THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell f TEXT BOOKS FOR ENGINEERS AND STUDENTS. BY VOLSON WOOD, Professor of Engineering in Stevens Institute of Technology. This upon this has since beei A TREATISE ON THE RESISTANCE OP MATERIALS, AND AN APPENDIX ON THE PRESERVATION OF TIMBER. Seventh edition. 8vo, cloth $3 work originally consisted chiefly of the Lectures delivered by Professor Wood subject to the classes in Civil Engineering in the University of Michigan, but it been twice revised. It is a text-book, and gives a brief sketch of the history of the development of the theories connected with the growth of the subject, and a large amount of experimental matter. An English reviewer of the work pays : " It is equal in grade to Kankine's work." A TREATISE ON THE THEORY OF THE CONSTRUCTION OF BRIDGES AND ROOFS. Designed as a Text-Book and for Practical Use. Illustrated with numerous wood -engravings. Sixth edition. 1 vol., 8vo $2 00 The development of the subject in this work is progressive in its character. It begins with the most elementary methods, and ends with the most general equations. The skel- eton forms of the triangular trusses, including the Warren girder: and quadrangular forms, including the Long, Pratt, Howe, Towne, Whipple, Past, and other forms. The examples are original and many are novel in character. THE ELEMENTS OF ANALYTICAL MECHANICS. With numer- ous examples and illustrations. For use in Scientific Schools and Colleges. Seventh edition, including Fluids. 8vo. cloth $8 00 The Calculus and analytical methods are freely nsed in this work. It contains many problems completely solved, and many others which are left as exercise* for the student. The last chapter shows how to reduce all the equation* of mechanic* from the principle of D'Alemberu PRINCIPLES OF ELEMENTARY MECHANICS. Fully illustrated. Ninth edition. 12mo, cloth $1 25 The chief aim of this work is to define, explain and enforce the fundamental principles of mechanics. The analysis in simple, but the work is comprehensive. The principles of energy are applied to the solution or mauy problems. There is a variety of examples' at the end of each chapter. A novel feature of the work is the " Exercises," which contain many peculiar and inteiesting question*. SUPPLEMENT AND KEY TO PRINCIPLES OF ELEMENTARY . MECHANICS. 12mo, cloth $1 25 This work not only contains a solution of all the examples and answers to the exorcises, but much additional matter which can hardly fail to interest any student of this science. THE ELEMENTS OF CO-ORDINATE GEOMETRY. In Three Parts. I. Cartesian Geometry and Higher Logic. II. Quarternions. III. Modern Geometry and an Appendix. 1 vol., 8vo, cloth. New edition, with additions $2 00 It was designed to make this a thoroughly practical book for class use. The more abstruse parts of the subject are omitted The properties of the conic sections are. so far as practicable, treated under common heads, thereby enabling the author to condense the work. The most elementary principles only of qnarternions and modern geometry are presented ; but in these parts, as well as in the first part, are numerous examples. A TREATISE ON CIVIL ENGINEERING. By Prof. D. H. Mahan. Revised and edited, with additions and new plates, by Prof. De Volson Wood. With an Appendix and Complete Index. 8vo, cloth. . .$5 00 TRIGONOMETRY. Analytical, Plane, and Spherical. With Logarithmic Tables. Fourth edition. By Prof De Volson Wood. 12mo, cloth. $1 00 THERMODYNAMICS. By Prof. De Volson Wood, C.E., M.E. 8vo, cloth. Second edition $4 00 THEORY OF TURBINES. By Prof. De Volson Wood. 8vo, cloth, $2 5C THERMODYNAMICS, HEAT MOTORS, REFRIGERATING MACHINES. JOHM S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. BY DE VOLSON WOOD, C.E., M.A., tATE PROFESSOR OF MECHANICAL ENGINEERING, STEVENS INSTITUTE OF TECHNOLOGY. EIGHTH EDITION. REVISED AND ENLARGED. NEW YORK : JOHN WILEY & SONS, 53 EAST IOTH STREET. 1900. COFTBIOHTED BY DB VOLSON WOOD Engineering Library TJ JOHJM S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. PREFACE. THE following work has been prepared to meet a want experienced by myself in my course of instruction in Thermodynamics. After reading several works upon the subject, including those of the founders of the science Rankine, Clausius, Thomson I was most favorably impressed with the spirit of Rankine's mode of discussing the subject. It is in keep- ing with the modern method of treating Analytical Mechan- ics, in which the analysis is founded upon ideal conditions established by definitions, and the resulting formulas modi- fied to represent the infinite variety of conditions in nature. But Rankine's giant-like processes are not adapted to the wants of the average student. Article 241 of his Steam Engine and other Prime Movers reaches the height of sublimity in regard to terseness, comprehensiveness, and ob- scurity. Without a proper preliminary, he crowds into a few words a principle which has cost other writers protracted labor and heroic efforts to establish. My aim has not been to bring down the subject to the comprehension of the reader, but to lead him up, by a more easy and uniformly graded path, to the same height, and at the same time familiarize him with the way by a free use of illustrations, exercises, historic references, and numerical examples. 733386 IV PREFACE. The body of the work contains a development of the es- sential principles of the subject, to which I have added an Addenda, for the purpose of enlarging upon the matter con- tained in some of the articles, more especially those pertain- ing to vapors. This enabled me to follow the thread of the subject more closely without turning aside to consider applications to a variety of substances, and to enlarge more freely upon those secondary matters when separated from the body of the text. Special attention is called to the graphical representation of internal work, as in Figs. 36 and 37, supposed to be new, as well as many of the exercises and the discussion of En- tropy, or the Thermodynamic Function. DE VOLSON WOOD. HOBOKEN, Sept., 1888 THIRD EDITION. THE treatment of the theoretical part of Thermodynamics, including its application to the steam engine, as far as page 180, is the same in this edition as in the first and second editions. Since the first edition there have been added the following subjects : Vapor Engine ; Sterling's Engine ; Ericsson's Hot- Air Engine ; Gas Engine ; Naphtha Engine ; Ammonia Engine ; Steam Injector ; Pulsometer ; Com- pressed- Air Engine ; The Compressor ; Steam Turbine ; Re- frigerating Machines ; Miscellaneous matter in an Addenda ; Combustion of Fuel ; Steam, Ammonia and other Tables. The Ammonia Tables have been computed from the formu- las of the author and are new. THE AUTHOR. August, 1889. CONTENTS. CHAPTER I. GENERAL PRINCIPLES. ARTICLE PAG2 1. HEAT is ENERGY 1 2. HEAT is NOT MATERIAL 1 3. HEAT A RESULT OP THE MOTION OP THE PARTICLES 1 4. VELOCITY OP HEAT 2 5. HEAT ENERGY MEASURED BY ITS EFFECTS 2 6. THERMAL UNIT 3 7. WORK 3 8. INTERNAL WORK 4 9. ACTUAL ENERGY 4 10. LATENT HEAT 5 11. GENERAL EXPRESSION 5 12. TEMPERATURE 5 13. THERMOMETERS 6 14. THE AIR-THERMOMETER 7 15. A PERFECT GAS 8 16. AN ABSOLUTE SCALE 9 17. ABSOLUTE ZERO OP TEMPERATURE 10 18. EQUATION OP A GAS 10 19. EQUATION OP A PERFECT GAS 10 20. MARIOTTI'S LAW 12 21. LAW OF GAY LUSSAC 12 22. SO-CALLED IMPERFECT GASES 12 23. THERMAL LINES . . .13 vi CONTENTS. ABTICLE " 24. ISOTHERMAL LINES ] 25. ADIABATIC LINES 16 26. CYCLE 17 27. HEAT ENGINE 17 28. CARNOT'S CYCLE 1? 29. SOURCE 19 30. WORK DONE 19 31. INDICATOR DIAGRAM 20 32. CARNOT'S CVCLE is REVERSIBLE 20 33. CONDITIONS OF A REVERSIBLE CYCLE 21 34. HEAT ABSORBED 22 35. MECHANICAL EQUIVALENT OF HEAT 24 36. FIRST LAW OF THERMODYNAMICS 27 37. THERMAL CAPACITIES 28 38. SPECIFIC HEAT AT CONSTANT PRESSURE 29 39. SPECIFIC HEAT AT CONSTANT VOLUME 31 40. TEMPERATURE CONSTANT DURING EXPANSION 32 41. THOMSON AND JOULE'S METHOD 41 42. HEAT ABSORBED IN TERMS OF EXTERNAL WORK 43 43. HEAT TRANSMUTED INTO WORK 44 44. GENERAL CASE 45 45. TEMPERATURE AND PRESSURE AS INDEPENDENT VARIABLES. 46 46. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS 48 CHAPTER II. PERFECT GASES. 47. DIFFERENCE OF SPECIFIC HEAT 49 48. SPECIFIC HEAT CONSTANT 50 49. PERFECTNESS OF GASES 51 50. To FIND THE SPECIFIC HEAT OF A GAS AT CONSTANT VOLUME. 53 51. RELATIVE SPECIFIC HEATS 53 52. TEMPERATURE CONSTANT DURING EXPANSION 54 53. VOLUME CONSTANT 55 CONTENTS. V ABTICLB PAGE 54. SIGNIFICATION OF R 56 55. ARBITRARY PATH DURING EXPANSION 57 56. GAS EXPANDING WITHOUT TRANSMISSION OF HEAT 61 57. ICE MACHINES 62 58. AIR COMPRESSOR 63 59. VELOCITY OF SOUND IN A GAS 71 60. TO FIND THE VALUE OF / 75 61. VELOCITY OF SOUND IN AIR '. 75 62. To FIND THE SPECIFIC HEAT OF A GAS BY MEANS OF JOULE'S EQUIVALENT 77 63. To FIND THE MECHANICAL EQUIVALENT OF HEAT BY MEANS OF THE SPECIFIC HEAT OF A GAS 78 64. RELATIONS OF J, R, y 79 65. OTHER METHODS OF DETERMINING y 79 66. FLOW OF GASES 81 67. WEIGHT OF GAS DISCHARGED . . .83 CHAPTER III. IMPERFECT GASES 68. GENERAL DISCUSSION 85 69. TEMPERATURE CONSTANT 8tf 70. CHANGE OF STATE OF AGGREGATION SS 71. LATENT HEAT OF FUSION 88 72. EXPERIMENTAL VERIFICATION 91 73. EXPANSION DURING FUSION 92 74. LATENT HEAT OF EVAPORATION 94 75. VAPOR 96 76. RELATIONS BETWEEN THE TEMPERATURE AND PRESSURE OF VAPOR 97 77. VOLUME OF VAPOR 98 78. WEIGHT OF VAPOR 99 79. EXPERIMENTAL DETERMINATION OF THE DENSITY OF SAT- URATED STEAM. . . , . 104 viii CONTENTS. ARTICLE V " 80. MEASUREMENT OF HEIGHTS 105 81. SUBLIMATION 106 82. EVAPORATION WITHOUT EBULLITION 106 83. SPECIFIC HEAT OP SOLIDS SENSIBLY THE SAME FOR PRESS- URE CONSTANT AND VOLUME CONSTANT 107 84. MECHANICAL MIXTURES 108 85. TOTAL HEAT OF EVAPORATION 110 86. EVAPORATIVE POWER Ill 87. SUPERHEATED STEAM 113 88. FREE EXPANSION 114 89. ABSOLUTE ZERO 116 90. GENERAL EXPRESSION FOR SPECIFIC HEAT 117 91. SPECIFIC HEATS AT CHANGE OF STATE OF AGGREGATION.. 117 92. MODIFIED EXPRESSION FOR THE SPECIFIC HEAT 118 93. APPARENT AND REAL SPECIFIC HEATS, 120 94. GENERAL EXPRESSION FOR THE DIFFERENCE OF THE SPECIFIC HEATS 122 95. SPECIFIC HEAT OF WATER 126 96. ANOTHER GENERAL EQUATION OF THERMODYNAMICS 126 97. OTHER GENERAL EQUATIONS 131 97A. THE THERMODYNAMIC FUNCTION OR ENTROPY 136 98. LIQUID AND ITS VAPOR 143 99. SPECIFIC HEAT OF SATURATED VAPOR 145 100. ADIABATICS OF IMPERFECT GASES 148 101. CONDENSERS 154 102. ISODIABATIC LINES . . 157 CHAPTER IV. HEAT ENGINES. 103. EFFICIENCY 159 104. PERFECT ELEMENTARY HEAT ENGINE 159 1 05. REGENERATORS ... i an CONTENTS. ix ARTICLE PAGE 106. SOME GENERAL PRINCIPLES 169 107. REMARKS 169 108. THE STEAM-ENGINEGENERAL STATEMENT 169 109. IDEAL STEAM DIAGRAM 171 110. STEAM-ENGINE ISOTHERMAL EXPANSION 172 111. STEAM ENGINE ADIABATIC EXPANSION, APPROXIMATE LAW. 175 112. ADIABATIC EXPANSION, THEORETICAL LAW 177 112a. GENERAL EQUATIONS FOR VAPOR ENGINES 180 113. CUT-OFF 200 114. SPECIAL ENGINES 205 115. MULTIPLE EXPANSIONS 210 116. CONDENSATION 212 117. EXPERIMENTS OF ENGINES 214 118. MISCELLANEOUS 217 119. HOT- AIR ENGINES STIRLING'S ENGINE 223 120. THEORY OF STIRLING'S 224 121. DESIGNING 230 122. ERICSSON'S HoT-AiR ENGINE 234 123. DESCRIPTION OF ERICSSON'S ENGINE 235 124. ANALYSIS 237 125. VALUE OF THE RATIO OF EXPANSION TO PRODUCE MAXIMUM MEAN EFFECTIVE PRESSURE 245 126. HEAT ABSORBED AT CONSTANT PRESSURE 246 127. GAS ENGINES 247 128. HISTORY 248 129. SOME DETAILS .' 251 130. THEORY 253 181. FURNACE 256 132. WORK AND EFFICIENCY 257 133. EXPANSION AND COMPRESSION CURVES 258 134. EXPERIMENTS 259 135. PETROLEUM ENGINE 266 136. EXPERIMENTS 267 137. EFFICIENCY OF NAPHTHA-ENGINE PLANT 270 138. EFFICIENCY OF FLUID IN NAPHTHA ENGINE 271 139. REMARKS 273 140. AMMONIA ENGINE . . 274 x CONTENTS. PAOB ARTICLE 141. BINARY- VAPOR ENGINE 142. PRODUCTS OF COMBUSTION THE WORKING FLUID 278 143. THE STEAM INJECTOR 2 ~9 144. THEORY OP STEAM INJECTOR & 145. APPROXIMATE FORMULAS FOR INJECTOR 288 146. INJECTOR AND DIRECT-ACTING PUMP 289 147. THE PULSOMETER 292 148. ANALYSIS OF THE PULSOMETER 293 149. COMPRESSED AIR ENGINE ". 29(5 150. ANALYSIS OF COMPRESSED-AIR ENGINE 296 151. AIR COMPRESSOR 301 152. ANALYSIS OF THE COMPRESSOR . 302 153. EFFICIENCY 305 154. FRICTION OF AIR IN PIPES 306 155. STEAM TURBINE 308 156. OUTWARD FLOW TURBINE 308 157. REACTION TURBINE 309 158. ANALYSIS OF OUTWARD FLOW TURBINE 314 CHAPTER V. REFRIGERATION. 159. REFRIGERATING MACHINE 818 160. PRACTICAL OPERATION 319 161. EFFICIENCY 321 162. CIRCULATING FLUID : 324 163. SOME PROPERTIES OF AMMONIA 325 164. LATENT HEAT OF EVAPORATION OF AMMONIA 328 165. SPECIFIC VOLUME OF LIQUID AMMONIA 833 166. SPECIFIC VOLUME OF AMMONIA GAS 333 167. ISOTHERMALS OF AMMONIA VAPOR 333 168. ADIABATICS OF AMMONIA VAPOR 834 169. SPECIFIC HEAT OF THE SATURATED VAPOR is NEGATIVE 835 170. SPECIFIC HEAT OF LIQUID AMMONIA 335 171. WORK OF THE COMPRESSOR 337 172. VOLUME OF CYLINDER FOR n POUNDS OF VAPOR. . . . . 340 CONTENTS. XI ABTICLE PAQE 173. VOLUME OF CYLINDER FOR REQUIRED REFRIGERATING EFFECT 340 174. DUTY 340 175. CASE OF SUPERHEATING 344 176. EFFICIENCY 347 177. TEST OF A COMPRESSOR SYSTEM , 348 178. TEST OF AN ICE-MAKING PLANT 852 179. ABSORPTION SYSTEM 353 180. TEST OF ABSORPTION PLANT 355 180a. SULPHUR DIOXIDE 357 CHAPTER VI. COMBUSTION. 181. ESSENTIAL PRINCIPLE. . 358 182.. HEAT OF COMBUSTION 360 183. INCOMBUSTIBLE MATTER 363 184. AIR REQUIRED FOR COMBUSTION 364 185. FORCED DRAFT 365 186. TEMPERATURE OF FIRE. 365 187. HEIGHT OF CHIMNEY 366 APPENDIX I. THE LUMINIFEROUS ETHER 369 APPENDIX II. SECOND LAW OF THERMODYNAMICS 389 ADDENDA. MISCELLANEOUS TOPICS 395 TABLES 453 INDEX 472 THERMODYNAMICS. CHAPTEK I. FUNDAMENTAL PRINCIPLES AND GENERAL EQUATIONS. 1. Heat is energy.^ Energy is a capacity for doing work, and it has been shown in many ways that heat, by its action upon substances, can do work. Thus, it may cause steam to drive a piston ; it causes solids and liquids to ex- pand, and changes the molecular condition of bodies, as when solids are fused or liquids vaporized. Heat is also recognized as a sensation. 2. Heat is not material. A body has the same weight when hot as when cold. Count Rumford, in 1T98, dis- covered that he could boil a large quantity of water by the heat produced in boring a piece of cannon. Sir Humphry Davy (about 1799) melted ice by rubbing two pieces to- gether without heat being imparted to them. 3. Heat consists of a motion of the particles of a body. The only known method of directly meas- uring energy is by a combination of mass and velocity ; thus, if m be the mass of a body and v its velocity, then will its kinetic energy be ^ mv 9 . The mass being constant while the body is heated we infer that its heat energy is produced by the velocity of its mass elements. These motions are in- visible, and hence their character can only be inferred ; it may be a motion of the ultimate atom, or of an atmosphere 2 THERMODYNAMICS. [4, 5.] about the atom, a vibratory or periodic motion of some kind, or a combination of simple motions. It is probably not a to-and-fro rectilinear motion of the molecule. A develop- ment of the theory of heat, fortunately, does not require a knowledge of this motion, or even a particular hypothesis, beyond the fact that there is a motion of some kind. Rankine constructed an hypothesis called " molecular vortices,'" from which he deduced many important consequences pertaining to heat. (See Edinburgh Trans., vol. xx. ; Philosophical Mag., 1851 and 1855.) 4. Velocity of heat. The perfect identity of the laws of radiant heat with those of light as to reflection, refraction, interference and absorption, and the identity of their velocities, being 186,300 miles per second, requires essentially the same theory as to their nature and mode of propagation. Electricity is also a form of energy and gov- erned by laws similar to those of heat. As light is trans- mitted by means of a subtle ether pervading all space, and called the luminiferous ether, so it is believed that the same ether transmits heat, electricity and magnetism. (See Ap- pendix.) 5. Heat-energy is measured only by its ef- fects. The kinetic energy of a mass may be computed if its mass and velocity are known, or it may be determined by the work it does in being brought to rest, but since the ve- locity of the particles producing heat cannot be measured, heat-energy can be measured only by its effects. Thus, if a ball of hot iron would just melt one pound of ice, and after being heated again would just melt two pounds of ice, then would the ball in the second case have contained twice the heat above the melting-point of ice that it did in the first case. Similarly, it requires about twice as much heat to raise the temperature of a given amount of water two degrees as it does one degree. The same principle applies [6, 7.J THE THEKMAL UNIT. 3 to other bodies of one substance having different weights and to bodies composed of different substances, or to hetero- geneous substances. For .scientific purposes some specific effect must be assumed as a standard, and considered as a unit. 6. The thermal unit is the heat necessary to raise the temperature of unity of weight of water at its maximum density one degree. Water is at its maximum density at 39.1 F. (4 C.). Ac- cording to the experiments of Kopp, its volume is 1.00012 at 32 F., 1.00000 at 39.1 F., 1.00011 at 46 F., and 1.04312 at 212 F. The British Thermal Unit (B. T. U.) is the heat neces- sary to raise the temperature of one pound of water from 39 F. to 40 F. The French Calorie is the heat necessary to raise the tem- perature of one kilogramme of water from 4 C. to 5 C., and is 3.968 times the B. T. U. Some writers, in defining the thermal unit, start the meas- urement with the temperature of melting ice, instead of at 39 F., and although there is but litile difference between the two values thus obtained, yet for scientific purposes and for physical reasons, the latter is preferable, and should be generally adopted. 7. Work. When heat-energy disappears as heat, it must, according to the principles of the conservation of energy, appear or exist in some other form of energy. When the heat in steam drives the piston of an engine, the steam loses heat by the operation, and an exact equivalent of the energy so disappearing reappears as work done or as energy in the moving parts of the engine, no allowance being made in this illustration for losses due to radiation or friction. To aid one in this conception conceive that one end of the cyl- inder is filled with small, perfectly elastic balls, bounding 4 THERMODYNAMICS. [8, 9.] and rebounding between the head of the cylinder and the piston ; they will, by their continued action, produce a pres- sure upon the piston. If the piston moves forward the energy of the balls will be diminished, as is shown in me- chanics in the discussion of the impact of elastic bodies, and this loss of energy will equal that imparted to the piston, or, if the piston moves uniformly, equal to the work done. In general, when heat-energy' disappears it is said that an equiv- alent amount of work has been done, although the entire work may not be visible energy. Some of it may produce molecular changes in the substance. In the preceding illus- tration, if the piston be forced inward against the rebound- ing balls, their velocity will be increased, and hence their energy will be increased by an amount equal to the work imparted to them by the piston. 8. Internal work is some kind of effect produced upon the molecular character of a substance. Thus, if one pound of water at 32 F. be mixed with one Ib. at 33 F. it will produce two pounds of water at 32 F., but if one pound of ice at 32 F. be mixed with 1 pound of water at 33 F. the temperature of the mixture will be 32 F. Indeed, it is found by experiment that it will require about 141 pounds of water at 33 to melt one pound of ice producing 145 pounds of water only a very little above 32 F., so that nearly all the heat between 32 F. and 33 F. in the 144 pounds of water is necessary to change solid water (ice) at 32 to liquid water at the same temperature. Similarly, a large amount of heat is absorbed in changing liquid water at 212 F. to gaseous water (steam) at the same temperature. This disgregation of the molecular structure is called internal work, or energy of a potential form. 9. The actual heat-energy of a substance is de- pendent upon its temperature. The heat absorbed by a substance may do external work, as in driving a piston, and [10, 11, 12.] LATENT HEAT. 5 internal work, as shown in the preceding article, and in ad- dition to both it may increase the temperature of the sub- stance, thus increasing its energy. The last is called actual energy. The actual energy is some function of the tempera- ture. 10. Latent heat is heat which produces effects other than that of change of temperature. Strictly speaking, it is not heat, but is a measure of the heat which has been de- stroyed in producing effects other than that of changing the actual energy of the substance. Thus, heat becomes latent in producing changes in the state of aggregation of the sub- stance, as infusion, vaporization or sublimation ; and as defined in Article 8, constitutes internal work. But it also becomes latent in doing external work by expansion, and if the tem- perature be maintained constant during expansion, the heat destroyed in doing the work is called the latent Jieat of expansion. 11. General expression. The total heat in a defi- nite weight of any substance is unknown, although if gases were perfect it might be computed, as will hereafter be shown ; but it is possible to find expressions for the heat ab- sorbed in passing from one known state to another, for we have Heat absorbed = change of actual energy -f- change of potential energy -\- external work / = total change of internal energy -\- exter- nal work j = change of actual energy -\- total work. In this expression the total internal energy includes all the heat involved both in changing the temperature and the in- ternal structure of the substance. 12. Temperature is a condition of relative heat. Ex- perience shows that when two bodies, one hotter than the 6 THERMODYNAMICS. [13.] other, are near each other, the hot body becomes cooler and the cooler one hotter. Heat of itself passes from a hotter to a colder body, and this process cannot be reversed except by an expenditure of mechanical energy. The hotter body i said to have a higher temperature than the colder one. . Temperature is not an indication of the quantity of heat absorbed by a body, nor of the amount of heat in a body, but of the intensity of the heat. Thus, if a pound of iron has the same temperature as a pound of water, the latter will contain about eight times as much heat as the former for each degree, as would be found by putting each pound into an- other quantity of some liquid at a different temperature. Temperature is a measure of the sensible heat that is, actual heat which can affect the senses. 13. Thermometers are instruments for measuring differences of temperature. The more common ones depend for their action upon the expansibility of a liquid such as mercury or alcohol. The liquid is confined in a tube as nearly cylindrical as possible, within which it expands. When the expansion of metals is employed for determining temperature, the instruments used are called pyrometers. The air thermometer depends for its action upon the pressure produced by heat at constant volume. All thermometers have two fixed points : one the melting point of ice, the other the boiling point of water at atmos- pheric pressure. The melting point of ice is a more nearly fixed point than the freezing point of water. In some carefully con- ducted experiments water has been reduced several de- grees below the ordinary freezing point, 32 F., before freez- ing. To secure such a result, the water must be kept in a condition of as perfect rest as possible. The boiling point of water depends upon the pressure to which it is subjected ; and since the pressure of the atmosphere is continually changing, as shown by the baromoter. the pressure of one [14.] THE AIR THERMOMETER. 7 atmosphere must be fixed- for scientific purposes. The value determined by Regnault, and now generally adopted, is 2116.2 pounds per square foot, or 14.7 pounds, very nearly, per square inch. In determining the boiling point of water the thermometer should be placed in the vapor near the water. The Fahrenheit scale has 180 equal divisions between the fixed points, and the zero of the scale is 32 such divisions below the melting point of ice. It is designated by F 1 ., or Fahr. The Centigrade scale has 100 equal divisions between the fixed points, its zero being at the lower or melting point of ice. It is indicated by C. It is sometimes called the Celsius scale. The Reaumur scale has 80 equal divisions between the fixed points, its zero being at the lower. To reduce the readings of one scale to those of another, the following equations may be used : (7=| (^-32) ; F = -f O -f 32 ; R = f C. The construction here implied assumes that liquids ex- pand equally for equal quantities of heat, and that the tubes containing them are uniform ; but neither of these conditions are exactly realized, the practical considerations of which belong to Thermometry. 14. The Air Thermometer. In order to gain an idea of an elementary air thermometer, conceive a small, perfectly cylindrical tube closed at the lower end to contain a quantity of air, limited at its upper end by a drop of mercury acting as a piston. Subject this instrument to the temperature of melting ice under the pressure of one atmos- phere, 29.922 inches of mercury, and mark the upper end of the air column ; then, next subject it to the temperature of boiling water under the same pressure and mark the upper end of the air column at this temperature. The two marks will be t\\Q fixed points before described. If the length of 8 THERMODYNAMICS. [15.] the column from the lower end to the lower mark be unity, then will its length to the upper mark be 1.3665 as found by Eegnault. The expansion is 0.3665 of its original volume. For the Fahrenheit scale the space between the fixed points would be divided into 180 equal parts, and hence each part would be A T 3 ip- = 0.00203611 of the distance below the lower fixed point. If the length below the lower fixed point be divided into equal parts of the same magnitude, the num- ber of such spaces will be, 1 180 0.00203611 0.3665 = 491.13. If these parts are numbered according to the natural numbers, 0, 1, 2, 3, etc., beginning with zero at the extreme lower end of the tube called the absolute zero of the air thermometer then would the temperature of melting ice be 491.13 F. from the absolute zero of the air thermometer, and that of boiling water 671.13 F. from the same zero. If air were a perfect gas, this would constitute an absolute scale, but as it is not, a correction is required in order to es- tablish such a scale. For air thermometers the pressure at constant volume is commonly used, instead of the volume at constant pressure as above described. 15. A perfect gas is defined to be such that, under a constant pressure, its rate of expansion would be exactly equal to its rate of increase of temperature, and, the volume being constant, increments of pressure will be equal for equal increments of heat. In other words, it would be a substance in which no internal work would be done by changes of temperature or pressure. No such substance is known it is ideal, subjected merely to a definition and to laws to be assigned and yet it is of great service in this science. The idea of a perfect gas was the result of ex- periments upon existing gases, as air, oxygen, hydrogen, etc., which, at first, were supposed to be represented by the per- [16. J AN ABSOLUTE SCALE. 9 feet law. In mechanics, at the present time, the bodies treated are, at first, the subjects of definition, and considered perfect, as perfect solids, perfect liquids, perfectly elastic^ etc., and the results obtained from these hypotheses made practical by the introduction of moduli the values of which are found by experiments. The same method is adopted in this science. 16. An absolute scale is one whose divisions would be indicated by a perfect gas thermometer. On such a scale the divisions would be exactly equal for equal incre- ments of heat down to the zero of the scale. Since a per- fect gas is unknown, the zero of the absolute scale can be determined only approximately by computation, as will be shown hereafter, where the best result yet obtained fixes it at 492. 66 F. below the melting point of ice. The letter F. here affixed implies that there are 180 divisions between the fixed points, as in Fahrenheit's scale. This zero on the centigrade scale io -| of 492.66 = 273.7 C. Temperature on the absolute scale will generally be indicated by the Greek letter r, and the temperature of melting ice by r . If T F. indicate the temperature from the zero of the Fah- renheit scale, and T C. from the zero of the centigrade scale, we will have r a = 492.66 F = 273.7 C. r -.= 460.66 F+ T F. 273.7 C+ T C. It is found that air is so nearly a perfect gas within the ranges of temperature and pressure for which it has been tested that it may be considered as such for all practical purposes, and will be so considered theoretically except in the determination of the place of the zero of the absolute scale. Further, the ordinary mercurial thermometer agrees sufficiently well with the air thermometer for the more or- dinary ranges of temperature met with in engineering prac- 10 THERMODYNAMICS. [17, 18, 19.] tice to be used in such cases. But for scientific purposes and for extreme cases in practice, the difference is too large to be ignored. Regnault found that when the air ther- mometer marked 630 F. above the melting point of ice, the mercurial thermometer indicated 651.9 above the same point, a difference of about 22. Liquids generally expand more rapidly the higher the temperature. 17. The absolute zero of temperature is the zero of the absolute scale, and corresponds to the condition of total deprivation of heat ; at which temperature no sub- stance could exercise any expansive power. This tempera- ture has never been reached, and the nearest approach to it has been produced by expansion in liquefying air, oxygen and nitrogen, reaching 373 F. (225 C.), or more than f the distance from the zero of Fahrenheit to the absolute zero. (Complex Rendus, Feb. 9, 1865; Jour. Frank. Inst., Sept. 1886, p. 213). The absolute zero is about 492.66491.13 = 1.5 degrees Fahr. below the zero of the air thermometer, as computed on the hypothesis of the same rate of contrac- tion of air below 32 as from 32 to 212. This law might change as the temperature was extremely reduced, but it would continue uniform for the ideally perfect gas. 18. The equation of a gas is an equation expressing a continuous relation between its volume, pressure and tem- perature throughout a finite range of the same. Let p be the pressure on a unit of area of the substance when the volume of one pound is v and absolute temperature is T, then, generally, P=f(v, r \ which may be considered as the equation to a surface, called the therm-odynamic surface. 19. Equation of a perfect gas According to the definition in Article 15, (p\ oo T, and (v) p oc r, [19.] EQUATION OF A PERFECT GAS. 11 where the subscripts represent the quantities which are constant while the others vary, and combining these in one expression, we have p v be the pressure of one atmosphere, r the absolute temper- ature of melting ice, and v the volume of one pound of the gas at that pressure and temperature, then will equation (1) become 1 = S> = *(*,), (2) which is the equation required. The values of p and T t have already been given and are independent of the nature of the gas, but v depends upon the density of the gas. A cubic foot of dry air weighs at sea level at the temperature of melting ice y\ = 0.080728 pounds ; hence, 1 v, = -- = 12.387 ; 0.080728 . p. v a _ 2116.2 X 12.387 _ 26214 _ g3 gl r " 492.66 ~~492^66 ~ when T O is measured from the zero of the absolute scale ; but if it be from the zero of the air thermometer, we have ^ 7 T O 491.13 and equation (2) becomes, for the absolute scale, pv = 53.21 r, (3) for the air thermometer, p v 53.37 t. (3 r ) For French units, let p* = the pressure of one atmosphere in kilogrammes per square metre/ 12 THERMODYNAMICS. [20, 21, 22.] -BO' = the volume of a kilogramme of the gas in cubic metres, TO' = the absolute temperature of melting ice on the centigrade scale ; then x ___ 2.2 (3.28) 3 _ ~r~ fr 1.8222 r which for air becomes in kilogramme metre centigrade units (k. m. c.), omitting the accents, pv = 29.20 T and in decimetre kilo. centigrade units, pv = 2.920 T. 20. Mariotte's law. If the temperature be con- stant, equation (2) shows that the volume varies inversely as the pressure ; a law discovered by direct experiments upon gases, and known as Mariotte's law, supposed by some to have first been discovered by that investigator, but by others this credit is given to Boyle. Fora time after the announce- ment of the law it was supposed to be perfect for the so- called permanent gases, but more refined experiments have shown that the actual law governing them is only a very close approximation to it. 21. Law of Gay Lussac (or of Charles). According to equation (2), if the pressure be constant, the volume will increase directly as the temperature, or *=***, P a law discovered by Gay Lussac (or, according to some, by Charles) by experiments upon actual gases, and known as the law of Gay Lussac. At first it was supposed to be the perfect law of the so-called permanent gases, but it is now known not to be exact though very nearly so. Originally, it was not stated in terms of the absolute temperature, as that term was not then known, but the law of the increments is the same on any thermometric scale. 22. The so-called imperfect gases include all such as cannot be represented with sufficient accuracy by [23.] THERMAL LINES. 13 equation (2). All known gases are imperfect, strictly speak- ing, but the permanent gases are so nearly perfect that they may, for engineering purposes, be considered a perfect. No single formula can represent exactly the law of imperfect gases, but the most comprehensive one, and one which may be made to represent actual substances with sufficient ac- curacy for practical purposes, was deduced by Kankine from his theory of Molecular Vortices, and is P^ = P^_^__ ^__^L_. <&>., (4) r r^ r r' 2 r 3 in which # OJ # # &c., are functions of the density to be determined by experiment ; but as the theory here referred to is not a recognized part of science, the formula is accepted only so far as it conforms to the results of experiment. (Kankine's So. Papers, 32.) For carbonic acid gas the form of the equation, as con- firmed by the experiments of Regnault, becomes p v =$? r - J_ = E f - A, (5) T TV TV in which p = 2116.2, v, = 8.15T2, p v = 17262, T O = 492.66, o = 481600; ...^,= 851-^2. (6) Sir William Thomson and Dr. Joule used, for imperfect gases, the formula, (7) in which for air the constants for French units are E = 2.8659, a = 771386, ft = 8M560, y = 214325840. (Phil. Trans. [1854], CXLIT., 360). 23. Thermal lines. Any line the co-ordinates of which represent the contemporaneous relation between the pressure, volume, and temperature of a body subjected to 14 THERMODYNAMICS. [24.] thermal conditions, is a thermal line. Ideally, it may be any line on a thermodynamic surface ; actually, the projec- tion of a thermal line on any one of the co-ordinate planes is called a thermal line, and geometrically it is called the path of the fluid, although the latter refers to the projec- tion on the co-ordinate plane, pv, uilless otherwise stated. Thermal lines on the plane pv constitute a diagram of energy. If the pressure p be constant, the line is called an isobar if the volume v be constant, it is called an iso- metric. Thermal lines were introduced into this science by M. Clapeyron. 24:. Isothermal lines represent the relation between the pressure and volume when the temperature is maintained constant. In equation (2) if r be constant we have j)v = Itr = m, -(8) for the equation of an isothermal of a perfect gas. It is an equilateral hyperbola referred to its asymptotes as shown in Fig. 1, in which O a is the axis of the hyperbola, the branches of which will be asymptotic respectively to the axes v and Op; O v being the axis of volumes and Op the axis of pressures. o PIG. 1. EXERCISES. 1. Construct an isothermal for air considered as a perfect gas. Assume a temperature of 60 F. or call it 520* P. absolute, then p v = 27669 v = 10, p = 2766.9 v = 100, p = 276.69 v = p, p = 166.4 v = 1000, p = 27.669 &c. &c. These numbers are so large we take ^n of their values as inches, or parts [24-1 ISOTHERMAL LINES. 15 of an inch, and construct the curve as in Fig. 2. But for the equilateral hyperbola it is unnecessary to compute any co-ordinates except for the ver- tex c ; for, having found c by making p = v = O a = a c, bisect a at d and make d e = 2 a c, &c. ; and make g = 2 a and g h = \ac, &c. 2. Construct an isothermal for air at the temperature of 1 F. absolute. 3. Find the vertex of the hyperbola of the isothermal for air whose tem- perature is T = 400 F. O" 4. Find It for the following gases : FIG. 2. For hydrogen, v = 178.83, R = nitrogen, v a = 12.75, R = oxygen, v = 11.20, R = 5. Find the value of R in French units for hydrogen. 6. Find the equation to the isothermal for carbonic acid gas for the temperature T = 60 F. 7. What is the volume of air, considered as a perfect gas, under the pressure of four atmospheres and absolute tem- perature of r = 800 ? 8. If the heat in one pound of carbon is 14500 B. T. U., how many pounds of carbon completely consumed are neces- sary to increase the temperature of 2000 pounds of water 45 F. ? 9. How many kilogrammes of water would be raised 25 C. by the heat in one pound of carbon ? 10. On a diagram of energy draw on the plane v t the locus of the path of a perfect gas when the pressure is con- stant. 11. Find the pressure per square inch of two pounds oi air when its volume is one half of a cubic metre and its ab- solute temperature is 500 C. 12. Show that all isothermals of a perfect gas are asymp- totic to each other as well as to the co-ordinate axes^? and v. 13. "What is the temperature of a pound of air when its 16 THERMODYNAMICS. L 25 -) volume is 5 cubic feet and pressure 35 pounds per square foot? 14. What is the weight of a cubic foot of air when the pres- sure is 50 pounds per inch and the temperature 160 F '* 25. Adiabatic or Tseiitropic lines represent the relations between the volume and pressure of a substance doing work by expansion without transmission of heat. Conceive a gaseous substance to be enclosed in a cylinder having a frictionless piston, it will, by driving the piston, do work. It will be conceived that the external pressure is infin- itesimally less than the internal during expansion. The temperature of the enclosed gaseous substance may depend upon several conditions. If heat be properly supplied the temperature may be maintained con- stant, producing isothermal expansion, which may be represented by the line A J, Fig. 3. Having performed that operation, bring the substance to its initial state A, and conceive the ex- pansion to take place without any o transmission of heat, to do which the vessel must be considered as imper- meable to the passage of heat, in which case the external work will be at the expense of the heat-energy of the sub- stance, and therefore the temperature will fall as expansion proceeds, and the pressure will also fall on account of the loss of temperature, as shown by equation (2), and the line A d representing the continuous relation between the vol- ume and pressure will be lower than the isothermal A b, and its slope downward greater for equal volumes. If the sub- stance be compressed from state A, the line A e will be above the isothermal 1) A c. The line e A d, representing the law of expansion or of compression without transmis- sion of heat, is by Kankine called an Adiabatic (from , to pass through), and by Gibbs, Clausius and [20, 27, 28.] CYCLE. 17 others, Isentropic, because the entropy (a term to be con- sidered later) remains constant in this kind of expansion. Adiabatics are asymptotic to the axes^> and v and also to the isothermals. 26. If a fluid, after a series of changes of pressures and volumes, returns to its initial state, the path of the fluid will be a re-entrant curve, as A and _Z?, Fig. 4, and in such cases the fluid is said to work in a cycle. 27. A heat engine is a machine for continuously transforming heat into work. Such engines in practice work -. FIG. 4. in cycles. 28. Carnot's cycle. This is a cycle performed by an imaginary heat engine, devised by M. Carnot in 1824:, and involves the most important fundamental principle of this science. The following is the operation : Let B, Fig. 5, be a piston moving in a frictionless cylin- der, all parts of which are perfectly impermeable to the pas- sage of heat except the base F. Let the base of the cylinder be one square foot, so that the height of the piston will correspond with the number of cubic feet below it, and let the cylinder contain one pound of air, or any other gas. Let // be an indefinitely large vessel containing heat at a given tem- perature, and L another indefinitely large vessel contain- ing heat at a lower temperature, the initial letters, II and Z, indicating the relative temperatures. The vessels are assumed to be indefinitely large, so that, in imparting heat to a finite body, they will maintain a sensibly uniform temperature. Let N and N' be plates, as large or larger than the base F of the cylinder, perfectly impermeable to FIG. 5. 18 THERMODYNAMICS. [28.J the passage of heat. Two of these are used simply for con- venience of arrangement, so that the operation to be de- scribed, passing in the direction indicated by the arrows, will be the more suggestive of a cycle. Conceive the base f of the cylinder to be placed against the vessel // ; the pound of air in the cylinder will quickly become of the same tem- perature as that of //. While in this condition let the pis- ton move outward against a resistance which is continually infinitesimally less than the pressure within, the tempera- ture will be constantly that of //, and the expansion will be isothermal. After the piston has been moved outward as far as desired in this manner, transfer the cylinder to the non-conducting cover 2V and allow the piston to move outward still further by a gradual reduction of the external pressure ; the press- ure and temperature of the substance will both fall, and since the walls of the cylinder are impermeable to the pas- sage of heat, the expansion will be adidbatic. Let the op- eration be continued until the temperature of the pound of gas in the cylinder has been reduced to that in the vessel L, At the end of the preceding operation let the cylinder l>e removed to the vessel Z, and the piston then forced inward ; the heat generated by the compression of the pound of air will escape as fast as generated, and is said to be rejected or emitted into the vessel Z, the temperature of which will not be sensibly changed ; hence the temperature of the pound of air will be constantly that of the vessel Z, and the compres- sion will be isothermal. Let the operation continue to such a point that when the cylinder is removed to the cover JV and the air compressed adiabatically until the temperature is raised to that in the vessel //, the volume will be the same as that at the beginning of the series of operations. To show these operations graphically, let O I, Fig. 6, rep- resent the volume and 5 B the pressure of the pound of gas [29, 30. J SOURCE. 19 FIG. 6. in the initial state ; then will B on the diagram represent this state. First operation. When the cylinder is in contact with the vessel 77, the expansion of the gas will be represented by the isothermal B C, O c being the final volume and c C the final pressure. Second operation. When the cylin- der is in contact with the cover N, the expansion will be represented by the adiabatic C D, O d being the final volume and d D the corresponding pressure. Third operation. The compres- sion, when the cylinder is in contact with the vessel Z, will be represented by the isothermal line D A. Fourth operation. The compression when the cylinder is on the cover N' will be represented by the adiabatic A 1$. These are the successive operations as indicated by Car- not ; but it is more convenient, in describing the process, to begin either at the state C or A, on account of limiting the third operation. Thus, when the cylinder is on the ves- sel // and in the state (7, let it be transferred to N and ex- panded along CD until the temperature is reduced to that of L ; then transferred to L and compressed along DA any desired amount ; thence transferred to N' and compressed until the temperature is raised to that of II \ then transferred to II and expanded along B C to the state C. 29. Source. The vessel from which the working sub- stance receives heat, as II in the above operation, is called the source. Similarly, the vessel receiving the heat emitted from the working substance, as L in the above operation, is called the refrigerator. In engineering science these are called, respectively, the furnace and condense?'. 30. Work done. During the expansion from state B 20 THERMODYNAMICS. [31, 32.] to state C work is done by the gas while forcing the piston outward, represented by the area I B C'c, and while expand- ing from C to D more work is done by the gas, represented by the area c CD d\ but during the compression from D to A work is done by the piston upon the gas, the amount being represented by the area d D A a, and work is still fur- ther done upon the gas in compressing it from A to B, rep- resented by the area a A I. The difference between these works will be the external work done by the cycle of operations. We have + IB Cc + cC Dd- dJ)Aa-aABl = A 31. Indicator diagram. The diagram A B C D would be described by an indicator on Garnet's imaginary engine ; and the area of an actual indicator diagram, taken from any engine, expressed in foot-pounds, is a measure of the heat destroyed in the cycle. It is in this sense that we speak of " foot-pounds of heat." 32. Cariiot's cycle is reversible. In a complete cycle, if all the heat taken in is at one uniform tempera- ture, and all the heat rejected is at a uniform lower tem- perature, the operation is called Gamut's cycle. Such a cycle is reversible, for all the operations may be performed in precisely the reverse order, the final result, however, being work done by the piston upon the gas in the cylinder, the energy of the gas thereby being increased by an amount represented by the area B A 1) C, Fig. 6, expressed in foot-pounds. A reversible engine is also called a perfect engine. Non-reversible cycle. As an example of a non-reversible cycle, after the sub- stance has expanded isothermally while in communication with the source, represented by the line B C, Fig. 7, let it be transferred directly to the refriger- 133.] CONDITIONS OF A REVERSIBLE CYCLE. 21 ator heat will be abstracted and the pressure may be reduced at constant volume, and hence without doing work, the operation being represented by the line CD. Then compress it isothermally when in communication with the refrigerator along the line D A ; then transfer it direct- ly to the source, raising the temperature and pressure to the initial state B. This cycle cannot be performed in precisely the reverse order ; for the pressure cannot be reduced from B to A when the engine is in communication with the source, nor raised from D to C when in communication with the refrigerator. 33. Conditions of a reversible cycle. In order that a cycle be reversible, the difference between the exter- nal pressure and the internal during a change of volume must be infinitesimal during expansion the external being infinitesimally less, and during compression infinites.imally greater than the internal ; also during the transfer of heat, the difference between the heat of the substance and that of the external body shall also be infinitesimal during absorp- tion being infinitesimally less than the source, and during emission infinitesimally greater than the refrigerator. The differences being infinitesimal, the quantities will in finite measures be equal. 33a. It follows from the conditions of the preceding article, that if a closed cycle be bounded by the isothermals and adiabatics of any substance, the cycle will be reversible when worked with that substance. Thus, if there be an adiabanc compression along A -B, Fig. 8, an isothermal ex- pansion along B A', adiabatic expan- TIG 8 sion A' B', isothermal expansion ' A" , and so on back to A, the cycle will be reversible. Also, the cycle A B CD, Fig. 7, may be made reversible by conceiving an indefinite number of sources of heat differing by THERMODYNAMICS. [34.] FIG. 9. d r and passing down CD by an indefinite number of indefi- nitely short isothermal compressions and a corresponding number of indefinitely short intermediate adiabatic expaii- ^ sions as indicated in Fig. 9 ; and a similar reversed operation in ascending from A to B. 34. The heat absorbed by a sub- stance in working from a state A to state B may be represented on a diagram of energy by the area included between the path of the fluid and the adiabatics pass- ing through A and B respectively, ex- tended indefinitely in the direction of the expansion, Fig. 10. Let A be the initial and B the final states for the expan- sion ABv,-vv l AD = AB CD = (p l A B (p y (p l D C (p where (p, and l and ^ A B tp, in foot-pounds, to find which requires an experiment with the substance in order to deter- mine its thermal capacity under constant pressure. The specific heat at constant pressure is the amount of heat absorbed in increasing the temperature of a unit-mass of the substance one degree, the pressure being constant and the specific heat constant throughout the degree. In English units, it is the number of thermal units (Art. 6) absorbed in rais- ing the temperature of one pound of the substance one degree Fahrenheit. To represent it on a diagram of Vl energy, the line A B, perpendicular to the ^9-axis, must be limited by two isothermals, as r and r -f- 1, differing by unity of temperature ; then will the dynamic specific heat at and from r be represented by the indefinitely extended area cp t A B l and B q> as shown in Article 34. Let k, = the specific heat for a constant volume at the temperature T in ordinary thermal units, KV = its equivalent dynamic specific heat, d h v = the thermal units absorbed in raising the tempera- ture d T, dH v = the foot-pounds of heat in d A v ; then d A v k v d r, = Jk v d r = K, d -c ; (16) If K v be constant, then in Fig. 12, JT V = v = R r, provided r a T I = T 3 r a , &c. 7. In the preceding exercise, show that a right line parallel to the j>axis will also be divided into equal parts. 8. Show that a line drawn through the origin of co-ordi- nates is not divided into equal parts by the successive equi- lateral hyperbolas of Exercise 6. 4O. Let the temperature be constant during expansion. SECOND LAW. In this case the path of the [40. J TEMPERATURE DURING EXPANSION. 33 fluid will be an isothermal, as A B, Fig. 13 ; and the heat absorbed during the expansion from v l to -y, will, according to Article 34, be represented by the area

n conceive it to be divided into an indefinite number of strips of equal areas by isothermals of the given substance, as dc,ji t <...y z, &c. ; they will repre- sent equal quantities of heat, and if an elementary engine be worked in the successive cycles A B c d, d c ij, &c., the re- sultant works done will also be equal. These are Carnofs cycles, since all the heat absorbed will be at one temperature, and that which is rejected, at one lower temperature. Let the successive equal quantities of heat thus transmuted into external work constitute a scale of temperatures known as Thomson's Absolute Thermometric Scale (Phil. May., xi. (1856) '216. Thomson's Papers, p. 100). The characteristic quality of this scale is equal quantities of heat when worked in Carnot's cycle will do equal quantities of external work 'Independently of the nature of the working substance. At first, any amount of heat or area, as ij d c, may be taken, arbitrarily, as a unit, and a repetition of this unit will con- stitute a scale of natural numbers, as 7, 8, 9, tfcc., the zero of which may be placed arbitrarily. Having assigned its place and the unit of heat, the quantity of heat involved in any number of such operations becomes known. Thus, the heat necessarily destroyed in performing the operations numbered 7, 8 and 9 will be three times the unit initially assumed. Fractional parts of the scale will correspond to fractional parts of the unit. The scale may be so numbered that the two fixed points shall correspond with 32 F. and 212 F. Conceive that the zero of the scale corresponds with the total deprivation of heat from the substance, and that in raising the pressure from to t\ A, there are T of the arbitrary units. Let each unit be divided into an indefinite [40.J TEMPERATURE DURING EXPANSION. 37 number of equal parts by isothermals, each represented by d r ; then will the number of parts in each unit be 1 -f- d T, and the number in r units will be number of strips = The area of any one of the infinitesimal strips, as A B c d, being known, we have tp, AB, J Vi **'* J^ in which the subscript r, of the bracket implies, as explained above, that r within the parenthesis is to be considered con- stant during the integration. As a thermal capacity, Article 37, the latent heat of expansion is r ( P as if d v were unity, dr being the rate at which heat is absorbed per unit of volume. Differentiating equation (21), considering r as constant and v variable, gives [40.] TEMPERATURE DURING EXPANSION. 39 which is the same as equation (20), where d v is the abscissa of b in reference to v^, Fig. 13. 'The scale of temperatures above used is not practical, except for the purposes of analysis, since heat cannot be actually divided with accuracy by any known means according to the process described ; and it remains to be shown how the re- sult can be made of practical value. Conceive a quantity of heat equal to that absorbed by a pound of the substance, cp^AB (p to be absorbed by such a quantity of & perfect gas as will give the same temperature, and let the temperatures be measured by an ideally perfect gas thermometer graduat- ed from absolute zero and having T equal divisions up to the temperature here considered ; then will equal divisions on this scale correspond with equal quantities of actual heat in the perfect gas so that, if the gas be cooled by abstracting equal, successive quantities of heat, the successive tempera- tures will be indicated by equal divisions on the scale. In this manner, the heat in a perfect gas might be divided into equal parts. Let the temperature of the given substance be reduced an amount d r on this scale by working the heat in Carnof s cycle, the same amount of heat will be transmuted into work as must be abstracted from the perfect gas in re- ducing its temperature the same amount, and so on. Con- ceive isothermals of the substance to be drawn on the dia- gram of energy, differing by d t of the perfect gas thermom- eter ; there will be * -f- d t such divisions between zero and r, as in the former case. These isothermals may be con- ceived to be described, geometrically, from the equation of the gas given in terms of the scale of the perfect gas ther- mometer, or, physically, by supplying heat to the expanding gas so that the temperature will remain constant as indicated by this thermometer and noting the contemporaneous press- ures and volumes. These processes, perfectly done, would give the same isothermals ; and since the number is made the same as in the earlier part of this article where the strips 40 THERMODYNAMICS. [40.] were, arbitrarily, 'made equal, and since the lowest, or zero- isothermals, coincide, also the highest, or r-isothermals, it is inferred that the successive isothermals in the two cases co- incide. It follows, then, that if the area ^ABtp^ be inter- sected by isothermals differing by an absolute constant tem- perature, the areas between the successive isothermals will be equal; and if the number representing the difference of temperatures be commensurable with the number represent- ing the highest temperature, the entire area (p^ABcp^ will be divided into equal parts. By making the difference indefinite- ly small, or d T, the question of commensurability disappears. But a perfect gas is unknown ; it has, however, been found, as stated in Articles 14 and 16, that the air thermometer dif- fers but little from that of a perfect gas thermometer, the temperature of melting ice being 491.13 F. above the absolute zero of the air thermometer, and about 492.66 F. above the zero of the absolute scale, a difference of about sfa of the entire 491, a quantity too small to be measured in actual practice, and can be determined only by the most refined experiments. The position of the zero of the abso- lute scale cannot be determined exactly, but, accepting the results of Thomson and Joule, if the zero of the air ther- mometer be made to coincide with the melting point of ice, then by adding 492.66 F. to the reading of the air ther- mometer, the sum will be the value of the temperature on the absolute scale, almost exactly. Equations (20) and (21) are theoretically exact, and hence are practically so for volumes, pressures and temperatures determined by the best methods known. The following reasoning may aid the reader in satisfying himself of the equality of the strips. Conceive the area ^ it will be less than that representing the heat absorbed. If the working fluid is a perfect gas the areas abed and efg h (Fig. 15) will be equal, but if the gas be imperfect all the small areas in , A b n, Fig. 13, during isothermal expansion, the part d b transmuted into work by working in one of his cycles, was ^ (r l T^) of the heat absorbed, where /* is a func- tion of the higher temperature only and hence independent of the nature of the working substance, and r, T a , the fall in temperature of the working substance. In this case, let r l r 2 = d r, then will // d r be the fractional part of fp l A b n transmuted into work. Let J^be the latent heat of expansion in thermal units, then will M d v be the heat units in q> t A b n, and in foot- pounds we have J M d v = (p l A b n, and the heat transmuted into the work b d will be b d = jn J M d v d T. But we also have bd= 42 THEEMODYNAMICS. [41.] and making these equal gives SJf=l(*J>\ j* \drJ Carnot did not find the form of the function //, In re- gard to it Thomson says : " It has an absolute value, the same for all substances for any given temperature, but which may vary with the temperature in a manner that can only be determined by experiment" (Thomson's Papers, p. 187). Thomson, whose resources ever seem sufficient for the oc- casion, set about its determination, the processes for which are described in the Philosophical Magazine, and more recent- ly in Thomson's Mathematical and Physical Papers, cover- ing many pages. Early in the investigation, Joule suggest- ed that the value of n might be " inversely as the tempera- ture from zero" (Thomson's Papers, p. 1 199); and these experimenters established the truth of this suggestion by that celebrated series of experiments known as " the experi- ments with porous plugs." Hence, we have p = -; r as already found. The quantity /* is known as " Garnet's function," the title given to it by Sir William Thomson. The value 1 -s- /i = T OT the absolute temperature of melting ice, was found to be 273.68 C. (ibid., p. 391). Thomson's absolute scale may be thus defined : The num- bers expressing degrees of absolute temperatures are propor- tional to the quantities of heat absorbed and emitted at those temperatures in a reversible cycle. Thus, if // = g> t A B <#, = the heat absorbed, Fig. 13, and it be divided into r equal parts, then will one part be H -j- r; and if h be the [42.] TEMPEEATUEE DUEING EXPANSION. 43 heat emitted = gj t y z cp^ then ; will the number of equal parts in h be h + (H+ T) = ~T = t(^y); h _ t '"H ~ 7' The equal parts of heat in qj^ABcp^ may be conceived to be secured, physically, by a succession of perfect engines in which the refrigerator of one is the source of the next, and so on. It was in this manner that Carnot established his expression for efficiency. The amount of work done by heat depended only upon the difference of the temperatures of the source and refrigerator and some function of the higher temperature, as already given. 4:2. To express equation (21) in terms of the external work, from Fig. 13, we have U = v, A B v z = F p d v ; . . dU = p d v ; .-.*? = (*) d v, also written * d U', dr ' \d rJ dr hence, substituting in (21), 9l AB

, A B q> y . If r be the absolute temperature of any isothermal, as a o, and d v the expansion from state a to state o, then, accord- ing to equation (20), will the area m a o m t = r [vM d v ; 46 THERMODYNAMICS. [45.] and, according to Article 39, the area m y o b m^ = KV d r ; hence, ultimately, d , (24) which is a GENERAL differential EQUATION OF THERMODY- NAMICS. In this solution the polygon A n a o bp, &c., is inscribed in the figure

