to twice the distauce between the centres of the shafts. (See accurate formula below.) To find the Angle of the Arc of Contact of a Belt.— Divide the difference between the radii of the two pulleys in inches by the distance between their centres, also in inches, and in a table of natural sines find the angle most nearly corresponding with the quotient. Multiply this angle by 2, and add the product to 180° for the angle of contact with the larger pulley, or subtract it from 180° for the smaller pulley. Or, let R = radius of larger pulley, r = radius of smaller; L = distance between centres of the pulleys; a — angle whose sine is (R — r) ■+■ L. Arc of contact with smaller pulley = 180° — 2a; " " " " larger pulley = 180° + 2a. To find the Length of Belt in Contact with the Pulley.— For the larger pulley, multiply the angle a, found as above, by .0349, to the product add 3.1416, and multiply the sum by the radius of the pulley. Or length of belt in contact with the pulley = radius X O + .0349a) = radius x n(l -f jM. For the smaller pulley, length = radius X (7r-.0349a)= radius X ir\l - =.J • The above rules refer to Open Belts. The accurate formula for length of an open belt is, Length » „R(l + ^) + Tf(l - ^) + 2L cos a = R(n + .0349a) -f r(n - .0349a) + 2L cos a, in which R = radius of larger pulley, r = radius of smaller pulley, L = distance between centres of pulleys, and a = angle whose sine is (R - r) -f- L\ cos a = |"L 2 — (R — J') 2 - For Crossed Belts the formula is Length of belt = w fl(l + ^) + w(l + ^) + 2L cos p, = (R + r) X (7T + .03490) + 2L cos (3, in which |3 = angle whose sine is (R + r) -5- L ; cos $ — \ X 2 - (R + ?") 2 . To find the Length of Belt when Closely Rolled -The sum of the diameter of the roll, and of the eye in inches, x the number of turns made by the belt and by .1309, = length of the belt in feet To find the Approximate Weight of Belts —Multiply the length of belt, in feet, by the width in inches, and divide the product by 13 for single, and 8 for double belt. Relations of the Size and Speeds of Driving and Driven Pulleys.— The driving pulley is called the driver, D, and the driven pulley the driven, d. If the number of teeth in gears is used instead of diameter, in these calculations, number of teeth must be substituted wherever diameter occurs. R — revs, per min. of driver, r = revs, per min. of driven. D = dr-i-R; Diam. of driver = diam. of driven x revs, of driven -*- revs, of driver. d = DR -=- r; Diam. of driven = diam. of driver x revs, of driver -+- revs, of driven. R = dr + D; Bevs. of driver = revs, of driven x diam. of driven -*- diam. of driver. MISCELLANEOUS NOTES ON BELTING. 885 -= DR + d; Revs, of driven = revs, of driver x diam. of driver -f- diam. of driven. Evils of Tight Belts. (Jones and Laughlins.)— Clamps with powerful screws are often used to put on belts with extreme tightness, and with most injurious strain upon the leather. They should be very judiciously used for horizontal belts, which should be allowed sufficient slackness to move with a loose undulating vibration on the returning side, as a test that they have no more strain imposed than is necessary simply to transmit the power. On this subject a New England cotton- m ill engineer of large experience, says: I believe that three quarters of the trouble experienced in broken pul- leys, hot boxes, etc., can be traced to the fault of tight belts. The enormous and useless pressure thus put upon pulleys must in time break them, if they are made in any reasonable proportions, besides wearing out the whole out- fit, and causing heating and consequent destruction of the bearings. Below are some figures showing the power it takes in average modern mills with first-class shafting, to drive the shafting alone : Mill, No. Whole Load, H.P. Shafting Horse- power. ? Alone. Per cent of whole. Mill, No. Whole Load, H.P. Shafting Alone. Horse- Per cent power, of whole. 1 2 3 4 199 472 486 677 51 111.5 134 190 25.6 23.6 27.5 28.1 5 6 7 8 759 235 670 677 172.6 84.8 262.9 182 22.7 36.1 39.2 26.8 These may be taken as a fair showing of the power that is required in many of our best mills to drive shafting. It is unreasonable to think that all that power is consumed by a legitimate amount of friction of bearings and belts. I know of no cause for such a loss of power but tight belts. These, when there are hundreds or thousands in a mill, easily multiply the friction on the bearings, and would account for the figures. Sag oi* Belts.— In the location of shafts that are to be connected with each other by belts, care should be taken to secure a proper distance one from the other. This distance should be such as to allow of a gentle sag to the belt when in motion. A general rule may be stated thus: Where narrow belts are to be run over small pulleys 15 feet is a good avei-age, the belt having a sag of V/» to 2 inches. For larger belts, working on larger pulleys, a distance of 20 to 25 feet does well, with a sag of 2*^ to 4 inches. For main belts working on very large pulleys, the distance should be 25 to 30 feet, the belts working well with a sag of 4 to 5 inches. If too great a distance is attempted,the belt will have an unsteady flapping motion, which will destroy both the belt and machinery. Arrangement of Belts and Pulleys.— If possible to avoid it, con- nected shafts should never be placed one directly over the other, as in such case the belt must be kept very tight to do the work. For this purpose belts should be carefully selected of well-stretched leather. It is desirable that the angle of the belt with the floor should not exceed 45°. It is also desirable to locate the shafting and machinery so that belts should run off from each shaft in opposite directions, as this arrangement will relieve the bearings from the friction that would result when the belts all pull one way on the shaft. In arranging the belts leading from the main line of shafting to the counters, those pulling in an opposite direction should be placed as near each other as practicable, while those pulling in the same direction should be separated. This can often be accomplished by changing the relative posi- tions of the pulleys on the counters. By this procedure much of the friction on the journals may be avoided. If possible, machinery should be so placed that the direction of the belt motion shall be from the top of the driving to the top of the driven pulley, when the sag will increase the arc of contact. The pulley should be a little wider than the belt required for the work. 886 BELTING. The motion of driving should run with and not against the laps of the belts. Tightening or guide pulleys should be applied to the slack side of belts and near the smaller pulley. Jones & Laughlins, in their Useful Information, say: The diameter of the pulleys should be as large as can be admitted, provided they will not pro- duce a speed of more than 3750 feet of belt motion per minute. They also say: It is better to gear a mill with small pulleys and run them at a high velocity, than with large pulleys and to run them slower. A mill thus geared costs less and has a much neater appearance than with large heavy pulleys. M. Arthur Achard (Proc. Inst. M. E., Jan. 1881, p. 62) says: When the belt is wide a partial vacuum is formed between the belt and the pulley at a high velocity. The pressure is then greater than that computed from the tensions in the belt, and the resistance to slipping is greater. This has the advantage of permitting a greater power to be transmitted by a given belt, and of diminishing the strain on the shafting. On the other hand, some writers claim that the belt entraps air between itself and the pulley, which tends to diminish the friction, and reduce the tractive force. On this theory some manufacturers perforate the belt with numerous holes to let rhe air escape. Care of Belts.— Leather belts should be well protected against water and even loose steam and other moisture. Belts of coarse, loose leather will do better service in dry warm places; for wet or moist situations the finest and firmest leather should be used. (J. B. Hoyt & Co.) Do not allow oil to drip upon the belts. It destroys the life of the leather. Leather belting cannot safely stand above 110° of heat. Strength of Belting.— The ultimate tensile strength of belting does not generally enter as a factor in calculations of power transmission. The strength of the solid leather in belts is from 2000 to 5000 lbs. per square inch; at the lacings, even if well put together, only about 1000 to 1500. If riveted, the joint should have half the strength of the solid belt. The work- ing strain on the driving side is generally taken at not over one third of the strength of the lacing, or from one eighth to one sixteenth of the strength of the solid belt. Dr. Hartig found that the tension in practice varied from 30 to 532 lbs. per square inch, averaging: 273 lbs. Adhesion Independent of Diameter, (Schultz Belting Co.)— 1. The adhesion of the belt to the pulley is the same — the arc or number of degrees of contact, aggregate tension or weight being the same— without reference to width of belt or diameter of pulle3\ 2. A belt will slip just as readily on a pulley four feet in diameter as it will on a pulley two feet in diameter, provided the conditions of the faces of the pulleys, the arc of contact, the tension, and the number of feet the belt travels per minute are the same in both cases. 3. A belt of a given width, and making any given number of feet per minute, will transmit as much power running on pulleys two feet in diam eter as it will on pulleys four feet in diameter, provided the arc of contact, tension, and conditions of pulley faces are the same in both cases. 4. To obtain a greater amount of power from belts the pulleys may be covered with leather; this will allow the belts to run very slack and give 25$ more durability. Endless Belts.— If the belts are to be endless, they should be put on and drawn together by " belt clamps " made for the purpose. If the belt is made endless at the belt factory, it should never be run on to the pulleys, lest the irregular strain spring the belt. Lift out one shaft, place the belt on the pulleys, and force the shaft back into place. Belt Data.— A fly-wheel at the Amoskeag Mfg. Co., Manchester, N. H., 30 feet diameter, 110 inches face, running 61 revolutions per minute, carried two heavy double-leather belts 40 inches wide each, and one 24 inches wide. The engine indicated 1950 H.P., of which probably 1850 H.P. was transmitted by the belts. The belts were considered to be heavily loaded, but not over- taxed. — = 323 feet per minute for 1 H.P. per inch of width. Samuel Webber (Am. Mach., Feb. 22, 1894) reports a case of a belt 30 inches wide, % inch thick, running for six years at a velocity of 3900 feet per minute, on to a pulley 5 feet diameter, and transmitting 556 H.P. This gives a velocity of 210 feet per minute for 1 H.P. per inch of width. By Mr. Nagle's TOOTHED-WHEEL GEARINO. 88* table of riveted belts this belt would be designed for 332 H.P. By Mr. Taylor's rule it would be used to transmit only 123 H.P. The above may be taken as examples of what a belt may be made to do, but they shou d not be used as precedents in designing. It is not stated how much power was lost by the journal friction due to over-tightening of these belts. Belt Dressings.— We advise, when the belt is pliable, and only dry and husky, the application of blood-warm tallow. This applied, and dried in by heat of fire or sun, will tend to keep the leather in good working condition. The oil of the tallow passes into the tallow of the leather, serving to soften it, and the stearine is left on the outside, to fill the pores and leave a smooth surface. The addition of resin to the tallow for belts, if used in wet or damp places, will be of service and help preserve their strength. Belts which have become hard and dry should have an application of neat's-foot or liver oil, mixed with a small quantity of resin. This prevents the oil from injuring the belt and helps to preserve it. There should not be so much resin as to leave the belt sticky. (J. B. Hoyt & Company.) Belts should not be soaked in water before oiling, and penetrating oils should but seldom be used, except occasionally when a belt gets very dry and husky from neglect. It may then be moistened a little, andfhave neat's- foot oil applied. Frequent applications of such oils to a new belt render the leather soft and flabby, thus causing it to stretch, and making it liable to run out of line. A composition of tallow and oil, with a little resin or bees- wax, is better to use. Prepared castor-oil dressing is good, and maybe applied with a brush or rag while the belt is running. (Alexander Bros.) Cement for Cloth or Leather. (Molesworth.)— 16 parts gutta- percha, 4 india-rubber, 2 pitch, 1 shellac, 2 linseed-oil, cut small, melted to- gether and well mixed. Rubber Belting.— The advantages claimed for rubber belting are perfect uniformity in width and thickness; it will endure a great degree of heat and cold without injury; it is also specially adapted for use in damp or wet places, or where exposed to the action of steam; it is very durable, and has great tensile strength, and when adjusted for service it has the most per- fect hold on the pulleys, hence is less liable to slip than leather. Never use animal oil or grease on rubber belts, as it will greatly injure and soon destroy them. Rubber belts will be improved, and their durability increased, by putting on with a painter's brush, and letting it dry, a composition made of equal parts of red lead, black lead, French yellow, and litharge, mixed with boiled linseed-oil and japan enough to make it dry quickly. The effect of this will be to produce a finely polished surface. If, from dust or other cause, the belt should slip, it should be lightly moistened on the side next the pulley with boiled linseed-oil. (From circulars of manufacturers.) GEARING-. TOOTHED-WHEEL GEARING. Pitch, Pitch-circle, etc.— If two cylinders with parallel axes are pressed together and one of them is rotated on its axis, it will drive the other by means of the friction between the surfaces. The cylinders may be con- sidered as a pair of spur-wheels with an infinite number of very small teeth. If actual teeth are formed upon the cylinders, making alternate elevations and depressions in the cylindrical surfaces, the distance between the axes remaining the same, we have a pair of gear-wheels which will drive one an- other by pressure upon the faces of the teeth, if the teeth are properly shaped. In making the teeth the cylindrical surface may entirely disap- pear, but the position it occupied may still be considered as a cylindrical surface, which is called the " pitch-surface," and its trace on the end of the wheel, or on a plane cutting the wheel at right angles to its axis, is called the " pitch-circle " or " pitch-line." The diameter of this circle is called the pitch-diameter, and the distance from the face of one tooth to the corre- sponding face of the next tooth on the same wheel, measured on an arc of the pitch-circle, is called the " pitch of the tooth," or the circular pitch. If two wheels having teeth of the same pitch are geared together so that their pitch-circles touch, it is a property of the pitch-circles that their diam- eters are proportional to the number of teeth in the wheels, and vice versa,' GEARING. thus, if one wheel is twice the diameter (measured on the pitch-circle) of the other, it has twice as many teeth. If the teeth are properly shaped the linear velocity of the two wheels are equal, and the angular velocities, or speeds of rotation, are inversely proportional to the number of teeth and to th« diameter. Thus the wheel that has twice as many teeth as the other will revolve just half as many times in a minute. The '•pitch,' 1 or distance measured on an arc of the pitch-circle from the face of one tooth to the face of the next, consists of two parts— the " thick- ness'" of the tooth and the "space" between it and the next tooth. The space is larger than the thickness by a small amount called the " back- lash," which is allowed for imperfections of workmanship. In finely cut gears the backlash may be almost nothing. The length of a tooth in the direc- tion of the radius of the wheel is called the "depth," and this is di- vided into two parts: First, the "addendum," the height of the tooth above the pitch line; second, the "dedendum," the depth below the pitch line, which is an amount equal to the addendum of the mating gear. The depth of the space is usually given a little "clearance" to allow for inaccuracies of workmanship, especially in cast gears. Referring to Fig. 153, pi, pi are the pitch-lines, al the addendum-line, rl the root-line or dedendum -line, cl the clearance-line, and b the back- lash. The addendum and dedendum are usually made equal to each other. .-.-.,, No. of teeth 3.1416 Diametral pitch =- Circular pitch diam. of pitch-circle in inches diam. X 3.1416 3.1416 circular pitch' No. of teeth — diametral pitch" Some writers use the term diametral pitch to i ■ircular pitch No. of teeth , but the first definition is the more common and the more 3.1416 convenient. A wheel of 12 in. diam. at the pitch-circle, with 48 teeth is 48/12 = 4 diametral pitch, or simply 4 pitch, The circular pitch of the same , . . 12 X 3.1416 r . orA 3.1416 ___. . wheel is ^ = .7854, or — - — = .7854 in. 48 4 Relation of Diametral to Circular Pitch. Diame- Circular Diame- Circular Circular Diame- Circular Diame- tral Pitch. tral Pitch. Pitch. tral Pitch. tral Pitch. Pitch. Pitch. Pitch. 1 3.142 in. 11 .286 in. 3 1.047 15/16 3.351 m 2.094 12 .262 2fc 1.257 Vs 3.590 2 1.571 14 .224 2 1.571 13/16 3.867 2J4 1.396 16 .196 1% 1.676 H 4.189 m 1.257 18 .175 m m 1.795 11/16 4.570 m 1.142 20 .157 1.933 y 8 5.027 3 1.047 22 .143 m 2.094 9/16 5.585 Wz .898 24 .131 1 7/16 2.185 y 2 6.283 4 .785 26 .121 1% 2.285 7/16 7.181 5 .628 28 .112 1 5/16 2.394 % 8.378 6 .524 30 .105 m 2.513 5/16 10.053 7 .449 32 .098 1 3/16 2.646 X 12.566 8 .393 36 .087 m 2.793 3/16 16.755 9 .349 40 .079 1 1/16 2.957 % 25.133 10 .314 48 .065 1 3.142 1/16 50.266 Since circular pitch diam. X 3.1416 No. of teeth ' which always brings out the diameter as a number with an inconvenient eirc. pitch X No. of t ee th 37T416 TOOTHED-WHEEL GEARING. 889 fraction if the pitch is in even inches or simple fractions of an inch. By the diametral-pitch system this inconvenience is avoided. The diameter may- be in even inches or convenient fractions, and the number of teeth is usually an even multiple of the number of inches in the diameter. Diameter of Pitch-line of Wheels from 10 to 100 Teeth of 1 in. Circular Pitch. S3 K CD is 0® Si .3 h o - i.s ft % U d i= H ft H ft H ft fcH a h o H ft 10 3.183 26 8.276 41 13.051 56 17.825 71 22.600 86 27.375 11 3.501 27 8.594 42 13.369 57 18.144 72 22.918 87 27.693 18 3.820 28 8.913 43 13.687 58 18.462 73 22.236 88 28.011 13 4.138 29 9.231 44 14.006 59 18.781 74 23.555 89 28.329 14 4.456 30 9.549 45 14.324 (JO 19.099 75 23.873 90 28.648 15 4.775 m 9.868 46 14.642 61 19.417 76 24.192 91 28.966 16 5.093 32 10.186 47 14.961 62 19.735 77 24.510 92 29.285 17 93 18 5.730 84 10.823 49 15.597 64 20.372 79 25.146 94 29.921 19 6.048 35 11.141 50 15.915 65 20.690 SO 25.465 95 30.239 20 6.366 36 11.459 51 16.234 66 21.008 81 25.783 96 30.558 81 6.685 37 11.777 52 16.552 67 21.327 S2 26.101 97 30.876 22 7.003 3H 12.096 53 16.870 68 21.645 83 26.419 98 31.194 23 7.321 39 12 414 54 17.189 69 21.963 84 26.738 99 31.512 24 7.639 40 12.732 55 17.507 70 22.282 85 27.056 1 00 31.831 25 7.958 For diameter of wheels of any other pitch than 1 in., multiply the figures in the table by the pitch. Given the diameter and the pitch, to find the num- ber of teeth. Divide the diameter by the pitch, look in the table under diameter for the figure nearest to the quotient, and the number of teeth will be found opposite. Proportions of Teeth. Circular Pitch = 1. 1. 2. 3. 4. 5. .30 .37 .33 .30 .40 .43 .40 .60 .73 .66 .70 .80 .75 .70 .10 .07 .45 .47 .45 .475 .55 .53 .55 .525 .09 .07 .10 .05 .47 .45 .70 Depth of tooth above pitch-line. . " " below pitch-line. . Working depth of tooth Total depth of tooth Clearance at root Thickness of tooth Width of space Backlash Thickness of rim — .70 .75 .05 .45 .54 .10 .485 .515 .03 10.* Depth of tooth above pitch-line.. " " " below pitch-line. Working depth of tooth Total depth of tooth Clearance at root Thickness of tooth Width of space Backlash 25 to .33 ,35 to .42 .6 to .75 .35+.0 .65+ '.6 .48 to .485 52 to .515 04 to . .52+ .03'' .04+. 06'' .687 ,04 to .05 .48 to .5 -j .52 to .5] .0 to .04 1-^-P 1.157-^-P 2h-P 2.157^-P .157h-P 1.51 -^-Pto 1.57h-P 1.57 h-P to 1.63 -hP to 6-=-P * In terms of diametral pitch. Authorities.— 1. Sir Wm. Fairbairn. 2, 3. Clark,, R. T. D.; "used by en- gineers in good practice.' 11 4. Molesworth. 5, 6. Coleman Sellers : 5 for cast, 6 for cut wheels. 7, 8. Unwin. 9, 10. Leading American manufacturers of cut gears. The Chordal Pitch (erroneously called "true pitch" by some authors) is the length of a sti-aight line or chord drawn from centre to centre of two adjacent teeth. The term is now but little used, 890 GEARIKG. Chordal pitch = diam. of pitch-circle X sine of - 180° Chordal No. of teeth pitch of a wheel of 10 in. pitch diameter and 10 teeth, 10 x sin 18° = 3.0903 in. Circular pitch of same wheel = 3.1416. Chordal pitch is used with chain or sprocket wheels, to conform to the pitch of the chain. Formulae for Determining the Dimensions of Small Gears. (Brown & Sharpe Mfg. Co.) P = diametral pitch, or the number of teeth to one inch of diameter of pitch- circle; D' — diameter of pitch circle . D = whole diameter N — number of teeth — . ... V = velocity d' = diameter of pitch-circle.. d = whole diameter n = number of teeth v = velocity [Larger Wheel. Smaller Wheel. These wheels run together. a = distance between the centres of the two wheels; b = number of teeth in both wheels; t = thickness of tooth or cutter on pitch-circle; s = addendum; D" — working depth of tooth; / = amount added to depth of tooth for rounding the corners and for clearance ; D"-\-f — whole depth of tooth; n = 3.1416. P' = circular pitch, or the distance from the centre of one tooth to the centre of the next measured on the pitch-circle. Formulae for a single wheel: iV+2. DxN. r ~~ D~' U ~ N+2 ' *>*$- 1 P' 2s; s = — = ±-=£l *-'£ »T?' N = PD' N = PD D' D -2; S ~ N ~ iV+2 r = ^. D £5+l, / = -; J 10' «H-/=i0+i) P=^; D=D' + P' ,«« = y 2 p'- Formulae for a pair of wheels: b = 2aP; n = PD'V V D _2a(N+2). b ' AT = — : v = PD'V, 'n ' 2a(n + 2): d- b -, NV n = ; V V = NV. m n ' 6 a = -2P ; V = nv m D'-\-d' Ct =— IT* bV jy = 2a V m v + V ; 2a F The following proportions of gear wheels are recommended by Prof. Cole* man Sellers. {Stevens Indicator, April, 189£.) TOOTHED-WHEEL GEARING. 891 Proportions of Gear-wheels. Circular Pitch. Outside of Pitch-line. PX .3 Inside of Pitch-line. Width of Space. "3 is* > '5 o lve ot ne. O s c O c6 £=3 fell Us a 3 rt £ s tf > S-3 a H l 20.50 79.50 1.00 32.50 8.00 227. 1,816 1.00 0.75 1.000 2 6S.00 32.00 .40 62.44 14.00 436. 6,104 3.33 1 20 .186 3 69.00 31.00 .39 30 00 92.30 196. 18,090 10.00 1.50 .050 4 71.20 28.80 .36 28.00 92.60 168. 15,556 8.60 2.50 .035 5 73.96 26.04 .33 48. 0C 73.30 17.5 1,282 0.71 2 80 .380 6 75.66 24.34 .31 53. 0C 56.60 370 20,942 11.60 1.80 .036 7 77.00 23.00 .29 44. 3C 55.00 310. 17,050 9.4C 2.75 .029 8 81.03 18.97 .24 61.00 48.50 426. 20,000 11.60 3.75 .018 No. 1 was Weston's triplex block; No. 3, Weston's differential; No. 4, Weston's imported. The others were from different makers, whose names are not given. All the blocks were of one-ton capacity. Proportions of Hooks.— The following formulae are given by Henry R. Towne, in his Treatise on Cranes, as a result of an extensive experimental and mathematical investi- gation. Thev apply to hooks of capaci- ties from 250 lbs. to 20,000 lbs. Each size of hook is made from some commercial size of round iron. The basis in each case is, therefore, the size of iron of which the hook is to be made, indicated by A in the diagram. The dimension D is arbitrarily assumed. The other di- mensions, as given by the formulae, are those which, while preserving a proper bearing-face on the interior of the hook for the ropes or chains which may be passed through it, give the greatest re- sistance to spreading and to ultimate rupture, which the amount of material in the original bar admits of. The sym- bol A is used to indicate the nominal ca- pacity of the hook in tons of 2000 lbs. The formulae which determine the lines ot the other parts ot the hooks ot the j several sizes are as follows, the measure- F ments being all expressed in inches: IG 164. D = .5 A -f 1.25 E = .64 A + 1.60 F = .33 A -f .85 G = .75 D. H= 1.08.4. O = .363 A + .66 i"= 1.334 Q = .64 A + 1.60 J= 1.204 K = 1.134 L = 1.054 M = .504 N= .80B - .16 U= .8664 The dimensions A are necessarily based upon the ordinary merchant sizes of round iron. The sizes which it has been found best to select are the following: Capacity of hook: 1 1^ 2 3 4 5 6 8 10 tons. Dimension A: % 11/J6 H 11/36 I14 \% 1% 2 2% 2y % 2V8 ®A in. 908 HOISTING. Experiment has shown that hooks made according to the above formulae will give waj r first by opening of the jaw, which, however, will not occur except with a load much in excess of the nominal capacity of the hook. This yielding of the hook when overloaded becomes a source of safety, as it constitutes a signal of danger which cannot easily be overlooked, and which must proceed to a considerable length before rupture will occur and the load be dropped. POWER OF HOISTING-ENGINES. Horse-power required to raise a Load at a Given „ .. „ ,, Gross weight in lbs . . -> . ^ , . Speed. — H.P. = 33000 '""' X s P eed in ft - P er min - To tn,s add 25$ to 50% for friction, contingencies, etc. The gross weight includes the weight of cage, rope, etc. In a shaft with two cages balancing each other use the net load + weight of one rope, instead of the gross weight. To find the load ivhich a given pair of engines ivill start.— Let A = area of cylinder in square inches, or total area of both cylinders, if there are two; P = mean effective pressure in cylinder in lbs. per sq. in.; S= stroke of cylinder in inches; C = circumference of hoisting-drum in inches; L = load lifted by hoisting- rope in lbs. ; F — friction, expressed as a diminution of the load. Then £ = ^1^- P. An example in ColVy Engr., July, 1891, is a pair of hoisting-engines 24" X 40", drum 12 ft. diam., average steam-pre-sure in cylinder = 59.5 lbs.; A = 904.8; P- 59.5; S = 40; C- 452.4. Theoretical load, not allowing for friction, AP2S -h C = 9589 lbs. The actual load that could just be lifted on trial was 7988 lbs., making friction loss F — 1601 lbs., or 20 + per cent of the actual load lifted, or 1G$£% of the theoretical load. The above rule takes no account of the resistance due to inertia of the load, but for all ordinary cases in which the acceleration of speed of the cage is moderate, it is covered by the allowance for friction, etc. The re- sistance due to inertia is equal to the force required to give the load the velocity acquired in a given time, or, as shown in Mechanics, equal to the product of the mass by the acceleration, or R — — — , in which R — resist- ance in lbs. due to inertia; W — weight of load in lbs. ; V = maximum veloc- ity in feet per second; T — time in seconds taken to acquire the velocity V\ g = 32.16. Effect of Slack Rope upon Strain in Hoisting.— A. series of tests with a dynamometer are published by the Trenton Iron Co., which show that a dangerous extra strain may be caused by a few inches of slack rope. In one case the cage and full tubs weighed 11,300 lbs. ; the strain when the load was lifted gently was 11,525 lbs.; with 3 in. of slack chain it was 19.0-25 lbs , with 6 in. slack 25.750 lbs., and with 9 in. slack 27,950 lbs. Limit of Depth for Hoisting.— Taking the weight of a cast-steel nbisting-rope of \% inches diameter at 2 lbs. per running foot, and its break- ing strength at 84,000 lbs., it should, theoretically, sustain itself until 42,000 feet long before breaking from its own weight. But taking the usual factor of safety of 7, then the safe working length of such a rope would be only 6000 feet. If a weight of 3 tons is now hung to the rope, which is equivalent to that of a cage of moderate capacity with its loaded cars, the maximum length at which such a rope could be used, with the factor of safety of 7, is 3000 feet, or 2x -f 6000 = Mi^°_. .-.x = 3000 feet. This limit may be greatly increased by using special steel rope of higher strength, by using a smaller factor of safety, and by using taper ropes. (See paper by H. A. Wheeler, Trans. A. I. M. E., xix. 107.) Large Hoisting Records.— At a colliery in North Derbyshire dur- ing the first week in June, 1890, 6309 tons were raised from a depth of 509 yards, the time of winding being from 7 a.m. to 3.30 p.m. At two other Derbyshire pits, 170 and 140 yards in depth, the speed of winding and changing has been brought to such perfection that tubs are drawn and changed three times in one minute. (Proc. Inst, M, E., 1890.) POWER OF HOISTING-ENGINES. 909 At the Nottingham Colliery near Wilkesbarre, Pa., in Oct. 1891, 70,152 tons were shipped in 24.15 days, the average hoist per day being 1318 mine cars. The depth of hoist was 470 feet, and all coal came from one opening. The engines were fast motion, 22 x 48 inches, conical drums 4 feet 1 inch long. 7 feet diameter at small end and 9 feet at large end. (Eng'g Neics, Nov. 1891.) Pneumatic Hoisting. (H. A. Wheeler, Trans. A. I. M. E., xix. 107.)- A pneumatic hoist was installed in 1876 at Epinac, France, consisting of two continuous air-tight iron cylinders extending from the bottom to the top of the shaft. Within the cylinder moved a piston from which was hung the cage. It was operated by exhausting the air from above the piston, the lower side being open to the atmosphere. Its use vas discontinued on ac- count of the failure of the mine. Mr. Wheeler gives a description of the sys- tem, but criticises it as not being equal on the whole to hoisting by steel ropes. Pneumatic hoisting-cylinders using compressed air have been used at blast-furnaces, the weighted piston counterbalancing the weight of the cage, and the two being connected by a wire rope passing over a pulley-sheave above the top of the cylinder. In the more modern furnaces steam-engine hoists are generally used. Counterbalancing of Winding-engines. (H. W. Hughes, Co- lumbia Coll. Qly.)— Engines running unbalanced are subject to enormous variations in the load; for let W ~ weight of cage and empty tubs, say 6270 lbs.; c = weight of coal, say 4480 lbs.; r = weight of hoisting rope, say 6000 lbs. ; %•' = weight of counterbalance rope hanging down pit, say 6000 lbs. The weight to be lifted will be: If weight of rope is unbalanced. If weight of rope is balanced. At beginning of lift: W+c + r- Woy 10,480 lbs. W + c + r - ( W+ r'), At middle of lift: Y 4480 lbs. *F+c-r-|'~(V-f 0or 4480 lbs. W+c + r -+ | _(tT+|+^)' At end of lift: W+c— {W+ r) or minus 1520 lbs. W+ c + r' — (W-\- r), That counterbalancing materially affects the size of winding-engines is shown by a formula given by Mr. Robert Wilson, which is based on the fact that the greatest w r ork a winding-engine has to do is to get a given mass into a certain velocity uniformly accelerated from rest, and to raise a load the distance passed over during the time this velocity is being obtained. Let W — the weight to be set in motion: one cage, coal, number of empty tubs on cage, one winding rope from pit head-gear to bottom, and one rope from banking level to bottom. v = greatest velocity attained, uniformly accelerated from rest; g = gravity = 32.2; t — time in seconds during which v is obtained; L = unbalanced load on engine; E — ratio of diameter of drum and crank circles; P = average pressure of steam in cylinders; N = number of cylinders; S — space passed over by crank-pin during time t ; C = %, constant to reduce angular space passed through by crank, to the distance passed through by the piston during the time t; A = area of one cylinder, without margin for friction. To this an ad- dition for friction, etc., of engine is to be made, varying from 10 to 30$ of A. 1st. Where load is balanced, PNSC. 2d. Where load is unbalanced: The formula is the same, with the addition of another term to allow for the variation in the lengths of the ascending and descending ropes. In this case 910 HOISTING. hi — reduced length of rope in t attached to ascending cage; h 2 — increased length of rope in t attached to descending cage; w — weight of rope per foot in pounds. Then '=["( %->+:-K*t)-.*^*> PNSC. Applying the ahove formula when designing new engines, Mr. Wilson found that 30 inches diameter of cylinders would produce equal results, when balanced, to those of the 36-inch cylinder in use, the latter being unbal- anced. Counterbalancing may be employed in the following methods : (a) Tapering Rope.—A.t the initial stage the tapering rope enables us to wind from greater depths than is possible with ropes of uniform section. The thickness of such a rope at any point should only be such as to safely bear the load on it at that point. With tapering ropes we obtain a smaller difference between the initial and final load, but the difference is still considerable, and for perfect equaliza- tion of the load we must rely on some other resource. The theory of taper ropes is to obtain a rope of uniform strength, thinner at the cage end where the weight is least, and thicker at the drum end where it is greatest; (b) The Counterpoise System consists of a heavy chain working up and down a staple pit, the motion being obtained by means of a special small drum placed on the same axis as the winding drum. It is so arranged that the chain hangs in full length down the staple pit at the commencement of the winding; in the centre of the run the whole of the chain rests on the bottom of the pit, and, finally, at the end of the winding the counterpoise has been rewound upon the small drum, and is in the same condition as it was at the commencement. (c) Loaded-wagon System. — A plan, formerly much employed, was to have a loaded wagon running on a short incline in place of this heavy chain; the rope actuating this wagon being connected in the same manner as t.ie above to a subsidiary drum. The incline was constructed steep at the com- mencement, the inclination gradually decreasing to nothing. At the begin- ning of a wind the wagon was at the top of the incline, and during a portion of the run gradually passed down it till, at the meet of cages, no [pull was exerted on the engine— the wagon by this time being at the bottom. In the latter part of the wind the resistance was all against the engine, owing to its having to pull the wagon up the incline, and this resistance increased from nothing at the meet of cages to its greatest quantity at the conclusion of the lift. (d) The Endless-rope System is preferable to all others, if there is suffi- cient sump room and the shaft is free from tubes, cross timbers, and other impediments. It consists in placing beneath the cages a tail rope, similar in diameter to the winding rope, and, after conveying this down the pit, it is attached beneath the other cage. (e) Fiat Ropes Coiling on Reels —This means of winding allows of a cer- tain equalization, for the radius of the coil of fascending rope continues to increase, while that of the descending one continues to diminish. Conse- quently, as the resistance decreases in the ascending load the leverage increases, and as the power increases in the other, the leverage diminishes. The variation in the leverage is a constant quantity, and is equal to the thickness of the rope where it is wound on the drum. By the above means a remarkable uniformity in the load may be ob- tained, the only objection being the use of flat ropes, which weigh heavier and only last about two thirds ihe time of round ones. (/) Conical Drums.— Results analogous to the preceding may be obtained by using round ropes coiling on conical drums, which may either be smooth, with the successive coils lying side by side, or they may be provided with a spiral groove. The objection to these forms is. tliat, perfect equalization is not obtained with the conical drums unless the sides are very sleep, and con- sequently there is great risk of the rope slipping ; to obviate this, scroll drums were proposed. They are, however, very expensive, and the lateral displacement of the winding rope from the centre line of pulley becomes very great, owing to their necessary large width. (g) The Koepe System of Winding. — An iron pulley with a single circular groove takes the place of the ordinary drum. The winding rope passes from one cage, over its head-gear pulley, round the drum, and, after pass- CRAKES. 911 ing over the other head-gear pulley, is connected with the second cage. The winding rope thus encircles about half the periphei-y of the drum in the same manner as a driving-belt on an ordinary pulley. There is a balance rope beneath the cages, passing round a pulley in the sump; the arrange- m nt may be likened to an endless rope, the two cages being simply points of attachment. BELT-CONVEYORS. Grain-elevators. — American Grain-elevators are described in a paper by E. Lee Heidenreich, read at the International Engineering Con- gress at Chicago (Trans. A. S. C. E. 1893). See also Trans. A.-S. M. E. vii, 660. Bands for carrying Grain. — Flexible-rubber bands are exten- sively used for carrying grain in and around elevators and warehouses. An article on the grain-storage warehouses of the Alexandria Dock, Liverpool (Proc. Inst. M. E., July, 1891), describes the performance of these bauds, aggregating three miles in length. A band 16^ inches wide, 1270 feet long, running 9 to 10 feet per second has a carrying capacity of 50 tons per hour. See also paper on Belts as Grain Conveyors, by T. W. Hugo, Trans. A. S. M. E . vi. 400. Carrying-bands or Belts are used for the purpose both of sorting coal and of removing impurities. These carrying-bands may be said to be confined to two descriptions, namely, the wire belt, which consists of an endless length of woven wire; and the steel-plate belt, which consists of two or three endless chains, carrying steel plates varying in width from 6 inches to 14 inches. (Proc. Inst. M. E., July, 1890.) CRANES. Classification of Cranes. (Henry R. Towne, Trans. A. S. M. E., iv. 288. Revised in Hoisting, published by The Yale & Towne Mfg. Co.) A Hoist is a machine for raising and lowering weights. A Crane is a hoist with the added capacity of moving the load in a horizontal or lateral direction. Cranes are divided into two classes, as to their motions, viz., Rotary and Rectilinear, and into four groups, as to their source of motive power, viz.: Hand. — When operated by manual power. Poiver. — When driven by power derived from line shafting. Steam, Electric. Hydraulic, or Pneumatic. — When driven by an engine or motor attached to the crane, and operated by steam, electricity, water, or air transmitted to the crane from a fixed source of supply. Locomotive. — When the crane is provided with its own boiler or other generator of power, and is self-propelling ; usually being capable of both rotary and rectilinear motions. Rotary and Rectilinear Cranes are thus subdivided : Rotary Cranes. (1) Swing-cranes. — Having rotation, but no trolley motion. (2) Jib-cranes. — Having rotation, and a trolley travelling on the jib. (3) Column-cranes. — Identical with the jib-cranes, but rotating around a fixed column (which usually supports a floor above). (4) Pillar-cranes.— Having rotation only; the pillar or column being sup- ported entirely from the foundation. (5) Pillar Jib-cranes.— Identical with the last, except in having a jib and trolley motion. (6) Derrick-cranes.— Identical with jib-cranes, except that the head of the mast is held in position by guy- rods, instead of by attachment to a roof or ceiling. (7) Walking-cranes.— Consisting of a pillar or jib-crane mounted on wheels and arranged to travel longitudinally upon one or more rails. (8) Locomotive-cranes.— Consisting of a pillar crane mounted on a truck, and provided with a steam-engine capable of propelling and rotating the crane, and of hoisting and lowering the load. Rectilinear Cranes. (9) Bridge-cranes. — Having a fixed bridge spanning an opening, and a trolley moving across the bridge. (10) Tram-cranes.— Consisting of a truck, or short bridge, travelling lon- gitudinally on overhead rails, and without trolley motion. (11) Travelling-cranes. — Consisting of a bridge moving longitudinally on overhead tracks, and a trolley moving transversely on the bridge. 012 HOISTING. (12) Gantries.— Consisting of an overhead bridge, carried at each end by a trestle travelling on longitudinal tracks on the ground, and having a trolley moving transversely on the bridge. (13) Rotary Bridge-cranes. — Combining rotary and rectilinear movements and consisting of a bridge pivoted at one end to a central pier or post, and supported at the other end on a circular track ; provided with a trolley moving transversely on the bridge. For descriptions of these several forms of cranes see Towne's "Treatise on Cranes." Stresses in Cranes.— See Stresses in Framed Structures, p. 440, ante. Position of the Inclined Brace in a Jib-crane. — The most economical arrangement is that in which the inclined brace intersects the jib at a distance from the mast equal to four fifths the effective radius of the crane. (Hoisting.) A Large Travelling-crane, designed and built by the Morgan Engineering Co., Alliance, (J., for the 12-inch-gun shop at the Washington Navy Yard, is described in American Machinist, June 12, 1890. Capacity, 150 net tons; distance between centres of inside rails, 59 ft. 6 in.; maximum cross travel, 44 ft. 2 in.; effective lift, 40 ft. ; four speeds for main hoist, 1, 2, 4, and 8 ft. per min. ; loads for these speeds, 150, 75, 37^£, and 18% tons respec- tively ; traversing speeds of trolley on bridge, 25 and 50 ft. per minute ; speeds of bridge on main track, 30 and 60 ft. per minute. Square shafts are employed for driving-. A i 50-ton Pillar-crane was erected in 1893 on Finnieston Quay, Glasgow. The jib is formed of two steel tubes, each 39 in. diam. and 90 ft. long. The radius of sweep for heavy lifts is 65 ft. The jib and its load are counterbalanced by a balance-box weighted with 100 tons of iron and steel punchings. In a test a 130-ton load was lifted at the rate of 4 ft. per minute, and a complete revolution made with this load in 5 minutes. Eng'g News, July 20, 1893. Compressed-air Travelling-cranes,— Compressed-air overhead travelling-cranes have been built by the Lane & Bodley Co., of Cincinnati. They are of 20 tons nominal capacity, each about 50 ft. span and 400 ft. length of travel, and are of the triple-motor type, a pair of simple reversing-engines being used for each of the necessary operations, the pair of engines for the bridge and the pair for the trolley travel being each 5-inch bore by 7-inch stroke, while the pair for hoisting is 7-inch bore by 9-inch stroke. Air is furnished by a compressor having steam and air cylinders each 10-in. diam. and 12-in. stroke, which with a boiler-pressure of about 80 pounds gives an air- pressure when required of somewhat over 100 pounds. The air-compressor is allowed to run continuously without a governor, the speed being regulated by the resistance of the air in a receiver. From a pipe extending from the receiver along one of the supporting trusses communication is continuously maintained with an auxiliary receiver on each traveller by means of a one- inch hose, the object of the auxiliary receiver being to provide a supply of air near the engines for immediate demands and independent of the hose connection, which may thus be of small dimension. Some of the advantages said to be possessed by this type of crane are: simplicity; absence of all mov- ing parts, excepting those required for a particular motion when that motion is in use; no danger from fire, leakage, electric shocks, or freezing; ease of repair; variable speeds and reversal without gearing; almost entire absence of noise; and moderate cost. Quay-cranes.— An illustrated description of several varieties of sta- tionary and travelling cranes, with results of experiments, is given in a gaper on Quay-cranes in the Port of Hamburg by Chas. Nehls, Trans. A. S. . E., Chicago Meeting. 1893. Hydraulic Cranes, Accumulators, etc.— See Hydraulic Press- ure Transmission, page 616, ante. Electric Cranes.— Travelling-cranes driven by electric motors have largely supplanted cranes driven by square shafts or flying-ropes. Each of the three motions, viz., longitudinal, traversing and hoisting, is usually ac- complished by a separate motor carried upon the crane. WIRE-ROPE HAULAGE. Methods for transporting coal and other products by means of wire rope, though varying from each other in detail, may be grouped in five classes: I. The Self-acting or Gravity Inclined Plane. II. The Simple Engine-plane. WIRE-ROPE HAULAGE. 013 III. The Tail-rope System. IV. The Endless-rope System. V. The Cable Tramway. The following 1 brief description of these systems is abridged from a pamphlet on Wire-rope Haulage, by Wm. Hildenbraud, C.E., published by John A. Roebling's Sons Co., Trenton. N. J. I. The Self-acting Inclined Plane.— The motive power for the self-acting inclined plane is gravity; consequently this mode of transport- ing coal finds application only in piaces where the coal is conveyed from a higher to a lower point and where the plane has sufficient grade for the loaded descending cars to raise the empty cars to an upper level. At the head of the plane there is a drum, which is generally constructed of wood, having a diameter of seven to ten feet. It is placed high enough to allow men and cars to pass under it. Loaded cars coming from the pit are either singly or in sets of two or three switched on the track of the plane, and their speed in descending is regulated by a brake on the drum. Supporting rollers, to prevent the rope dragging on the ground, are generally of wood, 5 to 6 inches in diameter and 18 to 24 inches long, with H- t0 %-inch iron axles. The distance between the rollers varies from 15 to 30 feet, steeper planes requiring less rollers than those with easy grades. Considering only the reduction of friction and what is best for the preserva- tion of rope, a general rule may be given to use rollers of the greatest possible diameter, and to place them as close as economy will permit. The smallest angle of inclination at which a plane can be made self-acting will be when the motive and resisting forces balance each other. The motive forces are the weights of the loaded car and of the descending rope. The resisting forces consist of the weight of the empty car and ascending rope, of the rolling and axle friction of the cars, and of the axle friction of the supporting rollers. The friction of the drum, stiffness of rope, and resistance of air may be neglected. A general rule cannot be given, because a change in the length of the plane or in the weight of the cars changes the proportion of the forces; also, because the coefficient of friction, depending on the condition of the road, construction of the cars, etc., is a very uncer- tain factor. For working a plane with a %-inch steel rope and lowering from one to four pit cars weighing empty 1400 lbs. and loaded 4000 lbs., the rise in 100 feet necessary to make the plane self-acting will be from about 5 to 10 feet, decreasing as the number of cars increase, and increasing as the length of plane increases. A gravity inclined plane should be slightly concave, steeper at the top than at the bottom. The maximum deflection of the curve should be at an inclination of 45 degrees, and diminish for smaller as well as for steeper inclinations. II. The Simple Engine-plane.— The name "Engine-plane" is given to a plane on which a load is raised or lowered by means of a single wire rope and stationary steam-engine. It is a cheap and simple method of conveying coal underground, and therefore is applied wherever circum- stances permit it. Under ordinary conditions such as prevail in the Pennsylvania mine region, a train of twenty-five to thirty loaded cars will descend, with reason- able velocity, a straight plane 5000 feet long on a grade of 1% feet in 100, while it would appear that 2J4 feet in 100 is necessary for the same number of empty cars. For roads longer than 5000 feet, or when containing sharp curves, the grade should be correspondingly larger. III. The Tail-rope System.— Of all methods for conveying coal underground by wire rope, the tail-rope system has found the most applica- tion. It can be applied under almost any condition. The road may be straight or curved, level or undulating, in one continuous line or with side branches. In general principle a tail-rope plane is the same as an engine- plane worked in both directions with two ropes. One rope, called the " main rope," serves for drawing the set of full cars outward; the other, called the " tail-rope," is necessary to take back the empty set, which on a level or undulating road cannot return by gravity. The two drums may be located at the opposite ends of the road, and driven by separate engines, but more frequently they are on the same shaft at one end of the plane. In the first case each rope would require the length of the plane, but in the second case the tail rope must be twice as long, being led from the drum around a sheave at the other end of the plane and back again to its starting- 9U HOtSTlttG. point. When the main rope draws a set of full cars out, the tail-rope drum runs loose on the shaft, and the rope, being attached to the rear car, un- winds itself steadily. Going in, the reverse takes place. Each drum is provided with a brake to check the speed of the train on a down grade and prevent its overrunning the forward rope. As a rule, the tail rope is strained less than the main rope, but in cases of heavy grades dipping out- ward it is possible that the strain in the former may become as large, or even larger, than in the latter, and in the selection of the sizes reference should be had to this circumstance. IV. The Endless-rope System.— The principal features of this system are as follows: 1. The rope, as the name indicates, is endless. 2. Motion is given to the rope by a single wheel or drum, and friction is obtained either by a grip-wheel or by passing the rope several times around the wheel. 3. The rope must be kept constantly tight, the tension to be produced by artificial means. It is done in placing either the return-wheel or an extra tension wheel on a carriage and connecting it with a weight hanging over a pulley, or attaching it to a fixed post by a screw which occasionally can be shortened. 4. The cars are attached to the rope by a grip or clutch, which can take hold at any place and let go again, starting and stopping the train at will, without stopping the engine or the motion of the rope. 5. On a single-track road the rope works forward and backward, but on a double track it is possible to run it always in the same direction, the full cars going on one track and the empty cars on the other. This method of conveying coal, as a rule, has not found as general an in- troduction as the tail-rope system, probably because its efficacy is not so apparent and the opposing difficulties require greater mechanical skill and more complicated appliances. Its advantages are, first, that it requires one third less rope than the tail-rope system. This advantage, however, is partially counterbalanced by the circumstance that the extra tension in the rope requires a heavier size to move the same load than when a main and tail rope are used. The second and principal advantage is that it is possible to start and stop trains at will without signalling to the engineer. On the other hand, it is more difficult to work curves with the endless sys- tem, and still more so to work different branches, and the constant stretch of the rope under tension or its elongation under changes of temperature frequently causes the rope to slip on the wheel, in spite of every attention, causing delay in the transportation and injury to the rope. V. Wire-rope Tramways.— The methods of conveying products on a suspended rope tramway find especial application in places where a mine is located on one side of a river or deep ravine and the loading station on the other. A wire rope suspended between the two stations forms the track on which material in properly constructed "carriages" or "buggies' 1 is transported. It saves the construction of a bridge or trestlework, and is practical for a distance of 2000 feet without an intermediate support. There are two distinct classes of rope tramways: 1. The rope is stationary, forming the track on which a bucket holding the material moves forward and backward, pulled by a smaller endless wire rope. 2. The rope is movable, forming itself an endless line, which serves at the same time as supporting track and as pulling rope. Of these two the first method has found more general application, and is especially adapted for long spans, steep inclinations, and heavy loads. The second method is used for long distances, divided into short spans, and is only applicable for light loads which are to be delivered at regular intervals. For detailed descriptions of the several systems of wire-rope transporta- tion, see circulars of John A. Roebliug's Sons Co., The Trenton Iron Co., and other wire-rope manufacturers. See also paper on Two-rope Haulage Systems, by R. Van A. Norris, Trans. A. S. M. E., xii. 626. In the Bleichert System of wire-rope tramways, in which the track rope is stationary, loads of 1000 pounds each and upward are carried. While the average spans on a level are from 150 to 200 feet, in crossing rivers, ravines, etc., spans up to 1500 feet are frequently adopted. In a tramway on this system at Granite, Montana, the total length of the line is 9750 feet, with a fall of 1225 feet. The descending loads, amounting to a constant weight of about 11 tons, develop over 14 horse-power, which is sufficient to haul the empty buckets as well as about 50 tons of supplies per day up the line, and SUSPENSION CABLEWAYS OR CABLE HOISTS. 915 also to run the ore crusher and elevator. It is capable of delivering 250 tons of material in 10 hours. SUSPENSION CABLEWAYS OR CABLE HOISTS. (Trenton Iron Co.) In quarrying, rock-cutting, stripping, piling, dam -building, and many other operations where it is necessary to hoist and convey large individual loads economically, it frequently happens that the application of a system of derricks is impracticable, by reason of the limited area of their efficiency and the room which they occupy. To meet such conditions cable-hoists are adapted, as they can be efficiently operated In clear spans up to 1500 feet, and in lifting individual loads up to 15 tons. Two types are made — one in which the hoisting and conveying are done by separate running ropes, and the other applicable only to inclines, in which the carriage descends by gravity, and but one running rope is re- quired. The moving of the carriage in the former is effected by means of an endless rope, and these are commonly known as "endless -rope" cable- hoists to distinguish them from the latter, which are termed "inclined" cable-hoists. The general arrangement of the endless-rope cable-hoists consists of a main cable passing over towers, A frames or masts, as may be most conve- nient, and anchored firmly to the ground at each end, the requisite tension in the cable being maintained by a turnbuckle at one anchorage. Upon this cable travels the carriage, which is moved back and forth over the line by means of the endless rope. The hoisting is done by a separate rope, both ropes being operated by an engine specially designed for the purpose, which may be located at either end of the line, and is constructed in such a way that the hoisting-rope is coiled up or paid out automatically as the carriage is moved in ai.d out. Loads may be picked up or discharged at any point along the line. Where sufficient inclination can be obtained in the main cable for the carriage to descend by gravity, and the loading and unloading is done at fixed points, the endless rope can be dispensed with. The carriage, which is similar in construction to the carriage used in the endless-rope cableways, is arrested in its descent by a stop-block, which may be clamped to the main cable at any desired point, the speed of the descending carriage being under control of a brake on the engine-drum. Stress In Hoisting-ropes on Inclined Planes. (Trenton Iron (Jo. ) a c • ri C • a G • £8§ o.2 o &J3.S III &£ o 3"j wM"o <1 ^|o ft. ft. ft. 5 2° 52' 140 55 28° 49' 1003 110 470 44/ 1516 10 5° 43' 240 60 30° 58' 1067 120 50° 12' 1573 15 8° 32' 336 65 33° 02' 1128 130 52° 26' 1620 20 11° 10' 432 70 35° 00' 1185 140 54° 28' 1663 25 14° 03' 527 75 36° 53' 1238 150 56° 19' 1699 30 16° 42' 613 80 38° 40' 1287 160 58° 00' 1730 35 19° 18' 700 85 40° 22' 1332 170 59° 33' 1758 40 21° 49' 782 90 42° 00' 1375 180 60° 57' 1782 45 24° 14' 860 95 43° 32' 1415 190 62° 15' 1801 50 26° 34' 983 100 45° 00' 1450 200 63° 27' 1822 The above table is based on an allowance of 40 lbs. per ton for rolling fric- tion, but an additional allowance must be made for stress due to the weight of the rope proportional to the length of the plane. A factor of safety of 5 to 7 should be taken. In hoisting the slack-rope should be taken up genliy before beginning the lift, otherwise a severe extra strain will be brought on the rope. The best rope for inclined planes is composed of six strands of seven wires each, laid about a hempen centre. The wires are much coarser than those of the 114-wire rope of the same diameter, and for this reason the 42-wire rope is better adapted to withstand the rough usage and surface wear encountered upon inclined planes. A DQuble-suspension Cableway, carrying loads of 26 tons, erected near 916 Williamsport, Pa., by the Trenton Iron Co., is described by E. G. Spilsbury in Trans. A, I. M. E. xx. 766. The span is 733 feet, crossing the Susquehanna River. Twoisteel cables, each 2 in. diam., are used. On these cables runs a carriage supported on four wheels and moved by an endless cable 1 inch in diam. The load consists of a cage carrying a railroad-car loaded with lum- ber, the latter weighing about 12 tons. The power is furnished by a 50-H.P. engine, and the trip across the river is made in about three minutes. A hoisting cableway on the endless-rope system, erected by the Lidger- wood Mfg. Co., at the Austin Dam, Texas, had a single span 1350 ft. in length, with main cable 2J<£ in. diam., and hoisting-rope \% in. diam. Loads of 7 to 8 tons were handled at a speed of 600 to 800 ft. per minute. Tension required to Prevent Slipping of Wire on Drum. (Trenton Iron Co. )— The amount of artificial tension to be applied in an endless rope to prevent slipping on the driving-drum depends on the char- acter of the drum, the condition of the rope and number of laps which it makes. If Tand S represent respectively the tensions in the taut and slack lines of the rope; W, the necessary weight to be applied to the tail-sheave; JS, the resistance of the cars and rope, allowing for friction ; n, the number of half-laps of the rope on the driving-drum; and/, the coefficient of fric- tion, the following relations must exist to prevent slipping: T= Sef" v , W=T+S, and R = T - S; e fnir . j from which we obtain W — ! — R, e fnir _ ! in which e = 2.71828, the base of the Naperian system of logarithms. The following are some of the values of / : Dry. Wet. Greasy. Rope on a grooved iron drum 120 .085 .070 Rope on wood-filled sheaves 235 .170 .140 Rope on rubber and leather filling 495 .400 .205 e fwr _i_ j The values of the coefficient — , corresponding to the above values efmr _ j of /, for one up to six half-laps of the rope on the driving-drum or sheaves, are as follows: / n - Number of Half-laps on Driving-wheel. 1 2 3 4 5 6 .070 9.130 4.623 3.141 2.418 1.999 1.729 .085 7.536 3.833 2.629 2.047 1.714 1.505 .120 5.345 2.777 1.953 1.570 1.358 1.232 .140 4.623 2.418 1.729 1.416 1.249 1.154 .170 3.833 2.047 1.505 1.268 1.149 1.085 .205 3.212 1.762 1.338 1.165 1.083 1.043 .235 2.831 1.592 1.245 1.110 1.051 1.024 .400 1.795 1.176 1.047 1.013 1.004 1.001 .495 1.538 1.093 1.019 1.004 1.001 The importance of keeping the rope dry is evident from these figures. When the rope is at rest the tension is distributed equally on the two lines of the rope, but when running there will be a difference in the tensions of the taut and slack lines equal to the resistance, and the values of T and S mav be readily computed from the foregoing formula?. Taper Ropes of Uniform Tensile Strength.— Prof. A. S. Herschel in The Engineer, April, 1880, p, 267, gives an elaborate mathe- matical investigation of the problem of making a taper hoisting-rope of uniform tensile strength at every point in its length. Mr. Charles D. West, commenting on Prof. Hersche^s paper, gives a similar solution, and derives therefrom the following formula, based on a breaking strain of 80,000 lbs. per sq. in. of the rope, core included, with a factor of safety of 10: F = 3680[log G - log g] ; log G = —^ + log g; in which F = length in fathoms, and G and g the girth in inches at any two sections F fathoms apart. WIRE-ROPE TRANSMISSION. 917 Example —Let it be required to find the dimensions of a steel-wire rope to draw 6720 lbs.— cage, trams, and coal— from a depth of 400 fathoms. Area of section at lower end = 6720 -*- 8000 = .84 sq. in.; therefore girth = 3J4 in. at bottom. Log G = 400 -r- 3 - log 3.25 = .11 I + .51188 = .62057; therefore G = 4.174, or, say, 4 3/16 in. girth at top. The equations show that the true form of rope is not a regular taper or truncated cone, but follows a logarithmic curve, the girth rapidly increas- ing towards the upper end. Relative Effect of Various-sized Sheaves or Drums on the Life of Wire Ropes. (Thos. E. Hughes, GolVy Eng., April, 1893.) Cast-steel Ropes for Inclines. Made of 6 strands, of 7 wires each, laid around a hemp core. Diameters of Sheaves or Drums in feet, showing perceut- Diam. of ages of life for various diameters. Rope in inches. 1001 901 801 751 601 501 251 m 16 14 12 11 9 7 4.75 w% 14 12 10 8.5 7 6 4.5 12 10 8 7.25 6.5 5.5 4.25 10 • 8.5 7.75 7 6 5 4 l 8.5 7.75 6.75 6 5 4.5 3.75 % 7.75 7 6.25 5.75 4.5 3.75 3.25 & A 7 6.25 5.5 5 4.25 3.5 2.75 6 5.25 4.5 4 3.25 3 2.5 ¥> 5 4.5 4 3.5 2.75 2.25 1.75 The use of iron ropes for inclines has been generally abandoned, steel ropes being more satisfactory and economical. Cast-steel Hoisting-ropes. Made of 6 strands, of 19 wires each, laid around a hemp core. Diam. of Rope in Diameters of Sheaves or Drums in feet, showing percent- ages of life for various diameters. inches. 100*. 901 801 751 60^. 501 25$. 1H 14 12 10 8.5 7 6 4.5 !« 12 10 8 7 6 5.25 4 25 10 8.5 7.5 6.75 5.5 5 4 M 9 7.5 6.5 6 5 4.5 3.75 i 8 7 6 5.5 4.5 4 3.50 % • 7.5 6.75 5.75 5 4.25 3.5 3 H 5.5 4.5 4 3.75 3.25 3 2.25 % 4.5 4 3.75 3.25 3 2.5 2 H 4 3 3 2.75 2.25 2 1.5 % 3 2 1.5 WIRE-ROPE TRANSMISSION". The following data and formulae are taken from a paper by Wm. Hewitt, of the Trenton Iron Co., 1890. (See also circulars of John A. Roebling's Sons Co., Trenton, N. J.; "Transmission of Power by Wire Ropes, 1 ' by A. \V. Stahl, Van Nostrand's Science Series No. 28; and Reuleaux's Constructor.) The Section of Wire Rope best suited, under ordinary conditions, for the transmission of power is composed of 6 strands of 7 wires each, laid together about a hempen centre. Ropes of 12 and 19 wires to the strand are also used. They are more flexible, and may be applied with advantage un- der conditions which do not allow the use of large transmission wheels, but admit of high speed. They are not as well adapted to stand surface wear, however, on account of the smaller size of the wires. 918 WIRE-ROPE TRANSMISSION". Section of Rim. Section of Arm. Fig. 1&, Tlie Driving-wheels (Fig. 165) are usually of cast iron, and are made as light as possible consistent with the requisite strength. Various materials have been used for filling the bot- tom of the groove, such as tarred oakum, jute yarn, hard wood, India-rubber and leather. The filling which gives the best satisfaction, however, consists of segments of leather and blocks of India-rubber, soaked in tar and packed alternately in the groove, and then turned to a true surface. In long spans, intermediate supporting wheels are frequently used, and it is usually sufficient to support only the slack or follow ing side of the rope; but whatever the distance that the power is transmitted, the driving side of the rope will require a less number of sup- ports than the slack side. The sheaves sup- porting the driving side, however, should in all cases be of equal diameter with the driving- wheels. With the slack side smaller wheels may be used, but their diameter sbould not be less than one half that of the driving-sheaves. The system of carrying sheaves may generally be replaced to advantage by that of intermediate stations. The rope thus, instead of running the whole length of the transmission, runs only from one station to the other; and it is advisable to make the stations equidistant, so that a rope may be kept on hand, ready spliced, to put on the wheels of any span, should its rope give out. This method is to be preferred where there is sometimes a jerking motion to the rope, as it prevents sudden movements of this kind from be- ing transmitted over the entire line. Gross horse-power transmitted = iV = .0003702D 2 v D = diameter of rope in inches (= 9 times diameter of single wire); v = velocity of rope in feet per second; k = safe stress per square inch on wires = for iron 25,700 lbs.; E - modulus of elasticity - 28,500,000 for iron; R - ED radius of driving-wheels in inches. The term — — = the stress per square inch due to bending of wires around sheaves. Loss due to centrifugal force = N t = . 00004 24 D 2 ?; 3 : Loss due to journal friction of driving-wheels = N 2 = . 0000045 ( 1 6502V,, -\-ivv) ; " " " " " intermediate-wheels = .0000045( W -f- w)v; in which W = total weight of rope; w — weight of wheel and axle. Net horse-power transmitted, (h -— ") in which = N - JV, - N. 2 : D*v [.0003675(fc - JjS)- .0000424?; - .D000045mw. For a maximum value of iVthe diameter of the wheels should be approxi- !. mately from 185 to 192 times the diameter of the rope, and f»r the latter j ratio of diameters an approximate formula for the actual horse-power transmitted is N = 3 0148 D 3 V, in which V — number of revolutions of wheels per minute. The proper deflections when the rope is at rest are obtained from the for- 1 mula Deflection = .00005765 span 2 , and are as follows: Span in feet.. 50 100 150 200 250 300 350 400 450 Deflection. . . \%" 7" V Z%" 2' 3%" 3' 734" 5' 2J4" 7'%" 9' 2%" 11 '81 j It has been found in practice that when the deflection of the rope at rest is less than 3 inches the transmission cannot be effected with satisfaction, and shafting or belting is to be preferred. This deflection corresponds to a span of about 54 feet. It is customary to make the under side of the rope the driving side. The maximum limit of span is determined by the maxi- mum deflection that may be given to the upper side of the rope when in motion. Assuming that the clearance between the upper and lower sides of the rope should not be less than two feet, and that the wheels are at least 10 feet in diameter, we have a maximum deflection of the upper side of 8 feet, which corresponds to a span of about 370 feet. Much greater spans than this are practicable, in cases where the contour of the ground is such that the upper side of the rope may be made the WIRE-ROPE TRANSMISSION". 