LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class 
 
A TREATISE ON HYDRAULICS 
 
AGENTS 
 
 AMERICA . THE MACMILLAN COMPANY 
 
 64 & 66 FIFTH AVENUE, NEW YORK 
 
 CANADA . THE MACMILLAN COMPANY OF CANADA, LTD. 
 27 RICHMOND STREET WEST, TORONTO 
 
 INDIA . . MACMILLAN AND COMPANY, LTD. 
 MACMILLAN BUILDING, BOMBAY 
 309 Bow BAZAAR STREET, CALCUTTA 
 
A TREATISE 
 
 ON 
 
 HYDKAULICS 
 
 BY 
 
 WILLIAM CAWTHORNE UNWIN, LL.D., F.E.S. 
 
 \ 
 
 KMERITUS PROFESSOR OF THE CENTRAL TECHNICAL COLLEGE ; M. INST. CIVIL ENGINEERS 
 
 HON. M. INST. MECHANICAL ENGINEERS ; HON. M. AM. INST. OF MECHANICAL ENGINEERS 
 
 FOREIGN HON. M. AM. ACADEMY OF ARTS AND SCIENCES 
 
 M, INST. OF ELECTRICAL ENGINEERS 
 
 OF THE 
 
 ( UNIVERSITY ) 
 
 OF 
 
 LONDON 
 ADAM AND CHARLES BLACK 
 
 1907 
 
PREFACE 
 
 IN the present treatise the author returns to a subject which 
 occupied his attention at intervals during a long period, and 
 which always seemed attractive. Not only has Hydraulics 
 formed part of his teaching, but he was much engaged in the 
 early sixties in designing water turbines and centrifugal 
 pumps, and has had on many occasions since to consider 
 questions of flow, storage, and measurement of water. In 
 18*76, he wrote the article " Hydraulics " for the ninth edition 
 of the Encyclopedia JSritannica, which he has reason to think 
 has been useful to many engineers. 
 
 Strictly rational hydrodynamics, so far as it has been 
 developed, is concerned mainly with fluids deprived of viscosity, 
 and leads to results flagrantly at variance with the action of 
 actual fluids. Hence in dealing with the practical problems 
 of hydraulics the engineer has recourse to comparatively 
 simple mechanical principles and simplified assumptions 
 which furnish rough formulae, which can be modified by 
 empirical constants so as to be true to the necessary approxima- 
 tion over any required range of conditions. There now exists 
 an enormous mass of experimental data relating to hydraulic 
 problems, which has been accumulated during a period extend- 
 ing over two centuries, and which is of very varying trust- 
 worthiness and importance. It is really on the results of 
 these investigations that the engineer relies in deciding the 
 questions which arise in many branches of professional work, 
 and theoretical formulae only render partial assistance in 
 reducing to intelligibility and order the mass of empirical 
 
 v 
 
 175546 
 
vi HYDKAULICS 
 
 observations. The difficulty in treating hydraulics appears 
 to the author to lie in the need of giving a sufficient account 
 of experimental investigations to enable a student to realise 
 the limitations of formulae, and the degree of confidence which 
 can be placed in calculations, without getting involved in a 
 cumbrous and confusing amount of empirical details. 
 
 Full references have been given to primary sources of 
 information, in order that students may supplement the neces- 
 sarily brief statements in the text, by consulting the fuller 
 details in original memoirs. 
 
 As to what is special in the present treatise, the author 
 thinks it important that the problems concerning the flow of 
 incompressible fluids, and the closely related problems dealing 
 with compressible fluids, should be treated together. The 
 practical importance of the latter class of problems has 
 increased considerably in recent years. 
 
 To most of the chapters numerical examples have been 
 added, selected from those which the author has set for his 
 students during many years past. 
 
 July 1907. 
 
CONTENTS 
 
 INTRODUCTION 
 
 UNITS OF MEASUREMENT 
 
 PAGE 
 
 Conversion of English and metric measures Units of volume 
 Heaviness of water Change of density with change of 
 temperature Atmospheric pressure Acceleration due to 
 gravity Transformation of equations . 1 
 
 CHAPTER I 
 
 PROPERTIES OF FLUIDS 
 
 Perfect fluid Gaseous and liquid fluids Compressibility of 
 liquids Viscous fluids Free surface Fluid pressure 
 Pressure on a plane Pressure on curved surfaces Pipe 
 bends Hydraulic press 1 . . . -. 9 
 
 CHAPTER II 
 
 DISTRIBUTION OF PRESSURE IN A LIQUID VARYING WITH 
 THE LEVEL 
 
 Pressure column Free surface level Watt's hydrometer 
 Mercury siphon gauge Pressure on surfaces varying as 
 the depth from the free surface Pressure on a wall 
 Pressure on a flap valve Centre of pressure Graphic 
 determination of pressure on surfaces Buoyancy Equi- 
 librium of floating bodies . . . . .20 
 
 CHAPTER III 
 
 PRINCIPLES OF HYDRAULICS 
 
 Two modes of motion of water Uniform and varying motion 
 Steady and unsteady motion Volume and mean velocity of 
 flow Bernoulli's theorem The term Head Variation of 
 
 vii 
 
viii HYDEAULICS 
 
 PAGE 
 
 pressure across the stream lines Eadiating current Free 
 circular vortex Free spiral vortex Forced vortex Venturi 
 meter Principle of the Conservation of Momentum Abrupt 
 changes of section Labyrinth piston packing . . 36 
 
 CHAPTEE IV 
 
 DISCHARGE FROM ORIFICES 
 
 Experimental observations Coefficients of velocity and resistance 
 Coefficient of contraction Coefficient of discharge Use 
 of orifices in measuring water Measurement of head Bell- 
 mouths and conoidal orifices Sharp-edged orifices Self- 
 adjusting orifices Flow of liquids other than water 
 Imperfect contraction Partially suppressed contraction 
 Minimum coefficient of contraction Application of theorem 
 of Bernoulli to flow from orifices Discharge from a fire 
 nozzle Flow when the head varies with time Cylindrical 
 mouthpiece Convergent mouthpieces Divergent mouth- 
 piece Influence of temperature on flow from orifices . 61 
 
 CHAPTEE V 
 
 NOTCHES AND WEIRS 
 
 Large vertical rectangular orifices Notches or weirs Velocity of 
 approach Partially submerged orifices Drowned weirs 
 Broad-crested weirs Rafter's experiments on broad-crested 
 weirs Triangular notches Notch with no end contractions 
 Francis's formula Bazin's researches on weirs Practical 
 gauging by weirs Separating weirs . . .95 
 
 CHAPTEE VI 
 
 STATICS AND DYNAMICS OF COMPRESSIBLE FLUIDS 
 
 Heaviness of gases Specific heats of gases Boyle's law Dalton's 
 law Charles's law Mercurial barometer Measurement of 
 heights by barometer Flow of air from orifices under small 
 differences of pressure Expansion of compressible fluids 
 Modification of theorem of Bernoulli for compressible fluids 
 Flow from orifices when the variation of density is taken 
 into account . . . . . .118 
 
 CHAPTEE VII 
 
 FLUID FRICTION 
 
 General laws Froude's experiments Friction of discs rotated in 
 
 water Influence of temperature . . . .132 
 
CONTENTS ix 
 
 CHAPTER VIII 
 
 FLOW IN PIPES 
 
 PAGE 
 
 Non-sinuous motion of water Flow in pipes when the motion is 
 turbulent Steady flow in pipes of uniform diameter Chezy 
 formula Pipe connecting two reservoirs Inlet resistance 
 Darcy's investigation Maurice Levy's formula Later deter- 
 minations of the coefficient for pipes Herschel's gaugings of 
 a steel pipe Timber stave pipes Fire-hose pipes Practical 
 calculations of flow in pipes Secondary losses of head in 
 pipes Valves, cocks, and sluices . . . .146 
 
 CHAPTER IX 
 
 DISTRIBUTION OF WATER BY PIPES 
 
 Town supply Supply main Break-pressure reservoirs Main of 
 different diameters Main when the discharge decreases along 
 the length Pipe connecting two reservoirs and delivering 
 water en route Branched pipe Hydraulic gradient of pipe 
 of variable diameter Cost of mains Corrosion and incrusta- 
 tion Pipe aqueducts Vyrnwy, East Jersey, and Coolgardie 
 aqueducts Pumping main Suction pipe of pumps Water 
 hammer ....... 177 
 
 CHAPTER X 
 
 LATER INVESTIGATIONS OF FLOW IN PIPES 
 
 Kutter's formula Defects of Chezy formula Experimental data 
 available Method of dealing with the data Correction for 
 temperature Variation of resistance with diameter Values 
 of m General formula of flow and values of constants 
 Distribution of velocity in cross section of pipe Influence 
 of temperature on resistance . . . .199 
 
 CHAPTER XI 
 
 FLOW OF COMPRESSIBLE FLUIDS IN PIPES 
 
 Flow in pipes under small differences of pressure Flow of 
 lighting gas in mains Coefficient of friction in gas mains 
 Flow of air in a long pipe when the variation of density is 
 taken into account Variation of velocity and pressure in 
 long air mains Coefficient of friction Distribution of 
 velocity ....... 221 
 
x HYDKAULICS 
 
 CHAPTEE XII 
 
 UNIFORM FLOW OF WATER IN CANALS AND CONDUITS 
 
 PAGE 
 
 Steady flow of water in channels of constant slope and section 
 Darcy's research on the value of for open channels 
 Ganguillet and Kutter's formula Bazin's later investigation 
 of flow in channels Channels of circular section Egg-shaped 
 channels or sewers Trapezoidal channels Minimum section 
 Discharge of a channel with different depths of water 
 flowing Parabola of discharge Distribution of velocity in 
 cross section Depression of point of greatest velocity 
 Vertical velocity curve Transverse velocity curve Ratio 
 of mean and surface velocities Aqueducts Eoman aqueducts 
 Loch Katrine, Thirlmere, and New Croton aqueducts 
 River bends 231 
 
 CHAPTER XIII 
 
 GAUGING OF STREAMS 
 
 Water-level gauge Mean velocity calculated from longitudinal 
 slope Measurement of transverse sections Float gauging 
 Float paths Screw current meter Harlacher screw current 
 meter Current meter of J. Amsler Laffon Calibrating the 
 screw current meter Pitot tube and Darcy gauge Surface 
 and mean velocities in a stream Velocities on one vertical 
 Surface or rod float gauging Calculation of discharge 
 from vertical velocity curves and from contours of equal 
 velocity Gauging streams by chemical means . .264 
 
 CHAPTER XIV 
 
 IMPACT AND REACTION OF FLUIDS 
 
 Jet deviated in one direction Jet impinging on a solid of revolu- 
 tion Special cases Jets impinging on plane and on hemi- 
 spherical cups Pressure of a steady stream of limited section 
 on a plane Pressure of an unlimited stream on a solid at 
 rest Stanton's experiments Pressure on solids of various 
 forms Pressure on planes oblique to the direction of the 
 stream Wind pressure Evidence of high pressures in 
 storms Forth Bridge experiments . . . .291 
 
CONTENTS xi 
 
 APPENDIX 
 
 PAGE 
 
 TABLE I. FUNCTIONS OF NUMBERS FROM 0-1 TO 10-0 . 317 
 
 II. VELOCITY AND HEAD . . . .318 
 
 III. SLOPE TABLE . . . . .319 
 
 IV. TABLE TO FACILITATE CALCULATIONS ON PIPES . 320 
 V. DISCHARGE OF PIPES AT DIFFERENT VELOCITIES 
 
 IN GALLONS PER HOUR. ; . .321 
 
 VI. DISCHARGE OF PIPES AT DIFFERENT VELOCITIES IN 
 
 CUBIC FEET PER SECOND . . . 322 
 
 VII. Loss OF HEAD IN NEW CAST-IRON PIPES . 323 
 
 VIII. Loss OF HEAD IN INCRUSTED CAST-IRON PIPES . 324 
 
 INDEX 325 
 
INTRODUCTION 
 
 UNITS OF MEASUREMENT 
 
 1. IN (practical hydraulics the most convenient units of 
 measurement are the foot, the pound, and the second. In this 
 treatise these units are used throughout, except in a few cases 
 where other units are specially mentioned. 
 
 It happens that a great number of memoirs on hydraulics 
 are in French or German, very important researches having 
 been carried out abroad. In such memoirs metric units are 
 employed. Hence a student of hydraulics finds it necessary to 
 become more or less familiar with formulae expressed in either 
 English or metric units, and often has to convert formulae 
 from one system to the other. For that reason some particu- 
 lars of the conversion factors from metric to English units are 
 given. The convenient units in the metric system are the 
 metre, the kilogram, and the second. 
 
 To avoid confusion, the secondary units employed should 
 be the square foot (square metre), cubic foot (cubic metre), 
 foot per second (metre per second), pound per square foot 
 (kilogram per square metre). But in certain cases, especially 
 in dealing with air and steam, the pound per square inch 
 (kilogram per square centimetre) is almost in universal use, 
 though in some respects inconvenient. 
 
 The following table gives the relation of the English 
 and metric units, and the logarithms of the factors for 
 conversion : 
 
HYDEAULICS 
 
 INTROD. 
 
 1 
 
 O 
 
 CD 
 
 o 
 
 CM 
 
 O 
 
 00 
 
 CO 
 CO 
 
 8 
 
 GO 
 
 i I 
 
 CO 
 
 o 
 
 CO 
 
 
 i-H 
 
 s 
 
 
 F 1 
 
 CO 
 
 rt< 
 
 Tf 
 
 
 CD 
 
 Qq 
 
 o 
 
 
 i-H 
 
 O5 
 
 03 
 
 >o 
 
 9 
 
 
 CO 
 
 o 
 
 00 
 
 op 
 
 1 1 
 
 
 CO 
 
 
 3 
 
 6 
 
 
 pH 
 
 6 
 
 6 
 
 6 
 
 IrH 
 
 ^ 
 
 
 Ir-" 
 
 ICM 
 
 * 
 
 00 
 
 
 
 CD 
 
 00 
 
 
 
 
 
 oo 
 
 rJH 
 
 t-j 
 
 o 
 
 "^f 
 
 O 
 
 
 
 
 ^ 
 
 Qq 
 
 *o 
 
 
 
 CM 
 
 ft 
 
 00 
 
 CD 
 
 
 
 
 00 
 
 J^ 
 
 ^N 
 
 CM 
 
 
 o 
 
 CO 
 
 s 
 
 CM 
 
 1^ 
 
 CO 
 
 
 CM 
 
 G<l 
 
 CO 
 
 CM 
 
 
 CM 
 
 p 
 
 pj 
 
 CO 
 
 6 
 
 o 
 
 CM 
 
 w 
 
 * 
 
 6 
 
 Tt< 
 
 
 6 
 
 
 
 s 
 
 
 14 
 
 CO 
 
 
 
 
 
 I ( 
 
 
 
 
 
 
 ' 
 
 
 
 1 
 
 
 
 ft 
 
 3 
 ^d 
 
 
 
 | 
 
 1 
 
 
 
 ' 
 
 
 - 
 
 1 
 
 1 
 
 02 
 
 1 
 
 g 
 
 
 I 
 
 1 
 
 
 
 1 
 
 _^ 
 
 
 ! 
 
 1 
 
 1 
 
 1 
 
 
 "S 
 
 3 
 g 
 
 
 
 1 
 
 1 
 
 
 1 
 
 -4-3 
 
 O 
 
 o 
 
 3 
 
 I 
 
 ~ 
 
 o "o 
 
 I 
 
 
 
 
 1 
 
 ^3 
 
 g 
 
 CQ 
 
 3 
 
 3 
 
 3 
 
 1 
 
 .S 
 
 O^ P-i 
 
 a 
 
 o 
 
 
 
 3 
 
 3 
 
 1 
 
 a 
 
 g 
 
 a 
 
 cr 1 
 
 OQ 
 
 IH 
 
 QQ Cu 
 
 
 
 1 
 
 3 
 
 .a 
 
 metres i 
 
 _d 
 1 
 
 3 
 d 
 
 02 
 
 i 
 1 
 
 o 
 
 a 
 
 E 
 
 02 d 
 
 OB 
 
 I 
 
 al 
 
 ll 
 
 ft 
 
 to ^3 
 
 
 1 
 
 i 
 
 _0 
 
 1 
 
 43 
 
 1 
 
 ll 
 
 f 
 
 d 
 1 
 
 & 
 
 r3 ft 
 
 rS ft 
 
 
 1 
 
 m 
 
 O 
 
 3 
 
 1 
 
 3 
 
 W 
 
 3 
 
 
 3 
 
 w 
 
 a 
 
 o 
 
 o 
 
 ! 1 
 
 a. 
 
 
 
 ^ 
 
 J>- 
 
 o 
 
 
 CO 
 
 CD 
 
 g 
 
 ^ 
 
 00 
 
 Q^ 
 
 CO 
 
 
 o 
 
 (M 
 
 
 
 CO 
 
 
 *fi 
 
 oo 
 
 CD 
 
 o 
 
 IO 
 
 00 
 
 
 
 ^4 
 
 
 oo 
 
 O 
 
 m 
 
 Tt< 
 
 < 
 
 Tt* 
 
 CO 
 
 ^f 
 
 I 1 
 
 rH 
 
 00 
 
 
 CO 
 
 
 3 
 
 IrH 
 
 
 ICM 
 
 M 
 
 liH 
 
 IfH 
 
 6 
 
 ICM 
 
 
 6 
 
 r 1 
 
 1 
 
 00 
 
 C5 
 
 CO 
 
 CO 
 
 oo 
 
 >o 
 
 CM 
 
 CM 
 
 CO 
 
 
 CO 
 
 
 1 
 
 o 
 
 CO 
 
 i 
 
 oo 
 
 CM 
 
 9 
 
 CO 
 
 
 CO 
 
 oo 
 
 CO 
 
 i i 
 
 00 
 
 CO 
 
 
 
 o 
 
 
 CM 
 
 oo 
 op 
 
 Oi 
 r- 1 
 
 
 
 i 
 
 6 
 
 6 
 
 
 
 6 
 
 6 
 
 6 
 
 1 1 
 
 o 
 
 
 "* 
 
 CO 
 r- 1 
 
 
 ' 
 
 
 
 ' 
 
 1 
 
 
 o> 
 
 ft 
 
 n 
 
 1 
 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 I 
 
 -s 
 
 & 
 
 3 
 
 
 3 
 
 3 
 
 
 
 1 
 
 1 
 
 1 
 
 8 
 
 a 
 a 
 
 1 
 
 3 
 .3 
 
 
 3 
 
 d 
 
 3 
 
 .a 
 
 
 
 g 
 
 "S 
 
 a 
 
 3 
 
 s 
 
 a 
 
 4 ,. 
 
 If 
 
 rd 
 " 
 
 2 
 -t-a 
 
 43 
 
 8 
 
 
 
 
 c3 
 
 ^ 
 
 H 
 
 
 i^ 
 
 
 d 
 
 <D 
 
 tS 
 
 o 
 
 
 0) 
 
 a 
 ^ 
 % 
 
 1 
 
 3 
 
 rH 
 
 43 
 
 
 
 O- 1 
 
 M 
 
 1 
 
 ~s 
 s 
 
 ^0 
 
 d 
 
 o 
 
 Mass 
 Pounds avoirdupois 
 
 i 
 i 
 
 4 i 
 
 1 s. 
 
 43 
 
 <O 
 
 & 
 
 3 " 
 
 1 r " 
 
 ll < 
 ^ ft K 
 
 I 
 
 1? 
 
 fS 
 
 Pressure 
 Pounds per square i 
 
 per square centim 
 
 II 
 ^ 
 
 Jl 
 
 n3 ^ 
 |ft 
 
 fS 
 
 Heaviness 
 Pounds per cubic f 
 per cubic metre 
 
INTROD. 
 
 UNITS OF MEASUEEMENT 
 
 2. Units of volume. In most hydraulic calculations 
 the convenient unit of volume is the cubic foot (or cubic 
 metre). But in water-supply engineering it has been 
 customary to use the gallon as the volume unit. The imperial 
 gallon is defined to be the volume of 10 Ibs. of distilled water 
 at 62 F. Hence, if in general calculations the cubic foot of 
 water is taken to weigh 62*4 Ibs., it must also be taken to 
 be equivalent to 6 '2 4 gallons. In the metric system the 
 kilogram is the weight of a cubic decimetre of water at 
 39'l F. Hence a cubic metre of water of maximum density 
 weighs 1000 kilograms, and this value is taken in general 
 calculations on pressure, etc., though at ordinary temperatures 
 the weight is slightly less. In the United States the wine 
 gallon, now disused in England, is the ordinary unit of 
 volume, and is equal to 0'8333 imperial gallon. 
 
 CONVERSION TABLE 
 
 
 Multiplier. 
 
 Logarithm. 
 
 Cubic feet into imperial gallons 
 Cubic feet into U.S. gallons 
 Cubic feet into cubic metres 
 
 6-24 
 7-49 
 0-02832 
 
 0-7952 
 0-8744 
 2-4521 
 
 Imperial gallons into cubic feet 
 U.S. gallons into cubic feet .... 
 Cubic metres into cubic feet .... 
 
 0-1603 
 0-1336 
 35-31 
 
 T-2048 
 T-1256 
 1-5479 
 
 U.S. gallons into imperial gallons 
 Imperial gallons into U.S. gallons 
 
 0-8333 
 
 1-200 
 
 1-9208 
 0-0792 
 
 To convert imperial gallons per 24 hours into cubic feet per second 
 
 divide by 539,200. 
 To convert U.S. gallons per 24 hours into cubic feet per second divide by 
 
 647,100. 
 
 3. Heaviness of water. In ordinary hydraulic calcula- 
 tions it is usual to disregard the small variations of density 
 of water due to changes of pressure and temperature. In 
 this treatise the weight of a cubic foot of water will be 
 denoted by G- and will be taken at 62*4 Ibs. In calculations 
 on the metric system the weight of a cubic metre is generally 
 taken at 1000 kilograms from the simplicity which this 
 
HYDKAULICS 
 
 INTEOD. 
 
 introduces into calculation. Eiver and spring water is not 
 sensibly denser than pure water unless in exceptional cases 
 or when carrying mud or sewage. Sea water is usually 
 taken at 64 Ibs. per cubic foot, though its density varies 
 somewhat in different localities. 
 
 Generally V cubic feet of water weigh G-V Ibs. in gravita- 
 tion units. In treatises on theoretical hydromechanics absolute 
 units are employed. Then if M is the mass in poundals, the 
 weight is W = M.g Ibs. where g is the acceleration due to 
 gravity in the locality considered. Hence if p is the density 
 or mass of unit volume its weight is gp, and V units of volume 
 weigh gpV Ibs. 
 
 ANALYSES OF SOME TYPICAL WATERS IN PARTS PER 100,000 
 
 
 Total Solids in 
 Solution. 
 
 Temporary 
 Hardness. 
 
 Total 
 Hardness. 
 
 
 Rain water . 
 
 2-9 
 
 
 0-3 
 
 
 Loch Katrine 
 
 3-1 
 
 
 1-0 
 
 From Moorland 
 
 Manchester . 
 
 6-2 
 
 o-i 
 
 3-7 
 
 
 
 Liverpool . 
 
 8-5 
 
 o-i 
 
 3-7 
 
 )> 
 
 London, from 
 
 25-8 
 
 
 19-2 
 
 Thames and Lea 
 
 to 
 
 29-4 
 
 
 19-9 
 
 
 
 London 
 
 40-3 
 
 
 28-7 
 
 Chalk wells 
 
 Northampton 
 
 57-8 
 
 8-6 
 
 10-3 
 
 Well in Lias 
 
 Sea water . 
 
 3898-0 
 
 49-0 
 
 797-0 
 
 
 Change of volume and density of water with change 
 of temperature. Water expands and decreases in density 
 as the temperature rises, and though in ordinary hydraulic 
 calculations this is disregarded without serious error, it is 
 otherwise when dealing with water raised to steam temperatures. 
 In the following short Table a is the relative density, that of 
 pure water at 39'3 F. being taken as unity. G is the weight 
 per cubic foot. Eoughly, if the density of pure water is unity, 
 that of river water is on the average 1/0 00 3, that of spring 
 water 1/00 1, and that of sea water 1/025. 
 
INTROD. 
 
 UNITS OF MEASUKEMENT 
 
 DENSITY OP PURE WATER AT DIFFERENT TEMPERATURES 
 
 Tempera- 
 ture Fahr. 
 
 Relative 
 Density. 
 
 Weight of a 
 cub. ft. 
 in Ibs. 
 
 Tempera- 
 ture Fahr. 
 
 Relative 
 Density. 
 
 Weight of a 
 cub. ft, 
 in Ibs. 
 
 t 
 
 a- 
 
 G 
 
 t 
 
 (T 
 
 G 
 
 
 
 
 
 
 
 
 32 
 
 99987 
 
 62-416 
 
 130 
 
 98608 
 
 61-555 
 
 39-3 
 
 1-00000 
 
 62-424 
 
 135 
 
 98476 
 
 61-473 
 
 45 
 
 99992 
 
 62-419 
 
 140 
 
 98338 
 
 61-386 
 
 50 
 
 99975 
 
 62-408 
 
 145 
 
 98193 
 
 61-296 
 
 55 
 
 99946 
 
 62-390 
 
 150 
 
 98043 
 
 61-203 
 
 60 
 
 99907 
 
 62-366 
 
 155 
 
 97889 
 
 61-106 
 
 65 
 
 99859 
 
 62-336 
 
 160 
 
 97729 
 
 61-006 
 
 70 
 
 99802 
 
 62-300 
 
 165 
 
 97565 
 
 60-904 
 
 75 
 
 99739 
 
 62-261 
 
 170 
 
 97397 
 
 60-799 
 
 80 
 
 99669 
 
 62-217 
 
 175 
 
 97228 
 
 60-694 
 
 85 
 
 99592 
 
 62-169 
 
 180 
 
 97056 
 
 60-586 
 
 90 
 
 99510 
 
 62-118 
 
 185 
 
 96879 
 
 60-476 
 
 95 
 
 99418 
 
 62-061 
 
 190 
 
 96701 
 
 60-365 
 
 100 
 
 99318 
 
 61-998 
 
 195 
 
 96519 
 
 60-251 
 
 105 
 
 99214 
 
 61-933 
 
 200 
 
 96333 
 
 60-135 
 
 110 
 
 99105 
 
 61-865 
 
 205 
 
 96141 
 
 60-015 
 
 115 
 
 98991 
 
 61-794 
 
 210 
 
 95945 
 
 59-893 
 
 120 
 
 98870 
 
 61-719 
 
 212 
 
 95865 
 
 59-843 
 
 125 
 
 98741 
 
 61-638 
 
 
 
 
 For temperatures greater than those in the Table, Kankine's 
 approximate rule may be used : 
 
 124-85 
 
 461 
 
 500 
 
 500 t 
 
 The following are values at a few temperatures calculated by 
 this rule : 
 
 t G 
 
 50 62-42 
 
 100 62-02 
 
 200 60-08 
 
 250 58-75 
 
 300 57-29 
 
 350 55-78 
 
 400 54-21 
 
 It will be seen that in dealing with volumes of water at 
 
HYDEAULICS 
 
 INTROD. 
 
 steam temperatures there would be great error in neglecting 
 the change of density with change of temperature. 
 
 4. Intensity of pressure. Very various units of intensity 
 of pressure are adopted in different cases, depending in part 
 on the different methods by which the pressure is measured. 
 The following Table gives equivalent values of various units 
 and the logarithms of the conversion factors : 
 
 UNITS OF INTENSITY OF PRESSURE 
 
 
 Multiplier. 
 
 Logarithm. 
 
 Atmospheres into Ibs. per square inch . 
 square foot . 
 kilograms per square centi- 
 metre ....... 
 
 14-7 
 2116-3 
 
 1-0335 
 
 1-1672 
 3-3256 
 
 0-0143 
 
 Feet of water at 53 into Ibs. per square inch . 
 square foot . 
 Pounds per square inch into feet of water 
 
 0-4333 
 
 62-4 
 2-308 
 
 T-6368 
 1-7952 
 0-3632 
 
 square foot 
 Kilograms per square centimetre into Ibs. 
 per square inch ..... 
 
 0-01603 
 14-223 
 
 2-2049 
 1-1530 
 
 Inches of mercury at 32 into Ibs. per square inch 
 square foot 
 
 0-4912 
 
 70-73 
 
 1-6912 
 1-8496 
 
 5. Atmospheric pressure. In most cases a liquid mass 
 has at some point a free surface exposed to atmospheric pressure 
 which is transmitted throughout the mass. In any given 
 case the atmospheric pressure can be deduced from the baro- 
 metric height at the given place and time. On the average, 
 at sea -level, the atmospheric pressure is 29*92 inches of 
 mercury at 32, 33'9 feet of water, 14- 7 Ibs. per sq. inch, or 
 2116-3 Ibs. per sq. foot. 
 
 Many forms of pressure gauge indicate only the difference 
 between the pressure at a point and atmospheric pressure. 
 Pressures so observed are termed gauge pressures. The gauge 
 pressure plus the atmospheric pressure is termed the absolute 
 pressure. 
 
 6. Acceleration due to gravity. The acceleration due 
 to gravity, denoted by g, varies with latitude and elevation. 
 In practical calculations it is usual to disregard this variation 
 
INTROD. 
 
 UNITS OF MEASUREMENT 
 
 in ordinary cases. In this treatise g will be taken at 32*18 
 ft. per sec. per sec., or at 9'8088 metres per sec. per sec. 
 
 ENGLISH MEASURES 
 
 g= 32-18 
 = 64-36 
 = 5-673 
 = 8-023 
 ^20 = 5-349 
 
 Logarithm 
 1-5076 
 1-8086 
 0-7538 
 0-9043 
 0-7283 
 
 METRIC MEASURES 
 
 
 Logarithm 
 
 0= 9-8088 
 
 0-9916 
 
 20=19-6176 
 
 1-2927 
 
 70= 3-1319 
 
 0-4958 
 
 720= 4-4292 
 
 0-6463 
 
 72^= 2-9528 
 
 0.4702 
 
 The following table gives an idea of the amount of the 
 variation of g with latitude and elevation : 
 
 VALUES OF g AND 
 
 
 
 Elevation above Sea-Level in Feet. 
 
 Latitude. 
 
 Typical Locality. 
 
 
 
 2500 
 
 5000 
 
 
 
 2500 
 
 5000 
 
 
 
 
 Values of g in Feet. 
 
 Values of \J(1g). 
 
 60 
 
 North Canada 
 
 32-215 
 
 32-21 
 
 32-20 
 
 8-027 
 
 8-026 
 
 8-025 
 
 55 
 
 North Britain 
 
 32-200 
 
 32-19 
 
 32-18 
 
 8-025 
 
 8-024 
 
 8-023 
 
 40 
 
 f Mediterranean 1 
 \ Philadelphia j 
 
 32-154 
 
 32-15 
 
 32-14 
 
 8-019 
 
 8-019 
 
 8-018 
 
 30 
 
 fNorth India | 
 \New Orleans / 
 
 32-124 
 
 32-12 
 
 32-11 
 
 8-016 
 
 8-015 
 
 8-014 
 
 20 
 
 Cuba . 
 
 32-099 
 
 32-09 
 
 32-08 
 
 8-012 
 
 8-011 
 
 8-010 
 
 At Greenwich 
 At Paris 
 
 = 32-191 ; 
 = 32-183; 
 
 8-024. 
 
 7. Transformation of an equation from one system of 
 units to another. Eational homogeneous equations are valid 
 in all systems of units, but a large proportion of hydraulic 
 .equations are empirical and require different numerical 
 coefficients for different units. For instance, let 
 
 be an equation in which M, A, B are in feet, and x and y are 
 numerical coefficients. It is required to find the values of x 
 and y when M, A, B are in metres. The equivalents of M, A, 
 
8 HYDKAULICS INTROD. 
 
 B metres in feet are 3'28M, 3'28A, 3-28B. Inserting these in 
 the equation, 
 
 3-28M = x V{3-28A(1 + y x/3'28B)} 
 
 /Q.. 
 
 3-28 
 
 M = x- V{A(1 + y ^3-28 v/B)} 
 
 = -552z x/{A(l + l-81y x/B)}, 
 
 so that the new constants for a formula in metric measures 
 are 552#, and 1*8 ly. 
 
 Hydraulic problems are most conveniently solved by the 
 use of tables of four-figure logarithms and antilogarithms. 
 Most hydraulic formulae are affected by empirical constants 
 which are accurate only to one per cent, or at most in some 
 cases one per thousand. Hence in the answers it is un- 
 necessary and useless to keep more than three, or at most four, 
 significant figures. The short Tables I. and II. in the 
 Appendix will often be useful in obtaining rapidly approxi- 
 mate answers. 
 
 PROBLEMS 
 
 1. A boiler is found to contain 72,000 Ibs. of water at a tem- 
 
 perature of 55 F. How many pounds will it contain at a 
 temperature of 350 F. ? 63,472. 
 
 2. How many gallons of water per foot run will a pipe 30 inches in 
 
 diameter contain ? 30-63. 
 
 3. Convert ten atmospheres of pressure into pounds per square inch, 
 
 and into feet of water. 176-4; 407-1. 
 
 4. On a mountain the barometric pressure is observed to be 24 inches 
 
 of mercury at 32 F. Find the pressure in pounds per square 
 inch? 11-79. 
 
CHAPTER I 
 
 PROPERTIES OF FLUIDS 
 
 8. FLUIDS are substances, the parts of which possess an almost 
 unlimited mobility, which oppose almost no resistance to the 
 separation of one part from another, or which offer practically 
 no resistance to distortion of form. A mass of fluid poured 
 into a vessel takes immediately the shape of the vessel and 
 exhibits no rigidity of form. 
 
 A perfect fluid may be defined as a substance which yields 
 continually to the slightest tangential stress, so that if it is at 
 rest there can be no tangential stress. It is easily deduced 
 from this that the pressure of a perfect fluid is normal to any 
 surface immersed in it, or that the pressure of one part of a 
 fluid on another part is normal to the interface which separates 
 them. The stress at the surface or interface must be a 
 pressure, not a tension, or there would be separation. Further, 
 at any point in a fluid the pressure is the same in all direc- 
 tions, or to put it in another way, the pressure on any small 
 element of surface is independent of its orientation. 
 
 Gaseous and liquid fluids. Fluids are divided into 
 liquids, or incompressible fluids and gases, or compressible 
 fluids. Very great changes of pressure change the volume of 
 liquids only by an extremely small amount, and if the pressure 
 on them is reduced to zero they do not sensibly dilate. On 
 the other hand, in gases or compressible fluids the volume 
 alters sensibly for small changes of pressure, and if the 
 pressure is indefinitely diminished they dilate without limit. 
 In practical hydraulics water is treated as absolutely incom- 
 pressible, so that its density or weight per cubic foot is 
 considered to be independent of the pressure within the limits 
 
10 HYDKAULICS 
 
 CHAP. 
 
 of accuracy usually required. In dealing with gases the 
 changes of volume which accompany changes of pressure must 
 always be taken into account, or in other words, the density is 
 always expressed as a function of the pressure. 
 
 9. Compressibility of liquids. All liquids are slightly 
 compressible, and up to high pressures the compression is pro- 
 portional to the pressure. Let AV be the decrement of 
 volume of V cubic feet for an increment of pressure AP in Ibs. 
 per square foot. Then AV/V is the compression per unit 
 volume, and 
 
 AP 
 
 is called the co-efficient of elasticity of volume. For water, 
 according to Grassi's observations,^ increases from 42,000,000 
 at 32 F. to 48,000,000 at 128 F. The average value of k 
 may be taken at 44,000,000 in ordinary cases, and then the 
 compression is about '00 005V for each atmosphere of pres- 
 sure. Thus one cubic foot of water subjected to a pressure of 
 1000 Ibs. per square inch, or about 64 tons per square foot, 
 would decrease in volume by the amount 
 
 One cubic foot weighing 6 2 '4 Ibs. uncompressed would 
 become '9968 cubic foot when compressed. The weight of 
 the compressed water would be 62'4/0'9968 = 62'6 Ibs. per 
 cubic foot. It is obvious that the ordinary assumption that 
 water is incompressible involves insignificant errors in ordinary 
 cases. 
 
 10. Viscous fluids. Actual fluids do oppose a small re- 
 sistance to separation of parts and to distortion of form, and 
 there may exist in them temporarily tangential stresses. Such 
 fluids are termed viscous fluids. 
 
 In an elastic solid a distorting force produces immedi- 
 ately a definite deformation, which is permanent so long as 
 the distorting force acts. In a viscous body the distortion 
 increases as long as the force acts, and an indefinitely large dis- 
 tortion is produced in time by a distorting force however small. 
 Alcohol is less, and oil more viscous than water. Certain 
 
! PEOPEETIES OF FLUIDS 11 
 
 substances, such as pitch or sealing-wax, are properly fluids 
 with a very high viscosity. Under the action of gravity a 
 block of pitch will flatten and flow in all directions like water, 
 only the action is very slow. The resistance to distortion of a 
 viscous body is proportional to the velocity of the relative 
 motion of the parts, and becomes zero when the velocity is 
 indefinitely small. 
 
 An interesting experiment due to Lord Kelvin illustrates 
 the action of bodies so viscous as to have the appearance of 
 solids. Let a disc of cobbler's wax, about three inches thick, 
 be fixed in a vessel of water below the surface, and let some 
 bullets be placed on the wax, and some corks below it. 
 Under the action of the weight of the bullets and the buoy- 
 ancy of the corks the wax will slowly yield. After some 
 weeks it will be found that the bullets have sunk through the 
 wax and the corks have risen above it. The disc of wax, 
 however, will be found continuous and unperforated, having 
 closed up during the passage of the solid bodies. 
 
 In ordinary fluids the viscosity is small, and in many 
 problems may be neglected without sensible error. On the 
 other hand, when the relative motion of parts of the fluid is 
 rapid, it produces very considerable effects, and in such cases 
 the problems are of so great complexity that usually they 
 have to be dealt with by empirical methods. As water is the 
 most generally diffused liquid, and the one which has generally 
 to be considered in engineering problems, it will be taken as the 
 representative liquid. The great mass of experimental in- 
 vestigation as to the behaviour of 
 liquids under the action of forces has 
 related to water. 
 
 11. Free surface of a liquid. 
 The surface of a liquid at rest is hori- 
 zontal. For if not, an inclined surface 
 can be taken cutting the water surface 
 in two points a and 6. The weight W 
 of the mass above ab will have a com- 
 ponent acting down the incline, which Fig. l. 
 could only be resisted by a tangential 
 
 stress. But as there is no tangential stress in a liquid at rest 
 its surface must be horizontal. In a very large water surface, 
 
12 
 
 HYDKAULICS 
 
 CHAP. 
 
 such as that of a sea, the directions of gravity at distant 
 points are not parallel. In that case the water surface is at 
 all points perpendicular to the direction of gravity. That 
 a liquid surface is a plane appears from the fact that it reflects 
 objects undistorted like a plane mirror ; and that it is 
 horizontal appears from the fact that a plumb line and 
 its reflection are in one straight line. If a vessel filled with 
 water is moving with uniform velocity the water surface is 
 still horizontal, for gravity is the only force acting on molecules 
 at the surface. But if the vessel moves with acceleration 
 the particles are subjected to a force equal and opposite to the 
 accelerating force due to their inertia, and the water surface is 
 then perpendicular to the resultant force acting on the 
 molecules. For instance, if a vessel has a constant acceleration 
 p per sec. per sec., the inertia of a molecule of weight W Ibs. 
 
 C 
 
 Fig. 3. 
 
 is P = pW/g. The surface of the water (Fig. 2) is perpendicular 
 to the resultant K of P and W, which makes with the vertical 
 the angle a, such that 
 
 tan a = =-?-. 
 W g 
 
 If a vessel revolves uniformly about a vertical axis, the friction 
 of the water against the vessel will cause it after a time to 
 revolve with uniform angular velocity also like a solid. Let 
 co be the angular velocity, W the weight of a particle at the 
 surface at P, where the radius is r. The velocity of the 
 particle at P is cor, and its radial acceleration is C = Wrco 2 /g. 
 
PKOPEKTIES OF FLUIDS 
 
 13 
 
 The resultant E of C and W (Fig. 3) makes with the vertical 
 an angle 0, such that 
 
 , C ro) 2 
 
 Produce E to meet the axis of rotation in b. The subnormal 
 ab is 
 
 ab = r cot = -^, 
 
 CO 
 
 a constant, which is a property of the parabola. 
 
 12. Fluid pressure. Pascal's law. Fluid pressure is 
 not only normal to any surface on which it acts, and independent 
 of the orientation of the 
 surface, but it is exerted 
 equally in all directions 
 throughout a fluid mass. 
 Suppose a vessel fitted with 
 pistons of equal area. Any 
 inward force P applied to 
 one of them, A, is instantly 
 transmitted, and acts as an 
 outward force P on all the 
 others. 
 
 In any actual fluid the 
 upper layers press by their 
 weight on the lower layers. Fig. 4. 
 
 Hence, as will be discussed 
 
 presently, the pressure in a fluid mass varies with the level. But 
 there are light fluids, such as air, in which for a considerable 
 difference of level there is only a small difference of pressure, 
 and in heavier fluids such as water the general pressure may be 
 so great, that the differences of pressure, due to such differences 
 of level as there are in the mass considered, are relatively insig- 
 nificant. If the pressure at a given level in a mass of water 
 is 100 Ibs. per square inch, or 14,400 Ibs. per square foot, then 
 for points 10 feet above and below that level the pressures are 
 -13,776 and 15,024 Ibs. per square foot, the whole difference 
 being about 4 per cent. In some practical problems this differ- 
 ence can be neglected. With lighter fluids, such as air or steam, 
 the variation of pressure with level is much less, and in a 
 
14 HYDEAULICS CHAP. 
 
 large class of problems is disregarded. In the atmosphere a 
 difference of 1000 feet in level corresponds to less than 4 
 per cent difference of pressure. Hence it is convenient to 
 divide problems on fluid pressure into two classes, in one of 
 which the pressure is regarded as uniform throughout the 
 mass, as if the fluid were weightless ; in the other the varia- 
 tion of pressure with level is taken into account. On a level 
 plane in a fluid the pressure is always uniform. 
 
 Fluid pressures are most conveniently measured in 
 hydraulic calculations in Ibs. per square foot. But there are 
 other units of intensity of pressure the relations of which 
 have been given in 4. Pressures are experimentally 
 determined by instruments termed gauges, and usually these 
 measure the excess of the fluid pressure above the atmospheric 
 pressure at the time and place. To find the absolute pressure 
 the barometric pressure must be added to the gauge pressure 
 (see 5). In hydraulic problems the difference of pressure 
 at two points in the fluid is alone the question, at both of 
 which the atmospheric pressure is the same. Then the 
 atmospheric pressure may be disregarded. But in some cases, 
 for instance the question of the flow of gas in mains, the two 
 points considered may be far apart and different in level, and 
 then the difference of barometric pressure at the two points 
 cannot be disregarded. 
 
 13. Uniform fluid pressure on a plane. Consider a 
 plane MN inclined at to the vertical and subjected to a 
 uniform pressure p. Let MN be projected on two planes at 
 right angles, for simplicity suppose horizontal and vertical 
 planes. If A = U is the area of MN, the area of its horizontal 
 projection is A h = A sin 6, and that of its vertical projection 
 is A V = A cos 6. The resultant normal pressure on MN is 
 ~P=pA=pbl. The vertical component of P is V = P sin 
 0=pA sin 0=pA h . Similarly the horizontal component of 
 P is H = P cos 0=pA cos 0=pA v . Hence the resultant 
 pressure on the plane MN in any given direction is the 
 intensity of pressure p multiplied by the projected area of 
 MN normal to that direction. This is true whatever the 
 shape of the plane. 
 
 The pressure being uniformly distributed on the surface 
 MN, the resultant acts through the centre of figure or mass 
 
PROPERTIES OF FLUIDS 
 
 15 
 
 centre of MN. Also its components act through the mass 
 centres of the projections of MN. 
 
 Corollary. On a horizontal plane in a fluid the pressure 
 is always uniform and normal to the surface, and its resultant 
 acts through the mass centre of the surface. 
 
 A 
 
 
 
 A* 
 
 <> 
 
 
 J 
 
 
 -'N 
 
 r. 5. 
 
 14. Uniform pressure on curved surfaces. If the surface 
 of area A on which a uniform fluid pressure p acts is not 
 plane, the total amount of pressure on the surface is pA, but 
 the pressure acts in different directions at different parts of 
 the surface, and the resultant pressure on the surface has a 
 different value. 
 
 Let ACB be any curved surface, which for simplicity may 
 be taken to be one foot in length perpendicular to the paper. 
 The uniform internal pressure is p. The total pressure on 
 any small element ab is p X ab, and the horizontal and 
 vertical components of this are pxbe and p x ae. But the 
 horizontal component will be exactly balanced by the horizontal 
 
16 
 
 HYDKAULICS 
 
 CHAP. 
 
 pressure on the vertical projection cf of the part cd of the sur- 
 face. The only unbalanced part of the pressure on db is p x 
 C ae, and the resultant vertical 
 
 pressure on the whole curved 
 surface ACB is p 2 ~a~e, that 
 is p x the horizontal pro- 
 jected area of the curved 
 surface that is, if the ring 
 is one foot in length, pi Ibs. 
 Hence the resultant pressure on any curved surface cut off by 
 a plane is normal to the plane, and equal to the intensity of 
 pressure multiplied by the area of the projection of the 
 surface on the plane. 
 
 Fig. 7. 
 
 Example 1. Consider a hollow cylinder of diameter d feet subjected 
 to a uniform internal pressure p Ibs. per square foot. Let abed be a 
 diametral plane dividing the cylinder into halves. The resultant pressure 
 P on each half is normal to abed, and equal to 
 
 P=j0xarea abed 
 
 because Id is the area of the projection abed of the semicylinder. 
 
 Example 2. Some pumps have trunks of 
 half the area of the piston. 
 
 Let D be the diameter of the piston ab in 
 feet, d that of the trunk cd, and let p v p 2 be the 
 pressures on front and back of the piston in 
 
 Ibs. per square foot. Then ^i=Pi^ 2 ac te 
 forward on the back of the piston, and 
 P 2 =_p 2 ^(D 2 - d 2 ) acts backwards on the annular 
 face of the piston. The resultant force driving the piston is 
 
PKOPEETIES OF FLUIDS 17 
 
 bs- If the piston faces are recessed or curved in 
 any way the resultant driving pressure is not altered. 
 
 15. Abutment at dead ends or bends of pipes. The 
 
 ends of pipes when blanked oft' are subject to an endways 
 
 thrust which, if not resisted by an abut- 
 
 ment, would draw the adjacent pipe joints. 
 
 Let d be the diameter of the pipe in 
 
 inches, h the greatest statical pressure in 
 
 the pipe in feet of head, for instance the 
 
 difference of level of the surface of water 
 
 in the supply reservoir and the pipe end. 
 
 Then as, from 4, the pressure is 0'4333 h Ibs. per square 
 
 inch, the total thrust on the pipe end is 
 
 P = jx -433d% = 0-34d% Ibs. 
 4 
 
 This is often a considerable force. In a 3 6 -inch pipe under 
 200 feet of head the thrust would be 88,180 Ibs., or nearly 
 forty tons. Under certain circumstances, such as the sudden 
 shutting of a valve on a branch near the pipe end, an 
 additional thrust due to dynamical action might be produced. 
 Consider next a pipe bend (Fig. 10), and let dOb = 0,d = 
 the pipe diameter, and h the head in feet. The wedge abed is 
 
 acted on by the thrusts P = P = ^ x 6 2 '4^ = 49^A Ibs. along 
 
 the axis of each pipe. The resultant thrust tending to displace 
 the bend is 
 
 R =2? sin - = 98d% sin Ibs. 
 2 2 
 
 Thus for a 3 6 -inch pipe with a head of 200 feet, bent at an 
 angle of 120, so that 0=60, 
 
 R = 98 x 9 x 200 x sin 30 = 88,200 Ibs. 
 
 If the water is flowing round the bend there is additional 
 thrust due to the deviation of the water, which will be 
 discussed in a later chapter. It is usual to provide a 
 masonry or concrete block to resist the thrust in such cases. 
 
 The result can be arrived at in another way. If we 
 suppose the pipe divided into two troughs of semicircular 
 
 2 
 
18 
 
 HYDRAULICS 
 
 CHAP. 
 
 section by the line ef, all the upward-acting pressures act on 
 the upper, and all the downward-acting pressures on the lower 
 trough. The projections of the troughs on a horizontal plane 
 
 Fig. 10. 
 
 are shown below. The difference of their areas is the area of 
 
 the two ellipses, the major axes of which are d and their minor 
 /a 
 
 axes d sin -. Hence the upward thrust is 
 
 2 
 
 R = 
 
 sn 
 
 16. 
 
 pistons 
 pistons 
 smaller 
 exerted 
 fluid is 
 P 2 = Pj 
 
 2 radius Oe 
 
 Hydraulic press. Suppose a vessel fitted with two 
 of area a and A normal to the direction in which the 
 If a downward pressure P x is exerted on the 
 
 move. 
 
 piston a much greater upward pressure P 2 will be 
 on the larger. The intensity of pressure in the 
 PI/, and the upward pressure on the large piston is 
 /a. This is the principle of the hydraulic press, in 
 
i PEOPEKTIES OF FLUIDS 19 
 
 which pressure produced by the plunger of a small pump is 
 transmitted to a very large ram. 
 
 Obviously the small piston will move a greater distance 
 
 a/rea a i!' 1 IM] a/raA 
 
 Fig. 11. 
 
 than the larger. If v lt v 2 are the piston velocities, vjv 2 = A/a. 
 The volume v^a displaced by the small piston is equal to the 
 volume v 2 A described by the large piston. Generally the 
 friction of the pistons is not inconsiderable, and this modifies 
 somewhat the ratio of the efforts given above. 
 
 Example. The pump plunger of a large press is j inch in diameter, 
 and the press ram is 20 inches in diameter. Then A/a = 20 2 /(f) 2 = 710. 
 Suppose a man exerts by a lever a force of P t = 150 Ibs. on the plunger. 
 Then the upward force exerted by the press ram is 710 x 150 = 106,500 
 Ibs., or 48 tons. That is neglecting the friction of plunger and ram. To 
 move the press ram one inch the plunger must move through 710 inches. 
 Forging presses have been made on this principle capable of exerting an 
 effort of 10,000 tons. 
 
 PROBLEMS 
 
 1. Treating water as incompressible, find the pressure in tons per 
 
 square foot on the bed of the Atlantic, the depth being 5 miles, 
 weight of sea water 64 Ibs. per cubic foot. 754. 
 
 2. With the conditions in the last question, find the weight of a 
 
 cubic foot of water at the bed of the Atlantic, taking the com- 
 pression of the water into account. 66'6 Ibs. per cubic foot 
 
 3. A pipe 24 inches in diameter has a right-angled bend. The 
 
 pressure in the pipe is 150 feet of head. Find the force tend- 
 ing to displace the bend. 18*6 tons. 
 4 Show that the surface of water in the buckets of a water-wheel 
 revolving uniformly are parts of cylindrical surfaces having 
 the same axis. 
 
CHAPTER II 
 
 DISTRIBUTION OF PRESSURE IN A LIQUID VARYING WITH 
 THE LEVEL 
 
 17. Pressure column. Free surface level. Let a small 
 vertical pipe AB be introduced into a mass of liquid. The liquid 
 will rise in the pipe to some level 00, such that the weight 
 of the column BA balances the pressure on its mouth. This is 
 true whether the liquid is at rest or in motion, provided the 
 
 mouth of the pipe is parallel to the 
 direction of motion so that the liquid 
 does not impinge on it. The height 
 AB = h measures the pressure at A. 
 Let co be the area of the cross section 
 of the pipe, p the intensity of pres- 
 sure at A, and G the weight of a 
 cubic unit of fluid 
 
 . (1). 
 
 If h is in feet, p in Ibs. per sq. ft., 
 G= 62*4. For metre-kilogram units 
 Fig. 12. G=1000. The result is expressed 
 
 by saying that h is the height due 
 
 to the pressure p, or conversely p the pressure due to the 
 height h. The level 00 is the free surface level. 
 
 In general, atmospheric pressure will be acting on the free 
 surface at 00. Consequently h measures the gauge pressure, 
 not the absolute pressure at A (5). Let p a be the atmo- 
 spheric pressure in Ibs. per sq. ft. Then p a /Gr is the height 
 in feet of water equivalent to atmospheric pressure, and the 
 absolute pressure at A is p = Gh +p a Ibs. per sq. ft., or h +p a /G- 
 
 20 
 
CHAP. II 
 
 DISTRIBUTION OF PRESSURE 
 
 21 
 
 feet of water. p a /Q is about 3 3 '9 feet on the average. If a 
 line XX is taken at a height p a /Gr above 00, the absolute 
 pressure at A is h a feet of water, the layer between XX and 
 OO representing a layer of water the weight of which is 
 equivalent to atmospheric pressure. In many hydraulic 
 problems only differences of pressure at two points are con- 
 cerned, and atmospheric pressure may then be ignored. 
 
 18. Relative level of liquids of different density. 
 Suppose two liquids of density G v G 2 are placed in a bent 
 tube. At the level of the plane 
 of separation 00 the pressure 
 must be the same in both arms. 
 Hence the pressure of the two 
 columns above that level must be 
 the same 
 
 H, 
 i.o 
 
 0,70, = V*i (2)- 
 As atmospheric pressure is the 
 same on both columns it does 
 not need to be taken into con- 
 sideration. 
 
 Fig. 13. 
 
 Watt's hydrometer. A bent tube connects two beakers 
 containing fluids of different densities G I} G 2 . 
 If a partial vacuum is formed in the bent 
 tube the liquids will rise to different heights 
 h v h 2 . Let p be the pressure in the bent 
 tube and p a the atmospheric pressure on the 
 free surface in the beakers. The pressure 
 due to the weight of the columns in each 
 leg must be equal to the difference of pres- 
 sure^ p . Hence 
 
 I 
 
 H , 
 
 13 i G 1 /G 2 = ^ 2 /A 1 . . (3). 
 
 If the density of one of the fluids is known, 
 y - that of the other can be determined by 
 measuring the height of the columns. 
 
 Mercury siphon gauge. Pressure is 
 often measured by a siphon gauge AB containing mercury 
 
 L r -l 
 
 Fig. 14. 
 