.t ? latent heat ? latent heat of expansion ? thermal capacity ? What is a perfect gas ? imperfect gas ? Does the coefficient of expao- sion vary with different gases ? For what is it least ? What is the " ab- solute zero"? Can it be realized ? Of what value is it in theory'; What is thermodynamics ? What is a general equation of thermodynam- ics ? Eliminate dr from equations (A), and deduce a third equation for dH. CHAPTER II. PERFECT GASES. 47. Difference of specific heats. From the equation of a perfect gas, equation (2), we find (*}=*=., \d T ) V T ' (dv \ = R^ __ v_ \dr) p " T ' and these in equations (A) give dll= K v dr -\-pdv. From equation (2) j? d v + v d p = Rdr, which in equation (27), after placing the second members equal, give K^ K, = E ; (28) hence, the difference of the two specific heats for a perfect gas is constant. 48. Specific heat constant. In a perfect gas no internal work is done during a change of state, hence, at constant volume, no work will be done by the absorption of heat, and all the heat absorbed will be sensible at all temper- atures ; hence, the specific heat of a perfect gas at constant volume will be constant, and equation (28) shows that, in this case, the specific heat at constant pressure will also be constant. It is found that the specific heat for sensibly per- fect gases at constant volume is independent of the volume. 60 PERFECT GASES. [48.] Let JT V = C v and K v = <7 P for sensibly perfect gases, and equations (27) become which are the general equations of sensibly perfect gases. Equation (28) becomes R = C\ - C v . (29) When Clausius first established the preceding equation, he concluded that both specific heats were constant for per- fect gases at all pressures and temperatures, although this view opposed the one then prevalent that the specific heat was a function of the density of the gas. Soon after, how- ever, the experiments of Regnault confirmed the conclusion of Clausius by showing that it was practically constant for the so-called permanent gases, as air, oxygen, hydrogen and nitrogen. Iiegnault found the following results for air at constant pressure (Relation des Experiences, ii., 108). Heat required to raise the temperature of one pound of air 1 C. at constant pressure, Or between 30 C. and -f 10 C. 0.23771 thermal units, " C. " -)- 100 C. 0.23741 " " " C. " + 200 C. 0.23751 " " which show that it is not strictly uniform, neither is the law of change apparent. There is, apparently, a minimum value, but it is not safe to assert that such is the fact, much less to assign its place. Other experimenters find values differing slightly from these. The departure from the mean is so small, we may, for all ordinary purposes, consider the specific heat as constant. Kegnault also determined the specific heat of air under different pressures from 1 to 12 atmospheres, and of hydro- [49.] THE PERFECTNESS OF A GAS. 51 gen from 1 to 9 atmospheres, and found the specific heats of each to be sensibly constant within these respective ranges. 49. The perfectiiess of a gas may also be tested by comparing its agreement with the equation of a perfect gas. Thus, Regnault found for atmospheric air, if the vol- ume be constant, the following : Density, or pressure in atmospheres at 0C. Ratio of pressure at constant volume at 100 C. to that at C. 0.1444 1.36482 0.2294 1.36513 0.3501 1.36542 0.4930 1.36587 1.0000 1.36650 2.2084 1.36760 2.8213 1.36894 4.8100 1.37091 If the gas were perfect we would have for a constant vol- ume -y, from equation (2), ^ _p^ p, _ R , r _ T x _ r, - T, fr ~ Pi ~ V iPi 7 T ! in which the range, T 2 T,, of temperature being constant, and equal 100 C. in the preceding table, and T, = 273.7, the ratio of * would also be constant. The preceding table shows a slight increase in this ratio with the increase of the density from 0.1444 to 33.3 times that value. The depart- ure, however, from uniformity is so small that, for ordinary purposes, air may be treated as a perfect gas in this re- spect. When the pressure was constant, it was found that the volume increased as follows : PERFECT GASES. [49.] Pressure. Increase of volume for an increase of 100 C., the original volume being unity in each case. Atmospheric air. Carbonic acid. Hydrogen. 'IGQmm. 2525 0.36706 0.36944 0.37099 0.38455 0.36613 0.36616 In these experiments the increase of volumes was greater for the same range of temperatures when the pressure was greater, although for hydrogen the rate was almost exactly constant. If the gas were perfect we would have for a con- stant pressure^,, the equation the left member of which should be constant if the gas were perfect, the range of temperatures and the initial tempera- ture being constant. In some other experiments the same mass of different gases was subjected to different pressures with the following results, v being the volume of one pound of the mass : Density or Hydrogen. Nitrogen. Atmospheric air. p pv \ P pv P 1.9975 3.9860 7.9457 15.8045 pv 2 4 8 16 2.00081.0004 4.0061 1.0015 8.0339 1.0042 16.16161.0101 1.9995 3.9918 7.9641 15.8597 0.9992 0.9979 0.9955 0.9912 0.998782 0.996490 0.993212 0.987780 This table shows that these three gases follow nearly the gaseous law expressed by the equation^? v = R T that for hydrogen p v increases slightly with increase of pressure, while for nitrogen and atmospheric air this product decreases with increase of pressure. [50, 51.] TO FIND C v . 53 The following table gives the expansion of several gases under constant pressure from C. to 100 C., and the in- creased tension for the same range of temperatures under constant volumes, the initial pressure being one atmosphere, as determined by Regnault. Substance. Increase in volume under constant pressure for Increase of pressure under constant volume for 100 C. 1F. 100 C. 1F. Hydrogen 0.3661 0.002034 0.36700.002039 0.36700.002039 0.3669,0.002038 0.3710 0.002061 0.3667 0.3665 0.3668 0.3667 0.3688 0.002037 0.002036 0.002039 0.002037 0.002039 Atmospheric air Nitrogen Carbonic oxide " acid . . Protoxide of nitrogen Sulphurous acid Cyanogen. 0.3719!o.002066 0.3676|0.002032 0.3903 ! 0.002168;0.3S45 0.002136 0.3877 0.002154 0.38290.002127 5O. To find. O v . The specific heat of any substance at constant volume has not been found to any degree of ac- curacy by direct experiment, but its value for sensibly per- fect gases may be computed from equation (29), for \ve have <7 v = c\ - R, which is the required equation. Regnault found for air, the mean value c n = 0.2375 T. U. (30) .-. C\ = 184.77 = 0.2375 X 778. Also, equation (3'), R = 53.37 difference = C v = 131.40 ; .-. c v = 0.1689 = 131.40 -^- 778. h Equation (3') is here used because the determinations were made with the air thermometer. 51. Relative specific heats. Since both specific 64 PERFECT GASES. [52.] heats are constant for perfect gases, their ratio will be con- stant, which we will represent by y ; then ' For air we have This ratio was originally found by means of the velocity of sound in the gas, in a manner soon to be explained, Art. 60, From equations (30) and (31) we find C 9 = r B = Y .&Lv = Dv , (32) r -i y-i r where D is a constant for sensibly perfect gases ; hence, for another gas we have that is, the specific heats of two sensibly perfect gases are di- rectly as their specific volumes. But the specific volumes are inversely as the specific weights, or densities, of the gas, or V = w = id' hence, that is, the specific heats of two perfect gases are inversely as their densities. 52. Let the temperature be constant during expansion, find the heat absorbed. For this condition dr = o 1 53. ' LET THE VOLUME BE CONSTANT. 55 in the first and second of equations (JB\ and from the first we have H = d v, (35) which may be integrated if p be a known function of v. The equation of the path of the fluid will be equation (2), making T = T^ PV = TV the value of p from which substituted in the preceding equation gives ^ = R T I log . (36) The first member of this equation may be represented by the area (p^AB(p Fig. 19, and the last member by v t A B v which is the external work done during the expansion ; hence, in a per- fect gas the external work done during an isothermal expansion equals the heat absorbed a necessary result, there be- FIG jg ing no internal work. Since the area A B b is common, it follows that, for a perfect gas, (p, I B (p, = v, A b v,. 53. Let the volume be constant. v = and v = V H and equations (B} give But the equation of the gas gives, v, p, = R r,, v, p, = B r and the condition of the problem gives v l = v 9 ; Then will (37) = (<7 P - R) (T, - 56 PERFECT GASES. [54.] which placed equal to equation (37) gives as before found in equation (29). 54. Let the pressure be constant. Significance of R y equation (29). Let the heat absorbed be d H = C v dr, (38) and equation (B], becomes and since p is constant during the absorption of heat, as indi- cated by the condition in equation (38), we have by integrating the last equation between the limits r and r + 1 for temper- ature, and v t and v t for volumes, observing that -y, -y, will be the horizontal distance between the isothermals T and r -\- I at the upper extremity of the ordinate p, we have that is, the value of R is the energy \ \ exerted by one pound of the gas A\ \c in expanding at constant pressure while the temperature increases one degree. In Fig. 20, if the isothermals through A and B respectively differ by one FIGSO degree, A B being horizontal, B C vertical, we have O p = m, A B ra 3 , C v = m t C B ra,, and by the second law, -y, A C v, = m l A C m t . ' . <7 P O v = m l A B m, m, CB m, = m,AB Cm t which is the external work done during the expansion at [55.J LET THE PATH BE ARBITRARY. 57 constant pressure from the state A on one isothermal to the state B on the isothermal one degree higher, as stated above. For air this becomes 53.21 foot-pounds (Eq. (3) ), or 53.21 -r- 778 = 0.069 of a thermal unit. 55. Let the path he arhitra- ry. Then will the first of equations The second term is, Fig. 21, FIG. 21. p d v = v l A B v n and may be separated into two parts. Through A draw the isothermal A C y and the adiabatios A m, C ni T , B m z ; then v 1 A C v^ = m A C mr ; /. / p d v = m A C m,T -f- A 23 C. But II m A B m 2 ; v = m AS m, - in A C m T - A B C = mr G B m, = C v (r a - r,), (Eq. (37)) ; that is, to find the increased energy of the substance in passing from state A to state B due to the absorption of heat, through the initial state A of the substance represented on a diagram of energy pass an isothermal, and note the point C where it intersects the ordinate to the second state, then will the area between G JB and two adiabatics drawn respectively through C and indefinitely extended in the direction of increased volume, represent the increased energy of the substance. Let the isothermal A C be prolonged to an intersection PERFECT GASES. [55.] with B

3 CD

d H = = Or d T+ p d v. But from equation (40), .- a./^r = y -y v v = a v, Fig. 24 ; 158.] AN AIR-COMPRESSOR. 6V T O = 7-1 7-1 . . 7 = 1.405 very nearly. If 7 = 1.406, as previously found, then = 64735; = 64566, 7- 1 which is about -^ of this value less than the preceding. If the terminal temperature be the zero of the air thermometer, then the value of Hia. equation (a) would be H = 131.40 X 491.13 = 64534, which agrees very nearly with the preceding value, where 7 = 1.406. 4. Eequired the height to which a ball weighing one pound could be projected upward in a vacuum by the heat- energy in one pound of air under the pressure of one at- mosphere at the temperature of melting ice. (Use the value in (b) of exercise 3.) How many times the height of a homogeneous atmosphere ? 5. Required the entire heat-energy in one pound of hy- drogen at sea level at the temperature of melting ice. 6. Required the heat-energy in two pounds of air under the pressure of one atmosphere at the temperature of 100 F. 7. If a gas be forced into or out of a receiver of constant volume, without transmission of heat, can equations (42) be transformed so as to give the rela- tion between weights, temperatures and pressures ? 8. For a perfect gas, verify the fact that the external work v l A B v Fig. 25, equals (p l AB(pAB being an iso- thermal, A 9?, and B q> t adiabatics, by finding the area tp 1 ~b B

i will be Y v Pi v l = pv , PERFECT GASES. of B (p 3 , p t v t =pv; p=ec = PLAFig.25, from the former, from latter, and area g>ibBcp,= i-dr* 'J*. v Y y-1 y I-Y Also, /" rf v ^i i jQi v. v area viAbv t =l p, v, - y~^~l ' 'F, v and since ^4 5 is an isothermal, we have pi PI = p* v, hence area

dt Let jj' (if +)-*(& -- dt dx' in the Mecanique Analytiquz, tome II., has discussed the problem, of the movement of a heavy liquid in a very long canal ; M. Navier published a Memoire on the flow of elastic fluids in pipes, in the Acddemie des Sciences, tome IX. ; and M. Poisson wrote several Memoires on the propa- gation of wave movements in an elastic medium, and the theory of sound, for which see Journal de I'Ecole Poly technique, 14th chapter, and of the Academie des Sciences, tomes II. and X. These eminent mathe- maticians established the basis of the analysis for the solution of the problem. More recently we have M. Lame's Lecons ur I'Elasticite, des Corps solves, and Lord Rayliegh's Treatise on Sound, both of which are works o* ?reat merit. [59-] VELOCITY OF A WAVE. ' 73 tben (*I\ = *Jf + a -*J- (44) \dt) dt*^ dxdt' where the parenthesis indicates a partial differential coefficient, and __ dx / d td x ' dx* (45) and equations (43), (44), (45), or, more directly, the last equations on p. 72 give (*n =a (!*Z\ (46) \dt / \dx J The total differential of V = f(x, t) is by substituting (46), and integrating, ^ V I dy fAr>\ -rr+a-7 , (*() where F is any arbitrary function. Similarly subtracting a from (43), F 1 f( f\ ^^ ^ ,y /4g) Adding and subtracting (47) and (48), we have the respective equations But and substituting from above, gives integrating, = -L F (x + a d (x + a - ~ f(x - a f> d (x - a f) ; a (t 6 (t + a - = E d JL. ^ dx The lamina d x will be urged forward or backward by the differ- ence of the elastic forces on opposite sides of it, and as the quantities are infinitesimal, this difference will be dp^ or * = **=* IT* Let D be the density of the lamina, then its mass will be M D d x, and we have from equation (21), page 18, of Analytical Mechanics, which is a partial differential equation of the motion of any lamina, the integral of which is given in works on Differential Equations. One of the methods is as follows : Let E -L. D = a* and adding a $ dxdt to both members, we have Let in the Mecanique Analytiqua, tome II., has discussed the problem, of the movement of a heavy liquid in a very long canal ; M. Navier published a Memoire on the flow of elastic fluids in pipes, in the Academie des Sciences, tome IX. ; and M. Poisson wrote several Memoires on the propa- gation of wave movements in an elastic medium, and the theory of sound, for which see Journal de I'Ecole Poly technique, 14th chapter, and of the Academie des Sciences, tomes II. and X. These eminent mathe- maticians established the basis of the analysis for the solution of the problem. More recently we have M. Lame's Lemons snr I'Elattidte des Corps solves, and Lord Rayliegh's Treatise on Sound, both of which are works G* ^eat merit. [59.] VELOCITY OF A WAVE. ' 73 then (*-I\ = * + a -^J- (44) \dt ) d* dxdt ' where the parenthesis indicates a partial differential coefficient, and a a -J(, (45) and equations (43), (44), (45), or, more directly, the last equations on p. 72 give ^) =a (|Z\ (46) The total differential of.V = f(x, t) is by substituting (46), fU-i (d x + a d t) and integrating, where Fis any arbitrary function. Similarly subtracting a - ' y from (43), dxdt r = f(x-af) = ^L-a^. (48) Adding and subtracting (47) and (48), we have the respective equations d -jL = $ F(x + a + i f(x - a t), But and substituting from above, gives d y = 2^ F (a? + a t) d (x + a t) - ^ f(x-at)d(x-af); integrating, y = V (-c + a t) ~

function may be suppressed, and we have y = 1>(x + at), (50) and differentiating, *l\ dx) V (* + a, 0, which is the rate of dilation (the expansion or contraction of a prism of the air), and which is the velocity of a particle, and dividing the latter by the former, 57 = * <> which is the velocity of the wave ; hence, u = a = '; (56) by means of which y may be found when the velocity of sound in a gas of given weight and tension are known. We have w 1 1 T P p V p o V o T which reduces the determination of y to that of the velocity, u, of sound in a gas at known temperature r;g,p v , being known. 61. The velocity of sound has been determined by direct experiment with the following results : Velocity per second. In dry air at C. Centimetres. Feet. MM. Bravais and Martins 33237 1090.5 Hr. Moll, Yan Beck, and Kuytenbrouwer 33226 1090.1 The French Academy, 1738 33200 1089.2 " " " 1822 33120 1086.6 M. Kegnault 33070 1085.0 M. Le Koux . . . 33066 1084.9 Mean. ... . 33153 1087.7 76 PERFECT GASES. [61.] Again, if v be the velocity of sound, A the wave length and n the number of waves (or the frequency of the waves) in a second, then v = n A. - By this means it has been found for the velocity of sound in dry air at C. Centimetres. Feet. By M. Dulong .................. _____ 33300 1092.5 " Hr. Seebeck ...................... 33277 1091.8 " M. Schneebeli .................... 33206 1089.4 " Hr. Wertheim ..................... 33313 1093.0 Mean ..................... 33274 1091.6 Hr. Kayser determined very accurately the wave length A, by means of Professor Kundt's dust figures, correspond- ing to a given tone of vibration, or frequency ??, by which means he found for Centimetres. Feet. the velocity of sound in air at C ...... 33250 1090.9 Mean of the eleven determinations ...... 33206 1089.4 The height of a homogeneous atmosphere of dry air at C. as determined by Regnault is 20214 feet; hence, the theoretical velocity of sound in air, neglecting the effect -of wave condensation, would be v = /26214 X 32.2 = 918.7 ft. per second ; /1089 4\ a ' r = ~ = L4061 = 1 - 40 This value of y is on the supposition that air is a perfect gas, and the error resulting from this hypothesis is scarcely appreciable, and certainly cannot affect the result so much as the errors of observation. The mean of such a large number of good determinations is probably more reliable than any single observation taken at random. This was the original mode of finding the value of y, and the only one known before the determination of Joule's equivalent. It [62, 63. J SPECIFIC HEATS OF PERFECT GASES. 77 is nearly the same for all the permanent gases, as air, hydrogen, oxygen and nitrogen. 62. To find the specific heats of the sensibly perfect gases. We found in equation (32) that .-. C v = B. (60) For air we have 53.37 = 184.83 ; ~r\j\j .-. c p = 184.83 -r- 778 = 0.23757. The mean of the values given by Regnault is 0.23751 {Relation desJExp.ftome II., p. 101), the first four figures of which agree with the results of our computation. Professor Rankine was the first to make this computation, in 1850, using, however, y 1.4, which gave c v = 0.24. Up to that time this was the most accurate determination of the specific heat of air ; and when, soon afterward, the very accurate and entirely reliable experiments of Regnault gave very nearly the same result, Rankine's determination was considered a crucial test of the correctness of the dynamical theory of heat. 63. To find the mechanical equivalent of heat by means of the specific heat of a gas. From equa- tion (59) we have C = Jc f = ^jlt; " J = 7-1"? Having found y by means of equation (58) without a knowledge of the value of J, and R and c p having been found by Regnault, we have, by substitution, 1406 53.37 J = "406 X 02375 = 7 ' 8 foot -P o ds 78 PERFECT GASES. [63.] on the scale of the air thermometer. If R = 53.21 we find J 776 for the probable mechanical equivalent on the absolute scale. The former equals 426.8 kilogram - metres. This was substantially the method originally used by Mayer of Germany, by which means, in 1842, he found 365 kilogram-metres, and Holtzman, in 1845, found 374. The fact that Dr. Mayer assumed air to be a perfect gas, and made no attempt to prove the correctness of the assumption, added to the fact that the value he obtained was scarcely a rough approximation, has, in the eyes of some historians, deprived him of the honor of being the first to determine this important constant. Joule justly has the credit of first determining it accurately. Mayer did not work out the equations as above, but solved the problem in the most elementary manner, the process for which is worthy of special study. Thus, he considered that it re- quired a certain amount of heat to increase the temperature of a given amount of gas at constant volume all of which simply made the gas hotter, and if the gas expanded against an external resistance it required more heat in order to maintain the higher temperature thus reviving the idea of Rumford and Davy. Let one pound of air under the pressure of one atmosphere, p pounds per square foot, occupy the volume v. The increase of volume for one degree of temperature, p being constant, will be v -r- r, and hence the external work done will be = R (equation (2)). The heat absorbed at constant volume will be c y , and, at constant pressure, c p , and the difference of these amounts of heat does the work fi, provided no internal work is done ; hence, J= R ____ R ~ c v - c v ~ (y 1 as before. [64, 65.] OTHER METHODS OF DETERMINING 79 64. The constants J, 7?, y, are so related as to serve as mutual checks upon each other, but this relation does not determine the exact values of any one of them. When de- termined directly they are subject to small errors, due chiefly to errors ~of observation, but the results are believed to be correct within of one per cent, and in some cases the error is probably much less. 65. Other methods of determining y. This constant has been found by the principle of adiabatic ex- pansion. Thus, equations (41) give log p logp, log T: log r l ~ log v 1 log v i log-c-\- log r l To secure data for use in these equations, MM. Clement and Desormes used a 20-litre glass globe closed by a stop- cock A, Fig. 29, and connected with a vertical glass tube B, dipped into water, which acted as a manometer. By means of an air-pump attached at D a partial vacuum was produced in the vessel, after which, by opening the cock at A a very short time, air would rush in and produce the pressure of the external atmosphere, and by com- pressing that already in the vessel, raise the temperature ; and after the cock was closed it cooled to that of the sur- rounding temperature, and the pressure diminished. In the preceding equations let p = p = the atmospheric pressure, p l the initial and p its final pressure ; then the temperature being the same at the beginning and end of the experiment, FIG. 29. 80 PERFECT GASES. [65.] we have v ' _ - v ~ pC or log v 1 log v = log p log p l ; lff p It was found in one of the experiments that p = 1.0136,^, = 0.9953, p = 1.0088; . . y = 1.3524.* MM. Gay Lussac and Wilter modified this experiment by forcing air into the vessel and allowing it to escape adiabati- cally until the pressure in the vessel equalled that of the external air. They found j) = 1.0096,^, = 1.0314, j> = 1.0155; . -. Y = 1.3745.| In this manner M. Ilirn found 1.3845, M. Dupre, 1.399, Hr. Weisbach, 1.4025, M. Masson, 1.419 for air. For carbon dioxide M. Masson found 1.30. The discrepancy in the results arises chiefly from the fact that the changes in pressure are not adiabatic, but the inertia of the inflowing gas produced a compression exceed- ing the normal value, resulting in a reaction tending to force a portion of the air out again, producing an oscillating effect, as shown by M. Gazing who also found, by similar means, the same value, 1.41, for air, oxygen, nitrogen, hydrogen and carbon monoxide. Hr. Kohlrausch substituted an aneroid for the manometer used above, because more sensitive to pressure, and found Y = 1.296 (Poffff. Annalen, 1869, CXXXVL, 618). This * Jour, de Physique, LXXXIX. (1819), 428. t Ann. de C%. et dt, Phys., 1821, XIX., 436. \Ibid., 1862, LXVI.,206. [06. J FLOW OF GASES. 81 result was not considered good on account of the small quantity of air used in the experiment, although the method is considered an improvement on the preceding. Dr. Kontgen in 1872 made a series of experiments with a more perfect apparatus, containing a much larger quan- tity of the gas, the mean of ten good experiments giving Y = 1.4053 for air (Hid. (1873), CXLVIIL, 580). The difficulty in these experiments of obtaining the ob- servations for strictly adiabatic changes generally results in too small a value for this constant. 66. Flow of gases. The flow of perfect gases as af- fected by the principles of thermodynamics was investi- gated by Messrs. Joule and Thomson (Proc. Hoy. .Soc., 1856) and Weisbach (Civilingeneur, 1856). See the au- thor's Analytical Mechanics, page 389. Let w the weight of a unit of volume, p = the pressure at any point of the issuing jet, Y = the velocity at the point where p is measured ; then, for a unit of section and distance d s the mass moved will be w d s -j- y and the work done \>y d p will be , w d s d p d s = d(Y)i g YI r^ p 2 g J w If the cooling due to the expansion during discharge follows the adiabatic law, then from equations (41) we have 82 PERFECT GASES. [66.J which substituted above gives i Zl = ? r*'d = _ L _p i r l _T,-\. 2g~ w,J 1 - Y - 1 w, L T,J ' JT 2 /vjY (62) where p l is the tension just within the reservoir, and^,, that just outside. But the equation of a perfect gas reduces to p -P r ' It . which in English units becomes = 14.933 V ^? r 4 (l - -') ; (63) TO Tj/ and this for air becomes F = 14.933 V 53.21 r, (l -*} i = 108.93 |/ TI f i _ ! V nearly. i If the flow be into a vacuum, p^ = o ; . . T, = o, and F= 108.93 |/ir ; which at the temperature of melting ice becomes F= 108.93 i/492^6 = 2417 feet per second. Making T a = o in equation (62) and comparing with equa- tion (56) of Article 60, shows that the velocity of discharge into a vacuum will be J 2 7-1 times the velocity of sound in the gas at the melting point [67.] THE WEIGHT OF GAS. of ice, which for air becomes 2.214 X 1089.4 = 2417 feet per second, which is less than half a mile per second. 67. The weight of gas escaping per second will be r - in which Q = the volume escaping measured outside the reservoir, w^ = the weight of unity of volume outside the reservoir, S = the section of the orifice, and Jc the coefficient of efflux. Equation (64) is a maximum for 2 which for air becomes A = 0.831, L = 0.527, ^ = 1.577 *, Pi v * The values of k as found by Professor Weisbach are : Conoidal mouthpieces, of the form of the contracted vein, with effective pressures of from 0.23 to 1.1 atmospheres ............. 0.97 to 00.99 Circular orifices in thin plates ............... 0.55 to 0.79 Short cylindrical mouthpieces ................ 0.73 to 0.84 The same rounded at the inner end ........... 0.92 to 0.93 Conical converging mouthpieces, the angle of convergence being 7 9' .................. 0.90 to 0.99 EXERCISES. 1. For a perfect gas, if the temperature of the gas at the outside of the orifice equals that of the reservoir, what will be the velocity of exit ? 84 PKKKECT GASES. I 67 -] 2. What is the initial velocity with which hydrogen will flow into a vacuum from a vessel in which the tempera- ture is 60 F. ? 3. What weight of air will flow from a very large vessel in one second in which the internal pressure is 4 atmospheres and temperature 100 F., the external, one atmosphere and temperature 40 F., flowing through a short cylindrical tube | inch in diameter, the coefficient of discharge being 0.8 ? Consider the vessel so large that the pressure may be con- sidered constant during the discharge. SUGGESTIONS FOR REVIEW. What does R represent ? In Fig. SO, may A be taken anywhere on the i- isothermal ? Draw several verticals between two isothermals differing by unity and show what areas must be equivalent if Cv be constant. In Fig. 22, if the gas be compressed from state B to state A, show what changes take place in the heat. Do the principles applicable to expansion also hold for compression ? What is a perfect gas? What is a thermal capac- ity ? Define the two more common specific heats. Can there be more than two specific heats ? Illustrate. If a pound of air occupies 10 cubic feet, and another pound 40 cubic feet, both at the same temperature, which will absorb the more heat in having its temperature raised one degree ? De- scribe Mayer's method of determining the mechanical equivalent of heat. What gas has the greatest specific heat ? If the mechanical equivalent were the heat necessary to raise the temperature of one pound of air one degree, about what would be its numerical value ? Describe methods of finding the value of y. What is the smallest value of y given in the text ? the largest ? the value adopted ? How was the last one determined ? Will the greatest wlnine of a gas escape from an orifice when the velocity of exit is greatest ? Will the greatest weigM of gas escape when the veloc Ity is greatest ? Why not ? CHAPTEE III. IMPERFECT FLUIDS. 68. General discussion. Equations (A) are the general equations for the heat absorbed by an imperfect fluid, and for convenience are brought forward. They are dH= p dr - r (-^) dp. j In the first of these equations the last term is the entire work, both external and internal, due to an expansion dv, so that if p' be such a pressure that when multiplied by d v would equal the entire work done, we have (66) which we call a virtual pressure. In Fig. 30, if v l A = p be the external pressure at the vol- ume v and temperature r, then will some ordinate, as v l a, represent p ', and hence ^d r> will be the real virtual pressure, being such an ideal pressure as would when multiplied by d t give the internal work due to expansion only. If the path A B be arbitrary, we have, generally, IMPERFECT FLUIDS. [69.] v, A B v, = lp d v. Vi a I v, = fr (d (r). + V W, (67) the indicated integral being the latent heat due to expan- sion, and q> (r) a function of the temperature, being the latent heat due to an increase of temperature. If K v in the first of equations ( A] is variable, then will a part of the heat absorbed do internal work due to a change of temperature. It appears, then, that the internal work may be considered in two distinct parts : one due entirely to change of volume, the other entirely to change of temperature. 69. Let the temperature be constant during expansion. This is a case of isothermal expansion, and we have d r = o and T = r,, and the first of equations (A) be- FIG. 31. (68) in which (f. } is to be found from the \drJv equation to the gas. In Fig. 31 let a b be the path of the fluid, which will be an isothermal of the substance, then will i>, be the external work done dur- ing expansion, and let aefb represent the internal work, then will v l eft\ represent the total work done, and will be the latent heat of expansion ; and we have ordinate to e f=p' = T, !-y*-J */*.- /&*)<* [69.] LET TEMPERATURE BE CONSTANT. 87 v t al>v t = I pdv, EXERCISES. 1. If equation (7) be the equation of an imperfect gas, find the total work done during an isothermal expansion from #, to 2 v t . (Use equation (68).) 2. If a &, Fig. 31, be the isothermal of the gases in equa- tion (7), and c d what it would be when , ft and y are each zero, give the equations of a &, c d, ef t and the values of v l a, a c and c e. 3. Given equation (7) to find the external work -y, a 1> v^ Fig. 31, and the internal, a efl>. (Equations (69) and (70)). 4. If the equation of the gas be p v = 12 r ^, show that a c = c e, Fig. 31, a 1) being the isothermal of the gas, and c d what the isothermal becomes when I is zero. If one pound of carbonic acid gas at 300 F. expand isothermally from 10 cubic feet to 20 cubic feet, find the total work done, also the external and internal work. (Use equations (6), (69) and (70).) 5. What will be the total internal work of expanding two pounds of carbonic acid gas indefinitely at the constant -temperature of 200 F., the initial volume being 8 cubic feet ? (The limits of integration in equation (70) will be oo and 8.) Ans. 364 foot-pounds. 6. The initial volume of a pound of carbonic acid gas being 8 cubic feet, how much must it be compressed at the constant absolute temperature of 600 F., so that the inter- nal work shall equal the external work ? 7. If equation (4) be the equation of the gas, in which 88 IMPERFECT FLUIDS. [70, 71.] 1 2 3 = j #, = jj # = 3? find the equation of an isother- -y v v mal, the external work done during an isothermal expan- sion, and the total work done. Ans. for the total work, Sr, log v -*- f-J , + -i?-i --^-i,&c. v, Lr, < r l v? r? v* T I v, ' 7O. Change of state in regard to aggregation. Let the temperature and pressure be constant, required the heat absorbed. For this case dr = o, and r fy\ will be independent of v, hence These conditions are realized during three physical changes fusion, vaporization and sublimation. 71. Latent heat of fusion, or of liquefaction. Substances may be melted changing from a solid to a liquid state under the constant pressure of the atmosphere, or other pressure, and at a fixed temperature for that pressure ; and during this change of state heat is absorbed which does not affect the thermometer, and hence, according to the definition, is called latent. Its value can be found only by direct experiment. Having this value of // for any sub stance, which, for distinction, call If f (noticing that/" is tho initial letter of fusion), we may find d = r(v,-v (72) dp U f for which the rate of change of temperature per unit of pressure may be calculated. If the volume v l of the sub- stance in the initial, or solid state, exceeds that in the ter- minal, or liquid state, v u then will be negative, and the d p [71.] LATENT HEAT OF FUSION. temperature of fusion will be lowered by an increase of pressure, a principle first pointed out by Professor James Thomson (Edinburgh Trans.^Vol. XVL). Water, antimony, cast iron and some other substances, are more bulky in the solid than in the liquid state ; and the melting point of all such substances is lowered by pressure. The latent heat of fusion of ice is 144 B. T. JL, as deter- mined by experiment or 144 X 778 = 112032 foot-pounds ; and this is the work which must be expended upon one pound of ice at 32 F. in reducing it to liquid water at the same temperature, which work is necessary to completely break down the crystalline structure of the ice. Conversely, it is the equivalent of the heat-energy which must be emitted from a pound of water and absorbed by surrounding objects in changing water from the liquid to the solid state at 32 F. Solids, under a definite pressure, have a corresponding definite melting point, or point of fusion. The following are some examples of the latent heat of fusion : Melting point Latent heat of fusion. Deg. Fahr. B. T. U. Foot-pounds Hf. Ice 32 144* 112032 Zinc 793 50 6 39367 Sulphur .... Lead 224 635 16.9 13148 9 7 7547 Mercury .... Tin -40 455 5.1 3968 {500 as given by Rankine. 26.6 " " " Box on Heat, Spermaceti . . Cast iron .... 124 2000 p. 13t j 148 (Eankine). 1 46.4 (Box). 233 as given by Clements. * Phil. Mag., 1871, XLL, 182. Peclet found 135, Person, 144.04, and 144 appears to be the most reliable. f We have not determined which (if either) is correct. 90 IMPERFECT FLUIDS. [71.] If the specific heat of water were constant, 144 pounds of water at any temperature above 33 F. would have its tem- perature reduced one degree in just melting one pound of ice at 32. The mixture after melting would reduce the temperature a little more. The expansive force resulting from congealing water was well illustrated by Major Williams in 1786, at Quebec, Can- ada, by filling an iron shell with water and driving an iron plug weighing over 2 pounds into the fuse hole, and sub- jecting it to an out-door temperature of 18 F. ; when, upon freezing, the plug was fired out and projected over 400 feet (Trans. Koy. Soc. Edinburgh, II., 23). EXERCISES 1. If for water we have r = 492.66 F. v, = 0.01602 cu. ft. per pound ; and for ice r = 492.66 F. -y, 0.0174 cu. ft. per pound, and H t = 112032 ; how much will the melting point of ice be lowered by a pressure of one atmosphere, 2116.2 pounds per square foot? (Use equation (72).) Ans. 0.0128 F. 0.0071 C. 2. Kequired the pressure per square foot necessary to lower the melting point of ice 1 F. We have _&J>__ H f 112032 d r ~ T (w, - t>.) ~ 492.66 X 0.00138 ~ 164784 Ibs " In this exercise d T is considered unity and dp = 164784 , or the sec- ond member may be considered constant and the left member integrated between limits, giving p * ~ P '= 164784. The notation of the preceding exercise may be treated in a similar manner. [72.] EXPERIMENTAL VERIFICATION. 91 3. If 1140 pounds pressure per square inch will lower the melting point of ice from 32 F. to 31 F., diminishing the volume of one pound 0.00138 of a cubic foot ; required the latent heat of fusion of one pound of the ice. Ans. 143.2. 4. Required the external and internal work in melting ice at 32 F. at atmospheric pressure. The external work will be that done in lowering the atmosphere through a distance equal to the decrease of volume in changing the state of ag- gregation, or 2116.3 X 0.00138 = 3 foot-pounds, nearly. The total work will be by using the result in exercise 2, T ^dv = 492.66 X 164784 X 0.00138 = 112032 ft.-lbs., nearly, dr and 3 pounds more due to atmospheric pressure. From this it appears that the work is nearly all internal, and is more than 36500 times the external work. 5. The pressure required to reduce the melting point of ice 1 F. being 16-4784 pounds per square foot when the ini- tial temperature is r = 492.66 F. ; find the diminution of volume of one pound in changing from congealed to liquid water. 6. Required the pressure necessary to reduce the melting point of ice to 18 C., assuming that the above formula is valid so far below C. 7. What is the highest temperature at which ice can exist indefinitely in a vacuum ? 72. Experimental verification. Sir William Thomson, by a delicate and beautiful experiment, proved that the melting point of ice was lowered by pressure (Phil. Mag. (1850), III., XXXVII., 123). A delicate thermometer, constructed for the purpose, was enclosed in a vessel with water and lumps of clear ice and an air gauge for measuring the pressure. At atmospheric pressure the ice would- not melt if below 32 F., but it was found that when the con- 92 IMPERFECT FLUIDS. [73.} tents of the vessel was subjected to pressure the thermome- ter fell as the water assumed the temperature of the melting point of ice corresponding to the increased pressure ; and the observed results corresponded well with those calculated. Professor Mousson made the following experiment : A prism of steel, Fig. 32, was used, having a cylindrical bore 0.71 cent. (0.28 inch), closed at the lower end by a copper cone forced in by a strong screw, and the upper end by a long slightly conical copper plug a pressed down by a steel piston by means of a strong screw, and when in an inverted position a small brass rod b was dropped in and the FIG. 32. bore filled with water. After being exposed to cold at 9.5 C. the protruding ice was removed, the copper cone inserted and screwed up, and the whole reversed and put into a freezing mixture at 18 C., after which the upper plug was forced in at a pressure roughly estimated at not less than 13250 atmospheres. When the lower plug was removed the brass rod dropped out first, showing that the ice had been melted, permitting the rod to fall to the lower end. The pressure was more than five times that required by theory to melt the ice, but the tem- perature at which it melted is unknown. 73. It is natural to infer \\iQOppositeprinciple that the melting point of those substances which expand when fused will be raised by compression, and this principle has been verified by Mr. Hopkins (Rep. B. A. (1854), II., 56), as well as by others. In Mr. Ilopkins's experiments the instant of fusion was determined by means of a small iron ball sup- ported by the substance when solid, but which fell when the substance liquefied ; and when supported it deflected a needle which was suspended just outside the vessel, but the deflection ceased when the ball fell. The temperature was determin- ed by that of the oil in a bath in which the whole was im- [73.] EXPEKIMENTAL VERIFICATION. 93 mersed ; and the effective pressure was taken as the half sum of the pressures which forced the piston inward and that required to just permit it to return to its initial position, tims elminating the effect of friction. The following results were found : Melting points in degrees centigrade. Pressure Atmospheres. Spermaceti. Wax. Sterrane. Sulphur. 1 51.1 64.7 67.2 107.2 538.4 60.0 74.7 68.3 135.3 820.5 80.3 80.3 73.9 140.6 The melting point of wax will be given nearly by the em- pirical formula t e = 64.68 + 0.0188 p. M. Person gives the following empirical formula as the results of his experiments on the latent heat of fusion of non-metallic substances (Ann. de Chem. et de P/iys., Nov., 1849). I = (c 1 - c) (T + 256) (73) in which c = the specific heat of the substance in the solid state, o' = " " " . " liquid state, T = its temperature of fusion in degrees Fahr., and I = the latent heat of fusion of one pound in B. T. IT. at the pressure of the atmosphere. EXERCISES. 1. If the specific heat of water be unity, of ice 0.504, and the temperature of fusion, T 7 , be 32 F., required the latent heat of fusion. 2. If ice be subjected to such a pressure that it will melt 94 IMPERFECT FLUIDS. [74.] at F., find its latent heat of fusion, the law being assumed to hold good to that point ; and the amount less than at 32 F. 3. What must be the temperature of ice that there will be no latent heat of fusion, and to what pressure must it be subjected ? Ans. T = - 256 F. Pressure = 256 + 3 - + 1 = 22500 atmospheres. 0.0128 If this could be realized there would be no mechanical distinction be- low this temperature between solid and liquid water. 4. Assuming that experimental results may be used to the extent implied in the questions here given, find the increase of volume due to the fusion of spermaceti at 124 F. For the ratio of d p to d T considered as finite, see the preceding table ; then we have Ik _ 148 X 778 Vt Vl " dp- 537.4X2116' T d~r * 140-124 5. The latent heat of fusion of solid water at 32 F. being 144 B. T. U. and its specific heat in the liquid state being unity, find its specific heat in the solid state. (Equation (73).) 6. The specific heat of ice being 0.504 at 32 F., and the la- tent heat of fusion 144 B. T. U., find the specific heat of water. 7. The specific heat of solid spermaceti being 0.32 and considering its latent heat of fusion as 46.4 B. T. U. at 120 F., find its specific heat in the liquid state. (Equation (73).) 74. Latent heat of evaporation. It is found by experiment that there is a definite boiling point for liquids corresponding to the pressure to which they are subjected, and from this condition they will pass into a vapor at that temperature and pressure. Hence, equation (71) is direct- ly applicable to this case, and indicating this particular latent [74.] LATENT HEAT OF EVAPORATION. 95 lieat by I1 6 (e being the initial letter of evaporation), we have and J A e = H v By determining the factors of equation (74) by experi- ment, the value of I1 6 may be computed. M. Regnault de- termined the value of A e directly for water at a" series of boiling points from its freezing point to about 375 F., which may be represented with great precision by the empirical formula, in English units, JJ e = [1091.7 - 0.695 (T- 32) - 0.000000103 (T - 39.1) 3 ] X 778, (75) or in French units, 7/ e = [606.5 0.695 T - 0.00000033 (T 4) 3 ] 426.8. (76) (Me'moire Academie des Sciences, 1847. Trans. Roy. Soc., Edinburgh, Yol. XX.) In practice it will be sufficiently exact to use the follow- ing: e = 966 0.7 (T 212) = 1092 - 0.7 (T - 32) = 1114.4 - 0.7 T = 1436.8 - 0.7 T, B. T. U., (77) or its equivalent, // e = 751548 - 544.6 (T - 212) = 867003 - 544.6 T = 1117880 544.6 T, foot-pounds, (78; = a 1) r. The latent heat of evaporation of some other substances is given in the Addenda. The exact value of the latent heat of evaporation of one pound of water at the pressure of one atmosphere, as found by Eegnault, is 966.1 B. T. U. = 751624 foot-pounds. This is the work necessary to simply change water from its liquid state when at 212 F. under the pressure of one at- 96 IMPERFECT FLUIDS. L'3-i mosphere to the condition of vapor at the same temperature and pressure. Since water in the form of steam occupies more space than as a liquid, the molecules must be farther apart in the former than in the latter state, and hence, with- out considering their exact condition in the two cases, it ap- pears that it requires 751624 foot-pounds of energy to sim- ply separate the molecules of a pound of water sufficiently to produce steam. This amount of heat is absorbed and dis- appears without affecting the temperature ; and the same amount reappears, or passes to other bodies, when it returns to a liquid at that pressure. Latent heat of evaporation of one pound of certain substances at the pressure of one atmosphere. Substance. Boiling point. Deg. F. Latent heat. B. T. U. Foot-pounds. Water 212 172.2 95.0 114.8 316 141 966.1 364.3 162.8 156.0 184.0 236.0 751624 283425 125658 121368 143152 l.3608 Alcohol Ether Bisulphide of carbon Oil of turpentine Naphtha 75. Vapor is any substance in a gaseous condition at the maximum density for that tempfffittire or pressure. If a vapor be in contact with the liquid, as steam in the same vessel as water, there is a certain pressure which is at once the least pressure under which the substance can exist in the liquid state and the greatest pressure at which it can exist in the gaseous state at that temperature. Under such condi- tions the vapor is called saturated vapor (or saturated steam\ and the pressure, the pressure of saturation. If at the tem- perature of saturation the pressure be slightly increased some [76.] PRESSURE AND TEMPERATURE OF A VAPOR. 97 of the vapor will be condensed until equilibrium be restored, and, conversely, if the pressure be diminished more vapor will be generated. To illustrate, if a cylinder containing a piston were placed directly over a steam boiler having one end open to the boiler, and the piston be forced inward, a portion of the steam would be condensed, and if the piston be drawn outward more steam would be generated the tem- perature and pressure remaining constant while the volume varied. This is known as Dalton's law (Stewart on Heat, p. 143). 76. The relation between pressure and tem- perature of a vapor can be determined only by experi- ment, as has been done by Regnault (Memoirs de V Acade- mie des Sciences, 1847 ; Comptes Rendus, 1854). His results are represented quite accurately by the following empirical formula, given by Rankine, and first published in the Edinburgh New Philosophical Journal for July, 1849 (Phil. Mag., Dec,, 1854). com. log p = A. -- (80) where A, B, C, are constants to be determined by experi- ment. That author remarks that this formula is sufficiently accurate for temperatures from 22 F. to 446 F. From (80) we find ' (81) B = p . + X 2.3026, (82) X 2.3026. (83) The following are the values of the constants for the 98 IMPERFECT FLUIDS. [77.] vapor of water (saturated steam) for degrees Fahrenheit and pressures \npounds per square foot : A = 8.28203, log B = 3.441474, log C = 5.583973. These values differ slightly from those given by Rankine, because he used r = 493.2 F., while here r = 492.66 F. Kegnault's experiments also furnish the data for determin- ing the constants for Alcohol, Ether, Bisulphide of Carbon and Mercury. 7 7 . Volume of vapor. From equations (74) and (83) we have V. V, = r dp (84) d r which for saturated steam becomes, equation (78), 1117880 - 544.6 r MT + - from which r a , the volume of a pound of the steam, may be determined. It will be found hereafter that the volume of a pound of saturated steam at 212, the pressure of one at- mosphere being 29.9218 inches of mercury, is 26.50 cubic feet, nearly, and the volume of one pound of water at its maximum density, 39.1 F., is 0.01602 cubic feet ; hence, a pound of saturated steam under the pressure of one atmos- phere occupies 1654 times the volume of one pound of water at its maximum density. This increase is illustrated by Fig. 33, in which the small square at the upper left-hand corner represents the volume of one pound of water, i\ 0.01602, and the entire large square the volume occu- 178.] WEIGHT OF VAPOR. 99 pied by the steam produced from it at 212 F., v % = 26.58, and the large square minus the small one represents the increase of vol- ume in changing the water to va- por, Vy V^ If the boiling point increases in tem- perature, the vol- ume of steam de- creases ; still with- in the range of ordinary practice the volume of water is so small FIG - 33 - compared with that of the steam generated from it, the former may be neglected, and we have with sufficient accuracy (86) 78. Weight of vapor per cubic foot, sometimes called the density of the vapor. Since a pound of the vapor occupies 7U W 1M lip, md p or, p or,p or,p PIG. = 14. = 60, = 100, = 160, = 200, 33a. 7, v, v V V = 26.59, = 7.100, = 4.403, = 2.830, = 2.294, then n = n = n = n = 1 1 1 1 .06470 ; .065837 ; .065954 ; .06551. Mean n = 1.06550 ; .-.pv 1 -'"" = constant, (88) or, p tfl = 484, (89) p being pounds per square inch and v cubic feet per pound. EXERCISES. 1. Required the latent heat of evaporation of water at 20 F., 60 F., 200 F., 400 F., and explain why it is less [78.] WEIGHT OF VAPOK. 101 at the higher temperatures. (Equation (78), or tables of saturated steam.) 2. If Kegnault's law can be trusted so far, find the tem- perature at which the latent heat of evaporation will be zero for steam. Ans. T = 2052 F. or T = 1592 F., or about the temperature of the melting point of brass, above which there would be no difference between the liquid and vaporous forms. This is called the critical tem- perature, and has been determined experimentally for some substances (Phil. Trans. (1869), CLIX., 575). 3. If the latent heat of evaporation of one pound of water at the melting point of ice could be utilized in projecting that pound vertically upward, how many miles high would it ascend in a vacuum, considering gravity constant, and g = 32.2 ? 4. Through what height must a 100-pound ball descend in a vacuum so that its energy if entirely utilized for the purpose would just evaporate one pound of water at and from 212 F. 5. Find the value of JL for saturated steam at 212 F. d r If we resort to tables of saturated steam, we find that Rankine's Table VI. (Steam Engine) is not suitable for this purpose, because the tempera- tures are for differences of 9 F. Several other tables give the pressures for every degree. From one of these we find that at 211 F., the pressure is 14.406 Ibs. per sq. in. 212 .. " 14696 " " " " 213 " " " " 14.991 " " " " Hence, from 211 to 212 we have ~ = 0.290 X 144 = 41.76, approximately, from 212 to 213, we have = 0.295 X 144 = 42.48* approximately, 102 IMPERFECT FLUIDS. [78.] the mean of which gives which is nearer correct, but cannot be exact. To find the correct value directly, use equation (82), and we have dp* 2116.2 r B 2(7 "I 3026 d T ~ 672.66 L 672. 66 (672.66)' 2 J -M1UX MX|^ = 42.06, which is the value required. 6. Find the volume of a pound of saturated steam at 212 F., and the weight of a cubic foot of it. From equation (84) we have ff e *' *%P (I T Water increases in volume 0.04775 per unit from that at its maximum density to 212 F. ; hence, v, = 0.01602 X 1.04775 = 0.01678 = 0.017 with sufficient accuracy. Then . = 0.017 + pffvvYlp = 26 ' 585 CU " ft " 6 < 2.66 X 42.06 and w = = 0.03762 Ibs. per cu. ft. (Rankine gives the following empirical formula for the vol- ume of liquid water at any absolute temperature. If steam followed the law o f perfect gases, we could now [78.] WEIGHT OF VAPOR. 103 find the volume of a pound of it at any temperature. For we would have pv 2116.2 x 26.5 - ~~ .-.*= 83.37 - (89a) 7. Find the volume of one pound of saturated steam at 1200 F. 8. Find the volume of saturated steam at 212 F. gener- ated from one cubic foot of water. If fl a = 26.6, as in Exercise 6, then will the required volume be 1659 cubic feet. Heretofore writers have used Rankine's results, giving 1644 or 1645. Observation has not fixed the exact value. 9. Find the external and internal work in changing one pound of water into steam at and from 212 F., and their ratio. The volume of one pound of steam will be v y = 26.585, " " " " " water at 212 F., vi = 017, 2 - , = 26.568. The entire work will be r j (v, - v,) = 672.66 X 42.06 X 26.568 = 751672. 966.1 X 778 = 751626. These results would be identical if all the quantities were determined with precision. The external work consists in forcing the pressure of one atmosphere through 26.568 feet, or 2116.2 X 26.568 = 56219 foot-pounds, which deducted from the former, gives 751672 56219 = 695453 foot-pounds for the internal work ; hence, the internal work will be 695453 56219 times the external for the conditions given in the problem. 78a. Isothermal of a liquid, its vapor and its gas. 104 IMPERFECT FLUIDS. [ 79 -] To illustrate, conceive that one pound of the liquid is subjected to an immense pressure at some moderate tem- perature, T, say 2 1 2 F. Let A , Fig. d, represent this state, the ordinate to which will represent the pressure, and the ab- scissa the volume. Let the pressure be gradually removed while the tempera- ture is maintained constant the volume will increase slightly and the path of the fluid may be represented by A B, the abscissa of exceeding that of A. Let B be the state at which the liquid will boil at the temperature r ; then at that pressure the volume will increase, the temperature remaining constant, and B C will represent the path. At C let the pound be entirely vaporized and the pressure-gradually removed, while the temperature, T, is maintained constant ; then will the path be the isothermal C r. The entire broken line A B C r is an isothermal of the fluid. Let the operation be re- peated at a higher temperature the boiling point will be reached at a state above and a little to the right of B, so that a curve F E passed through such points may be called the locus of boiling points. Continuing the operation as be- fore, and the state at which the pound will be evaporated will be represented by a point above and to the left of C, and the curve traced through all such points will be the curve of saturation E C G, already described. These curves will meet at some point E, and the temperature of that state is called the critical temperature. 79. An experimental determination of the density of saturated steam was first made by Fairbairn and Tate in 1860 (Phil. Trans. (London), CL., (1860), 185; CLIL, (1862), 591). The densities as thus found differed from those previously found by Rankine from T F to ^ of the experimental values, thereby giving larger values above 242 [80.] MEASUKEMENT OF HEIGHTS. 105 F., and below some were larger and others smaller than the experimental ones (Miscellaneous Scientific Papers, p. 423). In these experiments, the steam was in a statical condition, while in Regnault's experiments the steam was in rapid mo- tion from the boiler to the condenser differences of con- dition which would naturally affect the results. The proper value of J would also affect the result ; but the results ob- tained by the two methods agree as nearly as one might ex- pect under the circumstances. Rankine's Tables are in his " Prime Movers," and those of Fairbairn and Tate, above the pressure of one atmosphere, are in Richard's Steam In- dicator, Weisbach on the Steam Engine, and other works. The apparatus employed by Fairbairn and Tate for deter- mining the temperature of saturation consisted of two glass globes connected by a bent tube below them. The tube was filled with mercury, above which in the globes was the liquid, one containing more than the other, then the globes and tube were placed inside of a small boiler containing the same liquid, and the whole heated. So long as the steam is saturated the mercury in the tube will remain stationary, but the instant that the smaller volume of water is all changed to vapor (some water still remaining in the other), the mer- cury will rise in that end of the tube nearest the globe in which all the water has been evaporated, as after that, that steam becomes superheated, and the rate of increase of pressure is for saturated than for superheated vapor. At the instant of change the volume of the steam will be the volume of the space above the mercury, and the temperature and pres- sure of the steam in the globes will be the same as that in the boiler, and hence may be readily measured. 8O. Measurement of heights. The principles * Theori* Mecanique de la Chaleur, % ed. (1875), II. 106 IMPERFECT FLUIDS. [81, 82.] here discussed furnish the means of determining altitudes. Thus, if at any point on a hill or mountain the temperature of boiling water be observed, the pressure of the atmosphere at that place and time may be computed by means of equa- tion (80), then will the height above the level of the sea be h = 60346 log ^ feet, (90) in which p Q is the pressure of the atmosphere at sea-level, 2116.2 pounds per square foot. (See the author's Elementary Mechanics, p. 328.) 8 1 . Sublimation consists of a change from the solid to the vaporous state without passing through the liquid state. Experimental data in regard to this change are want- ing, so that we are unable to make use of any analysis rep- resenting this change. 82. Evaporation without ebullition. Experi- ment indicates that evaporation takes place at all temperatures for a great variety of substances, if not for all. Snow and ice evaporate at temperatures below freezing, and many solids emit sufficient vapor to be detected by the sense of smell. The evaporation of water in the atmosphere is the most im- portant part of this subject. Water is elevated in the form of mist in the atmosphere, forming clouds, by which means water is redistributed over the earth. Evaporation takes place at the surface of bodies of water and is unaffected by the conditions below, except so far as they modify the sur- face. It is generally greatest at higher temperatures, al- though other conditions modify this law. It is modified by the hygrometrical conditions of the atmosphere, and is greater during a wind than during a calm. It is greater during summer than winter, and generally greater during June and [83.] PKESSURE AND VOLUME BOTH CONSTANT. 107 July than for any other months of the year, but it is some- times, though rarely, greater during August than for any other month. It is greatly affected by locality, there being sometimes a difference of several inches within a few miles. It has been observed that one half an inch in depth has been evaporated from a large pond in twenty-four hours in lati- tude about 42 north, but this is an extreme case. The amount is also very different for different years, the maxi- mum exceeding the minimum by more than fifty per cent. Twenty to thirty -five inches of evaporation per year is very common, in which cases the amount for June would range from about 2.8 inches to 5.4 inches, " July " " " " 3.0 " " 5.8 " " August " " '< " 2.3 " 5.3 " It is recorded that in July, 1875, near Boston, the evapo- ration was 7.21 inches. (Much valuable information on this subject was collected and published for the use of the court in The Case in the Supreme Court of the State of New York General Term Fifth Department for an Applica- tion by the City of Rochester to acquire certain rights to draw water from Hemlock Lake, Yol. II., about 1884. Also certain Transactions of the American Society of Civil Engineers, New York, particularly a paper by Mr. D. Fitz- gerald, Transactions, Yol. XV., 581, (1886).) 83. Assume the pressure and volume both constant during the absorption of heat. These conditions make dp = o, and d v = 0, in equations (A) ; hence, d H = K v d r = K v d r ; .-. K, = K v . Strictly speaking, these conditions are realized for only a few exceptional cases, as water at its maximum density. Generally, the volume changes during the absorption oi heat under constant pressure, but for solids and liquids, the change of volume under constant pressure, and of pressure at constant volume is so small they may.be considered as 108 IMPERFECT FLUIDS. [84.] constant, and more especially so when it is considered that the work due to these changes for these conditions is usually very small compared with the energy expended in making the substance hot. If C be the mean specific heat of a solid or a liquid for a large range of temperatures, we have prac- tically, JT V = K V = a Tables of specific heats of solids and liquids give values for C only. Specific heat of a few solids and liquids. Snbstance. Specific heat C. Authority. Wrought iron 0.11379 Ttegnault. Cast iron 12983 Copper 09511 Ice 0.504 Person. Spermaceti 0.320 Irvine. Alcohol 0622 Dalton Sulphur 0.20259 Regnault. 84. Mechanical mixtures may produce a change of the state of aggregation, as when warm water melts ice. The general equations of thermodynamics are discontinuous for such cases ; but having determined expressions for the heat absorbed by fusion and evaporation, we may write an ex- pression for the heat absorbed in passing from the solid to the gaseous state. Thus, let I and T > 60 ; 8>I, T = 60 ; S > I, T < 60 (impossible), S < I, T < 60, &c. 10. How many pounds of water at 212 will be necessary by mixing with 5 pounds of alcohol at 40 F. to just make the latter boil under a pressure of one atmosphere ? 85. Total heat of evaporation is a conventional term to indicate the heat absorbed in raising a liquid from a fixed temperature in the liquid state to the boiling point and evaporating it at the latter temperature. It is the sum of the sensible and latent heats above the fiseed temperature. It is also called the total heat of the vapor, and in reference to water the total heat of steam. To find it, we use the first of equations (A), or in which the first term of the second member is to be in- tegrated from T^ to r t , while the second term of that mem- [86.] EVAPORATIVE POWER. Ill ber is zero during that operation, and then the value of the last term is found while r^ remains constant, and hence is H^ the latent heat of evaporation. The total value of II will be the sum of these values ; hence, making J5T V = C, the dynamic specific heat, we have H= 67(r 1 -r a ) + // ei . (93) The lower fixed temperature is that of melting ice, unless otherwise specified ; hence, in English units r, = TO Tl = r +T- 32, where T is the temperature on Fahrenheit's scale, and 11= tf(r-32) + J7 e , (94) and for water C = 778, and substituting the value of JJ e , (page 95), we have H = 778 [1091.7 + 0.305 (T - 32)] = 849342 + 237 (T - 32) = 841758 + 237 T. (95) By means of this formula a table may be computed that will give the " total heat of steam" above the melting point of ice. 86. Evaporative power. If the temperature at which water is fed to a boiler be T F., the foot-pounds of heat which must be supplied in order to evaporate it will be II = 778 [1091.7 + 0.305 (T 32) (T, - 32)]. (96) In determining the efficiency of a boiler, or the amount of water evaporated by a pound of fuel, it is customary to reduce the amount of evaporation which actually takes place from the temperature of the feed water at the temper- ature of the steam to an equivalent amount at and from 212 F. (100 C.). In the latter case 966 heat units are absorbed, and making this the unit of evaporative power, the evaporative power in any other case will be, nearly, 112 IMPERFECT FLUIDS. [86.] 1092 + .3 (T - 32) - (T, - 32) 966 0.3 (T - 212) -f (212 - T,) 966 a form due to Rankine, who properly called the expression a factor of evaporation. By assuming a series of values for T and T v a table may be formed of the factors by means of which the given conditions may readily be reduced to that of the above unit, and the actual evaporative power will be, in foot-pounds, 778 X 966 X tabular number. The preceding expression reduces to , 148.4 + 0.3 T - T, 966 (97) by means of which the following table has been computed. FACTORS OF EVAPORATION. Boiling Initial temperature of the feed water, T y degrees F. point. VF. | 40 60 80 100 120 140 160 180 300 212 212 1.18 1.15 1.13 1.11 1.09 1.07 1.05 1.03 1.01 1.00 230 1.181.16 1.141.12 1.10 1.08 1.061.04 1.02 1.01 250 1.191.17 1.15 1.12 1.10 1.08 1.06 1.04 1.02 1.01 270 1.19.1.18 1.15 1.12 1.10 1.08 1.06 1.04 1.02 1.02 290 1.19,1.19 1.161. 141. 12il. 10 1.071.04 1.03 1.02 310 1.201.20 1.161.141.1211.10 1.08 i .(:. 1.04 1.03 330 1.211.21 1.17(1.151.13 1.11 1.0911.07 1.05 1.03 350 1.22 1.21 1.191.17 1.15 1.12 1.10 1 .OS 1.06 1.04 370 1.231.21 1.19,1.17 1.15 1.12 1.10 1.08 1.06 1.04 390 1 . 24 1 . 22 1.191.17 1.15 1.13 1.11 1.09 1.07 1.05 410 1.24 1.22 1.201.18 l.lf) 1.14 1.12 1.10 1.08 1.06 430 1.24 1.22 1.2111.19 1.171.15 1.13 1.11 1.09 1.07 [87. J SUPERHEATED STEAM. 113 87. Superheated steam. "When the temperature of a vapor for a given pressure is higher than the boiling point for that pressure, the vapor is said to be superheated, and is sometimes called " steam gas." Yapor may be super- heated by separating it from its liquid and subjecting it to a still higher temperature. Let the vapor be generated at t l degrees and afterward heated to r^, degrees, then will the heat absorbed above T I degrees be, r T< * II =-. // e + / /r p d r = II, + K^ (r. - r t ). (98) If the vapor be steam generated from water at T n degrees Fah. evaporated at 'L\ degrees, and superheated at constant pressure to T^ degrees, H= 778 [(T, -To ) -f- 1121.7 - 0.695 T, + 0.48(r a - r,)] = 778 [1121.7 0.175 TI - T a -f- 0.48 TiJ. (99) EXERCISES. 1. If one pound of coal will evaporate 10 pounds of water at and from 212, how many pounds would it evaporate from 80 F. at 310 F. ? 2. If, when the feed water is at 32 F. and the boiling point at 410 F., one pound of coal evaporates 7 pounds of water, how much ought it to evaporate at and from 212 F.? - 3. Experiment proves that one pound of good coal, com- pletely burned, will develop 14500 heat units (B. T. U.), how many pounds of water could one pound of such coal evaporate at and from 212 F. if all its heat of combustion were utilized for that purpose, under the pressure of one atmosphere ? Ans. 15.0 Ibs. -1. Under what physical conditions could one pound of 114 IMPERFECT FLUIDS. [88. j such coal as mentioned in Exercise 3 evaporate 20 pounds of water ? 5. If the feed water be 32 F., what must be the tempera- ture and pressure that the " factor of evaporation" shall be 2.0? 6. Find the B. T. U. required to produce one pound of saturated steam at 212 from water at 32 ; and steam gas at the same temperature from the same water, and compare the results. (Those who have not time to pursue the more abstruse part of the sub- ject may omit to Art. 98, p. 143, except Art. 95, p. 126.) 88. Free expansion of gases. When the exter- nal pressure is much less than the expansive force of the gas during expansion, the expansion is said to \>efree. This principle has been used for determining the difference be- tween the absolute zero of the perfect scale and that of the air thermometer. When a gas rushes freely from a vessel under pressure into another of lower pressure, the only work done will be that of the friction through the passage and among its own particles, which process will generate heat ; but during the expansion in the second vessel the gas will be cooled, and if there were no transmission of heat to or from external bodies and the gas were perfect the final tempera- ture should be the same as the initial. Joule made an experiment upon air, by immersing in a vessel of water two other vessels connected by a pipe, one of which was filled with air at 22 atmospheres and the other exhausted of air ; after which, by opening a stop- cock in the connecting pipe, the air rushed from one vessel to the other, but no apparent change of temperature was observed (Phil. Mag. (3), (1845), XXYL, 376). M. Hirn, in 1865, made a more delicate experiment for the same purpose, but without detecting any change in tempera- ture (Tkeorie Mecanique de la Chaleur, 3 6me , I., 298). Sir William Thomson in 1851 executed a much more 88.] FEEE EXPANSION OF GASES. 115 delicate experiment. A porous plug, composed of a bunch of fine silk or, in some cases, of cotton was inserted in a long tube, and the difference of pressures on either side was regulated by the amount of silk or cotton in the plug. The air was forced through a box with a perforated cap stuffed with cotton-wool, so as to prevent fluctuations, and this pump was worked some time, so as to secure steady action before records were made. The pressure and tem- perature were kept nearly constant during the experiment. The part of the tube beyond the plug was immersed in a vessel of water, observations upon which determined the amount of cooling (Thomson's Mathematical Papers, pp. 333-455 ; Phil. Mag. (1852), (4), IT., 481). Since heat will generally be abstracted, H will be negative, and the second of equations (A) becomes -dH= K^dr - rj^dp. Adding and subtracting v dp, we have d II = K v d r (r ( - -- v J dp v dp ; dp. (100) considering K v as constant. But integrating v d p by parts, between the limits of p, v 9 and v l PV gives v dp = v^p, v,p, +v*P d v - The last term would be negative if the order of the limits were reversed. But the work done upon the gas in forcing it through the plug will be nearly the external work for the sensibly perfect gases ; or and substituting these in equation (100), gives 116 IMPERFECT FLUIDS. [89.] f(r ~ - *>) dp = R, (r, - T,) + v,p t - v,p, (101) But this cannot be reduced, since r depends upon the zero of the perfect scale, which we do not know ; we, how- ever, know by experiment that it is not far from the zero of the air thermometer ; hence, if t be the temperature from the absolute zero of the air thermometer, we have from equa- tion (2) p v Po v -j - j- - R, very nearly ; dv _ 7? ' di = 1? Rt dv d v ^ = jr vei r, r, = tf, t n almost exactly, for the range of temperatures in an experiment, and con- sidering the specific heat as constant, or J5T P = <7 P ; these will reduce equation (101) to ~ ~~' (102) which shows that the absolute zero of the perfect scale is lower than the zero of the air thermometer. 89. The absolute zero, found by experiments upon air, oxygen and nitrogen, by the aid of equation (102), has a mean value of 492.66 F. = 273.7 C. below the melting point of ice (Thomson's Papers, p. 392), while the zero of the air thermometer is 491.13 F. below. so that the absolute zero is about 1.53 F. below that of the [90, 91.] SPECIFIC HEAT. 117 air thermometer. The difference is so small compared with the distance from the melting point of ice as to render it probable that the approximation is very close, supposed to be within -- of the exact value. SPECIFIC HEAT. 9O. General expression for specific heat. The first of equations (A) gives which is a general expression for the sp. heat of any sub- stance at the volume v and temperature T, the path of the fluid being arbitrary. Unless otherwise stated, the path is assumed to be either parallel to, or perpendicular to, the -y-axis, giving rise to the conditions j? constant or v constant ; hence, dp = o or d v 0, which in equations (A) give the respective equations as previously established in Articles 38 and 39. 91. Specific heat at a change of state of aggregation. It has been shown that for fusion, evapo- ration and sublimation the change of state takes place at a fixed temperature corresponding to a given pressure for each substance ; hence, for these cases we have d-c = o, d H > 0; similarly, JT T = Jr,-J?=co; or the specific heats AT the change of state are infinite, 118 IMPERFECT FLUIDS. [92.] 92. Modified expression for the specific heat. The specific heat of substances which are capable of a large expansion conceived to be indefinitely large admit, ac- cording to the theoretical views of Rankine, of being ex- pressed in two parts a constant and a variable part. To show this, let b, Fig. 34, be the initial state of one pound of the substance, b e an isother- mal for the substance, c d a consecutive isothermal, K v the specific heat at the volume v, K v - the specific heat at the volume v". From the state b let the temperature be increased d r, thus in- creasing the pressure an amount b c ; from the state c ex- pand it. doing the external work v c d v", at d abstract heat, so as to reduce the temperature d r, reducing the pressure to v" e, thence compress it ' isothermally to J, doing work i)" e b v upon the substance, then will the resultant external work be f*v" Icde I dpdv, Jv and, according to the second law, omitting the parentheses indicating partial differentials, ^ -V FIG. 34. / v Also, m, 5 c m t =. J" v d r, = K^- dr; [92.] SPECIFIC HEAT. 119 And, m, b c m, -j- m 3 c d m 6 m, b em 4 m t edm s b cde\ hence, substituting and reducing, Removing v" indefinitely to the right, we have v" = , and then JT V . becomes K^ , which, according to a theory of Rankine, is constant for a constant state of aggregation, according to which the preceding equation becomes v, (105) in which the last term is the rate at which the internal work is done at the volume v and temperature r, due to a change of temperature. It may be found more directly from equa- tion (67) by differentiating it, considering r as the independ- ent variable, giving and dividing by d r gives the heat doing internal work per unit of temperature, which is the same as that above when the corresponding limits are assigned. Equation (105) being the specific heat at the volume v and temperature r, we have, for the heat absorbed at constant volume, ' JT J d'** the last term of which may be integrated when the equa- tion of the gas is given. Similarly, if p and r be the independent variables, we 120 IMPERFECT FLUIDS. [93.] would find K 9 = C + R - r /*() dp, (107) Jo and for an increase of temperature r, r^ the heat absorbed at constant pressure would be H= f K v dT= J 93. The apparent specific heat is the total heat absorbed by unity of weight in producing an increase of temperature one degree, and includes that necessary to make the substance hot as well as that doing the internal work, and also the external in the case of constant pressure. It is represented, for expansible gases, by the second mem- bers of equations (105) and (107). The real specific heat is that part of the apparent specific heat which directly makes the substance hotter. It is rep- resented by C in equation (105), and is called actual energy. The apparent specific heat is the sum of the actual and potential energies in the particular specific heat. Equations (105) and (107) are the dynamic specific heats, and to find them in ordinary units they must be divided by J. The theory of Rankine, referred to above, is " The real specific heat of each substance is constant at all densities so long as the substance retains its condition, solid, liquid, or gaseous" (Prime Movers, p. 307). The correctness of this theory has been questioned by Clausius, and in one place Kankine says "it is prolatdy constant" (ibid., p. 250). The theory is useful in showing the different effects probably produced by the absorbed heat ; but it is of no service ex-. cept in expansible gases, and cannot be used in those cases except where the equation of the fluid is known and it is known for only a few substances. [93.] SPECIFIC HEAT. 121 EXERCISES. 1. Find the specific heat at volume v and temperature r for the gases represented by the equation^? v = R t We have , a a + f dv 2 a From this result it appears that the specific heat at constant volume decreases as the temperature increases, for all gases represented by the above equation, and approaches C as a limit. 2. Find the heat necessary to raise the temperature of one pound of the gas of the preceding exercise from r l to r t at the constant volume v. Equation (106) becomes From this it appears that G -\ is the mean specific heat between r> "0 ^aTi and r,. S. Find the heat absorbed by one pound of CO t in rais 122 IMPERFECT FLUIDS. [94.J ing its temperature from 500 F. to 600 F. at the volume 8.5 cubic feet. From equation (6), page 13, we have R = 35, a = 481600, and the preceding equation gives (C being 132), / H = (132 + ~) 100 = (132 + 0.377) 100 = 13238 foot-pounds. It will here be observed that the term due to internal work is very small compared with the actual energy, and may properly be omitted, especially when we consider ia addition thereto that it is less than the errors of ob- servation determining the number 132. 4. Find the apparent specific heat of the gases represented by the equation p v = R r. (Use equations (105) and (107).) 94. General expression for the difference of specific heats. In equation (103) the left member is the specific heat at constant pressure, if p be constant ; hence, To make this more apparent, use both of equations which are dv, subtracting, gives In Figure 35 let A r and B C be two consecutive isother- inals and A 1> the path of the fluid, A B perpendicular to the L94.1 SPECIFIC HEAT. 123 axis of volumes, A 7 parallel thereto, then, as shown by equa- tion (20), will r ( - 1 d v = m. B b m, : d rl in which d v in the former expression is the abscissa of b from A = A a, and dp = a b in the latter is the or- FIG 3g dinate of b from A C. If d v in the former be made the abscissa of C in reference to A, that is, d v = A C. then would T ( -JL \ d v be the area m^ B C m 4 , \a rJ which is the sum of the two preceding expressions, anc 1 hence the value of the expression in the [ ] of the former equation. To indicate that d v extends thus far, it is nec- essary to express v as a function of r, since it will be limited by two isothermals, and will be written thus : then which, substituted, gives -=, \drJ \ Ans. Ap A v = - \p-\ 3. Find the difference between the specific heats of car- bonic acid gas at the melting point of ice at the pressure of one atmosphere. 4. Find the difference between the specific heat of water at constant pressure and at constant volume when at its maximum density. It is at its maximum density under the pressure of one atmosphere at the temperature of 39.1 F., and at this state ldv\ _ 5. Find the difference between the specific heats of water at 39 F. and 77 F. under the pressure of one atmosphere, [94.J SPECIFIC HEAT. 125 the coefficient of expansion at 77 F. being 0.00014 of its volume per degree, and the coefficient of compression being 0.000046 of its volume for one atmosphere. The volume of one pound of water at maximum density being 0.016 cubic feet, we have *= ^016X0.00014). = A,- Assuming that the rate of expansion for 38 (= 77 39) is uniform, the volume at 77 F. will be 0.016 -}- 0.016 X 0.00014 X 33 0.01609, nearly ; hence, for a pressure of one pound per square foot we have X 0.000046 351 _ dp) ~ 2116.2 10 13 ' . K - K v = 537.6 X aihr = 7.68 ; A similar computation at 122 gives Adopting Regnault's values for the specific heat at constant pressure, we find i c =1.0000 at39 1 ;= 1.0000 at 77 j C ? = 1 ' 0016 7 \c v = 0.9917 at 122 It will be observed that while the value of c p increases with the temper- ature c v decreases, and hence the difference increases more and more as the temperature increases. The exact numerical results here found are not to be relied upon, but they are approximately correct, and indicative of a general law. 6. Show, both analytically and geometrically, that the specific heat for constant pressure exceeds that at constant volume. 7. Show that the term r i -^ d v disappears for gases whose equations (if any) are p v = R r -f- cp (v). 126 IMPERFECT FLUIDS. [95, 96.] 95. Water is the only substance whose specific heat has been accurately found for a large range of temperatures. According to Regnault's experiments, there is a small increase in the -specific heat with every increase of temperature above 32 F., but according to the very accurate experiments of Professor Rowland, the specific heat of water decreases from 4 C. to 27 C., and may be approximately represented by the empirical formula c=l- 0.00052 (t - 4) + 0.000003 (t - 4)', (111) in which t should not exceed 30 C. Rankine's formula for representing Regnault's experi- ments is c = 1 + 0.000000309 (T - 39.1)" (Fall.), (112) = 1 + 0.000001 (t - 4)'* (Cent.). M. Bosscha represents Regnault's experiments by the formula c = 1 + 0.00022 t (Cent). (113) When the law of the specific heat is known, the number of thermal units absorbed in raising a pound of the sub- stance from T, to T y degrees will be h= f *cdT, -I JT V and the mean specific heat will be h -* 2 * \ 96. Another form of the general equation. Substituting the value of A' v , equation (105) in the first of equations (A) gives r I ^drdv J 00 dH= C dr + r drdv + rdv. (114) * Trans. R. S. E., XX. (1851), 441. [96.] GENERAL EQUATION. 127 This is analyzed by Rankiue as follows (Prime Movers, p. 312) : " I. The variation of the actual heat of unity of weight of the fluid C v dr. " II. The heat which disappears in producing work by mutual molec- ular actions depending on change of temperature and not on change of volume. " III. The latent heat of expansion, T ~ d r, that is, heat which dis- appears in performing 1 work, partly by the forcible enlargement of the vessel containing the fluid and partly by mutual molecular actions de- pending on expansion." The integral of the last equation would if determined, give the heat absorbed, in foot-pounds, in producing the changes of temperature and volume ; but the integral can- not be performed without the equation of the path of the fluid and the equation of the fluid. Still another form. Subtracting pdv, the external work done, from both members of the preceding equation gives --p rf,(115) in which each member is the excess of the heat absorbed above the exter- nal work done. The several terms may be represented by Fig. 36, in which A B is the path of the fluid, A C an isothermal through A, and A a c C represents the internal work done during the isothermal expansion A C. Then H = m l A B m 3 , fp d v = v, A B v., = v, A Cv t + C. I I T . d r dv = areas represented ~by the dotted J J d T between C m^ and B m 3 , 128 IMPERFECT FLUIDS. [96] C (T, r,) = m, C B m 3 areas represented by the dot- ted lines, C (r, *,) + / ft ~^ d T dv = m, C B m s , / / (IT v = A a c C, v i a c v t = m l A C m y ; H ip d v m, A B m 3 v t A B v t It is apparent that the second term of the second member of equation (115) is the