919 drivei', as in crossing 1 gullies or valleys, and there is nothing to interfere with obtaining the proper deflections. Some very long transmissions of power have been effected in tins way without an intervening support. There is one at Lockport, N. Y., for instance, with a clear span of about 1700 feet. In a later circular of the Trenton Iron Co. (1892) the above figures are somewhat modified, giving lower values for the power transmitted by a given rope, as follows: The proper ratio between the diameters of rope and sheaves is that which will permit the maximum working tension to be obtained without overstrain- ing the wires in bending. For rope of 7-wire strands this ratio is about 1 : 150; for rope of 12-wire strands, 1 : 115; and for rope of 19-wire strands, 1:90; which gives the following minimum diameter of sheaves, in inches, corresponding to maximum efficiency. Diam. rope, in inches. Ya 5/16 % k/i6 M 9/16 % 11/16 % % 1 m 7-wire strands.. 12 '• " 19 " " 37 47 36 56 | 66 43 50 34 1 39 75 57 45 84 65 51 56 103 78 6"2 112 86 68 ioi 79 115 90 'ioi Assuming the sheaves are of equal diameter, and not smaller than con- sistent with maximum efficiency as determined by the preceding table, the actual horse-power transmitted approximately equals 3.1 times the square of the diameter of the rope in inches multiplied by the velocity in feet per second. From this rule we deduce the following: Horse-power of Wire- rope Transmission, Velocity, in feet ( per second. f 20 30 40 I 50 1 60 70 80 ; Diam. Rope, in inches. Horse-power Transmitted. 1/4 4 6 8 10 12 14 16 5/16 G 9 12 15 18 21 24 3/8 9 13 17 22 26 31 35 7/16 12 18 24 30 36 42 47 1/2 16 23 31 39 47 54 62 9/16 20 29 39 49 59 69 78 5/8 24 36 48 61 73 85 97 11/16 29 44 59 73 88 103 117 3/4 35 52 70 87 105 122 140 7/8 48 71 95 119 142 166 190 1 62 93 124 155 186 217 248 The proper deflection to give the rope in order to secure the necessary tension is h = .0000695S 2 . h — the deflection with the rope at rest, and S = the span, both in feet. Durability of Wire Kopes.-At the Risdou Iron Works, San Fran- cisco, a steel wire rope 2*4 inches in circumference running over 10-foot sheaves at 5000 ft. per minute lias transmitted 40 H.P. for six years without renewing the rope. At the wire-mills a steel wire rope 234 i' 1 - i° circumfer- ence running over 8-foot sheaves has been running steadily for a period of three years at a velocity of 4500 ft. per minute, transmitting 80 H.P. In "inclined Transmissions, when the angle of inclination is great, the proper deflections cannot be readily determined, and the rope be- comes more sensitive to the ordinary variations in the deflections, so that tightening sheaves must be resorted to for producing the requisite tension, as in the case of very short spans. When the horizontal distance between the two wheels is less than 60 ft., or when the angle of inclination exceeds 30 to 45 degrees, it will be found desirable to use tightening sheaves. Tightening 1 pulleys should be placed on the slack side of the rope. The Wire-rope Catenary. (From an article on Wire-rope Transmission, by M. Arthur Achard, Proc. Inst. M. E., Jan. 1881.)- The wires have to bear two distinct molecular strains : First, the tension 6' 920 WIRE-HOPE TRANSMISSION. resulting from the maximum tension T necessary to transmit the motion, whose value in pounds per square inch is S — ^p—^-i d being the diameter of the wires and i their number ; second, the strain produced by flexure upon the pulley, which is approximately Z — E—, R being the radius of the pulley and E the modulus of elasticity of the metal. The approximate values allowed in practice for iron-wire ropes are S = 14,220 lbs. per square inch, and Z = 11,380 lbs. per square inch. £ -f Z should not exceed say 11 tons (24,640 lbs.) per square inch. The curve in which the rope hangs is a catenary; and it is upon the form of the particular catenary in which it hangs, whether more or less deep, as well as upon its lineal weight, that the tension to which it is subjected de- pends. By fixing the weight of the rope and its length, the forms Avhich its two spans assume in common, when at rest, is determined, and consequently their common tension ; which latter must be such as to produce in running the two unequal tensions, T and t, necessary for the transmission of the power. The driving force = T — t. Moreover, the tension in either span is not the same throughout its whole length; it is a minimum at the lowest point of the curve and goes on increas- ing towards the two extremities. The calculation of the tension at the low- est point is very complicated if based upon the true form of the catenary; but by substituting a parabola for the catenary, which is allowable in almost all cases, the calculation becomes simple. If the two pulleys are on the same level, the lowest point is midway between them, and the tension at this point is S = ^-, p being the lineal weight, or pounds per foot, of the rope, I its horizontal projection, which is approximately equal to the distance between the centres of the pulleys, and h the deflection in the middle. The catenary possesses the remarkable mechanical property that the difference between the tensions at any two points is equal to the weight of a length of rope corresponding to the difference in level between the two points. The pi 2 tensions therefore at the two ends will be S t = S -f ph = -=- -{-ph. By substituting for S] in the above equation the required values of Tand //, and solving it with relation to 7i, the deflections /h 1 . Taking the sectional area w of the rope in square inches, and its weight p in pounds per foot run, the ratio w -s- p differs little from a mean value of 0.24. The safe limit of working tension usually assigned for iron-wire ropes is 5 = 14,220 lbs. per square inch. Hence xos -*- p = 0.24 x 14,220 = 3410; and we have theapprox- imate equation ^ — |- /i l = 3410, which is useful as giving a relation between the length I and deflection h t for the driving-span of a rope. In the case of leather, w -=- p = 2.53 approximately, and it is impossible to give 0 2 . m which S = the tension in lbs ; W = weight of the rope in lbs.; I = span, and h = deflection, in feet. Diameter and Weigbt of Pulleys for Wire Rope, Ordi- nary : Diameter, ft 18 14.9 12.4 7.0 Single groove, lbs .. . 6232 5180 2425 798 Double groove, lbs.. 8267 6988 4078 1164 Table of Transmission of Power by "Wire Ropes. (J. A. Roebling's Sons Co., 1886.) — ? = tea 1 °a goo ft . go K A •2u_i c §£•2 £o- 80 100 120 140 80 23 23 23 23 23 8 8 3 fi 4 7 8 8 8 8 140 80 100 120 140 20 19 19 19 19 100 23 % 5 9 80 j 20 119 120 23 % 6 9 100 j 20 119 140 23 ¥a 7 9 120 J 20 1 19 80 22 7/16 9 9 140 J20 119 100 22 7/16 11 10 80 (19 118 120 22 7/16 13 10 100 (19 118 140 22 7/16 15 10 120 i 19 118 80 21 H 14 10 140 (19 118 100 21 X 17 12 80 J 18 117 120 21 % 20 12 100 (18 117 140 21 H 23 12 120 J 18 117 16 80 20 9/16 20 12 120 100 20 9/16 25 14 80 1? 120 20 9/16 30 14 100 {? go 9/16 % 9/16 % 9/16 % 9/16 % % 11/16 % H/16 % 11/16 % 11/16 11/16 % 11/16 % 11/16 % % 1 IK 1 V/8 39 45 j 47 1 48 j 58 1 60 j 69 1 73 ( 82 1 84 ( 64 1102 ( 112 1119 ( 93 1 99 I 116 I 124 (140 1 149 173 j 141 1 148 (176 1185 liOng-distance Transmissions. (From Circular of the Trenton Iron Co., 1892.)— In very long: transmissions of power the conditions do not always admit of obtaining the proper tensions required in the ordinary sys- tem, or "flying transmission of power," as it is termed. In other words, to obtain the proper conditions, it would necessitate numerous and expensive intermediate stations. In case, for instance, it is desired to utilize the power of a turbine to drive a factory, say a mile away, the best method is to em- ploy a larger rope than would ordinarily be used, running jt at a moderate 922 EOPE-DRIVIKG. speed. The rope may be in one continuous length, supported, at intervals of about 100 ft., on sheaves of comparatively small diameter, since the greater rigidity of these ropes preserves them from undue bending strains. Where sharp angles occur in the line, however, sheaves must be used of a size corresponding to the safe limit of tension due to bending. The rope is run under a high working tension, far in excess of what in the ordinary system would cause the rope to slip on the sheaves. The working tension may be four or five times as great as the tension in the slack portion of the rope, and in order to prevent slipping, the rope is wrapped several times about grooved drums, or a series of sheaves at each end of the line. To provide for the slack due to the stretch of the rope, one of the sheaves is placed on a slide worked by long-threaded bolts, or, better still, on a car- riage provided with counterweights, which runs back and forth on a track. The latter preserves a uniform tension in the slack portion of the rope, which is very important. Wire-rope tramways are practically transmissions of power of this kind, in which the load, however, instead of being concentrated at one terminal, is distributed uniformly over the entire line. Cable railways are also trans- missions of this class. The amount of horse-power transmitted is given by the formula N =[4. 7552)2 - .000006 (W+ g + gj]v; in which D = diameter of the rope in inches; v = velocity in ft. per second; W = weight of the rope; g — weight of the terminal sheaves and axles, and g . . V ^ irrt [>« > s \ - f/ o fcP v 6 4 2 \ s \ \ \ \\ X 2 ^ 1 3 1 2 3 4 5 6 7 8 E 1 jO 1 120 130 140 Velocity of Drving Rope in feet per second. Fig. 166. PL 2 by the parabolic formula S - -^=r + PD, S being the assumed strain T on the driving side, and t, calculated by equation (1), on the slack side. The tension t varies with the speed. Horse-power of Transmission Rope at Various Speeds. Computed from formula (2), given above. CW 43 <* CO ©■■ Speed of the Rope in feet per minute. CD °. Kc .23 q3 a «8 s — 1500 2000 2500 3000 3500 4000 4500 5000 6000 7000 8000 U, 1.45 1.9 2.3 2.7 3 3.2 3.4 3.4 3.1 2.2 20 2.3 3.2 3.6 4.2 4.6 5.0 5.8 5.3 4.9 3.4 24 3.3 4.3 5.2 5.8 6.7 7.2 7.7 7.7 7.1 4.9 30 4 4.5 5.9 7.0 8.2 9.1 9.8 10.8 10.7 9.3 6.9 36 i'° 5.8 7.7 9.2 10.7 11.9 12 8 13.6 13.7 12.5 8.8 42 8$ 9.2 12.1 14.3 16.8 18. 6 20.0 21.2 21.4 19.5 13.8 54 13.1 17.4 20.7 23 1 26.8 28.8 30.6 30.8 28.2 19.8 60 V^i 18 23.7 28.2 32.8 36.4 39.2 41.5 41.8 37.4 27.6 72 2 23.2 30.8 36.8 42.8 47.6 51.2 54.4 54 8 50 35.2 84 The following notes are from the circular of the C. W. Hunt Co., New York : For a temporary installation, when the rope is not to be long in use, it might be advisable to increase the work to double that given in the table. For convenience in estimating the necessary clearance on the driving and on the slack sides, we insert a table showing the sag of the rope at different speeds when transmitting the horse-power given in the preceding table. When at rest the sag is Dot the same as when running, being greater on the driving and less on the slack sides of the rope. The sag of the driving side when transmitting the normal horse-power is the same no matter what size of rope is used or what the speed driven at, because the assumption is that the strain on the rope shall be the same at all speeds when transmitting the SAG OF THE ROPE BETWEEN PULLEYS. 925 assumed horse-power, but on the slack side the strains, and consequently the sag, vary with the speed of the rope and also with the horse -power. The table gives the sag for three speeds. If the actual sag is less than given in the table, the rope is strained more than the work requires. This table is only approximate, and is exact only when the rope is running at its normal speed, transmitting its full load and strained to the assumed amount. All of these conditions are varying in actual work, and the table must be used as a guide ouly. Sag of tlie Rope between Pulleys. Distance Driving Side. between Pulleys in feet. All Speeds. 40 feet 4 inches 60 " 10 " 80 1 " 5 " 100 2 " " 120 2 " 11 " 140 3 " 10 " 160 Is 1 Slack Side of Rope. 80 ft. per sec. 60 ft. per sec. 40 ft per sec. Ofeet 7 inches 1 " 5 " 2 " 4 " 5 " 3 " 7 " 2 " 9 " 3 " Ofeet 9 inches 1 " 8 " 1 " 10 " 4 " 5 " 6 " 3 " 8 " 9 " 11 " 3 " Ofeet 11 inches 1 " 11 " 3 " 3 " 5 " 2 " 7 « 4 « 9 " 9 " 14 " " The size of the pulleys has an important effect on the wear of the rope — the larger the sheaves, the less the fibres of the rope slide on each other, and consequently there is less internal wear of the rope. The pulleys should not be less than' forty times the diameter of the rope for economical wear, and as much larger as it is possible to make them. This rule applies also to the idle and tension pulleys as well as to the main driving-pulley. The angle of the sides of the grooves in which the rope runs varies, with different engineers, from 45° to 60°. It is very important that the sides of these grooves should be carefully polished, as the fibres of the rope rubbing on the metal as it comes from the lathe tools will gradually break fibre by fibre, and so give the rope a short life. It is also necessary to carefully avoid all sand or blow holes, as they will cut the rope out with surprising rapidity. Much depends also upon the arrangement of the rope on the pulleys, es- pecially where a tension weight is used. Experience shows that the increased wear on the rope from bending the rope first in one direction and then in the other is similar to that of wire rope. At mines where two cages are used, one being hoisted and one lowered by the same engine doing the same work, the wire ropes, cut from the same coil, are usually arranged so that one rope is bent continuously in one direction and the other rope is bent first in one direction and then in the other, in winding on the drum of the engine. The rope having the opposite bends wears much more rapidly than the other, lasting about three quarters as long as its mate. This difference in wear shows in mauila rope, both in transmission of power and in coal- hoisting. The pulleys should be arranged, as far as possible, to bend the rope in one direction. The wear of the rope is independent of the distance apart of the shafts, since the wear takes place only on the pulleys; hence in transmitting power any distance within the limits of rope-driving, the life of the rope will be the same whether the distance is small or great, but the first cost will be in proportion to the distance. Tension on the Slack Part op the Rope. Speed of Diameter of the Rope and Pounds Tension on the Slack Rope. Rope, in feet per second. H Vs Va % 1 m M 1% 2 20 10 27 40 54 71 110 162 216 283 30 14 29 42 56 74 115 170 226 296 40 15 31 4?. 60 79 123 181 240 315 50 16 33 49 65 85 132 195 259 339 60 18 36 53 71 93 145 214 285 373 70 19 39 59 78 101 158 236 310 406 80 21 43 64 85 111 173 255 340 445 90 24 48 70 93 122 190 279 372 487 926 ROPE-DRIVING. For large amounts of power it is common to use a number of ropes lying side by side in grooves, each spliced separately. For lighter drives some engineers use one rope wrapped as many times around the pulleys as is necessary to get the horse-power required, with a tension pulley to take up the slack as the rope wears when first put in use. The weight put upon this tension pulley should be carefully adjusted, as the overstraining of the rope from this cause is one of the most common errors in rope driving. We therefore give a table showing the proper strain on the rope for the various sizes, from which the tension weight to transmit the horse-power in the tables is easily deduced. This strain can be still further reduced if the horse-power transmitted is usually less than the nominal work which the rope was proportioned to do, or if the angle of groove in the pulleys is acute. Diameter of Pulleys and Weight of Rope. Diameter of Smallest Diameter Length of Rope to Approximate Rope, of Pulleys, in allow for Splicing, Weight, in lbs. per in iuches. inches. in feet. foot of rope. Vi 20 6 .12 % 24 6 .18 H 30 7 .24 % 36 8 .32 1 42 9 .49 VA 54 10 ,60 m 60 12 .83 m 72 13 1.10 2 84 14 1.40 1W 1%" .98 W 2" .72 .844 1.125 1.3 44 38 33 28 25 145 170 193 228 256 430 500 600 675 780 43 50 60 67 7S 242 280 347 380 446 34 41 49 54 63 With a given velocity of the driving-rope, the weight of rope required for transmitting a given horse-power is the same, no matter what size rope is adopted. The smaller rope will require more parts, but the weight will be the same. Miscellaneous Notes on Rope-driving.— W. H. Booth commu- nicates to the Amer. Machinist the following data from English practice with cotton ropes. The calculated figures are based on a total allowable tension on a 1%-inch rope of 600 lbs., and an initial tension of 1/10 the total allowed stress, which corresponds fairly with practice. Diameter of rope 1*4" Wd' Weight per foot, lbs 5 .6 Centrifugal tension = F 2 divided by 64 53 " for V= 80 ft. per sec, lbs. 100 121 Total tension allowable 300 360 Initial tension 30 36 Net working tension at 80 ft. velocity 170 203 Horse-power per rope " " 24 28 The most usual practice in Lancashire is summed up roughly in the fol- lowing figures: 154-inch cotton ropes at 5000 ft. per minute velocity = 50 H. P. per rope. The most common sizes of rope now used are 1% and \% in. The maximum horse-power for a given rope is obtained at about 80 to 83 feet per second. Above that speed the power is reduced by centrifugal tension. At a speed of 2500 ft. per minute four ropes will do about the same work as three at 5000 ft. per min. Cotton ropes do not require much lubrication in the sense that it is re- quired by ropes made of the rough fibre of manila hemp. Merely a slight surface dressing is all that is required. For small ropes, common in spin- ning machinery, from J^ to §4 inch diameter, it is the custom to prevent the fluffing of the ropes on the surface by a light application of a mixture of black-lead and molasses,— but only enough should be used to lay the fibres,— put upon one of the pulleys in a series of light dabs. Reuleaux's Constructor gives as the " specific capacity " of hemp rope in actual practice, that is, the horse-power transmitted per square inch of cross-section for each foot of linear velocity per minute, .004 to .002, the cross-section being taken as that due to the full outside diameter of the rope. For a 1%-in. rope, with a cross section of 2.405' q. in., at a velocity of 5000 ft. per min., this gives a horse-power of from 24 to 48, as against 41.8 by Mr. Hunt's table and 49 by Mr. Booth's. MISCELLANEOUS NOTES ON ROPE-DRIVING. 927 Reuleaux gives formulae for calculating sources of loss in hemp-rope transmission due to(l) journal friction, (2) stiffness of ropes, and (3) creep of ropes. The constants in these formulae are, however, uncertain from lack of experimental data. He calculates an average case giving loss of power due to journal friction = 4%, to stiffness 7.8$. and to creep 5%, or 16.8$ in all, and says this is not to be considered higher than the actual loss. T. Spencer Miller (Eug'g News, Dec. 6, 1890) says: In England hemp and mauila ropes have been largely superseded by ropes of cotton; and I am satisfied that one reason for this is that dry manila ropes wear out too fast, while lubricated ropes give too low a coefficient of friction. The angle of 45° for the groove lias been in use for 33 years, having been first introduced by Jas. Combe in Belfast, Ireland; but if we are to use tallow-laid or other lubricated ropes, we should certainly use a sharper angle in the groove, especially in the American system, which employs a continuous rope with many wraps. Mr. Hunt's formula, Tension of driving side of rope -s- tension of slack side of rope = 2, implies a coefficient of friction of only .10. But I have obtained a coefficient of friction of .26, and have found one authority giving .28. Reuleaux advises for single-line transmission 30° angle of groove. Ramsbottom, an English engineer, and Yale & Towne use a 30° groove in transmission-wheels of travelling-cranes, and I hope to see the best Ameri- can practice use 30° or 35° as a standard groove angle. The work done in pulling out a greasy mauila rope from a 30° groove is not worth considera- tion, although we hear a great deal about the loss of power on this account. I am strongly in favor of using the continuous-rope system, and also of using smaller ropes than are recommended in Mr. Hunt's paper. The most perfect small transmission I have ever seen (about 20 H.P.) em- ploys 5/1(5- in. manila rope on wheels 30 in. in diameter, using a tension car- riage. Rather than use large ropes I think it wiser to replace small ones oftener, for by so doing a great gain may be made in efficiency, thus saving fuel. A large majority of failures in the continuous-rope plan have occurred where the driving and driven sheaves were of widely different diameters, as for example, driving dynamos, or driving a line-shaft from an engine fly- wheel. As ordinarily installed the ropes will not pull alike, and by calcula- tion or by experiment we may find one rope pulling twice or three times as much as the others on the sheave. An installation designed by the writer employs an engine-driving sheave about three times the diameter of the driven sheave. To equalize the pull on the different ropes the grooves of the large driving-sheaves were made with an angle of 30° and those of the small sheaves with an angle of 45°. This change of groove angle has entirely remedied the unequal pulling com- plained of. It has been observed that in sheaves of the same diameter, by the use of a proper tension weight, the ropes may all pull alike; while, where the sheaves are of unequal diameter, the pull is unequal. The only difference of condi- tions in the two cases lies in the different arc of contact of the rope on the two sheaves, which leads to a greater frictional hold of the rope on the large sheave. To equalize the frictional hold on the two sheaves we may sharpen the angle of the small sheave or increase the angle of the large sheave. The Walker Mfg. Co. adopts a curved form of groove instead of one with straight sides inclined to each other at 45°. The curves are concave to the rope. The rope rests on the sides of the groove in driving and driven pul- leys. In idler pulleys the rope rests on the bottom of the groove, which is semicircular. The Walker Mfg. Co. also uses a '•' differential " drum for heavy rope drives, in which the grooves are contained each in a separate ring which is free to slide on the turned surface of the drum in case one rope pulls more than another. A heavy rope-drive on the separate, or English, rope system is described and illustrated in Power, April, 1892. It is in use at the India Mill at Darwen, England. This mill was originally driven by gears, but did not prove success- ful, and rope-driving was resorted to. The 85.000 spindles and preparation are driven by a 2000 horse-power tandem compound engine, with cylinders 23 and 44 inches in diameter and 72- inch stroke, running at 54 revolutions per minute. The fly-wheel is 30 feet in diameter, weighs 65 tons, and is arranged with 30 grooves for 1%-inch ropes. These ropes lead off to receiv- ing-pulleys upon the several floors, so that each floor receives its power direct from the fly-wheel. The speed of the ropes is 50b9 feet per minute, and five 7-foot receivers are used, the number of ropes upon each being proportioned^ 928 FRICTIOK AND LUBRICATION". to the amount of power required upon the several floors. Lambeth cotton ropes are used. FRICTION AND LUBRICATION. Friction is defined by Rankine as that force which acts between two bodies at their surface of contact so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. Coefficient of Friction.— The ratio of the force required to slide a body along a horizontal plane surface to the weight of the body is called the coefficient of friction. It is equivalent to the tangent of the angle of repose, which is the angle of inclination to the horizontal of an inclined plane on which the body will just overcome its tendency to slide. The angle is usually denoted by 0, and the coefficient by /. / = tan 6. Friction of Rest and of Motion.— The force required to start a body sliding is called the friction of rest, and the force required to continue its sliding: after having: started is called the friction of motion. Rolling Friction is the force required to roll a cylindrical or spheri- cal body on a plane or on a curved surface. It depends on the nature of the surfaces and on the force with which they are pressed together, but is essentially different from ordinary, or sliding, friction. Friction of Solids.— Rennie's experiments (I8x!9) on friction of solids, usually unlubricated and dry, led to the following conclusions: 1. The laws of sliding friction differ with the character of the bodies rubbing together. 2. The friction of fibrous material is increased by increased extent of surface and by time of contact, and is diminished by pressure and speed. 3. With wood, metal, and stones, within the limit of abrasion, friction varies only with the pressure, and is independent of the extent of surface, time of contact and velocity. 4. The limit of abrasion is determined by the hardness of the softer of the two rubbing parts. 5. Friction is greatest with soft and least with hard materials. 6. The friction of lubricated surfaces is determined by the nature of the lubricant rather than by that of the solids themselves. ' Friction of Rest. (Rennie.) Pressure, lbs. Values of /. per square inch. Wrought iron on Wrought on Steel on Brass on Wrought Iron. Cast Iron. Cast Iron. Cast Iron. 187 .25 .28 .30 .23 224 .27 .29 .33 .22 336 .31 .33 .35 .21 448 .38 .37 .35 .21 560 .41 .37 .36 .23 672 Abraded .38 .40 .23 784 Abraded Abraded .23 Law of Unlubricated Friction.— A. M. Wellington, Eng'g Neivs, April 7, 1888, states that the most important and the best determined of all the laws of unlubricated friction may be thus expressed: The coefficient of unlubricated friction decreases materially with velocity, is very much greater at minute velocities of -f, falls very rapidly with minute increases of such velocities, and continues to fall much less rapidly with higher velocities up to a certain varying point, following closely the laws which obtain with lubricated friction. Friction of Steel Tires Sliding on Steel Rails. (Westing- house & Gal ton.) Speed, miles per hour Coefficient of friction Adhesion, lbs. per ton (2240 lbs.) 10 0.110 246 15 25 .051 45 50 .047 .040 114 90 929 Rolling Friction is a consequence of the irregularities of form and the roughness of surface of bodies rolling one over the other. Its laws are not yet definitely established in consequence of the uncertainty which exists in' experiment as to how much of the resistance is due to roughness of surface, how much to original and permanent irregularity of form, and how much to distortion under the load. (Thurston.) Coefficients of Rolling Friction.— If R = resistance applied at the circumference of the wheel, W — total weight, r = radius of the wheel, and / = a coefficient, R = fW -¥■ r. f is very variable. Coulomb gives .06 for wood, .005 for metal, where W is in pounds and r in feet. Tredgold made the value of /for iron on iron .002. For wagons on soft soil Morin found/ — .065, and on hard smooth roads .02. A Committee of the Society of Arts (Clark, R. T. D.) reported a loaded omnibus to exhibit a resistance on various loads as below: Pavement Speed per hour. Coefficient. Resistance. Granite. 2.87 miles. .007 17.41 per ton. Asphalt 3.56 " .0121 27.14 Wood 3.34 " .0185 41.60 Macadam, gravelled 3.45 " .0199 44.48 granite, new. . 3.51 " .0451 101.09 Thurston gives the value of /for ordinary railroads, .003, well-laid railroad track, .002; best possible railroad track, .001. The few experiments that have been made upon the coefficients of rolling friction, apart from axle friction, are too incomplete to serve as a basis for practical rules. (Trautwine). Laws of Fluid Friction.— For all fluids, whether liquid or gaseous, the resistance is (1) independent of the pressure between the masses in contact; (2) directly proportional to the area of rubbing-surface; (3) pro- portional to the square of the relative velocity at moderate and high speeds, and to the velocity nearly at low speeds; (4) independent of the nature of the surfaces of the solid against which the stream may flow, but dependent to some extent upon their degree of roughness; (5) proportional to the den- sity of the fluid, and related in some way to its viscosity. (Thurston.) The Friction of Lubricated Surfaces approximates to that of solid fric- tion as the journal is run dry, and to that of fluid friction as it is flooded with oil. Angles of Repose and Coefficients of Friction of Build- in ir Materials. (From Rankine's Applied Mechanics.) Dry masonry and brickwork Masonry and brickwork with damp mortar Timber on stone Iron on stone . . Timber on timber • " " metals Metals on metals Masonry on dry clay " " moist clay Earth on earth " " dry sand, clay, and mixed earth. Earth on earth, damp clay " " " wet clay " " " shingle and gravel 35° to 16%° 26^° to 11^° 31° to \1%° 14° to sy 2 ° 27° 18M° 14° to 45° 21° to 37° 45° > to 48° / = tan 0. about .4 .7 to .3 .5 to .2 .6 to .2 .25 to .15 .51 .33 .25 to 1.0 .38 to .75 1.0 .81 1.67 to 1.4 1.35 2.5 1.43 to 3.3 2 to 5 1.67 to 5 4 to 6.67 1.96 3. 4tol 2.63 to 1.33 1 3.23 1.23 to 0.9 Friction of Uf otion.— The following is a table of the angle of repose . the coefficient of friction / = tan 0, and its reciprocal, 1 -=-/, for the ma- terials of mechanism— condensed from the tables of General Morin (1831), and other sources, as given by Rankine; 930 FRICTION AND LUBRICATION". Surfaces. Wood on wood, dry .... " " " soaped.. Metals on oak, dry " " " wet " " " soapy.. . '• " elm, dry Hemp on oak, dry " " " wet Leather on oak ... " " metals, dry.. " " " wet.. " " " greasy " oily... Metals on metals, dry. .. " " " wet... Smooth surfaces, occa- sionally greased Smooth surfaces, con- tinuously greased Smooth surfaces, best results Bronze or lignum vitee, constantly wet 14° to 26^° uy 2 ° to 2° 2G\i° to 31° 13^° to 14° 1114° 11^° to 14° 28° 18^° 15° to 19^° 20° 13° &A° 8^° to 11° 16^>° m° to 2° .25 to . .2 to . 5 to .24 to . .2 .2 to A .53 .33 .27 to . .56 .36 .23 .07 to . .05 .03 to .( .05? Coefficients of Friction of Journals. (Morin.) Cast iron on cast iron. , Cast iron on bronze.. . . Cast iron on lignum-vitse . Wrought iron on cast iron ) " " " bronze.. \ Iron on lignum vitse \ Bronze on bronze. . Oil, lard tallow. Unctuous and wet. Oil, lard, tallow. Unctuous and wet. Oil, lard. Oil, lard, tallow. Oil, lard. Unctuous. Olive-oil. Lard. Intermittent. Continuous, .07 to .08 .14 .07 to .08 .16 .07 to .08 .11 .19 .10 .09 .03 to .054 .03 to .054 .09 .03 to .054 Prof. Thurston says concerning the above figures that much better results are probably obtained in good practice with ordinary machinery. Those here given are so greatly modified byvariations of speed, pressure, and tem- perature, that they cannot be taken as correct for general purposes. Average Coefficients of Friction. Journal of cast iron in bronze bearing; velocity 120 feet per minute; temperature 70° F.; intermittent feed through an oil-hole. (Thurston on Friction and Lost Work.) Oils. Pressures, pounds per square inch. 8 16 32 48 Sperm, lard, neat's-foot,etc. Olive, cotton -seed, rape, etc. Cod and menhaden. . .... Mineral lubricating-oils. . . . .159 to .250 .160 " .283 .248 " .278 .154 " .261 .138 to .192 .107 " .245 .124 " .167 .145 " .233 .086 to .141 .101 " .168 .097 " .102 .086 " .178 .077 to .144 .079 " .131 081 " .122 .091 " .222 With fine steel journals running in bronze bearings and continuous lubri- cation, coefficients far below those above given are obtained. Thus with sperm-oil the coefficient with 50 lbs. per square inch pressure was .0034; with 200 lbs., .0051; with 300 lbs., ,0057, FRICTION". 931 For very low pressures, as in spindles, the coefficients are much higher. Thus Mr, Woodbury found, at a temperature of 100° and a velocity of 600 feet per minute, Pressures, lbs. per sq. iu 1 2 3 4 5 Coefficient 38 .27 .22 .18 .17 These high coefficients, however, and the great decrease in the coefficient at increased pressures are limited as a practical matter only to the smaller pressures which exist especially in spinning machinery, where the pressure is so light and the film of oil so thick that the viscosity of the oil is an import- ant part of the total frictional resistance. Experiments on Friction of a Journal Lubricated by an Oil-bath (reported by the Committee on Friction, Froc. Inst. M. E., Nov. 1883) show that the absolute friction, that is, the absolute tangential force per square inch of bearing, required to resist the tendency of the brass to go round with the journal, is nearly a constant under all loads, within or- dinary working limits. Most certainly it does not increase in direct propor- tion to the load, as it should do according to the ordinary theory of solid friction. The results of these experiments seem to show that the friction of a perfectly lubricated journal follows the laws of liquid friction much more closely than those of solid friction. They show that under these circum- stances the friction is nearly independent of the pressure per square inch, and that it increases with the velocity, though at a rate not nearly so rapid as the square of the velocity. The experiments on friction at different temperatures indicate a great diminution in the friction as the temperature rises. Thus in the case of lard-oil, taking a speed of 450 revolutions per minute, the coefficient of fric- tion at a temperature of 120° is only one third of what it was at a tempera- ture of 60. The journal was of steel, 4 inches diameter and 6 inches long, and a gun- metal brass, embracing somewhat less than half the circumference of the journal, rested on its upeer side, on which the load was applied. When the bottom of the journal was immersed in oil, and the oil therefore carried under the brass by rotation of the journal, the greatest load carried with rape-oil was 573 lbs. per square inch, and with mineral oil 625 lbs. In experiments with ordinary lubrication, the oil being fed in at the cen- tre of the top of the brass, and a distributing groove being cut in the brass parallel to the axis of the journal, the bearing would not run cool with only 100 lbs. per square inch, the oil being 1 pressed out from the bearing-surface and through the oil-hole, instead of being carried in by it. On introducing the oil at the sides through two parallel grooves, the lubrication appeared to be satisfactory, but the bearing seized with 380 lbs. per square inch. When the oil. was introduced through two oil-holes, one near each end of the brass, and each connected with a curved groove, the brass refused to take its oil or run cool, and seized with a load of only 200 lbs. per square inch. With an oil-pad under the journal feeding rape-oil, the bearing fairly car- ried 551 lbs. Mr. Tower's conclusion from these experiments is that the friction depends on the quantity and uniformity of distribution of the oil, and may be anything between the oil-bath results and seizing, according to the perfection or imperfection of the lubrication. The lubrication may be very small, giving a coefficient of 1/100; but it appeared as though it could not be diminished and the friction increased much beyond this point with- out imminent risk of heating and seizing. The oil-bath probably represents the most perfect lubrication possible, and the limit beyond which friction cannot be reduced by lubrication ; and the experiments show that with speeds of from 100 to 200 feet per minute, by properly proportioning the bearing- surface to the load, it is possible to reduce the coefficient of friction to as low as 1/1000. A coefficient of 1/1500 is easily attainable, and probably is fre- quently attained, in ordinary engine-bearings in which the direction of the force is rapidly alternating and the oil given an opportunity to get between the surfaces, while the duration of the force in one direction is not sufficient to allow time for the oil film to be squeezed out. Observations on the behavior of the apparatus gave reason to believe that with perfect lubrication the speed of minimum friction was from 100 to 150 feet per minute, and that this speed of minimum friction tends to be higher ■I, with an increase of load, and also with less perfect lubrication. By the I speed of minimum friction is meant that speed in approaching which from rest the friction diminishes, and above which the friction increases. 932 FRICTION AND LUBRICATION. Coefficients of Friction of Journal with Oil-bath.— Ab- stract of results of Tower's experiments on friction (Proc. Inst. M. E., Nov. 1683). Journal, 4 in. diam., 6 in. long; temperature, 90° F. Lubricant in Bath. Nominal Load, in pounds per square inch. 625 520 415 310 205 153 i 100 Coefficients of Friction. Lard-oil : 157 ft. per miu.. 471 " Mineral grease : 157 ft. per min.. 471 ' Sperm-oil : 157 ft. per min 471 " " Rape-oil : 157 ft. per min 471 kl " Mineral-oil : 157 ft. per min 471 " " .., Rape-oilfed by syphon lubricator: 157 ft. per miu 314 " " Rape-oil, pad under journal: 157 ft. per min 314 " " .001 .002 (573 lb.) 001 .0009 .0017 .0014 .0022 .001 .0015 .0012 .0018 0009 0016 ,0012 .002 .0014 .0029 .0020 .0042 .0022 .004 .0034 .0066 .0011 .0019 .0016 .0027 .0008 .0016 .0014 .0024 .0014 .0024 .0021 .0035 .0056 .0068 .0098 .0077 .0099 .0099 .0105 .0078 .0042 .009 .0076 .0151 .003 .0064 .004 .007 .004 .007 .0125 .0152 .0099 .0133 Comparative friction of different lubricants under same circumstances, temperature 90°, oil-bath: Sperm-oil 100 per cent. I Lard 135 per cent. Rape-oil 106 " Olive-oil 135 Mineral oil 129 " | Mineral grease 217 " Coefficients of Friction of Motion and of Rest of a Journal.— A cast-iron journal in steel boxes, tested by Prof. Thurston at a speed of rubbing of 150 feet per minute, with lard and with sperm oil, gave the following: Pressures per sq. in., lbs 50 Coeff., with sperm 013 " lard 02 The coefficients at starting were: 100 250 .005 .0085 With sperm . . With lard. . . . .07 .07 500 .004 .0053 .15 .10 750 .0043 .0066 .185 .12 1000 .009 .0125 The coefficient at a speed of 150 feet per minute decreases with increase of pressure until 500 lbs. per sq. in. is reached ; above this it increases. The coefficient at rest or at starting increases with the pressure throughout the ranee of the tests. Value of Anti-friction Metals. (Denton.)— The various white metals available for lining brasses do not afford coefficients of friction lower than can be obtained with bare brass, but they are less liable to "overheating," because of the superiority of such material over bronze in ability to permit of abrasion or crushing, without excessive increase of friction. Thurston (Friction and Lost Work) says that gun-bronze, Bnbbitt, and other soft white alloys have substantially the same friction ; in other words, the friction is determined by the nature of the unguent and not by that of the rubbing-surfaces, when the latter are in good order. The soft metals run at higher temperatures than the bronze. This, however, does not nec- essarilv indicate a serious defect, but simply deficient conductivity. The value of the white alloys for bearings lies mainly in their ready reduction to a smooth surface after any local or general injury by alteration of either surface or form. morin's laws of frictioh. 933 Cast-iron for Bearings. (Joshua Rose.)— Cast iron appears to be an exct-piion to the general rule, that the harder the metal the greater the resistance to wear, because cast iron is softer in its texture and easier to cut with steel tools than steel or wrought iron, but in some situations it is far more durable than hardened steel; thus when surrounded by steam it will wear better than will any other metal. Thus, for instance, experience has demonstrated that piston-rings of cast iron will wear smoother, better, and equally as long as those of steel, and longer than those of either wrought iron or brass, whether the cylinder in which it works be composed of brass, steel, wrought iron, or cast iron; the latter being the more note- worthy, since two surfaces of the same metal do not, as a rule, wear or work well together. So also slide-valves of brass are not found to wear so long or so smoothly as those of cast iron, let the metal of which the seating is composed be whatever it may; while, on the other hand, a cast iron slide- valve will wear longer of itself and cause less wear to its seat, if the latter is of cast iron, than if of steel, wrought iron, or brass. Friction of Metals under Steam-pressure.— The friction of brass upon iron under steam-pressure is double that of iron upon iron. (G. H. Babcock, Trans. A. S. M. E., i. 151.) Morin's "Laws of Friction."— 1. The friction between two bodies is directly proportioned to the pressure; i.e., the coefficient is constant for all pressures. 2. The coefficient and amount of friction, pressure being the same, is in- dependent of the areas in contact. 3 The coefficient of friction is independent of velocity, although static friction (friction of rest) is greater than the friction of motion. Eng'g News, April 7, 1888, comments on these "laws" as follows : From 1831 till about 1876 there was no attempt worth speaking of to enlarge our knowledge of the laws of friction, which during all that period was assumed to be complete, although it was really worse than nothing, since it was for the most part wholly false. In the year first mentioned Morin began a se- ries of experiments which extended over two or three years, and which resulted in the enunciation of these three " fundamental laws of friction," no one of which is even approximately true. For fifty years these laws were accepted as axiomatic, and were quoted as such without question in every scientific work published during that whole period. Now that they are so thoroughly discredited it has been attempted to explain away their defects on the ground that they cover only a very lim- ited range of pressures, areas, velocities, etc., and that Morin himself only announced them as true within the range of his conditions. It is now clearly established that there are no limits or conditions within which any one of them even approximates to exactitude, and that there are many conditions under which they lead to the wildest kind of error, while many of the con- stants were as inaccurate as the laws. For example, in Morin's "' Table of Coefficients of Moving Friction of Smooth Plane Surfaces, perfectly lubri- cated," which may be found in hundreds of text-books now in use. the coeffi- cient of wrought iron on brass is given as .075 to .103, which would make the rolling friction of railway trains 15 to 20 lbs. per ton instead of the 3 to 6 lbs. which it actually is. General Morin, in a letter to the Secretary of the Institution of Mechanical Engineers, dated March 15, 1879, writes as follows concerninghis experiments on friction made more than forty years before: " The results furnished by my experiments as to the relaiions between pressure, surface, and speed on the one hand, and sliding friction on the other, have always been regarded by myself, not as mathematical laws, but as close approximations to the truth, within the limits of the data of the experiment? themselves. The same holds, in my opinion, for many other laws of practical mechanics, such as those of rolling resistance, fluid resistance, etc." Prof. J. E. Denton (Stevens Indicator, July, 1890) says: It has been gen- erally assumed that friction between lubricated surfaces follows the simple law that the amount of the friction is some fixed fraction of the pressure be- tween the surfaces, such fraction being independent of the intensity of the pressure per square inch and the velocity of rubbing, between certain limits of practice, and that the fixed fraction referred to is represented by the co- efficients of friction given by the experiments of Morin or obtained from ex- perimental data which represent conditions of practical lubrication, such as those given in Webber's Manual of Power. By the experiments of Thurston, Woodbury, Tower, etc., however, it appears that the friction between lubricated metallic surfaces, such as ma- 934 FRICTIOH AND LUBRICATION. chine bearings, is not directly proportional to the pressure, is not indepen- dent of the speed, and that the coefficients of Morin and Webber are about tenfold too great for modern journals. Prof. Denton offers an explanation of this apparent contradiction of au- thorities by showing, with laboratory testing machine data, that Moriu's laws hold for bearings lubricated by a restricted feed of lubricant, such as is afforded by tne oil-cups common to machinery; whereas the modern ex- periments have been made with a surplus feed or superabundance of lubri- cant, such as is provided only in railroad -car journals, and a few special cases of practice. That the low coefficients of friction obtained under the latter conditions are realized in the case of car journals, is proved by the fact that the tem- perature of car-boxes remains at 100° at high velocities; and experiment shows that this temperature is consistent only with a coefficient of friction of a fraction of one per cent. Deductions from experiments on train resistance also indicate the same low degree of friction. But these low co-efficients do not account for the internal friction of steam-engines as well as do the co- efficients of Morin and Webber. In American Machinist, Oct. 23, 1890, Prof. Denton says: Morin's measure- ment of friction of lubricated journals did not extend to light pressures. They apply only to the conditions of general shafting and engine work. He clearly understood that there was a frictional resistance, due solely to the viscosity of the oil, and that therefore, for very light pressures, the laws which he enunciated did not prevail. He applied his dynamometers to ordinary shaft-journals without special preparation of the rubbmg-surfaces, and without resorting to artificial methods of supplying the oil. Later experimenters have with few exceptions devoted themselves exclu- sively to the measurement of resistance practically due to viscosity alone. They have eliminated the resistance to which Morin confined his measure- ments, namely, the friction due to such contact of the rubbing-surfaces as prevail with a very thin film of lubricant between comparatively rough sur- faces. Prof. Denton also says (Trans. A. S. M. E., x. 518): " I do not believe there is a particle of proof in any investigation of friction ever made, that Morin's laws do not hold for ordinary practical oil-cups or restricted rates of feed." Laws of Friction of well-lubricated Journals.— John Goodman (Trans. Inst. C. E. 1886, Eiufg News, Apr. 7 and 14, 1888), review- ing the results obtained from the testing-machines of Thurston, Tower, and Stroudley, arrives at the following laws: Laws of Friction: Well- lubricated Surfaces. (Oil-bath.) 1. The coefficient of friction with the surfaces efficiently lubricated is from 1/6 to 1/10 that for dry or scantily lubricated surfaces. 2. The coefficient of friction for moderate pressures and speeds varies ap- proximately inversely as the normal pressure; the frictional resistance va- ries as the area in contact, the normal pressure remaining constant. 3. At very low journal speeds the coefficient of friction is abnormally high; but as the speed of sliding increases from about 10 to 100 ft. per min , the friction diminishes, and again rises when that speed is exceeded, varying approximately as the square root of the speed. 4. The coefficient of friction varies approximately inversely as the temper- ature, within certain limits, namely, just before abrasion takes place. The evidence upon which these laws are based is taken from various mod- ern experiments. That relating to Law 1 is derived from the " First Report on Friction Experiments, 1 ' by Mr. Beauchamp Tower. , Method of Lubrication. Coefficient of Friction. Comparative Friction. Oil-bath .00139 .0098 .0090 1 00 Siphon lubricator Pad under journal 7.06 6.48 With a load of 293 lbs. per sq. in. and a journal speed of 314 ft. per min. Mr. Tower found the coefficient of friction to be .0016 with an oil- bath, and LAWS OF FRICTION". 935 .0097, or six times as much, with a pad. The very low coefficients ob- tained by Mr. Tower will be accounted for by Law 2, as he found that the frictional resistance per square inch under varying loads is nearly constant, as below: Load in lbs. per sq. in 529 468 415 363 310 258 205 153 100 Frictional resist, persq. in. .416 .514 .498 .472 .464 .438 .43 .458 .45 The frictional resistance per square inch is the product of the coefficient of friction into the load per square inch on horizontal sections of the brass. Hence, if this product be a constant, the one factor must vary inversely as the other, or a high load will give a low coefficient, and vice versa. For ordinary lubrication, the coefficient is more constant under varying loads; the frictional resistance then varies directly as the load, as shown by Mr. Tower in Table VIII of his report (Proc. Inst. M. E. 1883). With respect to Law 3, A. M. Wellington (Trans. A. S. C. E. 1884), in ex- periments on journals revolving at very low velocities, found that the friction was then very great, and nearly constant under varying conditions of the lubrication, load, and temperature. But as the speed increased the friction fell slowly and regularly, and again returned to the original amount when the velocity was reduced to the same rate. This is shown in the following table: Speed, feet per minute: 0+ 2.16 3.33 4.86 8.82 21.42 35.37 53.01 89.28 106.02 Coefficient of friction : .118 .094 .070 .069 .055 .047 .040 .035 .030 .026 It was also found by Prof. Kimball that when the journal velocity was in- creased from 6 to 110 ft. per minute, the friction was reduced 70%; in another case the friction was reduced 67% when the velocity was increased from 1 to 100 ft. per minute; but after that point was reached the coefficient varied approximately with the square root of the velocity. The following results were obtained by Mr. Tower: Feet per minute. .. 209 262 314 366 419 471 Nominal Load per sq. in. Coeff . of friction . . .0010 .0013 .0014 .0012 .0014 .0015 .0013 .0015 .0017 .0014 .0017 .0019 .0015 .0018 .0021 .0017 .002 .0024 520 lbs. 468 " 415 " The variation of friction with temperature is approximately in the inverse ratio, Law 4. Take, for example, Mr. Tower's results, at 262 ft. per minute: Temp. F. 110° 100° 90° 80° 70° 60° Observed Calculated .0044 .00451 .0051 .00518 .006 .00608 .0073 .00733 .0092 .00964 .0119 .01252 This law does not hold good for pad or siphon lubrication, as then the co- efficient of friction diminishes more rapidly for given increments of tem- perature, but on a gradually decreasing scale, until the normal temperature has been reached; this normal temperature increases directly as the load per sq in. This is shown in the following table taken from Mr. Stroudley's experiments with a pad of rape oil: Temp. F 105° 110° 115° 120° 125° 130° 135° 1 140° 145° .022 .0180 .0160 .0140 0020 .0125 .0015 .0115 .0010 .0110 .0005 .0106 .0004 .0102 Decrease of coeff. . .0040 .0020 .0002 In the Galton-Westinghouse experiments it was found that with velocities below 100 ft. per min., and with low pressures, the frictional resistance varied directly as the normal pressure; but when a velocity of 100 ft. per min. was exceeded, the coefficient of friction greatly diminished; from the same experiments Prof. Kennedy found that the coefficient of friction for high pressures was sensibly less than for low. Allowable Pressures on Bearing-surfaces. (Proc. Inst. M. E., May, 1888.)— The Committee on Friction experimented with a steel ring of 936 FRICTION" AND LUBRICATION. rectangular section, pressed between two cast-iron disks, the annular bear- ing-surfaces of which were covered with gun-metal, and were 12 in. inside diameter and 14 in. outside. The two disks were rotated together, and the steel ring was prevented from rotating by means of a lever, the holding force of which was measured. When oiled through grooves cut in each face of the ring and tested at from 50 to 130 revs, per min., it was found that a pressure of 75 lbs. per sq. in. of bearing-surface was as much as it would bear safely at the highest speed without seizing, although it carried 90 lbs. per sq. in. at the lowest speed. The coefficient of friction is also much higher than for a cylindrical bearing, and the friction follows the law of the friction of solids much more nearly than that of liquids. This is doubtless due to the much less perfect lubrication applicable to this form of bearing compared with a cylindrical one. The coefficient of friction appears to be about the same with the same load at all speeds, or, in other words, to be independent of the speed; but it seems to diminish somewhat as the load is increased, and may be stated approximately as 1/20 at 15 lbs. per sq. in., diminishing to 1/30 at 75 lbs. per sq. in. The high coefficients of friction are explained by the difficulty of lubricat- ing a collar-bearing. It is similar to the slide-block of an engine, which can carry only about one tenth the load per sq. in. that can be carried by the crank-pins. In experiments on cylindrical journals it has been shown that when a cylindrical journal was lubricated from the side on which the pressure bore, 100 lbs. per sq. in. was the limit of pressure that it would carry; but when it came to be lubricated on the lower side and was allowed to drag the oil in with it, 600 lbs. per sq. in. was reached with impunity; and if the 600 lbs. per sq. in., which was reckoned upon the full diameter of the bearing, came to be reckoned on the sixth part of the circle that was taking the greater pro- portion of the load, it followed that the pressure upon that part of the circle amounted to about 1200 lbs. per sq. in. In connection with these experiments Mr. Wicksteed states that in drill- ing-machines the pressure on the collars is frequently as high as 336 lbs. per sq. in., but the speed of rubbing in this case is lower than it was in any of the experiments of the Research Committee. In machines working very slowly and intermittently, as in testing-machines, very much higher pres- sures are admissible. Mr. Adamson mentions the case of a heavy upright shaft carried upon a small footstep-bearing, where a weight of at least 20 tons w,as carried on a shaft of 5 in. diameter, or, say, 20 sq. in. area, giving a pressure of 1 ton per sq. in. The speed was 190 to 200 revs, per min. It was necessary to force the oil under the bearing by means of a pump. For heavy horizontal shafts, such as a fly-wheel shaft, carrying 100 tons on two journals, his practice for getting oil into the bearings was to flatten the journal along one side throughout its whole length to the extent of about an eighth of an inch in width for each inch in diameter up to 8 in. diameter; above that size rather less flat in proportion to the diameter. At first sight it appeared alarming to get a continuous flat place coming round in every revolution of a heavily loaded shaft; yet it earned the oil effectually into the bearing, which ran much better in consequence than a truly cylindrical journal without a flat side. In thrust-bearings on torpedo-boats Mr. Thornycroft allows a pressure of never more than 50 lbs. per sq. in. Prof. Thurston (Friction and Lost Work, p. 240) says 7000 to 9000 lbs. pressure per square inch is reached on the slow-working and rarely moved pivots of swing bridges. Mr. Tower says (Proc. Inst M. E., Jan. 1884): In eccentric-pins of punch- ing and shearing-machines very high pressures are sometimes used without seizing. In addition to the alternation in the direction, the pressure is ap- Elied for only a very short space of time in these machines, so that the oil as no time to be squeezed out. In the discussion on Mr. Tower's paper (Proc. Inst. M. E. 1885) it was stated that it is well known from practical experience that with a constant load on an ordinary journal it is difficult and almost impossible to have more than 200 ibs. per square inch, otherwise the bearing would get hot and the oil go out of it; but when the motion was reciprocating, so that the load was alternately relieved from the journal, as with crank-pins and similar jour- nals, much higher loads might be applied than even 700 or 800 lbs. per square inch. FRICTION OF CAR-JOURNAL BRASSES. 937 Mr. Goodman (Proc. Inst. C. E. 1886) found that the total frictional re- sistance is materially reduced by diminishing the width of the brass. The lubrication is most efficient in reducing the friction when the brass subtends an angle of from 1J0° to 60°. The film is probably at its best be- tween the angles 80° and 110°. In the case of a brass of a railway axle-bearing where an oil-groove is cut along its crown and an oil-hole is drilled through the top of lhe brass into it, the wear is invariably on the off side, which is probably due to the oil escap- ing as soon as it reaches the crown of the brass, and so leaving the off side almost dry, where the wear consequently ensues. In railway axles the brass wears always on the forward side. The same ob- servation has been made in marine engine .journals, which always wear in exactly the reverse way to what they might be expected. Mr. Stroudley thinks this peculiarity is due to a film of lubricant being drawn in from the un- der side of the journal to the aft part of the brass, which effectually lubri- cates and prevents wear on that side; and that when the lubricant reaches the forward side of the brass it is so attenuated down to a wedge shape that there is insufficient lubrication, and greater wear consequently follows. Prof. J. E, Denton (Am. Alack., Oct. 30, 1890) says: Regarding the pres- sure to wnich oil is subjected in railroad car-service, it is probably more severe than in any other class of practice. Car brasses, when used bare, are so im- perfectly fitted to the journal, that during the early stages of their use the area of bearing may be but about one square inch. In this case the pressure per square inch is upwards of 6000 lbs. But at the slowest speeds of freight service the wear of a brass is so rapid that, within about thirty minutes the area is either increased to about three inches, and is thereby able to relieve the oil so that the latter can successfully prevent overheating of the journal, or else overheating takes place with any oil. and measures of relief must be taken which eliminate the question of differences of lubricating power among the different lubricants available. A brass which has been run about fifty miles under 5000 lbs. load may have extended the area of bearing-surface to about three square inches. The pressure is then about 1700 lbs. per square inch. It may be assumed that this is an average minimum area for car-ser- vice where no violent and unmanageable overheating has occurred during the use of a brass for a short time. This area will very slowly increase with any lubricant. C. J. Field (Poirer, Feb. 1893) says: One of the most vital points of an en- gine for electrical service is that of main bearings. They should have a sur- face velocity of not exceeding 350 feet per minute, with a mean bearing- f>ressure per square inch of projected area of journal of not more than 80 bs. This is considerably within the safe limit of cool performance and easy operation. If the bearings are designed in this way, it would admit the use of grease on all the main wearing-surface, which in a large type of engines for this class of work we think advisable. Oil-pressure in a Bearing.— Mr. Beauchamp Tower (Proc. Inst. M. E , Jan. 1885) made experiments with a brass bearing 4 inches diameter by 6 inches Ions:, to determine the pressure of the oil between the brass and the journal. The bearing was half immersed in oil, and had a total load of 8008 lbs. upon it. The journal rotated 150 revolutions per minute. The pressure of the oil was determined by drilling small holes in the bearing at different points and connecting them by tubes to a Bourdon gauge. It was found that the pressure varied from 310 to 625 lbs. per square inch, the great- est pressure being a little to the " off " side of the centre line of the top of the bearing, in the direction of motion of the journal. The sum of the up- ward force exerted by these pressures for the whole lubricated area was nearly equal to the total pressure on the bearing. The speed was reduced from 150 to 20 revolutions, but the oil-pressure remained the same, showing that the brass was as completely oil-borne at the lower speed as at the higher. The following was the observed friction at the lower speed: Nominal load, lbs. per square inch . . . 443 333 211 89 Coefficient of friction 00132 .00168 .00247 .0044 The nominal load per square inch is the total load divided by the product of the diameter and length of the journal. At the same low speed of 20 revo- lutions per minute it was increased to 676 lbs. per square inch without any signs of heating or seizing. Friction of Car-journal Brasses. (J. E. Denton, Trans. A. S. M. E , xii. 405.) — A new brass dressed with an emery-wheel, loaded with 5000 lbs., may have an actual bearing-surface on the journal, as shown by the polish 938 FKICTIOH AND LUBttlCATiOtf. of a portion of the surface, of only 1 square inch. With this pressure of 5000 lbs. per square inch, the coefficient of friction may be 6%, and the brass may be overheated, scarred and cut but, on the contrary, it may wear down evenly to a smooth bearing, giving a highly polished area of contact of 3 square inches, or more, inside of two hours of running, gradually decreasing the pressure per square inch of contact, and a coefficient of friction of less than 0.5%. A reciprocating motion in the direction of the axis is of importance in reducing the friction. With such polished surfaces any oil will lubricate, and the coefficient of friction then depends on the viscosity of the oil. With a pressure of 1000 lbs per square inch, revolutions from 170 to 320 per minute, and temperatures of 75° to 113° F. with both sperm and parraffine oils, a co- efficient of as low as 0.11$ has been obtained, the oil being fed continuously by a pad. Experiments on Overheating of Bearings.— Hot Boxes. (Denton.)— Tests with car brasses loaded from 1100 to 4500 lbs. per square inch gave 7 cases of overheating out of 32 trials. The tests show how purely a matter of chance is the overheating, as a brass which ran hot at 5000 lbs. load on one day would run cool on a later date at the same or higher pres- sure. The explanation of this apparently arbitrary difference of behavior is that the accidental variations of the smoothness of the surfaces, almost in- finitesimal in their magnitude, cause variations of friction which are always tending to produce overheating, and it is solely a matter of chance when these tendencies preponderate over tbe lubricating influence of the oil. There is no appreciable advantage shown by sperm-oil, when there is no ten- dency to overheat— that is, paraffine can lubricate under the highest pres- sures which occur, as well as sperm, when the surfaces are within the condi- tions affording the minimum coefficients of friction. Sperm and other oils of high heat-resisting qualities, like vegetable oil and petroleum cylinder stocks, only differ from the more volatile lubricants, like paraffine, in their ability to reduce the chances of the continual acci- dental infinitesimal abrasion producing overheating. The effect of emery or other gritty substance in reducing overheating of a bearing is thus explained : The effect of the emery upon the surfaces of the bearings is to cover the latter with a series of parallel grooves, and apparently after such grooves are made the presence of the emery does not practically increase the friction over the amount of the latter when pure oil only is between the surfaces. Tbe infinite number of grooves constitute a very perfect means of insuring a uniform oil supply at every point of the bearings. As long as grooves in the journal match with those in the brasses the friction appears to amount to only about 10% to 15% of the pressure. But if a smooth journal is placed between a set of brasses which are grooved, and pressure be applied, the journal crushes the grooves and becomes brazed or coated with brass, and then the coefficient of friction becomes upward of 40$. If then emery is applied, the friction is made very much less by its presence, because the grooves are made to match each other, and a uniform oil supply prevails at every point of the bearings, whereas before the application or the emery many spots of the latter receive no oil between them. Moment of Friction and Work of Friction of Sliding- surfaces, etc. Moment of Fric- Energy lost by Friction tion, inch-lbs. in ft. -lbs. per miu. Flat surfaces fWS Shafts and journals YzfWd .2618/TTdn Flat pivots YsfWr A745fWrn Collar-bearing %fW *\ ~ *\ .1745/PTn *" 2 ,, ~ *'*' Conical pivot % fWr cosec a A745fWrn cosec a Conical 3ournal %fW r r sec a A745fWrn sec a Truncated-cone pivot HfW 3 ' 2 T r * . l?45/T T r2 T Tx v aj r 2 sin a r 2 sin a Hemispherical pivot fWr .2618/PFr Tractrix, or Schiele's " anti- friction " pivot fWr .2618/PTr. PIVOT-BEARINGS. 939 In the above / — coefficient of friction; W = weight on journal or pivot in pounds; r = radius, d = diameter, in inches; S = space in feet through which sliding takes place; r 2 = outer radius, r 3 = inner radius; n = number of revolutions per minute; a = the half-angle of the cone, i.e., the angle of the slope with the axis. To obtain the horse-power, divide the quantities in the last column by fWdn 33,090. Horse-power absorbed by friction of a shaft = o Kn - n - The formula for energy lost by shafts and journals is approximately true for loosely fitted bearings. Prof. Thurston shows that the correct formula varies according to the character of fit of the bearing; thus for loosely fitted journals, if U = the energy lost, 2fTrr Tir . , , .2618/ Wdn . . „ U = — Wn inch-pounds = foot-lbs. Vi + P V\ +P For perfectly fitted journals U = 2MfnrWn inch-lbs. = . 3325/ Wdn, ft.-lbs. For a bearing in which the journal is so grasped as to give a uniform pressure throughout, U — fit^rWn inch-lbs. — AU2 fWdn, ft.