22 
 
 HYDEAULICS 
 
 CHAP. 
 
 and open at one end to the atmosphere. Let Fig. 14a repre- 
 sent a water main C in which the pressure is to be determined, 
 and let b be the atmospheric pressure in 
 inches of mercury, h the difference of level 
 of the mercury columns in the siphon 
 gauge in inches. The absolute pressure 
 at A is b inches of mercury, that at B is 
 b + h inches of mercury. If the specific 
 gravity of mercury is 13'57, the absolute 
 pressure at B is 
 
 (b + 
 
 = 1 -131(6 + h) feet of water. 
 
 Fig. 14a. 
 
 If, as is often the case, the siphon gauge 
 is at a considerable height H feet above 
 the centre C of the main, the absolute 
 pressure at C is 
 
 1*131(6 + h) + H feet of water, 
 and the gauge pressure, or pressure in excess of atmospheric 
 pressure, is M3U + H. 
 
 19. Pressure on surfaces varying as the depth from 
 the free surface. In any heavy fluid the pressure must 
 increase with the depth reckoned from the actual or virtual 
 free surface. 
 
 Let A be a small vertical surface of area sq. ft. at a 
 depth h ft. The intensity of pressure at that depth is p Gh 
 Ibs. per sq. ft. The total pressure on 
 
 the surface is pa Ghco Ibs. Take *^=--===T=-=I==^-^L=S.- 
 a surface B equal and parallel to A ! 
 at a distance h, and complete the i* 
 prism AB. Its volume is hco, and if 
 composed of fluid its weight is Ghco 
 Ibs. 
 
 Hence the horizontal pressure on 
 a small vertical surface at the depth 
 h is equal to the weight of a prism of 
 fluid of length h and cross section 
 
 equal to the area of the surface. It will easily be seen 
 that the restriction to a vertical surface is not necessary. 
 
 B 
 
II 
 
 DISTRIBUTION OF PKESSUEE 
 
 23 
 
 But in any case the resultant pressure thus estimated is 
 normal to the surface. 
 
 When the surface is not small it cannot be regarded as 
 all at the same depth. But for each small element of the 
 surface the rule applies. Consider a strip abed of a vertical 
 wall, of width db = b, and height ad = h, supporting water 
 pressure. Take de = h and complete the wedge dbcdef. At 
 any depth the intensity of pressure is proportional to the 
 horizontal thickness of the wedge at that depth. The total 
 
 Fig. 16. 
 
 pressure on the wall is the weight of a wedge abcdef of fluid. 
 The volume of the wedge is ^bh 2 and the pressure on the 
 wall is 
 
 P = JG6AMbs. . . . (4). 
 
 Further, since the distribution of pressure is represented 
 by the wedge, the resultant pressure acts through the mass 
 centre of the wedge, that is at h/3 above the base. 
 
 As the pressure varies uniformly the mean pressure in this 
 case is 
 
 p m = Gbh?/bh = $Gh Ibs. per sq. ft., 
 
 which is the pressure at the mass centre of abed. 
 
 The rule is general. The mean pressure on any immersed 
 
24 
 
 HYDEAULICS 
 
 CHAP. 
 
 plane is the pressure at its mass centre due to its depth from 
 the free surface, and the resultant pressure normal to the 
 surface is the mean pressure multiplied by the area of the 
 surface. The point of the surface at which the resultant 
 pressure acts is not in general the mass centre, and 
 this will be determined presently. In the case of a curved 
 surface the total pressure is also the pressure due to the depth 
 of the mass centre multiplied by the area of the surface, but 
 this result has little meaning. As the pressure acts every- 
 where normal to the surface the total pressure consists of 
 components acting in different directions. The resultant 
 pressure on a curved surface will be found presently. 
 
 Example. A vertical semicircular plate of radius r feet and area 
 <o = J^rr 2 , supports water on one side level with its straight edge. The 
 depth of the mass centre of the semicircle is 4r/37r. The mean pressure 
 on the surface is p m = 4Gr/37r Ibs. per square foot. The resultant pressure 
 on the surface is 
 
 Water at different levels on two sides of a wall. In 
 
 cases of this kind it is convenient to consider a strip of the 
 wall one foot in width (Fig. 
 
 b 
 
 Fig. 17. 
 
 Let h lt h 2 be the depths of water. The distribution of 
 pressure on each side is given by the dotted triangles with 
 
ii DISTRIBUTION OF PEESSUKE 25 
 
 bases equal to k lt Ji 2 , and the total pressures P 1} P 2 are equal 
 to the weight of wedges of water one foot thick of the area of 
 these triangles. Hence the pressures per foot run of wall are 
 Pj = |GV and P 2 = GA 2 2 Ibs. These pressures act at hJS 
 and h 2 /3 above the base of the wall, and the overturning 
 moment about the toe A of the wall is 
 
 -V)ft.-lbs. . . . (5). 
 
 Let the wall be h feet high and & feet thick, and let G TO be 
 the weight per cubic foot of masonry. The weight of the 
 wall is GfJ)h Ibs., and the moment about A resisting over- 
 turning is QJbh x ^b -JG m & 2 A. If the moment of stability is 
 to be 2^- times the overturning moment 
 
 <> 
 
 In this case as the total atmospheric pressure is the same 
 on both sides of the wall it is neglected without any error. 
 
 20. Pressure on a flap valve covering the end of a 
 pipe of circular section (Fig. 18). 
 
 Let d be the diameter of the pipe in feet and the angle 
 of inclination of the flap to the vertical. The projection of the 
 
 flap on a vertical plane is a circle of area A y = -d?. Its pro- 
 jection on a horizontal plane is an ellipse, the principal axes 
 of which are d and d tan 6. Hence its area is A h -d? tan 6. 
 
 The mean head on the flap is h measured to its centre of 
 figure. The horizontal and vertical components of the pressure 
 on the flap are equal to the mean pressure multiplied by the 
 areas of the vertical and horizontal projections. That is, the 
 vertical component is 
 
 P v = Gh x A h = jGM 2 tan Ibs., 
 and the horizontal component is 
 
26 
 
 HYDEAULICS 
 
 CHAP. 
 
 The resultant pressure normal to the flap is 
 P = ^/P + p^2 = Q^ A lbs> 
 
 9 m (7) 
 
 It will be shown presently that the horizontal component acts 
 at a point h + d?/16h below the water surface, which is more 
 nearly equal to h as h is greater. If the horizontal component 
 
 Fig. 18. 
 
 is drawn at this depth, the point where it intersects the flap 
 is the centre of pressure at which the resultant pressure on the 
 flap acts. 
 
 21. Centre of pressure on any vertical surface. Let 
 AB (Fig. 19) be any surface of area A square feet, the vertical 
 projection of which is given on the right. Let h lt h 2 be the 
 depths of A and B from the free surface. Let D be the mass 
 centre of the surface at depth h m and E the centre of pressure 
 at depth z. The resultant pressure on the surface is 
 
 P - Gh m A. Ibs. 
 Consider a horizontal strip of the surface between the ^ depths 
 
II 
 
 DISTRIBUTION OF PEESSUEE 
 
 h and h + dh and of width b. Its area is bdh, and the pressure 
 on it is Gbhdh. The moment of this, about a horizontal axis 
 through C, is Gbh 2 dh. The total moment of the pressure on 
 the surface about C is therefore 
 
 ft 
 
 G I 
 
 A 
 
 bh*dh = GI, 
 
 where I is the moment of inertia of the surface about a 
 horizontal axis through C, normal to the plane of the figure. 
 
 Fig. 19. 
 
 But this must be equal to the moment of the resultant 
 pressure about the same axis. Hence 
 
 (8), 
 
 or if I = & 2 A where k is the radius of gyration of the surface 
 about the axis through C, 
 
 The moment of inertia of a surface about an axis through 
 the mass centre of the surface is known for various surfaces. 
 Let I be the moment of inertia of the surface about an axis 
 through its mass centre and normal to the plane of the figure. 
 
28 
 
 HYDKAULICS 
 
 CHAP. 
 
 B 
 
 A 
 
 X 
 *"l 
 
 Y- + 
 
 c 
 
 G 
 
[I 
 
 DISTRIBUTION OF PEESSURE 
 
 29 
 
 CENTRE OF PRESSURE AND TOTAL PRESSURE 
 
 Surface. 
 
 Area A. 
 
 Depth of 
 Mass Centre 
 from Free 
 Surface x. 
 
 Depth of Centre 
 of Pressure from 
 Free Surface z. 
 
 Total Pressure 
 on Surface P. 
 
 Rectangle A . 
 
 
 
 h 
 
 2 
 
 > 
 
 2 
 
 Rectangle B . 
 
 bh 
 
 *i 
 
 * 4- CL 
 
 (..-:> 
 
 3 2a + h T 
 
 Triangle C 
 
 j 
 
 h 
 3 
 
 h 
 2 
 
 6 
 
 Circle D . 
 
 ., 
 
 a + r 
 
 r 2 
 
 -- 
 
 T 4(r + a) 
 [a = 
 
 Semicircle E . 
 
 -7TT 
 
 4r 
 
 3 
 
 J- 
 
 Ellipse G, one 
 axis vertical 
 
 7rr i r 2 
 
 a + r l 
 
 r 2 
 
 ^,,, 
 
 Trapezium F . 
 
 h(B + 6) 
 
 kE + 2b 
 
 ^B + 36 
 
 -G/i 2 (B + 26) 
 
 3 B + 6 
 
 2B + 26 
 
30 HYDEAULICS 
 
 Then by the well-known rule 
 
 CHAP. 
 
 
 AA m 
 
 . (10). 
 
 Example. Let the surface be a circle of diameter d. 
 
 ^d* + - 
 64 4 
 
 22. Pressure and centre of pressure on any plane 
 surface. Let AB be the surface in a plane normal to the 
 
 plane of the figure inclined 
 &_ -^ at 6 to the horizontal. 
 
 >x x ^ <? - Take OB for the X axis 
 
 /^ v x O*^ N and an axis through per- 
 
 pendicular to the plane of 
 the figure for the Y axis. 
 Let A be the area of the 
 surface ; E the pressure on 
 it ; OA = a?! ; OB = x 2 . 
 Let c be the mass centre 
 of the surface and d the 
 centre of pressure, and let Oc = x c and Od = x d . 
 
 Consider a strip of the surface between x and x + dx of 
 breadth y. Its depth below the water surface is x sin 0, and 
 the total pressure on it is Gx sin Oydx. Hence the whole 
 pressure on AB is 
 
 Fig. 20. 
 
 R = G sin 
 
 r* j 
 
 xydx. 
 
 But xydx = Ax c 
 
 R = GAz c sin 0, 
 
 where Gx c sin 6 is the intensity of pressure at the mass centre 
 of the surface. Taking moments about the Y axis, 
 
 /* / 7 < 
 
 J x l 
 
UNIVERSITY 
 
 OF 
 
 ii 
 
 DISTRIBUTION OF PRESSURE 
 
 31 
 
 But 
 
 fx 2 
 
 x z ydx is 
 J x l 
 
 the moment of inertia I of the surface 
 
 about the Y axis, 
 
 GI sin (9 
 
 (11). 
 
 But I = tfA where k is the radius of gyration of the surface 
 about the Y axis, 7. 2 
 
 x d = - .... (12). 
 
 The lateral position of the centre of pressure is found thus : 
 the mass centre and centre of pressure of the surface are in 
 the same vertical plane, parallel to the plane of the figure. 
 
 When surfaces are not vertical it is often convenient to 
 find the component pressures on their horizontal and vertical 
 projections separately and combine them. 
 
 The Table on p. 29 gives the pressure and depth of 
 centre of pressure for various vertical surfaces. 
 
 '23. Graphic determination of the pressure on surfaces. 
 
 Case of a curved face of a retaining wall or dam. Let 
 Fig 20 a represent the vertical section of a curved wall, 
 
32 
 
 HYDKAULICS 
 
 CHAP. 
 
 ABCD, which may be treated as polygonal without serious 
 error if the divisions are taken small enough. It is con- 
 venient in such cases to consider one foot length of the wall. 
 
 The curved face being divided into lengths AB, BC, CD, 
 each equal to a, the area of these faces will be a also. Let 
 hi, h 2 , h 3 , A 4 be the depths of A, B, C, D below the free 
 surface. Take Aa normal to AB and equal to ^ ; B5 normal 
 to AB and equal to A 9 . Join ab. Then Aa&B represents 
 
 in magnitude and distri- 
 n ^x^7 bution the normal pressure 
 on AB. The total pressure 
 on AB is the weight of 
 a prism of water Aa&B 
 one foot thick. That is 
 Pj = lG<x(A 1 -f 7t 2 ), and it 
 acts through the mass 
 centre C of Aab~B normally 
 to AB. Similarly the pres- 
 sures P 2 = jGa(/i 2 + 7& 3 ), 
 and P 3 = ^Ga (h B -f- A 4 ) can 
 be found in position and 
 x ^ / / direction. Draw the force 
 
 I / / polygon (Fig. 206) with 
 
 \ i i / sides equal on any scale 
 
 \ ', / / and parallel to P 1? P 2 , P 3 . 
 
 \ i / / The closing line gives the 
 
 \ \if resultant B in magnitude 
 
 and direction. Choose a 
 
 Fig. 20Z>. pole and draw rays to 
 
 the angles . of the force 
 
 polygon. Next draw the funicular polygon mnopq with sides 
 mn, no, op,pq parallel to the rays, taken in order, and intersect- 
 ing the pressures P x , P 2 , P 3 at n, o, p. Produce the first and 
 last lines of the funicular polygon to meet in x. Then x 
 is a point through which the resultant B of the pressure 
 acts. B can be drawn through x and parallel to B in the 
 force polygon. The resultant pressure on ABCD is therefore 
 found in magnitude, position, and direction. 
 
 24. Loss of weight of immersed bodies. Buoyancy. 
 Principle of Archimedes. Let Fig. 21 represent a body 
 
ii DISTRIBUTION OF PRESSURE 33 
 
 immersed in water. Consider a prism ab of small cross 
 section at a depth h. Since the vertical projections of the 
 
 two ends of this prism are 
 
 equal, and the pressure due to 
 the depth h is the same on 
 each, the horizontal forces on 
 the prism must balance ; and 
 since the body can be divided 
 into such prisms the horizontal 
 forces on the whole body must 
 balance also. Next consider 
 a small vertical prism cd. If 
 a* is the horizontal cross 
 section, and h lf h 2 the depths 
 
 of the ends below the free surface, the resultant pressure acting 
 on it is an upward force Go>(A 2 h^. But this is equal and 
 opposite to the weight of a prism cd of water. Since the 
 body can be divided into a set of similar vertical prisms, the 
 whole upward pressure on it must be the weight of a volume 
 of water equal to the volume of the body. If W re the 
 weight of the body not immersed, and V its volume, the 
 upward pressure is GV, and the resultant downward force 
 W GV. The body loses, when immersed, a weight equal 
 to the weight of water displaced. The upward pressure GV is 
 termed the buoyancy, and it acts through the mass centre of 
 the water displaced, a point termed the centre of buoyancy. 
 If the body is homogeneous, the centre of buoyancy coincides 
 with the mass centre of the body, provided it is wholly 
 immersed. If the body is not wholly immersed, or is hollow 
 or of varying density, the centre of buoyancy will not generally 
 coincide with the mass centre of the body. 
 
 Note that if GV is greater than W the body will float. 
 As part of it rises out of the water, the volume V of water 
 displaced diminishes. The plane of notation when the body 
 comes to rest is such that GV = W where V is not now the 
 volume of the body, but the volume of the water displaced, 
 the buoyancy then exactly balancing the weight. 
 
 25. Equilibrium of floating bodies. If a body floats on 
 water the weight W of the body and the buoyancy B are 
 equal. But W acts at the mass centre b of the body, and B 
 
34 
 
 HYDRAULICS 
 
 CHAP. 
 
 at the mass centre a of the displaced water. If these are not 
 on the same vertical there is a couple Wx tending to turn the 
 
 body, and it must move 
 till a is on a vertical 
 through I. The line pass- 
 ing through a and b when 
 the body has taken a 
 position of rest is called 
 the axis of notation. If 
 the axis of notation is 
 known, as in the case 
 of various symmetrical 
 bodies, the depth of nota- 
 tion is easily found. Thus 
 
 if the body is a prism of section A perpendicular to the axis 
 of notation, W its weight, and D the depth immersed, 
 
 D = W/GA. 
 
 Stability of floating bodies. Metacentre. A body floats 
 in an upright position if a plane through the axis of flotation 
 divides it into sym- 
 metrical parts. The 
 body is stable if when 
 slightly displaced it 
 returns to its former 
 position, unstable if 
 a small displacement 
 tends to increase. 
 Let Fig. 2 3 represent 
 a floating body, and 
 let W be its weight, 
 V its displacement, 
 so that W = GV. 
 Let B be the centre 
 
 of buoyancy when the body floats upright, and G its mass centre. 
 If the body is displaced, the centre of buoyancy moves out to 
 some point B x . The weight W and buoyancy GV then form a 
 couple tending to rotate the body. Let M be the intersection 
 of GV with the axis of flotation through B and G. This 
 point is termed the metacentre. If M is above G the body 
 
ii DISTRIBUTION OF PEESSUEE 35 
 
 will turn so that G sinks and M rises, and the action tends to 
 annul the displacement. If M is below G the body is un- 
 stable. If M and G coincide equilibrium is indifferent. 
 
 When M is above G the righting couple is Wx, where x is 
 the horizontal distance between the metacentre M and the 
 mass centre G. If MG = c and <f) is the angle of displacement, 
 the righting couple is We sin </>. It increases, therefore, with 
 c. On the other hand the rapidity of rolling increases with c, 
 and therefore thers is a limit to the metacentric height which 
 is desirable. But these are questions beyond the scope of the 
 present treatise. 
 
 PROBLEMS. 
 
 1. If mercury is 13^ times heavier than water, find the height in 
 
 inches of a mercury column corresponding to a pressure of 
 100 Ibs. per square inch. 205-1. 
 
 2. A masonry dam vertical on the water side supports water of 100 
 
 feet depth. Find the pressure per square foot at 25 and 75 
 feet from the water surface, and the total pressure on one foot 
 length of dam. 1560 and 4680 Ibs. per square foot ; 3 1 2,000 Ibs. 
 
 3. Find the resultant pressure on a circular plate 5 feet in diameter, 
 
 with its top edge 10 feet below the water surface (1) When 
 the plate is vertical ; (2) When the plate is inclined at 30 to 
 the horizontal Also the position of the centre of pressure 
 when the plate is vertical 
 
 15,310 and 13,790 Ibs. ; 12-625 feet from surface. 
 
 4. A dock entrance is closed by a caisson 50 feet wide at bottom and 
 
 60 feet wide at the water surface, 24 feet above the bottom. 
 Find the total pressure on the caisson when the dock is empty. 
 
 958,280 Ibs. 
 
 5. Two lock-gates are each 10 feet wide, and support water 10 feet 
 
 deep in the head bay, the lock being empty. The gates meet 
 at an angle of 120. Find the total pressure on each gate, 
 and the thrust at the hollow quoins. 31,200 Ibs. ; 31,200 Ibs. 
 
 6. A ship weighs 1000 tons. Find its displacement in sea water. 
 
 350,000 cubic feet 
 
 7. If the ship in the last question is vertical-sided near the water- 
 
 line, and has a section of 1500 square feet at the water-line, by 
 how much would the draught change in passing from sea to 
 fresh water ? 6 feet. 
 
 8. A homogeneous log is 3 feet wide, 2 feet deep, and 20 feet long. 
 
 Its density is half that of water. It carries at its centre a load 
 of 2000 Ibs. Find its depth of immersion. 18-4 inches. 
 
 9. A dam supporting water pressure is vertical for 20 feet below the 
 
 water surface, slopes at 1 in 5 from 20 feet to 30 feet> and at 
 1 in 3 from 30 feet to 40 feet. Find, graphically, the magni- 
 tude and position of the resultant water pressure. 
 
CHAPTEE III 
 
 PRINCIPLES OF HYDRAULICS 
 
 26. Hydraulics is the science of liquids or incompressible 
 fluids in motion, and comprises 
 
 (a) The laws of discharge from orifices, and sluices, and 
 over weirs. The application of these is chiefly to the measure- 
 ment of the flow of water. 
 
 (6) The laws of flow in pipes, canals, and rivers. The 
 application of these is partly to water measurement, partly 
 to the design of pipes and channels. 
 
 (c) The laws of impact of water streams on surfaces, the 
 most important applications of which are to the design of 
 some types of water motors. 
 
 (cT) The laws of the resistance of water to the motion of 
 bodies immersed or floating in it. The application of these 
 is to ship design. 
 
 Pure theoretical hydrodynamics has proceeded but little 
 beyond the consideration of the action of a perfect fluid 
 without viscosity. The conclusions reached are in no case 
 correct for actual fluids, and in some cases are in startling 
 contradiction with the facts of experience. In practical 
 hydraulics it is impossible to proceed on strictly theoretical 
 lines. There are rational principles which serve for the solu- 
 tion of some elementary problems. In more complex cases 
 dynamical reasoning serves as a basis or guide in generalising the 
 results of experiment. But usually in hydraulics theoretical 
 conclusions have to be checked and modified by the results of 
 observation. In rigid dynamics rational solutions of problems 
 are obtained based on the accurate determination of a few 
 fundamental physical constants. In hydrodynamics the 
 conditions are generally so complex that no such simple 
 
CHAP, in PRINCIPLES OF HYDRAULICS 37 
 
 rational conclusions can be found. In the strict sense 
 hydraulics is not a science. It is embarrassed by tangles of 
 formulae, which, initially based on imperfect reasoning, have 
 been modified and adjusted to conform more or less accurately 
 to the results of experiments, themselves affected to some 
 extent by observational errors. On the other hand, it must 
 be recognised that during more than two centuries a very 
 large mass of experimental observation on the motion of water 
 in different circumstances has been accumulated. For the 
 practical purposes of the engineer, the empirical laws of 
 hydraulics used with proper insight into their limitations 
 are sufficient and trustworthy as solutions of practical problems. 
 
 27. The two modes of motion of water. The first 
 fundamental difficulty in hydraulics is that water moves in 
 two different and characteristic ways. When water is acceler- 
 ated or retarded the inertia forces acting on the mass are the 
 same as for any other heavy body. But from the extreme 
 mobility of the parts they readily take relative motions which 
 absorb energy, which is rapidly destroyed by internal retarding 
 forces commonly termed frictional resistances, though they are 
 essentially different from the friction of solids. In certain 
 cases these frictional resistances vary directly as the trans- 
 lational velocity of flow, in others they vary nearly as the 
 square of that velocity. It is clear that in the two cases 
 there must be an essential difference in the character of the 
 motion. Using floating threads, or Professor Osborne Reynolds' 
 method of coloured fluid streams, it is found that in one class 
 of cases the particles follow very direct and constant paths 
 or stream lines ; in the other the particles eddy about in 
 constantly changing paths of great sinuosity. Professor 
 Reynolds has pointed out that the surface of a slow current 
 of clear water sometimes presents a plate -glass appearance, 
 reflections of objects on the surface being undistorted. That 
 appearance corresponds to non-sinuous or stream-line motion. 
 At other times the surface presents a sheet-glass appearance, 
 reflections being blurred or distorted. That is due to eddy 
 motions slightly disturbing the water surface. In a river 
 in flood the continual breaking up of the surface by eddies is 
 obvious enough. 
 
 Now in stream -line motion of the water (Fig. 24, a) the 
 
38 
 
 HYDEAULICS 
 
 CHAP. 
 
 (a) 
 NONS1NUOUS 
 
 resistance is due to the laminse sliding on each other with very 
 small differences of relative velocity. The relative motion is 
 opposed by the viscosity of the liquid ; the resistance is of the 
 nature of a shearing resistance, and is proportional to the 
 velocity of sliding. On the other hand, in eddying or 
 turbulent motion (Fig. 24, V) the relative velocities are very 
 much greater, energy is expended in giving motion to the 
 eddies, and this energy is gradually dissipated as the eddies 
 die out in consequence of their mutual friction. The kinetic 
 
 energy of an eddy is propor- 
 tional to the square of its 
 velocity, and as this must have 
 a definite relation to the general 
 velocity of translation of the 
 stream it is intelligible that 
 the resistance varies nearly as 
 the square of the velocity. In 
 a stream in turbulent motion 
 there is a continual generation 
 of eddies and stilling of them 
 again by fluid friction, and 
 consequently a continual degra- 
 dation of mechanical energy 
 into heat throughout the fluid 
 mass. The theory of stream- 
 line motion is much more 
 perfect than the theory of 
 turbulent motion ; indeed, in 
 
 the strict sense there is no rational theory of turbulent 
 motion but only empirical laws deduced from experiment. 
 Unfortunately, almost all cases of practical importance to the 
 engineer are cases of turbulent motion. 
 
 In cases of eddying motion, such as that shown in Fig. 
 24, b, the motion may be analysed into two parts : (a) a general 
 average motion of translation, and (5) an eddying motion 
 superposed which has no resultant motion. It is the former 
 only with which the engineer is in general concerned, and to 
 which the empirical laws of flow apply. 
 
 As an instance of how eddying may come in to modify the 
 action of water, an interesting experiment by Mr. Church, at 
 
 Or) 
 
 TURBULENT 
 
 Fig. 24. 
 
Ill 
 
 PEINCIPLES OF HYDEAULICS 
 
 39 
 
 ^", :,"W XXX ^ 
 
 the Cornell University, may be taken. He tried the discharge 
 through two orifices, A and B (Fig. 25). These were exactly 
 of the same size, except that B had a smoothly formed con- 
 traction at the inlet ; but it was found that B discharged 
 about 10 per cent more than A. Now, 
 why should contracting the section increase 
 the discharge ? The reason is simple, viz., 
 that in B the change of section of the 
 water stream is fairly gradual, and there 
 is not much tendency to disturb the 
 stream -line motion and generate eddies. 
 But in A the abrupt inlet angle generates 
 eddies, and so destroys part of the head 
 available for producing the velocity of flow. 
 But if the velocity of discharge is reduced 
 10 per cent the kinetic energy of the jet 
 is reduced about 20 per cent, or nearly 
 one-fifth of the energy is absorbed by the 
 eddies due to the sharp corner. That is a 
 case where the influence of eddies is com- 
 paratively small. In flow through a long 
 pipe it is much greater. Take a pipe of 
 12 inches diameter with a virtual slope of 1 in 1000. If in 
 such a pipe non-sinuous motion were possible the velocity 
 would be 72 feet per second. But the actual velocity, the 
 motion being turbulent, " is only l foot per second. The 
 difference shows the enormous amount of mechanical energy 
 expended in eddy-making. 
 
 28. Uniform and varying motion. Let ab (Fig. 26) 
 represent a path along which fluid particles are moving. If 
 the velocity of a particle a is constant 
 along the path the motion is uniform, 
 Fig 26 if not it is varying. In the ordinary 
 
 cases of turbulent motion it is said to 
 
 be uniform if the general velocity of translation is constant, 
 and varying if it is not constant. In a canal of constant 
 section the motion along the canal is usually uniform. In a 
 river the section of which varies the motion is varying, that 
 is, it is faster where the section is smaller, and slower where 
 it is greater. 
 
40 HYDKAULICS 
 
 CHAP. 
 
 Steady and unsteady motion. This introduces an idea 
 special to hydraulics and of great importance. Consider a 
 definite bounded space (Fig. 27) through which water is flow- 
 ing along definite stream lines. If in that space the velocity 
 
 is constant from minute 
 
 > > ~ ^_ ^ to minute, or from hour 
 
 > * x^ "X ~ to hour, the motion is 
 
 > JL \ steady. If it changes 
 
 ^ ^ 7 T the motion is said to 
 
 be unsteady. In turbu- 
 
 \ j lent motion, if at a 
 
 ^ Y - \ J given point the general 
 
 ^ > X / motion of translation is 
 
 constant in velocity and 
 direction, the eddies 
 
 Fig. 27. being disregarded, it is 
 
 said to be steady. If 
 
 not it is unsteady. At a given point on a river bank in 
 normal conditions the velocity and direction of motion are the 
 same from day to day. But when rising in flood or subsiding 
 afterwards, the velocity varies from minute to minute by some 
 small amount, and the motion is unsteady. 
 
 In ordinary streams and rivers in which the motion is 
 turbulent, the velocity and direction of motion at any point 
 varies from moment to moment. But if the eddies are dis- 
 regarded the average velocity over short periods varies very 
 little either in direction or velocity. If the variations are 
 regarded as periodic, then the general motion, apart from the 
 temporary fluctuations, is treated as steady motion. The 
 motion is regarded as equivalent to simple stream-line motion, 
 except that the energy absorbed and dissipated in eddies has 
 to be allowed for by experimental corrections. 
 
 29. Volume of flow. Let A (Fig. 28) be any ideal plane 
 surface, of area o>, in a stream, normal to the direction of 
 motion, and let V be the velocity of the fluid. Then the 
 volume flowing through the surface A in unit time is 
 
 Q = o>V . . . . (1). 
 
 Thus, if the motion is rectilinear, all the particles at any 
 instant in the surface A will be found after one second in a 
 
in PKINCIPLES OF HYDEAULICS 41 
 
 similar surface A', at a distance V, and as each particle is 
 followed by a continuous thread of other particles, the volume 
 of flow is the right prism AA' having a base & and length V. 
 
 Fig. 29. 
 
 If the direction of motion makes an angle 6 with the 
 normal to the surface, the volume of flow is represented by 
 an oblique prism A A' (Fig. 29), and in that case 
 
 Q = o>V cos . . . (2). 
 
 Mean velocity of flow. In most practical cases the 
 velocity V will be different at different parts of the cross 
 section A of the stream. In a river, for instance, the velocity 
 is greater towards the middle and top surface, and less towards 
 the bottom and sides. If v is the velocity at some small 
 element da) of the section, the volume of flow is 
 
 Q=/fo. . . (3), 
 
 where the integration extends to the whole surface &> of the 
 cross section. The mean velocity over the section is 
 
 V -f (4) 
 
 V m \*h 
 
 co 
 
 and in a large number of practical problems it is this mean 
 velocity which is required. Obviously the volume of flow is 
 
 Q = V, nW . (5). 
 
 If V m is inclined at 6 to the surface, 
 
 Q = V 7n o>cos6>. 
 
 Principle of continuity. Consider a fixed bounded space 
 through which liquid is flowing. If for any given time the 
 space is continuously filled with fluid the inflow and outflow 
 in that time must be equal, for the volume in the space is 
 
42 HYDEAULICS CHAP. 
 
 constant. If inflow is reckoned + and outflow , the volume 
 of flow for all the boundaries is 
 
 2Q = 0. . . (6). 
 
 In general the condition that the space should be con- 
 tinuously filled is that the pressure must be a thrust every- 
 where throughout the space. If water contains air in solution 
 as is ordinarily the case, the air is disengaged, and there is a 
 break in continuity if the thrust falls below a certain value, 
 depending on the amount of air in solution. 
 
 Let A I} A 2 be two cross sections of a stream flowing in 
 rigid boundaries, and V 1} V 2 the normal velocities at those 
 sections. Then from the principle of continuity 
 
 that is, the normal velocities are inversely as the areas of the 
 cross sections. This is true of the mean velocities if at each 
 section the velocity of the stream varies. In a river of vary- 
 ing slope the velocity varies with the slope. It is easy, there- 
 
 fore, to see that in parts of 
 large cross section the slope 
 is smaller than in parts of 
 small cross section. 
 
 If we conceive a space 
 in a liquid bounded by 
 normal sections at A p A 2 , 
 Fig. so. an( i between A 1? A 2 by 
 
 stream lines (Fig. 30), then, 
 as there is no flow across the stream lines, 
 
 as in a stream with rigid boundaries. 
 
 30. Application of the principle of the conservation of 
 energy to stream-line motion. Bernoulli's theorem. 
 
 Let AB (Fig. 31) be any one elementary stream in a 
 steadily moving fluid mass. Then from the steadiness of the 
 motion AB is a fixed path in space, and the fluid in it may 
 
Ill 
 
 PEINCIPLES OF HYDKAULICS 
 
 43 
 
 be regarded as flowing in a tube. Let 00 be the free surface 
 level, and XX any horizontal datum plane. Let co be the area 
 of a normal cross section, v the velocity, p the pressure, and z 
 the elevation above the datum plane at A, and to lt v^p^ z lt the 
 corresponding quantities at 
 B, and let Q be the flow in __ __ ___ ___ ^___ __ ___ ^ _J? '_ 
 
 unit time. Suppose that 
 in a short time t, AB comes 
 to A'B'. Then AA' = vt 
 and BB' = v-J, and the vol- 
 umes of fluid AA', BB', 
 the equal inflow and out- 
 flow = Q = covt = to^t. If 
 all frictional or viscous re- 
 sistances are absent the Fi s- 31 - 
 work of the external forces 
 will be equal to the change of kinetic energy. 
 
 The normal pressures on the surface of AB, except at the 
 ends, are everywhere perpendicular to the direction of motion 
 and do no work. Hence the external forces to be reckoned 
 are the pressures on the ends and gravity. The work of 
 gravity when AB falls to A'B' is the same as if AA' were 
 transferred to BB'. That is 
 
 Work of gravity = GQ(z - zj foot-pounds. 
 
 The work of the pressures on the ends, reckoning that at B 
 negative because it opposes motion, is (pressure x volume 
 described) 
 
 The change of kinetic energy in the time t is the difference of 
 the kinetic energy of AA' and BB', for in the space A'B the 
 energy is unchanged when the motion is steady. 
 
 The mass of AA' or BB' is Q,t, and the change of kinetic 
 
 t/ 
 
 energy in t seconds is 
 
 Equating work expended and change of kinetic energy, 
 
44 HYDRAULICS CHAP. 
 
 Dividing by GQt the weight of fluid and rearranging, 
 
 B^tr^" (8) ' 
 
 or as A and B are any two points, 
 
 /)2 rn 
 
 + ~ + z = constant = H foot-pounds . . (9), 
 
 where the quantities are reckoned per pound of fluid. The 
 three terms on the left are quantities of energy, and correspond 
 to the three forms in which energy may exist in a fluid in 
 motion, due to elevation, pressure, and velocity. They are 
 commonly called the heads due to elevation, pressure, and 
 velocity respectively, head being defined as energy per pound 
 of fluid. H is the total energy per pound. If h is the height 
 from the point considered measured up to the free surface, 
 
 y 2 v 
 
 - + ^ - h = H foot-pounds . . (10). 
 
 2g (* 
 
 The theorem may be expressed thus : The total head or total 
 energy per pound of fluid, relatively to a given horizontal 
 datum plane, is uniformly distributed along a stream line. 
 
 The term head in hydraulics. The term head is an old 
 millwright's word. A mill was said to have a good head of 
 water if it possessed a waterfall which, from its volume of flow 
 and height, was capable of developing a good amount of power 
 when used on a water-wheel. The term is now scientifically 
 understood as just defined. Since a pound of water falling 
 through a height h acquires li foot-pounds of energy, height 
 and head of elevation are numerically equal. Hence the term 
 head is often used loosely as equivalent to height, but this is 
 misleading. The term head should be restricted to cases in 
 which energy is considered. 
 
 Consider water flowing through a frictionless pipe AB 
 (Fig. 32), and that for the present viscosity effects such as the 
 production of eddies are negligible. Let pressure columns be 
 introduced at A and B. Let z, p, v, be the elevation, pressure, 
 and velocity at A, and z 1} p lf v 1} the same quantities at B. The 
 water will rise in the pressure columns to heights p/Gr and pi/G, 
 so that the heights of A' and B r above the datum XX are 
 
Ill 
 
 PRINCIPLES OF HYDRAULICS 
 
 45 
 
 z+p/G and Zi+pi/Q- Join A'B' and draw A'D horizontal. 
 A'B' is called the line of hydraulic gradient, or slope of the 
 pressure-column tops when the liquid is flowing. The fall of 
 the free surface level DB' = p/G + z (pjGr + z^, and this by 
 the theorem above is equal to (v^ v 2 )/2g. Consequently if 
 distances A'A" = v*/2g and B'B" = v*/2g are set up, A"B" is a 
 horizontal line at a height H above the datum XX. The 
 atmospheric pressure is assumed to be the same at both 
 pressure columns ; if it is not, the heads due to atmospheric 
 pressure at A and B must be reckoned as part of the pressure 
 
 Fig. 32. 
 
 heads. The modification of this when friction has to be 
 considered will be given later. 
 
 It will be seen from Bernoulli's equation that the three 
 forms of head which make up the total head are convertible. 
 Thus for points on the same level, if the velocity increases the 
 pressure must diminish, and vice versa. If the pipe is of 
 uniform section so that the velocity is uniform, then if the 
 elevation increases the pressure diminishes, and vice versa. 
 
 31. Illustrations of the theorem of Bernoulli. In a 
 lecture to the mechanical section of the British Association in 
 1875, the late Mr. W. Froude gave some experimental illustra- 
 tions of the principle of Bernoulli. Mr. Froude remarked that 
 it was a common but erroneous impression that a fluid exercises 
 in a contracting pipe A (Fig. 33) an excess of pressure against 
 the entire converging surface which it meets, and that, con- 
 
46 
 
 HYDKAULICS 
 
 CHAP. 
 
 versely, as it enters an enlargement B, a relief of pressure is 
 experienced by the entire diverging surface of the pipe. 
 Further, it is commonly assumed that when passing through a 
 
 contraction C, there is in the narrow neck an excess of pressure 
 due to the squeezing together of the liquid at that point. 
 These impressions are in no respect correct ; the pressure is 
 smaller as the section of the pipe is smaller, and conversely. 
 Fig. 34 shows a pipe so formed that a contraction is 
 
 Fig. 34. 
 
 followed by an enlargement, and Fig. 35 one in which an 
 enlargement is followed by a contraction. The vertical pressure 
 columns show the decrease of pressure at the contraction, and 
 increase of pressure at the enlargement. The line abc in 'both 
 figures shows the variation of free surface level, supposing the 
 pipe frictionless. In actual pipes, however, work is expended 
 
Ill 
 
 PKINCIPLES OF HYDEAULICS 
 
 in Motional eddies ; the total head diminishes in proceeding 
 along the pipe, and the free surface level is a line such as 
 a\c l} falling below abc. 
 
 Mr. Froude further points out that, if a pipe contracts 
 
 Fig. 35. 
 
 and enlarges again to the same size, the resultant pressure on 
 the converging part exactly balances the resultant pressure on 
 the diverging part, so that there is no tendency to move the 
 pipe bodily when water flows through it. Thus the conical 
 part AB (Fig. 36) presents the same projected surface as HI, 
 
 Fig. 36. 
 
 and the pressures parallel to the axis of the pipe, normal to 
 these projected surfaces, balance each other. Similarly the 
 pressures on BC, CD, balance those on GH, EG. In the same 
 way, in any combination of enlargements and contractions there 
 is a balance of the pressures parallel to the axis of the pipe, 
 provided the area and direction of the ends are the same. If, 
 however, the eddy loss is taken into account the balance is 
 
48 
 
 HYDEAULICS 
 
 CHAP. 
 
 imperfect, and there is a drag in the direction of the motion 
 of the water. 
 
 Let Fig. 37 represent two cisterns A and E provided with 
 a converging pipe B and a diverging pipe D. The water will 
 flow from A, cross the gap C, and fill E, till the level in it is 
 nearly the same as in A. The pressure head h at the datum 
 line XX in A becomes a velocity head v 2 /2g at the gap, and 
 is reconverted into a pressure head nearly equal to h in E. 
 There is a small loss due to inexact correspondence of the 
 orifices and to eddy loss. In the jet crossing the gap there is 
 
 Fig. 37. 
 
 no pressure except the atmospheric pressure acting uniformly 
 throughout the system. 
 
 31 A. Variation of pressure across the stream lines in 
 two-dimensional motions. 1 Let AB, CD be two stream lines 
 in the plane of the figure (Fig. 3 *7a). Along the stream lines 
 the variation of pressure and velocity is determined by 
 Bernoulli's theorem. Normal to the plane of the figure, since 
 the stream lines are parallel, the distribution of pressure is 
 hydrostatic. There remains the direction in the plane of the 
 figure and along the radius of curvature, that is the direction 
 PQ. Let PQ be particles moving along the stream lines at 
 a distance PQ = ds, and let z be the elevation above a datum 
 
 1 See Cotterill, " On the Distribution of Energy in a Mass of Fluid in Steady 
 Motion," Phil. Mag., February 1876. 
 
Ill 
 
 PKINCIPLES OF HYDEAULICS 
 
 49 
 
 plane, p the pressure, and v the velocity at Q. At Q the 
 total head or energy per pound of fluid is 
 
 Differentiating, the increment of head between Q and P is 
 
 ._ . dp vdv 
 But dz = ds cos <f), 
 
 dH = | + ^%<fecos4> . . (11), 
 
 where the last term disappears when the motion is in a 
 horizontal plane. 
 
 Fig. 37a. 
 
 Imagine a small cylinder of section co described round PQ 
 as an axis. This will be in equilibrium under the action of 
 its weight Gcods ; the pressures on its ends pco and (p + dp)o> ; 
 and its centrifugal force acting along the radius of curvature 
 
 C^ -rJ 9 
 
 and equal to , where r is the radius of curvature at Q. 
 
 Taking components parallel to PQ, 
 
50 HYDKAULICS CHAP. 
 
 p 2 
 
 - Go> cos 
 
 y 
 
 Introducing this in (11), the increment of head between Q and 
 
 <H-nf . . . (13). 
 
 Corollary. If the stream lines are straight and parallel 
 in a horizontal plane, r is infinite and the increment of head 
 across the stream lines is vdvjg. Comparing this with (11), 
 dp/G = 0, or the pressure is uniform in a direction normal to 
 the stream lines. If the stream lines are straight and parallel 
 in a vertical plane dIL = vdv/g, and comparing this with (11), 
 dpjQc = ds cos <f> = dz, or p/Q + z = constant, that is, the pressure 
 along a vertical varies hydrostatically, or in the same way as 
 in a fluid at rest. 
 
 32. Radiating current. Suppose water supplied steadily 
 at the centre and flowing outwards between two parallel plates 
 at a distance d apart (Fig. 38). From the uniformity of 
 conditions the stream lines will be straight and radial. Con- 
 ceive two cylindric sections of the current at radii r t and r Z) 
 where the velocities are v 1 and v 2 , and the pressures p l and p 2 . 
 Since the flow across each section must be the same, 
 
 Q = 
 
 The velocity varies inversely as the radius, and would be 
 infinite at the centre if the radial flow could extend so far. 
 The motion being steady, 
 
 H= pi + v = p 2+ v 
 
 G + 2g G 2g 
 
 _2 r i Vj 2 
 
 -G + r/ 2? 
 
 aHft-^Yl-^l (14) 
 
 G ~2< r.V ' ' ' I4)l 
 
Ill 
 
 PEINCIPLES OF HYDKAULICS 
 
 51 
 
 or in another form 
 
 (Ha). 
 
 Hence the pressure increases from the interior outwards in a 
 way indicated by the pressure columns in Fig. 38. In the 
 plane of the figure the curve 
 through the pressure column 
 tops, or curve of the free 
 surface, is a quasi-hyperbola 
 of the form xy 2 = c 3 . This 
 curve is asymptotic to the 
 vertical axis of the current 
 and to a horizontal line H 
 feet above the plane from 
 which the pressures are meas- 
 ured It is worth noting 
 that if the discharge is into 
 the air the pressure p z /Gr at 
 the circumference is atmo- 
 spheric pressure. All the pres- 
 sures at less radii are smaller 
 than atmospheric pressure. 
 Hence the total pressure 
 above the top plate is greater 
 than that below it, and if the 
 top plate is loose it would 
 tend to approach the lower 
 plate and not to recede from Fig. 33. 
 
 it. 
 
 Free circular vortex. A free circular vortex is a re- 
 volving mass of water, in which the stream lines are concentric 
 circles, and in which the total head for each stream line is 
 the same. Hence, if by any slow radial motion portions of 
 the water strayed from one stream line to another, they would 
 take freely the velocities proper to their new positions under 
 the action of the existing fluid pressures only. 
 
 For such a current, the motion being horizontal, we have 
 for all the circular elementary streams 
 
 4)2 
 
 H = p + = constant ; 
 
 \3T &y 
 
52 HYDKAULICS CHAP. 
 
 /g ^ H = |? + ^ = o .... (15). 
 
 Consider two stream lines at radii r and r + dr (Fig. 38). 
 Then in eq. (13) r = r and ds = dr, 
 
 v 2 _ vdv 
 dr+ =0, 
 
 grr gr 
 
 <&;_ d?' 
 
 7" "r 1 
 
 v~- (16), 
 
 r 
 
 precisely as in a radiating current ; and hence the distribution 
 of pressure is the same, and formulae 14 and 14a are applic- 
 able to this case. 
 
 Free spiral vortex. As in a radiating and circular 
 current the equations of motion are the same, they will also 
 apply to a vortex in which the motion is compounded of these 
 motions in any proportions, provided the radial component of 
 the motion varies inversely as the radius as in a radial current, 
 and the tangential component varies inversely as the radius 
 as in a free vortex. Then the whole velocity at any point 
 will be inversely proportional to the radius of the point, and 
 the fluid will describe stream lines having a constant inclina- 
 tion to the radius drawn to the axis of the current. That is, 
 the stream lines will be logarithmic spirals. When water is 
 delivered from the circumference of a centrifugal pump or 
 turbine into a chamber, it forms a free vortex of this kind. 
 The water flows spirally outwards, its velocity diminishing 
 and its pressure increasing according to the law stated above, 
 and the head along each spiral stream line is constant. 
 
 33. Forced vortex. If the law of motion in a rotating 
 current is different from that in a free vortex, some force 
 must be applied to cause the variation of velocity. The 
 simplest case is that of a rotating current in which all the 
 particles have equal angular velocity, as for instance when 
 they are driven round by radiating paddles revolving uniformly. 
 Then in equation (13), considering two circular stream lines 
 of radii r and r + dr (Fig. 39), we have r = r, ds = dr. If the 
 angular velocity is a, then v = ar and dv = adr. Hence 
 
Ill 
 
 PKINCIPLES OF HYDEAULICS 
 
 53 
 
 , TT a 2 r, a*rdr 2a 2 r 7 
 dH = dr + - = dr. 
 
 g g g 
 
 Comparing this with eq. (11), and putting dz 0, because the 
 motion is horizontal, 
 
 dp 
 
 dp_a?Ti 
 
 P = aW 
 G~ 2<7 
 
 dr. 
 
 constant 
 
 (17). 
 
 Let p lt r lt ^j be the pressure, radius, and velocity at one 
 cylindrical section, p 2 , r 2 , v 2 
 those at another ; then 
 
 (18). 
 
 That is, the pressure increases 
 from within outwards in a 
 curve which in radial sections 
 is a parabola, and surfaces of 
 equal pressure are paraboloids 
 of revolution (Fig. 39). This 
 case corresponds to a crude 
 form of centrifugal pump. 
 Apart from a small head pro- 
 ducing the radial flow, the lift 
 
 of the pump is p feet, 
 
 where p 2 and p 1 are the pres- 
 sures at the outlet and inlet 
 of the pump disc. 
 
 34. Venturi meter. An Fig. 39. 
 
 extremely beautiful application 
 
 of this principle has been made by Mr. Clemens Herschel, in 
 the construction of what he has termed the Venturi meter 
 for measuring water flowing in pipes. Suppose in any water- 
 
54 
 
 HYDKAULICS 
 
 CHAP. 
 
 main a contraction is made (Fig. 40), the change of section 
 being very gradual to avoid the production of eddies. The 
 ratio p of the sections at inlet and throat is in actual meters 
 between 5 to 1 and 20 to 1, and is very carefully determined 
 by the maker of the meter. Then the ratio of the velocity v 
 in the main and the velocity u at the throat is definitely 
 known. Now suppose glass tubes, "piezometer tubes" they 
 are sometimes called, are inserted, in which the water ascends 
 to a height which measures the pressure. Since the velocity 
 is greater at the throat than in the main, the pressure will 
 
 
 !| JS. 
 
 -*-. . 
 
 i 
 
 -*-. FRICTION 
 
 
 
 A,; 
 
 lb_ *. 
 
 d 
 
 H 
 
 I"*- TH 
 
 j. 
 
 B_j 
 
 - 
 
 l-fZ" HEAD 
 1 ^ 
 
 =^-^---=.-^=. 
 
 ROAT 
 
 f p 
 
 
 
 
 / 
 
 y 
 
 
 
 / ' - 
 
 / / 
 / / 
 / / 
 
 Fig. 40. 
 
 be less and the pressure head h 2 will be less than h it and this 
 is a quantity easily observed. Using Bernoulli's equation, 
 
 or putting u = pv, where p is the ratio of the cross sections, 
 
 t 
 + 
 
 (19), 
 
 from which the velocity at the throat can be determined if 
 the Venturi head h-^ h 2 is observed, and the ratio p of 
 the sections is known. But if u and the area at the throat 
 
in PKINCIPLES OF HYDEAULICS 55 
 
 are known, the discharge of the meter is known. Let ft be 
 the section of the pipe, then fl/p is the section at the throat. 
 For simplicity let h 1 h 2 = h. Then the discharge is 
 
 Hence, by a simple observation of the piezometric heights, 
 the flow in the main at any moment can be determined. 
 Notice that if a third piezometer is introduced where the 
 water has regained its original section and velocity, the piezo- 
 metric height will be the same as at first, except for a small 
 loss due to the fact that the motion is not quite non-sinuous, 
 and that some eddies are generated in the meter. 
 
 In order to get the pressure head at the throat very exactly, 
 Mr. Herschel surrounds the throat with an annular passage 
 communicating with the throat by small holes, sometimes 
 formed in vulcanite plugs to prevent corrosion. 
 
 Although constructed to secure as far as possible non- 
 sinuous motion, the eddy motion cannot be entirely prevented 
 in the Yenturi meter. The main effect of this is to cause a 
 loss of head between the two ends of the meter, varying between 
 1 foot and 5 feet according to the velocity through the meter. 
 But the eddying also affects the difference of head at inlet and 
 throat, from which the discharge through the meter is 
 calculated ; consequently, even with this meter, an experi- 
 mental coefficient must be introduced, determined by tank 
 measurement. However, the range of this coefficient is 
 surprisingly small. Mr. Herschel found coefficients ranging 
 between 0*97 and TO for throat velocities varying between 8 
 feet per second and 28 feet per second, or inlet velocities 
 varying between 0'9 foot per second and 3'1 feet per second. 
 
 Putting eq. (20) in the form 
 
 Q = dl v/ | c. ft. per sec. . (20a), 
 
 where c is the coefficient of the meter, the mean value of c is 
 0'972, and it is rather smaller for small values and greater for 
 large values of the Venturi head h. It is stated to be desirable 
 that the throat velocity should be 15 to 40 feet per second. 
 If the Venturi head is measured by a mercury siphon gauge, let 
 
56 HYDKAULICS CHAP. 
 
 h m be the difference of level in the gauge in inches, and let 
 13 '5 9 be the density of mercury. Then the Venturi head in 
 feet of water is 
 
 m . . (21). 
 
 Mr. Kent of Holborn has constructed two meters for 
 94-inch mains at the reservoir works at Staines. The coned 
 parts are of riveted steel plates, and have a total length of 84 
 feet. The throat ratio is 1 to 7, and they can register a flow 
 varying from 400,000 to 6,000,000 gallons per hour. Two 
 still larger meters are being constructed for a pumping station 
 at Divi in the Madras Presidency. The main pipes are 120 
 inches in diameter. The upstream cones are of steel plate 
 bedded in concrete, and the downstream cones of concrete only. 
 Each meter can register from 1 to 11 million gallons per 
 hour. Various forms of recording apparatus have been used 
 with the meter. In one, a line proportional in length to the 
 discharge is drawn on the recording drum at every quarter 
 hour or other predetermined interval. In another, a line is 
 drawn showing the Venturi head at each instant. An in- 
 tegrating arrangement is also used, the total flow for any given 
 time being shown by a counter. 
 
 35. Principle of the conservation of momentum. If a 
 force P acts on a body of weight W, or mass m = W/g, moving 
 in the direction of P, the change of velocity from v 1 to v 2 in 
 time t is given by the relation 
 
 R-fa-k)-j<*-i) (22), 
 
 where Pt in second-pounds is termed the impulse of the force, 
 and m(v 2 vj the change of momentum. Thus the impulse 
 of a force is equal to the change of momentum in the direction 
 of the force. Conversely, if the body suffers a decrease of 
 momentum due to a change of velocity from v 2 to v 1} it must 
 exert an impulse of ~Pt second-pounds in the direction of the 
 change of momentum. The principle of momentum is of 
 special use in hydraulics, because it can be applied irrespectively 
 of the mutual action of the particles and of their actual motions, 
 only their velocity components in the direction considered 
 being required. 
 
in PRINCIPLES OF HYDRAULICS 57 
 
 36. Relation of pressure and velocity in a stream in 
 steady motion when the changes of section of the stream 
 are abrupt. When a stream changes section abruptly, rotating 
 eddies are formed which dissipate energy. The energy absorbed 
 in producing rotation is at once abstracted from that effective 
 in causing the flow, and sooner or later it is wasted by 
 frictional resistances due 
 to the rapid relative 
 motion of the eddying 
 parts of the fluid. The 
 energy thus lost is com- 
 monly termed energy lost 
 in shock. Suppose Fig. 
 41 to represent a stream 
 
 having such an abrupt F DD' 
 
 change of section. Let 
 AB, CD be normal sec- 
 
 tions at points where ordinary stream -line motion has not 
 been disturbed and where it has been re-established. Let 
 o), p, v be the area of section, pressure, and velocity at AB, 
 and G)J, p lt V-L corresponding quantities at CD. Then if no work 
 were expended internally, and assuming the stream horizontal, 
 
 K-8+5J 11 <> 
 
 But if work is expended in producing irregular eddying motion, 
 the head at the section CD will be diminished. 
 