v R r , verify the statement that the internal work is the same whether the path be A C and C J2, Fig. 36, or the indefinitely ex- tended isothermals r a and r b . (Equations (117), (118), (119).) 2. If the equation be the general one given by Kankine, p v = 12 T a ~ &c., a.,, a., a., &c., being TV T 2 V constant, verify the fact that the internal work is the same, whether A C and C B be the path, or whether the path be along two isothermals through A and , respectively, in- definitely extended. 3. Test, the same principle for the ideal gas whose equa- tion is p V* = c T, c being a constant. 4. Will the principle stated in Exercise 2 be true if the equation of the gas were p = v T ? 5. Find the internal work done by ex- pansion at constant pressure from v l to v, when the equation of the gas i&pv = n ^] f R r _ ^L. ( (117), or (118) and (119).) Ans. 2 a\ L.T LT, v l r 9 v y _\ V, 1/4 FIG. 38. This is the value of the last term of equation (108). The last term of equation (108) is not easily found directly, since 130 IMPERFECT FLUIDS. [96.] it contains internal work due botli to a change of volume and temperature. Since r 2 increases with i\ when p is con- stant, it follows from the preceding Answer that the internal work increases with the expansion and approaches the f) . limit _- , which is a function of the initial state only, and TI Vl decreases with the increase of temperature. It may be in- ferred from this although we have by a more complete analysis found that the last term of equation (107) is not only essentially negative but decreases, numerically, with in- crease of temperature ; hence, the internal work of all ex- pansible substances whose equation approximates to the fonn p v = R T -- increases with increase of temperature un- der constant pressure. 6. Find the real dynamic specific heat of carbonic acid gas at the pressure of one atmosphere. We have <7 P = CV + R = 132 + = 167 foot-pounds. 7. Find the internal work of expanding C O^ from v, = 8.5 cu. ft. at r l 500 F. to T, = 600 F. at a constant pressure. From equation (6) we have 5 500__481600__ =20471b 8.5 500 X (8.5)* _35_XJ>00 481600 v t 600 , ' . . r>i = 10.22 cubic feet. Substituting in the Answer of Exercise 5, gives 2 X 481600 [ 500 x a5 ~ eoo X 1 10.22] = 70 ^-pounds, nearly. But this is the work for an increase of 100 degrees of temperature ; hence, the average will be 0.70 foot-pounds per degree of temperature. This is less than ^ ff of the heat producing actual energy of ihe substance [97.] GENERAL EQUATIONS. 131 per degree of temperature, as will be found by comparing this result with that given in the preceding Exercise. From the preceding analysis it appears that for ordinary engineering purposes the specific heat of all substances may be considered constant for a constant state of aggregation ; and the most important element involving the imperfection of the fluid is that due to a change of state of aggregation. It, however, furnishes a wide field for scientific investigation. 97. Other forms of the general equations. In Fig. 39 let the path A b be intersected by equidistant iso- thermals, of which T through A and r -f- d r through b are consecutive. Through A draw the horizontal A a and through b the vertical b a ; then will the heat absorbed in passing from A to a at constant pressure be excepting that d r is not independent, FIG 39 but is dependent upon d v, the abscissa of b in reference to A. and hence we have Similarly the heat absorbed from a to 1 at constant vol- ume will be hence, ultimately, the heat absorbed in working from A to will be the sum of these, or a form given by Zeuner ( Theorie Mecanique de la CkaZeur, p. 547)-. Substituting {-\ found from equation (109) and (-3-^ 132 IMPERFECT FLUIDS. [97.] from the same, reduces this to which may be found directly from equations ( A) by elimi- nating d T between them. This form is given by Clausius (on Heat, p. 179). Again, from equation (A} t we have, for r constant, and from equation (105) L\ *L (. vv 3 v* v+ FIG. 40. from which, the second members being the same, we have dff, d.II, Similarly, if r 3 be the temperature of the isothermal d c, r. of v z, we would have clH, dff, dII 3 dH._ T l T 2 T 3 T 4 or, considering heat emitted as essentially negative, all the terms may be written with the plus sign, and, generally, when a succession of operations are performed in Carnofs cycle, we have dH or, ultimately, -0. 134 IMPERFECT FLUIDS. [97.] But any cycle may be divided into an indefinite number of strips by adiabatics drawn across it, and by drawing iso- thermals from their intersections with the path to adjacent adiabatics, an in- scribed polygon may be constructed whose area may be made to differ from that of the given cycle by less than any assignable quantity ; hence, ulti- mately, if the integral extend through- FIG. 41. ou t the entire cycle, = 0. (122) f d -f If the integral be separated into two parts, one along A a B, during which heat is absorbed, the other along the B J A, during which heat is emitted, we have r s *n f A a_B =0 JA r Jn T Equation (122) is Thomson's generalization of the second law. It was first published by Clausius in 1854 (Poyy. Ann., Vol. XCIIL, p. 500 ; Clausius on Heat, p. 90). (The differential of a function of two or more independent variables is said to be an exact differential. Let Mdx+JTdy be such an expression, in which M and TV may be functions of a? and y ; then it is shown by the calculus that the differ- ential of M in regard to y equals the differential of N in regard to x, or, dM ^d_N_ d y dx This principle has been successfully applied to many prob- lems on heat, and of the early investigators, Thomson led in this mode of analysis.) Since it has been shown that the integral of - - is zero [97.] GENEEAL EQUATIONS. 135 for a complete cycle, it is an exact differential, and may be represented by a single symbol, as q>, and we have = ^j hence, from (^)j we have -2-^K- (^}d (123) t v T \ dt ) Applying the preceding principle, we have d_ /Kv\ _ d_ ( d P\ dv\r)~dr \d T/ or , ^ (d*p\ . d JK V = T \~j3j dv, which is the differential of equation (105). In differen- tiating the left member, r is not a function of v, since v is constant during the change of temperature. Finally, let d = the internal energy of the substance both actual and potential, L = the latent heat of expansion as a thermal capacity, and other notation as previously given, then dff= d E-\-p dv = KT, d r -\-Ldv ; (124) . *.<^- 7 = K v d r -j- (L p)dv. (125) But when any substance is worked in a complete cycle the resultant internal work is zero ; hence, dE is an exact differential, and we have, omitting the parentheses of the partial differentials, d __ * dv dr dr Similarly, for r and p independent variables 136 IMPEKFECT FLUIDS. [97a."j Since is an exact differential, therefore will its value V 7 I -" j f-\ OCX dr-\ dv v 1 ^") also be an exact differential ; hence, d.Xv = d.. (129) dv T dr T> dv d T T observing that r in the left member is not a function of v ; hence, d T = in that member, and by (126) dp = dL_dJC I = L (13()) 9 7 a. The Thermodyiiamic Function, or En- tropy. The function tp Kankine calls the therinodynamic function. The differential of q>, or d (p, is the heat ab- sorbed for each degree of absolute temperature between zero P "VB FIG. 42. and T, while the substance is worked along any path from a point on one adidbatic to a point on the adjacent adiabatic. First let the path be an isothermal, as A J2, Fig. 42, whose temperature is T. Intersect the path by an indefinite number of ordinates having jetween them the constant distance dv, and from [97a.] THE THEKMODYJSTAMIC FUNCTION. 137 the points of intersection #, &, a', I', &c., draw adiabatics; and across tliese draw the isothermals K D, J f] G H, &c., the successive ones differing by one degree, in which case the temperature of C D will be t 1, of E F, T 2, &c. Then will d = abdc = c df e = ef h g, &c. And for an expansion from state A to state , we have d y = A a c C = CceEEegG, &c., and similarly for other expansions. But C c e E does not equal c d f e, &c. For the sensibly perfect gases equation (131) becomes d cp = d v, according to which d

cdfe > d c' e' f, &c. } and for v indefinitely large the area representing d

will be indefinitely large. The indefinite integral of the preceding equation is q> = R log v -\- C, 138 . IMPERFECT FLUIDS. [97a.' a and for v = 1, we have g) = C (p, (say) ; . .

be divided by iso- thermals of which the successive ones shall differ by one degree of temperature, and adiabatics drawn in the manner just described, all the small areas thus formed will ~be equal. Since Fig. 42 is used to illustrate two modes of divi- sion, it should be observed that the two sets of adiabatics will not coincide, but that there will be a less number be- tween the initial and terminal adiabatics, A (p, and B

r _ P' . dv' T' p J 9 ' hence, the increments of tfte volume vary inversely as the pressures for those volumes, or directly as the volumes. The increments, then, become indefinitely large as the adiabatics approach indefinitely near the axis v and dimin- ish indefinitely as they approach the axis O 2). Let these adiabatics be nwribered in the order of the natural numbers, beginning with any arbitrary number, as 7 for A - T,} (q> B -

A ). (132a) The natural zero-adiabatic is the ordinate Op, but be- tween that and the initial adiabatic of any problem, as A ,, tiiere will be an infinite number of adiabatics including areas equal to A a c C between the isothermals B A and D C', hence from this zero the number between A and B would be expressed by the difference between two infinites, thus, GO 00 , which is indeterminate. An arbitrary zero-adiabatic may be assigned, but it is unnecessary so to do, since it is only the difference of the thermodynamic functions that is sought. The form of the expression in the second member of equation (132) is similar to that expressing the area of a rectangle. Thus, suppose that in measuring a rectangular field, A B D 6 Y , a point is arbitrarily assumed in the side A B prolonged, from which the corner A is a feet, and B, 1) feet ; then will the length of that side be 1) a feet long in- dependently of the position of the point. Similarly, if the corner D be x yards from a point in the line of B D, and B, y yards from the same point, the area of the rectangle will be A B DC = (b a)(x y) ft.-yds., the unit area being one foot wide and one yard long. If differences -of temperature only were used the position of the absolute zero would be of little consequence. Equation (132a) furnishes the following definition : The difference between the numerical values of the thermodynam- ic functions corresponding to two adiabatics is equal to the quotient of the number of foot-pounds of heat absorbed 142 IMPERFECT FLUIDS. [97a.J or rejected in passing along any isothermal from one of these adiabatics to the other, divided by the absolute temper- ature corresponding to that isothermal. If the path of the fluid is not an isothermal, the same principle is applicable, but T will vary, and T d

no heat will be absorbed or emitted, and for this case d = 0; . . (p = constant ; that is, the entropy or the thermodynamic function of an adiabatic is constant, and this is the characteristic equation of an adiabatic. For this reason we have fre- quently used the symbol q> to mark the adiabatic. This property may be put in contrast with a property of an isothermal in the following manner : That property of a substance which remains constant throughout such changes as are represented by an isothermal line is the temperature. The constant property is that of constant heat. That property of a substance which remains constant throughout the changes represented by an adiabatic line is the Thermodynamic function, or Entropy. This constant property is the constant rate at which heat must be ab- sorbed by a substance per unit of absolute temperature when the path of the fluid is from any point on an adiabatic to a point on the adjacent one. 98. Liquid and its vapor, combined. To find the differential expression for the heat absorbed, we first find the heat necessary to evaporate the d x part of one pound, that is, a weight d as. For d v volume we have and if the volume of a pound of saturated vapor be v^ then d v - = d x. v * If TFbe the pounds of fluid of which d TFbe evaporated, then if d II be the heat absorbed for an increase d r of 144 IMPERFECT FLUIDS. - temperature of the liquid and d x the weight evaporated, we have r dW* 7/ e , or = Cdr + dx 7/ e , (133) where x is the fractional part of a pound of the substance vaporized. Integrating this, observing that the conditions of the problem require that the higher temperature r l must be that at which the quantity x is evaporated, we have H=C(r l -T) + x.H. l . (134) If the substance be water, we have C = J = 778 foot-pounds, 77 e = 1117880 - 544.6 r. Equation (134) is sometimes used in calorimeter tests for determining the amount of water in steam. Thus, to find the per cent of water, we have from equation (134) 100(1-*) = 100 . -.-. (See Addenda.) EXERCISES. 1. By condensing the steam from a boiler into a reservoir of water it was found that 600000 foot-pounds of heat had been imparted to one pound of the steam above the temper- ature of the feed water ; the temperature of the feed water being 100 F. and the steam from the boiler 320 F., how much liquid water did the steam contain ? Here we have r, - r, = 220, C(r l -r,) = 171160 ft.-lbs., H. = 692000 " " nearly, at 320 F., {99.J SPECIFIC HEAT. 145 which is the foot-pounds of heat that the steam should have contained, above 100 F. if it had all been evaporated, but the test showed that // = 600000 ; . . difference 263160 ; -.,- ,.-. 2. If the feed water be 100 F. and the temperature of the steam be 338 F. and the heat absorbed above that of the feed water, II = 900000 foot-pounds, required the amount of water suspended in the steam. Here, C (T, - T,) = 238 x 778 = 185000 H e = 683000 ft-lbs. at 338, sum = 868000 " " which not being so much as was measured, the steam must have been superheated. 99. The specific heat of saturated vapor is not that at constant pressure nor that at constant volume, but it is the heat necessary to raise the temperature of one pound of the substance one degree when the steam remains continually at the point of saturation. Conceive the tem- perature of the entire mass to be increased an amount d r and the volume an amount d v ; then will the heat exist in the three following parts : 1. The heat absorbed by the liquid. The liquid not evaporated will be W w, using the notation of the pre- ceding article, and the heat absorbed by it will be (W-w}Cdr. 2. The heat absorbed by the vapor. Let 8 be the dy- namic specific heat of the saturated vapor of constant weight, then will the heat absorbed by it be w Sdr. 3. An additional amount, d w, of the liquid will be evaporated both on account of the enlargement, d v, and the 146 IMPERFECT FLUIDS. [99.] increase of temperature, d T, and the amount of heat ab- sorbed will be dw - J7 e . Hence, equation (^1), becomes W dII=(W-'w} Cdr + wSdr + dw n e ; . . d n = (1 - x) Od r + x S d r + JI e d x, (135) which will be the heat absorbed by one pound of the entire substance under the conditions imposed. All the quantities in this equation have been determined except S. To find -6', let a pound of liquid (water, for instance) be evapo- rated at state A / AD being the volume when the o-th part of the pound is evaporated, neg- lecting the volume of the liquid ; A F the volume when the entire pound is evaporated, for which x = 1. Let state B be d r higher than A, B m the volume when the sctli part is vapor, in n be the arbi- trary increment of increase of the a*th volume, and B E the volume of a pound of the vapor for the pressure O B, then d x m n -v- B E. Draw the adiabatics and join n D. It is proposed to find the heat absorbed along the path n D. In working around the area A B n D A (or any other cycle) the heat absorbed minus the heat emitted in foot pounds will equal the area of the cycle. We have A D - xv, t =. C d r (heat absorbed),