-lbs. Resistance of railway trains and wagons due to friction of trains: f X 2240 Pull on draw-bar = - — ^— pounds per gross ton, in which R is the ratio of the radius of the wheel to the radius of journal. A cylindrical journal, perfectly fitted into a bearing, and carrying a total 'oad. distributes the pressure due to this load unequally on the bearing, the maximum pressure being at the extremity of the vertical radius, while at the extremities of the horizontal diameter the pressure is zero. At any point of the bearing-surface at the extremity of a radius which makes an angle 6 with the vertical radius the normal pressure is proportional to cos 6. If p — normal pressure on a unit of surface, to — total load on a unit of length of the journal, and r = radius of journal, . . K „ to COR 9 w cos = l.olrp, p = < „ — . 1 .oir PIVOT-BEARINGS. The Scliiele Curve.— W. H. Harrison, in a letter to the Am. Machin- ist. 1891, says the Schiele curve is not as good a form for a bearing as the segment of a sphere. He says: A mill-stone weighing a ton frequently bears its whole weight upon the flat end of a hard-steel pivot V/Q' diameter, or one square inch area of bearing; but to carry a weight of 3000 lbs. he advises an end bearing about 4 inches diameter, made in the form of a seg- ment of a sphere about ^ inch in height. The die or fixed bearing should be dished to fit the pivot. This form gives a chance for the bearing to adjust itself, which it does not have when made flat, or when made with the Schiele curve. If a side bearing is necessary it can be arranged farther up the shaf t. The pivot and die should be of steel, hardened; cross-gutters should be in the die to allow oil to flow, and a central oil-hole should be made in the shaft. The advantage claimed for the Schiele bearing is that the pressure is uni- formly distributed over its surface, and that it therefore wears uniformly. Wilfred Lewis (Am. Mach., April 19, 1894) says that its merits as a thrust- bearing have been vastly overestimated; that the term '•anti-friction'" applied to it is a misnomer, since its friction is greater-, than that of a flat step or collar of the same diameter. He advises that flat thrust-bearings should always be annular in form, having an inside diameter one half of the external diameter Friction of a Flat Pivot-bearing.— The Research Committee on Friction (Proc. Inst. M. E. 1891) experimented on a step-bearing, flat- ended, 3 in. diam., the oil being forced into the bearing through a hole in its centre and distributed through two radial grooves, insuring thorough lubrication, The step was of steel and the bearing of manganese-bronze. 940 FMCTIOK AND LUBRICATION. At revolutions per min 50 128 194 290 353 The coefficient of friction varied j .0181 .0053 .0051 .0044 .0053 between I and .0221 .0113 .0102 .0178 .0167 With a white-metal bearing at 128 revolutions the coefficient of friction was a little larger than with the manganese-bronze. At the higher speeds the coefficient of friction was less, owing to the more perfect lubrication, as shown by the more rapid circulation of the oil. At 128 revolutions the bronze bearing heated and seized on one occasion with a load of 260 pounds and on another occasion with 300 pounds per square inch. The white-metal bearing under similar conditions heated and seized with a load of 240 pounds per square inch. The steel footstep on manganese-bronze was after- wards tried, lubricating with three and with four radial grooves; but the friction was from one and a half times to twice as great as with only the two grooves. (See also Allowable Pressures, page 936.) Mercury-hath. Pivot.— A nearly frictionless step-bearing may be obtained by floating the bearing with its superincumbent weight upon mer- cury. Such an apparatus is used in the lighthouses of La Heve, Havre. It is thus described in Eng'g, July 14, 1893, p. 41: The optical apparatus, weighing about 1 ton, rests on a circular cast-iron table, which is supported by a vertical shaft of wrought iron 2.36 in. diameter. This is kept in position at the top by a bronze ring and outer iron support, and at the bottom in the same way, while it rotates on a removable steel pivot resting in a steel socket, which is fitted to the base of the support. To the vertical shaft there is rigidly fixed a floating cast-iron ring 17.1 in. diam- eter and 11.8 in. in depth, which is plunged into and rotates in a mercury bath contained in a fixed outer drum or tank, the clearance between the vertical surfaces of the drum and ring being only 0.2 in., so as to reduce as much as possible the volume of mercury (about 220 lbs.), while the horizon- tal clearance at the bottom is 0.4 in. BALL-BEARINGS, FRICTION ROLLERS, ETC. A. H. Tyler (Encfg, Oct. 20, 1893, p. 483), after experiments and com- parison with experiments of others arrives at the following conclusions: That each ball must have two points of contact only. The balls and race must be of glass hardness, and of absolute truth. The balls should be of the largest possible diameter which the space at disposal will admit of. Any one ball should be capable of carrying the total load upon the bearing. Two rows of balls are always sufficient. A ball-bearing requires no oil, and has no tendency to heat unless over- loaded. Until the crushing strength of the balls is being neared, the frictional re- sistance is proportional to the load. The frictional resistance is inversely proportional to the diameter of the balls, but in what exact proportion Mr. Tyler is unable to say. Probably it varies with the square. The resistance is independent of the number of balls and of the speed. No rubbing action will take place between the balls, and devices to guard against it are unnecessary, and usually injurious. The above will show that the ball-bearing is most suitable for high speeds and light loads. On the spindles of wood-carving machines some make as much as 30.000 revolutions per minute. They run perfectly cool, and never have any oil upon them. For heavy loads the balls should not be less than two thirds the diameter of the shaft, and are better if made equal to it. Ball-hearings have not been found satisfactory for thrust-blocks, for the reason apparently that the tables crowd together. Better results have been obtained from coned rollers. A combined system of rollers and balls is described in Encfo. Oct. 6, 1893, p. 429. Friction-rollers. —If a journal instead of revolving on ordinary bearings be supported on friction-rollers the force required to make the jour- nal revolve will be reduced in nearly the same proportion that the diameter of the axles of the rollers is less than the diameter of the rollers themselves. In experiments by A. M. Wellington with a journal 3Vs in. diam. supported on rollers 8 in. diam., whose axles were Yji in. diam., the friction in starting from rest was J4 the friction of an ordinary 3J^-in. bearing, but at a car speed of 10 miles per hour it was % that of the ordinary bearing. The ratio of the diam. of the axle to diam. of roller was 1%: 8, or as 1 to 4.6, FRICTION OF STEAM-ENGINES. 941 Bearings for Very High Rotative Speeds. (Proc. Inst. M. E., Oec. 1888, p. 48-'.) — Iu the Parsons steam-turbine, which has a speed of as high as 18,000 i ev. per min., as it is impossible to secure absolute accuracy of balance, the bearings are of special construction so as to allow of a certain very small amount of lateral freedom. For this purpose the bearing is sur- rouudel by two sets of steel washers 1/16 inch thick and of different diam- eters, the larger fitting close in the casing and about 1/32 inch clear of the bearing, and the smaller fitting close on the bearing and about 1/32 inch clear of the casing. These are arranged alternately, and are pressed together by a spiral spring. Consequently any lateral movement of the bearing causes them to slide mutually against one another, and by their friction to check or damp any vibrations that may be set up in the spindle. The tendency of the spindle is then to rotate about its axis of mass, or prin- cipal axis as it is called; and the bearings are thereby relieved from exces- sive pressure, and the machine from undue vibration. The finding of the centre of gyration, or rather allowing the turbine itself to find its own centre of gyration, is a well-known device in other branches of mechanics: as in the instance of the centrifugal hydro-extractor, where a mass very much out of balance is allowed to find its own centre of gyration ; the faster it ran the more steadily did it revolve and the less was the vibration. An- other illustration is to be found in the spindles of spinning machinery, which run at about 10,000 or 11.000 revolutions per minute: they are made of hardened and tempered steel, and although of very small dimensions, the outside diameter of the largest portion or driving whorl being perhaps not more than 1J4 in., it is found impracticable to run them at that speed in what might be called a hard-and-fast bearing. They are therefore run with some elastic substance surrounding the bearing, such as steel springs, hemp, or cork. Any elastic substance is sufficient to absorb the vibration, and permit of absolutely steady running. FRICTION OF STEAM-ENGINES. Distribution of the Friction of Engines.— Prof . Thurston in Irs " Friction and Lost Work," gives the following: 1. Main bearings. , 47.0 Piston and rod 32.9 Crank-pin 6.8 Cross-head and wrist-pin 5.4 Valve and rod 2.5 Eccentric strap. - 5.3 Link and eccentric Total 100.0 100.0 100.0 No. 1, Straight-line, 6" X 12", balanced valve; No. 2, Straight-line, 6" X 12", unbalanced valve; No. 3, 7" X 10", Lansing traction locomotive valve-gear. Prof. Thurston's tests on a number of different styles of engines indicate that the friction of any engine is practically constant under all loads. (Trans. A. S. M. E., viii. 86; ix. 74.) In a Straight-line engine, 8" X 14", I.H.P. from 7.41 to 57.54, the friction H. P. varied irregularly between 1.97 and 4.02, the variation being independent of the load. With 50 H.P. on the brake the I.H.P. was only 52.6, the friction being only 2.6 H.P., or about 5%. In a compound condensing-engine, tested from to 102.6 brake H.P., gave I.H.P. from 14.92 to 117.8 H.P., the friction H.P. varying only from 14.92 to 17.42. At the maximum load the friction was 15.2 H.P., or 12.9^. The friction increases with increase of the boiler-pressure from 30 to 70 lbs., and then becomes constant. The friction generally increases with in- crease of speed, but there are exceptions to this rule. Prof. Denton (Stevens Indicator, July, 1890), comparing the calculated friction of a number of engines with the friction as determined by measure- ment, finds that iu one case, a 75-ton ammonia ice-machine, the friction of the compressor, 17J^ H.P., is accounted for by a coefficient of friction of 7}?&% on all the external bearings, allowing 6$ of the entire friction of the machine for the friction of pistons, stuffing-boxes, and valves. In the case of the Pawtucket pumping-engine, estimating the friction of the external bearings with a coefficient of friction of 6% and that of the pistons, valves, and stuff- ing-boxes as in the case of the ice-machine, we have the total friction disiributed as follows : 2. 3. 35.4 35.0 25.0 21.0 5.1| 4.1 f 13.0 26.4/ 4.0) 22.0 9.01 942 FRICTION AND LUBRICATION. Horse- Per cent power, of Whole. Crank-pins and effect of piston-thrust on main shaft.. 0.71 11.4 Weight of fly-wheel and main shaft 1.95 32.4 Steam-valves 0.23 3.7 Eccentric 0.07 1.2 Pistons 0.43 7.2 Stuffing-boxes, six altogether 0.72 11.3 Air-pump 2.10 32 . 8 Total friction of engine with load 6.21 100.0 Total friction per cent of indicated power ... 4.27 The friction of this engine, though very low in proportion to the indicated power, is satisfactorily accounted for by Morin's law used with a coefficient of friction of 5%. In both cases the main items of friction are those due to i the weight of the fly-wheel and main shaft and to the piston-thrust on crank-pins and main-shaft bearings. In the ice-machine the latter items are the larger owing to the extra crank pin to work the pumps, while I in the Pawtucket engine the former preponderates, as the crank-thrusts are partly absorbed by the pump-pistons, and only the surplus effect acts on the crank -shaft. Prof. Denton describes in Trans. A. S. M. E., x. 392, an apparatus by which he measured the friction of a piston packing-ring. When the parts of the piston were thoroughly devoid of lubricant, the coefficient of friction was found to be about 7*4%; with an oil-feed of one drop in two minutes the coefficient was about 5$; with one drop per minute it was about 3%. These rates of feed gave unsatisfactory lubrication, the piston groaning at the ends of the stroke when run slowly, and the flow of oil left upon the surfaces was found by analysis to contain about 50$ of iron. A feed of two drops per minute reduced the coefficient of friction to about 1%, and gave practically perfect lubrication, the oil retaining its natural color and purity. LUBRICATION. Measurement of the Durability of Lubricants. (J. E. Den- ton, Trans. A. S. M. E., xi. 1013.)— Practical differences of durability of lubri- cants depend not on any differences of inherent ability to resist being "worn out" by rubbing, but upon the rate at which they flow through and away from the bearing-surfaces. The conditions which control this flow are so delicate in their influence that aU attempts thus far made to measure dura- bility of lubricants may be said to have failed to make distinctions of lubri- cating value having any practical significance. In some kinds of service the limit to the consumption of oil depends upon the extent to which dust or other refuse becomes mixed with ir, as in railroad-car lubrication and in the case of agricultural machinery. The economy of one oil over another, so far as the quality used is concerned— that is, so far as durability is concerned— is simply proportional to the rate at which it can insinuate itself into and flow out of minute orifices or cracks. Oils will differ in their ability to do this, first, in proportion to their viscosity, and, second, in proportion to the ca- pillary properties which they may possess by virtue of the particular ingre- dients' used in their composition. Where the thickness of film between rub- bing-surfaces must be so great that large amounts of oil pass through beaiings in a given time, and the surroundings are such as to permit oil to be fed at high temperatures or applied by a method not requiring a perfect fluidity, it is probable that the least amount of oil will be used when the vis- cosity is as great as in the petroleum cylinder stocks. When, however, the oil must flow freely at ordinary temperatures and the feed of oil is restricted, as in the case of crank-pin bearings, it is not practicable to feed such heavy oils in a satisfactory manner. Oils of less viscosity or of a fluidity approximating to lard-oil must then be used. Relative Value of Lubricants. (J. E.Denton, Am. Much., Oct. 30, 1890.)— The three elements which determine the value of a lubricant are the cost due to consumption of lubricants, the cost spent for coal to overcome the fiictional resistance caused by use of the lubricant, and the cost due to the metallic wear on the journal and the brasses. In cotton-mills the cost of the power is alone to be considered; in rolling-mills and marine engines the cost of the quantity of lubricant used is the only important factor; but in railroads not only do both these elements enter ihe problem as tangible LUBRICATION. 943 factors, but the cost of the wearing away of the metallic parts enters in ad- dition, and furthermore, the latter is the'greatest element of cost in the case. The Qualifications of a Good. Lubricant, as laid down by W. H. Bailey, in Proc. Inst. C. E., vol. xlv., p. '672, are: 1. Sufficient body to keep the surfaces free from contact under maximum pressure, 2. The greatest possible fluidity consistent with the foregoing condition. 3. The lowest possible coefficient of friction, which in bath lubrication would be for fluid friction approximately. 4. The greatest capacity for storing and carrying away heat. 5. A high temperature of decomposition. 6. Power to resist oxidation or the action of the atmosphere. 7. Freedom from cor- rosive action on the metals upon which used. Best Lubricants for Different Purposes. (Thurston.) Low temperatures, as in rock-drills j T .,. ^ ; „ /i ,.„ 1 i„u„i„„ f; „„ ^;i„ driven by compressed air: } Ll S ht mineral lubncating-oils. Very great pressures, slow speed... -j G Stt' nt s s oapstone ' and other Solid Heavy pressures, with slow speed. . . -j T * e re £ s ™' a ' nd lard ' tallow ' and other Heavy pressures and high speed. . . . { Sp £Xus'. ca8tor_oi1 ' and heavy min " Light pressures and high speed -j ^X^-seed* petroleum ' ° live ' rape ' Ordinarv machinerv i Lard-oil, tallow-oil, heavy mineral oils, Ordinary machineiy -, ftnd the heavier vegetable oils. Steam -cylinders Heavy mineral oils, lard, tallow. Watches and other delicate media ( Clarified sperm, neat's-foot, porpoise, w n a S s ana otner aellcate mecna-; oliv6) and ljght mineral lubricating nifem - ( oils. For mixture with mineral oils, sperm is best: lard is much used; olive and cotton-seed are good. Amount of Oil needed to Run an Engine.— The Vacuum Oil Co. in 1892, in response to an inquiry as to cost of oil to run a 1000-H.P. Corliss engine, wrote: The cost of running two engines of equal size of the same make is not always the same. Therefore while we could furnish figures showing what it is costing some of our customers having Corliss engines of 1000 H.P., we could only give a general idea, which in itself might be considerably out of the way as to the probable cost of cylinder- and engine-oils per year for a particular engine. Such an engine ought to run readily on less than 8 drops of 600 W oil per minute. If 3000 drops are figured to the quart, and 8 drops used per minute, it would take about two and one half barrels (52.5 gallons) of 600 W cylinder-oil, at 65 cents per gallon, or about $85 for cylinder-oil per year, running 6 days a week and 10 hours a day. Engine-oil would be even more difficult to guess at what the cost would be, because it would depend upon the number of cups required on the engine, which varies somewhat according to the style of the engine. It would doubtless be safe, however, to calculate at the outside that not more than twice as much engine-oil would be required as of cylinder-oil. The Vacuum Oil Co. in 1892 published the following results of practice with " 600 W " cylinder-oil: Corliss compound engine J 20 and 33 X 48; 83 revs - P er min - ; 1 dro P of oil oornss compound engine, ( per mjn tQ 1 drop in twQ minutes " triple exp. " 20, 33, and 46 X 48; 1 drop every 2 minutes. Porter Allen " -I 20 and 36 X 36; 143 revs - V ev min -' 2 drops of oil ( per min., reduced afterwards to 1 dropper min. tj-m u J 15 X 25 X 16; 240 revs, per min.; 1 drop every 4 cau 1 minutes. Results of tests on ocean-steamers communicated to the author by Prof. Denton in 1892 gave: for 1200-H.P. marine engine, 5 to 6 English gallons (6 to 7.2 U. S. gals.) of engine-oil per 24 hours for external lubrication; and for a 1500-H.P. marine engine, triple expansion, running 75 revs, per min., 6 to 7 English gals, per 24 hours. The cylinder-oil consumption is exceedingly variable, — from 1 to 4 gals, per day on different engines, including cylinder- oil used to swab the piston-rods. Quantity of Oil used on a Locomotive Crank-pin.— Prof. Denton. Trans. A. S. M. E., xi. 1020, says: A very economical case of practical oi'-consiunption is when a locomotive main crank-pin consumes about six 944 FRICTION AND LUBRICATIOH. cubic inches of oil in a thousand miles of service. This is equivalent to a consumption of one milligram to seventy square inches of surface rubbed over. The Examination of Lubricating-oils. (Prof. Thos. B. Still- man, Stevens Indicator, July, 1890.) — The generally accepted conditions of a good lubricant are as follows: 1. " Body " enough to prevent the surfaces, to which it is applied, from coming in contact with each other. (Viscosity.) 2. Freedom from corrosive acid, either of mineral or animal origin. 3. As fluid as possible consistent with " body." 4. A minimum coefficient of friction. 5. High "flash 1 ' and burning points. 6. Freedom from all materials liable to produce oxidation or " gumming." The examinations to be made to verify the above are both chemical and mechanical, and are usually arranged in the following order : 1. Identification of the oil, whether a simple mineral oil, or animal oil, or a mixture. 2. Density. 3. Viscosity. 4. Flash-point. 5. Burning -point. 6. Acidity. 7. Coefficient of friction. 8. Cold test. Detailed directions for making all of the above tests are given in Prof. St ill man's article. Weights of Oil per Gallon.— The following are approximately the weights per gallon of different kinds of oil (Penn. R. R. Specifications): Lard-oil, tallow -oil, neat's-foot oil, bone-oil, colza-oil, mustard-seed oil, rape-seed oil, paraffine-oil, 500° fire-test oil, engine-oil, and cylinder lubricant, 7$4 pounds per gallon. Well-oil and passenger-car oil, 7.4 pounds per gallon; navy sperm-oil, 7.2 pounds per gallon; signal -oil, 7.1 pounds per gallon; 300° burning-oil, 6.9 pounds per gallon; and 150° burning-oil, 6.6 pounds per gallon. Penna. R. R. Specifications for Petroleum Products. 1889. — Five different grades of petroleum products will be used. The materials desired under this specification are the products of the dis- tillation and refining of petroleum unmixed with any other substances. 150° Fire-test Oil.— This grade of oil will not be accepted if sample (1) is not "water-white" in color; (2) flashes below 130° Fahrenheit; (3) burns below 151° Fahrenheit; (4) is cloudy or shipment has cloudy barrels when received, from the presence of glue or suspended matter; (5) becomes opaque or shows cloud when the sample has been 10 minutes at a temper- ature of 0° Fahrenheit. The flashing and burning points are determined by heating the oil in an open vessel, not less than 12° per minute, and applying the test flame every 7°, beginning at 123° Fahrenheit. The cold test may be conveniently made by having an ounce of the oil, in a four-ounce sample bottle, with a ther- mometer suspended in the oil, and exposing this to a freezing mixture of ice and salt. It is advisable to stir with the thermometer while the oil is cooling. The oil must remain transparent in the freezing mixture ten minutes after it has cooled to zero. 300° Fire-test Oil.— This grade of oil will not be accepted if sample (1) is not "water white " in color; (2) flashes below 249° Fahrenheit; (3) burns below 298° Fahrenheit; (4) is cloudy or shipment has cloudy barrels when received, from the presence of glue or suspended matter; (5) becomes opaque or shows cloud when the sample has been 10 minutes at a temper- ature of 32° Fahrenheit. The flashing and burning points are determined the same as for 150° fire- test oil. except that the oil is heated 15° per minute, test-flame being applied first at 212° Fahrenheit. The cold test is made the same as above, except that ice and water are used. Paraffine-oil. — This grade of oil will not be accepted if the sample (1) is other than pale-lemon color; (2) flashes below 249° Fahrenheit; (3) shows viscosity less than 40 seconds or more than 65 seconds when tested as described under " Well Oil " at 100° Fahrenheit throughout the year; (4) has gravity at 60° Fahrenheit, below 24° Baume, or above 29° Baume; (5) from October 1st to May 1st has a cold test above 10° Fahrenheit. The flashing-point is determined same as for 300° fire-test oil. The cold test is determined as follows: A couple of ounces of oil is put in a four-ounce sample bottle, and a thermometer placed in it. The oil is then frozen, a freezing mixture of ice and salt being used if necessary. When the oil has become hard, the bottle is removed from the freezing mixture and the frozen oil allowed to soften, being stirred and thoroughly mixed at the same time by means of the thermometer, until the mass will run from one end of SOLID LUBRICANTS. 945 the bottle to the other. The reading of the thermometer when this is the case is regarded as the cold test of the oil. Well Oil.— This grade of oil will not be accepted if the sample (1) flashes, from May 1st to October 1st, below 249° Fahrenheit, or from October 1st to May 1st below 200° Fahrenheit; (2) has a gravity, at 60° Fahrenheit, below 28° Bauine. or above 30°; (3) from October 1st to May 1st has a cold test above 10° Fahrenheit; (4) shows any precipitation in 10 minutes when 5 cubic centimetres are mixed with 95 cubic centimetres of 88° gasoline; (5) shows a viscosity less than 55 seconds, or more than 100 seconds, when tested as described below. From October 1st to May 1st the test must be made at 100° Fahrenheit, and from May 1st to October 1st at 110° Fahrenheit. For summer oil the flashing-point is determined the same as for paraffine- oil; and for winter oil the same, except that the test-flame is applied first at 193° Fahrenheit. The cold test is made the same as for parafflne-oil. The precipitation test is to exclude tarry and suspended matter. It is easiest made by putting 5 cubic centimetres of the oil in a 100-cubic -cen- timetre graduate, then filling to the mark with gasoline, and thoroughly shaking. The viscosity test is made as follows: A 100 cubic -centimetre pipette of the long bulb form is regraduated to hold just 100 cubic centimetres to the bottom of the bulb. The size of the aperture at the bottom is then made such that 100 cubic centimetres of water at 100° Fahrenheit will run out the pipette down to the bottom of the bulb in 34 seconds. Pipettes with bulbs varying from \% inches to \]4, inches in diameter outside, and about 4]4 inches long give almost exactly the same results, provided the aperture at the bottom is the proper size. The pipette being obtained, the oil sample is heated to the required temperature, care being taken to have it uniformly heated, and then is drawn up into tbe pipette to the proper marK. The time occupied by the oil in running out, down to the bottom of the bulb, gives the test figures. 500° Fire-test Oil— This grade of oil will not be accepted if sample (1) flashes below 415° Fahrenheit; (2) shows precipitation with gasoline when tested as described for well -oil. The flashing-point is determined the same as for well-oil, except that the test flame is applied first at 438° Fahrenheit. SOLID LUBRICANTS. Graphite in a condition of powder and used as a solid lubricant, so called, to distinguish it from a liquid lubricant, has been found to do well where the latter has failed. Rennie, in 1829, says : " Graphite lessened friction in all cases where it was used." General Morin, at a later date, concluded from experiments that it could be used with advantage under heavy pressures; and Prof. Thurston found it well adapted for use under both light and heavy pressures when mixed with certain oils. It is especially valuable to prevent abrasion and cutting under heavy loads and at low velocities. Soapstone, also called talc and steatite, in the form of powder and mixed with oil or fat, is sometimes used as a lubricant. Graphite or soap- stone, mixed with soap, is used on surfaces of wood working against either iron or wood. Fifore-grapliite.— A new self-lubricating bearing known as fibre- graphite is described by John H. Cooper in Trans. A. S. M. E., xiii. 374, as the invention of P. H. Holmes, of Gardiner, Me. This bearing material is composed of selected natural graphite, which has been finely divided and freed from foreign and gritty matter, to which is added wood-fibre or other growth mixed in water in various proportions, according to the purpose to be served, and then solidified by pressure in specially prepared moulds ; after removal from which the bearings are first thoroughly dried, then satu- rated with a drying oil. and finally subjected to a current of hot, dry air for the purpose of oxidizing the oil, and hardening the mass. When finished, they may be " machined " to size or shape with the same facility and means employed on metals. Metaline is a solid compound, usually containing graphite, made in the form of small cylinders which are fitted permanently into holes drilled in the surface of the bearing. The bearing thus fitted runs without any other lubrication. 946 THE FOUHDRY. THE FOUNDRY. CXJPOL.A PRACTICE. The following notes, with the accompanying table, are taken from an article by Simpson Bolland in American Machinist, June 30, 1892. The table shows heights, depth of bottom, quantity of fuel on bed, proportion of fuel and iron in charges, diameter of main blast-pipes, number of tuyeres, blast - pressure, sizes of blowers and power of engines, and melting capacity per hour, of cupolas from 24 inches to 84 inches in diameter. Capacity of Cupola.— The accompanying table will be of service in deter- mining the capacity of cupola needed for the production of a given quantity of iron in a specified time. First, ascertain the amount of iron which is likely to be needed at each cast, and the length of time which can be devoted profitably to its disposal; and supposing that two hours is all that can be spared for that purpose, and that ten tons is the amount which must be melted, find in the column, Melt- ing Capacity per hour in Pounds, the nearest figure to five tons per hour, which is found to be 10,760 pounds per hour, opposite to which in the column Diameter of Cupolas, Inside Lining, will be found 48 inches ; this will be the size of cupola required to furnish ten tons of molten iron in two hours. Or suppose that the heats were likely to average 6 tons, with an occasional increase up to ten, then it might not be thought wise to incur the extra ex- pense consequent on working a 48-inch cupola, in which case, by following the directions given, it will be found that a 40-inch cupola would answer the purpose for 6 tons, but would require an additional hour's time for melting whenever the 10-ton heat came along. ; The quotations in the table are not supposed to be all that can be melted in the hour by some of the very best cupolas, but are simply the amounts which a common cupola under ordinary circumstances may be expected to melt in the time specified. Height of Cupola.— By height of cupola is meant the distance from the base to the bottom side of the charging hole. Depth of Bottom of Cupola.— Depth of bottom is the distance from the sand-bed, after it has been formed at the bottom of the cupola, up to the under side of the tuyeres. All the amounts for fuel are based upon a bottom of 10 inches deep, and any departure from this depth must be met by a corresponding change in the quantity of fuel used on the bed ; more in proportion as the depth is increased, and less when it is made shallower. Amount of Fuel Required on the Bed. — The column " Amount of Fuel re- quired on Bed. in Pounds" is based on the supposition that the cupola is a straight one all through, and that the bottom is 10 inches deep. If the bot- tom be more, as in those of the Colliau type, then additional fuel will be needed. The amounts being given in pounds, answer for both coal and coke, for, should coal be used, it would reach about 15 inches above the tuyeres ; the same weight of coke would bring it up to about 22 inches above the tuyeres, which is a reliable amount to stock with. First Charge of Iron. — The amounts given in this column of the table are safe figures to work upon in every instance, yet it will always be in order, after proving the ability of the bed to carry the load quoted, to make a slow and gradual increase of the load until it is fully demonstrated just how much burden the bed will carry. Succeeding Charges of Fuel and Iron. — In the columns relating to succeed- ing charges of fuel and iron, it will be seen that the highest proportions are not favored, for the simple reason that successful melting with any greater proportion of iron to fuel is not the rule, but, rather, the exception. When- ever we see that iron has been melted in prime condition in the proportion of 12 pounds of iron to one of fuel, we may reasonably expect that the talent, material, and cupola have all been up to the highest degree of excellence. Diameter of Main Blast -pipe. —The table gives the diameters of main blast-pipes for all cupolas from 24 to 84 inches diameter. The sizes given opposite each cupola are of sufficient area for all lengths up to 100 feet. CUPOLA PRACTICE. 947 43d Xipud _ 1,500 2,000 2,500 3,000 3,500 4,000 4,820 5,640 6,460 7,550 8,640 9,730 10,760 11,790 12.820 13.850 14,880 15,910 16,940 18,340 19,770 21,200 22,630 24,060 26,070 27,980 29,890 31.800 33,710 35,620 37.530 -"a jo -d'H = HrH rtW(M«cowC'^ . io io io in oo i-i r-l r^ N Ql « 7( J) II CO TO CO ra M CO Tf MaA^Ojg JUBAajJlHS JO S3Zl S il 0?CN!C?MeOCOrf^TfinininCCcOOl>i>t-OOQOaOGOOOGCOS05050^05CiO uaA^gjooH -i'3 3" 'cTH sno^'niOAaH 250 210 250 333 241 282 325 235 270 304 212 256 277 170 ISO 192 208 225 240 142 150 163 175 187 200 140 148 ieo •pajtnb -a.i jaMojg joog jo sazig ■z. •jHNSJi-H ilrti-iMM«MM«'*'*'fl''*lOI01fllOlOin!0!0!0!e©fflt-Nt- •pa.imba.1 •ssa-id'-jstqg o COCDCONJ>t-XQOCCOOO««W^itTt<'*if!tii*i*'*tO!OtOtOtO!0«0 || 1 111 ^11 ^^^^^^^^^ ^1^1\ffi!\C-i\«>\0!\5S!\O7\53^ ^ ^ ^ ^ ^ ^ \e}\S)\SQ\91^}\Si 5* OJ OJ ?> OJ O? ">! ~> ?! "N ~> 7} ~> CO CO CO CO CO CO CO CO CO CO CO CO CO « S . -■■■■ ■ ' ■ ■■- ■ '. ' - ■■:■■.-■'■-■'■■:..■: .. :.'.''■' .;- £ !?J(NC0C0^Tt'T)iCOC0COCOC0C000Q00CQC000000aC00OOO9*ff}0JC\frt»iO!? 1 WKico?:co-i i> i> co oo oo 948 THE FOUNDRY. Tuyeres for Cupola. — Two columns are devoted to the number and sizes of tuyeres requisite for the successful working of each cupola ; one gives the number of pipes 6 inches diameter, and the other gives the number and dimensions of rectangular tuyeres which are their equivalent in area. From these two columns any other arrangement or disposition of tuyeres may be made, which shall answer in their totality to the areas given in the table. When cupolas exceed 60 inches in diameter, the increase in diameter should begin somewhere above the tuyeres. This method is necessary in all common cupolas above 60 inches, because it is not possible to force the blast to the middle of the stock, effectively, at any greater diameter. On no consideration must the tuyere area be reduced; thus, an 84-inch cupola must have tuyere area equal to 31 pipes 6 inches diameter, or 16 flat tuyeres 16 inches by 13J^ inches. if it is found that the given number of flat tuyeres exceed in circumference that of the diminished part of the cupola, they can be shortened, allowing the decreased length to be added to the depth, or they may be built in on end; by so doing, we arrive at a modified form of the Blakeney cupola. Another important point in this connection is to arrange the tuyeres in such a manner as will concentrate the fire at the melting-point into the smallest possible compass, so that the metal in fusion will have less space to traverse while exposed to the oxidizing influence of the blast. To accomplish this, recourse has been had to the placing of additional rows of tuyeres in some instances— the "Stewart rapid cupola' 1 having three rows, and the "Colliau cupola furnace" having two rows, of tuyeres. Blast -pressure. — Experiments show that about 30,000 cubic feet of air are consumed in melting a ton of iron, which would weigh about 2400 pounds, or more than both iron and fuel. When the proper quantity of air is sup- plied, the combustion of the fuel is perfect, and carbonic-acid gas is the result. When the supply of air is insufficient, the combustion is imperfect, and carbonic-oxide gas is the result. The amount of heat evolved in these two cases is as 15 to 4^ showing a loss of over two thirds of the heat by im- perfect combustion. It is not always true that we obtain the most rapid melting when we are forcing into the cupola the largest quantity of air. Some time is required to elevate the temperature of the air supplied to the point that it will enter into combustion. If more air than this is supplied, it rapidly absorbs heat, reduces the temperature, and retards combustion, and the fire in the cupola may be extinguished with too much blast. Slag in Cupolas.— A certain amount of slag is necessary to protect the molten iron which has fallen to the bottom from the action'of the blast ; if it was not there, the iron would suffer from decarbonization. When slag from any cause forms in too great abundance, it should be led away by inserting a hole a little below the tuyeres, through which it will find its way as the iron rises in the bottom. In the event of clean iron and fuel, slag seldom forms to any appreciable extent in small heats ; this renders any preparation for its withdrawal un- necessary, but when the cupola is to be taxed to its utmost capacity it is then incumbent on the melter to flux the charges all through the heat, car- rying it away in the manner directed. The best flux for this purpose is the chips from a white marble yard. About 6 pounds to the ton of iron will give good results when all is clean. When fuel is bad, or iron is dirty, or both together, it becomes imperative that the slag be kept running all the time. Fuel for Cupolas.— -The best fuel for melting iron is coke, because it re- quires less blast, makes hotter iron, and melts faster than coal. When coal must be used, care should be exercised in its selection. All anthracites which are bright, black, hard, and free from slate, will melt iron admirably. The size of the coal used affects the melting to an appreciable extent, and, for the best results, small eupolas should be charged with the size called l 'egg," a still larger grade for medium-sized cupolas, and what is called " lump " will answer for all large cupolas, when care is taken to pack it carefully oti the charges. Charging a Cupola.— Ohas. A. Smith (Am. Mach , Feb. 12, 1891) gives the following: A 28-in. cupola should have from 300 to 400 pounds of coke on bottom bed; a 36-in. cupola, 700 to 800 pounds; a 48-in. cupola, 1500 lbs.; and a 60-in. cupola should have one ton of fuel on bottom bed. To every pound of fuel on the bed, three, and sometimes four pounds of metal can be added with safety, if the cupola has proper blast; in after-charges, to every CUPOLA PKACTICE. 949 pound of fuel add 8 to 10 pounds of metal; any well-constructed cupola will stand ten. F. P. WolcottM»u. Mach., Mar. 5, 1891) gives the following as the practice of the Colwell Iron-works, Carteret, N. J.: " We melt daily from twenty to forty tons of iron, with an average of 11.2 pounds of iron to one of fuel. In a 36-in. cupola seven to nine pounds is good melting, but in a cupola that lines up 48 to 60 inches, anything less than nine pounds shows a defect in arrangement of tuyeres or strength of blast, or in charging up." "The Moulder's Text-book, 1 ' by Thos. D. West, gives forty-six reports in tabular form of cupola practice in thirty States, reaching from Maine to Oregon. Cupola Charges in Stove-foundries. (Iron Age, April 14, 1892.) No two cupolas are charged exactly the same. The amount of fuel on the bed or between the charges differs, while varying amounts of iron are used in the charges. Below will be found charging-lists from some of the prom- inent stove-foundries in the country : lbs. A— Bed of fuel, coke 1,500 First charge of iron 5,000 All other charges of iron . . 1,000 First and second charges of coke, each 200 Four next charges of coke, each 150 Six next charges of coke, each 120 Nineteen next charges of coke, each 100 Thus for a melt of 18 tons there would be 5120 lbs. of coke used, giving a ratio of 7 to 1. Increase the amount of iron melted to 24 tons, and a ratio of 8 pounds of iron to 1 of coal is obtained. lbs. -Bed of fuel, coke 1,600 First charge of iron 1,800 First charge of fuel 150 All other charges of iron, each 1,000 Second and third charges of fuel 130 All other charges of fuel, each 100 For an 18- ton melt 5060 lbs. of coke would be necessary, giving a ratio of 7.1 lbs. of iron to 1 pound of coke. lbs. -Bed of fuel, coke 1,600 First charge of iron 4,000 First and second charges of coke . . 200 All other charges of iron . . All other charges of coke . lbs. 2,000 150 In a melt of 18 tons 4100 lbs. of coke would be used, or a ratio of 8.5 to 1. lbs. I lbs. D— Bed of fuel, coke 1,800 | All charges of coke, each 200 First charge of iron 5,600 | All other charges of iron 2,900 In a melt of 18 tons, 3900 lbs. of fuel would be used, giving a ratio of 9.4 pounds of iron to 1 of coke. Very high, indeed, for stove-plate. lbs. All other charges of iron, each 2,000 All other charges of coal, each 175 E— Bed of fuel, coal 1,900 First charge of iron 5,000 First charge of coal 200 In a melt of 18 tons 4700 lbs. of coal would be used, giving a ratio of 7.7 lbs. of iron to 1 lb. of coal. These are sufficient to demonstrate the varying practices existing among different stove-foundries. In all these places the iron was proper for stove- plate purposes, and apparently there was little or no difference in the kind of work in the sand at the different foundries. Results of Increased ©riving. (Erie City Iron-works, 1891.)— May— Dec. 1890: 60-in. cupola, 100 tons clean castings a week, melting 8 tons per hour; iron per pound of fuel, 7^ lbs. ; percent weight of good castings to iron charged, 75%. Jan. -May, 1891 : Increased rate of melting to 1 \}4 tons per hour; iron per lb. fuel, 9^; per cent weight of good castings, 75; one week, 1334 tons per hour, 10.3 lbs. iron per lb. fuel; per cent weight of good cast- ings, 75.3. The increase was made by putting in an additional row of tuyeres and using stronger blast, 14 ounces. Coke was used as fuel. (W. O. Webber, Trans. A. S. M. E. xii. 1045.) 95 ) THE FOUKDLiY. I Buffalo Steel Pressure-Mowers. t Speeds and Capacities t as applied to Cupolas. 1 03 . 5.S leg o °«£ Ceo sw 03 O ft i o a 6 03 °c« o ft ftw" 3t, «_, 03 O ft o £ o 3 6 11 03 c-2 - fc 5.8 m « 3301 12. 46 11 73 12 1639 21978 5861 23.9 14 1777 23838 6357 30.3 68 12 88 12 1639 32395 8636 35.2 14 1777 35190 9384 43.7 In the table are given two different speeds and pressures for each size of blower, and the quantity of iron that may be melted, per hour, with each. In all cases it is recommended to use the lowest pressure of blast that will do the work. Run up to the speed given for that pressure, and regulate quan- tity of air by the blast-gate. The tuyere area should be at least one ninth of the area of cupola in square inches, with not less than four tuyeres at equal distances around cupola, so as to equalize the blast throughout. Va- riations in temperature affect the working of cupolas materially, hot weather requiring increase in volume of air. (For tables of the Sturtevant blower see pages 519 and 520.) Loss in Melting Iron in Cupolas.— G. O. Vair, Am. Mach., March 5, 1891, gives a record of a 45-in. Colliau cupola as follows: Ratio of fuel to iron, 1 to 7.42. Good castings 21,314 lbs. New scrap 3,005 " Millings 200 " Loss of metal 1,481 " Amount melted 26,000 lbs. Loss of metal, 5.69$. Ratio of loss, 1 to 17.55. Use of Softeners in Foundry Practice. (W. Graham, Iron Age, June 27, 18S9.)— In the foundry the problem is to have the right proportions of combined and graphitic carbon in the resulting casting; this is done by getting the proper proportion of silicon. The variations in the proportions of silicon afford a reliable and inexpensive means of producing a cast iron of any required mechanical character which is possible with the material employed. In this way, by mixing suitable irons in the right proportions, a required grade of casting can be made more cheaply than by using irons in which the necessary proportions are already found. If a strong machine casting were required, it would be necessary to keep the phosphorus, sulphur, and manganese within certain limits. Professor Turner found that cast iron which possessed the maximum of the desired qualities contained, graphite, 2.59$; silicon, 1.42$; phosphorus, 0.39$; sul- phur, 0.06$; manganese, 0.58$. A strong casting could not be made if there was much increase in the amount of phosphorus, sulphur, or manganese. Irons of the above percent- ages of phosphorus, sulphur, and manganese would be most suitable for this purpose, but they could be of different grades, having different percentages of silicon, combined and graphitic carbon. Thus hard irons, mottled and white irons, and even steel scrap, all containing low percentages of silicon and high percentages of combined carbon, could be employed if an iron having a large amount of silicon were mixed with them in sufficient amount. This would bring the silicon to the proper proportion and would cause the combined carbon to be forced into the graphitic state, and the resulting SHRINKAGE OP CASTINGS. 951 easting would be soft. High-silicon irons used in this way are called " soft- eners.' 1 The following are typical analyses of softeners: Ferro-silicon. Softeners, American. Scotch Irons, No. 1. Foreign. American. Well- ston. Globe Belle- fonte. Eg- linton Colt- ness. Silicon Combined C. Graphitic C Manganese . . Pliosphorus. . Sulphur 10.55 1.84 0.52 3.86 0.04 0.03 9.80 0.09 1.12 1.95 0.21 0.04 12.08 0.06 1.52 0.76 0.48 Trace 10.34 0.07 1.92 0.52 0.45 Trace 6.67 2^57 0^50 Trace 5.89 0.30 2.85 1.00 1.10 0.02 3 to 6 0.25 3. 0.53 0.35 0.03 2.15 (1.21 3.76 2.80 0.62 0.03 2.59 ' V.70* 0.85 0.01 (For other analyses, see pages 371 to 373.) Ferro -silicons contain a low percentage of total carbon and a high per- centage of combined carbon. Carbon is the most important constituent of cast iron, and there should be about 3.4$ total carbon present. By adding ferro-silicon which contains only 2% of carbon the amount of carbon in the resulting mixture is lessened. Mr. Keep found that more silicon is lost during the remelting of pig of over 10% silicon than in remelting pig iron of lower percentages of silicon. He also points out the possible disadvantage of using ferro-silicons contain- ing as high a percentage of combined carbon as 0.70$ to overcome the bad effects of combined carbon in other irons. The Scotch irons generally contain much more phosphorus than is desired in irons to be employed in making the strongest castings. It is a mistake to mix with strong low-phosphorus irons an iron that would increase the amount of phosphorus for the sake of adding softening qualities, when soft- ness can be produced by mixing irons of the same low phosphorus. (For further discussion of the influence of silicon see page 365.) Shrinkage of Casting's. — The allowance necessary for shrinkage varies for different kinds of metal, and the different conditions under which they are cast. For castings where the thickness runs about one inch, cast under ordinary conditions, the following allowance can be made: For cast-iron, % inch per foot. " brass, 3/16 " " " " steel, % " " " " mal. iron, % " " " For zinc, 5/16 inch per foot. " tin, 1/12 " " " " aluminum, 3/16 " " " " Britannia, 1/32 " " " Thicker castings, under the same conditions, will shrink less, and thinner ones more, than this standard. The quality of the material and the manner of moulding and cooling will also make a difference. Numerous experiments by W. J. Keep (see Trans. A. S. M. E., vol. xvi.) showed that the shrinkage of cast iron of a given section decreases as the percentage of silicon increases, while for a given percentage of silicon the shrinkage decreases as the section is increased. Mr. Keep gives the follow- ing table showing the approximate relation of shrinkage to size and per- centage of silicon: Sectional Area of Casting. Percentage of Yz" n 1" D 1" X 2" 2" D 3" □ 4" D Silicon. Shrinkage in Decimals of an inch per foot of Length. 1. .183 .158 .146 .130 .113 .102 1.5 .171 .145 .133 .117 .098 .087 2. .159 .133 .121 .104 .085 .074 2.5 .147 .121 .108 .092 .073 .060 3. .135 .108 .095 .077 .059 .045 3.5 .123 .095 .082 .065 .046 .032 952 THE FOUNDRY. Mr. Keep also gives the following " approximate key for regulating foun- dry mixtures" so as to produce a shrinkage of Y & in. per ft. in castings of different sections: Size of casting \i 1 2 3 4 in. sq. Silicon required, per cent 3.25 2.75 2.25 1.75 1.25 percent. Shrinkage of a ^-iu. test-bar. .125 .135 .145 .155 .165 in. per ft. "Weight of Castings determined from Weight of Pattern. (Rose's Pattern-maker's Assistant.) A Pattern weighing One Pound, made of— Will weigh when cast in *« Zinc. Copper. gJJ« »•* Mahogany — Nassau Honduras " Spanish. Pine, red ... " white.. " yellow 10.7 12.9 8.5 12.5 16.7 14.1 10.4 12.7 8.2 12.1 16.1 13.6 lbs. 12.8 15.3 10.1 14.9 19.8 16.7 lbs. 12.2 14.2 19.0 16.0 lbs. 12.5 14.6 19.5 16.5 Moulding Sand. (From a paper on "The Mechanical Treatment of Moulding Sand." by Walter Bagshaw, Proc. Inst. M. E. 1891.)— The chemical composition of sand will affect the nature of the casting, no matter what treatment it undergoes. Stated generally, good sand is composed of 94 parts silica, 5 parts alumina, and traces of magnesia and oxide of iron. Sand con- taining much of the metallic oxides, and especially lime, is to be avoided. Geographical position is the chief factor governing the selection of sand; and whether weak or strong, its deficiencies are made up for by the skill of the moulder. For this reason the same sand is often used for both heavy and light castings, the proportion of coal varying according to the nature of the casting. A common mixture of facing-sand consists of six parts by weight of old sand, four of new sand, and one of coal-dust. Floor-sand requires only half the above proportions of new sand and coal-dust to renew it. Ger- man founders adopt one part by measure of new sand to two of old sand; to which is added coal-dust in the proportion of one tenth of the bulk for large castings, and one twentieth for small castings. A few founders mix street-sweepings with the coal in order to get porosity when the metal in the mould is likely to be a long time before setting. Plumbago is effective in preventing destruction of the sand; but owing to its refractory nature, it must not be dusted on in such quantities as to close the pores and prevent free exit of the gases. Powdered French chalk, soapstone, and other sub- stances are sometimes used for facing the mould; but next to plumbago, oak charcoal takes the best place, notwithstanding its liability to float occasion- ally and give a rough casting. For the treatment of sand in the moulding-shop the most primitive method is that of hand-riddling and treading. Here the materials are roughly pro- portioned by volume, and riddled over an iron plate in a flat heap, where the mixture is trodden into a cake by stamping with the feet; it is turned over with the shovel, and the process repeated. Tough sand can be obtained in this manner, its toughness being usually tested by squeezing a handful into a ball and then breaking it; but the process is slow and tedious. Other things being equal, the chief characteristics of a good moulding-sand are toughness and porosity, qualities that depend on the manner of mixing as well as on uniform ramming. Toughness of Sand.— In order to test the relative toughness, sand mixed in various ways was pressed under a uniform load into bars 1 in. sq. and about 12 in. long, and each bar was made to project further and further over the edge of a table until its end broke off by its own weight. Old sand from the shop floor had very irregular cohesion, breaking at all lengths of projections from ^ in. to 1J^ in. New sand in its natural state held together until an overhang of 2% in. was reached. A mixture of old sand, new sand, and coal-dust Mixed under rollers broke at 2 to 2*4 in. of overhang. " in the centrifugal machine " " 2 " 2J4 " " " through a riddle " " 1% " 2^ " " " SPEED OF CUTTING-TOOLS IN LATHES, ETC. 953 Showing as a mean of the tests only slight differences between the last three methods, but in favor of machine-work. In many instances the frac- tures were so uneven that minute measurements were not taken. Dimensions of Foundry ladles.— The following table gives the dimens ons. inside the lining, of ladles from 25 lbs. to 16 tons capacity. All the ladles are supposed to have straight sides. (Am. Mach., Aug. 4, 1892.) Capacity. Diam. Depth. in. in. 54 56 52 53 49 50 46 48 43 44 39 40 34 35 31 32 27 28 24^ 25 22 22 Capacity. Diam. Depth. i tons 14 " 12 " 10 " 8 " 6 " 4 " 3 " 2 " 1 " %ton .., Y2 " ... H " ... 300 pound: 250 200 150 100 75 50 35 10% 10 ^y 2 13^ 10V6 THE MACHINE-SHOP. SPEED OF CUTTING-TOOLS IN LATHES, MILLING MACHINES, ETC. Relation of diameter of rotating tool or piece, number of revolutions, and cutting-speed : Let d = diam. of rotating piece in inches, n = No. of revs, per min.; S = speed of circumference in feet per minute; 3.82S 12 Approximate rule : No. of revs, per min. — 4 X speed in ft. per min. -r- diam. in inches. Speed of Cut-for Lathes and Planers. (Prof. Coleman Sellers, Stevens'" Indicator, April, 1892.)— Brass may be turned at high speed like wood. Bronze.— A speed of 18 feet per minute can be used with the soft alloys- say 8 to 1, while for hard mixtures a slow speed is required— say 6 feet per minute. Wrought Iron can be turned at 40 feet per minute, but planing-machines that are used for both cast and forged iron are operated at 18 feet per minute. Machinery Steel. — Ordinary, 14 feet per minute; car-axles, etc., 9 feet per minute. Wheel Tires.— 6 feet per minute; the tool stands well, but many prefer to run faster, say 8 to 10 feet, and grind the tool more frequently. Lathes. — The speeds obtainable by means of the cone-pulley and the back gearing are in geometrical progression from the slowest to the fastest. In a well-proportioned machine the speeds hold the same relation through nil the steps. Many lathes have the same speed on the slowest of the cone and the fastest of the back-gear speeds. The Speed of Counter-shaft of the lathe is determined by an assumption of a slow speed with the back gear, say 6 feet per minute, on the largest diameter that the lathe will swing. Example. — A 30-inch lathe will swing 30 inches =, say, 90 inches. circumfer- ence = 7' 6" ; the lowest triple gear should give a speed of 5 or 6 per minute. In turning or planing, if the cutting-speed exceed 30 ft. per minute, so much heat will be produced that the temper will be drawn from the tool. The speed of cutting is also governed by the thickness of the shaving, and by the hardness and tenacity of the metal which is being cut; for instance, in cutting mild steel, with a traverse of % in. per revolution or stroke, and with a shaving about % in. thick, the speed of cutting must be reduced to about 8 ft. per minute. A good average cutting-speed for wrought or e$s$ 954 THE MACHINE-SHOP. iron is 20 ft. per minute, whether for the lathe, planing, shaping, or slotting machine. (Proc. Inst. M. E., April, 1883, p. 248.) Table of Cutting-speeds. Feet per minute. Revolutions per minute. 76.4 152.8 229.2 305 6 382.0 50.9 101.9 153.8 203.7 254.6 38.2 76.4 114.6 152.8 191.0 30.6 61.1 91.7 122.2 152.8 25.5 50.9 76.4 101.8 127.3 21 .8 43.7 65.5 87.3 109.1 19.1 38.2 57.3 76.4 95.5 17.0 34.0 50.9 67.9 84.9 15.3 30.6 45.8 61.1 76.4 13 9 27.8 41.7 55.6 69.5 12.7 25.5 38.2 50.9 63.6 10.9 21.8 32.7 43.7 54.6 9.6 19.1 28.7 38.2 47.8 8.5 17.0 25.5 34.0 42.5 7.6 15.3 22 9 30.6 38.2 6.9 13.9 20.8 27.8 34.7 6.4 12.7 19.1 25.5 31.8 5.5 10.9 16.4 21.8 27.3 4.8 9.6 14.3 19.1 23.9 4.2 8.5 12.7 17.0 21.2 3.8 7.6 11.5 15.3 19.1 3.5 6.9 10.4 13.9 17.4 3.2 6.4 9.5 12.7 15.9 2.7 5.5 8.2 10.9 13.6 2.4 4.8 7.2 9.6 11.9 2.1 4.2 6.4 8.5 10.6 1.9 . 3.8 5.7 7.6 9.6 1.7 3.5 5.2 6.9 8.7 1.6 3.2 4.8 6.4 8.0 1.5 2.9 4.4 5.9 7.3 1.4 2.7 4.1 5.5 6.8 1.3 2.5 3.8 5.1 6.4 1.2 2.4 3.6 4.8 6.0 1.1 2.1 3.2 4.2 5.3 1.0 1.9 2.9 3.8 4.8 .9 1.7 2.6 3.5 4.3 .8 1.6 2.4 3.2 4.0 .7 1.5 2.2 2.9 3.7 7 1.4 2.0 2.7 3.4 .6 1.3 1.9 2.5 3.2 .5 1.1 1.6 2.1 2.7 .5 .9 1.4 1.8 2.3 .4 .8 1.2 1.6 2.0 .4 .7 1.1 1.4 1.8 .3 .6 1.0 1.3 1.6 458.4 305.6 229.2 183.4 152.8 13i 114.6 101.8 91. 83.3 76.4 65.5 57.3 50.9 45 41 19.1 16.4 14.3 12.7 11.5 10.4 9.5 5. 5.2 4.8 4.4 4.1 3.8 3.: 2.' 2.4 2.1 1.9 534.8 611.2 687.6 350.5 407.4 458.3 267.4 305.6 343 8 213.9 244.5 275.0 178.2 203.7 229.1 152.8 174.6 196.4 133.7 152.8 171.9 118.8 135.8 152.8 106.9 122.2 137.5 97.2 111.1 125.0 89.1 101.8 114.5 76.4 87.3 98.2 66.9 76.4 86.0 59.4 67.9 76.4 53.5 61.1 68.8 48.6 55.6 62.5 44.6 50.9 57.3 38.2 43.7 49.1 33.4 38 2 43.0 29.7 34.0 38.2 26.7 30.6 34.4 24.3 27.8 31.2 22.3 25.5 28.6 19.1 21.8 24.6 16.7 19.1 21.5 14.8 17.0 19.1 13.3 15.3 17.2 12.2 13.9 15.6 11.1 12.7 14.3 10.3 11.8 13.2 9.5 10.9 12.3 8.9 10.2 11.5 8.4 9.5 10.7 7.4 8.5 9.5 6.7 7.6 8.6 6.1 6.9 7.8 5.6 6.4 7.2 5.1 5.9 6.6 4.8 5.5 6.1 4.5 5.1 5 7 3.7 4.2 4.8 3.2 3.6 4.1 2.8 3.2 3.6 2.5 2.8 3.2 2.2 2.5 2.9 764.0 509.3 382.0 305.6 254.6 218.3 191.0 169.7 152.8 138.9 127.2 109.2 95.5 84.9 76.4 69.5 63.7 54.6 47.8 42.5 38.1 34.7 31.8 27.3 23.9 21.2 19.1 17.4 15.9 14.7 13.6 12.7 11.9 10.6 Speed of Cutting with Turret Lathes.— Jones & Lamson Ma- chine (Jo. give the following cutting-speeds for use with their flat turret lathe: Ft. per minute. ( Tool steel and taper on tubing 10 Threading < Machinery , 15 | Very soft steel 20 Tiirninp- Cut wnich reduces the stock to )4 of its original diam. . 20 hi .o iiiTiPPv J Cut wll 'eh reduces the stock to % of its original diam. . 25 sreli i Cut which reduces the stock to % of its original diam. . 30 to 35 (. Cut which reduces the stock to 15/16 of its original diam. 40 to 45 Turning very soft machinery steel, light cut and cool worj?. , , 50 to 6Q GEARING OF LATHES. 955 Forms of Metal-cutting Tools.— " Hutte" the German Engi- neers' Pocket-book, gives the following cutting-angles for using least power: Top Rake. Angle of Cutting-edge. Wrought iron 3° 51° Cast iron 4° 51° Bronze 4° 66° The American Machinist comments on these figures as follows : We are not able to give the best nor even the generally used angles for tools, because these vary so much to suit different circumstances, such as degree of hardness of the metal being cut, quality of steel of which the tool is made, depth of cut, kind of finish desired, etc. The angles that cut with the least expenditure of power are easily determined by a few experiments, but the best angles must be determined by good judgment, guided by expe- rience. In nearly all cases, however, we think the best practical angles are greater than those given. For illustrations and descriptions of various forms of cutting-tools, see articles on Lathe Tools in App. Cyc. App. Mech., vol. ii., and in Modern Mechanism. Cold Chisels.— Angle of cutting- faces (Joshua Rose): For cast steel, about 65 degrees; for gun-metal or brass, about 50 degrees; for copper and soft metals, about 30 to 35 degrees. Rule for Gearing Lathes for Screw-cutting. (Garvin Ma- chine (Jo.) — Read from Hie lathe index the number of threads per inch cut by equal gears, and multiply it by any number that will give for a product a grear on the index; put this gear upon the stud, then multiply the number of threads per inch to be cut by the same number, and put the resulting gear upon the screw. Example.— To cut 11^ threads per inch. We find on the index that 48 into 48 cuts 6 threads per inch, then 6 X 4 = 24, gear on stud, and 11' j X 4 = 46, gear on screw. Any multiplier may be used so long as the products include gears that belong with the lathe. For instance, instead of 4 as a multiplier we may use 6. Thus, 6 X 6 = 36, gear upon stud, and llj^ x 6 = 69, gear upon screw. Rules for Calculating Simple and Compound Gearing where there is no Index. {Am Mach.)—lt the lathe is simple- geared, and the stud runs at the same speed as the spindle, select some gear for the screw, and multiply its number of teeth by the number of threads per inch in the lead-screw, and divide this result by the number of threads per inch to be cut. This will give the "number of teeth in the gear for the stud. If this result is a fractional number, or a number which is not among the gears on hand, then try some other gear for the screw. Or, select the gear for the stud first, then multiply its number of teeth by the number of threads per inch to be cut, and divide by the number of threads per inch on the lead-screw. This will give the number of teeth for the gear on the screw. If the lathe is compound, select at random all the driving-gears, multiply the numbers of their teeth together, and this product by the num- ber of threads to be cut. Then select at random all the driven gears except one; multiply the numbers of their teeth together, and this product by the number of threads per inch in the lead-screw. Now divide the first result by the second, to obtain the number of teeth in the remaining driven gear. Or, select at random all the driven gears. Multiply the numbers of their teeth together, and this product by the number of threads per inch in the lead- screw. Then select at random all the driving-gears except one. Multiply the numbers of their teeth together, and this result by the number of threads per inch of the screw to be cut. Divide the first result by the last, to obtain the number of teeth in the remaining driver. When the gears on the com- pounding stud are fast together, and cannot be changed, then the driven one has usually twice as many teeth as the other, or driver, in which case in the calculations consider the lead-screw to haze twice as many threads per inch as it actually nas ; and then ignore the compounding entirely. Some lathes are so constructed that the stud on which the first driver is placed revolves only half as fast as the spindle. This can be ignored in the calculations by doubling the number of threads of the lead-screw. If both the last condi- tions are present ignore them in the calculations by multiplying the number of threads per inch in the lead-screw by four. If the thread to be cut is a fractional one, or if the pitch of the lead-screw is fractional, or if both are fractional, then reduce the fractions to a common denominator, and use the numerators of these fractions as if they equalled the pitch of the screw 950 THE MACHINE-SHOP. to be cut, and of the lead-screw, respectively. Then use that part of the rule given above which applies to the lathe in question. For instance, suppose it is desired to cut a thread of 25/32-inch pitch, and the lead-screw has 4 threads per inch Then the pitch of the lead-screw will be J4 inch, which is equal to 8/32 inch. We now have two fraction, 25/32 and 8/32, and the two screws will be in the proportion of 25 to 8, and the gears can be figured by the above rule, assuming the number of threads to be cut to be 8 per inch, and those on the lead-screw to be 25 per inch. But this latter number may be further modified by conditions named above, such as a reduced speed of the stud, or fixed compound gears. In the instance given, if the lead-screw had been 2^ threads per inch, then its pitch being 4/10 inch, we have the fractions 4/10 and 25/32, which, reduced to a common denominator, are 64/160 and 125/160, and the gears will be the same as if the lead-screw had 125 threads per inch, and the screw to be cut 64 threads per inch. On this subject consult also " Formulas in Gearing," published by Brown & Sharpe Mfg. Co.. and Jamieson's Applied Mechanics. Change-gears for Screw-cutting Lathes. — There is a lack of uniformity among lathe-builders as to the change-gears provided for screw- cutting. W. R. Macdonald, in Am. Mach., April 7, 1892, proposes the follow- ing series, by which 33 whole threads (not fractional) maybe cut by changes of only nine gears: 70 110 130 Spindle. 40 50 60 70 4 4/5 7 1/5 9 3/5 14 2/5 16 4/5 26 2/5 28 4/5 31 1/5 5 1/7 6 6/7 8 4/7 10 2/*" 18 20 4/7 2.2 3/~ 110 120 130 2 2/11 3 3/11 4 4/11 5 5/11 6 6/11 7 7/11 13'i/ii 14 2/11 13 1 11/13 2 10/13 3 9/13 4 8/13 5 7/13 6 6/13 10 2/13 11 1/13 Whole Threads. 2 11 22 3 12 24 4 13 26 5 14 28 6 15 30 7 16 33 8 IK 36 9 20 39 10 21 42 Ten gears are sufficient to cut all the usual threads, with the exception of perhaps ll^j, the standard pipe-thread; in ordinary practice any fractional thread between 11 and 12 will be near enough for the customary short pipe- thread; if not, the addition of a single gear will give it. In this table the pitch of the lead-screw is 12, and it may be objected to as too fine for the purpose. This may be rectified by making the real pitch 6 or any other desirable pitch, and establishing the proper ratio between the lathe spindle and the gear-stud. Metric Screw-threads may be cut on lathes with inch-divided lead- ing-screws, by the use of a change-wheel with 127 teeth; for 127 millimetres equal 5 inches (126 X .03937 = 4.99999 in.). Rule for Setting the Taper in a Lathe. (Am. Mach.)— No rule can be given which will produce esact results, owing to the fact that the centres enter the work an indefinite distance. If it were not for this cir- cumstance the following would be an exact rule, and it is an approximation as it is. To find the distance to set the centre over: Divide the difference in the diameters of the large and small end of the taper by 2, and multiply this quotient by the ratio which the total length of the shaft bears to the length of the tapered portion. Example: Suppose a shaft three feet long is to have a taper turned on the end one foot long, the large end of the taper being two inches and the small end one inch diameter. — — - X - = \% inches. Electric Drilling-machines -Speed of Drilling Holes in Steel Plates. (Proc. Inst. M. E., Aug. 1887, p. 329.;— In drilling holes in the shell of the S.S. "Albania," after a very small amount of practice the men working the machines drilled the %-inch holes in the shell with great rapidity, doing the work at the rate of one hole every 69 seconds, inclusive of the time occupied in altering the position of the machines by means of differ- ential pulley-blocks, which were not conveniently arranged as slings for this purpose. Repeated trials of these drilling-machines have also shown that, when using electrical energy in both holding-on magnets and motor MILLING-CUTTERS. 95? amounting to about % H.P., they have drilled holes of 1 inch diameter through l^j inch thickness of solid wrought iron, or through \% inch of mild steel in two plates of 13/16 inch each, taking exactly 1% minutes for each hole. Speed of Twist-drills.— The cutting-speeds and rates of feed recom- mended by the Morse Twist -drill and Machine Company are given in the following table. Revolutions per minute for drills 1/16 in. to 2 in. diam., as usually applied: Diameter Speed Speed Speed Diameter Speed Speed Speed of for for for of for for for Drills. Steel. Iron. Brass. Drills. Steel. Iron. Brass. inch. inch. 1/16 940 1280 1560 1 1/16 54 75 95 % 460 660 785 v& 52 70 90 3/16 SIC 420 540 1 3/16 49 66 85 x 4 230 320 400 VA 46 62 80 5/16 190 260 320 1 5/16 44 60 75 % 150 220 260 Ws 42 58 72 7/16 130 185 230 1 7/16 40 56 69 H 115 160 200 1H 39 54 66 9/16 100 140 180 1 9/16 37 51 63 % 95 130 160 1% 36 49 60 11/16 85 115 145 1 11/16 34 47 58 H 75 105 130 m 33 45 56 1-3/16 70 100 120 1 13/16 32 43 54 % 65 90 115 1% 31 41 52 15/16 62 85 110 1 15/16 30 40 51 1 58 80 100 2 29 39 49 To drill one inch in soft cast iron will usually require: For %-in. drill, 125 revolutions; for J^-in. drill, 120 revolutions; for %-m.. drill, 100 revolutions; for 1-in. drill, 95 revolutions. The rates of feed for twist drills are thus given by the same company: Diameter of drill 1/16 y± ' % \£ % 1 \\& Revs, per inch depth of hole. 125 125 120 to 140 1 inch feed per min. MILLING-CrTTEBS. George Addy, (Proc. Inst. M. E., Oct. 1890, p. 537), gives the following: Analyses of Steel.