 Suppose the mass ABCD comes in a short time t to A'B'C'D'. 
 The resultant force parallel to the axis of the stream is 
 
 where p is put for the unknown pressure on the annular space 
 between AB and EF. The impulse of that force is 
 
 The horizontal change of momentum in the same time is the 
 difference of the momenta of CDC'D' and ABA'B', because the 
 amount of momentum between A'B' and CD remains unchanged 
 if the motion is steady. The volume of ABA'B' or CDC'D', 
 being the inflow and outflow in the time t, is Q = avt = at^t, 
 
58 HYDKAULICS 
 
 CHAP. 
 
 and the momentum of these masses is Qvt and Qfl^. The 
 
 P 
 change of momentum is therefore Q%j - v). Equating this 
 
 to the impulse, 
 
 Assume that PQ=%>, the pressure at AB extending unchanged 
 through the portions of fluid in contact with AE, BF which 
 lie out of the path of the stream. Then (since Q = 
 
 P 
 
 
 This differs from the expression obtained for cases where no 
 
 sensible internal work is done, by the last term on the right. 
 
 (u _ )2 
 
 That is, has to be added to the total head at CD, which 
 z ff 
 
 is ~ + <p to make it equal to the total head at AB, or ^ *' 
 
 is the head lost in shock at the abrupt change of section. 
 But v v 1 is the relative velocity of the two parts of the 
 stream. Hence, when an abrupt change of section occurs, the 
 
 (m - V ) 2 
 
 head due to the relative velocity is lost in shock, or 2 l 
 
 foot-pounds of energy is wasted for each pound of fluid. 
 Experiment verifies this result, so that the assumption that 
 Po=p appears to be admissible. 
 If there is no shock, 
 
 Pl _p t^-V 
 G~G 20 
 If there is shock, 
 
 Pi_P_ v i( v i ~ v) 
 G~O~ g 
 
 Hence the pressure head at CD in the second case is less than 
 in the former by the quantity 
 
in PKINCIPLES OF HYDEAULICS 59 
 
 or, putting co^ = wv, by the quantity 
 
 so -.-)* <> 
 
 The labyrinth piston packing. Pistons for pumps are 
 sometimes made with a series of circumferential 
 recesses without any other packing. The passage 
 between the cylinder and piston then consists of n 
 wide spaces of cross section A and n + 1 spaces of 
 smaller cross section a. Let Q be the amount of 
 leakage per second. Then the velocity in the narrow Fig 42< 
 passages is Q/a, and that in the wide passages is 
 Q./A. At each change of velocity in passing from a narrow 
 to a wide passage there will be a loss of head 
 
 fy 
 
 And as the energy in the last narrow passage is also wasted 
 the whole loss of head is 
 
 1 _ 1\ 2 l_ 
 which when A is large compared with a tends to the limit 
 
 2g a? ' 
 
 As the total difference of head between the two sides of the 
 piston which produces the leakage is a fixed quantity, the 
 greater the head wasted the smaller the leakage. The larger 
 n and the smaller a the less will be the leakage. There are 
 in addition some resistances in the small passages which are 
 not included in this reckoning. 
 
 PROBLEMS 
 
 1. A pipe AB, 100 feet long, has an inclination upwards of 1 in 4. 
 The head due to the pressure at A is 50 feet, the velocity is 
 4 feet per second, and the section of the pipe is 3 square feet. 
 Find the head due to the pressure at B, where the section is 
 Ij square feet. 25 feet. 
 
60 HYDRAULICS CHAP, m 
 
 2. The injection orifice of a condenser is at 12 feet below the surface 
 
 of supply tank. The condenser gauge shows a pressure of 5 
 inches of mercury. Neglecting frictional resistances, find the 
 velocity at which water will enter the condenser. 
 
 50-9 ft. per sec. 
 
 3. A Venturi meter has a diameter of 4 feet in the large part and 
 
 1-25 feet in the throat. With water flowing through it, the 
 pressure head is 100 feet in the large part and 85 feet at the 
 throat. Find the velocity in the small part and the discharge 
 through the meter. Coefficient of meter taken as unity. 
 
 38-3 c. ft. per sec. 
 
 4. Ten cubic feet of water are discharged by a pipe per second under 
 
 a total head of 100 feet. Find h.p. of the stream. 113. 
 
 5. Water flows radially outwards between two parallel plates. At 
 
 2 feet radius the pressure head is 10 feet and -the velocity is 
 10 feet per second. Find the pressure and velocity at 4 feet 
 radius. 10 ft. per sec.; 14 -7 ft. 
 
 6. Ten cubic feet of water per second flow through a pipe of 1 square 
 
 foot area, which suddenly enlarges to 4 square feet area. Taking 
 the pressure at 100 Ibs. per square foot in the smaller part of 
 the pipe, find (1) the head lost in shock ; (2) the pressure in 
 the larger part ; (3) the work expended in forcing the water 
 through the enlargement; (4) the rise of temperature of the 
 water at the enlargement. 
 
 0-87 ft.; 136 Ibs. per sq. in.; 545 ft. -Ibs. per sec.; 0'07 F. 
 
 7. A centrifugal pump with radial vanes has diameters of 1 foot 
 
 inside and 2 feet outside. It revolves 360 times per minute. 
 Find the pressure height produced in the pump. 16'6 ft. 
 
 8. A Venturi meter is 3 feet in diameter at each end and 1 foot in 
 
 diameter at the throat. Find the Venturi head when the inlet 
 velocity is 3 feet per second. Coefficient 0'97. 10-53 ft. 
 
 9. Find the energy stored per cubic foot of water in an accumulator 
 
 loaded to 700 Ibs. per square inch. 100,800 ft. -Ibs. 
 
 10. In a Venturi meter the diameters at inlet and throat are 12 
 
 inches and 5 inches. With water flowing through the meter, 
 the Venturi head is observed to be 6 inches of mercury. Find 
 the discharge. 2-9 c. ft. per sec. 
 
CHAPTER IV 
 
 DISCHARGE FROM ORIFICES 
 
 37. Experimental observations. Some simple laws govern- 
 ing the discharge from orifices are directly indicated by 
 simple observations. Suppose a 
 reservoir arranged as shown in 
 Fig. 43, with a horizontal orifice 
 h feet below the free surface and 
 a vertical jet. That this condition 
 may be permanent, and the flow 
 steady, water must be supplied 
 continuously at the free surface 
 at the rate at which it is dis- 
 charged by the jet. The jet rises 
 very nearly to the free surface 
 level in the reservoir, and the 
 small difference h r may reasonably 
 be attributed to small resistances 
 of the air or orifice. Neglecting 
 this small quantity, particles 
 which rise freely to a height h Fig. 43. 
 
 must have issued from the orifice 
 with a velocity given by the relation 
 
 v = *J(2gh) ft. per sec. . . . (1). 
 
 This relation was first discovered by Torricelli and Bernoulli. 
 If the orifice is of a proper conoidal form, the section of the 
 jet at the orifice is equal to the area of the orifice, and the 
 elementary streams forming the jet are normal to the orifice. 
 Let co be the area of the orifice. Then (29) the discharge 
 must be, neglecting the small resistances, 
 
 61 
 
62 HYDEAULICS CHAP. 
 
 Q = wfl = 
 
 = 8*023o> N/A c. ft. per sec. (la). 
 
 The actual velocity and discharge will be slightly less than 
 this if the resistances are considered. 
 
 In the case of a horizontal orifice the head is the same 
 at all parts of the orifice. But equations (1) and (la) are 
 
 used also for the more ordinary 
 case in which the orifice is 
 vertical, and the head varies at 
 different parts of the orifice, 
 and it is necessary to inquire 
 how far this is justifiable. In 
 the case of vertical orifices the 
 head h is taken to be the head 
 measured to the centre of the 
 orifice. Consider a conoidal 
 Fig 44t rectangular orifice such that the 
 
 section of the jet is identical 
 
 with the area of the outlet of the orifice (Fig. 44). Let Hj 
 be the head at the top edge, and H 2 that at the bottom edge 
 of the orifice, and B its breadth. The area is B(H 2 - Hj) 
 and the mean head is h = ^(H 2 -f Hj). Putting these values 
 in eq. (la), 
 
 Q = B(H 2 -H 1 )v/MH 2 + H 1 )}, 
 
 and the velocity of discharge, the same at all parts of the 
 orifice, on the assumption that the variation of head is 
 negligible, is 
 
 Consider a horizontal lamina issuing between the levels H and 
 H + <m. Its area is BrfH, and the discharge is 
 The discharge of the whole orifice is 
 
 /H, 
 
 ; H*< 
 
 J H! 
 
 Q = Bv/2? H'dH 
 
 -^ 1 } . . (2). 
 
 Hence the mean velocity when the variation of head is taken 
 into the reckoning is 
 
iv DISCHARGE FROM ORIFICES 63 
 
 o _ H * - H i* 
 
 v, 3 (H t -H 1 )^(H 1 
 Let Hj = pH 2 , 
 
 * 2 _2v/2 1-p 1 
 
 Comparing this with the velocity found if the variation of 
 head is neglected, 
 
 . (3). 
 
 0-0 0-9428 
 
 0-2 0-9797 
 
 0-5 0-9952 
 
 It is clear that v 2 is always a little less than v lt but the 
 difference becomes less as p increases and is negligibly small 
 if p>0*5. Hence, except when the head on the top of the 
 orifice is less than half the head on the bottom, the approxi- 
 mate equation (1) or (la) may be used without sensible error 
 in place of the more complicated equations (2) and (2 a). The 
 practically important case when H x = will be dealt with 
 later. 
 
 38. Coefficients of velocity and resistance. The ap- 
 proximate formula just given may be made exact for any 
 given conditions by introducing an experimental coefficient. 
 The actual velocity of discharge is 
 
 v a = c v */tyh . . (4), 
 
 where c v is a coefficient termed the coefficient of velocity, 
 which experiment shows to vary little for a given type of 
 orifice. For well-formed simple orifices c v is 0'97 to 0'98, 
 and rather greater for very great heads. The velocity of dis- 
 charge can be expressed in another way. If h e is the actual 
 height to which the molecules rise, v a = \/2gh e . If the loss 
 of head A r = c r A e , where c r is a coefficient termed the co- 
 efficient of resistance, 
 
 < 5 > 
 
64 HYDKAULICS 
 
 Equating the two expressions for v at 
 
 CHAP. 
 
 (6). 
 
 Thus if c v = 0-97, c r = 0*0628 ; and if c v = 0'98, c r = 0'0412. 
 
 The work of gravity on each pound of water descending 
 from the free surface level to the orifice is h ft.-lbs., and if 
 unresisted the water would acquire v^/Zg ft.-lbs. of kinetic 
 energy. The actual energy of the jet is only v a 2 /2g = h e ft.- 
 lbs. per pound. Hence h r = c r v a */2g ft.-lbs. per pound is the 
 energy wasted in overcoming resistances. With the values of 
 c r given above, from 6^ to 4-J- per cent of the head is wasted. 
 
 Fig. 45. 
 
 Coefficient of contraction. When a jet issues from an 
 orifice it may either spring clear from the inner edge of the 
 orifice as at a or b (Fig. 45), or it may adhere to the sides of 
 the orifice as at c. The former condition always obtains if the 
 orifice is bevelled to a sharp edge as at a, and generally for 
 cylindrical orifices such as I if the thickness of the plate is 
 not more than the diameter of the orifice. If the plate 
 thickness is 1^- times the diameter of the orifice or more, the 
 condition shown at c obtains, and it is convenient to dis- 
 tinguish orifices of that kind as mouthpieces. At c the jet 
 issues " full bore," or of the same diameter as the orifice, but 
 in the other cases the jet contracts to a diameter smaller than 
 
IV 
 
 DISCHAEGE FKOM OEIFICES 
 
 65 
 
 the orifice in consequence of the convergence of the streams 
 which make up the jet. 
 
 Let & be the area of the orifice and c c o> the contracted area 
 of the jet. Then c c is a coefficient to be determined experi- 
 mentally, called the coefficient of contraction, which is found 
 to be nearly constant for certain types of orifice. For sharp- 
 edged or virtually sharp-edged orifices, such as those shown in 
 a and 5, the average value of c c is 0'64, but with different 
 kinds of orifice its value may range from 0*5 to 1/0. With 
 c c =0'64 the diameter of the contracted 
 section of a circular jet is 0*8 of the 
 diameter of the orifice. 
 
 It may be noted that as the stream 
 lines are curved when approaching the = 
 contracted section there is a centrifugal = 
 pressure across the stream lines (Fig. 46). 
 Hence the pressure is greater and the 
 velocity less towards the centre of the 
 converging jet. At the contracted section 
 the stream lines become parallel, the 
 pressure is uniform, and probably the velocity nearly uniform. 
 Coefficient of discharge. The discharge Q = <ov is 
 
 diminished partly by reduc- 
 tion of velocity, partly by 
 contraction of section. Hence 
 the actual discharge is 
 
 Fig. 46. 
 
 = cv x co> = 
 
 or if c c c v = c, which is termed 
 the coefficient of discharge, 
 
 Qa = ftuV2^ . (7). 
 For sharp-edged plane ori- 
 fices c averages about 0'975 
 x 0-64 = 0*62. But exact 
 values for different cases will 
 be given presently. 
 
 39. Experimental de- 
 
 termination of c v , c c , and c. To determine the coefficient of 
 contraction, the section of the jet must be measured at a distance 
 from the orifice equal to about half its diameter. Fig. 4*7 shows 
 
 5 
 
 *"* 47 - 
 
66 
 
 HYDKAULICS 
 
 CHAP. 
 
 an arrangement of set-screws which can be set to touch the 
 jet, and the distance between them afterwards measured. 
 When the orifice is not circular the measurement is difficult, 
 
 because the section 
 of the jet is not ex- 
 actly similar to the 
 orifice. 
 
 The coefficient of 
 velocity is most easily 
 found by measuring 
 the parabolic path of 
 a horizontal jet. Let 
 OAB (Fig. 48) be 
 the path of the jet. 
 Take OX, OYas hori- 
 zontal and vertical 
 co-ordinate axes. Let 
 h be the head over 
 the centre of the 
 
 orifice, and x, y the co-ordinates of any point A. If v a is the 
 horizontal velocity of the jet, and t the time in which a particle 
 falls from to A, 
 
 1 .* //gx\ 
 
 -*; y = -^', *Vfe> 
 
 consequently 
 
 Fig. 48 
 
 As a check, other co-ordinates, such as x 1} y l} should be measured. 
 In principle, the coefficient of velocity could be found by 
 measuring the height h e (Fig. 43) to which a vertical jet rises 
 under a head h. Then 
 
 V (*)' 
 
 but, except for moderately small heads, the measurement is 
 difficult. 
 
 In practical hydraulics the coefficient of discharge is much 
 more important than the others, and it can be determined with 
 very great accuracy by tank measurement. In Fig. 49 is 
 shown an arrangement of a measuring tank for gauging the 
 
IV 
 
 DISCHAEGE FKOM OKIFICES 
 
 flow from an orifice or notch. The orifice is placed at the end 
 of the reservoir A, and discharges into the waste channel C, 
 and the water flows to waste at F. A trough on rollers B can 
 be slid under the jet, and then delivers the water into the 
 measuring tank D. In the tank is a stilling screen S, and an 
 outlet valve E. Means are provided for very accurately 
 measuring the water-level at the beginning and end of a 
 convenient interval of time, and the area of the tank must be 
 carefully determined. Let the water be discharged into the 
 tank for t seconds, during which the level in the reservoir of 
 
 Fig. 49. 
 
 area A rises H 2 - H! feet, and let h be the head at the orifice, 
 and w its area. 
 
 /TT TT \ A 
 
 Q = l ~i - = coi \/2gh cubic feet per second, 
 _(H 2 -H 1 )A 
 
 (8). 
 
 All the required measurements can be made with great 
 accuracy, especially if the tank is large enough to contain the 
 flow during ten or fifteen minutes. 
 
 40. Use of orifices in measuring water. The Eomans 
 used orifices of bronze to deliver regulated quantities of water 
 from the aqueducts to consumers. The unit of discharge was 
 that from an orifice 0*907 inches diameter, and was termed a 
 quinaria. Fifteen sizes were used, the largest being 8*964 inches 
 diameter, and delivering 97 quinarise. The discharge was 
 assumed to be proportional to the area of the orifice, and 
 
68 HYDKAULICS 
 
 CHAP. 
 
 although it was known that the discharge depended in some 
 way on the head, the arrangements adopted to secure approxi- 
 mate uniformity of head in different cases are not known and 
 appear to have been imperfect (Frontintis, De Aquis, translated 
 by Herschel). 
 
 In the case of the irrigation works of Northern Italy the 
 water was supplied to estates through orifices, termed modules, 
 for which the height and head were legally fixed, and the 
 width varied according to the amount of water required. This 
 is an almost exact way of delivering a measured quantity of 
 water. The Sardinian unit module was an orifice 0'656 feet 
 square with a head of 0'656 above the top edge, delivering 
 about 2 cubic feet per second. 
 
 An old measure of the discharge of the same kind was the 
 so-called water inch, defined by some of the older French 
 hydraulicians as the discharge of an orifice one inch in 
 diameter, with a head of one line above the top edge. In the 
 mining district of California a similar method was used in 
 supplying water to different mines from a supply channel. 
 The unit of discharge was termed the miner's inch, and was 
 the discharge through one square inch of orifice with a head 
 of 6ijr inches, or about 1'5 cubic feet per minute. But as the 
 form of the orifice and the head were not defined as carefully 
 as in the Italian regulations, the value of the miner's inch 
 varied a good deal in different districts. Later legal defini- 
 tions of the miner's inch were adopted, varying in different 
 cases from 1:5 to 1-2 cubic feet per minute. 
 
 In delivering compensation water from reservoirs to 
 streams in this country an orifice is used, the head on which 
 is regulated so as to be constant. The arrangement is such 
 that any riparian owner interested in the flow in the stream 
 can at any time see whether the proper head, and therefore the 
 proper discharge, is maintained. 
 
 41. Measurement of the head over an orifice. The 
 most convenient way of measuring the head over an orifice in 
 a tank is by a gauge-glass, scale, and vernier (Fig. 50). A 
 bar AA is rigidly attached to the tank, having a slot in 
 which the scale BB slides. The scale has at the bottom an 
 adjusting screw by which its zero can be set exactly to the 
 level of the centre of the orifice. A slider C, with a finger 
 
IV 
 
 DISCHAKGE FKOM OEIFICES 
 
 69 
 
 projecting across the gauge-glass, has also a vernier reading 
 on the scale. The scale is most conveniently divided into 
 feet, tenths, and hundredths of a foot. The vernier then 
 reads to O'OOl foot. The zero of the scale can be properly 
 
 Fig. 50. 
 
 Fig. 51. 
 
 fixed by very carefully levelling a surface plate between the 
 orifice and scale, and transferring the centre of the orifice to the 
 scale by a scribing block. 
 
 Another method of measuring the head is by using a float. 
 If the float has a cord passing over a pulley, a finger attached 
 
70 HYDKAULICS CHAP. 
 
 to the pulley will give a magnified motion which can be read 
 on a dial. In this case the zero of the scale can be deter- 
 mined by bringing the water-level exactly to the lower edge 
 of the orifice and noting the reading on the finger of the 
 dial. 
 
 A more accurate method of determining the exact water- 
 level is the use of the hook gauge, invented by Mr. V. 
 Boyden in 1840. It consists of a fixed frame with sliding 
 scale and vernier (Fig. 51). The vernier is fixed to the frame, 
 and the scale slides vertically. The scale carries at its lower 
 end a hook with a fine point, and the scale carrying the hook 
 can be raised or lowered very slowly by a fine-pitched screw. 
 If the hook is depressed below the water surface and then 
 raised gradually by the screw, the moment of its reaching the 
 water surface will be very clearly marked by a sudden re- 
 flection from a small capillary elevation of the water surface 
 over the point of the hook. In good light differences of level 
 of 0*0001 of a foot are easily detected by the hook gauge. 
 The gauge is specially useful in measuring the head over 
 weirs which requires to be determined very accurately. The 
 point of the hook should be set by levelling very exactly at 
 the level of the weir crest, and a reading taken. Then the 
 difference of any reading of the water-level and this reading is 
 the head on the weir. It is generally convenient to place the 
 hook gauge in a small cistern, communicating with the stream 
 passing over the weir by a pipe. The water-level in such a 
 cistern fluctuates less than in the stream, and the gauge is 
 more easily read. 
 
 42. Coefficients for bellmouths or conoidal orifices. 
 When a bellmouth is formed so as to contract gradually, and 
 finally become cylindric, when in fact it has nearly the form 
 of a contracting jet, the contraction occurs within the mouth- 
 piece and there is no further contraction beyond it. The 
 section of the jet is then equal to the area co of the smaller 
 end of the mouthpiece. c c = 1 , and c v for moderate heads is 
 about 0'97, which is also the value of c, 
 
 Q = c v < t >^/(2gh) . . . (9). 
 
 For such an orifice Weisbach has found the following 
 values of the coefficients with different heads : 
 
IV 
 
 DISCHARGE FROM ORIFICES 
 
 Head over Orifice in Feet = h. 
 
 66. 
 
 1-64. 
 
 11-48. 
 
 55-77. 
 
 337-93. 
 
 Coefficient of velocity = c v 
 Coefficient of resistance = c r . 
 
 959 
 087 
 
 967 
 069 
 
 975 
 052 
 
 994 
 012 
 
 994 
 012 
 
 Fig. 52 shows a conoidal mouthpiece of approximate!}' 
 correct form. 
 
 Fig. 52. 
 
 43. Coefficients for sharp-edged orifices with complete 
 contraction. Orifices are used in measuring the rate of flow 
 of water. If the flow is discharged through an orifice, 
 and the head at the orifice measured, the rate of flow can be 
 determined, provided the coefficients of the orifice are known. 
 The orifice which can be constructed and measured with most 
 accuracy is a circular hole in a comparatively thin plate. In 
 certain cases a rectangular aperture is more convenient. 
 Sometimes the edges of the hole are bevelled (Fig. 45, a) ; but 
 this is not important, and the edge is more liable to injury. 
 The form b is most generally used, the edge being kept as 
 square as possible. A large amount of experimental research 
 has been done in determining the coefficients of orifices of this 
 type of various sizes and under different heads, and the results 
 have been tabulated, so that for most cases the coefficients can 
 be selected. Perhaps the most complete collection of such 
 experiments on the discharge of sharp-edged orifices is to be 
 
72 
 
 HYDKAULICS 
 
 CHAP. 
 
 found in Hamilton Smith's Hydraulics (London, 1886), where 
 the results are discussed and plotted in curves. In cases where 
 great accuracy is important it is desirable that the coefficients 
 for the particular orifice used should be determined by direct 
 experiment. Differences in the condition of the edge and the 
 position of the orifice relatively to the walls of the reservoir 
 cause variations of the coefficient which cannot be indicated in 
 any tables. 
 
 Broadly, for large sharp-edged orifices in plane surfaces, 
 and not near lateral boundaries, under moderately large heads, 
 the coefficient of discharge has a fairly constant value not 
 differing much from c=0'595. The value of the coefficient 
 is greater as the head is smaller, and as the area of the orifice 
 is smaller. For small orifices under comparatively small heads 
 it may have the value c=0'650, an increase of 9 per cent. 
 The following tables contain values selected from Hamilton 
 Smith's reductions, modified where necessary to be applicable 
 in the ordinary formula 
 
 Q = <W2p . . . (10). 
 
 For large vertical orifices under small heads there is a decrease 
 of c. 
 
 COEFFICIENT OF DISCHARGE c OF SQUARE SHARP-EDGED ORIFICES 
 
 IN EQ. (10) 
 
 
 Length of Side of Square in Feet. 
 
 Head over Centre 
 
 
 in Feet, h. 
 
 
 
 
 
 
 
 
 0-02. 
 
 0-03. 
 
 0-05. 
 
 0-10. 
 
 0-40. 
 
 i-oo. 
 
 
 
 
 
 
 
 
 0-4 
 
 
 
 637 
 
 621 
 
 
 
 0-5 
 
 
 648 
 
 633 
 
 619 
 
 597 
 
 ... 
 
 0-7 
 
 656 
 
 642 
 
 628 
 
 616 
 
 600 
 
 582 
 
 1-0 
 
 648 
 
 636 
 
 622 
 
 613 
 
 602 
 
 592 
 
 1-5 
 
 641 
 
 629 
 
 618 
 
 610 
 
 604 
 
 599 
 
 2-0 
 
 637 
 
 626 
 
 615 
 
 608 
 
 605 
 
 600 
 
 3-0 
 
 632 
 
 622 
 
 612 
 
 607 
 
 605 
 
 602 
 
 4-0 
 
 628 
 
 619 
 
 610 
 
 606 
 
 605 
 
 602 
 
 5-0 
 
 626 
 
 617 
 
 610 
 
 606 
 
 604 
 
 602 
 
 7-0. 
 
 621 
 
 615 
 
 608 
 
 605 
 
 604 
 
 602 
 
 10-0 
 
 616 
 
 611 
 
 606 
 
 604 
 
 603 
 
 601 
 
 20-0 
 
 606 
 
 605 
 
 603 
 
 602 
 
 601 
 
 600 
 
IV 
 
 DISCHAKGE FKOM OKIFICES 
 
 73 
 
 COEFFICIENT OF DISCHARGE c FOR CIRCULAR SHARP-EDGED ORIFICES 
 
 IN Eg. (10) 
 
 
 Diameter of Orifice in Feet. 
 
 Head over Centre 
 
 
 
 in Feet, h. 
 
 1 
 
 
 
 
 
 
 0-02. 
 
 0-03. 
 
 0-05. 
 
 0-10. 
 
 0-40. 
 
 1-00. 
 
 0-4 
 
 
 
 631 
 
 618 
 
 
 
 0-5 
 
 
 643 
 
 627 
 
 615 
 
 592 
 
 
 0-7 
 
 651 
 
 637 
 
 622 
 
 611 
 
 596 
 
 579 
 
 1-0 
 
 644 
 
 631 
 
 617 
 
 608 
 
 597 
 
 586 
 
 1-5 
 
 637 
 
 625 
 
 613 
 
 605 
 
 599 
 
 592 
 
 2-0 
 
 632 
 
 621 
 
 610 
 
 604 
 
 599 
 
 594 
 
 3-0 
 
 627 
 
 617 
 
 606 
 
 603 
 
 599 
 
 597 
 
 4-0 
 
 623 
 
 614 
 
 605 
 
 602 
 
 598 
 
 596 
 
 5-0 
 
 620 
 
 613 
 
 605 
 
 601 
 
 598 
 
 596 
 
 7-0 
 
 616 
 
 609 
 
 603 
 
 600 
 
 598 
 
 596 
 
 10-0 
 
 611 
 
 606 
 
 601 
 
 598 
 
 597 
 
 595 
 
 20-0 
 
 601 
 
 600 
 
 598 
 
 596 
 
 596 
 
 594 
 
 Mr. Mair carried out a series of careful tests on the 
 coefficient of discharge of circular orifices in conditions per- 
 mitting exceptional accuracy of observation (Proc. Inst. Civil 
 Engineers, Ixxxiv., 1885-86). The following Table gives a 
 selection of the results : 
 
 VALUES OF c IN EQ. (10) 
 
 
 Diameter of Orifice in Inches. 
 
 
 1 
 
 4 
 
 2 
 
 2 J 
 
 3 
 
 Head in Feet 
 
 
 
 
 
 
 
 Diameter of Orifice in Feet. 
 
 
 083. 
 
 125. 
 
 167. 
 
 208. 
 
 250. 
 
 
 
 
 
 
 
 75 
 
 0-616 
 
 0-616 
 
 0-616 
 
 0-607 
 
 0-609 
 
 1-5 
 
 0-610 
 
 0-611 
 
 0-610 
 
 0-603 
 
 0-605 
 
 2-0 
 
 0-609 
 
 0-609 
 
 0-609 
 
 0-604 
 
 0-605 
 
 The results agree closely with the relation 
 
 0-0098 
 
 c = 0-6075 + 
 
 - 0-0037^, 
 
 where h is in feet and d in inches. In the case of a 2-inch 
 
74 
 
 HYDKAULICS 
 
 orifice a minute rounding of the square edge altered the coeffi- 
 cient from 0*612 to 0*622 under the same conditions exactly. 
 
 Mr. Ellis measured indirectly by a weir the discharge 
 from a sharp-edged orifice 2 feet square, under heads varying 
 from 2 to 3j feet. For h = 2 feet, c = 0*611. For the larger 
 heads c was not sensibly different from 0*60 (Trans. Am. Soc. 
 Civil Engineers, 1876). 
 
 Rectangular orifices. Experiments of Poncelet and 
 Lesbros. For rectangular orifices there is a variation of the 
 coefficient of discharge c both with the height a and the 
 width b of the orifice. But for ratios of I /a not exceeding 20, 
 it appears that c depends chiefly on the smaller dimension of 
 the orifice independently of the other. The following are a 
 few values selected from the results obtained by Poncelet and 
 Lesbros : h 2 is the head at the top edge of the orifice, so that 
 
 the head to the centre of the orifice is h z + -. The discharge 
 is therefore 
 
 The sides of the channel of approach were at least 2^6 
 from the vertical edges, and the bottom at least 2^a from 
 the lower edge of the orifice. The head was measured not 
 immediately at the orifice, but at some distance back, where 
 the water was nearly at rest. 
 
 COEFFICIENTS OF DISCHARGE c FOR EECTANGULAR ORIFICES 
 IN EQ. (11) 
 
 Head over 
 
 Width, 6 = 0-656 Feet. 
 
 Width, 6 = 1-968. 
 
 of Orifice, 
 
 Height in Feet, a = 
 
 Height, a = 
 
 Feet. 
 
 0656. 
 
 164. 
 
 328. 
 
 656. 
 
 0656. 
 
 656. 
 
 066 
 
 659 
 
 615 
 
 596 
 
 572 
 
 643 
 
 
 164 
 
 658 
 
 625 
 
 605 
 
 585 
 
 641 
 
 597 
 
 328 
 
 654 
 
 630 
 
 611 
 
 592' 
 
 639 
 
 602 
 
 656 
 
 648 
 
 630 
 
 615 
 
 598 
 
 635 
 
 605 
 
 1-64 
 
 640 
 
 628 
 
 617 
 
 603 
 
 630 
 
 607 
 
 3-28 
 
 633 
 
 626 
 
 615 
 
 605 
 
 626 
 
 605 
 
 4-92 
 
 619 
 
 620 
 
 611 
 
 602 
 
 623 
 
 602 
 
 6-56 
 
 612 
 
 613 
 
 607 
 
 601 
 
 620 
 
 602 
 
 9-84 
 
 610 
 
 606 
 
 603 
 
 601 
 
 615 
 
 601 
 
IV 
 
 D1SCHAEGE FROM ORIFICES 
 
 44. Submerged sharp-edged orifices. If the orifice is 
 drowned below the tail water the conditions of discharge are 
 in no important way altered, except that the effective head is 
 the difference of level of the free surface of the head and tail 
 water. As there is often some disturbance in the tail water 
 near the orifice the level of the tail water should be taken at 
 a point where the disturbance has subsided. So far as is 
 known, the coefficient of discharge is the same as for an orifice 
 discharging in the air. Some experiments by Hamilton Smith 
 show that this must be very nearly the case. 
 
 COEFFICIENT OF DISCHARGE c IN EQ. (10) OF ORIFICES DROWNED TO 
 
 THE EXTENT OF 0'57 TO 0'73 FEET (HAMILTON SMITH) 
 
 Circular, d=Q'Q5. 
 
 Circular, d=Q'I. 
 
 Square, 0'05 x 0'05. 
 
 Square, O'l x O'l. 
 
 Effective 
 Head, 
 h. 
 
 c. 
 
 Effective 
 Head, 
 h. 
 
 c. 
 
 Effective 
 Head, 
 h. 
 
 c. 
 
 Effective 
 Head, 
 h. 
 
 c. 
 
 4-08 
 
 602 
 
 3-97 
 
 599 
 
 4-06 
 
 607 
 
 3-95 
 
 605 
 
 2-16 
 
 604 
 
 2-00 
 
 601 
 
 2-21 
 
 609 
 
 2-32 
 
 604 
 
 44 
 
 618 
 
 25 ' -605 
 
 35 
 
 620 
 
 21 
 
 612 
 
 45. Orifice at the end of a channel. When the orifice 
 is at the end of a channel the cross section of which H is not 
 very large compared with the area &> of the orifice, the 
 velocity of approach to the orifice increases the discharge. In 
 that case the discharge is 
 
 Q = . / -, 
 
 /f- 
 
 VV-G 
 
 )J 
 
 02); 
 
 the head h is measured at some distance back from the 
 orifice. The value of c in this case is not well determined. 
 
 46. Self-adjusting orifices for constant discharge. 
 The Spanish module. In a number of cases, especially in 
 the case of the distribution of irrigation water, it is required 
 to deliver from a canal or reservoir a constant supply of 
 water, notwithstanding variations of level in the canal or 
 reservoir. A number of devices for this purpose have been 
 
76 
 
 HYDKAULICS 
 
 CHAP. 
 
 invented, and the Spanish module used on the canal of 
 Isabella II., which supplies Madrid with water, may be taken 
 as a type. The module, Fig. 53, consists of two chambers, 
 the upper being in free communication with the canal and the 
 lower discharging by a culvert to the fields. In the floor 
 between the chambers there is a sharp-edged orifice in a 
 bronze plate. Hanging in this is a bronze plug of varying 
 
 Fig. 53. 
 
 diameter suspended from a float. If the water-level falls the 
 plug gives a larger opening, and conversely if the water rises 
 the plug fills a greater part of the orifice. Thus if the plug 
 is properly formed a constant discharge with varying head is 
 obtained. The theory of the module is very simple. Let K 
 (Fig. 54) be the radius of the fixed orifice, r the radius of the 
 plug at a distance h from the plane of flotation of the float, 
 and Q the required constant discharge of the module. Then 
 
iv DISCHARGE FROM ORIFICES 
 
 Taking c = 0'63, 
 
 77 
 
 r - 
 
 
 
 15-88 Jhy 
 
 A value of R is chosen such that for the lowest head the 
 expression in brackets is not negative, and then values of r 
 can be found for various 
 values of h, and with 
 these the curve of the 
 plug can be drawn. 
 The module in Fig. 53 
 discharges 1 c. metre 
 per sec. The fixed 
 opening is 0*2 metre 
 diameter, and the 
 greatest head above 
 the orifice is 1 metre. Fig. 54. 
 
 47. Flow from 
 
 orifices of liquids other than water. The same laws apply 
 to all liquids, provided the head is measured in feet of the 
 liquid itself. If a liquid of density G m Ibs. per cubic foot 
 issues under a pressure p Ibs. per square foot the correspond- 
 ing head is p/Gr m . Thus if mercury weighs 711 Ibs. per cubic 
 foot, a pressure of 50 Ibs. per square inch, or 7200 Ibs. 
 per square foot, corresponds to a head of 7200/711 = 10*12 
 feet of mercury, and under this pressure the velocity of issue 
 from an orifice would be N /(64'4 x 10-12)= v /(650'4) = 25'5 
 feet per second nearly. From a few experiments by Weisbach, 
 the coefficients of velocity and contraction for mercury are not 
 very different from those for water. 
 
 Hamilton Smith, with a circular orifice 0'02 feet diameter, 
 found for mercury c = 0'62 for a head of 0'5 feet; 0'607 for 
 a head of 1 foot ; 0'595 for a head of 3 feet. For lubricating 
 oil, with the same orifice, c = 0'75 for a head of 0'5 feet; 0'735 
 for a head of 1 foot ; 0'72 for a head of 3 feet. 
 
 48. Imperfect contraction. If the sides of the channel 
 bounding the stream approaching the orifice are near the edges 
 of the orifice they interfere with the convergence of the 
 elementary streams which causes the contraction. Roughly, 
 
 
HYDEAULICS 
 
 CHAP. 
 
 it may be said that the influence of the lateral boundaries is 
 sensible if their distance from the edge of the orifice is less 
 than 2 j- times the corresponding width of the orifice. If a 
 circular orifice of area o> is at the end of a cylindrical pipe of 
 area O, then the coefficient of discharge c f to be used in eq. 
 (10) is greater than the coefficient c for the ordinary case in 
 which the contraction is perfect in about the following ratio : 
 
 12 
 
 
 
 0-1 
 0-3 
 0-5 
 0-75 
 0-9 
 
 c 
 
 1-000 
 1-014 
 1-059 
 1-134 
 1-303 
 1-470 
 
 Partially suppressed contraction. If an orifice has 
 
 round part of its edge a rim, or 
 if over part of the edge the 
 orifice touches lateral bound- 
 aries, the convergence of the 
 streams at that part is pre- 
 vented and the coefficient of 
 contraction increased (Fig. 55). 
 If n is the length of the rim 
 measured round the edge of the 
 orifice, and p the whole periphery, 
 then the coefficients of contraction 
 are as follow : 
 
 
 Circular Orifices. 
 
 Rectangular Orifices. 
 
 n 
 
 
 
 = 
 
 c c = 
 
 c c = 
 
 P 
 
 
 
 0-25 
 
 0-640 
 
 0-643 
 
 0-50 
 
 0-660 
 
 0-667 
 
 0-75 
 
 0-680 
 
 0-691 
 
 But] there are few experiments on this point, and the values 
 given can only be taken as a general guide. 
 
 49. Inversion of the jet. When an orifice in a vertical wall 
 
IV 
 
 DISCHAKGE FROM OEIFICES 
 
 79 
 
 has dimensions not small compared with the head, the jet after 
 leaving the orifice passes through remarkable changes of cross 
 section. These were first investigated by Bidone (G. Bidone, 
 Experiences sur la forme des veines, Turin, 1829) and Magnus, 
 and later by Rayleigh (Proc. Roy. Soc. xxix. 71). Messrs. 
 Strickland and Farmer have also made careful observations in 
 the laboratory at Montreal (Trans. R. S. Canada, 1898). 
 The jet from a square orifice (Fig. 56) 
 converges to the vena contracta, where 
 the section is approximately octagonal. 
 Beyond this point sheets spread out 
 perpendicular to the sides of the orifice. 
 The spreading of these sheets reaches 
 a limit in consequence of the action of 
 the surface tension, which then gradu- 
 ally causes the sheets to subside into 
 the central portion of the jet. The 
 distance from the contracted section 
 to this point, which may be considered 
 a wave-length, depends on the head. 
 Beyond this point a second set of 
 sheets is squeezed out, but in directions 
 bisecting the angles between the first 
 
 sheet, and these are subjected to the same action as the 
 first sheets. Similarly a third or fourth set of sheets may 
 be developed till the jet breaks up into spray. The explana- 
 tion of these changes of form given by Messrs. Strickland and 
 Farmer is that they are due to the lateral motion of the 
 filaments converging towards the orifice. Hence any filament 
 except the central one has a transverse component of velocity 
 which causes it to press on and displace neighbouring filaments. 
 It is also true that filaments issuing at different heights from 
 the orifice when vertical have different horizontal velocities 
 and tend to describe parabolic paths of different range, and 
 this must cause mutual pressure. 
 
 50. Minimum coefficient of contraction. In one special 
 case the coefficient of contraction can be determined rationally. 
 Let Fig. 57 represent a vessel with vertical sides, 00 
 being the free surface level. The liquid is discharged by a 
 re-entrant mouthpiece with thin edges. The jet is formed by 
 
 Fig. 56. 
 
80 
 
 HYDKAULICS 
 
 CHAP. 
 
 filaments converging all round through angles of 180 with 
 the axis of the jet, and as this is the greatest possible 
 convergence, the contraction will be greatest and the co- 
 efficient of contraction a minimum. Let fl be the area of 
 the mouthpiece AB, a that of the contracted jet aa. Suppose 
 that in a short time t, the mass OOaa comes to O'O'aV. 
 
 The impulse of the ex- 
 ternal forces estimated 
 horizontally will be 
 equal to the horizontal 
 momentum produced 
 ( 35). 
 
 The pressure on 00 
 will be balanced by that 
 on OE, and so for other 
 parts of the mass ex- 
 cept EF and the surface 
 AaaE of the jet. Let 
 p a be the atmospheric 
 pressure and h the depth 
 of the centre of EF from 
 00. The horizontal 
 pressure exerted by the 
 vessel on the water at EF is (p a + G-A)fl. The horizontal 
 pressure of the atmosphere on the surface AaaB, which 
 is the pressure on its vertical projection, is p a l. Hence 
 the resultant pressure acting horizontally is (p a + GA)H 
 p a l = Ghl. Since the motion is steady there is no 
 change of horizontal momentum in the time t between OO 
 and aa. The momentum generated is the momentum 
 of aaa'af. If v is the velocity of the jet, the volume aaa'a! 
 discharged in the time t is covt. Its mass is (Gnovf)/g and 
 its momentum (GawPfy/g. Equating impulse and change of 
 momentum ( 35), 
 
 - 
 
 ff 
 
 gh 
 
 Fig. 57. 
 
 But neglecting the very small resistances, 
 
IV 
 
 DISCHARGE FROM ORIFICES 
 
 81 
 
 (O 1 
 
 (13). 
 
 Borda found by experiment c c = *5149; Bidone, 
 c c = 0*5547; and Weisbach, c c = *5324, results which do not 
 differ greatly from the theoretical value. The thickness of 
 the edge of the mouthpiece affects the results. The reaction 
 of the jet on the vessel is the pressure GM1 In the case of a 
 simple orifice the velocity of the converging filaments in 
 contact with the vessel in the neighbourhood of C and D 
 reduces the pressure there, and hence the pressure on OE is 
 not balanced by that on OC, and the reaction is greater than 
 Grhfl. It is easily seen to follow from the equation that the 
 contraction is less, but the exact amount is not calculable. 
 
 51. Application of the principle of Bernoulli to the 
 discharge from orifices. A jet is composed of elementary 
 streams, each of which starts 
 into motion at some point in the A_ 
 reservoir where the velocity is 
 zero, and gradually acquires the 
 velocity of the jet. Let Mm 
 (Fig. 58) be such an elementary 
 stream, M being a point where 
 the velocity is insensibly small, 
 and ra a point in the contracted 
 section of the jet where the 
 filaments have become parallel 
 and exercise uniform mutual 
 pressure. Take the free surface 
 
 AB for datum line, and let p lt v ly h lt be the pressure, velocity, 
 and depth below datum at M ; p, v, h, the corresponding 
 quantities at ra. Then 
 
 ?. 58. 
 
 But at M, since the velocity is insensible, the pressure is 
 the hydrostatic pressure due to the depth ; that, is v l 0, 
 Pi=p a + Grh l . At m, p=p a , the atmospheric pressure round 
 the jet. Hence, inserting these values, 
 
 6 
 
82 
 
 HYDKAULICS 
 
 CHAP. 
 
 or 
 
 = 8-025 
 
 That is, neglecting the viscosity of the fluid, the velocity 
 of filaments at the contracted section of the jet is simply the 
 velocity due to the difference of level of the free surface in the 
 
 reservoir and the orifice. If 
 the orifice is small in dimen- 
 sions compared with h, the 
 filaments will all have nearly 
 the same velocity, and if h 
 is measured to the centre of 
 the orifice, the equation above 
 gives the mean velocity of 
 the jet. 
 
 Case of a submerged 
 orifice. Let the orifice dis- 
 charge below the level of the tail water (Fig. 59). Then at 
 M, v, = 0, p l = Gr&i +p a ; at m, p = Gh B +p a . 
 
 Fig. 59. 
 
 g = h 2 -h B = h (15), 
 
 where h is the difference of 
 level of the head and tail 
 water, and may be termed the 
 effective head producing flow. 
 
 Case where the press- 
 ures are different on the 
 free surface and at the 
 orifice. Let the fluid flow 
 from a vessel in which the 
 
 ._*__ . 
 
 h 
 
 Fig. 60. 
 
 pressure is p into a vessel in which the pressure is p 
 (Fig. 60). Let h Q be the height from the centre of the 
 orifice to the free surface in the first vessel. The pressure p Q 
 will produce the same effect as a layer of fluid of thickness 
 
IV 
 
 
 DISCHAKGE FKOM OEIFICES 83 
 
 added to the head water ; and the pressure p will produce 
 
 the same effect as a layer of thickness added to the tail 
 
 water. Hence the effective difference of level, or effective head 
 producing flow, will be 
 
 and the velocity of discharge will be 
 
 v 
 
 . '. (16). 
 
 We may express this result by saying that differences of pres- 
 sure at the free surface and at the orifice are to be reckoned 
 as part of the effective head. 
 
 Hence in all cases thus far treated the velocity of the jet 
 is the velocity due to the effective head, and the discharge, 
 allowing for contraction of the jet, is 
 
 Q = 
 
 . (17), 
 
 where &> is the area of the orifice, ceo the area of the contracted 
 section of the jet, and h the effective head measured to the 
 centre of the orifice. If h and G> are taken in feet, Q is in 
 cubic feet per second. 
 
 52. Discharge from a fire nozzle. Mr. John R 
 Freeman has made very accurate tests of the discharge from 
 the nozzles used with 
 
 orifice 
 
 & 
 
 hose in delivering water 
 in streams at fires. He 
 has found the coeffi- 
 cients for such nozzles 
 so constant that he 
 suggests their use in 
 measuring the dis- 
 charge of pumps and 
 in similar cases (Trans. 
 Am. Soc. of Civil 
 
 Engineers, 1891). Fig. 61 shows the arrangement adopted. 
 For three nozzles tried the coefficient of discharge was 0*995, 
 with heads of 12 to 120 feet. The head was corrected for 
 
 ^S Stilling^ VJtTi tfon 
 
 Fig. 61. 
 
84 
 
 HYDEAULICS 
 
 CHAP. 
 
 velocity of approach, but the correction was very small except 
 for low heads. The nozzles n were If to 2j inches 
 diameter. They were smoothly tapering, with sides converging 
 at 5 to 7J degrees to the axis, and polished for 3 or 4 
 diameters back from the outlet. The pressure in the supply 
 chamber was taken at a piezometer orifice made carefully 
 flush with the inside of chamber. With the tin cone removed 
 and a square corner to the brass flange in which the nozzle 
 was screwed, coefficients of 0'985 to 0'990 were obtained. 
 
 53. Flow from a vessel when the effective head 
 varies with the time. Various useful problems arise relating 
 to the time of emptying and filling vessels, reservoirs, lock 
 chambers, etc., where the flow is dependent on a head which 
 
 Head wcJler ueveu 
 
 
 j 
 jt =.-_^-- 
 
 
 
 
 t 
 si fcL/l 
 
 
 
 
 
 
 
 
 
 TaiL water- level? 
 
 
 
 
 
 Fig. 62. 
 
 increases or diminishes during the operation. The simplest 
 of these problems is the case of filling or emptying a vessel 
 of constant horizontal section, such as a river lock. Suppose 
 the lock chamber, which has a water surface of H square feet, 
 is emptied through a sluice in the tail gates, of area co, placed 
 below the tail-water level. Then the effective head producing 
 flow through the sluice is the difference of level in the 
 lock chamber and tail bay. Let H (Fig. 62) be the initial 
 difference of level, h the difference of level after t seconds. 
 Let dh be the fall of level in the chamber during an interval 
 dt. Then in the time dt the volume in the chamber is altered 
 by the amount ldh, and the outflow from the sluice in the 
 same time is co)^/2ghdt. Hence the differential equation 
 connecting h and t is 
 
 tth = 0. 
 
IV 
 
 DISCHAEGE FEOM OKIFICES 
 
 85 
 
 For the time t during which the initial head H diminishes 
 to any other value h, 
 
 = _flf /2H_ /2h] 
 ~co>(V 9 V 9) 
 
 For the whole time of emptying, during which h diminishes 
 from H to 0, 
 
 Comparing this with the equation for flow under a constant 
 head, it will be seen that the time is double that required for 
 the discharge of an equal volume under a constant head H. 
 The time of filling the lock through a sluice in the head 
 
 ..X 
 
 Fig. 63. 
 
 gates is exactly the same if the sluice is below the tail-water 
 level. But if the sluice is above the tail-water level, then 
 the head is constant till the level of the sluice is reached, and 
 afterwards it diminishes with the time. 
 
 54. Cylindrical mouthpiece. When water is discharged 
 through a short cylindrical mouthpiece, the axis of which is 
 normal to the side of the reservoir (Fig. 63) and its length 
 2 to 3 times its diameter, there is an internal contraction 
 
86 HYDKAULICS CHAP. 
 
 at EF due to the convergence of the streams at the inlet, but 
 the jet then expands to fill the mouthpiece and issues full 
 bore. Let H be the cross section GH of the mouthpiece and 
 co the cross section EF of the interior contraction. Then 
 G>/n = c c is the coefficient of contraction. Let p and v be 
 the pressure and velocity at GH ; p 1} v lt the pressure and 
 velocity at EF ; Q, the discharge per second. Then 
 
 Q = wflj = ttv 
 
 % = v/c c . 
 
 Let h be the head over the axis of the jet, and c the co- 
 efficient of discharge of the mouthpiece, which, as there is no 
 external contraction, is also the coefficient of velocity. Then 
 
 v = c\/2gh .... (19). 
 
 Between EF and GH there is the loss of head fa vfjZg 
 due to the change of velocity from v 1 to v ( 37), and a fric- 
 tional loss c r v 2 /2g which is negligible for very short mouth- 
 pieces. Hence the total head at GH is less than that at EF 
 by these losses. 
 
 -vY v 2 } 
 
 + %r 
 
 But v l = vje r and v = 
 
 g'-V-IVr-lWV* (20). 
 
 Suppose a small vertical pipe dipping into a reservoir at a 
 lower level (Fig. 64) introduced into the mouthpiece at the 
 internal contraction. The pressure p acts on the free surface 
 of the lower reservoir as well as at the outlet of the mouth- 
 piece, and P! is the pressure inside the mouthpiece. Hence 
 the water will rise in the tube to a height KL = Jif = (p Pi)/Gr- 
 If h f is greater than the distance X between the axis of 
 the jet and the surface of ' the lower reservoir, the water will 
 be continuously pumped up from the lower reservoir and 
 discharged at the level of the mouthpiece. This arrangement 
 is a jet pump in its crudest form, in which one body of water 
 descending a distance h pumps up another body of water a 
 height X. Putting for the moment c = 0'82, c c = 0'64, and 
 neglecting the small quantity c r , 
 
 h' = 
 
IV 
 
 DISCHAEGE FKOM OKIFICES 
 
 which is the greatest value of X at which pumping will occur. 
 The values assumed will be seen presently to be about average 
 values of the coefficients. 
 
 In order that the continuity of the stream may not be 
 broken, the lowest pressure must not be negative, that is, p l 
 
 : x 
 
 //Kj5 
 
 V//v/;r 
 
 X 
 
 
 
 i 
 
 
 K-*- , 
 
 
 
 
 i 
 
 V 
 
 
 L <b x 
 
 
 -~~^ =- 
 
 
 
 y 
 
 / 
 
 
 / // // 
 
 //// / / 
 
 Fig. 64. 
 
 must be greater than 0. Let the atmospheric pressure height 
 33-9 feet of water. The condition of flow full bore is 
 
 h< 
 
 33-9 
 
 (21). 
 
 With the values of the coefficients assumed above, h must be 
 less than 33*9/0*75 = 45 feet, or the jet will not discharge 
 full bore. 
 
 Let c v be the coefficient of velocity corresponding to the 
 resistances between CD and EF (Fig. 63). Then 
 
 and the head wasted between CD and EF ( 36) is 
 
88 HYDKAULICS CHAP. 
 
 (L iW-K-I-lY^ 
 
 U? /5fr-tfW /a/ 
 
 There are therefore three losses of head between CD and GH, 
 two of which have already been given, and the effective head 
 producing the velocity v is h less these three losses. 
 
 1 \ 2 2 
 
 c c 
 
 and putting v c 
 
 C 2 W Cc 
 
 and the coefficient of discharge for the mouthpiece is 
 
 1 ^ 
 
 Taking c c = 0'64, c v = 0'97, and neglecting c r , 
 
 c = 0-824. 
 
 Weisbach made experiments on some cylindrical mouth- 
 pieces of different diameters, and lengths about three diameters, 
 and found the following values of c, which do not differ much 
 from the value just calculated : 
 
 Diameter = 0-032 0'066 0'098 0'131 feet. 
 c= -843 -832 -821 -810 
 
 The coefficient varies somewhat with the length of the mouth- 
 piece. Its average value may be taken to be as follows : 
 
 Length 
 
 ^ ^--=1 2 to 3 12 
 
 Diameter 
 
 c=0-88 0-82 0-77 
 
 55. Convergent mouthpieces. With these there is an 
 external contraction at the outlet as well as the internal 
 contraction. Two cases may be distinguished; the inner 
 
iv DISCHAKGE FKOM ORIFICES 89 
 
 angle may be sharp as at A (Fig. 65), or well rounded as at B. 
 
 e 
 
 Fig. 65. 
 
 In the latter case the loss due to the internal contraction is 
 diminished. The discharge is 
 
 (23), 
 
 where 1 = j^ 2 is the area at the external end. The length 
 of the mouthpiece is about 3d. 
 
 Angle 6 
 c for case B 
 c for case A 
 
 
 
 0-97 
 0-83 
 
 5f 
 0-95 
 0-94 
 
 0-92 
 0-92 
 
 0-88 
 0-85 
 
 45 
 0-75 
 
 90 
 0-63 
 
 56. Divergent conoidal mouthpiece. Suppose a mouth- 
 piece with a convergent inlet and divergent outlet so designed 
 that there is nowhere any 
 abrupt change of velocity in A 
 the stream passing through 
 it, as in Fig. 66. The inlet 
 may be of the form of a 
 contracted stream from a 
 sharp-edged orifice, and the 
 
 divergent part should ex- |_ 
 
 pand very gradually, becom- 
 ing cylindrical at the end. 
 
 Let ft>, v, p, be the area 
 of section, velocity, and pres- 
 sure at CD, and H, v lf p 1} 
 the same quantities at EF. 
 
 Let the atmospheric pressure be pj&= 3 3 '9 feet of water, and 
 let h be the head over the mouthpiece. 
 
 Then the velocity at EF is 
 
 Fig. 66. 
 
90 
 
 HYDKAULICS 
 
 CHAP. 
 