i x // e (emitted),

4 in n q> 6 = d x II e (absorbed). g> 3 D n cp 6 = d II (which may be absorbed or emitted depending upon the slope of n D / we will consider it as emitted, then if in any case it is absorbed the algebraic sign will change). x H -- d r = A D dp = x v dp = the area D A B n D . . dll= CdT + x ( dH J*L) d T -f- //. d x ; (136) \ dr r J dr \ dr r ) d r which is the specific heat of a fluid in which the (1 a?th) part is liquid and the a?th part of it is vapor, the path being arbitrary. If the weight of vapor remain constant during the change of temperature, then d x = 0. If the entire pound be dry saturated vapor during the change of state, then a. = 1 and d no = 0, and d H -r- dr will be the specific heat of the vapor kept at the point of saturation throughout the change of state ; and the resulting value will be S in equation (135) ; .. d r dr or in heat units substituting Si rom (138) in (135a) will give (136). For water c 1, A e = 1436 -8 7 r ; 1436-8 .:.>!- -^-, (HO) which will be negative for all values of r less than 1436 F. above absolute zero, or 976 F. above the zero of Fahren- heit's scale. The negative value may be thus explained : If saturated steam be expanded in a non-conducting cylinder, a portion of it will condense, giving up its For an analytical solution see Sir William Thomson's Math, and Phys.. Paper*, Vol. I., pp. 141-207 ; Phil. Mag. (1852), IV.; Trans. R. Soc. Ed., 1851. 148 IMPERFECT FLUIDS. [100.] heat to the remainder of the steam, thus maintaining the temperature corresponding to the pressure of saturation ; and if it be compressed in such a cylinder, heat must be abstracted if the pressure and temperature continually correspond to those of saturation. If heat be not abstracted in the latter case the steam will be superheated, and the temperature will exceed that corresponding to the pressure of saturated steam. In regard to this Rankine said : " This conclusion (the liquefaction of steam) was arrived at contemporaneously and independently by M. Clausius and myself. Its accuracy was subsequently called in question, chiefly on the ground of experiments which show that steam after being wire- drawn, that is to say, by being allowed to escape through a narrow orifice, is superheated, or at a higher temperature than that of liquefaction at the reduced pressure. Soon after- ward, however, Professor William Thomson proved that these experiments are not relevant against the conclusion in question, by showing the difference between the free ex- pansion of an elastic fluid, in which all the power due to the expansion is expended in agitating the particles of the fluid, and is reconverted into heat, and the expansion of the same fluid under a pressure equal to its own elasticity, when the power developed is all communicated to external bodies, such, for example, as the piston of an engine" (Misc. Sc. Papers, p. 399). Professor Clausius said : " The conclusion that the spe- cific heat of saturated steam is negative was drawn by Ran- kine and by myself independently at about the same time (Theory of Heat, p. 135).* 1OO. Adiabatics of imperfect gases. This con- dition requires that II = 0, . . d H = in equations (A\ giving * Both papers were read in February, 1850-Rankine's in Edinburgh, and Clausius' in Berlin. 1 100.] ADIABATICS OF IMPERFECT GASES. 149 K In order to integrate the first of these, K^ and must be known functions oi r and v. Ji v not only depends upon the volume but is not a known function of r. Even grant- ing that its general expression is given by equation (105), its determination requires a knowledge of the equation of the fluid, and that can be known only empirically, and hence would apply only for the range of the experiments upon which the formulas were based. We have, however, found for carbonic acid gas, and for all other fluids investi- gated, that the specific heat at constant volume for a con- stant state of aggregation is, without a large error, constant within the range of ordinary experience ; and similarly for K v ; hence, representing these by (7 V and (7 P , respectively, we have (Ml) in which y must be constant for the range through which the specific heats are considered constant. Assuming equation (4) as the general equation of fluids, and considering that &o a, a, . dv\ dr J60 IMPERFECT FLUIDS. [100.; in which > , & J, are constants to be determined by experi- ment, we have T? v = R t ~r &c. * V T V t V R . o, . 2 &, and (141) becomes & , y d v =- -TJ aj), which are the differential equations to the adiabatics for imperfect gases. From (143), v can be eliminated by means of equation (142), resulting in an equation involving r and p only ; and r from (143) s by means of the same equation ; but the resulting equations will be too complex to admit of integration, and therefore the general finite equation to adia- batics is unknown. It is customary to assume that the equation of the adia- batics for such superheated vapors as are used for engi- neering purposes, as steam, is of the same form as that for the sensibly perfect gases, at least, within the limits used in practice ; and hence may be represented by the equation p v* = c, (144) in which y must be found for the particular substance, and the particular state of that substance. To find y for steam considered as a perfect gas, we found in Article 78 the volume of a pound of steam at 212 JF. under the pressure of one atmosphere to be 26.5 cubic ADIABATICS OF IMPERFECT GASES. 151 feet ; hence, if it followed the gaseous law down to 32, the volume at the latter temperature would be v = 26.50 + 1.366 = 19.39 cu. ft.; .-. p v = 19.39 X 2116.2 = 41033 ; A" p = 0.48 X 778 = 373.44, /C = 373.44 - 83.28 = 290.16, .-.y = f? =1.3, nearly; (145) -Zly hence, the equation of the adiabatic for steam, considered as a sensibly perfect gas, will be p v 1 - 3 = p l v, '' (146) This value of y is used for superheated steam at all tem- peratures. But steam as used in the steam-engine is generally more or less saturated, for which case Rankine used - a - for the approximate value of y^ so that for such cases the equation of the adiabatic will be pi^ p^^ (147) Rankine was the first writer to give even an approximate equation to the adiabatic of saturated steam. M. G. Schmidt, in his Theorie des Machines d Vapeur, 1861, assumed that steam comported like a perfect gas, and so assumed y 1.4, a value entirely without foundation, as shown by equations (145) and (147), and which that author later abandoned. In 1863 Grashof reviewed the question, and found the mean value of y = 1.1354. Still later, Professor Zeuner, by a series of experiments in which the initial pressures varied from 1 to 4 atmospheres, final pressures from to 2 atmospheres, and in which the specific quantity of initial vapor (or the per cent of the fluid 152 IMPERFECT FLUIDS. [100.] in the cylinder that was vapor before cut-off) was 0.70, 0.80, 0.90 ; found results from which he concluded that : The value of y is dependent chiefly upon the initial specific volume of the vapor. That it is nearly constant for the same initial state of the vapor for all the pressures observed from one to four at- That the value of y may he represented by the empirical formula y = 1.035 + 0.100 a?,, (148) in which a?, is the initial specific quantity of the vapor. This formula is limited to values of a?, between 0.7 and 1 (The- orie Mecanique do la Chaleur (1869), (329-335). In this formula, if x l = 0.76, that is, if 24 parts in 100 of the fluid is initially water, it gives y = 1.111, which is the con- stant value proposed by Rankine. If equation (148) can be extended to values of a?, much less than 0.7, it appears that the adiabatic for saturated steam approximates more and more nearly to the isothermal of the perfect gas in which y = 1 ; and for values of a?, less than 0.50, the two curves will nearly coincide within the ranges of expansions used in ordinary practice. Hence the curve of adiabatic expansion of wet saturated steam approximates to that of the equi- lateral hyperbola. But when we consider the complex nature of the problem the temperature of the surrounding walls being modified by the nature of the metal, its thickness, its exposure exter- nally ; the time of the exposure internally depending upon the piston speed ; rendering it practically impossible to realize exact adiabatic expansion it is too much to expect any em- pirical formula to cover all the cases of approximate adia- batic expansion which might arise ; and we conclude, as did Zeuner, that the empirical formula of Eankine, equation (147), is sufficiently exact for theoretical or practical pur- poses when the initial steam contains but little water. [100.] ADIABATICS OF IMPERFECT GASES. 153 To find the equation to the adiabatic for saturated vapor when liquid is present, make d H = in equation (136), and it reduces to Integrating between initial and general limits gives -.tf^-S-ii-fLS G FIG. 43. 1 d r l d r f dp i ri 7 T\^?T . . x v = (x l v l -j- + C log -!- ) -: V ar, f / dp (149) For steam C becomes J. By means of equation (86), if c and A e be ordinary thermal units, t, = = y " d r T and (149) may be written ( U9) In this solution the specific volume of the liquid is neg- lected, since the volume of both the liquM and the vapor is essentially that of the vapor. In equation (149) a?, v l = O G, Fig. 43, will be the volume of one pound of the satu- rated steam and water at the beginning of expansion, and a? v the volume of the steam and water at any point of the expansion B C, corresponding to the temperature T or pressure p. Eliminating ^ by means of equation (82), dp and then r by means of (81), the result will be the equation of the adiabatic C\ but the second member will be too 154 IMPERFECT FLUIDS. [101.] complex to be of practical value ; and the approximate equation of Raiikine (14:7) will be used instead. An im- portant theoretical deduction may be made from the equa- tion in its present form ; thus, if the steam be dry at B, the point of cut off, a?, will be unity, and making x v = u, we have with the aid of (86), (150) which is positive for values of T V less than 1436 F., the same limit that makes equation (140) negative. This shows that the volume of steam and water will be less than the specific volume, v, of steam only at the temperature T. This is due to condensation, as stated in Article 99. Let the initial volume of steam be one cubic foot, its weight will be = u\, and let r = = the variable ratio v i v* of expansion, then will equations (150) and (87) reduce to (152) by means of which the ratio of expansion may be com- puted. 1O1. Condensers. A condenser consists of a vessel kept at a comparatively low temperature by means of cold water, for the purpose of condensing steam. In the jet con- denser a liquid spray is forced into the vessel, and for the surface condenser the cold liquid circulates about the vessel or through tubes in the vessel, producing a cold surface. When the piston of the engine is very near the end of its stroke, a communication being made between the steam end of the cylinder and the condenser through the exhaust passage, the steam rushes into the condenser, and the greater [101-1 CONDENSERS. 155 part of it is suddenly condensed the pressure falling to two pounds per square inch, more or less. Using the sub- script 2 to indicate the conditions at the end of the stroke, and 3 for those in the condenser at the end of the operation, and discarding the effect of molecular changes under varying pressures, thus assuming that the heat abstracted will be the difference in the heats in the initial and terminal strokes (which will be approximately correct), equation (148) will gi ve, H = J(T, - T 3 ) + a?, v,^l - x,v,^ , (153) t\ v a for water and for the Fahrenheit scale, and is the heat abstracted from a pound of steam in reducing its tempera- ture from T 7 , to T 3 degrees. The steam end of the cylinder will remain practically at constant volume during this change, and neglecting, as be- fore, the specific volume of the water from which the steam is generated, and assuming that the volume of the space within which the change of temperature takes place is constant during the change, we have a?, v, = x 3 v 3 = u a (154) and the preceding equation becomes H=J (T, - TO + n, & - ?*\ . (155) In a continuously working engine a constant mass of vapor remains in the condenser at the end of each stroke, the amount condensed being equal to that exhausted, and H* may be neglected in (155), for which case we have Jf=J(T z - T 3 ] + u, _ The pressure of the vapor in the condenser determines its temperature, and that will be the inferior limit of tem- perature at which the steam will be worked. 156 IMPERFECT FLUIDS. [101.] EXERCISE. Determine, approximately, the amount of water that must pass through the condenser of a steam-engine per pound of steam exhausted, having given T y = 300 F. the temperature of the steam in the cylinder as it exhausts into the condens- er, a- 2 = 0.90 the fractional part of the steai and water in the cylinder that may be considered as pure saturated steam, the pressure in the condenser two pounds per square inch absolute, the water entering the condenser at 60 F., and leaving it at 100 F. By means of a table of the properties of saturated steam, Or by eqs. (78), (85) and (95), we find, using approximations to the larger numbers. Temperature of the condenser for 2 Ibs. per sq. in., T 3 = 126 From the problem, 7\ = 300 T t - T t = 174 ; . ' . J(T* - T 3 ) = 174 X 778 = 135400 ft.-lbs. Total heat of the steam at 300 will be 778 X 1173 = 912600 Heat in the water above 32, 778 X 270 = 210000 Difference, H f , = 702600. The table gives, for the specific volume of the steam at 300, v., = 6.2 cu. ft. ; ...f = 1 13 000. Total heat of steam at 126% 778 X 1120 = 871400 Heat in the water above 32, 778 X 94 = 73100 Difference, H tt = 798300 The tabular value for v t is v s = 172 cu. ft.; These values in equation (155) give H= 135400 -f 604600 = 740000 ft.-lbs. The water supplied to the condenser being raised through 100 60 = 40 degrees, the quantity required will be [102.] ISODIABATIC LINES. 157 740000 9 = 77g x 4Q = 24 pounds, nearly. Equation (155a) gives 24.7 Ibs. ; that is, a condensing engine running with stearn at 52 pounds gauge pressure will require about 25 pounds of water for the condenser for every pound of steam condensed if the temperature of the water be raised 40 degrees. If a greater difference of temperature of the water at arriving and leaving be allowed, it would require less water, or if the gauge pressure be higher, it will require more water for the same difference of temperature. The numerical computation of (155) will be facilitated by a table of the latent heat of evaporation per cubic foot, since 1O2. Isodiabatic Lines. Let CN and B M, Fig. 44, be any two isothermals cut by an arbitrary path A D. In pass- ing from A to D a certain amount of heat will be absorbed, represented by the area between D A and two adiabatics drawn from A and D respectively, as shown in Article 34. It is possible to find another path, C B, in working along which the same amount of heat will be emitted as was ab- sorbed along A D. To prove this, conceive an indefinite number of isothermals between C N and B M, and at the points of division with A D draw adiabatics ; then find a point near C, which call z, on the isothermal next below C J), such that when joined with C the area included be- tween z C and two adiabatics through z and C\ respectively, will equal that be- tween the corresponding pair at D. Proceed in this manner with the next isothermal, and so on to B\ then will the area between B C and the adiabatics through B and C respectively equal the area between A D and the adiabatics through A and D respectively, which was to be proved. The lines D A and B C are called isodiabatics in reference to each other (Rankine's Misc. Sc. Papers, p. 345 ; Steam- Engine, p. 345). 158 IMPKUFECT FLUIDS. [102.] To find the analytical condition, conceive C JV and B M to be consecutive isothermals, then the heat absorbed in passing from A to D will be, from equation (A\ d 11= Crdr and that emitted along C B, d 11= C' v dr + which, according to the conditions of the problem, are to be equal, giving ()<= ( which relation is independent of the specific heat of the sub- stance. For sensibly perfect gases we have fdp\ _ p _ R \d^l T T (dp\ _P> _ B (dr'J ~^~^' and by substituting above and integrating, we have v = B v or = A. a constant ; (157) P, that is, the ratio of the pressures, or of the volumes, at tlie respective points where the successive isothermals cut the curves A D and B C must le constant. CHAPTER IY. HEAT ENGINES GENERAL PRINCIPLES. 103. Efficiency. Heat engines, in practice, work in cycles, and when running under uniform conditions, the suc- cessive cycles will be identical, in which case the total effect will be that produced in one cycle multiplied by the num- ber of cycles. It is, therefore, important to investigate the properties of one cycle. The efficiency of a plant is the ratio of the work which the plant can produce to that of the energy supplied. Thus, if the plant consist of a furnace and engine, it is the ratio of the work it can do to the theoretical energy of the fuel supplied to the furnace. The efficiency of an engine is the ratio of the work it can do to the energy of the heat absorbed. In case of an hydraulic machine, it is the ratio of the work it can do to the theoretical energy of the waterfall. The measure of the efficiency does not involve the mag- nitude of the machine, and, hence, is only an incidental element in proportioning the engine. If one pound of air when worked in a cycle will produce a given amount of work, two pounds will produce twice as much when worked in a similar cycle. The proportions of an engine having a given efficiency depend upon the amount of work to be done in one cycle. 104. Perfect elementary heat engine. An engine receiving all its heat at one temperature and rejecting heat at one lower temperature, must pass through its series 160 HEAT ENGINES. [104.] M, of changes of pressure and volume according to Carnot's cycle. Such an engine is reversible. No such engine can be constructed or operated, but as it would give the high- est theoretical efficiency of any engine working between the temperatures of the source and refrigerator, it serves as a theoretical standard of comparison, and is referred to as a Perfect Ele- M mentary Heat Engine. Let AI A, B t J?,, Fig. 45, represent a Carnot's cycle, according to which the engine receives all its heat at the temperature T,, being the temperature of the isothermal A, A^ ; and rejects heat only at the temperature r a , being the temperature of the isothermal B, B v Then will the heat absorbed in expanding from state A, to A, at the con- stant temperature r l be, according to equation (-4),, page 48, since d r will be zero, H, = -f r l and the heat absorbed along the adiabatic A t B^ will be v v v 4 FIG. 45. and the heat rejected along the isothermal B^ B^ and along the adiabatic B^ A t , = - r/r v ,Zr- r\(*P\dv; JT, Jv, ^l*' and the sum of these will give the heat transmuted into ex- ternal work, since the cycle is complete ; hence, [104.] PERFECT ELEMENTARY HEAT ENGINE. 161 (158) The efficiency, according to the preceding article, will be E- H * ~ H * - r *- r *- T *~ T * (159) "HT ~^~ -j\~+m^' Since equations (A) are general, and applicable to all substances, the result must be equally general ; hence, the efficiency of the perfect elementary engine depends only upon the highest and lowest temperatures between which, it is worked, and is independent of the nature of the working substance. If iron, or any other solid, could be worked between the temperatures r, and r 2 , according to Carnot's cycle, it would be just as efficient as if the substance were the most perfect gas. The range of volumes through which solids expand and contract is small, so that the work done in a cycle would be comparatively small, and the changes of temper- ature are so slow as to preclude the use of such substances in the construction of heat engines. But this fact does not affect the efficiency of the cycle. The highest temperature at which the engine works can- not exceed that of the source, for it is an axiom that heat cannot of itself flow from a hot body to one still hotter, a principle stated by Clausius (Theory of Heat, p. 78). Neither can it be worked at a lower temperature than that of the refrigerator, for it is held as an axiom that a heat engine cannot be worked at a lower temperature than that of the coldest of surrounding bodies, a principle stated by Thomson (Math, and Phys. Papers, p. 181). These axioms are the same in substance, and originally were stated independently by the respective authors. If any of the heat absorbed is at a lower temperature than r 1? while all is rejected at r the efficiency will be less 162 HEAT .ENGINES. [104.] than if it were all absorbed at the higher temperature. To show this, let T 3 be the constant temperature of the second source, then we would have = *> ^ and the efficiency would be ff t + IF, which is less than the value of equation (159) so long as H\\e> less than ff t . A reversible engine has the highest efficiency for the heat utilized, and the perfect elementary heat engine has the highest efficiency of any engine work- ing between the same limits of temper- ature. The principle of efficiency is applied in the same manner, whatever be the FIG 46 path of the fluid. Thus, if the cycle be A a B d A, Fig. 46, A M and B N adiabatics indefinitely extended, then, according to Article 34, we have H, = MAaBN, II, = MAdBN-, . v _ MAaBN- MAdBN AaBdA __ U,- f.Aa. MAaBN , If the indicator card of the steam-engine were A B C J, Fig. 47, in which A B is the steam line of constant tem- [104.] PERFECT ELEMENTARY HEAT ENGINE. 163 perature, r^ B C the expansion line of no transmission of heat extended until the pressure falls to that of the back pressure, C J the back pressure line of constant temperature, T^ and J A the compression line of no transmission of heat being made to pass through the ini- tial state, A, then will the efficiency be r.-r, _ T t - T, r t as before. In Fig. 45 a constant quantity of air is supposed to remain in the cylinder of the engine during the changes forming the cycle, but in the steam-engine the heat is carried into the cylin- der with the steam, so that the mass of steam increases with the stroke from A to JB, Fig. 47 ; from I> to C the mass re- mains constant ; at Cfhe exhaust is open, communicating with the refrigerator, and remains open until the piston reaches /, at which point the exhaust is closed, and the mass of steam remaining in the cylinder at J remains constant throughout the compression J A. At the completion of the cycle the fluid in the cylinder at the state A will have the initial pressure and volume, but since the changes of state are not effected with a constant mass of fluid the operation will not be that of a Carnot's cycle, and the above expression for efficiency will not be applicable. The only theoretical mode of improving the efficiency of the elementary engine is to increase the range of tem- peratures between which it is worked. It does not follow from this principle that different substances worked between the same limits of pressure will be equally efficient, for pressures are not proportional to the absolute temperatures, except for the sensibly per- fect gases. If, however, the operation be in a Carnot's cycle, the temperatures corresponding to the pressures being found, equation (159) will be applicable. 164 HEAT ENGINES. [104.] It might be urged that some work would be expended in forcing the mass of steam into and out of the cylinder, thereby producing less external work than the same heat would do in case the changes were produced with a con- stant mass of fluid in the engine. In regard to this point, it is sufficient to observe that, if the argument be valid, the energy so absorbed is too insignificant compared with the heat energy of the fluid, to be considered. Actual engines do not produce the indicator diagrams here assumed, and, hence, must be made the subject of special investigation. EXERCISES. 1. In an ideal elementary engine working one pound of air, if the lowest pressure be that of one atmosphere, 2116.2 Ibs. per square foot at B n Fig. 45, the absolute temperature of the refrigerator r a = 550 (T, = 89.34 F.), that of the source r, = 950 (T, = 489.34 F.), and the volume swept through by the piston during each single stroke 12 cubic feet; find the greatest and least volumes of the air in the cylinder, the power developed in one end of the cylinder during one cycle or double stroke of the piston the heat absorbed, and the efficiency. To find the largest volume, v t , we have, equation (3), v _ 53 - 21 T * _ 53.21 X 550 _ 13 83 cu ft p, 2116.2 To find p, and v a the adiabatic A, B equation (41), gives 7^1 3.463 = 2116.2 (^ } = 14045 Ibs. 2.463 = 13.83 = 3.60 cu. ft. [104.J PERFECT ELEMENTARY HEAT ENGINE. 165 To find the least volume, # 15 the problem gives v 4 v l = 12 ; . . Vl = 13.83 12 = 1.83 cu. ft. And the isothermal A^ A 3 gives, equation (3), p, v, = 53.21 X 950 = 50550 ft.-lbs. . . p l = 27630 Ibs. per sq. ft. = 191.9 Ibs. per sq. in. Similarly, = ?!* = = ?!? = 1.97; 1\ v 3 p, v, .'.p,= 41 62 Ibs. v 2 = 7.03 cu. ft. The heat absorbed will be, equation (36), p i Vl log -* = 34207 ft.-lbs. The heat rejected will be ^ X 34207 = 19804 ft.-lbs. ; Lu and, hence, the work done in one cycle will be 34208 - 19804 = 14404 foot-pounds, independent of the time. The efficiency will be 14404 34208 = ' 42 ' according to which more than half the energy of the heat is rejected by the engine. The ratio of the greatest to the least volumes is and of pressures, - 4 = Ti, nearly, = 13. 1\ 166 HEAT EXGIXES. [105.] 2. In the preceding exercise, what must be the area of the piston in order to operate one pound of air between the limits assigned, the stroke of the piston being six feet. '3. In Exercise 1, if the engine make 20 revolutions per minute, what will be the horse-power developed on one side of the piston-? 4. If, in Exercise 1, two pounds of air had been used, and the lowest pressure that of one atmosphere, the tem- peratures being the same as those given in the exercise, what would have been the greatest and least volumes of air, the volume swept through by the piston being 24 cu. ft.? Would the efficiency be the same ? Would the work have been the same for the same volume swept through by the piston ? 5. If in an elementally air engine the highest pressure be 150 pounds per square inch, the highest temperature 450 F., the lowest pressure 14.7 pounds per square inch, and lowest temperature 60 F., what will be the volume swept through by the piston per pound per stroke ? 1O5. Regenerators consist of a chamber well filled with thin plates of metal so arranged as to present a large surface to the fluid and offer as little resistance to its passage as possible. The fluid, after escaping from the engine by passing through this chamber to the refrigerator, gives up a portion of its heat to the metal plates, the re- frigerator finally absorbing the heat which is permanently rejected ; after which, by passing back through the chamber and being at a lower temperature than during its former passage, it absorbs heat from the plates, thus requiring a less amount from the source in order to raise it to the required temperature. During the flow of the air from the cylinder the plates act as a refrigerator, by abstracting heat from the gas ; but during the return of the gas they act in the oppo site sense, and hence become regenerators. [105.] REGENERATORS. 167 If the temperature changed by insensible degrees in the regenerator, the efficiency would be unaffected, but such not being the case, they cause a loss of 5 or 10 per cent, even when well proportioned. Their great advantage consists in reducing the size of the cylinder, as will appear in the fol- lowing exercise. Assume that heat is absorbed at one temperature and re- jected at another, as in the preceding case, but that the change of pressure from one isothermal to the other is effected at constant volume by passing the air through a regenerator, in which case the indicator diagram will be represented by Fig. 48. If r l be the temperature of the isothermal B C, r^ of A _>, C v the specific heat at constant vol- ume, the heat absorbed in passing from A to B to C, if the operation were reversible, would be, equa- tion (5),, page 50, II, ^ C\ (r, - r.) + ^ ^ heat rejected, .-.//,-//, = B (r, - r,) which is the mechanical energy expended. But in deter- mining the efficiency, the loss of heat in passing to and fro through the regenerator must be added to U 1 ; since that amount of heat must be drawn from the source and is not accounted for in the preceding value of H^ and, rep- resenting this by the expression #v ft - *,}, 168 HEAT ENGINES. [104.] in which n is a fraction, <7 V = 0.169 X 778 = 131, we have for U the efficiency = ^ + 13J n ^ _ ^ in which n will be -fa, or -^, or whatever fraction repre- sents the lieat lost by the regenerator. EXERCISE. In an air engine with a regenerator producing changes at constant volume, let p t . v t ; p v, ; r,, r,, be the same as in the first of the preceding exercises ; determine the ratio of the pressures and volumes. Considering the engine as perfect, the work done will be the same as in Exercise 1, page 164, for the expansion during the absorption of heat must be the same. We will have, p, = 27630, j>, = 14045 ; V| = 1.83 = v n v, = 3.6. = v a Fig. 48 ; then p< -.-= p, L* =: li x 14045 = 8132 Ibs., p s =p l - = X 27630 = 15996 Ibs.; r i iy .-.= 1.97, v i = 3.40. P. Comparing these results with the exercise referred to, it appears that the greatest volume in that case was nearly 4 times the greatest volume in this ; hence, the volume of the cylinder with the regenerator, under the conditions imposed, need be only about one-fourth as large as without [106-8.] STEAM-ENGINE. 169 106. Air engines have been made in which the changes of temperature have been effected at nearly con- stant pressure, and others in which the change takes place at nearly constant volume. These conditions require special forms of mechanism, which will be considered further on ; but the work performed in a cycle may be computed from the indicator card, as in Articles 104 and 105. 107. Heat engines, whether of air, or steam, or other vapor, are assumed to transform a certain amount of heat energy into work independent of the mechanism in- volved. That is, aside from the friction of the engine wastes due to leaks arid clearance, it is immaterial whether the engine be single-acting, double-acting, reciprocating, os- cillating, rotary, disk, trunk, compound, or any other of the many forms of engines used ; the work done will be the same in all the engines by the same fluid worked between the same limits of temperature. Therefore, considering the engine as a Jceat engine only, we have only to consider the thermal changes produced in the working fluid during a complete cycle, involving the tem- perature of the feed water, and the initial and final tem- peratures in the cylinder. But &.s,apiece of mechanism, the several forms have their mechanical advantages, which must be considered in the light of practical mechanism. All the details of the engine, such as the valve mechanism, the size of the bearings, the strength of the parts, compactness, etc., belong to constructive mechanism, and are treated of in works which consider these engines as machines. In order to analyze a heat engine it is necessary to know the law according to which it receives and rejects heat ; and since, in actual engines, all these laws are not known, as- sumptions in regard to them are made which are supposed to be approximately correct. 108. Steam-engine. Steam in the cylinder works under such a variety of conditions that a complete analysis 170 HEAT ENGINES. [108.] requires the consideration of several hypotheses. Thus, steam may be superheated, in which case it will expand, approximately, like a perfect gas ; or it may be saturated, in which case, by expanding without transmission of heat, it may remain constantly at the point of saturation ; or by means of a steam jacket, the steam, by being constantly supplied with heat, may be considered as dry saturated steam. The curve of expansion may be too complex to be analyzed with great exactness. "When steam enters the cylinder it may, . and generally will, be hotter than the walls of the cylinder, and give up heat to the walls, thus reducing the pressure, even if it does not actually condense any of the steam ; and as the steam becomes cooler by expansion, the walls of the cylinder will give up heat to the steam, thus raising its pressure at the latter part of the stroke. The water in the cylinder, if any, may also be re-evaporated. In either case the restored heat taking place near the end of the stroke does not compensate for the loss at the beginning, for the former can act through only a small part of the stroke, and as soon as the exhaust opens the restored heat escapes with the stea/n and is lost. Water in the cylinder may result from condensation of the saturated steam, as shown in Article 99, or it may be carried over from the boiler with the steam in the form of very small drops, as a spray. If the cylinder be jacketed the walls will be kept at a more nearly uniform temperature, and thus condensation in the cylinder be prevented, which is a great gain in the working of the engine. Condensation in the steam jacket does not affect the working of the engine. The refinements result- ing from these numerous conditions are beyond the reach of analysis, because the laws governing their actions are un- known. This fact, however, is not seriously prejudicial to analysis, for the hypotheses assumed agree so nearly with actual cases as to give results, not only approximately cor- rect, but so nearly correct as to be reliable in ordinary [109.] IDEAL STEAM DIAGRAM. 171 practice. If, however, it becomes necessary to investigate these refinements, or so-called exceptional conditions, the problem of the steam-engine in this respect ceases to be analytical, and is essentially empirical. It must not be in- ferred that theory, even in this case, is useless, or is to be ignored, for it is only by theory that exceptions are known. Theory gives the first grand approximation to the truth, when, by comparing the results with actual cases, the de- fects in the theory become known, and thus, in turn, furnish the means of correcting or amending the original theory ; after which a second and nearer approximation may be made, and so on, bringing the results of theory and of practice more nearly to an agreement. A consideration of these many conditions demands a special treatise ; we will consider only a few special cases. 1O9. Ideal steam diagram. Let A B CEF be an ideal diagram of a steam-engine, A B being the steam line at constant tem- perature and pressure, BC the expan- sion line, C E the fall in pressure at the end of the stroke, due to the sudden opening of the exhaust passage, E T^the * ' ' ' C back pressure line, H the line of ab- O 0- solute zero of pressures ; then O A = GB FIG - 49. will be the total forward pressure up to the point of cut-off, C H the forward pressure at the end of the stroke, II E = J^the back pressure. The admission line A B is an isothermal of constant pressure, and in this respect resembles the case described in Articles 74 and 77, in which a liquid was evaporated under constant pressure at a constant temperature. In that case more and more liquid was evaporated, producing more and more steam, as the volume increased, while here more and more steam enters from the source as the volume in- creases. We might proceed, as with the perfect engine, to 172 HEAT ENGINES. [HO.j find the heat absorbed and rejected throughout the cycle, and take the sum ; but it is customary to tind the results directly in terms of pressure and volume. The ideal diagram is one freed from all irregular and dis- turbing causes, such as late opening for admission, initial expansion, wire drawing at the point of cut-off, slow closing of the port, irregularities in the expansion line B C, too early opening of the exhaust near C, a want of sufficient opening at E, and of compression near F '; but such a dia- gram represents the greater part of the work done, and by applying theory to it a result approximately correct will be obtained. 1 1 0. Isothermal expansion. Assume that the steam is superheated and the cylinder steam jacketed, then will the expansion line be nearly isothermal. Assume it to be exactly so, and let p l = A G B, Fig. 49, be the initial pressure, j). t II C, the terminal pressure, p t = II E, the back pressure, p = any ordinate to B C, , = G = the volume occupied by one pound of steam in the cylinder up to the point of cut-off, v t = O H, the volume of one pound at full stroke, r = v, -f- v l = ratio of expansion, i) = any volume between G and H, p m =. the mean absolute pressure, being such an ideal pressure as would if exerted throughout the stroke produce the same work as that of the variable pressures, p 6 = the mean effective pressure. The equation of C will be, page 103, 2> v = p l v, p r l = pi T, (162) [HO.] ISOTHERMAL EXPANSION. , 173 Saturated steam after being generated in a boiler is con- ceived to be superheated in a separate vessel. We have area G- B C II ' = / ' p dv = p, v t log* , (163) J Vi 0, and 0^5 tf// = ^ iV| .(! + %. r); also, p m v, = A B Pi r P*=Pm-P*> (166) The .effective energy exerted by one pound of steam against the piston = AB GEFA = U=(p m - j9 3 ) *v (167) To find the heat absorbed per pound of steam, let H = the heat absorbed. It may be represented by a diagram thus : Let L be the initial state of the feed water as to pressure and tempera- ture, T t ; A the state at the temperature of boiling, T^ JIT the state of dry saturated vapor at the pressure p 1 and temperature T^ ; B the state of the superheated steam at FIG - 50 - the pressure p l and temperature T 6 ; B C the isothermal expansion curve, then will the temperature at C be T^ the same as at B. (The temperatures marked T 9 and T 3 are for use in the two following exercises.) 174 HEAT ENGINES. [HO.] Through LAMB and C draw adiabatics, then H 1 = - r, (l + log. ^)] + ^-=p H w (182) For the work per pound of steam working full cycle, U= A B CEF = J^ - r 2 (l + log. ^)] _L ^^ H + (p. ~ P*} u v (183) The heat expended per pound of steam admitted to the cylinder will be the same as in the preceding Article, or K = J(T 1 - T) + ff w (184) The efficiency will be [Messrs. Gantt and Maury determined the Efficiency of Fluid Vapor Engines according to this hypothesis using these equations for Water, Alcohol, Ether, Bisulphide of Carbon and Chloroform (Thesis, Ste- vens Institute of Technology, 1884 ; Van Nostrand's Engineering Mag- azine, 1884 (2), pp. 413-432)]. EXERCISE. = 14400 Ibs. ; p 3 = 360 Ibs. ; T, = 110 F., as in the preceding exercise, and p^ = 1115 Ibs., as found in that 178 HEAT ENGINES. [112.] If the ratio of expansion were given, p t could be found only by a tedious approximation ; therefore, we have as- signed the final pressure. We have, r v = 788.26 ; . . 7 1 , = 327.66, as before. r, = 640; .-.T 9 = 180, " r. = 590 ; .'.21= 134, r t = 570 ; . . T 4 = 110, " j9, = 14400 Ibs. " '* p t = 1115 Ibs. " " d, = 4.375 cu. ft. " " q ' (88)> J + ) = 9.55, Eq. (152), 17, JJ e j \ T, T, / Wj = ^ (778 hg. ^ + ^), ^ (150), -* J e3 \ *i T i ' = rv, = 41.76, 0, -w, = 6.64. U = 171507 ft-lbs., Eq. (183). The preceding exercise gives, H = 857706 ft. -Ibs. Efficiency of fluid Steam condensed due to expansion only v -^r = - 137 ' or nearly 14 per cent. 'est U 171507 [112.] EXPANSION OF SATURATED STEAM. 179 Mean total forward pi^essure j> ra = 4170 + 360 = 4530 Ibs. It will be seen that there is little or no advantage in using the exact, but more complex, formulas of this Article over the approximate ones of the preceding Article. The efficiencies found in the three preceding cases are : For superheated steam, expanding isothermally (Ilia) 0.2C5 For saturated steam, expanding adiabatically, approximate law (181a) 0.204 theoretical law (185a) 0.200 The effect on the efficiency by superheating is too small to be of practical importance. As this fact appears to be contrary to the popular opinion, it is well to observe that the superheated steam in Article 109 is not used in the most economical manner ; for a much larger amount of heat is thrown away at the end of the stroke than in the example of saturated steam, so that if it were utilized in heating feed water, or worked in another engine, or used for any other useful purpose, the efficiency of the plant would be in- creased. Or if it had been expanded down to that of the terminal pressure of the other cases, p t = 1115 Ibs., it would have shown a greater efficiency ; but to accomplish this result the ratio of expansion must be greater, other data being the same. These considerations have reference to the efficiency of the fluid only, but in considering the efficiency of the plant, the size and cost of the engine enter as elements of the problem. Thus, to do the respective works, 231757 and 174931, deduced'in two of the preceding exercises, with two engines making the same number of revolutions in the same time, according to the conditions assumed, the volume of the cylinder of the one supplied with superheated steam must be larger than that supplied with saturated steam in the ratio of the volume of a pound 180 HEAT ENGINES. [113.] of superheated steam at admission to that per pound of sat- urated steam, or, as 5 ' 2T -120- 375 " but the ratio of the works done will be 231757 174931 - hence, per cubic foot of the cylinder capacities the former engine will do " = 1.10 time. the work of the latter. The engine using isothermal expansion and doing 231757 foot-pounds of work per pound of steam, if it uses tlu? pound per minute, will do 33000 horse-powers per pound of steam ; and, per hour, it will require 1980000 __=:b.o4 pounds per horse-power. The engine which expands adiabatically, doing 174:931 foot-pounds of work, would require 1980000 -iiiwr = 11J " po " lkKi per horse-power per hour. These results are for perfect con- ditions, no allowance having been made for wastes, clearance, or initial condensation of steam. It is a very good plant that does not consume more than seventeen pounds of feed water per indicated horse-power per hour, although re- liable records of some good tests show less than this amount. Some multiple expansion engines have been reported as [112-] NO EXPANSION. 181 consuming about thirteen pounds, as determined from the indicator card, but that mode of determining the weight of steam does not allow for the condensation of steam. The only reliable way is to weigh the water used. Thirty to forty pounds is more common in practice. The heat of combustion of a pound of pure carbon is 14500 B.T.U., and if it could all be utilized for the purpose it would evaporate 14500 -j- 966 = 15 pounds of water at and from 212; hence, if the feed water be at 110 F. and boiling point at 327 F., as in the two preceding exercises, it would, according to the table on page 112, evaporate 15 -~ 1.14 = 13.15 pounds ; and to develop one I.H.P. per hour it would require 11.32 * 13.15 = 0.861 pounds of coal. This does not allow for waste in producing steam. If the efficiency of the furnace be 0.70, it would require 0.861 -7- 0.70 = 1.31 pounds of coal. Case of no expansion. In many simple direct-act- ing steam pumps, the full pressure of steam is maintained throughout the stroke. For this case r = 1 in equation (177), and the indicated work will be &= fa -?,)* ( 185 &) when v l is the volume of a pound of the vapor at the pres- sure p r The work done during the forward pressure will be the external work performed during evaporation at the pressure^,, and is sometimes called the external latent heat of vaporization. That part of the apparent latent heat which performs disgregation work will be lost at the exhaust. The volume of the cylinder, the piston making n single strokes per minute for m horse-powers, will be cu.ft. (185.) 182 HEAT ENGINES. [113.] "Water consumed per indicated horse-power per hour, jrr 33000 X 60 ~ -- pounds. EXERCISES. 1. In a direct-acting steam pump, let the uniform gauge pressure be 70 pounds and back pressure 16 pounds, feed water 60 F. ; required the work done per pound of steam, effi- ciency, volume of the cylinder for one horse-power if there be 50 double strokes per minute, and the water consumed. jp, = 70 + 14.7 = 84.7, Pi- P* = 68 -7, r l = 775.6, Eq. (81) ; or, T t = 315 F. ; v, = 5.14, Eq. (86) or (89) ; U - 68.7 X 144 X 5.14 = 50849 ft. lbs.,Eq. (1855) ; W = 38.9 Ibs. per hour, Eq. (lS5d) ; V = ^ C u. ft. = 52.4 cu. in., Eq. (185c) ; H = 893844, Eq. (179) ; E = 0.057, Eq. (185), or, the theoretical efficiency of the fluid will be about 5.7 per cent. In actual practice the loss from condensation and radi- ation in these pumps is considerable, and the clearance is not only relatively large but somewhat irregular especially in the smaller sizes, and it is found that the water consump- tion ranges from 75 to 125 pounds per horse-power per hour, with the possibility of being outside these limits in either direction. The mean average efficiency for ilie fluid admitted (steam and water) will be for small pumps of this class about -J of the theoretical, or, E = 0.019, approximately, and, U 17000 ft. Ibs., approximately, also, W = 120 Ibs. of water per horse-power per hour. [112.J NO EXPANSION. 183 But the size of the cylinder need not be correspondingly increased, for the condensed steam will occupy but little volume. The efficiency of the furnace, boiler and connections may be taken at 50 per cent, giving for the entire plant E' = 0.0095, or about 1 per cent of the theoretical heat of the fuel burned in the furnace. It has been found .by actual measurements that the average duty (or the work which 100 pounds of coal can do) in direct-acting pumps feeding 75 to 100 horse-power boilers, with coal of good quality, may, in the absence of direct experiment, be taken as 10000000 foot-pounds. This is 100000 foot-pounds per pound of coal, or 100000 -f- 778 = 128.5 thermal units, which is about T ^ of the heat of com- bustion of the average of commerical coal. The efficiency of such a plant, then, is actually about 1 per cent of the heat in coal of good quality. Such a plant will require from 9 to 15 pounds of coal per indicated horse-power per hour. 2. In the preceding Exercise, if the stroke be five inches, what will be the diameter of the cylinder ? 3. If, in a direct-acting steam pump, the gauge pressure be 100 pounds, back pressure 16 pounds, feed water 90 F., find the efficiency of the fluid, and compare the result with that in Exercise 1. 4. If, in Exercise 3, the gauge pressure be 40 pounds, required the efficiency of the fluid. 5. Explain the several causes of the loss of the 99 per cent (more or less) of the heat of combustion as found in these Exercises. What effect has the temperature of the feed water upon the efficiency ? 184 HEAT ENGINES. [112a.] 1 12a. General equations of vapor engines. Consider only one pound of fluid in the cylinder, and let B C be the curve of saturation, and E F any adiabatic in which there is only a fraction of the pound of vapor throughout the expansion. A B, Fig. 50, will represent the volume of a pound of vapor at FIG. 50. the absolute pressure O A = p, and absolute temperature r,, G /the volume at the absolute temperature r and pressure O G = p. Let x t = A E +- A B = the fractional part of the fluid at the state Et\\&i is vaporized, v, = A B, *, v t = A E, x = G II -r- G /, v = G /, volume corresponding to the pressure O G, x v = G II, c, the specific heat of the liquid, A e , the latent heat of evaporation at temperature r in ordi- nary heat units, which will be A ei at temperature r r Then will the equation of the adiabatic E F be, Eqs. (86) and (149), GH= xv = (clog e -IL+^'A) , .() ^ f ^ \ e which may be put under the more symmetrical form xh e , r x. h e . T, - + c log o = - - + G Iog 6 !- = a constant, (fy in which r is any arbitrary temperature. Since the vapor is to be continually saturated, this equation is limited to the conditions that a?, must not be negative, and must be less than unity, and at the same time a?, for any amount of expansion, must be less than unity. [112.] VAPOK ENGINES. 185 Let subscript be used for the terminal state F, then a? Q A e , 7 r, as.h 6 . T. -^ + c log* - = -- + G log, The difference between the initial and terminal weights of vapor will be and this may be negative, zero, or positive. We will designate those vapors whose specific heats are negative as " steam-like vapors," and those which are positive as " ether- like vapors," steam and ether being typical of their respec- tive classes. If the fluid be water, then c = 1, and let x, = 0.436 at T = 800 F. (absolute), A e = 1436.8 - 0.7 r. Then equa- tion (a) gives for T = 900, x = 0.404, t = 600, as = 0.450, r = 800, x = 0.436, T = 500, x = 0.436, T =-- 700, x 0.450, T 400, x = 0.407, T = 650, * = 0.453, T = 200, a? = 0.277 ; from which it appears that steam increased with the expan- sion as the temperature fell from 900 to 650, or from 340 to 190 on the Fahrenheit scale ; and after that it decreased continually with the temperature. This change of the weight of steam can take place only by the evaporation of water initially in the presence of the vapor, and by condensa- tion later in the expansion. The converse is also true, that if, in the initial state, only a fraction of the fluid be vapor, the liquid may at first be evaporated by adiabatic compres- sion, but it may reach a state beyond which it will be con- densed by adiabatic compression. Thus, in the example above given, if at 600 F. (absolute) 45 per cent, of the fluid be vapor, it will increase to 45.3 per cent., after which it will condense indefinitely with adiabatic compression. 186 HEAT ENGINES. [112aj. If at 650 there be 45.3 per cent, of steam, the vapor will condense both by adiabatic compression and expansion from that state. This may be illustrated by the annexed diagram, Fig. 50, in which the relations are greatly exaggerated. Let D E F represent successive states of con- stant steam weight, and A B C an adiabatic of part liquid and vapor. These curves may intersect each other at two points a and I ; above a the weight of vapor in the adi- batic will be less than at , and the adiabatic will lie to the left of D E F, and below I it will lie below the curve of con- stant steam weight. The adiabatic is less curved than the curve of constant steam weight. To find the minimum weight of vapor such that, by con- tinued compression of steam-like vapor, the liquid will le continually evaporated. In equation (b) first find the value of r that will make the left member a minimum when x = 1. Neglecting all powers of r above the first in the latent heat of evaporation, Regnault's experiments give A e a , = --* where a and are constants depending upon the particular fluid. Using this value, it will be found that the required function is a minimum for a that is, r will be near the " temperature of inversion," which, in the case of steam, is about 1436 F. (absolute), or 976 F. actual. Since the law of the latent heat of evapo- tH2.] VAPOR ENGINES. 187 ration here given is not exact, ; nd, even if it were, naulfs experiments would not warrant the extension to such high temperatures, we will discard fractions, and treat the entire number, 1436, as if it were exact. Since the adiabatic law is not applicable above this state, the maximum condensation by adiabatic expansion will be found by beginning at this state and expanding down to the required temperature. In equation (<:), letting x^ 1, T I .- 1436, 7< ei = 14360.7 T, c = 1, then : 14-36 2.3(>26% 10 - .+ 0.3 1 x = 1 - 1436 _ Abso. Temp. Per cent, of Steam. Per cent, of Water. Temp. Deg. F. If r = 800, x = 0.808, 1 - x = 0.192, 340. = TOO, x = 0.753, 1 - x = 0.247, 240. = 672, x = 0.725, 1 - x = 0.265, 212. = 600, x = 0.692, 1 x = 0.308, 140. It thus appears that if 72 per cent, of the fluid be satu- rated steam, or 26^ percent, of it be water at 212 F., the steam will condense continually by adiabatic expansion, or the water be continually evaporated by adiabatic compres- sion. If there be less than twenty-six per cent, of water at 212, the water will all become evaporated before the tem- perature reaches the critical temperature, and, after passing- that state, compression will produce superheating. Every adiabatic having more than 72 per cent, of steam at 212 is tangent to some curve of constant steam weight ; and hence, with the exception of the adiabatic tangent to the curve of saturation, will have a state of maximum steam weight, at which point the curves of constant steam weight and the adiabatic will have a common tangent. From this state condensation of steam will result from compression as well as from expansion. The adiabatic which is tangent to 188 HEAT ENGINES. the curve of saturation passes through the state of the tem- perature of inversion. According to the preceding table, at T = 672, if 72 per cent, is steam, compression will produce evaporation up to 1436. If at r = 672 we assume 70 per cent, of steam, we find the following results : T = 672, a; 0.70. r = 1100, x o.849. T = 700, x = 0.73. T = 1200, x = 0.855. r = 800, a? = 0.76. T = 1250, x = 0.860. r = 900, x = 0.794. T = 1300, a? = 0.859. T = 1000, x = 825. T = 1400, x 0.84. It will be seen that the amount of steam will be a maxi- mum at 1250 F. absolute. PIG. 50c. In. Fig. 50c the dotted lines are curves of equal steam weights, and the full lines except the curve of saturation are adiabatics, one of which is tangent to the curve of satura- tion ; another tangent to the curve whose constant steam weight is 86 per cent., the point of tangency being at the [112.] VAPOR ENGINES. 189 temperature of 790 F. ; another is tangent to the curve of 50 per cent, of steam at 240 F. ; and the fourth tangent to the curve of 45.3 per cent, of steam at 190 F. absolute. In order to show the properties on a small scale, it is necessary to exaggerate the relations, thus distorting what would be the correct figure. An examination of ether will show that the results here deduced for steam are not necessarily applicable to other vapors. In " ether-like vapors " the temperature of inver- sion is below ordinary temperatures ; and for such if a?, = 1, condensation will result from adiabatic compression for temperatures above that of inversion. Thus, for ether, omitting terms above the first power of r, we have from Regnault' s experiments, 7> e = 93.3214 + 0.3870 r. c =: 0.517. Hence, from equation (139), page 147, s = 0.517 = specific heat of the saturated vapor. If s 0, then r = 180 (absolute), or 280 F. ; and this is the temperature of inversion. Assuming any tem- perature above this, as r^ = 520, and a?, = 1 in equation (a), then 0.5664 - 2.3026 % 10 -^ "~ ^ + 0.3870 ' From this it appears that x will diminish as r increases, and finally become zero for r = 915, nearly. There appears to be no proportion of vapor to liquid such that they will be the same at two different states on an adia- batic, as has been found for steam. It may be shown that for any value of x will decrease as r increases, showing 190 HEAT ENGINES. [ 113ft.] that reevaporation does not take place during adiabatic com- pression. If the fluid be initially all liquid, then x l = 0, which in equation (a) gives for the equation of A J, Fig. 50#, TV , T I xv = c-j- Iog 9 -, () This expression may in some cases have a maximum, from which it" appears that if the fluid be initially all liquid, under adiabatic expansion the liquid may be evaporated until the temperature is so reduced as to produce the maxi- mum weight of vapor, after which the vapor will condense. Thus, for steam < = 1, and if r, = 800, x will be a maxi- mum for T = 350 (absolute), nearly, at which state x will be 0.21 ; or 2-t per cent, of the liquid will have become vapor. At 300, x = 0.239 ; for r = 200, x = 0.21. All these latter temperatures are, howeveit, so much below any used in practice, that it is not probable that the formula for evaporation will be applicable ; and we may assert, that, within practical limits, steam will be continually generated under adiabatic expansion, if in the initial state the fluid be entirely water. With ether, if initially liquid, evaporation will increase with adiabatic expansion until it all becomes saturated vapor, after which it will superheat ; provided that the liquid be- comes vapor before the temperature of inversion is reached. The numerical values of these results will be modified in some cases considerably if higher powers of the temper- ature be included in the analysis. The ratio of expansion will be [112a-] VAPOK ENGINES. 191 If, at. the cut-off, B, tlie fluid be all vapor, as it may be for steam-like vapors, then a?, = 1, and reducing by means of equation (a) we have which is the equivalent of equation (152), page 154. For ether-like vapors, if the final state is that of vapor only, then a? 2 = 1, and substituting a?, from equation (a) gives _ v, A,. . The weight of ether vapor at J5, the beginning of expan- sion, in order that the pound of fluid shall be all vapor at C, the end of the expansion, will be , in equation (a) when a? = 1, or In practice, the adiabatic expansion of steam-like vapors may be approximately realized, but there is well-nigh an insuperable difficulty in securing the adiabatic expansion of saturated ether-like vapors ; for, in the former case, if steam be in the state of saturation at the instant of the cut-off, it will continue to be saturated during expansion ; but, with the latter, if no ether liquid be present at the instant of cut- off, the vapor will superheat during expansion, and instead of realizing equation (a), the curve of expansion will be of the form p -y n = a constant, in which n will be the ratio of the specific heat at constant- pressure to that at constant volume. We will continue to consider the vapor as saturated. 192 HEAT ENGINES. [112.] To find the work A E F D, Fig. 50a, p. 184, the expan- sion E F ' being adiabatic, the vapor being saturated throughout expansion, we have Z7, = A EFD =GIL dp = = J\<- (r, -r,- r.loff which becomes equation (182) if go, = 1. If, in this expression, the value of a?, from equation (a) be substituted, and subscript be attached to those variables which are without subscripts, we will have ,,] ; Equation (&) is better adapted for the discussion of steam- like vapors, and equation (I) for ether-like vapors ; for in the former x, may be unity, and in the latter a? a may be unity. Eliminating log,. - from these equations by means of equation (a) gives U, = j[c (r, - r,) + , A., - a, A e , ], (m) in which a?, and a% are limited as before. If, during the return stroke, the fluid be refrigerated so as to maintain the constant temperature r v the pressure will be uniform and equal CD ; and if at some point, as t -.?,),*> where p^ = J), p 3 = M, absolute pressures. If r t be the temperature of the feed water, the heat expended will be H = Jc (r, - r^ + x, 7/ ei , where H ei = Jh ei . Hence the efficiency will be J L c ( T I r r % ^ ) + T ' ~ ^ Xi Tlei \ + (P* Pa) X* v* E= - -- (0) J [ c (r, - r 4 ) + a, A el J From this result it appears that in the case of actual engines, the specific heat of the working fluid and the latent heat of evaporation both affect the efficiency. If the feed 194 HEAT ENGINES. [112a water be at the temperature of the exhaust, then r 4 = r 9 and the preceding expression may be reduced to ft - T, c r, E= r, c (r, - r,) + J, h el By retaining ,r, and a 1 ,, equation (o) is applicable both to " steam-like" and " ether-like"" vapors, only observing that neither x t nor a% can exceed unity, and that they are related to each other through equation (a). To find the work done during adiabatic expansion when the initial state A is that of liquid only, make a?, = in the value of U n or a?, = in equation (#), giving = ADJ=Jc and if the temperature at / be r t then will A M L be found by substituting r s for r t in the preceding equation. Actual engines do not expand down to the back pressure, neither is the pound of fluid retained in the cylinder ; but at the end of the expansion the exhaust is opened, and the vapor escapes until the exhaust is closed at the point L in the back stroke. The adiabatic A .L will then be for only a fraction of a pound of fluid. To find its equation let z be the fraction of the pound of fluid, including both liquid and vapor, then equation (a) gives T , 3C, h K . If the fluid be all liquid at A, then x, = 0, and ,. _ ??-.,^.3, . _ . which reduces to equation (f ), if z = 1 as it should. [H2.] VAPOK ENGINES. 195 But in practice there is clearance and the fluid will not be reduced to a liquid at state A. Representing the clear- ance by P A, Fig. 50r7 ; then will z = ' P A + A , where A B is the p A B volume of a pound ; and equation (/) will be the equation of the adiabatic. It will, however, be more convenient Q to use the approximate equation N p vV = p, vV, as has been done in the following equation (it). Some practical considerations. A steam or hot- air jacket is sometimes used to prevent liquefaction of steam in the cylinder by keeping the walls of the cylinder hot. Liquefaction of steam in the jacket produces no bad effect, although it represents cost for fuel. The entire action is too complex to admit of definite computation. The velocity of the steam through the steam pipe, if not more than 100 feet per second, does not, according to D. K. Clark, produce any appreciable loss by frictional resistance. The loss of pressure in passing through the ports into the cylinder is, in practice, from 3 to 10 pounds, and in excep- tional cases even more. Wire-drawing of steam is the reduction of pressure due to friction. This does not represent a corresponding w r aste of energy, for it produces heat, thus superheating it that is, produces a temperature higher than the boiling point corre- sponding to its pressure, the pressure being lower than at the boiler ; but the entire energy is never restored in this way. It is better to cut-off earlier with throttle open than to throttle and cut-off later, to produce the same work. Superheating may be produced by wire-drawing, by a steam-jacket, by circulating a hot fluid through flues in the steam-chest, by heating the pipes conducting the steam to the 196 HEAT ENGINES. [I12a.] engine, by heating in the cylinder by hot pipes, or by inject- ing some superheated vapor into a body of saturated vapor. The object of superheating is to prevent condensation, to diminish the back pressure by producing steam of less den- sity, and to increase the efficiency of the fluid. When steam is superheated to such an extent that it may, without material error in practice, be treated as perfectly gaseous, it is sometimes called steam gas. Experiments of Him and others show that a very moderate amount of super- heating produces steam gas ; from which it is inferred that the formulas for steam gas will be practically correct for ordinary superheated steam. EXERCISE. Find the work per pound of ether working in an engine without clearance or compression, expansion complete, be- tween the absolute pressures of 100 and 14.7 pounds per square inch ; the ratio of expansion and the efficiency, the fluid being, entirely saturated vapor at the end of the expan- sion, and the temperature of the liquid ether 60. Since expansion is complete, the final pressure, 14.7 pounds, will equal the back pressure. The specific heat of liquid ether is c = 0.517. To find the initial and terminal temperatures we have, equation (81), 1 JA - loff u p & B_, C r 4 C' 1 26' in which for ether, A = 7.5041. B = 2057.8, log B = 3.313425. C = 104950, lo'j C = 5.217355. Hence, r, = 670, r a = 558 ; . . T, 216 F., T, = 98 F. Latent heat of evaporation as determined by Regnault, A = 171.24 0.0487 T 0.000473 T* L 112 -l VAPOR ENGINES. 19? hence, for terminal state, since T^ = 98. . . .h^ =. 16^5 for initial state, since T, = 216 ____ h ei = 139 Work, ABCD, Fig. 50a, o? t = 1 in Eq.(Z),ft.-lbs. U = 22200 Initial weight of vapor, Eq. (/), Ibs ........... a?, = . 0.92 Efficiency, Eq. (o), making a?, = 0.92, p, = p 3 . E 0.137 Volume of a Ib. of liquid ether, cu. ft ........ v = 0.0223 Vol.of Ib.of vapor at p, = 100 lbs.,Eq.(84),cu. ft. v t = 2.98 " " " "^, = 14.7" " " " " v t = 12.52 Ratio of expansion, Eq. (/) ................. r = 4.6 Tf tfA0y &e a clearance, and sufficient fluid be retained to just fill the clearance by compression, as indicated by L A, Fig. 50^?, this fluid will act as a cushion, and the energy in it will be stored and restored with each stroke, and will not form any part of the working fluid. In this case, freeing the diagram of the effects of the cushion fluid, as in Stir- ling's engine, page 224, u t will be represented by a line equal to J C, in which case equation (n) becomes ap- plicable by subtracting from it a trilinear area, of which the hypothenuae is the curved line J L, and the base the projection of L J on F E. But the mean effective pres- sure will be diminished because the effective work per revolution will be less, the back pressure being greater. Ratio of expansion with clearance. A C E Z, Fig. 50J, being the diagram described by the indicator, the cut-olf being at B, A B will be the apparent steam line, and P B the real steam line. Let F E r' ^ = the apparent ratio of expansion, C r = -- = the real ratio of expansion, PA c = = the ratio of the volume of clearance to . the piston displacement, s F E = the stroke of the piston ; then 198 HEAT ENGINES. (fi The adiabatic A L will be for a fractional part of one pound of the vapor, and if A B be proportional to the volume of one pound of the vapor, and A J terminates at the end of the clearance, then will the weight of vapor re- quired to produce A L be AP of one pound = c r ; and the volumes will be in the same ratio at equal pressures. Hence, N L _ c r' v t PA-^^/' "")* 00 which determines the point where compression must begin. The mean effective pressure will be diminished, but, like many other elements, the exact amount cannot be deter- mined theoretically, except by a full solution of the problem ; still a sufficiently near approximation may be found by means of equation (176), page 175, which is in which r* will be between 1 and 2 for all ratios of expan- sion used in practice, and when p t is large compared with Pv P* w iH y ary, approximately, inversely as r. Hence, if PS be the mean effective pressure with clearance, and p* [112.] VAPOR ENGINES. 199 without, we have, approximately, P* = ? P*' W The piston displacement per minute in doing the same work as without clearance will be increased in the ratio r' -r- r. If the steam is completely exhausted during each return stroke, the real volume of steam will be P B = (1 -f c r') A , (w) or 1 -f- c r' times the apparent volume. The mean absolute pressure will also be diminished, to find an approximate expression for which, conceive that the piston displacement equalled the volume of the cylinder, including the clearance ; then would the work done be p m (1 + c} u But the work done in passing over the clearance would be PI c u , ; and if p m ' be the mean absolute pressure, we have Pm u* = pm (1 + c) u, p, c u, ; . jv -P* = C (PI - p^> (o The expenditure of heat per pound of steam per stroke without clearance, or with cushion space just filled by the compression of vapor, being H = JJ ei .+ <7(r,-7 2 ), (y) with clearance and complete exhaustion with each return stroke will be, equations (w) and (y\ H (1 +cr') = [# + C(r, -.r,)] (1 + c /). (z) The efficiency of the fluid will also be diminished / for effective work with clearance _ p m f jp 3t effective work without clearance p m p 3 ' 200 HEAT ENGINES. [112a.] and the expenditure of steam, and consequently of heat, is greater with clearance, as shown by equation (3), and hence the diminution of the efficiency will be E- E' = -. -- 2=4 o nearly. Although the effects due to these and other practical con- siderations cannot be thoroughly analyzed, yet their dis- cussion shows that they do not oppose, or revolutionize, the general theory of the vapor engine they simply modify its results. A more complete knowledge of vapor engines requires special experiments and a study of the engine itself under varied conditions. Theory teaches much, and we are thankful that we know so much, and regretful that we know so little. 113. Cut-off, With a given plant, if the cut-off be early more work may be done with a given amount of fuel than if the cut-off be late in the stroke ; and it is proposed to find the cut-off which shall give the most work per pound of steam admitted to the cylinder. This problem may be called the point of 'cut-off 'that will produce the greatest Effi- ciency of Fluid. With a given plant, it may be proposed to find the point of cut-off such that the owner may realize the greatest profit by selling the power produced. This condition will involve the first cost of the plant, attendance, repairs and deteriora- tion. The deterioration may be such that the cost of the entire plant will be absorbed in the course of a few years, or if sold during this time, it will be the difference between the original cost and the amount received by the sale. In this case, if the cut-off be early fuel may be saved, but the other changes may make the cost of the power delivered more per dollar expended than if the cut-off were later, thus making it a problem of maxima and minima. This may be called the Owners Problem. Again, in the plant of the preceding case, the parts may be improperly proportioned : but if a definite amount of [113.] CUT-OFF. 201 work is to be produced, the designer may be required to proportion the plant so that the boilers shall be of the proper size for working most economically for producing the re- quired amount of steam, and the engine so proportioned that by cutting off properly the power produced shall cost the least per dollar expended. If cut-off be too late in this case, more steam will be required, requiring larger boilers and more fuel, while the engine may be smaller, thus cost- ing less ; or if cut-off be too early, requiring less steam and smaller boilers, the cylinder and every part of the engine must be larger, costing more, so that this is a problem of maxima and minima, and may be designated as the Design- er's Problem. These and similar problems have received the general title, The Most Economical Point of Cut-off. A general solution of the owner's problem was made by Rankine, and is made the basis of the solution of the other two. It is substantially as follows : Let p l = the initial absolute pressure in the cylinder per square foot, p m the mean absolute pressure, F = the resistance of the engine other than the use- ful load, including friction and back press- ure, h = the cost of producing unity of weight of steam in unity of time (one hour), which consists of the cost of fuel, repairs, wages of firemen, in- terest on cost of boilers, and depreciation ; & = interest on the cost of the engine, plus engi- neer's wages, plus cost of repairs, plus depre- ciation of value of engine, plus cost of waste and oil, reduced to cost per square foot per hour ; A. = area of the piston in square feet. I = length of stroke in feet. )2 HEAT ENGINES. [113.] n = the number of times the cylinder takes steam in unity of time (one hour), once per revolution for single-acting engines, and twice for dou- ble-acting ; I n = the number of linear feet swept through by the piston in unity of time one hour ; y = the volume of unity of weight of steam at the pressure p^ taken from a table of " saturated steam " or from equation (86), or (89) ; W = weight of steam used by the engine in unity of time one hour, v t = the volume swept through by the piston in one stroke, i>, = the volume swept through to the point of cut- off, r = the ratio of expansion = v t -4- v t Z = an auxiliary quantity = p m v, -r- p t v t = the ratio of the work done at full stroke, working expansively, to the work done up to the point of cut-off. Then ' ass) .-.p m =*i Z. (187) Tc A = the interest on the cost of the engine per hour for the steam used, and * ^ the interest for each pound of steam used per hour ; hence, the total cost per pound per hour will be [113.] CUT-OFF. 203 The volume of TF pounds of steam will be v TF, and at full stroke, r v TF; .-. Aln = rv TF; (188) and the preceding expression becomes z t 7 rv *+'*** The useful work per stroke will be U=(p m -F)v which per pound of steam per hour becomes (189) which by means of Equations (187) and (188) becomes hence, the work done per unit of cost (one dollar) of steam will be Z-*L r v(p,Z~Fr] p. - which is to be a maximum in reference to r as a variable, and will be a maximum when the factor Z- F -r TT7T is a maximum. Equation (186) shows that Z is & function of p m and r ; but the form of the function p m is not defi- nitely known, depending, as has been previously stated, upon the behavior of the steam in the cylinder whether it be dry, saturated, superheated, wire-drawn, &c. If the 204 HEAT ENGINES. [113.] mean forward pressure be found in any manner as a func- tion of r, a graphical solution may be made as follows : Draw two axes, O X and Y, and construct the locus J. B, Fig. 50, rep- resenting Equation (!$ so long as it remains above the condition of saturation, and actual engines confirm this result. Calorimeter tests oi steam-jacketed engines have shown a total loss from lique faction of from 10 to 20 per cent. The third has been discussed by Professor Cotterell in his work on The Steam-Engine, ' pp. 246-269, in which he shows that this may be the principal cause of the loss of efficiency of the fluid. The l:ws of conduction and radi- ation are not sufficiently well known to enable one to estab- lish a complete theory of liquefaction in this regard ; and if they were, the variations of temperature due to expansion, as well as the varying temperature of the walls from the be- ginning to the end of the stroke, would greatly complicate the problem. If the liquefied water be deposited upon the inner sur- face of the cylinder, as it will be in the third case, it will facilitate the conduction of heat, and the result will be very different from the condition in which the water remains [117.] EXPLOSIVE GAS ENGINE. 213 distributed throughout the steam, as it is supposed to be in the first and second cases named above. The reduction of temperature, due to the action of the walls, would also have an influence upon the theory in- volved in the second case, in a manner which has not yet been considered. Initial condensation is that which takes place during the admission of steam, and is due chiefly to exposure to sur- faces colder than the steam, and is- independent of the case investigated by Rankine. It can be reduced by keep- ing the walls at nearly the temperature of the entering steam, and hence may be nearly prevented by a steam-jacket, and in other cases may be reduced by late cut-off arid high speed. The economy of high expansion is so well es- tablished by theory and confirmed by experience, when condensation is avoided, that other means than that of a late cut-off will be sought for preventing liquefaction. Theory does not enable us to compute the amount of condensation for any particular case ; it must, therefore, be determined by direct experiment. A few examples are given Jn the following notes. NOTES. 117. Experiments on steam-engines: (a.) Hirn's experiments. By far the most complete set of experiments scientifically conduced were those under the direction of M. Hirn, by MM. O. Hallauer, "W. Grosse- teste, and Dwelshauverse Dery, begun in 1873, and extending over several years. The results of the experiments are pub- lished in the Bulletin Special of the Societe IndustrieUe de Mulhouse, 1876. Smith on Steam Using contains a sum- mary of these experiments, pp. 188-285. (5.) Navy experiments. Mr. B. F. Isherwood, while chief 214 HEAT ENGINES. [117.] engineer of the United States Navy, made extensive experi- ments upon the boilers and engines of several steamships of the navy, under the general direction of the Department of the Navy, which were published in two large volumes, entitled Experimental Researches in Steam Engineering, Philadelphia, 1863, 1865. Mr. Isherwood made the first attempt, so far as known, to determine the amount of liquefaction of steam in un- jacketed cylinders for various ratios of expansion. The experiments were made on the engines of the steamer Michigan, in 1861, and showed a large amount of liquefac- tion, increasing at low speeds and high expansions. These were discussed by Rankine, and published in the Proceed- ings of the Institution of Engineers, Scotland, 1861-62. (f.) Navy experiments continued. Messrs. Emery and Loring, in 1874, experimented on the engines of the United States revenue steamers Hache, Rmh, Dexter, and Galla- tin, the reports for which were published by the Govern- ment, and also in Engineering, Yols. XIX. and XXI. From these and other experiments, some have concluded that, for unjacketed cylinders, it will be sufficient to allow for the liquefaction of steam in the cylinder, due to all For full stroke, 12 per cent of the total feed water, 2 expansions, 23 4 ' 36 6 ' 54 8 ' 67 10 72 12 ' 75 (American Engineer, 1884, May 23, p. 207.) For dry steam at high temperatures these allowances are probably much too large, although they may sometimes be realized with saturated vapor. (d} Experiments at the Stevens Institute. Messrs. Gately [117.] EXPERIMENTS ON STEAM-ENGINES. 215 and Kletzscli experimented upon a Harris-Corliss engine hav- ing an 18-incli cylinder, 42 -inch stroke, for the purpose of determining the laws of condensation under different condi- tions. The engine was not jacketed, but was covered with lagging and a non-conducting substance. They found that if y = cut-off = 0.13, then, cylinder condensation = x = 0.50 " = 0.225 " " " = 0.41 " = 0.33 " " " " = 0.34 " = 0.45 " " " = 0.27 " = 0.59 " " " = 0.225 which values are well represented by the equation (a? + 0.12) (y + 0.44) = 0.3543. If z = the area of the surface exposed in square feet and i the per cent of condensation, as before, the experiments gave a? s 1.026 x 4.Y7 z = 221.36. Yarying boiler pressures gave p = pressure 80.00 pounds, x = per cent cjl. condensation = 35.24 = 66.85 = 52.33 = 37.00 = 22.30 = 37.83 = 36.84 = 41.43 = 41.19 which may be represented by the equation x 45 0.1266 p. (Graduation Thesis, 1884 ; Jour. Frank. Iwt., 1885, Oct., Nov. and Dec.) Messrs. Blauvelt and Haynes, by calorimeter tests upon the engines of the steamship Hudson of the Cromwell Line, found in some cases only 10 per cent of liquefaction when the cut-oil was -j^. The pistons were 48 inches in diam- eter, stroke 6 feet; steam-jacketed cylinder; 800 horse- power engine. (Thesis, 1886.) The experiments of Mr. James S. Merritt upon a small direct-acting steam pump, at various piston speeds, the di- 216 HEAT ENGINES. [118.] ameter of the steam cylinder being eight inches, and the stroke about 9J inches, gave the following results : Water consumed Double strokes per minute. Indicated horse- power. per I. H. P. per hour. Steam condensed in cylinder ; per cent of the feed water. Calculated Actual pounds. pounds. 10 1.25 46 114 60 20 2.36 50 89 43 80 3.63 48 100 52 40 4.92 48 75 36 50 5.34 51 86 40 (Thesis, ISS6.). These results are not as uniform as is desirable in such an experiment. The one for 20 strokes appears to be excep- tional. Messrs. McElroy and Parsons, in their test of the boilers ind compound engines of the tug Blue Bonnet found that 37 per cent of the feed water was unaccounted for by the indicator cards. The cylinders were, respectively, 19" X 22" and 22" X 22". The cards indicated that twenty pounds of water per horse-power per hour were consumed. Boiler pressure, 70 Ibs. (Thesis, 1887.) (e) Cylinder condensation. A series of experiments to determine the amount of cylinder condensation were made by Major English, England, and published in Engineering, 1887. It is, however, questionable whether they are of much practical value, since the conditions do not conform to actual practice. 118. Miscellaneous: (/") The weight of steam required by an engine is in- dicative of its efficiency. Eighteen pounds per horse-power per hour is very good practice. Thirty-five to fifty pounds is common in ordinary practice. (g) At Calumet, Mich., an engine was tested working at LI 18.] MISCELLANEOUS. 217 about 600 horse-power, for over ten days, which ran with 16.3 pounds of feed water. (Am. Soc. Mechanical Engi- neers, Discussion, Hartford meeting, p. 14.) (A) In a three days' test of the steamship Para, having triple-expansion engines, the actual weight of steam con- sumed per I. II. P. per hour was 13.4 Ibs. Adding 15 per cent for initial condensation gives 15.4 Ibs.; coal, 1.54 Ibs. per indicated horse-power ; evaporative power of the coal, 12.0 Ibs. from and at 212 F. The Stella consumed 13.7 Ibs. of steam per I. II. P., with 1.36 Ibs. of coal whose evaporative power was 13.9 Ibs. of water from and at 212 F. (Proc. Institution Mech. Eng., 1886-87, pp. 492-506.) (i) Large Ocean Steamers. City of Rome. Umbria and Mruria. Servia. Length feet 542.5 500 515 Breadth " 52.0 57 52 11230 9860 10 960 Indicated H P 11890 14321 10 300 Speed miles per hour 18.2 20 2 17 Coal per day tons 185 315 205 Coal per I H. P 2.2 2 1 2 n i j ) Diana, ins (3@46 1 @ 71 1 @ 72 Cylinders \ ( Stroke " (3 @ 86 72 2 @ 105 72 2 @ 100 78 Steam pressure Ibs 90 110 (j } In the year 1840, the time of crossing the Atlantic Ocean in a steamship was about 13 days. The recent short passages have been : City of Rome 6 d. 18 h. Om. Oregon 6 d. 10 h. 35 m. Etruria 6 d. 5 h. 31 m. (Scribner's Magazine, 1887, p. 315.) In 1887, beginning May 28th, the time of the Umbria was 6 d., 4 h., 12 m. In 1888, beginning May 24th at Queenstown, the time of the Etruria was 6 d., 1 h., 55 m. 218 HEAT ENGINES. [118.] Distance, 3028 miles ; average speed, 20.7 miles per hour. On June 1st the average speed was 24.08 miles per hour. (K) Count de Pambour represented that the friction of an engine increased with the load, the expression for the resist- ance being of the form R = J? + x B, in which x is a coefficient, greater than unity, R the varia- ble load, and 7? the resistance of the unloaded engine. But Mr. Charles T. Porter, about 1871, wrote that " ex- periments with a friction brake have shown no appreciable difference between the losses of power in friction when very small and very large loads were driven by the same engine. (American Machinist, Dec. 25, 1886, p. 8.) The result of Porter's experiments has more recently been confirmed by several other experimenters. (Trans. Am. Sac. Mech. Eng., Vol. VII., pp. 86-113.) (7) The Stiletto is a torpedo boat 95 feet long, weighs 28 tons, develops 450 H. P., having 16 H. P. per ton of dis- placement, works with 150 pounds pressure, ran 30 miles in 77 minutes, or an average of 23.7 miles per hour, passing the Mary Powell, hitherto the fastest boat on the Xorth River. The highest speed attained is not given. (The Mech, Eng., June 27, 1885.) (m) The compound locomotives tried on the Boston and Albany Railroad failed as economizers of fuel, and were changed to the ordinary form. (n) Steam pressure used in marine engines has gradually increased at an average rate of more than two pounds per year for many years. The following values are given in Scribner's Magazine for May, 1887. [118.] MISCELLANEOUS. 219 1825 2 to 3 1830 5 1835 . 8 1840 10 1845 14 1850 21 1855 . 25 1860 30 1865 40 1870 50 1875 60 1880 70 1882 80 1886 150 to 160 The writer does not clearly state why these particular values are given. Some, though relatively few, steamships carry 160 pounds pressure, and if this be the highest, then, on the same plan, the highest for preceding years ought to have been given. Many locomotive boilers carry 160 pounds pressure, and have done so for several years. (0) Ideal efficiency. It is sometimes convenient for refer- ence to have an ideal maximum efficiency for steam powers. Assume, then, that the steam pressure is 200 pounds per square inch by the gauge, and that the back pressure is one pound per square inch absolute, and that the forward pressure decreases to one pound absolute ; then will the temperature corresponding to the higher pressure be 388 F. and to the lower 102 F. ; and the maximum efficiency when: working between these temperatures will be 388 102 _ 286 388 + 460 = 848 = ' 325 ' which is somewhat less than \. To realize this result would, according to the approximate adiabatic law, require about 118 expansions, which fact alone shows that such an efficiency is far beyond existing possibilities in a working engine. The waste of heat in the furnace of, say, 25 per cent, loss of pressure between the boiler and the engine, loss of heat at the exhaust, and other losses reduce the 220 HEAT ENGINES. [11H.] efficiency of steam plants below 15 per cent of the heat energy of the fuel. (p) Quadruple-expansion engines. In the American Machinist of December 3d, 1887, is an article taken from London Engineering, describing a system of quadruple- expansion marine engines, which have been placed on sev- eral new steamers. The boilers for these engines are de- signed to carry 180 Ibs. pressure. It is claimed that these are 6 to 8 per cent more efficient than triple-expansion en- gines. (q) Efficiency of plant. Take the case of the steamship Ohio, of the International Steamship Company, which has triple expansion engines of 2100 I. H. P., the gross tonnage of the vessel being 3325 tons. The engines were guaranteed to consume not more than 1.25 Ibs. of coal per I. H. P. per hour. The cylinders were 31 in., 46 in., 72 in., and 51 in. stroke. The trial trip developed an I. H. P. for 1.23 Ibs. of coal.* The caloritic capacity of this coal is not known to us, but it is quite certain that on such a trial the best of coal would be used. If the heat of combustion was 15000 thermal units which is a very high value there would have been ex- pended 1.23 X 15000 = 18450 thermal units per horse-power per hour, or 18450 X 778 = 14354100 foot-pounds of energy to produce one horse-power, or 33000 X 60 = 1980000 foot-pounds of work per hour, in which case the M si 1980000 A1 o ft ejficiency of plant = 1435410Q = - 138 > or nearly 14 per cent. * We Mechanical Engineer, Sept. 10th, 1887, pp. 49. 50, taken from Jndustrtot [118.J MISCELLANEOUS. 221 If the coal contained 14000 thermal units a fair value for very good coal the , . 7 1980000 efficiency of plant = 133{moo = 0.148, or nearly 15 per cent. The boilers would naturally be in excellent condition for such a trial, and if they, including the steam connections up to the steam-chest, gave an efficiency of 75 per cent, then we would have for of the engines only, ' ' = 0.184 in former case 75 = 0.197 " latter " 75 If the efficiency of the boiler and connections were 70 per cent a fair value the efficiency of the engines would be 0.197 in the former case and 0.211 in the latter. It is a remarkably good plant that will produce an indi cated horse-power with 1.23 pounds of the best coal. While it is well known that such a trial may be so con- ducted as to give a result too favorable to the contractors by not giving proper credit to the heat generated just before starting, or by letting the fires run too low at the close, or by not standardizing the indicator, &c. yet, on the other hand, the proprietors of the vessel would naturally check all the conditions so as to determine for themselves if the terms of the contract were fulfilled. We therefore feel some confidence that marine steam plants have been made that have developed an actual efficiency of some 14 or 15 per cent ; and certainly the conventional 10 per cent efficiency used by popular writers is exceeded in some cases. Tests of commercial coal taken at random show that the heat of combustion frequently falls below 12000 thermal units per pound. Ocean steamships have been reported as 222 HEAT ENGINES. [118.] developing an I. II. P. per hour with 2.1 pounds of coal, and in some cases with less than that amount ; hence if this coal contained 12000 thermal units, the efficiency would be 19800000 196000000 ~ or over 10 per cent. If the initial pressure in the cylinder were 160 pounds to the square inch absolute (about 145 gauge pressure) and back pressure 3 pounds, and the steam saturated, then would the initial temperature be 363 F. and the lower 142 F., and if this heat were used in a perfect elementary engine between these limits, the efficiency would be and TO per cent of this is 0.1876, which is nearly one of the values found above. (r) The torpedo boat Ariete, built for the Spanish Gov- ernment, has twin screws and compound engines, 14f", 24^", by 15" stroke. Average boiler pressure, 152 Ibs., revolu- lutions, 395 per minute, developing 155 II. P., and steam- ing 24.9 knots (28.8 miles) for two hours continuously. One measured mile was run at the rate of 26 knots (30.1 miles) per hour. (s) Steamer Anthracite, in 1880, had triple expansion en- gines. Pounds of water per I. H. P. per hour ............ 21.68 Evaporation per Ib. of coal at and from 212 ...... 9.27 u " " combustible u " ...... 11.25 Steam-pressure in the boiler, gauge, Ibs ............ 216.5 " " " 1st cylinder, gauge, Ibs ...... 201.6 Terminal pressure in the 3d cylinder, gauge, Ibs. . . . 9.55 Expansion, times .............................. 25.7 Coal per I. H. P. per hour, Ibs ................... * 2.61 [119.] STIRLING'S HOT-AIR ENGINE. 223 HOT-AIR ENGINES. 119. Stirling's (or Lauberean's) Hot-air En- gine. This engine was invented by Dr. Robert Stirling about the year 1816, and improved by his son, Mr. James Stirling ; for the details of which see Proceedings of tlie Institution of Civil Engineers, 1845. It was further im- proved by M. Laubereau, the form of which is shown in Fig. 51. It consists of two cylinders of different diame- ters, having a free communication be- tween them. The smaller piston, B, Fig. 52, is the working piston, and drives the engine. The larger piston or plunger, A, is made chiefly of plaster of Paris or other non-conductor of heat, and is somewhat smaller than the bore of the cylinder, so that the air may pass FIG. 51. freely past it. Also an annular space about the cylinder is filled with thin plates or small wires which heat quickly as the hot air passes among them, and as quickly give up their heat to the cold air on its return. This device is the regenerator referred to in Article 105. The top of the cylinder at C may be made double to admit of the passage of water ; or, what is bet- ter, the upper end of the cylinder may be filled with an extensive coil of small copper tubes through which water is made to flow by means of a force-pump worked by the engine, the object being to maintain a low temperature in that end of the cylinder, and thus cool the air at that end, and hence is called the refrigerator. The object of the plunger is to transfer a mass "of air from one end of the cyl- inder to the other and back again, and so on alternately, which is accomplished by the reciprocating motion of the plunger. The plunger is sometimes called the displacing piston. 224 HEAT ENGINES. [130.] The plunger, J., is operated by a cam so constructed and arranged that when the piston, , is near its upper dead point, the plunger will be driven very quickly to the lower end of its stroke and remain there until the piston descends to near its lower dead point, when the plunger will be driven quickly to the upper end of its stroke. The cylinder containing the plunger is called tfte receiver. The mass of air in the entire engine is constant, and is so maintained by a small air-pump worked by the engine, forcing air into the passage D. The mass of air in the lower part of the receiver, and which is referred to as being below the plunger, when the plunger and piston are at th', (204) = p b v b , (2), p. 11, \ [120.] THEORY OF STIRLING'S ENGINE. 229 If * be 'the ratio of the mass of working air to that of the cushion air, and r^ the absolute temperature of the iso- thermal F J, we have E r % = pt . s v t = p K . s V K pi . s Vi, &c. (206) From these we find Volumes per pound of the working air v& = vi = 53.21 -^-; ^ (210) W T v b = v c = rv,,= 53.21-^- P* \ Volumes of cushion air, per pound of working air, v t = A E = ( (215) 230 HEAT ENGINES. [121.] Also Mass of cushion air v { _ r, Mass of working air -y c r t Volume swept through, by the piston per pound of working air per stroke v, - v e = [l + (q - 1) -^ -- --] v b - (217) If there be given, instead of the ratio of expansion, the ratio of the volume swept through by the working piston to that swept through by the plunger, or 3 2_, then r may be found from equation (217), giving r= L. -. (218) If q is not given, find an approximate value of r by mak- ing q = r, giving r = !_H^_ . _!-_ + i . ( 219 ) the correct value of which will be somewhat less than the value thus found, and q will be somewhat greater. Assume q about 2 to 5 per cent more than the value of r found from (219), and find the correct value of r from equation (218). If the piston remained on its dead points while the plunger was moving, and the plunger on its dead point while the piston was moving, the indicator diagram, and the diagram representing the changes in the working fluid, would be as shown in Fig. 54. The approximate analysis of this case presents no serious difficulty. 121. Ill designing an engine of the Stirling type, the horse-power to be delivered and the number of revolu- [121.] IN DESIGNING. 231 tions per minute must be known, in addition to the data al- ready assumed. The number of revolutions will be limited by the piston speed and the length of stroke. The average piston speed may be between 100 and 200 feet per minuted One of Stirling's engines, having a four-foot stroke, was run, in actual practice, at about 28 revolutions per minute, giving an average piston speed of about 221 feet per minute. An air engine, reported upon by M. Tresca, had a stroke of 0.4 m. (1.3 ft.) and made about 90 revolutions per min- ute, giving a piston speed of about 120 feet per minute. The large air engines in the steamer Ericsson had an average piston speed of 108 feet per minute. Let ] = the number of revolutions per minute, S = the average piston speed, I = the length of stroke of the piston, h = number of horse-power required of the engine, TF" =. the work required of the engine per minute ; then, S=2JVl (220) IF =33000 A. (221) Let w = the number of pounds of working air required ; then, since the work done by one pound per revolution will be theoretically, the value of U in equation (201), we have : ' : ' But the actual work U will be less than the theoretical, and we will assume it to be 0.7, the theoretical. (In de- signing it is better to assume too small a fraction rather than too large.) Then 33000 h ~- 0.7 X 122.5 (r.-.rOfag.rX^' If r be assumed, the weight of air in one cubic foot will be, (205), (210),, 282 HEAT ENGINES. [121.] (223) The initial pressure, p w may be assumed, since it can be produced and maintained by the air pump in connection with the heat derived from the furnace. The volume of the lower part of the recewer will be, (223), (222), w cu. ft. (223a) Assume the stroke of the plunger to be y I, in which y is a fraction, say , f , or ^ ; and A its area ; then I A - Rrr * w .'. A = 53.21 !!JJJf. (224) ''/ '" PA The volume swept through by the piston per pound of working air per stroke being given by equation (217) and the stroke, Z, having been assumed, we have for the section, B, of the working cylinder, = q - w, (225) in which q will exceed r, and will be assumed. If -ll i. be given, instead of r, first find r approxi- ^b mately by equation (219) ; then assume q greater than r, and find r by (218), after which proceed as before. The mean effective pressure for the single-acting engine is such an uniform pressure as would, if acting throughout the upward stroke, do the same work as is done by the fluid during one revolution of the engine. Since w pounds of air (121. ] IN DESIGNING. 233 does the work w U during this time, we have EXERCISES. 1. Let T, = 600 F. ; T, = 120 F. ; p & = 120 Ibs. per sq. in. ; stroke of working piston, 2 feet ; 30 revolutions per q\ qj minute ; = ^ = the ratio of piston displacement v b to plunger displacement ; and 5 horse-power be developed. Find r l = 1060 ; r^ = 580, omitting decimals. _!L = 1.83; -^- = 0.55, nearly. r = 1.275, approximately, (219). Assume q = 1.30 ; then r = 1.25, (218). U = 5693 ft.-lbs., (201). E = 0.453, (202). E' = 0.317, if 0.7 E. 2\ = 17280 Ibs. per sq. ft. (given). v b = 4.08 cu. ft, (2 10) a . v q v e = 2.04 feet. Pounds of air, w = 1.0, nearly, (222), theoretical, or, w 1.43 Ibs., (222), practical. 2 336 Section of plunger, A ~ sq. ft., (224) ; and if y = , then Diameter of plunger = 2.46 feet. Section of piston B =, 1.17 sq. ft., (225). Diameter of piston B = 1.23 feet. Mean effective pi^essure, p^ = 2755 Ibs. per sq. ft., (226). 234 HEAT ENGINES. [122.] Find also the numerical values of v m %, v w u ft v p p^ and p v 2. In a double-acting engine made by Stirling, hav- ing a piston 16 inches diameter and a stroke of 4 feet, making 28 revolutions per minute, it was found by calcula- tion, and also by means of a friction brake, that the work done per minute on the piston was 1670000 foot-pounds. There were passed through the refrigerator 250 Ibs. of water per minute, and its temperature was increased 18 F. while passing. Assuming that the heat, except that doing mechanical work, was absorbed by the water passing through the refrigerator ; find the foot-pounds of heat ab- stracted by the water, the efficiency of the fluid, provided seven tenths of the heat were abstracted by the refrigerator ; the horse-power of the engine ; and the foot-pounds of heat unaccounted for by the absorption of heat by the water in the refrigerator. 3. In the engine described in the preceding Exercise, 83 pounds of coal were used per hour, possessing an estimated thermal capacity of 11580 B. T. U. ; find the efficiency of the plant ; also of the furnace, if that of the engine be 0.3. Ans. Efficiency of plant, 0.133. Efficiency of furnace, 0.44. From this it will be seen that the efficiency of the furnace is considerably less than that of the steam boiler, which, in good condition, may be assumed to be between 0.60 and 0.75. The efficiency of the plant is nearly equal to that of the most efficient steam plants of the present day. See page 201. But exact comparisons cannot be made, for the thermal capacity of the coal is not known with sufficient exactness, nor the comparative physical properties of air and steam in regard to conductivity. 122. Ericsson's Engine, in which the changes of temperature are made at constant pressure. About the year 1833 John Ericsson constructed in London a so-called [123. J DESCRIPTION. 235 " caloric engine," whicli attracted much attention, especially from scientific men ; but it was not a commercial success. His efforts at producing large engines of this class cul- minated in making in New York, in 1853, a vessel of 2200 tons, called the Ericsson, in which the motors consisted of four immense caloric engines.* (For dimensions, see Exer- cise 1, following.) After experimenting with these weak giants giants in size, but weak in power they were aban- doned ; but he produced another hot-air engine, f which was extensively introduced in various parts of the world ; still, after a few years, many of them were removed and replaced by steam engines. Their great bulk, the noise attendant upon their working, and the rapid destruction of their , fur- naces, were prejudicial to their general use. More recently Captain Ericsson has designed a small hot- air pumping engine, which is being extensively used, the principles of which we will consider. 123. Description. Fig. 55 is an external view of a small hot-air, Ericsson pumping engine, and Fig. 56 is a sectional view of the same. Within a cylinder of uni- form bore are two pistons, A and B, of which B is the driving piston and operates the mechanism in a manner so clearly shown as not to need explanation. The lower piston, A, which we will generally call the plunger, is made of some substance which is practically a non-conductor of heat. Its office is to transfer a body of air from the space below it to the space above, and back again, and so on alter- nately, and for this reason is known as the displacing, or transferring, piston. In the position shown, the plunger A is at the upper end of its stroke, and the piston- 7?, being governed in its speed by the crank I, is moving most rapidly, and is driven by the expansion of the air in the lower part of * Journal of Arts and Science, Sept., 1883. f Contribution to the Centennial Exhibition, 1875, by John Ericsson, pp. 425-38. 236 HEAT ENGINES. [123.] the receiver fl. The plunger remains nearly stationary, de- scending but little while the piston completes its upward stroke. The air in the upper part of the receiver is cooled by the water which has been raised by the pump r circulat- ing in the annular space xx, so that the water raised for other useful purposes acts as a refrigerator of the engine. During the earlier part of the return stroke of the piston, the plunger descends a little faster than the piston, main- FIG. 55. FIG. 56. taining a nearly uniform pressure upon the piston while air is transferred from below the plunger to the space above, the volume and temperature both decreasing. When the piston has reached about the position shown in Fig. 56 on its downward stroke, the plunger will have reached the lower end of its stroke and all the working air will have been transferred above and its temperature maintained at its inferior limit while it is compressed by the completion [124.] ANALYSIS. 237 of the downward stroke of the piston JB ; after which the plunger will rise to the position assumed at the beginning of this description, during which the working air will be trans- ferred to the lower part of the receiver, and its temperature and volume both increased at nearly constant pressure. The mass of air in the engine is constant. 1134. Analysis. Fig. 57 is a copy of an indicator diagram taken from a small engine of this class in Stevens Institute of Technology. It will be seen that the changes of temperature at constant pressure are clearly indicated, FIG. 57. and the isothermals, being nearly straight lines, show that the variation of pressure is small compared with the change of volume. Assuming that the change of state from that of constant pressure to that of constant temperature is instantaneous, the diagram of one pound of the working fluid may be rep- resented by D E F ' G, Fig. 58. F will represent the state of the working fluid at its highest temperature, r ]? greatest volume and least pressure ; hence the plunger and piston will both be at the upper ends of their strokes in the ideal case ; and the mass of air below the plunger will be considered as working air, and all the other air cushion air. I -Z^will represent the volume of one pound of working air at its highest temperature, and corresponds to the space below the plunger. Let F Fi to the same scale corre- spond to all the space above the working air ; then will 238 HEAT ENGINES. 1124.] correspond to the entire volume of the cylinder per pound of working air. The cushion air being supposed to remain at the inferior limit of temperature, take / L = F FI and construct the isothermal L K for the temperature T 2 , to represent the path of the cushion air. Make D D, = A B T ._ H K = E Et, G G, = I L ; then will D, E, F, G, be the real indicator diagram of the engine. If D l falls to the left of F, the piston at the lower end of its stroke will pass into the space occupied by the working air at its greatest volume. Let r t be the absolute temperature of the isothermal E F, and T S that of D G, and C p the dynamic specific heat of air at constant pressure; then will the heat absorbed per pound of air be, from state D to state E, along E F, along F G, along GD, H t = R T, log,, r ; hence, the work done per pound of air per revolution, if [124. J ANALYSIS. 239 the conditions were perfect, would be the sum of these, or U = 122.5 (r, - T.) % 10 r. (227) Efficiency of fluid with perfect regenerator y E =^< (228) or the efficiency would be the same as that of the perfect elementary engine. There being no regenerator, the effi- ciency of fluid, if working perfectly without radiation, will be (228) If all the losses due to radiation and the refrigerator be represented by n C v (^ rj, then the efficiency would be TJ . , Q . . H l -f- 184r7i(r, r 2 ) The value of n is not known, but will exceed unity in this class of engines, especially with very slow speed. In this analysis, the pressure at state G will be assumed to equal that of the atmosphere, although it may be some- what less, as shown in Fig. 57 ; then if Pi be the pressure per square foot of the atmosphere, r a its absolute temperature, and v & the volume of a pound ; then p* v* = R r a , (2) ; (229) ..-= 53.21 -^~ and -w, with subscripts, as in Article 120, represent respectively the pressures and volumes at the corresponding states ; then A =P^ P**t = XT,; (230) 240 1IKAT FNUIXKS. [134.] from which it appears that if r^ exceeds r a , V K will exceed y a . v & may be taken, roughly, at 12 cubic feet. If r be the ratio of expansion, we have, making r r 2 in Eq. (2), ^_ = J^_ = r= *L =: J!l = _rL. (231) v d j} K p t v e v k From the figure and equation (231), we have Pressures Pt=l> g = Pt, = p gl ; p A = p* = PK = p n = rp g . (232) Volumes working air, Cushion air ; let q be the ratio of the entire volume of air in the cylinder to the greatest volume of working air, then 9l = v n v, = (q 1) v r ; t-k = A = -?-=- ~ r. ; (234) Total volumes (235) r r Volume swept through by the piston per pound of work- ing air per revolution 2 K - v dl ) = 2 ( q (r - 1) + 1 - -I-\ ^L. (236) Mean effective pressure per unit area of the piston, or energy expended per foot of volume swept through by the piston, distributed over two strokes r. = 5L (237) -^ 2 K - dl ) [124.] ANALYSIS. 241 Mean total forward pressure Mean back pressure PB = P* - p v (239) Greatest vol. working air v f Piston displacement v fl v A ' Let .ZT be the number of revolutions per minute, /$, the average piston speed, Z, the length of stroke of the working piston, TF, the work in foot-pounds developed by the piston per minute, HP, the horse-power developed per minute, w, the pounds of working air per revolution, A, the area of the working piston ; then S= 2 Nl; W = 33000 HP ; (241) TF = w U N =2p f lA N. (242) If the isothermals are so nearly right lines that they may be considered as straight, the indicator diagram may be treated as a trapezoid ; hence, its area, referring to Fig. 58, will be GFx H 1= G,F, x ///; or, (v t - O (p A - p,} = 53.21 (T, - r,) (r - 1), (243) for the work done per pound of air per revolution. EXERCISES. 1. In the steamer Ericsson, there were four single-acting working cylinders, producing an aggregate of 300 horse- power, as determined by an indicator. The pistons were 14 feet in diameter ; length of stroke, 6 feet ; revolu- 242 HEAT ENGINES. [124.J tions, 9 per minute ; fuel, 1.87 pounds of coal per horse- power per hour, the total heat of combustion of each pound being estimated at MOOO thermal units. Let T, = 420 F., T s = 120 F. From this data find Mean eff. pressure, Ibs. per sq. in. per double stroke. . 1.1. Average piston speed, feet per minute 108. Vol. swept through by piston per I. H. P., ft. per min. 222. Heat of combustion of one Ib. coal 14000 X 778 ft.-lbs. = 10892000. Duty, 1 Ib. coal.. 33000 X 60 -j- 1.87 ft.-lbs. = 1059000. Efficiency of plant.. .^ *J^ , 0.097!. Theoretical efficiency of fluid, (228) E = 0.340. Actual efficiency if 0.8 of theoretical 0.272. Probable efficiency of furnace.0.0971 -=- 0.272 = 0.357. Notwithstanding the good efficiency of the plant, the ex- cessive size of the cylinders and other practical considera- tions prevented the general introduction of this class of engines for large powers. 2. Let the bore of the air cylinder of the pumping engine shown in Figs. 55 and 56 be 6 inches, stroke of the working piston, B, 2 inches, stroke of the plunger, 0, 5| inches, diameter of the plunger, 5f inches (length of the plunger about 20 inches), being the dimensions of this engine in the Institute. The piston at the lower end of its stroke passes into the plunger space about one inch, and near the middle of the stroke, as shown in Fig. 56, there is about T V of an inch between the piston and plunger, thus re- ducing the cushion air to a minimum. The furnace extends upward about one half the length of the plunger, above which the cushion air surrounds the plunger when at the upper end of its stroke ; and as only that is effective work- [124.] ANALYSIS. 243 ing air which is subjected l>oth to the refrigerator and fur- nace, the working air will be less in volume than that of the plunger- displacement ; but the relation cannot be deter- mined with accuracy. As nearly as we can determine in this engine, we have. = 0.75, Eq. (240). v f Assume r a = 520 ; T, = 130 F. ; T, = 720 F. ; total air volume per pound of working air at its greatest vol- ume, q = 1.2; and 50 revolutions per minute. Find : r, = 590 ; r t = 1180 ; r l r^ = 590 - = 0.5. Greatest vol. of a pound of working air, (233), cn.it ........................ v f = 29.67. Greatest total volume, (235), .......... ' v tl = 35.60. Least total volume,' (235),, or (243). . . . v dl = 11.35. Volume swept through by the piston per pound of working air per stroke, (243#), cu. ft ............................ fl v d , = 24.25. Katio of expansion, (235), or (236) ...... r = l.S. Work per Ib. of air per revolution, (227), ft.-lbs ...................... . U = 18450. M.E. P., (237), (2430), (233),, Ibs. per sq. ft. double stroke ................ j?> = 414.6. M. E. P. for the single working stroke of air, Ibs. per sq. ft ................ p^ 829.2. M. E. P., for the single working stroke of air, per sq. in .................... P* = 5.76. Area of working piston, sq. in .......... 28.2744. Work per revolution, ft.-lbs. . 28 - 27 ^ x 2 j 829.7 = 33.92. Work per minute, ft.-lbs ............. 50 X 33.92 = 1696. Horse-power ........... ......... 1696 -r- 33000 = 0.051. 244 HEAT ENGINES. [124.] Efficiency, if n = 1 in Eq. (228a) ...... E = 0.128. Efficiency of plant if eff. of furnace be 0.3 0.037. Pounds of working air, (241) .......... w = 0.0018. The pump described in this Exercise is used by the students in their experimental course, and from one of those I make the following abstract ; At 50 rev. per m. indicated M. E. P. was, Ibs. per sq. in . . 5.42. ^ . 5.42 X 28.2744 X 2| X 50 1. work perm., it.-lbs. . ^ - = 1597. Indicated horse-power .......... 1597 -f- 33000 = 0.0484. Weight of gas, cu. ft. per hour ..................... 15.7. "Weight of one cubic foot of the gas ............. 0.04584. Calorific power, E. T. U. per Ib .................. 13650. " " " " " " cu. ft ................ 625.7. Ind.ef. of ft ,d fluid, 157 627x 778 = 0.0125, or about 1^ per cent. This result shows the great loss of heat in this engine without a regenerator. Either the furnace has an efficiency of only about 0.1, or if its efficiency be 0.3, then n in equa- tion (228) should be between 3 and 4. If the efficiency of the furnace be about 0.2, then n must be 2 or more. The effective power as determined by the water pumped was 640 foot-pounds per minute ; hence the loss by leakage and friction was estimated to be 1597 - 640 _ 1597 or nearly 60 per cent. Efficiency of plant, including fuel, engine and pump ' or only one half of one per cent of the theoretical heat of combustion of the fuel was utilized by the plant. " This was one of the best of fifteen experiments. [125.] RATIO OF EXPANSION. 245 The manufacturers guarantee that this size of pump will raise 200 gallons of water per hour 50 feet high with 18 cubic feet of gas. This would give an effectual work of 200 X 5^- X X 62 = 1383 foot-pounds. This is more 60 than twice the amount found by the experiment above cited ; but a part of the difference may be due to the fact that more gas is required than was consumed in the experi- ment, the quality of the gas, the condition of the engine, etc. ; and the remainder if any to the art of advertising. A very small power steam-engine with furnace and boiler may require from 8 to 12 pounds of coal per horse- power per hour, giving an indicated efficiency of some 2 per cent, more or less. The hot-air pumping engine is used not on account of its superior efficiency, but on account of its greater economy and safety there being no danger of ex- plosion, and requiring but little expense for attendance. 125. Ratio of expansion to give a maximum mean effective pressure. A general solution cannot be made. We will assume some elements, and thus illustrate the process for a particular case. Let q = 1.3 ; Vfl ~ ^ dl = 0.8 ; m = -' ; and let 12 cubic V ( T 1 feet of air weigh a pound. Then, (236) m = 0.5 r 0.3. With these conditions, and equations (227) and (237), we have 122 5 P* = -^- ^ (1.3 - 0.5 r) logj, (244) which is a maximum for r = 1.69, as may be found by trial. The corresponding value of m will be m = 0.545. 246 HEAT ENGINES. [126.] The value of p, in (244) will be zero for ' - = = 8.6, . . under which condition, the engine, if f rictionless, would run without doing work, and would simply change the states of the working fluid. The mean effective pressure is unaffected by the initial temperature ; simply the range of tempera- tures being involved in the value of m, which may finally be expressed as a function of r, as above. The work per minute will depend upon the number of revolutions, and heat must be supplied in sufficient quantity and with suffi- cient rapidity to maintain the assumed temperatures. REMARK. The application of the analysis in the two preceding articles is very delicate, for there are so many physical conditions that cannot be definitely determined, and a small change in any one of them may produce a large change in the results. Thus, it will be seen that changing the value of q from 1.2, as in the preceding Exercise, to 1.8 in this Article, changes r from 1.75 to 1.69. (The value used in the Ex ercise is 1.8, being the nearest entire tenth, but its value is exactly 1.75.) The "log r" will be changed in a greater ratio, thus affecting the final result in a corresponding manner. The solution given shows that the fraction is large when the engine is run at its best effect ; and if it be assumed as 0.5 it would produce much less work per minute. The greater the difference of the temperatures the greater the work done per pound of working air, and the greater the ratio of the absolute tempera- tures the greater the efficiency, other things being equal. The working of the engine is quite as delicate as the analysis, it being much affected by friction in the cylinder and stuffing boxes, and the con- dition of the furnace. 126. Heat received and rejected only at con- stant pressure. An engine involving this principle was proposed by Joule and Thomson (Phil. Trans., 1885), but, so far as known, has not been constructed. In this engine the expansion and compression of the fluid would be adiabatic. A B C D, Fig. 59, page 238, would be the indicator diagram of such an engine, in which A B and D C are parallel to the axis O v, and B C and A D are adiabatics. [127.] A GAS ENGINE. 247 GAS ENGINES. . A gas engine is a hot-air engine in which the cylinder containing the working air is also the furnace, heat being produced by the rapid combustion of the fuel in the cylinder so rapid as to be called an explosion. The fuel is an inflammable gas. When the piston is moving for- ward in its stroke, air and gas are drawn into the cylinder, and, at the proper time, the gas is ignited, an explosion takes place, the air is suddenly heated and a high pressure produced ; after which a part of the energy thus developed is imparted to the piston during the remainder of the stroke, and the other part is forced out of the cylinder at the ex- haust. The two most prominent systems which have been de- veloped are : one in which the charge is fired with every revolution, when the cylinder is about half full of air and gas ; the other at each alternate revolution, when the piston is near its remote dead point. In the former, the energy developed can act on the piston during only about one half of a single stroke ; while in the latter it will act during nearly the whole stroke ; so that the latter ought to be, as it is found to be in practice, much more efficient than the former. Fig. 60, page 238, illustrates an ideal diagram of the former engine. In nearly all the more recent gas engines the piston draws in the charge of gas and air during a full forward stroke, then compresses it during the next backward stroke ; and when just past the next dead point the gas is ignited and the piston is driven by the energy thus developed dur- ing the next forward stroke, and during the next backward stroke the products of combustion are forced out ; thus requir- ing two revolutions to complete a cycle. These are trunk engines. Gas engines are made which take a charge at 248 HEAT ENGINES. [128.] both ends of the cylinder and thus resemble double-acting engines, although, in reality, there is only one explosion during each revolution. Others, like the Clerk engine, compress the charge in an auxiliary cylinder which is fired in one end of the working cylinder with every revolu- tion. Thus, while the steam-engine has been improved by passing from single acting to double, quadruple, &c., acting during each revolution, the gas engine has been im- proved by passing from double to single acting during each revolution, and, finally, to one action during a bi-revolution. 128. History. The origin of the gas engine is not definitely known. It appears to be an outgrowth of an effort to use gunpowder as the fuel, which substance was sug- gested for this purpose as early as 1680 by the celebrated Huyghens. The gas engine proper was first patented in England more than a century later, 1794, and, although in the years following there were many improvements and many patents, yet it became of no practical value until about I860, during which year M. Lenoir constructed in Paris the first gas engine that was actually introduced into public use ; and during the five years immediately following several hundred were used in France. It was patented in England by J. II. Johnson. It was of the non-compression type, and in its external appearance re- sembled the ordinary double- acting steam-engine. The charge was fired at each end during each revolution. It contained no new principle, and its success was the result of the care and thoroughness with which the details were worked up. A. section is shown in Fig. 61. Fig. 62 is an indicator diagram taken from a two horse-power engine of this class, (128.J HISTORY. 249 FIG. 62. as shown in the Journal of the Franklin Institute^ Vol. LI., 1866, Feb., p. 176. The length of the line A , Fig. 62, represents the length of stroke of the engine, while the line itself is the atmos- pheric line. Three lines are traced representing the action at one end of the cylinder during six revolutions. From A to b the charge was taken in at atmospheric pressure, but from b to W 1 1 Wll l'( 1 lllOt5(>M R pU\\'l Oil tllC piston rod engages a ratchet on the main shaft, thus imparting to the latter a rotary motion, which is rendered nearly uniform by the fly- wheel. The idea of compressing the charge before explosion was FIG. 63. 250 HEAT ENGINES. [128.] mentioned as early as 1801, but the system now generally used was patented by Barnett, an Englishman, in 1838, and by Million, a Frenchman, in 1861, and further developed by M. Beau de Rochas in France and Sir C. W. Siemens in England, both in 1862. The advantages of compression became fully recognized by this time, and the principle has been incorporated into nearly all gas engines constructed since that date. In 187G M. Otto produced his " Otto Silent" engine, which, for smoothness and quietness of running, and the economy in the use of the gas fuel, far exceeded all pre- vious inventions of this class of engines, and in less than ten years after its invention it is claimed that 15,000 were sold. No new principle was incorporated, the success being en- tirely dependent upon the skilful use of the principles de- veloped by others. Fig. 6i is an external view of an " Otto,'' used in making experiments in Stevens Institute. Successful gas engines of many varieties are now used. At the American Institute Fair, in the fall of 1887, six different types were exhibited by as many different invent- ors ; among which was an " Otto" containing the most recent improvements, some of which were exceedingly [129. J SOME DETAILS. 251 ingenious, and a " Baldwin" of recent invention, which for silent running and uniformity of motion seemed to be all that could be desired. All these engines outwardly re- sembled the modern horizontal steam-engine. Some en- gines of this class are duplex, some vertical, but mostly horizontal. The great improvement made in the gas engine is strik- ingly illustrated by the fact that the first successful ones, Lenoir's, consumed about 100 cubic feet of gas per indicated horse-power per hour, while an Otto has consumed less than 20 cubic feet for the same power. Some of the earlier engines consumed more than 100 cubic feet, and the latter ones more generally require about 24 cubic feet. The low- est figure given above was for a rich gas and an 8 H. P. engine. 129. Some details. Between the piston at its dead point and the end of the cylinder is a space not swept over by the piston, called the combustion chamber ; the volume of which is 0.4, more or less, of the entire volume of the cylinder. The fly-wheel is large compared with the power developed to insure more uniform running. The speed is also regu- lated in part by a governor, which operates differently in different engines. In some it cuts off a part of the supply of gas with each charge; in others it cuts off an entire charge until the speed is properly reduced ; and in still others it closes 'the exhaust so as to retain a part, or all, of the products of combustion of the previous explosion, thus preventing a full charge of both air and gas being taken in. The gas is ignited in various w r ays. A flame of gas, ex- ternal to the cylinder, communicating with the interior by a small orifice covered by the piston until the charge is taken in and then uncovered during its regular stroke, has proved to be efficient. The orifice may be so small as to remain open during the explosion, but in the more recent engines 252 HEAT ENGINES. [129.] the orifice is covered by a valve. The flame may be carried a very short distance in a cavity of a valve. Incandescent metal, so rendered by a flame or by an electric current, has been successfully used. In some cases ignition has been pro- duced by an electric spark ; and in still others by chemical action. This is an exceedingly important detail, and has given inventors much trouble, as its action must not only be certain, but must act promptly at a detinital part of the stroke, and, sometimes, more frequently than 90 times per minute. A space is provided for the circulation of water about the cylinder, through the piston, and also through the cylinder heads. This is rendered necessary to prevent injury to the metal from the high temperatures due to the explosion. Mr. Dugald Clerk made experiments upon gas exploded in a closed vessel, determining the time of explosion and the pressures resulting, from which the temperature was com- puted by means of equation (1), page 11. He gives the following results : MIXTURES OF AIR AND OLDHAM COAL GAS. TEMPERATURE BEFORE EXPLOSION, 17" C. Mixture. Max. press, above atmos. Inponnd* per gq. in. Temp, of explosion calculated from observed pressure. Theoretical temp, of explosion if all heat were evolved. Gas. Air. 1 vol. 14 vols. 40 806 C. 1786 C. 1vol. 13 vols. 51.5 1033 C. 1912 C. 1 vol. 12 vols. 60 1202 C. 2058 C. 1 vol. 11 vols. 61 1220 C. 2228 C, 1 vol. 9 vols. 78 1557 C. 2670 C. 1 vol. 7 vols. 87 1733 C. 3334 C. 1 vol. 6 vols. 90 1792 C. 3808 C. 1vol. 5 vols. 91 1812 C. 1vol. 4 vols. 80 1595 C. I (The Gas Engine, by D. Clerk, p. 111.) The computed temperatures may not have been the highest [130. THEORY. 253 at the instant of explosion, and, as Mr. Clerk says, are merely averages ; and it may be taken, that coal gas mix- tures with air give upon explosion temperatures ranging from 800 C. (1500 F. nearly) to nearly 2000 C. (3600 F.), depending upon the dilution of the mixture. Since cast iron will melt when subjected to a prolonged heat oi about 2000 F. (p. 89), the heat of explosion would de- stroy the working surface if it were not cooled by some artificial means; but with the means employed, cylinders have been used for years, and a wearing surface main- tained as perfect as in the steam-engine. The glow result- FIG. 65. ing from the explosion has been observed by inserting in the cylinder a small tube containing a strong glass through which one could look. 13O. Theory. We will consider the bi-re volution compression system. Fig. 65 is an actual indicator diagram taken from a 10 horse-power Otto engine during an experi- ment in the Institute, except that we have added the part A\ C B to represent the combustion chamber, and reduced the linear dimensions one half. It is a fair sample of many others that were taken. A D is the atmospheric line, 1 D the stroke of the piston, A 1 the clearance, 2 5 the com- pression line, 5 C the explosion line, C 7 the expansion line. The explosion is nearly, but not quite, instantaneous, as 254 HEAT ENGINES. [l30.J shown by the line 5 C the piston moving n very short distance before the explosion is complete. During the tak- ing in of the charge the pressure follows the line 12; during compression, the line 25 ; during explosion, the line 5 C\ during expansion, the line C 7; during exhaust, 78; during the fourth stroke the products of combustion are forced out, following 81. In subjecting these operations to analysis, we proceed, as before, to construct an ideal indicator diagram, in which we assume that the explosion takes place instantaneously at the remote dead point that the expansion and compres- sion lines are adiabatic that the fall of pressure at the end of the stroke takes place without change of volume and that the change is taken in and expelled at atmospheric pressure ; thus producing a diagram like 0, 1,2, 3, Fig. 66, in which C O represents the pressure of the atrnos- FIG. 66. phere. Letting the subscripts corresponding to the corners of the diagram represent the corresponding states ; then will the heat energy developed by the explosion be the area 3012 indefinitely extended to the right between the adiabatics 03 and 12 prolonged; the value of which for each pound of the substance will be 77, - C v (r t - r ), as in Exercise 3, page 64. The heitf rejected in passing from state 2 to state 3 will be, similarly, II, = ^v(r,-r 3 ); hence the efficiency will be Tl ~l2 (T - ~ ^ = l ~ I* - r'- (245) Since # = v t and v t = # we have from equation (42), if y [130. J THEORY. 255 be constant, and the same for the curve of expansion as for compression, *" r. ! 5 (246) according to which it appears that the efficiency depends only upon the ratio of the temperatures just before and just after compression or, generally, upon the temperatures at the extremities of either adiabatic, but otherwise is inde- pendent of the temperature of the explosion. The work per pound will be To find this work in terms of p and v, we have pv = I?T&$ in equation (2), and J2 = (y 1) (7 V , as in the answer to Ex- ercise 7, page 59 ; . ' . C v r = ^ v , as at the top of page 65, and similarly for p^ -y,, r^ &c. ; hence, ffib. = ^3 (P, v* - A < - P. , +p t <0. (248) This may be further reduced by means of equation (42), and making U^ = L\ -r- v s = the work done per cubic foot of the mixture, r = v t -f- the ratio of expansion, and^?j = p 9 ?' y , we have gr fi = P>-P ^ Y ~ 1 ~ 1 . (249) The mean effective pressure on a square foot of the pis- ton, or the energy developed per foot of volume, distrib- uted over four strokes, will be (250) 256 HEAT ENGINES. Let N be the number of revolutions per minute ; S, the average piston speed ; I, the length of stroke ; ffP, the horse-power; TF, the work in foot-pounds developed by the piston per minute ; w, the pounds of the mixture at each explosion ; A, the area of the piston ; then^ 8= *Nl; W = 33000 HP = ZpJAN (251) Work done per pound per single stroke will be W Volume swept over by the piston per stroke Al = m ^ff P - (253) Volume swept over by tlie piston per pound of the mixture per stroke .-,= *L. (254) 4^ e 131. The furnace. The smaller the combustion chamber, compared with the volume swept over by the pis- ton, the more efficient will be the charge, as shown by equa- tion (246), since T S -f- r will be smaller the smaller v -r- v 3 as shown by equation (42). But, on the other hand, the smaller v = v l is when the explosion takes place, the greater will be p l and r l after the explosion, as shown by equation (2) ; but as the temperature of explosion is very high in practice, there will be a practical inferior limit to the size of the combustion chamber, which must be deter- mined by a protracted use of the engine. In the " Otto,'' with which experiments were made at the Institute, the combustion chamber was 0.38 of the entire volume of the cylinder, and about the same relation exists in some other engines. [132.] THEORETICAL ENERGY OF THE GAS, 257 The efficiency of the explosion, or, as we may say, the effi- ciency of the furnace, is not perfect. Experiments made by Mr. D. Clerk in closed vessels of fixed volume, on the supposition that the absolute temperature varied as the absolute pressure at constant volume, found that the heat evolved varied from 50 to 60 per cent of the theoretical (The Gas Engine, p. 182) ; the latter being nearly the highest value found in any case, while in many cases it is considerably less than the former. Thus, for a mixture of air and Oldham gas, he found Fraction of gas, vol. ^, T V, ^ TT i> T Heating efficiency 0.40, 0.48, 0.50, 0.46, 0.40, 0.37. This shows that the furnace efficiency diminishes with the richness of the gas when the gas exceeds y 1 ^- of the vol- ume of the mixture (ibid,., p. 113). The table in Article 129 shows that the efficiency in that case varied from a little below to a little above 50 per cent, when the initial tempera- ture was 17 C. The initial temperature in the engine will be considerably above this, which, added to the facts that the pressure in the cylinder may be less than that of the atmosphere a part of the products of combustion will be retained possible leakage and imperfect action make it advisable, in the absence of actual measurements, to con- sider the efficiency of the explosion as not more than 0.45 of that indicated by the chemical composition. The cause of the large difference between the theoretical and actual heat developed is not well known. It is found that, gen- erally, the best results are obtained when the volume of air is 6 or 7 times that of the gas, so that the volume of the gas for each charge will be | or of the volume swept over by the piston in one stroke. 132. To find the work and efficiency in terms of the theoretical energy of the gas. Let & be the energy of the gas in thermal units, devel- 258 HEAT ENGINES. [133.J oped by the explosion of one pound of the gas as deter- mined from its chemical composition, J\ c the dynamic equivalent, m the coefficient of reduction of its efficiency, J,, r,, P,, /> the temperatures and pressures, respectively, which would result at the states 1 and 2, if the efficiency, m, were unity, and the expansion adiabatic ; and assuming that the entire energy of the explosion is communicated to the mixture, we have JT e =Jt,= C,(r,-r.); (255) and establishing an equation in the same manner as (247) we have, for the indicated work per pound of the mixture per stroke of the explosion, or per bi-revolution of the engine, (257) hence, the indicated efficiency of the plant will be (258) 133 The expansion and compression, curves. Since there is a flow of heat from the working fluid to the water jacket and the reverse, the curves will not be strictly adiabatic, neither will they be isothermal. Their character may be more accurately determined from a study of an actual diagram. Assuming that they follow the law p v 1 = constant^ Messrs. Brooks and Stewart found, for the curve between 6 and 7, Fig. 65, x = 1.363 ; and for the compression curve, |134.] EXPERIMENTAL RESULTS. 259 x = 1.335. Professors Ayrton and Perry, by an experi- ment which they confess was not as accurate as the above, found for the expansion curve, x = 1.479, and for the com- pression curve, x = 1.304.* These results show that it is much more nearly adiabatic than isothermal. The value of y used in our analysis should be less than 1.4, the value for air, because the presence of the hydrocarbon of the mix- ture will reduce the ratio of the specific heats ; but since the quantity of air predominates, we may, in the absence of actual measurements, use y = 1.4. Air behaves so nearly like a perfect gas that this value would be practically con- stant, even for the highest temperatures, if the expansion were adiabatic. 134. Experimental results. Messrs. Brooks and Steward, during the summer of 1883, made a thorough test of an Otto engine,f from which we make the following abstract. Dimensions of the cylinder, 8 inches diameter, 14 inches stroke. The air and gas used in mixtures were both measured by a gas metre, and it was found that when the volume of air used was 7.1 times that of the gas, the best indicated results were obtained. The diagram taken during the 19th test is shown in Fig. 65, with the linear dimensions reduced to one half their original value. During this test it was found that : Vol. air a ? Weight of air -. #Q ^^ = 6.63 ; T _ . y ' = 13.68. Vol. gas Weight oj gas About one haL the heat of explosion was carried away by the water jacket. The temperatures were computed by means of equation (2), the volumes and pressures being * Phil. Mag., 1884, (2), 65. f Graduation Tliesis at Stevens Institute of Technology, 1883 ; Van Nostrand's Engineering Magazine, 1884, Feb., pp. 90-104. 260 HEAT ENGINES. [134.] measured from the indicator diagram. The specific heats of the mixture were computed to be c p = 0.268 ; c v = 0.196 ; . . y = 1.37. The complete combustion of the gas, determined from its chemical composition, gave 9070 calories per kilog., or 617.5 B. T. U. per cubic foot. From 23.5 to 25.6 cubic feet of this gas were used per indicated horse-power. The mean effective pressure was about 58 pounds per square inch per stroke of the explosion ; and if there was an explosion for every fourth stroke the average pressure was Ity Ibs. for every stroke. The average number of revolutions was nearly 155 per minute. Temperature of the exhaust gases, from 720 F. to 778 F., as determined by a pyrometer. The indicated efficiency was 18 per cent of the total heat of combustion of the gas, rnd the effectual efficiency for the plant, as determined by a brake, was 14^ per cent. The experiments upon the " Otto," of various sizes and in distant parts of the world, have been numerous, giving efficiencies of 15, 16, 17, and 18 per cent. From these results it will be seen that the explosive gas- engine gives the highest indicated efficiency for the plant of any system thus far considered. For intermittent work and cost of attendance, it has an advantage over the steam- engine. On the other hand, the engines are much larger for the same power, and are thus objectionable for very large powers, and the cost of fuel for steady running is greater, since a heat unit in the form of gas costs considerably more in the market than a heat unit in the form of coal. Cir- cumstances must decide what class of engines is most econ- nomical for a particular case. The following is an analysis by Professor T. B. Stillman, of Stevens Institute, of the gas used by Messrs. Brooks and Stewart. The gas was taken from the mains supplied by the Hoboken Gas Company : [134.] EXPERIMENTAL RESULTS. 261 By volume. H Hydrogen 395 CH 4 Marsh gas 373 N Nitrogen 082 C 3 H 6 , Average. . . .Heavy hydrocarbons 066 CO Carbonic oxide 043 O Oxygen 014 HjOj.COj.HiiS, &c.. Impurities, &c 027 1.000 By weight its composition is found to be : Cu. Densi- Kilos, per Wt p. metres. ties.* en. m. unit. H .395 X .087 = .035 .058 CH 4 .373 X .694 = .258 .426 N .082 X 1.215 = .099 .163 C 3 H 6 , Av'e .066 X 1.84 = .121 .200 CO .043 X 1.215 = .052 .086 O .014 X 1.388 - .019 .031 H 3 O,, &c. .027 X ~.8 = .022 .036 1.000 X .606 - .606 1.000 HEATING POWER OF THE GAS. Upon complete combustion the gas develops heat per cubic metre, as follows : Calories. Calories. fromH 29060 X .035 = 1020 " CH 4 11710 X .258 = 3020 " C 3 H 6 ,&c. 11000 X .121 = 1330 " CO 2400 X .052 = 125 per cu. m. 5495c. and per kilog. gas = 9070 calories. Expressed in British measures, one cubic foot of gas develops 617.5 heat units. AlR NECESSARY FOR COMPLETE COMBUSTION AND THE PRODUCTS OF COMBUSTION. In order to determine the amount of air to be supplied for complete * Scho'ttler: Die Gasmaschinc, p. 77. By "density" is meant the weight of one cubic metre in kilogrammes. As will be seen from the above, one cubic metre of the gas in question weighs 0.606 kilos. 262 HEAT ENGINES. [134.] combustion, it is necessary to ascertain the quantity of oxygen that is taken into chemical combination by the several combustible constituents of the gas. 2H + O = H,O by volume 2+1=2 by weight 2 + 16 = 18 CH 4 + 40 =. C0 a + 2H,O by volume 2 -j- 4 = 2 +4 by weight 16 -j- 64 = 44 +36 C,H. + 90 = 3CO a + 3H a O by volume 2 + 9 =6 +6 by weight 42 + 144 = 132 +54 CO + O = CO, by volume 2 + 1 =2 by weight 28+16 =44 The combining proportions per unit of the several constituents is : By volume 1H + iO = !H a O 1CH 4 + 20 = ICO, + 2H a O 1C S H. + 4*O = SCO, + 3H a O ICO + |O = ICO, By weight 1H +80 = 9H 2 1CH 4 + 4O = VCO, + JH,0 1C.H. + ^O = ^CO, + ?H a O ICO + K> = V-CO, The volume of oxygen required for the combustion of 1 volume of gas is : H^ .395 X i = .197 CH .373 X 2 = .746 C,H .066 X 4i = .297 CO .043 X 4 = .022 1.262 less O in gas .014 014 1.248 Taking oxygen as 21 per cent in atmospheric air, the volume of air re- quired is 1-248 = 5.94 per volume gas, or the entire volume is 6.94 times the volume of gas. [134.] EXPERIMENTAL RESULTS. 263 Since air weighs 1.251 kilos, per cu. metre, the ratio by weight is ' From the combustion of 1 unit weight of gas with 12.26 air there re- sults 13.26 units weight of a mixture the composition of which will be : S(CH 4 ) 426 X V = 1.171 ) (C 3 H 6 ) 200 X V- = .629 f (CO) 086 X -V- = -135 ) i 1.93 (H) ................ 058 X 9= .522) H 8 0-< (CH 4 ) .............. 426 X J = .958V 1.74 (C 3 H 6 ) ............. 200 X = .257) N j from the air... ........ , ...... 9.407) q .. * \ in gas itself ..................... 163 f Impurities in gas ...................................... 0.03 13.27 Per unit weight of mixture the composition will be : CO, .......................................... 146 H a O ............................................ 131 N .............................................. 721 Impurities ............................... ...... 002 1.000 The volume which 13.27 kilos, of products of combustion will occupy is found from the known volumes of the constituent gases as follows : en. m. per kilos. kilo. cu. m. CO 2 1.93 X -524 = 1.011 H S 1.74 X 1,28 - 2.227 N 9.57 X .823 = 7.876 Impurities .03 X -.9 = .027 11.141 The products of combustion then occupy 11.141 cu. m. to every kilog. of gas. To find the ratio per cu. metre of gas we have simply to multi- ply by .606 the number of kilos, in a cubic metre, and we get 6.751 as the result. As there is necessary 6.94 cu. m. of mixture of air and gas to every cu. m. gas, it is seen that by combustion a contraction of 2.7 per cent takes place. When there is an excess of air present, as is always the case in prac- tice, the contraction becomes less in proportion, and may be considered to be about 2 per cent. In the following thermodynamic computations no account is taken of this contraction. 264 HEAT ENGINES. [134.] SPECIFIC HEATS AND THEIR RATIO. The specific heats of the products of combustion are determined from the specific heats of the several component gases as follows : Specific heat at constant pressure (water = 1). { .2169 X .146 (CO,) _ I .4805 X -131 (H 2 O) = .uozy Cp - 1 .2438 X .721 (N) = .1758 [ ~A X .002 (impurities) = .0008 Specific heat at constant volume (water 1). f .1714 X .146 (CO,) = .02501 r _ .3694 X .131 (H 2 0) = .0484 I 1QRS Cv ~ .1727 X .721 (N) = .1245 f - iyt [ -.3 X .002 (impurities) = .0006 J The ratio of these specific heats is the exponent of adiabatic expansion, and is found to be : Since there is always an excess of air present, these values will be somewhat modified by that fact. From the metre records of test 19 the ratio of air to gas by volume was found to be 6.63 to 1 ; by weight the ratio is 6.63 X 1.251 1 X .606 Since for complete combustion only 12.26 parts of air by weight are needed, there are 1.42 parts in excess. The specific heats of air being (7 P = .2375 and C v = .1684. the effect of the excess of air will be to reduce the specific heat slightly. (.2712 X 13.26) 4- (.2375 X 1.42) Cp = 14.68 _ (.1985 X 13.26) + (.1684 X 1.42) 14.68 C P .268 7 = T\ = 196 = 7 EXERCISES. 1. Required the horse-power and efficiency of a gas-en- gine whose cylinder is 8 inches diameter, stroke 14: inches, revolutions 160 per minute, charge every fourth stroke, combustion chamber 0.38 of the volume of the cylin- der, heat of combustion of one pound of the gas 1G32>> [134.] EXPERIMENTAL RESULTS. 265 B. T. IL, volume of working air 7 times that of the gas, weight of gas say -^ of that of the air, efficiency of the furnace 0.60 of the theoretical, y = 1.38. (These are approximately the conditions of the engine and test in Brooks and Stewart's experiments.) We find- Area of the piston, sq. in = 56.75. Piston displacement per stroke, cu. ft. = 0.46. Yol. of air taken in each fourth stroke, of 0.46 = 0.402. Pounds of air for each charge 0.402 X 0.08 = 0.032. Pounds of gas for 80 charges.. . T y X 0.032 X 80 =. 0.150. Work per Ib. gas, eq. (257), ft.-lbs 0.60 X 778 X 16326 [1 - 0.38 - 38 ] = 2344970. IHP. for the 0.15 Ibs. gas 2344970 X 0.15 = 10A Indicated efficiency, (258) E = 0.185. If m = 0.55 we would have : Indicated HP. for 0.15 Ib. gas = 9.7. Indicated efficiency E= 0. 169. These last results agree very nearly with the measured re- sults of Brooks and Stewart the horse-power being the same and the efficiency about one per cent less than their best result. 2. Required the horse-power, efficiency, pounds of air and of gas per minute, of a gas-engine having a 10-inch cylin- der, 16-inch stroke, making 150 revolutions per minute, charge every fourth stroke, combustion chamber 0.4 of the volume of the whole cylinder, heat of combustion of one pound of the gas 18000 B. T. U., volume of working air 6$ times that of the gas, weight of the gas 0.55 that of an equal volume of air, 12J cubic feet of air to weigh a pound, efficiency of furnace, 0.50, and y = 1.4. HEAT ENGINES. [135.J THE PETROLEUM ENGINE. 135. Naphtha Engine. In petroleum engines, the working fluid is either petroleum, or some of its products. Bray ton's petroleum engine was a modification of that in- ventor's gas-engine. The products of combustion entered the cylinder, and in this respect was similar to the gas- engine ; but combustion was gradual, the indicator diagram showing that it was made at nearly constant pressure up to the point of cut-off. Mr. Clerk concludes from his ex- periments with a 5 HP. engine that it utilizes about 6 per cent of the heat of the petroleum. Professor Thurston, from an experiment with one of these engines, concluded that 32.06 cubic feet of gas were consumed per IHP. ; but Mr. Clerk concludes that the same experiment shows a consumption of 55.2 cu. ft. per IHP. per hour. (The Gas Engine, p. 158.) "We will consider only a recent form, called the Naphtha Engine, of which Fig. 67 is an external view. In this en- gine the products of combustion do not enter the cylinder, but the same substance is used for the fuel and working fluid. A small plunger pump near D, but not shown in the figure, worked by the engine, forces some liquid naphtha into the boiler F at each revolution, a part of which is conducted from the boiler down the tube H to the coiiil ms- tion chamber, or furnace, below the valve chest A, and is there burned. At E'\& an opening into the tube II through which air may be forced to increase the rate of combustion. The heat of the burning naphtha vaporizes that remaining in the boiler, and the vapor thus generated is used in the engine in precisely the same manner as if it were steam ; and the law of its action in the engine is precisely the same as steam, as may be inferred from Fig. 68, which is a copy of an indicator diagram taken by Doty and Beyer, [136.; EXPERIMENTS. 267 FIG. 67. in the experiments referred to be- low. The ratio of expansion, as here shown, is about 2, but it may be varied at pleasure. The drop, when the exhaust opens at the end of the stroke, is sudden, and the back pres- sure and compression lines are good. The depression of the steam line, showing initial expansion, is prob- ably due to the setting of the valve, since the diagram from one of the cylinders was free from this de- fect. If the diagram be freed of its irregularities and of compression, it would be analyzed precisely like the steam- engine, and the solution in Articles 110, 111, and 112 would be applicable. Probably Article 110 represents the case more nearly, since the cylinders are very near the boiler, and receive heat continually from it. But the analysis cannot be carried out numerically, since the physical properties of Naphtha are not sufficiently well known. The latent heat of evaporation at varying pressures is not known, nor the value of R in the equation p v = R r if indeed it is constant for the vapor. We will, however, after giving the results of some experiments, make an ap- proximate solution. 136. Experiments. Messrs. Doty and Beyer made experiments upon a naphtha engine, of which the following is a summary of their report :* Three single-acting trunk engines were connected to the FIG. 68. * Graduation Thesis of Paul Doty and Richard Beyer, Stevens Institute of Technology, Hoboken, 1888. The Iron Age, July, 188a 268 HEAT ENGINES. [136.] crank shaft, the cranks making successive angles of 120 with each other. The lower ends of the coils constituting the boiler were connected with the pump, and the upper ends entered a common chamber. Diameter of cylinders, each 3 inches. Stroke of pistons, each 4 Piston displacement, each 37 cu. in. Admission ports, each fa x 2-jV inches. Exhaust ports, each -fa x 2-iV Diameter of pump 1^ " Stroke of pump If " Travel of main valves, each f " Clearance of cylinders, each. . f 1.86 cu. in. or 5J per cent of piston displacement. Boiler, seven spirals of four coils each. Coils, copper tubing outside diam " Height and diameter of coils, each 12 " Burner had 26 openings, diam. of each W " Heating surface 12 sq. ft. RESULTS. Average revolutions per minute, number 280.7. Total Indicated HP. from the three cylinders 2.81. Mean effective pressure, Ibs. per sq. in. , about 35. Naphtha burned per IHP. per hour, Ibs 3.53. Price of naphtha, June 5th, cents per gallon 10. Cost per IHP. per hour, cents 6.2. Heating surface, square feet per IHP 4.3. Water used to condense the exhaust naphtha, Ibs. per IHP. per hour : 25594. Increase of temperature of condensing water, degrees F 3.9. Naphtha passing through condenser per hour, Ibs 421. Temperature of the stack, degrees F., about 685. Specific gravity of the naphtha, that of water being unity 0.683. One gallon weighed, pounds . . 5.69. Assuming that the vapor was saturated, three elements of this data give, for 2559-i X 3 9 the latent heat of evaporation - 1 - - = 237 thermal units, at atmospheric pressure. [136.] EXPERIMENTS. 269 In order to determine the relation between the tempera- ture and pressure of saturated vapor, these experimenters devised a special thermometer with which they determined the temperature of the vapor in the steam chest, and at the same time determined the pressure by means of a pressure gauge. These measurements gave for 35 Ibs. gauge pressure a temp, of 225 F. ; " 47 " " " " " " 242 F. ; " 60 " " " " " " 258 F. Substituting in equation (80), page 97, these values of p reduced to their equivalent in pounds per square foot, and the corresponding temperatures reduced to the absolute scale, and finding the values of A., J3, C, we have **.,= 8.4818 -_.; (259) in which log^B = 2.949092 ; log^C = 5.796469. Making^ = 2116.2 this formula gives r = 602.62 or T = 141.96 F. Naphtha has not a fixed boiling point. In an experiment it began to boil at 60 C., and as the more volatile parts passed off, the temperature gradually in- creased, and, in the course of twenty minutes, it raised to 68 C., giving a mean of 64 C. = 147 F. The value found by the formula agrees, approximately, with the lower observed temperature, 60 C. = 140 F. To find the volume of one pound of the saturated vapor of naphtha at atmospheric pressure, we have, from equation (84), page 98, . = 0-0234 +. -8 = 7.69 cu. ft., in which 0.0234 is the value assigned for one pound of liquid naphtha at 60 F. 270 HEAT ENGINES. [137.] It was found that the naphtha, exposed to an atmospheric pressure of 29.982 inches, at a temperature of 70 F., evaporated at the rate of 0.092094 Ibs. per square foot per hour. Some of the values found by these experimenters differ largely from those given by others. Thus, the boiling point, as found by them, is between 116 F. and 167 F., but Box On Heat, page 14, gives 306 F. on the authority of Ure. The latter must be an error. They found the latent heat of evaporation to be 237 ; while Box, page 18, gives 184 on the authority of Ure. If lire's figures were reversed, giving 184 F. for the boiling point and 306 for the latent heat of vaporization, they would appear more rational ; still we can- not say what is correct. They give 0.683 for the specific gravity at 60 F., while Rankine, in Table II. of the Steam Engine, gives 0.848 at 32 F. ; and both cannot be correct. The discrepancies may be due in part to the difference in the chemical composition of the different specimens. The relation between the pressure and volume, given in equation (259), is considered only approximate, not only on account of the heterogeneous character of the substance, but also be- cause it depends upon three experiments only, whereas there should be a larger range of experiments in order to test its accuracy and reliability. The results will, however, be sub- jects for comparison for future experimenters. 137. Efficiency of the Naphtha Engine. In the absence of a determination of the calorific power of the naphtha used in the above experiment, we will assume that the heat of combustion is 22000 thermal units per pound, since its value would be 22274 if the composition were C. H M . For, 6 X 12 X 14544 = 1047168 14 X 1 X 62032 = )1915616( 22274. [138.] EFFICIENCY OF FLUID. 271 There was consumed 3.53 pounds of naphtha per IHP. per hour, hence the indicated efficiency of the plant, includ- ing fuel, furnace, and engine, was T7 33000 X 60 * ~ 3.53 X 22000 X 778 or nearly 3 per cent. It is a good 3 horse-power en- gine, that including fuel, furnace, boiler and engine yields this efficiency. If the particular naphtha used were richer in hydrogen than that assumed, or rather if its chemi- cal composition gave 23000 thermal units, the efficiency would be reduced to 3.1 per cent. The cost of running, 6.2 cents per IHP. per hour, is not a measure of the efficiency, but of the economy. A steam-engine, run by the waste fuel of a saw-mill, may cost nothing *for fuel ; while the same engine run with anthracite coal may cost many dollars daily for this item, while as a heat engine the efficiency should be the same in the two 138. Efficiency of fluid. Any solution of this part of the problem will necessarily be approximate, since some of the data must be assumed ; and yet such a solution may give some idea of its probable efficiency, and hence of the efficiency of the furnace and boiler. Regnault found the specific heat of petroleum to be 0.434, and we will assume it to be the same for liquid naphtha. The latent heat of evaporation at atmospheric pressure is 237 B. T. U. per pound, as found above ; but the law of change with tem- perature and pressure is not known. As this value approxi- mates more nearly to that of acetic acid than any other sub- stance now before us (see Article 74 of Addenda), we will assume, although otherwise quite arbitrary, that h e = 237 - 0.1 T. (2GO) Let the initial temperature of the liquid be 58 F. ; the 272 HEAT ENGINES. [138.] initial absolute pressure in the cylinder 60 Ibs., at which the temperature will be 239 F., Eq. (259) ; ratio of expansion, 2 ; back pressure, 16 pounds ; and, in order to make as few other assumptions as possible, consider-the law of expansion to be p = 21033 (203) ft.-lbs. per pound of naphtha per stroke, or 26.40 thermal units. The cut-oif being one half, the weight of vapor in the three cylinders when half full will be X 37 ' 5 = .01493 Ibs. 2 X 1728 X 2.18 Hence per pound per revolution the work done will be 21033 X 0.01493 = 314 foot-pounds. The mean work per revolution in the preceding experi- ment was which results agree sufficiently well when we consider that in the ideal diagram there is no compression or clearance. Mean effective pressure, theory, (176), p 9 = 60 (5 4.16) 16 = 34.4 Ibs. per sq. in. [13y.]- REMARKS. 273 Mean effective pressure from the experiment 330 (work per revolution) -^ e 3jL?_ij_5 (piston dixjjlacement) = 35.2 Ibs. per sq. inch, which is a very fair agreement under the circumstances. The heat supplied per pound will be, in thermal units, equations (93) and (260), 7i = c(T l - T t ) + A. = 0.434 X 181 -f 237 - 24 = 291. (264) Efficiency of fluid, (263), (264) 2 -~= o^, which we will call nine per cent. Efficiency of furnace - 033 =0.36, or 36 per cent ; that is, 36 per cent of the theoretical heat of combustion of the naphtha, as determined by its chemical composition, is utilized by the boiler. This is a good result for so small a boiler. 139. Remarks. In the design of this engine about four square feet of heating surface per horse-power was allowed. This is about one fourth of what would be al- lowed in the design of a steam boiler. In the former engine the boiler is completely filled with the flame of the burning fluid, and thus is made quite efficient. The naphtha engine has met with much favor for propel- 274 HEAT ENGINES. [140.] ling steam launches and yachts. Their great compactness is apparent from the preceding report ; pressure is raised very quickly not only on account of the low boiling point of naphtha, but especially on account of its very volatile and highly inflammable character when used for fuel. As soon as the supply is cut off, the flame ceases, and no vapor is left in the boiler to be blown off or cooled down, as in the steam- engine. As a fuel it is more conveniently stored in pipes and vessels in the lower part of the boat than coal can be. It appears to be quite as efficient as a steam plant of this power if not more so. But every power has disadvan- tages ; and this is objectionable for general purposes on ac- count of the very volatile and inflammable character of the fluid, endangering as it might all combustible material in its vicinity, and becoming more dangerous the greater the quantity stored ; still it is especially adapted to launches. 14O. Ammonia engines are those in which am- monia vapor is used instead of steam. These engines are condensing a condition which is rendered necessary on ac- count of the nature of the substance. Aqua-ammonia is in- troduced into the boiler ; vapor is generated in precisely the same manner as steam, and, after it is used in the engine, it is condensed and pumped back into the boiler, thus using it over and over. Much interest has recently been excited in these engines from the fact that ammonia has been substituted for steam, using the same boiler and engine, and doing, it is claimed, the same work with less fuel. It is asserted by some writers that the science of Thermodynamics teaches that the effi- ciency of an engine is independent of the nature of the work- ing fluid when used between the same limits of temperature ; and, hence, the above fact leads such to look with suspicion upon the correctness of this part of the science. The fact is, the bove statement is correct in only one very restricted case, and that case is never realized in practice ; we will, [140.J AMMONIA ENGINES. 