— The following are analyses of milling-cutter blanks, made from best quality crucible cast steel and from self-hardening " Ivanhoe V steel : Carbon Silicon Phosphorus Manganese Sulphur Tungsten , Iron, by difference . . Crucible Cast Steel, per cent. 1.2 0.112 0.018 .. . 0.36 0.02 Ivanhoe Steel, per cent. 1.67 0.252 0.051 2.557 0.01 4.65 90.81 100.000 100.000 The first analysis is of a cutter 14 in. diam., 1 in. wide, which gave very good service at a cutting-speed of 60 ft. per min. Large milling-cutters are sometimes built up, the cutting-edges only being of tool steel. A cutter 22 in. diam. by 5)4 in. wide has been made in this way, the teeth being clamped between two cast-iron flanges. Mr. Addy recommends for this form of tooth one with a cutting-angle of 70°, the face of the tooth being set 10° back of a radial line on the cutter, the clearance -angle being thus 10°. At the Clarence Iron-works, Leeds, the face of the tooth is set 10° back of the radial line for cutting wrought iron and 20° for steel. Pitch of Teeth.— For obtaining a suitable pitch of teeth for milling- cutters of various diameters there exists no standard rule, the pitch being usually decided in an arbitrary manner, according to individual taste. 958 THE MACEIKE-SHOP. For estimating the pitch of teeth in a cutter of any diameter from 4 in. to 15 in., Mr. Addy has worked out the following rule, which he has found capa- ble of giving good results in practice: Pitch in inches = 4/(diam. in inches x 8) X 0.0625 = .177 Vdiam. J. M. Gray gives a rule for pitch as follows: The number of teeth in a milling-cutter ought to be 100 times the pitch in inches; that is, if there were 27 teeth, the pitch ought to be 0.27 in. The rules are practically the same, for if d = diam., n = No. of teeth, p = pitch, c = circumference, c = pn; d = ^ = — ^ = 31.83p2; p = \/MUd = .177 Vd; No. of teeth, n, = 3.14d-=-p. Number of Teeth in Mills or Cutters. (Joshua Rose.)— The teeth of cutters must obviously be spaced wide enough apart to admit of the emery- wheel grinding one tooth without touching the next one, and the front faces of the teeth are always made in the plane of a line radiating from the axis of the cut'.er. In cutters up to 3 in. in diam. it is good practice to provide 8 teetli per in. of diam., while in cutters above that diameter the spacing may be coarser, as follows: Diameter of cutter, 6 in. ; number of teeth in cutter, 40 7 k ' " " " " " 45 8 " " " " " " 50 Speed of Cutters.— The cutting speed for milling was originally fixed very low; but experience has shown that with the improvements now in use it may with advantage be considerably increased, especially with cutters of large diameter. The following are recommended as safe speeds for cut- ters of 6 in. and upwards, provided there is not any great depth of material to cut away: Steel. Wrought iron. Cast iron. Brass. Feet per minute 36 48 60 120 Feed, inch per min. .. J^ 1 \% 2% Should it be desired to remove any large quantity of material, the same cutting-speeds are still recommended, but with a finer feed. A simple rjle for cutting-speed is: Number of revolutions per minute which the outer spindle should make when working on cast iron = 240, divided by the diam- eter of the cutter in inches. Speed of Milling-cutters. (Proc. Inst. M. E., April, 1883, p. 248.)— The cutting-speed which can be employed in milling is much greater than that which can be used in any of the ordinary operations of turning in the lathe, or of planing, shaping, or slotting. A milling-cutter with a plentiful supply of oil, or soap and water, can be run at from 80 to 100 ft. per min., when cutting wrought iron. The same metal can only bs turned in a lathe, with a tool-holder having a good cutter, at the rate of 30 ft. per min., or at about one third the speed of milling. A milling-cutter will cut cast steel at the rate of 25 to 30 ft. per min. The following extracts are taken from an article on speed and feed of milling-cutters in Eng'g, Oct. 22, 1891: Milling-cutters are successfully em- ployed on cast iron at a speed of 250 ft. per min. ; on wrought iron at from 80 ft. to 100 ft. per min. The latter materials need a copious supply of good lubricant, such as oil or soapy water. These rates of speed are not ap- proached by other tools. The usual cutting-speeds on the lathe, planing, shaping, and slotting machines rarely exceed about one third of those given above, and frequently average about a fifth, the time lost in back strokes not being reckoned. The feed in the direction of cutting is said by one writer to vary, in ordi- nary work, from 40 to 70 revs, of a 4-in. cutter per in. of feed. It must always to an extent depend on the character of the work done, but the above gives shavings of extreme thinness. For example, the circumference of a 4-in. cutter being, say, 12^ in., and having, say, 60 teeth, the advance corre- sponding to the passage of one cutting-tooth over the surface, in the coarser of the above-named feed-motions, is 1/40 X 1/60 = 1/2100 in.; the finer feed gives an advance for each tooth of onlv 1/70 X 1/60 = 1/4200 in. Such fine feeds as these are used only for light finishing cuts, and the same author- ity recommends, also for fini-hing, a cutter about 9 in. in circumference, or nearly 3 in. in diameter, which should be run at about 60 revs, per min. to cut tough wrought steel, 120 for ordinary cast iron, about 80 for wrought MILLING-MACHINES. 959 iron, and from 140 to 160 for the various qualtities of gun-metal and brass. With cutters smaller or larger the rates of revolution are increased or diminished to accord with the following table, which gives these rates of cutting-speeds and shows the lineal speed of the cutting-edge: Steel. Wrought Iron. Cast Iron. Gun-metal. Brass. Feet per minute... 45 60 90 105 120 These speeds are intended for very light finishing cuts, and they must be reduced to about one half for heavy cutting. The following results have been found to be the highest that could be at- tained in ordinary workshop routine, having due consideration to economy and the time taken to change and grind the cutters when they become dull: Wrought iron— 36 ft. to 40 ft. per min.; depth of cut. 1 in.; feed, % in. per min. Soft mild steel — About 30 ft. per min.; depth of cut, J4 m -\ feed, % in. per min. Tough gun-metal— 80 ft. per min. ; depth of cut, % in. ; feed, % in. per min. Cast-iron gear-wheels— 26J^ ft. per min.; depth of cut, J^ in.; feed, % in. per min. Hard, close-grained cast iron— 30 ft. per min.; depth of cijt, 2J^ in.; feed, 5/16 in. per min. Gun-metal joints, 53 ft. per min.; depth of cut, l%in.; feed, % in. per nun. Steel-bars— 21 ft. per min.; depth of cut, 1/32 in.; feed, % in. per min. A stepped milling-cutter, 4 in. in diam. and 12 in. wide, tested under two conditions of speed in the same machine, gave the following results: The cutter in both instances was worked up to its maximum speed before it gave way, the object being to ascertain definitely the relative amount of work done by a high speed and a light feed, as compared with a low speed and a heavy cut. The machine was used single-geared and double-geared, and in both cases the width of cut was lOJ^ in. Single-gear, 42 ft. per min.; 5/16 in. depth of cut; feed, 1.3 in. per min. =: 4.16 cu. in. per min. Double-gear, 19 ft. per min.; %in. depth of cut; feed, % in. per rain. = 2.40 cu. in. per min. Extreme Results witli Milling-machines. — Horace L. Arnold (Am. Mack., Dec. 28, 1893) gives the following results in flat-surface milling, obtained in a Pratt & Whitney milling-machine : The mills for the flat cut were 5" diam., 12 teeth, 40 to 50 revs, and 4%" feed per min. One single cut was run over this piece at a feed of 9" per min., but the mills showed plaiuly at the end that this rate was greater than they could endure. At 50 revs, for these mills the figures are as follows, with 4%" feed: Surface speed, 64 ft., nearly; feed per tooth, 0. 00812": cuts per inch, 123. And with 9" feed per min.: Surface speed, 64 ft. per min.; feed per tooth, 0.015"; cuts per inch, 66%. At a feed of 4%" per min. the mills stood up well in this job of cast-iron surfacing, while with a 9" feed they required grinding after surfacing one piece; in other words, it did not damage the mill-teeth to do this job with 123 cuts per in. of surface finished, but they would not endure 66% cuts per inch. In this cast-iron milling the surface speed of the mills does not seem to be the factor of mill destruction: it is the increase of feed per tooth that prohibits increased production of finished surface. This is precisely the re- verse of the action of single-pointed lathe and planer tools in general: with such tools there is a surface-speed limit which cannot be economically ex- ceeded for dry cuts, and so long as this surface-speed limit is not reached, the cut per tooth or feed can be made anything up to the limit of the driv- ing power of the lathe or planer, or to the safe strain on the work itself, which can in many cases be easily broken by a too great feed. In wrought metal extreme figures were obtained in one experiment made in cutting key ways 5/16" wide by %" deep in a bank of 8 shafts 1J4" diam. at once, on a Pratt & Whitney No. 3 column milling-machine. The 8 mills were successfully operated with 45 ft. surface speed and 19^ in. per min. feed; the cutters were 5" diam., with 28 teeth, giving the following figures, in steel: Surface speed, 45 ft. per min.; feed per tooth. 0.02024"; cuts per inch, 50, nearly. Fed with the revolution of mill. Flooded with oil, that is, a large stream of oil running constantly over each mill. Face of tooth radial. The resulting keyway was described as having a heavy wave or cutter-mark in the bottom, and it was said to have shown no signs of being heavy work on the cutters or on the machine. As a result of the experiment it was decided for economical steady work to run at 17 revs., with a feed of 4" per min., flooded cut, work fed with mill revolution, giving the following figures: Surface speed, 22!4 ft- V er niin.; feed per tooth, 0.0084"; cuts per inch, 119, 960 THE MACHINE-SHOP. An experiment in milling a wrought iron connecting-rod of a locomotive on a Pratt & Whitney double-head milling-machine is described in the Iron Age, Aug. 27, 1891. The amount of metal removed at one cut measured 3)4 in. wide by 1 3/16 in. deep in the groove, and across the top % in. deep by 4% in. wide. This represented a section of nearly Ay% sq. in. This was done at the rate of 1% in. per min. Nearly 8 cu. in. of metal were cut up into chips every minute. The surface left by the cutter was very perfect. The cutter moved in a direction contrary to that of ordinary practice; that is, it cut down from the upper surface instead of up from the bottom. Milling "with" or "against" tlie Feed.— Tests made with the Brown & Sharpe No. 5 milling-machine (described by H. L. Arnold, in Am. Mach., Oct. 18, 1894) to determine the relative advantage of running the milling-cutter with or against the feed—" with the feed " meaning that the teeth of the cutter strike on the top surface or "scale" of cast-iron work in process of being milled, and "against the feed " meaning that the teeth begin to cut in the clean, newly cut surface of the work and cut up- wards toward the scale — showed a decided advantage in favor of running the cutter against the feed. The result is directly opposite to that obtained in tests of a Pratt & Whitney machine, by experts of the P. & W. Cot In the tests with the Brown & Sharpe machine the cutters used were 6 inches face by 414 an( i 3 inches diameter respectively, 15 teeth in each mill, 42 revolutions per minute in each case, or nearly 50 feet per minute surface speed for the 4J^-inch and 33 feet per minute for the 3-inch mill. The revo- lution marks were 6 to the inch, giving a feed of 7 inches per minute, and a cut per tooth of .011". When the machine was forced to the limit of its driving the depth of cut was 11/32 inch when the cutter ran in the " old " way, or against the feed, and only J4 inch when it ran in the " new " way, or with the feed. The endurance of the milling-cutters was much greater when they were run in the " old " way. Spiral Milling-cutters.— There is no rule for finding the angle of the spiral; from 10° to 15° is usually considered sufficient; if much greater the end thrust on the spindle will be increased to an extent not desirable for some machines. Milling-cu tters with Inserted Teeth.— When it is required to use milling-cutters of a greater diameter than about 8 in., it is preferable to insert the teeth in a disk or head, so as to avoid the expense of making solid cutters and the difficulty of hardening them, not merely because of the risk of breakage in hardening them, but also on account of the difficulty in obtaining a uniform degree of hardness or temper. Milling - machine versus Planer. — For comparative data of work done by each see paper by J. J. Grunt, Trans. A. S. M. E., ix. 259. He says : The advantages of the milling machine over the planer are many, among which are the following : Exact duplication of work; rapidity of pro- duction—the cutting being continuous; cost of production, as several machines can be operated by one workman, and he not a skilled mechanic; and cost of tools for producing a given amount of work. POWER REQUIRED FOR MACHINE TOOL.S. Resistance Overcome in Cutting Metal. (Trans. A. S. M. E., viii. 308.) — Some experiments made at the works of William Sellers & Co. showed that the resistance in cutting steel in a lathe would vary from 180,000 to 700,000 pounds per square inch of section removed, while for cast iron the resistance is about one third as much. The power required to remove a given amount of metal depends on the shape of the cut and on the shape and the sharpness of the tool used. If the cut is nearly square in section, the power required is a minimum; if wide and thin, a maximum. The dulness of a tool affects hut little the power required for a heavy cut. Heavy Work on a Planer.— Win. Sellers & Co. write as follows to the American Machinist : The 120'' planer table is geared to run 18 ft. per minute under cut, and 72 feet per minute on the return, which is equivalent, without allowance for time lost in reversing, to continuous cut of 14.4 feet per minute. Assuming the work to be 28 feet long, we may take 14 feet as the continuous cutting speed per minute, the .8 of a foot being much more than sufficient to cover time loss in reversing and feeding. The machine carries four tools. At %" feed per tool, the surface planed per hour would be 35 square feet. The section of metal cut at %" depth would be .75" X ,125" X 4 — -375 square inch, which would require approximately 30,000 lbs, POWER REQUIRED FOR MACHINE TOOLS. 961 pressure to remove it. The weight of metal removed per hour would be 14 X 12 X .375 X .26 x 60 = 1082.8 lbs. Our earlier form of 36" planer has removed with one tool on %" cut on work 200 lbs. of metal per hour, and the 120" machine has more than five times its capacity. The total pulling power of the planer is 45,000 !bs. Horse-power Required to Run Lathes. (J. J. Flather, Am. Mach., April 23, 1891.)— The power required to do useful work varies with the depth and breadth of chip, with the shape of tool, and with the nature and density of metal operated upon; and the power required to run a ma- chine empty is often a variable quantity. For instance, when the machine is new, and the working pai'ts have not become worn or fitted to each other as they will be after running a few months, the power required will be greater than will be the case after the x-unning parts have become better fitted. Another cause of variation of the power absorbed is the driving-belt; a tight belt will increase the friction, hence to obtain the greatest efficiency of a machine we should use wide belts, and run them just tight enough to prevent slip. The belts should also be soft and pliable, otherwise power is consumed in bending them to the curvature of the pulleys. A third cause is the variation of journal-friction, due to slacking up or tightening the cap-screws, and also the end-thrust bearing screw. Hartig's investigations show that it requires less total power to turn off a given weight of metal in a given time than it does to plane off the same amount; and also that the power is less for large than for small diameters. The following table gives the actual horse-power required to drive a lathe empty at varying numbers of revolutions of main spindle. HORSE-POWER FOR SMALL LATHES. Without Back Gears. I With Back Gears. Revs, of Spindle per min. 132.72 219.08 365.00 47.4 125.0 54.6 82.2 H.P. required to drive empty. .145 .197 .310 .159 .259 .206 .339 .455 .210 .326 Revs, ot Spindle per min. 24.33 38.42 4.84 12.8 19.2 6.61 14.8 H.P. required to drive empty. .126 .141 .274 .132 .187 .230 .157 .206 .249 .035 .063 .087 20" Fitchburg lathe. Smallla the (13^">, Chem- nitz. Germany. New machine. 17^" lathe do. New machine. If H.P. = horse-power necessary to drive lathe empty, and N= number of revolutions per minute, then the equation for average small lathes is H.P.o = 0.095 -f 0.0012IV. For the power necessary to drive the lathes empty when the back gears are in, an average equation for lathes under 20" swing is H.P.o = 0.10 + 0.006iV. The larger lathes vary so much in construction and detail that no general rule can be obtained which will give, even approximately, the power re- quired to run them, and although the average formula shows that at least 0.095 horse-power is needed to start the small lathes, there are many Amer- ican lathes under 20" swing working on a consumption of less than ,05 horse-power, 962 THE MACHINE-SHOP. The amount of power required to remove metal in a machine is determin- able within more accurate limits. Referring to Dr. Hartig's researches, H.P.! = CW, where C is a constant, and Wthe weight of chips removed per hour. Average values of C are .030 for cast-iron, .032 for wrought-iron, .047 for steel. The size of lathe, and, therefore, the diameter of work, has no apparent effect on the cutting power. If the lathe be heavy, the cut can be increased, and consequently the weight of chips increased, but the value of C appears to be about the same for a given metal through several vaiwing sizes of lathes. Horse-power required to remove Cast Iron in a 20-inch Lathe. (J. J. Hobart.) m d H o> > o a .o o s a Q fc 1 22 2 15 3 17 4 2 5 4 fi 1 7 1 Side tool Diamond . . . Round nose . Left - hand round nose Square -faced tool y^' broad . a A a •g « i CD CD > I- O • a i2W 1-S 3 CD 3* o O 93 cd« P-j CD Mo CD-3 • 5 s te0 g g$a CD j=Xi &C3 go Jfg'jj ^1^5 £%a > O £§■§ ^Ifl .025 .020 .028 .018 .027 The above table shows that an average of .26 horse-power is required to turn off 10 pounds of cast-iron per hour, from which we obtain the average value of the constant = .024. Most of the cuts were taken so that the metal would be reduced \i" in diameter; with a broad surface cut and a coarse feed, as in No. 5, the power required per pound of chips removed in a given time was a maximum; the least power per unit of weight removed being required when the chip was square, as in No. 6. Horse-power required to remove Metal in a 29-inch Lathe. (R. H. Smith.) u a £a Metal. •o a CD .5 CD 3 a . a' 3 o o .a ,3 ■o eg CD pq a CD 1 ws-g 03 ^ CD— ^ g a > 53 c^o -3 rd a > a o o a o3 ee-a O o CD 3 a '5 £<£ ®_\ > £ a 3 o CD Q ** > ", B. G 0.41 0.867 0.47 0.462 0.53 0.91 0.16 0.24 0.63 1.14 0.24 84 1.47 0.62 0.41 1.33 1.24 0.53 0.67 1.08 0.28 0.44 0.95 0.28 0.66 0.18 0.28 93 1.52 7.12 4.41 0.79 4.12 2.70 4.24 3.03 4.63 5.00 3.20 6.91 3.23 5.64 0.96 0.49 3.68 2.11 2.73. 2.25 2.00 2.45 1.55 3.11 0.56 0.18;0.15*-0.34t 0.207; 0.16-0.466 0.12; 0.12 to 0.31 0.05; 0.03 to 0.33 0.187; 0.12to0.66 0.37; 0.39 to 0.81 0.23 to 3.40 0.086 to 0.26 0.07; 0.07 to 0.12 0.21 ; 0.01 to 0.47 0.26; 0.15 to 0.73 0.12; 0.12 to 0.40 0.27 0.60 0.39 0.15; 0.15 to 0.43 0.62 0.62 0.44;0.1*-0.44t 0.30; 0.12*-0.80t 0.46 0.09; 0.05 to 0.25 0.22; 0.15 to 0.65 0.57; 0.43 to 0.94 0.01; 0.003-0.13 0.26; 0.26 to 0.55 0.10 0.11 0.12; 0.10-0.12*; 0.10to0.25t 37 0.67 1.00 0.16 0.61 .54 3.35 1 42 1.25 O.74J-0.17§ 1.45 4.18 0.70 1.16 0.19 0.34 1.67; 0.65 to 2.0 1.42 0.61 2.17 1.30 2.00 0.32 0.24 0.40' Screw-cutting lathe 26", B. G Lathe, 80" face plate, will swing 108", T. G Large facing lathe, will swing 68", T. G Wheel lathe 60" swing Small shaper, Richards (9^" X 22") Shaper (15" stroke Gould & Eberhardt). .. Large shaper, Richards (29" X 91") Crank planer (capacity 23" X 27" X 28J^" stroke). . Planer (capacity 36' x 36" X 11 feet) Small drill press Upright slot drilling mach. (will drill 2}4" diam.).. . . Large drill press Radial drill press Slotter (91^" stroke) ... . Slotter (15" stroke) Universal milling mach (Brown & Sharpe No. 1).... Milling machine (13" cutter-head, 12 cutters) Small head traversing milling machine (cutter-head Gear cutter will cut 20" diameter Horizontal boring machine for iron, 22^" swing Large plate shears— knives 28" long, 3" stroke Large punch press, over-reach 28", 3" stroke, \y^' stock can be punched Small punch and shear comb'd, 7%" knives, 1}^" str. Circular saw for hot iron (30^" diameter of saw). . . Plate-bending rolls, diam. of rolls 13", length 9^ ft. Wood planer 13}^" (rotary knives, 2 hor'l 2 vert. . . . Wood planer 24" (rotary knives) Circular saw for wood (23" diameter of saw) Circular saw for wood (35" diameter of saw) Hor*l wood-boring and mortising machine, drill 4" diam., mortise 8J^ deep X HJ^" long Tenon and mortising machine Tenon and mortising machine . Tenon and mortising machine Edge-molder and shaper. (Vertical spindle) Wood-molding mach. (cap. 714 X 2y»). Hor. spindle 680 ft. per minute Emery wheel HJ/£" diameter x M"- Saw grinder. . * With back gears. + Without back gears. % For side cutters. B. G., back-geared. T. G., triple-gear surface cutters. §With I ed. ABRASIVE PROCESSES. 965 Horse-power consumed in Machine-shops.— How much power is required to drive ordinary machine-tools? and h'ow many men can be employed per horse-power'? are questions which it ismipossible to answer by any fixed rule. The power varies greatly according to the conditions in each shop. The following table given by J. J. Flather in his work on Dyna- mometers gives an idea of the variation in several large works. The percen- tage of the total power required to drive the shafting varies from 15 to 80, and the number of men employed per total H.P. varies from 0.62 to 6.04. Horse-power; Friction; Men Employed. Name of Firm. Lane & Bodley J. A. Fay & Co Union Tron Works Frontier Iron & Brass W'ks Taylor Mfg. Co Baldwin Loco. Works W. Sellers & Co. (one de- partment) Pond Machine Tool Co — Pratt & Whitney Co Brown & Sharpe Co Yale&TowneCo Ferracute Machine Co T. B. Wood's Sons , Bridgeport Forge Co Singer Mfg. Co Howe Mfg. Co. Worcester Mach. Screw Co Hartford " " Nicholson File Co Horse-power. o Bay Chaleur (New | Brunswick), f Liverpool or Melling. Kind of Grit. Texture of Stone. Color of Stone, All kinds, from finest to coarsest Medium to finest Medium to fine All kinds, from hardest to softest Soft and sharp Soft, with sharp Blue or yellowish gi-ay Uniformly light blue Reddish: For Wood-working Tools. Wickersley Liverpool or Melling. Bay Chaleur (New ( Brunswick)/ f Huron. Michigan . .. Medium to fine Medium to fine \ Medium to finest Fine Very soft Soft, with sharp grit Soft and sharp Soft and sharp Grayish yellow- Reddish Uniform light blue Uniform light blue For Grinding Broad Surfaces, as Saws or Iron Plates. Newcastle Coarse to med'm The hard ones Yellow Independence Coarse Hard to medium Grayish white Massillon Coarse Hard to medium Yellowish white TAP DRILLS. Taps for Machine-screws. (The Pratt & Whitney Co.) Approx. Approx. Diameter, Wire No. of Threads Diameter, Wire No. of Threa ds fractions Gauge. to inch. fractions Gauge. to inch. of an inch. of an inch. No. 1 60,72 No. 13 20,24 2 48, 56, 64 H 14 16, 18, 20, 22 24 3 40, 48, 56 15 18, 20, 24 7/64 4 32, 36, 40 17/64 16 16, 18, 20, 22 5 30, 32, 36, 40 9/32 18 16, 18, 20 9/64 6 30, 32, 36, 40 19 16, 18, 20 7 24, 30, 32 5/16 20 16, 18, 20 5/32 8 24, 30. 32, 36, 40 22 16, 18 9 24, 28, 30, 32 % 24 14, 16, 18 3/16 10 20, 22, £4, 30, 32 26 16 11 22, 24 28 16 7/32 12 20, 22, 24 30 16 The Morse Twist Drill and Machine Co. gives the following table showing the different sizes of drills that should be used when a full thread is to be tapped iu a hole. The sizes given are practically correct. TAP DRILLS. 971 ? T-i CO O? «C SW(00«M©N(OM CO «£ „ m £J S3 ., T-l CO T-Kr-KlO lOMrtOlCSlCClOrH e^M t- r-l r-Ki-H oa 1-1 ic}\OJ .^Eftff* . O* O* 01 OJ O? OltOOl OJtOOJ «©« OJtOOJ OJtOCO COtOCO CO «5 CO CO i-l CO COt-iCO CO-— 'CO •ao\^o^sji\\i-i noico \m sfijt- " OS nsoi-i i-i co \-rm co t- ^ci o ^~ H\iO CO 8- --i\05 lO i-i ctfsi-i fr- 1-1 ,-Ki-i 03 i-i uSsOJ i-i O* CCKOJ i-l C> t>SO* i-i CO CO ■C5~ ■OJ • (M :»: • ;^ : ;«; ■ -Ol • : :§ : : i 02 . lis OJO?> T* "* -<* OJ tO 0? — — eo ^ — " ~" N ^oi:u oo • -0000 • •■>#■<# • • oj o* • • ojoj KM ■ -T-Irt • -MH ■ ■«!-! • -HH ooaoQOco«o0!aoo(Kooa OJ tO OJ O? tO 0? O) tO 0! O* 5D OJ «tDM O? tO O* NtfCO — i CO \$CC0 i-i CO \C!TO — CO VXCO rt CO \tCC —i CO \XCO i-i CO 0* SO c? ri osmi cot- m t-cscj m-ico m co £- anon \\\ « i-i i-i i-i ti 8!n« OJi-iW OJi-iCO i-i i-l CO 972 THE MACHINE-SHOP. . TAPER BOL.TS, PINS, REAMERS, ETC. Taper Bolts for Locomotives,- Bolt-threads, American stan- dard, except stay-bolts and boiler-studs, V threads, 12 per inch; valves, cocks, and plugs, V threads, 14 per inch, and J^-inch taper per 1 inch. Standard bolt taper 1/16 inch per foot. Taper Reamers.— The Pratt & Whitney Co. makes standard taper reamers for locomotive work taper 1/16 inch per foot from y% inch diam. ; 7 in. length of flute to 1% inch diam.; 16 in. length of flute, diameters ad- vancing by 16ths. P. & W. Co.'s standard taper pin reamers taper J4 in. per foot, are made in 14 sizes of diameters, 0.135 to 1.009 in.; length of flute 1 5/16 in. to 12 in. Dimensions of the Pratt & Whitney Company's Standard-taper Socket. Reamers for Morse No. Diameter Small End, inches. Diameter Large End, inches. Gauge Diam.da'ge end, inches Gauge L'ngth, inches. Length Flute, inches. Total L'ngth. Taper per foot, inches. 1 2 3 4 5 6 0.374 0.574 0.783 1.027 1.484 2.117 0.525 0.749 0.982 1.283 1.796 2.566 0.481 0.699 0.950 1.232 1.746 2.500 2^ 2Y 2 3 5/16 4 5 3 4 5 6 5J4 m/ 2 0.605 0.600 0.605 0.615 0.625 0.634 Standard Steel Taper-pins, the Pratt & Whitney Co.: Number: 12 3 4 5 Diameter large end: .156 .172 .193 .219 .250 Approximate fractional sizes: 5/32 11/64 3/16 7/32 M Lengths from H H % % % To* 1 114 ^A 1M 2 Diameter small end of standard taper-reamer : + .125 .146 .162 .183 .208 .240 .279 The following sizes are made by 289 .341 .409 .492 .591 19/64 11/32 13/32 3M 1 .331 5K 1M 6 .581 Standard Steel Mandrels. (The Pratt & Whitney Co.)— These mandrels are made of tool-steel, hardened, and ground true on their cen- tres. The ends are of a form best adapted to resist injury likely to be caused by driving. They are slightly taper. Sizes, % in. diameter by 3% in. long to 3 in. diam. by 14^ in. long, diameters advancing by 16ths. PUNCHES AND DIES, PRESSES, ETC. Clearance between Pnncli and Die. —For computing the amount of clearance that a die should have, or, in other words, the difference in size between die and punch, the general rule is to make the diameter of die-hole equal to the diameter of the punch, plus 2/10 the thickness of the plate. Or, D -■ d X .2f, in which D = diameter of die-hole, d = diameter of punch, and t = thickness of plate. For very thick plates some mechanics prefer to make the die-hole a little smaller than called for by the above rule. For ordinary boiler-work the die is made from 1/10 to 3/10 of the thickness of the plate larger than the diameter of the punch; and some boiler-makers advocate making the punch fit the die accurately. For punching nuts, the punch fits in the die. (Am. Machinist.) Kennedy's Spiral Punch. (The Pratt & Whitney Co.)— B. Martell, Chief Surveyor of Lloyd's Register, reported tests of Kennedy's spiral punches in which a %-inch spiral punch penetrated a %-ineh plate at a pres- sure of 22 to 25 tons, while a flat punch required 33 to 35 tons. Steel boiler- plates punched with a flat punch gave an average tensile strength of 58,579 * Taken \%' from extreme end, each size overlaps smaller one about y%" . Taper J4" to the foot. + Lengths vary by J4" each size, FORCING AND SHRINKING FITS. 973 lbs. per square inch, and an elongation in two inches across the hole of 5.2$, while plates punched with a spiral punch gave 63,929 lbs., and 10.6$ elonga- tion. The spiral shear form is not recommended for punches for use in metal of a thickness greater than the diameter of the punch. This form is of great- est benefit when the thickness of metal worked is less than two thirds the diameter of punch. Size of Blanks used in the Drawing-press. Oberlin Smith (Jour. Frank. Inst., Nov. 1886) gives three methods of finding the size of blanks. The first is a tentative method, and consists simply in a series of experiments with various blanks, until the proper one is found. This is for use mainly in complicated cases, and when the cutting portions of the die and punch can be finally sized after the other work is done. The second method is by weighing the sample piece, and then, knowing the weight of the sheet metal per square inch, computing the diameter of a piece having the required area to equal the. sample i n weight. The third method is by computation, and the formula is x = 4/d 2 + 4dh for sharp-cornered cup, where x — diameter of blank, d — diameter of cup, h = height of cup. For round-cornered cup where the corner is s mall, say r adius of corner less than J4 height of cup, the formula is x — ( Vd' 2 + 4dh) — r, about; r being the radius of the corner. This is based upon the assumption that the thickness of the metal is not to be altered by the drawing- operation. Pressure attainable by the Use of the Drop-press. (R. H. Thurston, Trans. A. S. 31. E., v. 53.)— A set of copper cylinders was prepared, of pure Lake Superior copper; they were subjected to the action of presses of different weights and of different heights of fall. Companion specimens of copper were compressed to exactly the same amount, and measures were obtained of the loads producing compression, and of the amount of work done in producing the compression by the drop. Comparing one with the other it was found that the work done with the hammer was 90% of the work which should have been done with perfect efficiency. That is to say, 90% of the work done in the testing-machine was equal to that due the weight of the drop falling the given distance. „...-,, - , Weight of drop X fall X efficiency Formula: Mean pressure in pounds = - : -. compression. For pressures per square inch, divide by the mean area opposed to crush- ing action during the operation. Flow of Metals. (David Townsend, Jour. Frank. Inst., March, 1878.) —In punching holes ?/16 inch diameter through iron blocks 1% inches thick, it was found that the core punched out was only 1 1/16 inch thick, and its volume was only about 32% of the volume of the hole. Therefore, 68% of the metal displaced by punching the hole flowed into the block itself, increasing its dimensions. FORCING AND SHRINKING FITS. Forcing Fits of Pins and Axles by Hydraulic Pressure. —A 4-iuch axle is turned .015 inch diameter larger than the hole into which it is to be fitted. They are pressed on by a pressure of 30 to 35 tons. (Lec- ture by Coleman Sellers, 1872.) For forcing the crank-pin into a locomotive driving-wheel, when the pin- hole is perfectly true and smooth, the pin should be pressed in with a pres- sure of 6 tons for every inch of diameter of the wheel fit. When the hole is not perfectly true, which may be the result of shrinking the tire on the wheel centre after the hole for the crank-pin has been bored, or if the hole is not perfectly smooth, the pressure may have to be increased to 9 tons for every inch of diameter of the wheel-fit. (Am. Machinist.) Shrinkage Fits.— In 1886 the American Railway Master Mechanics' Association recommended the following shrinkage allowances for tires of standard locomotives. The tires are uniformly heated by gas-flames, slipped over the cast-iron centres, and allowed to cool. The centres are turned to a diameter equal to the inside diameter of the tire plus the shrinkage allow- ance: Diameter of tire, in 38 44 50 56 62 66 Shrinkage allowance, in... .040 .047 .053 .060 .066 .070 This shrinkage allowance is approximately 1/80 inch per foot, or 1/960. A common allowance is 1/1000. Taking the modulus of elasticity of steel at 974 THE MACHINE-SHOP. 30,000,000, thestrain caused by shrinkage would be 30,000 lbs. per square inch, which is well within the elastic limit of machinery steel. SCREWS, SCREW-THREADS, ETC.* Efficiency of a Screw.— Let a = angle of the thread, that is, the angle whose tangent is the pitch of the screw divided by the circumference of a circle whose diameter is the mean of the diameters at the top and bottom of the thread. Then for a square thread -n ~. • 1 — / tan a Efficiency = , , , y — , 1 + / cotan a in which / is the coefficient of friction. (For demonstration, see Cotterill and Slade. Applied Mechanics, p. 146.) Since cotan = 1 -=-tan, we may substitute for cotan a the reciprocal of the tangent, or if p = pitch, and'c = mean cir- cumference of the screw, Efficiency = ■. Example.— Efficiency of square-threaded screws of y 2 in. pitch. Diameter at bottom of thread, in 1 2 3 4 " top " '« » ... iy 2 2y 2 sy 2 4y 2 Mean circumference " " "....3.927 7.069 10.21 13.35 Cotangent a = c -s- p =7.854 14.14 20.42 26.70 Tangent a = p -4- c '. = .1273 .0661 .0490 .0?75 Efficiency if /= .10 =55.3* 41.2* 32.7* 27.2* '«;/=. 15 = 45* 31.7* 24.4* 19.9* The efficiency thus increases with the steepness of the pitch. The above formulae and examples are for square-threaded screws, and consider the friction of the screw-thread only, and not the friction of the oollar or step by which end thrust is resisted* and which further reduces the efficiency. The efficiency is also further reduced by giving an inclination to the side of the thread, as in the V-threaded screw. For discussion of this subject, see paper bv Wilfred Lewis, Jour. Frank. Inst. 1880; also Trans. A. S. M. E., vol. xii. 784. Efficiency of Screw-bolts.— Mr. Lewis gives the following approx- imate formula for ordinary screw-bolts (V threads, with collars): p = pitch of screw, d = outside diameier of screw, F = force applied at circum- ference to lift a unit of weight, E = efficiency of screw. For an average case, in which the coefficient of friction may be assumed at .15, f = p±A E = -^ 3d ' p + d For bolts of the dimensions given above, J^-in. pitch, and outside diam- eters 1^2, 2J^, 3^, and 4% in., the efficiencies according to this formula would be, respectively, .25, .167, .125, and .10. James McBride (Trans. A. S. M. E.. xii. 781) describes an experiment with an ordinary 2-in. screw-bolt, with a V thread, 4]4 threads per inch, raising a weight of 7500 lbs., the force being applied by turning the nut. Of the power applied 89.8* was absorbed by friction of the nut on its supporting washer and of the threads of the bolt in the nut. The nut was not faced, and had the flat side to the washer. Prof. Ball in his " Experimental Mechanics " says: "Experiments showed in two cases respectively about % and % of the power was lost. 11 Trautwine says: "In practice the friction of the screw (which under heavy loads becomes very great) make the theoretical calculations of but little value.' 1 Weisbach says: " The efficiency is from 19* to 30*." Efficiency of a Differential Screw.- A correspondent of the American Machinist describes an experiment with a differential screw- punch, consisting of an outer screw 2 in. diam., 3 threads per in., aud an inner screw 1% in. diam., "&/% threads per inch. The pitch of the outer screw * For U. S. Standard Screw-threads, see page 204. KEYS. 975 being ^ in. and that of the inner screw 2/7 in., the punch would ad- vance in one revolution y% — 2/7 = 1/21 in. Experiments were made to de- termine the force required to punch an 11/16-in. hole in iron J4 in. thick, the force being applied at the end of a lever arm of 47% in. The leverage would be 47% X 2tt X 21 = (5300. The mean force applied at the end of the lever was 95 lbs., and the force at the punch, if there was no friction, would be 6300 X 95 = 598,500 lbs. The force required to punch the iron, assuming a shearing resistance of 50,000 lbs. per sq. in., would be 50,000 x 11/16 x tt X V± = 27,000 lbs., and the efficiency of the punch would be 27,000 -=- 598,500 = only 4.5^. With the larger screw only used as a punch the mean force at the end of the lever was only 82 lbs. The leverage in this case was 47% X %ir X 3 = 900, the total force referred to the punch, including friction, 900 X 82 =3 73,800, and the efficiency 27,000 -=- 73,800 = SQ.7%. The screws were of tool-steel, well fitted, and lubricated with lard-oi! and plumbago. Powell's New Screw-thread.— A. M. Powell (Am. Mach., Jan. 24, 1895) has designed a new screw-thread to replace the square form of thread, giving the advantages of greater ease in making fits, and provision for " take up " in case of wear. The dimensions are the same as those of square- thread screws, with the exception that the sides of the thread, instead of being perpendicular to the axis of the screw, are inclined 14^° to such per- pendicular; ihat is, the two sides of a thread are inclined 29° to each other. The formulae for dimensions of the thread are the following: Depth of thread = ^ -=- pitch; width of top of thread = width of space at bottom = .3707 -=- pitch; thickness at root of thread = width of space at top = .6293 -r- pitch. The term pitch is the number of threads to the inch. PROPORTIONING PARTS OF MACHINES IN A SERIES OF SIZES. (Stevens Indicator, April, 1892.) The following method was used by Coleman Sellers while at William Sellers & Co.'s to get the proportions of the parts of machines, based upon the size obtained in building a large machine and a small one to any series of machines. This formula is used in getting up the proportion-book and ar- ranging the set of proportions from which any machine can be constructed of intermediate size between the largest and smallest of the series. Rule to Establish Construction Formulae.— Take difference between the nominal sizes of the largest and the smallest machines that have been designed of the same construction. Take also the difference be- tween the sizes of similar parts on the largest and smallest machines se- lected. Divide the latter by the former, and the result obtained will be a " factor, 1 '' which, multiplied by the nominal capacity of the intermediate machine, and increased or diminished by a constant " increment," will give the size of the part required. To find the " increment :" Multiply the nomi- nal capacity of some known size by the factor obtained, and subtract the result from the size of the part belonging to the machine of nominal ca- pacity selected. Example.— Suppose the size of a part of a 72-in. machine is 3 in., and the corresponding part of a 42-in. machine is 1%, or 1.875 in.: then 72 — 42 — 30, and 3 in. - 1% in. = W s in. = 1.125. 1.125 -+- 30 = .0375 = the " factor," and .0375 X 42 = 1.575. Then 1.875 - 1.575 = .3 = the "increment'' 1 to be added. Let D = nominal capacity; then the formula will read: x = D X .0375 + .3. Proof: 42 X .0375 -4- .3 = 1.875, or \% the size of one of the selected parts. Some prefer the formula: aD + c — a\ in which D = nominal capacity in inches or in pounds, c is a constant increment, a is the factor, and x — the part to be found. KEYS. Sizes of Keys for Mill-gearing. (Trans. A. S. M. E., xiii, 229.)— E. G. Parkhurst's rule : Width of key = Y 8 diam. of shaft, depth = 1/9 diam. of Shaft; taper % in. to (he foot. Custom in Michigan saw-mills : Keys of square section, side = \\ diam. of shaft, or as nearly as may lie in even sixteenths of an inch. J. T. Hawkins's rule : Width = % diam. of hole; depth of side abutment in shaft = % diam. of hole. W. S. Huson's rule : J^-inch key for 1 to 1*4 in. shafts, 5/16 key for 1J4 to 1^4 in. shafts, % in. key for \y% to 1% in. shafts, and so on. Taper % in. to the foot. Total thickness at large end of splice, 4/5 width of key. 976 THE MACHItf E-SHOP. Unwin (Elements of Machine Design) gives : Width = y^d -f % in. Thick- ness = %d -\- % in., in which d = diam. of shaft in inches. When wheels or pulleys transmitting only a small amount of power are keyed on large shafts, he says, these dimensions are excessive. In that case, if H.P. = horse- power transmitted by the wheel or pulley, N = revs, per min, P = force acting at the circumference, in lbs., and R = radius of pulley in inches, take 3/IOO HJ\ 3/pjg 1 = y n or y 630 ■ Prof. Coleman Sellers (Stevens Indicator, April, 1892) gives the following : The size of keys, both for shafting and for machine tools, are the propor- tions adopted by William Sellers & Co., and rigidly adhered to during a pe- riod of nearly forty years. Their practice in making keys and fitting them is, that the keys shall always bind tight sidewise, but not top and bottom; that is, not necessarily touch either at the bottom of the key-seat in the shaft or touch the top of the slot cut in the gear-wheel that is fastened to the shaft ; but in practice keys used in this manner depend upon the fit of the wheel upon the shaft being a forcing fit, or a fit that is so tight as to re^ quire screw-pressure to put the wheel in place upon the shaft. Size of Keys for Shafting. Diameter of Shaft, in. Size of Key, in. 1M 1 7/16 1 11/16 5/16 -x % 115/16 2 3/16 7/16 x J^ 2 7/16 9/16 x % 2 11/16 2 15/16 3 3/16 3 7/16 ll/16x% 3 15/16 4 7/16 4 15/16 13/16 x % 6 7/16 5 15/16 6 7/16 15/16x1 6 15/16 7 7/16 7 15/16 8 7/16 8 15/16.. 1 1/16x1}$ Length of key-seat for coupling = 1J4 x nominal diameter of shaft. Size of Keys for Machine Tools. Diam. of Shaft, in. Size of Key, sq. in. 4 to 5 7/16 13/16 5^ to 6 15/16 15/16 7 to 8 15/16 1 1/16 9 to 10 15/16 1 3/16 11 to 12 15/16 1 5/16 13 to 14 15/16 1 7/16 Diam. of Shaft, in. Siz f n ° f s * ey ' 15/16 and under % 1 to 13/16 3/16 1M tol 7/16 34 \y 2 tol 11/16 5/16 1% to 2 3/16 7/16 2M to 2 11/16 9/16 2% to 3 15/16 11/16 John Richards, in an article in Cassier's Magaz me, writes as follows: There are two kinds or system of keys, both proper and necessary, but widely dif- ferent in nature. 1. The common fastening key, usually made in width one fourth of the shaft's diameter, and the depth five eighths to one third the width. These keys are tapered and fit on all sides, or, as it is commonly de- scribed, " bear all over." They perform the double function in most cases of driving or transmitting and fastening the keyed-on member against movement endwise on the shaft. Such keys, when properly made, drive as a strut, diagonally from corner to corner. 2. The other kind or class of keys are not tapered and fit on their sides only, a slight clearance being left on the back to insure against wedge action or radial strain. These keys drive by shearing strain. For fixed work where there is no sliding movement such keys are com- monly made of square section, the sides only being planed, so the depth is more than the width by so much as is cut away in finishing or fitting. For sliding bearings, as in the case of drilling-machine spindles, the depth should be increased, and in cases where there is heavy strain there should be two keys or feathers instead of one. The following tables are taken from proportions adopted in practical use. Flat keys, as in the first table, are employed for fixed work when the parts are to be held not only against torsional strain, but also against move- ment endwise ; and in case of heavy strain the strut principle being the strongest and most secure against movement when there is strain each way, as in the case of engine cranks and first movers generally. The objections HOLDING-POWER OF KEYS AND SET-SCREWS. 977 to the sj'Stem for general use are, straining the work out of truth, the care and expense required in fitting, aud destroying the evidence of good or bad fitting of the keyed joint. When a wheel or other part is fastened with a tapering key of this kind there is no means of knowing whether the work is well fitted or not. For this reason such keys are not employed by machine- tool-makers, and in the case of accurate work of any kind, indeed, cannot be, because of the wedging strain, and also the difficulty of inspecting com- pleted work. I. Dimensions of Flat Keys, in Inches. Diam. of shaft ... Breadth of keys Depth of keys 1 U;, m m 2 •>u 3 3^> 4 5 6 7 J 4 5/16 %7/16 to n % 1 m m 1L, o/32 y/io M9/82 ;>/l6 *8 v/ie y s % 11/16 13/16 58 7 8 II. Dimensions op Square Keys, in Inches. Diam. of shaft Breadth of keys.. Depth of keys 1 7/32 m m 2 2Yo 3 3Yo 5/32 9/32 11/32 13/32 15/32 17/32 9/16 3/16 Ya 5/16 % 7/16 to 9/16 Va 11/16 III. Dimensions of Sliding Feather-keys, in Inches. Diam. of shaft Breadth of keys. Depth of keys : . . m m \% 2 21/4 % 2Yo 3 VA 4 Va Vi 5/16 5/16 % Vo 9/16 9/16 % % 7/16 7/16 to to Va u u 4y 2 P. Pryibil furnishes the following table of dimensions to the Am. Machin- ist. He says : On special heavy work and very short hubs we put in two keys in one shaft 90° apart. With special long hubs, where we cannot use keys with noses, the keys should be thicker than the standard. Diameter of Shafts, Width, Thick- Diameter of Shafts, Width, Thick- inches. inches. ness, in. inches. inches. ness.m. % tol 1/16 \y 8 to 1 5/16 3/16 3/16 3 7/16 to 3 11/16 Va Va 5/16 Ya 3 15/16 to 4 3/16 1 11/16 1 7/16 tol 11/16 Va 5/16 4 7/16 to 4 11/16 15* u 1 15/16 to 2 3/16 % 4% to 5% m 15/16 2 7/16 to 2 11/16 Va to 5% to 6% m 1 2 15/16 to 3 3/16 H 9/16 W% to 7% m m Keys longer than 10 inches, say 14 to 16", 1/16" thicker; keys longer than 10 inches, say 18 to 20", %" thicker; and so on. Special short hubs to have two keys. For description of the Woodruff system of keying, see circular of the Pratt & Whitney Co. ; also Modern Mechanism, page 455. HOIiDING-POWER OF KEYS AND SET-SCREWS. Tests of the Holding-power of Set-screws in Pulleys. (G. Lanza, Trans. A. S. M. E., x. 230.)— These tests were made by using a pulley fastened to the shaft by two set-screws with the shaft keyed to the holders; then the load required at the rim of the pulley to cause it to slip was determined, and this being multiplied by the number 6.037 (obtained by adding to the radius of the pulley one-half the diameter of the wire rope, and dividing the sum by twice the radius of the shaft, since there were two set-screws in action at a time) gives the holding-power of the set-screws. The set-screws used were of wrought-iron, % of an inch in diameter, and ten threads to the inch; the shaft used was of steel and rather hard, the set- screws making but little impression upon it. They were set up with a force of 75 lbs. at the end of a ten-inch monkey-wrench. The set-screws used were of four kinds, marked respectively A, B, C, and D. The results were as follows : 978 DYNAMOMETERS. A, ends perfectly flat, 9/16-in. diameter, 1412 to 2294 lbs. ; average 2064. B, radius of rounded ends about y% inch, 2747 " 3079 " " 2912. C, " " " " " J4 " 1902 " 3079 " " 2573. D ends cup-shaped and case-hardened, 1962 " 2958 " " 2470. Remarks. — A. The set-screws were not entirely normal to the shaft ; hence they bore less in the earlier ti ials, before they had become flattened by wear. B. The ends of these set-screws, after the first two trials, were found to be flattened, the flattened area having a diameter of about J4 inch. G. The ends were found, after the first two trials, to be flattened, as in B. D, The first test held well because the edges were sharp, then the holding- power fell off till they had become flattened in a manner similar to B, when the holding-power increased again. Tests of the Holding-power of Keys. (Lanza.)— The load was applied as in the tests of set-screws, the shaft being firmly keyed to the holders. The load required at the rim of the pulley to shear the keys was determined, and this, multiplied by a suitable constant, determined in a sim- ilar way to that used in the case of set-screws, gives us the shearing strength per square inch of the keys. The keys tested were of eight kinds, denoted, respectively, by the letters A, B, C, D, E, F, G and H, and the results were as follows : A, B, D and F, each 4 tests; E, 3 tests ; C, G, and H, each 2 tests. A, Norway iron, 2" x A" X 15/32", 40,184 to 47,760 lbs.; average, 42,726. B, refined iron, 2" X Va" X 15/32", 36,482 " 39,254; " 38,059. C, tool steel, 1" X M" X 15/32", 91,344 & 100,056. D, machinery steel, 2" x Va" X 15/32", 64,630 to 70.186; " 66,875. E, Norway iron, \y s " x %" X 7/16", 36,850 " 37,222; " 37,036. F, cast-iron, 2" X M" X 15/32", 30,278 " 36,944; ■ " 33,034. G, cast-iron, \y B " X %" X 7/16", 37,222 & 38,700. H, cast-iron, 1" X W X 7/16", 29,814 & 38,978. In A and B some crushing took place before shearing. In E, the keys be- ing only 7/16 in. deep, tipped slightly in the key- way. In H, in the first test, there was a defect in the key-way of the pulley. DYNAMOMETERS. Dynamometers are instruments used for measuring power. They are of several classes, as : 1. Traction dynamometers, used for determining the power required to pull a car or other vehicle, or a plough or harrow. 2. Brake or absorption dynamometers, in which the power of a rotating shaft or wheel is absorbed or converted into heat by the friction of a brake; and, 3. Transmission dynamometers, in which the power in a rotating shaft is measured during its transmission through a belt or other connection to another shaft, without being absorbed. Traction Dynamometers generally contain two principal parts: (1) A spring or series of springs, through which the pull is exerted, the exten- sion of the spring measuring the amount of the pulling force; and (2) a paper- covered drum, rotated either at a uniform speed by clockwork, or at a speed proportional to the speed of the traction, through gearing, on which the ex- tension of the spring is registered by a pencil. From the average height of the diagram drawn by the pencil above the zero-line the average pulling force in pounds is obtained, and this multiplied by the distance traversed, in feet, gives the work done, in foot-pounds. The product divided by the time in minutes and by 33,000 gives the horse-power. The Prohy brake is the typical form of absorption dynamometer. (See Fig. 167, from Flather on Dynamometers and the Measurement of Power.) Primarily this consists of a lever connected to a revolving shaft or pulley in such a manner that the friction induced between the surfaces in contact will tend to rotate the arm in the direction in which the shaft revolves. This rotation is counterbalanced by weights P, hung in the scale-pan at the end of the lever. In order to measure the power for a given number of revolu- tions of pulley, we add weights to the scale-pan and screw up on bolts bb, until the friction induced balances the weights and the lever is maintained THE ALDEN ABSORPTION-DYNAMOMETER. 9?9 in its horizontal position while" the revolutions of shaft per minute remain constant. For small powers the beam is generally omitted— the friction being mea- sured by weighting a band or strap thrown over the pulley. Ropes or cords are often used for the same purpose. Instead of hanging weights in a scale-pan, as in Fig. 16T, the friction may be weighed on a platform-scale; in this . case, the direction of rotation being j; ,_ ml, the same, the lever-arm will be on the i 1 *™ -3B^ opposite side of the shaft. | /~* \ \ In a modification of this brake, the pnf -i-at -t $• brake-wheel is keyed to the shaft, ^— / I A and its rim is provided with inner ^ eJ / \ flanges which form an annular trough / \ for the retention of water to keep the /AxAx pulley from heating. A small stream J MpP\\ of water constantly discharges into the trough and revolves with the p IG J57 pulley— the centrifugal force of the particles of water overcoming the action of gravity; a waste-pipe with its end flattened is so placed in the trough that it acts as a scoop, and removes all surplus water. The brake consists of a flexible strap to which are fitted blocks of wood forming the rubbing-surface; the ends of the strap are con- nected by an adjustable bolt-clamp, by means of which any desired tension may be obtained. The horse-power or work of the shaft is determined from the following: Let W — work of shaft, equals power absorbed, per minute; P = unbalanced pressure or weight in pounds, acting on lever-arm at distance L; L = length of lever-arm in feet from centre of shaft; V = velocity of a point in feet per minute at distance L, if arm were allowed to rotate at the speed of the shaft; N == number of revolutions per minute; H.P. = horse-power. Then will W = PV = 2nLNP. Since H.P. = PV +- 33,000, we have H.P. = 2irLNP -4- 33,000. If L = — , we obtain H.P. = -tj™-. 33 -h 2n is practically 5 ft. 3 in., a value often used in practice for the length of arm. If the rubbing-surface be too small, the resulting friction will show great irregularity— probably on account of insufficient lubrication— the jaws be- ing allowed to seize the pulley, thus producing shocks and sudden vibra- tions of the lever-arm. Soft woods, such as bass, plane-tree, beech, poplar, or maple are all to be preferred to the harder w r oods for brake-blocks. The rubbing-surface should be well lubricated with a heavy grease. The Alden Absorption-dynamometer. (G. I. Alden, Trans. A. S. M. E., vol. xi. 958; also xii, 700 and xiii. 429.)— This dynamometer is a friction-brake, w r hich is capable in quite moderate sizes of absorbing large powers with unusual steadiness and complete regulation. A smooth cast- iron disk is keyed on the rotating shaft. This is enclosed in a cast-iron shell, formed of two disks and a ring at their circumference, which is free to revolve on the shaft. To the interior of each of the sides of the shell is fitted a copper plate, enclosing between itself and the side a water-tight space. Water under pressure from the city pipes is admitted into each of these spaces, forcing the copper plate 'against the central disk. The chamber enclosing the disk is filled with oil. To the outer shell is fixed a weighted arm, which resists the tendency of the shell to rotate with the shaft, caused by the friction of the plates against the central disk. Four brakes of this type, 56 in. diam., were used in testing the experimental locomotive at Purdue University (Trans. A. S. M. E., xiii. 429). Each was designed for a maximum moment of 10,500 foot-pounds with a w r ater-press- ure of 40 lbs. per sq. in. The area in effective contact with the copper plates on either side is rep- resented by an annular surface having its outer radius equal to 28 inches, and its inner radius equal to 10 inches. The apparent coefficient of friction between the plates and the disk was 3}4%. 980 DYNAMOMETERS. W. W. Beaumont (Proc. Inst. C. E. 1889) has deduced a formula by means of which the relative capacity of brakes can be compared, judging from the amount of horse-power ascertained by their use. If W = width of rubbing-surface on brake-wheel in inches; V— vel. of point on circum. of wheel in feet per minute; K — coefficient; then K = WV -f- H.P. Capacity of Friction-brakes.— Prof. Flather obtains the values of K given in the last column of the subjoined table : 21 19 20 40 33 150 24 180 475 125 | 250 f 401 125 f i Brake- • pulley. < « . .- a* c % 5 © A *£ h^ bD o a d a a P3 Ed Q J 150 7 5 33" 148.5 7 5 33.38" 146 7 5 32.19" 180 10.5 5 32" 150 10.5 5 32" 150 10 9 142 12 6 38.31" 100 24 5 126.1" 76.2 24 7 191" 290 1 250 f 24 4 63" 322 | 290 f 13 4 27%" Design of Brake. Royal Ag. Soc, compensating McLaren, compensating " water-cooled and comp Garrett, " " " Schoenheyder, water-cooled Balk Gately & Kletsch, water-cooled . . . Webber, water-cooled — Westinghouse, water-cooled 741 749 282 465 847 The above calculations for eleven brakes give values of K varying from 84 7 to 1385 for actual horse-powers tested, the average being K = 655. Instead of assuming an average coefficient, Prof. Flather proposes the following : Water-cooled brake, non-compensating, K = 400; W — 400 H.P. -4- V. Water-cooled brake, compensating, K = 750; W = 750 H.P. -?- V. Non-cooling brake, with or without compensating device, K = 900; W = 900 H.P. -s- V. Transmission Dynamometers are of various forms, as the Batchelder dynamometer, in which the power is transmitted through a " train-arm " of bevel gearing, with its modifications, as the one described by the author in Trans. A. I. M. E., viii. 177, and the one described by Samuel Webber in Trans. A. S. M. E., x. 514: belt dynamometers, as the Tatham; the Van Winkle dynamometer, in which the power is transmitted from a revolving shaft to another in line with it, the two almost touching, through the medium of coiled springs fastened to arms or disks keyed to the shafts; the Brackett and the Webb cradle dynamometers, used for measuring the power required to run dynamo-electric machines. Descrip- tions of the four last named are given in Flather on Dynamometers. Much information on various forms of dynamometers will be found in Trans. A. S. M. E., vol. vii. to xv., inclusive, indexed under Dynamometers. OPERATIONS OF A REFRIGERATING-MACHINE. 981 ICE-MAKING OR REFRIGERATING- MACHINES. References.— An elaborate discussion of the thermodynamic theory of the action of the various fluids used in the production of cold was published by M. Ledoux in the ..4717! ales des Mines, and translated in Van NostrantVs Maga- zine in 1879. This work, revised and additions made in the light of recent ex- perience by Professors Denton, Jacobus, and Riesenberger, was reprinted in 1892. (Van Nostrand's Science Series, No. 46.) The work is largely mathe- matical, but it also contains much information of immediate practical value, from which some of the matter given below is taken. Other references are Wood's Thermodynamics, Chap. V., and numerous papers by Professors Wood, Denton, Jacobus, and Linde in Trans. A. S. M. E., vols. x. to xiv. ; Johnson's Cyclopaedia, article on Refrigeratiug-machines; also Eng'g. June 18, July 2 and 9, 1886; April 1, 1887; June 15, 1888; July 31. Aug. 28, 1889; Sept. 11 and Dec. 4, 1891 ; May 6 and July 8, 1892. For properties of Ammonia and Sulphur Dioxide, see papers by Professors Wood and Jacobus, Trans. A. S. M. E., vols. x. and xii. For illustrated articles describing refrigerating-machines, see Am. Mach., May 29 and June 26, 1890, and Mfrs. Record, Oct. 7, 1892; also catalogues of builders, as Frick & Co., Waynesboro, Pa. ; De La Vergne Refrigerating-ma- chine Co , New York; and others. Operations of a Refrigerating-machine.— Apparatus designed for refrigerating is based upon the following series of operations: Compress a gas or vapor by means of some external force, then relieve it of its heat so as to diminish its volume; next, cause this compressed gas or vapor to expand so as to produce mechanical work, and thus lower its tem- perature. The absorption of heat at this stage by the gas, in resuming its original condition, constitutes the refrigerating effect of the apparatus. A refrigerating-machine is a heat-engine reversed. From this similarity between heat-motors and freezing-machines it results that all the equations deduced from the mechanical theory of heat to deter- mine the performance of the first, apply equally to the second. The efficiency depends upon the difference between the extremes of tem- perature. The useful effect of a refrigerating-machine depends upon the ratio between the heat-units eliminated and the work expended in compressing and expanding. This result is independent of the nature of the body employed. Unlike the heat-motors, the freezing-machine possesses the greatest effi- ciency when the range of temperature is small, and when the final tempera- ture is elevated. If the temperatures are the same, there is no theoretical advantage in em- ploying a gas rather than a vapor in order to produce cold. The choice of the intermediate body would be determined by practical considerations based on the physical characteristics of the body, such as the greater or less facility for manipulating it, the extreme pressures required for the best effects, etc. Air offers the double advantage that it is everywhere obtainable, and that we can vary at will the higher pressures, independent of the temperature of the refrigerant. But to produce a given useful effect the apparatus must be of larger dimensions than that required by liquefiable vapors. The maximum pressure is determined by the temperature of the con- denser and the nature of the volatile liquid: this pressure is often very high. When a change of volume of a saturated vapor is made under constant ' pressure, the temperature remains constant. The addition or subtraction of heat, which produces the change of volume, is represented by an increase or a diminution of the quantity of liquid mixed with the vapor. On the other hand, when vapors, even if saturated, are no longer in con- tact with their liquids, and receive an addition of heat either through com- pression by a mechanical force, or from some external source of heat, they comport themselves nearly in the same way as permanent gases, and be- come superheated. It results from this property, that refrigerating-machines using a liquefi- able gas wiil afford results differing according to the method of working, 982 ICE-MAKING OR REFRIGERATING MACHINES. and depending upon the state of the gas, whether it remains constantly sat- urated, or is superheated during a part of the cycle of working. The temperature of the condenser is determined by local conditions. The interior will exceed by 9° to 18° the temperature of the water furnished to the exterior. This latter will vary from about 52° F., the temperature of water from considerable depth below the surface, to about 95° F., the tem- perature of surface-water in hot climates. The volatile liquid employed in the machine ought not at this temperature to have a tension above that which can be readilv managed by the apparatus. On the other hand, if the tension of the gas at the minimum temperature is too low, it becomes necessary to give to the compression-cylinder large dimensions, in order that the weight of vapor compressed by a single stroke of the piston shall be sufficient to produce a notably useful effect. These two conditions, to which may be added others, such as those de- pending upon the greater or less facility of obtaining the liquid, upon the dangers incurred in its use, either from its inflammability or unhealthful- ness, and finally upon its action upon the metals, limit the choice to a small number of substances. The gases or vapors generally available are: sulphuric ether, sulphurous oxide, ammonia, methylic ether, and carbonic acid. The following table, derived from Regnault, shows the tensions of the vapors of these substances at different temperatures between — 22° and -f- 104°. Pressures and Boiling-points of Liquids available for Use in Refrigerating-machines. Temp, of Ebullition. Tension of Vapor, in lbs. per sq. in., above Zero. Deg. Fahr. Sul- phuric Ether. Sulphur Dioxide. Ammonia. Methylic Ether. Carbonic Acid. Pictet Fluid. - 40 10.22 13.23 16.95 21.51 27.04 33.67 41.58 50.91 61.85 74.55 89.21 105.99 125.08 146.64 170.83 197.83 227.76 - 31 — 22 - 13 5.56 7.23 9.27 11.76 14.75 18.31 22.53 27.48 33.26 39.93 47.62 56.39 66.37 77.64 90.32 11.15 13.85 17.06 20.84 25.27 30.41 36.34 43.13 50.84 59 56 69.35 80.28 92.41 "25l!e" 292.9 340.1 393.4 453.4 520.4 594.8 676.9 766.9 864.9 971.1 1085.6 1207.9 1338.2 - 4 5 14 23 32 41 50 59 68 86 95 1.30 1.70 2.19 2.79 3.55 4.45 5.54 6.84 8.38 10.19 12.31 14.76 17.59 13.5 16.2 19.3 22.9 26.9 31.2 36.2 41.7 48.1 55.6 64 J. 73.2 104 82.9 The table shows that the use of ether does not readily lead to the produc- tion of low temperatures, because its pressure becomes then very feeble. Ammonia, on the contrary, is well adapted to the production of low tem- peratures. Methylic ether yields low temperatures without attaining too great pres- sures at the temperature of the condenser. Sulphur dioxide readily affords temperatures of — 14 to — 5, while its pressure is only 3 to 4 atmospheres at the ordinary temperature of the condenser. These latter substances then lend themselves conveniently for the production of cold by means of mechanical force. The "Pictet fluid" is a mixture of 97$ sulphur dioxide and 3% carbonic acid. At atmospheric pressure it affords a temperature 14° lower than sulphur dioxide. Carbonic acid is as yet (1895) in use but to a limited extent, but the rela- tively greater compactness of compressor that it requires, and its inoffensive THE AJvJMoKlA ABSoitMlOtf-MACHlKE. 983 character, are leading to its recommendation for service on shipboard, where economy of space is important. Certain ammonia plants are operated with a surplus of liquid present dur- ng compression, so that superheating is prevented. This practice is known as the "cold system " of compression. Nothing definite is known regarding the application of methylic ether or of the petroleum product chymogene in practical refrigerating service. The inflammability of the latter and the cumbrousness of the compressor required are objections to its use. "Ice-melting Effect. "— It is agreed that the term "ice-melting effect 1 ' means the cold produced in an insulated bath of brine, on the as- sumption that each 142.2 B.T.U.