 #! = c v \/(2gh) . . . (24), 
 and the effective head producing this velocity is 
 
 So that the head wasted in friction and eddies in the mouth- 
 piece is /:. _ 2 \fc 
 
 This wasted head may be taken to consist of two parts: % 
 wasted in the converging, and z 2 wasted in the diverging part 
 of the mouthpiece. Then if atmospheric pressure is taken 
 
 into the reckoning the total head at CD is h+^ z^, and 
 that at EF is h -\- P -~ -z l - z 2 . Consequently if pJG = 33'9, 
 
 v z p 
 2g + & 
 
 />! 2 i 
 
 33-9 
 
 (246), 
 
 33-9, 
 
 or if the jet discharges into the atmosphere p l = p a , and 
 
 Then the discharge is 
 
 (25), 
 
 which is independent of the area at the throat CD. But 
 
 there is one obvious 
 limit to this. As the 
 velocity is greater at 
 CD than EF the pres- 
 sure must be less, that 
 is, less than atmo- 
 spheric pressure. If 
 the ratio of the sec- 
 tions p = fl/o> is great 
 enough p becomes zero 
 or negative, and flow 
 full bore is impossible. 
 
 The stream breaks away from the mouthpiece as in Fig. 67. 
 
 But v = pv l} and inserting this in eq. (24&), 
 
 Fig. 67. 
 
iv DISCHAEGE FKOM OEIFICES 91 
 
 p becomes zero if 
 
 83-9 
 
 From experiments on bellmouths, z-^ may be taken as about 
 The value of z 2 may be considerably greater. In 
 an expanding stream there is great instability and tendency 
 to break up into eddies, which waste energy. If the mouth- 
 piece is short, the stream breaks into eddies ; if long, the 
 friction of the surface gives rise to eddies. The following 
 short table is calculated for the limiting cases ^ 2 = and 
 2 2 = 0'9A. 
 
 LIMITING VALUES OF p 
 
 h = 1 5 10 20 50 
 
 Whenz = 6-06 2-83 2-13 1'66 1-30 
 
 When z" = 0'9h 26*4 8-0 4'5 2*8 1-7 
 
 Venturi experimented on a mouthpiece of this kind, and 
 concluded that the discharge would be a maximum when the 
 diverging part was of a length equal to nine times its least 
 diameter and the angle of the cone a little more than 5. 
 Francis (Lowell Hydraulic Experiments) obtained results with 
 a similar mouthpiece. 
 
 The diameter at CD was 0'102 feet; at EF, 0'321 feet ; 
 p = 9*9 ; the length of the diverging cone 4 feet ; the mouth- 
 piece was drowned, and the difference of level of head and tail 
 water was from 0*1 to 1*4 feet. The mean coefficient of 
 velocity (or discharge) was c w =0'23, so that from eq. (24a) 
 the effective head was 0'23 2 A = '053A. Consequently "947A 
 was the head wasted during the passage of the water through 
 the mouthpiece. This corresponds to the total head lost 
 between inlet and outlet of a Venturi meter, h being the 
 height due to velocity at inlet or outlet. 
 
 57. Influence of temperature on the flow from orifices. 
 Experiments were made by the author (Phil. Mag., 1878) 
 with a conoidal mouthpiece 0'394 inches diameter, a head of 
 
92 HYDKAULICS 
 
 CHAP. 
 
 1 to 1|- foot. Neglecting the expansion of the reservoir and 
 orifice, the coefficient is 
 
 Temperature F. Value of c. 
 
 190 0-9871 
 
 130 0-9740 
 
 60 0-9418 
 
 With a sharp-edged orifice also 0*394 inches diameter and 
 the same heads, and also neglecting any correction for expan- 
 sion of the reservoir and orifice 
 
 Temperature F. Value of c. 
 
 205 -5936 
 
 140 -5964 
 
 62 -5980 
 
 The results show that the influence of temperature is very 
 small. The correction for expansion of the reservoir and 
 orifice would be very small. 
 
 Mr. Mair repeated these experiments on a much larger 
 scale. With a conoidal orifice 1-J inch in diameter and a head 
 of T75 feet, the following values were obtained: 
 
 Temperature F. Value of c. 
 
 170 0-981 
 
 110 0-967 
 
 55 0-961 
 
 With a sharp-edged orifice 2j inches diameter and 1*75 
 feet head, the following were the results : j 
 
 Temperature F. Value of c. 
 
 179 0-607 
 
 110 0-604 
 
 57 0-604 
 
 In the case of the conoidal orifice the increase of tempera- 
 ture appears to reduce sensibly the frictional loss. In the 
 case of the sharp-edged orifice the influence of temperature is 
 very small. 
 
 PROBLEMS 
 
 1. The pressure in the pump cylinder of a fire-engine is 14,400 Ibs. 
 per square foot ; assuming the resistances of the valves, hose, 
 and nozzle are such that the coefficient of resistance is 0*7, find 
 the velocity of discharge. 93-5 feet per second. 
 
iv DISCHARGE FKOM ORIFICES 93 
 
 2. The pressure in the hose of a fire-engine is 13,000 Ibs. per square 
 
 foot ; the jet rises to a height of 150 feet. Find the coefficients 
 of velocity and resistance. 0*849 and 1*39. 
 
 3. A horizontal jet issues under a head of 9 feet At 6 feet from the 
 
 orifice it has fallen vertically 15 inches. Find the coefficient 
 of velocity. 0-89. 
 
 4. Required the coefficient of resistance corresponding to a coefficient 
 
 of velocity = 0'96. State what percentage of the energy due 
 to the head is wasted. 0-085. 7 -8 per cent. 
 
 5. A fluid of one-quarter the density of water is discharged from a 
 
 vessel, in which the pressure is 60 Ibe, per square inch (absolute), 
 into the atmosphere, where the pressure is 15 Ibs. per square 
 inch. Find the velocity due to the head. 163-5 ft. per second. 
 
 6. Find the diameter of a circular orifice to discharge 2000 cubic 
 
 feet per hour under a head of 5 feet. Coefficient 0-62. 
 
 3-03 inches. 
 
 7. A cylindrical cistern contains water 16 feet deep, and is 1 square 
 
 foot in cross section. On opening an orifice of 1 square inch 
 in the bottom, the water-level fell 7 feet in one minute. Find 
 the coefficient of discharge. 0'598. 
 
 8. A miner's inch is defined to be the discharge through an orifice in 
 
 a vertical plane of 1 square inch area, under an average head 
 of 6j inches. Find the supply of water per hour in gallons. 
 Coefficient 0-62. 571. 
 
 9. A vessel fitted with a piston of 10 square feet area discharges 
 
 water under a head of 9 feet. What weight placed on the 
 piston would double the rate of discharge ? 270 Ibs. 
 
 10. Required the discharge from a thin-edged vertical sluice opening 
 
 3 feet wide and 1 foot deep. Depth of water to lower edge of 
 orifice = 7 feet, coefficient of discharge = 0-62. 
 
 50'7 cubic feet per second. 
 
 11. The discharge from an orifice 10 feet below the water surface is 
 
 18 cubic feet per minute. What will be the discharge when 
 the head is 25 feet? 28*45 cubic feet per minute. 
 
 12. Show that about ^ of the energy due to the head is wasted at a 
 
 cylindrical mouthpiece. Coefficient 0-83. 
 
 The loss is 31 per cent. 
 
 13. A jet has a diameter of 3 inches when issuing vertically under a 
 
 head of 9 feet. Find its diameter at 6 feet above the orifice. 
 
 3-95 inches. 
 
 14. What must be the size of a sluice in a lock gate to empty the lock 
 
 in ten minutes? Area of water-surface of lock 15 feet by 
 100 feet. Lift 6 feet. The sluice is below the tail water, and 
 the coefficient of discharge is 0*75. 2*03 square feet. 
 
 15. A vessel is of such a form that its horizontal area is A -f BOJ -f Cx 2 
 
 at x feet above- the bottom. Show that if there are h feet 
 initially in the vessel, and it empties through an orifice of area 
 <o, the time of emptying is given by the equation 
 
94 HYDEAULICS CHAP, iv 
 
 16. Coal gas weighs 0-04 Ibs. per cubic foot. Treating it as a liquid, 
 
 find the velocity of discharge from an orifice due to a pressure 
 of 1 inch of water. Coefficient of velocity 0*96. 
 
 87-8 feet per second. 
 
 17. A tank 1000 square feet in area discharges through an orifice 
 
 1 square foot in area. Calculate the time required to lower 
 the level in the tank from 50 feet to 25 feet above the orifice. 
 Coefficient 0'6. 2591 seconds. 
 
 18. A vertical-sided lock is 60 feet long and 15 feet wide. Lift 
 
 15 feet. Find the area of a sluice below tail water to empty 
 the lock in ten minutes. Coefficient 0'5. 2 '8 9 5 square feet. 
 
 19. A Spanish module has an orifice 18 inches in diameter, and the 
 
 head in the upper chamber varies from 1-5 to 4 feet. Design 
 the plug so that the discharge shall be 7 cubic feet per second. 
 
CHAPTEK V 
 
 NOTCHES AND WEIRS 
 
 58. Large vertical rectangular orifices. When the head 
 over the top edge of the orifice is less than half the 
 height of the orifice, the variation of head has an influence 
 too great to be neglected ( 37). If, as in most cases, there is 
 contraction of the jet the theory of flow presents some difficulty. 
 In the plane of the orifice the issuing streams are not normal 
 
 
 
 
 
 
 
 =j 
 
 &^=^ 
 
 ^ =3 
 
 ^ T ^ 
 
 1 
 
 J 
 
 i 
 
 t j 
 
 V ' ^ 
 
 
 .j 
 
 r - J 
 
 y N* 
 
 J N 
 
 '///\ 
 
 Fig. 68. 
 
 to the plane or parallel to each other. At the contracted 
 section the streams are parallel and normal to the section, but 
 the dimensions of the section cannot in general be directly 
 observed. However, let the contracted section, which in the 
 case of a rectangular orifice must itself be very approximately 
 rectangular, be considered. Let h 1} h z be the heads over its 
 top and bottom edges and 6 its width. Consider a lamina 
 
 95 
 
96 HYDKAULICS CHAP. 
 
 between the levels li and h + dh. Its cross section is bdh, and 
 neglecting small resistances its velocity is fJ(2gJi\ and its 
 discharge I ^/(2gh)dh. Hence the whole discharge of the 
 orifice is 
 
 Ch 9 f 
 h 2 dh 
 
 -^ 1 } (1), 
 
 where the numerical factor on the right is a coefficient 
 depending only on the form of the contracted cross section. 
 Now let Hj, H 2 be the heads at top and bottom edges, and B 
 the width of the orifice itself. Let 
 
 c= 
 
 Then the discharge, in terms of the dimensions of the 
 orifice, is 
 
 Q = |cB N /2^{H 2 f -H 1 f } . . . (2), 
 
 which is commonly given as the theoretical formula for 
 vertical rectangular orifices, and C is often stated to be the 
 coefficient of contraction. But C is clearly not the coefficient 
 of contraction, the value of which must be 
 
 Equation (2) is only rational if C is understood to be a 
 coefficient the value of which will vary with the proportions 
 of the orifice, and experiment shows this to be the case. 
 
 59. Notches or weirs. A practically very important 
 case is that in which Hj = and the jet is discharged from 
 an open notch or orifice extending up to the free surface. 
 Weirs in rivers are cribwork or masonry constructions, 
 primarily intended to raise the surface-level of the river up- 
 stream, while permitting the passage of floods. Notches for 
 measuring purposes are weirs fitted with a plate in which an 
 open notch is formed through which the water passes. The 
 
NOTCHES AND WEIRS 
 
 notch is usually rectangular, but sometimes triangular or 
 trapezoidal. As the water surface falls when approaching 
 the notch, the head h over the bottom of the notch, or over 
 the crest of the weir, should be measured some distance back 
 from the weir beyond the origin of the surface curve. 
 The jet or stream passing over a weir may be termed the 
 weir sheet. For an ordinary sharp-edged weir or notch the 
 sheet is of the form shown in Fig. 69, A, B. The weir sheet 
 contracts at the two ends and at its top and bottom surfaces. 
 If the length b of the weir is equal to the width of the 
 channel of approach there are no end contractions, and the 
 weir is termed a weir with suppressed end contractions. If 
 the tail-water level is above the crest of the weir it is termed 
 a drowned weir. If the crest of the weir is broad or rounded, 
 
 B 
 
 [777 
 
 Fig. 69. 
 
 or if the upstream or downstream faces of the weir are 
 sloped, the phenomena of discharge are complex, the water 
 sheet in some cases springing clear, and in some cases 
 adhering to the weir (Fig. 69, C). 
 
 The equation of discharge for rectangular weirs is found 
 by putting Hj = in eq. (2). Also let h be the head above 
 the crest and I the length of the notch or weir. Then 
 
 2 
 
 (3), 
 
 where c is a coefficient of discharge, which varies considerably 
 in different cases. This is the formula which has been most 
 generally used in computing weir discharge, and it is trust- 
 
 7 
 
98 
 
 HYDKAULICS 
 
 CHAP. 
 
 worthy for practical purposes if the value of c is selected 
 from observations in similar conditions. The following small 
 tables give values selected from those obtained by Hamilton 
 Smith from plottings of various experiments by Francis, 
 Fteley and Stearns, Lesbros, and others. It will be seen that 
 c varies more for weirs with end contractions than for weirs 
 with no end contractions. 
 
 COEFFICIENTS OF DISCHARGE FOR WEIRS WITH COMPLETE 
 CONTRACTION (HAMILTON SMITH) 
 
 Head on 
 
 Values of c when the Length of the Weir is in Feet. 
 
 Weir Crest 
 
 
 in Feet. 
 
 1 
 
 2 
 
 3 
 
 5 
 
 7 
 
 10 
 
 19 
 
 0-15 
 
 625 
 
 634 
 
 638 
 
 640 
 
 640 
 
 641 
 
 642 
 
 0-2 
 
 618 
 
 626 
 
 630 
 
 631 
 
 632 
 
 633 
 
 634 
 
 0-3 
 
 608 
 
 616 
 
 619 
 
 621 
 
 623 
 
 624 
 
 625 
 
 0-5 
 
 596 
 
 605 
 
 608 
 
 611 
 
 613 
 
 615 
 
 617 
 
 0-7 
 
 590 
 
 598 
 
 603 
 
 606 
 
 609 
 
 612 
 
 614 
 
 1-0 
 
 ... 
 
 590 
 
 595 
 
 601 
 
 604 
 
 608 
 
 611 
 
 1-5 
 
 ... 
 
 ... 
 
 585 
 
 592 
 
 596 
 
 601 
 
 608 
 
 COEFFICIENTS OF DISCHARGE FOR WEIRS WITH SUPPRESSED END 
 CONTRACTIONS (HAMILTON SMITH) 
 
 Head on 
 
 Values of c when the length of the Weir is in Feet. 
 
 Weir Crest 
 
 
 in Feet 
 
 
 
 
 
 
 
 
 3 
 
 5 
 
 7 
 
 10 
 
 15 
 
 19 
 
 0-15 
 
 649 
 
 645 
 
 645 
 
 644 
 
 644 
 
 643 
 
 0-2 
 
 642 
 
 638 
 
 637 
 
 627 
 
 636 
 
 635 
 
 0-3 
 
 636 
 
 631 
 
 629 
 
 628 
 
 627 
 
 626 
 
 0-5 
 
 633 
 
 627 
 
 624 
 
 621 
 
 620 
 
 619 
 
 0-7 
 
 635 
 
 628 
 
 624 
 
 620 
 
 619 
 
 618 
 
 1-0 
 
 641 
 
 633 
 
 628 
 
 624 
 
 621 
 
 619 
 
 1-5 
 
 ... 
 
 641 
 
 636 
 
 630 
 
 625 
 
 622 
 
 60. Velocity of Approach. So far it has been assumed 
 that the stream approaching the weir was of large section 
 compared with the jet over the weir, and that the head h was 
 measured where the water was nearly still. In many cases 
 the weir is at the end of a channel of limited section, and 
 
NOTCHES AND WEIES 
 
 99 
 
 the head must be measured where the water has a velocity 
 too great to be negligible. In that case the observed head 
 has to be corrected for velocity of approach before using it in 
 the weir formula. 
 
 Let Fig. 70 represent a vertical rectangular orifice at the 
 end of a channel in which the velocity of approach is u. Let 
 
 Fig. 70. 
 
 b be the width of orifice, and 7^ h 2 be the heads over the top 
 and bottom edges of the orifice measured at a point in the 
 channel where the mean velocity is u. It is obvious that 
 somewhere upstream there must have been a fall of free 
 surface 
 
 in producing the velocity u. Hence the true heads over the 
 edges of the orifice, reckoned from still water level, are 
 A! + {) and h 2 + f). Putting these values in eq. (2), 
 
 In the case of a notch or weir of length I, ^ = 0, and 
 may be written h, 
 
 &)*-&*} 
 
 Q = 
 
 (5), 
 
 which is the equation most generally used for weirs when 
 velocity of approach must be allowed for. It is not from the 
 theoretical point of view entirely satisfactory, because in the 
 section where h is measured the velocity varies, and it is 
 uncertain in what proportion different portions of the stream 
 
100 
 
 HYDE.AULICS 
 
 CHAP. 
 
 go to make up the jet over the weir. It is probable that {) 
 should be affected by an empirical coefficient a to allow for 
 this. In most cases f) is small compared with h, and the 
 last term in the bracket is very small. Hence for simplicity 
 some writers take 
 
 at))*} . . (6), 
 
 which is easier to compute. It appears that a = about 1*5. 
 An analysis of Francis and Fteley and Stearns' experiments 
 led Hamilton Smith to the conclusion that a should be taken 
 1*33 for weirs with no end contractions, and 1*4 for weirs 
 with end contractions. It will be seen later that new experi- 
 ments by Bazin have led to a better method of dealing with 
 velocity of approach. The following table will give an idea 
 of the importance of velocity of approach in weir calcula- 
 tions : 
 
 VALUES OF ft 
 
 Velocity of 
 Approach 
 u. 
 
 u* 
 20 
 
 il* 
 
 32c/ 
 
 -1 
 
 Velocity of 
 Approach 
 u. 
 
 V? 
 
 2<7 
 
 1 u* 
 32g 
 
 "J 
 
 Feet per 
 second. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet per 
 second. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 0-2 
 
 0006 
 
 0008 
 
 0009 
 
 0-8 
 
 0099 
 
 0133 
 
 0139 
 
 0-3 
 
 0014 
 
 0019 
 
 0020 
 
 0-85 
 
 0112 
 
 0150 
 
 0157 
 
 0-4 
 
 0025 
 
 0033 
 
 0035 
 
 0-9 
 
 0126 
 
 0168 
 
 0176 
 
 0-5 
 
 0039 
 
 0052 
 
 0054 
 
 0-95 
 
 0140 
 
 0187 
 
 0196 
 
 0-6 
 
 0056 
 
 0075 
 
 0078 
 
 1-0 
 
 0155 
 
 0207 
 
 0218 
 
 0-7 
 
 0076 
 
 0102 
 
 0107 
 
 1-2 
 
 0224 
 
 0298 
 
 0313 
 
 0-75 
 
 0087 
 
 0117 
 
 0122 
 
 1-5 
 
 0350 
 
 0466 
 
 0489 
 
 When the velocity of approach u is directly measured by a 
 current meter, for instance, eq. (5) or (6) presents no difficulty. 
 More commonly only the cross section fl of the channel of 
 approach is known. Then if Q is the discharge over the 
 weir, 
 
 If this value is introduced in eq. (5) or (6) it is very cumbrous. 
 It is better to proceed by approximation. Let Q' be the 
 
NOTCHES AND WEIKS 
 
 101 
 
 discharge if the velocity of approach is neglected, that 
 is, by eq. (3). Then u' Q'/fl is an approximate value of 
 u, and ty = u' 2 /2g is an approximate value of !). Putting 
 this in eq. (5) or (6) a second approximation Q" to Q is ob- 
 tained. A third approximation can be found, but this is 
 rarely necessary. 
 
 61. Partially submerged orifices. Drowned weirs. 
 When the tail-water level is above the lower and below the 
 upper edge of the 
 orifice, it divides 
 the orifice into two 
 parts in which the 
 conditions of flow 
 are different. Let 
 Fig. 7 1 represent 
 such an orifice, 
 where h 1} h 2 , h are 
 the depths below 
 the free surface of 
 the upper and lower 
 
 Fig. 71. 
 
 edges of the orifice 
 and the tail water, and b is the width of the orifice. An 
 elementary stream M. l m l issuing above the tail-water level 
 has the head h' y which for different parts of the orifice varies 
 from to h. An elementary stream M 2 m 2 issuing below the 
 
 tail- water level has a head h" h 
 
 f " 
 
 1 
 
 1 
 
 --* 
 
 h, which is the same 
 for all parts below the 
 tail-water level. If Q x 
 Q 2 are the discharges of 
 the upper and lower 
 parts of the orifice, 
 
 Fig. 72. 
 
 The important case 
 is that of a drowned weir, in which the tail -water level 
 is above the weir crest (Fig. 72). Then 7^ = 0, and the 
 discharge is 
 
102 
 
 HYDEAULICS 
 
 CHAP. 
 
 where I is the length of the weir, h 2 the head over the weir 
 measured upstream, and h the difference of level of head and 
 tail water. From some experiments by Fteley and Stearns 
 (Trans. Am. Soc. of Civil Engineers, 1883) the following 
 values of c are calculated : 
 
 d 
 
 h* 
 0-1 
 0-2 
 0-3 
 0-4 
 0-5 
 0-6 
 0-7 
 0-8 
 0-9 
 0-95 
 1-0 
 
 0-9 
 0-8 
 0-7 
 0-6 
 0-5 
 0-4 
 0-3 
 0-2 
 0-1 
 0-05 
 
 o-o 
 
 629 
 614 
 600 
 590 
 582 
 578 
 578 
 583 
 596 
 607 
 628 
 
 The weir was sharp edged, 5 feet in length, with end 
 contractions suppressed. The weir crest was 3 '2 feet above 
 the bottom of the channel ; h 2 varied from 0*3 to 0*8 feet. 
 
 62. Broad-crested weirs. Broad-crested weirs are un- 
 suitable for water measurement, but it is sometimes necessary 
 to estimate the flow at such weirs. The following is a theory 
 of the flow over broad-crested weirs, which is interesting. 
 
 
 
 
 
 Fig. 73. 
 
 Let Tig. 73 represent a weir with a crest of width d such 
 that the stream over it consists of rectilinear and parallel 
 elementary streams. Let the upstream edge be rounded so' 
 that there is no contraction there. Consider an elementary 
 stream aa', the point a being so far from the weir that the 
 velocity at that point is negligible. Let 00 be the free 
 
v NOTCHES AND WEIES 103 
 
 surface, and let a be fi" below OO and h' above a'. Let a! be 
 z below the free surface at that point. Let h be the head on 
 the weir crest, and e the thickness of the stream on the crest. 
 The pressure at a is Gh", and at a' is Gz. If v is the velocity 
 at a', 
 
 and if b is the length of the weir, 
 
 e)} . . . (8). 
 
 Now Q = for e = and for e = h. The discharge will be a 
 maximum for a value of e found by putting dQ,/de = 0. This 
 gives e = |7&. Inserting this value, 
 
 Q = 0-3856W(20*) . . . (9). 
 
 This is equivalent to taking c = 0*5 7*7 in the ordinary 
 weir formula eq. (3). Experiment shows that the discharge 
 of broad-crested weirs approaches and even falls below this 
 value if d is large. The formula is also applicable to large 
 masonry sluice passages with flat floors, over which the water 
 passes with a free surface. With h>1'5d the attachment 
 of the stream to the weir crest is unstable, and with h>2d 
 the stream springs clear from the upstream edge, and the 
 conditions approximate to those of a sharp-edged weir. 
 
 From various experiments the following values are derived. 
 If h is the head at the weir, d the width of crest, and c the 
 coefficient for a sharp -edged weir in the same conditions, 
 then the coefficient of discharge in the formula 
 
 . . . (9a) 
 may be taken as follows : 
 
 h/d=0'25 0-50 0*75 I'OO 1-25 1-50 
 
 C/c = 0-75 0-78 0-82 0'86 0'90 0-93 
 
 If c = 0-63, C = 0-47 0-50 0'52 0'54 0*57 0'59 
 
 The value c = 0'63 is a mean value for weirs with no end con- 
 tractions. 
 
 The following table gives results of experiments by Mr. 
 Blackwell : 
 
104 
 
 HYDKAULICS 
 
 CHAP. 
 
 
 - 
 
 S 3 
 ^ 
 
 r-ko .10 .cc .t^ 
 
 CDO5 -i-H -rj< -O 
 ' ' 
 
 .05 , . CO CO . CO CO iO 
 J^ rH i 1 -CDCOO 
 
 . (M - .!>!> . O O J> 
 
 O5 OS O Oi CO CD CO 
 
 t^ co cs o . r-i r^ , co , 
 cococoo -co<?q -os 
 
 iocor^i ico . co r i CM 
 
 Tt<T^COCOr-H r-H OS 05 
 
 * 
 
 " icOOCO 
 
 T#Tj<^TtTJ<OTj<T^^ 
 
 TH >0 . CO r-< . 
 
 io t^ . 10 r~- 
 
 j> co * co co 
 
 ********** 
 
 * ***** * 
 
 Jt^iOr- ICOCOOC5C51OO5 
 
 fl o 
 
 8-8 & 
 
 % oj O 
 
 g^^2 
 
 .2^3.2-^ 
 
 S S 
 
 fi Jg| 
 
 * * 
 CH co o 
 
 .to 
 
 CO CO 
 CO CD " IO 
 
 j> to o r- CM co 
 t- r- co i i o os 
 
 CO CO CD CD CO O 
 
NOTCHES AND WEIKS 
 
 105 
 
 63. Rafter's experiments on broad -crested weirs. 
 
 These experiments were made in 1898 at the Cornell Hydraulic 
 Laboratory (Trans. Am. Soc. of Civil Engineers, 1900). The 
 
 ?.74. 
 
 height of the weirs varied from 4^ to 5 feet, and the length 
 of crest was 8 '5 8 feet. The forms used are shown in Fig. 74. 
 In the form d the upstream edge was rounded to a radius 
 of 4 inches. 
 
 Form of 
 Weir. 
 
 Upstream 
 Slope. 
 
 Width 
 of Crest, 
 Feet. 
 
 Down- 
 stream 
 Slope. 
 
 Values of C for h= 
 
 0-5 
 
 1-0 
 
 1-5 
 
 2-0 
 
 5-0 
 
 a 
 
 to 2 
 
 0-33 
 
 Vert 
 
 626 
 
 687 
 
 713 
 
 704 
 
 692 
 
 
 to 2 
 
 0-66 
 
 , 
 
 602 
 
 642 
 
 670 
 
 683 
 
 692 
 
 
 to 5 
 
 0-66 
 
 )} 
 
 619 
 
 622 
 
 624 
 
 625 
 
 633 
 
 m 
 
 to 4 
 
 0-66 
 
 }j 
 
 
 642 
 
 646 
 
 650 
 
 650 
 
 ?j 
 
 to 3 
 
 0-66 
 
 )} 
 
 681 
 
 713 
 
 715 
 
 688 
 
 663 
 
 b 
 
 to 2 
 
 o-o 
 
 1 to 1 
 
 786 
 
 792 
 
 763 
 
 741 
 
 687 
 
 n 
 
 to 2 
 
 0-66 
 
 1 to 2 
 
 586 
 
 638 
 
 644 
 
 674 
 
 679 
 
 n 
 
 1 to 2 
 
 0-33 
 
 1 to 5 
 
 616 
 
 666 
 
 672 
 
 655 
 
 666 
 
 c 
 
 Vert 
 
 2-62 
 
 Vert 
 
 486 
 
 498 
 
 513 
 
 530 
 
 633 
 
 
 
 
 6-56 
 
 
 467 
 
 486 
 
 474 
 
 463 
 
 504 
 
 d 
 
 
 
 2-62 
 
 
 
 553 
 
 562 
 
 566 
 
 575 
 
 647 
 
 5) 
 
 
 
 6-56 
 
 ! -506 
 
 528 
 
 530 
 
 530 
 
 549 
 
 64. Triangular notches. The triangular notch (Fig. 75) 
 has this peculiar- , 
 
 ity, that whatever " ~X: ~ j/~ 
 
 the level in the 
 notch, the section 
 of the stream is 
 similar, that is, its 
 linear dimensions 
 have a fixed ratio. 
 Consider two tri- 
 angular notches of 
 
 .Jr. 
 
 Fig. 75. 
 
 the same angle, and in which the ratio of the linear dimensions 
 
106 HYDKAULICS CHAP. 
 
 is 1 to n. The streams through the notches must be made 
 up of similar and similarly situated elementary streams. 
 Taking any pair of corresponding elementary streams, their 
 cross sections must be as 1 to n 2 , their depths below the free 
 surface as 1 to n, and their velocities as 1 to ^/n. Con- 
 sequently the discharge of these two streams must be in the 
 ratio 1 to n . As this holds for all pairs of similarly situated 
 elementary streams, the total discharge of the notches must 
 
 be in the ratio 1 to ri*. But in any one notch, for two 
 different levels of the water the same must hold, and if 
 h lt h 2 are the heads measured to the vertex of the notch 
 
 the discharges must be in the ratio (h^h^. Hence, generally, 
 if h is the head at any time the discharge is 
 
 and this equation has a more rational basis than the ordinary 
 formula given above for rectangular weirs. It is easy to see 
 that as the surface width Z varies directly as h, the equation 
 can be put in the form 
 
 where c is a coefficient of contraction, ^clh is the section of 
 the contracted stream, and Jc is a constant expressing the 
 ratio of the mean velocity in the contracted stream to the 
 velocity due to the head. The value of k must be about 8/15. 
 Prof. James Thomson first indicated the probability that 
 the coefficient for a triangular notch would be nearly constant. 
 Writing the formula 
 
 Q = T V*M2^) . (10), 
 
 he found that for a right-angled notch, sharp-edged, c = 0'617. 
 For a right-angled notch I = 2h, and the formula becomes 
 
 . . . (lOa). 
 
 The notch is convenient for measuring a very variable flow 
 when the quantity is not very large. 
 
 65. Rectangular notch with no end contractions. 
 The length of the notch or weir is equal to the distance 
 between the walls of the channel of approach. It is desirable 
 that the side walls should extend a little beyond the crest 
 
NOTCHES AND WEIES 
 
 107 
 
 of the notch above its level, but provision must be secured 
 for the free access of air below the water stream passing over. 
 As there are no end contractions, and the top and bottom 
 contractions are the same for all vertical slices of the stream, 
 the discharge must be accurately proportional to the length 
 of the weir. 
 
 Taking any one vertical slice of the stream of width yh 
 and head h, its discharge must be, as in the case of the 
 
 triangular notch, proportional to 7i*, and as the stream, 
 whatever the head, can be considered as made up of l/yh such 
 slices, the whole discharge must be 
 
 which can be put in the form 
 
 = dh x k*J(2gh), 
 
 where c and k have the same meaning, as in the case of 
 the triangular notch, and k must be about 2/3. Then simply 
 
 Q = JcV2^* . . . (11), 
 
 where c may be expected to be constant for different values 
 of*. 
 
 The following are values of c deduced from some very 
 trustworthy experiments on weirs with no end contractions. 
 The values of h have been corrected for velocity of approach, 
 but the correction in all cases was small. 
 
 Length of 
 Crest. 
 
 Head h. 
 
 Discharge Q. 
 
 c. 
 
 Authority. 
 
 5-0 
 
 82 
 68 
 
 12-61 
 9-38 
 
 6304] 
 6276V-6284 
 
 Fteley and Stearns 
 
 5) 
 
 47 
 
 5-37 
 
 6272J 
 
 M 
 
 
 
 22 
 
 1-747 
 
 6365 
 
 w 
 
 
 
 10 
 
 586 
 
 6852 
 
 w 
 
 9-995 
 
 1-0048 
 
 33-49 
 
 6222 
 
 
 Francis 
 
 
 
 9834 
 
 32-56 
 
 6248 
 
 6239 
 
 n 
 
 18-996 
 
 7979 
 1-6184 
 9907 
 
 23-79 
 130-12 
 62-02 
 
 6246 
 6223 
 6195 
 
 6201 
 
 M 
 
 Fteley and Stearns 
 
 
 
 4690 
 
 20-18 
 
 6186 
 
 
 " 
 
108 
 
 HYDKAULICS 
 
 CHAP. 
 
 I 
 
 i 
 *. 
 
 The coefficient increases with very small heads. Exclud- 
 ing these cases, it will be seen that c is very nearly constant. 
 
 66. Sharp-edged weir with end contraction. Francis's 
 formula. The influence of the ends in causing contraction 
 
 extends only for 
 a certain distance, 
 and in a long weir 
 the discharge over 
 the middle part is 
 proportional to the 
 length, as in a weir 
 with no end con- 
 tractions. Let I be 
 the length of the 
 part where the dis- 
 charge is propor- 
 
 tional to the length, and \mh the length of the parts near the 
 ends influenced by end contraction (Fig. 76). Then the whole 
 length L = 1 -f mh. The two parts at the ends taken together 
 form a weir of length mh, in which the linear dimensions are 
 in fixed ratio. The discharge of this part must be given by a 
 relation of the form 
 
 o, - ^, 
 
 as in the case of a triangular weir. The middle part is 
 virtually a weir with no end contractions, and its discharge 
 must be given by a relation of the form 
 
 Fig. 76. 
 
 Hence the whole discharge is 
 
 which may also be written 
 
 2 
 
 (12), 
 
 in which c and I are constants. This is the rational basis of 
 a formula for weirs, arrived at in a purely empirical way by 
 Mr. Francis (Lowell Hydraulic Experiments, New York, 1868), 
 which has proved of great service in practical calculations. 
 
NOTCHES AND WEIES 
 
 109 
 
 For sharp-edged weirs with full end contractions Mr. 
 Francis found for b the value 0*1. The formula is not 
 applicable to short weirs in which L is less than 3h, nor to 
 cases in which h is very small. 
 
 VALUES OF c IN FRANCIS'S FORMULA 
 
 1 
 
 
 
 
 Length of 
 Weir. 
 
 h 
 
 Q 
 
 c 
 
 Authority. 
 
 9-997 
 
 1-5598 
 
 62-60 
 
 6012^) 
 
 
 Francis 
 
 >j 
 
 1-0007 
 
 32-58 
 
 6089 
 
 
 n 
 
 n 
 
 8007 
 
 23-43 
 
 6118 
 
 mean 
 
 > 
 
 n 
 
 6246 
 
 16-22 
 
 6146 
 
 606 
 
 
 
 3-999 
 
 1-0202 
 
 13-14 
 
 5964 
 
 
 
 
 
 
 6830 
 
 7-27 
 
 6027J 
 
 
 
 
 It will be seen that for a considerable range of conditions 
 c is very constant in Francis's formula. 
 
 67. Bazin's researches on weirs. Some very remark- 
 able researches on. flow over various types of weir have been 
 carried out by M. Bazin with exceptional resources, and under 
 conditions which secured the greatest accuracy of measure- 
 ment. The results are contained in a series of papers in the 
 Annales des Fonts et Chaussees, 1888, 1890, 1891, 1894, 
 1896, and 1898. The first object was to ascertain the 
 coefficients for a standard measuring - weir with no end 
 contractions, especially with reference to the influence of 
 different velocities of approach. This weir was afterwards 
 used in measuring the flow over other weirs of different forms. 
 
 The standard weirs had a height of 3*724 feet above the 
 bottom of the channel of approach, a vertical upstream face, 
 and a sharp-edged crest .formed by an iron plate J inch thick. 
 The length of the weir was equal to the width of the channel 
 of approach, so that end contractions were suppressed. 
 Chambers were formed in the side walls below the weir to 
 admit air below the water sheet and to ensure its free detach- 
 ment from the crest. The lengths of the weirs of standard 
 type experimented on were 1*64, 3*28, and 6*56 feet. 
 
 Taking the rectangular weir formula in its simplest form, 
 and putting ra = -|c, the discharge is 
 
 . . . (13). 
 
110 
 
 HYDKAULICS 
 
 CHAP. 
 
 For the same heads the coefficient m was very approxi- 
 mately the same for the four lengths of weir used. The 
 following table gives a selection of the values obtained from 
 an average of the results on all the weirs. The coefficient 
 for standard weirs will be denoted by ra : 
 
 STANDARD WEIRS, 3-72 FEET HIGH, WITH NO END CONTRACTIONS 
 Values of Coefficient m Q in Eq. (13) 
 
 Head. 
 Feet. 
 h 
 
 o 
 
 Head. 
 Feet. 
 h 
 
 ? 
 
 Head. 
 Feet. 
 h 
 
 ra 
 
 Head. 
 Feet. 
 h 
 
 mo 
 
 0-197 
 
 4432 
 
 656 
 
 4262 
 
 1-156 
 
 4273 
 
 1-575 
 
 4307 
 
 262 
 
 4372 
 
 722 
 
 4259 
 
 181 
 
 4277 
 
 1-640 
 
 4313 
 
 328 
 
 4336 
 
 787 
 
 4258 
 
 247 
 
 4281 
 
 1-706 
 
 4318 
 
 394 
 
 4310 
 
 853 
 
 4260 
 
 312 
 
 4286 
 
 1-772 
 
 4324 
 
 459 
 
 4292 
 
 919 
 
 4263 
 
 378 
 
 4291 
 
 1-837 
 
 4329 
 
 525 
 
 4278 
 
 984 
 
 4266 
 
 444 
 
 4297 
 
 1-903 
 
 4335 
 
 591 
 
 4269 
 
 1.050 
 
 4269 
 
 509 
 
 4302 
 
 1-969 
 
 4341 
 
 Next, the influence of velocity of approach was examined. 
 For this purpose the height of the weir above the bottom of 
 the approach channel was altered to 2 '4 6, 1*64, 1*15, and 
 0*787 feet. The following table gives a short selection of the 
 values of m for different heights of weir, and therefore different 
 velocities of approach : 
 
 STANDARD WEIRS OF DIFFERENT HEIGHTS 
 
 Values of the Coefficient m Q for Standard Weirs with no End Contractions 
 
 Head. 
 
 Height of Weir in Feet. 
 
 h 
 
 3-72 
 
 2-46 
 
 1-64 
 
 1-15 
 
 079 
 
 0-197 
 
 4432 
 
 4438 
 
 4445 
 
 4455 
 
 4468 
 
 394 
 
 4310 
 
 4326 
 
 4349 
 
 4396 
 
 4473 
 
 591 
 
 4269 
 
 4320 
 
 4377 
 
 4463 
 
 4579 
 
 787 
 
 4258 
 
 4345 
 
 4426 
 
 4549 
 
 4699 
 
 984 
 
 4266 
 
 4374 
 
 4184 
 
 4638 
 
 4822 
 
 1-181 
 
 4277 
 
 4407 
 
 4544 
 
 4731 
 
 4949 
 
 1-378 
 
 4291 
 
 4441 
 
 4605 
 
 
 
v NOTCHES AND WEIES 111 
 
 To find a general formula accordant with these results, 
 M. Bazin starts from the well-known eq. (6), 
 
 (14), 
 
 where u is the velocity of approach, and a is a constant having 
 usually a value about 1*5. p is a coefficient less than m , 
 and connected with it by the relation 
 
 or since the second term in the bracket is a small fraction, 
 
 (1A 2 \ 
 l + l ' 5a 2gh) neady " " ^' 
 
 If p is the height of the weir, the section of the stream 
 in the channel of approach is (p + h)l, and the velocity of 
 approach is u = Q//Q? + ^). Eeplacing Q by its value 
 
 where K is a new coefficient. With this relation, m in 
 eq. (13) can be found directly from the dimensions of the 
 weir without the need to calculate u. A careful discussion 
 of all the results leads Bazin to adopt the following values 
 of m, and he gives the preference to the second as more con- 
 venient : 
 
 The coefficient ^ varies only with the head, and its 
 average values are : 
 
112 HYDEAULICS CHAP. 
 
 Head Value of Coefficient 
 
 h p. 
 
 0-164 -4481 
 
 328 -4322 
 
 656 -4215 
 
 984 -4174 
 
 1-312 -4144 
 
 1-640 -4118 
 
 1-968 -4092 
 
 With these values the coefficient m in eq. (1*7) can be 
 found, and the discharge over any sharp-edged weir without 
 end contractions calculated, including the influence of the 
 velocity of approach. The formula then supersedes for such 
 weirs the less convenient formulae (5 or 6) previously given. 
 Further, the values of //, are very approximately given by the 
 relation 
 
 For heads from 0*33 to TO ft. with close approximation 
 
 which can be used when a possible error of 2 to 3 per cent 
 can be allowed. 
 
 In the case of weirs with vertical faces and flat crests of 
 a width d, such as weirs constructed of horizontal beams of 
 square timber, the weir sheet adheres to the crest if JKl'od; 
 it may adhere or spring clear from the upstream edge if 
 h>l'5d and<2d; and springs clear if Ji>2d. When the 
 sheet is adherent to the crest the coefficient of discharge 
 depends on the ratio h/d, and is approximately for weirs with 
 no end contractions 
 
 7 + 0-185 7 |] . (19), 
 
 where m Q is the coefficient for a standard weir of the 
 same height. Even with a head of 1'48 feet and a width of 
 crest of 6*6 feet, so that h/d = 0'22, the coefficient of discharge 
 was 0*337, which is little different from the value given by 
 the equation. If h>2d the coefficient of discharge is the 
 same as for a standard weir of the same height. A rounding 
 
NOTCHES AND WEIES 
 
 113 
 
 of the upstream edge of the crest modifies sensibly the 
 discharge. A rounding to a radius of 4 inches increased the 
 discharge 12 to 14 per cent. 
 
 From some experiments on drowned weirs, much too 
 extensive to be described here, Bazin obtained the following 
 expression for the coefficient of discharge : 
 
 m= I '05m, 
 
 (20), 
 
 where h is the head above the weir crest on the upstream 
 side, and h t that on the downstream side ; p is the height of 
 the weir, and z the difference h 7^ of the water-level above 
 and below the weir. The weirs were without end contractions. 
 Bazin made a very extensive series of researches on weirs 
 
 Fig. 77. 
 
 with inclined faces, and with crests either sharp, flat, or 
 rounded. A short abstract of these would be of little use ; 
 the original account must be referred to. The weir sheet 
 takes the following forms : (1) Free weir sheet, as in the case 
 of a sharp-edged weir, the sheet falling freely in the air. 
 For this condition the coefficient of discharge is best defined. 
 (2) Depressed sheet and sheet drowned underneath. If pro- 
 vision is not made for free access of air below the sheet, and 
 if the head does not exceed a certain limit, the sheet is 
 detached from the weir, and encloses a volume of air at less 
 than atmospheric pressure. The tail water rises in level 
 behind the sheet, and the sheet is depressed by the excess of at- 
 mospheric pressure on its outer face (Fig. 77, A). The discharge 
 is somewhat greater than for a free sheet. If the head increases, 
 the whole of the air beneath the sheet is expelled, and the 
 
 8 
 
114 H YDE AULICS CHAP. 
 
 sheet may be said to be drowned underneath (C). It rides over 
 an eddying mass of water in the space which, with a free sheet, 
 is occupied by air. The sheet drowned underneath may or 
 may not be affected by the tail water. If at the foot of the 
 weir there is a rapid followed by a brusque elevation or stand- 
 ing wave, the tail- water level does not influence the discharge. 
 On the other hand, if the tail water covers the foot of the 
 descending sheet, it may influence the discharge, although its 
 level is below the weir crest. (3) Adherent sheets (B). In certain 
 cases with small heads the sheet becomes directly adherent to 
 the downstream face of the weir, without any eddying mass 
 of water behind it. This condition corresponds often to a 
 marked increase of discharge. When the tail water rises 
 above the weir crest, the sheet drowned underneath preserves 
 its general form, until for a certain difference of head and tail 
 water level it breaks into waves. 
 
 68. Measurement of the head at weirs. It is assumed 
 in the preceding discussion that the head on the upstream 
 side of the weir is measured at a point above the origin of the 
 curve of surface fall towards the weir. Fteley and Stearns 
 concluded that the distance from the weir should be at least 
 two and a half times the height of the weir above the bottom 
 of the channel of approach, but no doubt this would be an 
 excessive distance if the height of the weir is large compared 
 with h. The exact measurement of the head is very important, 
 and a hook gauge (41) should be used, as accuracy is im- 
 portant. With h = O'l foot, an error of O'OOl foot, or about 
 a hundredth of an inch in the measurement of h, causes an 
 error of 1|- per cent in the calculated discharge. With 
 greater values of h the percentage error is less, but is not 
 unimportant. As the water-level fluctuates, a series of read- 
 ings at equal intervals of time should be observed and the 
 arithmetical mean taken. 
 
 69. Practical gauging by weirs. The most accurate 
 method of gauging the discharge of small streams, as in ascer- 
 taining the flow from a catchment basin, is to construct a weir 
 of timber or concrete across the stream. A single reading of 
 the head gives the means of calculating the discharge, and 
 observations are made once or twice a day for as long a period 
 as necessary. For small flows a triangular notch may be 
 
NOTCHES AND WEIES 
 
 115 
 
 used, but ordinarily the notch is rectangular. An auto- 
 matic registering apparatus may be used, motion being given 
 to a pencil by a float through the action of a cam designed to 
 allow for the variation of the coefficient of discharge. The 
 reduction of the results is simplified if a weir with no end 
 contractions is used, as the coefficient is nearly constant. The 
 crest of the weir should be a metal plate, flush with the 
 upstream face of the weir, with planed edge accurately levelled. 
 
 PLAN OF 
 CAST IRON KEY. 
 
 ^_ 1 y^-v / vv_- *['('*//.'' / * 
 
 '<? D ^*0 ^ ^^X) ^ ^ ^ 
 
 siiip 
 
 Fig. 78. 
 
 70. Separating weirs. When water is collected in 
 reservoirs for towns' supply from moorland districts, it is 
 desirable to separate the clear water of ordinary periods from 
 the discoloured water in periods of flood. The latter is diverted 
 to waste or sent to a reservoir used only to supply compensa- 
 tion water to the streams. This is effected by a separating 
 weir on the stream feeding the reservoir. Fig. 78 shows one 
 form of such a weir. With small or moderate flow the water 
 drops into the circular channel leading to the reservoir. In 
 
116 HYDKAULICS CHAP. 
 
 flood-time the water springs over the gap, and flows into a 
 channel beyond the weir. 
 
 PROBLEMS 
 
 1. Find the discharge through a rectangular notch, sharp-edged, and 
 
 with complete contraction. The notch is 3 feet wide, and the 
 head 1 J feet. Velocity of approach negligible. 
 
 13-23 cubic feet per second. 
 
 2. What will be the discharge of the same notch if the velocity of 
 
 approach is 3 feet per second ? 16'7 cubic feet per second. 
 
 3. Find the discharge over a sharp-edged weir 10 feet wide, with a 
 
 head of 9 inches. There are no end contractions. 
 
 21*55 cubic feet per second. 
 
 4. Find the discharge of the same weir by Bazin's formula, taking 
 
 the height of the weir to be 2 feet. 
 
 2 2 '9 6 cubic feet per second. 
 
 5. What must be the width of an overfall weir to discharge 24 cubic 
 
 feet per second, with 8 inches head ? Coefficient 0'62. 
 
 13-32 feet. 
 
 6. A district of 6500 acres (1 acre = 43,560 square feet) drains into 
 
 a reservoir. The maximum rate at which rain falls is 2 inches 
 in 24 hours. Supposing this rain to fall when the reservoir 
 is full, it would have to be discharged over the bye-wash weir. 
 Find the length of such a weir under the condition that the 
 head shall not exceed 18 inches. Coefficient of weir 0-66. 
 
 84-16 feet. 
 
 7. A sharp-edged weir, with full contraction, is 10 feet long, and 
 
 has 15 inches of water passing over it. Find the discharge by 
 Francis's formula. 46'27 cubic feet per second. 
 
 8. Find the discharge from a triangular right-angled notch with 
 
 2 feet head. 14-93 cubic feet per second. 
 
 9. A sharp-edged rectangular weir is to discharge daily 30,000,000 
 
 gallons of compensation water, with a normal head of 18 
 inches. The end contractions are suppressed, and the velocity 
 of approach negligible. Find the length of the weir. 
 
 8-99 feet. 
 
 10. Draw a curve of discharge from a right-angled triangular notch 
 
 for different heads. The discharge may be calculated for 2, 
 4, 8, and 12 inches head. Coefficient 0'6. 
 
 11. A lake discharges over a weir 5 feet high above the stream bed 
 
 and 10 feet wide. The water-level above the weir is 8 feet, 
 and below the weir 6 feet, above the stream bed. Find the 
 discharge, taking the coefficient of the weir c = 0'6. 
 
 68-1 cubic feet per second. 
 
 12. A weir is 30 feet long and has 18 inches head. The height of 
 
 the weir is 3 feet. The channel of approach is the same 
 width as the weir. Find the discharge. 
 
 198-6 cubic feet per second. 
 
NOTCHES AND WEIRS 117 
 
 13. If the weir in the last question had end contractions, and the 
 
 velocity of approach was taken into account, find the discharge. 
 
 185-6 cubic feet per second. 
 
 14. A weir is 8 feet wide, 2^ feet high, and has a flat crest Ij feet 
 
 wide. The head is 15 inches, and there are no end contrac- 
 tions. Find the discharge. 33-7 cubic feet per second. 
 
 15. To determine the quantity of water used by a turbine, a sharp- 
 
 edged weir, with full end contractions, was erected in the tail 
 race. The width of the weir was 12 feet, and the head 
 measured to still water level was 0'75 foot. Find the discharge 
 by Francis's formula. 22-1 cubic feet per second. 
 
CHAPTEE VI 
 
 STATICS AND DYNAMICS OF COMPRESSIBLE FLUIDS 
 
 71. THE present chapter deals with a few problems 
 relating to compressible fluids which are closely related to 
 those discussed in the preceding chapters. In compressible 
 fluids the density varies with ordinary differences of pressure 
 and temperature instead of being nearly constant as in the 
 case of liquids. But some reservations may be made. Gases 
 are so much lighter than water that the variation of pressure 
 with difference of level can often be disregarded. In some 
 cases, as for instance the flow of lighting gas in mains, the 
 difference of pressure causing flow is so small compared with 
 the absolute pressure that the variation of density can be 
 neglected without much error. On the other hand, in a large 
 number of cases the variation of density must be taken into 
 the reckoning, and then the formulae for compressible fluids are 
 more complicated than those for water. 
 
 Heaviness of gases. The density or weight per cubic 
 unit of volume, G, must be stated with reference to some 
 standard pressure and temperature. The most convenient 
 standards are 32 F., and one atmosphere, or 2116'3 Ibs. per 
 square foot. The volume V in cubic feet per pound is the 
 reciprocal of the weight G in pounds per cubic foot. V is 
 often termed the specific volume. 
 
 118 
 
CHAP. VI 
 
 COMPEESSIBLE FLUIDS 
 
 119 
 
 HEAVINESS OP GASES AT 32 F. AND ONE ATMOSPHERE 
 
 
 Approx. 
 
 Specific 
 
 Weight 
 
 Cubic 
 
 
 fi~ 
 
 
 Molecular 
 Weight 
 /* 
 
 Gravity 
 Air=l 
 s. 
 
 in Ibs. per 
 cubic feet 
 G . 
 
 feet per 
 pound 
 Vo- 
 
 P O V O . 
 
 bras 
 
 Constant 
 R. 
 
 Hydrogen 
 
 2 
 
 0-0693 
 
 0-00559 
 
 178-30 
 
 378819 
 
 768-1 
 
 Oxygen . 
 
 32 
 
 1-106 
 
 0-0895 
 
 11-17 
 
 23710 
 
 48-0 
 
 Nitrogen . 
 
 28 
 
 0-971 
 
 0-0786 
 
 12-71 
 
 26990 
 
 54-6 
 
 Carbon monoxide 
 
 28 
 
 0-955 0-0773 
 
 12-94 
 
 27380 
 
 55-5 
 
 Carbon dioxide 
 
 44 
 
 1-529 
 
 0-1238 
 
 8-08 
 
 17145 
 
 347 
 
 Air . . . 
 
 29 
 
 1-000 
 
 0-0810 
 
 12-35 
 
 26214 
 
 53-2 
 
 Steam gas 
 
 18 
 
 0-622 
 
 0-0502 
 
 19-91 
 
 42141 
 
 85-3 
 
 ffrom 
 
 
 0-485 
 
 0-0393 
 
 25-47 
 
 
 109-2 
 
 Coal gas | to ^ 
 
 ... 
 
 0-354 
 
 0-0287 
 
 34-89 
 
 
 149-6 
 
 Mond gas dry . 
 
 ... 
 
 0-808 
 
 0-0654 
 
 15-29 
 
 ... 
 
 65-6 
 
 Producer gas 
 
 ... 
 
 0-965 
 
 0-0781 
 
 12-80 
 
 ... 
 
 55-0 
 
 The weight per cubic foot at 32 F. and one atmosphere 
 = //,/358, and the corresponding volume per pound is 
 
 s 
 
 Specific heats of gases. For the simpler gases the 
 specific heats at constant pressure and volume appear to be 
 nearly independent of the pressure and temperature. For the 
 more complex gases it is now certain that they increase with 
 increase of pressure and temperature. For the calculations in 
 this chapter only the ratio of the specific heats, 7 = c p /c v , is 
 required, and it will be sufficient to assume that for air and 
 the so-called permanent gases 7=1*40; for steam gas and 
 carbon dioxide 7 = T28. 
 
 For air the following values are useful : 
 
 y - = 0-286; 
 
 7 
 
 - = 0-714 
 
 = 2-5; 
 
 -^-=3-5. 
 
 y-1 
 
 Boyle's law. At a constant 
 
 72. Gaseous laws. Boyle's law. At a constant tem- 
 perature the pressure of a gaseous mass varies inversely as the 
 volume. If P is the pressure in pounds per square foot, V the 
 volume of a pound in cubic feet, and G the weight of a cubic 
 foot in pounds ; then if the temperature is constant, 
 
 P/G = PV = constant . . . (1). 
 
120 HYDKAULICS CHAP. 
 
 If P , V are the values at 32 F. and one atmosphere, then 
 P V is a constant for each gas which has been determined 
 with great precision. 
 
 Dalton's law. In a mixture of gases the pressure is the 
 sum of the pressures which would be exerted by each gas 
 separately if it occupied the space alone. Let v lt v 2 . . . be 
 the fractions of a cubic foot of each of the gases in one cubic 
 foot of mixture at a pressure. P. Then the pressures due to 
 the different gases are 
 
 Pl = ?Vl', ^2 = PV 2 
 
 Let w lt w z . . .be the fractions of a pound of each of the 
 gases in one pound of the mixture, and /i^ /^ 2 . . . their 
 molecular weights. Then 
 
 (2). 
 