275 therefore, state distinctly some of the principles established by this science, bearing upon this part of the subject. The efficiency of an engine is independent of the fluid used when worked between the same limits of absolute tem- perature, PROVIDED ALL THE HEAT RECEIVED IS AT ONE TEM- PERATURE AND ALL THAT IS REJECTED IS AT ONE LOWER TEM- PERATURE, the mass of fluid in the engine being constant. (See p. 161.) If the substance could be worked in this way, steam, am- monia, alcohol, &c., would be equally efficient. The fact that there is a latent heat of evaporation of the substance would not affect the truth of the statement. If water could be worked according to this law, beginning with a tempera- ture of 60 F., evaporated at 250 F., and raised to 300 F., the efficiency would be the same as if the substance were a perfect gas, or 300 60 AQ1 , E= 300 + 460 ^ This will be proved for a vapor in the Addenda, Article 112, from which it may readily be inferred to be true for the more general proposition. This gives the absolute maximum efficiency of any heat engine. But no engine works according to this law. In practical vapor engines, the mass of working fluid is variable. In such engines, it has been shown on page 193, Eq. (o), that the effective work and the efficiency both depend upon the latent heat of evaporation and the specific heat of the sub- stance ; hence In practice, the efficiency of all vapor engines depends upon the nature of the working fluid, and involves both the latent heat of evaporation and the specific heat. In a more complete theory of the vapor engine, involving incomplete expansion, the mean specific heat of the fluid and the latent heats of evaporation at the lower and higher 276 HEAT ENGINES. [140.] temperatures are involved in such, a complex manner as to require a knowledge of their numerical values iu order to determine whether, theoretically, one fluid will yield a higher efficiency than another. Again, this science does not teach that the efficiency is independent of the working fluid when worked between the same limits of pressure. To illustrate this principle, observe that in Articles 110 and 111, the work, equations (167) and (177), is expressed in terms of pressure only ; hence, if the same pressures can be obtained at lower temperatures, there may be a gain of efficiency. Saturated steam at 100 Ibs. per sq. in. has a temp, of 327 F. a .. ],; .. a .. .- .. .. .- a 216 F Dif. TTT Saturated ammonia vapor at 100 Ibs. per sq. in. has a temp, of 57 F. Saturated ammonia vapor at 16 Ibs. per sq. in. has a temp, of 23 F. Dif. ~SO The heat expended per pound of vapor above the lower temperature will be tor steam, Eq. (184), H = 111 /+//.; and for ammonia H = 80 <7+// e . If the indicated work, U, is the same in both cases, since II e is less for ammonia than for water and C also less than J, the efficiency of fluid will be greater for the former than the latter. Again, the formula for maximum efficiency, "> is [140.] AMMONIA ENGINES. 277 applicable only to a constant mass of fluid when working in the engine / and is not, in any sense, applicable to the plant. The efficiency of the boiler and steam connections de- pends upon the absorption of heat by the fluid, radiation from the furnace, and radiation from the connections. It will be seen from the preceding remarks that a given pressure may be produced with ammonia at a much lower temperature than with steam. Such being the fact, a lower temperature may be maintained in the furnace, boiler, and connections, and hence less heat would be lost by radiation. Our knowl- edge of ammonia does not enable one to determine whether it would absorb more heat in the same time than would water; if it would, the per cent of heat escaping up the chimney would be less, and the direct efficiency of the boiler thus increased. Admitting that an ammonia plant is more efficient than a steam plant, we see that this science explains the cause. But, Again, this science teaches that a condensing engine is more efficient than a non-condensing one, other conditions being the same. In the cases above cited that have come to the knowledge of the author, where the ammonia plant proved superior to the steam plant, the steam-engines were non-condensing, while the ammonia engines which replaced them were con- densing. Had the original steam-engines been changed to condensing engines there would have been an increase of efficiency ; but a want of knowledge of certain physical constants of ammonia prevent this science from determin- ing with certainty whether the efficiency would have been still further increased by substituting it for steam. If the interest on the cost of a plant and the cost for re- pairs be considered, a small plant involving a condens- ing engine may not be as economical as with a non- condensing one, although the former may be the more efficient. 278 HEAT ENGINES. [141, 142.] Many conditions are involved in the choice of the work- )*Jig fluid, aside from its thermodynamic relations : as the first cost, its effect on the working parts, its safety or danger to life, and the character of the necessary mechanism. 141. A biliary vapor-eiigiiie is worked by means of two fluids, one more volatile than the other, the fluids being worked in separate cylinders. If a surface condenser of a steam-engine be cooled by ether, the ether may be vaporized by the heat given up by the steam, and made to work a vapor-engine in precisely the same manner as steam drives a steam-engine. The exhausted vapor of the vapor- engine may be passed through a surface condenser cooled by water, and pumped in its cooled condition into the con- denser of the steam-engine, when the former operation may be repeated. A binary-engine is, theoretically, no more efficient than a properly designed steam-engine consuming the same amount of heat, and practically is not as economical on account of unavoidable waste and extra cost of the mechanism. But a steam-engine wasteful of heat may be improved by the addition of an ether-engine. (Manuel du Conducteur des Machines a Vapeurs conibinees, M. du Tumbley, Lyons, 1850-51 ; Institution of Civil Engineers, February, 1859.) 142. The products of combustion sometimes form, the working fluid. In these engines the entire prod- ucts of combustion enter the cylinder. The principle of their analys'is is similar to the Ericsson's hot-air engine. One of the most serious objections to this class of engines is the fact that the solid parts of the working fluid, dust and grit, wear out the working parts, with which they come in contact, rapidly. A recent attempt has been made by MM. Bermier Brothers, Paris, to overcome this objection, but as yet their engine is only an experiment. (Scientific American Supplement, 1889, p. 11099.) [143.1 THE INJECTOR. 279 THE STEAM-INJECTOR. 143. The injector is a device for feeding steam boilers, in which steam is taken from the boiler, and, by passing through the instrument, takes water with it, carrying the water and condensed steam in a steady stream back into the boiler. Fig. 69 shows an improved form of one of this class of instruments. A valve W is secured to the rod , and has its seat on another valve X. A is a tube con- taining these valves, and the passage of steaui through the FIG. 69. tube is controlled by the valve X. A hollow spindle, be- ginning with W and terminating at C, passes through the valve X, and may be moved, independently of the latter, for a short distance, by means of the lever II, thus admitting steam to the spindle without moving the valve JT, but a further movement of the lever will unseat the latter valve. The chamber M M contains a piston JV N, which terminates 2SO HEAT ENGINES. 1144.J in a gradually contracting nozzle at a point 0, just beyond C. By a slight movement of the handle II, steam issues from the orifice O, and a partial vacuum will be formed in JV^Y, into which water will be forced by outside pressure, and then forced through the delivery tube D, and at first es- cape through the waste orifice jP, and as soon as a solid stream escapes, a further movement of the lever II closes the orifice P by the valve A", and opens the valve X, and a continuous flow of water will then pass the check-valve into the boiler. If too much water passes C some will enter the chamber O and force the piston N N back, thus throttling the water, and if sufficient water is not admitted the reduced pressure at will cause the valve to move forward and permit more water to flow in. 144. Theory of the steam -inject or. Let 7F = the weight of water required of the injector per unit of time, TF = weight of steam required to force TF into the boiler. The heat in the steam above that of the feed-water when forced into the boiler will be, in ordinary heat units, considering the specific heat of the water as uniform and equal to unity, equation (134), Wh = [(r l -T t )+xh e ] TF, (265) and this will be the heat lost by this amount of steam in the injector and which is assumed to be imparted to the feed- water. The heat imparted to the water, above that in the reser- voir from which it is taken, will be (r. ~ r t ) TF , (266) where r t = the absolute temperature of the feed-water in the tank, r a = the absolute temperature of the water just after it has passed the injector, T, = the absolute temperature of the steam in the boiler. [144.] THEORY OF THE STEAM-INJECTOR. 281 It will be shown hereafter that the work of lifting the water from the reservoir to the injector and of forcing it into the boiler together require only a small fractional part of the heat energy lost by the steam in having its tempera- ture lowered from that of the boiler to that of the mixture of steam and water ; and, neglecting these two elements, expressions (265) and (266) become equal, giving, in terms of ordinary scales of temperature, TF '~ = -T=T~+ah W ' (267) For our present purpose it will be sufficiently accurate to assume that the steam supplied to the injector is pure satu- rated steam, or x = 1, and that equation (77) is sufficiently exact, or h e = 1114.4 - 0.7" TV To find the velocity of the water in. the passage G, Fig. 70, let p = the absolute pressure per unit in the boiler, p = '' " " " " of the atmosphere, V= the velocity of the water, tf = weight of unity of volume of the water = 62.4 per cubic foot at ordinary temperatures, then (268) The value of 6 may be found with sufficient accuracy by means of the formula at the foot of page 102, thus d = - - - , (269) v, (_L_i_ 5o y which, for 150 F., or 610 absolute, gives d = 61.2 pounds, and this value might properly be used in equation 282 HEAT ENGINES. [144.] (268), but as 62.4 pounds, the weight at ordinary tempera- tures, will not produce an error of 1 per cent in the veloc- ity, and as by its use the resulting formula will be more generally applicable to ordinary cases, we retain the latter. Just after entering the chamber G, the water will be under atmospheric pressure, and p a = 2116.2 pounds per square foot, and 2 g = 64.4. "With these values, equation (268) reduces to F = 1.0158 V~p~ 2116.2ft. per sec. (270) If p be in pounds per square inch, V = 12.1896 V p - 14.7 ft. per sec. . (271) If p be in atmospheres, F = 46.7355 tf p I ft. per sec. (272) If the diameter of the suction pipe J^be n times that of the passage E, the velocity in it will be F, = . (273) To find the area of the opening E for the passage of the water ; con- sider that the steam passing through the injector will have been con- densed to liquid water, then will the volume of the water and condensed FIG. 70. steam passing the opening per sec- ond be 0.016 ( IF + TF ) cubic feet, and if k be the area of this section, then j. = 0.016 ( W + TT ) The diameter will be (275) [144. J THEORY OF THE STEAM-INJECTOR. 283 To fold tJie velocity of the steam issuing from the eiid of the passage C, it will be necessary to find the pressure in the condensing chamber. Let JP, be the pressure in the condensing chamber D, pv the pressure of the atmosphere on the water in the tank B, h = C B, the height of the condenser above the water in the tank, T 7 ,, the velocity of the water at F, entering the condens- ing chamber, then 17 = 2 g [^=^ - A], (276) in which h is negative because the water is raised instead of being a positive head. From this may be found The velocity of the steam Fj will be given by the general equation for T 7 , following equation (62), page 82, and after substituting for T 2 and T I their values in terms of p and -y, becomes in which y has the value in equation (1-48), page 152. If the steam contains no moisture, this becomes / r fr> \ 0-H89~i F, = 23.2687 V p v [I - ^ 2 J (279) The area of the cross-section at C will be T-F Volume of steam per sec. ~~T^~ and the diameter will be 284 HEAT ENGINES. 1 144.] The work done l>y the injector will be that of forcing the mixture of steam and water against the boiler pressure p sufficiently far to make a displacement for "IF-J- TT^, pounds of water. Since the steam will be subjected, externally, to- the atmosphere the resultant pressure against which the water is forced will be the gauge pressure, or p j> . Hence, if p be in pounds per square inch the work will be U = 144 (p - p } ( TF -f TF ) 0.016 ft. Ibs. (282) The efficiency as a force-pump will be Work done U Heat expended (r, r a -\- x A e ) TF J ' The efficiency of the plant. If 1 pound of coal is equivalent to q thermal units, and w pounds are required to generate TF pounds of steam from the temperature of the feed-water, then F' U E = r , and if all the heat of the coal could be utilized for generat- ing steam and the steam were pure saturated, E' would be the same as E. But there is always a waste of heat in the furnace and boiler. If the q thermal units would evaporate n pounds of water at and from 212 F., if there were no waste of heat, and in an actual boiler a pound of coal did evaporate n^ pounds under the same conditions ; then if T< be the temperature of the feed-water and H the total heat of steam at the temperature T l of the boiler, then TF(H T 4 ) = -$66 n, w = 966 n ^ w = qw -' ; (285) The value of n may be, theoretically, from 11 to 15, de- pending upon the composition of the coal, and n l from 6 to [144.] THEORY OF THE STEAM-INJECTOR. 285 11, depending upon the composition of the coal and the effi- ciency of the furnace. The duty will be the work done per 100 pounds of coal, or, D = 100 X 144 X gauge pressure X Volume of water injected per Ib. of steam. Pounds of (steam evapo- rated per Ib. of coal. Effect of rejecting the work of raising the water and of forcing it into the boiler in the above analysis. In the following exercise it will be seen that if the gauge pressure be 90 pounds per square inch, and other conditions as there given, there will be expended (331 120 -f 794) X 0.05 = 50 thermal units in supplying 0.833 pounds of water to the boiler. To raise this weight of water 20 feet by suction a distance too large to be realized in practice would re- quire 0.833 X 20 -=-- 778 = 0.02 thermal units, which is only ^Vir f ^^? an( ^ hence may be omitted in the computation. The work of forcing 0.833 pounds into the boiler will be, equation (282), in thermal units, U = 144 X 90 X 0.833 X 0.016 -=- 778 = 0.22, which is also so small compared with 50 that it may be omitted. The theory above given, in which these two items are omitted, is, then, sufficiently accurate for engi- neering purposes. EXERCISE. If the steam pressure in a boiler is 90 pounds gauge per square inch, height of suction 4 feet, and the boiler is re- quired to make 3000 pounds of steam per hour ; required 286 HEAT ENGINES. U**-] the area of the section k of the passage E 'for the water, the velocity of the steam F", at 6', the diameter of the suction- pipe, its section, being 5 times that of the section k (which is an average of actual values), the steam containing 10 per cent of moisture, the feed-water in the tank being 60, the temperature of the mixture of water and condensed steam 120 before it is forced into the boiler ; also the ratio of the velocity of the steam to that of the water, and the weight of water to that of the steam. We have p = 104.7, h = 4, r - 60, T a = 120, n = 5. From steam, table, or equation (81), page 97, find T, = 330.9 F. TT. = ^. = 0.833 Ibs. per second. 36WJ h, = 1114.4 - 0.7 X 330.9 = 882.6, Eq. (77). an = 0.9 ; . ' . J-, A = 794.34. V = 12.1896 V90 = 115.63 ft.,vel of water at E and G, Fig. 70, Eq. (271). k _ 0.016 (0.05 + 0.833) _ 0>00123 sq ft _ .017568sq. in., Eq. (274). 115.63 d = 0.149 in., diameter of water passage EOT 0, Eq. (275). p, = 2116.2 - 4 X 62.4 - 4-4" 1845 ' 87 lbs " P 61 " ** ft ' = 12.818 Ibs. per sq. in. at F, Eq. (277). v = 4.217, volume of one pound of steam at 104.7 Ibs., Eq. (86). V l = 23.2687 1/104.7 X 144 X 4.217 |~1 - /. 18818 \o-"""| = 2572.5 ft, per sec. velocity of the steam at C. Eq. (278). Ft = 4.22 X 0.05 X 144 i- 2572.5 = 0.0110 sq. in., Eq. (280). rf, = 0.12, diameter of steam nozzle, Eq. (281). Vtl. offiteam _ 2572.3 _ Vel. of water ~ 115.63 ~~ No allowance has been made in this computation for contraction or. f fictional resistances, and hence the diameters must be made larger than here found in order to deliver the assumed amounts. The diameters should be about 1.1 to 1.2 times those here found. [144.] THEOKY OF THE STEAM-1XJECTOK. 237 U= 144 X 90 X (0.833 -f 0.05) X 0.016 = 183.099 ft. Ibs., Eq. (282). = 0.235 thermal units. E = 0.235 - - = 0.0046, Eq. (283), or the efficiency is less than one- 1005 X 0.05 half of one per cent. If 10 pounds of coal evaporates 1 pound of water at and from 212 F., it will require, under the conditions of this exercise, to evaporate 0.05 pound of water, w = - = 0.00581 11,, Eq (285). 966 X 10 If the coal be equivalent to pure carbon, it would evaporate, with- out loss of heat, 14500 + 966 = 15 Ibs. at and from 212 F., and if one pound in the plant actual!}' would evaporate 10 Ibs., then would the effi- ciency of the plant be E' = 183 -" 1? = 0.00418 X - = 0.00279, Eq. (286)- 778 X 0.05 (1183 - 60) 13 3 Duty = 1296000 X (16.6 X 0.016) x 10 = 3442176 ft. Ibs., Eq. (287). TABLE I. GIVING CERTAIN RELATIONS WHEN THE DELIVERY INTO THE BOILER IS 1 POUND OP WATER PER SECOND, NEGLECTING LIFT AND WORK OF FORCING THE WATER INTO THE BOILER ; TEMPERATURE OF THE FEED-WATER BEING 60 F., AND OF THE MIXTURE AND STEAM BE- FORE ENTERING THE BOILER 160 F. Gauge Pressure. Diameter of Steam Nozzle in Inches. Eq. (281.) Diameter of Water Nozzle in Inches. Eq. (275.) Velocity of Steam, Ft. per Second. Eq. (279.) Velocity of Steam andWater, Ft. per Second. Eq. (271.) Ratio of Velocities. Col. (4) -* (5) Ratio of Weight of Water to Steam. Eq. (267.) Ratio of Volume of Steam to Water. 30 0.28 0.21 2007 9 66.7 30. 10.3 55.9 40 24 0.20 2178.8 77.1 28. 10.3 46.2 50 0.22 0.19 2213.5 86.2 25. 10.4 39.4 60 20 0.18 2428.8 94.4 25. 10.5 34.4 70 0.18 0.178 2522.3 101.2 25. 10.5 30.4 80 0.17 0.172 2554.1 108.0 24. 10.5 37.6 90 0.167 0.166 2590.6 115.6 22. 10.5 25.2 100 0.159 0.160 2735.8 121.8 22. 10.5 22.8 120 0.142 0.154 2842.7 133.5 21. 10.6 19.6 140 0.133 0.149 2922.3 144.2 20. 10.6 17.2 160 127 0.143 2999.7 154.2 19. 10.6 15.3 (1) (2) (3) (4) (5) (6) (7) 1 (8) 288 HEAT ENGINES. As the temperature 160 F. of the mixture of steam and water is near the higher limit of reliable working of the injector, we take another case of lower temperature. TABLE II. GIVING RESULTS FROM THE SAME DATA AS FOR TABLE I , EXCEPT THAT THE TEMPERATURE OF THE MIXTURE IS ASSUMED TO BE 140 F. Gauge Pressure. Diameter of Steam Nozzle in Inches. Eq (281 , Diameter of Water Nozzle in Inches. Eq (275) Velocity of Steam, Ft. per Second. Eq. (279.) Velocity of Steam andWater, Ft. per Second. Eq. (271.) Ratio of Velocities. Col. (4) - (5). Ratio of Weight of Water to Steam. Eq. (267.) Ratio of Volume of Steam to Water. 30 .25 .18 2030 66.7 30.4 13.21 44.2 40 .218 .202 2211 77.1 28.0 13.29 36.3 50 .197 .190 2326 86.2 27.0 13.29 31.1 60 .158 .180 2431 94.4 25.6 13 33 27.1 70 .150 .175 2526 101.2 24.9 14.52 22.0 90 .146 .164 2677 115.6 23.1 13.42 19.6 100 .137 .160 2743 121.9 22.5 13.60 17.7 120 .125 .152 2845 133 5 21.3 13.49 15.4 140 .116 .147 2934 144 2 20.3 13.53 13.5 160 .108 .14^ 3008 154.2 19.5 13.58 12.0 (1) (2) (3) i (4) (5) (6) (7) (8) 145. Approximate Formulas. Certain general inferences will be apparent if we assume average condi- tions. Let T 4 = 60 F., T t = 150 R, and x = 1, then, equation TF _ 964.4 , 1 |_ __ T W 90 ^300 '* (288) If the gauge pressure be 80 pounds, then T 7 , = 323 F., and W = 11.79; that is, under ordinary conditions, 1 pound of steam will inject about 12 pounds of water into the boiler; or 13 pounds, including its own weight. [146.] INJECTOR COMPARED. 289 "For the diameter of the Cylindrical water-passage, E, equation (275), /0.016 X| 12.1896 /; (289) that is, the diameter will vary directly as the square root of the weight of water injected per second, and inversely as the fourth root of the gauge boiler pressure. The velocity of the steam, according to the preceding tables, will be about half a mile per second. The velocity of the water will be about 100 feet per sec- ond. The duty will be, if gauge pressure = 80, and 9 pounds of steam be generated per pound of coal, D = 1152000 X 12 X 0.016 X 9 nearly = 2000000 nearly. Since there will be some frictional resistance and radiation, and since 9 pounds of water are rarely evaporated at 80 pounds gauge, the duty would be somewhat less than 2000000. Efficiency, equation (283), which is about -^ of 1 per cent. The efficiency of the plant would be about -- of 1 per cent as a pump. 146. Injector compared with Direct-Acting Pump. By comparing these results with those on page 182 it will be seen that the efficiency and duty of the in- jector are much less than that of a direct-acting pump being about ^ as efficient. This is for service as a pump. But as a heat device, if there be no radiation nor lift of feed-water the efficiency of the injector will be perfect ; similarly, if 290 HEAT ENGINES. [146.] all the exhaust heat from the direct-acting pump be re- turned to the boiler, and there be no radiation, the heat efficiency of the pump will also be perfect ; and hence in either case would cost nothlntj for fuel. In both cases the furnace (or boiler) heats the water from the temperature of the feed to that of the boiler. If there be no losses from radiation, the difference in the cost for fuel in running the two devices will be that which furnishes the steam for run- ning the pump for doing the same work, if this steam be wasted at the exhaust. To illustrate : the work done by 1 pound of steam in the approximate cases above is that of forcing 13 pounds of water against 30 pounds pressure, and is U 144 X 80 X 13 X 0.01 o = 2396 ft. Ibs. One pound of steam in the direct-acting pump will, at about 70 or 80 Ibs. boiler pressure, do the actual work of 10,000 foot-pounds ; hence, to do 2396 foot-pounds will require 2396 -T- 10000 = 0.24 Ibs., nearly, of steam ; hence, it requires, in this case, about 24 hun- dredths as much steam to feed the boiler with a direct-act- ing pump as with an injector. But this steam is saved by the injector, and, we assume, is wasted by the pump. If 1 pound of coal generate S pounds of steam under a pressure of 80 Ibs. gauge, this waste will require 0.24 -=- 8.5 = 0.0282 pounds of coal for every 12 pounds of feed-water forced into the boiler. To evaporate this 12 pounds of water will require 12 -f- 8.5 = 1.41 pounds of coal ; hence, the fractional part of the fuel required by the pump will be 0.0282 -f- 1.41 = 0.02, or about 2 per cent of the fuel burned in the furnace. [146.] INJECTOR COMPARED. 291 Or, if the engine requires 30 pounds of feed-water per horse-power per hour, it will require 0.0282 X n = 0.0705 pounds of coal to work the feed-pump per hour per horse- power ; and if the plant requires 3 pounds of coal per horse- power per hour, then will the fractional part of the fuel re- quired by the feed-pump be = 0.0201, 35000 or about 2 per cent of the fuel burned, as before found. The low efficiency of the injector, as a pump, is due to the fact that the high velocity of the steam is very suddenly reduced to a comparatively low one by its impact against the non-elastic water, and the kinetic energy lost by the steam will be as the difference of the squares of the velocity before impact and that after. Considering the velocity of the steam as 25 times that of the mixture, and the weight of the mixture as 13 times that of the steam, the kinetic energy of the mixture will be 13 G>V) 2 = 0.0208 of the initial energy of the steam ; or 98 per cent of the initial energy is lost by the change of velocity at E. The 2 per cent remaining is gradually diminished on account of the decreasing velocity in the passage from G to V. The thermodynamic theory of the inspirator is the same as that of the injector. Steam-injectors are also used as pumps where intermit- tent action is required, as in the hold of a ship, and in mines ; also as ejector- condensers when attached to the escape-pipe of a condensing engine to avoid the use of an air-pump ; also as a gas-pump where it was more efficient than as a water-pump ; also as a steam-blower ; and also in the well-known case of the locomotive exhaust. THE PULSOMETER. [147.] THE PULSOMETER. 147. The Pulsometer is a pump consisting prin- cipally of two bottle-shaped chambers, A, A, joined together side by side, with tapering necks bent toward each other, uniting in one common upright passage, into which a small ball, C, is fitted so as to oscillate with a slight rolling motion be- tween seats formed in the junc- tion. These chambers also connect by means of openings with the verti- cal induction passage, D, having valves, E, E, and their seats, F, F. The delivery passage, II. which is common to both chamber*, is also constructed so that in the openings that communicate with each cylinder are placed valve- seats fitted for the reception of the same style of valves, G, G, as in the induction passage. J represents the air chamber, cast with and between the necks of chambers A, A, and connects only with the in- duction passage below the valves E, E. A small brass air check-valve is screwed into the neck of each chamber, A, A, and one into the vacuum chamber /, so that their stems hang downward. Those in the chamber allow a small quantity of air to enter above the water, to prevent the steam from agitating it on its first entrance. Conceive that the left chamber is full of water; steam passes to the left of the valve (7, and acting by its pressure directly upon the upper surface of the water, forces the water through the valve G and into the air chamber J. During this operation the chamber A is being filled, and water by its momentum finally drives the valve C to the FIG. 71. [148.] HEAT ENGINES. 293 left, thus cutting off steam communication with the kfc chamber in which the steam condenses, forming a vacuum, when water will be forced through the valve E by at- mospheric pressure into the left chamber, while the steam is forcing the water out of the right chamber. All the steam entering the pump is condensed and forced out with the other water; and the temperature of the discharged water will be higher than that entering the pump. 148. Analysis. The work done by the pulsometer will be that of lifting the water from the . source to the pump by the operation usually called u suction," and of lifting this water and the condensed steam to the point of delivery neglecting losses, such as friction, contractions, etc. Let TT be the weight of water raised in a unit of time, TT r , the weight of steam used in the same time, T^ the temperature of the water at the source, T lt " " " " mixture, r, " " " " steam, A e , the latent heat of evaporation of the steam, A , the height of the pump above the source, A,, " " " " delivery above the pump, A, the total height = 7? + A,. Considering the specific heat of water as constant and equal to unity, the heat lost by the steam will oe W (T-T t + A.), and the heat gained by the water will be ^ 8 (2\ - r.), and no allowance being made for radiation, these will be equal ; 294 THE PULSOMETER. [148.] (301) Observing the boiler pressure, and the temperature of the water before and after mixture, the ratio of the weight of the steam to that of the water may be determined. The work will be U= TT.A. + (TT. + IT)/*, (302) If the temperature of the feed-water be the same as that of the source, or T , then will the heat expended be H = J W(T - T t + A e ) ; (303) hence the efficiency will be E- U - W '** + ( W * + F ) A ' (304 -H- JW^-T.+h.)' If the work of lifting the condensed steam and f notional resistances be neglected, then ' (305) EXERCISES 1. By actual measurement 105000 gallons of water were raised in ten hours with 274 pounds of coal a height of 38 feet, and drawn horizontally 600 feet. If 10 per cent be allowed for resistances, find the work done in ten hours, the weight of water raised per pound of coal and the horse- power ; and if a pound of coal evaporated 7^- pounds of water, find the pounds of coal required per horse-power per hour, the weight of water raised per pound of steam, the increase of temperature of the water pumped, assuming its initial value to be 60 F., the gauge pressure 50 pounds, and the efficiency, the feed-water also being 60 F. If a gallon be 231 cubic inches, and a cubic foot be 62.2 pounds, then [148.] HEAT ENGINES. 295 Weight of water, 105000 X T 2 T Ve X 62.2 Ibs. = 873000. Work for 10 hours, 873000 X 38 X 1.10 ft. Ibs. = 36491400. " " 1 hour, . ..... . . 3649140. Horse-power, . . . ' . . 1.84. Coal per horse-power per hour, Ibs., . . . 14.8. Water raised per pound of coal, Ibs., . . ... 3186. Pounds of steam, 274 X 7$, . . . . 2000. Water raised per pound of steam, Ibs., . . 436.5. Work done per pound of steam, ft. Ibs., . . - * 18246. Heat in the steam above 60 F., B. T. IL, , 1137. Increased temp, of water, 1137 -j- 436.5, Deg. F., 2.6. Efficiency, 1?587__ = . 0.0187 J ' 1137 X 778 Efficiency, Eq. (305), . -. .... . 0.0180 or less than 2 per cent. (See page 452.) The assumption in regard to the evaporating power of the furnace would make the efficiency of the furnace about 56 per cent, making the efficiency of the entire plant over 1 per cent. Diameter of discharge pipe, if the coefficient of discharge be 0.8 and velocity 4 feet per second, d = A three-inch pipe was used. Pressure producing a velocity of 5 feet per minute against the atmosphere and a head of 38 feet of water, pounds per square inch " 31 2. If the temperature of the source be 60 F., of the mixture 65 F., the gauge pressure 60 pounds, lift by suction 5 feet and lift above the pump 15 feet; required the.number of pounds of water raised per pound of steam, the efficiency ; also the horse-power if 300 pounds of steam are used per hour. (These quantities are ideal.) COMPRESSED AIR-ENGINE. [149, 150.] COMPRESSED AIR-ENGINE. 149. A Compressed Air-Engine is an engine in which the working fluid is common air under a high tension. The air is usually compressed by a machine called an air- compressor, to a tension of from 40 to 1000 pounds to the square inch, and stored in an air reservoir, called a receiver, from which it is taken for driving an engine. Any ordinary steam-engine may be run by compressed air ; the only prac- tical difficulty being the tendency of the moisture in the air to freeze, and thus choke the exhaust. The freezing may be prevented by causing a circulation of warm air about the exhaust passage through channels especially provided ; and without this the evil may be mitigated in a measure by a proper form of the exhaust passage gradually enlarging it as it goes outward and making it smooth, so that the ice, if formed, will not adhere so firmly. 150. Analysis. We assume that the cylinder is filled with air of uniform pressure and temperature up to the point of cut-off, that then it expands according to an assumed law, then exhausts and a uniform back pressure during the back stroke ; also that there is no clearance. The diagram cleared from irregularities and clearance will be similar to A B C E FA, Fig. 72. Let jp, be the absolute pressure O A, Pv the absolute pressure // C at the end of expansion, p the absolute back pressure HE, v, the volume of a pound of air at the pressure p a B K J^fc Q FIG. 72. (a) Adiabatic expansion incomplete. per pound at full pressure will be The work done v,. [ 150 -1 HEAT ENGINES. 297 The work done per pound during expansion will be, Ex- ercise 3, page 64, or the second equation in Article 56, T * d r --= C v (T I - r,). The negative work during the return stroke will be hence, in the cycle, the work done per pound will be U = A B GEF = C v ( Tl - T.) + Pl Vl - p, Vv (306) Since the fluid is considered perfect, we have, equations (2) and (29), p, v, = R r, = (C v - C v } * p. v, = R r 2 = (#, - <7 V ) r 2 ; . . V = C v (r, - r 2 ) + (C\ - Q (r, - r^. (307) (J) Adiabatic expansion complete. The back pressure will be along CD, and j9 3 = p r^ = r 3 , and 17 = A B CD = O p (r 1 r s ) = Jy c v (T, - r s ). (308) (c) If there be no expansion, p^ = p T t = r a , and U=(C p - O v ) ('l - ^) r,. (309) Equations (307) and (309) may be put in a more symmetrical form by introducing an auxiliary T X , thus : C^ (r, - r,) + (<7 P - Q (^ - r a g) = C, (r, - r,) ; ... I: _' + 3L^. s J + LZl- 1 . (310) ft^V ^ - ^ ^ 7 ^ 8 Equation (309) will reduce to precisely the same value ; hence (307) and (309) become U= <7 P l - T,. (311) 298 COMPRESSED AIR-ENGINE. [150.] Making a table of values of , having for argument ^?, the computation TI p t for the work may be much facilitated. Let y 1.41, then : g p T a Pt Pi T a 2 0.855 9 742 3 0.807 10 0.739 4 0.783 11 0.736 5 0.768 12 0.734 6 0.758 13 0.732 7 0.732 14 0.731 8 0.746 15 0.730 Final temperatures. Knowing the initial temper- ature, the final may be found from equation (41), which is -MS) 1 (312) FINAL TEMPERATURES, THE INITIALS BEINO r, = 68 P. OB r, = 528V66. PI Pt Final Temp. Deg. F. Pi Pt Final Temp. Deg. F. 2 - 28 9 -181 3 - 76 10 -189 4 -107 11 -197 5 -129 12 -203 6 -148 13 -209 7 -160 14 -214 8 171 15 -219 Such low temperatures are fatal to successful working if moisture be present in the working air, as ice would be formed in the exhaust. Either the initial temperature must be considerably higher or the range of pressures must be small, or adiabatic expansion avoided, unless the air be thoroughly dry. (d) Let the expansion l>e isothermal and incomplete. [150.] HEAT ENGINES. 299 Then (313) and U, = A B C E F = p l v l -f- i 2) d v p s v^ = Pi Vi [~1 +. I g^ _ Pi]. (314) L v, p,_\ For the same cut-off, the isothermal will lie wholly above the adiabatic, and L\ in equation (314) will exceed U in equation (306). In order to secure isothermal expansion it is necessary to heat the air in the cylinder during expan- sion. It is not, however, practical to maintain a uniform temperature in this way. The very low final temperature has, however, in practice been prevented by working the air in one cylinder through a part of the full range of pressures, then exhausting into a receiver and there heating it, after which it is worked in a second cylinder. (e) 'Expansion isothermal and complete. In this case the terminal pressure H (7, Fig. 72, will equal the back pressure H E\ or p 9 = p^ in equation (314) ; .-. U = DAB C = G B OH = p, v, log* r = 122.5 r t log lo r. (315) Weight of air per minute. Let W be the number of pounds of working air necessary to deliver N horse-powers per minute, then, sinee V is the work in foot-pounds per pound of air, W U = 33000 N-, Volume of the cylinder. If p M -y , be respectively the pressure and specific volume of the air before compression, and if the temperature of the air entering the engine be the same as before compression, then p^ v, = p v a> and the final 300 COMPRESSED AIR-ENGINE. [150.] specific volume v, may be found when the law and amount of expansion are fixed. The terminal volume of TF pounds will be TF # and this will equal the volume swept through by the piston per minute, if there be no clearance. Let V be the volume of the cylinder and n the number of single strokes of the piston per minute, then for a double-acting engine, In V= Wv, = WE A; . F= 88000 a 2 n Up, Efficiency. In order to determine the efficiency, the full cycle of operations must be known, and this involves the law of compression, which will be considered in the discus- sion of the air-compressor. We know, however, if air were compressed according to any law and expanded according to the same law, there being no escape of heat by radiation between the states of expansion and compression, that the efficiency would be unity ; but there would be no resultant work, even neglecting the friction of the engine. The above formulas being for perfect conditions must be modified in order to conform to practice. Pernolet deter- mined that the moisture in the air, when converted into va- por, did not materially affect the theoretical results of con- sidering the air as dry. The weight of air as determined from equation (316) must be increased to allow for clear- ance, leakage, and imperfect working, as is done with the steam-engine ; and this must be still further increased in determining the weight of air before it enters the compres- sor, to allow for the imperfect working of the compressor. Compressed air-engines are frequently used where if steam were used there would be excessive condensation, as, in mines and other underground work, for driving drills, pumps, hoisting engines and locomotives ; also for small in- termittent powers. [151.] HEAT ENGINES. "801 THE COMPRESSOR. 151. All air-compressor is a kind of air-pump for receiving air at ordinary conditions, and after compressing it to a higher tension, forcing it into a vessel called a re- ceiver. It is not a motor, but requires a motor for driving it. The principles of construction are substantially the same as for an hydraulic pump, although in detail clear- ance spaces must be as small as possible and the valves be so made as to work with certainty. The valves are the most important details, and have received a large amount of at- tention from inventors and practical men. The best condi- tion for the proper working of the air valves, both inlet and exit, is to have them open and close by moving vertically and automatically ; and for this reason the compression cyl- inder has often been placed vertically, although vertically moving valves are used with horizontal cylinders. In the latter case, at least -two valves at each end of the cylinder are commonly used one for inlet, the other for outlet. When the cylinders are vertical, the compression cylinders are frequently single acting, and are driven by a double- acting steam cylinder. The steam cylinder may be vertical or horizontal. In some cases the axes of the cylinders have been inclined to each other, but the horizontal types are most common. Other fluids than air may be compressed in such a compressor. Fig. T3 is a view of a duplex air-compressor made by the author and worked in a silver mine in Colorado. The two steam cylinders are at the left hand, and the other two are the com- FIG - pression cylinders. The cranks are so set that the steam in one cylinder will 302 THE COMPRESSOR. [152.] be at full pressure when the piston in the air cylinder on the other side will be near the end of its stroke where re- sistance is greatest. 152. Analysis. During the back stroke of the piston the air flows into the cylinder ; assume that it has the uniform pressure D, Fig. 74. During the return stroke the pressure rises from C to B, and the air is then forced into a receiver at a press- ure which we assume to be uniform and equal to O A. FIG. 74. Let p 9 ' t v t ' rj represent state P\-> v \i T i " " the subscripts denoting the states ordinarily used in this work, and the accents dis- tinguishing them from the notation of an engine. a. Adiabatic compression. The work will be, equation (308), U r = ABCD=C (r/ - r,'). (318) For air C v = 184.77 (p. 53). We have, equation (42), page 61, (319) where y = 1.4. From this the final temperature due to compression may be found. Thus : FINAL TEMPERATURES, THE INITIAL TEMPERATURE = 68 F., OR r ML Pt 1 Final Temp. Deg.F/ Final Temp. Deg. F. 2 186 538 3 266 10 569 4 329 11 599 5 382 18 625 6 427 13 650 7 468 14 675 8 505 15 700 [152.] HEAT ENGINES. 303 The temperature exceeds that of boiling water under the corresponding pressure before the tension reaches four atmospheres. b. Isothermal compression. Here/>/ v/ = pj v,' ; and CV = A B C D = pi' / - p,' ,' + /"*,'* pdv = 122.5 r, Iog 10 ^< (320) which is equation (315). Equations (31 8), (319), (320), give which is less than unity for all practical cases ; hence iso- thermal compression requires less work than adidbatic com- pression between the same pressures. Isothermal compression is secured approximately by in- jecting water into the cylinder in the form of a fine spray in sufficient quantity to absorb the heat due to compression. The same result is secured less efficiently by performing part of the compression in one cylinder, and allowing the air to cool in a receiver, after which the compression is com- pleted in another cylinder. If air is compressed adiabati- cally, the heat lost between the compressor and the motor represents lost energy. If no heat were thus lost, adiabatic compression would be desirable. Energy lost by radiation. If T I be the final temperature in the compressor and T 3 the temperature at the motor, then will the energy lost per pound be / The work done by the compressor per pound of air will be / C 'B II. The FIG. 75. air then loses heat and enters the engine at A under the press- ure p l =j9j 'and a temperature TJ and expands adiabatically to D, the temperature being reduced to r^ where it is exhausted. The work of the engine will be ID A H. The resultant work will be A B C D. If no heat were lost, the tempera- ture at A would equal that at B, and that at D equal that at C, or r l = r/ y r. 2 = rj ; . . E = 1, or the efficiency would be perfect. In this case, however, A D will fall on B C, and the resultant work will be zero. The compression might be along D A and expansion along B C. Equation (325) expresses the efficiency if the air enters the motor at a less pressure than that of HA, and exhausts it at a higher or lower pressure than that of C. In this case the cycle will not be complete. Mass of fluid constant. In some operations, especially in refrigerating machines, the mass of working fluid is constant, the operation B A being effected by abstracting heat, and D C by supplying heat. In this case, if A D and B C are adiabatics, the heat supplied along D C will be, per 306 THE COMPRESSOR. [134.1 #, = c (*; - o, and abstracted along B A, //,= C 9 (r;-Tfo also H* ' ; 1 and the efficiency of fluid will be -F_ CKr.'-rJ-CKT.'-r,) r/ - r/ (326 , = X~ #p (*/-*,) ~~^~" the same as for the perfect elementary engine ; and is the fraction of the work which is transmuted into heat. If the operation were in a reversed direction, the result would be positive and would be the fraction of the heat absorbed which would be transmuted into work. 154. Friction of air in pipes. The experiments at Mont Cenis gave the formula 7^=0.00936^, (327) a in which d is the diameter of the pipe in inches, I the length in feet, n the velocity in feet per second, and F the loss of pressure in pounds. EXERCISES. 1. Required the volume of the cylinder of a double-act- ing air-compressor making 50 revolutions per minute to deliver to a compressed air-engine, making 100 revolutions, sufficient air to give 5 indicated horse-powers, allowing fifty per cent lost in the power of the compressed air. Let the initial temperature of the air be 60 F., and compressed iso- thermally to 5 atmospheres absolute, the initial pressure in the engine also 5 atmospheres at the same temperature. [154.] HEAT ENGINES. 307 Find the weight of water which must be injected per stroke at 55 F., that the temperature may not exceed 65 F. ; the volume of the cylinder of the engine, the point of cut-off of the engine that the expansion shall be complete, the final temperature at the end of the stroke, and the efficiency of the system. First find the work which one pound of compressed air will do. We have TI = 60 + 460 = 520, Pl = 14.7 X 5 = 73.5, p, = 14.7. Final temperature, r,, 520 X (i)?. degrees absolute ............. 328. degreesF ................................. 132. Difference of temperatures, r, r., .......................... 192. Work per Ib. of air, Eq. (308), 184.77 X 192, ft.-lbs ............ 35476. Work required per minute, 33000 X 5, ft.-lbs .................. 165000. Air required per minute, Ibs ............ ..................... 4.65. Vol. cyl., Eq. (317), 4 65 X 53.21 X 328 4- (200 X 2116.3)cu. ft- 0.192. Diameter, if stroke is li times the diameter, inches ............ 6.55. Ratio of expansion, Ei = (^Yr 4 = (5)* ..................... 3.15. Vl \PJ ' Air to be supplied to compressor, 4.65 X 2, Ibs. per minute ____ 9.30. * per stroke, Ibs ............... 0.093. Vol. air cyl., Eq. (324), * ' ^ * .............. ^ Diameter, if stroke is 1^ times the diameter, inches ............ 12.1. Work per Ib., Eq. (320), 122.5 X 520 X 0.699, ft.-lbs .......... 44526. Work per minute, 44526 X 9.3, ft.-lbs ........................ 414092. Efficiency, 165000 -f- 414092 = ............................... 0.40. Water injected, 0.24 X 192 -j- 10, Ibs. per Ib. of air ............. 4.61. Ibs. per stroke 4.61 X 0.093 ................... 0.43. cu. in, per stroke 0.43 X 0.016 X 1728 ......... 11.89. If the temperature be limited to 100 F., it would require less than three cubic inches of water per stroke. 2. In the preceding exercise if the compression were adiabatic, find the final temperature of compression, the final pressure being 5 atmospheres, initial temperature of the air, 65 F. ; also the efficiency. 308 STEAM TURBINES. [156.] THE STEAM TURBINE. 155. Steam turbines act on the same general prin- ciples as hydraulic turbines ; an essential difference bei:ig that water is considered non-compressible, while steam ana other vapors are compressible. A more general term for this class of turbines would be elastic vapor turbines. They may be reacting, like the Barker mill, Whitelaw or Scottish turbine, parallel flow, outward or inward flow. One is described in the Pneumatics of Hero of Alexandria. Rankine also mentions an inward-flow turbine which was used at the Glasgow City Saw Mills, and was considered equal in efficiency to an ordinary high-pressure engine (Steam Engine, p. 538). The claim, however, is not sus- tained by any authentic experiments. Very few of these turbines appear to have been in use until quite recently ; now they are being used to drive electrical dynamos, chiefly on account of the very small space occupied by them and the ease with which they may be located wherever desired. In many cases they are wasteful of steam on account of the clearance spaces permitting a part of the steam to pass through the engine without doing work, but one quite re- cently invented by Messrs. Dow appears to be a great im- provement on previous engines of this class. 156. Balanced outward-flow steam turbine. The turbine shown in Fig. 76 is the joint invention of J. II. Dow and H. II. Dow, of Cleveland, Ohio. A A represents the casing, or stationary part of the engine ; B B the rotat- ing wheel firmly secured to the shaft C, and containing the buckets or floats shown in the section D D, which are ar- ranged in concentric circles ; and concentric with these and between them are rings projecting from the stationary part of the engine through which are cut steam passages or guides. Steam entering through the stationary part at E, [is?.; HEAT ENGINES. 309 FIG. 76 passes both sides of the rotating disk FF, into the an- nular cavity near the centre, thence outward through passages in the an- nular spaces be- tween th<3 buckets, and through the buckets, finally es- caping at the outer circumference at D, and is conducted away at the exhaust at G. . It is balanced laterally by means of the disk F F, which is firmly secured to the shaft C\ so that if there be a lateral movement, however small, the space on one side of the disk will be reduced and on the other side enlarged, so that the increased amount of steam entering the latter will force it back to its normal position. It is claimed that this move- ment may be limited to 0.002 of an inch. The energy of the steam is gradually absorbed by the wheel as it passes through it, thereby diminishing, its pressure and .causing ex- pansion, similar to that in a multiple-expansion engine, there being six compoundings in this wheel. 157. Analysis. Reaction Turbine. These may be constructed like the Barker mill, Scottish or Whitelaw tur- bines, or other hydraulic turbines of this class. The section of the orifices is very much smaller than that of the arms. The reaction of the steam as it escapes from the arms im- parts to them a rotary motion, and, consequently, as the fluid passes outward in the arms a rotary motion is imparted 310 STEAM TURBINES. [157.] to it in common with the arms ; the fluid escaping in a backward direction relative to the motion of the orifices. The velocity of exit will depend upon the three elements : 1. The pressure at the orifices due to the boiler pressure, as if the arms were at rest ; 2. The additional pressure due to rotation, as if the orifices were closed ; and 3. The velocity of the orifice relative to the earth. The velocity of discharge relative to the earth will be the resultant due to these three causes acting simultaneously. On account of the compressibility of the fluid and the cen- trifugal action, the density of the steam will increase from the axis of rotation outward. The centrifugal force of the liquid, if any, in the vapor will cause the liquid to flow outward more rapidly than the vapor, and thus greatly com- plicate the solution ; and it would be still further compli- cated by considering the change of temperature and of re- evaporation in passing outward along the arm. We will assume that the steam is dry saturated or slightly super- heated and the temperature uniform. Let A, be a head producing the pressure at the entrance to the arms where the weight of a unit of volume is w l and pressure p. t . At any distance p let the pressure be p and w the weight of unity of volume ; then, since the weight will be directly as the pressure, we have p l = A, w a p = A, w. (328) The variation of pressure will be due to the centrifugal force of an element whose thickness is d p and base unity ; (329) [157.] HEAT ENGINES. 311 where co is the angular velocity per second. These equa- tions give Fa* ~P, (330) where p, is the pressure at the orifice, V 3 the velocity of the orifice, and Fthe velocity due to the head h r If F, = 0, p l = p^ as it should. The pressure due to the centrifugal force will be The interior pressure ready to produce velocity will be p, ; now, if the orifice be opened into the atmosphere, the resultant pressure will be j? a p^ when p n is the pressure of one atmosphere. The velocity of exit will be found from equation (278), page 283, after making y = 1.3, as given in equation (145), page 151 ; hence (332) Without rotation, the velocity relative to the orifice, or the earth, would be / T / />> \0.2307~| F, = 16.705 A/ p> v, [ 1 - [^ J . (333) The orifices have a velocity ca p = F, opposite to F a ; hence the velocity of discharge relative to the earth will be F=F, -F 3 . (334) The pounds of steam discharged per second will be, equa- tion (64), value of R, p. 103 ; 312 STEAM TURBINES. [157.] W= w t ksV'= 1.8295 **JP,(^} T ^rA (335) "in which T& is the coefficient of discharge, p l the pressure in the arm at theoriiice and isj9, in equation (330), r a the tem- perature outside and r, the temperature in the arm, w y the weight of unity of volume at the section of greatest con- traction, and W the weight discharged per second at that point. The work done by the reaction per second will be or per pound, U=l(V,-V t ) F,. 9 The energy expended will be that in the steam above the temperature of the feed-water, and per pound will be, equations (93) and (77), H = 778 (1114.4 + 0.305 T, TJ, (337) where T^ is the temperature at the boiler in degrees Fahr- enheit and T y that of the feed. The efficiency will be The horse-powers will be W 77 H P = (339) 33000 EXERCISE. 1. In a reaction turbine having orifices 12 inches from the axis of rotation, if the boiler pressure be 50 pounds (gauge), section of all the orifices 0.02 of a square inch, coeffi- t 157 -] HEAT ENGINES. 313 cient of discharge .50, velocity of the orifices one fourth the theoretical velocity due to the steam pressure when the engine is at rest; find the work per second, the efficiency, the temperature of the feed-water being 60 F., the horse- power and the pounds of water used per horse-power per hour. From Eq. (84) or from steam tables, volume of 1 Ib. of steam at 64.7 Ibs. absolute, will be, cu. ft. 6.58. Weight of a cu. ft., Ibs 0.152. Yel. of steam, no rotation (333), F, ft. per sec. . . 2223. Telocity of orifices of 2223, F s 555.7. Revolutions per second 89.3. " '' minute 5358. Pressure at axis where steam enters the arms 64.7. Pressure at orifices, (330), p. Ibs. per sq. in 140. Tol. of 1 Ib. steam at orifices 6.58 X 64. T ~ 140, 3.04. Toi. discharge relative to orifice, F 2 , (332) 2639. " " " the earth, V, - F 3 2084. Work per pound, U, (336), ft. Ibs 35952. Energy expended, H, (337), (T, = 297, T, = 60) 890732. Efficiency (338), per cent 4.0. Discharged per sec. if T, = 220, Eq. (335), Ibs., 0.0195. Horse-powers 1.2,7. Steam per H. P. per hour, Ibs 55. This result' appears to make the engine quite as efficient as small non-condensing engines. Theoretically, the efficien- cy would be high if the velocity of the orifices were half that of the theoretical velocity of discharge, or twice that assumed above, but the resistance of the air to the motion of the arms would be so great as to consume the power of the engine. At the velocity in this example the speed of the orifices is over seven miles per minute. At 200 feet per second, or about ten times the average highest piston speeds, the efficiency would be very low, not even considering 314 STEAM TUKBINES. [158.] prejudicial resistances. To be efficient the speed must be high. 158. Outward-flow turbine of Fig. 76. The best speed for the turbine requires that the fluid shall be discharged with the least velocity just sufficient to escape from the wheel. To accomplish this the steam must ex- pand down nearly to the pressure of the atmosphere. If the wheel is so constmcted and operated that the steam will expand without transmission of heat, the method of Article 112 will be applicable, and work done per pound of steam would be (340) if there were no losses from friction, contraction, eddies or clearances. To determine the speed requires a definite knowledge of its construction. A properly constructed wheel must run at a definite speed for maximum efficiency, and it cannot be correctly analyzed for speeds differing much from that, on account of eddies or whirls being induced, the effect of which cannot be formulated. Let 0, Fig. 78, be the centre of the wheel, a g the inner rim, 5 e the outer, c a a guide, a I a bucket. If there were no friction or eddies, the analysis for several concentric circles of buckets would be the same as if all the work were done in one series of buckets ; so we treat the case as if the several series were devel- oped into one. Let the initial ele- [158. j HEAT ENGINES. 315 ment of the bucket at a be tangent to the radius af* of the wheel, a d a tangent to the inner rim, a h tangent to the guide c a ; also d a h = of, the angle between the terminal element of a guide and the inner rim of the wheel, r, = O a, the inner radius of the wheel, F", the velocity of the steam at a in the direction a ^, -y t , the tangential component of F, V T , the radial component of F", and the same letters with accents to indicate similar quanti- ties at >, the point of discharge ; also &>, the angular velocity of the wheel, and M = TF -r- , or per pound, Z7 = fo r - = V n g U. (34-3) On account of condensation, clearance and friction, n for non-condensing engines is from to ^. If the wheel does not run at best velocity, v t f r' will not be zero ; let it be rj t\ r, in which ij will have a different value for every different velocity ; also GO r will not equal v if let it be G> r ; then J The number of revolutions per minute will be JT=8i - (846) EXERCISES. 1. Consider a Dow steam turbine run with steam at 70 pounds boiler pressure (gauge), using 600 pounds of steam per hour, the efficiency being one fourth the theoretical. Assume that the gauge pressure at the engine is 67 pounds, or about 4 per cent less than the boiler pressure, that the terminal pressure is 17 pounds per square inch, the inner radius, r = 1J. We have [158.] HEAT ENGINES. 317 r, = 773, T S = 679, H ei = 778 X S94 = 696000, u t = 21.9, eq. (150), -M, -f- v, = 4.05 = ratio of expansion. Work per Ib. of steam, Eq. (340), ft. Ibs., approx . . 88000. Work per Ib. as per hypothesis, ft. Ibs ............ 22000. Steam per H. P. per hour, 1980000 -f- 25075 Ibs . 90. Yel. of inner rim if rj = 0. 1, n = , e = 4, ft. per sec. 223 Revolutions per minute ....................... 20452. Horse-power, 600 -=- 90 .......................... 6.7. An engine made by the Messrs. Dow had turbine wheels 5f inches in diameter ; shaft, % inch diameter ; depth of buck- ets, T 3 T inch ; depth of guides also T \ inch ; weight of moving parts, 7 pounds, 7 ounces ; weight, including casing, 68 pounds ; highest measured speed with 70 pounds steam, 35000 revolutions per minute ; so that the velocity of the circumference was nearly nine miles per minute. An approximate computation of its regular daily perform- ance at 70 pounds pressure gave about 8 horse-power with about 75 pounds of steam per horse-power per hour, the speed being about 25000 revolutions per minute. Accurate measurements will doubtless modify these results. 2. Required the number of revolutions per minute neces- sary to burst a cast-iron disk from the centrifugal force, the modulus of tenacity, T being 20000 and the diameter of the disk 6 inches, there being a hole 1 inch in diameter at the centre for the shaft, weight of a cubic inch J of a pound. Assume the centre of gravity of each half to be at 7- = 1.3 inches from the centre. * = - 30 JX* ^000 XI. CHAPTER Y. REFRIGERATION. 159. A refrigerating machine is a device for pro- ducing relative cold. It has been repeatedly shown in the preceding pages that in any fluid doing work by expansion, without transmission of heat, the temperature is lowered. Advantage may be taken of this fact to produce a low tem- perature. Let m JV, Fig. 79, be the volume of a pound of the fluid when the cylinder of a compressor is full ; let it be compressed adiabatically to B and at constant pressure to A ; thence expanded adiabatically to J and at constant pressure to C. If the fluid be a compressible gas, the temperature will de- crease from B to A and increase from J to C ; but if it be a vapor the temperature will be constant at constant pressure some or all of the vapor being condensed during com- pression, and evaporation taking place during expansion. In both cases heat must be abstracted from the working fluid during the operation A B the heat being carried away by the cooling substance ; and absorbed by the work- ing fluid during the operation J C being taken from sur- rounding substances. The latter result is the one sought, and is made practical by placing the articles to be chilled in a room whose walls are made practically impermeable to the passage of heat, and abstracting heat from the room by repeated operations like the one just described, the heat so [160.] PKACTICAL OPERATION. 319 carried out by the working fluid being imparted to objects outside said room. 16O. Practical operation. The practical operation is shown in Fig. 80, which represents a vapor plant. Omit- FIG. 80. ting minor details, it is as follows : The working fluid is taken into one end of the compressor A during the back stroke of the piston, the operation being represented by J C, Fig. 79, the volume of a pound being m N when the cylinder is full ; during the forward stroke of the piston the fluid is com- pressed, the operation being represented by C B, and at B the valve is opened and the fluid forced into the coils of the condenser J?, Fig. 80. Water flows over the coils, reducing the temperature, if the fluid be a permanent gas, and liquefy- ing it if it be a vapor, the operation being represented by B A. 320 REFRIGERATION. L 1(50 -J The heat absorbed by the water is wasted unless the water is used for other purposes. At the left of Fig. 80 is the refrigerating room C, which should be enclosed on all sides, including roof and floor, with several inches in thick- ness of sawdust, felt, or other non-conductor of heat. This room contains many coils of pipe through which the fluid is made to flow, the coils being in the centre of the room, or, as is often the case, arranged about its walls. The fluid passes from the condenser B to this room, where, by properly adjusted cocks, it expands against a pressure, reducing the temperature and pressure until the latter is that of the initial in the compressor ; the operation being represented by A J, Fig. 79. During the back or return stroke of the piston the fluid flows into the compressor at constant pressure, the pressure being maintained by the heat in the refrigerating room, the operation being J C. If the fluid be a gas, the heat of the refrigerating room increases the heat of the gas, the temperature at J being lower than that of the room ; but for a vapor the pressure and temperature are maintained constant by the evaporation of the liquid, its volume being increased from I) Jto D C. It will be seen that only a part of the changes here de- scribed are made in the compressor ; however, the inaicator diagram C B A J represents the changes passed through by the circulating fluid, and represents the work done by the compressor. Let the adiabatics B C'and A /be extended indefinitely to the right ; then will the heat taken from the working fluid and carried away by the condenser be G A B F\ and that taken from the refrigerating room will be G J C F. The operation is in effect that of taking the heat out of the re- frigerating room, adding heat to that by the compressor, and finally causing both heats to be carried away by the water which passes through the condenser. The operation of all refrigerating machines is essentially f 161 J EFFICIENCY. 321 the same in principle : condense the gas or vapor, deprive it of heat diminishing its volume, lower its temperature by doing work, then expand it ; during the last operation heat is supplied by the articles to be cooled, and produces the re- frigerating effect. The heat of the refrigerating room is carried out by the circulating fluid to the condenser, where it is carried away by the water of the condenser. During its passage thither heat is added to the fluid by the work done upon it by the com- pressor raising its temperature, and by removing both heats at the condenser, the circulating fluid is put into a condition to take up heat again as it passes through the refrigerating room, so that the mass of circulating fluid may be constant. The mechanical operation of transferring the heat may be illustrated by the removal of water from a chamber at a lowel level than that of surrounding objects. For instance, conceive a mine having springs of water, and that the water is to be kept at a low level ; or conceive a room nearly but not quite water-tight submerged in a lake, and that the water in the room is to be kept at a low level. By placing a pump in the room the water may be raised, as fast as it accumulates, to a higher level than that of surrounding objects, from which point it will flow away naturally. If it be not raised sufficiently high it will not flow away. In the refrigerating apparatus the compressor raises the temperature of the fluid to a higher value than that of surrounding objects, thus en- abling the heat to flow away ; and by exposing it for a suffi- cient time it would escape by radiation without the use of water ; but the condensing water hastens the process. 161. Efficiency. It will be seen that a refrigerating machine is a heat engine reversed. Instead of transmuting heat into work, work is transmuted into heat. LetN 1 = f^A G, Fig. 79, be the heat carried away by the condenser 77, = F C J G the heat taken from 322 REFRIGERATION. [ML] the refrigerating room and absorbed by the circulating fluid ; then the work done by the compressor upon the fluid will be //, - H,. The general expression for the efficiency is Energy obtained (or work done) Energy expended If the energy obtained be the heat removed from the re- frigerating room, and the energy expended be the work done on the fluid, then representing this efficiency by E^ we have E, = jTJ. (347) In practice this will exceed unity, a result due to the peculiar unit to which the energy sought is referred. In most cases the energy obtained is a part of the energy ex- pended, which is not the case in the above assumption. If the energy obtained be referred to the heat expended, the expression will be less than unity. Thus, let E' be the efficiency of the furnace compared with the heat of combustion of the fuel, E", the efficiency of the engine, compared with the heat energy delivered to it by the furnace, E'", the efficiency of the compressor referred to the en- gine as unity, J5j, the efficiency of the refrigerating system compared with the compressor as unity ; then will the efficiency of the system be E = E'. E". E'". E,. (348) If the cycles were Carnot's, and no losses from clearance, friction or leakage in the engine and compressor, and the efficiency of the furnace be 0.70, then E = 0.70 ^Ll: . _!__, (349) EFFICIENCY. 323 in which r, is the absolute temperature of the steam at the furnace, r t the temperature of the refrigerator of the en- gine, T 4 the temperature of the condenser, and T, the tem- perature of the refrigerating room. The efficiency realized is far less than this. It appears that the efficiency of a re- frigerating machine will increase as the temperature of the condenser decreases, and also as the temperature of the re- frigerating room, increases. This is also apparent from general considerations, for the higher the temperature of the refrigerating room is allowed to be, the greater amount of heat will be carried away by a pound of the circulating fluid in expanding at constant pressure, and the lower the tem- perature of the condenser the less the work required of the compressor in raising the temperature from r 9 to r 4 . EXERCISE. 1. Let the efficiency of the boiler be 0.75, of the steam utilized by the engine 0.15, of the engine compared with one without friction or other waste 0.50, of the compressor compared with one without waste 0.70, temperature of the refrigerating fluid when it leaves the condenser 75 F., when it leaves the refrigerating room 5 F., and that 15 per cent of the latter energy is lost, required the efficiency of the plant. And if 1 pound of coal fed to the furnace de- velops 12300 thermal units 'when completely burned, how many pounds of ice at 32 F. may be formed from water also at 32 F. for each pound of coal burned ? We have E= 0.75 X 0.15 X 0.50 X 0.70 X = ~ X 0.85 = 0.222, or the efficiency is 22.3 per cent ; that is, for each thermal unit contained in the coal fed to the furnace, 0.223 of a thermal unit will be taken from the refrigerating room. 324 REFRIGEKATION. [162.] If eacli pound of coal contains 12300 thermal units, then for each pound burned there will be 12300 X 0.222 = 2731 thermal units taken from the refrigerating room, and as 144 thermal units are required to congeal 1 pound of water at 32 (page 89), there may be congealed 2731 ^ 144 = 18.96 pounds. In this solution it is assumed that a Carnot's cycle is per- formed. x If 25 per cent of the energy were lost instead of 15, the result would have been 16.8 pounds, and this is in the vicinity of actual values. Later we will show how purely theoretical results may be found. If this engine developed a horse-power with 3 pounds of coal per hour, then would 66.40 pounds of ice be made per horse-power per hour from water at 32 F. Compared with the work done by the compressor on the circulating fluid, the efficiency would be = 8.646, that is, for every thermal unit of work done by the com- pressor more than 5.6 thermal units would be removed from the refrigerating room. 162. The circulating fluid. Thermodynamically, any fluid may be the working fluid ; but there are certain phy- sical and practical considerations which determine a choice. It must admit of a low temperature without congealing. Air offers the advantage of being abundant, without cost, and admitting of any desired range of temperature ; but its den- sity being small, the required apparatus must be correspond- ingly large. If vapors are used they must be capable of vaporizing at low temperatures. Among the substances [163.] SOME PROPERTIES OF AMMONIA. 325 used are ammonia, JV 77 3 , sulphur dioxide, $, <9, methylic ether, C, 7/ 6 <9, and sulphuric ether ; the first two of which are the most common, and of these we will consider ammo- nia especially. The general formulas will be applicable to any vapor. Generally brine water thoroughly saturated with salt circulates in the coils, the brine being cooled in a tank by the ammonia, as above described. This saves a large amount of ammonia. Brine may be produced that will not congeal until the temperature is below zero Fahrenheit. 163. Some properties of ammonia. Certain properties of ammonia have been determined by Regnault, but his determination of the latent heat of vaporization and the specific heat of liquefied ammonia were lost during the reign of the Commune, in 1870 ; and these we will deter- mine by computation founded on the results of experiment and certain thermodynamic principles. In delation des Experiences, Vol. II., pp. 598-607, are the results of Regnault' s experiments upon temperature and corresponding pressure of saturated ammonia. These we have plotted in Fig. 81, the ordinates to the dots represent- ing the pressures, and the abscissas, temperatures. If the law be represented by Rankine's formula, equation (80), p. 97, the value of C will be so small that its effect will be in- appreciable, and the formula 2196 com. logp = 8.4079 - -p- ; (350) >r, if p be pounds per square inch, 2196 foffnP = 6.2495 - ^ represents the results of the experiments with much accu- 326 REFRIGERATION. [163.] racy from about 20 F. to 100 F., or from about 18 pounds per square inch to 215 pounds.* 640 -40-20 20 40 60 80 100120140160180 Temperatures, Degrees Fahr. FIG. 81. The specific heat of ammonia gas is 0.50836, which is a little more than for steam (Rel. des Exp., II., p. 162). Density of lique/ed ammonia, that of water being unity. * In the Transactions of the American Society of Mechanical Engineers OOftA for 1889, I used the formula log p = 6.2469 - ' and showed the dif- ference between the computed and observed values. This formula is nearer correct for higher pressures. [163.] SOME PROPERTIES OF AMMONIA. 327 Temp. Density. Dif. Authority. At 15.5 C.... 0.731 Faraday. -10 ....0.6492 1 -63 - 5 ....0.6429 65 ....0.6364 D'Andreeff: An. (3), 56, 317 (" Smithsonian Miscellaneous Col- . . . .0.6298^^ y lections," Vol. XXXII., 1888). 10 ....0.6230 70 15 0.6160 71 J These may be expressed very nearly by the formula 3 0.6364: - 0.0014 t = 0.6502 - 0.000777 r, when t is degrees centigrade and T degrees Fahrenheit. Density of the gas. Regnault gives, for the theoretical density of the gas, 0.5894 (Bd. des Exp., Yol. II., p. 162), but he also says : " The real density of ammonia gas is cer- tainly higher than the theoretical; the only experimental density of which I have knowledge gives 0.596 " (Ibid., Yol. III., p. 193). We will use the latter value. Volume of abound of the gas at the melting-point of ice. We have Weight of litre of air at C., 760 m 1.293187 grammes, or weight of cu. metre of air at C., 760 m ..1.293187 kilog. (Ibid., Yol. L, p. 162.) Hence the weight of one litre of the gas at C., 760""% will be 1.293187 X 0.596 = 0.770739 grammes, and the volume of one gramme of the gas will be 0^739= 328 REFRIGERATION. [164. , Reducing this to the equivalent of one pound and cubic- feet gives 35.3161 1.2973 22(M6 = 20.7985 cu. ft. per Ib. = v f Value of R. & r _ 2116.3 X 20.7985 X _ ~^~ 492.66 This is 89.343 -=- 778 = 0.11483 of a thermal unit; hence, at this state, equation (28), p. 49, = 0.50836 - 0.11483 = 0.39352 ; (353) and, equation (31) y = 1.292; (354) and. although y will not be constant, it will practically be so for the superheated gas. 164. To find the latent heat of evaporation of Ammonia. From equation (74), p. 95, in which v, is the volume of a pound of the liquid ; and as this is small compared with the volume of a pound of the vapor it may be omitted, and we have, omitting also the subscripts, k e = rv j~ -T- 778. (355) From equation (350) we have ^2. = 2196 X 2.3G26 2L . . A. = 6.49922 ^. (356) At the state when PJL = ?!_H? = 80 343, we have T r A. = 580.66. [164.; LATENT HEAT OF EVAPORATION. 329 This result must be for a state where v > e a ; for the general theory of imperfect gases shows that for the same volume p + r is less for a small- er pressure, and in this case at the pressure p a the gas is superheated, and at the point of saturation p will be less than p ; hence tfie latent heat of evaporation of ammonia mmt be less than 580.66 when the specific volume is 20.7985 cubic feet* The general value of p v + T will be found from the equation of the gas. In Vol. II. of Expei-iences, p. 152, Regnault has given the results of his experiments upon the elastic resistance of ammonia at the constant temperature of 8.1 C. (46.58 F.). These give the relations between the pressures and volumes of the actual isothermal A C, Fig. 82 ; the isothermal of the gas pass- ing through A, if perfect, being A B. These experiments reduced to volumes in cubic feet per pound, and pressures in pounds per square foot, are given in the following table : \ FIG. 82. TABLE. RELATIONS BETWEEN VOLUMES AND PRESSURES OF AMMONIA GAS AT THE TEMPERATURE 46.58 F.f V olumes cu. ft. per ID. Lbs. per sq. ft. Lbs. per sq. in. 24.3716 1862.706 12.93 23.157 1958 976 13.60 21.944 2064.096 14.33 20.7985 2178.960 15.13 19.563 2311.200 16.05 18.365 2458 800 17.08 17 160 2618.784 18.19 15.961 2822.544 19.60 14.762 3042.288 21.13 13.557 3303.648 22.94 12.355 3617.768 2512 11.1412 3996.820 27.76 * This shows that the latent heat found by Ledoux is erroneous (Ice-Makimj Machines, by M. Ledoux, Van Nostrand's Science Series, No. 40). For the volume 20.8 cubic feet Ledoux gives about 600 B. T. U. t Trans. Am. Soc. Mech. Engineers, 1889. 330 KEFKIGEKATION. [164.] Assuming in equation (4), p. 13, a c = o, and neglecting all terms after ,, it may be written in the form The first, sixth, and last experiments of the preceding table give for the products p v, and the corresponding values v, r (24.3716)" b = pv = 45397. = v T = 45156. r (18.365) = 44529. r(l 1.141)" In these equations r = 507.24, and they give a = 91 005, b = 16921 r, n = 0.97. Letting a s 91, b = 16920 r, n = 0.97. 7% equation of the gas will be and hence, equation (356), The latent heat of ammonia is . 5065.7 16920 = 592.52(1-1^*) (358) We now proceed to find the latent heat for certain states of the fluid. In Fig. 83, a represents the state of ammonia gas at the temperature of melting ice under the pressure of one atmosphere, for which the volume, as found above, is 20.7985 cubic feet per pound ; that is, o h = 20.7985, h a = 2116.3, r = 492.66. Let state represent the pressure and volume of the first experiment in the preceding table, for which o t = 24.3716, t s = 1862.7 Ibs. per sq. ft State e is the last in the table, for which oj = 11.141, je = [164.] LATENT HEAT OF EVAPORATION. 331 From s to e is the actual isothermal of the gas as determined by Reg- nault's experiments for the temperature of 46.68 F. As this was the tempera- ture of the water sur- rounding the tube, the temperature of the gas may have been somewhat less ; but we use this value as exact. The isothermal prolonged intersects the curve of saturation in m. To find the latent heat of evaporation at c, having the same volume as at a, it is necessary to find the temperature at this point. Make v = 20.7985 and equation (357) gives _ 91 P ~ 20.7985 16920 (20.7985) = 4.3753 r - 42.843 ; which substituted in equation (350) gives log (4.3753 T 42.843) = 8.4079 - * ; Eq. (350) and equation (358) gives A. = 578.96. It was found above for this volume, 20.7985, that if p .. T were con- stant down to the point of saturation, the latent heat would be 580.66, a value exceeding that just found by only 1.70 thermal units a difference in the right direction, and of reasonable amount. This test being satis- factory, we now apply it to other states. . ' .p = 1823.7 Ibs. per sq. foot. = 12.7 Ibs. per sq. inch ; 332 REFRIGERATION. f !64.] For the state immediately below , on the curve of saturation, we have v = 24.372, and with the same equations as in the preceding case, there results T= 420.4; . '. 1 = 40.26F., p = 1531.1 Ibs. per sq. ft. = 10.6 per sq. inch, h e = 579.67 thermal units. For state m, r = 507.24, or r=46 8 .58F., and from the same equations p = 11988 per sq. ft. = 83.25 per sq. inch, v = 3.41 cubic feet, h e = 526.47 thermal units. For the state for which we find r = 4687; .-. T= 8M F., p = 5279 per ft, = 36.8 per inch, A e = 550.52. Assuming the form of expression adopted by Regnault for the latent heat of evaporation of all substances, h e =d-eT-fT*, (359) and, using in it the data for the three last cases just given, we have 526.47 = d - 46 .58 e - 2169 7 /, 550.52 = d - 8.1 e - 6561 /, 579.67 = d + 40 e 1600 /. These give d = 555.50, e = 0.61302, / = 0.000219, and equation (359) gives the following as A more convenient formula for the latent heat of evapo- ration of ammonia : h e = 555.5 - 0.613 T - 0.000219 T. (360) The latent heat of ammonia vapor in the table at the end of the volume has been computed by means of this formula. [165, 166, 167.] AMMONIA VAPOR. 333 165. Specific volume of liquefied ammonia. If the volume of a pound of water be 0.016 of a cubic foot, then will the volume of a pound of liquid ammonia be, equation (351), *, = - - 016 (361) 0.6502 - 0.000| T This formula is sufficiently accurate for temperatures be- tween 5 F. and 100 F. A mean value gives about 41 pounds per cubic foot. 166. Specific volume of ammonia gas. From equation (84), page 98, ' By the aid of equation (350), after omitting the sub- script s , we have * = 63558 ! + "" (362) The volumes in the table of the Properties of Saturated Ammonia were computed from this equation. Since v l is small compared with -y, it may generally be omitted. 167. Isothermals of ammonia vapor. If the vapor be saturated, the isothermal will be parallel to the axis of v, as A B, Fig. 74. If the vapor be superheated, the equation will be (357), after making r constant. It will be 16920 ,o ft oN V v = 91 r. , . (363) v The general equation of vapors in which the last term is a function of v only, will be pv = ar-*-. (364) if v* 334 KEFKIGERATION. 168. Adiabatics of ammonia vapor. If the vapor be continually saturated, the equation of the adiabatic will be (a) or (J), page 184, or u = x v = (o log + ^AL T --, (365) in which u is the volume of the vapor and liquid when only the 07th part of the liquid is vaporized ; but as, in our analy- sis, the volume of the liquid compared with the vapor is neglected, it really represents the volume of the orth part of a pound of vapor ; c is the specific heat of liquid ammonia, the experimental value of which is c = 1.22876. If the vapor le superheated, then the first of equations (A), page 48, and equation (364) give dH=K v dr + rt d v. v But for an adiabatic d t dv = a r a v T, where v t and T, are inferior limits, and (366) To obtain an equation between p and v eliminate T between (366) and (364), giving [169, 170.] LIQUID AMMONIA. 335 For ammonia gas, these become , = 911- (*)"""_"? ( v 9 \v / v (371) the last of which is in terms of p and r as variables. 169. The specific heat of the saturated vapor of ammonia of constant weight is negative. Equation (139), page 147, gives, omitting terms containing r\ 837.5 s = c . T If c = 1, this will be negative for values of T less than 837, or 377 F. ; hence, for the range of temperatures ordinarily used in engineering practice, the specific heat of saturated ammonia is negative, and the saturated vapor will condense with adiabatic expansion, and the liquid will evaporate with the compression of the vapor, and when all is vaporized will superheat. Thus, in Fig. 84, if B Cs be the curve of saturation, and the vapor be compressed adiabatically from any point, as (7, on the curve of saturation, the adiabatic C I will rise above B C, and if it be expanded from the same point it will fall, below Cs. Equation (370) is the equation of C I, and (365) of CK, the part below C. 170. Specific heat of liquid ammonia. As- sume the volume m M, Fig. 84, of the pound of liquid to be constant at all pressures, and let M D be the absolute pres- 336 KEFRIUKRATION. [170.] sure at the absolute temperature T, B Cs, the curve of satu- ration, D II, A G, B F, IK, adiabatics. Let the vapor be expanded from D at the pressure p and temperature T until it is ail evaporated at state C, thence com- pressed adiabatically to /, thence com- FIG. 84. pressed at constant pressure to A, where it is liquefied, thence by the abstraction of heat let the pres- sure be reduced to D ; then II DAG-}- GAIK= IIDCK+ DCIA. Let the temperature of A B be T -j- d r, and of I,r-\-d T\ for the vapor from B to I will be superheated, its tempera- ture increasing with increase of volume ; then, if c be the specific heat of the liquid, HD A G = Jcdr, DCIA = vdp, G A IK = GABF+FB1K, Equation (360) gives, since d T = d T, - ( j^ = 0.6130 + 0.000438 T, From equation (350) find dP = 6.49922^. dr T* . J Differentiating (371), after which change dr to dr' and drop all subscripts in the second member, and (372) becomes .-. + 0.000438 T + [171.] COMPRESSOR AND CONDENSER. 337 As this investigation depends upon a comparatively small range of volumes, from 11 to 24 (table 011 page 329), and assumptions in regard to the forms of the functions in equa- tions (350) and the first equation on page 330, it may not be very reliable for a large range of temperatures. We will compute a value for a volume within the limits of the table, and will take values given on page 331, viz. : v = 20.7985, T = 426.6, T = - 34 .,p = 1823.7 Ibs., for which we find c = 1.093. The mean of eight determinations made by Dr. Hans von Strombeck, chemist, gave c = 1.22876. The experimental value was determined when the tem- perature of the liquid was about 80 F. KEFRIGEKATING SYSTEM WITH VAPOR CONTINUALLY SATU- RATED. 171. Work of the compressor and condenser. In Fig. 85 let E F be any adiabatic, p r, apply to A -B, p^ r 2 to D C, a?, the fraction of a pound of vapor at state _*, x^ at A Fj sc 3 at A (when it is not zero), J 4 at J, v l the volume of a pound G of vapor at B, v^ the volume of a pound under the pressure j? a ; the odd figures for subscripts belong- ing to the upper line, and the FIG . g5. even to the lower. The work of compression from F to E, and of forcing it into the condenser from Eio A, will be, equation (m\ page 192, U'= A EFD = J \c (T,- r t ) -f 338 REFRIGERATION. [1~1-J In the expansion chamber the vapor expands against a re- sistance, reducing the pressure from p l to p^ and tempera- ture from r t to r,, doing the work A D J, and, assuming an adiabatic change, the expression for the work will be found by writing x t for a?, and a% for a?, in the preceding equation, since all the other quantities will remain the same ; (375) After the vapor is forced into the condensing chamber, its specific volume is reduced by condensation from A Eto A (and for the sake of generalizing the analysis we assume for the present that it is not reduced entirely to a liquid at A, but that a% has a finite value), and the heat emitted from the circulating fluid and absorbed by the condenser will be the area between EA and the indefinitely extended adiabat- icsA J m&EF, or, and the heat absorbed by the circulating fluid (ammonia or brine), which is taken from the refrigerating room, will be -7/, = -e/te-<>A M ; (376) then, U - U" = K> - //,. Since the cycle is Carnot's, we have fo - a?,) A e . fa -ap A ef . The efficiency, referred to the work done by the com pressor, will be (878) [171.] COMPRESSOR AND CONDENSER. 339 which is independent of the amount of liquid evaporated, as it should be. More heat, however, will be removed by the work of the compressor if all the liquid be evaporated that is possible and leave it continually saturated ; that is, if the fluid be vapor only at B ; and also if it be entirely condensed at A. These conditions require that a?, = 1 and x 3 = ; and the equations for largest effect become U 1 = J c> ( TI - r t ) + J fll - x, /. (379) V" = J^c(r l -T^-x, - a> 4 ) A ea . (382) H, - H y = J \~h n - (a>, - x,} A e ,l = V - U".(3S3} Eq. (365), aj, =. ( c % e ^ + ^J . (384) For a?, = 0, aj 4 = c -p- % e (385) "ea 7 a = 2.3026 c ^- fc<7.. . These formulas solve the theoretical part of the problem. The temperature of the circulating fluid in the condenser will be several degrees above that of the condensing water, depending upon the amount of condensing surface say about 10 F. above ; and the temperature of the con- densing water will depend upon external circumstances, and may be 40 F. in winter and 60 F; to 70 F. jn sum- mer. Also the temperature of the refrigerating room will be higher than that of the circulating fluid say some 10 F. depending upon the rapidity of the circulation and the amount of surface of the pipes. For the manufacture of ice, the refrigerating room is kept at a temperature of 340 REFRIGERATION. 1172, 173, 174.] about 15 F., and that of the brine may be between F. and 5 F. 172. Volume of the compressor cylinder per n pounds of ammonia per stroke. Let V be the required volume in cubic feet and v the volume of a pound of ammonia gas at the lower tempera tu re, equation (3b'2), then V = n v. (386) 173. Volume of the compressor cylinder to produce a given refrigerating iffect. Let A 9 be the thermal units abstracted per pound of ammonia, q = nji^ the required number of thermal units to be ab- stracted which will also be a measure of the refrigerating power, V, the volume swept over by the piston or pistons con- sidered single-acting per revolution, -V, the number of revolutions per minute, v, the volume of a pound of ammonia gas at the lower temperature, then N V . (3 ' 7) 174. Duty. The duty of a refrigerating plant nuiv be referred to the number of thermal units required to UK h one pound of ice. The latent heat of fusion of ice at the pressure of one atmosphere is 144 thermal units (page 89), and if A, be the thermal units abstracted by the circulating fluid per pound, then will the duty be Ice-capacity, Ibs. = -A-. (388) J.TT [174.] DUTY. 341 If referred to one pound of coal fed to the furnace, then Pounds of] circulating fluid pei- Specific heat uuiu pci I 3ity _ _ hour _ per Ib. of "fuel, Ibs. I44"x pounds of fuel per Range of tempera- Lure If w' pounds of coal evaporates W pounds of steam per hour, then Pounds of steam per Ib. of fuel (390) w which may be used in equation (389). EXERCISES 1. How many thermal units will be removed from the refrigerating room per pound of ammonia, the highest average temperature of the liquid being 62 F., and that after expansion 1 F., the vapor being continually satu- rated, and the specific heat of the liquid 1.06 ? Omit 0.66 in the absolute temperature. FromEq. (360), ^ = 556.1. " " (384), (1.06X 2.3026%, g + ^)^-, * = 0.9808. \ 40" 0*-* / 000. 1 " " (385), 1.06 X 2.3026 -^- log !j|? x< = 0.1125. oob. 1 4o " " (382), 7< 3 = 455.16. That is, each pound of ammonia will carry away 455.06 thermal units. There must be taken into the compressor j-a = 93.08 per cent of vapor, and therefore 6.92 per cent of liquid. There must be x t = 11.25 per cent of liquid vaporized during the rise of temperature from 1 F. to 62 F. 2. In the preceding exercise if the ammonia cools a brine whose specific heat is 0.8, the lowest temperature of the brine being 8 F., the highest 15 F., how many thermal units will be removed by the brine per pound, and how many pounds of brine will be required to absorb the heat of one pound of the ammonia ? 342 REFRIGERATION. [174.] The heat imparted to the brine raises its temperature 15 8 = 7" F. ; hence each pound will absorb 0.8 X 7 = 5.6 thermal units. The ammonia imparts to the brine, equation (382), /, = (j-, - se t ) 556.1 = 454.06. Let z be the pounds of brine necessary to absorb this heat in raisini: its temperature one degree, then 0.8 X 7 z = 454.06 ; . . z = 81.17 pounds. When brine is used, larger pipes or greater velocity of the brine will be required than if ammonia only were used, and a lower ti'iuperature <>f the ammonia will be required. 3. What will be the efficiency in Exercise 1 i Equation (378) gives E _ 459_ 7 2 El ~ 63~~ That is, for every thermal unit of work dorw by tlie comprtwnr. 7.2S thermal unit* will be removed from the cold room. If the work done by the compressor be expressed in foot-pounds, and the heat removed be in thermal units, then E, = ~ = 0.0093. That is, for every foot-pound of work done by the compressor. 0.0093 of a thermal unit will be abstracted from the cold room. 4. How many pounds of water will be required by the condenser per pound of ammonia if condensed to a liquid at 63 F M the temperature of the water being increased from 50 F. to 64 F. ? The heat surrendered by the ammonia will be7'j v + p' v' - p" v" A " n The last reduction may be effected by finding from (368) and (366) p' v' p" v" = (l ("' T") b ( ~r n ~"~ If the vapor be saturated at state/", and/ B be the adia- batic of superheated vapor, the value of the workfJ? A I) will be found by substituting v,, t p v respectively, for v", r", p", giving If the vapor be superheated at state C, its volume would be given by equation (364), p and r being given. If it be saturated, *y, will be given by equations (362) and (350) when either p or r are given, this -c being r 2 in (369), (370) and (371), when the initial state of the adiabatic of the superheated vapor is on the curve of saturation. Since C I is an adiabatic, equation (369) will give the volume v' at state /, and (371) the pressure, for the temper- 346 REFRIGERATION. [175.] ature T', This pressure is assumed to be constant during condensation ; during the first part, from / to B, the con- densation is produced by a reduction of the temperature and volume, until the temperature is that of saturation, r,, un- der the pressure /> while from B to A the temperature and pressure are both constant, and the reduction of volume is effected by the condensation of vapor to a liquid. Assuming that the specific heat of the vapor is constant at constant pressure, then, for complete liquefaction, the heat abstracted in the operation / A will be A, = ^(r'-r,) + A ei . (396) Assume that heat is abstracted from the liquid at con.-r;mr volume from state A, reducing the pressure from p^ to p n and temperature from r t to r t ; the path of the fluid will be A D, and the heat so abstracted will be H D A G. If this heat be abstracted from the circulating fluid in the cold room, then will the room absorb that amount of heat, and in the evaporation and expansion afterward an equal amount of heat must be supplied from the cold room at the lower tem- perature before any useful amount can be absorbed by the circulating fluid ; and if H D j g be the amount so absorbed we have II D A G = II Djg. The heat emitted will be HD A G = c(r t - r f ). Let II D j g be the latent heat of evaporation in the oj.th part of a pound of vapor at the temperature r, ; then *. * = c (T, - r t ). (397) (This value of x is the same as x 4 in (380), when u" = ; but exceeds * 4 in Eq. (385).) [176.] THE EFFICIENCY. 347 The refrigeration per pound per revolution will be A, = g j C K, = A M ~ c(r,- T,) + c' p (r" - r& (398) = * + ( r 2 - r.) - c (r, - r.) + e p (r" - T,), = h _ c > (r, - r.) + c p (r" - r,), (399) in which r is the absolute temperature of melting ice, and h the total heat of vapor above r , Article 85, Eq. (93). Then equations (396) and (397), A, - A, - A ei + p (r- - r 2 ) + C (r, - r 2 ) A IK - A., - P (r" - r s ) h,\ (400) which is the equivalent of equation (391) or of (395). 176. The efficiency will be EXERCISE. If the inferior absolute pressure of the ammonia be 29' pounds per square inch, temperature when it leaves the cooling coils and enters the compressor, 36 F., then com- pressed adiabatically to a temperature of 117 F., then con- densed to saturation and to a liquid at the constant pressure corresponding to the 117 F., then admitted to the cooling coils and the temperature reduced to that corresponding to the initial absolute pressure, 29 pounds, then evaporated and heated to the initial state ; find the thermal units of refriger- ation and of condensation, the specific heat of the liquid being 1.08 ; also the efficiency. 348 REFRIGERATION. Retain results only to the nearest tenth. Inferior temperature, Eq. (350) r a = 458.7. Hence, T* = 1.9. Absolute temperature at state C. 460.6 + 36 r" = 496.6. Superheating r" r 2 = 37.9. Greatest volume of a pound, Eq. (357) for p = 4176, v" = 9.8. At /, absolute temperature, 117 + 460.6 r' = 577.6. " volume, Eq. (369) v' = 2.4. " pressure, Eq. (370) p' = 118. From B to A, absolute temperature, Eq. (350). ... r, = 527.8. Hence, T, = 67.2. Fall of temperature from A to D Ti T* 69.1. Heat absorbed during this fall of temperature 1.08 X 69.1 = 74.6. Latent heat of evaporation at 1. 9 P.. Eq. (360). fl rt = 557. " " " " " 66.2, " " h,t = 515. Heat rejected during condensation, Eq. (396) hi = 540.8. Refrigeration per lb., Eq. (398) Jit = 500.7. Work done by the compressor, per lb., Eq. (400). U = 31198. Efficiency, Eq. (401) E, = 12.5. 177. Experimental Results. The following data and results are taken from the report of a test of a I)e La Yergne refrigerating plant by Messrs. R. M. Anderson and C. H. Page, Jr., having a nominal ice-melting capacity of about 110 tons in 24 hours. It was in its every -day work- ing condition and was run at about two-thirds its ordinary capacity. Only the ice plant was involved in the experiment. It consisted of two single-acting vertical compressor cylinders, driven by a horizontal double-acting Corliss engine, as shown in Fig. 80, a feed pump and a condenser. The test was during 11 hours and 30 minutes. Two tests were made of the boilers, as the first indicated such high efficiency it was considered advisable to check the results. It will be seen that the second test also gave a high efficiency. The second test was during 12 hours. All the instruments used in the test were carefully standardized. (Thesis 1887.) FUEL, FURNACE AND BOILERS. There were two double return flue boilers arranged to run, automati- cally, between 60 Ibs. and 70 Ibs. pressure (gauge). [176.] EXPERIMENTAL RESULTS. 349 Fuel. Lebigh nut (anthracite), heat of combustion, B. T. U.. . . 12229.6. Coal for 12 hours, Ibs 4422. Wood for starting 377 Ibs. 0.4 coal equivalent, Ibs 150.8. Coal, total equivalent, Ibs 4572.8. Unburnt coal, Ibs 125.0. Coal consumed, Ibs 4447.8. Combustible, total, Ibs 3577.8. Coal burned per hour, Ibs 360.65. Heat in 360.65 Ibs. of coal, 360.65 X 12229.6 B. T. U 4532901. Coal per H. P. per h. (2d test, H. P. was 102.92) 3.6013. Furnace, grate area, sq. ft 39. Ratio of heating surface to grate surface 35.63. Boilers, heating surface, sq. ft 1389. Water fed per hour, Ibs 3559.7.. Evaporation per Ib. coal fired, Ibs 9.341. " " " " from and at 212 9.957. " " " consumed 9.601. " " " " from and at 212 10.226. " " combustible, from and at 212 12.703. Average gauge pressure, Ibs 66.0. Temperature of fire-room, deg. F 81. Average temperature of flue boiler, deg. F 319.28. "feed-water " "... 180.2. Total heat in 3559.7 Ibs. steam above 180.2 F., B. T. U. . . 3661463. Efficiency of furnace and boiler, per cent, 8< y^ 300 = 80.7. ENGINE (CORLISS). Piston, diameter of, inches 32. " stroke " " 36. " speed per minute, mean, ft 194.988. Revolutions per minute, average 31. 720. Average indicated horse-power 91.13. Water consumed per IHF. per hour, Ibs 34. 104. Steam per IHP. per hour, Ibs 25.79. Steam condensed in the engine, per cent 24.35. Average ratio of expansion 5.807. Steam consumption by the feed-pump per hour, Ibs 50. 25. " " engine per hour, Ibs 3509.45. Total heat in 3509.45 Ibs. steam above 180. 2 F., B. T. U. . . - 3609479. Heat changed to work per h., B. T. U., 102.92 X 1980000 * 778 = 261930. Efficiency of fluid 8609479 = ' 07ai - 350 REFRIGERATION. [176.] Coal per I.H.P. per hour for engine and pump 3.60. Combustible for I.H.P. per hour, Ibs. 2.91. COMPRESSOR. Number of cylinders, single-acting 2. Length of stroke, inches. 36. Diameter of pistons, each, inches 18. Area head end of pistons, each, sq. in 254.47. Average number of revolutions per m 31.720. Piston displacement per stroke, cu. ft 5.301. " hour, both, cu. ft 20179. Volume of sealing oil per hour, cu. ft 143.8. Volume filled with gas per hour, both, cu. ft 20036. Indicated horse-power, mean 76.0892. Heat eq. of work by compressor per hour, B. T. U 193645. Efficiency of compressor from coal = 0.0407. " " mechanism 0.755. Temperature of ammonia entering compressor, Deg. F 57. 7. leaving " 116.1. Absolute pressure entering compressor, Ibs 28.88. leaving " " 132.01. CfornyoressoT*' J3 35.014Q FIG. 88. Fig. 88 is an exact copy of one of the indicator cards taken from one of the compressor cylinders. The lines nearly vertical at the upper part of the diagram are due to the oscillations of the indicator spring. [177.] EXPERIMENTAL RESULTS. 351 REFRIGERATION. Pressure in cooling coil, Ibs. absolute 28.88. Hence, temp, of liquid, Deg. F 2.0. Temp, compressed gas, Deg. F 116.1. " of gas entering the compressor, Deg. F 57.7. Rise of temperature due to compression, Deg. F 58.4. Latent heat of evaporation of 1 Ib. at 2.0 F. Eq. (360) 556.7. Superheating in cooling coils, 37.9 F. , B. T. U 19.27. Fall of temp, of liquid in cold room, Deg. F. 69.4. Heat imparted to cold room by this fall, 1.08 X 69.1 74.95. Heat removed from cold room per Ib., 556.7 + 19.27 - 74.95 = 501.02. Ammonia evaporated per hour, Ibs 1669.5. " " B. T. U 851311. " Ice-melting capacity" per hour, Ibs 5995. for 24 hours, tons (each 2000 Ibs.). . . 71 .95. per IHP. per hour, Ibs 65.79. " Ib. of coal (3.6013 per HP.), Ibs.. . 18.26. " 10 Ibs. steam (182.6 -- 9.601) Ibs. . 19.02. " Ib. combustible (65.79 * 2.91), Ibs. 22.61. " Ib. ammonia evap., Ibs 3.59. EFFICIENCIES. Efficiency of furnace and boiler (see above) 0.807. " " steam utilized by engine (see above) 0.0721. " " engine referred to coal 0.0582. " " compressor referred to engine (see above). 0.755. " " " " " coal (775 X 0.0582) 0.0439. QPjl O-J -J " " refrigeration referred to compressor 193645 = 4>39 - " IHP. of engine 3.31. " boiler.... 3.67 X 0.0643= 0.238. "coal 0.236X0.807= 0.192. that is, for every thermal unit in the coal there was abstracted 0.19 of a thermal unit from the cold room. In actual ice-making, only about 30 to 45 per cent of the pounds of ice-melting capacity can be produced as hard ice suitable for commercial purposes, giving, in this case, be- tween 5.0 and 7.5 pounds of hard ice. The experiments of Professor M. Schroter gave from 19.1 pounds to 37. 4 pounds of ice (net) per hour per indicated '352 REFRIGERATION. [178.] horse-power of the engine. (One result is given as 48.8 pound?, but the test was of too short duration to be reliable.) If 4 pounds of coal were used per H. P. per hour, then there would be produced about 4.8 to 9.3 pounds, net, per pound of coal consumed. Ledoux remarks that manufacturers estimate about 50 pounds per horse-power per hour measured on the driving- shaft ; hence, if the delivered power be 0.80 of the indi- cated, this would be equivalent to about 45 pounds of ice per indicated horse-power. M. Schroter's tests, and the following, by Mr. Shreve, show that this is too high, if commercial ice is intended. The amount of ice made depends upon many conditions : as, clearances in the cylinders, friction of mechanism, speed of engine, losses along the pipes, losses in opening the valves,, leakage, losses by unavoidable radiation, losses at cans by water unfrozen and ice cleavages ; and, these being con sidered, it seems advisable, in designing, to assume less than one-third the pounds of ice-melting capacity for the probable pounds of commercial ice to be produced. 178. Test of an ice-making plant. An ice- making plant of the Cincinnati Ice Manufacturing & Cold Storage Co. was tested in 1888, by Messrs. A. L. Shreve and L. W. Anderson, chiefly to determine the amount of solid ice which could be manufactured in 24 hours with the plant running under normal conditions. The plant con- sisted of two 25-ton (nominal) and one 50-ton ice-machines, boilers, pumps, etc., used in actual ice-making. While the machinery was doing its regular work, at a certain hour, the steam pressure was observed to be 75 pounds (gauge), and all the conditions of the furnace, engines, and plant generally were observed, and the conditions continued as nearly uniform as possible for 24 hours, during which time 108.87 tons (of 2000 Ibs.) of ice were drawn, from which [179.] THE ABSORPTION SYSTEM. 353 should be deducted nearly a ton on account of the terminal condition being slightly different from the initial. The actual ice weighed out at 75 pounds' steam pressure, then, may be taken as 216,000 pounds. The engines were not tested at the same time, so that the ice per horse-power was not definitely known ; but an inde- pendent test of the boilers and engines was made prior to the " capacity " test given above, from which it appears that at 75 pounds' pressure, the three engines developed about 415 indicated horse-power ; according to which 21 pounds of solid ice were weighed per indicated horse-power per hour. The actual amount may have been several per cent more or less. The machines were new and bearings large, and the engine and compressor absorbed an average of about 52 per cent of the indicated horse-power of the engine. ABSORPTION SYSTEM. 179. The absorption system depends upon the fact that water will absorb many times its volume -of am- monia gas ; at 59 F. it will absorb 727 times its volume. This is a chemical action, and therefore generates heat. Ac- cording to Favre and Silberman, 925.7 B. T. II. will be developed for each pound of gas absorbed. This action is substituted for the compressor in raising the temperature of the ammonia after leaving the cold room. According to Oarius, the coefficient of solubility of ammonia gas in water is represented by the empirical formula (t being Deg. C.) ft = 1049.62 - 29.4963 t + 0.676873 f - 0.0095621 t* ; according to which the solubility diminishes as the temper- ature increases and soon reaches a condition at which it ceases to act ; and to insure continuous working the absorp- tion chamber is cooled by water externally. The process is illustrated in Fig. 89. A solution of aqua- ammonia being in the generator A and heated by means of steam passing through coils in this chamber, the vapor of 354 REFRIGERATION [179.] ammonia is generated and rises, passing through tortnous ways in the analyzer C ; thence to the condenser D. The steam which rises in the 'analyzer C is partly condensed as it approaches the upper end of the vessel and falls back to the generator, and that which passes into the coils over D is led back by the ammonia drip, so that nearly pure ammonia gas enters the condenser T). Here the ammonia is at its highest pressure and temperature, and its state may be represented by /?, Fig. 00, or by the upper right-hand corner of the indicator diagram of Fig. 88. The ammonia B gas passes through coils of pipes ~\ in the condenser, about which | \c circulates water ; the ammonia being condensed to a liquid under a constant pressure, the path of the fluid being represented by B A, Fig. 90. The liquid passes to the lower part of the coils, or to a receiver especially provided, and thence through a cock, by which the reduction of pressure is regulated as {180.] TEST OF ABSORPTION PLANT. 355 it passes into the cooler /?. The reduction of pressure is represented by A D, and of evaporation during the expan- sion by D J. The liquid and a small amount of saturated vapor is now in the cooler H, where the liquid vaporizes, absorbing heat, the operation being represented by J (7, Fig. 90, or the corresponding line in Fig. 88. The brine, which circulates in the cold room, is cooled by the cold am inonia as it passes through the cooler. During the process of cooling the brine, the liquid am- monia becomes a gas. The gas passes from the cooler II to the absorber K, the pressure in K being kept a little lower than in H by the ammonia pump, shown at the right, which draws its supply from K and forces it into 'the generator A. Into the absorber A", water is forced and absorbs a large volume of the gas, as stated above, generating heat, the oper- ation for which being represented by C B, Fig. 90, or the compression line in Fig. 88. In this system pure water is not used in the absorber, but instead thereof, water is drawn from the lower part of the generator A by the pipe Z, con- t lining but little ammonia, the mixture being called "weak ammoniacal liquor/' By the absorption of the gas, strong aqua-ammonia is formed, which is pumped back into the generator, and the operation repeated. It will be seen that the operation completes a cycle, and that the changes in the states of the ammonia are similar to those in which a compressor is used ; hence, if there were no losses of heat, except those described, the efficiency would be the same. The vessel K is kept sufficiently cool to facilitate the chemical action by means of water flowing over it. ISO. Test of absorption plant. Professor J. E. Dentoii made a seven days' continuous test of an absorp- tion plant with the following results.* Every element en- * Tram. Am. Soc. Mech. Eug.,Vol. X ., May, 1889. 356 REFRIGERATION. [180. } tering into the problem was, as far as practicable, directly measured. Average pressures, above atmosphere, generator, Ibs. per. sq. in. 150.77. " " " steam, " " " 47.70. absorber, " " " 23.4. Average temperatures, Deg. F. , Generator 272. " " Condenser, inlet 54J. outlet. 80. " " " range 25$. " Brine, inlet 21.20. " outlet 16.14. range 506. " Absorber, inlet 80. " outlet 111. range 31. Heater, upper outlet to generator 212. " lower " absorber. 178. " inlet from absorber 132. " Inlet from generator 272. Water returned to main boilers.. 260. Steam per hour for boiler and ammonia pump, Ibs 1986. Brine circulated per hour, cu. ft 1633.7. " " pounds 119260. " Specific heat 0.800. " Heat eliminated per pound, B. T. U 4.104. " Cooling capacity per 24 hours, tons of melting ice 40.67. per pound of steam, B T. U 243. " Ice-melting capacity per 10 Ibs. of steam, Ibs. 17.1. Calorics, refrigerating effect per kilo, of steam consumed 135. Heat rejected at condenser per hour, B. T. U 918000. "absorber " " " 1116000. " consumed by generator per Ib. of steam condensed, B. T. U. 932. Condensing water per hour, Ibs 36000. Condensing coil, approx. sq. ft. of surface 870. Absorber ' " " " " " 350. Steam " " " " " " 200. Pump, Ammonia, dia. steam cyl. , in 9. " " " ammonia cyl., in 3fc. [180.] TEST OF ABSORPTION PLANT. 357 Pump, Ammonia, stroke, in .. 10. " revolutions per minute 22. " Brine, steam cyl., diam., in i. brine cyl., " b'. " stroke, in 10 revolutions per m 70. Effective stroke of puuips 0.8 of full stroke. 18Oa. Sulphur Dioxide, (or Sulphurous Acid). The following relations have been found for this acid : Specific heat of the gas, 0.15483. " " " liquid, 0.4. Eelation between the pressure and temperature of the saturated vapor, 1439.0 235629 log p = 5.2330 5 (a) Equation of the gas, or superheated vapor, pv =. 23.87 r 2 5 J; 9 - (b) Latent heat of vaporization, A e = 171.26 0.25605 T 0.0013795 T\ (c) Volume of a pound of the saturated vapor, 778 ~~ 2.3026 Volume of a pound of the liquid, 0.016 1.484 0.0015659 T (Trans. Soc. Mech. Eng., 1890.) CHAPTER VI. COMBUSTION 181. Essential principle. Combustion, chemi- cally speaking, is the combination of chemical elements producing heat. Burning, popularly speaking, is the ro suit of a rapid combination of oxygen with other ele- ments. Carbon and hydrogen are the chief elements of the fuel used for engineering purposes. Sulphur, another ele- ment, is frequently present, but is comparatively of little value. When two substances unite chemically, forming a sub- stance different from either, it is said that a. chemical ofni'tij exists between them. The difference between n mechanical mixture and a chemical combination may be illustrated in the case of gunpowder. The process of manufacture makes a mechanical mixture of charcoal, sulphur, and nitre, but if the gunpowder be fired a chemical combination results and a large volume of gas is produced, generating a large amount of heat and developing a strong elastic force ; and the original substance entirely disappears and new sub- stances composed of different combinations of the original elements are formed. Definite proportions. In every chemical compound a definite and unvarying proportion of its elements exists among themselves. For instance, in water there is always 8 times as much oxygen by weight as there is of hydrogen, so that in l"i> [181.] ESSENTIAL PRINCIPLE. 359 pounds of water there is 88.8 pounds of oxygen and 11.1 pounds of hydrogen. Any chemical compound of oxy- gen and hydrogen in other proportions would be a sub- stance entirely different from water ; or, if in the 100 parts there were some other element, as carbon, the substance would also be different from water. The chemical equivalent or atomic weight is expressed by a definite number, and the chemical principle of definite proportions may be expressed in the form of the two follow- ing laws : 1. The proportions by weight in which substances com- bine chemically can all be expressed by their chemical equivalents, or by simple multiples of their chemical equiva- lents. 2. The chemical equivalent of a compound is the sum of the chemical equivalents of its constituents. Perfect gases at a given pressure and temperature com- bine in proportion to their volume. Neglecting fractions the following are the chemical equivalents for the principal elementary constituents with which we have to deal in fuel and air : TABLE I. Chemical e juivalent. By weight. By volume. o 16 1 N 14 1 H 1 1 c 12 ? s 32 The composition of a compound substance is indicated by writing the symbol of the elements one after the other, and 360 COMBUSTION. [182.] affixing to each symbol, in the form of a subscript tue num- ber of its equivalents which enter into one equivalent of the compound. Thus, water contains two chemical equivalents of hydrogen to one of oxygen, -and is indicated by the ex- pression H a O ; and the constituents by weight will be 2 II -f- 16 O. Similarly, C O s , carbonic acid, contains one equiva- lent of carbon and two of oxygen, and by weight 12 C -f- 32 O. The following table gives the composition of several sub- stances : TABLE II. Name. iU Proportions of elenientH by weight. ft 1 I Air H,0 NH, CO C O-, C,H 4 CH 4 so SH, 8,C N77H H2- H3- C 12- C 12- C 12- C 12- S 32- S 32- S 64- -O23 - O 16 -X 14 - O 16 - O32 -H2 -114 >- O32 -C 12 100 18 17 28 44 14 16 64 84 76 N 79 O 21 H 2 -(- H3 + N C + C + O2 C + H2 C+H4 100 2 2 2 2 2 2 2 2 2 Water Ammonia Carbonic oxide. . . .... Carbonic acid Oletiant gas Marsh gas or fire-damp. Sulphurous acid Sulphuretted hydrogen. Bisulphide of carbon. Air is not a chemical compound, but a mechanical mixture of nitrogen and oxygen. 182. The heat of combustion of one pound of a substance combining with sufficient oxygen to completely burn it has been found by experiment. The usual process is to surround a small furnace with a quantity of water so arranged as to prevent the escape of heat ; the increased [182.] THE HEAT OF COMBUSTION. 361 temperature produced in the water by the burning of each pound of the fuel in the furnace being a measure of the " heat of combustion." The results of the experiments of Favre and Silberman are given in the following table : TABLE III SHOWING THE TOTAL QUANTITIES OF HEAT EVOLVED BY THE COM- PLETE COMBUSTION OF ONE POUND OF COMBUSTIBLE WITH OXYGEN ; ADAPTED FROM THE RESULTS OBTAINED BY FAVKE AND SlLBETt- MAN. THE UNIT OF WEIGHT IN THIS TABLE BEING ONE POUND, AND THE UNIT OF TEMPERATURE ONE DEGREE FAHR. (FROM 39 TO 40). Substance. Formula. Product. Unite of heat. GAbES. H HaO 62 032 Carbonic oxide Marsh gas CO C H 4 COa C O 2 & H s O 4,325 23 513 Oleflant "-as C 2 H 4 C Oa & Ha O 21 343 LIQUIDS. Oil of turpentine ClO H,6 COa&Ha 19 533 C 2 H 6 O C Oa & Ha O 12 931 Spermaceti (solid) C,, H 64 Oo C Oa & Ha O 18 616 C Sa 0' Oa & S O' 6 122 SOLIDS. Carbon (wood charcoal) C (CO (C O 2 4,451 14,544 14 485 13 972 Native Graphite 14 035 s S Oa 4048 Phosphorus (observed by Andrews.). . p Pa0 5 10,715 The heat units in a pound of fuel will be nearly the sum of the heat units of combustion of its constituents. Take, for example, oleh'ant gas : According to Table II. the chemical equivalents by weight are 14, of which 12 are carbon 362 COMBUSTION. [182.] and 2, hydrogen ; or, j of the compound is carbon and \ hydrogen. Then, from Table III., we find \ Ib. H gives } of 62032 = 8862 B. T. U. f Ib. carbon gives f of 14554 = 12475 " . " " ' Total ........................ = 21337" " " which is only 6 thermal units less than the value given in the table, as deduced from experiment. If the principal constituents are carbon, hydrogen, and oxygen, it is found that the total heat of combustion in B. T. U. will be given nearly by the following formula : A = 14500 (C + 4.28 (// - i 0) ), in which 4.28 = , reduces the hydrogen to an equiva- 14500 lent of carbon, and is deducted from the hydrogen, for it is assumed that the oxygen present unites with the hydro- gen, forming water. Such substances are called hydro- carbons. The total heat of combustion is usually computed from its chemical analysis, as shown on page 261. The following table gives the total heat of combustion of certain fuels (see Journal of United Semice Institution, Eng., Vol. XL, 1867 ; Box On Heat, p. 60). The speci- mens were of the best quality, and are too high for ordinary practice. Commercial coal of similar grade would be about 0.7 to 0.8 of these values. Commercial Lehigh (anthracite), analyzed at the Institute, gave 12229 B. T. U. [183. J THE INCOMBUSTIBLE MATTER. 363 TABLE IY. TOTAL HEAT OF COMBUSTION OF FUEL. FUEL. Carbon. 1 I Equivalent to pure carbon. II 1 E 14 12 14 13.2 123 15.75 15.9 15.4 153 14.25 16.0 15.15 15.6 13.65 12.15 10.0 7.25 7.5 5.8 22.7 22.5 Total heat of combustion. I. CHARCOAL from wood ' from peat. . c. 0.93 0.94 0.88 0.82 0.915 0.90 0.87 0.80 0.77 0.88 0.81 0.84 0.77 0.70 0.58 0.50 0.84 0.85 H. 0.035 0.04 0.04 0.054 0.05 0.052 0.052 0.056 0.052 0.05 006 0.16 0.15 0. 0.026 0.02 0.03 0.016 0.06 0.054 0,04 0.08 0.15 0.20 0.31 C'. 0.93 0.80 0.94 0.88 0.82 105 1.06 1.025 1 02 0.95 1.075 1.01 1.04 0.91 0.81 0.66 0.50 1.52 1.49 h. 13500 11600 13620 12760 11890 15225 15310 14860 14790 18775 15837 14645 15080 13195 11745 9660 7000 7245 5600 21930 21735 II COKE good " middling . . bad ... III. COAL- 1 Anthracite 2 Dry bituminous . . 3 " < 4 " " 5 " " 6. Caking 8 Cannel 10. Lignite IV. PEAT dry " containing 25$ moisture .. V WOOD Dry " containing 20$ moisture. . VI. MINERAL OIL from to 183. The incombustible matter is called ash. The principal ingredients of ash are shown in the following analysis, which is from the geological survey of Ohio : Bituminous coal. Percentage of ash, 5.15. Silica 58.75 Alumina 35.30 Sesquioxide of iron 2.09 Lime 1.20 364 COMBUSTION. [184.] Magnesia 0.68 Potash ami sochv 1 .08 Phosphoric acid 0. 13 Sulphuric acid 0,24 Sulphur, combined 0.41 The proportions vary greatly with different fuels. 99.88 184. Air required for combustion. Consider pure carbon. The chemical equivalent of oxygen is, ac- cording to Table I., |f of that of carbon. If the carbon be completely burned, C O t is formed, so that the proportion by weight will be ff of oxygen to 1 of carbon. According to Table II., 0.23 of the air by weight is oxygen ; hence Weight of air jter U>. of carbon = ?$ -f- 0.23 = 12 Ibs ., nearly. If the compound contains carbon, hydrogen and oxygen, we will have, nearly, Weight of ,,}.<>,> //>. fuel = A = 12 C + 30 (// t i O). The following table, computed from this formula, is given by Kankine. TABLE V. I. CHARCOAL from wood ' ' from peat 0.93 80 11.16 9 6 II. COKE good 0.94 1 1 28 III. COAL anthracite .- 915 035 0026 12.13 dry bituminous 087 0.05 0.04 12 06 caking 0.85 05 0.06 11.73 75 005 0.05 1058 cannel 84 06 08 11.88 dry long flaming 77 05 0.15 10. ?2 lignite , 70 005 0.20 9.30 IV. PEAT dry 58 006 0.31 7.68 V. WOOD dry . 50 6 00 VI. MINER \L OIL 85 015 o 15 65 [185, 186.] TEMPERATURE OF FIRE. 365 Besides the air necessary to furnish the oxygen, some is required for dilution, so as to secure a more free access of air to the fuel. The above table shows that about 12 pounds of air is required per pound of fuel, and experiment indi- cates that from 1 to 2 times this amount is required in the furnace for combustion and dilution. An excess of air causes a waste of heat by transporting an unnecessary amount of heat up the chimney, and a deficiency causes imperfect com- bustion. Forced draft requires less air than a natural draft. 185. Forced draft. Most American sea-going steam- ers have boilers designed to burn anthracite coal with natural draft, and 5 to 5| pounds of coal is burned per hour per square foot of grate. Torpedo-boats, with bituminous coal and forced draft of 6 inches of water, may burn 96 pounds per square foot of grate per hour. In a furnace in which 19 pounds of anthracite coal were burned, a forced draft by means of a screw revolving in the chimney was introduced, causing a burning of 38^ pounds of coal, and a production of 80 per cent more steam than in the former case. In 18-17 Robert L. Stevens introduced the plan of air-tight fire-rooms, by which means a forced draft was produced by forcing air into the room occupied by the firemen. 186. Temperature of fire. If the volume be con- stant, as in case of an explosion in a closed vessel, then will the rise of temperature be given by equation (37), page 55 ; but if the pressure be constant, as in the ordinary furnace, then will it be given by equation (38). Take the case of pure carbon burned with 2-1 pounds of air. The total heat of combustion will be, Table III., 14544 thermal units. This heat is expended in heating 25 pounds of matter, of which 24 pounds is air, whose specific heat is 0.238, and one pound of carbonic acid gas, whose specific heat is 0.217. Call the specific heat of the mixture 0.237 ; then 366 COMBUSTION. [187.] The only measurement with a pyrometer which has come to my notice gives a much lower temperature than is found by this formula. 187. Height of chimney. The height of the chimney to produce a natural draft must be such that the difference between the weight of a column of the hot gases, having one square foot for its base and height equal to the height of the chimney, and that of a column of equal height of external air, shall produce the required velocity of air in the chimney. Let w be the weight of fuel burned in the furnace per second, F , the volume of the air at 32 supplied per pound of fuel burned, T O , the absolute temperature at 32, A, the area of the cross-section of the chimney, m, A -r- perimeter of chimney, T,, the absolute temperature of the gases discharged from the chimney, w, the weight of a cubic foot of the hot gases, I, the length from the furnace to the top of chimney, u, the velocity of the current in the chimney per second ; then, if 24 pounds of air be supplied per pound of fuel, F = 25 X 12 = 300 cu. ft. u A w n V, ^ = volume of gases per second ; 4*, w = Is (0.0807 [187. J HEIGHT OF CHIMNEY. 367 Accordiiig to Peclet, the head to produce the velocity u for 20 Ibs. coal burned per sq. ft. of grate per hour will be Having A, the height of the chimney may be found. Let H be the height of chimney, r s , the absolute temperature of the external air ; then Weight of the column of air = 0.0807 . Is H. " gases = f 0.0807 + -y j -*-H. fi "a column of gases of length h = (0.0807 +^)1-*; . . (o.0807 + -L) - (H + h) = 0.0807 1 H \ 300/ Tj T y 0.96 ~ l -1 The weight discharged per second will be wu = ^ [0.0807 + = constant |/0.96 L - This will be a maximum for that is, the lest chimney draft takes place when the absolute temperature of the gas in the chimney is to that of the ex- ternal air as 25 to 12. 368 COMBUSTION. [187.; This in the preceding equation gives h = //. In the solution for a maximum, I is considered constant, an assumption which affects the result by only a small amount. The height of a chimney is often determined by sur- rounding circumstances, and sometimes by imagined con- ditions of future use ; and in such cases are not subject to computation. In ordinary practice chimneys are from forty to one hun- dred and twenty feet. Above one hundred feet the effect of additional height is comparatively small. The tallest chimney of which we have knowledge is 441.6 feet high, eleven feet and a half in diameter at the base, and ten feet at the top. It was built by the Mechernich Lead Mining Co. (Van Nostrand's Eng. Mag., 1886, page 264). For dimensions of large chimneys, see Van Xostrand's Eng. Mag., September, 1883, page 216 ; also Trans. Am. Soc. Civ. Engineers, 1885 ; also No. 1, Science Series, by D. Van Nostrand. APPENDIX I. (Extract from an article by the author in the Philosophical Magazine for November, 1885.*) THE LUMINIFEROUS ETHER. Two properties of the luminiferous ether appear to be known and measurable with a high degree of accuracy. One is its ability to transmit light at the rate of 186300 miles per second, f and the other its ability to transmit from the sun to the earth a definite amount of heat energy. In regard to the latter, Herschel found, from a series of experiments, that the direct heat of the sun, received on a body at the earth capable of absorbing and retaining it, is competent to melt an inch in thickness of ice every two hours and thirteen minutes. This is equivalent to nearly 71 foot-pounds of energy per second. In 1838 M. Pouillet found that the heat energy transmitted from the sun to the earth would, if none were absorbed by our atmosphere, raise 1.76 grammes of water 1 C. in one minute on each square centimeter of the earth normally exposed to the rays of the sun. { This is equivalent to 83.5 foot-pounds of energy per second, and is the value used by Sir William Thomson in determining the probable density of the ether. Later determinations of the value of the solar constant by MM. Sorret, Crova, and Violle have made it as high as 2.2 to 2.5 calories. But the most recent, as well as the most reliable, determination is by Pro- fessor S. P. Langley, who brought to his service the most refined ap- paratus yet used for this purpose, and secured his data under favorable conditions ; from which the value is found to be 2.8 calories | with some uncertainty still remaining in regard to the first figure of the deci- * Published in Van Nostrand's Engineering Magazine, January, 1886. Also Snenct Series, No 85. f Professor Michelson found the velocity of light to be 299740 kilometers per sec- ond in air, and 299828 kilometers in a vacuum, giving an index of refraction of 1 000265. (Journal of Arts and Science, 1879, Vol. XVIii., p. 390.) * Camples Rendus, 1838, Tom. VII., pp. 24-26. Trans. Roy. Soc. of Edinburgh, Vol. XXI., Part I. I Am. Journ. of Arts and Science, March, 1883, p. 195. Also Comntes Eendus. 370 APPENDIX I. mal. We will consider it as exactly 2.8 in this analysis, according to which, there being 7000 grains in a pound and 15.432 grains in a gramme, we have for the equivalent energy 2.8 X 15.432 9 772 X 144 7000 X 5 X (U551T60 per second for each square foot of surface normally exposed to the sun's rays, which value we will use. Beyond these facts, no progress can be made without an assumption. Computations have been made of the density, and also of the elasticity, of the ether founded on the most arbi- trary, and in some cases the most extravagant, hypotheses. Thus, Her- Bchel estimated the stress (elasticity) to exceed 17 X 10 9 = (17,000,000,000) pounds per square inch ; * and this high authority has doubtless caused it to be widely accepted as approximately correct. But his analysis was founded upon the assump- tion that the density of the ether was the same as that of air at sea-level, which is not only arbitrary, but so contrary to what we should ex]xrt from its non-resisting qualities as to leave his conclusion of no value. That author also erred in assuming that the tensions of gases were as the wave-velocities in each, instead of the mean square of the velocity of the molecules of a self-agitated gas ; but this is unimportant, as it happens to be a matter of quality rather than of quantity. Herschel adds. " Consid- ered according to any hypothesis, it is impossible to escape the conclusion that the ether is under great stress." We hope to show that this con- clusion is not warranted ; that a great stress necessitates a great density ; but that both may be exceedingly small. A great density of the ether not only presents great physical difficulties, but, as we hope to show, is inconsistent with the uniform elasticity and density of the ether which it is believed to possess ; and every consideration would lead one to ac- cept the lowest density consistent with those qualities which would enable it to perform functions producing known results. In a work on the Physics of Ether, by S. Tolver Preston, it is esti mated that the probable inferior limit of the tension of the ether is 500 tons per square inch, a very small value compared with that of Herschel's. But the hypothesis upon which this author founded his analysis was The tension of the ether exceeds the force necessary to separate the atoms of oxygen and hydrogen in a molecule of water ; as if the atoms were forced together by the pressure of the ether, as two Magdeburg hemispheres are forced together by the external air when there is a vacuum between them. This assumption is also gratuitous, and is re- jected for want of a rational foundation. Young remarks : " The luminiferous ether pervading all space is not * Familiar Lecture*, p. 282. THE LUMINIFEROUS ETHER. 871 only highly clastic, but absolutely solid." * We are not certain in what sense this author considered it as solid ; but if it be in the sense that the particles retain their relative positions, and do not perform excursions as they do in liquids, jt is a mere hypothesis, which may or may not have a real existence. If it be in the sense that the particles suffer less resistance to a transverse than to a longitudinal movement, there are some grounds for the statement, as shown in circularly-polarized light. Bars of solids are more easily twisted than elongated, and, generally, the shearing re- sistance is less than for a direct stress. It certainly cannot be claimed that the compressibility of the ether (in case we could capture a quantity of it) is less than that of solids. Sir William Thomson made a more plausible hypothesis, by assuming that " the maximum displacement of the molecules of the ether in the transmission of heat energy was ^ of a wave length of light, the average of which maybe taken as ^^y of an inch." Hence the displacement was assumed to be ^5^5^05 of an inch ; by means of which he found the weight of a cubic foot to be X 10 ~ 20 of a pound, f We also notice that Hr. Belli estimated the density of the ether to be X 10~ 1S of a pound ; \ but M. Herwitz, assuming this value to be too small and Thomson's as too large, arbitrarily assumed it as 10 ~ 18 of a pound per cubic foot ; but arbitrary values are of small account unless checked by actual results. We propose to treat the ether as if it conformed to the Kinetic Theory of Gases, and determine its several properties on the conditions that it shall transmit a wave with the velocity of 186300 miles per second, and also transmit 133 foot-pounds of energy per second per square foot. This is equivalent to considering it as gaseous in its nature, and at once com- pels us to consider it as molecular ; and, indeed, it is difficult to conceive of a medium transmitting light and energy without being molecular. The Electromagnetic Theory of Light suggested by Maxwell, as well as the views of Newton, Thomson, Herschel, Preston, and others, are all in keeping with the molecular hypothesis. If the properties which we find by this analysis are not those of the ether, we shall at least have deter- mined the properties of a substance which might be substituted for the ether, and secure the two results already named. It may be asked, Can the Kinetic theory, which is applicable to gases in which waves are propa- gated by a to-and-fro motion of the particles, be applicable to a medium in which the particles have a transverse movement, whether rectilinear, circular, elliptical or irregular ? In favor of such an application, it may be stated that the general formulae of analysis by which wave motion in general, and refraction, reflection and polarization in particular, aie dis- cussed, are fundamentally the same ; and in the establishment of the * Young's Works, Vol. I., p. 415. t Phil. Mag., 1855 [4] IX., p. 39. t Cf . Fartschritte der Physik, 1859. 372 APPENDIX I. equations the only hypothesis in regard to the path of a particle is It will move along the path of least resistance. The expression F* x < -^ (5 is generally true for all elastic media, regardless of the path of the indi- vidual molecules. Indeed, granting the molecular constitution of the ether, is it not probable that the Kinetic theory applies more rigidly to it than to the most perfect of the known gases ? * The 133 foot-pounds of energy per second is the solar heat energy in a prism whose base is 1 square foot and altitude 186300 miles, the distance passed over by a ray in one second ; hence the energy in 1 cubic foot will be d) Where results are given in tenth-units of high order, as in the last ex- pression, it seems an unnecessary refinement to retain more than two or three figures to the left hand of the ten* ; and we will write such expres- sions as if the}' were the exact results of the computations. If V be the velocity of a wave in an elastic medium whose coefficient of elasticity, or in other words, its tension, he and density 6, both for the same unit, we have the well-known relation And for gases we have *. where y = 1.4 ; and the differential of the latter substituted in the former gives The tension of a gas varies directly as the kinetic energy of its mole- cules per unit of volume. If J be the mean square of the velocities of the molecules of a self -agitated gas, we have e = ? F. (4) Assuming, with Clausius, that the heat energy of a molecule due to the action of its constituent atoms, whether of rotation or otherwise, is a multiple of its energy of translation, we have for the energy in a unit of volume producing heat, * See also remarks by G. J. Stoney, PMl. Mag., 1868 [4] XXXVI., pp. 188, 188. THE LUMINIFEKOUS ETHER. 373 where y is a factor to be determined. If c be the specific heat of a gas, w its weight per cubic foot at the place where g = 32.2, J. Joule's me- chanical equivalent, r its absolute temperature ; then the essential energy of a cubic foot of the medium will becwrj; and observing that w = gd, we have 4 y 6 v 1 = c g 6 T J, (5) which, reduced by (4), gives (6) the second member of which is constant for a given gas. To find its value we have Hydrogen. Air. Oxygen. Specific heat,* ..... 3.4093 0.2375 0.2175 Velocity of sound, feet per second, at ) ^ ^ ^ T = 4O.^ , . . . . . ) and g = 32.2, y = 1.4, J = 772. These, substituted in the second member of (6), give a; y for hydrogen, ..... 6.599 " air ....... 6.706 " oxygen, ..... 6.596 3)19.901 Mean, 6.63 This value, which is nearly constant for the more perfect gases, we pro- pose to call the modulus of the gas, and represent it by /* ; and for the pur- poses of this paper we will use P = 6.6. This relation of the product x y being a constant, has, so far as we are informed, been overlooked by physicists, and is worthy of special notice, since it determines the value of one of the factors when the other has been found. Kronig, Clausius,f and Maxwell give for x the constant number 3, but variable values for y. \ We are confident that the value of x is not strictly constant ; or if it is, it exceeds 3, since the effect of the viscosity of a gas would necessitate a larger velocity to produce a given tension than if it were perfectly free * Stewart on Heat, p. 29. t Phtt. Mag., 1857 [4] XIV., p. 123. J Theory of Heat, pp. 314 and 317. Maxwell states that the value for y Is probably equal to 1.634 for air and several of the perfect gases. This would make x = 4 nearly. 374 APPENDIX I. from internal friction. For our purpose, it will be unnecessary to find the separate values of x and y ; but if we have occasion to use the former in making general illustrations, we will call it 3, as others have done heretofore. If the correct value of x exceeds 3, it will follow that the velocity of the molecules exceeds the values heretofore computed. * Ac- cording to Thomson, Stokes showed that in the case of circularly polar- ized light the energy was half potential and half kinetic ;f in which case y = 2, and therefore x = 3.3. The energy in a cubic foot of the ether at the earth being given by (1) and (5), we have, by the aid of (4), . . _ _ 4 X 1.4 X 2 __ _ 2__ (9) ~ 3 X 1C 7 X 6.6 X (186300 X 5280)* 35 X 10* 4 which is the mass of a cubic foot of the ether at the earth, and which would weigh at the place where g = 32.2 about tc = Aj of a pound, (10) compared with which Thomson's value is less than 4000 times this value. Thomson remarked that the density could hardly be 100,000 times as small a limit so generous as to include far within it the value given in (9). According to equation (10), a quantity of the ether whose volume equals that of the earth, would weigh about ^ of a pound. If a particle describes the circumference of a circle in the same time that a ray passes over a wave-length ?-, the radius of the circle will be, using equation (4), or the displacement from its normal position will be about ^ of a wave- length, or about mr'uoff f an * ncn at the earth. Eliminating V between (2) and (8) gives (11) for the tension of the ether per square foot at the earth, and is equiva- lent to about 1.1 of a pound on a square mile. The tension of the atmos- phere at sea-level is more than 30,000,000,000 times this value. It some- * Maxwell gives for the mean square of the velocities, or, in other words, the velocity whose square is the mean of the squares of the actual velocities of the molecules, in feet per second at 493.2 F. above absolute zero, hydrogen 6282, oxygen 1572, carbonic oxide 1276, carbonic acid 1570. Phil. Mag., 1873, p. 68. Our equation (4) gives for air 1593. t Phil. Mag., 1855 [4] IX., p. 37. THE LUMINIFEKOUS ETHER. 375 what exceeds the tension of the most perfect vacuum yet produced by artificial means, so far as we are informed. Crookes produced a vacuum of .02 millionth of an atmosphere * without reaching the limit of the capacity of the pumps ; and Professor Rood produced one of 5-5^75 ffnnnr of an atmosphere f without passing the limit of action of his apparatus. The latter gives a pressure per square foot of 14 7 X 144 390000000" = Tr 1 1!TJ of a P und - This > in round numbers, is 140 times the value given in equation (11). ' Even at this great rarity of the atmos- phere, the quantity of matter in a cubic foot of the air would be some 200 million million times the quantity in a cubic foot of the ether such is the exceeding levity of the ether. Admitting that the ether is subject to attraction according to the Newtonian law, and of compression according to the law of Mariotte, we propose to find the relation between tlie density of the ether at the surface of an attracting sphere and that at any other point in space, providing that the sphere be cold and the only attracting body, and the gas con- sidered the only one involved. Let o a> <( -itic heats of different gases arc as the squares of the wave- velocities in the respective substances, the other elements being the same, if the specific heat of air be 0.23, we should have for the specific heat of the ether as before. The correct value of the specific heat of air, 0.2375, would give over 47 X 10", and nearly 48 X 10 11 ; but these differences are quite immaterial in this connection, the object being to check the former result. On the other hand, in order that common air might be able to transmit a wave with the known velocity of light, its sped lie heal being taken con- stantly at 0.23, its temperature would be, according to equation (20), = 4 X 10 U degrees F. (= 400,000,000,000,000 F.). If the sun were composed of a substance having such specific heat, it could radiate heat at its present rate for more than, a hundred millions of centuries without its temperature being reduced 1 F., exclusive of any supply from external sources, or from a contraction of its volume. We know only such substances in the sun as we are able to experiment with in the laboratory; and if there be an exceptional substance in it, we have no means at present of determining its physical properties. It is, more- over, a question whether the ether constitutes an essential part of bodies. We conceive of it only as the great agent for transmitting light and heat throughout the universe. On account of the enormous value of the specific heat, it will require an inconceivably large amount of heat (mechanically measured) to in- crease the temperature of one pound of it perceptibly. Thus, it heat from the sun, by passing through a pound of water at the earth, would raise the temperature 100 F. and maintain it at, say, 600 F., absolute, it would, under similar conditions, raise the temperature of one pound of the ether, if its power of absorption be the same as that of water, iauouooooao of a degree. The distance of the earth from the sun being 210 times the radius of the latter, the amount of heat passing a square foot of spherical surface at the sun will be about 45000 times the heat received on a square foot at the earth normally exposed to its rays, so that, under the conditions imposed, the temperature would not be a billionth of a degree F. higher at the sun than at the earth. This, then, is a condition favorable to a sensibly uniform temperature, even if heated by the sun's rays. We are now inclined to admit that the ether is not perfectly diathermanous to THE LUMINiFEROUS ETHEK. 383 the sun's rays, but that its temperature, however small, may be due directly to the absorption of the heat of central suns ; for we begin to realize the fact that the ether may possess many of the qualities of gases, such as a molecular constitution, and hence also mass, elasticity, specific heat, compressibility, and expansibility, although the magnitude of these properties is anomalous. We have already considered its compressibility at the surface of the sun, due to the weight of an infinite column, and found it to be exceedingly small ; now, it may be possible that the expan- sion due to the excess of temperature of a small fraction of one degree at the surface of the sun over that at remote distances will diminish the density as much, or about as much, as pressure increased it, thereby making the density even more exactly uniform than it otherwise would be. According to what we know of refraction, it is impossible for a ray of light to be refracted in passing through the ether only at least, not by a measurable amount ; for not only are the density and elasticity practically uniform, but their ratio is, if possible, even more constant as shown by equations (16) and (16'). But the freedom of the ether mole- cules may be constrained, or their velocity impeded, by their entangle- ment with gross matter, such as the gases and transparent solids ; in which case refraction may be produced in a ray passing obliquely through strata of varying densities.* Neither is it believed that the ether does, or can, reflect light ; for if it did, the entire sky would be more nearly luminous. The rays in free space move in right lines. The masses of the molecules in different gases being inversely as their specific heats, and as the specific heat of hydrogen is 3.4, and the com- puted mass of one of its molecules { J X 10 ~ 29 f of a pound, we have for * Professor Michaelson concludes from his experiments that the luminiferous ether has no perceptible motion in reference to the earth, in other words, it is at the surface of the earth carried along with the earth the same as the atmosphere. (Paper read at the meeting of the American Association for the Advancement of Science, 1887.) t Stoney concludes that " it is therefore probable that there are not fewer than some- thing like a unit eighteen (10 18 ) of molecules in a cubic millimeter of a gas at ordinary temperature and pressure" (PhU. Mag., 1868 [4] XXXVI., p. 141). According to the Kinetic theory, the number of molecules in a given volume under the same pressure and temperature is the same for all gases. The weight of a cubic foot of hydrogen at the temperature of melting ice and under constant pressure being 0.005592 of a pound, and as a cubic foot equals 28,315,000 cubic millimeters, the probable mass of a molecule of hydrogen will be 0.005592 11 32.2 X 28315000 X 10 35 ~~ 18 X 10" Maxwell gives -^ 5 of a gramme = - lb., which is about 3/5 the value given above (PhU. Mag., 1873 [4], XLVL, p. 468). The difference in these results arises chiefly from the calculated number of molecules in a cubic foot of gas under ordinary conditions. Thomson gives as the approximate 384 APPENDIX I. the computed mass of a molecule of the luminiferous ether, 3.4 _1 __ {23) 22 X 10 40 The mass of a cubic foot of the ether, equation (to), divided by the mass of a molecule, gives the number of molecules in a cubic foot, which will be which call 10 1 *. This number, though large, is greatly exceeded by the estimated number of molecules in a cubic foot of air under standard con- ditions, which, according to Thomson, does not exceed 17 X 10", a number nearly 17,000,000,000 times as large as that in equation (24) ; and yet, at moderate heights, the number of molecules in a given volume of air will be less than that of the ether. Assuming that air is compressed according to Boyle's law, and is sub- jected to the attraction of the earth, equation (15) will give the law of the decrease of the density. Taking the density of air at sea-level at ^ of a pound per cubic foot, -- l' physical limitations or of the imperfection of the data on which it is founded. For instance, a uniform temperature is assumed, and, im- pliedly, an unlimited divisibility of the molecules. The latter is neces- sary in order to maintain a law of continuity. But modern investiga- tions show that not only air, but all the gases, are composed of molecules of definite magnitudes whose dimensions can be approximately deter- mined ; and hence if there IK? only a few molecules in a cubic foot, and much less if there be but one molecule in a cubic mile, it cannot be claimed that the gas will be governed by the same laws as at the surface of the earth. We conclude, then, that a medium whose density is such that a volume of it equal to about twenty volumes of the earth would weigh one pound, and whose tension is such that the pressure on a square mile would be about one pound, and whose specific heat is such that it would require as much heat to raise the temperature of one pound of it 1 F. as it would to raise about 2,300,000,000 tons of water the same amount, will satisfy the requirements of nature in being able to transmit a wave of light or heat 186300 miles per second, and transmit 133 foot-pounds of heat-energy from the sun to the earth, each second per square foot of surface normally exposed, and also be everywhere practically non-resist- ing and sensibly uniform in temperature, density and elasticity. This medium we call the Luminiferous Ether. ADDENDA. Granting that the temperature of the ether, however low, is produced by the heat from central suns passing through it, we may determine the effect upon it of a change of temperature of the source of heat. * Phil. Trans. Roy. Soc., London, 1881, Part II., p. 389. THE LUMKMl-KKOUS ETHER. c'.87 The law for perfect gases is continuing our notation e v = R T (36) where R is e v -f r 0) these values being initial. Since will necessarily be constant we see that e will vary as T, where r is the temperature of the ether, and equation (21) becomes e - = constant, as it should, since the mean density cannot change, the volume being constant. This equation reveals no new truth, but is consistent with the conditions which we anticipate in nature. The only way in which the density can change by a diminution of elasticity of the ether, is to cause it to be more dense near the attractive bodies, and more rare in space more remote from them ; or, in other words, the ether would not be so nearly uniform as at present. Assuming the density as uniform while the elasticity changes, it appears from equation (2) that the velocity of light through it will vary as the square root of the elasticity. Thus, if the heat of our sun dimin- ishes so as to become one fourth as intense as at present, and if the elasticity of the ether also becomes one fourth as much as at present, then will the velocity of light be one half as great as at present. We may find the conditions which would cause a gas of the pressure of our atmosphere at sea-level and of the same specific heat, to be as nearly uniform throughout space as is the ether. This will be found with sufficient accuracy for our purpose by finding such a value for <5 as will make the numerator in equation (15), - ^^ , the same as given in (16), where e a = 2116 the tension of the air per square foot. We will find The volumes being inversely as the densities, the last result combined with equation (36) shows that the required rarity (or density) may be secured by a temperature 10 15 times that of the present temperature. If the absolute temperature be 500 when the pressure of the air per square foot is 2000 pounds, then if it be heated to something like 500,000,000,000,000 F., . the tension would be nearly uniform throughout space. A volume of such air of the size of the earth would weigh less than jfo of a pound at a place where g = 32. 2. APPENDIX II. SECOND LAW OF THERMODYNAMICS. THE second law of thermodynamics has, by different writers, been stated in a variety of ways, and, apparently, with ideas so diverse as not to cover a common principle. For the convenience of the student in considering this subject, we here quote some of the expressions which have been given by certain authors. Maxwell, in his Theory of Heat, p. 153, says, " Admitting heat to be a form of energy, the second law asserts that it is impossible, by the unaided action of natural processes, to transform any part of the heat of a body into mechanical work, except by allowing heat to pass from that body into another at a lower temperature. Clausius, who first stated the principle of Carnot in a manner consistent with the true theory of heat, expresses this law as follows : " ' It is impossible for a self -acting machine, unaided by any external agency, to convert heat from one body to another at a higher temperature.' " Thomson gives it a slightly different form : " ' It is impossible, by means of inanimate material agency , to derive me- chanical effect from any portion of matter by cooling it below the tempera- ture of the coldest of surrounding objects.' " The last quotation may be found in Phil. Mag., 1852, IV. ; Thomson's Mathematical and Physical Papers, p. 179 ; and Clausius's statement on p. 181. Clausius considers this principle as " a new fundamental principle," and states it thus : " Heat cannot pass from a colder to a hotter body without compensation." (Mechanical Theory of Heat, Browne's transla- tion, p. 78.) It appears, so far as we can judge, that Maxwell has, gratuitously, claimed for these writers the above statement for the second law ; for not only they, but Rankine included, consider those statements as ax- ioms. In regard to Rankine's views, see Miscellaneous Scientific Papers, p. 449 ; Steam-Engine, p. 224. There would be a certain propriety in calling this the second law. and if necessary establish a third, for it is the first principle in the order of de- velopment involved in the physical operation of realizing Carnot's cycle, in which the expansion being isothermal requires a supply of heat from 390 APPENDIX II. a source, and experience shows that the temperature of the source must at least equal that of the working substance, and in reality be iufinitesi- mally higher, since heat from a colder body will not make a hot body hotter. But the question is not what might have been the second law, but what is it ? We quote from Rankine : " The internal work is incapable of direct measurement. Here it is that the second law becomes useful ; for it informs us how to deduce the whole amount of work done internal and external from the knowl- edge which we have of the external work. That law is capable of being stated in a variety of forms, expressed in different ways, although virtu- ally equivalent to each other. The most convenient form for the present purpose appears to be the following : To find the whole work, internal find external, multiply the ab&olnt / //-- perature at which the cJuinge of dimensions takes place by tJie rait /< / nt aU tlie heat which remains unconverted into icork at a lower fixed t< /////< ra- ture, tlie fraction of the whole heat expended which is converted into exter- nal work is expressed by dividing the difference between tliose tempt- rir r~ ^273-7 /'37S-7 -^273-7 t t 160 1600.0970 X d 20 20 + 0.0294 X d 180 1800.1363 40 40 + 0.0398 ' 200 2000.1772 60 60 + 0.0361 ' 220 2200.2202 80 80 + 0.0220 ' 240 2400.2627 100 100 260 2600.3099 120 120 00280 ' 280 2800.3562 140 140 0.0607 ' 300 3000.4030 (Thomson's Papers, p. 100.) 396 ADDENDA. 22. Numerous equations have been proposed to represent the results of experiments upon gaseous substances. Rankine's equation, (4), p. 13, is the most general, and is sufficiently accurate for all substances used in engineering practice. It seems a useless labor to construct equa- tions that will represent with extreme accuracy the experiments made by any person, for the results of different experimenters will differ, and a formula that will agree nearly with one set will not agree with others. If the experiments are reliable, like those of Reguault, the formula, when plotted, should exhibit the law indicated by the ex- periments, and give approximately the values found by experiment. The formula pertaining to steam will be given in Article 78. The following are equations for carbonic acid gas. Rankine also Thomson and Joule gave an equation of the form (Phil. Tram., 1854, p. 336 ; 1862, p. 579.) Him gave (P + r) (v x) = R r where x = "la somme de volumes des atomes ;" r = " la somme des action internes." (Theorie Mecanique de la, Chaleur, 2" ed., i., p. 195 ; 3' ed., ii., p. 211.) Racknel, in 1871 and 1872, gave the formula where a is a constant to be determined by experiment. J. D. Van der Waals gave - - ' v b v 3 in which if the unit of pressure is one atmosphere, and the unit of volume that which a kilo, of carbonic acid occupies under the pressure of one atmosphere at the melting point of ice, then R = 0.003673, a = 0.00874, b = 0.0023. Over de Continuiteit van den Gas en Vloeistoestand, Leinden, 1873, p. 76, Op. cit., p. 76. Clausius, in 1880, gave - P T c * r - in which if the pressure be in kilogrammes per square metre, and vol- ume in cubic metres, we have per kilogramme of carbonic acid, ADDENDA. 397 E = 19.273, c = 5533, a = 0.000426, p = 0.000494. This formula gave results agreeing remarkably well with those of observation. (Phil. Mag., 1880, (1), 401.) 23, 24, 25. Thermal lilies. The more common thermal lines are defined in the body of the book ; but the following are sometimes used : Isopiestic, or Isobar lines are lines of equal pressure, and, therefore, on the plane p v, are parallel to the axis of v. Isometric lines are lines of equal volume, and their pro- jections on the plane p v are parallel to the axis of p. Isengeric, or Isodynamic lines are lines of equal energy. In this case the internal energy remains constant, and all the heat absorbed during the change of state is transmuted into external work. See top of page 129. If the gas be perfect, the isengeric coincides with an isothermal. Isentropic lines are lines of equal entropy, and hence coincide with adiabatics. 4O. Page 33. In order that the algebraic expressions may be serviceable in numerical problems, the volume -y,, Fig. 13, must represent a definite mass of the working sub- stance ; and we assume a unit-mass / and in English meas- ures let it be one- pound. Clausius, Zeuner, and others, in some cases, include the internal work in the expression internal energy ; but we prefer to apply the term work to all that part of the heat absorbed which is destroyed put out of existence for the time being transmuted into an- other form of energy ; and if any part remains, call it a change of internal energy. The second law is sometimes called, briefly, the revers- ible engine or, more fully, an expression of the facts in- volved in ike simple reversible engine. Isothermal expan- d98 ADDENDA. sion and compression are the first fundamental principles of this law ; the second being the axiom of Thomson that no engine can be worked with mechanical profit at a lower temperature than that of the coldest of surrounding objects ; and since absolute cold cannot be produced in surrounding objects, it follows that only a fractional part of the heat absorbed can be transmuted into external work. It is worthy of remark that during isothermal expansion the heat of the working fluid does no work ; it is merely an agent for transmuting actual heat energy into work. If the working fluid be a perfect gas, it will transfer the heat directly from the source to the piston of the engine. If the working fluid be an imperfect gas, a part of the heat from the source will be transferred from the source to the piston of the engine and transmuted into external work, and the remaining part will be transmuted into internal work, being the work necessary to overcome the resistance of the particles in being separated during expansion. The actual, or kinetic, energy of the working fluid remains con- stant during isothermal expansion. Page 34. Sir William Thomson has proposed two scales of absolute temperatures. In the first scale it was pro- posed to consider the difference of the temperatures of the source and refrigerator as constant when the work done by a perfect engine on abstracting a unit of heat from the source is constant, whatever be the temperature of the heat absorbed. To get an idea of this principle, observe that in Fig. a, the successive divisions represent equal works done in suc- cessive elementary engines ; but the heat absorbed, ^, in doing the work A B c d, is more than (p l d c tp^ in doing the equal work d c ij. But the preceding principle re- quires that the heat absorbed along y z must equal that along A B, while the elementary works done in the cycles must be the same ; therefore, to represent this case y z must ADDENDA. 399 be prolonged so that the area under it and between the two adiabatics through its extremities shall equal cp^AB (f>^ and the elementary strip under y z must be so narrow that its area shall equal A B c d. To find the symbolic expression, let H = (p^ A B (p 3 = the heat ab- sorbed, in foot-pounds, dII=ABcd = the heat trans- muted into work, d t = the difference of absolute temperatures between the source and refrigerator, = the fractional part of the heat absorbed that is transmuted into work per unit of tempera- ture ; , V Va v FIG. a. then the work = t If d t d d II But according to the principle above stated, p is not only independent of the temperature, but the ratio of the left member is to be constant ;* hence jn is constant, and we have by integrating between the limits 2I l and H t for heats, and t l and t t for temperatures, that is, if the differences on the scale of the thermometer * Or integrating, we have : Work = H, ^ = ^(1 6 ). But, in this scale, the work is a function of the difference of temperatures, whatever be the temperature of Hi, and this condition requires that /* should be constant. This relation is clearly shown in equation (a). 4 2 equals the increment A B c d of the external work v l A B v,. This is equivalent to asserting that the difference of the internal works done during the perform- ance of the external works v l d c v^ and v l A B v^ is zero ; or, at most, a difference of the second order compared with the difference of external works. As a verification of this principle for a particular case, let the equation of the gas be p = -R -- ^r~ i 1 tnen w ^ tne area of one of any one of the strips in 99, A B