* represents one pound of ice, this being the latent heat of fusion of ice, or the heat required to melt a pound of ice at 32° to water at the same temperature. The performance of a machine, expressed in pounds or tons of " ice-melt- ing capacity," does not mean that the refrigei ating-machine would make the same amount of actual ice, but that the cold produced is equivalent to the effect of the melting of ice at 32° to water of the same temperature. In making artificial ice the water frozen is generally about 70° F. when sub- mitted to the refrigerating effect of a machine; second, the ice is chilled from 12° to 20° below its freezing-point; third, there is a dissipation of cold, from the exposure of the brine tank and the manipulation of the ice-cans: there- fore the weight of actual ice made, multiplied by its latent heat of fusion, 142.2 thermal units, represents only ahout three fourths of the cold produced in the brine by the refrigerating fluid per I.H.P. of the engine driving the compressing-pumps. Again, there is considerable fuel consumed to operate the brine-circulating pump, the condeusing-water and feed-pumps, and to reboil, or purify, the condensed steam from which the ice is frozen. This fuel, together with that wasted in leakage and drip water, amounts to about one half that required to drive the main steam-engine. Hence the pounds of actual ice manufactured from distilled water is just about half the equiv- alent of the refrigerating effect produced in the brine per indicated horse- power of the steam-cylinders. When ice is made directly from natural water by means of the "plate system," about half of the fuel, used with distilled water, is saved by avoid- ing the reboiling. and using steam expansively in a compound engine. Ether-machines, used in India, are said to have produced about G lbs. of actual ice per pound of fuel consumed. The ether machine is obsolete, because the density of the vapor of ether, at the necessary working-pressure, requires that the compressing-cylinder shall be about 6 times larger than for sulphur dioxide, and 17 times larger than for ammonia. Air-machines require about 1.2 times greater capacity of compress- ing cylinder, and are, as a whole, more cumbersome than ether machines, but they remain in use on ship-board. In using air the expansion must take place in a cylinder doing work, instead of through a simple expansion-cock which is used with vapor machines. The work done in the expansion-cylin- der is utilized in assisting the compressor. Ammonia Compression-machines. — "Cold " vs. "Dry " Systems of Compress ion. —In the "cold" system or "humid" system some of the ammonia entering the compression-cylinder is liquid, so that the heat de- veloped in the cylinder is absorbed by the liquid and the temperature of the ammonia thereby confined to the boiling-point due to the condenser-pres- sure. No jacket is therefore required about the cylinder. In the " dry" or " hot" system all ammonia entering the compressor is gaseous, and the temperature becomes by compression several hundred de- grees greater than the boiling-point due to the condenser-pressure. A water- jacket is therefore necessary to permit the cylinder to be properly lubri- cated. Relative Performance of* Ammonia Compression- and Absorption-machines, assuming no Water to be En- trained with tbe Ammonia-gas in the Condenser. (Denton and Jacobus, Trans. A. S. M. E., xiii.) — It is assumed in the calculation for both machines that 1 lb. of coal imparts 10,000 B.T.H. to the boiler. The * The latent heat of fusion of ice is 144 thermal units (Phil. Mag., 1871, xli.. 182); but it is customary to use 142. (Prof. Wood, Trans. A. S. M. E., xi. 834.) ■ . . , ■ , . . , • 984 ICE-MARlHG OR REFRIGERATING MACHINES. condensed steam from the generator of the absorption-machine is assumed to be returned to the boiler at the temperature of the steam entering the generator. The engine of the compression-machine is assumed to exhaust through a feed-water heater that heats the feed-water to 21v2° F. The engine is assumed to consume 2034 lbs - °f water per hour per horse-power. The figures for the compression- machine include the effect of friction, which is taken at 15$ of the net work of compression. Condenser. Refrigerat- ing Coils. Pounds of Ice-melting Effect per lb. of Coal. O CD ft fa 20 . o Com Dress. Absorption- Machine. machine.* £ .a £ cffl eS fn 3 a gfa amm. xhausts osphere eater, temp, ater. MaT-p 2 3 fa 0) s 1 "3 8 fa Oh 2| = 2 2'B'o So .a a P. £> CO So P < M W 61.2 110.6 5 33.7 61.2 38.1 71.4 38.1 33.5 969 59.0 106.0 5 33.7 59.0 39.8 74.6 38.3 33.9 967 59.0 106.0 5 33.7 130.0 39.8 74.6 39.8 35.1 931 59.0 106.0 -22 16.9 59.0 23.4 43.9 36.3 31.5 1000 86.0 170.8 5 33.7 86.0 25.0 46.9 35.4 28.6 988 86.0 170.8 5 33.7 130.0 25.0 46.9 36.2 29.2 966 86.0 170.8 -22 16.9 86.0 16.5 30.8 33.3 26.5 1025 86.0 170.8 -22 16.9 130.0 16.5 30.8 34.1 27.0 1002 104 227.7 5 33.7 104.0 19.6 36.8 33.4 25.1 1002 104.0 227.7 -22 16.9 104.0 13.5 25.3 31.4 23.4 1041 The Ammonia Absorption-machine comprises a generator which contains a concentrated solution of ammonia in water; this gener- ator is heated either directly by a fire, or indirectly by pipes leading from a steam-boiler. The condenser communicates with the upper part of the gen- erator by a tube; it is cooled externally by a current of cold water. The cooler or brine-tank is so constructed as to utilize the cold produced; the up- per part of it is in communication with the lower part of the condenser. An absorption-chamber is filled with a weak solution of ammonia; a tube puts this chamber in communication with the cooling-tank. The absorption-chamber communicates with the boiler by two tubes: one leads from the bottom of the generator to the top of the chamber, the other leads from the bottom of the chamber to the top of the generator. Upon the latter is mounted a pump, to force the liquid from the absorption -cham- ber, where the pressure is maintained at about one atmosphere, into the gen- erator, where the pressure is from 8 to 12 atmospheres. To work the apparatus the ammonia solution in the generator is first heated. This releases the gas from the solution, and the pressure rises. When it reaches the tension of the saturated gas at the temperature of the condenser there is a liquefaction of the gas, and also of a small amount of steam. By means of a cock the flow of the liquefied gas into the refrigerat- ing-coils contained in the cooler is regulated. It is here vaporized by ab- sorbing the heat from the substance placed there to be cooled. As fast as it is vaporized it is absorbed by the weak solution in the absorbing-chamber. Under the influence of the heat in the boiler the solution is unequally sat- urated, the stronger solution being uppermost. The weaker portion is conveyed by the pipe entering the top of the absorb- ing-chamber, the flow being regulated by a cock, while the pump sends an equal quantity of strong solution from the chamber back to the boiler. * 5$ of water entrained in the ammonia will lower the economy of the ab- sorption-machine about 15$ to 20$ below the figures given in the table. SULPHUR-DIOXIDE MACHINES. 985 The working of the apparatus depends upon the adjustment and regula- tion of the flow of the gas and liquid; by these means the pressure is varied, and consequently the temperature in the cooler may be controlled. The working is similar to that of compression-machines. .The absorption- chamber fills the office of aspirator, and the generator plays the part of compressor. The mechanical force producing exhaustion is here replaced by the affinity of water for ammonia gas; and the mechanical force required for compres- sion is replaced by the heat which severs this affinity and sets the gas at liberty. (For discussion of the efficiency of the absorption system, see Ledoux's work; paper by Prof. Linde, and discussion on the same by Prof. Jacobus, Trans. A. S. M. E., xiv. 1416, 1436; and papers by Denton and Jacobus, Trans. A. S. M. E. x. 792; xiii. 507. Sulphur-Dioxide Machines.— Results of theoretical calculations are given in a table by Ledoux showing an ice-melting capacity per hour per horse-power ranging from 134 to 63 lbs., and per pound of coal ranging from 44.7 to 21.1 lbs., as the temperature corresponding to the pressure of the vapor in the condenser rises from 59° to 104° F. The theo- retical results do not represent the actual. It is necessary to take into ac- count the loss occasioned by the pipes, the waste spaces in the cylinder, loss of time in opening of the valves, the leakage around the piston and valves, the reheating by the external air, and finally, when the ice is being made, the quantity of the ice melted in removing the blocks from their moulds. Manufacturers estimate that practically the sulphur-dioxide apparatus using water at 55° or 60° F. produces 56 lbs. of ice, or about 10,000 heat-units, per hour per horse-power, measured on the driving-shaft, which is about 55$ of the theoretical useful effect. In the commercial manufacture of ice about 7 lbs. are produced per pound of coal. This includes the fuel used for re- boiling the water, which, together with that wasted by the pumps and lost oy radiation, amounts to a considerable portion of that used by the engine. Prof. Denton says concerning Ledoux's theoretical results: The figures given are higher than those obtained in practice, because the effect of superheating of the gas during admission to the cylinder is not considered. This superheating may cause an increase of work of about 25$. There are other losses due to superheating the gas at the brine-tank, and in the pipe leading from the brine-tank to the compressor, so that in actual practice a sulphur-dioxide machine, working under the conditions of an absolute pressure in the condenser of 56 lbs. per sq. in. and the corresponding tem- perature of 77° F., will give about 22 lbs. of ice-melting capacity per pound of coal, which is about 60$ of the theoretical amount neglecting friction, or 70$ including friction. The following tests, selected from those made by Prof. Schroter on a Pictet ice-machine having a compression-cylinder 11.3 In. bore and 24.4 in. stroke, show the relation between the theoretical and actual ice-melting capacity. Temp, in degrees Fahr. corresponding to pressure of vapor. Ice-melting capacity per pound of coal, assuming 3 lbs. per hour per H.P. No. of Test. Condenser. Suction. Theoretical friction included.* Actual. Per cent loss due to cylinder super- heating, or differ- ence between cols. 4 and 5. 11 12 13 14 77.3 76.2 75.2 80.6 28.5 14.4 -2.5 -15.9 41.3 31.2 23.0 16.6 33.1 24.1 17.5 10.1 19.9 22.8 23.9 39.2 The Refrigerating Coils of a Pictet ice-machine described by Ledoux had 79 sq. ft. of surface for each 100,000 theoretic negative heat-units produced per hour. The temperature corresponding to the pressure of the dioxide in the coils is 10.4° F., and that of the bath "(calcium chloride solu- tion) in which they were immersed is 19 4°. * Friction taken at figure observed in the test, Avhich ranged from 23$ to §6$ of the work of the steam-cylinder, 986 ICE-MAKING OR REFRIGERATING MACHINES. d I ^ 5^ gtt-4 g,Q t» ©*°^2 •N 3 o g co S3 CD =£; M u a •9jnmi9d -ui9X jo QSn^a \I o 08 Sequins -SB 'A\[013dT?Q SurjT9iu-90i JO UOJ, J9d 'J9JUAV - Suisuapuo^ "08 HI 3 O o d ■uop -OjJjj qj?A\ 'Japaii^o-urBaig jo j'H '.i9d anoq J9d reoo jo sqi g Sujuinssu l \voj jo qj J9d £%\6vdvQ Suijt9ui-9Di 02 a cococ? ft r3 m %a9maav\d -sia uo-jsij jo qooj oiqno J9d XjioBd'BO Sarqaiu - 90j a "#tH>0 03 ft .£"3 If CD'S ftft •uoijou^ q^TAV •J9AVOd -9SJOH J9d anoq J9J a CD 1 § •uoi^ou^ qj!M •uoiss9aduioo jo ©< j[jo^jo-q[--jja9 ( i £ o 9% a -ft 3 CD a Cv O ci D. s 3 ft 'o o o ft o 3 u $ ft CD P. a 3 c o w 5 \I9A10d ft -uregjs pajBOipuj £ jo 'uoijou^ qj!A\. rH - ft 000 coSco lis 15 m f •uoTjoiJ^jnoqiiAi £ .3 ft 000 gi- ll B3 qD9dO]9A -9 9ai^^9^; jo jgquin^ ft' as oJ oj M9SU9pUO0 jb p9')0 , ej}sqy jBajj ^ ft H M 0000 a CD u - p9SS9.ld g -uioq sb£) jo jqSj9AV s 5 a o o to 3 •TIOISS9jdlUO0 JO „ pn[j()'BSB0JO9-mj'BJ9dtu9j J ^ ft ft C3iT-lS0 T3 CD CD •Sp00-StU?'B.l9SlJJ *T -9£[ ut 9JUSS9.IJ - 9jniosqy + CD • J* CD 3. 3 cS JO -p •S]lOO iods?9 SuiJ'BJ9SlJJ9H UI D 9.inss9jj 04 Sul IJ0Q 9JnjBJ9dU!9J J ft CD ft too© ?ooc 1 ,§^*o tm llll &•■: s IS-S^ £:!l «C3^ 5i*>§ w ^ 3^5^ tD-73 CO O CD =3 SS &. CD CDt3 § s a^ *» 22 ® 3 gjft- S * CD O • CD &C* 3 .-s ame 1 nterin y. In mete press CO CD O.S S 55^g = han the actual, for Is in superheating that indicated by t by means of a sp e suction and conde CD > .t3 u *» are high 3 cylinder agree w ured dire Iculated a C 3 O w 3 3 O^ oj oS s- CD s „ 03 ults f a the amo achin it hea - a 43 ^ oj B73 s-- g BS_§ 00 a" Sc Sf 2cd3-c.9 AMMONIA COMPRESSION-MACHINES. 987 03 bJO a *d o Q •sanoq f% m ifcjioud -i?0 3ui%\aw-a6i jo uoj, J9d ajnujm .i9j "3 (19) .89 .92 .94 .96 .98 1.00 63 tf P Ifi 63 B 63 BS O ' B b <° . 63 w B J £-, 10 i Oi B 5 Oc go B& H 63 ££ .4 M° H B Q B 2i Eh < °B is §^ s < <1b s 03 B J s » c |~ O 63 ' K 85 (19) .97 .99 1.02 1.04 1.07 1.09 •^4}OBdBO Sill -3I9UI-90I JO UOJ, J9J (18) 1290 1320 1350 1380 1410 1440 (18) 1,390 1,420 1,470 1,500 1,540 1,570 •duidx Jo 9Sa«a 0O8 Suuuns -SB '(J119lU90B[dS|Q 1104s jj jo -4 j •na'j9 < j "3 (17) .2872 .2882 .2890 .2898 .2904 .2910 (17) .1611 .1620 .1628 .1636 .1643 .1649 "o c3 P, cS o be a I •jngrago'BidsiQ uo%si£ jo '%}. no .led 73 a H (16) .000223 .000219 .000215 .000211 .000206 .000202 (16) .000116 .000114 .000111 .000109 .000107 .000105 pie 1" uotjou^ q^iAV (15) 39.8 33.4 28.7 25.0 22.0 19.6 (15) 23.4 20.6 18.4 16.5 14.9 13.5 •nor; -ouji ^noqijAi (J (14) 45.8 38.4 33.0 28.7 25.3 22.6 (14) 26.9 23.7 21.1 18.9 17.1 15.5 •uoi^ou^ q^AV (13) 119.3 100.2 86.1 75.0 66.1 58.9 (13) 70.2 61.8 55.1 49.4 44.7 40.6 •noq -01 j^ ^noqijAi (12) 137.0 115.2 99.0 86.2 76.0 67.7 (12) 80.7 71.0 63.3 56.8 51.4 46.6 .5 •nopoijg; Saipnpui ''d'H J9 d anon J^d Eh (11) 16,960 14,250 12,240 10,660 9,400 8,380 (11) 9,980 8,790 7,840 7,030 6,360 5,780 •uoiqouj£ ^ax -prqoui "psp'uadxg JlJOAi jo qi-jj J9 pacity per 24 hours } 24 X pounds X specific heat X range of temp. ) >■ of brine circulated per hour. 142.2 X 2000 The analogy between a heat-engine and a refrigerating -machine is as fol- lows: A steam-engine receives heat from the boiler, converts a part of it into mechanical work in the cylinder, and throws away the difference into the condenser. The ammonia in a compression refrigerating machine re- ceives heat from the brine-tank or cold-room, receives an additional amount of heat from the mechanical work done in the compression-cylinder, and throws away the sum into the condenser. The efficiency of the steam-engine = work done -f- heat received from boiler. The efficiency of the refrigerat- ing-machine = heat received from the brine-tank or cold-room -*- heat re- quired to produce the work in the compression-cylinder. In the ammonia Cold Water J i «. ■X- Compressor " „o Brine Outlet V fr — *- Condenser 309 c 82° 239°' 10° 3° v 64° Brine Tank Ammonia Coils Cold Room •*• Heat received from compression. 1 1 »• Warm Water Heat rejected \X- 14° Heat received from brine Inlet DIAGRAM OF AMMONIA COMPRESSION MACHINE. 1 54° i J Torce Pump DIAGRAM OF AMMONIA ABSORPTION MACHINE. absorption-apparatus, the ammonia receives heat from the brine-tank and additional heat from the boiler or generator, and rejects the sum into the condenser and into the cooling water supplied to the absorber. The effi- ciency = heat received from the brine h- heat received from the boiler. §90 ICE-MAKlHG Oil ftEFElGERATING MACHINES. TEST-TRIALS OF REFRIGEBATING-MACHINES. (G. Linde, Trans. A. S. M. E., xiv. 1414.) The purpose of the test is to determine the ratio of consumption and pro- duction, so that there will have to be measured both the refrigerative effect and- the heat (or mechanical work) consumed, also the cooling water. The refrigerative effect is the product of the number of heat-units (Q) abstracted from the body to be cooled, and the quotient — c —, — ; in which Tc = abso- lute temperature at which heat is transmitted to the cooling water, and T = absolute temperature at which heat is taken from the body to be cooled. The determination of the quantity of cold will be possible with the proper exactness only when the machine is employed during the test to refrigerate a liquid; and if the cold be found from the quantity of liquid circulated per unit of time, from its range of refrigeration, and from its specific heat. Sufficient exactness cannot be obtained by the refrigeration of a current of circulating air, nor from the manufacture of a certain quantity of ice, nor from a calculation of the fluid circulating within the machine (for instauce, the quantity of ammonia circulated by the compressor). Thus the refrig- eration of brine will generally form the basis for tests making any pretension to accuracy. The degree of refrigeration should not be greater than neces- sary for allowing the range of temperature to be measured with the neces- sary exactness; a range of temperature of from 1° to 6° Fahr. will suffice. The con dense :• measurements for cooling water and its temperatures will be possible with sufficient accuracy only with submerged condensers. The measurement of the quantity of brine circulated, and of the cooling water, is usually effected by water-meters inserted into the conduits. If the necessary precautions are observed, this method is admissible. For quite precise tests, however, the use of two accurately gauged tanks must be ad- vised, which are alternately filled and emptied. To measure the temperatures of brine and cooling water at the entrance and exit of refrigerator and condenser respectively, the employment of specially constructed and frequently standardized thermometers is indis- pensable; no less important is the precaution of using at each spot simul- taneously two thermometers, and of changing the position of one such thermometer series from inlet to outlet (and vice versa) after the expiration of one half of the test, in order that possible errors may be compensated. It is important to determine the specific heat of the brine used in each instance for its corresponding temperature range, as small differences in the composition and the concentration may cause considerable variations. As regards the measurement of consumption, the programme will not have any special rules in cases where only the measurement of steam and cooling water is undertaken, as will be mainly the case for trials of absorption-ma- chines. For compression-machines the steam consumption depends both on the quality of the steam-engine and on that of the ref rigerating-machine, while it is evidently desirable to know the consumption of the former sep- arately from that of the latter. As a rule steam-engine and compressor are coupled directly together, thus rendering a direct measurement of the power absorbed by the refrigerating-machine impossible, and it will have to suffice to ascertain the indicated work both of steam-engine and compressor. By further measuring the work for the engine running empty, and by compar- ing the differences in power between steam-engine and compressor resulting for wide variations of condenser-pressures, the effective consumption of work Le for the refrigerating-machine can be found very closely. In gen- eral, it will suffice to use the indicated work found in the steam-cylinder, especially as from this observation the expenditure of heat can be directly determined. Ordinarily the use of the indicated work in the compressor- cylinder, for purposes of comparison," should be avoided; firstly, because there are usually certain accessory apparatus to be driven (agitators, etc.), belonging to the refrigerating-machine proper; and secondly, because the external friction would be excluded. Heat Balance. — We possess an important aid for checking the cor- rectness of the results found in each trial by forming the balance in each case for the heat received and rejected. Only such tests should be re- garded as correct beyond doubt which show a sufficient conformity in the heat balance. It is true that in certain instances it may not be easy to account fully for the transmission of heat between the several parts of the machine and its environment by radiation and convection, but generally TEMPERATURE RANGE. 991 (particularly for compression-machines) it will be possible to obtain for the neat received and rejected a balance exhibiting: small discrepancies only. Report of Test.— Reports intended to be used for comparison with the figures found for other machines will therefore have to embrace at least the following observations : Refrigerator: Quantity of brine circulated per hour Brine temperature at inlet to refrigerator -.. Brine temperature at outlet of refrigerator t Specific gravity of brine (at 64° Fahr.) Specific heat of brine Heat abstracted (cold produced) Qe Absolute pressure in the refrigerator Condenser : Quantity of cooling water per hour Temperature at inlet to condenser Temperature at outlet of condenser t Heat abstracted Q x Absolute pressure in the condenser Temperature of gases entering the condenser COMPRESSTON-MACHINE. Compressor : Indicated work Lt Temperature of gases at inlet. . Temperature of gases at exit.. Steam-engine : Feed-water per hour Temperature of feed-water Absolute steam-pressure before steam-engine Indicated work of steam-engine Le Condensing water per hour Temperature of da Total sum of losses by radiation and convection ± Q 3 Heat Balance : Qe + ALo = Qi ± Q z . Absorption-machine. Still : Steam consumed per hour Abs. pressure of heating steam. Temperature of condensed steam at outlet Heat imparted to still Q'e Absorber : Quantity of cooling water per hour Temperature at inlet ... Temperature at outlet Heat removed Q 2 Pump for Ammonia Liquor : Indicated work of steam-engine Steam-consumption for pump.. Thermal equivalent for work of pump ALp Total sum of losses by radiation and convection ± Q 3 Heat Balance : Qe + Q'e = Qi + Qz± Q 3 . For the calculation of efficiency and for comparison of various tests, the actual efficiencies must be compared with the theoretical maximum of effi- ciency {-ry-J max. = — — corresponding to the temperature range. Temperature Range. — As temperatures (T and To) at which the heat is abstracted in the refrigerator and imparted to the condenser, it is cor- rect to select the temperature of the brine leaving the refrigerator and that of the cooling water leaving the condenser, because it is in principle impos- sible to keep the refrigerator pressure higher than would correspond to the lowest brine temperature, or to reduce the condenser pressure below that corresponding to the outlet temperature of the cooling water. Prof. Linde shows that the maximum theoretical efficiency of a com- pression-machine may be expressed by the formula Q T AL ~ Tc - T ' in which Q = quantity of heat abstracted (cold produced) ; AL = thermal equivalent of the mechanical work expended; L = the mechanical work, and A = 1 -s- 778 ; T = absolute temperature of heat abstraction (refrigerator) ; Tc = " " " " rejection (condenser). If u = ratio between the heat equivalent of the mechanical work AL, and the quantity of heat Q' which must be imparted to the motor to produce the work i, then 992 ICE-MAKING OR REFRIGERATING MACHINES. AL Q' and^: uT It follows that the expenditure of heat Q' necessary for the production of the quantity of cold Q in a compression-machine will be the smaller, the smaller the difference of temperature To, — T. Metering tlie Ammonia*— For a complete test of an ammonia re- frigerating-machine it is advisable to measure the quantity of ammonia cir- culated, as was done in the test of the 75-ton machine described by Prof. Denton. (Trans. A. S. M. E., xii. 326.) PROPERTIES OF SULPHUR DIOXIDE AND AMMONIA GAS. Ledoux's Table for Saturated Sulphur-dioxide Gas. Heat-units expressed in B.T.U. per pound of sulphur dioxide. ®.2 5 = fe 3-9 x •]• Deg. F. -22 -13 - 4 95 104 Lbs. 5.56 7.23 9.27 11.76 14.74 18.31 22.53 27.48 33.25 39.93 47.61 56.39 66.36 77.64 90.31 Ip2 B.T.U. 157.43 158.64 159.84 161.03 162.20 163.36 164.51 165.65 16790 168.99 170.09 171.17 172.24 173.30 -1 &i* -19.56 -16.30 -13.05 - 9.79 - 6.53 - 3.27 0.00 3.27 6.55 9.83 13.11 16.39 19.69 22.98 26.28 B.T.U. 176.99 174.95 172.89 170.82 168.73 166.63 164.51 162.38 160.23 158.07 155.89 153.70 151.49 149.26 147.02 13.59 13.83 14.05 14.84 15.01 15.17 15.32 15.46 15.59 15.71 15.82 15.91 161.12 158.84 156.56 154.27 151.97 149.68 147.37 145.06 142.75 140.43 138.11 135.78 133.45 131.11 Cu. ft. 13.17 10.27 8.12 6.50 5.25 4.29 3.54 2.93 2.45 2.07 1.75 1.49 1.27 1.09 .91 > be .123 .153 .190 .340 .407 .483 .570 1.046 Density of Liquid Ammonia. (DAndreff, Trans. A. S. M. E., x. 641.) At temperature C -10—5 5 10 15 20 F +14 23 32 41 50 59 68 Density...., 6492 .6429 .6364 .6298 .6230 .6160 .6089 These may be expressed very nearly by 8 = 0.6364 - 0.0014*° Centigrade; 8 = 0.6502 - 0.000777T Fahr. Latent Heat of Evaporation of Ammonia. (Wood, Trans. A. S. M. E.,x. 641.) he = 555.5 - 0.613T - 0.0002192 12 (in B.T.U., Fahr. deg.); Ledoux found he = 583.33 - 0.5499T - 0.0001173T 2 . For experimental values at different temperatures determined by Prof. Denton, see Trans. A. S. M. E., xii. 356. For calculated values, see vol. x. 646. Density of Ammonia Gas.— Theoretical, 0.5894; experimental, 0.596. Regnault (Trans. A. S. M. E.. x. 633). Specific Heat of Liquid Ammonia. (Wood, Trans. A. S. M. E., x 645 )— The specific heat is nearly constant at different temperatures, and about equal to that of water, or unity. From 0° to 100° F., it is c = 1.096 - .0012T, nearly. In a later paper by Prof. Wood (Trans. A.S, M. E,, xii. 136) he givesahigher value, viz., c = 142136 + 0.00043ST, PKOPERTIES OF AMMONIA VAPOR. 993 Dr. Von Strombeck, in 1890, found from the mean of eight experiments, at a temperature about 80° F., c = 1.22876,— about % greater than that cal- culated from this formula. In Prof. Wood's Thermodynamics (edition of 1894) in addition to the above figures he gives the mean of six determinations by Ludeking and Starr, 0.886. This, says Prof. Wood, leaves the correct result in doubt, and one may con- sider it as unity until determined by further experiments. Properties of the Saturated Vapor of Ammonia. (Wood's Thermodynamics.) Temperature. Pressure, Abso' 11 *" Heat of Volume Volume Weight of a cu. Vaporiza- tion, ther- mal units. of Vapor per lb., cu. ft. of Liquid per lb., cu. ft. Degs. F. Abso- lute, F. Lbs.per sq. ft. Lbs.per sq. in. ft. of Vapor, lbs. - 40 420.66 1540.7 10.69 579.67 24.372 .0234 .0410 - 35 425.66 1773.6 12.31 576.69 21.319 .0236 .0468 - 30 430.66 2035.8 14.13 573.69 18.697 .0237 .0535 - 25 435.66 2329.5 16.17 570.68 16.445 .0238 .0608 - 20 440.66 2657.5 18.45 567.67 14.507 .0240 .0689 - 15 445.66 3022.5 20.99 564.64 12.834 .0242 .0779 - 10 450.66 3428.0 23.80 561.61 11.384 .0243 .0878 - 5 455.66 3877.2 26.93 558.56 10.125 .0244 .0988 460.66 4373.5 30.37 555 . 50 9.027 .0246 .1108 h 5 465.66 4920.5 34.17 552.43 8.069 .0247 .1239 - 10 470.66 5522.2 38.34 549.35 7.229 .0249 .1383 - 15 475.66 6182.4 42.93 546.26 6.492 .0250 .1544 - 20 480.66 6905.3 47.95 543.15 5.842 .0252 .1712 - 25 485.66 7695.2 53.43 540.03 5.269 .0253 .1898 - 30 490.66 8556.6 59.41 536.92 4.763 .0254 .2100 - 35 495.66 9493.9 65.93 533.78 4.313 .0256 .2319 - 40 500.66 10512 73.00 530.63 3.914 .0257 .2555 - 45 505.66 11616 80.66 527.47 3.559 .0259 .2809 - 50 510.66 12811 88.96 524.30 3.242 .0261 .3085 - 55 515.66 • 14102 97. y3 521.12 2.958 .0263 .3381 - 60 520.66 15494 107.60 517.93 2.704 .0265 .3698 - 65 525.66 16993 118.03 514.73 2.476 .0266 .4039 - 70 530.66 18605 129.21 511.52 2.271 .0268 .4403 - 75 535.66 20336 141.25 508.29 2.087 .0270 .4793 - 80 540.66 22192 154.11 505.05 1.920 .0272 .5208 - 85 545.66 24178 167.86 501.81 1.770 .0273 .5650 - 90 550.66 26300 182.8 498.11 1.632 .0274 .6128 - 95 555.66 28565 198.37 495.29 1.510 .0277 .6623 -100 560.66 30980 215.14 492.01 1.398 .0279 .7153 -105 565.66 33550 232.98 488.72 1.296 .0281 .7716 -110 570.66 36284 251.97 485.42 1.203 .0283 .8312 -115 575.66 39188 272.14 482.41 1.119 .0285 .8937 -120 580.66 42267 293.49 478.79 1.045 .0287 .9569 -125 585.66 45528 316.16 475.45 0.970 .0289 1.0309 - iSu 590.66 48978 340.42 472.11 0.905 .0291 1.1049 -135 595.66 52626 365.16 468.75 0.845 .0293 1.1834 -140 600.66 56483 392 22 465.39 0.791 .0295 1.2642 -145 605.66 60550 420.49 462.01 0.741 .0297 1.3495 -150 610.66 64833 450.20 458.62 0.695 .0299 1.4388 -155 615.66 69341 4S1.54 455.22 0.652 .0302 1.5337 - -160 620.66 74086 514.40 451.81 0.613 .0304 1.6343 r- 165 625.66 79071 549.04 418.39 0.577 .0306 1.7333 Specific Heat of Ammonia Vapor at the Saturation Point. (Wood, Trans. A. S. M.;E , x. 644.)— For the range of temperatures ordinarily used in engineeering practice, the specific heat of saturated am- monia is negative, and the saturated vapor will condense with adiabatic ex- pansion, and the liquid will evaporate with the compression of the vapor, and when all is vaporized will superheat. Regnault (Rel. cles. Exp., ii. 162) gives for specific heat of ammonia-gas 0-50836, (Wood, Trans. A. S. M. E., xii. 133,) 994 ICE-MAKING OR REFRIGERATING MACHINES. Properties of Brine used to absorb Refrigerating Effect of Ammonia. (J. E. Denton, Trans. A. S. M. E., x. 799.)— A soluuon of I Liverpool salt in well-water having a specific gravity of 1.17, or a weight per cubic foot of 73 lbs., will not sensibly thicken or congeal at 0° Fahren- heit. (It is reported that brine of 1.17 gravity, made with American salt, begins to congeal at about 24° Fahr.) The mean specific heat between 39° and 16° Fahr. was found by Denton to be 0.805. Brine of the same specific gravity has a specific heat of 0.805 at 65° Fahr., according to Naumann. Naumann's values are as follows {Lehr- und Handbuch der Tliermochemie, 1882): Specific heat 791 .805* .863 .895 .931 .962 .978 Specific gravity. 1.187 1.170 1.103 1.072 1.044 1.023 1.012 * Interpolated. Chloride-of-calcium solution has been used instead of brine. Ac- 1 cording to Naumann, a solution of 1.0255 sp. gr. has a specific heat of .957. A solution of 1.163 sp. gr. in the test reported in Eng'g, July 22, 1887, gave a specific heat of .827. ACTUAL PERFORMANCES OF ICE-MAKING MACHINES. The table given on page 996 is abridged from Denton, Jacobus, and Riesen- berger's translation of Ledoux on Ice-making Machines. The following shows the class and size of the machines tested, referred to by letters in the table, with the names of the authorities: Class of Machines. Authority. Dimensions of Compres- sion-cylinder in inches. A. Ammonia cold-compression.. B. Pictet fluid dry -compression. C. Bell-Coleman air D. Closed cycle air E. Ammonia dry-compression . F. Ammonia absorption \ Ren wick & ( Jacobus. Denton. 28.0 10. 12.0 16.5 24.4 23.8 18.0 30.0 Performance of a 75-ton Ammonia Compression- machine. (J. E. Denton, Trans. A. 8. M. E., xii, 326. )— The machine had two single-acting compression cylinders 12" X 30", and one Corliss steam - cylinder, double-acting, 18 ,/ X 36". It was rated by the manufacturers as a 50-ton machine, but it showed 75 tons of ice-refrigerating effect per 24 hours during the test. The most probable figures of performance in eight trials are as follows : Ammonia Brine c iuo^ I'll 3 ..; F a §3 o O 3 & © (S s g£ 13 ££ o O oo 150 28 24 2.00 30 3.61 37.5 4.51 393 513 640 150 7 14 1.69 17.5 2.11 21.5 2.58 240 300 366 105 28 34.5 4.10 43 5.18 54 6.50 591 725 923 105 7 22 2. C5 27.5 3.31 34.5 4.16 376 470 591 The non -condensing engine is assumed to require 25 lbs. of steam per horse-power per hour, the non-compound condensing 20 lbs., and the com- densing 16 lbs., and the boiler efficiency is assumed at 8.3 lbs. of water per lb. coal under working conditions. The following conclusions were derived from the investigation : 1. The capacity of the machine is proportional, almost entirely, to the weight of ammonia circulated. This weight depends on the suction- pressure and the displacement of the compressor-pumps. The practical suction-pressures range from 7 lbs. above the atmosphere, with which a temperature of 0° F. can be produced, to 28 lbs. above the atmosphere, with which the temperatures of refrigeration are confined to about 28° F. At the lower pressure only about one half as much weight of ammonia can be cir- culated as at the upper pressure, the proportion being about in accordance with the ratios of the absolute pressures, 22 and 42 lbs. respectively. For each cubic foot of piston-displacement per minute a capacity of about one sixth of a ton of " refrigerating effect " per 24 hours can be produced at the lower pressure, and of about one third of a ton at the upper pressure. No other elements practically affect the capacity of a machine, provided the cooling- surface in the brine-tank or other space to be cooled is equal to about 36 sq. ft. per ton of capacity at 28 lbs. back pressure. For example, a differ- ence of 100$ in the rate of circulation of brine, while producing a propor- tional difference in the range of temperature of the latter, made no practical difference in capacity. The brine-tank was 10^ X 13 X 19^ ft., and contained 8000 lineal feet of 1-in. pipe as cooling-surface. The condensing-tank was 12 X 10 X 10 ft., and coutained 5000 lineal feet of 1-in. pipe as cooling-surface. 2. The economy in coal-consumption depends mainly upon both the suc- tion-pressures and condeusing-pressures. Maximum economy, with a given type of engine, where water must be bought at average city prices, is obtained at 28 lbs. suction-pressure and about 150 lbs. condensing-pressure. Under these conditions, for a non-condensing steam-engine, consuming coal at the rate of 3 lbs. per hour per I.H.P. of steam-cylinders, 24 lbs. of ice- refrigerating effect are obtained per lb. of coal consumed. For the same condensing-pressure, and with 7 lbs. suction-pressure, which affords tem- peratures of 0° F., the possible economy falls to about 14 lbs. of " refrigerat- ing effect " per lb. of coal consumed. The condensing-pressure is determined by the amount of condensiug-water supplied to liquefy the ammonia in the condenser. If the latter is about 1 gallon per minute per ton of refrigerating effect per 24 hours, a condensing-pressm - e of 150 lbs. results, if the initial tem- perature of the water is about 56° F. Twenty-five per cent less water causes the condensing-pressure to increase to 190 lbs. The work of compression is thereby increased about 20$, and the resulting "economy" is reduced to about 18 lbs. of " ice effect " per lb. of coal at 28 lbs. suction-pressure and 11.5 at 7 lbs. If, on the other hand, the supply of water is made 3 gallons per minute, the condensing-pressure may be confined to about 105 lbs. The work of compression is thereby reduced about 25$, and a proportional increase of economy results. Minor alterations of economy depend on the initial temperature of the condensing-water and variations of latent heat, but these are confined within about 5$ of the gross result, the main element of control being the work of compression, as affected by the back pressure and con- densing-pressure, or both. If the steam engine supplying the motive power may use a condenser to secure a vacuum, an increase of ecouomy of 25$ is available over the above figures, making the lbs of " ice effect" per lb. of 996 ICE-MAKING OR REFRIGERATING MACHINES. coal for 150 lbs. condensing-pressure and 28 lbs. suction -pressure 30.0, and for 7 lbs. suction-pressure, 17.5. It is, however, impracticable to use a con- denser in cities where water is bought. The latter must be practically free of cost to be available for this purpose. In this case it may be assumed that water will also be available for condensing the ammonia to obtain as low a condensing-pressure as about 100 lbs., and the economy of the refrig- erating-machine becomes, for 28 lbs. back pressure, 43.0 lbs. of "ice effect " per lb. of coal, or for 7 lbs. back-pressure, 27.5 lbs. of ice effect per lb. of coal. If a compound condensing-engine can be used with a steam-con- sumption per hour per horse-power of 16 lbs. of water, the economy of the refrigeratiug-machine may be 25$ higher than the figures last named, mak- ing for 28 lbs. back pressure a refrigerating effect of 54.0 lbs. per lb. of coal, and for 7 lbs. back pressure a refrigerating effect of 34.0 lbs. per lb. of coal. Actual Performance of Ice-making Machines, 1^* B c 9 ,0 3 t» emperature corresponding to Pressure, in degrees Fahr. o , i-'O a fl*3 v oa a a > V B "z 6 ft 3 a aS .2 .■a a !H a3 £ c Sous cu a . __ ? 9 EH < H E-i s ft 02 O U II '3 O 3 o ft. $1 |'5.2 £ft£ 3«; - *7J 9B o u ^2 Q O a o W a o o o ft CO c o i o '43 o "3 ft a =-S © IS *a S, ei s 2 S a o £ 3 > o t x ir» & ft ^Sffift CD tn * «$ oS ~ >>0>~ ^acq •-N SN . 1 i %™? oS u%> 03™ £ £-"os 3 P-rt 3 frO — a-g 3 n T3« §£?! C V. a $ ^>>J 3 c^ 1 sl!-£ 5 5* 3 3 "-J* pagj x a o3 .Eoo X a o3 .3 c .3 a M a ■- 93 .3 o.S v X a £ u esHCQ sSWffl osWPQPh Jhw § r5 s 161 lbs. Av. high ammonia press, above atmos 151 lbs. 152 lbs. 147 lbs. Av. back ammonia press, above atmos Av. temperature brine inlet 28 " 8.2 " 13 " 27.5 " 36.76° 6.27° 14.29° Av. temperature brine outlet. 28.86° 2.03° 2.29° '28.45° Av. range of temperature 7.9° 4.24° 12.00° 7.91° Lbs. of brine circulated per minute 2281 2173 943 2374 Av. temp, condensing-water at inlet 44.65° 56.65° 46.9° 54.00° Av. temp, condensing-water at outlet 83.66° 85.4° 85.46° 82.86° Av. range of temperature 39.01° 28.75° 38.56° 28.80° Lbs. water circulated p. min. thro' cond'ser 442 315 257 601.5 Lbs. water per min. through jackets 25 44 40 14 Range of temp rature in jackets 24.0° 16.2° 16.4° 29.1° Lbs. ammonia circulated per min *28.17 14.68 16.67 28.32 Probable temperature of liquid ammonia, entrance to brine-tank *71.3° *G8° *63.7° 76.7° Temp, of amm. corresp. to av. back press. +14° - 8° - 5° 14° Av. temperature of gas leaving brine-tanks 34.2° 14.7° 3 0° 29.2° Temperature of gas entering compressor. . *39° 25° 10.13° 34° Av. temperature of gas leaving compressor 213° 263° 239° 221° Av. temp, of gas entering condenser Temperature due to condensing pressure.. 200° 218° 209° 168° 84.5° 84.0° 82.5° 88.0° Heat given ammonia: By brine, B T.U. per miniute 14776 7186 8824 14647 By compressor, B.T.U. per minute 2786 2320 2518 3020 By atmosphere, B.T.U. per minute. ... 140 147 167 141 Total heat rec. by amm., B.T.U. per min. 17702 9653 11409 17708 Heat taken from ammonia: By condenser, B.T.U per min 17242 608 9056 712 9910 656 17359 406 By jackets, B.T.U. per min By atmosphere, B.T.U. per min 182 338 250 252 Total heat rej. by amm., B.T U. per min. . . 18032 10106 10816 18017 Dif. of heatrec'd and rej., B.T.U. per min. 330 453 407 309 % work of compression removed by jackets. 22# 31 % 26% Vi% Av. revolutions per min 58.09 57.7 57.88 58.89 Mean eff. press, steam-cyl., lbs. per sq. in.. 32.5 27.17 27.83 32.97 Mean eff. press, amm.-cyl., lbs. per sq. in . . 65.9 53.3 59.86 70.54 Av. H.P. steam-cylinder 85.00 65.7 71.7 54.7 73.6 59.37 88.63 71.20 Av. H.P. ammonia-cylinder Friction in per cent of steam HP 23.0 24.0 20.0 19.67 Total cooling water, gallons per min. per ton per 24 hours 0.75 1.185 0.797 0.990 Tons ice-melting capacity per 24 hours 74.8 36.43 44.64 74.56 Lbs. ice-refrigerating eff. per lb. coal at 3 lbs. per H.P. per hour 24.1 14.1 17.27 23.37 Cost coal per ton of ice-refrigerating effect at $4 per ton $0,166 ■,$0,283 $0,231 $0,170 Cost water per ton of ice-refrigerating effect at $1 per 1000 cu. ft... SO. 128 $0,200 $0,136 $0,169 Total cost of 1 ton of ice-refrigerating eff... $0,294 $0,483 $0,467 $0,339 Figures marked thus (*) are obtained by calculation; all other figures are obtained from experimental data ; temperatures are in Fahrenheit degrees. ARTIFICIAL ICE-MANUFACTURE. 999 Ammonia Compression-machine. Actual Results obtained at the Munich Tests. (Prof. Liu.de, Trans. A. S. M. E., xiv. 1419.) No. of Test . . , ' 2 3 4 5 Temp, of refrig- (_ Inlet, deg. F 43.194 28.344 13.952 -0.279 28.251 erated brine ) Outlet, t deg. F. . . 37.054 22.885 8.771 -5.879 23.072 Specific heat of brine 0.861 0.851 0.843 0.837 0.851 Quantity of brine circ. per h., cu. ft. 1,039.38 908.84 633.89 414.98 800.93 Cold produced, B.T.U. per hour — 342,909 2C3.950 172,776 121,474 220,284 Quant, of cooling water per h.. c. ft. 338.76 260.83 187.506 139.99 97.76 I.H.P. in steam-engine cylinder (Le). 15.80 16.47 15.28 14.24 21.61 Cold pro- ) Per I.H.P. in comp.-cyl. 24,813 18,471 12,770 10,140 11,151 duced per >Per I H.P. in steam-cyl. 21.703 16,026 11,307 8,530 10,194 h.. B.T.U. ) Per lb. of steam 1,100.8 785.6 564.9 435.82 512.12 Means for Applying the Cold. # (M. C. Bannister, Liverpool Eng'g Soc'y, 1890.) — The most useful means for applying the cold to various uses is a saturated solution of brine or chloride of magnesium, which remains liquid at 5° Fahr. The brine is first cooled by being circulated in contact with the refrigerator-tubes, and then distributed through coils of pipes, arranged either in the substances requiring a reduction of tempera- ture, or in the cold stores or rooms prepared for them; the air coming in contact with the cold tubes is immediately chilled, and the moisture in the air deposited on the pipes. It tben falls, making room for warmer air, and so circulates until the whole room is at the temperature of the brine in the pipes. In a recent arrangement for refrigerating made by the Linde British Re- frigeration Co., the cold brine is circulated through a shallow trough, in which revolve a number of shafts, each geared together, and driven by me- chanical means. On the shafts are fixed a number of wrought-iron disks, partly immersed in the brine, which cool them down to the brine tempera- ture as they revolve; over these disks a rapid circulation of air is passed by a fan, being cooled by contact with the plates; then it is led into the cham- bers requiring refrigeration, from which it is again drawn by the same fan; thus all moisture and impurities are removed from the chambers, and de- posited in the brine, producing the most perfect antiseptic atmosphere yet invented for cold storing; while ihe maximum efficiency of the brine tem- perature was always available, the brine being periodically concentrated by suitable arrangements. Air has also been used as the circulating medium. The ammonia-pipes refrigerate the air in a cooling-chamber, and large wooden conduits are used to convey it to and return it from the rooms to be cooled. An advantage of this system is that by it a room may be refrigerated more quickly than by brine-coils. The returning air deposits its moisture in the form of snow oh the ammonia-pipes, which is removed by mechanical brushes. ARTIFICIAL. ICE-MANUFACTURE. Under summer conditions, with condensing water at 70°, artificial ice-ma- chines use ammonia at about 190 lbs. above the atmosphere condenser- pressure, and 15 lbs. suction-pressure. In a compression type of machine the useful circulation of ammonia, allowing for the effect of cylinder- heating, is about 13 lbs. per hour per in- dicated horse-power of the steam cylinder. This weight of ammonia pro- duces about 32 lbs. of ice at 15° from water at 70°. If the ice is made from distilled water, as in the "can system," the amount of the latter supplied by the boilers is about 33$ greater than the weight of ice obtained. This excess represents steam escaping to the atmosphere, from the re-boiler and steam-condenser, to purify the distilled water, or free it from air; also, the loss through leaks and drips, and loss by melting of the ice in extracting it from the cans. The total steam consumed per horse-power is, therefore, about 32 X 1.33 = 43,0 lbs. About 7.0 lbs. of this covers the steam -consump- tion of the steam-engines driving the brine circulating-pumps, the several 1000 ICE-MAKING OK REFRIGERATING MACHINES. cold-water pumps, and leakage, drips, etc. Consequently, the main steam- engine must consume 36 lbs. of steam per hour per I.H.P,, or else live steam must be condensed to supply the required amount of distilled water. There is, therefore, nothing to be gained by using steam at high rates of expansion in the steam-engines, in making artificial ice from distilled water. If the cooling water for the ammonia-coils and steam-condenser is not too hard for use in the boilers, it may enter the latter at about 1?5° F., by restricting the quantity to \y% gallons per minute per ton of ice. With good coal %% lbs. of feed-water may then be evaporated, on the average, per lb. of coal. The ice made per pound of coal will then be 32 -h -—— — 6.0 lbs. This cor- responds with the results of average practice. If ice is manufactured by the "plate system," no distilled water is used for freezing. Hence the water evaporated by the boilers may be reduced to the amount which will drive the steam-motors, and the latter may use steam expansively to any extent consistent with the power required to compress the ammonia, operate the feed and filter pumps, and the hoisting machinery. The latter may require about 15$ of the power needed for compressing the ammonia. If a compound condensing steam-engine is used for driving the com- pressors, the steam per indicated steam horse-power, or per 32 lbs. of net ice, may be 14 lbs. per hour, ffhe other motors at 50 lbs. of steam per horse- power will use 7.5 lbs. per hour, making the total consumption per steam horse-power of the compressor 21.5 lbs. Taking the evaporation at 8 lbs., the feed-water temperature being limited to about 110°, the coal per horse- power is 2.7 lbs. per hour. The net ice per lb. of coal is then about 32 -s- 2.7 = 11.8 lbs. The best results with "plate-system 1 ' plants, using a compound steam-engine, have thus far afforded about 10^ lbs. of ice per lb. of coal. In the " plate system " the ice gradually forms, in from 9 to 14 days, to a thickness of about 14 inches, on hollow plates 10 x 14 feet in area, in which the cooling fluid circulates. In the "can system " the water is frozen in blocks weighing about 300 lbs. each, and the freezing is completed in from 50 to 60 hours. The freezing- tank area occupied by the "plate system" is, therefore, about four times, and the cubic contents about twelve times, as much as is required in the " can system." The investment for the "plate" is about one third greater than for the "can" system. In the latter system ice is being drawn throughout the 24 hours, and the hoisting is done by hand tackle. In the " plate system " the entire daily product is drawn, cut, and stored in a few hours, the hoisting being performed by power. The distribution of cost is as follows for the two systems, takingthe cost for the "can " or distilled-water system as 100, which represents an actual cost of about $1.25 per net ton: Can System. Plate System. Hoisting and storing ice 14.2 2.8 Engineers, firemen, and coal-passer 15.0 13.9 Coal at $3.50 per gross ton 42.2 20.0 Water pumped directly from a natural source at 5 cts. per 1000 cubic feet... 1.3 2.6 Interest and depreciation at 10$ 24.6 32.7 Repairs 2.7 3.4 100.00 75.4 A compound condensing engine is assumed to be used by the " plate sys- tem." Test of the New York Hygeia Ice-making Plant.— (By Messrs. Hupfel, Griswold, and Mackenzie; Stevens Indicator, Jan. 1894.) The final results of the tests were as follows: Net ice made per pound of coal, in pounds 7. 12 Pounds of net ice per hour per horse-power 37.8 Net ice manufactured per day (12 hours) in tons 97 Av. pressure of ammonia-gas at condenser, lbs. per sq. in. ab. atmos. 135.2 Average back pressure of amm.-gas, lbs. per sq. in. above atmos.. . . 15.8 Average temperature of brine in freezing-tanks, degrees F 19.7 Total number of cans filled per week 4389 Ratio of cooling-surface of coils in brine-tank to can-surface 7 to 10 MARINE ENGIKEERIKG. lOOi Ratio of brine in tanks to water in caus 1 to 1 .2 Ratio of circulating water at condensers to distilled water 26 to 1 Pounds of water evaporated at boilers per pound of coal 8.085 Total horse-power developed by compressor-engines 444 Percentage of ice lost in removing from cans 2.2 APPROXIMATE DIVISION OF STEAM IN PER CENTS OF TOTAL AMOUNT. Compressor-engines 60. 1 Live steam admitted directly to condensers 19.7 Steam for pumps, agitator, and elevator engines , 7.6 Live steam for reboiling distilled water 6.5 Steam for blowers furnishing draught at boilers 5.6 Sprinklers for removing ice from cans 0.5 The precautions taken to insure the purity of the ice are thus described: The water which finally leaves the condenser is the accumulation of the exhausts from the various pumps and engines, together with an amount of live steam injected into it directly from the boilers. This last quantity is used to make up any deficit in the amount of water necessary to supply the ice-cans. This water on leaving the condensers is violently reboiled, and afterwards cooled by running through a coil surface-cooler. It then passes through an oil-separator, after which it runs through three charcoal-filters and deodorizers, placed in series and containing 28 feet of charcoal. It next passes into the supply-tank in which there is an electrical attachment for detecting salt. Nitrate-of-silver tests are also made for salt daily. From this tank it is fed to the ice-cans, which are carefully covered so that the water cannot possibly receive any impurities. MAKINE ENGINEERING. Rules for Measuring Dimensions and Obtaining- Ton- nage of Vessels. (Record of American & Foreign Shipping. American Shipmasters' Assn., N. Y. 1890.)— The dimensions to be measured as follows: I. Length, L.— From the fore side of stem to the after side of stern-post measured at middle line on the upper deck of all vessels, except those hav- ing a continuous hurricane-deck extending right fore and aft, in which the /ength is to be measured on the range of deck immediately below the hurri- cane-deck. Vessels having clipper heads, raking forward, or receding stems, or rak- ing stern-posts, the length to be the distance of the fore side of stem from aft-side of stern-post at the deep-load water-line measured at middle line. (The inner or propeller-post to be taken as stern-post in screw-steamers. II. Breadth, B.— To be measured over the widest frame at its widest part; in other words, the moulded breadth. III. Depth, D.— To be measured at the dead-flat frame and at middle line of vessel. It shall be the distance from the top of floor-plate to the upper side of upper deck-beam in all vessels except those having a continuous hurricane-deck, extending right fore and aft, and not intended for the American coasting trade, in which the depth is to be the distance from top of floor-plate to midway between top of hurricane deck-beam and the top of deck-beam of the deck immediately below hurricane-deck. In vessels fitted with a continuous hurricane deck, extending right fore and aft. and intended for the American coasting trade, the depth is to be the distance from top of floor-plate to top of deck-beam of deck immedi- atelv below hurricane-deck. Rule for Obtaining Tonnage. — Multiply together the length, breadth, and depth, and their product by .75; divide the last product by 100; T y 7? v T) v 75 the quotient will be the tonnage. — — : — = tonnage. The U. S. Custom-liouse Tonnage Law, May 6, 1864, provides that "the register tonnage of a vessel shall be her entire internal cubic capacity in tons of 100 cubic feet each.'' 1 This measurement includes all the space between upper decks, however many there may be. Explicit direc- tions for making the measurements are given in the law. The Displacement of a "Vessel (measured in tons of 2240 lbs.) is the weight of the volume of water which it displaces. For sea-water it is equal to the volume of the vessel beneath the water-line, in cubic feet, divided by 35, which figure is the number of cubic feet of sea-water at 60° 100& MAH1KE EKGitfEERlim. F. in a ton of 2240 lbs. For fresh water the divisor is 35.93. The U. S. reg' ister tonnage will equal the displacement when the entire internal cubic capacity bears to the displacement the ratio of 100 to 35. The displacement or gross tonnage is sometimes approximately estimated as follows: Let L denote the length in feet of the boat, B its extreme breadth in feet, ami D the mean draught in feet; the product of these three dimensions will give the volume of a parallelopipedon in cubic feet. Put- ting V for this volume, we have V = L X B X D. The volume of displacement may then be expressed as a percentage of the volume V, known as the " block coefficient.' 1 '' This percentage varies for different classes of ships. In racing yachts with very deep keels it varies from 22 to 33; in modern merchantmen from 55 to 75; for ordinary small boats probably 50 will give a fair estimate. The volume of displacement in cubic feet divided by 35 gives the displacement in tons. Coefficient of Fineness.— A term used to express the relation be- tween the displacement of a ship and the volume of a rectangular prism or box whose lineal dimensions are the length, breadth, and draught of the ship. Coefficient of fineness = f R w ; D being the displacement in tous of 35 cubic feet of sea-water to the ton, Lthe length between perpendiculars, B the extreme breadth of beam, and W the mean draught of water, all in feet. Coefficient of Water-lines.— An expression of the relation of the displacement to the volume of the prism whose section equals the midship section of the ship, and length equal to the length of the ship. Coefficient of water-lines = ~-. — — -j T -: r . Seaton area of immersed water section X L gives the following values: Coefficient Coefficient of of Fineness. Water-lines. Finely-shaped ships 0.55 0.63 Fairly-shaped ships 0.61 0.67 Ordinary merchant steamers for speeds of 10 to 11 knots 0.65 0.72 Cargo steamers, 9 to 10 knots 0.70 0.76 Modern cargo steamers of large size 0.78 0.83 Resistance of Ships.— The resistance of a ship passing through water may vary from a number of causes, as speed, form of body, displace- ment, midship dimensions, character of wetted surface, fineness of lines, etc. The resistance of the water is twofold : 1st. That due to the displace- ment of the water at the bow and its replacement at the stern, with the consequent formation of waves. 2d. The friction between the wetted sur- face of the ship and the water, known as skin resistance. A common ap- proximate formula for resistance of vessels is Resistance = speed 2 x /(/displacement 2 x a constant, or R = S*D§ x C. If D = displacement in pounds, S = speed in feet per minute, R = resist- ance in foot-pounds per minute, R = CS^Di. The work done in overcom- ing the resistance through a distance equal to S is R X S = CS 3 D%\ and if Eis the efficiency of the propeller and machinery combined, the indicated horse-power I.H.P. = ~^~ - If S = speed in knots, D = displacement in tons, and (7 a constant which includes all the constants for form of vessel, efficiency of mechanism, etc., I.H.P.= *£. The wetted surface varies as the cube root of the square of the displace- ment; thus, let L be the length of edge of a cube just immersed, whose dis- placement is D and wetted surface W. Then D = L 3 or L = y'D, and W = 5xi 2 = 5X( |/2>) a . That is, IF varies as Z>§. MARINE ENGINEERING. 1003 Another approximate formula is I.H.P. = area of immersed mid ship section x « A" The usefulness of these two formulae depends upon the accuracy of the so-called "constants " Cand K, which vary with the size and form of the ship, and probably also with the speed. Seaton gives the following, which may be taken roughly as the values of C and K under the conditions ex- pressed : General Description of Ship, Ships over 400 feet long, finely shaped . . 300 Ships over 300 feet long, fairly shaped . . Ships over 250 feet long, finely shaped . . . Ships over 250 feet long, fairly shaped . . . Ships over 200 feet long, finely shaped. . . Ships over 200 feet long, fairly shaped . . Ships under 200 feet long, finely shaped . Ships under 200 feet long, fairly shaped. Speed, Value knots. of C. 15 to 17 240 15 " 17 190 13 " 15 240 11 " 13 260 11 " 13 240 9 " 11 260 13 " 15 200 11 " 13 240 9 " 11 260 11 " 13 220 9 " 11 250 11 " 12 220 9 " 11 240 9 " 11 220 11 " 12 200 10 " 11 210 9 " 10 230 9 " 10 200 Value of K. 620 500 650 700 650 700 580 660 700 620 680 600 640 620 550 580 620 600 Coefficient of Performance of Vessels. -The quotient ^/(displacement) 2 X (speed in knots) 3 tons of coal in 24 hours gives a quotient of performance which represents the comparative cost of propulsion in coal expended. Sixteen vessels with three-stage expansion- engines in 1890 gave an average coefficient of 14,810, the range being from 12,150 to 16,700. In 1881 seventeen vessels with two-stage expansion-engines gave an aver- age coefficient of 11.710. In 1881 the length of the vessels tested ranged from 260 to 320, and in 1890 from 295 to 400. The speed in knots divided by the square root of the length in feet in 1881 averaged 539; and in 1890, 0.579; ranging from 0.520 to 0.641. (Proc. Inst. M E.. July, 1891, p. 329.) Defects of the Common Formula for Resistance.— Modern experiments throw doubt upon the truth of i he statement that the resistance varies as the square of the speed. (See Robt. Mansel's letters in Engineer- ing, 1891 ; also his paper on The Mechanical Theory of Steamship Propulsion, read before Section G of the Engineering Congress, Chicago, 1893.) Seaton says: In small steamers the chief resistance is the skin resistance. In very fine steamers at high speeds the amount of power required seems excessive when compared with that of ordinary steamers at ordinary speeds. In torpedo-launches at certain high speeds the resistance increases at a lower rate than the square of the speed. In ordinary sea-going and river steamers the reverse seems to be the case. Rankine's Formula for total resistance of vessels of the "wave- line" type is: R = ALBVH1 -f 4 sin 2 + sin* 0), in which equation is the mean angle of greatest obliquity of the stream- lines, A is a constant multiplier, B the mean wetted girth of the surface ex- posed to friction, L the length in feet, and V the speed in knots. The potver demanded to impel a ship is thus the product of a constant to be determined by experiment, the area of the wetted surface, the cube of the speed, and the 1004 MARINE ENGINEERING. quantity in the parenthesis, which is known as the "coefficient of augmen- tation. 1 ' The last term of the coefficient may be neglected in calculating the resistance of ships as too small to be practically important. In applying the formula, the mean of the squares of the sines of the angles of maximum obliquity of the water-lines is to be taken for sin 2 0, and the rule will then read thus: To obtain the resistance of a ship of good form, in pounds, multiply the length in feet by the mean immersed girth and by the coefficient of augmen- tation, and then take the product of this "augmented surface,' 1 as Rankine termed it, by the square of the speed in knots, and by the proper constant coefficient selected from the following: For clean painted vessels, iron hulls A — .01 For clean coppered vessels A — .009 to .008 For moderately rough iron vessels A — .011 -f- The net, or effective, horse-power demanded will be quite closely obtained by multiplying the resistance calculated, as above, by the speed in knots and dividing by 326. The gross, or indicated, power is obtained by multiplying the last quantity by the reciprocal of the efficiency of the machinery and propeller, which usually should be about 0.6. Rankine uses as a divisor in this case 200 to 260. The form of the vessel, even when designed by skilful and experienced naval architects, will often vary to such an extent as to cause the above con- stant coefficients to vary somewhat; and the range of variation with good forms is found to be from 0.8 to 1.5 the figures given. For well-shaped iron vessels, an approximate formula for the horse-power SV 3 required is H.P. = , in which S is the "augmented surface." The ex- SV 3 pression ^5" nas been called by Rankine the coefficient of propulsion. In the Hudson River steamer " Mary Powell," according to Thurston, this coefficient was as high as 23,500. DaV 3 The expression TT _. has been called the locomotive performance. (See M.ir. Rankine's Treatise on Shipbuilding, 1864; Thurston's Manual of the Steam- engine, part ii. p. 16; also paper by F. T. Bowles, U.S.N., Proc. U. S. Naval Institute, 1883.) Rankine's method for calculating the resistance is said by Seaton to give more accurate and reliable results than those obtained by the older rules, but it is criticised as being- difficult and inconvenient of application. Dr. Kirk's Method.— This method is generally used on the Clyde. The general idea proposed by Dr. Kirk is to reduce "all ships to so definite and simple a form that they may be easily compared; and the magnitude of certain features of this form shall determine the suitability of the ship for speed, etc. The form consists of a middle body, which is a rectangular parallelopiped, and fore body and after body, prisms having isosceles triangles for bases, as shown in Fig. 168. D E This is called a block model, and is such that its length is equal to that of the ship, the depth is equal to the mean draught, the capacity equal to the displacement volume, and its area of section equal to the area of im- MARINE ENGINEERING. 1005 mersed midship section. The dimensions of the block model may be obtained as follows: Let AG - HB = length of fore- or after-body = F\ GH = length of middle body = M ; KL — mean draught = H ; area of immersed midship section EK = zz - = B. Volume of block = (F + M) x B x H; Midship section = B X H; Displacement in tons — volume in cubic ft. -v- 35. AH = AG + GH = F+M = displacement x 35 -*- (B X H). The wetted surface of the block is nearly equal to that of the ship of the same length, beam and draught ; usually 2% to 5% greater. In exceedingly fine hollow-line ships it may be 8$ greater. Area of bottom of block = (F-\- M) X B; Area of sides = 2M x H. Area of sides of ends = A a/ F 2 + (^) 2 X H"; Tangent of half angle of entrance = —=-■ = — . From this, by a table of natural tangents, the angle of entrance may be obtained: Angle of Entrance Fore-body in of the Block Model, parts of length. Ocean-going steamers, 14 knots and upward. 18° to 15° .3 to .36 12 to 14 knots 21 to 18 .26 to .3 cargo steamers, 10 to 12 knots.. 30 to 22 .22 to .26 E. R. Mumford's Method of Calculating Wetted Surfaces is given in a paper by Archibald Denny, Eng'g, Sept. 21, 1894. The following is his formula, which gives closely accurate results for medium draughts, beams, and finenesses: S=(LXDX 1.7) + {LXBXC), ill which & = wetted surface in square feet; L = length between perpendiculars in feet; D = middle draught in feet: B — beam in feet; C = block coefficient. The formula may also be expressed in the form S = L(1.7D -\- BC). In the case of twin-screw ships having projecting shaft-casings, or in the case of a ship having a deep keel or bilge keels, an addition must be made for such projections. The formula gives results which are in general much more accurate than those obtained by Kirk's method. It underestimates the surface when the beam, draught, or block coefficients are excessive; but the error is small except in the case of abnormal forms, such as stern-wheel steamers having very excessive beams (nearly one fourth the length), and also very full block coefficients. The formula gives a surface about 6$ too small for such forms. To Find tUe Indicated Horse-power from the Wetted Surface. (Seaton.) — In ordinary cases the horse-power per 100 feet of wetted surface may be found by assuming that the rate for a speed of 10 knots is 5, and thar the quantity varies as the cube of the speed. For exam- ple: To find the number of I.H.P. necessary to drive a ship at a speed of 15 knots, having a wetted skin of block model of 16,200 square feet: The rate per 100 feet = (15/10)3 x 5 = 16.875. Then I.H.P. required = 16.875 X 162 = 2734. 1006 MARINE ENGINEERING. When the ship is exceptionally well-proportioned, the bottom quite clean, and the efficiency of the machinery high, as low a rate as 4 I.H.P. per 100 feet of wetted skin of block model may be allowed The gross indicated horse-power includes the power necessary to over- come the friction and other resistance of (he engine itself and the shafting, and also the power lost in the propellor. In other words, I.H.P. is no meas- ure of the resistance of the ship, and can only be relied on as a means of deciding the size of engines for speed, so long as the efficiency of the engine and propellor is known definitely, or so long as similar engines and propellers are employed in ships to be compared. The former is difficult to obtain, and it is nearly impossible in practice to know how much of the power shown in the cylinders is employed usefully in overcoming the resistance of the ship. The following example is given to show the variation in the efficiency of propellers: Knots. I.H.P. H.M.S. " Amazon," with a 4-bladed screw, gave. 12.064 with 1940 H.M.S. " Amazon,' 1 with a 2-bladed screw, increased pitch, and less revolutions per minute 12.396 " 1663 H.M.S. "Iris,' 1 with a 4-bladed screw 16.577 " 7503 H.M.S. " Iris," with 2-bladed screw, increased pitch, less revolutions per knot 18.587 " 7556 Relative Horse-power Required for Different Speeds oi Vessels. (Horse-power for 10 knots = 1.) — The horse-power is taken usually to vary as the cube of the speed, but in different vessels and at different speeds it may vary from the 2.8 power to the 3.5 power, depending upon the lines of the vessel and upon the efficiency of the engines, the propeller, etc. 1*1 ft5 4 6 .239 8 .535 10 1. 12 1.666 14 2.565 16 3.729 18 5.185 20 6.964 22 9.095 24 11.60 26 14.52 28 HPCC .0769 17.87 #2-9 .0701 .227 .524 1. 1.697 2.653 3.908 5.499 7.464 9.841 12.67 15.97 19.80 .S3 .0640 .216 .512 1. 1 . 72S 2.744 4.096 5.83.2 8. 10.65 13.82 17.58 21.95 , S 3-1 .0584 .205 .501 1. 1.760 2.s>;s 4.293 6.185 8.574 11.52 15.09 19.34 S3 "2 .0533 .195 .490 1. 1.792 2.935 4 500 6.559 9.189 12.47 16.47 21.28 26 97 S3 -3 .0486 . 1 S5 .479 1. 1 S.25 3 036 4 716 6 957 9 849 13 49 17 98 23 41 29.90 flS-4 .0444 .176 1. 1.859 3 139 1 943 7.378 10.56 14.60 19.62 25.76 33.14 S 35 .0105 .167 .458 1. 1.893 3.247 5.181 7 824 11.31 15.79 21.42 28.34 36.73 21.67 24.19 27. 30.14 33.63 37.54 41.90 Example in Use op the Table.— -A certain vessel makes 14 knots speed with 587 I.H.P. and 16 knots with 900 I.H.P. What I.H.P. will be required at 18 knots, the rate of increase of horsepower with increase of speed remain- ing constant ? The first step is to find the rate of increase, thus: 14* : 16 x :: 587 : 900. x log 16 - x log 14 = log 900 - log 587; a;(0.204120 - 0.146128) = 2.954243 - 2.768638, whence x (the exponent of S in formula H.P. cc S x ) = 3 2. From the table, for S 3 2 and 16 knots, the I H.P. is 4.5 times the I.H.P. at 10 knots, .-. H.P. at 10 knots = 900 h- 4.5 = 200. From the table, for S 3 " 2 and 18 knots, the I.H.P. is 6.559 times the I.H.P. at 10 knots: .-. H.P. at 18 knots = 2.10 X 6.559 = 1312 HP. Resistance per Horse-power for Different Speeds. (One horse-power = 33.000 lbs. resistance overcome through 1 tt. in 1 rain.)— The resistances per horse-power for various speeds are as follows: For a speed of 1 knot, or 6080 feet per hour = 101^ ft. per min., 33,000 -=- 101Jg = 325.658 lbs. per horse-power; and for any other speed 325.658 lbs. divided by the speed in knots; or for 1 knot 325.66 lbs. 6 knots 54.28 lbs. 2 knots 162.83 " 7 " 46.52 " 3 " 108.55 " 8 " 40.71 " 4 " 81.41 " 9 " 36.18 " 5 " 65.13 " 10 " 32.57 " 11 knots 29.61 lbs. 12 " 27.14 " 13 " 25.05 " 14 " 23.26 " 15 " 21.71 " 16 knots 20.35 lbs. 17 " 19.16 " 18 " 18.09 " 19 " 17.14 " 20 li 16.28 " MARINE ENGINEERING. 1007 Results of Trials of Steam-vessels of Various Sizes. (From Seaton's Marine Engineering.) Length, perpendiculars Breadth, ext reme Mean draught water Displacement (tons) Area Immersed mid. section. Wetted skin W3 Length, fore-body. Angle of entrance. Displacement X 35 Length x Imm. mid area"'" Speed (knots) Indicated horse-power I.H.P. per 100 ft. wetted skin .... I.H.P. per 100 ft. wetted skin, re duced to 10 knots I.H.P. ' Immersed mid area X S 3 6 a a wo* 1 A *& H o ^ 08 90' 0" 10' 6" 2' 6" 29 73 24? 