 Charles's law. Under constant pressure all gases expand 
 alike. Thus between 32 and 212 F. one cubic foot expands 
 to 1'3654 cubic feet, or, putting it another way, a gas expands 
 1/493 of its volume at 32 for each degree rise of temperature. 
 Let V be the volume of one pound at 32 and V its volume 
 at t, the pressure being the same. 
 
 If temperatures are reckoned from 461 on the Fahrenheit 
 scale, in which case they are termed absolute temperatures, 
 the equation takes a simpler form. Let T, T be the absolute 
 temperatures corresponding to and 32. 
 
 V/V = T/T . . (4). 
 
 The laws of Boyle and Charles can be combined to give 
 the general relation of pressure, volume, and temperature in 
 gases. For, let P , V , T be the pressure, volume, and tem- 
 perature of one pound at 32 F., and P, V, T the same quantities 
 under other conditions. By Charles's law, if T changes to 
 T, and V to V 7 , the pressure remaining constant, 
 
 V' = V T/T . 
 
vi COMPRESSIBLE FLUIDS 121 
 
 But by Boyle's law the product of pressure and volume is 
 constant if the temperature does not change. Let P now 
 change to P and V to V at constant temperature. Then 
 
 PV = P V' = P V T/T . 
 
 Let P V /T = K, which is called the gaseous constant. Then 
 PV = RT . . . . (5) 
 
 is the general equation connecting pressure, temperature, and 
 volume. Values of R are given in the table on p. 119. 
 
 73. The Mercurial Barometer. In the mercurial baro- 
 meter the pressure due to the height of the column h (Fig. 79) 
 balances the atmospheric pressure. If G TO = 848*8 
 is the weight of mercury in pounds per cubic foot, 
 p a = atmospheric pressure in pounds per square 
 foot, and h is in feet, 
 
 p a = Gji = 848-8/1 . . (6). 
 
 If A is given in inches, and p a is required in 
 pounds per square inch, p a = 0*49 12h. 
 
 As mercury expands with rise of temperature, 
 the actual barometer readings should be corrected 
 to 32 F., the expansion of the brass scale being 
 also allowed for. The correction depends on the Fig. 79. 
 height of the barometer at the time, and tables 
 are obtainable giving the correction. If t is the temperature 
 at the time of an observation, the correction for a barometer 
 with brass casing is approximately 
 
 - 32) - -OOOOl^ - 62) 
 
 
 1 + -0001(*-32) 
 
 the scale being assumed correct at 62 F. The correction is 
 in inches if h is taken in inches. 
 
 Let G rt be the weight of air in pounds per cubic foot. If 
 the atmosphere were homogeneous instead of decreasing in 
 density upwards, the height of the atmosphere would be 
 
 H = ^ = ^feet . . . (7). 
 
 ^a ^a 
 
 For mercury G m = 848. The mean barometric pressure at 
 
122 HYDKAULICS CHAP. 
 
 sea-level is 29*92 inches, or 2*493 feet, and the weight of air 
 at 32 and that pressure is Gr a = 0-08073 Ibs. per cubic foot. 
 Hence 
 
 74. Variation of pressure with elevation. Application 
 of the barometer to determine heights. Let the atmo- 
 sphere be at 32 F., and let G be its density at h feet above 
 the point where the pressure is one atmosphere. If p is the 
 pressure at a height h, the pressure at h + dh will be less by 
 the weight of a layer of thickness dh. That is, 
 
 dp= -Gdh. 
 
 But at constant temperature .p/G=j9 /G , where p Q) G are 
 values at 32 and one atmosphere. 
 
 Integrating, since p = p Q , when h = 0, 
 
 The quantity pjGr Q is the height H of a homogeneous 
 atmosphere at 32 F. above a point where there is standard 
 
 YATkC3C1TIT*k O T rt f\ f\-n CC1 4~tT 
 
 pressure and density 
 
 The height above a point where the height of a homogeneous 
 atmosphere is H is 
 
 where p, p are the barometric pressures. , If p lt p 2 are the 
 barometric pressures at two stations at heights 7i 1} h 2 above 
 the point where the pressure is one atmosphere, 
 
 . . . . (86). 
 
 2 
 
 Pi/p 2 is a ratio, the pressures may be taken in any units, 
 
vi COMPEESSIBLE FLUIDS 123 
 
 for instance inches of mercury. Putting H=26190, and 
 substituting common for natural logarithms, 
 
 h 2 -h l = 60300 Iog 10 -^ ft. 
 Pz 
 
 Let t lf t 2 be the temperatures at the two stations. The 
 mean temperature of the air between the stations is approxi- 
 mately = 0(^1 + ^2)' But a column of air 1 foot high at 32 
 
 expands to 
 
 -32 
 
 at t. Hence the true height between the stations corrected 
 for temperature is k(k 2 h^. 
 
 EXAMPLE. The observed barometric heights at two stations were 
 30 and 27 inches, and the corresponding air temperatures 65 and 50 F. 
 
 h 2 -h l = 60300 log (30/27) = 6437 ft. 
 The mean temperature was 57*5. 
 
 Tc = 1 + (57 '5 - 32)/493 = 1 '05. 
 Corrected difference of level = 6437 x 1-05 = 6759 ft. 
 
 75. Flow of air through orifices under small differences 
 of pressure. In some cases the air is discharged from a 
 vessel in which the pressure is rather more than an atmo- 
 sphere into the atmosphere. In that case the difference of 
 pressure causing flow is small, and the variation of density 
 of the air is very small also. For instance, if the difference 
 of pressure is one pound per square inch the pressure ratio is 
 15 '7 to 14' 7 Ibs. per square inch, or 1*07 nearly, and as in 
 the cases under consideration there is no material change of 
 temperature, this is the ratio of variation of density also. In 
 many practical cases the variation of pressure and density 
 is even smaller than this. In such cases the flow may be 
 treated as if the fluid were incompressible. 
 
 Let p lt p. 2 be the absolute pressures in pounds per square 
 foot inside and outside the reservoir from which the air flows. 
 
 Tj the absolute initial temperature F. 
 
 v the velocity acquired by the air. 
 
 Vj the volume of a pound of air at pressure p l and 
 temperature Tj. 
 
124 HYDKAULICS CHAP. 
 
 Gj the weight of a cubic foot in the same conditions. 
 Then neglecting the variation of density, the head produc- 
 ing flow is (p 1 jp 2 )/Gi' 
 
 . ( ~ ~ \ 
 
 ft. per sec. . . (9). 
 
 = /j 
 
 If o> is the area of the orifice in square feet and c the 
 coefficient of discharge, the volume discharged per second is 
 
 Q = CM = co> 2L c. ft. per sec. . (10). 
 
 The weight G x of a cubic foot of air at pressure p^ and 
 temperature Tj is 
 
 Gr 1 =p l /53'2T 1 Ibs. per c. ft. 
 
 Hence the weight in pounds discharged per second is 
 
 Ibs. 
 
 . (ii). 
 
 /PI(P 
 V" 
 
 Tl 
 
 Ibs. 
 
 When dealing with small differences of pressure, it is 
 common to measure the pressures in inches of water column. 
 One inch of water = 5 '2 02 Ibs. per square foot. Hence if the 
 pressures are in inches of water, 
 
 A l( ^ 
 
 (12). 
 
 Professor Durley has carried out careful experiments on 
 the discharge of sharp-edged orifices, -f$ to 4^ inches diameter, 
 with differences of pressure from 1 inch to 6 inches water 
 column. The following table gives the coefficients of dis- 
 charge obtained : 
 
VI 
 
 COMPKESSIBLE FLUIDS 
 
 125 
 
 VALUES OF c. 
 
 
 Heads in Inches of Water. 
 
 Diameter 
 
 
 
 
 in Inches. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 Tff 
 
 0-603 
 
 0-606 
 
 0-610 
 
 0-613 
 
 0-616 
 
 
 0-601 
 
 0-603 
 
 0-605 
 
 0-606 
 
 0-607 
 
 2 
 
 0-600 
 
 0-600 
 
 0-600 
 
 0-600 
 
 0-600 
 
 3 
 
 0-599 
 
 0-598 
 
 0-597 
 
 0-596 
 
 0-596 
 
 4 
 
 0-598 
 
 0-596 
 
 0-594 
 
 0-593 
 
 0-592 
 
 The channel of approach to the orifice was at least twenty 
 times the area of the orifice, so that the velocity of approach 
 was negligible. 
 
 *76. Expansion of compressible fluids. Two cases are 
 important. If the expansion takes place without change of 
 temperature, heat must be supplied during expansion ; Boyle's 
 law is applicable, and the product PV is constant. Such 
 expansion is termed isothermal or hyperbolic. If no heat is 
 supplied or lost during expansion, it is shown in treatises on 
 thermodynamics that the product PV 7 is constant where 7 is 
 the ratio of the specific heats at constant pressure and constant 
 volume. The expansion is termed adiabatic, and as external 
 work is done at the 
 expense of the internal y 
 energy of the fluid the 
 temperature falls. 
 
 Let one pound of air 
 expand from V x to V 2 , 
 the pressure changing 
 from Pj to P 2 . Then 
 
 r = V 2 / Y! is the volume 
 ratio of expansion, and 
 p = P 2 /P 1 may conveni- 
 ently be called the pres- 
 sure ratio of expansion. 
 The relation of pressure Fig. 80. 
 
 and volume during ex- 
 pansion is given graphically by a curve CD. During any 
 
126 
 
 HYDKAULICS 
 
 CHAP. 
 
 small change from V to V + dV the work of expansion is ~PdV. 
 Hence the whole work of expansion from the state given by 
 PjVj to that given by P 2 V 2 , reckoned per pound of fluid, is 
 
 f v 2 
 
 U= PdV ft.-lbs. 
 J V, 
 
 Work of isothermal expansion. Since in this case 
 PV is constant, the expansion curve CD is a hyperbola. 
 P = P 1 V ] /V. Hence 
 
 u = Pl vJ T = Woge 
 
 J V, l 
 
 = P^ log e r = P 1 V 1 log e - ft.-lbs. 
 
 (13). 
 
 Work of adiabatic expansion. In this case PV V = 
 constant. 
 
 U 
 
 -, 
 
 V 
 
 V 7 
 
 iVf-JL _I1 
 
 '-HV 1 v/' 1 / 
 
 7 - 
 
 - ft.-lbs. . (14). 
 
 = R(T 1 -T 8 ) 
 7-1 
 
 It is convenient to remember the following relations in 
 adiabatic expansion : 
 
 . (15). 
 
VI 
 
 COMPEESSIBLE FLUIDS 
 
 It is also useful to state the thermodynamic result that the 
 change of temperature in adiabatic expansion is given by the 
 relation 
 
 T 2 
 
 :- 
 
 (16). 
 
 77. Modification of the theorem of Bernoulli for 
 compressible fluids. Suppose 
 that in a short time t the mass 
 AB comes to A'B'. Let P p w lt 
 ^i Gj, "V\> T!, be the pressure, 
 section, velocity, weight per 
 cubic foot, volume per pound, 
 and absolute temperature at A. 
 Let P 2 , o> 2 , v 2 , G 2 , V 2 , T 2 , be the 
 same quantities at B. The 
 motion being steady, the weight 
 of fluid passing A and B in a 
 given time must be the same. If W is the flow in pounds per 
 
 second, 
 
 W = G 
 
 Fig. 81. 
 
 If z l5 z 2 are the heights of A and B above the datum plane, 
 the work of gravity is 
 
 Gr 1 <ta 1 v l (z 1 - z 2 ) = W(z l - z 2 ) ft.-lbs. per sec. 
 The work of the pressures on the sections at A and B is 
 
 i-A - P 2 a, 2 v 2 = i- 1 - 1 W ft,lbs. per sec. 
 
 The work of expansion is 
 
 w| v Wv 
 
 = WU ft.-lbs. per sec. 
 
 The change of kinetic energy is the difference of the energy of 
 W Ibs. entering at A and W Ibs. leaving at B. That is, 
 
 W 
 
 (v 2 2 - v^) f t.-lbs. per sec. 
 
 Equating the work done to the change of kinetic energy, and 
 for simplicity dividing by W, 
 
128 HYDKAULICS CHAP. 
 
 P P - 
 
 An expression similar to that for liquids, except that the work 
 of expansion U appears. The result may be stated thus : the 
 total head at A, plus the work of expansion between A 
 and B, is equal to the total head at B. Since A and B are 
 any two points, it may be said that the total head along a 
 stream line increases by the work of expansion (or decreases 
 by the work of compression) to that point. If difference of 
 level is neglected and the expansion is adiabatic, eq. (14), 
 
 . (18). 
 
 78. Flow of compressible fluids from orifices when 
 the variation of density is taken into account. When the 
 flow is due to pressure differences which are not small com- 
 pared with the absolute pressures in the fluid, the work of 
 expansion is not negligible. Suppose the fluid flowing from 
 a point in a reservoir where the pressure is P 1? and where it 
 is sensibly at rest, through an orifice into a space where the 
 pressure is P 2 , and where it has acquired the velocity v 2 . 
 Neglecting any difference of level, and introducing a coefficient 
 of velocity c v to allow for the resistance of the orifice, from 
 eq. (18), 
 
 . (19). 
 
 Approximate equations. When the pressure difference 
 is small, let 8 = (P x - Pa)/?!, so that p = P 2 /P X = 1-3, where 
 S is a small fraction. 
 
vi COMPEESSIBLE FLUIDS 129 
 
 Then eq. (19) becomes 
 
 *2 = <y{^n^ 2 } - (20), 
 
 the approximate equation previously obtained on the assump- 
 tion that the fluid could be treated as incompressible for small 
 pressure differences. A closer approximation is obtained by 
 taking another term in the expansion of 
 
 y-l 
 
 . (21), 
 
 an equation given by Grashof. 
 
 Weight of fluid discharged from an orifice. Let a> 
 
 be the area of the orifice, and c c the coefficient of contraction. 
 Let PJ, Vj be the pressure and volume per pound in the 
 reservoir ; P 2 , V 2 the same quantities in the space into which 
 the fluid is discharged. Let r be the volume and p the pres- 
 sure ratio of expansion in the stream issuing from the orifice. 
 The volume discharged per second, reckoned, at the lower 
 pressure, is 
 
 Q 2 = c c fl 2 a> cubic feet, 
 and the weight is 
 
 W = ^lbs. per second. 
 
 V 2 
 
 But V 2 = rV lf and putting c = c/ c , by eq. (18) 
 
 But r = l/p\ 
 
 and this is a maximum when P 2 /Pi = p is 
 
130 
 
 HYDKAULICS 
 
 CHAP. 
 
 which may be called the critical pressure ratio. If 7= 1'4, as 
 for air, the discharge is greatest for < = 0'528. The maximum 
 discharge, putting in the value of p just found, is 
 
 (24), 
 
 and for 7= 1*4 this becomes 
 
 the external pressure being then a little more than half 
 the pressure in the reservoir. When P 2 /Pi is less than <, the 
 critical value of the pressure ratio, or in other words if P x is 
 greater than </>P 2 , the weight of fluid discharged diminishes, a 
 result which is paradoxical and extremely improbable. It 
 must therefore be inquired if there is any defect in the 
 reasoning. There is one assumption which is unverified, 
 namely, that the expansion is completed at the contracted 
 section of the jet, and that the pressure at that section is P 2 . 
 
 Experiments, first made by Mr. 
 E. D. Napier with steam, showed 
 that for P 2 /Pi less than < the 
 pressure at the contracted section 
 was greater than the external 
 pressure P 2 , and that the fluid 
 continued to expand after the 
 contracted section was passed. 
 Hence the section at which the 
 pressure is P 2 is a section greater 
 than c c a), and may even be greater 
 The jet when P 2 /Pj is less than 
 
 Fig. 82. 
 
 than the area of the orifice. 
 
 <f> takes a form like that shown in Fig. 82. 
 
 The centrifugal force of the curved elementary streams 
 near the contracted section makes the mean pressure there 
 greater than P 2 . Experiment shows further that whenever 
 P 2 /Pi is less than <j> the discharge is found by substituting 
 </>Pj for P 2 in the general eq. (22). Hence for such cases 
 the discharge is found by using eq. (24) instead of eq. (22). 
 
 Discharge of air from orifices. For air, 7=1-4 and 
 <=0'528. Two cases occur. (a) When P 2 /Pi is greater 
 than (j>, and putting p for P 2 /Pu 
 
vi COMPKESSIBLE FLUIDS 131 
 
 (6) When P 2 /Pj is less than <, 
 
 W=3-885c<o ffi . . . (25a). 
 
 It appears that for sharp-edged circular orifices, c = 0'64 ; 
 for short cylindrical mouthpieces without rounding at the 
 inner edge, c = 0'81 to 0*83; for short conoidal mouthpieces, 
 c = 0'97 ; and for coned blast nozzles, c = 0'86. 
 
 The discharge of steam under great differences of pressure 
 is complicated by variations of wetness in the steam and 
 other circumstances. Careful experiments by Mr. Eosenhain 
 are described in Proc. Inst. Civil Engineers, cxl. 199. For 
 dry steam and P 2 /Pi less than <, 
 
 W = 0-1995c c o>P 1 ' 97 Ibs. nearly . . (26), 
 or, what is the form of the equation more generally given, 
 
 where V x is the specific volume of the steam at the 
 pressure P x . 
 
CHAPTEK VII 
 
 FLUID FEICTION 
 
 79. WHEN a liquid flows in contact with a solid surface, or 
 when a solid of shipshape form moves in a liquid at rest, 
 there is a resistance to motion which is termed fluid friction, 
 though it is wholly different in character from the friction of 
 solids. At very low velocities the motions of the fluid near 
 the solid may be stream-line motions, and the resistance is 
 due to the shearing action of filaments moving with different 
 velocities. Such conditions hardly ever obtain in cases of 
 practical interest to the engineer. Whenever the velocity is 
 not very small, eddies are generated which absorb energy 
 afterwards dissipated in consequence of the viscosity of the 
 fluid. The frictional resistance in this case is measured by 
 the momentum imparted to the water in unit time when a 
 solid moves in still water, or abstracted from the motion of 
 translation and dissipated when a current flows over a surface. 
 The laws of fluid friction may be stated thus : 
 
 (1) The frictional resistance is independent of the pressure 
 in the fluid. 
 
 (2) Under certain restrictions to be stated presently the 
 frictional resistance is proportional to the area of the immersed 
 surface. 
 
 (3) At very low velocities the frictional resistance is pro- 
 portional to the velocity of the fluid relatively to the surface. 
 At all velocities above a certain critical value depending on 
 the general conditions, that is, in all cases in which the motion 
 of the fluid is turbulent, the frictional resistance is nearly 
 proportional to the square of the velocity. 
 
 Also in cases where the motion is turbulent : 
 
 132 
 
CHAP. VII 
 
 FLUID FRICTION 
 
 133 
 
 (4) The frictional resistance increases very rapidly with 
 the roughness of the solid surface. 
 
 (5) The frictional resistance is proportional to the density 
 of the fluid. 
 
 These laws can be expressed mathematically for the case 
 of turbulent motion in this way. Suppose a thin board of 
 total area a>, wholly immersed, to move through a fluid at 
 rest with a velocity v. Let / be the frictional resistance 
 reckoned per square foot of the surface at a velocity of one 
 foot per second. Then the total resistance of the board is 
 
 R=/Wlbs. . . . . (1), 
 
 where / is a constant for a given quality of surface and a 
 fluid of given density. It is convenient to express this in 
 another way. Let f = (2#/)/G, where f is termed the 
 coefficient of friction. Then 
 
 R 
 
 . (2). 
 
 As the board moves through the fluid the resistance is 
 overcome through a distance of v feet per second. Hence 
 the work expended in overcoming friction is 
 
 U = 
 
 
 ft.-lbs. per sec. 
 
 (3). 
 
 The following are average values of the coefficient of 
 friction for water, obtained from experiments on large plane 
 surfaces moved in an indefinitely large mass of water : 
 
 
 
 Frictional 
 
 
 Coefficient 
 
 ^Resistance in 
 
 
 of Friction 
 
 Ibs. per 
 
 
 
 
 square foot 
 
 
 
 ft* 
 
 New well-painted iron plate 
 
 00489 
 
 00473 
 
 Painted and planed plank (Beaufoy) . 
 
 00350 
 
 00339 
 
 Surface of iron ships (Rankine) . 
 
 00362 
 
 00351 
 
 Varnished surface (Froude). 
 
 00258 
 
 00250 
 
 Fine sand surface 
 
 00418 
 
 00405 
 
 Coarser sand surface 
 
 00503 
 
 00488 
 
 80. Mr. Froude's experiments. The most valuable direct 
 
134 HYDKAULICS 
 
 CHAP. 
 
 experiments on fluid friction are those carried out by Mr. W. 
 Froude at Torquay. 1 The method adopted was to tow a thin 
 board in a still water canal, the velocity and resistance being 
 simultaneously recorded. The boards were generally 3/16 inch 
 thick and 19 inches deep, with a sharp cutwater, and from 
 1 to 50 feet in length. The boards were covered with various 
 substances, such as paint, varnish, tinfoil, sand, etc., to deter- 
 mine the influence of different roughnesses of surface. The 
 results obtained by Mr. Froude may be summarised as 
 follows : 
 
 (1) The friction per square foot of surface varies very 
 greatly for different surfaces, being generally greater as the 
 sensible roughness of the surface is greater. Thus, when the 
 surface of the board was covered as mentioned below, the 
 resistance for boards 5 feet long, at 1 feet per second, was : 
 
 Tinfoil or varnish . . 0'25 Ib. per square foot. 
 Calico . . . 0-47 
 
 Fine sand .... 
 Coarser sand 
 
 (2) The power of the velocity to which the friction is 
 proportional varies for different surfaces. Thus, with short 
 boards 2 feet long : 
 
 For tinfoil the resistance varied as v 2 ' 16 
 For rough surfaces v 2 ' 00 
 
 With boards 5 feet long : 
 
 For varnish or tinfoil the resistance varied as v 1 " 88 
 For sand -y 2 ' 00 
 
 (3) The average resistance per square foot of surface was 
 much greater for short than for long boards ; or, what is the 
 same thing, the resistance per square foot at the forward part 
 of the board was greater than the friction per square foot of 
 portions more sternward. Thus, at 1 feet per second : 
 
 Mean Resistance in Ibs. 
 per Square Foot. 
 
 Varnished surface . . 2 feet long 0-41 
 
 i, 50 0-25 
 
 Fine sand surface 2 0-81 
 
 . 50 0-405 
 
 1 British Association Reports, 1875. 
 
VII 
 
 FLUID FKICTION 
 
 135 
 
 This remarkable result is explained thus by Mr. Froude : 
 " The portion of surface that goes first in the line of motion, 
 in experiencing resistance from the water, must in turn com- 
 municate motion to the water in the direction in which it is 
 itself travelling. Consequently, the portion of surface which 
 succeeds the first will be rubbing, not against stationary water, 
 but against water partially moving in its own direction, and 
 cannot therefore experience so much resistance from it." 
 
 The following table gives a general statement of the 
 numerical values obtained by Mr. Froude. In all the experi- 
 ments in this table the boards had a fine cutwater and a fine 
 stern end or run, so that the resistance was entirely due to 
 the surface. The table gives the resistance per square foot 
 in pounds, at the standard speed of 600 feet per minute, and 
 the power of the speed to which the friction is proportional, 
 so that the resistance at other speeds is easily calculated 
 
 
 Length of Surface, or Distance from Cutwater, in Feet. 
 
 
 Two Feet. 
 
 Eight Feet. 
 
 Twenty Feet. 
 
 Fifty Feet. 
 
 
 A 
 
 B 
 
 C 
 
 A 
 
 B 
 
 C 
 
 A 
 
 B 
 
 C 
 
 A 
 
 B 
 
 C 
 
 Varnish 
 
 2-00 
 
 41 
 
 390 
 
 1-85 
 
 325 
 
 264 
 
 1-85 
 
 278 
 
 240 
 
 1-83 
 
 250 
 
 226 
 
 Paraffin 
 
 1-95 
 
 38 
 
 370 
 
 1-94 
 
 314 
 
 260 
 
 1-93 
 
 271 
 
 237 
 
 ... 
 
 ... 
 
 
 Tinfoil . 
 
 2-16 
 
 30 
 
 295 
 
 1-99 
 
 278 
 
 263 
 
 1-90 
 
 262 
 
 244 
 
 1-83 
 
 246 
 
 232 
 
 Calico . 
 
 1-93 
 
 87 
 
 725 
 
 1-92 
 
 626 
 
 504 
 
 1-89 
 
 531 
 
 447 
 
 1-87 
 
 474 
 
 423 
 
 Fine sand 
 
 2-00 
 
 81 
 
 690 
 
 2-00 
 
 583 -450 
 
 2-00 
 
 480 
 
 384 
 
 2-06 
 
 405 
 
 337 
 
 Medium sand 
 
 2-00 
 
 90 
 
 730 
 
 2-00 
 
 625 -488 
 
 2-00 
 
 534 
 
 465 
 
 2-00 
 
 488 
 
 456 
 
 Coarse sand . 
 
 2-00 
 
 1-10 
 
 880 
 
 2-00 
 
 714 
 
 520 
 
 2-00 
 
 588 
 
 490 
 
 ... 
 
 
 ... 
 
 Columns A give the power of the speed to which the resistance is 
 approximately proportional 
 
 Columns B give the mean resistance per square foot of the whole 
 surface of a board of the lengths stated in the table. 
 
 Columns C give the resistance in pounds of a square foot of surface at 
 the distance sternward from the cutwater stated in the heading. 
 
 It may be noticed that although the friction per square 
 foot decreases as the surface is longer in the direction of 
 motion, yet the decrease, which is considerable between 2 feet 
 and 8 feet, is small between 20 feet and 50 feet. Hence for 
 surfaces more than 50 feet long it makes little difference 
 
136 HYDKAULICS 
 
 CHAP. 
 
 whether the friction is supposed to diminish at the same rate 
 or not to diminish at all. If the decrease of friction stern - 
 wards is due to the generation of a current accompanying the 
 moving plane, there is not at first sight any reason why the 
 decrease should not be greater than that shown by the experi- 
 ments. The current accompanying the board might be assumed 
 to gain in volume and velocity sternwards, till the velocity 
 was nearly the same as that of the moving plane and the 
 friction per square foot nearly zero. That this does not 
 happen appears to be due to the mixing up of the current 
 with the still water surrounding it. Part of the water in 
 contact with the board at any point, and receiving energy of 
 motion from it, passes afterwards to distant regions of still 
 water, and portions of still water are fed in towards the 
 board to take its place. In the forward part of the board 
 more kinetic energy is given to the current than is diffused 
 into surrounding space, and the current gains in velocity. 
 At a greater distance back there is an approximate balance 
 between the energy communicated to the water and that 
 diffused. The velocity of the current accompanying the board 
 becomes constant or nearly constant, and the friction per 
 square foot is therefore nearly constant also. 
 
 81. Friction of discs rotated in water. In many 
 hydraulic machines, turbines, and centrifugal pumps, surfaces 
 rotate in water, and the friction is an important cause of loss 
 of energy. A disc rotated in water is virtually a surface of 
 indefinite length in the direction of motion, and experiments 
 carried out in this way by the author, Proc. Inst Civil JEng. 
 Ixxx. 1885, permitted considerable variation of the conditions. 
 Fig. 83 shows a section of the apparatus. It consisted of 
 a wooden frame on which was placed a cast-iron cistern C. 
 A cast-iron bracket B at the top of the frame carried a three- 
 armed crosshead bb, from which an inner cistern AA was sus- 
 pended by three fine wires. The crosshead could be adjusted 
 to any position and clamped by the nut a. Adjusting- 
 screws in the arms of the crosshead permitted the cistern AA 
 to be levelled. The discs which were to be rotated in water 
 were 10, 15, and 20 inches diameter ; one is shown in position 
 at DD keyed on a vertical shaft SS. This shaft was centred 
 on conical ends and driven by a catgut band running on 
 
VII 
 
 FLUID FEICTION 
 
 137 
 
 pulleys P. The rotating disc is contained in the submerged 
 copper cylinder AA. The flat bottom of this is fitted with 
 
 Fig. 83. 
 
 very little play round the gun-metal support of the spindle. 
 Above the disc was a flat cover EE parallel to the flat bottom 
 of the cistern. The height of the chamber in which the 
 
138 
 
 HYDKAULICS 
 
 CHAP. 
 
 disc revolved could be varied, the disc being always placed in 
 the centre of the chamber. A thick india-rubber ring bolted 
 round the cover EE made a water-tight connection with the 
 cylinder. 
 
 To measure the friction of the disc, the reaction tending 
 to turn the cistern AA was measured, for the reaction on the 
 chamber must be equal and opposite to the effort required to 
 turn the disc. To the suspended cylinder was attached an 
 index-finger moving over a graduated scale. This was adjusted 
 to zero when the apparatus was at rest. When the disc 
 
 234 
 
 REVOLUTIONS PER SECOND 
 
 Fig. 84. 
 
 rotates, the copper cylinder tends to rotate in the same 
 direction. To measure the effort to rotate which is equal to 
 the effort turning the disc, a fine silk cord attached to an arc 
 on the cistern was carried over the pulley e to a scale-pan G-. 
 Weights in the scale-pan balanced the friction and kept the 
 index at zero. The rotations were observed by timing the 
 rotations of the worm-wheel W by a chronograph. A clip 
 brake K on the shaft was useful in adjusting the speed. 
 
 Fig. 84 shows a plotting of one set of results on brass 
 discs of three sizes. It will be seen that the observations 
 plot in quite regular curves. The three upper curves are for 
 a 20 -inch disc of polished brass with lj, 3, and 6 -inch spaces 
 
vii FLUID FKICTION 139 
 
 between the disc and the flat ends of the cistern. The resist- 
 ance diminishes a little as the spaces are narrower. The 
 other curves are for a 15 -inch and 10 -inch disc of brass. 
 
 82. Theoretical expression for the friction of a disc 
 rotating in liquid. Let it be supposed that the general 
 law of fluid friction which applies to large plane surfaces 
 moved uniformly in water may be used to determine the 
 friction of a disc. That is, supposing CD to be the area of 
 any small portion of the disc moving with the velocity v, let 
 it be assumed that the friction of that portion of the surface is 
 fcov n ; where / is a constant differing for different surfaces, 
 and n a constant which at the velocities used in these experi- 
 ments does not differ greatly from 2. 
 
 Let a be the angular velocity of rotation, K the radius of 
 the disc. Consider a ring of the surface between the radii r 
 and r -f dr. Its area is 2irrdr f its velocity is ar, and the 
 friction of this portion of the surface is therefore, on the 
 assumption above, 
 
 / x 2-rrrdr x a n r n . 
 
 The moment of the friction of the ring about the axis of 
 rotation is then 
 
 and the total moment of friction for the two sides of the disc 
 is then 
 
 1. 
 
 >n+3 
 
 If N is the number of rotations per second, since a = 2?rN, 
 
 on+2 _n 
 
 M = 2 *__ 3 
 
 n + 6 
 The work expended in rotating the disc is in ft.-lbs. per sec. 1 
 
 1 If 7i = 2, from which it never differs much, this formula becomes 
 Work expended in friction = 623/N 3 R 5 ft.-lbs. per sec., 
 
 where/ varies from 0'002 to 0*003 for ordinarily rough surfaces, and increases 
 to 0'007 for the rough surface of a metal disc covered with coarse sand. 
 
140 
 
 HYDEAULICS 
 
 CHAP. 
 
 EESULTS OF EXPERIMENTS 
 
 
 
 
 Thickness 
 
 
 Lowest 
 
 Number 
 of Experi- 
 
 Nature of Disc and Surface. 
 
 Virtual 
 Radius of 
 
 of Water 
 Space on 
 
 Tempera- 
 ture 
 
 Speed in 
 Rotations 
 
 ment. 
 
 
 Disc. 
 
 each side 
 
 Fahr. 
 
 per 
 
 
 
 
 of Disc. 
 
 
 Second. 
 
 1 
 
 Clean polished brass 
 
 Foot. 
 0-8488 
 
 Inches. 
 
 55-0 
 
 1-425 
 
 2 
 
 
 
 3 
 
 53-0 
 
 1-459 
 
 3 
 
 59 55 
 
 " 
 
 6 
 
 55-0 
 
 1-415 
 
 4 
 
 Painted cast iron . . . 
 
 $ 5 
 
 ]i 
 
 60-5 
 
 1-380 
 
 5 
 
 55 55 
 
 5 5 
 
 3 
 
 61-0 
 
 1-385 
 
 6 
 
 59 59 .... 
 
 
 6 
 
 59-0 
 
 1-787 
 
 7 
 
 Painted and varnished cast iron 
 
 59 
 
 3 
 
 59-0 
 
 1-449 
 
 8 
 
 99 99 
 
 
 6 
 
 63-0 
 
 1-469 
 
 9 
 
 Tallowed brass ..... 
 
 5 , 
 
 3 
 
 64-5 
 
 1-958 
 
 10 
 
 99 
 
 
 3 
 
 67-0 
 
 1-950 
 
 11 
 
 Cast iron ...... 
 
 5) 
 
 H 
 
 55-0 
 
 1-440 
 
 12 
 
 . 
 
 
 3 
 
 54-0 
 
 1-419 
 
 13 
 
 . 
 
 
 6 
 
 55-0 
 
 1-409 
 
 14 
 
 covered with fine sand 
 
 
 3 
 
 56-5 
 
 1-541 
 
 15 
 
 ,, ,, coarse sand . 
 
 > 9 
 
 1* 
 
 62-5 
 
 1-146 
 
 16 
 
 95 99 99 
 
 
 3 
 
 62-0 
 
 1-113 
 
 17 
 
 99 95 59 
 
 55 
 
 6 
 
 62-0 
 
 1-387 
 
 2 
 
 Clean polished brass 
 
 0-8488 
 
 3 
 
 53-0 
 
 1-459 
 
 18 
 
 59 99 
 
 
 3 
 
 52-0 
 
 1-785 
 
 16 
 
 Cast iron covered with coarse sand . 
 
 
 3 
 
 62-0 
 
 1-113 
 
 19 
 
 99 99 99 95 
 
 5 
 
 
 53-0 
 
 1-086 
 
 2 
 
 Clean polished brass 
 
 0-8488 
 
 3 
 
 53-0 
 
 1-459 
 
 20 
 
 59 55 
 
 0-6353 
 
 3 
 
 62-0 
 
 2-816 
 
 21 
 
 99 59 
 
 0-4320 
 
 3 
 
 54-0 
 
 5-230 
 
 22 
 
 Clean polished brass 
 
 0-8488 
 
 3 
 
 41-2 
 
 1-935 
 
 2 
 
 99 9 ) 
 
 5j 
 
 3 
 
 53-0 
 
 1-459 
 
 23 
 
 99 55 
 
 
 3 
 
 70-4 
 
 1-984 
 
 24 
 
 99 95 .... 
 
 5 5 
 
 3 
 
 130-5 
 
 2-840 
 
 2 
 
 Clean polished brass 
 
 0-8488 
 
 3 
 
 53-0 
 
 1-459 
 
 25 
 
 55 95 
 
 " 
 
 3 
 
 59-5 
 
 1-383 
 
 Remarks 
 
 2. "Water a little coloured. 4. Water not quite clear. 5. "Water a little 
 
 coloured. 
 9. The surface of the tallow on the disc seemed to alter a little during 
 
 immersion. 
 11, 12, 13. Cast iron a little rusty. 
 
 14. Disc coated with white lead and varnish, and covered with fine sand. 
 
 Surface about as rough as ashlar stone. 
 
 15, 16, 17. Sand-coated cast-iron disc, the sand very coarse, and mixed with 
 
 small gravel pebbles. 
 
VII 
 
 FLUID FEICTION 
 
 141 
 
 ON ROTATING Discs 
 
 Highest 
 
 Mean 
 
 
 Mean Friction per 
 
 
 Speed in 
 
 Value of 
 
 Mean 
 
 Value of c : Square Foot 
 
 
 Rotations 
 
 n for each 
 
 Value 
 
 corrected at 10 Feet 
 
 
 
 per 
 
 kind of 
 
 ofc. 
 
 to 60 i per Second 
 
 
 Second. 
 
 Surface. 
 
 
 Fahr. j =/10 
 
 
 5-875 
 
 1-85 
 
 0-1102 
 
 0-1089 
 
 0-2018 
 
 
 4-501 
 
 H 
 
 0-1149 
 
 0-1130 
 
 0-2093 
 
 
 5-531 
 
 n 
 
 0-1256 
 
 0-1241 
 
 0-2299 
 
 
 4-686 
 
 1-86 
 
 0-1169 
 
 0-1170 
 
 0-2182 
 
 
 5-382 
 
 n 
 
 0-1242 
 
 0-1245 
 
 0-2321 
 
 
 5-112 
 
 > 
 
 0-1329 
 
 0-1326 
 
 0-2473 
 
 
 4-892 
 
 1-94 
 
 0-1106 
 
 0-1103 
 
 0-2200 
 
 
 5-470 
 
 
 
 0-1160 
 
 0-1169 
 
 0-2331 
 
 
 4-237 
 
 2-06 
 
 0-0975 
 
 0-0986 
 
 0-2167 
 
 
 5-160 
 
 1-86 
 
 ... 
 
 ... 
 
 ... 
 
 
 5-010 
 
 2-00 
 
 0-1029 
 
 o-ioi7 
 
 0-2129 
 
 
 5-324 
 
 |j 
 
 0-1101 
 
 0-1085 
 
 0-2273 
 
 
 4-990 
 
 
 0-1176 
 
 0-1162 
 
 0-2432 
 
 
 4-456 
 
 2 ! 05 
 
 0-1572 
 
 0-1557 
 
 0-3395 
 
 
 3-300 
 
 1-91 
 
 0-3004 
 
 0-3019 
 
 0-5874 
 
 
 3-604 
 
 ii 
 
 0-3261 
 
 0-3277 
 
 0-6376 
 
 
 3-655 
 
 n 
 
 0-3658 
 
 0-3676 
 
 0-7153 
 
 
 4-501 
 
 1-85 
 
 O'll49 
 
 0-1130 
 
 0-2093 
 
 Chamber clean. 
 
 4-975 
 
 1-95 
 
 0-1235 
 
 0-1212 
 
 0-2436 
 
 Chamber coated with rough sand. 
 
 3-604 
 
 1-91 
 
 0-3261 
 
 0-3277 
 
 0-6376 
 
 Chamber clean. 
 
 2-735 
 
 2-17 
 
 0-3381 
 
 0-3325 
 
 07986 
 
 Chamber covered with coarse sand. 
 
 4-501 
 
 1-85 
 
 0-1149 
 
 0-1130 
 
 0-2093 
 
 } 
 
 7-598 
 
 ii 
 
 0-0324 
 
 0-0326 
 
 ... 
 
 J- Diameter varied. 
 
 7-849 
 
 
 
 0-0048 
 
 0-0048 
 
 
 J 
 
 5-668 
 4-501 
 5-630 
 5-133 
 
 1-85 
 it 
 ii 
 ii 
 
 0-1215 
 0-1149 
 0-1112 
 0-1003 
 
 ... 
 
 0-2251 
 0-2128 
 0-2061 
 0-1859 
 
 I Temperature varied. Friction 
 j- per square foot unconnected for 
 1 temperature. 
 
 4-501 
 
 1-85 
 
 0-1149 
 
 0-1130 
 
 0-2093 
 
 In water. 
 
 4-708 
 
 1-93 
 
 0-1195 
 
 ... 
 
 0-2-364 
 
 In syrup, sp. gr. 1*061. 
 
 Remarks 
 
 18, 19. The top and bottom of the chamber were coated with coarse sand, like 
 
 the disc in experiments 15, 16, 17. 
 20. The disc was slightly greasy. 
 22. About two pailfuls of ice placed in water outside the copper chamber. 
 
 24. Water taken from an engine boiler. It was rather dirty from sediment 
 
 produced by boiling. 
 
 25. Haifa hundredweight of sugar dissolved in water in the cistern. 
 
142 HYDKAULICS CHAP. 
 
 on+S -j.n+2 JTn+1 
 
 The experiments give directly the moment of friction M 
 corresponding to any speed N for each disc. But for any 
 given disc 
 
 M = cN w . . . . (6), 
 
 where c is a constant. Hence for any pairs of values of M 
 and N obtained in the experiments on a given disc, 
 
 logM^logMg 
 
 "logN 1 -logN 2 
 
 The mean value of n thus obtained is given for each of the 
 .surfaces tried. When the mean value of n has been obtained 
 from pairs of results in which the speed was different, values 
 of c for each speed were obtained by the formula 
 
 log c = log M - n log N, 
 
 and the mean values of c thus found are given in the table, 
 page 140. The values of n for different pairs of speeds never 
 varied very greatly for any given disc in like conditions, 
 nor did the values of c vary greatly for different speeds. 
 Further, the variations from the mean value followed no 
 regular law, so that they may be attributed to errors of 
 observation, or to unavoidable small fluctuations of speed 
 during the observations. 
 
 In the formulas above, / is the friction per square foot at 
 unit velocity, but for any given kind of surface in like 
 conditions 
 
 J ~ 
 
 Variation of resistance with diameter of disc. Three 
 sets of experiments with discs 0'8488, 0'6353, and 0'4320 
 foot virtual radius, rotating in the same chamber of fixed size, 
 gave moments of resistance in the ratios 
 
 1 : 0-2887 : 0'0425, 
 
 or for discs of different diameters in a chamber of constant 
 size the resistance varies as the (n+2'82)th power of the 
 
VII 
 
 FLUID FKICTION 143 
 
 radius. The theoretical formula above (4) is strictly applicable 
 to discs in chambers the linear dimensions of which are pro- 
 portional to the diameter of the disc, in which case the 
 resistances are as the (7i+3)th power of the radius. The 
 difference of the two cases is not very great, and is consistent 
 with the experimental result that the resistance with a given 
 disc is greater as the chamber is larger. 
 
 Influence of temperature on the resistance. The 
 four results with a bright brass disc, experiments 2, 22, 23, 
 and 24, show that the friction diminishes with unexpected 
 rapidity as the temperature increases. The diminution is 
 sensible even for a few degrees difference of temperature, and 
 hence it appears that a correction for temperature ought to 
 be introduced in experiments on the flow of water in pipes 
 and channels. The diminution between 41 and 130 Fahr. 
 is about 18 per cent, or 1 per cent for 5 increase of 
 temperature. 
 
 The experiments were not numerous enough to determine 
 exactly the law of variation of friction with temperature, and the 
 apparatus was not adapted for securing a constant temperature 
 during a prolonged experiment. The results agree fairly with 
 the empirical formula 
 
 ct = 0-1328(1 -0-00210 . . . (9), 
 
 where c t is the value of c for a bright brass disc at the 
 temperature t. 
 
 In the experiments 1 to 17 the temperature varied in 
 different instances from 53 to 62. The factor 
 
 1-0-0021 x 60 
 
 1-0-0021* 
 
 has been used to reduce the values of c to a standard tempera- 
 ture of 60. The correction is in any case small, and does 
 not affect the conclusions drawn from the results. 
 
 Influence of roughness of surface. The results of the 
 experiments are altogether in accord with those of Mr. Froude 
 as to the influence of the roughness of the surface. Even the 
 numerical values of the frictional resistance obtained in these 
 experiments differ very little from those obtained by him for 
 
144 
 
 HYDEAULICS 
 
 CHAP. 
 
 long surfaces. Taking Mr. Froude's results for planks 50 feet 
 long, and comparing them with those obtained in the present 
 experiments, the resistances in pounds per square foot at 
 1 feet per second are : 
 
 MR. FROUDE'S EXPERIMENTS. 
 
 Tinfoil surface . 
 Varnish . 
 Fine sand 
 Medium sand , 
 
 0-232 
 0-226 
 0-337 
 0-456 
 
 PRESENT EXPERIMENTS. 
 
 Bright brass . 0-202 to 0-229 
 
 Varnish . . 0-220 0-233 
 
 Fine sand . . 0*339 
 
 Very coarse sand . 0-587 0*715 
 
 Power of the velocity to which resistance is pro- 
 portional. There is in this also a remarkable agreement 
 between the present experiments and those of Mr. Froude. 
 For the smoother surfaces the resistance varies as the 1*8 5th 
 power of the velocity; for the rougher surfaces as a power 
 of the velocity ranging from 1*9 to 2*1. Mr. Froude's results 
 are precisely the same. 
 
 Influence of the size of chamber on the resistance. 
 In all these experiments, without a single exception, the 
 friction of the disc increased when the chamber in which it 
 rotated was made larger. The author is disposed to attribute 
 this to the stilling of the eddies by the surface of the stationary 
 chamber. The stilled water is fed back to the surface of the 
 disc, and hence the friction depends not only on its own surface, 
 but on that of the open chamber in which it rotates. The 
 discs were rotated in chambers 3, 6, and 12 inches deep, and 
 the surfaces of these chambers would be about 1000, 1200, 
 and 1600 square inches. In the larger chambers the kinetic 
 energy of the water may be supposed to be more rapidly 
 destroyed than in the smaller, in consequence of the larger 
 area of stationary surface. The water being more rapidly 
 stilled, and the stilled water fed back to the disc in greater 
 quantity, the resistance of the disc is increased. 
 
 Effect of roughening the surface of the chamber. In 
 experiments 18 and 19 the upper and lower surfaces of the 
 chamber were covered with coarse sand. Koughening the 
 surface of the chamber materially increased the friction of 
 the disc. This may be explained in precisely the same way 
 as increase of friction due to increasing the size of the 
 chamber. 
 
vii FLUID FKICTION 145 
 
 PROBLEMS 
 
 1. The resistance of a ship is 1 Ib. per square foot of immersed surface 
 
 at 10 knots. Find the H.P. required to drive a ship having 
 8000 square feet of immersed surface at 15 knots. One knot 
 = 6086 feet per hour. 829'9. 
 
 2. The disc-shaped covers of a centrifugal pump are 2 feet diameter 
 
 outside and 1 foot diameter inside. Find the work expended 
 in friction in rotating the pump at 360 revolutions per minute 
 /= 0-0025, and n = 2. 326 ft.-lbs. per second. 
 
 10 
 
CHAPTEK VIII 
 
 FLOW IN PIPES 
 
 83. Non- sinuous motion of water. When water from a 
 reservoir which has been at rest long enough for eddies to die 
 out issues from a sharp-edged orifice, the stream is perfectly 
 clear and smooth on the surface even at high velocities. Any 
 disturbance of the water in the reservoir shows itself in 
 striation of the jet due to the presence of eddies disturbing 
 the stream-line motion in the jet. The jet from a cylindrical 
 mouthpiece is always troubled from the formation of eddies 
 at the inner edge. In capillary tubes, which have been 
 experimented on by Poisseuille and others, the motion is 
 generally non-sinuous and free from eddies up to considerable 
 velocities. But in ordinary water mains the motion is gener- 
 ally sinuous and turbulent. 
 
 Professor Osborne Eeynolds investigated the conditions in 
 which sinuous and non-sinuous motion occurred in pipes 
 (Trans. Eoy. Soc. 1884). A steady stream of water was set 
 up through a glass tube with a flared mouth so that there 
 was no inlet disturbance. Into the stream a small jet of 
 coloured liquid was introduced. 
 
 So long as the velocity was low enough the coloured water 
 showed as a straight undisturbed stream line flowing through 
 the tube with the other water. If the velocity was raised 
 there came a point at which the coloured liquid suddenly 
 mingled with the rest of the water, and on viewing the water 
 by an electric spark it was seen that the water contained a 
 mass of more or less distinct coloured curls or eddies. With 
 water at constant temperature and the tank as still as possible 
 the critical velocity at which the stream lines broke up and 
 
 146 
 
CHAP, vni FLOW IN PIPES 147 
 
 eddies were formed varied almost exactly inversely as the 
 diameter of the pipe and directly as the viscosity. Very small 
 disturbing causes, such as a disturbance of the water in the 
 tank or fine sediment in the water, caused the break-up to 
 occur at lower velocities. Hence the critical velocity deter- 
 mined in this way is the higher limit of stable stream-line 
 flow in pipes. The coefficient of viscosity for water decreases 
 as the temperature rises, and is given by the equation 
 
 0-017 
 77 ~ 1 + 0-034* + 0-00023* 2 
 
 where t is the temperature centigrade. The denominator of 
 this fraction may be termed the relative fluidity, and will be 
 denoted by/. 
 
 The higher critical velocity as determined by Osborne 
 Eeynolds by the colour-band method is given by the equation 
 
 v e = 0-2458 jj ft. per sec. . (2), 
 
 where d is the diameter of the pipe in feet. 
 
 HIGHER CRITICAL VELOCITY 
 
 d= J 1 -lj 2 inches 
 
 d= -0417 '0833 -1250 -1667 feet 
 t; c atOC= 5-90 2-95 1'97 1 -4 7 ft. per sec. 
 
 Later experiments by Professor Coker, Mr. Clement, and 
 Mr. Barnes have shown that under certain favourable con- 
 ditions stream-line flow may subsist to considerably higher 
 velocities than those observed by Keynolds, and throw a little 
 doubt on the law that the higher critical velocity varies 
 inversely as the diameter. 1 
 
 In another series of experiments Osborne Keynolds 
 allowed water initially disturbed to flow through a long 
 smooth pipe. It was found that if the velocity was below a 
 certain limit the disturbances died out in a short length of 
 the pipe, and the motion then became non-sinuous. Measur- 
 ing the resistance to flow in a length of the pipe beyond the 
 disturbed part, it was found that when the motion was non- 
 sinuous the resistance varied very exactly as the velocity, but 
 
 1 Trans. Royal Society, 1903. Proceedings Royal Society, vol. btxiv. 
 
148 HYDEAULICS CHAP. 
 
 that when the motion was turbulent it varied as the 
 1'7 2th power of the velocity, or nearly as the square of the 
 velocity. If the velocity in the pipe is slowly increased, the 
 point at which the eddies cease to die out and there is 'a 
 deviation from the law that the resistance varies as the 
 velocity can be observed, and this velocity may be termed the 
 lower critical velocity. This also was found to vary inversely 
 as the diameter of the pipe and directly as the viscosity. 
 The lower limit of critical velocity found by Osborne Eeynolds 
 is given by the equation 
 
 v c = 0'03 8 7 T h. per sec. . . (2a). 
 jd 
 
 LOWER CRITICAL VELOCITY 
 
 d= J 1 Ij 2 inches 
 
 d= 0-0417 0-0833 0-1250 0-1667 feet 
 v c at 0C= -928 -465 -310 -232 ft. per sec. 
 
 Later experiments by Professor Coker and Mr. Clement 
 gave the relation 
 
 v c = 0-0199 ft. per sec. . . (26), 
 
 or about half the values obtained by Osborne Eeynolds. The 
 reason of the difference has not been explained. 
 
 It will be seen that in somewhat wide limits for small 
 pipes the motion may be sinuous or non-sinuous, but that 
 above the lower limit very small causes of disturbance render 
 the motion turbulent. Practically, for the larger pipes and 
 the velocities with which an engineer has to deal, the motion 
 is always turbulent. 
 
 Let d be the diameter of a horizontal pipe, and p the 
 difference of pressure in a length I ; the velocity of flow when 
 the motion is in rectilinear stream lines is given by the 
 relation 
 
 19. 
 
 (3), 
 
 where p is in grams per square centimetre and the units are 
 C.G.S. units. A more convenient form is this. Let h be the 
 difference of pressure in a horizontal pipe in a distance I 
 measured in feet of liquid of density p. 
 
vin FLOW IN PIPES 149 
 
 v = 1711-, centimetre units, 
 
 = 52150^^ foot units. 
 
 Taking for water p = 0'999 and for mercury p = 13'6, then 
 for h in feet of water 
 
 62100-5*1 
 
 and for h in inches of mercury 
 
 . (4). 
 
 84. Practical theory of flow in pipes when the motion 
 is turbulent. In all ordinary cases with which the engineer 
 has to deal, the water has in addition to its forward motion of 
 translation a distributed eddying motion. It is beyond hope 
 to have a theory which will give rationally the velocity of 
 flow and discharge of pipes in such conditions. It is not only 
 that the eddying motion of the water is so complicated that 
 in the strict sense there is no exact theory, but in addition 
 one of the factors in any formula of flow must express the 
 exact roughness of the surface of the pipe on which the 
 production of eddies depends. There is no scientific measure 
 of roughness, and very small apparent differences in the 
 quality of the pipe surface cause considerable differences in 
 the resistance. 
 
 Permissible velocities in pipes. Theoretically any given 
 discharge can be obtained either by varying the pipe diameter 
 or the head producing velocity of flow, but practically the 
 range of discharge for a given pipe is much limited. If the 
 velocity in the pipe is small it must be of large size and 
 expensive. If great, it is difficult to obtain sufficient pressure 
 in the distant parts of a district supplied, in hours of large 
 consumption, and the risk to the mains from sudden variations 
 of flow, causing what is termed hydraulic shock, is great. A 
 fair rough rule for pipes used in town's supply is the follow- 
 ing. Let v be the velocity in a pipe of diameter d (foot 
 units), then 
 
150 HYDRAULICS 
 
 CHAP. 
 
 d = 6 9 12 18 24 36 inches 
 
 = 0-5 0-75 1-0 1-5 2 3 feet 
 
 v = 2-7 3-1 3-4 4-2 4-9 6 '3 feet per second. 
 
 Of course, cases occur where higher velocities can be 
 permitted. In short supply pipes to turbines, velocities of 
 7 to 10 feet per second are not unusual. The reason for 
 adopting somewhat lower velocities in small mains is that 
 otherwise the rate of fall of pressure would be excessive. 
 
 85. Steady flow in pipes of uniform diameter. If a 
 long pipe connects two reservoirs at different levels, water 
 will flow from the upper to the lower, and the conditions 
 being constant the velocity and rate of discharge will be 
 constant also. Steady flow being established, since the water 
 starts from rest and comes back to rest, the work of gravity 
 on the descending water is exactly balanced by the work of 
 the resistances, of which much the largest is fluid friction. 
 Let Q be the discharge in cubic feet per second, H the cross 
 section and d the diameter of the pipe, v the mean forward 
 velocity of the water. 
 
 Q = iv = d 2 v cubic feet per second . (5). 
 