n . Equations (B) give vdp-\-ypdv = an(Y 1) V- 1 ef 0. Let v = x, p-y, P=-, Q = a n (y - 1) x "- = B x n- ADDENDA. 405 and the equation becomes where P and Q are functions of x. This is a differential equation of the first order and first degree. To find the integral, first let Q = 0, then - f j Pdx where G. is a function of x instead of a constant of integration. Differ- entiating, -fet* Y-=-PG x e dx -fPdx /Pdx .'.Qdx=e .dC,; .'. C x =qe dx; fPdx f Pdx P = ^TT-y" Exercise 16 may also be reduced to the linear form, giving Exercise 17 gives the differential equation dp^ _ na(y l) v a ~ l - yp dv nb(\ y)p u ~ 1 v 71. Page 89. The melting point or freezing point of liquid carbon disulphide 116 C. " absolute alcohol - 130.5 C. Alcohol becomes viscid at 129 C. (Phil. Mag., 1884, (1), 490.) 12. Page 90. By experiment it has been found that the melting point of ice is raised 0.0066 C. by a reduction of pressure from 760 mm. to 5 mm. (Phil. Mag., 1887, (2), 295.) 406 ADDENDA. 74. Of liquids and saturated vapor. Regnault found the latent, heat of evaporation by determining the total heat and the heat of the liquids independently, and taking their difference. Total heat of liquids, being the number of ther- mal units necessary to raise the' temperature of a unit-mass from that of melting ice to t degrees centigrade, as deter- mined by Regnault, at atmospheric pressure. Substance. Number of Thermal Units. Water .............. q = < + 0.00002 < 8 + 0.0000003 t 3 . Alcohol ............ q = 0.54754 1 +0.001122 1* + 0.000002 1 3 . Ether .............. q = 0.52901 1 + 0.0002959 <*. Chloroform ......... q = 0.23235 t + 0.0000507 *. } (1) Chloride of carbon . .q .= 0.19788 1 +0. 0000906 1*. Acetic acid .......... q = 0.506403 1 + 0.000397 ?-. Bisulphide of carbon q = 0. 23523 1 + 0.000088 <*. The general law of these equations may be represented by the empir- ical equation q = a, t + 6, P + < r of B. T. TL, q = g q = | [| (T -32) + 0.00002 [ (T - 32)] + 0.0000003 [$ (T- 32)] 3 ]; and similarly for the other liquids. Substituting and re- ducing, we have ADDENDA. 407 Substance. No. of B. T. U. at temp. T. "Water, q = 31.991656 + 0.99957333 T+ 0.000002222 T* + 0.0000000926 I Alcohol, g = - 16.903214 + 0.509543 T + 0.00056407 T 1 + 0.00000061 7284 T*. Ether, q = - 16.759986 + 0.518489 T + 0.00016439 T 3 . Chloroform, q - 7.406358 + 0.230547 7 + 0.00002817 T. Chloride of carbon, q = - 6.280619 + 0.194659 T + 0.00005033 7 12 . Acetic acid, q - 15.979005 + 492287 T + 0.00022055 T*. Bisulphide of carbon, q = - 7.480711 + 0.232314 1 + 0.00004555 T. The general law will be The figures in equations (3) were obtained by carrying out the decimals to many more places, and then retaining the above to the nearest unit for the right-hand figure. The specific heat at any temperature will be the differential coefficients of the preceding expressions, which, for water, will be ^-2 = 1 + 0.00004 t + 0.0000009 f, (5) per degree centigrade, and ",= 0.999573 + 0.000004444 T -\- 0.00000027768 T, (6) per degree Fahrenheit. These results for water are not exactly the same as Ran- kine's or Bosscha's, given in Article 95, but any one of them is sufficiently exact for ordinary practice. 74, 85. Total heat of vapor. This expression in- cludes the heat imparted to the liquid in raising its tem- perature from that of the melting point of ice to that at which the vapor is generated added to the heat necessary to evaporate the liquid at the higher temperature. The latent 408 ADDENDA. heat of evaporation includes both internal and external work ; the external being the work of enlarging the volume at the pressure corresponding to the higher temperature, and may be represented by the work done by a pound of satu- rated steam in pushing a piston against a constant resistance up to the point of cut-off in an engine, and the internal, that of overcoming the mutual attractions between the molecules. Let h be the number of heat units necessary to raise one kilogram of a liquid from C. to a temperature t, and vaporize it at that temperature ; then Regnault's ex- periments may be represented by the following empirical formulae : Number of heat unite in the " total heat of vapor " in French thermal unite. Water h = 606.50 + 0.305 t. Ether h = 94.00 + 0.45000 t - 0.00055556 P. Acetic acid h = 140.50 + 0.36644 t - 0.000516 P. Chloroform h = 67.00 + 0. 1375 t. Chloride of carbon . . h = 52.00 -j- 0.14625 t - 0.000172 P. Bisulphide of carbon, h = 90.00 + 0.14601 t - 0.0004123 t*. J For English units, if h be the number of heat units in one pound of the substance on the Fahr. scale, then for water we would have A = f h = f [606.5 + 0.305 X \ (T 32)] = 1091.7 4- - 3 5 (T 32) = 1081.94 4- 0.305 T-, . . H = 841829 4- 237.29 T, which differs slightly from Eq. (95), page 111, because some fractions were omitted in determining the latter. In this manner we find the following results : Number of B. T. U. in the " total heat of vapor " of 1 lb., Fahr. scale, above 0* F. Water h = 1081.94 + 0.305 T. Ether h = 154.4839481 + 46974324 T- 0.000308633 T>. Acetic acid. . h = 240.880363 + 0.3847866 T - 0.000286666 T 2 . I (?) Chloroform., h = 116.2 4- 0.1375 T. Ch. of carbon h = 88.82215 + 0.1523655 T - 0.00009555 T*. B. of carbon, h = 157.093127 + 0. 16066955 T - 0.000229055 T. J General equation A = o, + 6, T-c* T'. (8) ADDENDA. 409 74. Latent heat of evaporation. Subtracting the " heat of the liquid " from the " total heat of the vapor " gives the latent heat of evaporation ; hence ^ e = A - q ; and making the substitutions from above, we have the following results : Latent heat of evaporation, being the No. of French Substance. heat units in one kilo, of the vapor at the boiling point. Water h e = 606.5 - 0.695 t - 0.00002 1* - 0.0000003 1* Ether h e = 94.0 -i 0.07901 t - 0.0008514 P. Acetic acid h e = 140.0 - 0.13999 t - 0.0009125 P. . Chloroform h e = 67.0 - 0.09485 t - 0.0000507 f. Chlo. of carbon. . h. = 52.0 - 0.05173 t - 0.0002526 < 2 . Bisulp. of carbon h e = 90.0 - 0.08922 t - 0.0004938 P. In English units these become : Latent heat of evaporation, being the heat necessary Substance. to evaporate one pound of the substance at the boiling point, in B. T. U. Water A. = 1121.7 - 0.6946 T - 0.000002222 7" - 0.0000000926 T 3 . Ether A. = 171.24 - 0.0487 T - 0.000473 T\ Acetic acid A. = 256.86 - 0.1075 T - 0.000507 T\ Chloroform A e = 123.60 - 0.0930 T - 0.000282 T*. Chlo. carbon A. = 95.103 - 0.0423 T - 0.0001403 T*. Bisulph carbon. .A. = 164.57 - 0.0716 T - 0.0002746 7" s . Alcohol h, = 527.04 - 0.92211 T - 0.000679 I \ General equation : Ji t 4 _ J 4 T c 4 T* d t T 3 . (11) These for English units and absolute temperature on the Fahrenheit scale become : Latent heat of evaporation in B. T. U. , absolute Substance. temperature, r. Water A. = 1442.474 - 0.751472 T + 0.0012538 r 2 - ] 0.0000000926 r\ Ether A. = 93.3214 + 0.3870 T - 0.000473 r 2 . Acetic acid A. = 197.925 + 0.3595 r - 0.0005070 r 5 . \ (12) Chloroform A, = 160.4924 - 0.0571 r - 0.0000282 f\ Chlo. of carbon. . A, = 85.0245 + 0865 r - 0.0001403 r*. Bisulp. carbon. . . A. = 140.1806 + 0.1810 r - 0.0002743 t>. J General equation : fe e a& _j_ J 5 r + c 6 r 2 -+- rf r 3 . (13) 410 ADDENDA. The effect of retaining the smaller decimals will be ap- parent by comparing the above results for water with the corresponding ones on page 95. Those on the latter page are considered sufficiently accurate for practice. None of them can be relied upon for temperatures much outside of those in the experiments upon which they are founded. The two equations are not very different within the range of temperatures ordinarily used in practice. To find the heat which does the disgregation worl\ we must find the external work done during evaporation. This may be done as follows : The pressure for the absolute tem- perature of the vapor is, equation (80), page 97, B C logp = A ----? "We have computed the following constants by means of Regnault's experiments. They are for degrees Fahrenheit and pounds per square foot : Fluid. A. log B. log C. Steam 8.28203 3.44L474 5.583973 Ether 7.5641 3.3134249 5.2173549 Alcohol 8.6817 3.4721707 5.4354440 Bisulph. Carbon..7.4263 3.3274293 5.134414(5 Chloroform 4.3807 B is 3.288394 negative. 6.1899631 Snip. Dioxide... 7.3914 3.1580608 5.3667327 Naphtha 6.4618 2.949092 5.796469 Ammonia 8.4079 3.34154 Mercury 7.9711 3.74293 Far steam. For ether. B = 2763.59, B = 2057. 3, C = 383683. C = 164950. Now find the volume of a pound of the vapor by means of equation (84), or the approximate one, (86). Thus, ADDENDA. 411 V r dp ' ^ dp. Tr T rr The value of h e is given by equations (10) and (11), and substituting, gives a - I T - c T* /. 1 + ^)2, 23026 This is the outer work. The disgregation work will be p = II, -p (v,- v,}. (11) Zeuner, by this laborious process, computed the disgre- gation work for a range of temperatures, and for various substances, and assumed, arbitrarily, that they followed the law p = a t - I, t c,f, and determined the constants by means of his previous computations, and obtained the following results : For French units. Ether .......... p= 86.54 - 0.10648 1 - 0.0007160 f. Acetic acid ..... p = 131.63 0.20184 0.0006280 t. Chloroform ____ p= 62.44 0.11282 1 0.0000140 f. Ohio, of carbon p = 48.57 - 0.06844*5 - 0.0002080 f. Bisulp. of carbon p = 82.79 0.11446 1 - 0.0004020 f. For saturated steam, the outer work may be found verv nearly from a table of the properties of saturated steam, by multiplying together the corresponding pressures and volumes. The product will be p u,, Eq. (14). The temper- atures being given in such a table, we may find the total latent heat of evaporation by means of equation (78), page 95. Eeduce the pressures to pounds per square foot, if neces- sary; and substitute in equation (14), to find the disgre- gation work. 412 ADDENDA. The latent heat of evaporation, as commonly used, might be called the APPARENT latent heat of evaporation / and the disgregation work, the REAL latent heat of evaporation. Page 96. The latent heat of evaporation is reported differently by different authors. Thus I find that one author gives for oil of turpentine, 123, another, 133, and still an- other, 184; and I'have not ascertained which is correct. For ether, 164.0, 162.8. Alcohol, 304.8, 372.7, 385. Naph- tha, boiling point, 306, 141 F. ; latent heat of evaporation, 184, 236. Densities of some vapors compared with that of air when near their boiling points : Atmospheric air 1.000 Steam 0.6235 Alcohol vapor 1.6138 Sulphuric ether vapor 2.5860 Vapor of oil of turpentine 3.0130 Vapor of mercury 6.976 The densities of vapors at the boiling points of the liquids are approximately inversely as their latent heats of evapo- ration. Thus, Density of vapor of alcohol = 1.6138 _ . _ Density of steam = 0.6235 ~ Latent heat of evaporation of steam = 966.1 _ Latent heat of evaporation of alcohol = 372.7 ~ 76. Eankine, in his article On the Centrifugal Theory of Gases,* deduces an equation of the form, logp a - -, for the relation between the pressure p and absolute tem- perature r of saturated vapor. It was found, according to * Mis. Sc. Papers, p. 43 ; Phtt. Mag., Dec , 1851. ADDENDA. 413 Regnault's experiments and others, to be accurate for a limited range of temperatures only. Kankine then proceeded to find an empirical formula that would represent more accurately a greater range of temperatures, and was led, by analogy, to try a third term containing the inverse square of T. thus giving B (T logp = A - - -,. which was found to represent, quite satisfactorily, the re- sults of experiments upon steam, mercury, alcohol, ether, turpentine, and petroleum. Some fifty, or more, formulas have been devised to ex- press the relation between the pressure and temperature of saturated steam ; all of which are sufficiently accurate for certain small ranges of temperature and pressure. Ran- kine's, given above, is the most accurate for a large range. Some of the most celebrated of the other formulas are : Dulong and Arago's, for pressures above four atmospheres p - (0.4873 + 0.012244 <) 5 Ibs. per sq. in., t being the temperature centigrade. Mallet's, from 1 to 4 atmospheres, P=(TiTiSF Ibs. persq. in. (15) Tredgold's is the same, except that 175 is substituted for 111.78. Pambour's, from 1 to 4 atmospheres, t C. Roche's p = a^ (17) Regnault made three equations ; the first applying from 30 C. to C. log p = a + b a", millimetres, (18) in which a = - 0.08038 ; log b = 9.6024724 - 10 ; log a = n = 32 + t. 414 ADDENDA. From to 100 C. l-jgp = a - b a 1 + c p, millimetres, (19) in which a = 4.7384380 ; log b = 0:6116485 ; log c = 8.1340339 - 10 ; log a = 9.9967449 - 10 ; log ft = 0.0068650. From 100 to 220 C. log p = a - b a" + c p a , (20) in which a = 5.4583895 ; log b = 0.4121470 ; log c = 7.7448901 - 10 ; log a- 9.99741212 - 10 ; l and for the specific volume of saturated steam, 10821 h K = * + ?(-%..) ' (36) Also, for alcohol ; logp = 7.448 - ^- 9 - (27) For ether ; log p = 6.9968 - ^~. (28) For carbonic acid ; = 8.4625 - ?^H (29) The author, considering steam as an imperfect fluid or gas both when saturated and superheated, applied Ran- kine's general equation for imperfect gases with the follow- ing results : The general equation being p V = T-a -^-%- &c, (30) and first assuming that the resulting equation might, pos- sibly, be of the same form as his for carbonic acid gas, I made a = 0, a l = , and all the remaining terms zero, giv- pv = Er-. (31) But this gave no satisfactory result. I then assumed I >?>> and all succeeding terms zero, as Rankine considered that a , a a &c., were inverse functions of 0, thus giving p v = E r - ~ (32) 416 ADDENDA. To find the constants R, 5, n, requires three contempo- raneous values of the variables p, v, T. I determined values of y by assuming several values for p and T by equation (85), page 98, and compared the results with the recent tables of Professor Peabody, and in no case did they differ by more than .02 and in most cases they agreed exactly to the second decimal figure and this too notwithstanding he used Regnault's equations for the relation between pres- sures and temperatures, while I used Rankine's, being equa- tion (80), page 97 ; but using for constants my own values on page 98, instead of those computed by Rankine. So close an agreement was not anticipated under these condi- tions ; and where the difference was greatest it might pos- sibly have been less had I used more decimals. I there- fore use Peabody's tables with confidence. I have, how- ever, used v, = 26.58 for the specific volume of saturated steam at 212, and pressure* of 14.7 Ibs. per square inch as computed on page 102 ; but all other values I have taken from the tables. Using as arguments the three sets of values : Pl = 14.7 X 144, p, = 100 X 144, p, = 160 X 144, \. >TK.\M ii MI* fr- fii. f. I*T kilo. |*T Hi. 1 -', III M i:, < 7 |U 141 Ml 474.H 101,0 744 PrMTOfM, Ib*. per Kf|iur* foot. Errorn ( ompntad T.,,,,.!.,. l.-l.ul.ir. 214 2110.2 -f-^ K ::,{ + Thcuc reunite rlo not agree o nearly as for satiint. ! and the errors are all in one .-<-ii-<-. tin- ln-iii^ ;ill t/o lar^e. This we would ;intiri|;it- the more the steam in Htijierheatcd, the more iii-;ulv will it behave like a perfect gan, awl conform nmn- nt-;irly to c /rro. ADDENDA. 419 Page 104, the external work done during the evaporation of one pound of the liquid will be p X B C, at the tem- perature T. When the abscissa between F E and E C is zero, as at E, the corresponding temperature will be the critical temperature for that substance. Neglecting the external work due to the enlargement of tin 1 volume of the liquid, the critical temperature will be that temperature which will reduce the apparent as well as the real latent heat of evaporation to zero. For steam, the first two terms of equation (12) of this Addenda gives r = 1919 ; or T 1459 F. The critical temperature of a few substances has been found by experiment. Thus, it is for Deg.C. Carbon tetraehloride 292.5 Carl>on disulphide 270.1 Acetone 240.1 (Fogy. Ann., cli., (1874), 303.) Theory gives higher values than these. The critical temperatures and pressures for twenty-one substances is given iu the Philosophical Magazine, 1884, (2\ page 214. Avenarius showed by experiment that over a certain temperature fixed for each substance there is no distinction between the liquid and vapor states, so that pressure alone will not cause a gas to liquefy. 79. In Rankine's tables the absolute zero was assumed at 401.2 below the zero of Fahrenheit's scale, while those at the end of this work are computed with 400.00. 85. See Article 75 of this Addenda. 9(>. Clausius claims to be the first to announce that in- ternal work is a function of the initial and terminal states onJy. (Clausius On Heat, page "">.) 420 ADDENDA. 97. It will be a good exercise for the student to give geometrical interpretations of the equations on page 132. 97a. " On, the dimensions of temperature in length, mass and time ; and on the absolute C. G. S. unit of temperature " (Phil. Mag., 1887, (2), 96). It is shown that, in accordance with Thomson's absolute scale, the unit temperature would be that of a perfect gas whose mean kinetic energy per molecule was one erg. If E = the mean kinetic energy of a molecule of the gas, T = absolute temperature of a perfect gas, k = a constant, p = pressure per unit, = volume of a pound of the gas, n = the number of molecules in a pound ; then, E = k r, and making k = 1, E= r. But, equation (2) of Appendix, gives, if t>, = | for 1 pound, xp f i = t>* = 2 . E; hence, making x = 3, and omitting the subscript, we have p v = n E ; *=* OT = 2 - 5xl -'' = r for the value of the temperature at C., or 273" C. absolute, C. G. 8., and tfie absolute unit = 273 -*- 2.5 X 10-'* = about 10 18 C. degrees. " Having seen that temperature is of the same dimensions as energy, and knowing that the same is true of heat, it follows that entropy, whose dimensions are heat -f temperature, is a purely numerical quantity ; and the unit of entropy is therefore independent of all other physical units. In fact, the entropy of a perfect gas increases by unity, when (without alter- ing its temperature) it receives by conduction a quantity of heat equal to the mean energy of one of its molecules." 98. "Priming or superheating. Equation (135), page 144, may be put in a more customary form as fol- lows : Let T, = the temperature of the water at the boiling point under the given pressure in degrees Falir., ADDENDA. 421 T^ = the temperature of the feed water, which is as- sumed to be the same as that of the higher temperature of the water supplying the calo- rimeter. T 3 = the initial temperature of the water supplied to the calorimeter. h = the total heat of steam, being the heat units necessary to raise the temperature of one pound of water from 32 F. to the boiling point and evaporating it at that point. Its value may be found in special tables, or from equation (13-i) after making x = 1. * h 1 = the heat units in one pound of the water at the boiling point above 32 F., which will be T, - 32, nearly, but may be found more accurately from Article 95, or more directly from suit- able tables : A, = the heat units in one pound of the feed water above 32 F., h the heat units in one pound of the steam above the temperature of the feed water as deter- mined from a calorimeter, w = the weight of the steam condensed in the calo- rimeter, W = the weight of the water supplied to the calo- rimeter ; then *= (T t -T,\ (1) 7< e = h - h, = h - (T> - 32). (3) Dividing both numerator and denominator of equa- tion (135) by J = C, and- substituting the above values gives 422 ADDENDA. Per cent of priming = 100 e + TI ~ TQ ~ (4) = 100 h -- /<, If this equation becomes negative there will have been superheating. In equation (5), li A, is the heat supplied above the temperature of the feed water to produce one poun which is the same as for the perfect elementary engine, equation (159), page 161. This result might have been anticipated, since the cycle is Carnot's. 97. To represent geometrically certain rela- tions. Equations (A), page 48, or (123), page 135, give ADDENDA d II (dp 429 dr -- . r If r be constant during the expansion d v, we have d r 0, and Dividing both sides by d v gives d_v\ _ fdp\ dvJr~ \d r); The factor (-=2- I is the rate of change of pressure per \d T/V unit of temperature, and therefore if the rate were uniform during the change of unity of temperature from r to t -f- 1, it would be the increase of pressure due to an in- crease of one degree of temperature. Draw two parallel lines to represent two isothermals differing by unity. As in the cal- culus, these lines may be tangents to actual isother- mals. Let a w and g p be a the isothermals, differing by one degree of tempera- ture. At a let the pres- sure be p ; then will dp\ -^- ] = a c. d T / v If the abscissa of J in ref- erence to a be a e = d v, then a cfb ac ae = l-JL\ d v. 430 ADDENDA. Let the straight line a g through a be an adiabatic (tangent to an actual adiabatic). Divide a p into parts each equal to d i), and through the points of division draw lines parallel to a c, and through their points of intersection with a w draw lines parallel to a g ; the spaces thus formed will be equal ; and equal to a cfl, since they have the same base and altitude. Hence Dividing by d v, Let a i represent unity of volume, or a i = 1 ; then id which is the right member of the preceding equation. f -~J is the number of d \ ldp\ as deduced above, analytically. It will be observed that the subscript of the parenthesis on one side of the equation is the same as the independent variable on the other side ; that is, r is a subscript on the left side, and the independent variable on the other. Writ- ing the reciprocal of the preceding equation, and observing a similar order of subscripts, we have, (d T\ f - 135V ADDENDA. 431 the geometrical interpretation of which we will proceed to show. The equation