903 171' 9" 18' 9" 6' 9^" 280 99 3793 130' 0" 31' 0" 8' 10" 370 148 3754 286' 0" 34' 3" 6' 0" 800 200 8222 230' 0" 29' 0" 13' 6" 1500 340 10,075 45' 0" 72' 00" 42' 6" 143' 0" 79' 6" 12° 40' 11° 30' 23° 50' 13° 21' 17° 0" 0.481 0.576 0.608 0.489 0.671 22 01 460 50.9 15.3 798 21.04 10.74 371 9.88 17.20 1490 18.12 10.01 503 5.00 4.78 5.87 7.97 3.56 4.90 223 192 172.8 293.7 266 '556? 445 495 683 6«J0 w o 327' 0" 35' 0" 13' 0" 1900 336 15,782 129' 0" 11° 26' 0.605 17.8 4751 30.00 «?> °^V co^ s co'5 kT tog o 270' 0" 300' 0" 300' 0" 370' 0" 392. 0" 4.2' 0" 46' 0" 46' 0" 41' 0" 39 0" 18' 10" 18' 2" 18' 2" 18' 11" 21' 4" 3057 3290 3^90 4635 5767 632 700 700 656 738 16,008 18,168 18,163 22,633 26,235 101' 0' 135' 6" 135' 6" 123' 0" 118' 0" 18° 44' 16° 16' 16° 16' 16° 4' 16° 30' 0.629 0.548 0.548 0.668 0.698 14.966 18.573 15.746 13.80 12.054 4015 7714 3958 2500 1758 25.08 42.46 21.78 11.04 6.7 7.49 6.634 5.58 4.20 3.83 175.8 183.7 218.2 292 320 527.5 581.4 690.5 689 735 tf'm Length, perpendiculars.. Breadth, extreme Mean draught water Displacement (tons) Area Imm. mid. section ™ ■ f Wetted skin "u a 5 ■§ i Length, fore-body pq ^ [ Angle of entrance Displacement X 35 Lsngth x Imm. mid area Speed (knots) Indicated horse-power I.H.P. per 100 ft, wetted skin . . I.H.P. per 100 ft. wetted skin, r duced to 10 knots Dl X .S* I.H.P. ■■"' Immersed mid area X S 3 I.H.P. 450' 0" 45' 2" 23' 7" 8500 926 32,578 129' 0" 17° 16' 0.714 15.045 4900 15.04 289.3 642.5 ioos MARINE EJSTGIKEERIHG. Results of Progressive Speed Trials in Typical Vessels. (Eng'g, April 15, 1892, p. 463.) eg O 6 CO A - ■ * In „ U £0 '5d: o E- H-3 ■So ~ m =« Length (in feet) 135 230 265 300 360 375 525 Breadth " 4V Draught (mean 14 5' 1" 27 8' 3" 41 16' 6" 43 16'2" 60 23' 9" 65 25' 9" 63 ) on trial 21' 3" 103 735 2800 3330 7390 9100 11550 I.H.P.— 10 knot 14 " 18 " 20 " 110 260 870 1130 450 1100 2500 3500 700 2100 6400 10000 800 2400 6000 9000 1000 3000 7500 11000 1500 4000 9000 12500 2000 4600 10000 14500 Speed Ratio of speed 3 10 1. Ratio of H.P.= 1 1 1 1 1 1 1 14 2.744 " " = 2 36 2.44 3 3 3 2.67 2.3 18 5.832 44 44 _ 7.91 5.56 9.14 7.5 7.5 6. 5 20 8. 44 44 _ 10.27 7.78 14.14 11.25 11 8.42 7.25 Admiralty coeff . f 10 knots. 200 181 284 279 380 290 255 C-f* S3 i! 4 8 << 232 203 259 255 347 298 304 147 190 181 217 295 282 297 I.H.P. ^ u 156 186 159 198 276 278 281 The figures for I.H.P. are " round." The " Medusa's " figures for 20 knots are from trial on Stokes Bay. and show the retarding effect of shallow water. The figures for the other ships for 20 knots are estimated for deep water. More accurate methods than those above given for estimating the horse-power required for any proposed ship are: 1. Estimations calculated from the results of trials of " similar" vessels driven at " corresponding " speeds; " similar " vessels being those that have the same ratio of length to breadth and to draught, and the same coefficient of fineness, and " corre- sponding" speeds those which are proportional to the square roots of the lengths of the respective vessels. Froude found that the resistances of such vessels varied almost exactly as wetted surface x (speed) 2 . 2. The method employed by the British Admiralty and by some Clyde shipbuilders, viz., ascertaining the resistance of a model of the vessel, 12 to 20 ft. long, in a tank, and calculating the power from the results obtained. Speed. On Canals.— A great loss of speed occurs when a steam-vessel passes from open water into a more or less restricted channel. The average speed of vessels in the Suez Canal in 1882 was only b% statute miles per hour. {Enq'g. Feb. 15, 1884, p. 139.) Estimated Displacement, Horse-power, etc. -The table on the next page, calculated by the author, will be found convenient for mak- ing approximate estimates. 2 The figures in 7th column are calculated by the formula H.P. = S 3 Ds -*- c, in which c — 200 for vessels under 200 ft. long when C = .65, and 210 when C = .55; c = 200 for vessels 200 to 400 ft. long when C = .75, 220 when C = .65, 240 when C — .55; c = 230 for vessels over 400 ft. long when C = .75, 250 when C = .65, 260 when C = .55. The figures in the 8th column are based on 5 H.P. per 100 sq. ft. of wetted surface. The diame ters of screw in the 9th column are from for mula D — 3.31 |/I.H.P., and in the 10th column from formula D = 2.71 |/l7H.P. To find the diameter of screw for any other speed than 10 knots, revolu- tions being 100 per minute, multiply the diameter given in the table by the 5th root of the cube of the given speed h- 10. For any other revolutions per minute than 100, divide by the revolutions and multiply by 100. To find the approximate horse-power for any other speed than 10 knots, multiply the horse-power given in the table by the cube of the ratio of the given speed to 10, or by the relative figure from table on p. 1006. MARINE ENGINEERING. 1009 Estimated Displacement, Horse-power, etc., of Steam- vessels of Various Sizes. ■g Displace- Estimat (1 Horse- Diam. of Screw for 10 lis g«] 5 "S •3 =° LBDx C Wetted Surface Ul.lD + BC) Sq. ft. power at 10 knots. knots sp revs, pe eed and 100 Calc. from Dis- Calc. irom Wetted ►j 51 " 35 If Fitch = If Pitch = 3 1.5 .55 tons. placem't. Surface. Diam. 1.4 Diam. 12 .85 48 4.3 2.4 4.4 3.6 »■{ 3 1.5 .55 1.13 64 5.2 3.2 4.6 3.8 4 2 .65 2.38 96 8.9 4.8 5.1 4.2 20 j 3 1.5 .55 1.41 80 6.0 4.0 4.7 3.9 4 2 .65 2.97 120 10.3 6.0 5.3 4.3 24 -j 3.5 1.5 .55 1.98 104 7.5 5.2 5 4.1 4.5 2 .65 4.01 152 12.6 7.6 5.5 4.5 30 j 4 2 .55 3.77 168 11.5 8.4 5.4 4.4 5 2.5 .65 6.96 224 18.2 11.2 5.9 4.8 40 ] 4.5 2 .55 5.66 235 15.1 11.8 5.7 4.7 6 2.5 .65 11.1 326 24.9 16.3 6.3 5.2 50 -j 6 3 .55 14.1 420 27.8 21.0 6.4 5.4 8 3.5 .65 26 558 43.9 27.9 7.1 5.8 60 | 8 3.5 .55 26.4 621 42.2 81.1 7.0 5.7 10 4 .65 44.6 798 62.9 39 9 7.6 6.2 «.{ 10 4 .55 44 861 59.4 43.1 7.5 6.1 12 4.5 .65 70.2 1082 85.1 54.1 8.1 6.6 80 -j 13 4.5 .55 67.9 1140 79.2 57.0 7.9 6.5 14 5 .65 104.0 1408 111 70 4 8.5 7.0 90 1 13 5 .55 91.9 1408 97 70.4 8.3 6.8 16 6 .65 160 1854 147 92.7 9 7.3 I 13 5 .55 102 1565 104 78.3 8.4 6.9 100^ 15 5.5 .65 153 1910 143 95.5 8.9 7.3 1 17 6 .75 219 2295 202 115 9.6 7.8 ( 14 5.5 .55 145 2046 131 102 8.8 7.2 120^ 16 6 .65 214 2472 179 124 9.4 r i'.6 1 18 6.5 .75 301 2946 250 147 10 8.2 I 16 6 .55 211 2660 169 133 9.2 7.4 140 -{ IS 6.5 .65 306 3185 227 159 9.8 8.0 ( •JO 7 .75 420 3766 312 188 10.5 8.5 I 17 6.5 .55 278 3264 203 163 9.6 7.8 160^ 19 7 .65 395 3880 269 194 10.1 8.3 1 2i 7.5 .75 540 4560 368 228 10.8 8.8 ( •20 7 .55 396 4122 257 206 10.1 8.2 180^ 2-2 7.5 .65 552 4869 337 243 10.6 8.7 1 24 8 .75 741 5688 455 284 11.3 9.2 22 7 .55 484 4800 257 240 10.1 8.2 200^ •25 8 .65 743 5970 373 299 10.8 8.8 | •28 9 .75 1080 7260 526 363 11.6 9.5 I 28 8 .55 880 7250 383 363 10.9 8.9 250^ 32 10 .65 1486 9450 592 473 11.9 9.7 ( 36 12 .75 2314 11850 875 593 12.8 10.5 ( 32 10 .55 1509 10380 548 519 11.7 9.6 3(XK 36 12 .65 2407 13140 806 657 12.6 10.4 j 40 14 75 3600 17140 1175 857 13.6 11.1 ( 38 12 '.55 2508 14455 769 723 1-2.5 10.2 350 i 42 14 .65 3822 17885 1111 894 13.5 11.0 I 46 16 .75 5520 21595 1562 1080 14.4 11.8 44 14 .55 3872 19200 1028 960 13.3 10.8 400^ 48 16 .65 • 5705 23360 1451 1168 14.2 11.6 1 52 18 .75 8023 27840 2006 1392 15.2 12.4 I 50 16 .55 5657 24515 1221 1226 13.7 11.2 450^ 54 18 .65 8123 29565 1616 1478 14.5 11.9 ( 5S 20 .75 11157 31875 2171 1744 15.4 12.6 52 18 .55 7354 29600 1454 1480 14.2 11.6 500 -j 56 •20 .65 10400 35200 1966 1760 15.1 12.4 60 •22 .75 14143 41200 2543 2060 15.9 13.0 ( 56 20 .55 9680 36245 1747 1812 14.7 12 550-^ 60 •jo .65 13483 42735 2266 2137 15 5 12.7 i 64 34 .75 18103 49665 2998 24S3 . 16.4 13.4 ( 60 22 .55 12446 42900 2065 2145 15.2 12.5 600^ 64 •24 .65 17115 50220 2656 2511 15.4 13.1 1 68 •26 .75 22731 58020 3489 2901 16.9 13.8 1010 MARINE ENGINEERING. THE SCREW-PROPELIEB. The "pitch" of a propeller is the distance which any point in a blade, describing a helix, will travel in the direction of the axis during one revolu- tion, the point being assumed to move around the axis. The pitch of a propeller with a uniform pitch is equal to the distance a propeller will advance during one revolution, provided there is no slip. In a case of this kind, the term "pitch' 1 is analogous to the term "pitch of the thread" of an ordinary single-threaded screw. Let P = pitch of screw in feet, R = number of revolutions per second, V = velocity of stream from the propeller = P x R, v = velocity of the ship in feet per second, V — v = slip, A = area in square feet of section of stream from the screw, approximately the area of a circle of the same diameter, A X V = volume of water projected astern from the ship in cubic feet per second. Taking the weight of a cubic foot of sea-water at 64 lbs., and the force of gravity at 32, we have from the common formula for force of accel- V\ W v-i JV eration, viz.: F = M-f = — -+, or F = — v x , when t = 1 second, v x being t g t 9 the acceleration. 64 AV Thrust of screw in pounds = — — — (V - v) = 2AV(V — v). Rankine (Rules, Tables, and Data, p. 275) gives the following: To calculate the thrust of a propelling instrument (jet, paddle, or screw) in pounds, multiply together the transverse sectional area, in square feet, of the stream driven astern by the propeller; the speed of the stream relatively to the snip in knots; the real slip, or part of that speed which is impressed on that stream by the propeller, also in knots; and the constant 5.66 for sea- water, or 5.5 for fresh water. If S = speed of the screw in knots, s = speed of ship in knots, A — area of the stream in square feet (of sea-water), Thrust in pounds = A X S(S - s) X 5.66. The real slip is the velocity (relative to water at rest) of the water pro- jected sternward; the apparent slip is the difference between the speed of the ship and the speed of the screw; i.e., the product of the pitch of the screw by the number of revolutions. This apparent slip is sometimes negative, due to the working of the screw in disturbed water which has a forward velocity, following the ship. Nega- tive apparent slip is an indication that the propeller is not suited to the ship. The apparent slip should generally be about 8% to 10$ at full speed in well- formed vessels with moderately fine lines; in bluff cargo boats it rarely exceeds 5$. The effective area of a screw is the sectional area of the stream of water laid hold of by the propeller, and is generally, if not always, greater than the actual area, in a ratio which in good ordinary examples is 1.2 or there- abouts, and is sometimes as high as 1.4; a fact probably due to the stiffness of the water, which communicates motion laterally amongst its particles. (Rankine's Shipbuilding, p. 89.) Prof. D. S. Jacobus, Trans. A. S. M. E., xi. 1028, found the ratio of the ef- fective to the actual disk area of the screws of different vessels to be as follows : Tug-boat, with ordinary true-pitch screw 1 .42 " " screw having blades projecting backward 57 Ferryboat" Bergen," with or-) at speed of 12.09 stat. miles per hour. 1.53 diuary true-pitch screw ) " " 13.4 " . " " " 1.48 Steamer " Homer Ramsdell," with ordinary true-pitch screw 1.20 Size of Screw.— Seaton says: The size of a screw depends on so many things that it is very difficult to lay down any rule for guidance, and much must always be left to the experience of the designer, to allow for all the circumstances of each particular case. The following rules are given for ordinary cases. (Seaton and Rounthwaite's Pocket-book): 101339 P — pitch of propeller in feet = _ — -, in which S = speed in knots, R — revolutions per minute, and x = percentage of apparent slip 112.6S For a slip of 10$, pitch = - R THE SCREW PROPELLER. 1011 D — diameter of propeller = K / , ' " . 3 , K beii being a coefficient given in the table below. If K / I.H.P. = 20, D - 20000 JU -p- Total developed area of blades = CJU ' ' ', in which C is a coefficient to be taken from the table. Another formula for pitch, given in Seaton's Marine Engineering, is P = — a/ ' D2 ' , in which C = 737 for ordinary vessels, and 660 for slow- speed cargo vessels with full lines. Thickness of blade at root •V- X k, in which d = diameter of tail- shaft in inches, n = number of blades, b = breadth of blade in inches where it joins the boss, measured parallel to the shaft axis; k = 4 for cast iron, 1.5 for cast steel, 2 for gun-metal. 1.5 for high-class bronze. Thickness of blade at tip: Cast. iron MD 4- A in.; cast steel .03D -f- .4 in.: gun-metal .03 D + .2 in. ; high-class bronze .02Z) +. 3 in., where D = diameter of propeller in feet. Propeller Coefficients. 83 hi M ^5 i P ajs P. .5 < d o goj Description of Vessel. 3 02 = CQ u CO > 0) 3 > "3.S.3 P Bluff cargo boats 8-10 One 4 17 -17 5 19 -17.5 Cast iron Cargo, moderate lines. . . 10-13 " 4 18 -19 17 -15.5 " " Pass, and mail, fine lines. 13-17 " 4 19.5-20.5 15 -13 C. I. or S. " " " " " 13-17 Twin 4 20.5-21-5 14.5-1-2.5 " " " " " " very fine. 17-22 One 4 21 -22 12.5-11 G. M. or B " " " " " 17-22 Twin 3 22 -23 10.5- 9 n " " Naval vessels, " " 16-22 4 21 -22.5 11.5-10.5 " " " " " " " 16-22 " 3 22 -23.5 8.5- 7 " " " Torpedo-boats, " ' ; 20-26 One 3 25 7- 6 B. or F S. C.I., cast iron; G. M., gun-metal; B., bronze From the formulee D =i 20000 / I.K.P. V (P X P) 3 ^400 X I.H.P. , if P = D S., steel; F. S., fo rged steel. , • 737 Vl.H.P. and R = 100, we obtain D = -j/400 X I. H.P. = 3 31^1 -H.P. If P = 1.4D and E = 100, then D = ^145.8 X I.H.P. = 2.71 fl.H.P. From these two formulae the figures for diameter of screw in the table on page 1009 have been calculated. They may be used as rough approximations to the correct diameter of screw for any' given horse-power, for a speed of 10 knots and 100 revolutions per minute. For any other number of revolutions per minute multiply the figures in the table by 100 and divide by the given number of revolutions.. For any other speed than 10 knots, since the I.H.P. varies approximately as the cube of the speed, and the diameter of the screw as the 5th root of the I.H.P., multiply the diameter given for 10 knots by the 5th root of the cube of one tenth of the given speed. Or, multiply by the following factors: For speed of knots: 4 5 6 7 8 9 11 12 13 14 15 16 \/(S ~r- 10)3 = .577 .660 .807 .875 .939 1.059 1.116 .170 l.: . 1.275 1.327 1012 MARINE ENGINEERING. 18 19 20 21 22 23 2 605 1.648 1.1 26 27 28 ! 1.774 1.815 1.855 ms +- 10)3 = 1.375 1.423 1.470 1.515 1.561 1.605 1.648 1.691 1.' For more accurate determinations of diameter and pitch of screw, the formulae and coefficients given by Seaton, quoted above, should be used. Efficiency of tlie Propeller.— According to Rankine, if the slip of the water be s, its weight W, the resistance R, and the speed of the ship v, R ■■ Ws Rv -■ Wsv This impelling action must, to secure maximum efficiency of propeller, be effected by an instrument which takes hold of the fluid without shock or disturbance of the surrounding mass, and, by a steady acceleration, gives it the required final velocity of discharge. The velocity of the propeller over- coming the resistance R would then be v + (v + s) _ v s, and the work performed would be _,/ , s\ Wvs . Ws* the first of the last two terms being useful, the second the minimum lost work; the latter being the wasted energy of the water thrown backward. The efficiency is *=* + ■(«+!); and this is the limit attainable with a perfect propelling instrument, which iimit is approached the more nearly as the conditions above prescribed are the more nearly fulfilled. The efficiency of the propelling instrument is probably rarely much above 0.60, and never above 0.80. In designing the screw-propeller, as was shown by Dr. Froude, the best angle for the surface is that of 45° with the plane of the disk; but as all parts of the blade cannot be given the same angle, it should, where practi- cable, be so proportioned that the " pitch-angle at the centre of effort" should be made 45°. The maximum possible efficiency is then, according to Froude, 77$. In order that the water should be taken on without shock and discharged with maximum backward velocity, the screw must have an axially increas- ing pitch. The true screw is by far the more usual form of propeller, in all steamers, both merchant and naval. (Thurston, Manual of the Steam-engine, part ii., p. 176.) The combined efficiency of screw, shaft, engine, etc., is generelly taken at 50$. In some cases it may reach 60% or 65$. Rankine takes the effective H.P. to equal the I.H.P. -s- 1.63. Pitch-ratio and Slip for Screws of Standard Form, Pitch-ratio. Real Slip of Screw. Pitch- ratio. Real Slip of Screw. .8- 15.55 1.7 21.3 .9 16.22 1.8 21.8 1.0 16.88 1.9 22.4 1.1 17.55 2.0 22.9 1.2 18.2 2.1 23.5 1.3 18.8 2.2 24.0 1.4 19.5 2.3 24.5 1.5 20.1 2.4 25.0 1.6 20.7 2.5 25.4 THE PADDLE-WHEEL. 1013 Results of Recent Researches on the efficiency of screw-propel- lers are summarized by S. W. Baruaby, in a paper read before section G of the Engineering Congress. Chicago, 1893. He states that the following gen- eral principles have been established: (a) There is a definite amount of real slip at which, and at which only, maximum efficiency can be obtained with a screw of any given type, and this amount varies with the pitch-ratio. The slip-ratio proper to a given ratio of pitch to diameter has been discovered and tabulated for a screw of a standard type, as below (see table on page 1012): (b) Screws of large pitch-ratio, besides being less efficient in themselves, add to the resistance of the hull by an amount bearing some proportion to their distance from it, and to the amount of rotation left in the race. (c) The best pitch-ratio lies probably between 1.1 and 1.5. (d) The fuller the lines of the vessel, the less the pitch-ratio should be. (e) Coarse-pitched screws should be placed further from the stern than fine-pitched ones. (/) Apparent negative slip is a natural result of abnormal proportions of propellers. (g) Three blades are to be preferred for high-speed vessels, but when the diameter is unduly restricted, four or even more may be advantageously employed. (h) An efficient form of blade is an ellipse having a minor axis equal to four tenths the major axis. (i) The pitch of wide-bladed screws should increase from forward to aft, but a uniform pitch gives satisfactory results when the blades are narrow, and the amount of the pitch variation should be a function of the "width of the blade. (j) A considerable inclination of screw shaft produces vibration, and with right-handed twin-screws turning outwards, if the shafts are inclined at all, it should be upwards and outwards from the propellers. For results of experiments with screw-propellers, see F. C. Marshall, Proc. Inst. M. E. 1881; R. E. Froude, Trans. Institution of Naval Architects, 1886; G. A. Calvert, Trans. Institution of Naval Architects 1887; and S. W. Bar- uaby, Proc. Inst. Civil Eng'rs 1890, vol. cii. One of the most important results deduced from experiments on model screws is that they appear to have practically equal efficiencies throughout a wide range both in pitch-ratio and in surface-ratio; so that great latitude is left to the designer in regard to the form of the propeller. Another im- portant feature is that, although these experiments are not a direct guide to the selection of the most efficient propeller for a particular ship, they sup- ply the means of analyzing the performances of screws fitted to vessels, and of thus indirectly determining what are likely to be the best dimensions of screw for a vessel of a class whose results are known. Thus a great ad- vance has been made on the old method of trial upon the ship itself, which was the origin of almost every conceivable erroneous view respecting the screw-propeller. (Proc. Inst. M. E., July, 1891.) THE PADDLE-WHEEL. Paddle-wheels with Radial Floats. (Seaton's Marine En- gineering.) — The effective diameter of a radial wheel is usually taken from the centres of opposite floats; but it is difficult to say what is absolutely that diameter, as much depends on the form of float, the amount of dip, and the waves set in motion by the wheel. The slip of a radial wheel is from 15 to 30 per cent, depending on the size of float. Area of one float = ..'.- ' / X C. D is the effective diameter in feet, and C is a multiplier, varying from 0.25 in tugs to 0.175 in fast-running light steamers. The breadth of the float is usually about *4 its length, and its thickness about % its breadth. The number of floats varies directly with the diam- eter, and there should be one float for every foot of diameter. (For a discussion of the action of the radial wheel, see Thurston, Manual of the Steam-engine, part ii., p, 182.) Feathering Paddle-wheels. (Seaton.) — The diameter of a feathering-wheel is found as follows : The amount of slip varies from 12 to 20 per cent, although when the floats are small or the resistance great it 1014 MARINE ENGINEERING. is as high as 25 per cent; a well-designed wheel on a well-formed ship should not exceed 15 per cent under ordinary circumstances. If K is the speed of the ship in knots, S the percentage of slip, and B the revolutions per minute, Diameter of wheel at centres = n ^ n . 3.14 X B The diameter, however, must be such as will suit the structure of the ship, so that a modification may be necessary on this account, and the revolutions altered to suit it. The diameter will also depend on the amount of " dip " or immersion of float. When a ship is working always in smootli water the immersion of the top edge should not exceed V% the breadth of the float; and for general service at sea an immension of ^ the breadth of the float is sufficient. If the ship is intended to carry cargo, the immersion when light need not be more than 2 or 3 inches, and should not be more than the breadth of float when at the deepest draught; indeed, the efficiency of the wheel falls off rapidly with the immersion of the wheel. I.H P Area of one float = — — — X O. C is a multiplier, varying from 0.3 to 0.35; D is the diameter of the wheel to the float centres, in feet. The number of floats = ^(Z) + 2). The breadth of the float = 0.35 X the length. The thickness of floats — 1/12 the breadth. Diameter of gudgeons = thickness of float. Seaton and Rounthwaite's Pocket-book gives: Number of floats = , Vb where B is number of revolutions per minute. a v, . . \. I.H.P. X 33000 X K Area of one float (in square feet) = — — — — , JS X \J-) X B) where N = number of floats in one wheel. For vessels plying always in smooth water K = 1200. For sea-going steamers K — 1400. For tugs and such craft as require to stop and start frequently in a tide-way K — 1600. It will be quite accurate enough if the last four figures of the cube (D X BY be taken as ciphers. For illustrated description of the feathering paddle-wheel see Seaton 's Marine Engineering, or Seaton and Rounthwaite's Pocket-book. The diam- eter of a feathering -wheel is about one half that of a radial wheel for equal efficiency. (Thurston.) Efficiency of Paddle-wheels.— Computations by Prof. Thurston of the efficiency of propulsion by paddle-wheels give for light river steamers with ratio of velocity of the vessel, v, to velocity of the paddle -float at centre of pressure, V, or — , = -, with a dip = 3/20 radius of the wheel, and a slip of 25 per cent, an efficiency of .714; and for ocean steamers with = % radius, an efficiency of .685. JET-PROPULSION. Numerous experiments have been made in driving a vessel by the reaction of a jet of water pumped through an orifice in the stern, but they have all resulted in commercial failure. Two jet-propulsion steamers, the' " Waterwitch," 1100 tons, and the 'Squirt," a small torpedo-boat, were built by the British Government. The former was tried in 1867, and gave an efficiency of apparatus of only 18 per cent. .The latter gave a speed of 12 knots, as against 17 knots attained by a sister-ship having a screw and equal steam-power. The mathematical theory of the efficiency of the jet was discussed by Rankine in Tlie Engineer, Jan. 11, 1867, and he showed that the greater the quantity of water operated on by a jet-propeller, the greater RECEXT PRACTICE IK MARtHE ENGtKES. 1015 is the efficiency. In defiance both of the theory and of the results of earlier experiments, and also of the opinions of many naval engineers, more than $■200,000 were spent in 1888-90 in New York upon two experimental boats, the '• Prima Vista ' 1 and the " Evolution," in which the jet was made of very small size, in the latter case only %-incli diameter, and with a pressure of 2500 lbs. per square inch. As had been predicted, the vessel was a total failure. (See article by the author in Mechanics, March, 1891.) The theory of the jet-propeller is similar to that of the screw-propeller. If A = the area of the jet in square feet, Fits velocity with reference to the orifice, in feet per second, v — the velocity of the ship in reference to the earth, then the thrust of the jet (see Screw-propeller, ante) k -lAV (V - v). The work done on the vessel is 2AV(V— v)v, and the work wasted on the rearward projection of the jet is ^ X 2AV(V - v) 2 . The efficiency is n JTy/T ^ r 1 — ; — TT^-rp rs = r= • This expression equals unity when 2AV(V — v)v -\- Al (V- v) 2 \ + v V = v, that is, when the velocity of the jet with reference to the earth, or V — v, = 0; but then the thrust of the propeller is also 0. The greater the lvalue of Fas compared with v. the less the efficiency. For V = 20v, as was proposed in the " Evolution, " the efficiency of the jet would be less than 10 per cent, and this would be further reduced by the friction of the pumping mechanism and of the water in pipes. The whole theory of propulsion may be summed up in Rankine's words: "That propeller is the best, other things being equal, which drives astern the largest body of water at the lowest velocity." It is practically impossible to devise any system of hydraulic or jet propul- sion which can compare favorably, under these conditions, with the screw or the paddle-wheel. Reaction of a Jet.— If a jet of water issues horizontally from a ves- sel, the reaction on the side of the vessel opposite the orifice is equal to the weight of a column of water the section of which is the area of the orifice, and the height is twice the head. The propelling force in jet-propulsion is the reaction of the stream issuing from the orifice, and it is the same whether the jet is discharged under water, in the open air, or against a solid wall. For proof, see account of trials by C. J. Everett, Jr., given by Prof. J. Burkitt Webb, Trans. A. S. M. E., xii. 904. RECENT PRACTICE IN MARINE ENGINES. (From a paper by A. Blechynden on Marine Engineering during the past Decade, Proc. Inst. M. E., July, 1891.) Since 1881 the three-stage-expansion eneine has become the rule, and the boiler-pressure has been increased to 160 lbs. and even as high as 200 lbs. per square inch. Four-stage-expansion engines of various forms have also been adopted. Forced. Draught has become the rule in all vessels for naval service, and is comparatively common in both passenger and cargo vessels. By this means it is possible considerably to augment the power obtained from a given boiler: and so long as it is kept within certain limits it need result in no injury to the boiler, but when pushed too far the increase is sometimes purchased at considerable cost. In regard to the economy of forced draught, an examination of the ap- pended table (page 1018) will show that while the mean consumption of coal in those steamers working under natural draught is 1.573 lbs. per indicated horse-power per hour, it is only 1.336 lbs. in those fitted with forced draught. This is equivalent to an economy of 15%. Part of this economy, however, may be due to the other heat-saving appliances with which the latter steamers are fitted. Boilers.— As a material for boilers, iron is now a thing of the past, though it seems probable that it will continue yet awhile to be the material for tubes. Steel plates can be procured at 132 square feet superficial area and iy^ inches thick. For purely boiler work a punching-machine has be- come obsolete in marine-engine work. The increased pressures of steam have also caused attention to be directed to the furnace, and have led to the adoption of various artifices in the shape of corrugated, ribbed, and spiral flues, with the object of giving increased strength against collapse without abnormally increasing the thickness of the plate. A thick furnace-plate is viewed by many engineers with great 1016 MARINE ENGINEERING. suspicion; and the advisers of the Board of Trade have fixed the limit of thickness for furnace-plates at % inch ; but whether this limitation will stand in the light of prolonged experience remains to be seen. It is a fact generally accepted that the conditions of the surfaces of a plate are far greater factors in its resistance to the transmission of heat than either the material or the thickness. With a plate free from lamination, thickness being a mere secondary element, it would appear that a furnace-plate might be increased from % hich to % irch thickness without increasing its resist- ance more than 1J4#. So convinced have some engineers become of the soundness of this view that they have adopted flues % i"ch thick. Piston-valves.— Since higher steam -pressures have become common, piston-valves have become the rule for the high-pressure cjdiuder, and are not unusual for the intermediate. When well designed they have the great advantage of being almost free from friction, so far as the valve itself is concerned. In the earlier piston-valves it was customary to fit spring rings, which were a frequent source of trouble and absorbed a large amount of power in friction; but in recent practice it has become usual to fit spring- less adjustable sleeves. For low-pressure cylinders piston-valves are not in favor; if fitted with spring rings their friction is about as great as and occasionally greater than that of a well-balanced slide-valve; while if fitted with springless rings there is always some leakage, which is irrecoverable. But the large port-clear- ances inseparable froin the use of piston -valves are most objectionable; and with triple engines this is especially so, because with the customary late cut-off it becomes difficult to compress sufficiently for iusuring econo- my and smoothness of working when in " full gear, 1 ' without some special device. Steam-pipes.— The failures of copper steam-pipes on large vessels have drawn serious attention both to the material and the modes of con- struction of the pipes. As the brazed joint is liable to be imperfect, it is proposed to substitute solid drawn tubes, but as these are not made of large sizes two or more tubes may be needed to take the place of one brazed tube. Reinforcing the. ordinary brazed tubes by serving them with steel or copper wire, or by hooping them at intervals with steel or iron bands, has been tried aud found to answer perfectly. Auxiliary Supply of Fresh Water— Evaporators.— To make up the losses of water due to escape of steam from safety-valves, leakage at glands, joints, etc., either a reserve supply of fresh water is carried in tanks, or the supplementary feed is distilled from sea-water by special apparatus provided for the purpose. In practice the distillation is effected by passing steam, say from the first receiver, through a nest of tubes inside a still or evaporator, of which the steam-space is connected either with the second receiver or with the condenser. The temperature of the steam inside the tubes being higher than that of the steam either in the second receiver or in the condenser, the result is that the water inside the still is evaporated, and passes with the rest of the steam into the condenser, where it is condensed and serves to make up the loss. This plan localizes the trouble of the de- posit, and frees it from its dangerous character, because an evaporator can- not become overheated like a boiler, even though it be neglected until it salts up solid; and if the same precautions are taken in working the evapo- rator which used to be adopted with low-pressure boilers when they were fed with salt water, no serious trouble should result. Weir's Feed-water Heater.— The principle of a method of heating feed-water introduced by Mr. James Weir and widely adopted in the marine service is founded on the fact that, if the feed-water as it is drawn from the hot-well be raised in temperature by the heat of a portion of steam introduced into it from one of the steam-receivers, the decrease of the coal necessary to generate steam from the water of the higher temperature bears a greater ratio to the coal required without feed-heating than the power which would be developed in the cylinder by that portion of steam would bear to the whole power developed when passing all the steam through all the cylinders. Suppose a triple-expansion engine were working under the following conditions without feed-heating: boiler-pressure 150 lbs.; I.H.P. in high-pressure cylinder 398, in intermediate and low-pressure cylinders to- gether 790, total 1188. The temperature of hot-well 100° F. Then with feed- heating the same engine might work as follows: the feed might be heated to 220° F., and the percentage of steam from the first receiver required to heat it would be 10.9$; the I.H.P. in the h.p. cylinder would be as before 398, and in the three cylinders it would be 1103, or 93$ of the power developed without RECENT PRACTICE IN MARINE ENGINES. 1017 feed-heating. Meanwhile the heat to be added to each pound of the feed- water at 220° F. for converting it into steam would be 1005 units against 1125 units with feed at 100° F., equivalent to an expenditure of only 89.4$ of the heat required without, feed-heating. Hence the expenditure of heat in relation to power would be 89.4 -:- 93.0 = 96.4$, equivalent to a heat economy of 3.6$. If the steam for heating can be taken from the low-pressure receiver, the economy is about doubled. Passenger Steamers fitted with Twin Screws. Vessels. £ £ <*> S3 n Cylinders, two sets in all. I** 1 0> A Diameters. Stro. City of New York \ " " Paris i Majestic (_ Teutonic ( Feet 525 565 500 463^ 440 415 460 Feet 63M 58 55^ 51 48 54J^ Inches 45, 71, 113 43, 68, 110 40, 67, 106 41, 66, 101 32, 51, 82 34, 54, 85 34^, 57^, 92 In. 60 60 66 66 54 51 60 Lbs. 150 180 160 160 160 160 170 I.H.P. 20,000 18,000 11,500 12,500 10,125 10,000 11,656 Columbia Empress of In di a ) " " Japau V " " China ) Orel Scot Comparative Results of Working of Marine Engines, 1872, 1881, and 1891. Boilers, Engines, and Coal. 1872. 1881. 1891. 52.4 4.410 55.67 376 2.110 77.4 3.917 59.76 467 1.828 158.5 Heating-surface per horse-power, sq. ft Revolutions per minute, revs 3.275 63.75 529 Coal per horse-power per hour, lbs 1 522 Weight of Three - stage - expansion Engines in Nine Steamers in Relation to Indicated Horse-power and to Cylinder-capacity. * Weight of Machinery. Relative Weight of Machinery. Per Indicated Horse - Engine-room per cu. ft. of Cylinder- capacity. S£ be Type of W c 2 53 o o p I o power. Machinery. 6 Engine- room. Boiler- room. Total tons. tons. tons. lbs. lbs. lbs. tons. tons. 1 681 662 1313 226 220 446 1.30 3.75 Mercantile 2 638 619 1257 259 251 510 1.46 4.10 " 3 134 128 262 207 198 405 1.23 3.23 " 4 38.8 46.2 85 170 203 373 1.29 3.30 " 5 719 695 1414 167 162 329 1.41 3.44 " 6 75.2 107.8 183 141 202 343 1.37 3.37 " 7 44 61 105 77 108 185 1.21 2.72J Naval horizontal 8 73.5 109 182.5 78 116 194 1.11 2.78 do. 9 202 429 691 62.5 102 165 0.82 2.70 | Naval vertical 1018 MARINE ENGINEERING. ! " WWKW W pdK KK •d'H'I J8CI - ~ . • "■..--•;. MTIOTJ .I9d 9)V.lg jo -y bs .iad !>ii.inq l^oo •aj-ejS jo -^j bs aad VTH'I 'd'H'I I^SoSoiod •13d I O^hUmotw suormiOAan •3 rassa.Kl k *S '5 "5 "* S 5? .^ ^S 0000 -tuua^s •aoBj.ms I*- III! -JUS SlIJIOOO i ^jHTjrt^HOfeo'rHr-'oOHfNINNsrri; 'OOOHOOHICC ^ ^ CONSTRUCTION OF BUILDINGS. 1019 Dimensions, Indicated Horse - power, and Cylinder - capacity of Three - stage - expansion .Engines in Nine Steamers. o ^ 53 Single or Twin Screws. Cylinders. = CT, K=Q, W=QE, P = CE. As these relations contain no coefficient other than unity, the letters may represent any quantities given in terms of those units. For example, if E represents the number of volts electro-motive force, and R the number of ohms resistance in a circuit, then their ratio _£"-=- R will give the number of amperes current strength in that circuit. The above six formulae can be combined by substitution or elimination, so as to give the relations between any of the quantities. The most impor- tant of these are the following : Q = ^t, K = j^t, W=CEt = ^t= C*Rt = Pt, _ E* ^ 9r> W QE STANDARDS OF MEASUREMENT. 1025 The definitions of these units as adopted at the International Electrical Congress at Chicago in 1893, and as established by Act of Congress of the United States, July 12, 1894, are as follows: The ohm is substantially equal to 10 9 (or 1 ,000,000,000) units or resistance of the C.G.S. system, and is represented by the resistance offered to an un- varying electric current by a column of mercury at 32° F., 14.4521 grammes in mass, of a constant cross-sectional area, and of the length of 100.3 centi- metres. The ampere is 1/10 of the unit of current of the C.G.S. system, and is the practical equivalent of the unvarying current which when passed through a solution of nitrate of silver in water in accordance with standard speci- fications deposits silver at the rate of .001118 gramme per second. The volt is the electro-motive force that, steadily applied to a conductor whose resistance is one ohm, will produce a current of one ampere, and is practically equivalent to 1000/1434 (or .6974) of the electro-motive force be- tween the poles or electrodes of a Clark's cell at a temperature of 15° C, and prepared in the manner described in the standard specifications. The coulomb is the quantity of electricity transferred by a current of one ampere in one second. The farad is the capacity of a condenser charged to a potential of one volt by one coulomb of electricity. The joule is equal to 10,000,000 units of work in the C.G.S. system, and is practically equivalent to the energy expended in one second by an ampere in an ohm. The watt is equal to 10,000,000 units of power in the C.G.S. system, and is practically equivalent to the work done at the rate of one joule per second. The henry is the induction in a circuit when the electro-motive force in- duced in this circuit is one volt, while the inducing current varies at the rate of one ampere per second. The ohm, volt, etc., as above defined, are called the "international !: ohm, volt, etc., to distinguish them from the " legal " ohm, B.A. unit, etc. The value of the ohm, determined by a committee of the British Associa- tion in 1863, called the B.A. unit, was the resistance of a certain piece of copper wire preserved in London. The so-called " legal " ohm, as adopted at the International Congress of Electricians in Paris in 1884, was a correc- tion of the B.A. unit, and was defined as the resistance of a column of mercury 1 square millimetre in section and 106 centimetres long, at a tem- perature of 32° F. 1 legal ohm = 1.0112 B.A. units, 1 B.A. unit = 0.9889 legal ohm; 1 international ohm = 1.0136 " " 1 " " - 0.9866 int. ohm; 1 " " = 1.0023 legal ohm, 1 legal ohm = 0.9977 ■« " Derived Units. 1 megohm = 1 million ohms; 1 microhm = 1 millionth of an ohm; 1 milliampere = 1/1000 of an ampere; 1 micro-farad — 1 millionth of a farad. Relations of Various Units. 1 ampere =1 coulomb per second ; 1 volt-ampere =1 watt = 1 volt-coulomb per second; ( = .7373 foot-pound per second, 1 watt < — .0009477 heat-units per second (Fahr.), / = 1/746 of one horse-power; 1 = .7373 foot-pound, 1 joule •< = work done by one watt in one second, ( = .0009477 heat-umt; 1 British thermal unit = 1055.2 joules; l = 737.3 foot-pound per second, 1 kilowatt, or 1000 watts ■< = .9477 heat-units per second, { = 1000/746 or 1.3405 horse-powers; 1 Kilowatt-hours, I = 1.3405 horse-power hours, 1000 volt-ampere hours, •< = 2,654,200 foot-pounds, 1 British Board of Trade unit, ( = 3416 heat-units; * , „,„ „^„ r ^„ i = 746 watts = 146 volt -amperes, 1 horse-power -> = 83 ,000 foot-pounds per minute. The ohm, ampere, and volt are defined in terms of one another as follows: Ohm, the resistance of a conductor through which a current of one ampere will pass when the electro-motive force is one volt. Ampere, the quantity 1026 ELECTRICAL ENGINEERING. 00 s 6 1) > H 1,055 watt seconds. 778 ft.-lbs. 107.6 kilogram metres. .000293 K. W. hour. .000393 H.P. hour. .0000688 lbs. carbon oxi- dized. .001036 lbs. water evap. from and at 212° F. $?& 53 &v ?>^ 2? ^oo 7.233 ft.-lbs. .00000365 H.P. hour. .00000272 K. W. hour. .0093 heat-units. 14,544 heat-units. 1.11 lb. Anth'cite coal ox. 2.5 lbs. dry wood oxidized. 21 cu. ft. illuminating-gas. 4.26 K. W. hours. 5.71 H.P. hours. 11,315,000 ft.-lbs. 15 lbs. of water evap. from and at 212° F. .283 K. W. hour. 379 H.P. hour. 965.7 heat-units. 103,900 k. g. m. 1,019.000 joules. 751,300 ft.-lbs. .0664 lb. of carbon oxi- dized. 5 3 3 ? • si gcc E W 53 53 h aa § II big lib. Carbon Oxidized with per- fect Effi- ciency = Su-I ii 5 * o a 0j a O) 746 watts. .746 K. W. 33,000 ft.-lbs. per minute. 550 ft.-lbs. per second. 2,545 heat-units per hour. 42.4 heat-units per minute. .707 heat-units per second. .175 lbs. carbon oxidized per hour. 2.64 lbs. water evap. per hour from and at 212° F. 1 watt second. .000000278 K. W. hour. .102 k. g. m. .0009477 heat-units. .7373 ft -lb. 1.356 joules. .1383 k.g. m. .000000377 K. W. hours. .001285 heat-units. .0000005 H.P. hour. 1 joule per second. .00134 H.P. 3.412 heat-units per hour. .7373 ft.-lbs. per second. .0035 lbs. water evap. per hr. 44.24 ft.-lbs. per minute. 8.19 heat-units per sq. ft. per minute. 6371 ft.-lbs. per sq. ft. per min- ute. .193 H.P. per sq. ft. S II II o •-a ~% II II "la > 53 o 5 a> 6 s a> > a "3 > '3 cr W 1,000 watt hours. 1.34 horse-power hours. 2,654,200 ft.-lbs. 3,600,000 joules. 3,412 heat-units. 367,000 kilogram metres. .235 lb. carbon oxidized with perfect efficiency. 3.53 lbs. water evap. from and at 212° F. 22.15 lbs. of water raised from 62° to 212° F. .746 K. W. hours. 1,980,000 ft.-lbs. 2,545 heat-units. 273,740 k.g. m. .175 lb. carbon oxidized with perfect efficiency. 2.64 lbs. water evaporated from aud at 212° F. 17.0 lbs. water raised from 62° F. to 212° F. 1,000 watts. 1.34 horse-power. 2,654,200 ft.-lbs. per hour. 44,240 ft.-lbs. per minute. 737.3 ft.-lbs. per second. 3,412 heat-units per hour. 56.0 heat-units per minute. .948 heat-unit per second. .22751b. carbon oxidized per hour. 3.53 lbs. water evap. per hour from and at 212° F. I p II =1 6 "3 II ia FLOW OF WATER AttD ELECTRICTY. 102? Of current which will flow through a resistance of one ohm when the electro- motive force is one volt. Volt, the electro-motive force required to cause a current of one ampere to flow through a resistance of one ohm. Units of the Magnetic Circuit.— (See Electro-magnets, page 1058.) For Methods of making Electrical Measurements, Test- ing, etc., see Munroe & Jamieson's Pocket-Book of Electrical Rules, Tables, and Data; S. P. Thompson's Dynamo-Electric Machinery; and works I on Electrical Engineering. Equivalent Electrical and Mechanical Units.— H. Ward Leonard published in The Electrical Engineer. Feb. 25, 1895, a table of use- ful equivalents of electrical and mechanical units, from which the table on page 1026 is taken, with some modifications. ANALOGIES BETWEEN THE FLOW OF WATER AND ELECTRICITY. Water. Electricity. Head, difference of level, in feet. (Volts; electro-motive force; differ- Difference of pressure per sq. in., in •< ence of potential or of pressure; E. lbs. ( or E.M.F. lS rt^SS&fi5^ ^?h I °Se^s S d1ISl? asKSTol arer^^^^s^dr] S^^^SS^^ Rate of flow, as cubic ft. per second, f A 5XnlftVoTS?renTrl?e offlol'-'i gallons per minute, etc., or volume \ ~_lSmbM second' divided by the time. In the mining { ampeie - l coulomb pei second, regions sometimes expressed in | volts E " miners' inches." I Amperes = ^^ ; C = — ; E = CR. Quantity, usually measured in cubic 1 feet or gallons, but is also equiva- | Coulomb, unit of quantity, Q, = rate lent to rate of flow X time, as J- of flow X time, as ampere-seconds, cubic feet per second for so many j 1 ampere-hour = 3600 coulombs, hours. J 'Joule, volt-coulomb, W, the unit of Work, or energy, measured in foot- pounds; product of weight of fall- ing water into height of fall: ''" work, — product of quantity by the electro-motive force — volt-ampere- second. 1 joule= .7373 foot-pound. pumping, product of quantity in -J If C (amperes) = rate of flow, and cubic feet into the pressure in per square foot against which the water is pumped. E (volts) = difference of pressure between two points in a circuit, energy expended = CEt, = C' 2 Et, since E = CR. Power, rate of work. Horse-power,ft.- , lbs. of work done in 1 min.-s- 33,000. I Watt, unit of power, P, = volts X In falling water, pounds falling in | amperes, = current or rate of flow one second -+- 550. In water flowing \~ X difference of potential. in pipes, rate of flow in cubic feet I 1 watt = .7373 foot-pound per second per second X pressure resisting the | = 1/746 of a horse-power. flow in lbs. per sq. ft. -r- 550. J Analogy between the Ampere and the Miner's Inch. (T. O'Connor Sloane.) — The miner's inch is defined as the quantity of water which will flow through an aperture an inch square in a board two inches thick, under a head of water of six inches. Here, as in the case of the am- pere, we have no reference to any abstract quantity, such as gallons or pounds. There is no reference to time. It is simply a rate of flow. We may consider the head of water, six inches, as the representative of electri- cal pressure; i.e., one volt. The aperture restricting the flow of water may be assumed to represent the resistance of one ohm; the flow through a re- sistance of one ohm under the pressure of one volt is one ampeie; the flow through the resistance of a one-inch hole two inches long under the pressure of six inches to the upper edge of the opening is one miner's inch. The miner's inch-second is the correct analogue of the ampere-second; the one denotes a specific quantity of water, 0.194 gallon; the other a specific quantity of electricity, a coulomb. 1028 ELECTRICAL ENGINEERING. ELECTRICAL. RESISTANCE. Laws of Electrical Resistance.— The resistance, R, of any con- ductor varies directly as its length, /, and inversely as its sectional area, s, or R cc — . s Example.— If one foot of copper wire .01 in. diameter has a resistance of .10323 ohm, what will be the resistance of a mile of wire .3 in. diam. at the same temperature ? The sectional areas being proportional to the squares of the diameters, the ratio of the areas is .3 2 : .01 2 = 900 to 1. The lengths are as 5280 to 1. The resistances being directly as the lengths and inversely as the sectional areas, the resistance of the second wire is .10323 x 5280 -r- 900 = .6056 ohm. Conductance, c, is the inverse of resistance. R = I sR If c and c 2 represent the conductances, and R and i? 2 the respective resistance of two substances of the same length and section, then c : e% : : R 2 : R. Equivalent Conductors.— With two conductors of length I, Z x , of conductances c, c 1? and sectional areas .s, s lf we have the same resistance, and one may be substituted for the other when — = — — . CS c'lSi The specific resistance, also called resistivity, a, of a material of unit length and section is its resistance as compared with the resistance of a standard conductor, such as pure copper. Conductivity, or specific con- ductance, is the reciprocal of resistivity. I al If two wires have lengths I, l x , areas s, s l7 and specific resistances a, a x , their actual resistances are R= — , R t = , and — s s x R x als x dills' Electrical Conductivity of Different Metals and Alloys. — Lazare Weiler presented to the Societe Internationale des Electriciens the results of his experiments upon the relative electrical conductivity of certain metals and alloys, as here appended : 1. Pure silver 100 2. Pure copper 100 3. Refined and crystallized copper 99.9 4. Telegraphic silicious bronze 98 5. Alloy of copper and silver (50$) 86.65 6. Pure gold 78 7. Silicide of copper, 4$ Si 75 8. Silicide of copper, 12$ Si. . . 54.7 9. Pure aluminum 54.2 10. Tin with 12% of sodium... 46.9 11. Telephonic silicious bronze 35 12. Copper with 10% of lead .... 30 13. Pure zinc .. 29.9 14. Telephonic phosphor - bronze 29 15. Silicious brass, 25$ zinc 26.49 16. Brass with 35$ of zinc 21.5 17. Phosphor tin 17.7 18. Alloy of gold and silver (50$) 16.12 19. Swedish iron 16 20. Pure Banca tin 15.45 21. Antimonial copper 12.7 22. Aluminum bronze (10$) .... 12.6 23. Siemens steel 12 24. Pure platinum 10.6 25. Copper with 10$ of nickel.. 10.6 26. Cadmium amalgam (15%). 10.2 27. Dronier mercurial bronze.. 10.14 28. Arsenical copper (10$) 9.1 29. Pure lead .. 8.88 30. Bronze with 20$ of tin 8.4 31. Pure nickel 7.89 32. Phosphor-bronze, 10$ tin .. 6.5 33. Phosphor-copper, 9$ phos.. 4.9 34. Antimony 3.88 The above comparative resistances may be reduced to ohms on the basis that a wire of soft copper one milimetre in diameter at a temperature of 0° C. has a resistance of .02029 international ohms per metre; or a wire .001 inch diam. has a resistance of 9.59 international ohms per foot. ELECTRICAL RESISTANCE. 1029 Relative Conductivities of Different Metals at 0° and 100° C. (Mattiiiessen.) Conductivities. c. F. Metals. Conductivities. Metals. At 0° C. " 32° F. At 100° " 212° At 0° C. " 32° F. At 100° C. " 212° F. 100 99.95 77. 9G 29.02 23.72 18.00 16.80 71.56 70.27 55.90 20.67 16.77 Tin 12.36 8.32 4.76 4.62 1.60 1.245 8.67 Copper, hard Gold, hard Zinc, pressed 5 86 3 33 Antimony. . Mercury, pu Bismuth ■e. . 3.26 Platinum, soft. .. Iron, soft Conductors and Insulators in Order of their Value. Conductc All metals Well- burned ehai Plumbago Acid solutions Saline solutions Metallic ores Animal fluids Living vegetable Moist earth Water >rs. coal substance 3 Insulators (Non-conductors). Dry Air Ebonite Shellac Gutta-percha Paraffin India-rubber Amber Silk Resins Dry Paper Sulphur Parchment Wax Dry Leather Jet Porcelain Glass Oils Mica According to Culley, the resistance of distilled water is 6754 million times as great as that of copper. Resistance Varies witli Temperature.— For every degree Cen- tigrade the resistance of copper increases about 0.4#, or for every degree F. 1%. Thus a piece of copper wire having a resistance of 10 ohms at 32° would have a resistance of 11.11 ohms at 82° F. The following table shows the amount of resistance of a few substances used for various electrical purposes by which 1 ohm is increased by a rise of temperature 1° F., or 1° C. Rise of R. of 1 Ohm when Heated- Material. 1°F. Platinoid 00013 Platinum-silver 00018 German silver (see below) .00024 Gold, silver 00036 Cast iron 00044 Copper 00222 1° C. .00021 .00031 .00044 .00065 .00080 .00400 Annealing.— The degree of hardness or softness of a metal or alloy affects its resistance. Resistance is lessened by annealing. Matthiessen gives the following relative conductivities for copper and silver, the com- parison being made with pure silver at 100° C. : Metals. Temp. C. Copper 11° Silver 14.6° Hard. 95.31 95.36 Annealed. Dr. Siemens compared the conductivities of copper, silver, and brass with pure mercury at 0° C, with the following results: Metal. Hard. Annealed. Copper 52.207 55.253 Silver 56.252 64.380 Brass 11.439 13.502 Edward Weston (Proc. Electrical Congress 1893, p. 179) says that the re- sistance of German silver depends on its composition. Mathiessen gives it as nearly 13 times that of copper, with a temperature coefficient of .0004433 per degree C. Weston, however, has found copper-nickel-zinc alloys (German 1030 ELECTRICAL ENGINEERING. silver) which had a resistance of nearly 28 times that of copper, and a tem- perature coefficient of about one half that given by Matthiessen. Kennelly and Fessenden (Proc. Elec. Cong., p. 186) find that copper has a uniform temperature coefficient of 0.40t$ per degree C, between the limits of 20° and 250° C. Standard of Resistance of Copper Wire. (Trans. A. I. E. E., Sept. and Nov. 1890.)— Matthiessen's standard is: A hard-drawn copper wire, 1 metre long, weighing 1 gramme has a resistance of 0.1469 B.A. unit at 0° C. (1 B.A. unit = 0.9889 legal ohm = 0.9866 international ohm.) Resist- ance of hard copper = 1.0226 times that of soft copper. Relative conducting power (Matthiessen): silver, 100; hard or unannealed copper, 99.95; soft or annealed copper, 102.21. Conductivity of copper at other temperatures than 0°C, Ct = C' (l - .00387* -f .000009009* 2 ). The resistance is the reciprocal of the conductivity, and is Rt = R (l -f- .00387* + .00000597* 2 ). A committee of the Am. Inst. Electrical Engineers recommend the follow- ing as the most correct form of the Matthiessen standard, taking 8.89 as the sp. gr. of pure copper : A soft copper wire 1 metre long and 1 mm. diam. has an electrical resist- ance of .02057 B.A. unit at 0° C. From this the resistance of a soft copper wire 1 foot long and .001 in. diam. (mil-foot) is found to be 9.720 B.A. units at 0° C. Standard Resistance at 0° C. B.A. Units. LegalOhms. ^Ss^' Metre-millimetre, soft copper 02057 .02034 .02029 Cubic centimetre " " 000001616 .000001598 .000001593 Mil-foot " " 9.720 9.612 9.590 1 mil-foot, of soft copper at 10°. 22 C. or 50°. 4 F. . . 10 9.977 " " " " " 15°. 5 " 59°. 9 F... 10.20 10.175 " " " " " 23°.9 " 75° F... 10.53 10.505 For tables of the resistance of copper wire, see pages 218 to 220, also pp. 1034, 1035. Taking Matthiessen's standard of pure copper as 100$. some refined metal has exhibited an electrical conductivity equivalent to 103$. Matthiessen found that impurities in copper sufficient to decrease its density from 8.94 to 8.90 produced a marked increase of electrical resistance. ELECTRIC CURRENTS. Olim's Law.- This law expresses the relation between the three fun- damental units of resistance, electrical pressure, and current. It is : „ electrical pressure _, E ; E Current = r— (?=-=•; whence E = CR, and R = —.. resistance R C In terms of the units of the three quantities, volts ... . , volts Amperes = ■ ■ ■ ; vOlts = amperes X ohms; ohms = . ohms amperes Examples: Simple Circuits.— 1. If the source has an effective electrical pressure of 100 volts, and the resistance is two ohms, what is the current ? ^ E 100 * n C = -^r = -Q- = 50 amperes. 2. What pressure will give a current of 50 amperes through a resistance of 2 ohms ? E = CR = 50 X 2 = 100 volts. 3. What resistance is required to obtain a current of 50 amperes when the pressure is 100 volts ? R — -— — — - = 2 ohms. The following examples are from R. E. Day's " Electric Light Arithmetic:" 1. The internal resistance of a certain Brush dynamo-machine is 10.9 ohms, and the external resistance is 73 ohms; the electro-motive force of the ma- chine being 839 volts. Find the strength of the current flowing in the circuit. E = 839; R = 73 + 10.9 = 83.9 ohms; C = E -v- R = 839 -*- 83.9 = 10 amperes. ELECTRIC CURRENTS. 1031 2. Three arc lamps in series have a resistance of 9.36 ohms, while the re- sistance of the leading wires is 1.1 ohm, and that of the dynamo is 2.8 ohms. Find what must be the electro-motive force of the machine when the strength of the current produced is 14.8 amperes. ij = 2.8 -f- 9.36 + 1.1 = 13.26 ohms; C = 14.8 amperes; E = C X B = 13.26 X 14.8 = 196.3 volts. 3. Calculate from the following data the average resistance of each of three arc lamps arranged in series. The electro-motive force of the machine is 244 volts and its resistance is 3.7 ohms, while that of the leading wires is 2 ohms, and the. strength of current through each lamp is 21 amperes. If x represent the average resistance in ohms of each lamp, then the total resistance of the circuit is B = Sx + 2 -f 3.7. But by Ohm's law B - E -s- C, .'. 3a -f- 5.7 = 244/21 = 11.61 ohms, whence x = 1.97 ohms, nearly. 4. Three Maxim incandescent lamps were placed in series. The average resistance, when hot, of each lamp was 39.3 ohms, and that of the dynamo and leading wires 11.2 ohms. What electro-motive force was required to maintain a current of 1.2 amperes through this circuit ? In this case we have B = 3 X 39.3 + 11.2 = 129.1 ohms, and C = 1.2 ampere; and therefore, by Ohm's law, E = C X B = 1.2 X 129.1 = 154.9 volts. _. The resistance of the arc of a certain Brush lamp was 3.8 ohms when a current of 10 amperes was flowing through it. What was the electro-motive force between the two terminals ? E - C X B = 10 X 3.8 = 38 volts. 6. Twenty-five exactly similar galvanic cells, each of which had an aver- age internal resistance of 15 ohms, were joined up in series to one incandes- cent lamp of 70 ohms resistance, and produced a current of 0.112 amperes. What would be the strength of current produced by a series of 30 such cells through 2 lamps, each of 30 ohms resistance ? The data of the first part of the problem enable us to determine the average electro-motive force of each cell of the battery. Let this be repre- sented by E; then we have 25.EJ = C X B = .112 X (25 X 15 + 70) = .112 X 445; Then from the data in the second part of the problem, we have, by Ohm's law, Divided. Circuits.— If the circuit has two paths, the total current in both divides itself inversely as the resistances. If B and B, are the resistances of the two branches, and C and C x the cur- C B rents, C X B = Cj X i?i, and — = -£-', whence °~ b ' Ci -rT' r -~c~' ^".cr In the case of the double circuit, one circuit is said to be in shunt to the other, or the circuits are in multiple arc or in parallel. Conductors in Series.— If conductors are arranged one after the other they are said to be in series, and the total resistance is the sum of their several resistances. B = B^ + B% -\- B a . Internal Resistance.— In a simple circuit we have two resistances, fhat of the circuit B and that of the internal parts of the source, called in^ 1032 ELECTRICAL ENGINEERING. ternal resistance, r. The formula of Ohm's law when the internal resistance j tt is considered is C = -j—-. — . K -f- r Total or Joint Resistance of Two Branches.— Let C be the I total current, and C u C 2 the currents in branches whose resistances respect- ively are R u i? 2 . Then C* = C x + C 2 ; C = ^ ; C x = — -; C\ = ~\ or, \i E = 1, *C = ^r = s- +^5-, whence K = D *, ' , which is the joint resistance of K K 1 xr, 2 Kx -f- K a J? x and i?„. Similarly, the joint resistances of three branches have resistances respect- ively of R^R,, R t , is R = BiBt *££\ BaRt - When the branch resistances are equal, the formula becomes #i n ~ * X n n where R x = the resistance of one branch, and n — the number of branches. Kirchhoff's I-*LO«OJ-*^6 -:--:;- - . . . . : ' ' - ' : :.:.-: Sq 1000==:=: = = = — 00 = 00 000000000000000000000 08J - lllssssss OOOO'OOOOt oooooooo< K?!-H'*C.C'.NCr.('-C - - r- -+• oc t^ 01 C<» Oi 1 - ■ . -■■■■■■-. ■■ r : ■ - . ■ - ::..-.:.: :. . -■ -. : • .. :. : - . -..:.. r - . - .-: - . -—00000000000000000000000000 T.HW«iCiC«CiH-*10HH '. " '- - . ~ ■:■'■_'■: ': - - : - : ■ : " ■" ' - " : 0000000 ------------------------- 0000000 OOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOrlrHi-IlN 00 t~^- :: ■•-. .'. ..-. - 2 ' '■ ' : v y '■ i r b z z z z z 'z z z~ O -•: :: C — -- - "" X '. • I - ~ = = O — i- i- ^ 0? O O --H O? H r* ~- — L- I- O "-0 O o © o o © © © © o © o © © o o © © © o ©■ © © © © o*ffl*M?. ;- n^^oioaix/ o_ o_ o; ;t_-. >-«M^il08«H^rtHHOOOOOI ^ r. c v: o i_- -r- %r 2 ' ~ ~ Tt ^ ^ '-5 lQ rz! ■— . O © W © 00 -*H OC t^ t-~ V. — — ' © * ' -r — 0? u- :c ' Oanx^ 07 © --C - ro i " -.r '^ ~ .-. - ■ - ~ - -.; r- ~ ■- ~. vc —i c- t~ -HO) JOHN'JOHCIN' ■ ' ■■•: — ' - — - — — . /■:-. — r ~ r. 7 7 / --h(,i--i.-;hhc.oOO HBIXCl'I-.S'.Hi-II^IOC--.' ■ - r - - ' - -h -< o © © o - . - : - .. : ;--*-noocc— ' = squai*e iuches. C = 2^r= is the greatest curreut which can be used in the magnet coils of a shunt machine having a certain pressure in order that they do not heat above a certain temperature. Thus for a rise of temperature of 50° F. above the surrounding air, C = 2^=, - .224 — . Substituting for E its equivalent CR, we get °=\/'- m i- If 80° F. is the maximum difference of temperature, C -223E = - 36 E = - 6( y B The formula can be used for series machines when C is known, for writing C*R - 1/2232-5, we get R = ^L. With a permissible rise of 50° F. or 80° F., we have respectively, .2245 . „ fl The surface area of the coil in square inches may be found from 22SW _ 22SCE _ 223C 2 i? ~ T '" T ~ T ' ELECTRIC CURRENTS. 1037 For a rise of temperature of 50° F. or 80° F., respectively, the surface will be 223 W 223 W S = ^ r =iAtW; and .9 = ^ : 2.8W. Fusion of Wires.— W. H. Preece gives a formula for the current re- quired to fuse wires of different metals, viz.: C = adh in which d is the diameter in inches and a a coefficient whose value for different metals is as follows: Copper 10214; aluminum 7585; platinum 5172; German silver 5230; platinoid 4750; iron 3148; tin, 1642: lead, 1379; alloy of 2 lead and 1 tin, 1318. .Diameters of Various Wires which will be Fused by a given Current. Formula, d =(- ) 3 ; a = 3148 for iron. 1642 for tin = 1379 for lead = 10244 for copper = Tin Wire. Lead Wire. Copper Wire. Iron Wire. Current, in amperes. Diam. inches. £T£- Diam. inches. ISR* Diam. inches. Approx. S.W. G. Diam. inches. Approx. S.W. G. 1 .0072 36 .0081 35 .0021 47 .0047 40 2 .0113 31 .0128 30 .0034 43 .0074 36 3 .0149 28 .0168 27 .0044 41 .0097 33 4 .0181 26 .0203 25 .0053 39 .0117 31 5 .0210 25 .0236 23 .0062 38 .0136 29 10 .0334 21 .0375 20 .0098 33 .0216 24 15 .0437 19 .0491 18 .0129 30 .0283 22 20 .0529 17 .0595 17 .0156 28 .0343 20.5 25 .0614 16 .0690 15 .0181 26 .0398 19 30 .0694 15 .0779 14 .0205 25 .0450 18.5 35 .0769 14.5 .0864 13.5 .0227 21 .0498 18 40 .0840 13.5 .0944 13 .0248 23 .0545 17 45 •0909 13 .1021 12 .0268 22 .0589 16.5 50 .0975 12.5 .1095 11.5 .0288 22 .0632 16 60 .1101 11 .1237 10 .0325 21 .0714 15 70 . 1220 10 .1371 9.5 .0360 20 .0791 14 80 .1334 9.5 .1499 8.5 .0394 19 .0864 13.5 90 .1443 9 .1621 8 .0426 18.5 .0935 13 100 .1548 8.5 .1739 7 .0457 18 .1003 12 120 .1748 7 .1964 6 .0516 17.5 .1133 11 140 .1937 6 .2176 5 .0572 17 .1255 10 160 .2118 5 .2379 4 .0625 16 .1372 9.5 180 .2291 4 .2573 3 .0676 16 .1484 9 200 .2457 3.5 .2760 2 .0725 15 .1592 8 250 .2851 1.5 .3203 .0841 13.5 .1848 6.5 300 .3220 .3617 00.5 .0950 12.5 .2086 5 Current in Amperes Required to Fuse Wires According to the Formula C — ad$- No. Diameter, ■d§- Tin. Lead Copper Iron. S.W. G. inches. a = 1642. a = 1379. a = 10244 a = 3148. 14 .080 .022627 37.15 31.20 231.8 71.22 16 .064 .016191 26.58 22.32 165.8 50.96 18 .048 .010516 17.27 14.50 107.7 33.10 20 .036 .006831 11.22 9.419 69.97 21.50 22 .028 .004685 7.692 6.461 48.00 14.75 24 .022 .003263 5.357 4.499 33.43 10.27 26 .018 .002415 3.965 3.330 24.74 7.602 28 .0148 .001801 2.956 2.483 18.44 5.667 30 .0124 .001381 2.267 1 904 14.15 4.347 32 .0108 .001122 1.843 1.548 11.50 3.533 1038 ELECTRICAL ENGINEERING. ELECTRIC TRANSMISSION. Cross-section of Wire Required for a Given Current.— Constant Current (Series) System. —The cross-sectional area of copper necessary in any circuit for a given constant current depends on the differ- ence between the pressure at the generating station and the maximum pressure required by all the apparatus on the circuit, and on the total length of the circuit. The following formulae are given in "Practical Electrical Engineering: 1 ' If V = pressure in volts at generators; v = sum of all the pressures (in volts) required by apparatus supplied in the circuit; n = total length (going and return) of circuit in miles; C = current in amperes; r = resistance of 1 mile of copper-conductor of 1 square inch sectional area in ohms; a = required cross sectional area of copper in square inches,— nrC a = — . V — v If we take the temperature of the conductor when the current has been flowing for some time through it, as 80° F., _.._. . , 0.0455nC r — 0.04oo ohm, and a = — ^ . It generally happens, however, that we are not tied down to a particular value of V, as the pressure at the generators can be varied by a few volts to suit requirements. In this case it is usual to fix upon a current density and determine the cross-sectional area of copper in accordance with it. If D = current density in amperes per square inch determined upon, C The current density is frequently taken at 100C amperes to the square inch, but should in general be determined by economical considerations for every case in question. Allowable Current Density in Insulated Cables. — Experiments on insulated cables in casing gave the results shown below, but they need con- firmation or correction of the current densities permissible in different sizes of insulated cables run underground. C and D are the current in amperes and the current density in amperes per square inch, respectively, which will raise the temperature of the conductor by the number of degrees Fahr. indicated by the suffix. No. S.W.G.* of Strands, each Wire. Area of Strand in c J8 D 18 C50 D 50 square inches. 0.0072 18 2,500 28 3.900 0.0357 59 1,400 95 2,700 0.0975 126 1,300 205 2,100 Constant Pressure (Parallel System).— To determine the loss in pressure in a feeder of given size in the case of two-wire parallel distribution. Let a — cross-sectional area of copper of one conductor of the feeder in square inches; n = length of feeder (going and return) in miles; C = current in amperes; V — v = loss of pressure in feeder in volts; r — resistance of 1 mile of copper conductor of 1 square inch sec- tional area in ohms. * Standard (British) Wire-gauge. ELECTRIC TRANSMISSION. 1039 If the temperature of the conductor with this current flowing in it is assumed to be 80° F., 0455n(7 r — 0.0455 ohm, and V - v = — . a Three-wire Feeder.— In the case of a three-wire feeder, let p x q x and p, 2 q 2 represent the two outer conductors, and let p'q' represent the middle conductor, p x , p', p 2 being at the feeding-point and q x , q\ g 2 at the generat- ing station, and let a = cross-sectional area of each of the outer conductors in square inches; a' — cross-sectional area of middle conductor; n = length in miles of each conductor of feeder ; V x = pressure between p x and p' in volts at generating station; V 2 — pressure between p' and p 2 in volts at generating station; v x — pressure between q x and q' in volts at feeding-point; r 2 = pressure between q' and g 2 in volts at feeding-point; C x = current in p x q x in amperes; C 2 = current in ptfi in amperes; r --■ resistance of 1 mile of copper conductor of 1 square inch sectional area in ohms. Then Oi.C.-Cj, TT i C 2 C x - c 2 , V x - v x - nr ) — H ; — ; — - !