 As the same quantity of water passes every section in 
 unit time the velocity must be the same, that is if we under- 
 stand by v the mean velocity of translation along the pipe. 
 In fact, the velocity is greater at the centre of the cross 
 section and less towards the sides of the pipe, and on this 
 general condition eddying motions are superposed. But the 
 mean velocity along the pipe is constant, and for simplicity 
 the complications must be disregarded. 
 
 The Chezy formula for flow in pipes. A very simple 
 theory furnishes an approximate formula which has been of 
 very great service in hydraulics, and which with tabulated 
 values of experimental coefficients is still employed more 
 generally than any other in hydraulic calculations. Let 
 Fig. 85 represent a short portion of a long pipe through 
 which water is steadily flowing. The water enters and 
 leaves at the same velocity, and consequently the work of 
 external forces must be equal to the work in overcoming 
 friction. Let dl be the length of the portion of pipe 
 
VIII 
 
 FLOW IN PIPES 
 
 151 
 
 considered, z and z + dz the elevations of the end sections 
 
 above any horizontal datum XX, p and p + dp the pressures 
 
 at the ends, ft the area of cross section, ^ the circumference, 
 
 and Q the discharge 
 
 per second. Then, 
 
 in passing through 
 
 the length dl, GQ 
 
 Ibs. of water descend *%^ '^"/ p-ty 
 
 a distance dz feet, 
 
 and the work of 
 
 gravity is 
 
 Z*dZ 
 
 Fig. 85. 
 
 a positive quantity 
 
 if dz is negative, and vice versa. The resultant pressure on 
 the two ends in the direction of motion is dp, and the work 
 of this pressure is 
 
 also positive if the pressure is decreasing along the pipe and 
 dp is negative. The only remaining force doing work on the 
 water is the frictional resistance. The area of the pipe surface 
 is %dl, and using the expression obtained above [ 79, eq. (2)] 
 and putting v for the velocity of the water the frictional 
 work is 
 
 or, since Q = h>, 
 
 a quantity always negative because it is work done against a 
 resistance. Adding these portions of work together and 
 dividing by GQ, 
 
 G 
 
 Integrating, 
 
 . (6). 
 
 Let A and B (Fig. 86) be two sections at distances l lt / 2 
 
152 
 
 HYDKAULICS 
 
 CHAP. 
 
 from any given point, so that the length of pipe now con- 
 sidered is L = Z 2 l lf and let p lf z 1 be the pressure and elevation 
 at A!, p 2 , z 2 , the same quantities at B. Then, if v is the mean 
 velocity along the pipe, 
 
 If pressure columns are introduced at A and B, the water 
 
 Fig. 86. 
 
 will rise to the levels C and D, such that AC = ^ and 
 
 P 
 BD=-p. It is assumed that the atmospheric pressure is the 
 
 same at C and D. In a very long pipe this might not be the 
 case. Consequently 
 
 . . (8). 
 
 The quantity h is the difference of free surface-level at the 
 two points of the pipe considered, and is termed the virtual 
 fall of the pipe. The quantity A/L is termed the virtual slope 
 of the pipe, and this will be denoted by i. The line CD 
 passing through the pressure - column tops is called the 
 hydraulic gradient. The quantity fl/^ which appears in this 
 and some other equations is termed the hydraulic mean 
 radius of the pipe, and will be denoted by ra. 
 
VIII 
 
 FLOW IN PIPES 153 
 
 The general equation for flow in pipes can now be written 
 more simply 
 
 2 fl h 
 
 For pipes of circular section and diameter d,m = fl/^ = d/4. 
 For such pipes the general equation of flow is 
 
 t = *L.= d l (9) 
 
 1 20 4 L 4 
 
 This equation, with a constant value for f, is the well-known 
 Chezy formula. It is still extremely useful if values of f, 
 varying with certain conditions, are used instead of a constant 
 value. 
 
 The following forms of this equation are useful in practical 
 applications. The virtual fall or head lost in the length L is 
 
 * = = 0-0622 feet . . (9a). 
 
 The velocity of flow is 
 
 = \A 2 4 E) = 4 ' 012 \A? B feet per sec - 
 
 The discharge is 
 
 Q = ^d 2 v = 3 ' 15 / v /(7f") cubic feet P er sec - 
 The diameter for a given discharge is 
 
 feet . . 
 
 The head lost for a given discharge is 
 
 h = 0-1008^ f 
 d b 
 
 A form of the equation which is in common use is this : 
 
 h = 0-1008^ feet . . . (9). 
 
 and by some writers this form only is termed the Chezy 
 equation. The constant c is given by the relation 
 
 y20 
 T 
 
154 
 
 HYDEAULICS 
 
 CHAP. 
 
 86. Case of a pipe connecting two reservoirs. Inlet 
 resistance taken into account. Let Fig. 8 7 represent a pipe 
 connecting two reservoirs at different levels. If the reservoir 
 levels are constant the velocity in the pipe and the rate of 
 discharge are constant. The total head causing flow is the 
 difference of level H, and this is expended in three ways. 
 (1) To give the initial energy to the water corresponding to 
 the velocity v there must be expended a head v z /2g. At the 
 outlet of the pipe this kinetic energy is wasted in shock and 
 eddies, so that this is part of the head lost. (2) There is 
 some resistance due to the form of the inlet, which may be 
 written % Q v 2 /2g, where f = about 0*5 for a cylindrical inlet, and 
 
 Fig. 87. 
 
 about 0'05 if the inlet is bell-mouthed. (3) The friction in 
 
 the length L has been found to be t-r . ^ feet of head, 
 
 a v 
 
 eq. (9 a). Adding these together, 
 
 -{<"*#}. 
 
 (10), 
 
 an equation which should always be used for short pipes. 
 
 As a matter of fact, water mains are not straight but 
 curved, to follow the variations of level of the ground. Hence 
 their length is really greater than the horizontal projection, 
 and the hydraulic gradient is not strictly a straight line. 
 But in most practical cases the differences of level of the pipe 
 are so small compared with its length that there is no error 
 of practical importance in taking L to be the length of the 
 horizontal projection of the pipe, or in assuming the hydraulic 
 gradient to be straight. 
 
VIII 
 
 FLOW IN PIPES 
 
 155 
 
 87. Inlet Resistance. The inlet to a pipe may be flush 
 with the reservoir wall, as at A, Fig. 88 ; re-entrant and with 
 square edges, B; re-entrant with sharp edges, C; or bell- 
 
 ^^sss/f/ s7~) 
 
 Fig. 88. 
 
 mouthed, D. Values of the coefficient of resistance 
 1 + f are given in the following table : 
 
 Form of Inlet. 
 A 
 B 
 C 
 D 
 
 0-5 
 
 0-56 
 
 1-30 
 
 0-02 to 0-05 
 
 1-5 
 1-56 
 2-30 
 1-02 to 1-05 
 
 an( i 
 
 The inlet resistance is equivalent to the frictional resistance 
 of a length of pipe given by the equation 
 
 (11). 
 
 VALUES OF l Q /d. 
 
 f 
 
 l-Hfo = 
 
 1-05. 
 
 1-5. 
 
 1-56. 
 
 2-3. 
 
 005 
 0075 
 010 
 
 53 
 35 
 
 26 
 
 75 
 50 
 38 
 
 78 
 52 
 39 
 
 115 
 
 77 
 58 
 
 If this length is added to the actual length of the pipe 
 the inlet resistance will be allowed for. 
 
 In practical calculations about water mains the length L 
 is usually very large, and (1 + f X is small enough compared 
 with 4f L to be neglected. Thus let L = 1000 ft; d = 1-5 ft. ; 
 =0075; ? = 0-5; H=10 ft. The velocity, by eq. (10), 
 is 5 '4 7 ft. per sec., but if the inlet resistance is neglected the 
 
156 HYDKAULICS CHAP. 
 
 velocity is 5 '6 7. The error is here not immaterial, but if the 
 length of the pipe is 10,000 feet and H= 100, the velocity, 
 from eq. (10), is 5*65, and if the inlet resistance is neglected 
 the velocity is 5*6*7, where the difference is in practical cases 
 negligible. 
 
 88. Pressure in the pipe when the water is flowing. 
 The vertical from the pipe to the hydraulic gradient is the 
 pressure in the pipe at that point in feet of water, in excess 
 of atmospheric pressure. If h is the height to the gradient, 
 h + 34: feet is the pressure, including atmospheric pressure. 
 Hence there could not be negative pressure in the pipe unless 
 it rose more than 34 feet above the hydraulic gradient. With 
 negative pressure the flow would of course be interrupted. 
 But all ordinary water contains air, which would be disengaged, 
 and would interfere with flow if the pressure fell much below 
 atmospheric pressure. Hence, as a practical rule, pipes are 
 not laid so as to rise above the hydraulic gradient. Further, 
 at all anticlinal bends air valves are placed so that the air in 
 the pipe when it is being filled may escape, and also any air 
 carried into the pipe afterwards, which would accumulate at 
 the top of vertical bends and interrupt the flow. Unless the 
 pipe is below the hydraulic gradient these valves cannot act. 
 
 89. Darcy's experimental investigation of the resistance 
 to flow in pipes. 1 An extremely important series of measure- 
 ments of the flow in pipes with different heads was carried 
 out by M. H. Darcy, then Engineer of the Paris Water 
 Supply, under the auspices of the French Government. The 
 general bearing of the results may be stated thus : 
 
 (1) The frictional resistance varies considerably with the 
 nature and degree of roughness of the surface of the pipes. 
 This is in accordance with Froude's results already described, 
 80. 
 
 (2) The greater part of the experiments were made on 
 new and clean pipes, some of them asphalted. A few were 
 made on old and somewhat incrusted pipes. It was found 
 that the resistance of old and incrusted pipes was double that 
 of new and clean pipes. 
 
 (3) The simple Chezy formula 
 
 1 Recherches exptrimentales relatives au mouvement de I'eau dans les tuyaux. 
 Paris, 1857. 
 
VIII 
 
 FLOW IN PIPES 15*7 
 
 very well expressed the results of the tests, if special varying 
 values were given to the coefficient f 
 
 (4) The coefficient f varies with the velocity of flow, with 
 the diameter of the pipe, and with the roughness of the surface 
 of the pipe. As, for practical reasons, there is not a wide 
 variation of velocity in water mains, the dependence of f on 
 the velocity may be disregarded in most practical calculations, 
 On the other hand, the diameters of pipes range from 2 inches 
 to 6 inches, and the variation of f with the diameter is very 
 important. 
 
 Generally, at ordinary velocities and with cast iron or steel 
 pipes laid in the ordinary way, 
 
 D 
 
 (13), 
 where the constants have the following values : 
 
 ft 
 Drawn wrought-iron or clean 
 
 cast-iron pipes . . . . '00497 '084 
 Pipes altered by light incrusta- 
 tions -0100 -084 
 
 Or, in an easily remembered form, 
 Clean and smooth pipes, 
 
 Incrusted pipes, 
 
 [TABLE. 
 
158 
 
 HYDKAULICS 
 
 CHAP. 
 
 VALUES OF DEDUCED FROM DARCY'S FORMULA 
 
 Diameter of Pipe. 
 
 Values of . 
 
 In Inches. 
 
 In Feet. 
 
 New Pipes. 
 
 Incrusted Pipes. 
 
 4 
 
 0-333 
 
 00622 
 
 01252 
 
 5 
 
 0-417 
 
 00597 
 
 01202 
 
 6 
 
 0-500 
 
 00580 
 
 01168 
 
 7 
 
 0-583 
 
 -00568 
 
 01144 
 
 8 
 
 0-667 
 
 00560 
 
 01126 
 
 9 
 
 0-75 
 
 00553 
 
 01112 
 
 12 
 
 1-00 
 
 00539 
 
 01084 
 
 15 
 
 1-25 
 
 00530 
 
 01067 
 
 18 
 
 1-50 
 
 00525 
 
 01056 
 
 21 
 
 1-75 
 
 00521 
 
 01048 
 
 24 
 
 2-00 
 
 00518 
 
 01042 
 
 27 
 
 2-25 
 
 00515 
 
 01037 
 
 30 
 
 2-50 
 
 00514 
 
 01034 
 
 33 
 
 2-75 
 
 00512 
 
 01031 
 
 36 
 
 3-00 
 
 00511 
 
 01028 
 
 42 
 
 3-50 
 
 00509 
 
 01024 
 
 48 
 
 4-00 
 
 00507 
 
 01021 
 
 60 
 
 5-00 
 
 00505 
 
 01017 
 
 It may be noted that, except for pipes less than about 
 1 2 inches in diameter, the variation of is not very great, and 
 in many approximate calculations a constant value of may 
 be assumed without very large error. 
 
 (5) There is a variation of f with the velocity, and for 
 cases where the velocities were large Darcy proposed the 
 expression 
 
 and gave the following values for the constants (foot units), 
 for clean pipes : 
 
 a = 0-004346 
 0^ = 0-0003992 
 /* = 0-0010182 
 /?! = 0-000005205 
 
 No doubt Darcy underrated the importance of the influence 
 of velocity on the frictional resistance, and his formula taking 
 
VIII 
 
 FLOW IN PIPES 
 
 159 
 
 account of it is extremely inconvenient. It can be taken 
 into account in a simpler way, which will be given later. 
 
 90. Maurice Levy's formula for pipes. Darcy's experi- 
 ments were made on pipes not more than 20 inches in 
 diameter, and within that limit his formula has considerable 
 authority. M. Maurice Levy came to the conclusion, from 
 experience, that in the case of large pipes Darcy's formula 
 makes the resistance greater than it really is, and leads to 
 the use of pipes unnecessarily large. M. Levy, on partially 
 theoretical grounds, obtained the following formulae for metric 
 measures : 
 
 For new and clean cast-iron pipes, 
 0=36-4 <s/{ri(l+ 
 
 For pipes incrusted, 
 
 0=20-5 
 
 where r is the radius of the pipe. Eeducing to English foot 
 units and substituting the diameter for the radius, these 
 equations become : 
 
 For new and clean pipes, 
 
 =135(1 + 0-4 
 For incrusted pipes, 
 
 = 42-8(1 + 1-17 
 
 . (Ua). 
 
 Where in the Chezy formula, eq. (12), the value of f is : 
 
 For new and clean pipes, 
 
 0-007408 
 
 For incrusted pipes, 
 
 1 + 0-4 
 
 0-02335 
 + 1-17 J 
 
 . (16). 
 
 The following table gives values of f calculated by Levy's 
 rule for comparison with those of Darcy: 
 
160 
 
 HYDKAULICS 
 
 CHAP. 
 
 VALUES OF FROM LEVY'S FORMULA 
 
 Diameter of Pipe. 
 
 Values of 
 
 Inches. 
 
 Feet. 
 
 New Pipes. 
 
 Incrusted Pipes. 
 
 4 
 
 0333 
 
 00602 
 
 0139 
 
 5 
 
 0-417 
 
 00589 
 
 0133 
 
 6 
 
 0-500 
 
 00577 
 
 0128 
 
 7 
 
 0-583 
 
 00567 
 
 0123 
 
 8 
 
 0667 
 
 00558 
 
 0119 
 
 9 
 
 0-75 
 
 00550 
 
 0116 
 
 12 
 
 1-00 
 
 00529 
 
 0108 
 
 15 
 
 1-25 
 
 00512 
 
 0101 
 
 18 
 
 1-50 
 
 00497 
 
 0096 
 
 21 
 
 1-75 
 
 00485 
 
 0092 
 
 24 
 
 2-00 
 
 00474 
 
 0089 
 
 27 
 
 2-25 
 
 00463 
 
 0085 
 
 30 
 
 2-50 
 
 00454 
 
 0082 
 
 33 
 
 275 
 
 00445 
 
 0079 
 
 36 
 
 3-00 
 
 00438 
 
 0077 
 
 42 
 
 350 
 
 00424 
 
 0073 
 
 48 
 
 4-00 
 
 00412 
 
 0070 
 
 60 
 
 5-00 
 
 00391 
 
 0065 
 
 91. Later determinations of the values of ' 
 
 Imperfect as is the theory on which the Chezy formula is 
 based, it is so convenient that it will continue to be used 
 in engineering calculations. The difficulty in using it is the 
 uncertainty in choosing the proper value of in different cases. 
 In a wide range of cases in which the flow in pipes has been 
 measured by competent observers, f has varied from 0*003 to 
 0*016. Even in cases in many respects identical there is 
 considerable variation. Mr. Gale and Mr. Stearns both 
 measured the flow in asphalted cast-iron pipes 4 feet in 
 diameter, and found =0*0031 and 0*0051 respectively. 
 
 In 1886 the author examined all the more carefully 
 made experiments on flow in pipes, including those of Darcy. 
 By classifying pipes according to the quality and condition of 
 their surfaces the range of variation of f can be greatly 
 limited. Using a relation between v, d, and i which allows 
 for the influence both of diameter and velocity, which will 
 be explained in Chapter X., it was possible to tabulate values 
 of f for most of the conditions which arise in practice. 
 
VIII 
 
 FLOW IN PIPES 
 
 161 
 
 The following tables give the values of the coefficient 
 in the Chezy formula 
 
 for different kinds of pipe, of different diameters, and with 
 different velocities of flow, deduced in this way. 
 
 VALUES OF 
 
 When d in Feet is 
 
 For Velocities in 
 
 Feet per Second. 
 
 1-2. 
 
 2-3. 
 
 3-4. 
 
 4-5. 
 
 
 Clean Wrought-Iron Pipes. 
 
 0-5-0-75 
 
 0057 
 
 0050 
 
 0046 
 
 0043 
 
 0-75-1-0 
 
 0054 
 
 0047 
 
 0043 
 
 0040 
 
 1-0-1-5 
 
 0050 
 
 0043 
 
 0040 
 
 0037 
 
 1-5-2-0 
 
 0046 
 
 0040 
 
 0037 
 
 0035 
 
 2-0-3-0 
 
 0043 
 
 0038 
 
 0034 
 
 0032 
 
 3-0-4-0 
 
 0040 
 
 0035 
 
 0032 
 
 0030 
 
 
 Asphalted Cast -Iron Pipes. 
 
 0-5-0-75 
 
 0064 
 
 0059 
 
 0056 
 
 0054 
 
 0-75-1-0 
 
 0062 
 
 0057 
 
 0054 
 
 0052 
 
 1-0-1-5 
 
 0059 
 
 0054 
 
 0052 
 
 0050 
 
 1-5-2-0 
 
 0056 
 
 0052 
 
 0049 
 
 0047 
 
 2-0-3-0 
 
 0054 
 
 0050 
 
 0047 
 
 0045 
 
 3-0-4-0 
 
 0052 
 
 0048 
 
 0045 
 
 0043 
 
 
 New Cast-Iron Uncoated Pipes. 
 
 0-5-0-75 
 
 0058 
 
 0056 
 
 0055 
 
 0054 
 
 0-75-1-0 
 
 0054 
 
 0053 
 
 0052 
 
 0051 
 
 1-0-1-5 
 
 0051 
 
 0050 
 
 0049 
 
 0048 
 
 1-5-2-0 
 
 0048 
 
 0047 
 
 0046 
 
 0046 
 
 2-0-3-0 
 
 0046 
 
 0044 
 
 0043 
 
 0043 
 
 3-0-4-0 
 
 0043 
 
 0042 
 
 0041 
 
 0041 
 
 
 Incrusted Cast-Iron Pipes. 
 
 
 For all Velocities. 
 
 0-5-0-75 
 
 0119 
 
 0-75-1-0 
 
 0113 
 
 1-0-15 
 
 0107 
 
 1-5-2-0 
 
 0101 
 
 2-0-3-0 
 
 0095 
 
 3-0-4-0 
 
 0090 
 
 11 
 
162 
 
 HYDEAULICS 
 
 CHAP. 
 
 If an expression of the form adopted by Darcy is used, 
 then the results given above agree fairly closely with the 
 following values, 
 
 VALUES OF 
 
 Kind of Pipe. 
 
 Values of a for Velocities in Feet per 
 Second. 
 
 Values 
 of/3. 
 
 1-2. 
 
 2-3. 
 
 3-4. 
 
 4-5. 
 
 Drawn wrought iron 
 Asphalted cast iron 
 Clean cast iron 
 
 00375 
 00492 
 00405 
 
 00322 
 00455 
 00395 
 
 00297 
 00432 
 00387 
 
 00275 
 00415 
 00382 
 
 0-37 
 0-20 
 0-28 
 
 Incrusted cast iron 
 
 At all velocities a = 0-00855 
 
 0-26 
 
 These values show that, as was generally believed from 
 practical experience, the influence both of diameter and 
 velocity is greater than Darcy supposed. 
 
 92. Herschel's gaugings of flow in riveted steel pipes. 
 Mr. Clemens Herschel, between 1892 and 1896, made numerous 
 gaugings of flow in riveted mains of exceptionally large 
 diameter. The volume of flow was measured by the Venturi 
 meter, a method which may be regarded as very satisfactory. 
 The pipes were asphalted, and some were made with taper 
 lengths and others with cylinder lengths alternately large 
 and small. No very clear difference was found between the 
 two as regards resistance. Mr. Herschel has plotted his 
 results, taking velocities for ordinates, and values of c in the 
 equation v = c \fmi } where m is the hydraulic mean radius, as 
 abscissae. From the curves drawn through the plotted points 
 he has deduced values of c for various velocities. From these, 
 for comparison with the values of f in the tables above, the 
 following values for steel riveted pipes have been deduced : 
 
VIII 
 
 FLOW IN PIPES 
 
 163 
 
 VALUES OF FOR NEW STEEL RIVETED PIPES 
 
 Diameter in 
 
 For Velocities in Feet per Second. 
 
 
 
 
 
 
 
 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 48 
 
 0063 
 
 0055 
 
 0051 
 
 0050 
 
 0051 
 
 0052 
 
 48 
 
 0068 
 
 0064 
 
 0062 
 
 0059 
 
 0058 
 
 0058 
 
 42 
 
 0070 
 
 0056 
 
 0051 
 
 0051 
 
 0052 
 
 0053 
 
 42 
 
 0063 
 
 0059 
 
 0057 
 
 0056 
 
 0055 
 
 0055 
 
 36 
 
 0087 
 
 0071 
 
 0060 
 
 0053 
 
 0047 
 
 0042 
 
 Broadly, these results confirm the general law given above. 
 The value of f diminishes as the velocity increases, and increases 
 as the diameter diminishes. But there are anomalies. There 
 are several cases where f is greater at 6 feet per second than 
 at 4 feet per second. What is more anomalous still is that 
 the 4 8 -inch pipe at 6 feet per second has a greater coefficient 
 than the 3 6 -inch pipe. These anomalies must be due to 
 errors of observation. Further, as a whole, the coefficients are 
 somewhat larger than they might be expected to be. There 
 is a series by Darcy and one by Hamilton Smith on riveted 
 pipes which give smaller coefficients if the difference of 
 diameter is allowed for. However, of course, in comparing 
 these with the results on cast-iron pipes, the roughness due 
 to the rivet-heads and joints must be considered, and the 
 resistance can only be determined by direct experiment on 
 riveted pipes. 
 
 After some of these pipes had been in use four years some 
 further gaugings were made, and the discharge was found to 
 have diminished considerably. The following are coefficients 
 for the 4 8 -inch main, one set corresponding to the upper part 
 of the main near the supply reservoir, the other to the lower 
 part. 
 
 VALUES OF FOR OLD RIVETED STEEL PIPES 
 
 Diameter in 
 Inches. 
 
 At Velocities in Feet per Second. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 481 
 48 2 
 
 0106 
 
 0068 
 
 0080 
 0060 
 
 0075 
 0058 
 
 0073 
 0060 
 
 0072 
 0060 
 
 0072 
 0060 
 
 1 Supply reservoir to Porapton Notch. 
 
 2 Pompton Notch to service reservoir. 
 
164 HYDKAULICS CHAP. 
 
 It is clear, the author thinks, that during the four years 
 slimy deposits had accumulated in the main and increased the 
 resistance to flow. As would be expected, these were almost 
 entirely in the first length of main from the supply reservoir 
 to Pompton Notch. In the remainder of the main the 
 coefficients are not sensibly different from those obtained in 
 the previous gaugings. 
 
 Messrs. Marx, Wing, and Hoskins made gaugings in 1897 
 and in 1899, by a calibrated Venturi meter, of a remarkable 
 supply pipe 6 feet in diameter, part of which was riveted steel 
 and part of wood staves, at the Pioneer Electric Power 
 Company, Ogden, Utah (Trans. Am. Soc. of Civil Engineers, 
 xl. 471, and xliv. 34). The results on the steel part of the 
 pipe plotted in curves furnish the following values for f. 
 
 COEFFICIENT f FOR SIX-FOOT KIVETED STEEL PIPE 
 
 v= 1-0 1-5 2-0 2-5 3-0 4'0 5O 5'5 
 
 1897 gauging 
 
 =0053 -0052 -0053 '0055 -0055 -0052 
 1899 gauging 
 
 =0097 -0076 -0067 '0063 -0061 -0060 -0058 -0058 
 
 The increase of resistance with time is very marked at the 
 low velocities if the measurements at these can be trusted. 
 It seems probable, however, that in the earlier gauging the 
 resistance at low velocities was under-estimated, or the resist- 
 ance at high velocities over-estimated. 
 
 93. Timber stave pipes. In the western part of the 
 United States remarkable pipe lines have been constructed of 
 wood staves hooped with steel bands. The wood used is redwood 
 or sequoia, which when wet appears to have great durability. 
 The staves break joint, and at their ends a thin piece of steel 
 is jammed in a saw-cut. By slightly humouring the staves 
 bends of large radius are easily obtained. The staves are 
 usually 1 J inch thick, accurately shaped by machinery. The 
 steel hoops are spaced at different distances according to the 
 pressure, and are drawn tight by a screwed end and nut. 
 These pipes can be put together in difficult country where 
 transport of metal pipes would be very costly. 
 
 The results of the gaugings, by Messrs. Marx, Wing, and 
 Hoskins, of the part of the pipe at Ogden constructed of wood 
 
VIII 
 
 FLOW IN PIPES 
 
 165 
 
 staves'and 6 feet in diameter ( 92), gave the following values 
 for the coefficient f : 
 
 VALUES OP t FOB SIX-FOOT WOOD STAVE PIPE 
 
 
 
 v= 1-0 1-5 2-0 3-0 4-0 5-0 5-5 
 
 1897 gauging 
 
 =0064 -0053 -0048 -0043 -0041 
 1899 gauging 
 
 =0048 -0046 -0045 '0044 -0043 -0043 -0043 
 
 In these, as in the results on riveted pipe, there seerns 
 some doubt as to the accuracy of the observations at the 
 lowest velocity. The variation of f with velocity would be 
 expected to be greater than in the 1899 gaugings for 
 frictional surface as smooth as that of a wood pipe. 
 
 94. Fire hose pipes. Very careful experiments on the 
 discharge through fire hose have been made by Freeman 
 (Trans. Am. Soc. of Civil Engineers, xxi. 303). For 2^-inch 
 hose pipes of different makes the following were the values of 
 the coefficient obtained : 
 
 VALUES OP 
 
 
 Velocity in Feet per Second. 
 
 4. 
 
 6. 
 
 10. 
 
 15. 
 
 20. 
 
 Unlined canvas . 
 
 0095 
 
 0095 
 
 0093 
 
 0088 
 
 0085 
 
 Hough rubber-lined cotton . 
 Smooth . 
 
 0078 
 0060 
 
 0078 
 0058 
 
 0078 
 0055 
 
 0075 
 0048 
 
 0073 
 0045 
 
 Let a hose pipe of diameter d connect with a nozzle of 
 diameter 8 ; let I be the length of hose pipe, H the head of 
 water at inlet, v the velocity in hose pipe, V the velocity of 
 jet from nozzle, and Q the discharge. 
 
 The head expended at the nozzle is h = V 2 /(2#c 2 ), where e v 
 is the coefficient of velocity for the nozzle, which may vary 
 from 0-95 to 0'98. But V = ^/S*. Hence 
 
166 HYDKAULICS 
 
 CHAP. 
 
 The head producing flow in the hose pipe is H h, and 
 therefore 
 
 -h)d 
 
 v = 
 
 . . (17). 
 
 95. Practical calculations of flow in pipes. In the 
 
 following calculations it is assumed that there are no special 
 obstructions due to valves, bends, etc., and that the pipe is 
 so long that only the frictional resistance requires to be 
 taken into account. In long mains the resistance of ordinary 
 bends is negligible. The fundamental equations are : 
 
 - (i), 
 
 Q = -^ . . (3). 
 
 From these equations the following are easily derived, and 
 for convenience are repeated here from 85 : 
 
 . (26), 
 . (Sa), 
 
 0-632 . . . (4), 
 
 k = 0-1008^ . . (44). 
 
vin FLOW IN PIPES 167 
 
 Kough preliminary calculations can be made by the follow- 
 ing approximate formulae obtained by taking a fixed value of f. 
 They are least accurate for small pipes : 
 
 For new and clean pipes, 
 
 For old and incrusted pipes, 
 v = 40 
 Q = 31 
 
 yO 2 
 ?, 
 
 When the dimensions of a pipe are given and the velocity 
 and discharge are required there is no great difficulty. If 
 Darcy's value of f is used it can be found from eq. (1), 
 and the calculations are straightforward. If a value of f 
 depending both on the diameter and velocity is to be used, 
 an approximate value of v can be obtained from eq. (5) or (6), 
 and then the value of f can be selected from the tables and 
 v and Q re-calculated. There is rather more difficulty when 
 the discharge is given and the diameter is required. Some- 
 times from past experience an engineer can assign probable 
 values for d and v, or they can be found approximately by 
 eq. (5) or (6). Then f can be found from Darcy's formula or 
 from the tables, and a new value of the diameter calculated by 
 eq. (4). The engineer has to consider whether he will allow for 
 an increase of resistance as the pipe becomes old and incrusted. 
 The rate at which a pipe becomes rougher from corrosion 
 depends on the quality of the water. In some cases the 
 interior of the pipe remains clean for a long time. In some 
 other cases the corrosion is rapid. A common rule of thumb 
 to provide for corrosion is to calculate the diameter of pipe 
 required when clean, to add one inch, and choose the nearest 
 larger commercial size. 
 
 9 6. Tables of flow in pipes. Tables are published giving 
 the velocity and discharge of pipes of different diameters with 
 different heads. Generally these are calculated on a fixed 
 
168 HYDKAULICS CHAP. 
 
 value of f, and the results are therefore only approximate. 
 They are of assistance, however, in settling pipe proportions. 
 The following may be mentioned : 
 
 (1) Hydraulic and other Tables, by Thomas Hennell, Spon, 
 1884. This is based on the Chezy formula. 
 
 (2) Tables for Calculating the Discharge of Water in Pipes, 
 by A. E. Silk, Spon, 1899. Based on a modified Darcy formula. 
 
 (3) Diagrams of Pipe Discharge, by E. B. and G. M. Taylor, 
 Batsford. These are based on Kutter's formula with a rough- 
 ness coefficient 0*013. 
 
 (4) Tables for the Solution of Ganguillet and Kutter's 
 Formula, by CoL E. C. S. Moore, RE., Batsford. 
 
 (5) Mr. E. 0. W. Eoberts has designed a very convenient 
 small circular slide-rule for facilitating calculations on flow in 
 pipes. The graduations are based on Kutter's formula. The 
 slide-rule is made by Mr. Gr. Kent of Holborn. 
 
 97. Secondary losses of head in pipes. In very long 
 mains the so-called skin friction or resistance of the pipe 
 surface, which is determined by the equations in 95, is so 
 large compared with any other losses that the latter are dis- 
 regarded in ordinary practical calculations or covered by 
 assuming a rather larger value of f. It is, however, sometimes 
 necessary to consider these smaller losses, especially in the 
 case of comparatively short mains. The inlet loss has already 
 been considered ( 87), and can be taken into the reckoning 
 without difficulty. The other losses due to changes of diameter 
 
 of the pipe, changes of direction, 
 valves, etc., are generally of the 
 nature of losses by shock. All losses 
 are properly ultimately due to fluid 
 friction, but it is rather convenient to 
 speak of these losses as shock losses, 
 as distinguished from the skin friction 
 losses previously discussed. 
 
 Abrupt enlargement of section. 
 Let d lt <o lf KI be the diameter, 
 Fi 8- 89 - section, and velocity in the narrower, 
 
 and d 2 , o> 2 , v 2 the same quantities in 
 
 the wider part of the pipe, Fig. 89. The head lost in shock 
 ( 36) is 
 
o c 
 UNIVERSITY 
 
 OF 
 
 VIII 
 
 But 
 
 FLOW IN PIPES 
 
 169 
 
 *$ - a*), 
 
 where f e is a coefficient depending only on the ratio of the 
 sections or diameters at the enlargement. 
 
 ^ = 1-2 1-5 1-7 2-0 3-0 
 
 - = 1-1 1-22 1-30 1-41 1-73 
 & = 0-04 0-25 0-49 I'OO 4'00 
 If p lf p 2 are the pressures in the two parts of the pipe, 
 .P2-.Pi V 
 
 Abrupt contraction of section. When a stream passes 
 from a larger to a smaller section abruptly 
 a contraction is formed at aa (Fig. 90), 
 and the stream then enlarges to fill the 
 pipe, eddies being formed as at an abrupt 
 enlargement. Let co be the section, and T; 
 the velocity, where regular steady motion 
 is re-established. At the contraction aa the Fig. 90. 
 
 section of the stream is c c a) and the velocity 
 is v/c c , where c c is a coefficient of contraction. Then the head 
 lost in turbulent motion is 
 
 = ic~ (19), 
 
 where . is a coefficient. If c c = 0'63, as in a free jet, = 0'345. 
 
170 
 
 HYDKAULICS 
 
 CHAP. 
 
 The value of the coefficient is not well ascertained. Weisbach 
 obtained as the result of experiments the empirical relation 
 
 0-077 
 
 e- 
 
 (20). 
 
 For a quite sharp edge at the change of section c c = 0'62 to 
 0'64. For a rounded edge c c = 0'7 to 0'8. 
 
 Gradual enlargement. The resistance in this case can 
 only be ascertained by experiment. Fliegner found the head 
 lost to be (Fig. 91) 
 
 Elbows. The loss of head at elbows appears to be due 
 
 Fig. 91. 
 
 Fig. 92. 
 
 to the formation of a contraction and abrupt increase of 
 section (Fig. 92). Weisbach, from experiments on a very 
 small pipe, obtained the expression 
 
 = 0'95 
 
 2-05 
 
 = 20 
 
 40 
 0-14 
 
 60 
 0-37 
 
 80 
 075 
 
 90 
 1-0 
 
 100 
 1-27 
 
 . (22). 
 
 120' 
 
 1-87 
 
 This is a loss additional to the pipe friction in the parts 
 constituting the elbow. 
 
 98. Resistance at bends. Till lately the resistance at 
 bends has been supposed to be a shock loss due to contraction 
 and abrupt enlargement of the stream at the bend. On this 
 hypothesis, and using the results of some experiments on small 
 
VIII 
 
 FLOW IN PIPES 171 
 
 bends, Weisbach found the following empirical expression for 
 the head lost at a bend (Die experimental Hydraulik, p. 156). 
 Let 6 be the angle subtended by the bend at the centre 
 of curvature in degrees, v the velocity, r the radius, and d the 
 diameter of the pipe, and E the radius of curvature measured 
 to the centre line of the bend. Then p r/K is the curvature. 
 The head lost is 
 
 , . (23). 
 
 = 0-131 + 
 
 p = 0-l 0-2 0-3 0-4 0-5 0-6 0'7 0'8 0*9 1-0 
 
 H/d= 5 2-5 1-67 1-25 1-0 '83 '71 '62 -55 -5 
 
 f& = -13 -14 -16 -21 -29 -44 -66 '98 1-41 1-98 
 
 No great confidence has been placed in these results, as 
 they are based on very limited and small experiments. 
 Recently Mr. Alexander (Proc. Inst. Civil Engineers, clix. 341) 
 has made some very careful experiments on small varnished 
 wood bends (d = 1^- inch) with considerable variation of radius 
 of curvature and velocity of flow. In spite of the small scale 
 of these experiments they throw some light on the nature of 
 the resistance at bends. The most important point is this, 
 that the total resistance at a bend is made up of the skin 
 resistance of a straight length of pipe of the same length as 
 the bend, and an additional resistance due to the curvature 
 which is not a shock resistance but merely an augmentation of 
 the skin friction. Hence the total resistance at a bend can 
 be expressed by the relation 
 
 H-fc^Jfeet - (24), 
 
 where I is the length of the bend measured along its centre 
 line, and d the diameter of the pipe. It appeared in the 
 experiments that the resistance per foot length of bend did 
 not regularly decrease with the curvature, but was a 
 minimum when p = 5, or when the radius of curvature was 
 2|- times the diameter of the pipe. Mr. Alexander has given 
 some empirical expressions for loss of head at bends, but they 
 are inconvenient, and it is sufficient for practical purposes to 
 proceed in a simpler way. Assuming the result that the 
 
172 HYDKAULICS CHAP. 
 
 bend resistance is merely an augmented skin friction resistance, 
 so that it can be expressed by the equation (24), the value 
 of f 6 may be found from such experiments as are available. 
 The most valuable experiments are some by Messrs. Williams, 
 Hubbell and Fenkel, on large bends of asphalted cast iron, and 
 of these the best are on bends in pipes of 30 inches in 
 diameter (Proc. Am. Soc. of Civil Engineers, xxvii. 314). 
 The coefficients are deduced for right-angled bends in which 
 Z = 7rK/2. For any other bends the resistance will be 
 proportional to the angle subtended at the centre of curvature, 
 so that if /! is the length of such a bend the coefficient will 
 be greater or less than those given below in the ratio l-^jl. 
 
 VALUES OF BEND COEFFICIENT fa FOR RIGHT-ANGLED BENDS 
 
 Weisbach, small pipes. 
 
 p = -025 -05 -1 -17 -25 -33 0'5 
 
 R/d = 20 10 5 32 1-5 1-0 
 
 fa = -001 -002 -004 -008 -012 -018 -046 
 
 Williams, Hubbell and Fenkel, 30-inch main. 
 
 p = -021 -031 -050 -083 -125 -21 
 
 R/d = 24 16 10 6 4 2-4 
 
 fa = -009 -0092 -0118 -015 -0155 -018 
 
 For small values of the curvature the coefficient of 
 resistance of Weisbach's small pipes is much less than that of 
 the 30 -inch pipe, but for large values of the curvature it is 
 not very different. It may be suspected that for the small 
 pipes with small curvature the motion of the water was 
 possibly approximately non-sinuous. 
 
 The results may be put in another way. Let l t be the 
 length of a straight pipe the resistance of which is equal to 
 that of a right-angled bend of length I along the centre line. 
 Then if % is the proper coefficient corresponding to the 
 diameter, velocity, and roughness in the ordinary formula for 
 pipe friction, 
 
 /ILi-X 1* 
 C 20 d ^ b 2g d ' 
 
 *! = #/ - - (25). 
 
 Taking f= 0*005 for a 30-inch asphalted pipe, the 
 lengths equivalent to a right-angled bend are as follows : 
 
VIII 
 
 FLOW IN PIPES 
 
 173 
 
 />=-021 
 
 031 
 
 050 
 
 083 
 
 125 
 
 21 
 
 E/d= 24 
 
 16 
 
 10 
 
 6 
 
 4 
 
 2-4 
 
 ljl= 1-8 
 
 1-84 
 
 2-36 
 
 3-0 
 
 3-1 
 
 3-6 
 
 1= 94 
 
 63 
 
 39 
 
 24 
 
 16 
 
 9 
 
 Z 1= 169 
 
 115 
 
 92 
 
 72 
 
 49 
 
 32 
 
 -J = 75 
 
 52 
 
 53 
 
 48 
 
 33 
 
 33 
 
 feet. 
 
 It cannot be said that knowledge of the resistance at 
 bends is satisfactory ; more experiments on an adequate scale 
 are necessary. But it is fairly certain that the additional 
 resistance at a bend over that of a straight pipe of equal 
 length is not, for practical calculations, a very large or serious 
 quantity when the resistance of long mains is in question. 
 
 99. Valves, cocks, and sluices. These contract the 
 section of the pipe, and there is a further contraction of the 
 stream passing the sluice, and an abrupt enlargement of the 
 section of the stream causing loss of head by shock. The loss 
 of head may be expressed by the relation 
 
 <y 2 
 *. = 6 2l (26), 
 
 where v is the velocity in the pipe beyond the sluice where 
 regular motion is re-established. 
 
 Pipe of rectangular section. Section at the sluice, ci ; 
 in pipe beyond the sluice, CD. 
 
 _i : 
 
 = 1-0 
 
 9 0-8 0-7 0-6 0-5 
 
 CO 
 
 
 
 f = 
 
 = o-o o- 
 
 09 0-4 0-95 2-08 4-02 
 
 : 
 
 = 0-4 
 
 0-3 0-2 0-1 
 
 (U 
 
 
 
 f.- 
 
 = 8-12 
 
 17-8 44-5 193-0 
 
 
 
 j 
 
 
 
 
 
 
 0^ 
 
 
 
 f 4_ 
 
 Fig. 93. 
 
 Fig. 94. 
 
 Sluice in cylindrical pipe. Let p = A/H be the ratio of 
 height of opening to the diameter of the pipe. 
 
174 HYDEAULICS CHAP. 
 
 P-l-0 ! I I i i i i 
 
 ^=1-0 0-95 0-86 0-74 0'61 '47 '32 -16 
 to 
 
 f s = OO 0-07 0-26 0-81 2-1 5-5 17*0 97'8 
 
 Some experiments by Kuichling on a 24-inch sluice in a 
 cast-iron main gave the following results : 
 
 p = 0-66 0-60 0-50 0-37 0'25 0-18 
 f s = 0-8 1-6 3-3 8-6 22-7 41-2 
 
 It will be seen how very largely the pressure beyond the 
 sluice is reduced when the valve is much closed. The form of 
 the valve casing has a good deal of influence on the resistance. 
 With various forms of casing the resistance when the valve or 
 sluice is full open may amount to from two to sixteen times 
 
 100. Flow in a main in which there are secondary 
 resistances. The equation for the velocity of flow becomes 
 too cumbrous if expressions for the secondary resistances are 
 inserted. It is best to proceed by approximation. Let H be 
 the total head in the length I. Then taking account only of 
 the inlet resistance and skin friction an approximate value of 
 the velocity v can be found from the equation 
 
 Knowing this approximate velocity, the losses of head due 
 to the secondary resistances can be calculated. Let h = the 
 sum of these losses. Then a more approximate value of v can 
 be found from the equation 
 
 PROBLEMS. 
 
 1. Find an expression for the relative discharge of a square and a 
 
 circular pipe of the same section and slope. 1-062 to 1. 
 
 2. A pipe is 6 inches in diameter, and is laid for a quarter of a mile at 
 
 a slope of 1 in 50 ; for another quarter of a mile at a slope of 
 1 in 100 ; and for a third quarter of a mile is level. The 
 surface of the supply reservoir is 20 feet above the inlet, and 
 that of the lower reservoir 9 feet above the outlet. Using 
 Darcy's coefficient for clean pipes, find the discharge. Also 
 
VIII 
 
 FLOW IN PIPES 175 
 
 draw the hydraulic gradient and mark the pressure in the pipe 
 at each quarter mile. 0'824 cubic foot per second. 
 
 3. A pipe, 2000 feet long, discharges Q cubic feet per second. Find 
 
 how much the discharge would be increased, if for the last 
 1000 feet a second equal pipe was laid alongside the first, and 
 the water allowed to flow equally through both. Show by a 
 sketch how the hydraulic gradient would be altered. 
 
 Ratio of discharge ^ to /s/5. 
 
 4. A reservoir, the level of which is 50 feet above datum, discharges 
 
 into a reservoir 30 feet above datum through a 12 -inch pipe 
 5000 feet in length. Using Darcy*s coefficient for clean pipes, 
 find the discharge. 2 '7 10 cubic feet per second. 
 
 5. The levels of the pipe in the last question are : at the upper 
 
 reservoir, 40 feet ; at 1000 feet, 25 feet ; at 2000 feet, 12 feet ; 
 at 3000 feet, 12 feet; at 4000 feet, 10 feet; at the lower 
 reservoir, 15 feet above datum. Sketch the line of hydraulic 
 gradient, and write down the pressure in the pipe at each of 
 these points. 
 
 6. A pipe, 9 inches in diameter, connects two reservoirs one mile 
 
 apart, the water -surf aces being 100 feet and 47 feet above 
 datum. Using Darcy's coefficient for in crusted pipes, find the 
 velocity and discharge. 
 
 3 - 3 feet per second ; 1*46 cubic feet per second. 
 
 7. A pipe is 1500 feet long and 6 inches in diameter. It is to 
 
 discharge 50 cubic feet of water per minute. Find the loss of 
 head in friction and virtual slope. Use Darcy's coefficient for 
 clean pipes. 19'6 feet; 0-013. 
 
 8. What is the head lost per mile in a pipe 2 feet diameter discharg- 
 
 ing 6,000,000 gallons in 24 hours : (a) when new ; (6) when 
 incrusted. 10'7 feet ; 21'5 feet. 
 
 9. A pipe is to supply 30,000 gallons per hour. The available head 
 
 is 80 feet per mile. Find the velocity and diameter (a) from 
 the approximate formula ; (6) from the tabular value of 
 corresponding to the approximate velocity and diameter. 
 
 571 foot and 5-2 feet per second. 
 577 foot and 5-10 feet per second. 
 
 10. A water main has a virtual slope of 1 in 850, and discharges 
 
 35 cubic feet per second. Find the velocity and diameter 
 (a) approximately by assuming = 0-0064 ; (6) more accurately 
 by selecting a coefficient from the tables for asphalted cast iron. 
 
 3-68 feet and 3-3 feet per second. 
 
 3-426 feet and 3*80 feet per second. 
 
 11. It is required to discharge water from a reservoir through a 
 
 horizontal pipe 6 inches diameter and 50 feet long. Head over 
 inlet 20 feet Find the discharge, taking into account the 
 inlet resistance. f= 0-0075. The inlet is flush with the 
 reservoir. 3-32 cubic feet per second. 
 
 12. Find the diameter of a new cast-iron pipe, having a fall of 10 feet 
 
 per mile, capable of delivering water with a velocity of 3 feet 
 per second. 0-1359 foot. 
 
176 HYDKAULICS CHAP, vm 
 
 13. A pipe 12 inches in diameter and 1 mile in length delivers water 
 
 from one reservoir to another with a difference of level of 
 60 feet. The surface area of the lower reservoir is 10,000 
 square feet, and the water-level is observed to be rising at the 
 rate of 11 inches per hour. Find the coefficient of friction 
 of the pipe. 0-0174. 
 
 14. A hydraulic main is 6 inches diameter; the velocity is 3 feet 
 
 per second ; and the pressure is 700 Ibs. per square inch. 
 What is the gross horse-power transmitted. 108 H.P. 
 
 15. Supposing the hydraulic main in the last question to be clean 
 
 cast iron, find the loss of pressure in pounds per square inch 
 per mile, and the percentage of the energy transmitted wasted 
 in friction. 14 '1 Ibs. per square inch ; 2-01 per cent. 
 
 16. A horizontal pipe is in three sections, each of 1000 feet in length, 
 
 and of diameters 10 inches, 12 inches, and 15 inches respec- 
 tively. The discharge is 5 cubic feet per second. Taking the 
 coefficient =0'01, find the loss of head in friction in each 
 length, and the change of pressure at each abrupt change of 
 diameter. Friction = 6 -90, 3'96, and 2-03 feet. 
 
 Pressure change, 0'555 and 0*288 foot. 
 
 1 7. Taking the pressure at the inlet of the pipe in the last question to 
 
 be 25 feet, draw the hydraulic gradient with a vertical scale 
 fifty times the horizontal. 
 
 18. A pipe connects two reservoirs 1000 feet apart with a difference 
 
 of surface-level of 20 feet. If a sluice at the outlet into the 
 lower reservoir is partially closed so that the discharge is 
 reduced to one-half, what will be the change in the hydraulic 
 gradient ? 
 
CHAPTEK IX 
 
 DISTRIBUTION OF WATER BY PIPES 
 
 101. Town supply. The amount of water supplied per head 
 in different towns varies very greatly. For ordinary domestic 
 purposes 12 gallons per head per day is a small supply, and 
 18 to 20 gallons an ample supply. For trade and manu- 
 facturing purposes 6 to 12 gallons per head per day is 
 generally sufficient. But in a great many towns the supply 
 is larger, and in some cases this is due to waste of water by 
 leakage from the mains. In some towns in the United States 
 the supply reaches 100 to 150 gallons per head per day. 
 The demand for water varies, being small at night and greatest 
 at certain hours in the day. In designing water-mains it is 
 usual to assume the maximum rate of flow to be double the 
 mean rate. In laying new mains a further allowance is made 
 for the prospective increase of population. 
 
 The greatest statical pressure in the mains is in ordinary 
 cases 200 to 300 feet of water, and with commercial fittings 
 a higher pressure is undesirable. The lowest pressure which 
 should be provided at points of delivery to consumers is 80 
 to 100 feet. If a district varies considerably in level it is 
 divided into zones, in each of which the difference of level does 
 not exceed 80 to 100 feet. An independent supply from a 
 service reservoir at least 200 feet above the lowest point in 
 the zone is provided. Such service reservoirs are fed by a 
 trunk main from the source of supply, and usually contain 
 three or more days' supply in case of accident to the main. 
 The distributing mains are calculated so that when losses of 
 head are allowed for there is adequate pressure at all points 
 of delivery during the hours of maximum demand. 
 
 The zones are divided into subdistricts, each with an 
 
 177 12 
 
178 
 
 HYDEAULICS 
 
 CHAP. 
 
 independent supply, and these districts vary in area with the 
 population. One reason for this is the desirability of control- 
 ling waste of water by waste-water meters, through which the 
 supply to limited districts can be passed and measured. The 
 smallest mains used are 3 inches in diameter, but generally 
 mains are not less than 4 or 6 inches in diameter. 
 
 Fig. 95. 
 
 102. Water-supply main. Fig. 95 shows the general 
 arrangement of a water-supply main connecting a storage 
 reservoir A and a service reservoir B. The line of hydraulic 
 gradient is drawn from the lowest level in A to the highest 
 in B, the condition in which the rate of flow will be least. 
 The pipe line follows generally the contour of the ground, but 
 
 Fig. 96. 
 
 is everywhere below the hydraulic gradient. At C is a stream, 
 where the pipe line may be carried under the stream by a 
 specially constructed steel pipe, termed a siphon, or over it on 
 a bridge aqueduct. At D is a valley, which may be crossed 
 by a siphon, or the pipe may be carried on piers. If high 
 ground occurs on the route it may be necessary to place the 
 pipes in a tunnel to avoid rising above the gradient. Another 
 
IX 
 
 DISTRIBUTION, OF WATER BY PIPES 179 
 
 way of dealing with rising ground between the inlet and outlet 
 is to adopt a main with pipes of two diameters. Thus, in Fig. 
 96, the rising ground at C prevents the adoption of a uniform 
 hydraulic gradient from A to B. Then a larger pipe must be 
 used from A to C, giving the required discharge on the flatter 
 gradient ; and a smaller pipe may be used from C to B, giving 
 the same discharge on the steeper gradient. 
 
 As to the pressure in the main when the outlet is full 
 open, the pressure in feet of water at any point is the vertical 
 intercept between the pipe line and the hydraulic gradient. 
 But if a valve at the outlet is closed and the water is 
 stationary in the main, the pressure is the vertical intercept 
 between the pipe line and the horizontal AF. Hence gener- 
 ally the strength of the pipe has to be calculated for this 
 latter pressure, if under any circumstances the outlet can be 
 
 Fig 97. 
 
 closed. Any regulation of the flow at the outlet increases 
 the pressure in the main. In certain cases, to reduce the cost 
 of the main, there is no valve at the outlet, and regulation of 
 flow is effected entirely by a valve at the inlet. In that case 
 the pressure at any point is never greater than the height to 
 the hydraulic gradient. 
 
 103. Break-pressure reservoirs. When a water-main 
 is of great length, and when there is a large fall H 3 between 
 the supply reservoir at A and the final service reservoir at 
 B, it is often necessary to introduce intermediate balancing 
 or break-pressure reservoirs, such as those shown in Fig. 97 
 at C and D. The general hydraulic gradient is the line AB 
 from the surface-level in A to the surface-level in B, and for 
 this gradient and the required discharge Q the diameter of 
 the pipes must be calculated. Now, if there are no inter- 
 mediate reservoirs, the pressures in the main at any point 
 
180 H YDKAULICS CHAP. 
 
 when the pipe is delivering the full discharge will be the 
 height from the pipe to the hydraulic gradient AB. So far 
 as this condition of things is concerned, intermediate reservoirs 
 are not necessary. But in the working of the main there 
 must be times when the delivery of the main is decreased, 
 and the pressure in the main will then be greater ; there 
 must be times when the delivery is stopped, and then the 
 pressure at any point in the main will be the hydrostatic 
 pressure due to the depth of the point below the surface-level 
 in the supply reservoir, or, what is the same thing, the height 
 from the pipe to the horizontal AE. Thus at D the hydro- 
 static pressure would be H 2 , and at B, H 3 . Hence, as respects 
 strength, the pipe must be calculated for the hydrostatic 
 pressure in the main when the delivery is stopped, and this 
 may involve inconvenient thicknesses of pipe and unnecessary 
 cost. By taking the pipe line so as to reach at C and D the 
 level of the hydraulic gradient, and introducing balancing 
 reservoirs there, into which one length of main discharges and 
 from which another receives its supply, the pressure conditions 
 are ameliorated. With full delivery the hydraulic gradient 
 is AB as before. But when the delivery is stopped, the 
 hydrostatic pressure in each length can never exceed that 
 due to the nearest higher reservoir. Thus at C the pressure 
 cannot exceed 7^ ; at D it cannot exceed h 2 ; and at B it 
 cannot exceed h 3 . 
 
 104. Loss of head in a main consisting of sections of 
 different diameters. Two cases may be considered, (a) The 
 discharge may be taken to be constant throughout the main. 
 (&) The velocity may be taken to be constant throughout, 
 portions of the flow being abstracted by branch mains at each 
 change of diameter. 
 