, and at t, p + 1, and a q, parallel to the axis of vol- umes ; then qt = l. The temperature at a being T, that at t will be r -f- I y- U; \ftpi . . change of temperature from a to t = ( -= } . \dph The area aghb = dcp; and if this area were extended to t, its value would be and this also equals tpb Assume any arbitrary area as a g I k for an increase of (p = 1. Since I -j ) is the increase of temperature for an increase of unity of pressure, it is the temperature of the isothermal t o above that of a k ; . . area a t n k = ( -, j ((p = 1). The expression (-3 j is the abscissa of v for (p unity, and hence is the abscissa of k in reference to a ; and this multi- plied by q t = 1, gives 432 ADDENDA. (dv 1 . -y = atn hence the equality of the expressions. The second equation of thermodynamics is , d II n dr */'v \dv /*' which may be geometrized in a similar manner. ADDENDA. 433 Thus, the right-hand member implies the increase of temperature for a change of unity of volume on an adia- batic. Let a q = v = 1, and from q erect a perpen- dicular to meet the adiabatic through a at t ; then will the increase of temperature at t above that at a be <4 v, . If the increase unity of cp be represented by a g Ik, then Also /d\ \d /dj\ \d t A B axis. Then the total differential of v, or d v (which is the left member of the equation), will be the abscissa of b in refer- ADDENDA. 441 ence to a, and the total d p will be the ordinate of I in reference to a ; hence dv a c, dp = b c. At a draw the tangent a A, and let a g = p = 1, and make a e = b G and draw e f and y h parallel to a d, then d that is, if the isothermal were the straight line a h, then by passing down it until p = 1 = a g, the abscissa g h in refer- ence to a will be as written above. It not being generally a straight line, take a e = I c = d p, then, by the simi- larity of triangles, ef=ae= dp. ag \dp) T Since a e is negative, being measured downward, we really have / \ap T j by construction. This is the first term of the second mem- ber. In the third term f ?) would be represented by a d, VZ T/ P since^ is constant, provided the isothermals were unity apart ; but as they are only d r apart we have, on the prin- ciple just given, hence, 442 ADDENDA. If v be constant, tlie path of the will be perpendicular to the axis of v. Let a of the preceding figure be moved to tlie right until it falls on c, as shown in the annexed figure, when total d v becomes zero, and we have d Page 132. The equation pjr dp = d. was deduced from equation (A), p. 48, by dividing both members by d v ; hence, in the right member the factor 1 should be retained in order to make the terms homogeneous, giving (d in which 1 represents unit}' of volume. In representing these quantities geometrically, we will for conven- ience use straight lines, a method strictly correct at a state. Let A be the initial state, Vt Vi = 1, be the increase of volume, A JSthe isothermal through A, or if the isothermal be a curved line, then A B will be tangent to the isothermal, A