- ; F 2 — v 2 = rar J a' j ' ' ■* (a a' J It will be noticed that if t^ = v 2 , and if CJis greater than C 2 , Vi is greater than V 2 by twice the loss of pressure in the middle wire; this result shows that the regulators must be in circuit with the two outer conductors. It is usual to make a' half a; then, if the greatest want of balance between t lie loads of the two sections of the three-wire system is m* per cent of the maximum load of the more heavily loaded section, and if C x is the maximum current in either of the outer conductors of the feeder under consideration, C 2 will not be less than G x \\ - tkr), and consequently C x — C 2 will not be mC greater than - x 100 We have then ... -f m TT nrC x 200 - m -_-; F 2 -t, 2= — X-^-; so that if v x and v 2 are each equal to V— the pressure required to be main- tained constant at the feeding-point — we can calculate V x and F" 2 for given values of n, a, and C x , employing the value of m, which we estimate should be the maximum it can have. These last expressions show that the difference in the pressures required at the station across the two sections of a three-wire feeder increases with the current carried by the feeder ; hence the regulators on each of the outer conductors should be equivalent to a variable resistance having at least nrm , -r^- ohms as a maximum. 100a It is usual to make the area of the middle conductor one half of that of each of the outer conductors, but this is not invariably the case. Sliort-circuiting.— From the law C= — it is seen that with any pres- sure E the current Cwill become very great if R is made very small. In short-circuiting the resistance becomes-small and the current therefore great. Hence the dangers of short-circuiting a current. Economy of Electric Transmission. (R. G. Blaine, Eng'g, June 5, 1891.)— Sir W. Thomson's rule for the most economical section of conductor * The value to be assigned to m may vary from 10 to 25, according to the case exercised in connecting customers to one section or the other, or both, and according to the local conditions. At a certain station supplying current on the three-wire low-pressure system to about 25,000 8-c.p. lamps, we were informed that in had never exceeded 7 or 8. 1040 ELECTRICAL ENGINEERING. is that for which the "annual interest on capital outlay is equal to the annual cost of energy wasted," and its practical outcome is that the area of 17 the copper conductor should be such that its resistance per mile = — (C being the current in amperes). Tables have been compiled by Professor Forbes and others in accordance with modifications of Sir W. Thomson's rule. For a given entering horse- power the question is merely one as to what current density, or how many amperes per square inch of conductor, should be employed. Sir W. Thom- son's rule gives about 393 amperes per square inch, and Professor Forbes's tables— for a medium cost of one electrical horse-power per hour— give a current density of about 380 amperes per square inch as most economical. When a given horse-power is to be delivered at a given distance, the case is somewhat different, and Professors Ay rton and Perry (Electrician, March, 1886) have shown that in that case both the current and resistance are variables, and that their most economical values may be found from the foK lowing formulas: and r = J>2 sin 4> nw (1 + sin <£) a ' in which C = the proper current in amperes; r = resistance in ohms per mile which should be given to the conductor; P = pressure at entrance in volts; n = number of miles of conductor; w = power delivered in watts; (J> = such an angle that tan = nt-i-P, t being a constant depending on the price of copper, the cost of one electrical horse-power, interest, etc.: it may be taken as about 17. In this case the current density should not remain constant, but should diminish as the length increases, being in all cases less than that calculated by Sir W. Thomson's rule. Example.— If the current for an electric railway is sent in at 200 volts, 100 horse-power being delivered, find the waste of power in heating the con- ductor, the distance being 5 miles and there being a return conductor. Here n = 10, t = 17, P = 200; tan 4> = 170 -=- 200 = .85, = 40° 22', sin - .6477. Hence most economical resistance 200 2 10 X 74600 or .1279 ohm in its total length. The most economical current, C C*R .6477 1.6477 2 ' .01279 ohm per mile, _ 74600 200 614.58 2 X .1271 1.6477 = 614.58 amperes, and W, = 64.75 horse-power. The following tables show the power wasted as heat in the conductor. Horse-power Wasted in Transmitting Power Electrically to a Given Distance, the Entering Power being Fixed. Pressure at Entrance, 200 Volts. Current Density, 380 Amperes per Square Inch. Horse-power Wasted, the Horse-power sent in.* Power is Transmitted being one Mile (there being a Return Conductor). Horse-power Wasted. Distance Five Miles. 10 1.663 8.318 20 3.327 16 636 40 6.654 33.27 50 8.318 41.59 80 13.308 66.54 100 16.636 83.18 200 33.272 166.36 * That is, horse-power at the generator terminals. ELECTRIC TRANSMISSION. 1041 Pressure at Entrance, 2000 Volts. Horse- power sent in. Horse-power Wasted. Distance One Mile (there being a Return Conductor). Horse- power Wasted. Dis- tance Five Miles. Horse- power Wasted. Distance Ten Miles. Horse-power Wasted. Distance Twenty Miles. 100 200 400 500 800 1000 2000 1.663 3.327 6.654 8.318 13.308 16.636 33.272 8.318 16.636 33.272 41.59 66.54 83.18 166.36 16.636 33.272 66.54 83.18 133.08 166.36 332.72 33.27 66.54 133.08 166.36 266.17 332.72 665.44 It will be seen from these numbers that when the current density is fixed the power wasted is proportional to the entering horse-power and the length of the conductor, and is inversely proportional to the potential. For a copper conductor the rule may be simply stated as W = 16.6! o# XI, E being the horse-power and P the pressure at entrance, and I the length of the conductor in miles. Horse-power Wasted in Electric Transmission to a Given Distance, the Power to be Delivered at the Distant End being Fixed. Pres- sure at Entrance, 200 Volts. Current and Resistance Calculated by Ayrton and Perry's Rules. Horse-power Wasted, the Distance to which Horse-power Horse-power Horse-power the Power is Transmitted Wasted. Wasted. Delivered, being One Mile (there Distance Five Distance Ten being a Return Miles. Miles. Conductor). 10 1.676 6.476 8.620 20 3.352 12.952 17.24 40 6.704 25.904 34.48 50 8.38 32.38 43.10 80 13.408 51.808 68.96 100 16.76 64.86 86.20 200 33.52 129.52 172.4 Pressure at Entrance, 2000 Volts. Horse-power Delivered. Horse-power Horse-power Horse-power Wasted. Distance Wasted. Distance Wasted. Distance One Mile. Five Miles. Ten Miles. 100 1.716 8.484 16.763 200 3.432 16.968 33.526 400 6.864 33.938 67.052 500 8.58 42.42 83.815 800 13.728 67.87 134.104 1000 17.16 84.84 167 63 2000 34.32 169.68 335.26 If H = horse-power sent in, w = power delivered in watts, C = current in amperes, r = resistance in ohms per mile, P = pressure at entrance in volts, and n = number of miles of conductor, (w + CV) -f- 746 = H; iv = 746H - ' 4(jHsin4> Hsin n + sin ' (4> = angle whose tangent = nt-i- P, and the value of t corresponding to a current density of 380 amperes per sq. in. is 16.636.) TABLE OF EL.ECTRICAL. HORSE-POWERS. Volts X Amperes 746 = H.P., or 1 volt-ampere = .0013405 H.P. Read amperes at top and volts at side, or vice versa. %i Volts or Amperes £c r> B u 1 00134 10 20 30 40 .0536 50 .0570 60 70 80 90 100 110 120 1 .0134 .0268 .0402 .0804 .0938 .1072 .1206 .1341 .1475 .1609 2 00268 .0268 .0536 .0804 .1072 .1341 .1609 .1877 .2145 .2413 .2681 .2949 .3217 3 (10402 .0402 .0804 .1206 .1609 .2011 .2413 .2815 .3217 .3619 .4022 .4424 .4826 4 00530 .0536 .1072 .1609 .21451 .2681 .3217 ,375:', .4290 .4826 .5362 .5898 .6434 6 00670 .0670 .1341 .2011 .2681 .3351 .4022 .4692 .5362 .6032 .6703 .7373 .8043 6 00804 .0804 ' .1609 .2413 .3217 .4022 .4826 .5630 .6434 .7239 .8043 .8847 .9652 7 ■ .0938 .1877 .2815 .3753 ,1602 .5630 .6568 .7507 .8445 .9384 1.032 1.126 8 oio;- .1072 .2145 .3217 1290 5362 .6434 .7507 .8579 .9652 1.072 1 180 1.287 9 oi-106 ,1206 .2413 .3619 .4826 .6033 .7239 .8445 .9652 1.086 1.206 1.327 1.448 10 01341 .1341 .2681 .4022 .5362 .6703 .8043 .9383 1.072 1.206 1.341 1.475 1.609 11 .01475 .1475 .2949 .4424 .5898 .7373 .8847 1.032 1.180 1.327 1.475 1.622 1.709 12 .1609 .3217 ■1826 .6434 .8043 .9652 1.126 1.287 1.448 1.609 1.769 1.930 13 .1743 .34S5 .5228 .6970 .8713 1.046 1.220 1.394 1.568 1.743 1.917 2.091 14 .01877 .1877 .3753 .5630 .7507 .9384 1.126 1.314 1.501 1.689 1.877 2.064 2.252 15 .02011 .2011 .4022 .6032 .8043 1.005 1.206 1.408 1.609 1.810 2.011 2.212 2.413 16 .02145 .2145 .4290 .6434 .8579 1.072 1.287 1.501 1.716 1.930 2.145 2.359 2.574 17 .2279 .4558 .6837 .9115 1.139 1.367 1.595 1.823 2.051 2.279 2.507 2-735 18 .02413 .2413 .4826 .7239 .9652 1.206 1.448 1.689 1.930 2.172 2.413 2.654 2.895 19 .02547 .2547 .5094 .7641 1.019 1.273 1.528 1.783 2.037 2.292 2.547 2.801 3.056 20 .02681 .2681 .5362 .8043 1.072 1.340 1.609 1.877 2.145 2.413 2.681 2.949 3.217 21 .02815 .2815 .5630 .8445 1.126 1.408 1 689 1.971 2.252 2.533 2.815 3.097 3.378 22 .02949 .2949 .5898 .8847 1.180 1.475 1.769 2.064 2.359 2.654 2.949 3.244 3.539 2.3 .0308.'! .3083 .6166 .9249 1.233 1.542 1.850 2.158 2.467 2.775 3.083 3.391 3.700 24 .03217 .3217 .6434 .9652 1.287 1.609 1.93,0 2.252 2.574 2.895 3.217 3 539 3.861 25 .03351 .3351 .6703 1.005 1.341 1.676 2.011 2.346 2.681 3.016 3.351 3.686 4.022 26 .03485 .3485 .6971 1.046 1.394 1.743 2.091 2.440 2.788 3.137 3.485 3.834 4.182 27 .03619 .:;0l! .7239 1.086 1.448 1.810 2.172 3.534 2.895 3.257 3.619 3.981 4.343 28 .0375; .3,753 .7507 1.126 1.501 1.877 2.252 2.6,27 3.003 3.378 3.753 4.129 4.504 29 .03887 .3887 .7775 1.166 1.555 1.944 2.332 2.721 3.110 3.499 3.887 4.276 4.665 30 .01022 .4022 .8043 1.206 1.609 2.011 2.413 2.815 3.217 3.619 4.022 4.424 4.826 31 .04156 .4156 .8311 1.247 1.662 2.078 2.493 2.909 3.324 3.740 4.156 4.571 4.987 32 .04290 .8579 1.287 1.716 2.145 2.574 3.003 3 432 3.861 4.290 4.719 5.148 33 ■ .4424 .8847 1.327 1.769 2.212 2.654 3.097 3.539 3.986 4.424 4.866 5.308 34 .0455b .4558 .911E 1.367 1.823 3.3,0 2.735 3.190 3.646 4.102 4.558 5.013 5.469 35 .04692 .4692 .9384 1.408 1.877 2.346 2.815 3.284 3.753 4.223 4.692 5.161 5.630 36 .0482t .4826 .9655 1.448 1.930 2.413 2.895 3.378 3.861 4.343 4.826 5.308 5.791 37 0!96l .4960 .'J92( 1.488 1.984 2.480 2.976 3.472 3.968 4.464 4.960 5.456 5.952 38 .05094 50!) 1.019 1.528 2.038 2.547 3.056 3.566 4.075 4.585 5.094 5.603 6.113 3S .0522b .522b 1.046 1.568 2.091 2.614 3.137 3.660 4.182 4.705 5.228 5.751 6.274 « .05365 .5365 1.072 1.609 2.145 2 681 3.217 3.753 4.290 4.826 5.362 5.898 6.434 41 .05491 .5496 1.099 1.649 2.198 2.748 3.298 3.847 4.397 4.946 5.496 6.046 6.595 45 .0503< .5631 1.126 2.252 2.815 3.378 3.941 4.504 5.067 5.630 6.193 6.756 ±: .0576 J .5761 1.153 1.729 2.300 2.882 3.458 4.035 4.611 5.187 5.764 6.341 6.917 41 .0589. ,589b 1.180 1.769 2.359 2.949 3.539 4.129 4.719 5.308 5.898 6.488 7.078 n .0603. .603; 1.206 1.810 2.413 3.016 3.619 4.223 4.826 5.439 6.032 6.635 7.239 u .0616 .616C 1.233 1.850 2.467 3.083 3.700 4.316 4.933 5.550 6.166 6.783 7.400 4 r .0630 .630( 1.260 1.890 2.520 3.150 3.780 4.410 5.040 5.070 6.300 6.930 7.560 a .0643 .013- 1.287 1.930 2.574 3.217 3.861 4.504 5.148 5.791 6.434 7.078 7.721 4 .0656 .656 1.314 1.970 2.627 3.284 3.941 4.598 5.255 5.912 6.568 7.225 7.882 5( .0670 .670 1.341 2.011 2.681 3.351 4.022 4.692 5.362 6.032 6.703 7.373 8.043 TABLE OF ELECTRICAL HORSE-POWERS. 1043 TABLE: OF ELECTRICAL HORSE-POAVERS- (Continued.) So Volts o • Amperes. < s 1 10 20 J 30 40 50 1 60 70 80 90 100 110 120 55 .0737?! .7373 1.475 2.212 2.949 3.686 4.424 5.161 5.89S 6.635 7.373 8.110 8.847 60 .0804J, -8043 Ml\ i } .8713 1.6(1! 2.413 3.217 4.022 4.826 5.630 6.434 7.289 8.043 8.847 9.652 65 1.743 2.614 3.485 4.357 5.228 6.099 6.970 7.842 8.713 9.584 10.46 70 .0938* .9384 1.87; 2.815 3.753 4.61C 5.630 6.568 7.507 8.445 9.884 10.32 11.26 75 .1005* 1.005 2.011 3.016 4.021 5.027 6.032 7.037 8.043 9.048 10.05 11.06 12.06 80 .10724 1.072 2.145 3.217 4.290 5.362 6.434 7.507 8.579 9.652 10.72 11.80 12.87 85 .11391 1.139 •.'.270 3.418 4.558 5.6971 6.836 7.976 9.115 10.26 11.39 12.53 13.67 90 .12065, 1.206 2.413 3.619 4.826 6.032 7.239 8.445 9.652 10.86 12.06 13.27 14.48 95 .12735 1.273 2.547 3.820 5.001 6.367 1 7.641 8.914 10.18 11.46 12.73 14.01 15.28 100 .13405 1.341 2.681 4.022 5.362 6.703J 8.043 9.384 10.72 12.06 13.41 14.75 16.09 200 .26810 2.681 5.362 8.043 10.72 13.41 16.09 18.77 21.45 24.13 26.81 29.49 32.17 300 .402151 4.022 8.013 12.06 16.09 20.11 24.13 28.15 32.17 36.19 40.22 44.24 48.26 400 .53620: 5.362 10.72 16.09 21.45 26.81 32.17 37.53 42.90 48.26 53.62 58.98 64.34 500 .67025! 6.703 13.41 20.11 26.81 33.51 40.22 46.92 53.62 60.32 67.03. 73.73 80.43 600 .80430 8.043 16.09 24.13 32.17 40.22 48.26 56.30 64.34 72.39 80.43 88.47 96.52 700 .93835 9.384 18.77 28.15 37.53 46.92 56.30 65.68 75.07 84.45 93.84 103.2 112.6 800 1.0724 10.72 21.45 32.17 42.90 53.62 64.34 75.07 85.79 96.52 107.2 118.0 128.7 900 1.2065 12.06 24.13 36.19 48.26 60.32 72.39 84.45 96.52 108.6 120.6 132.7 144.8 1 ,001) 1.3405 13.41 26.81 40 22 53.62 67.03 80.43 9.3.84 107.2 120.6 134.1 147.5 160.9 2,000 2.6810 26.81 53.62 80.43 107.2 134.1 160.9 187.7 214.5 241.3 268.1 294.9 321.7 3,000 4.0215 40.22 80.43 120.6 160.9 201.1 241.3 281.5 321.7 361.9 402.2 442.4 482.6 4,000 5.3620 53.62 107.2 160.9 214.5 268.1 321.7 375.3 429.0 482 6 536.2 589.8 643.4 5, 00(1 6.7025 67.03 134.1 201.1 268.1 335.1 1402.2 469.2 536.2 608.2 670.3 737.3 804.3 6.000 8.0130 80.43 160.9 241.3 321.7 402.2 1482.6 563.(1 643.4 723.9 804.3 8S4.7 965.2 7,000 9.3835 93.84 187.7 ,381.5 375.3 469.2 563.0 656.8 750.7 S44!5 938.4 1032 1126 8,000 10.724 107.2 214.5 321.7 429.0 536.2 1643.4 750.7 857.9 965.2 1072 1180 1287 9.000 12.06:. 120.6 241.3 361.9 482. 6 603.2 1723.9 844.5 965.2 1086 1206 1327 1448 10,000 13.405 134.1 268.1 402.2 536.2 670.3 804.3 938.3 1072 1206 1341 1475 1009 "Wire Table.— The wire table on the following page (from a circular of the Westinghouse El. & Mfg. Co.) shows at a glance the size of wire neces- sary for the transmission of any given current over a known distance with a given amount of drop, for 100-volt and 500-volt circuits, with varying losses. The formula by which this table has been calculated is D x 1000_ CX2L~ B, in which D equals the volts drop in electro-motive force, Cthe current, L the distance from the dynamo to the point of distribution, and R the line resist- ance in ohms per thousand feet. Example 1.— Required the size of wire necessary to carry a current of GO amperes a distance of 050 feet with a loss of b% at 100 volts. Referring to the table, under 60 amperes, we find the given distance, 050 feet. In the same horizontal line and under 5% drop at 100 volts, we find No. 000 wire, which is the size required. Example 2.— What size will be required for 10 amperes 2000 feet, with a drop of 10% at 500 volts. Under 10 amperes find 1930— the nearest figure to 2000— and in the same horizontal line under 10$ at 500 volts find No. 11, the size required. Wiring Formulae for Incandescent Lighting. (W. D. Weaver, Elec. World, Oct. 15, 1892.) — A formula for calculating wiring tables is . 2150LC A — - aE* aE where A = section in circular mils; W = watt rating of lamps; E— volt- age; L — distance to centre of distribution, in feet; N — number of lamps; a — percentage of drop; C = current in amperes. Example.— Volts, 50; amperes, 100; feet to centre of distribution, 100; drop, 2%. 2150 X 100 x 100 01Rn __ . , — 215,000 circular mils, or about 0000 B. 2 X 50 X S. gauge. 1044 ELECTRICAL ENGINEERING. f- R o o > § m | cd O e fc IO KS •* 50 t- t-HMOOO ■*oio5!ov £- to -* co eo s 5 o y. 3 ® q O iisia * gs ^ s ^TdSiOWt-OO^MMeQNWHH nisi i- ss - lO CO 0-5 C- -H .-I CO P 5 3i o Islss sssss : ^OCat-tOlOMMHH CO ** 1- i— t^ MOQCOIO 2 as jr. o ■* os i0 5)cat>!0 OOgOHNK^eCOOO.ONMJOOOO :!.■»::; . o 50 50 ;/: o 8° _ " ■ ." ;■-■-- 0§ = Stm oioiNot. S-*eo O O O O O O — 50 -H -H ^CC5)C'-0 to O CO CO O ..- .o — co >_- i - — o /; = — : ' " : ' -• i - — -f :o :" o o co o co co co co co v, cc — -r ( - ~. i - or. ^ co O t~- *-l £- 50 i-H-+£^Oi-l CCOC^m £-50 1-50 00 K ■- ;5 ^ CO OJOJtH^H »«l--#^ CO t~ 150 ■* CO OJIMrtrtrH — — 50 C JO —50 50 r^ i JO IX^Oi^ 7-- CO 0> CO O HiflO 1C 50 iC 50 50 CO JO to -H .O co" 'to -J-OVt) t-h CO CO JO CO CO 50 -+ Ct OOOOO OOK5NOO CO CC >* O — ! HOIOSO «GCO-i ;= 5o^5» ic oj os t- to txnno ■O IC At< 50 C! •H t-KOOOtO IC -* CO OJ OJ iMrHr-l §§ SSpSS gSclcls §2S^^" SScok o ©~b © o o" ©. co ioT- ©©1* -* =2 2£ £ - £ S 2 3 c? lilll !ssl s » s ^^ ssss^j^^- lllfihtlfts iisss llSIii SSSS5 §gS^« tO©tO03© COtOlO-^CO CN 04 i-l rH i-H lO» HHriH sssss s§ ^^WCO^iO ©t-00O>© 32333 tci-»oo Igggo. «.«-*o. S232S £S22§ CO ||S° toe- coos© HWCO^iO (DNCOOSO l^-p-gs^ -*lO«OC-00 fflOHWM -*»fl«OI>CO ° i< ^ U g ggOrHO, pQ go ooasOi-for cc^wot* °°2m ELECTRIC TRANSMISSION. 1045 The horse-power and efficiency of a motor being given, the size of the con- ducting wire in circular mils can be found from the following formula: _ 160,400,000 X H.P. X L ~ a£ 2 X efficiency Example.— Horse-power, 10; volts, 500; drop, %\ feed to distributing point, 600: efficiency of motor, 75$. 160,400,000X10X600 •„. . . .. . . __ D _ ■ _ A = ■ ' ' — enn ^ = 17,109 circular mils, or about No. 8 B. & S. 3 X 500 X 500 X 75 Cost of Copper for Long-distance Transmission. (Westinghouse El. & Mfg. Co.) Cost of Copper required for the Delivery of One Mechanical Horse- power at Motor Shaft with 1000, 2000, 3000, 4000, 5000, and 10,000 Volts at Motor Terminals, or at Terminals of Lowering Transformers. Loss of energy in conductors (drop), equals 20$. Distances equal one to twenty miles. Motor efficiency equals 90$. Length of conductor per mile of single distance, 11,000 feet, to allow for sag. Cost of copper equals 16 cents per pound. Miles. 1000 v. 2000 v. 3000 v. 4000 v. 5000 v. 10,000 v. 1 $2.08 $0.52 $0.23 $0.13 $0.08 $0.02 3 8.33 2.08 0.93 0.52 0.33 0.08 3 18.70 4.68 2.08 1.17 0.75 0.19 4 33.30 8.32 3.70 2.08 1.33 0.33 5 52.05 13.00 5.78 3.25 2.08 0.52 6 74.90 18.70 8.32 4.68 3.00 0.75 7 102.00 25.50 11.30 6.37 4.08 1.02 8 133.25 33.30 14.80 8.32 5.33 1.33 9 168.60 42.20 18.70 10.50 6.74 1.69 10 208.19 52.05 23.14 13.01 8.33 2.08 11 251.90 63.00 28 00 15.75 10.08 2.52 12 299.80 75.00 33.30 18.70 12.00 3.00 13 352.00 88.00 39.00 22.00 14.08 3.52 * 14 408 00 102.00 45.30 25.50 16.32 4.08 15 468.00 117.00 52.00 29.25 ' 18.72 4.68 16 533.00 133.00 59.00 33.30 21.32 5.33 17 600.00 150.00 67.00 37.60 24.00 6.00 18 675.00 169.00 75.00 42.20 27.00 6.75 19 750.00 188.00 83.50 47.00 30.00 7.50 20 833.00 208.00 92.60 52.00 33.32 8.33 A Graphical Method of calculating leads for wiring for electric lighting is described by Carl Hering in Trans. A. I. E. E., 1891. He furnishes a chart containing three sets of diagonal straight -line diagrams so con- nected that the examples under the general formula for wiring may be solved without calculation by simply locating three points in succession on the chart. The general principle upon which the chart is based is that for any formula containing three variable quantities, one of which is the product or the quotient of the other two, the "curves' 1 '' representing their relative values may always be represented by a series of straight diagonal lines drawn through the centre or zero-point. Such a set of lines will therefore enable one to make any calculations graphically for that formula. For instance, horse-power = volts X amperes; the constant 746 does not con- cern us at present. A series of diagonal lines properly spaced will there- fore give directly either the horse-power, the volts, or the amperes, when the other two are given. One scale is vertical, the other horizontal, and the diagonal lines (or the hyperbolas) each represent one unit (or a number of units) of the third scale. To make the "curves" straight lines the diagonals must be made 1046 ELECTRICAL ENGINEERING. Cost of Copper required to deliver One Mechanical Horse-power at Motor-shaft with Varying Percentages of Loss in Conductors, upon the Assumption that the Potential at Motor Terminals is in Each Case 3000 Volts. Distances equal one to twenty miles. Motor efficiency equals 90$. Length of conductor per mile of single distance, 11,000 feet, to allow for sag. Cost of copper equals 16 cents per pound. Miles. 10$ 15$ 20$ 25$ 30$ 1 $0.52 $0.33 $0.23 $0.17 $0.13 2 2.08 1.31 0.93 0.69 0.54 3 4.68 2.95 2.08 1.55 1.21 4 8.32 5.25 3.70 2.77 2.15 5 13.00 8.20 5.78 4.33 3.37 6 18.70 11.75 8.32 6.23 4.85 7 25.50 16.00 11.30 8.45 6.60 8 33.30 21.00 14.80 11.00 8.60 9 42.20 26.60 18.75 14.00 10.90 10 52.05 32.78 23.14 17.31 13.50 11 63.00 39.75 28.00 21.00 16.30 12 75.00 47.20 33.30 24.90 19.40 13 88.00 55.30 39.00 29.20 22.80 14 102.00 64.20 45.30 33.90 26.40 15 117.00 73.75 52.00 38.90 30.30 16 133.00 83.80 59.00 44.30 34.50 17 150.00 94.75 67.00 50.00 39.00 18 169.00 106.00 75.00 56.20 43.80 19 188.00 118.00 83.50 62.50 48.70 20 208.00 131.00 92.60 69.25 54.00 to represent one of the two quantities which is equal to the quotient of the other two, and not the one which is equal to the product of the other two, because the curves would then be hyperbolas. In the example given the diagonals must represent volts or amperes, but not horse-powers. The con- stants in such formula? affect only the positions of the diagonals; although they increase considerably the work of arithmetically calculating the results, they do not affect in the least the graphical calculations after the diagrams are once drawn. The general formula for wiring is : Cross-section = current for one lamp x No. of lamps x distance X constant loss in volts containing six quantities only, one of which is always constant, being equal to twice the mil-foot resistance of copper, if the cross-section is in circular mils. Calculations involving three of these five quantities may readily be made graphically by means of a single set of diagonal lines. In Mr. Hering's method the formula is split up into three smaller ones, each of which contains no more than three variable quantities. Each formula can then be calculated separately by a simple diagram, as de- scribed, thus permitting the whole formula to be calculated graphically. To do this, let the first diagram perform the calculation, _ current for one lamp loss in volts ' in which a; is a mere auxiliary quantity, perform the next calculation, y = x X number of lamps; and a third diagram the final calculation, cross-section = y X distance. Let a second similar diagram ELECTRIC TRANSMISSION. 1047 The constant may be combined with any one of these, it is immaterial which one. This triple calculation may at first seem to complicate matters on account of the new quantities, x and y. These, however, are easily elimiuated by the simple device of placing the three diagrams together, side by side, in such a position that the two x scales coincide, and similarly the two y scales. By doing this one has merely to pass directly from one set of diagonals to the next to perform the successive steps of the calculation, without being concerned about the intermediate auxiliary quantities. These intermediate quantities correspond, and are equal to the successive products or quotients which are obtained in the successive arithmetical multiplications and divisions of these five quantities in the formula, which cannot, of course, be eliminated in making the calculations arithmetically. Weight of Copper required for Long- distance Trans- mission.— W. F. C. Hasson (Trans. Tech. Socy. of the Pacific Coast, vol. x. No. 4) gives the following formula: D 2 W=~H..F. (100 - £), n 266.5, where IF is the weight of copper wire in pounds; D, the distance in miles; E. the E.M.F. at the motor in hundreds of volts; H.P., the horse-power delivered to the motor; L, the per cent of line loss. Thus, to transmit 200 horse-power ten miles with 10 per cent loss, and have 3000 volts at the motor, we have Vtt . 10 X 10 M (100 - W = 80*80 X2 °° X So" 10) X 266.5 = 53,300 lbs. Efficiency of Long-distance Transmission. (F. R. Hart, Potver, Feb. 1892.)— The mechanical efficiency of a system is the ratio of the power delivered to the dynamo-electric machines at one end of the line to the power delivered by the electric motors at the distant end. The com- mercial efficiency of a dynamo or motor varies with its load. The .maximum efficiency of good machines should not be under 90$ and is seldom above 92$. Under the most favorable conditions, then, we must expect a loss of say 9$ in the dynamo and 9$ in the motor. The loss in transmission, due to fall in electrical pressure or " drop " in the line, is governed by the size of the wires, the other conditions remaining the same. For a long-distance transmission plant this will vary from 5$ upwards. With a loss of 5$ in the line, the total efficiency of transmission will be slightly under 79$. With a loss of 10$ in the line, the efficiency would be slightly 'under 75$. We may call 80$ the practical limit of the efficiency with the apparatus of to-day. The methods for long-distance power transmission by electricity may be divided into three general classes: (1) Those using continuous current; (2) those using alternating current; and (3) regenerating or " motor-dynamo" systems. The subdivisions of each of these general classes are tabulated as follows: Continuous current Alternating current Regenerating systems One machine. Machines in parallel. One machine. Machines in parallel. Machines in series. 2 machines in series. Machines in multiple series. Machines in series. Without conversions. With conversions. Without conversions. With conversions. f Alternating continuous. I Alternating converter; line converter; alternating con- ■{ tinuous. I Continuous-continuous. I. Partial reconversion of any system. The relative advantages of these systems vary with each particular trans- mission problem, but in a general way may be tabulated as below. f Low voltage | High [ voltage [ Multiple-wire J Alternating single phase [ Alternating multiphase 1048 ELECTRICAL ENGINEERING. System. Advantages. Disadvantages. ( Low voltage. Safety, simplicity. Expense for copper. ( High voltage. Economy, simplicity. Danger, difficulty of building machines. p p o 3-wire. Low voltage on machines and saving in copper. Not saving enough in U Multiple-wire. Low voltage at machines and saving in copper. tances. Necessity for " balanced " system. Single phase. Economy of copper. Cannot start under load. Low efficiency. _p p Multiphase. Economy of copper, syn- chronous speed unnec- essary; applicable to very long distances. Complexity. Lower ef- ficiency of terminal apparatus. Not as yet "standard.' 1 < Motor-dynamo. High-voltage transmis- sion. Low-voltage de- livery. Expensive. Low efficiency. There are many factors which govern the selection of a system. For each problem considered there will be found certain fixed and certain unfixed conditions. In general the fixed factors are: (1) capacity of source of power; (2) cost of power at source; (3) cost of power by other means at point of delivery; (4) danger considerations at motors; (5) operation conditions; (6) construction conditions (length of line, character of country, etc.). The partly fixed conditions are: (7) power which must be delivered, i.e., the effi- ciency of the system; (8) size and number of delivery units. The variable conditions are: (9) initial voltage; (10) pounds of copper on line; (11) origi- nal cost of all apparatus and construction: (12) expenses, operating (fixed charges, interest, depreciation, taxes, insurance, etc.); (13) liability of trouble and stoppages; (14) danger at station and on line; (15) convenience in oper- ating, making changes, extensions, etc. Assuming that the cost of dyna- mos, motors, etc., will be approximately the same whatever the initial pressure, the great variation in the cost of wire at different pressures is shown by Mr. Hart in the following figures, giving the weights of copper required for transmitting 100 horse-power 5 miles : "Voltage. Drop 10 per cent. Drop 20 per cent. 2,000 16,800 lbs. 8,400 lbs. 3,000 7,400 " 3,700 " 10,000 620 " 310 " Efficiency of a. Combined Engine and Dynamo. — A com- pound double - crank Willans engine mounted on a single base with a dynamo of the Edison-Hopkinson type was tested in 1890, with results as follows : The low-pressure cylinder is 14 in. diam., 13 in. stroke; steam- pressure 120 lbs. It is coupled to a dynamo constructed for an output of 475 amperes at 110 volts when driven at 430 revolutions per minute. The arma- ture is of the bar construction, is plain shunt-wound, and is fitted with a commutator of hard-drawn copper with mica insulation. Four brushes are carried on each rocker-arm. Resistance of magnets 16. ohms Resistance of armature ... 0.0055 " I.H.P 83.3 E.H.P 72.2 Total efficiency 86.7 per cent Consumption of water per I.H.P. hour 21.6 pounds Consumption of water per E.H.P. hour 25 " The engine and dynamo were worked above their full normal output, which fact would tend to slightly increase the efficiency. The electrical losses were : Loss in magnet coils, 756 watts, equal to 1.4$; loss in armature coil, 1386 watts, equal to 2.6$; so that the electrical efficiency ELECTRIC TRANSMISSION. 1049 of the machine due to ohmic resistance alone was 96$. The remainder of the losses, a little over 8 horse-power, is clue to friction of engine and dynamo, hysteresis, and the like. Electrical Efficiency of a Generator and Motor.— A twelve- mile transmission of power at Bodie, Cal., is described by T. H. Leggett (Trans. A. I. M E. 1894). A single-phase alternating current is used. The generator is a Westinghouse 120 K. W. constant-potential 12-pole machine, speed 860 to 870 revs, per min. The motor is a synchronous constant-po- tential machine of 120 horse-power. It is brought up to speed by a 10-H.P. Tesla starting motor. Tests of the electrical efficiency of the generator and motor gave the following results : Test on Generator. Amperes Volts. Watts. 15.8 18.2 60 78 948 1419 6 Resistance of armature, 1.6618 ohms. 664.72 3032.32 Load 20 3414 68280 Apparent electrical efficiency of generator, 95.559$. Test on Motor. Amperes Volts. Watts. Self-excited field 52 62.4 3244.8 Resistance of armature, 1.4 ohms. 560 3804.08 20 3110 62200 Apparent electrical efficiency of motor, 93.883$. Efficiency of an Electrical Pumping-plant. (Eng. & M. Jour., Feb. 7, 1891.)— A pumping-plant at a mine at Normanton, England, was tested, with results given below: Above ground there is a pair of 20^ x 48-in. engines running at 20 revs, per min., driving two series dynamos giving 690 volts and 59 amperes. The cur- rent from each dynamo is carried into the mine by an insulated cable about 3000 feet long. There they are connected to two 50-h.p. motors which oper- ate a pair of differential ram-pumps, with rams 6 in. and 4^ in. diam. and 24 in. stroke. The total head against which the pumps operate is 890 feet. Connected to the same dynamos there is also a set of gearing for driving a hauling plant on a continuous-rope system, and a set of three-throw ram- pumps with 6-inch rams and 12-inch stroke can also be thrown into gear. The connections are so made that either motor can operate any or all three of the sets of machinery just described. Indicator-diagrams gave the fol- lowing results: Friction of engine 6.9 H.P. 9.4$ Belt and dynamo friction 4.8 " 6.5$ Leads and motor 6.7 " 9.4$ Motor belt, gearing and pumps empty 10.2 " 14.0$ Load of 117 gallons through 890 feet 31.5 " 43.1$ Water friction in pumps and rising main 12.9 " 17.6$ 73.0 H.P. 100.0$ At the time when these data were obtained the total efficiency of the plant was 43. 1$, but in a later test it rose to 47$. References on Power Distribution.— Kapp, Electric Transmis- sion of Energy; Badt, Electric Transmission Handbook; Martin and Wetzler, The Electric Motor and its Applications; Hospitalier, Poly phased Electric Currents, 1050 ELECTRICAL ENGINEERING. EL.ECTRIC RAILWAYS. Space will not admit of a proper treatment of this subject in this work. Consult Crosby and Bell, The Electric Railway in Theory and Practice, price $250; FairchiM, Street Railways, price $4.00; Merrill, Reference Book of Tables and Formulae for Street Railway Engineers, price $1.00. Test of a Street Railway Plant.— A test of a small electric-rail- way plant is reported by Jesse M. Smith in Trans. A. S. M. E., vol. xv. The following are some of the results obtained: Friction of engine, air-pump, and boiler feed-pump; main belt off 9.22 I.H.P. Friction of engine, air and feed pumps, and dynamo, brushes off. 11.34 I.H.P. Friction of dynamo and belt 2.12 I.H.P. Power consumed by engine, air and feed pumps and dynamo, with brushes on and main circuit open 14.34 I.H.P. Power required to charge fields of dynamo 3.00 I.H.P. Rated capacity of engine and dynamo each 150 I.H.P. Power developed by engine min. 21.27; max. 141.4; mean, 70.1 I.H.P. Volts developed by dynamo — range, 480 to 520; average, 501 volts Amperes developed by dynamo max. 200; min. 4.7; average, 67 amperes Average watts delivered by dynamo 33,567 Watts Average electrical horse-power delivered by dynamo 45 E.H.P. Average I.H.P. deFd to pulley of dynamo, estimating friction of armature shaft to be the same as friction of belt 59.8 I.H.P. Average commercial efficiency of dynamo 45 -*- 59.8 = 75.25$ Average number of cars in use during test 2.89 cars. Number of single trips of cars 64 Average number of passengers on cars per single trip 15.2 Weight of cars 14,500 lbs. Est. total weight of cars and persons 15,900 lbs. Average weight in motion 45,950 lbs. Average electrical horse -power per 1000 lbs. of weight moved. . 0.98 E.H.P. Average horse-power developed by engine per 1000 lbs. of weight moved 1.52 I.H.P. Average watts required per car 11,615 watts Average electrical horse-power per car 15.54 E.H.P. Average horse-power developed in engine per car 24.25 I.H.P. Length of road 10.5 miles. Average speed, including all stops, 21 miles in 1 .5 hours = 14 miles per hour. Average speed between stops, 21 m. in 1.366 hours = 15.38 miles per hour. Proportioning Boiler,Ems;iiie, and Generator for Power- stations. — Win. Lee Church (Street Railway Journal, 1892) gives a diagram showing the abrupt variations in the current required for an electric railway with variable grades. For this case, in which the maximum current for a minute or two at a time is 175 amperes, ranging from that to zero, and averaging about 50 amperes, he advises that the nominal capacity of the generator be 100 amperes. The reason of this is found in the fact that an electric generator can stand an overload, or even an excessive over- load, provided it does not have to stand it long. The question is simply one of heat. The overload here was seen to continue for only about one minute, during which time the generator could carry it with ease with no perceptible rise of temperature to injure the insulation. Had this load been continuous for an hour or so, as would occur in an electric-lighting station, a much higher relative generating capacity would be required, approximating the maximum load. An engine has no such capacity for excessive overload as a generator. In other words, the element of time does not enter into the engine problem, but it becomes a question of how much the engine can actually lift by main strength without taking the governor to an extreme which shall slow down the speed. In general terms, the engine should not be called to perform, even for a short time, more than 20$, or possibly 25$, above its rating. The engine capacity, therefore, would have a nominal rating greater than that of the generator, say about 25$ greater. The capacity of the engine should be determined without reference to condensation. This is for the obvious reason that a condenser may become choked, or disabled, or leaky, and the vacuum may be poor, or lost entirely under sudden fluctuations. The boiler has to deal only with the average of the total load. In this particular electric railways exactly resemble rolling-mills, ^aw - mills, ELECTRIC LIGHTING. 1051 and kindred industries, where the load is spasmodic, with variations lasting but a few seconds, or at most but a few minutes. The stored heat in the water of a boiler is enormous in quantity, and responds instantly to a release of pressure. That is to say, the boiler is an immense reservoir of power, and provided the drain upon it is not continued too Jong, it will stand exactions far beyond its nomiual capacity, and without any effect whatever upon the firing. The actual size of the boiler will depend upon the type of engine. With the compound engine described by Mr. Church, running non-condensing, an allowance of 30 pounds of water actually evaporated per I. H. P. per hour will give a margin for all contingencies. The engine duty under an average uni- form load is a very different tiling from the duty under a variable load rep- resented by the average. Under the uniform load, 23 pounds of water would be the actual engine performance, and the boiler could be propor- tioned with reference to this figure. Under the violent fluctuations of rail- way service, the average duty of the engine will rise to about 28 pounds, and if the maximum average load is taken, and the boiler proportioned for 30 pounds, there will be a sufficient margin. Other compound engines not possessing the feature which secures uniformity of duty will range up to at least 45 pounds under light loads, and often to 60 pounds, and represent an average duty not better than 35 to 40 pounds. The same is true of every form of non-compounded engine, whether high speed or low speed, both of which show a tremendous falling back of fuel duty under variable load. ELECTRIC LIGHTING. Quantity of Energy required to produce Light.— Accord- ing to Mr. Preece, the quantity of energy, measured in watts, required to pro- duce light equivalent to one candle-power, measured by the light given out by the standard candle, is as follows for different light-giving substances: Tallow 124 watts Coal gas 68 watts. Wax 94 " Cannel gas 48 " Spermaceti 86 " Incandescent lamp.. 15 " Mineral oils 80 " Arc lamp 3 " Vegetable oils 57 " And the relative costs of production are about 1 for the arc lamp; 6 for the incandescent lamp ; 5 for the mineral-oil lamp ; 10 for the gas-light; 67 for the spermaceti candle. Life of Incandescent Lamps. (Eng^g, Sept. 1, 1893, p. 282.)— From experiments made by Messrs. Siemens and Halske, Berlin, it appears that the average life of incandescent lamps at different expenditure of watts per candle-power is as follows: Watts per candle-power 1.5 2 2.5 3 3.5 Life of lamp, hours 45 200 450 1000 1000 Life and Efficiency Tests of Lamps. (P. G. Gossler, Elec. World, Sept. 17, 1892.)— Lamps burning at a voltage above that for which they are rated give a much greater illuminating power than 16 candles, but at the same time their life is very considerably shortened. It has been ob- served that lamps received from the factory do not average the same candle- power and efficiency for different invoices; that is, lamps which are received in one invoice are usually quite uniform throughout that lot, but they vary considerably from lamps made at other times. The following figures show the different illuminating-powers of a 16.c.p., 50-volt, 52-watt lamp, for various voltages from 25 to 80 volts: Volts: 25 34.8 40 48 50 52.5 55.6 59.5 62 68.2 72.5 80 Amperes * .561 .774 '.898 .968 1.055 1.097 1.161 1.226 1.29 1.419 1.484 1.58 Candles: .4 2.47 5.1 12.6 15.8 20.5 28.4 39.3 50.7 74.5 103.2 141 Watts - 14.03 26.94 35.92 46.34 52.75 57.57 64.55 72.92 79.98 96.78 107.5 126.4 Watts per c.p.: 35.1 10.81 7.04 3.68 3.34 2.81 2.30 1.96 1.58 1.30 1.04 .90 Street-lighting. (H. Robinson, M I.C.E., Eng'g News, Sept. 12, 1891.) —For street-lighting the arc-lamp is the most economical. The smallest 1052 ELECTRICAL EHGrlHEERINGr. size of arc-lamp at present manufactured requires a current of about 5 amperes; but for steadiness and efficiency it is desirable to use not less than 6 amperes. The candle-power of arc-lamps varies considerably, according to the angle at which it is measured. The greatest intensity with continuous- current lamps is found at an angle of about 40° below the horizontal line. The following table gives the approximate candle-power at various angles. The height of the lamps should be arranged so as to give an angle of not less than 7° to the most distant point it is intended to serve. Current in Amperes. 6 8 10 Lighting-power of Arc-lamps. Candle-power. ( At Angle At Angle At Angle Maximum at of 10°. of 20°. Angle of 40°. 207 322 460 350 546 780 495 770 1100 92 156 of 7 175 300 420 The following data enable the coefficient of minimum lighting-power in streets to be determined: Let P = candle-power of lamps; L = maximum distance from lamp in feet ; H = height of lamp in feet ; X = a coefficient. The light falling on the unit area of pavement varies inversely as the square of the distance from the lamp, and is directly proportional to the angle at which it falls. This angle is nearly proportional to the height of the lamp divided by the distance. Therefore A - i» X L or X PH '' LP' The usual standard of gas-lighting is represented by the amount of light falling on the unit area of pavement 50 feet away from a 12-c.p. gas-lamp 9 feet high, which gives a coefficient as follows: 12X9 50 3 ~ 0.000864. The minimum standard represents the amount of light on a unit area 50 feet away from a 24-c.p. lamp, 9 ft. high, and gives the coefficient .001728. Adopting the first of the above coefficients. Mr. Robinson calculates that the before-mentioned sizes of arc-lights will give the same standard of light at the heights and distances stated in Table A. Table B gives the corresponding distances, assuming the minimum standard to be adopted. Table A. Table B. Hgt. of Lamps. 30 ft.|s5 ft.|30 ft.|35 ft. Height 20 ft. | 25 ft. 1 30 ft. 1 35 ft. Current in Ampei-es. Max. distances served from lamp, in ft. Amperes. Max. distances served from Lamp. 6 8 10 160 185 205 175 202 225 190 220 243 202 235 260 6 8 10 130 150 170 144 165 190 155 180 205 166 193 220 The distances the lamps are apart would, of course, be double the dis- tances mentioned in Tables A and B. One arc-lamp will take the place of from 3 to 6 gas-lamps, according to the locality, arrangement, and standard of light adopted. A scheme of arc-lighting, based on the substitution of one arc-light on the average for 3L£ to 4 gas-lamps, would double the minimum standard of light, while the average standard would be increased 10 or 12 times. Candle-power of the Arc-light. (Elihu Thomson, El. World, Feb. 28, 1891.)— With the long arc the maximum intensity of the light is from 40° to 60° downward from the horizontal. The spherical candle-power is only a fraction of the rated c.p., which is generally taken at the maximum obtainable in the best direction. For this reason the term 2000 c.p. has little ELECTRIC WELDING. 1053 significance as indicating the illuminating-power of an arc. It is now gener- ally taken to mean an arc with 10 amperes and not le^s than 45 volts between the carbons, or a 450- watt arc. The quality of the carbons will determine whether the 450 watts are expended in obtaining the most light or not, or whether that light will have a maximum intensity at one angle or another within certain limits. The larger the current passing in an arc, the less is its resistance. Well-developed arcs with 4 amperes will have about 11 ohms, with 10 amperes 4.5 ohms, and with 100 amperes .45 ohm. It is not unusual to run from 50 to 60 lights in a series, each demanding from 45 to 50 volts, or a total of, say, 3000 volts. In going beyond this the difficulties of insulation are greatly increased. Reference Books on Electric Lighting.— Noll, How to Wire Buildings, $1.00; Hedges, Continental Electric-light Central Stations, $6.00; Fleming, Alternating Current Transformers in Theory and Practice, 2 vols., $8.00; Atkinson, Elements of Electric Lighting, $1.50; Algave and Boulard, Electric Light: its History, Production, and Application, $5.00. ELECTRIC WELDING. The apparatus most generally used consists of an alternating-current dynamo, feeding a comparatively high-potential current to the primary coil of an induction-coil or transformer, the secondary of which is made so large in section and so short in length as to supply to the work currents not exceeding two or three volts, and of very large volume or rate of flow. The welding clamps are attached to the secondary terminals. Other forms of apparatus, such as dynamos constructed to yield alternating currents direct from the armature to the welding-clamps, are used to a limited extent. The conductivity for heat of the metal to be welded has a decided influ- ence on the heating, and in welding iron its comparatively low heat conduc- tion assists the work materiallj\ (See papers by Sir F. Bramwell, Proc. Inst. C. E., part iv., vol. cii. p. 1; and Elihu Thomson, Trans. A. I. M.E., xix. 877.) Fred. P. Royce, Iron Age, Nov. 28, 1892, gives the following figures show- ing the amount of power required to weld axles and tires: AXLE-WELDING. Seconds. 1-inch round axle requires 25 H.P. for 45 1-inch square axle requires 30 H.P. for 48 1^4-inch round axle requires 35 H.P. for 60 lJ4-inch square axle requires 40 H.P. for 70 2-inch round axle requires 75 H.P. for 95 2-inch square axle requires 90 H.P. for 100 The slightly increased time and power required for w r elding the square axle is not only due to the extra metal in it, but in part to the care which it is best to use to secure a perfect alignment. TIRE-WELDING. Seconds. 1 X 3/16-inch tire requires 11 H.P. for 15 1J4 X %-inch tire requires 23 H.P. for ....?». 25 Wz X %-inch tire requires 20 H.P. for 30 1^2 X J^-inch tire requires 23 H.P, for 40 2 X J^-inch tire requires 29 H.P. for 55 2 X %-inch tire requires 42 H.P. for 62 The time above given for welding is of course that required for the actual application of the current only, and does not include that consumed by placing the axles or tires in the machine, the removal of the upset and other finishing processes. From the data thus submitted, the cost of welding can be readily figured for any locality where the price of fuel and cost of labor are known. In almost all cases the cost of the fuel used under the boilers for produc- ing power for electric welding is practically the same as the cost of fuel used in forges for the same amount of work, taking into consideration the difference in price of fuel used in either case. Prof. A. B. W. Kennedy found that 2^-inch iron tubes J4 incn thick were welded in 61 seconds, the net horse-power required at this speed being 23.4 (say 33 indicated horse-power) per square inch of section. Brass tubing re- 1054 Electrical ekgineerihg. quired 21 . 2 net horse-power. About 60 total indicated horse-power would be required for the welding of angle-irons 3x3x^ inch in from two to three minutes. Copper requires about 80 horse-power per square inch of section, and an inch bar can be welded in 25 seconds. It takes about 90 seconds to weld a steel bar 2 inches in" diameter. ELECTRIC HEATERS. Wherever a comparatively small amount of heat is desired to be auto- matically and uniformly maintained, and started or stopped on the instant without waste, there is the province of the electric heater. The elementary form of heater is some form of resistance, such as coils of thin wire introduced into an electric circuit and surrounded with a sub- stance, which will permit the conduction and radiation of heat, and at the same time serve to electrically insulate the resistance. This resistance should be proportional to the electro-motive force of the current used and to the equation of Joule's law : H = C*Rt X 0.24, where Cis the current in amperes; R. the resistance in ohms; t, the time in seconds; and h, the heat in gram -centigrade units. Since the resistance of metals increases as their temperature increases, a thin wire heated by current passing through it will resist more, and grow hotter and hotter until its rate of loss of heat by conduction and radiation equals the rate at which heat is supplied by the current. In a short wire, before heat, enough can be dispelled for commercial purposes, fusion will begin; and in electric heaters it is necessary to use either long lengths of thin wire, or carbon, which alone of all conductors resists fusion. In the majority of heaters, coils of thin wire are used, separately embedded in some substance of poor electrical but good thermal conductivity. The Consolidated Car-heating Co.'s electric heater consists of a galvanized iron wire wound in a spiral groove upon a porcelain insulator. Each heater is 30% in. long, 8% in. high, and 6% in. wide. Upon it is wound 625 ft. of wire. The weight of the whole is 23^ lbs. Each heater is designed to absorb two amperes of a 500-volt current. Six heaters are the complement for an ordinary electric car. For ordinary weather the heaters may be combined by the switch in different ways, so that five different intensities of heating- surface are possible, besides the position in which no heat is generated, the current being turned entirely off. For heating an ordinary electric car the Consolidated Co. states that from 2 to 12 amperes on a 500-volt circuit is sufficient. With the outside temperature at 20° to 30°, about 6 amperes will suffice. With zero or lower temperature, the full 12 amperes is required to heat a car effectively. Compare these figures with the experience in steam-heating of railway- cars, as follows- : 1 B.T.U. = 0.29084 watt-hours. 6 amperes on a 500-volt circuit = 3000 w r atts. A current consumption of 6 amperes will generate 3000 -4- 0.29084 = 10,315 B.T.U. per hour. In steam- car heating, a passenger coach usually requires from 60 lbs. of steam in freezing weather to 100 lbs. in zero weather per hour. Supposing the steam to enter the pipes at 20 lbs. pressure, and to be discharged at 200° F., each pound of steam will give up 983 B.T.U. to the car. Then the equivalent of the thermal units delivered by the electrical-heating system in pounds of steam, is 10,315 -h 983 = 10V£, nearly. Thus the Consolidated Co.'s estimates for electric-heating provide the equivalent of lOJ^j lbs. of steam per car per hour in freezing weather and 21 lbs. in zero weather. Suppose that by the use of good coal, careful firing, well designed boilers, and triple-expansion engines we are able in daily practice to generate 1 H.P. delivered at the fly-wheel with an expenditure of 2>£ lbs. of coal per "hour. We have then to convert this energy into electricity, transmit it by wire to the heater, and convert it into heat by passing it through a resistance-coil. We may set the combined efficiency of the dynamo and line circuit at 8b%, and will suppose that all the electricity is converted into heat in the resist- ance-coils of the radiator. Then 1 brake H.P. at the engine = 0.85 electrical H.P. at the resistance-coil = 1,683,000 ft. -lbs. energy per hour = 2180 heat- units. But since it required 2% lbs. of coal to develop 1 brake H.P., it fol- ELECTRICAL ACCUMULATORS OR STORAGE-LATTERIES. 1055 lows that the heat given out at the radiator per pound of coal burned in the boiler furnace will be 2180 ■+- 2^ = 872 H.U. An ordinary steam-heating system utilizes 9652 H.U. per lb. of coal for heating; hence the efficiency of the electric system is to the efficiency of the steam-heating svstem as 872 to 9652, or about 1 to 11. (Eug'g News, Aug. 9, '90; Mar. 30, '92;'May 15, '93.) ELECTRICAL, ACCUMULATORS OR STORAGE- BATTERIES. Storage-batteries may be divided into two classes: viz., those in winch the active material is formed from the substance of the element itself, either by direct chemical or electro-chemical action, and those in which the chemical formation is accelerated by the application of some easily reduci- ble salt of lead. Elements of the former type are usually called Plante, and those of the latter " Faure," or " pasted." Faraday when electrolyzing a solution of acetate of lead found that per- oxide of lead was produced at the positive and metallic lead at the negative pole. The surfaces of the elements in a newly and fully charged Plante cell consists of nearly pure peroxide of lead, Pb0 2 , and spongy metallic lead, Pb, respectively on the positive and negative plates. During the discharge, or if the cell be allowed to remain at rest, the sul- phuric acid (H 2 S0 4 ) in the solution enters into combination with the per- oxide and spongy lead, and partially converts it into sulphate. The acid being continually abstracted from the electrolyte as the discharge proceeds, the density of the solution becomes less. In the charging operation this action is reversed, as the reducible sulphates of lead which have been formed are apparently decomposed, the acid being reinstated in the liquid and therefore causing an increase in its density. The difference of potential developed by lead and lead peroxide immersed in dilute H2SO4 is, as nearly as may be, two volts. A lead-peroxide plate gradually loses its electrical energy by local action, the rate of such loss varying according to the circumstances of its prepara- tion and the condition of the cell. Various forms of both Plante and Faure batteries are illustrated in " Practical Electrical Engineering." In the Faure or pasted cells lead plates are coated with minium or litharge made into a paste with acidulated water. When dry these plates are placed in a bath of dilute H 2 S0 4 and subjected to the action of the current, by which the oxide on the positive plate is converted into peroxide of lead and that on the negative plate reduced to finely divided or porous lead. Gladstone and Tribe found that the initial electro-motive force of the Faure cell averaged 2.25 volts, but after being allowed to rest some little time it was reduced to about 2.0 volts. The following tables show the size and capacity of two types of Faure cells, known as the E. P. S. cells. (Eng- lish.) 66 E. P. S." Storage-cells, Li Type. Description of Cell. ^ ? "h "- H Working Rate. £ < Approximate Exter- nal Dimensions. = £ No. of Plates. Material of Box. Charge Dis- charge. a S ? 60 '5 w ^ D-d "1 23 1 Wood Glass Wood Glass Wood Glass Wood Glass Wood Glass lbs. 18 25 25 35 35 47 53 67 70 88 Amper. 10 to 13 10 " 13 16 " 22 16 " 22 25 " 30 25 " 30 38 " 46 38 " 46 50 " 60 50 " 60 Amper. 1 to 13 1 " 13 1 " 22 1 " 22 1 " 30 1 " 30 1 " 46 1 " 46 1 " 60 1 " 60 130 130 220 220 330 330 500 500 660 660 in. ?* Wi 141-4 19M 1X1, in. 13' 4 1314 1^ 11 ->4 I3L, n H in. I8I4 133, I8I4 1% 1st. x 13 : s ISI4 13?4 isi 4 loo 4 in. 2oy 2 15% 20^ 20^ 15% 201., lbs. 74 68 107 101 143 12S 228 211 286 265 1056 ELECTRICAL ENGINEERING. 4 E. P. S.» Cells, T Type. Description of Cell. . Working Rate i Approx. External Dimensions. ~5 o £ ■-* § O fe . So «M ffl" °-!r;' G No. of Material of Charge Dis- - £ ,c5 +j c3 **£< Plates. Box. ^H charge. -3 '5 bt,rtional to the strength of the magnetizing current, provided the core is >t saturated. 1060 ELECTRICAL EHGIKEERING. (2) The magnetic strength is proportional to the number of turns of wire * in the magnetizing coil ; that is, to the number of ampere turns. (3) The magnetic strength is independent of the thickness or material of the conducting wires. These laws may be embraced in the more general statement that the strength of an electro-magnet, the size of the magnet being the same, is proportional to the number of its ampere turns. Force in the Gap between Two Poles of a Magnet.— If P = force exerted by one of the poles upon a unit pole in the gap, and m = density of lines in the field (that is, that there are m absolute or C.G.S. units on each square centimetre of the polar surface of the magnet), the polar surface being large relative to the breadth of the gap, P — 2nm. The total force exerted upon the unit pole by both nortli and south poles of the magnet is 2P — 4nm, in dynes = B, or the induction in lines of force per square centimetre. If 8 = number of square centimetres in each polar surface, SB = total flow of force, or field strength = F; Sm = total pole strength = M, spread over each of the polar surfaces. We then have F = 4nM, as before; that is, the total field is 4n times the total pole strength. Total attractive force between the two opposing poles of a magnet, when the distance apart is small, = — - — , in dynes. This formula may be used to determine the lifting-power of an electro- magnet, thus: A bent magnet provided with a keeper is 3 cm. square on each pole, and the induction B — 20,000 lines per square centimetre. The attractive force 9 X 20000 2 of each limb on the keeper in dynes = , or in kilogrammes for 9 X 400 X 10 6 b0th limbS ' 25.12 x 981000 X 2 = 292 kilogrammes. The Magnetic Circuit.— In the conductive circuit we have C = — ; _ electro-motive force volts Current = r— = -= . resistance ohms In the magnetic circuit we have Number of lines, or loops, of force, or magnetism Current X conductor turns _ Ampere turns "~ Resistance of magnetic circuit _ Resistance of magnetic circuit' .l ,. i gilberts Or, in the new notation, webers = - — . Let N = No. of lines of force, Em - total magnetic resistance, At = At ampere turns, then N = — — . 4 The magnetic pressure due to the ampere turns = — irTO = 1.2577c, „ Air TO 1.257 TO where T= turns and C = amperes, whence N= Rm - - Rm — . If Rm = total magnetic resistance, and Ra, Ra. Rf the magnetic resist- ances of the air-spaces, the armature, and the field-magnets, respectively, Rrn = Ra + R A + R F ; and N = ^ ^ ° + R ^ Determining the Polarity of Electro-magnets.— If a wire is wound around a magnet in a right-handed helix, the end at which the current flows into the helix is the south pole. If a wire is wound around an ordinary wood screw, and the current flows around the helix in the direc- tion from the head of the screw to the point, the head of the screw is the south pole. If a magnet is held so that the south pole is opposite the eye of the observer, the wire being wound as a right-handed helix around it, the cuirent flows in a right-handed direction, with the hands of a clock. DYNAMO-ELECTRIC MACHINES. 1061 DYNAMO-ELECTRIC MACHINES. There are four classes of dynamo-electric machines, viz.: 1. The dynamo, in which mechanical energy of rotation is converted into the energy of a direct current. 2. The alternator, in which mechanical energy of rotation is converted into the energy of an alternating current. 3. The motor, in which the energy of a direct current is converted into mechanical energy of rotation. 4. The alternate-current motor, in which the energy of one or more alter- nating currents is converted into mechanical energy of rotation. For a steady direct current the product of the potential difference and the current strength is a true measure of the energy given off. With alternat- ing currents the product of voltage into current strength is greater than the true energy, since the conductor has the property of reacting upon itself, called "self-induction." Kinds of Dyiiaino-electric Machines as regards Man- ner of Winding. (Houston's Electrical Dictionary.) 1. Dynamo electric Machine. — A machine for the conversion of mechan- ical energy into electrical energy by means of magneto-electric induction. 2. Compound-wound Dynamo.— The field-magnets are excited by more than one circuit of coils or by more than a single electric source. 3. Closed-coil Dynamo.— The armature-coils are grouped in sections com- municating with successive bars of a collector, so as to be connected con- tinuously together in a closed circuit. 4. Open-coil Dynamo. — The armature-coils, though connected to the suc- cessive bars of the commutator, are not connected continuously in a closed circuit. 5. Separate-coil Dynamo.— The field-magnets are excited by means of coils on the armature separate and distinct from those which furnish cur- rent to the external circuit. 6. Separately-excited Dynamo. — The field-magnet coils have no connec- tion with the armature-coils, but receive their current from a separate machine or source. 7. Series-ivound Dynamo.— The field-current and the external circuit are connected in series with the armature circuit, so that the entire armature current must pass through the field-coils. Since in a series-wound dynamo the armature-coils, the field, and the ex- ternal-series circuit are in series, any increase in the resistance of the ex- ternal circuit will decrease the electro-motive force from the decrease in the magnetizing currents. A decrease in the resistance of the external cir- cuit will, in a like manner, increase the electro-motive force from the in- crease in the magnetizing current. The use of a regulator avoids these changes in the- electro-motive force. 8. Series and Separately-excited Compotind-wound Dynamo. — There are two separate circuits in the field-magnet cores, one of which is connected in series with the field-magnets and the external circuit, and the other with some source by which it is separately excited. 9. Shunt-wound Dynamo.— The field-magnet coils are placed in a shunt to the armature circuit, so that only a portion of the circuit generated passes through the field magnet coils, but all the difference of potential of the armature acts at the terminals of the field-circuit. In a shunt-dynamo machine an increase in the resistance of the external circuit increases the electro-motive force, and a decrease in the resistance of the external circuit decreases the electro-motive force. This is just the reverse of the series-wound dynamo. In a shunt-wound dynamo a continuous balancing of the current occurs. The current dividing at the brushes between the field and the external cir- cuit in the inverse proportion to the resistance of these circuits, if the resist- ance of the external circuit becomes greater, a proportionately greater current passes through the field-magnets, and so causes the electro-motive force to become greater. If, on the contrary, the resistance of the external circuit decreases, less current passes through the field, and the electro- motive force is proportionately decreased. 10. Series- and Shunt-wound Compound-wound Dynamo.— The field-mag- nets are wound with two separate coils, one of which is in series with the armature and the external circuit, and the other in shunt with the arma- ture. This is usually called a compound-wound machine. 11. Sliunt and Separately-excited Compound-wound Dynamo.— The field 1062 ELECTKICAL ENGINEERING. is excited both by means of a, shunt to the armature circuit and by a cur- rent produced by a separate source. Current Generated toy a Dynamo-electric Machine.— Unit current in the C.G.S system is that current which, flowing in a thin wire forming a circle of one centimetre radius, acts upon a unit pole placed in the centre with a force of 2n dynes. One tenth of this unit is the unit of current used in practice, called the ampere. A. wire through which a current passes has, when placed in a magnetic field, a tendency to move perpendicular to itself and at right angles to the lines of the field. The force producing this tendency is P = IcB dynes, in which I = length of the wire, c = the current in C.G.S. units, and B the in- duction in the field in lines per square centimetre. If the current C is taken in amperes, P = WB10 • If P k is taken in kilogrammes, P k = ~^ = 10.1937ZC£10-8 kilogrammes. Example. — The mean strength of field, B, of a dynamo is 5000 C.G.S. lines; a current of 100 amperes flows through a wire; the force acts upon 10 centi- metres of the wire = 10.1937 X 10 X 100 X 5000 X 10" 8 = .5097 kilogrammes. In the "English" or Kapp's system of measurement a total flow of 6000 C.G.S. lines is taken to equal one English line. Calling Be the induction in English, or Kapp's, lines per square inch, and B the induction in C.G.S. lines per square centimetre, Be = B -4- 930.04; and taking I" in inches and Pp in pounds, P p = 531 Gl"B E 10" 6 pounds. Torque of an Armature.— Pp in the last formula, = the force tending to move one wire of length I", which carries a current of C amperes through the field whose induction is Be English lines per square inch. The current through a drum-armature splits at the commutator into two branches, each half going through half of the wires or bars. The force exerted upon one of the wires under the influence of a pole-piece — y^Pp. If t = the number of wires under the pole-pieces, then the total force = \^Ppt. If r = radius of the armature to the centre of the conductors, expressed in feet, then the torque = l^Pptr, = % X 531 X Cl"B E X 10~ 6 X tr foot-pounds of moment, or pounds acting at a radius of 1 foot. Example.— Let the length I of an armature = 20 in., the radius = 6 in. or .5 ft., number of conductors = 120, of which t = 80 are under the influence of the two pole- pieces at one time, the average induction or magnetic flux through the armature-field Be = 5 English lines per square inch, and the current passing through the armature = 400 amperes; then Torque = Y 2 X 531 X 400 X 20 X 5 X 80 X .5 X 10~ 6 = 424.8. The work done in one revolution = torque X circumference of a circle of 1 foot radius = 424.8 X 6.28 = 2670 foot-pounds. Let the revolutions per minute = 500, then the horse-power 33000 = 40.5 H.P. Electro-motive Force of the Armature Circuit.— From the horse-power, calculated as above, together with the amperes, we can obtain the E.M.F., for CE = H.P. X 746, whence E.M.F. or E = H.P. X 746 -*- C. 40 5 V 746 If H.P, as above, = 40.5, and C = 400, E = ^ = 75.5 volts. The E.M.F. may also be calculated more directly by the following formulae given by Gisbert Kapp: C — Total current through armature; c, current through single armature conductor; e a = E.M.F. in armature in volts; t = Number of active conductors counted all around armature; p = Number of pairs of poles (p = 1 in a two-pole machine); n — Speed in revolutions per minute; F = Total induction in C.G.S. lines; Z ~ Total induction in English lines. DYNAMO-ELECTiUC MACHINES. 1063 Electro- motive ' force 1 e a - ZrnlO' j- for two-pole machines. for multipolar machines with series-wound armature. e a = pZmlO~ | Kilogramme-metres = 1.615FTC10" 10 \ for two-pole ma- i Foot-pounds =:7.05ZtC10' 6 ' chines. Torque -j Foot-pounds Kilogramme-metres = 3.2SFrcp 10 ) for multipolar ma- L Foot-pounds = U.lOZrcplO' 6 ) chines. Example.— t = 120, n — 500, length of armature I : in., diameter d = 12 in., cross-section = 20 x 12 = 240 sq. in., induction per sq. in. B E = sq. in., total induction Z = 240 X 5 = 1200; then 5 lines per sq. E= ZmlO- : 1200 X 120 X 500 X 10- « =.- 72 volts. A formula for horse-power given by Kapp is H.P. = 1/746 ZNtnlO- *Ca - 1/746 2abmNtnlO- 6 Ca. Ca = current in amperes, n = revs, per min., 2ab = sectional area of arm- ature-core, m — average density of lines per sq. in. of armature-core, Nt — total number of external wires counted all around the circumference, t = number of wires correspondirg to one plate in the commutator, N — num- ber of plates, Z = 2abm = total number of English lines of force. Kapp says that experience bas shown that the density of lines m in the core cannot exceed a certain limit, which is reached when the core is satu- rated with magnetism. This value is reached when m = 30. A fair average value in modern dynamos and motors is m = 20, and the area ab must be taken as that actually filled by iron, and not the gross area of the core. 2u English lines per sq. in. = 18,600 C.G.S. lines per square centimetre. Sil- vanus P. Thompson says it is not advisable in continuous-current machines to push the magnetization further than B = 17,000 C.G.S. lines per square centimetre. Thompson gives as a rough average for the magnetic field in the gap-space of a dynamo or motor 6300 lines per sq. cm., or 40,000 lines per sq. in., and the drag per inch of conductor .00351 lb. for each ampere of current carried. a * H - p - X 33,000 . Pounds average drag per conductor = ^ . \ in which Cis the it, per min. x o number of conductors around the armature. Strength, of tne Magnetic Field.