 (a) Constant discharge. Let Q be the discharge, d lt d 2 , d 3 
 the diameters, and l lt 1 2 , 1 3 the lengths of the sections of the 
 main. Then the velocities are 
 
 The losses of head due to friction are 
 
ix DISTKIBUTION OF WATEE BY PIPES 181 
 
 where, in approximate calculations, a common mean value can 
 be selected for f The total loss of head due to friction is 
 [ 95, eq. (46)] 
 
 f 1 1 7 1 
 
 (1)- 
 
 (b) Constant velocity in the main, the discharge diminish- 
 ing from section to section. Let Q x , Q 2 , Q 3 be the discharges 
 in the successive sections, d lt d 2 , d z the diameters, and l lt 1 2 , 1 3 
 the lengths of the sections, and let v be the common velocity 
 throughout the main. Then the diameters must be fixed by 
 the relations 
 
 i= / 
 
 V 
 
 1TV TTV 7TV 
 
 Introducing these quantities into the ordinary equation for 
 loss of head in friction, the total loss is [ 95, eq. (2&)] 
 
 - . . (2). 
 
 The secondary losses of head are neglected in these equations, 
 and usually have to be allowed for by an addition to H, 
 determined by experience in similar cases. 
 
 105. Equivalent main of uniform diameter. It sometimes 
 facilitates calculations of loss of head to substitute for a main 
 
 Fig. 98. 
 
 in sections of different diameter an equivalent uniform main 
 having the same discharge with the same loss of head. Let 
 A (Fig. 98) be a main of varying diameter having lengths 
 
182 
 
 HYDEAULICS 
 
 CHAP. 
 
 /... of diameters d v d# d s .... It is required 
 
 find the length I of an equivalent main B of diameter d. 
 Let v lt v 2 , v 3 ... be the velocities in A, and v the velocity in 
 B, with any discharge Q. Since the loss of head in B is to 
 be the same as that in A, from $95, eq. (2&), 
 
 "3 "3 
 
 d. 
 
 where a common mean value can be selected for f. But 
 
 Consequently 
 
 d-d? + d? + d/ + 
 
 (3), 
 
 which is the length of the equivalent main. 
 
 106. Main in which the discharge decreases uniformly 
 along the length. In street mains water is delivered into 
 branch mains or service pipes, so that the discharge pro- 
 
 Fig. 99. 
 
 gressively decreases. It is useful to consider a limiting case 
 in which the volume of flow in a main of uniform diameter 
 decreases proportionately to the length. Let AB (Fig. 99) be a 
 pipe supplied from a reservoir, and DE its hydraulic gradient. 
 Let Q cubic feet per second be supplied at A, and discharged into 
 
ix DISTRIBUTION OF WATER BY PIPES 183 
 
 service pipes uniformly along the route, so that the pipe loses 
 q = Q// cubic feet per second per foot run. Let C be any point, 
 AC = z, AB = /, h x = the virtual fall from A to C, ^ = the 
 virtual fall from A to B, and d = the diameter of the pipe. 
 The volume of flow at C is Q x = Q qx. In a short length 
 dx at C the head lost is [ 95, eq. (46)] 
 
 Hence between A and C the head lost is 
 
 = 0'1008|P(Q- 
 
 JQ 
 
 But 
 
 /(Q - qy-fdx = Qzfdx - ZQqfxdx + 
 
 f (Q - 
 JQ 
 
 . . (4). 
 But 
 
 AtB, 
 
 . / (5). 
 
 In other words, the total loss of head is precisely one-third 
 of what it would be if the flow was uniform along the pipe 
 instead of uniformly decreasing. The line of hydraulic 
 gradient in this case is a cubic parabola ; that is, assuming as 
 usual that lengths measured along the pipe do not sensibly 
 differ from their horizontal projections. 
 
 Determination of diameter of pipe which delivers water 
 uniformly en route. Suppose a pipe of uniform diameter d 
 receives Q cubic feet of water per second at the inlet and 
 delivers Q z cubic feet at x feet from the inlet, having distri- 
 
184 HYDKAULICS 
 
 CHAP. 
 
 buted qx cubic feet uniformly in that distance. From the 
 equation above, the loss of head in the distance x is 
 
 h x = 0-1008 
 
 Now let 
 
 Then in a simple form, similar to that for pipes in which the 
 discharge is uniform along the length, 
 
 Q' 2 . . . (6). 
 
 But Q' is greater than Q x + ^qx, and is less than Q x 
 -^qx\ that is, Q r lies between Q, x +Q'5qx and Q x + 
 
 As an approximation, let Q' = Q x + Q'55gx ; 
 
 ** 0-1008^(0, + 0-550B) 2 . . (7). 
 
 So that if the pipe is calculated for the discharge Q x at the 
 outlet end plus 0'55 of the delivery qx en route, like a pipe 
 of uniform discharge, it will satisfy the conditions. 
 [* 107. Pipe connecting a supply and a service reservoir, 
 and delivering water en route. Let / be the length of the 
 pipe, and h the difference of surface-level in the reservoirs. 
 During the night, when the consumption of water en route 
 is zero, the pipe delivers from A to B (Fig. 100) a quantity of 
 water given by the relation [95, eq. (4a)] 
 
 The hydraulic gradient is then the straight line AB. 
 
 When the consumption en route reaches the value ql, Qf 
 is received at A, and Q x = Q' ql is delivered at B. From the 
 equation above, 
 
 . (8). 
 
 If ql increases Q x diminishes till, when 
 
IX 
 
 DISTRIBUTION OF WATER BY PIPES 185 
 
 the discharge into the reservoir B ceases. The line of hydraulic 
 gradient is then a cubic parabola with a horizontal tangent at 
 B. When the service en route increases still more, the pipe is 
 
 fed at one end by the reservoir A and at the other end by the 
 reservoir B. The line of hydraulic gradient remains parabolic, 
 but its horizontal tangent is at some point C. 
 
 Let x l be the horizontal distance from A to C, and x 2 from 
 C to B, and let h x be the virtual fall from A to C. From 
 106, eq. (5), 
 
 f n*"r 3 
 
 h x = 0-1008 i 2 -*L; 
 and considering the section CB, 
 
 f rftcr 8 
 
 h x - h = 0-1008 j-, q -^- t 
 
 li O 
 
 also I = x 1 -f x. 2 . These three relations determine any three of 
 the quantities h, h x , d, q, x 1} x. 2 . It may be noticed that 
 
 - (9), 
 
 l-x. 
 
 108. Branched pipe connecting reservoirs at different 
 levels. Let A, B, C (Fig. 101) be three reservoirs connected by 
 pipes as shown. Let l lt ^, Q x , Vj be the length, diameter, 
 discharge, and velocity in the pipe AX ; 1 2 , d^ Q 2 , t? 2 the same 
 quantities for BX, and / 3 , d s , Q 3 , v 3 for XC. Suppose the 
 
186 
 
 HYDEAULICS 
 
 CHAP. 
 
 dimensions and positions of the pipes known and the discharges 
 required. If a pressure column is introduced at the junction 
 X the water will rise to a height XE, and aE, &E, cE will be 
 the hydraulic gradients of the pipes. If the surface-level at 
 E is above 6, the reservoir A supplies B and C. If the surface- 
 level at E is below 6, the reservoirs A and B supply C. Con- 
 
 _ ; D o,t/UTn_Lint. 
 
 Fig. 101. 
 
 sequently there are three cases (a) E above &, Qi = Q 2 + Q 3 ; 
 (V) E level with 6, Q l = Q 3 and Q 2 = ; (c) E below 6, Q x + 
 Q 2 = Q 3 . To determine which case has to be dealt with, 
 suppose XB closed by a sluice. Then there is a simple main 
 of two diameters. Let h a) h b , h c be the heights of the surface- 
 level in A, B, and C above datum, and h f the height of E, on 
 the assumption that XB is closed. Then by 95, eq. 
 
 *.-*' = 0-1008^', 
 
 But in the condition assumed Q l = Q 3 . 
 
 h a - h' 
 
 (10), 
 
 from which h f is easily calculated. If then ~h! is greater than 
 h b , opening the sluice in XB will allow water to flow into 
 oir B, and the case is (a). But if hf = h b , the case is (b) ; 
 and if hf is less than h b , opening the sluice will admit water 
 from B to C, and the case is (c). Having distinguished the 
 case, the problem can be solved by approximation, choosing a 
 new value of h between h' and h b , and recalculating Q 1? Q 2 , 
 and Q 3 . The problem is solved when, with the assumed value 
 
IX 
 
 DISTRIBUTION OF WATER BY PIPES 187 
 
 of h, the relations of the discharges are those stated above. 
 The approximation seems cumbrous, but is really easy. 
 
 109. Compound main. It is sometimes necessary to 
 supplement part of a main by one or more mains laid near it, 
 or between two points there may be several mains through 
 which water can flow. Such a system may be termed a 
 compound main. Suppose the points A and B are connected 
 by mains ra, n, and p. Let Q 1? Q 2 , Q 3 be the discharges of the 
 mains, d lt d 2 , d 3 their diameters, l l} 1 2 , 1 B their lengths, and h the 
 virtual fall or difference of level of the hydraulic gradient 
 between A and B. The total discharge of the mains, from 
 95, eq. (4a), is 
 
 It is sometimes convenient to calculate the diameter of a single 
 equivalent main having the same discharge as ra, n, and p with 
 the same virtual fall Let d be its diameter and / its length. 
 Then 
 
 If 
 
 110. Hydraulic gradient of a pipe of variable diameter. 
 
 -At a change of diameter, where the velocity changes from 
 
 Fig. 102. 
 
 ^2> there is: a change of pressure head (j>2 
 1; 2 2 )/2^, and also usually a loss of head in shock, the 
 
188 HYDEAULICS CHAP. 
 
 amount of which for different cases is discussed in S 97. 
 
 
 
 Suppose for simplicity the shock losses neglected and that a 
 mean value is selected for the pipe friction coefficient f. Let 
 Fig. 102 represent a main, the sections of which have 
 diameters d lt d 2 , d 3 ..., and lengths l lt 1 2 , l s ... ; and let Q be 
 the discharge. The losses of head due to pipe friction are 
 [ 95, eq. (46)], 
 
 At B there will be a gain of pressure head due to decrease 
 of velocity from v-^ to v 2 ; at C and D there will be loss of 
 pressure head due to increase of velocity from v 2 to v 3 and 
 from v s to v. The velocities can be calculated from the 
 diameters and the discharge, and the changes of head are 
 
 The pressure head lost in giving velocity at the inlet is 
 
 With these quantities the hydraulic gradient can be drawn, 
 and the total head lost, or virtual fall of the pipe, is 
 
 111. Cost of water-mains. The cost of water-mains 
 per foot run laid in the ground, with the ordinarily necessary 
 appendages, is nearly proportional to the diameter, and is 
 about 
 
 C = 5dto7d . . . (13), 
 
 where C is in shillings and d in feet. It can be deduced 
 
ix DISTRIBUTION OF WATEE BY PIPES 189 
 
 from this that it is more economical to deliver water from 
 one point to another by a single pipe than by several. Hence 
 more than one pipe should be used only if the limit of size 
 for a single pipe is reached. The cost of the pipes to convey 
 a given quantity of water from one point to another is less 
 as the total quantity to be conveyed is greater. The whole 
 cost of a distributing system between given points increases 
 about as the |th power of the volume of water distributed. 
 
 112. Corrosion and incrustation. With some qualities 
 of water, corrosion of iron mains occurs. The corrosion takes 
 the form of nodular or limpet-shaped masses, which in time 
 become confluent and reduce the discharging capacity of the 
 main, partly by reducing its cross section and partly by 
 increasing the roughness. With some other qualities of water 
 incrustations of matter derived from the water, such as 
 carbonate of lime, form on the pipe and have a similar effect. 
 
 In the case of some mains the discharge decreases rather 
 rapidly for some time after they are laid, in consequence of 
 corrosion and incrustation. The first case in which this was 
 noticed was at Torquay, where the main had not been coated 
 with asphalt, the idea being that the pure surface-water from 
 the Dartmoor hills would have little action on the pipes. 
 But in eight years the discharge had decreased 5 1 per cent. 
 At that time Mr. Appold suggested scraping the internal 
 surface of the main by scrapers driven through by the water 
 pressure. This plan was adopted, and after scraping, the 
 delivery increased 28 per cent. The plan has since been 
 adopted, in many cases, and the discharge has been increased 
 by scraping by from 28 to 82 per cent in different cases. 
 If scraping is adopted, however, it requires to be repeated, 
 for the protective coating of rust and incrustation is removed, 
 and thus, though slowly, the pipe is worn away. At Torquay 
 the nodules of rust are ^ to ^ inch in height after twelve 
 months (Ingham, Proc. Inst. Mech. Engineers, 1873, 1899). 
 In the case of Torquay the water from a granitic district has 
 a serious action on iron, possibly from containing an acid 
 derived from peat. The matter removed by scraping con- 
 tains about 38 per cent of oxide of iron, 43 per cent of sandy 
 matter deposited from the water, and 18 per cent of organic 
 matter. At Southampton, where the water is obtained from 
 
190 HYDKAULICS CHAP. 
 
 chalk wells, the incrustation consists of 98 per cent of 
 carbonate of lime, and a little sulphate of lime and iron oxide. 
 Well waters from the Old Ked Sandstone do not cause much 
 corrosion or incrustation. Soft water appears to have greater 
 action than hard water. 1 
 
 The best protection against corrosion is to coat the pipes 
 with what is known as Dr. Angus Smith's composition. The 
 pipes are heated in a cylindrical stove to about 600 F. and 
 then dipped in a bath of pitch and oil of such a consistency 
 as to produce a tough coating. Natural asphalt is preferred 
 by some, with enough creosote oil to give a tough coat. In 
 the case of steel pipes they should be cleaned in a sulphuric 
 acid bath followed by one of lime water to neutralise the acid, 
 and then dipped in the asphaltic composition kept at nearly 
 boiling temperature. 
 
 Slime deposits in pipes carrying unfiltered water. 
 A serious decrease of discharge occurred in the first length of 
 main of the Vyrnwy aqueduct, which has been traced to the 
 growth of an organic deposit, and no doubt the same cause 
 has operated in other cases. The organisms are brought into 
 the pipe with the water and attach themselves to the pipe. 
 Thread-like organisms with a gelatinous sheath develop, and 
 iron oxide is deposited in the sheaths, which continue to 
 thicken. Solid particles in the water are caught by the 
 gelatinous threads. Acidity other than carbonic acid always 
 characterises water which produces this slime, and an 
 appreciable quantity of iron in solution. Mr. G. F. Deacon 
 has succeeded in removing the slime deposit by a kind of 
 scraper with whalebone brushes which does not injure the 
 pipe (I. C. Brown, Proc. Inst. Meek. Engineers, 1903-4). 
 
 113. Pipe aqueducts. These are usually of cast iron, 
 sometimes of steel, and in "Western America of wood. Cast- 
 iron pipes do not exceed 48 inches diameter, are cast in 
 lengths of 9 or 12 feet, and have spigot and socket joints, 
 the joints being filled with lead. Sometimes the pipe lengths 
 have plain ends, and the joint is made by a collar forming a 
 double socket in which lead is run. The pipes are almost 
 always placed in a trench and covered to protect them from 
 
 1 Figures of various types of pipe-scrapers are given in Proc. Inst. C.E. 
 cxvi. p. 307. 
 
ix DISTRIBUTION OF WATEE BY PIPES 191 
 
 frost. As a protection against corrosion they are heated and 
 dipped vertically in a bath of pitch and oil, which forms a 
 smooth hard coating and reduces the frictional resistance to 
 the flow of water. Steel pipes are much thinner, and therefore 
 if corroded lose proportionately more strength and are more 
 liable to deformation by earth pressure. But in some cases 
 they cost less than cast iron, and can be made of larger size. 
 They are made from plates riveted, welded, or made with a 
 special locking-bar joint which is as strong as the solid plate. 
 They usually have collar joints run with lead. 
 
 A pipe aqueduct is carried up hill and down dale 
 necessarily below the line of hydraulic gradient, but otherwise 
 at any inclination adapted to the contour of the country, and 
 in some cases a greater velocity may be permitted in a pipe 
 than would be suitable for an open conduit. Changes of 
 direction are effected by special bend pipes, or short straight 
 lengths (about 3 feet) are jointed by double-socketed bevel 
 collars about 12 inches long, the sockets being inclined to 
 each other. 
 
 The appurtenances of a pipe line are : (1) Air valves, 
 which are placed at every summit in the pipe line to permit 
 the escape of air when the main is filled, and afterwards if any 
 air is carried into the main. They are also placed on long 
 stretches of nearly level main. They are generally ball-valves 
 lighter than water, which close the air vent so long as they 
 are immersed, but which drop and open the air vent if air 
 accumulates. (2) Scour valves are placed at the bottom of 
 all depressions for emptying the main or letting out sediment. 
 
 (3) Reflux valves on ascending parts of the main are flap 
 valves which open in the direction of flow, but which 
 automatically close if a burst occurs and the water flows back. 
 They diminish the damage done by escape of water at a burst. 
 
 (4) Momentum valves are also intended to limit the escape of 
 water at a burst. A disc is placed in the pipe on an arm, 
 counterweighted so that it is not moved by the ordinary flow 
 of water. If a burst occurs the accelerated flow presses back 
 the disc, and the arm releases a catch, and another set of 
 weights cause a disc throttle-valve in the pipe to close 
 gradually and arrest the flow of the water. (5) Sluice stop- 
 valves worked by hand or by a hydraulic cylinder for closing 
 
192 HYDKAULICS 
 
 CHAP. 
 
 the main or regulating the flow. In the case of large mains 
 the pressure on a large sluice-valve is very great, and the 
 force required to move the sluice when starting from the 
 closed position is very great. Thus on a 3 6 -inch valve, under 
 250 feet of head the pressure would be nearly 50 tons, and 
 the frictional resistance to moving the valve perhaps 7 tons. 
 To facilitate opening, the valve is sometimes divided into 
 three parts which can be opened separately. In other cases 
 the valve is made about one-third the area of the pipe. The 
 pipe is gradually contracted to the area of the valve and 
 gradually enlarged again. Then, though there is some loss of 
 head at the valve it is not very serious. 
 
 In a long main the flow is usually controlled by a sluice 
 at the lower end. In that case, although the pressure in the 
 main when water is flowing is only the pressure due to 
 the depth below the hydraulic gradient, yet when the sluice is 
 closed and the water at rest, the pressure is that due to the 
 depth below the supply reservoir. The strength of the pipes 
 must therefore be sufficient to sustain at all parts the statical 
 pressure due to the depth below top water-level in the reservoir. 
 In the case of the East Jersey main, Mr. Herschel has placed 
 the controlling sluice at the inlet to the main, directions for 
 regulating it being transmitted from the outlet end by 
 telephone. In that case the pressure in the main cannot 
 exceed at any point the pressure due to the depth below the 
 hydraulic gradient. The adoption of this plan permits a 
 material saving of thickness and cost in the pipes. 
 
 114. Examples of pipe aqueducts. (1) The Vyrnwy 
 aqueduct. This aqueduct carries 40 million gallons per day 
 from the reservoir at Vyrnwy to a service reservoir at 
 Liverpool, a distance of 68 miles. The water first passes 
 through the Hirnant tunnel of 7 feet diameter and 3900 yards 
 long, and for nearly the whole of the rest of the distance 
 through three lines of cast-iron pipes, each 42 to 39 inches 
 in diameter. As the statical head on the main would be 
 excessive if the pipe line was continuous, the total fall from 
 Vyrnwy to Prescot being 550 feet, balancing reservoirs have 
 been constructed at five points, breaking the pipe line into 
 stretches each having its own hydraulic gradient and a 
 maximum statical pressure due to the level in the reservoir 
 
DISTRIBUTION OF WATER BY PIPES 193 
 
 feeding it. The greatest pressure at any point is 317 feet of 
 head One of the 4 2 -inch pipe lines, after being laid twelve 
 years, with an hydraulic gradient of 4*5 feet per mile, dis- 
 charged 15 million gallons per day. This gives a velocity of 
 2'892 feet per second, and a coefficient f = 0*0 05*7 4. 
 
 (2) East Jersey steel aqueduct, for the supply of 
 Newark and other towns in New Jersey, U.S.A. This consists 
 of a steel riveted main, 48 inches in diameter and 21 miles 
 long, with a maximum pressure of 340 feet of head. It 
 delivers 50 million U.S. gallons per day, the velocity in the 
 main being about 6 feet per second. The chief peculiarity of 
 this main is that the cross-joints are riveted, so that the pipe 
 is a continuous riveted structure without provision for expan- 
 sion. It is calculated that the cross-joints are strong enough to 
 resist the stresses due to 45 F. change of temperature without 
 allowing for any assistance from the friction of the ground. 
 
 (3) The Coolgardie pipe line. The longest pipe line is 
 that through which water is pumped from a reservoir at Perth 
 to Coolgardie and Kalgoorlie, Western Australia. Coolgardie is 
 on a tableland which is one of the driest places in the world. 
 A daily supply of 5,600,000 gallons is pumped through a 
 30 -inch steel pipe of the locking-bar construction with collar 
 joints run with lead. There are eight pumping stations. 
 The distance from the storage reservoir to the service reservoir 
 at Coolgardie is 308 miles, and there is a rise of 1290 feet in 
 that distance. From the service reservoir the water gravitates, 
 the total length from Perth being 351^ miles. Most of the pipes 
 are J inch thick, which is sufficient for heads up to 250 feet. 
 They were coated with a mixture of one part asphalt and one 
 part coal-tar, and sprinkled on the outside with sand while 
 hot. In a test the following results were obtained, the pipes 
 being new and clean : 
 
 Hydraulic Gradient. 
 Feet per Mile. 
 
 Velocity. 
 Feet per Second. 
 
 Delivery. 
 Gallons per Day. 
 
 Value of f. 
 
 2-25 
 2-80 
 
 1-889 
 2-115 
 
 5,000,000 
 5,600,000 
 
 00480 
 00476 
 
 In arranging the pumping plant a loss of head of 3*76 
 
 13 
 
194 HYDKAULICS CHAP. 
 
 feet per mile has been allowed for, to provide against 
 contingencies. 
 
 115. Pumping main. It is a common case that water 
 has to be raised by pumping from a river to a reservoir, from 
 which it gravitates to the town supplied. In that case the 
 lift of the pumps H is known, the length of the pumping 
 main I, and the volume Q which must be pumped per second. 
 In deciding on the diameter of the rising main, it must be 
 considered that while the smaller the main the less its cost, on 
 the other hand the greater will be the cost of the pumping 
 engines and the annual cost of pumping, because the frictional 
 head to be overcome will be increased. Usually, for various 
 reasons, the velocity in the pumping main is restricted to from 
 1^ to 4 feet per second, but within these limits a diameter of 
 main can be found which is the most economical. 
 Let 
 
 / = length of main in feet. 
 Q = volume pumped in cubic feet per second. 
 d = diameter of main in feet. 
 H = total lift from river to reservoir. 
 h = frictional loss of head in main. 
 p = cost per I.H.P. of pumping engines, including the 
 
 capitalised cost of maintenance and working. 
 q = the cost of the main per foot of diameter and per 
 
 foot of length, including cost of laying. 
 N = total I.H.P. of the pumping engines. 
 7j = the mechanical efficiency of the engines. 
 The total cost of the installation of engines and main is 
 
 The frictional loss of head in the main is 
 
 Consequently 
 
 550?; 77 
 
 Inserting this value, 
 
 +qdl, 
 
ix DISTRIBUTION OF WATER BY PIPES 195 
 
 where d is the only variable. Differentiating and equating 
 to zero, 
 
 . (H). 
 
 In practice, d/^/Q, is from 0'75 to I'O. For instance, in the 
 Coolgardie main 
 
 116. Suction pipe of pumps. Let I be the height of 
 the water barometer or atmospheric pressure in feet of water, 
 and h the height from the water-level in the suction well to 
 the bucket of the pump, h must be less than I in any case, 
 or pumping is impossible. Let ft be the area of the pump 
 bucket and co the area of the suction pipe, r the radius of the 
 crank and n the number of revolutions per minute. The 
 average speed of the crank pin is u= 27rrn/6Q feet per second, 
 and the connecting rod being supposed long the velocity of the 
 pump bucket is v = u sin a, where a is the crank angle from 
 the lower dead point. The acceleration of the pump bucket 
 at the beginning of its stroke is fu^jr. The corresponding 
 acceleration of the water in the suction pipe is 
 
 fl w 2 
 
 Let I be the length of suction pipe. The weight of the water 
 which must be accelerated is GcoL The pressure acting on 
 the water to make it follow the piston is G(& A)a>, and this 
 will produce an acceleration 
 
 In order that the water may follow the pump bucket, 
 
 (b - h)g = ft tf 
 I > w r 
 
 Substituting for u its value above, the greatest speed of the 
 pump is given by the relation 
 
 If the speed exceeds this the water will separate from the 
 
196 
 
 HYDKAULICS 
 
 CHAP. 
 
 bucket at the beginning of the stroke and overtake it after- 
 wards with a shock. This may be prevented by increasing 
 the area co of the suction pipe, or to a great extent by placing 
 an air vessel on the suction pipe near the pump. 
 
 117. Water hammer. When a valve in a long water- 
 main is rapidly closed, the velocity of the column of water 
 behind the valve is retarded and its momentum is destroyed. 
 To change the momentum of the water, a backward force must 
 be exerted by the valve on the water, or conversely a forward 
 pressure is exerted by the water on the valve and pipe, which, 
 if the action is rapid enough, produces a shock termed water 
 hammer. This action is dangerous, and causes in many cases 
 
 fracture of the pipe. It is provided against by arrangements 
 which prevent a rapid closing of important valves. 
 
 If a steam-engine indicator is fitted to the pipe, with an 
 arrangement for moving the recording barrel uniformly, a 
 diagram such as is shown in Fig. 1 3 is produced, the abscissae 
 being time and the ordinates pressure. p is the statical 
 pressure in the pipe when the valve is closed ; p l is the initial 
 pressure with the water flowing before the valve begins to 
 close. If the valve begins to close at d, the pressure rises to 
 a maximum at e which is in excess of the statical pressure 
 p Q by an amount p. ab is the time of closing the valve. 
 Waves of pressure follow, gradually diminishing till the water 
 in the pipe comes to rest. 
 
 Professor Carpenter made some experiments on a pipe 1^ 
 
ix DISTRIBUTION OF WATER BY PIPES 197 
 
 inches in diameter, with a ^--inch bib-cock at the end. The 
 following were the pressures registered when the cock was 
 suddenly closed. There was a small air chamber near the 
 valve, which in one set of tests was filled with air and in 
 another with water. 
 
 GAUGE PRESSURES IN I^-INCH PIPE 
 
 
 Air Chamber. 
 
 Water Chamber. 
 
 Static pressure, Ibs. per sq. in. . 
 
 29 
 
 28 
 
 Number of impacts . 
 
 8 
 
 9 
 
 Maximum pressure . 
 
 67 
 
 76 
 
 Minimum pressure 
 
 6 
 
 9 
 
 Consider a column of water in the pipe of unit cross 
 section, extending back from the valve a distance I feet. The 
 weight of this column is GZ Ibs. If the initial velocity of 
 the water is v, the momentum of the column is Glv/g second- 
 pounds. If p m is the mean pressure per unit area exerted in 
 stopping the momentum, and t the time of closing the valve, 
 
 p m t = Glv/g. 
 
 The maximum excess pressure exerted is Po+ppi', and if 
 the mean pressure is taken to be half this, 
 
 p=2Gh/gt-p Q +p l , . (16). 
 
 The rate u at which a pressure wave is transmitted through 
 water is about 4500 feet per second. Hence iju seconds must 
 be occupied before the effect of closing the valve reaches the 
 distance I from the valve, and a further time l/u for the 
 pressure due to changing momentum at a distance I is trans- 
 mitted back to the valve. Hence if the time of closing the 
 valve is less than 21 /u, that time must be substituted for t 
 in the equation, and then 
 
 p = Gvulg-p + Pl . . (17). 
 
 Putting in the numerical quantities, and taking the pressures 
 in pounds per square inch and the velocities in feet per second, 
 the equations become 
 
198 
 
 HYDKAULICS 
 
 CHAP. IX 
 
 The first equation is to be used if t is greater than Z/2250 
 seconds. This equation gives p = 0, if 
 
 = or > 
 
 (18), 
 
 which is the condition to be satisfied in closing the valve if 
 there is to be no water hammer. The theory involves some 
 assumptions, and must be taken only as a general guide. 
 
 Some very elaborate experiments on water hammer in 
 pipes were made by Joukowsky at Moscow (Stoss in Wasser- 
 leitungsrohren, St. Petersburg, 1900). He used pipes 2, 4, 
 and 6 inches in diameter, and 2494, 1050, and 1066 feet in 
 length. The valve was closed in 0*03 second. Ten recording 
 gauges placed along the pipes showed that the maximum 
 pressures were substantially the same at all points. 
 
 The following table gives some of the results : 
 
 VALUES OF p+p Q -p 1 LBS. PER SQUARE INCH 
 
 4 -inch Pipe. 
 
 6 -inch Pipe. 
 
 Velocity v. 
 
 P+Po-Pi- 
 
 Velocity v. 
 
 P+Po-Pi- 
 
 0-5 
 
 31 
 
 06 
 
 43 
 
 1-9 
 
 115 
 
 1-9 
 
 106 
 
 2-9 
 
 168 
 
 3-0 
 
 173 
 
 4-1 
 
 232 
 
 5-6 
 
 369 
 
 9-2 
 
 519 
 
 7-5 
 
 426 
 
CHAPTEE X 
 
 LATER INVESTIGATIONS OF FLOW IN PIPES 
 
 118. THE different elementary streams which go to form the 
 flow through a pipe have different velocities parallel to the 
 axis of the pipe ; those near the sides are retarded by what is 
 often termed skin friction, and these in turn retard those 
 adjacent to them, and so on till the central elementary stream 
 is reached, which has the greatest velocity. It has not been 
 found possible to construct a rational theory of flow which 
 takes account of this distribution of velocity, except at very 
 low velocities. But experiment shows that the resistance to 
 flow involves a loss of energy or head which is proportional 
 to the area of the surface of the pipe and to some function of 
 the mean velocity parallel to the axis of the pipe. The 
 assumption on which the Chezy formula is based is that 
 
 the resistance varying directly as the square of the velocity. 
 In a memoir by Prony in 1804, discussing all the experiments 
 then made, that engineer suggested the expression 
 
 j * = + * .. (2), 
 
 in which, for metric measures, 
 
 a = 0-0000173, b = 0'000343, 
 and for English measures, 
 
 a -0-0000173, 6=0-000105, 
 corresponding to f='007l3 at 3 feet per second. This 
 
 199 
 
200 HYDEAULICS CHAP. 
 
 binomial expression is exceedingly inconvenient for calculation. 
 It meets the condition that at low velocities the resistance 
 varies as the velocity, and that at high velocities it varies 
 nearly as the square of the velocity; but it makes the transition 
 gradual, whereas it is now known to be abrupt. 
 
 119. Kutter's formula for pipes. Messrs. Ganguillet 
 and Kutter, in a laborious investigation on the results of the 
 gauging of streams, arrived at the following complicated 
 empirical formula. Let n be a coefficient of roughness, de- 
 pending on the character of the surface of the pipe, and m its 
 hydraulic mean radius, i the virtual slope, and v the mean 
 velocity ; then, for English measures, 
 
 v = 
 
 .. a 1-811 -00281 
 41-6 + + : 
 
 1+41-6 
 
 00281\ n 
 
 \/mi . . (3). 
 
 There is no good reason for thinking that this formula is 
 specially accurate for flow in pipes. Indeed, it is known not 
 to accord with experiment for small values of i or for small 
 diameters of pipe. But it has been adopted by some engineers, 
 and therefore requires to be mentioned. The usual value of n 
 assumed for clean pipes is '01 3. If in the term 0'00281/i, 
 which is usually relatively small, i is taken as 0*001, and n 
 is taken at 0*013, the formula reduces to the simpler form 
 
 18372 
 
 0-5773 
 
 j 
 v 
 
 m 
 
 or, to put it in a form comparable with the more usual 
 equations, 
 
 / 1-1546X 2 
 
 V 1 ' Jd) '+ di 
 - X 2 4 ' 
 
 where the first term on the left corresponds to f in the Chezy 
 formula. 
 
INVESTIGATIONS OF FLOW IN PIPES 201 
 
 d= 
 
 Inches. 
 
 3 
 
 6 
 
 12 
 24 
 36 
 48 
 
 Feet. 
 
 25 
 
 5 
 1-0 
 2-0 
 3-0 
 4-0 
 
 Kutter's value of 
 
 r- 
 
 0209 
 0132 
 0089 
 0063 
 0053 
 0047 
 
 120. Defects of the Chezy formula. 1 The Chezy 
 formula is extremely convenient, but involves, if reasonable 
 accuracy is required, the selection of the coefficient f amongst 
 a wide range of values. The variation of f depends on the 
 following conditions : 
 
 (1) In the case of most pipes the loss of head h does not 
 increase so fast as the square of the velocity v. Consequently 
 f must have values which decrease as the velocity is greater. 
 
 For instance, in a glass pipe on which Darcy experimented, 
 f changed from 0*010 for a velocity of half a foot per second, 
 to 0*0062 for a velocity of 7 feet per second, a decrease of 
 38 per cent. In a new cast-iron pipe f decreased from 0*0114 
 at half a foot per second to 0*0064 at 10 feet per second, or 
 a decrease of 50 per cent. 
 
 (2) Darcy showed that f decreases as the size of the pipe 
 is larger. Thus, taking Darcy's experiments on new cast- 
 iron pipes : 
 
 Velocities. 
 Feet per Second. 
 
 Values of for Diameters of 
 
 0-27 feet. 
 
 0-45 feet 
 
 0-62 feet 
 
 0-6 
 
 10-0 
 
 0114 
 0064 
 
 0073 
 0049 
 
 0059 
 0054 
 
 The results are not quite consistent, but they show a 
 considerable decrease in as d increases. 
 
 (3) The value of f changes with the condition of the 
 inside of the pipe. For asphalted, new, and corroded pipes 
 the values of were proportional to 1, 1^-, and 3 in some of 
 Darcy's experiments. 
 
 1 The discussion given here in abbreviated form was published by the 
 author in Industries in 1886. 
 
202 
 
 HYDKAULICS 
 
 CHAP. 
 
 (4) The experiments of Mr. Mair, agreeing with the 
 author's own experiments on discs, show that the resistance 
 decreases as the temperature increases. Thus, for a clean 
 brass pipe, 1|- inches diameter, Mr. Mair obtained the follow- 
 ing values : 
 
 At Velocities in 
 Feet per Second of 
 
 Values of f for Temperatures of 
 
 56 
 
 90 
 
 160 
 
 61 
 
 4 
 
 0047 
 0052 
 
 0042 
 0044 
 
 0035 
 0038 
 
 Alterations of at least 25 per cent for 100 F. 
 
 A coefficient which has four independent causes of varia- 
 tion, all of them so large, is not very useful for practical 
 purposes. To get over the difficulty, a formula must be found 
 with more than one constant derived from experiment, and 
 which expresses more nearly the true law of resistance. 
 
 It is many years since M. Barre de St. Venant proposed 
 a formula with two arbitrary constants. This is of the form 
 
 d h 
 
 where m and n are constants derived from the experiments. 
 
 12 
 
 M. de St. Venant deduced the values n = -=- and ra= 0*0002955 
 
 for metric and 0*0001265 for English measures. When this 
 is written in logarithmic form, 
 
 i i ( d h 
 
 log m + n log v log ( - y 
 
 (5), 
 
 we have, as St. Venant pointed out, the equation to a straight 
 line, of which m is the ordinate at the origin and n the ratio 
 of the slope. Hence, if the logarithms of a series of experi- 
 mental values of h and v are plotted, the determination of the 
 constants is reduced to finding the straight line which most 
 nearly passes through the plotted points. 
 
 In a remarkable memoir on the influence of temperature 
 on the movement of water in pipes by Hagen (Berlin, 1854), 
 
x INVESTIGATIONS OF FLOW IN PIPES 203 
 
 another modification of the St. Venant formula was given; 
 this is 
 
 r? 1 II ' <> 
 
 This involves three coefficients, derived from experiment. In 
 the experiments examined by Hagen, he found 
 
 so that 
 
 in which ra was nearly independent of variations both of 
 v and of d. But the range of values of d examined was 
 small. 
 
 It is obvious that this form of the equation of flow is 
 very advantageous, even regarded as an empirical formula, 
 for the three constants, n, x, and ra, can be taken so as 
 separately to allow for the three principal causes of variation 
 of resistance: the variation of velocity, of diameter, and of 
 roughness of surface. 
 
 In a very interesting paper in the Transactions of the 
 Eoyal Society, 1883, Professor Keynolds has made clearer the 
 causes of the change in the character of the motion of water, 
 from the regular stream-line motion at .low velocities to the 
 eddying motion which occurs in almost all the cases with 
 which the engineer has to deal. Further, partly by reasoning, 
 partly by induction from the form of the curves of experi- 
 ments when plotted, he has suggested the general equation 
 
 as applicable both to the case of undisturbed motion and of 
 eddying motion. The constant n having the value 1 for low 
 velocities and undisturbed motion, and a value ranging from 
 1*7 to 2 for greater velocities. Professor Keynolds's formula 
 reduces to the form 
 
 where P is a function of the temperature. Neglecting 
 
204 HYDKAULICS CHAP. 
 
 variations of temperature, Professor Keynolds's formula is 
 identical, for velocities not very small, with Hagen's formula ; 
 with the exception only that in Eeynolds's formula the indices 
 of d and of v are related, so that there are only two indepen- 
 dent constants instead of three. For the purpose of obtaining 
 the coefficients from experiment, Hagen's formula is the more 
 convenient. 
 
 121. The experimental data available. The earliest 
 experiments on flow in pipes were made by Couplet in 1732, 
 and since that time a considerable number of experiments 
 have been made. In selecting from these it must be borne 
 in mind that it is extremely desirable to exclude from in- 
 vestigation any experiments that are really untrustworthy. 
 No good result can be got by averaging accurate and erroneous 
 results. On the other hand, it would be absolutely wrong 
 in principle to exclude results from examination merely 
 because they did not appear to fit in well with some empirical 
 law. 
 
 All experiments may be at once excluded in which the 
 means of measuring the loss of the head or the quantity 
 discharged were unsatisfactory. All experiments may also 
 be excluded in which the condition of the surface of the pipe 
 was not noted. With these exclusions, the number of experi- 
 ments remaining to be examined is greatly reduced. 
 
 Of these experiments, by far the most complete and 
 valuable is the series of experiments on 17 pipes by Henry 
 Darcy. The care and insight with which these experiments 
 were made, and the skilful variation of the conditions of the 
 experiment, are worthy of the highest praise. Of all the 
 conditions to be noted in experimenting, there is only one 
 the importance of which did not occur to Darcy. In many 
 cases he neglected to observe the temperature of the water. 
 
 There is, however, one anomaly in Darcy's experiments 
 which cannot now be fully explained, and the nature of which 
 can perhaps best be seen in the plottings of some of his 
 results. Darcy measured the loss of head in two successive 
 50-metre lengths of his pipes. Now, in almost all cases his 
 results show a rather greater loss in the second 50 -metre 
 length than in the first, and this is really not intelligible. 
 On the whole, the author is inclined to think that the 
 
x INVESTIGATIONS OF FLOW IN PIPES 205 
 
 measurements in the first 50 -metre length are more reliable 
 than those in the second, and only the measurements of 
 head lost in the first 50 -metre length are used in these 
 reductions. 
 
 From Darcy's experiments have been taken the results 
 on new, cleaned, and incrusted cast-iron pipes, those on 
 wrought -iron gas -pipes, and those on lead pipes. These 
 pipes ranged in diameter from 0'0122 to 0*5 metre, or as 
 40 to 1. For each pipe the experiments began with a very 
 small loss of head, often only 0'02 metre in 50 metres. The 
 author has excluded the observations in which the loss of head 
 was less than O'l metre, partly because some of the experi- 
 ments with these very small heads correspond to conditions 
 of undisturbed motion, for which the law is different, and 
 partly because the errors in observing very small heads are 
 likely to be relatively large. Up to 6 metres of head the 
 heights were measured by a water column, and beyond that 
 by a mercury column. Now, as the observations with the 
 water gauge give ample range of velocities for the purpose 
 in hand, and as the observations with the mercury gauge at 
 high velocities were, as Darcy mentions, carried out with 
 great difficulty, the former only have been used in these 
 reductions. With a loss of head varying from O'l metre to 
 6 metres, the velocities ranged in different cases from O'l 
 metre per second to 5 metres per second, a very ample range 
 for examination. 
 
 Of other experiments available, the early (1771) experi- 
 ments of Bossut on the flow in tin pipes seem very trust- 
 worthy, and give values of the constants for a very clean 
 and smooth surface. These extend over a considerable range 
 of velocity. 
 
 Dr. Lampe's experiments on the Dantzig main are ex- 
 tremely useful, from the care with which they were carried 
 out, and the fact that they are on a large scale. 
 
 Of other experiments, the most valuable are the American 
 data collected in Mr. Hamilton Smith's Hydraulics. Of these, 
 there is a very valuable experiment by Mr. Stearns on a cast- 
 iron asphalted pipe, l metres in diameter. Mr. Hamilton 
 Smith's own experiments are also very useful, as filling up 
 and extending the series of results from other sources. This 
 
206 HYDKAULICS CHAP, x 
 
 makes the range of diameters of new pipes, on which experi- 
 ments are available, to extend from 0'266 metre to 1/219 
 metres. 
 
 Then there are some experiments on small wrought-iron 
 gas-pipes, which are useful for comparison with Darcy's, and 
 some experiments on large wrought-iron riveted water-mains. 
 
 122. Method of dealing with the experimental data. 
 The greater part of the experimental results are found origin- 
 ally in metric measures. Hence it was convenient to plot the 
 results in metric measures, and to obtain constants for a 
 formula in metric measures. These constants were finally 
 converted to English measures. 
 
 Taking Hagen's formula (6), and writing it logarithmic- 
 
 ally, 
 
 log /t = n log + log -jz + log- . . (8), 
 
 CL Zy 
 
 in which for any given pipe the second and third terms on the 
 right are constants. This is an equation to a straight line 
 having log {(ml)/(2gd x )}. for the ordinate at the origin, and a 
 slope of n to 1. For all the experimental data, arranged 
 in groups according to the type of pipe, values of log h were 
 plotted as abscissae and values of log v as ordinates, h and v 
 being taken in metric units. One of these plottings is given 
 on a reduced scale in Fig. 104. The values of n correspond- 
 ing to the average slope of the lines are given in the following 
 table 1 :- 
 
 1 For each of the Darcy pipes two lines are plotted, the full line correspond- 
 ing to observations in the first, and the dotted to those in the second 50-metre 
 length. 
 
 [TABLE. 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < -- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *' 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r\ 
 
 . 
 
 
 
 \ 
 
 
 
 
 
 V 
 
 | 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \' 
 
 s , 
 
 \ 
 
 
 
 ^ 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 5 
 
 
 
 \ 
 
 
 
 
 
 \; 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 ( \ 
 
 i 
 
 
 
 \ 
 
 
 
 
 
 X s - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 \ \ 
 
 
 
 \ 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \\ 
 
 , \ 
 
 
 
 
 
 
 5 
 
 
 
 KV 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 A 
 
 % 
 
 
 
 V ^ 
 
 
 
 \ 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 \'v 
 
 g 
 
 
 v . 
 
 \ 
 
 
 V 
 
 
 
 &. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 A 
 
 ^ 
 
 
 
 V 
 
 
 
 V 
 
 
 v 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 s 
 
 ^ 
 
 L 
 
 
 *\ 
 
 \ 
 
 
 
 
 
 \1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *A 
 
 A 
 
 t 
 
 
 I 
 
 N^ 
 
 \ 
 
 
 s 
 
 
 V 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * 
 
 V 
 
 \\ 
 
 - 
 
 
 \ 
 
 \ 
 
 i-. 
 
 1 
 
 s 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 s 
 
 V- 
 
 
 
 ^ 
 
 > 
 
 
 \; 
 
 be 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^> 
 
 
 ^ V 
 
 
 
 
 \ 
 
 5 
 
 
 | 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -s \ 
 
 5 
 
 v i 
 
 
 
 I 
 
 \ 
 
 
 \ 
 
 ft 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -; 
 
 \\ 
 
 \\ 
 
 ^ 
 
 
 \ 
 
 ^ 
 
 S 
 
 
 N^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 A 
 
 \-c 
 
 
 
 ^ 1 
 
 v 
 
 .5 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 % 
 
 \\ 
 
 \x 
 
 b 
 
 
 V 
 
 * 
 
 \ ^ 
 
 
 \" 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 v^ 
 
 
 
 v\V 
 tjj 
 
 \ 
 
 v 
 
 
 \^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9 
 
 \\ 
 
 \ > 
 
 L 
 
 
 V 
 
 \ 
 
 5 
 
 
 ^ 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7\ 
 
 A 
 
 v ^ 
 
 
 
 v" 
 
 
 C*7. 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 \ 
 
 \' 
 
 
 
 \ 
 
 j 
 
 -, 
 
 \ 
 
 ^ 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ '^ 
 
 \ 
 
 
 
 V 
 
 i 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 rD 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 v 
 
 
 
 \ 
 
 ^ 
 
 V 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 u 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c\ ^ 
 
 \ 
 
 
 
 S 
 
 ^ 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 SJ 
 
 \ 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 'v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N x 
 
 \ x \ 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 5. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 
 \ 
 
 . 
 
 
 
 
 
 
 
 
 
 
 Z" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 5 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \^ 
 
 
 
 \ 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N^ 
 
 
 
 V 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 1 
 
 - 
 
 
 \ 
 
 I 
 
 
 
 \', 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 X 
 
 s r 
 
 
 \ N 
 
 
 
 v i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 V 
 
 
 ( 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r* 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t, ' 
 
 a 
 
 
 
 
 
 
 
 
 ^ 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 a 
 
 
 
 
 
 
 
 
 
 i 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 \ 
 
 
 
 
 
 
 
 
 
 5 
 
 tk 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -- 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1. \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 4 
 
 
 t 
 
 
 .-j 
 
 
 * 
 
 
 2 
 
 
 
 
 
 
 
 
 N 
 
 
 ^ 
 
 
 "' 
 
 
 1 
 
 
 ^ 
 
 
 !M 
 
 
 r ; 
 
 
 o 
 
208 
 
 HYDEAULICS 
 
 VALUES OF THE INDEX OP VELOCITY 
 
 Surface of Pipe. 
 
 Authority. 
 
 Diameter 
 of Pipe in 
 Metres. 
 
 Values of n. 
 
 Tin-plate . 
 
 Bossut 
 
 036 
 
 1 ' 697 jl-72 
 
 
 
 
 054 
 
 1-730/ 
 
 Wrought iron (gas-pipe) 
 
 Hamilton Smith 
 
 0159 
 
 1-756] 
 
 .1 ->7K 
 
 
 11 
 
 0267 
 
 1-770 r '" 
 
 Lead .... 
 
 Darcy 
 
 014 
 
 1-866' 
 
 
 
 11 
 
 027 
 
 1-755 
 
 -1-77 
 
 
 11 
 
 041 
 
 1-760 
 
 
 Clean brass . 
 
 Mair 
 
 036 
 
 1-795 
 
 L-795 
 
 Asphalted . 
 
 Hamilton Smith 
 
 0266 
 
 1-760^ 
 
 
 
 Lampe 
 
 4185 
 
 1-850 
 
 -1 "85 
 
 
 Bonn 
 
 306 
 
 1-582 
 
 
 
 Stearns 
 
 1-219 
 
 880 
 
 
 Riveted wrought iron . 
 
 Hamilton Smith 
 
 2776 
 
 804^ 
 
 
 
 11 
 
 3219 
 
 892 
 
 1-87 
 
 
 11 
 
 3749 
 
 852^ 
 
 
 Wrought iron (gas-pipe) 
 
 Darcy 
 
 0122 
 
 
 
 
 
 0266 
 
 899 
 
 1-87 
 
 
 
 
 0395 
 
 838] 
 
 New cast iron 
 
 11 
 
 0819 
 
 950' 
 
 
 
 
 137 
 
 1-923 
 
 
 
 
 
 
 1 () 5 
 
 
 n 
 
 188 
 
 1-957 
 
 J. 7 V 
 
 
 11 
 
 50 
 
 1-950^ 
 
 
 Cleaned cast iron 
 
 11 
 
 0364 
 
 1-835' 
 
 
 
 
 0801 
 
 2-000 
 
 
 
 
 
 
 2 "00 
 
 
 11 
 
 2447 
 
 2-000 
 
 
 
 11 
 
 397 
 
 2-07 
 
 
 Incrusted cast iron 
 
 11 
 
 0359 
 
 1-980] 
 
 
 n 
 
 0795 
 
 1-990 V2-00 
 
 
 11 
 
 2432 
 
 1-990J 
 
 It will be seen that the values of the index n range from 
 1'72 for the smoothest and cleanest surface to 2*00 for the 
 roughest. The numbers after the brackets are rounded off 
 numbers, not exactly means, but numbers based partly on 
 judgment of the value of the different experiments, which 
 have been adopted in the following reductions. 
 
 Taking the values of n thus determined, the value of 
 m/d x , which should be a constant for any given pipe, is then 
 deduced. For each pipe the values of in/d x are averaged. It 
 is then possible to see how far the formula fits the experiments, 
 
x INVESTIGATIONS OF FLOW IN PIPES 209 
 
 at least so far as the variation of velocity with variation of 
 head is concerned. The following tables give the mean 
 values of m/d x found for each pipe, and the result of using the 
 formula 
 
 rS) .<> 
 
 to re-calculate h from the observed velocities. It will be seen 
 that, at least so far as variation of the velocity and head are 
 concerned, the formula fits the experiments with extraordinary 
 closeness. The re-calculated values of h approximate to the 
 observed values throughout the whole range of the experiments, 
 with differences which do not exceed the probable experimental 
 errors of observation. Metric measures are used in these 
 tables. 
 
 ASPHALTED CAST IRON 
 
 HAMILTON SMITH. 
 
 BONN WATERWORKS. 
 
 BONN WATERWORKS. 
 
 Diam. = 0'02661 m. 
 
 Diam. = 0'306 m. 
 
 Diam. = 0'306m. 
 
 .1-041 
 
 J= 0-0823 
 
 ^=0-0842 
 
 d x 
 
 d x 
 
 d* 
 
 
 * = 41 F. 
 
 1=lr & F . 
 
 V 
 
 h 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 6767 
 
 1-346 
 
 1-289 
 
 4779 
 
 0605 
 
 0535 
 
 4734 
 
 0605 
 
 054 
 
 9826 
 
 2-609 
 
 2-569 
 
 6449 
 
 0930 
 
 0932 
 
 6412 
 
 0975 
 
 094 
 
 1-451 
 
 5-169 
 
 5-285 
 
 7909 
 
 1345 
 
 1358 
 
 7986 
 
 1300 
 
 142 
 
 1-659 
 
 6-532 
 
 6-770 
 
 9523 
 
 1715 
 
 1916 
 
 9436 
 
 1810 
 
 193 
 
 LAMPE. 
 
 STEARNS. 
 
 LOCH KATRINE. 
 
 Diam. = 0-4185 m. 
 
 Diam. = l-219m. 
 
 Diam.=l-219m. 
 
 ^ = 0-0453 
 
 5-o-Mii 
 
 ^ = 0-0169 
 
 7i = l-85 
 
 7i = l'85 
 
 ro=l'85 
 
 < = 48'F. 
 
 * = 38 F. 
 
 t= (unknown) 
 
 V 
 
 h 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 4806 
 
 0296 
 
 0298 
 
 797 
 
 0159 
 
 0186 
 
 1-054 
 
 0474 
 
 0474 
 
 7565 
 
 0688 
 
 0688 
 
 1-140 
 
 0356 
 
 0360 
 
 
 
 
 8256 
 
 0815 
 
 0810 
 
 1-513 
 
 0610 
 
 0609 
 
 
 
 
 9419 
 
 0975 
 
 1-034 
 
 1-888 
 
 0925 
 
 0916 
 
 
 
 
 14 
 
210 
 
 HYDKAULICS 
 
 CHAP. 
 
 CLEAN TINPLATE PIPES 
 
 BOSSUT. 
 
 BOSSUT. 
 
 Diam. =0-03608 m. 
 
 Diam. = '05441 m. 
 
 |=0-6557 
 
 iTr? 
 
 2 = 50 F. 
 
 n t=lr\. 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Observ. 
 
 Calc. 
 
 
 Observ. 
 
 Calc. 
 
 3401 
 
 2700 
 
 2615 
 
 4435 
 
 2650 
 
 2594 
 
 3807 
 
 3220 
 
 3174 
 
 4956 
 
 3140 
 
 3139 
 
 4364 
 
 3980 
 
 4015 
 
 5608 
 
 3855 
 
 3884 
 
 5114 
 
 5380 
 
 5273 
 
 6431 
 
 5005 
 
 4916 
 
 5126 
 
 5210 
 
 529 
 
 6692 
 
 526 
 
 5269 
 
 5694 
 
 6410 
 
 634 
 
 7439 
 
 623 
 
 631 
 
 6324 
 
 7540 
 
 760 
 
 7912 
 
 710 
 
 702 
 
 6498 
 
 7915 
 
 796 
 
 8366 
 
 764 
 
 773 
 
 7598 
 
 1-0345 
 
 1-042 
 
 9685 
 
 987 
 
 994 
 
 8978 
 
 1-3470 
 
 1-389 
 
 1-091 
 
 1-194 
 
 1-222 
 
 9333 
 
 1-494 
 
 1-484 
 
 1-164 
 
 1-398 
 
 1365 
 
 1-314 
 
 2-649 
 
 2672 
 
 1-595 
 
 2-324 
 
 2-345 
 
 KIVETED WROUGHT IRON 
 
 HAMILTON SMITH. 
 
 HAMILTON SMITH. 
 
 Diam. = 0-2776 m. 
 
 Diam. =0 '3219m. 
 
 I 1 = 0-0822 
 d x 
 
 ^ = 0-0704 
 
 7i=l-87 
 
 7i = l'87 
 
 * = 55 F. 
 
 = 55 F. 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Observ. 
 
 Calc. 
 
 
 'Observ. 
 
 Calc. 
 