, e, = (-- 1. ADDENDA. 443 If the area i A B a : and the left member is the heat necessary to be added to that of 3 in order to increase the temperature one de- gree, and maintain it at that tempera- ture while it works under the pres- sure z'j a, through the space Vi a , v T d ^ provided the pressure is increased FIG. i. uniformly with increase of temperature. The right member represents the difference of the external works during the expansions at tempera- ture T and T + 1 plus the difference of the internal works during the same expansions. We have The difference of the internal works in expanding along the two isothermals, A B and a b, respectively, equals the difference of the in- ternal works along the paths A a and B b, respectively, and the latter is as given on page 119, K*. -S~B T -j -, (z'a Vi = 1) = t - r . 1. (IT 1 d T* The next equation on page 132 is 444 ADDENDA. This represents the heat absorbed at constant volume for an increase of one degree of temperature. It is, as shown by the right member, the heat which makes the substance hotter, represented by C, plus that which does internal work. It may be represented on a diagram of energy by the area tpi A B

i, A a n\. FIG. k. Conceiving the expansion to be isothermal, r will be constant during the ^-integration, and v constant for the r-integration. If the equa- tion of the gas be p v = R r - ~ t then for an increase of one de- gree of temperature * T (r + 1) Page 132. In the equation ADDENDA. 445 the value ( j is the heat absorbed for an increase of temperature of one degree (or strictly it is the rate at which heat is absorbed per degree of temperature) at constant volume, d ( - \ is an element- ary increase of this heat, and 3 ( - \ is the elementary amount for a v \ a T / v an increase of volume equal to unity. The value is more readily seen from the right member. Referring to equation (105), page 119, it will be seen that the right member is the heat absorbed in doing internal work for an expansion unity of volume, provided it is uniform through- out that volume. Therefore r ( -7-^ j may be considered as the ordinate at any point whose abscissa is of the shaded part tpi, A a HI, or of TI A a mi, the latter of which will be used in the solution of problems since r will then be constant. In the equation (page 132) dH\ __ T /dv dp)r (J7 the left member is the heat absorbed dur- ing isothermal expansion fora fall of pres- sure equal to unity. Let A B be an isother- mal, Bb = 1, then will FIG. I. Let c e be an isothermal one degree higher than A B; prolong A b to d, and draw B e parallel to A d; then Since b B is negative, we have Ad bB= j^ - ( Page 134. The differential of a function of any number of vari- ables may be found by well-known rules, the result being an exact dif- ferential ; but it is often difficult to find the primitive of a differential. A transformation is often necessary in order to render an equation or an 446 ADDENDA. expression an exact differential. Thus, the first member of the equa tion -r- y*) d x = > is not an exact differential as it stands, for it does not satisfy the condi- tioo d(a?y ) = diL+J^-; but multiplying the equation by 2 a? it be d x d y comes 2 a* y d y + 2 (1 + y) x d x = o, and then _ dx dy for it reduces to and the integral will be a* (1 + y) = e. If the expression does not equal zero we may write it d

the heats of the liquid corresponding ; x & is the part of one unit of weight of the fluid in the tube A which is dry steam, and 1 # a is the part which is water mingled with the steam ; *b is the corresponding quantity for the tube B ; 6 the weight of unity of volume of the liquid. It is assumed that neither tube gives heat to the steam or receives heat from them, and that the friction of the fluid on the sides of the wall can be neglected. The heat Q is supposed to be given at the orifice ; it is com- monly assumed to be zero, in which case the flow is said to be adiabatic. The value of .r a must be determined by experiment. o\> can then be determined by the equation : 450 2 REFRIGERATING MACHINE. lllliili "5 MO = c s? lil^fp \le9\ - h ill V 5 :W"3 iff !S := = ! : i if I I till = i j !! i 1 c* ^ If !*=< I | j E S I J J P| S ? H < fc. QJ o-^^ KEFKIGERATING MACHINE 451 ^2a05*. - O^c-^gSSj 3SSS3S S5S rt0f ' - 00 ro " SSSiSSS mi i si "SHI'S go ;siiii^js jwysiwh 13 ! i i I : |^ ||| ||| |1 E||| |'l | * B iJl^l *i*if!! i f ii^s^ii.- 111, ii -slilllll lilllSl' 8 * 5*' 5sgg IlilSlS I limit*?* I1 isij - - 452 REFRIGERATING MACHINE. TEST OF A PULSOMETEK BY C. G. ATWATER AND CHARLES B. HODGES, OP THE CLASS OF '91, STEVENS INSTITUTE, under the supervision of the Department of Tests. The pump was taken from the ordi- nary stock of machines on hand, and was known as No. 6. Tests were for three hours each. or THE TEST. DATA AND RESULTS. 1 2 3 4 Strokes per minute Bteam pressure in pipe before throttling, Ibs. " " ' after 71. 114. 19. 60. 110. 30. 57. 127. 43.8 64. 104.3 2C.1 ' temperature uf ter throttling, Deg. F. . 270.4 277.4 309. 850.1 ' amount of supei heating. Deg. F " total heat of, above highest tempera- ture of water, B. T. U 3.1 1118.67 3. 11U.44 17.4 1187. 1.4 1121.2 ' used as determined from temp., Ibs. .. 1617. 921 1518. 1019.9 ^Tater pumped Ibs 404786 180362 ,'> IN) 248063 " temp., before entering pump, beg. F. '' temperature after leaving pump,Deg.F. 44 rise of Deff F 75.15 79.62 4 47 80.6 86.1 5 5 76.3 83.79 7 49 70.25 74.8 4 55 44 heat absorbed, B. T. U 1815127. 1024993. 1704498. IHiMT. 11 head by gauge on lift, f t . 29.90 M.0i 54.05 29.9 44 4 * ** 4t suction * 12 26 12 2J 19 67 19 67 41 l4 " total (H) 42.16 66.31 73.72 4957 44 measure on lift. 253 50.3 503 25.30 14 44 44 44 suction 7 ft 16 3 1630 " " ' 4 total (h) 828 57.8 66.6 41.60 Efficiency of pump compared with total work, (h) + (H) 0.777 0877 0.911 0.839 Total work a per gauges, ft. Ibs Efficiency of the pulsometer 44 " plant exclusive of boiler UJM08W. 0.012 0.0093 12335940. 0.0155 0.0186 167951112. 0.0126 0.0115 lenaeto. 0.0138 0.0116 44 " " if that of boiler and fur- nace be 0.7 0.0065 0.0095 0.0080 0.0081 Duty of pump per 100 Ibs. coal if 1 Ib. evap- orates 8 Ibs. water 8409C20. 10712800. 8847200. %*KMO. Duty If 1 Ib. coal evaporates 10 Ibs. water. . . . 1051 1400. I3S9100.) 110U8500. 12036:100. Of the two tests having the highest lift (54.05 ft.), that was more effi- cient which had the smaller suction (12.26 ft.), and this also was the most efficient of all the tests. But in the other two tests having the same lift (29.9), that was most efficient which had the greater suction (19.67). The pressures used, 19, 30, 43.8, 26.1, were made to follow the order of total heads, but are not proportional thereto. They doubtless have much to do with the efficiency, but no attempt was made to determine the pressure which would give the highest efficiency. The first tost compared with the other three is somewhat paradoxical. Thus, it is peculiar that a pressure of 19 pounds of steam should produce a greater number of strokes and pump over 50 per cent, more water than 26.1 pounds of steam, the lift being the same. .REFRIGERATING MACHINE. 453 TEST OP THE NEW YORK HYEIA ICE-MAKING PLANT, BY A. G. HUPFEL, H. E. GKISWOLD, AND WILLIAM P. MACKENZIE, FOR GRADUATING THESIS, 1893. under the supervision of the Department of Tests of Stevens Institute : 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 Average pressure of ammonia gas at condenser in pounds per square inch above the atmosphere ...;... 135.2 Average back pressure of ammonia gas in pounds per square inch above the atmosphere 15.8 Average temperature of brine in freezing tanks in degrees Falir 19.7 Total number of cans filled per week 4,389 Ratio of cooling surface of coils in brine tank to can surface. 7 to 10 Ratio of brine in tanks to water in cans 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 EXERCISES. 1. Required the specific heat of air at 500 F. absolute, the path being p = 10 v (p, pounds per square foot ; , volume of a pound in cubic feet). The specific heat is the heat absorbed in raising one pound one degree. 2. If the equation of a superheated vapor be p v = R r Cp* (see p. 414) ; required the heat absorbed at the constant pressure pi in expanding from v l to tv 3. If the equation of the gas be p v = R r and of the path of the lluid, p = m v -\- n ; required the heat absorbed in expanding from state v l = 12, p t = 4000 to state n, = 24, p = 5000. 4. Find the internal and the external work of expanding the gas 454 REFRIGERATING MACHINE. pv = Rr t at the constant pressure pi from v l to ? 2 , and leave the final result without r, 5: Find the thermodynamic function for the gas whose equation is p v = R T s for isothermal expansion. 6. Find the velocity of discharge of a perfect gas from an orifice, the temperature remaining constant. Also the weight per second. 7. Write a formula for the pressure of a saturated vapor in terms of the volume of a pound. 8. Required the difference in the elevation of two stations at which water boils at atmospheric pressure respectively at 212 F. and 180 F. 9. A vessel containing two cubic feet of fluid, one-fourth of which by weight is steam and the remainder water ; required the work neces- sary to compress the vapor to water in one case adiabiatically, and in another isotliermally. 10. A frictiouless piston, in an upright cylinder, rests on a pound of water. Heat is absorbed under a pressure of 6 atmospheres absolute until the piston has swept over two cubic feet, then expansion is adiabatic until the pressure is reduced to two atmospheres absolute, then compressed isothermally until by adinbatic compression the vapor will be reduced to water at the initial pressure. Required (1) the heat ab- sorbed ; (2) the clearance ; (3) the entire stroke of the piston ; (4) the heat emitted during compression ; (5) the work done in the cycle ; ((>) t In- efficiency of the cycle. 11. Find difference of external works in expanding a gas adiabati- cally and isothermally from 0, to 2 r,. 12. Find heat absorbed by a liquid in raising the temperature from 40 to 200, if specific heat c = 1 -f a T J . If c = 1 - a (T - T ). If c= 1 -a(T-T#. 13. What is the specific heat of a gas at constant temperature ? 14. Find the thermodynamic function for air from state TI, p t to T t , p-t, independently of v. 15. A single-acting engine (vertical) has one pound of fluid (water) at the lower end, on which rests a frictiouless piston. By the absorption of heat the piston is raised against an absolute pressure of 9 atmospheres until the volume swept over by the piston is twice the volume of the dry saturated vapor at that pressure, then it expands adiabatically as a perfect gas until the pressure is reduced to 3 atmospheres, then re- frigerated at constant volume until the pressure is reduced to 2 atmos- pheres, then refrigerated at constant pressure until the volume is such that when compressed adiabatically it w r ill be reduced to a liquid at 7 atmospheres, and its temperature then raised under a pressure of 9 atmospheres to the initial state. KEFKIGEEATIJSG MACHINE. 455 Find: 1. Indicator diagram. 2. Volume at the beginning of adiabatic expansion. 3. Degrees of superheating. 4. Volume swept over by the piston. 5. Ratio of expansion. 6. Temperature at end of expansion. 7. Temperature at beginning of back stroke. 8. Degrees of superheating on back stroke. 9. Volume at end of refrigeration, so that when compressed adia- batically it will all be reduced to liquid at 7 atmospheres. 10. Temperature at end of back stroke when fluid is reduced to water. 11. Clearance. 12. Work done in the cycle. 13. Mean effective pressure. 14. Mean forward pressure. 15. Heat absorbed in the cycle. 16. Heat emitted. 17. Pounds of coal necessary to supply the heat for 100 *ycles, if each pound contains 14,500 heat units. 18. Pounds of water necessary to produce the refrigeration for 100 cycles temperature ranging from 60 to 80. 19. If length of stroke equals diameter of cylinder, find diameter of cylinder 20. The pounds of water necessary to develop a horse power if piston speed be 200 feet per minute. 21. The pounds of coal necessary to produce the horse power if a pound of coal has 1200 heat units and the efficiency of the furnace be 0.70. 16. A variety of exercises may be made similar to the preceding. We suggest the following : Let the volume of the vapor be the fraction of a pound. Let the expansion be entirely with saturated vapor and adia- batic. Let it be the curve of saturation, or a straight line entirely within the curve of saturation, or entirely without (superheated) ; or a line crossing the curve of saturation. Let the indicator card be entirely within the curve of saturation and bounded by any assumed lines. 17. A pound of water at 60 F. is within a closed vessel containing two cubic feet ; required the heat absorbed by the fluid in raising its temperature 335 F. From pp. 146 and 147 : d H= C 456 REFRIGERATING MACHINE. Let v = volume of mixture of liquid and vapor = 2 cubic feet. f a = specific volume of 1 pound vapor (see table). t>i = specific volume of liquid = 0.017 nearly for water. Then, r l x = t- L ^ = dr a b T He (Eq. (84)); II. - a b r, r*i d H, 77, d \ T (dlr)) Reduce and find, A n* I d n d r /r a dT l H= ' dri 18. If gas flows from one vessel into another through a short pipe, what must be the cross-section of the pipe that q pounds will flow in t seconds at constant temperature ? Let Q and Q be the volumes of the vessels, k the section, P' and p' the initial pressure, P' > p', P&udp the pressures at time t, /i the co- * efficient of velocity, Fthe velocity at the time t, and w the weight per cubic foot. Then, V = REFRIGERATING MACHINE. 457 These give which integrated between and t and o and q will give an equation from which k may be found. 19. Find the path of a fluid considered as a perfect gas when the heat absorbed xs n times the work done. Make npdv = d If in equations in p. 50, and find Y - n (y - 1) p v = B (a constant). Discuss making n = 1, 2, 5-9, oo . What value of n makes the path, a right line? Specific Heat of Aqua Ammonia. The mean of six deter- minations by Ludeking and Starr gives 0.886 (Am. Jour. Arts and Sc.). The value found by Hans von Strombuck was 1.22876 (page 337), which is nearly 50 per cent, larger than the above value. The value found by theory is nearly the mean of the two. The above experi- menters inform the author that they are not aware of any error in their own work, neither do either know of an error in the work of the other. This leaves the correct value in doubt, and one may con- sider it as unity until determined by further experiments. 458 LOGARITHMS OF NUMBERS. N 01231 56789 D 10 11 12 13 14 0000 043 086 128 170 414 453 492 531 569 792 828 864 899 934 1139 173 206 239 271 461 492 523 553 584 212 253 294 334 374 607 645 682 719 755 969 1004 1038 1072 1106 303 335 367 399 430 614 644 673 703 732 42 38 35 32 30 15 i<; 17 18 19 17G1 790 818 847 875 2041 068 095 122 148 304 330 355 380 405 553 577 601 625 648 788 810 833 856 878 903 931 959 9S7 2014 175 201 227 253 279 430 455 480 504 529 672 695 718 742 765 900 923 945 967 989 28 26 25 24 22 20 21 22 23 24 3010 032 054 075 096 222 243 263 284 304 424 444 464 483 502 617 636 655 674 692 802 820 838 856 874 118 139 160 181 201 324 345 365 385 404 522 541 560 579 598 711 729 747 766 784 892 909 927 945 962 21 20 19 19 18 25 26 27 28 29 3979 997 4014 4031 4048 4150 166 183 200 216 314 330 346 362 378 472 487 502 518 533 624 639 654 669 683 4065 4082 4099 4116 4133 232 249 265 281 298 393 409 425 440 456 548 5(54 579 594 609 698 713 728 742 757 17 16 16 15 15 30 31 32 33 34 4771 786 800 814 829 914 928 942 955 j69 5051 065 079 092 105 185 198 211 224 237 315 328 340 353 366 843 857 871 886 900 983 997 6011 5024 5038 119 132 145 159 172 250 263 276 289 302 378 391 403 416 428 14 14 13 13 13 35 36 37 38 39 5441 453 465 478 490 563 575 587 599 611 682 694 705 717 729 798 809 821 832- 843 911 922 933 944 955 502 514 527 53'J 551 623 635 647 658 670 740 752 763 775 786 855 866 877 888 899 966 977 988 999 coio 12 13 12 11 11 40 41 42 43 44 6021 031 042 053 064 128 138 149 160 170 232 243 253 263 274 335 345 355 365 375 435 444 454 464 474 075 083 096 107 117 180 191 201 212 222 284 294 304 314 325 385 395 405 415 425 484 493 503 513 522 11 10 10 10 10 45 46 47 48 49 6532 542 551 561 571 623 637 646 656 665 721 730 739 749 758 812 821 830 839 848 902 911 920 9:28 937 580 590 599 609 618 675 684 693 702 712 767 776 785 794 803 857 866 875 884 893 946 955 964 972 981 10 50 51 52 53 54 0990 098 7007 T016 7024 7076 084 093 101 110 160 168 177 185 193 243 251 259 267 275 324 332 340 348 356 7033 7042 7050 7059 70(57 118 126 135 143 152 202 210 218 226 235 284 292 300 308 316 364 372 380 388 396 8 8 8 LOGARITHMS OF NUAIBEKS. 459 N 01234 56789 D 55 56 57 58 59 7404 412 -419 427 435 482 490 497 505 513 559 566 574 582 589 634 642 649 657 664 709 716 723 731 738 443 451 459 466 474 520 528 536 543 551 597 604 612 619 627 672 679 686 694 701 745 752 760 767 774 8 8 8 7 7 60 61 62 63 64 7782 789 796 803 810 853 860 868 875 882 924 931 938 945 952 993 8000 8007 8014 8021 8062 069 075 082 089 818 8^5 832 839 840 889 896 90.^ 910 917 95.) 966 973 980 987 8028 8035 8041 8048 8055 096 102 109 116 122 7 65 66 67 68 69 8129 136 142 149 156 195 202 209 215 222 261 267 274 280 287 325 331 338 344 351 388 395 401 407 414 162 169 176 182 189 228 235 241 248 254 293 299 306 312 319 357 363 370 376 382 420 426 432 439 445 7 7 6 6 6 70 71 72 73 74 8451 457 463 470 476 513 519 525 531 537 573 579 585 591 597 633 639 645 651 657 692 698 704 710 716 482 488 494 500 506 543 549 555 561 567 603 609 615 621 627 663 669 675 681 686 722 727 733 739 745 6 6 6 6 6 75 76 77 78 79 8751 756 762 768 774 808 814 820 825 831 865 871 876 882 887 921 927 932 938 943 976 982 987 993 998 779 785 791 797 802 837 842 848 854 859 893 899 904 910 915 949 954 960 965 971 9004 9009 9015 9020 9025 6 6 6 5 80 81 82 83 84 9031 036 042 047 053 085 090 096 101 106 138 143 149 154 159 191 196 201 206 212 243 248 253 258 263 058 063 069 074 079 112 117 122 128 133 165 170 175 180 186 217 222 227 232 238 269 274 279 284 289 5 5 5 5 5 85 86 87 88 89 9294 299 304 309 315 345 350 355 360 365 395 400 405 410 415 445 450 455 460 465 494 499 504 509 513 320 325 330 335 340 370 375 380 385 390 420 425 430 435 440 469 474 479 484 489 518 523 528 533 538 5 5 5 5 5 90 91 92 93 94 9542 547 552 557 562 590 595 600 605 609 638 643 647 652 657 685 689 694 699 703 731 736 741 745 750 566 571 576 581 586 614 619 624 628 633 661 666 671 675 680 708 713 717 722 727 754 759 763 768 773 5 5 5 5 5 95 96 97 98 99 9777 782 786 791 795 823 827 832 836 841 868 872 877 881 886 912 917 921 926 930 956 961 965 969 974 800 805 809 814 818 845 850 854 859 863 890 894 899 903 908 934 939 943 948 952 978 983 987 991 996 5 5 4 4 4 460 LOGARITHMS OF NUMBERS. PROPORTIONAL PARTS. D 1 2 3 4 5 6 7 8 9 42 38 35 32 30 4.2 3.8 35 3.2 3.0 8.4 7.6 7.0 6.4 6.0 12.6 11.4 10.5 9.6 9.0 16.8 15.2 14.0 12.8 12.0 21.0 19.0 17.5 16.0 15.0 25.2 22.8 21.0- 19.2 18.0 29.4 26.6 24.5 22.4 21.0 33.6 30.4 28.0 25.6 24.0 37.8 34.2 31.5 28.8 27.0 28 26 25 24 22 2.8 2.6 2.5 2.4 2.2 5.6 5.2 5.0 4.8 4.4 8.4 7.8 7.5 7.2 (J.O 11.2 10.4 10.0 9.6 14.0 13.0 12.5 12.0 11.0 16.8 15.6 15.0 14.4 13.2 rj.o 18.2 17.5 16.8 15.4 22.4 20.8 20.0 19.2 17.6 25.2 23.4 22.5 21. G 19.8 21 20 19 18 17 2.1 2.0 1.9 .8 .7 4.2 4.0 8.8 3.6 3.4 6.3 6.0 5.7 5.4 5.1 8.4 8.0 7.6 7.2 6.8 10.5 10.0 9.5 9.0 8.5 12.0 12.0 11.4 10.8 10.2 14.7 14.0 13.3 12.6 11.9 10.8 16.0 15.2 14.4 13.6 18.9 18.0 17.1 16.2 15.3 16 15 14 13 12 .0 .5 A .3 .2 o.2 3.0 2.8 2.6 2.4 4.8 4.5 4.2 3.9 3.6 0.4 6.0 5.6 5.2 4.8 8.0 7.5 7.0 6.5 6.0 y.G 9.0 8.4 7.8 7.2 11.2 10.5 9.8 9.1 8.4 12.8 12.0 11.2 10.4 9.6 14.4 13.5 12.6 11.7 10.8 11 10 9 8 7 1.1 1.0 .9 .8 2.2 2.0 1.8 1.6 1.4 3.3 3.0 3.7 2.4 2.1 4.4 4.0 3.6 3.2 2.8 5.5 5.0 4.5 4.0 3.5 6.6 6.0 5.4 4.8 4.2 7.7 7.0 6.3 5.6 4.9 8.8 8.0 7.2 6.4 5.6 y.9 9.0 8.1 7.2 6.3 6 5 3 2 .6 .5 .4 .3 .2 1.2 1.0 .8 .6 .4 1.8 1.5 1.2 .9 .6 2A 2.0 1.6 1.2 .8 3.0 2.5 2.0 1.5 1.0 3.G 3.0 2.4 1.8 1.2 4.2 3.5 2.8 2.1 1.4 4.* 4.0 3.2 2.4 1.6 5.4 4.5 3.6 2.7 1.8 Modulus of the common system = 0.4342945. 1*10 -MOWS = 53^ e = 2.71828128 log l to 212 F. Specific Heat. Water, pure (at 39M F.). " sea, ordinary Alcohol pure w. 62.425 64.05 49 38 8. G. 1.000 1.026 791 E. 0.04775 0.05 1112 C. 1.000 0622 " proof spirit Ether 57.18 44 70 0.916 716 517 Mercury 848 75 13.596 0.018153 0033 Naphtha 52 94 II MS 434 Oil linseed 58 68 940 08 " olive 57.12 915 0.08 ' ' whale 57 62 0.923 " of turpentine Petroleum 54.31 54 81 0.870 878 0.07 .434 Ice 57 5 92 .504 Brass 487 to 533 7.8 to 8 5 .00216 Bronze 524 8.4 .00181 Copper 537 to 556 8 6 to 8.9 .00184 0951 Gold 1186 to 1224 19 to 19 6 0015 0298 Iron cast 444 7.11 .0011 480 7 69 0012 1138 Lead 712 11 4 0029 0293 1311 to 1373 21 to 22 0009 0314 Silver 655 10 5 002 0557 Steel 490 785 0012 Tin 462 74 0022 0514 Zinc 436 7 2 00294 0927 PROPERTIES OF GASES CONSIDERED PERFECT. 463 TABLE IY. PROPERTIES OP GASES CONSIDERED PERFECT. I. NAME op THE GAS. II. Chemical Compo- sition. III. Density. IV. V. Specific Heat at Constant Pressure VI VII. Specific Heat j.t C'nstantVohm e compared corn- Weight pared for Volume Weight for Vol- with umewith Water. Air. com- pared Weight for Weight wia Water. com- pared Volume for Vo'.mr.e with Air. Atmospheric Air O xyi;en 0, N a 1 1.1056 0.9713 0.0692 2.4502 5.4772 1.0384 0.9673 1.2596 1.5201 1.5241 0.6219 2.2113 1.1747 2.6258 0.5527 4.1244 0.9672 0.5894 2.6942 4.6978 1.1055 1.5890 2.5573 3.1101 2.2269 3.7058 3.4174 2.0036 3.0400 5.8833 4.7464 6.2667 6.6402 8.9654 0.2375 0.21751 0.24380 3.40900 12099 0.05552 0.2317 0.2450 0.1852 0.2169 0.2262 0.4805 0.1544 0.2432 0.1569 0.5929 0.1567 0.4040 0.5084 0.3754 0.5061 0.4580 0.4534 0.4797 0.4008 (1.2738 0.1896 0.2293 0.4125 0.4008 0.1322 0.1347 0.1122 0.1290 0.0939 1 1.013 0.997 0.993 1.248 1.280 1.013 0.998 0.982 1.39 1.45 1.26 1.44 1.20 1.74 1.38 2.72 1.75 1.26 4.26 10.01 2.13 3.03 5.16 5.25 2.57 2.96 3.30 3.48 5.13 3.27 2.69 2.96 3.61 3.54 0.1689 0.1551 0.1727 2.411 0.0928 0.0429 0.1652 0.1736 0.1304 0.172 0.181 0.370 0.123 0.184 0.131 0.468 0.140 0.359 0.393 0.350 0.491 0.395 0.410 0.453 0.379 0.243 0.171 0.209 0.378 0.378 0.120 0.120 0.101 0.119 0.086 1 1.018 0.996 0.990 1.350 1.395 1 018 0.997 0.975 1.55 1.64 1.36 1.62 1.29 2.04 1.54 3.43 2.06 1.37 5 60 13.71 2.60 3.87 6.87 6.99 3 21 3.76 4.24 4.50 6.82 4 21 3.39 3.77 4.67 4.59 Nitrogen Hydrogen Ho 01, Br a NO C O HC1 CO, N 2 O H 2 O SO, H a S cs C H 4 C H C1 3 C 2 H 4 NH, CH 6 C, Hi. CH 4 O C H 6 C H 10 C H 10 S Chlorine Bromine Nitric Oxide Carbonic Oxide Hydrochloric Acid... Carbonic Acid Nitric Acid Steam Sulphuric Acid Hydro-sulphuric Acid Carbonic Di-sulphide. Carburetted Hydrogen Chloroform Olefiant Gas Ammonia Gas Benzine Oil of Turpentine Wood Spirit Alcohol Ether Ethyl Sulphide Ethyl Chloride C H 5 C1 C H 5 Br C H.C1 S C H B C H 8 a SiCl 3 PC1 3 As C1 3 TiCl 4 SnCU Ethyl Bromide Dutch Liquid Aceton Butyric Acid Tri chloride of Silicon Tri-chloride of Phos- j phorus ... )' Tri-chloride of Arsenic Tetra-chloride of Ti- ) tanium . ... j Tetra-chloride of Tin . i04 TABLES. TABLE Y. SATURATED STEAM. p, Pressure per square inch. T, Temperature degrees F., Eq. (81), page 97. v. Volume of a pound of saturated steam in cubic feet, equation (84), in which ;> is pounds per square foot. tr, Weight of a cubic foot of steam = - //, The heat in one pound of liquid above 32 F. in thermal units, equation (9.~>), page 111. The values, however, are the direct results of Kegnault's experiments. A e . The latent heat of evaporation in thermal units, equation (10), page 409. h, The total heat of steam above 32 F. l>eing equal to h -f ?i e . SATURATED STEAM. P- T. V. w. h. Ae. h. 0.089 32.0 3b87. 000029 0.0 1091.7 1091.7 0.2 54.0 1482. 0.00066 22.1 1076.3 1098.4 1 102.0 335. 0.00299 70.0 1043.0 1113.1 5 162.3 73. 0.01366 130.7 1000.8 1131.5 10 193.2 38. O.OM31 161.9 979.0 1140:9 15 213.0 26.2 0.03826 181.8 965.1 i 1146.9 20 228.0 19.9 0.05023 196.9 954.6 1 1151 5 25 240.0 16.1 0.06199 209.1 946.0 j 1155.1 30 250.3 13.6 0.07360 219.4 938.9 1158.3 35 259.2 11.8 0.08508 228.4 932.6 1161.0 40 267.1 10.4 0.09644 236.4 927.0 1163.4 45 274.3 9.3 0.1077 243.6 922.0 1165.6 50 280.9 8.4 0.1188 250.2 917.4 1167.6 55 286.9 7.7 0.1299 256.3 913.1 1169.4 60 292.5 7.1 0.1409 261.9 909.3 1171.2 65 297.8 6.6 0.1519 267.2 905.5 1172.7 70 302.7 6.15 0.1628 272.2 902.1 1174.3 75 307.4 5.70 0.1736 276.9 898.8 1175.7 80 310.9 5.49 0.1843 281.4 895.6 1177.0 85 315.2 5.18 0.1951 285.8 892.5 1178.3 90 320.0 4.858 0.2058 290.0 889.6 1179.6 95 323.9 4.619 0.2165 294.0 886.7 1180.7 100 327.6 4.403 0.2271 297.9 884.0 1181.9 105 331.1 4.206 0.2378 301.6 881.3 1182.9 110 334.56 4.026 0.2488 305.2 878.8 1184.0 115 337.86 3.862 0.2589 308.7 876.3 1185.0 120 341.05 3.711 i;'.i.-, 312.0 S74.0 11S.0 SATUKATED STEAM. SATURATED STEALS Continued. 465 p- r. V. w. h. he. feu 125 130 135 344.13 347.12 350.03 3.572 3.444 3.323 0.2800 0.2904 0.3009 315.2 318.4 321.4 871.7 869.4 867.3 1186.9 1187.8 1188.7 140 145 150 352.85 355.59 358.26 3.212 3.107 3.011 0.3113 0.3218 0.3321 324.4 327.2 330.0 865.1 863.2 861.0 1189.5 1190.4 1191.2 155 160 165 360.86 363.40 365.88 2.919 2.833 2.751 0.3426 0.3530 0.3635 332.7 335.4 338.0 8593 857.4 855.6 1192.0 1192.8 1193.6 170 175 180 368.29 870.65 372.97 2676 2.603 2.535 0.3737 0.3841 0.3945 340.5 343.0 345.4 853.8 852.0 850.3 1194.3 11950 1195.7 185 190 195 375.23 377.44 379.61 2.470 2.408 2.349 0.4049 0.4153 0.4257 347.8 350.1 352.4 848.6 847.0 845.3 1196.4 1197.1 1197.7 200 205 210 381.73 383.82 385.87 2.294 2.241 2.190 0.4359 0.4461 0.4565 354.6 356.8 358.9 843.8 842.2 840.7 1198.4 1199.0 1199.6 215 220 225 387.88 389.84 391.79 2.142 2.096 2.051 0.4669 0.4772 0.4876 361.0 363.0 365.1 839.2 837.8 836.3 1200.2 1200.8 1201.4 230 235 240 393.69 395.56 397.41 2.009 1.968 1.928 0.4979 0.5082 0.5186 367.1 369.0 371.0 834.9 833.6 832.2 120.2.0 1202.6 1203.2 2-15 250 255 399.21 400.99 402.74 1.891 1.854 1.819 0.5289 0.5393 0.5496 372.8 374.4 376.5 830.9 829.5 8283 1203.7 1204.2 1204.8 260 265 270 404.47 406.17 40785 1.785 1.753 1.722 0.5601 0.5705 0.5809 378.4 380.2 381.9 826.9 825.6 824.4 1205.3 1205.8 1206.3 275 280 285 409.50 411.12 412.72 1.691 1.662 1.634 0.5913 0.6020 0.612 383.6 385.3 387.0 823.2 821.8 820.6 1206.8 1207.1 12076 290 295 300 414.32 415.87 417.42 1.607 1.580 1.554 0.622 0.633 0.644 - 388.6 390.3 391.9 819.5 818.3 817.2 1208.1 1208.6 1209.1 305 310 315 418.92 420.42 421.92 1.529 1.505 1.481 0.654 0.664 0.675 393.5 395.0 396.6 816.1 815.1 814.0 1209.5 1210.1 1210.6 320 325 330 335 42337 424.82 426.24 ' 427.64 1.459 1.437 1.415 1.395 0.685 0.696 0.707 0.717 398.1 399.6 401.1 402.6 813.0 811.7 810.6 809.5 1211.1 1211.3 1211.7 1212.1 466 SATURATED AMMONIA. "5 rf.2 l=ll-< -' 9 I H ><* ojgoo > Tj 1 t Oi O > IO t- 00 rl OCO SO 53 O ^ iffl 00 CO "* I-H 00 CD -^ TH i 00 ** ~H 00 -rt< -< 00 * >-! 00 3J '** sss ^^ ^^^ ISco SSo S'SJS SS ^i r t- SO 00 IO JO IO O 09 IO IO OJ ^ SO CXJ SO ' CO IO OJ > ai 00* t- O9* O 10' IO O .8 31 ig 1 iPi 5 10' O 10' O JO O JO O JO' O I p 10 O 10 O JO O 10 2S III III II SATURATED AMMONIA. 467 :5oi o 30 co NO O t- c-> OS O CO O 30 CO -^ CO CO CO CO CJ O5 O o' Tt< O f~ CO O' O CO O" SO OT OS o' C>"j ODCOt- OCQCO COWOS OTHTH OSr-HOl * * L- ^_ T^ I- CO O O CQ 30 CO CO O' l^ Tji i-i J> -^ T-H 30 O' i-I OO' Iff O? OO' o' CQ 30 Iff d 30' Iff SQ GO Iff r^ X SSr^ SS SSS SS^ ^^^ ^^^ ^^^ ^-^^ s's'i c8Si| sgg s'|| III s|| s'li ill s'Ji cs sO*OiO cOtOCO COOCO OOO OCOO OOO OOO OOO OOO ooo ooo =000 ooo Soo ooo ooo o_oo ooo o o iff o o' o" o o' o o* o' o' o' o* o ooo' ooo ooo' o o o 468 SATURATED VAPOR OF SULPHUR DIOXIDE. 51 II SK 0000 000 000 000 1-." ci 06 t- cb o id -* Tji 06 eo e* t- t- t- 00 1-1 in so eo co so "* -^< -^ **' S3 I ill ,_, r-i T-irti-c C* ?i Ti oo t^ co 00 co TH is T* oo o} O -^ 1O t~ i-HOOlO 1-HOOM* T-lt-T^ T-I 00 O (MOS-^ -^(??OS t-lO eoco^ otoco t-c-oo ososo -r;"^ SS^3^ ^^ T I TH T I r-* T I TH T I TH TH THTH(M GQC^CQ dC^C- l> 10* OS t-^ ^' -i-i 00 10 TH THTHO OOO O5O5 TJH Tji co co co co co ci d T^COC<} c^ioco cocQ^t* o c^co ot rji CS 1O TH l> -rj< TH O5 t- CO 1O' 1O' 1O' O 1O1OCO >C-GO OSO5O THWCO Tf IO OOOOC^ OOTHGQ lOOS-rH lOcMCO ^O5CO 1COOS OOCO THCQO aooooo oit-oo cjoo oscsco COOJCQ r~co o o od oi * 10 CD -^" -*' 06 COO-rH i>OOCO COOOOS IOJ>1O OiOit -rHCMOO OOCQ ^^lO !OCOi> -OOC5 O-rHCQ CO-^1O COOOOS OOi g g 3g g g OiO o'lOO" ICO' rt<-^i 1O1OCO COt- CO CO CO CO IO O IO o' O W 0100 10010 0100 100 10 0100 >00 lOiO CO5D- C-0000 OSOSO OTHTH MANUFACTURES. 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