— Kapp gives for the total num- ber of lines of force (Kapp's lines = C.G.S. lines -4- 6000) in the magnetic cir- cuit. Z — — — t—= — , in which Z= number of magnetic lines, X = the Ha -r BA -\- RF exciting pressure due to the ampere turns = AnTC, Ra, Ra, and Rf. = re- spectively the resistances of the air-spaces, the armature, and the field-mag- nets. . Kapp gives the following empirical values of Ra, Ra, and Rf, for dynamos and motors made of well-annealed wrought iron, with a permeability of /* = 940: J^ " 06' Rf -A» in which 5 = distance across the span between armature -core and polar surface, b = breadth of armature measured parallel to axis, A = length of arc embraced by polar surface, so that \b = the polar area out of which magnetic lines issue, a = radial depth of armature-core, so that ab = sec- tion of armature-core (space actually occupied by iron only being reckoned, AB = area of field-magnet core, I = length of magnetic circuit within ar- mature, L = length of magnetic circuit in field magnet; all dimensions in inches or square inches. 1064 ELECTRICAL ENGINEERING. For cast-iron magnets, Z — ^ — — -, — Qr • For double horse-shoe magnets of wrought iron, Kb ab AB . ' . . Z 0.8X and of cast iron, - = — ^ These formulae apply only to cases in which the intensity of magnetization is not too great— say up to 10 Kapp's lines per square inch. Silvanus P. Thompson gives the following method of calculating the strength of the field, or the magnetic flux, Mf, or the whole number of magnetic lines flowing in the circuit in C.G.S. lines: The magnetic resistance of any magnetic conductor is proportional direct- ly to its length and inversely to its cross-section and its permeability. Magnetic resistance = — , in which L = length of the magnetic circuit bp passing through any piece of iron, S = section of the magnetic circuit passing through any piece of iron. p = permeability of that piece of iron. In a dynamo-machine in which the resistances are three, viz.: 1. The field - magnet cores; 2. The armature-core; 3. The gaps or air-spaces between them,— let Lm, Sm, pm refer to the field -magnet part of the circuit; Las, Sas, pas refer to the air-space part of the circuit; La, Sa, pa refer to the armature part of the circuit; the lengths across each of the air-spaces being Las, and the exposed area of polar surface at either pole being Sas. Total magnetic resistance = ^ m • -f- - Q \- . a . Smpm Saspas Safia Magnetic flux, or total number of magnetic lines, = 1. 257 TwO Mf = Lm Las , La, '■ Smpm Saspas Sapa Tw — turns of wires, or number of turns in the spiral; C = current in amperes passing through spiral. Application to Designing of Dynamos. (S. P. Thompson.)— Suppose in designing a dynamo it lias been decided what will be a conven- ient speed, how many conductors shall be wound upon the armature, and tvhat quantity of magnetic lines there must be in the field, it then becomes necessary to calculate the sizes of the iron parts and the quantity of excita- tion to be provided for by the field-magnet coils. It being known what Mf is to be, the problem is to design the machine so as to get the required value. Experience shows that in every type of dynamo there is magnetic leakage; also, that it is not wise to push the saturation of the armature-core to more than 16,000 lines to the square centimetre at the most highly satu- rated part, and that the induction in the field-magnet ought to be not greater than this, even allowing for leakage. Leakage may amount to 34 of the whole: hence, if the magnet-cores are made of same quality of iron as the armature-cores, their cross-section ought to be at least 5/4 as great as that of the armature-core at its narrowest point. If the field-magnets are of cast iron, the section ought to be at least twice as great. Now, Ba (the induction in the armature-core) = Ma -s- Sa (or magnetic flux through armature h- cross-sectional area of the armature ; hence, if this is fixed at 16,000 lines per centimetre of cross-section, we at once get Sa — Ma -5- Ba. This fixes the cross-section of the armature-core. (Example: If Ma — 4,000,000 of lines, then there must be a cross-section equal to 250 + . . „ 4,000,000 OKA . square centimetres for =250.) DYNAMO-ELECTRIC MACHINES. 1065 Magnetic Length of Armature Circuit.— The size of wires oh the arma- ture is fixed by the number of amperes which it must carry without risk. Remembering that only half the current (in ring or drum armatures) passes through any one coil, and as the number is supposed to have been fixed be- forehand, this practically settles the quantity of copper that must be put on the armature, and experience dictates that the core should be made so large that the thickness of the external winding does not exceed 1/6 of the radial depth of the iron core. This settles the size of the armature -core, from which an estimate of La, the average length of path of the magnetic lines in the core, can be made. Length and Section or Surface Area of Air -space.— Experience further dictates the requisite clearance, and the advantage of making the pole- pieces subtend an arc (in two-pole machines) of at least 135° each, so as to gain a large polar area. This settles Las and Sas. Length of Field -magnet Iron Cores, etc.— As shown above, the minimum value of Sm is settled bj T leakage and materials; Lm therefore remains to be decided. It is clear that the magnet-cores must be long enough to allow of the requisite magnetizing coils, but should not be longer. As a rule, they are made so stout, especially in the yoke part, that they do not add much to the magnetic resistance of the circuit, then a little extra length as sumed in the calculation does not matter much. It now only remains to calculate the number of ampere-turns of excitation for which it will be needful to provide. It will now be more convenient to rewrite the formula of the magnetic circuit as follows: i x -kra _j_ o -Las . La ) .s. m ,, { Smum ' Sas.pas Sa.n-a \ A X Tmw = Ma- -— ; ; where A — amperes of current passing through the field-magnet coils; Tmw - total turns of the magnet wire; \ = leakage coefficient (say 5/4). Or, 4 X Tmtv = Ma ._, ■. Or, as before, ,, i n »„ Ax Tmw Ma=1 - 2D7 KR m + R ai -rRa > where Rm, Ras, Ra stand for the magnetic resistance of magnets, air- space, and armature, respectively. But we cannot use this formula yet, because the values of ju. in it depend on the degree of saturation of the iron in the various parts. These have to be found from the Hopkinson tables, given below; and, indeed, it is preferable first to rearrange the formula once more, by dividing it into its separate members, ascertaining separately the ampere-turns requisite to force the required number of magnetic lines through the separate parts, and then add. them together. 1. Ampere- turns required for magnet-cores = A ~ x ^^-*- 1.257. Sm n-m 2. Ampere-turns required for air-spaces = — -^ x 2— s -=- 1.257. Sas pas 3. Ampere-turns required for armature-core = ^ x — -*- 1.257. Sa na Now A.--— is the value of B in the magnet-cores, and reference to the table of permeability will show what the corresdonding value of /u-m must be. Similarly,— a will afford a clue to /u.a. When the total number of ampere- turns to be allowed for is thus ascertained, the size and length of wire will be determined by the permissible rise of temperature, and the mode of exciting the field -magnets, whether in series, or as a shunt machine, or with a compound-winding. 1066 ELECTRICAL EJSmiNEERLNGL Permeability.— Materials differ in regard to the resistance they otter to the passage of lines of force; thus iron is more permeable than air. The permeability of a substance is expressed by a coefficient fx,, which denotes its relation to the permeability of air, which is taken as 1. If H ~ number of magnetic lines per square centimetre which will pass through an air- space between the poles of a magnet, and B the number of lines which will pass through a certain piece of iron in that space, then ju. = B -*- H. The permeability varies with the quality of the iron, and the degree of satura- tion, reaching a practical limit for soft wrought iron when B =s about 18,000 and for cast iron when B = about 10,000 C.G.S. lines per square centimetre. The following values are given by Thompson as calculated from Hopkin- son's experiments: Annealed Wrought Iron. Gray Cast Iron. B H M B H ^ 5,000 2 2,500 4,000 5 800 9,000 4 2,250 5,000 10 500 10,000 5 2,000 6,000 21.5 279 11,000 6.5 1,692 7,000 42 133 12,000 8.5 1,412 8,000 80 100 13,000 12 1,083 9,000 127 71 14,000 17 823 10,000 188 53 15,000 28.5 526 11,000 292 37 16,000 52 308 17,000 105 161 18,000 200 90 19,000 350 54 Permissible Amperage and Permissible Depth of "Wind- ing for Magnets with Cotton-covered Wire. (Walter S. L)ix, El. Engineer, Dec. 21, 1892.)— The tables on pp. 1068, 1069, abridged from those of Mr. Dix, are calculated from the formula "\J\ 12 XW <«W XTXL M where C = current; W = emissivity in watts per square inch; co m f = ohms per mil-foot ; M = circular mils ; T = turns per linear inch ; L = number of layers in depth. The emissivity is taken at .4 watt per sq. in. for stationary magnets for a rise of temperature of 35° C. (63° F.)\ For armatures, according to Esson's experiments, it is approximately correct to say that .9 watt per sq. in. will be dissipated for a rise of 35° C. The insulation allowed is .007 inch on No. to No. 11 B. & S.; .005 inch on No. 12 to No. 24 ; and .0045 inch on No. 25 to No. 31 single ; twice these values for insulation of double-covered wires. Fifteen per cent is allowed for imbedding of the wires. The standard of resistance employed is 9.612 ohms per mil-foot at 0°. The running temperature of tables is taken at 25° -f 35° = 60° C. The column giving the depth for one layer is the diameter over insulation. Formulae of Efficiency of Dynamos. (S. P. Thompson in " Munro and Jamieson's Pocket-Book.") Total Electrical Energy (per second) of any dynamo (expressed in watts^ is the product of the whole E.M.F. generated by armature-coils into the whole current which passes through the armature. Useful Electrical Energy (per second), or useful output of the machine, is the product of the useful part of the E.M.F. (i.e., that part which is avail- able at the terminals of the machine) into the useful part of the current (i.e., that part of the current which flows from the terminals into the exter- nal circuit). DYNAMO-ELECTRIC MACHINES. 1067 Economic Coefficient or "electrical efficiency" of a dynamo is the ratio of the useful energy to the total energy. Commercial Efficiency of a dynamo is the ratio of the useful energy or output to the power actually absorbed by the machine in being driven. Let Ea = total E.M.F. generated in armature; Ee — useful E.M.F, available at terminals; Ca = total current generated in armature; C s = current sent round shuut-coils; Ce = useful current supplied to external circuit; Ra — resistance of armature-coils; Rm = resistance of magnet-coils in main circuit (series); Rs — resistance of magnet-coils in shunt; ' R e = resistance of external circuit (lamps, mains, etc.); Wa = Watts lost in armature; Wm— Watts lost in magnet-coils; Vl = lost volts; Te = total electrical energy (per second); Ue = useful electrical output; c — economic coefficient; p — commercial efficiency (percentage). When only one circuit (series machine) Ce = Ca. In shunt machines Cs should not be more than h% of Ce. Also, Ca = Ce + Cs. In all dynamos, Ra ought to be less than 1/40 as great as the working value of Re- in series (and compound) machines, Rm should be not greater than R a , and preferably only % as great. In shunt (and compound) machines, Rs should be not less than 300 times as great as Ra and preferably 1000 to 1200 times as great. Series Machine. Shunt Machine. Compound Machine (Short Shunt). w a C\R a Cf t R a C'a R a w m ClR m C%R s =El+R s ClR m +ClR 8 Vl C a R a Ca R a C a R a -\-C e R m T e E a C a = Cl{R a + R m + R e ) E aC a = 1 R,R P \ V R s +R e / 1 RJR m +R e ) \ E a C a = CI R a + S m e \ R s +R m +Rj v e E e C a =ClR e E e C e =ClR e E e C e =C%R e E R e C%R e C\R e E a R a +R m +R e C\R e +ClR a +C\R 8 C 2 e R e +ClR a +ClR s +ClR m p 100xE e C e -+- (H.P.X746) mxE e c e -+ (H.P.X746) 100x# e ^ -T'C ' Ih ^ ^ t-l ^4 ^H^ r^ CS 00 i>^ •JSOOO^MOCl'X L— •0 "Sap 09 ■4-B 4003 .iad suiqo 000150 000] SO 0001 S3 000313 000319 000289 000292 000305 000373 000443 000459 000551 000580 ( 073 000731 000838 000933 001015 001 101 001830 001465 00175 001845 00232;", 00370 (MM Hi',;, 00589 00743 00935 0118 0149 01874 02365 0249 •SUM .reino.no 30050 07 081 66373 500 1 1 52031 IS 100 71743 11209 33102 3,2400 37225 30251 21904 30817 17950 105 10 14400 13094 11881 10382 9035 8234 08S9 0580 5184 4225 4107 8257 2583 2018 1624 12S8 1024 810 642 509 484 •saqoui 'a.iBg'ui'Bta .284 .259 . 2570 .238 .2294 .22 .2043 .203 .1819 .18 .165 .162 .148 .1443 .134 .1385 .12 .1144 .109 .1019 .095 .0907 .083 .0808 .072 .065 .0041 .0571 .0.508 0458 .0103 .0359 .032 .0385 .0353 .0226 .023 6 1 -jm 0C oc^o^r-ccoossjo^so ^ jj & I -g ya I DYNAMO-ELECTRIC MACHINES. 1069 iilllli sssssss 5S§poS§ siSS^SS •A 1 ft o ft ft noonNO'*0'*0'*ojtQO«owio OTf^-tf-*t-'3>eOTOCOCOCOCO.rj'?->3-!0!C>*0* ft S «5 ClHOXOOOtTliNOtlO^rHMCOOil- rtOOO®!Oir!MMrHrH0305QOa)£«0!OlO o ft WrtO-HlONWMM^OOlfiWO^fflMW ft s < Oi-H»OSl-COi-'ar-iQOOS«OaiOMNt»TH ^^•oTp7^T-(OSQo«o»oec95-i-i«-iOOsc30QO iO 3 ft ft M :) rt ?;o -f X i- X c- O 3: i? - otot- «-r(:(rHOOC5o;xxa)t-t>N®!Dtoio ft Oi-iNfflC500Si>NKU>THMT-iiO!OMin ^mMM»«3fflO)NMnn«rtr«i-l«TH - ft ft ci i- i~ «n -* to — — cs ci t- i- ;o o -v tt to Oi ft £ CO*«3^IOCJ«X)OOOria»«)fl»t-N MOTtnnHi>offlOo-*:H-o?)ot-o ocoooc-<-~co»iOioin'^"9>cocococoffjc< Turns per linear inch. aoaDi-T-NooTHOfflsioot-istonfflO) coMcow^*niTtiooiointos3ffli>i^N o V ft p C4 CO t- iO «5 « O ij 5 « 33 ft Ciin-fcoco-*— ■5DCO rC 93 •- ?! C! ffl ■•■: iS S l« u s c :» T O CO o <*C S • - i- — W - - IQ T TO ^JTlllr-t^-.i-i-i^-.i-cOOOOOOOOO Ohms per foot at 60° 0. CS O TJ TO Ci Ci OS 7> O C! CO C! r« of ?! r- M M -r X X — 7 • -" X ---■-—.-..-.■.-•-:■::: 7 • sllllllllillllllli oooooooocooooooooo tOHMTf^ocoanoiOH-fi-tooO'* i- x • - -f tt ~ -ccc:!cOrtio-ooi •O O TO '-S 3-ti-'?!. r T> T! Ci CO C5 lO TC O oi-0'o:ix«-^i!i,-o^oi-to^w XOd.0iSf*^W03«MMe(HrtHH sg§ Qfflfl ■* OS fc- 00 OS TJHTO'- O 0! CO ^f"* GO Tf CO iOiOTOT'*! c ; X X CC-rtCOW«H :!:':!:>:H!5!«niHrlrlnHHHrlH be G3 M C*W T)< o SO £>00 OS O t-i a5 ft ci ao cs 1070 ELECTRICAL ENGINEERING. Alternating Currents, Multiphase Currents, Trans- formers, etc.— The proper discussion of these subjects would take more space than can be afforded in this work. Consult S. P. Thompson's " Dy- namo-Electric Machinery, 1 ' Bedell and Crehore on " Alternating Currents, 1 ' Fleming on " Alternating Currents," and Kapp on "Dynamos, Alternators and Transformers." The "Electric Motor. — The electric motor is the same machine as the dynamo, but with the nature of its operation reversed. In the dynamo mechanical energy, such as from a belt, is converted into electric current; in the motor the current entering the machine is converted into mechanical energy, which may be taken off by a belt. The difference in the action of the machine as a dynamo and as a motor is thus explained by Prof. F. B. Crocker, (Cassier's Mag., March. 1895): In the case of the dynamo there exists only one E.M.F., whereas in the motor there must always be two. One kilowatt dynamo, C — E -h R; 10 amperes = 100 volts -4- 10 ohms. _ . .. .. , „ E-e .. 100 volts - 90 volts. One kilowatt motor, C = — = — ; 10 amperes = — = * i?i 1 ohm C is the current; E, , the direct E.M.F.; e, the counter E.M.F.; R. the total resistance of the circuit; R x , the resistance of the armature. The current and direct E.M.F. are tiie same in the two cases, but the resistance is only one tenth as much in the case of the motor, the difference being replaced by the counter E.M.F. , which acts like resistance to reduce the current. In the case of the motor the counter E.M.F. represents the amount of the electrical energy converted into mechanical energy. The so-called electri- cal efficiency or conversion factor = counter E.M.F. -r- direct E.M.F. The actual or commercial efficiency is somewhat less than this, owing to fric- tion, Foucault currents, and hysteresis. For full discussions of the theory and practice of electric motors see S. P. Thompson's '"Dynamo-Electric Machinery," Kapp's "Electric Trans- mission of Energy," Martin and Wetzler's li The Electric Motor and its Applications," Cox's " Continuous Current Dynamos and Motors," and Crocker and Wheeler's "Practical Management of Dynamos and Motors." LIST OF AUTHORITIES QUOTED IN THIS BOOK. When a name is quoted but once or a few times only, the page or pages are given. The names of leading writers of text-books, who are quoted fre- quently, have the word "various' 1 '' affixed in place of the page-number,. The list is somewhat incomplete both as to names and page numbers. Abel, F. A., 642 Abendroth & Root Mfg. Co., 197, 198 American Screw Co., 209 Achard, Arthur, 886, 919 Addy, George, 957 Addyston Pipe and Steel Co., 187, 188 Alden, G. I., 979 Alexander, J. S., 629 Allen, Kenneth, 295 Allen, Leicester, 582 Andrews, Thomas, 384 Ansonia Brass and Copper Co., 327 Arnold, Horace L., 959 Ashcroft Mfg. Co., 752, 7?5 Atkinson, J. J., 532 Ayrton and Perry, 1040 . Babcock, G. H., 524, 933 Babcock & Wilcox Co., 538, 636 Baermann, P. H., 188 Bagshaw, Walter, 952 Bailey, W. H., 943 Baker, Sir Benjamin, 239, 247, 402 Balch, S. W., 898 Baldwin, Wm. J., 541 Ball, Frank H., 751 Barlow, W. H., 384 Barlow, Prof., 288 Barnaby. S. W., 1013 Barnes, D. L., 631, 861, 863 Barrus, Geo. H., 636 Bauer, Chas. A., 207 Bauschinger, Prof., 239 Bazin, M., 563, 587 Beardslee, L. A.. 238, 377 Beaumont, W. W., 979 Becuel, L. A., 644 Begtrup, J., 348 Bennett, P. D., 354 Bernard, M. & E., 330 Birkinbine, John, 605 Bjorling, P., 676 Blaine, R. G., 616, 1039 Blauvelt, W. H., 639, 649 Blechyuden, A., 1015 Bodmer, G. R., 753 Bolland, Simpson, 946 Booth, Wm. EL, 926 Box, Thomas, 475 Brings, Robert, 194, 478, 539, 672 British Board of Trade, 264, 266, 700 Brown, A. G., 723, 724 Brown, E. H., 388 Brown & Sharpe Mfg. Co., 219, 890 Browne, Ross E., 597 Brush, Chas. B., 566 Buckle, W., 511 Buel, Richard H., 606, 834 Buffalo Forge Co., 519, 529 Builders' Iron Foundry, 374 Burr. Wm. A., 565 Burr, Wm. H., 247, 259, 290, 381 Calvert, F. Crace, 386 Calvert & Johnson, 469 Campbell, H. H., 398. 459, 650 Campredon, Louis, 403 Carnegie Steel Co., 177, 272, 277, 391 Carpenter, R. C, 454, 615, 718, etc. Chad wick Lead Works, 201, 615 Chamberlain, P. M., 474 Chance, H. M., 631 Chandler, Chas. F., 532 Chapman Valve Mfg. Co., 193 Chauvenet, S. H., 370 Chase, Chas. P., 312 Chevandier, Eugene, 640 Christie, James, 394 Church, Irving P., 415 Church, Wm. Lee. 784. 1050 Clapp, Geo. H., 397, 403, 551 Clark, Daniel Kinnear, various Clarke, Edwin, 740 Claudel, 455 Clay, F. W.. 291 Clerk, Dugald, 847 Cloud, John W., 351 Codman, J. E., 193 Coffey, B. H., 810 Coffin, Freeman C, 292 Coggswell, W. B., 554 Cole, Romaine C„ 329 Coleman, J. J., 470 Cooper, John H., 876, 900 Cooper, Theodore, 262, 263, 359 Cotterill and Slade, 432, 974 Cowles, Eugene H., 329, 331 Cox, A. J., 290 Cox, E. T., 629 Cox, William, 575 Coxe, Eckley B., 632 Craddock, Thomas, 473 Cramp, E. S.,405 Crimp, Santo, 564 Crocker, F. B., 1070 Cummins, Wm. Russell, 772 Daelen, R. M., 617 Dagger, John H. J., 329 Daniel, Wm., 492 D'Arcy, 563 Davenport, R. W., 620 Day, R. E., 1030 Dean, F. W., 605, 689 1071 1072 LIST OF AUTHORITIES. Decoeur, P., 600 DeMeritens, A., 386 Denton, James E , 730, 761, 781, Dinsmore, R. E., 963 Dix, Walter S., 208, 1066 Dodge Manufacturing Co., 344 Donald, J. T., 235 Donkin, B., Jr., 491, 783 Dudley, Chas. B., 327, 333 Dudley, P. H., 401, 622 Dudley, W. D., 167 Dulong, M.,458, 476 Dunbar, J. H., 796 Durand, Prof., 56 Dwelshauvers-Dery, 662 Egleston, Thomas, 235, 641 Emery, Chas. E., 603, 613, 820 Engelhardt, F. E., 463 Ellis and Howland, 577 English, Thos.,753 Ericsson, John, 286 Eytelwein, 564 Fairbairn, Sir Wm, 240, 264, 301 Fairley, W., 531, 533 Falkenau, A., 509 Fanning, J. T., 564, 579 Favre and Silbermann, 621 Felton, C. E., 646 Fernow, B. E., 640 Field, C. J., 30, 937 Fitts, James H., 844 Flather, J. J., 961, 964 Flynn, P. J., 463, 559 Foley, Nelson, 700 Forbes, Prof., 1033 Forney, M. N., 855 Forsyth, Wm., 630 Foster, R. J., 651 Francis, J. B., 586, 739, 867 Frazer, Persifor, 624 Freeman, J. R., 581, 584 Frith, A. J., 874 Fulton, John, 637 Ganguillet & Kutter, 565 Gantt, H. L., 406 Garrison, F. L., 326, 331, 409 Garvin Machine Co., 955 Gause, F. T., 501 Gav, Paulin, 966 Gill, J. P., 657 Gilmore, E. P., 241 Glaisher, 483 Glasgow, A. G., 654 Goodman, John, 934 Gordon, F. W., 689, 740 Gordon, 247 Goss, W. F. M., 863 Gossler, P. G., 1051 Graff, Frederick, 385 Graham, W., 950 Grant, George B., 898 Grant, J. J., 960 Grashof, Dr., 284 Gray, J. McFarlane, 661 Gray, J. M., 958 Greene, D. M., 567 Greig and Eyth, 363 Grosseteste, W., 715 Gruner, L., 623 Hadfield, R. A., 391, 409 Halpin, Druitt, 789, 854 Halsey, Fred'k A., 490, 817 Harkness, Wm., 900 Harrison, W. H., 939 Hart, F. R., 1047 Hartig, J., 961 Hartman, John M., 364 Hartnell, Wilson, 348, 818, 838 Hasson, W. F. C, 1047 Hawksley, T., 485, 513, 564 Hazen, H. Allen, 494 Henderson, G. R, 347, 851 Henthorn, J. T., 965 Hering, Carl, 1045 Herschel, Clemens, 583 Hewitt, G. C, 630 Hewitt, Wm., 917 Hildenbrand, Wm., 913 Hill, John W., 17 Hiscox, G. D., 968 Hoadley, John C, 451, 688 Hobart', J. J., 962 Hodgkinson, 246 Holley, Alexander L., 377 Honey, F. R., 47, 52 Hoopes & Townsend, 210 Houston, Edwin J., 1061 Houston & Kennelly, 1058 Howard, James E., 242, 382, 385 Howden, James, 714 Howe, Henry M., 402, 407, 451, 516 Howe, Malverd A., 170, 312 Howland, A. H., 292 Hudson, John G., 465 Hughes, D. E., 396 Hughes, H. W., 909 Hughes, Thos. E., 917 Humphreys, Alex. C, 652 Hunsicker, Millard, 397 Hunt, Alfred E., 235, 317, 392, 553 Hunt, Chas. W., 340, 922 Huston, Charles, 383 Hutton, Dr., 64 Huyghens, 58 Ingersoll-Sergeant Drill Co., 503 Isherwood, Benj. F., 472 Jacobus, D. S., 511, 689, 726, 780 Johnson, J. B., 309, 314 Johnson, W. B.,475 Johnsou, W. R., 290 Jones,* Hoi'ace K., 387 Jones & Lamson Machine Co., 954 Jones & Laughlins, 867, 885 Kapp, Gisbert, 1033 Keep, W. J., 365, 951 Kennedy, A. B. W., 355, 525, 764 KernOt, Prof. 494 Kerr, Walter C, 781 Kiersted, W.. 292 Kimball, J. P., 499, 635, 637 Kinealy, J. H., 537 LIST OF AUTHORITIES. 1073 Kirk. A. C, 705 Kirk, Dr., 1004 Kirkaldy, David, 296 Kopp, H. G. C, 472 Kuichling, E , 578 Kutter, 559 Landretb, O. H., 712 Langley, J. W., 409, 410, 412 Lanza, Gaetano, 310, 369, 864, 977 La Rue, Benj. F., 248 Leavitt, E. D., 788 LeChatelier, M., 452 Le Conte, J., 565 Ledoux, M., 981 Leggett, T. H., 1049 Leonard, H. Ward, 1026 Leonard, S. H., 686 Lewis, Fred. H., 186, 189, 397 Lewis, I. N.,498 Lewis, Wilfred, 352, 362, 378, 899 Linde, G., 989 Lindenthal, Gustav, 385 Lloyd's Register, 264, 266, 700 Loss, H. V., 306 Love, E. G., 656 Lovett, T. D., 256 Lyne, Lewis F., 718 McBride, James, 974 MacCord, C. W., 898 Macdonald, W. R., 956 Macgovern, E. E., 545 Mackay, W. M., 542, 544 Mahler, M., 633 Main, Chas. T., 590, 780, 790 Mannesmann, L., 332 Manning, Chas. H., 675, 823 Marks, Win. D., 793, 811 Master Car Builders' Assoc, 376 Mattes, W. F., 399 Matthiessen, 1029 Mayer, Alfred M., 468 Mehrtens, G. G., 395, 405 Meier, E. D., 688 Meissner, C. A., 370 Melville, Geo. W., 674 Mendenhall, T. C.,23 Merriman, Mansfield, 241, 260, 282 Metcalf, William, 240, 412 Meyer, J. G. A., 795,856 Meystre, F. J.. 472 Miller, Metcalf & Parkin, 412 Miller, T. Spencer, 344, 927 Mitchell, A. E., 855, 856 Molesworth, Sir G. L., 562, 658 Molyneux and Wood, 736 Moore, Gideon E., 653 Morin, 435, 930, 933 Morison, Geo. S., 381, 393 Morrell, T. T., 407 Morris, Tasker & Co., 195, 196 Mumford, E. R., 1006 Murgue, Daniel, 521 Nagle, A. F., 292, 606, 878 Napier, 471, 669 Nason Mfg. Co., 4 8, 542 National Pipe Bending Co., 198 Nan, J. B., 367, 409 Newberry, J. S., 624 Newcomb, Simon, 432 New Jersey Steel & Iron Co., 253, 310 Newton, Sir Isaac, 475 Nichol, BC, 473 Nichols. 285 Nonis, R. Van A , 521 Norwalk Iron Works Co., 488, 504 Nystrom, John W., 265 Ordway, Prof., 469 Paret, T. Dunkin, 967 Parker, W., 354 Parsons, H. de B., 361 Passburg, Emil, 466 Pattinson, John, 629 Peclet, M.,471, 478, 731 Peltou Water Wheel Co., 191, 574, 585 Pence, W. D., 294 Pencoyd Iron Works. 179, 232, 868 Penned, Arthur, 555 Pennsylvania R. R. Co., 307, 375, 399 Philadelph'iaEngiueering Works, 526 Philbrick, P. H., 446 Phillips, W. B., 629 Phoenix Bridge Co., 262 Phoenix Iron Co., 181, 257 Pierce, C. S., 424 Pierce, H. M.,641 Pittsburg Testing Laboratory, 243 Piatt, John, 617 Pocock, F. A.. 505 Porter, Chas. T., 662, 787, 820 Potter, E. C, 646 Potts ville Iron & Steel Co., 250 Pouillet, 455 Pourcel, Alexandre, 404 Poupardin, M., 687 Powell, A. M , 975 Pratt & Whitney Co., 892, 972 Price, C. S., 638 Prony, 564 Pryibil, P., 977 Quereau, C. H„ 858, 862 Ramsey, Erskine, 638 Rand Drill Co , 490, 505 Randolph & Clowes, 198 Rankine, W. J. M., various Ransome, Ernest L., 241 Raymond, R. W., 631, 650 Reese, Jacob, 966 Regnault, M., various Reichhelm, E. P., 651 Rennie, John, 928 Reuleaux, various Richards, Frank, 488, 491, 500 Richards, John, 965, 976 Richards, Windsor, 404 Riedler, Prof., 507 Rites, F. M., 783, 818 Roberts-Austen, Prof., 451 Robinson, H., 1051 Robinson, S. W., 583 Rockwood, G. J., 781 John A. Roebling's Sons' Co., 214, 921 1074 List OF AUTHORITIES. Roelker, C. R., 26.5 Roney, W. R„ 711 Roots, P. H. & F. M.,526 Rose, Joshua, 414, 869* 970 Rothwell, R. P., 637 Rowland, Prof.; 456 Royce, Fred. P , 1053 Rudiger, E, A., 671 Ruggles, W. B., Jr., 361 Russell, S. Bent, 567 Rust and Coolidge, 290 Sadler, S. P., 639 Saint Venant, 282 Salom, P. G.,406, 1056 Sandberg, C. P., 384 Saunders, J. L., 544 Saunders, W. L., 505 Scheffler, F. A., 681 Schroter, Prof., 788 Schutte, L., &Co., 527 Seaton, various Sellers, Coleman, 890, 953, 975 Sellers, Wm, 204 Sharpless, S. P., 311, 639 Shelton, F. H., 653 Shock, W. H., 307 Simpson, 56 Sinclair, Angus, 863 Sloane, T. O'Connor, 1027 Smeaton, Wm., 493 Smith, Chas. A., 537, 874 Smith, C. Shaler, 256, 865 Smith, Hamilton, Jr., 556 Smith, Jesse M., 1050 Smith, J. Bucknall, 225, 303 Smith, Oberlin, 865, 973 Smith, R. H., 962 Smith, Scott A., 874 Snell, Henry I., 514 Stahl, Albert W., 599 Stanwood, J. B„ 802, 809, 813, 818 Stead, J. E., 409 Stearns, Albert, 465 Stein and Schwarz, 410 Stephens, B. F., 292 Stillman, Thos. B., 944 Stockalper, E., 493 Stromeyer, C. E., 396 Struthers, Joseph, 451 Sturtevant, B. F., Co., 487, 578 Stut, J. C. H., 844 Styffe, Knut, 383 Suplee, H. H., 769, 772 Suter, Geo. A., 524 Sweet, John E„ 826 Tabor, Harris, 751 Tatham & Bros., 201 Taylor, Fred. W., 880 Taylor, W. J., 646 Theiss, Emil, 818 Thomas, J. W., 369 Thompson, Silvanus P., 1064, 1066 Thomson, Elihu, 1052 Thomson, Sir Wm., 461, 1039 Thurston, R. H, various Tilghman, B. F., 966 Tompkins, C. R , 336 Torrance, H. C, 401 Torrey, Joseph, 582, 820 Tower, Beauchamp, 931, 934 Towne, Henry R., 876, 907, 911 Townsend, David, 973 Trautwine, J. O, 59, 118, 311, 482 Trautwine, J. C, Jr., 255 Trenton Iron Co., 216, 223, 230, 915 Tribe, James, 765 Trotz, E , 453 Trowbridge, John, 467 Trowbridge, W. P., 478, 513, 733 Tuit, J. E.,616 Tweddell, R. H., 619 Tyler, A. H., 940 Uchatius, Gen'l, 321 Unwin, W. Cawthorne, various Urquhart, Thos., 645 U. S. Testing Board, 308 Vacuum Oil Co., 943 Vair, G. O., 950 Violette, M , 640, 642 Vladomiroff, L., 316 Wade, Major, 321, 374 Wailes, J. W., 404 Walker Mfg. Co., 905 Wallis, Philip, 858 Wan-en Foundry & Mach. Co., 189 Weaver, W. D.. 1043 Webber, Samuel, 591, 963 Webber, W. O., 608 Webster, W. R, 389 Weidemann & Franz, 469 Weightman, W. H., 762 Weisbach, Dr. Julius, various Wellington, A. M., 290, 928, 935 West, Chas. D., 916 West, Thomas D., 328 Westinghouse & Galton, 928 Westinghouse El. & Mfg. Co., 1043 Weston, Edward, 1029 Whitham, Jay M., 472, 769, 792, 840 Whitney, A. J., 389 Willett, J. R., 538, 540 Williamson, Prof., 58 Wilson, Robert, 284 Wheeler, H. A., 908 White, Chas. F., 714 White, Maunsel, 408 Wohler, 238, 240 Wolcott, F. P., 949 Wolff, Alfred R , 494, 517, 528, 538 Wood, De Volson, various. Wood, H. A., 9 Wood, M. P., 386. 389 Woodbury, C. J. H., 537, 931 Wootten, J. E., 855 Wright, C. R. Alder, 331 Wright, A. W.,289 Yarrow, A. F., 710 Yarrow & Co., 307 Yates, J. A., 287 Zahner, Robert, 499 Zeuner, 827 INDEX. Abbreviations, 1 Abrasive processes, 965 Abscissas, 69 Absolute zero, 461 Absorption refrigerating machines, 984 Accelerated motion, 427 Acceleration, 423 work of, 430 Accumulators, electric, 1055 Air, 481-527 and vapor, weights of, 484 compressed, 499 density and pressure, 481 -pumps, 839 -thermometer, 454 Algebra, 33 Algebraical signs, 1 Alligation, 10 Alloys, 319-338 aluminum, 328 aluminum-silicon-iron, 330 antimony, 336 bismuth, 332 caution as to strength, 329 copper-nickel, 326 copper-tin, 319 copper-tin-zinc, 322 copper-ziuc, 321, 325 copper-zinc-iron, 326 for bearings, 333 fusible, 333 manganese-copper, 331 steels, 407 Alternating currents, 1070 Altitude by barometer, 483 Aluminum, 166 alloys of, 319-338 brass, 329 bronze, 328 bronze wire, 225 hardened, 330 properties and uses, 317 steel, 409 wire, 225 Ammonia ice-machines, 983 vapor, properties of, 993 Amperage permissible in magnets, Analyses of alloys (see Alloys) of asbestos, 233 of coals (see Coal) of fire-clay, 234 of magnesite, 233 of steel (see Steel) of water, 553 Analytical geometry, 69 Anemometer, 491 Angle-bars, sizes and weights, 179 weight and strength, 279 Angles, plotting without protract- or, 52 problems in, 37-38 Angular velocity, 425 Animal power, 433 Annealing, effect on conductivity, 1029 non-oxidizing, process of, 387 of steel, 394 tool-steel, 413 Annuities, 15-17 Annuiar gearing, 898 Anti-friction metals, 932 Anthracite, analyses of, 624 gas, 647 space occupied by, 625 value of sizes of, 632 Antimony, 166 alloys, 336 Apothecaries' measure and weight, 18, 19 Arc lamps, lighting power of, 1052 Arches, tie-rods for, 281 Area of circles, 103, 108 Arithmetic, 2 Arithmetical progression, 11 Armature circuit, E. M. F. of, 1062 Asbestos, 235 Asymptotes of hyperbola, 71 Atmosphere, moisture in, 4S3 Avoirdupois weight, 19 Axles, steel, specifications for, 401 strength of, 299 Babbitt metals, 336 Bagasse as fuel, 643 Balance, to weigh on an incorrect, 19 Ball bearings, 940 1075 1076 Bands and belts, theory of, 876 Bands for carrying grain, 911 Barometric readings, 482 Barrels (see Casks), 64 No. of in tanks, 126 Bazin's experiments on weirs, 587 Formula, flow of water, 563 Beams and channels, Trenton, 278 Beams, flexure of, 267 of uniform strength, 271 safe load on pine, 1023 safe loads, 269 strength of, 268 Bearing metal alloys, 333 Bearing-metals, anti-friction, 932 Bearings, ball, 940 for high-speeds, 941 pivot, 939 Bed-plates of engines, 817 Belt cement, 887 conveyors, 911 dressings, 887 Belting, 876-887 strength of, 302 Belts, open and crossed, 874, 884 Bends and curves, effect of on flow of water, 578 Bends, valves, etc., resistance of 672 Bessemerized cast iron, 375 Bessemer steel. 391 Bevel wheels, 898 Binomials, Theorem, 33, 35 Birmingham Gauge, 28 Bismuth, 1(56 alloys, 332 Blast-furnace boilers, 689 Blocks or pulleys, 438 strength of, 906 Blowers and fans, 511-526 experiments with, 514 for cupolas, 950 positive rotary, 526 steam -jet, 526 Blowing engines, 526 Blue heat, effect on steel, 395 Board measure, 20 Boiling-point of water, 550 Boiling points, 455 Bolts and nuts, 209, 211 Bolts, holding power of, 290 strength of, 292 Boiler compounds, 717 explosions, 720 Boilers, for steam-heating, 538 Boiler furnaces, height of, 711 heads 706 headsi strength of, 284-286 Boiler scale, 552 ship and tank plates, 399 the steam, 677-741 tubes, 196 tubes, holding power of, 307 Boilers, locomotive, 855 Brass alloys, 325 composition of rolled, 203 sheet and bar, 203 tubing, 198 Brass wire and plates, 202 Brick, fire, sizes of, 233-235 Brick, strength of, 302, 312 Bricks, absorption of water by, 312 Bricks, magnesia, 235 Brickwork, weight of, 169 Bridge members, working strain, 262 proportioning materials in, 381 trusses, 443 Brine, specific gravity, etc., 464, 994 Bronze (see Alloys), 319 Bronzes, ancient, 323 Building construction, 1019 materials, sizes and weights, 170- 184 Buoyancy, 550 Burr truss, 443 Cables, electric, insulated, 1033 wire, 222, 223 Cable-ways, suspension, 915 Cadmium, 167 Calculus, Differential, 72 Caloric engines, 851 Calorimeters, steam, 728 Calorimetric tests of coal, 636 Cam, the, 438 Canals, speed of vessels on, 1008 Canvas, strength of, 302 Carbon, burned out of steel, 402 effect of on strength of steel, 389 Car-heating by steam, 538 Casks, 64 Castings, steel, 405 weight of, from pattern, 952 iron, analyses of, 373 Cast iron, 365-375 and steel mixtures, 375 bad, 375 malleable, 375 specifications, 374 specific gravitv, 37'4 strength of, 369, 374 Catenary, construction of, 51 the wire rope, 919 Cement, weight of, 170 for belts 887 mortar, strength of, 313 Centigrade and Fahrenheit table, 449 Centre of gravity, 418 of gyration, 420 of oscillation, 421 of percussion, 421 Centrifugal fans, 511 force, 423 force in fly-wheels, 820 tension of belts, 876 Cera-perduta process, alloys for, 326 Chain-blocks, 907 cables, 308, 340 Chains, crane, 232 weight and strength, 307, 339 Channel beams, sizes and weight, 178, 180 Channels, steel, strength of, 275 Charcoal, 640 making results, 642 pig iron, 365, 374 weight of, 170 1077 Chemical elements, 163 Chimneys, 731-741 brick, 737 for ventilation, 533 stability of, 738 size of, 734 steel, 740 slieet iron, 741 table of sizes of, 735 Chords of circles, 57 Chrome steel, 409 Circle, equation of, 70 measures of, 57-58 Circles,- problems, 39-40 tables of, 103, 108 Circular arc, length of, 58 arcs, tables of, 114, 115 functions in calculus, 78 measure, 20 ring, 59 Circulating pump, 839 Circumference of circles, 103, 108, 113 Cisterns, cylindrical, 121, 126 Clearance in steam-engines, 752, 792 Coals, analyses of, 624-631 calorimetric tests, 636 evaporative power of, 636 heating value of, 634 relative value of, 633 Coal gas, illuminating, 651 hoisting, 343 products of distillation of, 639 washing, 638 weathering of, 637 Coefficient of elasticity, 237 Coefficients of friction, 928-932 Coiled pipes, 198 Coils, heating of, 1036 Coke, 637 Coking, experiments in, 637 Coke manufacture, by products, 639 Cold drawing steel, 305 Cold, effect of on iron and steel, 383 Cold rolling, effect of, 393 Columns, built, 256 cast iron, weight of 185 iron tests of, 305 strength of, 246-250 Combined stresses, 282 Combination. 10 Combustion, heat of, 456, 621 gases of, 622 theory of, 620 Composition of forces, 415 Compressed air, 499 cranes, 912 motors, 507 transmission, 488 Compressed steel, 410 Compression iw steam-engines, 751 Compression unit strains, 380 Compressive strength, 244 Compressive strength of iron bars, 304 Compressors, air, 503 Compound engines, 701-768 diameter of cylinder, 768 economy of, 780 Compound engines, work of steam in, 767 Compound interest, 14 Compound numbers, 5 Compound units of weight and measure, 27 Condenser, evaporative surface, 844 Condensers, 839-846 Condensing water, continuous use of, 844 Conduction of heat, 468 Conductivity, electrical, 1028 Conductors, electrical, 1029 Cone, measures of, 61 pulleys, 874 Conic sections, 71 Conoid, parabolic, 63 Connecting rods, 799 tapered, 801 Conservation of energy, 432 Construction of buildings, 1019 Convection of heat, 468 Conveyors, belt, 911 Co-ordinate axes, 69 Copper, 167 at high temperatures, strength of, 309 balls, hollow, 289 round bolt, 203 strength of, 300 tubing, 200 wire and plates, 202 wire tables of, 218-220 Copper wire, resistance of hot and cold, 1034, 1035 cost of for long - distance trans- mission, 1045 Cordage, 341,344, 906 Cork, properties of, 316 Corrosion of iron, 385 Corrosion of steam-boilers, 716, 719 Corrosive agents in atmosphere, 386 Corrugated iron, 181 Corrugated furnaces, 266, 709 Cosecant of an angle, 65 Cosine of an angle, 65 Cosines, tables of, 159 Cost of coal for steam-power, 789 of steam-power, 790 Cotangent of an angle. 65 Counterbalancing engines, 788 locomotives, 864 of wiuding engines, 909, Couples, 418 Coverings for steam-pipes, 469 Cox's formula for loss of head, 575 Cranes, classification of, 911 compressed air, 912 stress in, 440 Crank angles, 830 arms, 805, 806 pins, 801-804 Crucible steel, 410 Crushing strength of masonry ma- terials, 312 Cubature of volumes of revolution, 75 Cube root, 8 Cubes and cube roots, table of, 86 107* Cubic measure, 18 Cupolas, blast-pipes for, 519 blowers for, 519 practice, 946 Current motors, 589 Currents, electric, 1030 Cutting stone with wire, 966 Cycloidal teeth of gears, 892 Cycloid, construction of, 50 differential equation of, 79 Cylinders and pipes, contents of, 120, 121 condensation, 752, 753 engine, dimensions of, 792 hollow, resistance of, 264 hollow, strength of, 287-289 measures of, 61 Cylindrical ring, 62 Dangerous steam-boilers, 720 Dam, stability of a, 417 D'Arcy's formula, flow of w r ater, 563 Decimals, 3 Decimal equivalents of fractions, 3,4 Decimals, squares and cubes of, 101 Deck-beams, sizes and weights, 177 Delta metal, 326 Delta-metal wire, 225 Denominate numbers. 5 Deoxidized bronze, 327 Derricks, stresses in, 441 Diametral pitch, 888 Differential calculus, 72 gearing, 898 pulley, 439 screw, 439 screw, efficiency of, 974 windlass, 439 Discount and interest, 13 Draught of chimneys, 731 Drawing-presses, blanks for, 973 Drilling: holes, speed of, 956 machines, electric, 956 Drop-press, pressure of, 973 Drums for hoisting-ropes, 917 Drying and evaporation, 462 Drying in vacuum, 466 Dry measure, 18 Ductility of metals, 169 Dust explosions, 642 Dust-fuel, 642 Durability of iron, 385 Durand's rule, 56 Duty trials of pumping-engines, 609 Dynamo and engine, efficiency of, 1048 electric machines, 1061 Dynamos, designing of, 1064 efficiency of, 1066 Dynamometers, 978-980 Earths, weight of, 170 Earth-filling, weight of. 170 Eccentric loading of columns, 255 Eccentrics, steam engine, 816 Economizers, fuel, 715 Edison or circular mil wire gauge, 30-31 Effort, definition of, 429 Elastic limit, 236 elevation of, 238 Elasticity, modulus of, 237, 314 Elastic resilience, 270 Electric accumulators, 1055 generator, efficiency of, 1048 heating, 1054 lighting, 1051 motor, 1070 pumping plant, 1049 railways, 1050 transmission, 1038 transmission, economy of, *1039 welding, 1053 Electrical engineering, 1024 horse -powers, table of, 1042 resistance, 1028 standards of measurement, 1024 units, 1024 Electric conductivity of steel, 403 Electricity, analogy with flow of water, 1026 heating by, 546 Electro-chemical equivalents, 1057 magnetic measurements, 1058 magnets, 1058 magnets, polarity of, 1060 Electrolysis, 1057 Elements, chemical, 163 Elements of machines, 435 Ellipse, construction of, 45 equation of, 70 measures of, 59 Ellipsoid, 63 Elongation, measure of, 243 Emery, grades of, 968 wheels, 967-969 Endless screw, 440 Energy, conservation of, 432 of recoil of guns, 431 or stored work, 429 sources of, 432 Engine-frames or bed-plates, 817 Engines, blowing, 526 gas, 847 gasoline, 850 hoisting, 908 hot-air, 851 marine, sizes of steam -pipes, 674 naptha, 851 petroleum, 850 steam, 742-847 Epicycloid, 50 Equalization of pipes, 492 Equation of payments, 14 Equations, algebraic, 34 Equilibrium of forces, 418 Equivalent orifice, 533 Erosion by flow of water, 565 Evaporating by exhaust-steam, 465 Evaporation and drying, 462 by the multiple system, 465 from reservoirs. 463 latent heat of, 461 tabl3 of factors of, 695 Evaporators, fresh -water, 1018 Evolution, 7 Eye-bars, tests of, 304, 393 1079 Exhaust steam for heating, 780 Exhausters, steam-jet. 526 Expansion by heat, 459 of iron and steel, 385 of steam, 742 of steam, real ratios of, 750 of wood, 311 Explosive energy of steam-boilers, 720 Exponents, theory of, 36 Exponential functions, in calculus, 78 Factor of safety, 314 in steam boilers, 700 Factors of evaporation, tables, 695 Fahrenheit and centigrade table, 450 Failures of steel. 403 Falling bodies. 423-426 Fans and blowers, 511-526 Feed-pumps, 843 water, cold, strains caused by, 727 water, heater, Weir's, 1016 water, heaters, 727 water, purifying, 554 Fibre-graphite, 945 Fifth roots and fifth powers, 102 Fink roof-truss, 446 Fire brick, sizes of, 233-235 clay, analysis of 234 engines, capacities of, 580 Fireless locomotive, 866 Fireproof buildings, 1020 Fire-streams, 580 Fire temperature of, 622 Flagging, transverse strength'of, 313 Flanges, pipe, 192, 193, 676 Flat plates in steam-boilers, 701, 713 plar.es, strength of, 283 Flexure of beams, 267 Flooring material, weight of, 281 Floors, strength of, 1019, 1021 Flowing water, horse-power of, 589 Flow of air in pipes, 485 of air through orifices, 484 of compressed air, 489, 493 of gas in pipes, 657 of metals, 973 of steam in pipes, 669 of water from orifices, 555, 584 of water in house service-pipes. 578 of water over weirs, 586 Flues, collapse of, 265 corrugated. U. S. rules, 709 (see also Tubes and Boilers.) Fly-wheels, 817-824 arms of, 820 wire-wound, 824 wooden, 823 Flyun's formula, flow of water, 562 Foot, decimals of, in fractions of inch, 112 pound, unit of work, 428 Forced draught in marine practice, 1015 Force of a blow, 430 of acceleration, 427 unit of, 415 Forces, composition of, 415 Forces, equilibrium of, 418 parallel, 417 parallelogram of, 116 parallelopipedon of, 416 polygon of, 416 resolution of, 415 Forcing and shrinking fits, 973 Forging, hydraulic, 618, 620 tool steel, 413 Foundry, the, 946*956 Fractions, 2 Francis's formula for weirs, 586 Freezing of water, 550 French measures and weights, 21-26 Friction and lubrication, 92S-945 brakes, 980 of air in passages, 531 of steam-engines, 941 rollers, 940 work of, 938 Frictional heads, flow of water, 577 Fuel, 622-651 economizers, 715 gas, 646 pressed, 633 theory of combustion of, 620 weight of, 170 Fuels, classification of, 623, 624 Furnace, downward draught, 635, 712 kinds of, for different coals, 635 formulae, 702 Furnaces, corrugated, 266 for boilers, 711 gas-fuel, 651 use of steam in, 650 Fusible alloys, 333 plugs for boilers, 710 Fusibility of metals, 169 Fusing disk, Reese's, 966 Fusion of wires by electric currents, 1037 temperatures, 455 g, value of, 424 Gallons and cubic feet, table, 122 . Galvanized wire rope, 228 Gas-engines, 847 fired steam-boilers, 714 flow of in pipes, 657 fuel, 646 illuminating, 651 illuminating, fuel value of, 656 Gases, properties of, 479 waste, use under boilers, 689, 690 Gasoline-engines, 850 Gas-pipe sizes and weights, 188 Gas producers 649, 650 Gauges, wire and sheet metal, 26-31 Gearing, efficiency of, 899 frictional, 905 of lathes, 955 speed of, 905 toothed-wheel, 439, 887-906 Gear-teeth, strength of, 900-905 Geometrical problems, 37 progression, 11 propositions, 53 German silver, 326, 332 1080 1KDEX. German silver, strength of, 300 Girders for boilers, 703 Glass, skylight, 184 strength of, 308 Gold, 167 Gordon's formula, 247 Governors, 836 Grain, weight of, 170 Granite, strength of, 312 Grate and heating surface of a boil- er, 678 Graphite as a lubricant, 945 paint, 389 Gravity acceleration due to, 424 centre of, 418 specific, 163 Greatest common measure, 2 Greenhouse heating by hot water, 541 heating by steam, 542 Green's fuel economizers, 715 Grindstones, 968, 970 Gyration, centre and radius of, 420, 421 radius of, 247 Hawley down-draught furnace, 712 Heads of boilers, 706 Heat, 448-478 boiling-points, 455 conduction of, 468 convection of, 469 expansion by, 459 generated by electric currents, 1032 latent, 461 latent, of evaporation, 462 latent, of fusion, 461 melting-points, 455 of combustion, 456 radiation of, 467 specific, 457 storing of, 789 unit, 455, 660 Heaters, feed-water, 727 Heating a building to 70° F., 545 and ventilation, 528-546 blower system of, 545 by electricity, 546 by exhaust-steam, 780 by hot water, 542 of large buildings, 534 surface of boilers, 678 Helix, 60 Hodgkinson's formula, 246 Hoisting, 906-916 coal, 343 engines, power of, 908 pneumatic, 909 rope, 340 Hooks, hoisting, 907 Horse-gin, 434 work of a, 434 Horse-power, 429 constants, 757, 758 of flowing water, 589 of steam-boilers, 677, 679 of steam-engines, 755 power-hours, 429 Hose, friction losses in, 580 Hot-air engines, 851 water heating, 542 Howe truss, 445 Humidity in atmosphere, 483 Hydraulic apparatus, 616 engine, 619 Hydraulics, flow of water. 555-588 forging, 618 grade-line, 578 power, 617 pressure transmission, 616 ram, 614 Hydrometer, 165 Hygrometer, dry and wet bulb, 483 Hyperbola, equation of, 71 construction of, 49 Hyperbolic logarithms, 156 curve in indicator diagrams, 759 Hypocycloid, 50 I-beams, sizes and weights, 177 properties of, 274 spacing of, 273, 280 Ice and snow, 550 Ice-making machines, 981-1001 manufacture, 999 Illuminating gas, 651 Impact of bodies, 431 Incandescent lamps, 1051 Inches and fractions, decimals of a foot, 112 Inclined plane, 437 planes, hauling on, 913, 915 planes, motion on, 428 Incrustation and scale, 551, 716 India rubber, tests of, 316 Indicated horse-power, 755 Indicator diagrams, 754, 759 rigs, 759 Indirect heating surface, 537 Inertia, 415 moment of, 247, 419 of railroad trains, 853 Injectors, 725 Inspection of steam boilers, 720 Insulators, electrical, 1029 Integral calculus, 79 Integrals, integration, 73-74 Intensifier, the Aiken, 619 Interest and discount, 13 Involute gear- teeth, 892 construction of, 52 Involution, 6 Iridium, 167 Iron, 167 bars, sizes of, 170 bars, weight of, 171 corrosion of, 385 durability of, 385 and steel, classification of, 362 and steel, cold rolling of, 393 and steel, expansion of, 385 and steel, strength of, 296-300 and steel, strength at high tem- perature, 382 and steel, strength at low tem- perature, 383 INDEX. 1081 Irregular figure, area of, 55 Isothermal expansion, 742 Japanese alloys, 326 Jet propulsion of vessels, 1014 Jets, vertical water, 579 Joints, riveted, 354-363 Joule's equivalent, 456 Journal bearings, 810-815 friction, 931 Journals, engine, 810-815 Kerosene for scale in boilers, 718 Keys and set-screws. 977 for mill-gearing, 975 holding power of, 977 Kinetic energy, 429 King-post truss, 442 Kirkaldy's tests of materials, 296 Knot, or nautical mile, 17 Knots in ropes, 344 Kutter's formula, 559 Lacing of belts, 883 Ladles, foundry, sizes of, 953 Latent heat, 461 heat of evaporation, 461 heat of fusion, 459-461 Lathe-gearing, 955 tools, speed of, 953 Lap and lead of a valve, 829-833 Lead, 167 pipe, 200 Leakage of steam in engines, 761 Least common multiple, 2 Leather, strength of, 302 Leveling by barometer, 482 Levers, 435 Lime, weight of, 170 Limestone, strength of, 313 Limit gauges for screw-threads, 205 Lines of force, 1059 Links, engine, 816 Link-motion, 834 Liquation of alloys, 323 Liquid measure, 18 Liquids, weight and sp. gr., 164 Locomotives, 851-866 dimensions of, 860 tests of, 863 Logarithmic curve, 71 sines, etc., 162 Logarithms, hyperbolic, 156 differential of, 77 of numbers, 127-155 Logs, lumber, etc., weight, 232 Loop, the steam, 676 Loss of head in pipes, 573 Lubricants, 942 Lubrication, 942 Lumber, weight of, 232 Machines, elements of, 435 Machine-shop, the, 953-978 : shop practice, 953 shops, power used in, 965 screws, 208, 20'J tools, power required for, 960-965 tools, proportioning sizes of, 975 Maclaurin's theorem, 76 Magnesia bricks, 235 Magnesium, 168 Magnetic balance, 396 circuit, 1060 circuit, units of, 1058 field, strength of, 1063 Magnets, winding of, 1068 Malleability of metals, 169 Malleable castings, rules for, 376 cast iron, 375 Mandrels, sizes of, 972 Manganese, 168 bronze, 331 influence of on cast iron, 368 influence of on steel, 389 plating of iron, 387 steel, 407 Mannesmann tubes, 296 Manometer, air, 481 Man-power, 433 Manure as fuel, 643 Man-wheel, 434 Marble, strength of, 302 Marine engineering, 1001-1018 engine, horse-power of, 766 engine practice, 1015 engine, ratio of cylinders, 766, 773 Masonry materials, strength of, 312 materials, weight and sp. gr., 166 Materials, 163-235 strength of, 236-412 Maxima and minima, 76 Measures and weights, 17 Mechanical equivalent of heat, 456 powers (see Elements of Ma- chines), 435 stokers, 711 Mechanics, 415-447 Mekarski compressed air tramway, 509 Melting points of substances, 455 Memphis bridge, strains on steel, 381 Mensuration, 54 Mercury. 168 bath pivot, 940 Merrimau's formula for columns, 260 Metaline, 945 Metals, properties of the, 166 weight and sp. gr., 164 Metric conversion, tables, 22-26 measures and weights, 21-22 Meters, water, 579 Mil, circular, 18, 29 Milling cutters, 957 cutters for gears, 892 machines, results with, 959 Mill power, 589 Miner's inch, 18 inch measurement, 585 Mine-ventilating fans, 521 ventilation, 531 Modulus of elasticity, 237, 314 Moisture in steam, 728 Molesworth's fo.mula, flow of water, 562 Moment of force, 416 1082 Moment of inertia, 247, 419 statical, 417 Momentum, 428 Morin's laws of friction, 933 Mortar, strength of, 313 Motor, electric, 1070 Motors, compressed air, 507 Moulding sand, 952 Moving strut, 436 Mules, power of, 435 Multiphase currents, 1070 Mushet steel, 409 Nails, 213, 215 screws, etc., holding power, 289-291 Naphtha-engines, 851 Napier's rule for flow of steam, 669 Natural gas, 649 Newton's laws of motion, 415 Nickel steel, 406 Nozzles, measurement of water by. 584 Nuts and bolts, 209, 211 Ohm's law, 1030 Oil needed for engines, 943 Oils and coal as fuel, 646 lubricating, 944 Open-hearth steel, 391 Ores, weight of, 170 Oscillation, centre of, 421 Oxen, power of, 435 Ordinates, 69 it, value of, 57 Packing-rings, engines, 796 Paddle-wheels, 1013 Painting wood and iron structures, 388 Paint, qualities of, 389 Parabola, construction of, 48 equation of, 71 Parabolic conoid, 63 Parallel forces, 417 Parallelogram, 54 of forces, 416 Parentheses, 33 Partial payments, 15 Peat or turf. 643 Pelton-wheel table, 598 water-wheel, 597 Pendulum, 422 conical, 423 Percussion, centre of, 421 Permutation, 10 Perpetual motion, 432 Petroleum, 645 as fuel, 645, 646 burning locomotive, 865 distillates of, 645 engines, 850 products, specifications, 944 Phosphor-bronze, 327 bronze wire, 225 Phosphorus, influence of on cast iron, 367 influence of on steel, 389 Piezometer, the, 582 Pig iron, analysis of, 371 Pig iron, chemistry of, 370 grading of, 365 influence of silicon, etc., 365. tests of, 369 Pillars, strength of. 246 Pitot tube gauge, 583 Pipe, lead, 200, 201 riveted, 197 sheet-iron hydraulic, 191 spiral riveted, 197 Pipes, air-bound, 579 and cylinders, contents of, 120, 121 cast-iron, thickness of, 188, 190 cast-iron, weight of, 185, 186 coiled, 198 effect of bends in, 488 flow of air in, 485 flow of gas in, 657 flow of steam in, 669 flow of water in, 557 loss of head in, 573 steam, for steam-heating, 540 steam, sizes for engines, 673 water, riveted, 295 water, wrought-iron and steel, 295 wrought iron, 194 Piston-rods, 796-798 valves, 834 Pistons, steam-engine, 795 Pitch, diametral, 888 of gears, 887 of rivets, 357-359 of screw propellers, 1012 Pivot bearings, 939 Plane surfaces, mensuration, 54 Plane, inclined (see Inclined Plane) Plate iron, weight of, 175 steel, classification, 399 Plates, brass and copper, 202 square feet in, 123 strength of for boilers, 705 Platinum, 168 wire, 225 Pneumatic hoisting, 909 postal transmission, 509 Polyedrons, 62 Polygon of forces, 416 Polygons, construction of 42 table of, 55 tables of angles of, 44 Population of the United States, 12 Potential energy, 429 Powell's screw-thread, 975 Power, animal, 433 of a fall of water, 588 rate of work, 429 of ocean waves. 599 Power stations, electric, 1050 Powers of numbers, 7, 33 Pratt truss, 443 Pressed fuel, 632 Presses, punches, etc., 972 Prism, measures of, 60 Prismoid, 61 rectangular, 61 Prismoidal formula, 62 Problems, geometrical, 37-52 in circles, 39, 40 in lines and angles, 37, 38 INDEX. 1083 Problems in polygons, 42 in triangles, 41 Producer, gas, 649 Progression, arithmetical, and geo- metrical, 11 Prony brake, 979 Proportion, 5 Pulley, differential, 439 Pulleys, 873-875 arms of, 820 or blocks, 438 Pulsometer, 612 Pumps, 601-614 air, 841 air-lift, 614 boiler-feed, 605, 726 capacity of, 601 centrifugal, 606, 609 circulating, 842 efficiency of, 603, 608 horse-power of, 601 piston speed of, 605 sizes of, 603 speed of water through, 602 steam cylinders of, 602 suction of, 602 vacuum, 612 valves of, 605, 606 Pumpiug-engine, tests of, 783 Pumping-engines, duty trials, 609 leakage test, 611 Punched plates, strength of, 354 Punches and dies, 972 Punching and drilling steel, 395 steel, effect of, 394 Purifying feed-water, 554 Pyramid, 60 Pyrometers, 451-453 Pyrometry, 448-454 Quadratic equations, 35 Quadrature of surfaces of revolu- tion, 75 Quadruple-expansion engines, 772 Quantitative measurement of heat, 455 Quarter-twist belts, 883 Queen-post truss, 442 Radiating surface, rules for. 536 Radiation of heat, 467 Radius of gyration, 247, 420, 421 Railroad trains, resistance of, 851 Rail-steel, specifications, 401 Rails, maximum safe load on, 865 steel, strength of, 298 Railways, narrow-gauge, 865 Railway trains, speed of, 859 Ram, hydraulic, 614 Ratio and proportion, 5 Reamers, taper, 972 Recalescence of steel, 402 Receiver-space in engines, 766 Reciprocals of numbers, 80 use of, 85 Red- lead as a preservative, 3S9 Refrigerating machines, 981-1001 Registers and air-ducts, 539 Regnault's experiments on steam, 661 Resilience, 238 elastic, 270 Resistance, electrical, 1028 of copper wire, 1030 of ships, 1002 of trains, 851 to repeated stresses, 238 Resolution of forces, 415 Rhombus and Rhomboid, 54 Riveted joints, 299, 303, 354, 362 Riveting-machines, hydraulic, 618 of steam-boilers. 700 of steel plates, 394 pressures, 362 Rivets, diameter of, 360 sizes, etc., 211 Rivet -steel, 401 Roads, resistance of carriages on, 435 Rock-drills, air required by, 506 Roof-coverings, weight of, 184 trusses, 446 Roofing materials, 181, 184 Rope-driving, 922-927 wire-, 226-231 Ropes, 301, 338, 906 splicing 341, 345 Rotary blowers, 526 steam-engines, 791 Rotation, accelerated, 430 Rubber-belting, 887 Rule of three, 6 Rustless coatings for iron, 386 Safety, factors of, 721 Salinometer, strength of brines, 464 Salt, solubility of, 464 weight of, 170 Sand-blast, 966 moulding, 952 Sawdust as fuel, 643 Sawing metal, 966 Scale and incrustation, 551 in steam-boilers, 716 Schiele's anti-friction curve, 50 Schiele pivot-bearing, 939 Screw-bolts, efficiency of, 974 differential, 439 endless, 440 the, 437 thread, Powell's, 975 threads, 204, 208 thread, metric, 956 propeller, 1010 Screws and screw-threads, 974 holding power of, 290 machine, 208, 209 Secant of an angle, 65 Sectors and segments, 59 Sediment in steam-boilers, 717 Segments of a circle, table, 116 Segregation in steel ingots, 404 Separators, steam, 728 Set-screws, holding-power of, 977 Sewers, grade of, 566 Shaft-beariugs, 810 governor, 838 1084 IKDEX. Shafting, 867-872 Shafts, engine, 806-809 fly-wheel, 809 propeller, strength of, 815 Shapes of test-specimens. 243 Shearing, effect of, on steel, 394 strength of iron, 306 strength of woods, 312 resistance of rivets, 363 unit strains, 380 Shear poles, stresses in, 442 Sheet-iron and steel, weight of, 174 Shells, spherical, strength of, 286 Shingles, sizes and weight, 183 Shippiog measure, 19 Ships, resistance of, 1002 Shocks, resistance to, 240, 241 Shot, lead, 204 Shrinkage of castings, 951 Shrinking-fits, 973 Signs of trigonometrical functions, 66 arithmetical, 1 Silicon-bronze wires, 225, 328 Silicon, influence of on cast-iron, 3G5 influence of on steel, 389 Silver, 168 Simpson's rule, 56 Sine of an angle, 65 Sines, cosines, etc., table of, 159 etc., logarithmic, 162 Sinking-funds, 17 Siphon, the, 581 Slate, sizes and weights, 183 Slide-valve, 824-835 Smoke-prevention, 712 Snow and ice, 550 Soapstone as a lubricant, 945 Softeners, use of in foundry, 950 Softening hard water, 555 Solders, 338 Solid bodies, mensuration of, 60 of revolution, 62 measure, 18 Specifications for axles, steel, 400 for car-axles, 401 for cast iron, 374 for crank -pin steel, 400 for oils, 944 for plate steel, 399, 400 for rail steel, 401 for rivets, 401 for spring steel, 400 for steel, 397 for steel castings, 406 for steel rods, 400 Specific gravity, 163 gravity of alloys, 320, 323 gravity of cast iron, 374 gravity of steel, 403, 411 heat, 457 heat of air, 484 Speed of cutting tools, 953, 954 of vessels, 1006 Sphere, measures of, 61 Spherical polygon, volume of, 61 shells, strength of, 286 steam-engine, 792 triangle, area of, 61 Spherical zone, 62 Spheroid, 63 Spheres, table of, 118 weights of, 169 Spikes, sizes and weights, 212, 213 Spindle, surface and volume, 63 Spiral, construction of, 50 gears, 897 measures of, 60 Splicing of ropes, 341 wire ropes, 345 Spring steel, specifications, 400 strength of, 299 Springs for governors, 838 formula for, 347, 353 tables of, 349, 353 Square measure, 18 root, 8 Squares and cubes of decimals, 101 and square roots, table of, 86 Sui'face condenser, 840, 844 Sugar manufacture, 643 solutions, concentration of, 465 Sulphate of lime, solubility, 464 Sulphur dioxide refrigerating ma- chines, 985 influence of on cast iron, 367 influence of on steel, 389 Suspension cable-ways, 915 Stability, 417 Stand-pipes, design of, 292-294 Statical moment, 417 Stay-bolt iron, 379 Stay-bolts for boilers, 710 Stayed surfaces, strength of, 286 Stays for boilers, 703 Steam, 659-676 boiler, water tube, 688, 689 boilers, 677-740 boilers, air space in grate, 681 boilers, allowable pressures, 706 boilers, economy, of, 682 boilers, efficiency of, 683 boilers, explosive energy of, 720 boilers, factor of safety, 700 boilers, forced draught, 714 boilers, flues and passages, 680 boilers, gas-fired, 714 boilers, grate surface of, 678, 680 boilers, heating air supply to, 687 boilers, healing surface of, 678 boilers, horse-power of, 677 boilers, hydraulic test of, 700 boilers, incrustation and scale, 716 boilers, materials for, 700 boilers, measure of duty of, 678 boilers, performance of, 681 boilers, Philadelphia inspection rule, 708 boilers, proportions of, 678 boilers, rules for construction of, 700 boilers, safe working pressure, 707 boilers, strength of, 700 boilers, tests of, 685 boilers, tests, rules for, 690-695 boilers, tests with different coals, 688 boilers, tubulous, 686 ItfDEX. 1085 Steam boilers, using waste gases, 689 boilers, use of zinc in, 720 domes on boilers, 711 dry, identification of, 730 engine constants, 757 engine cylinders, 792-795 engine, spherical, 792 engines, 742-847 engines at Columbian exhibition, 774 engines, calculation of mean ef- fective pressure, 744 engines, counterbalancing, 788 engines, dimensions of parts of, 792-817 engines, efficiency of, 749 engines, feed-water consumption of, 753, 760, 775 engines, friction of, 941 engines, marine, 1015 engines, measures of duty, 748 engines, most economical point of cut-off, 777 engines, performance of, 775-789 engines, putting on center, 834 engines, relative economy of, 780 engines, rotary, 791 engines, triple expansion, 769 expansive working of, 747 flow of, 668 flow of in pipes, 669 heating, 536-540 jet- blowers, 527 loop, 676 loss of pressure in pipes, 671 mean pressures, 743 moisture in, 728 pipe coverings, 469 pipes, copper, 674 pipes, loss from uncovered, 676 pipes, marine, 1016 pipes, overhead, 537 pipes, size of for engines, 673 pipes, valves in, 675 pipes, wire-wound, 675 superheated, 661 supply mains, 539 table of properties of, 659 temperature, pressure, etc., 659- 662 turbines, 791 vessels, dimensions, horse-power, etc., 1009 vessels, trials of, 1007 water in, effect of on economy of engines, 781 work of iu a single cylinder, 746, 749 Steel, 389-414 aluminum, 409 analyses and properties of, 389 annealing, 413 Bessemer, 390-392 castings, 405 chrome, 409 compressed. 410 crucible. 410 effect of hammering, 412 Steel, effect of heat on, 412 effect of nicking, 402 electric conductivity of, 403 failures of, 403 hardening of soft, 393 manganese, 407 mushet. 409 open-hearth, 391, 392 segregation in, 404 specific gravity of, 403, 411 specifications for, 397 strength of, 389 tempering 412, 414 treatment of, 394 tungsten, 409 use in structures, 405 variation in strength of, 398 working stresses for, 264 Stone, strength of, 302, 312 Stoker, under-feed, 712 Stokers, mechanical, 711 Storage batteries, 1055 Storing steam-heat, 789 Strains in structural iron, 379 Straw as fuel, 643 Stream, horse-power of a, 589 Streams, measurement of, 584 Strength of boiler-heads, 285 of bolts, 292 of columns, 246, 250-261 compressive, 244 of flat plates, 283 of glass, 308 of materials, 236 of materials, Kirkaldy's tests, 296 of stayed surfaces, 286 of structural shapes, 272 of timber, 310 of unstayed surfaces, 284 torsional, 281 transverse, 266 tensile, 242 Stress and strain, 236 due to temperature, 283 Stresses, combined, 282 effect of. 236 in framed structures, 440 Structural iron, strains in, 379 shapes, elements of, 249 shapes, properties of, 272 shapes, sizes and weights, 177 steel, treatment of, 394 Structures, framed, 440 Struts, strength of, 246 Surface condensers, 840 Sugar concentration, 465 manufacture, 643 Tail-rope system of haulage, 913 Tan-bark as fuel, 643 Tangent of an angle, 65 Tangents, sines, etc., table of, 159 Tanks, cylindrical, 121, 126 rectangular, gallons in, 125 Tannate of soda, for boiler-scale, 718 Tap drills, 970, 971 Taper-bolts, pins, reamers, 972 in lathes, 956 Tapered wire ropes. 916 1086 IXDM. Taylor's rules for belting, 880 Taylor's theorem, 76 Tee-bars, 280 Tees, Pencoyd, sizes and weights, 180 Teeth of gears, forms of, 892 Telegraph-wire, 217 Temperature, effect on tenacity, 382 stresses in iron, etc., 283 Temperatures in furnace's, 451 judged by color, 454 Tempering steel, 414 Tenacity of metals, 169 of metals at different tempera- tures, 382 Tensile strength. 242 strength, increase by twisting, 241 Terra-cotta, 181 Test-pieces, comparison of small and large, 393 Testing materials, precautions in, 243 Thermal unit, British, 455 Thermodynamics, 478 Thermometers, 448 Three-wire currents, 1039 Tie-rods for brick arches, 281 Tiles, sizes and weights, 181 Timber measure, 20, 21 strength of, 310 Time, measure of, 20 Tin. 168 roofing, 181, 182 Tires, steel, strength of, 298 Tobin bronze, 327 Toggle-joint, 436 Tool-steel, heating, 412 Tonnage of vessels, 19, 1001 Torque of an armature, 1062 Torsional strength, 281 Tower spherical engine, 792 Tractive power of locomotives, 857 Tractrix, or Schiele's curve, 50 Trains, resistance of, 851, 853 Tramways, wire-rope, 914 Transformers, 1070 Transmission by hydraulic press- ure, 617-620 by wire rope, 917-922 eiectric, 1038 electric, efficiency of, 1047 of heat, 471-478 of power by ropes, 922-927 Transporting power of water, 565 Transverse strength, 266 Trapezium, 54 Trapezoid, 54 Trapezoidal rule, 55 Treatment of steel, 394 Triangle, mensuration of, 54 Triangles, problems in, 41 solution of, 68 Trigonometrical functions, 65-67 functions, table, 159 Trigonometry, plane, 65 Triple effect, multiple system, 463 expansion engine, 769 Troy weight, 19 I Truss, Howe and Warren, 445 king-post, 442 queen post, 442 Pratt or Whipple, 443 Trusses, roof, 446 Tungsten-aluminum alloys, 332 steel. 409 Tubes for steam-boilers, 704, 709 Mannesman n, 296 or flues, collapse of, 265 weights of, 169 wiought-iron, 196 Tubing, brass, 198 Turbines, steam, 791 Turbine-wheels, 591 wheels, tests of, 596 Turf or peat, 643 Turnbuckles, sizes, 211 Twin screw vessels, 1017 Twist-drills, sizes and speeds, 957 Type metal, 336 Unit of heat, 455 Unstayed surfaces, strength of, 284 Upsetting of steel, 394 Vacuum pumps, 612 Valve-diagrams, 825 motions, 825 rods, 815 Valves, engine-setting, 834 in st earn -pipes, 675 of pumps, 605, 606 Vapors, properties of, 480 used in refrigerating-machines, 982 Velocities, parallelogram of, 426 Velocity, angular, 425 Ventila'ting-ducts, discharge of, 530 fans, 517-525 Ventilation and heating, 528-546 blower system of, 546 by a steam-jet, 526 of large buildings, 534 Ventilators for mines, 521 Venturi meter, 583 Versed sine of an arc, 66 Vibrations of engines, preventing, 789 Vis-viva, 428 Warehouse floors, 1019 Warren girder, 445 Washers, sizes of, 212 Water, 547-554 analyses of, 553 compressibility of, 551 erosion by flowing, 565 expansion of, 547 flow in channels, 564 flow of, 555-588 flow of, experiments, 566 flow of, in pipes, tables, 558, 559, 567-570 gas, 648, 652 impurities of, 551 power, 588 power, value of, 590 pressure engine, 619 ' pressure of, 549 1087 Water, softening of hard, 555 specific heat of, 550 transporting power of, 565 weight of, 547 wheel, the Pelton, 597 Waves, power of ocean, 599 Weathering of coal, 637 Wedge, the, 437 volume of a, 61 Weights and measures, 17 of air and vapor, 484 Weight of bars, rods, plates, etc., 169 of brick-work, 169 of brass and copper, 197-203 of bolls and nuts, 209-211 of cast-iron pipes and columns, 185-193 of cement, 170 of flat rolled iron, 172 of fuel, 170 of iron bars, 171 of iron and steel sheets, 174 of ores, earths, etc., 170 of plate iron, 175 of roofing materials, 181-184 of steel blooms, 176 of structural shapes, 177-180 of tin plates, 182 of wrought-iron pipe, 194-197 of various materials, 169 Weir table, 587 Weirs, flow of water over, 586 Welding, electric, 1053 of steel, 394, 396 Welds, strength of, 300 Wheel and axle, 439 Whipple truss, 443 White-metal alloys, 336 Whitworth compressed steel, 410 Wiborgli air-pyrometer 453 Wind, 493 Winding engines, 909 of magnets, 1068 Windlass, 439 differential, 439 Windmills, 495 Wind pressures, 493 Wire cables, 222 copper and brass, 202 copper tables of, 218-220 gauges, 28-31 insulated, 221 Wire, iron and steel, 217 iron, size, strength, etc., 216 rope, 226, 231 rope haulage, 912 ropes, durability of, 919 ropes, splicing, 345 ropes, strength of, 301 ropes, tapered, 916 rope transmission, 917-922 rope tramways, 914 strength of, 301, 303 telegraph, 217 wound fly-wheels, 824 Wiring formula for incandescent lighting, 1043 Wires, current required to fuse, 1037 Wire -table, for 100 and 500 volts, 1044 table, hot and cold wires, 1034,1035 Wohler's experiments, 238 Wood as fuel, 639 composition of 640 expansion of, 311 heating value of, 639 strength of, 302, 306, 310, 312 weight of, 164, 232 Wooden fly-wheels, 823 Woodstone or xylolith,316 Woolf type of compound engine, 762 Woot ten's locomotive, 855 Work, energy, power, 428 of acceleration, 430 of men aud animals, 433 unit of, 428 Worm-gear, 440 gearing, 897 Wrought iron, 377-379 iron, chemistry of, 377 iron, specifications, 378 Xylolith or woodstone, 316 Yield point, 237 Z bars, properties of, 276 sizes and weights, 178 Zinc, 168 tubing, 200 use of, in steam-boilers, 720 Zeuner valve-diagram, 827 Zero absolute, 461 >