 1-436 
 
 425 
 
 413 
 
 1-401 
 
 334 
 
 337 
 
 1-858 
 
 667 
 
 667 
 
 2-121 
 
 714 
 
 732 
 
 2-111 
 
 847 
 
 848 
 
 2-635 
 
 1-109 
 
 1-098 
 
 2-639 
 
 1-279 
 
 1-287 
 
 3-262 
 
 1-659 
 
 1-638 
 
 3-054 
 
 1-654 
 
 1-691 
 
 
 
 
INVESTIGATIONS OF FLOW IN PIPES 211 
 
 KIVETED WROUGHT IRON Continued 
 
 HAMILTON SMITH. 
 
 HAMILTON SMITH. 
 
 Diam.= 0-3749 m. 
 
 Diam. = -6566m. 
 
 Diam. = 0'431 6 m. 
 
 1=0-549 
 
 ^ = 0-0260 
 
 ^ = 0-0440 
 
 7i = l"87 
 
 w = l-87 
 
 =1 "87 
 
 * = 55F. 
 
 t ? 
 
 t ? 
 
 V 
 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Observ. 
 
 Calc. 
 
 
 Observ. 
 
 Calc. 
 
 
 Observ. 
 
 Calc. 
 
 1-336 
 
 251 
 
 241 
 
 3-841 
 
 821 
 
 821 
 
 6-139 
 
 3-336 
 
 3-336 
 
 2-084 
 
 549 
 
 553 
 
 
 
 
 
 
 
 2229 
 
 613 
 
 627 
 
 
 
 
 
 
 
 2579 
 
 823 
 
 824 
 
 
 
 
 
 
 
 3-228 
 
 1-235 
 
 1-254 
 
 
 
 
 
 
 
 3-684 
 
 1-616 
 
 1-605 
 
 
 
 
 
 
 
 NEW CAST-IRON PIPES (UNCOATED) 
 
 DARCY. 
 
 DARCY. 
 
 Diam. =0'0819 m. 
 
 Diam. = 0-1 37 m. 
 
 |=0-3205 
 
 ~ x = 0-1454 
 
 V 
 
 h 
 Observ. Calc. 
 
 V 
 
 j 
 
 Observ. 
 
 i 
 Calc. 
 
 358 
 
 115 
 
 110 
 
 488 
 
 097 
 
 091 
 
 561 
 
 258 
 
 265 
 
 763 
 
 224 
 
 219 
 
 791 
 
 500 
 
 517 
 
 1-279 
 
 590 
 
 599 
 
 1-185 
 
 1-10 
 
 1-138 
 
 1-714 
 
 1-045 
 
 1-059 
 
 1-418 
 
 1-58 
 
 1-61 
 
 2-098 
 
 1-560 
 
 1-571 
 
 1-571 
 
 1-99 
 
 1-97 
 
 2-281 
 
 1-840 
 
 1-850 
 
 2-453 
 
 4-826 
 
 4-66 
 
 3-640 
 
 4-690 
 
 4-604 
 
 2-487 
 
 4-870 
 
 4-83 
 
 
 
 
 2-720 
 
 5-872 
 
 5-75 
 
 
 
 
212 
 
 HYDEAULICS 
 
 CHAP. 
 
 NEW CAST-IRON PIPES (UNCOATED) Continued 
 
 DARCY. 
 
 
 DARCY. 
 
 Diam. = 0'188 m. 
 
 Diam. = '50 m. 
 
 = 0-1192 
 
 
 ^ = 0-0382 
 
 d* 
 
 
 d x 
 
 t probably 70 
 
 F. 
 
 t probably 70 F. 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Observ. 
 
 Calc. 
 
 
 Observ. 
 
 Calc. 
 
 497 
 
 090 
 
 078 
 
 0-7932 
 
 0-120 
 
 124 
 
 758 
 
 180 
 
 177 
 
 7951 
 
 125 
 
 125 
 
 1-128 
 
 385 
 
 384 
 
 1-0412 
 
 210 
 
 211 
 
 1-488 
 
 640 
 
 660 
 
 1-1135 
 
 230 
 
 240 
 
 1-933 
 
 1-090 
 
 1-098 
 
 1-1197 
 
 260 
 
 243 
 
 2-506 
 
 1-855 
 
 1-822 
 
 1-1278 
 
 250 
 
 247 
 
 4323 
 
 5-276 
 
 5-274 
 
 
 
 
 WROUGHT-IRON (GAS) PIPE 
 
 HAMILTON SMITH. 
 
 HAMILTON SMITH. 
 
 Diam. = -01 594 m. 
 
 Diam. = '02676 m. 
 
 1=1-948 
 
 S=l-055 
 
 d x 
 
 d x 
 
 n = l'75 
 
 w=l-75 
 
 t about 60 F. 
 
 t about 60 F. 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Observ. 
 
 Calc. 
 
 
 Observ. 
 
 Calc. 
 
 314 
 
 656 
 
 653 
 
 292 
 
 375 
 
 312 
 
 481 
 
 1-375 
 
 1-379 
 
 433 
 
 614 
 
 622 
 
 700 
 
 2-650 
 
 2-657 
 
 656 
 
 1-288 
 
 1-286 
 
 873 
 
 3-892 
 
 3-911 
 
 968 
 
 2-516 
 
 2-537 
 
 1-031 
 
 5-265 
 
 5-245 
 
 1-203 
 
 3-715 
 
 3-716 
 
 1-182 
 
 6-661 
 
 6-653 
 
 1-424 
 
 5-035 
 
 4-991 
 
 
 
 
 1-623 
 
 6-350 
 
 6-272 
 
INVESTIGATIONS OF FLOW IN PIPES 213 
 
 CLEANED CAST IRON 
 
 DARCY. 
 
 DARCY. 
 
 DAROY. 
 
 Diam. =0-0801 m. 
 
 Diam. = 0-2447 m. 
 
 Diam. = 0-297 m. 
 
 - = 0-3912 
 
 - = 0-1082 
 
 - = 0-0770 
 
 d* 
 
 d* 
 
 d* 
 
 7i = 2-0 
 
 7i = 2-0 
 
 7i=2-0 
 
 t probably 45 F. 
 
 t probably 60 F. 
 
 *=70F. 
 
 V 
 
 h 
 
 9 
 
 h 
 
 V 
 
 h 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 385 
 
 148 
 
 148 
 
 949 
 
 245 
 
 248 
 
 823 
 
 125 
 
 133 
 
 614 
 
 370 
 
 376 
 
 1-420 
 
 565 
 
 556 
 
 1-155 
 
 255 
 
 262 
 
 624 
 
 375 
 
 389 
 
 1-904 
 
 1-000 
 
 1-000 
 
 1-652 
 
 535 
 
 536 
 
 864 
 
 795 
 
 745 
 
 2-206 
 
 1-350 
 
 1-343 
 
 2-390 
 
 1-170 
 
 1-122 
 
 1-248 
 
 1-51 
 
 1-553 
 
 2-572 
 
 1-840 
 
 1-825 
 
 2-799 
 
 1-570 
 
 1-539 
 
 1-526 
 
 2-30 
 
 2-324 
 
 4-497 
 
 5-505 
 
 5-581 
 
 3-160 
 
 2-022 
 
 1-962 
 
 INCRUSTED CAST IRON 
 
 DARCY. 
 
 DARCY. 
 
 DARCY. 
 
 Diam. =0-359 m. 
 
 Diam. =0-0795 m. 
 
 Diam. = -2432m. 
 
 5 = 1-8154 
 
 -=0-6898 
 
 ?=0-1873 
 
 d x 
 
 d* 
 
 d* 
 
 *:%. 
 
 t probably 45 F. 
 
 t probably 68 F. 
 
 V 
 
 h 
 
 V 
 
 h 
 
 V 
 
 h 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 
 Obs. 
 
 Calc. 
 
 130 
 
 081 
 
 078 
 
 251 
 
 Ill 
 
 Ill 
 
 452 
 
 098 
 
 098 
 
 253 
 
 300 
 
 296 
 
 446 
 
 355 
 
 350 
 
 707 
 
 235 
 
 239 
 
 381 
 
 665 
 
 672 
 
 678 
 
 800 
 
 808 
 
 1-106 
 
 565 
 
 584 
 
 551 
 
 1-405 
 
 1-406 
 
 931 
 
 1-535 
 
 1-524 
 
 1-547 
 
 1-130 
 
 1-142 
 
 633 
 
 1-85 
 
 1-855 
 
 1-142 
 
 2-265 
 
 2-294 
 
 1-833 
 
 1-580 
 
 1-604 
 
 
 
 
 
 
 
 2-073 
 
 2-020 
 
 2-050 
 
 123. The correction for temperature. The correction 
 for temperature is at present imperfectly known. No 
 experiments on the resistance at different temperatures, with 
 very rough surfaces, have been made ; but, in the absence of 
 information, it has been thought better to correct all the 
 values of m/d x to a common temperature of 60, in accordance 
 
214 HYDKAULICS 
 
 CHAP. 
 
 with Mr. Mair's results. Variation of temperature in the 
 different experiments examined ranges from 38 F. to 75 F. 
 In most of the experiments the temperature was between 50 
 and 70. For 10 difference from 60, the temperature 
 correction is under 3 per cent, so that it does not make a 
 great difference whether the temperature correction is applied 
 or not. In some of Darcy's experiments the temperatures 
 are not given, but they can be inferred with some degree of 
 approximation from the dates given. 
 
 124. Variation of resistance with the diameter of the 
 pipe. From the values of m/d* which have been obtained, the 
 value of x, the index of the diameter in the expression for 
 the head lost in the pipe, can be found. If m and x for any- 
 given kind of pipe are strictly constant, and if we plot 
 logarithmic values of d as ordinates, and m/d x as abscissae, 
 then the points found should lie on a straight line, the slope 
 of which is the required value of x. Broadly speaking, the 
 points corresponding to each set of experiments fell pretty 
 closely on a straight line, those for the pipes with rougher 
 surfaces lying higher than those for the pipes with smoother 
 surfaces. It is not surprising that the lines are more irregular 
 than those previously plotted, for this reason. The points 
 in these lines correspond, not to a series of experiments on 
 one pipe, but to different series of experiments on different 
 pipes. Small differences of roughness in these pipes would 
 quite account for such discrepancies as were found. 
 
 On examining the lines, it was found that in all cases 
 the slope is greater than 1 to 1, so that the index x of d, 
 in the formula of loss of head, must be greater than unity, 
 a result in accordance with Darcy's deductions from his 
 experiments. The slope is lowest (1*10 to 1) for the tin- 
 plate pipes of Bossut, which were very smooth, and in which, 
 probably, the joints did not affect the flow so much as in 
 other pipes. Generally, the slope does not exceed T2 to 1 ; 
 but there are one or two exceptions. 
 
 The riveted wrought-iron pipes of Hamilton Smith give 
 a slope of 1'39 to 1, which may possibly be due to the 
 different relative effect of the obstruction of the rivet-heads 
 and joints in pipes of different diameters of this kind. 
 Putting aside exceptional values of the index x, the fact 
 
x INVESTIGATIONS OF FLOW IN PIPES 215 
 
 that all the other results give values of x lying between 
 I'lO and 1'21 shows a very satisfactory constancy in the 
 coefficient. 
 
 According to Professor Keynolds's formula, the head lost 
 should vary as 
 
 That is, x should have the value 3 n. The following table 
 shows how far this reduction of the most trustworthy experi- 
 ments confirms this law : 
 
 Kind of Pipe. 
 
 n 
 
 3-71 X 
 
 Tinplate .... 
 
 1-72 
 
 1-28 
 
 1-100 
 
 Wrought iron (Smith) . 
 
 1-75 
 
 1-25 
 
 1-210 
 
 Asphalted pipes . 
 
 1-85 
 
 1-15 
 
 1-127 
 
 Kiveted wrought iron . 
 
 1-87 
 
 1-13 
 
 1-390 
 
 New cast iron 
 
 1-95 
 
 1-05 
 
 1-168 
 
 Cleaned cast iron 
 
 2-00 
 
 1-00 
 
 1-168 
 
 Incrusted cast iron 
 
 2-00 
 
 1-00 
 
 1-160 
 
 It will be seen that there is no great discrepancy between 
 the values of x and 37i, but there is no appearance of 
 relation in the two quantities. For the present, at least 
 it must be assumed that the value of x is independent [o 
 the value of n. 
 
 125. Values of the coefficient ra. It is now possible 
 to determine the values of the coefficient ra from the different 
 series of experiments, using the values of d x , calculated from 
 the values of x now assigned. It will be a general check 
 on the whole of the preceding reductions, if the values of ra 
 for each particular kind of pipe prove to be nearly constant. 
 Hence the values of ra for each of the twenty-eight series 
 of experiments which have been discussed are here given. 
 They are placed generally in the order of the index n, and 
 each set of pipes of one general character is placed in the 
 order of the diameters. 
 
216 
 
 HYDKAULICS 
 
 CHAP. 
 
 Kind of Pipe. 
 
 Diam. 
 in 
 Metres. 
 
 Value 
 of m. 
 
 Mean 
 Value of 
 m. 
 
 Authority. 
 
 Tinplate 
 
 0-036 
 0-054 
 
 01697 
 01676 
 
 j -01686 
 
 Bossut, 
 
 Wrought iron . 
 
 0-016 
 0-027 
 
 01302 
 01319 
 
 | -01310 
 
 Hamilton Smith. 
 
 Asphalted pipes 
 
 0-027 
 
 01749 
 
 
 Hamilton Smith. 
 
 
 0-306 
 
 02058 
 
 
 Bonn, W. W. 
 
 
 0-306 
 
 02107 
 
 
 Bonn, W. W. 
 
 
 0-419 
 
 01650 
 
 01831 
 
 Lampe. 
 
 
 1-219 
 
 01317 
 
 
 Stearns. 
 
 
 1-219 
 
 02107 
 
 
 Gale. 
 
 Riveted wrought iron 
 
 0-278 
 
 01370 
 
 > 
 
 
 
 0-322 
 
 01440 
 
 
 
 
 0-375 
 
 01390 
 
 01403 
 
 Hamilton Smith. 
 
 
 0-432 
 
 01368 
 
 
 
 
 0-657 
 
 01448 
 
 
 
 New cast iron . 
 
 0-082 
 
 01725 
 
 
 
 
 0-137 
 0-188 
 
 01427 
 01734 
 
 01658 
 
 Darcy. 
 
 
 0-500 
 
 01745 
 
 
 
 Cleaned cast iron 
 
 0-080 
 
 01979 
 
 \ 
 
 
 
 0-245 
 
 02091 
 
 \ -01994 
 
 Darcy. 
 
 
 0-297 
 
 01913 
 
 J 
 
 
 Incrusted cast iron . 
 
 0-036 
 
 03693 
 
 \ 
 
 
 
 0-080 
 
 03530 
 
 03643 
 
 Darcy. 
 
 
 0-243 
 
 03706 
 
 
 
 Here, considering the great range of diameters and 
 velocities in the experiments, the constancy of m is very 
 satisfactorily close. The asphalted pipes give rather variable 
 values ; but, as some of these were new and some old, the 
 variation is, perhaps, not surprising. The incrusted pipes 
 give a value of m quite double that for new pipes, but that 
 is perfectly consistent with what is known of fluid friction 
 in other cases. 
 
 126. General mean values of constants. The general 
 formula 
 
 h m v n 
 
 I d* 2g 
 
 (10) 
 
 will be found to agree with the results with convenient 
 
INVESTIGATIONS OF FLOW IN PIPES 217 
 
 closeness, if the following mean values of the coefficients 
 are taken, the unit being a metre : 
 
 Kind of Pipe. 
 
 m 
 
 X 
 
 n 
 
 Tinplate .... 
 
 0169 
 
 1-10 
 
 1-72 
 
 Wrought iron . 
 
 0131 
 
 1-21 
 
 1-75 
 
 Asphalted iron . 
 
 0183 
 
 1-127 
 
 1-85 
 
 Riveted wrought iron . 
 
 0140 
 
 1-390 
 
 1-87 
 
 New cast iron . 
 
 0166 
 
 1-168 
 
 1-95 
 
 Cleaned cast iron 
 
 0199 
 
 1-168 
 
 2-0 
 
 Incrusted cast iron 
 
 0364 
 
 1-160 
 
 2-0 
 
 The variation of each of these coefficients is within a 
 comparatively narrow range, and the selection of the proper 
 coefficient for any given case presents no difficulty, if the 
 character of the surface of the pipe is known. 
 
 It only remains to give the values of these coefficients 
 when the quantities are expressed in English feet. For 
 English measures the following are the values of the 
 coefficients : 
 
 Kind of Pipe. 
 
 m 
 
 X 
 
 n 
 
 Tinplate . ... 
 
 0265 
 
 10 
 
 1-72 
 
 Wrought iron . . 
 
 0226 
 
 21 
 
 1-75 
 
 Asphalted iron . . > 
 
 0254 
 
 127 
 
 1-85 
 
 Riveted wrought iron . 
 
 0260 
 
 390 
 
 1-87 
 
 New, cast iron . . . 
 
 0215 
 
 ' -168 
 
 1-95 
 
 Cleaned cast iron '. . 
 
 0243 
 
 168 
 
 2-0 
 
 Incrusted cast iron 
 
 0440 
 
 160 
 
 2-0 
 
 If formula (10) is put in the form 
 
 id'- 1 2g~' 
 
 it is seen that the coefficient f in the Chezy formula can be 
 deduced from these results by taking 
 
 Values of f thus obtained have been given in Chapter VIII. 
 91. Using these values in the Chezy formula the 
 results are nearly as accurate as if eq. (10) is used. 
 
218 
 
 HYDKAULICS 
 
 CHAP. 
 
 127. Distribution of velocity in the cross section of a 
 pipe. The mean velocity of translation along a pipe is 
 necessarily 
 
 Strictly, in consequence of the turbulence of the motion, the 
 velocity and direction of motion vary from moment to moment 
 at every point of the cross section. But at each point the 
 variations are temporary^ fluctuations about a fixed mean value. 
 The mean direction must be parallel to the axis of the pipe, 
 and at each point there must be a constant mean velocity in 
 that direction. Observation shows that these mean velocities 
 at different points are greater near the centre of the cross 
 section and less towards its boundary. Messrs. Williams, 
 Hubbel, and Fenkel found the mean velocity v m of the whole 
 cross section to be 0'84 of the central mean velocity, and the 
 mean velocity near the boundary to be '5 of the central mean 
 velocity. At a radius 0'75 of the radius of the pipe the 
 velocity was equal to the mean velocity v m of the whole cross 
 section. 
 
 The most exact research on the distribution of velocity in 
 pipes is one made by Bazin on a cement pipe 0*8 metre 
 diameter and 8 metres long (" Experiences nouvelles," 
 M6m. de I'Acaddmie des Sciences, xxxii., 1897). Let E be 
 the radius of the pipe, and r the radius at which the velocity is 
 observed ; let V be the maximum velocity at the centre, v the 
 velocity at radius r, and v m the mean velocity for the whole 
 cross section. Bazin obtained the following results : 
 
 r 
 R 
 
 V 
 
 V-v 
 
 v m 
 
 V 
 
 
 
 1-1675 
 
 
 
 0-125 
 
 1-1605 
 
 0070 
 
 0-250 
 
 1-1475 
 
 0200 
 
 0-375 
 
 1-1258 
 
 0417 
 
 0-500 
 
 1-0923 
 
 0752 
 
 0-625 
 
 1-0473 
 
 1202 
 
 0-750 
 
 1-0008 
 
 1667 
 
 0-875 
 
 0-9220 
 
 2455 
 
 0-937 
 
 0-8465 
 
 3210 
 
 1-000 
 
 0-7415 
 
 4260 
 
x INVESTIGATIONS OF FLOW IN PIPES 219 
 
 Let i be the virtual slope of the pipe. Then 
 
 At 
 
 where k varies from 33 to 42, and is on the average 38. 
 the sides, where r = K, the velocity is w V S 
 The mean velocity of the whole cross section is 
 
 On the average V/v m = 1*24 ; v m jN = 0*8 ; w/v m = 0'64, and 
 w/V = 0'51. At radius 0'74R the velocity is equal to v m . 
 
 Fig. 105. 
 
 Fig. 105 shows the velocities at different radii found by 
 Bazin. 
 
 128. Influence of temperature on the resistance in 
 pipes. In the experiments on discs, 82, it appeared that 
 the frictional resistance diminished as the temperature 
 increased. Froude found a similar result for boards towed 
 in water. Some experiments on flow of water at different 
 temperatures in a brass pipe 1^- inch diameter and 25 feet 
 long were made by Mr. J. G. Mair (Proc. Inst. of Civil 
 Engineers, Ixxxiv.). The head at inlet was taken at 1 2 inches 
 from the end of the pipe, to exclude loss at entry. The results 
 agree extremely closely with the equation 
 
 77? 
 
 ,,1-795 
 
 2g 
 
 The values of ra were as follows : 
 
220 HYDKAULICS CHAP, x 
 
 Temperature F. m 
 
 57 0-0178 
 
 70 169 
 
 80 166 
 
 90 161 
 
 100 157 
 
 110 151 
 
 120 147 
 
 130 145 
 
 160 133 
 
 The resistance is therefore 25 per cent less at 160 than at 
 57. The resistance varies directly as m, and m is given very 
 closely by the empirical relation 
 
 77i = 0-02(1 -0-002150- 
 
CHAPTER XI 
 
 FLOW OF COMPRESSIBLE FLUIDS IN PIPES 
 
 129. Notation. Let 
 
 P = absolute pressure in Ibs. per square foot. 
 T = absolute temperature F. 
 G = weight of one cubic foot of fluid in Ibs. 
 V = volume of one pound of fluid in cubic feet. 
 u, v, = velocities in feet per second. 
 W = weight of fluid per second in Ibs. 
 n = area of cross section of pipe in square feet. 
 d = diameter of pipe in feet. 
 L, /, = length of pipe in feet. 
 
 E = constant in gaseous equation. 
 
 When air flows along a pipe there is necessarily a fall of 
 pressure due to the resistance of the pipe, and consequently 
 the volume and velocity of the air increase going along the 
 pipe in the direction of motion. The effect of the resistance 
 is to create eddying motions which, as they subside, give back 
 to the air the heat equivalent of the work expended in pro- 
 ducing them. The result is that, apart from conduction 
 through the walls of the pipe, the flow is isothermal. 1 
 
 130. Flow in pipes under small differences of 
 pressure. In a large number of cases the pressure in a fluid 
 is one atmosphere or more, but the difference of pressure 
 causing flow is only a few inches of water. This is the case 
 in the distribution of lighting gas and in some cases of 
 compressed air transmission. Let P x , P 2 be the absolute 
 
 1 This was pointed out by the author in a discussion on Pneumatic 
 Transmission in 1875 (Proc. Inst. C. E. xliii.). The formula for air -flow in 
 this chapter was first given by the author in 1875 in a paper on the "Motion of 
 Light Carriers in Pneumatic Tubes " in the same volume. 
 
 221 
 
222 HYDEAULICS CHAP. 
 
 pressures at the inlet and outlet of a pipe. Then when 
 P! P 2 is small compared with P 1? the variation of density 
 during flow may be neglected without great error and the 
 hydraulic formulae are applicable. 
 
 Let d be the diameter, I the length of the pipe in feet, v 
 the velocity, P x P 2 the pressure difference causing flow in 
 Ibs. per square foot, and h the same pressure difference in feet 
 of the fluid itself. If G- is the weight of the fluid in Ibs. per 
 cubic foot, Pj P 2 = Gh. Then, as in 85, 
 
 (i). 
 
 2gdh\ l(2gd(P, - P 9 )l , 
 ~77T I - * / 1 ~TH n f * eefc P er second 
 4Y / \/ ^v ^ ' 
 
 Q = -d 2 v cubic feet per second 
 
 If T is the absolute temperature F., then, by the gaseous 
 equation 72, 
 
 a=p 1 /(RT). 
 
 If h w is the pressure difference measured in inches of 
 water, then 
 
 Example. Air initially at one atmosphere and 60F. (521 absolute) 
 flows through a 12 -inch pipe one mile long under a pressure difference of 
 10 inches of water. G = 21 16'3/(53'2 x521) = O0764 Ibs. per cubic foot. 
 Pj - P 2 = 5-2 x 10 = 52 Ibs. per square foot. The value of f may be taken 
 At 0-004. Then 
 
 22'77 feet per second. 
 
 The discharge is 0-7854 x 22-77 = 17-88 cubic feet per second, or 17-88 
 x 0-0764 = 1-367 Ibs. of air per second. 
 
 131. Flow of lighting gas in mains. Lighting gas is 
 distributed in cast-iron mains under pressure differences of 
 about 2 inches of water column per mile of main, or 2 x 5'2 
 = 10*4 Ibs. per square foot. The velocity is generally not 
 more than about 15 feet per second. In such conditions 
 the hydraulic formulae are applicable with very little error. 
 
 Pressures in gas mains are usually measured by water 
 
FLOW OF COMPEESSIBLE FLUIDS IN PIPES 223 
 
 siphon gauges open to the atmosphere. They indicate there- 
 fore the excess of pressure in the main over atmospheric pres- 
 sure. If h w is the gauge pressure in inches of water, and the 
 atmospheric pressure is 34 feet of water, then the absolute 
 pressure in the main is 34 + -^h w feet of water, or 62'4 (34 
 
 = 2121 + 5'2h w Ibs. per square foot. 
 Head lost in a horizontal main. Let Fig. 106 represent 
 a length I of horizontal main through which gas of density s 
 (air=l) is flowing. The difference y^y^ of the water 
 columns in the siphon gauges is the head lost in the length I. 
 
 B 
 
 y 
 
 Fig. 106. 
 
 Let G w G , G g be the weights in Ibs. per cubic foot of water, 
 air, and gas respectively. Then -G^ = sG a , where for ordinary 
 conditions of pressure and temperature G a = 0'08 nearly, and 
 Gr v = 62*4. Then if y lt y z are measured in inches of water, 
 the height of a column of gas equivalent to y^ y 2 is 
 
 feet 
 
 (2), 
 
 and this introduced in the hydraulic equations (1) will give 
 the velocity of flow and discharge. 
 
 Head lost in an inclined gas main. In a falling main 
 (Fig. 107) the atmospheric pressure is greater at B than at A 
 by the amount G a (z l z 2 ) Ibs. per square foot, or 
 
224 
 
 HYDKAULICS 
 
 CHAP. 
 
 water 
 
 a quantity which is negative for a rising main. Hence, 
 taking y l y 2 in feet, the head causing flow in feet of gas is 
 
 -z fl 
 
 (3). 
 
 Taking the values given above, and now supposing y l and y 2 
 given in inches of water, 
 
 / -I \ 
 
 . . (3a). 
 
 This is the value of h to be used in the hydraulic equations ( 1 ). 
 When there is much difference of level of A and B, the last 
 
 Fig. 107. 
 
 term is too large to be neglected. In some rising mains the 
 difference shown by the siphons is negative. 
 
 The coefficient of friction in gas mains. Unfortunately 
 there are very few experiments on the friction in gas mains, 
 
xi FLOW OF COMPKESSIBLE FLUIDS IN PIPES 225 
 
 and even those which are available are not very satisfactory. 
 A discussion by the author of such results as are available 
 (Proc. Inst. of Gas Engineers, 1904) led him to adopt pro- 
 visionally the following value : 
 
 This gives higher values of f than those deduced from tests 
 of air mains, but on the other hand gas mains are rather 
 more roughly jointed, and there were probably in the mains 
 tested some special resistances due to bends, etc. 
 
 VALUES OF f FOR GAS MAINS 
 
 Diameter of Pipe. f 
 
 2 inches 0-0082 
 
 6 -0057 
 
 12 -0050 
 
 18 -0048 
 
 24 -0047 
 
 Examples. Let 50,000 cubic feet of gas per hour, or 13-9 cubic feet 
 per second, of density s = 0-4, be conveyed in a horizontal main, and let 
 it be required to find the pressure head lost in friction per mile of main. 
 
 (a) Let the main be 8 inches or 0-666 foot in diameter. Then 
 =0-0044(1 +0-2 14) = 0-005 3. The cross section of the main is 0-349 
 square foot, and the velocity is 13-9/0-349 = 40 feet per second. Using 
 eq. (1), 
 
 h = 0*0053 
 
 of gas. Taking air in these conditions to weigh 0*08 Ib. per cubic foot, 
 the gas weighs 0-4 x 0-08 = 0-032 Ib. per cubic foot Hence the pressure 
 difference required per mile of main is 0-032 x 4178 = 133-8 Ibs. per 
 square foot, or 133-8/5-2 = 25-7 inches of water. 
 
 (6) If the main is sixteen inches or 1-333 feet in diameter, the other 
 conditions being the same, f =0*0044(1 + 0-107) = 0-00487. The cross 
 section of main is 1-395 square feet. The velocity is 13-9/1-395 = 10 feet 
 per second. 
 
 of gas, equivalent to 3-84 Ibs. per square foot, or 0'74 inch of water 
 per mile of main. 
 
 (c) If in the case of the 8 -inch main in (a), the outlet end is 150 feet 
 above the inlet, the frictional loss is the same, but there is a difference 
 of the pressures at the siphon gauges. Using eq. (3a), 
 
 15 
 
226 HYDEAULICS 
 
 CHAP. 
 
 = 162- 
 
 2/i ~ 2/2 = 24-33 inches of water. 
 
 (d) Similarly, if in the case of the 16-inch main in (&), the outlet 
 end is 1 50 feet above the inlet, 
 
 2/i - 2/2= - 0-647 inch of water. 
 That is, the upper siphon-gauge pressure would be greater than the lower. 
 
 132. Flow of air in a long uniform pipe, when the 
 variation of density is taken into account. In this case 
 the velocity increases along the pipe as the density diminishes. 
 The work of expansion of the fluid is not negligible. The 
 expansion will be taken to be isothermal. 
 
 For air, P/G=53'2T ( 72), and if the temperature is 
 60 F., so that T = 521, then P/G= 27710. 
 
 In steady flow the same weight of air must pass every 
 section in any given time. Let W be the weight of air 
 flowing per second, u the velocity, and O the area of 
 cross section. ~ p 
 
 ^-* T-?T^ ytjy 
 
 Consider a short length dl of the pipe, Fig. 108, between 
 transverse sections A A r Let d be 
 
 *--oll * the diameter, U the cross section, m 
 
 . -K the hydraulic mean radius. Let P 
 \f and u be the pressure and velocity 
 ; at A Q , P + dP, and u -f du the corre- 
 --^ spending quantities at A x . Let W 
 oo 1^1 b e tne weight of air flowing per 
 
 Fi 108 second units feet and pounds. 
 
 If in a short time dt the mass 
 
 A Aj_ comes to A^A'j, then A A' = udt and AjA'j = (u + du)dt. 
 Since in a short length the change of density is small the 
 head lost in feet of fluid is 
 
 or if H = v?/2g, the head lost in friction is 
 
 fHM/m feet . . . (6). 
 
xi FLOW OF COMPRESSIBLE FLUIDS IN PIPES 227 
 
 And since Wdt Ibs. flow in the time dt, the work expended in 
 friction is 
 
 - i-Wdl dt ft.-lbs. (7). 
 
 771 x ' 
 
 The change of kinetic energy in the time dt is the 
 difference of the kinetic energy of AjA'j, and A A' , that is 
 
 = ududt = WdHdt ft.-lbs. . (8). 
 
 9 
 
 The work of expansion of fludt cubic feet of air to 
 l(u + du}dt at a pressure initially P is SiPdudt. But 
 from (5) 
 
 RTW 
 
 nar 
 
 *OL- RTW 
 
 dP~ "" 
 
 and the work of expansion is 
 
 RTW 
 - ^^dPdt ft.-lbs. . (9). 
 
 The work of gravity is zero if the pipe is horizontal, and in 
 many other cases is negligible. 
 
 The work of the pressures on the ends of the mass is 
 
 P&udt - (P + dP)Q(u + du)dt 
 = - (Pdu + ud?)Qdt. 
 
 But if the temperature is constant, ~Pu is constant, and 
 ~Pdu + udP = Q. Hence the work of the pressures is zero. 
 Adding the quantities of work and equating them to the 
 change of kinetic energy, 
 
 dPdt - C- Wdldt 
 r m 
 
 prp TT 
 
 -dP + i-dl = Q 
 P m 
 
228 HYDRAULICS CHAP. 
 
 But 
 
 KTW 
 
 tf R 2 T 2 W 2 
 ~~ 
 
 For pipes of uniform section, II and m are constant, for 
 steady motion W is constant, and for isothermal flow T is 
 constant. Integrating, 
 
 012 2 P 2 I 
 
 log H + 2 + C = constant ' 
 
 For 
 
 I = 0, let H = H x and P = P L 
 
 I = L, let H = H 2 and P = P 2 
 
 where P x is the greater and P 2 the less pressure. By replacing 
 H,, H 2 , and W, 
 
 Hence the initial velocity in the pipe is 
 
 v%/{/*rir~ : ^\} (13) - 
 
 \. ^ - 1 - 1 / c. - 1 - 1 1 *- 1 \ S 
 
 When L is great, log Pj/P 2 is small compared with the other 
 term in the bracket. Then 
 
 ' ( } * 
 
 For pipes of circular section and diameter d in feet, m = d/4. 
 Let T= 521, then for air RT = 27710, and let p lf p 2 be the 
 pressures in Ibs. per square inch. Then 
 
FLOW OF COMPEESSIBLE FLUIDS IN PIPES 229 
 
 This equation is easily used. In some cases the approxi- 
 mate equation 
 
 ^ = (1-1 32 -0-726^) /(222900:0 . (13c) 
 
 may be more convenient. 
 
 If the terminal pressure p. 2 is required in terms of the 
 initial pressure p l} then 
 
 / , *... 9T 
 
 . . (U). 
 
 22290<W/ 
 
 133. Variation of pressure and velocity in long air 
 mains. The following cases have heen calculated to give an 
 idea of the way in which pressure and velocity vary in long 
 mains conveying air. The main is assumed to be 1 2 inches in 
 diameter, and the coefficient of friction to be f= 0'003. 
 
 AIR MAINS 
 
 CASE I. 
 Pressure (absolute) 
 in Ibs. per sq. in. 
 Velocity in main in 
 ft. per sec. 
 
 CASE II. 
 
 Pressure (absolute) 
 in Ibs. per sq. in. 
 Velocity in main in 
 ft. per sec. 
 
 At distances along main in miles. 
 
 
 
 115 
 25 
 
 115 
 50 
 
 1 
 
 112 
 25-6 
 
 104 
 55-1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 110 
 26-2 
 
 92 
 62-3 
 
 107 
 26-9 
 
 79 
 73-2 
 
 104 
 27-6 
 
 62 
 93-1 
 
 101 
 28-4 
 
 38 
 149-0 
 
 99 
 29-2 
 
 
 
 00 
 
 96 
 29-9 
 
 92 
 31-2 
 
 89 
 32-3 
 
 86 
 33'6 
 
 
 
 
 
 
 
 
 
 With an initial velocity of 2 5 feet per second the pressures 
 decrease and the velocities increase slowly. With an initial 
 velocity of 50 feet per second the variation of pressure and 
 velocity is much more rapid. Beyond 5 miles the pressure is 
 very small and the velocity enormous. 
 
 134. Coefficient of friction. The author obtained values 
 of the coefficient of friction from experiments made by 
 Professors Biedler and Gutermuth on the mains conveying 
 compressed air in Paris. 1 The mains were llf inches in 
 
 1 The details are given in Unwin, Development and Transmission of Power. 
 London, 1894. 
 
230 HYDKAULICS CHAP, xi 
 
 diameter, and in some tests the length of main tested was 
 10 miles. Experiments also were made by Mr. Stockalper 
 on the compressed air mains at the St. Gothard tunnel, which 
 were 0*492 and 0'656 feet in diameter. 1 
 
 = 
 
 Mean for 0'492 foot pipe . . . '004 49 
 0-656 ... -00377 
 
 0-980 ... -00290 
 
 These results agree with the relation 
 
 { = 0-0027 (l + A) . . (15 ). 
 
 Mr. Batcheller, who has developed and carried out the 
 remarkable systems of pneumatic transmission of parcels in 
 the United States, has also made careful experiments on the 
 resistance to the flow of air in mains. The pipes used were 
 cast-iron pipes bored smooth. 
 
 Air is supplied at a pressure of 6 Ibs. per square inch, 
 and a carrier weighing 1 Ib. 7 oz. passed through with the 
 air. For a main 6-J- inches or 0*51 foot diameter the mean 
 
 o 
 
 value of the coefficient of friction was 0*00435. By the 
 formula above it would be 0'00429. 
 
 The coefficient is applicable to gases of other densities. 
 
 135. Distribution of velocity in an air main. Threl- 
 fall has made experiments on the distribution of velocity in 
 air mains by means of a Pitot tube (Proc. Inst. of Electr. Eng. 
 1903; Proc. Inst. Mech. Eng. 1904). The average ratio of 
 mean to maximum central velocity was 0*873 constant at 
 different velocities. The velocity at 0*775 of the radius from 
 the centre was equal to the mean velocity. The highest 
 velocity tried was 60 feet per second. The velocity curve on 
 a diameter approximates to an ellipse. 
 
 1 Min. Proc. Inst. Civil. Eng. Ixiii. 29. 
 
CHAPTEK XII 
 
 UNIFORM FLOW OF WATER IN CANALS AND CONDUITS 
 
 136. IN flow through pipes the section of the stream of 
 water is determined by the cross section of the pipe, and the 
 velocity depends not on the actual slope of the pipe but on 
 that of the hydraulic gradient. When water flows along open 
 channels, its surface is parallel to the bed of the stream, or 
 nearly so, and the velocity depends on the actual slope of the 
 surface of the water. If the slope of the stream-bed varies, 
 the velocity of the stream varies also, being greater where the 
 slope is greater, and vice versa. Since in steady motion the 
 same quantity of water must pass every cross section in a 
 given time, the cross sections of the stream must vary inversely 
 as the velocity, being less where the slope is greater and greater 
 where the slope is less. 
 
 In artificial canals and conduits for conveying water the 
 slope is constant, and the cross sections of the channel are all 
 similar. In such cases the velocity is uniform, the cross 
 sections of the water stream normal to the direction of flow 
 are equal and similar, and the water surface is parallel to 
 the bed. 
 
 137. Steady flow of water in channels of constant 
 slope and section. Let aa'W (Fig. 109) be two normal 
 cross sections at a distance dl. Since aa'lib' moves uniformly, 
 the forces acting on it are in equilibrium. Let fl be the area 
 of cross section, ^ the wetted perimeter pq + qr + rs, and 
 w = n/x the hydraulic mean depth. Let v be the mean 
 velocity, i the slope be jab in feet per foot, W = Gldl the 
 weight of aa'bb'. 
 
 The external forces acting on aa'lfo' parallel to the direc- 
 
 231 
 
232 
 
 HYDKAULICS 
 
 tion of motion are (a) the pressures on CM' and bb f , which 
 are equal and opposite since the sections are equal and similar ; 
 (>) the component of W parallel to ab, that is Gfldl x the 
 cosine of the angle between W and ab, or Gfldl cos abc 
 = Gtlidl ; (c) the friction on the surface of the channel. 
 
 Fig. 109. 
 
 This is proportional to the wetted area ydl and to a function 
 of the velocity which may be written f(v), where f(v) is the 
 friction per square foot at the velocity v. Hence the fractional 
 resistance is xdlf(v). Equating the sum of the forces to zero 
 
 GQidl - x dlf(v) = 0. 
 
 &} - ? J 
 G X 
 
 But it has been shown in 79 that f(v) = fG ^-, and hence 
 
 -1,2 
 
 or if 
 
 . (i); 
 (2); 
 
 where 5" an( l c are coefficients depending on the size of the 
 channel and its roughness, and to a smaller extent on the 
 velocity. This is the Chezy formula previously found for 
 flow in pipes ( 85). 
 
 In the case of open channels there is a much greater 
 variation of size and of roughness than in the case of pipes, 
 and consequently a wide variation of values of f and c must 
 be expected in different cases. Imperfect as the theory above 
 is, as a theory of flow, the formula is very convenient in 
 
in UNIFOKM FLOW OF WATER 233 
 
 practical calculations, and it can be made to give accurate 
 results if the values of f and c are those found by experiment 
 in similar cases. Hence the practically useful problem is to 
 find means of selecting values of f and c in any given case. 
 
 138. Darcy's research on the value of f for open 
 channels. M. Darcy carried out an extremely important 
 series of gaugings of the flow in artificial channels of very 
 varied character, and M. Bazin, his successor, continued the 
 investigation after his death. The conclusion arrived at was 
 that the value of f depended chiefly on the roughness of the 
 channel and its size, being less for large channels and greater 
 for small ones. It appeared that the influence of size could 
 be provided for by taking for f the expression 
 
 an expression similar to that previously found for pipes. To 
 take account of the roughness of the channels, of which there 
 is no definite measure, Darcy adopted a classification of 
 channels according to their roughness. The following table 
 gives the values of a and yS for the different categories in 
 which channels were classed : 
 
 Kind of Channel. 
 
 a 
 
 ft 
 
 I. Very smooth channels, sides of smooth cement 
 or planed timber. 
 II. Smooth channels, sides of ashlar, brickwork, 
 planks. 
 III. Rough channels, sides of rubble masonry or 
 pitched with stone. 
 IV. Very rough canals in earth . 
 V. Torrential streams encumbered with detritus . 
 
 0-00294 
 0-00373 
 0-00471 
 
 0-00549 
 0-00785 
 
 o-io 
 
 0-23 
 0-82 
 
 4-10 
 5-74 
 
 The last values (Class V.) are not Darcy's, but are taken from experi- 
 ments by Ganguillet and Kutter on Swiss streams. 
 
 The following tables give the values of f calculated from 
 Darcy's eq. (3) for use in eq. (1), and the corresponding values 
 of c for use in eq. (2) : 
 
234 
 
 HYDEAULICS 
 
 DAROY'S VALUES OF 
 
 Hydraulic 
 
 Values of for Categories 
 
 Mean Depth 
 
 
 m 
 in Feet. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 0-5 
 
 00353 
 
 00545 
 
 01243 
 
 0505 
 
 0981 
 
 1 
 
 00323 
 
 00458 
 
 00857 
 
 0279 
 
 0529 
 
 2 
 
 00308 
 
 00414 
 
 00664 
 
 0167 
 
 0304 
 
 5 
 
 00300 
 
 00389 
 
 00546 
 
 0100 
 
 0169 
 
 10 
 
 00297 
 
 00380 
 
 00508 
 
 0077 
 
 -0123 
 
 20 
 
 00295 
 
 00376 
 
 00489 
 
 0065 
 
 0101 
 
 50 
 
 00294 
 
 00374 
 
 00477 
 
 0059 
 
 0087 
 
 oo 
 
 00294 
 
 00373 
 
 00471 
 
 0055 
 
 0079 
 
 VALUES OF c IN THE EQUATION v = c \/mi DEDUCED FROM 
 DARCY'S VALUES 
 
 Hydraulic 
 Mean Depth 
 
 in Feet. 
 
 Values of c for Categories 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 0-5 
 
 135 
 
 109 
 
 72 
 
 36 
 
 26 
 
 1-0 
 
 141 
 
 119 
 
 87 
 
 50 
 
 35 
 
 2-0 
 
 145 
 
 125 
 
 98 
 
 62 
 
 46 
 
 5-0 
 
 146 
 
 129 
 
 109 
 
 80 
 
 62 
 
 10-0 
 
 147 
 
 130 
 
 113 
 
 91 
 
 72 
 
 20-0 
 
 148 
 
 131 
 
 114 
 
 100 
 
 80 
 
 50-0 
 
 148 
 
 131 
 
 116 
 
 104 
 
 86 
 
 oo 
 
 148 
 
 131 
 
 117 
 
 108 
 
 90 
 
 139. Ganguillet and Kutter's Formula. In 1869, the 
 Swiss engineers Messrs. Ganguillet and Kutter undertook a 
 careful analysis of the results of gaugings of open channels 
 then available. Proceeding in a purely empirical way to fit 
 a formula to the results of gaugings, they arrived at the 
 following cumbrous formula : 
 
 41 . 6 
 
 0*0281 
 
 . (4), 
 
xii UNIFOKM FLOW OF WATER 235 
 
 in which n is a "coefficient of roughness," and the other 
 symbols have the same signification as above. They adopted 
 Darcy's method of classifying channels according to roughness, 
 and arrived at the values of n given in the following table : 
 
 KUTTER'S CONSTANT n. 
 
 n = 0-009. Well-planed timber. 
 = 0-010. Pure cement plaster, coated clean pipes. 
 = 0-011. Plaster in cement, iron pipes in best order. 
 = 0*012. Channels of unplaned timber. 
 
 = 0-013. Ashlar and good brickwork, iron pipes in ordinary condition. 
 = 0-015. Rough brickwork, incrusted iron. 
 = 0-017. Brickwork, ashlar, in bad condition, rubble in cement in 
 
 good order. 
 
 = 0-020. Rough rubble in cement, stone pitching. 
 = 0"025. Rivers and canals in perfect order, free from stones or weeds, 
 
 stone pitching in bad condition. 
 = 0*030. Rivers and canals in good order. 
 = 0-035. Rivers and canals in bad order. 
 = 0-050. Torrential streams encumbered with detritus. 
 
 In spite of its complication, Ganguillet and Kutter's formula 
 has been widely adopted, especially in India, where its use has 
 been facilitated by the publication of extensive tables. 
 
 A formula with so many arbitrary constants can of course 
 be made to agree with any selected set of results of gauging 
 more closely than a simpler formula. But the formula has 
 only the authority of the results used in obtaining it. If 
 some of these are untrustworthy, the formula must be untrust- 
 worthy also. Now the term 0'00281/i was introduced chiefly 
 to force the formula into agreement with certain gaugings 
 of the Mississippi, with very large values of ra and small values 
 of i. Those gaugings were made by the method of double 
 floats, and it is now known that the velocities so obtained are 
 probably greater than the true velocities. 
 
 00281 
 
 Let 41-6 + - -=k. 
 
 i 
 
 Then, as Bazin has shown, the formula can be put in the 
 form 
 
 Jmi 1_ kn ( 1 1 V 
 
 nv 1-811 *+ 1-811 WOT 1*811/ J 
 
236 
 
 HYDKAULICS 
 
 CHAP. 
 
 and if x/ra =1-811, or m=3'28 feet or one metre exactly, 
 then *Jmijv is equal to 7i/l*811 for all classes of channels. 
 That is, at this arbitrary limit \fmijv is independent of the 
 term involving the slope in all cases, and the influence of the 
 term in brackets is + or according as m is > or < one metre. 
 This result is improbable. Further, the comparison which 
 Bazin has made of the formula, with a more extensive list of 
 gaugings than were available when it was deduced, shows that 
 it departs widely in some cases from the results of experiment. 
 Calculation by Kutter's formula is a little facilitated if 
 the equation is put in the form 
 
 M = 1^41-6 
 
 >/m/M + 
 
 0-00281X 
 i ) 
 
 vW; 
 n V 
 
 
 \'mi. 
 
 (5). 
 
 
 
 j 
 
 
 
 Values of M for n 
 
 6 
 
 o-oio. 
 
 0-012. 
 
 0-015. 
 
 0-017. 
 
 0-020. 
 
 0-025. 
 
 0-030. 
 
 00001 
 
 3-2260 
 
 3-8712 
 
 4-8390 
 
 5-4842 
 
 6-4520 
 
 8-0650 
 
 9-6780 
 
 00002 
 
 1-8210 
 
 2-]852 
 
 2-7315 
 
 3-0957 
 
 3-6420 
 
 4-5525 
 
 5-4630 
 
 00004 
 
 1-1185 
 
 1-3422 
 
 1-6777 
 
 1-9014 
 
 2-2370 
 
 2-7962 
 
 3-3555 
 
 00006 
 
 0-8843 
 
 1-0612 
 
 1-3264 
 
 1-5033 
 
 1-7686 
 
 2-2107 
 
 2-6529 
 
 00008 
 
 0-7672 
 
 0-9206 
 
 1-1508 
 
 1-3042 
 
 1-5344 
 
 1-9180 
 
 2-3016 
 
 00010 
 
 0-6970 
 
 0-8364 
 
 1-0455 
 
 1-1849 
 
 1-3940 
 
 1-7425 
 
 2-0910 
 
 00025 
 
 0-5284 
 
 0-6341 
 
 0-7926 
 
 0-8983 
 
 1-0568 
 
 1-3210 
 
 1-5852 
 
 00050 
 
 0-4722 
 
 0-5666 
 
 0-7083 
 
 0-8027 
 
 0-9444 
 
 1-1805 
 
 1-4166 
 
 00075 
 
 0-4535 
 
 0-5442 
 
 0-6802 
 
 0-7709 
 
 0-9070 
 
 1-1337 
 
 1-3605 
 
 00100 
 
 0-4441 
 
 0-5329 
 
 0-6661 
 
 0-7550 
 
 0-8882 
 
 1-1102 
 
 1-3323 
 
 00200 
 
 0-4300 
 
 0-5160 
 
 0-6450 
 
 0-7310 
 
 0-8600 
 
 1-0750 
 
 1-2900 
 
 00300 
 
 0-4254 
 
 0-5105 
 
 0-6381 
 
 0-7232 
 
 0-8508 
 
 1-0635 
 
 1-2762 
 
 The formula can, however, hardly be used in practical work 
 without the aid of extensive tables. 
 
 140. Bazin's later investigation of the results of 
 experiments on flow in channels. M. Bazin has lately 
 returned to the study of the results of gaugings on flow in 
 channels, and has examined a more extensive series than has 
 previously been available (Annales des Fonts et Chausstes, 
 1897). He remarks that in the Darcy relation 
 
xii UNIFORM FLOW OF WATER 237 
 
 mi /, B\ 
 
 -^ =(!+). ... (6) 
 
 which proved to be suitable for pipes, the constants a and & 
 have no very wide range of values so long as the experiments 
 on pipes only are considered. But in the case of open 
 channels, with their great diversity of size and character of 
 surface, the constants a and ft have so wide a range of values 
 that the expression ceases to be sufficiently useful as a guide. 
 In addition, the form of the expression is defective. For if 
 m increases indefinitely, mi/v 2 = a, and this has a different 
 value for each class of channels. But it is reasonable to 
 suppose that in indefinitely large channels the influence of the 
 roughness of the stream bed must indefinitely diminish, so that 
 in very large channels mi/v 2 should tend to a value common 
 to all classes of channels. 
 
 After many trials, M. Bazin has adopted the following 
 relation, which obviates the difficulty just stated : 
 
 in which the constant a has the same value, 0*00635 (English 
 measures), for all classes of channels, and ft varies with the 
 character of the surface of the bed. 
 
 If the results of gauging are plotted so that the ordinates 
 y = ^/ m ijv, and the abscissae x= I/ ^/m, the expression may 
 be written 
 
 y = 0-00635 + fix, 
 or if 7 = ft j a 
 
 y = 0-00635(1 +yx) . . . (8), 
 
 the equation to a straight line. Eesults of this equation 
 plotted give a pencil of rays starting from a?=0,y = 0'00635. 
 The inclination measured by the angular coefficient 0*00635y 
 increases as the roughness of the bed increases. Fig. 110 
 shows a plotting. 
 
 The following are Bazin's values of the roughness coefficient 
 7 in eq. (8) : 
 
238 
 
 HYDKAULICS 
 
 CHAP. 
 
 BAZIN'S VALUES OP y 
 
 I. Very smooth. Smooth cement, planed timber . . 
 
 II. Smooth. Planks, ashlar, brick .... 
 
 III. Rough. Rubble masonry ..... 
 
 III. bis. Rough. Earth newly dressed, or pitched in whole 
 
 or part with stone ...... 
 
 IV. Very rough. Ordinary earth canals . . . 
 V. Excessively rough. Canals encumbered with weeds or 
 
 boulders ........ 
 
 y = 0'109 
 y = 0-290 
 y = 0'83 
 
 y = l'54 
 = 2'36 
 
 For practical calculations Bazin's new formula can be put 
 in the form 
 
 157-6 v^ 
 
 -. . . . (9). 
 1 + ' 
 
 N/ 
 
 m 
 
 In this form the equation is extremely convenient for 
 calculation. If m is known and v is to be found, the equation 
 
 030 
 
 0-2S 
 16-0 
 
 0-50 
 4-O 
 
 0-75 
 1-78 
 
 1-00 
 1-00 
 
 1-25 
 0-64 
 
 Fig. 110. 
 
 can be used quite straightforwardly. If v is given and the 
 dimensions of the channel are to be found, it is best to proceed 
 by approximation. Choose from tables or experience any 
 roughly probable value of m. With this calculate 1 -\-y/ ^/m, 
 and with this find a new value of m by eq. (9). With 
 this new value recalculate 1+y/^/m, and then find a more 
 
XII 
 
 UNIFOKM FLOW OF WATEE 
 
 239 
 
 approximate value of m by eq. (9). These two steps of 
 approximation are generally sufficient. 
 
 It will be seen that the Chezy form of equation 
 
 v = c 
 is identical with Bazin's, if 
 
 or 
 
 c = 
 
 157-6 
 
 1 + 
 
 The following tables give values of 1 H = calculated 
 
 vm 
 
 with Bazin's values of <y, for a series of values of m and 
 for all classes of channels. Also the corresponding values of 
 f and c in eq. (10). 
 
 In selecting values of f or c it should be remembered that 
 the roughness is often increased by organic growths after the 
 channel has been some time in use. Fitzgerald has given 
 some interesting observations on a large aqueduct at Sudbury. 
 The culvert is circular, 9 feet diameter with an invert of 
 13' 2 feet radius; it is lined with brick, with cement joints. 
 It has been found that if the surface of the brickwork is not 
 cleaned it accumulates in the course of a year so much organic 
 slime that the discharge flowing full is diminished 1 per cent 
 (Trans. Amer. Soc. Civil Engineers, xliv. 87). 
 
 [TABLE. 
 
240 
 
 HYDEAULICS 
 
 CHAP. 
 
 VALUES OF 1 + -4= 
 
 BAZIN'S EQUATION 
 
 Hydraulic 
 Mean Depth 
 in Feet. 
 
 Values of 1 + -^= for y = 
 
 V7TC 
 
 109 
 
 290 
 
 83 
 
 1-54 
 
 2-36 
 
 3-17 
 
 for Categories 
 
 
 I. 
 
 II. III. 
 
 III. bis 
 
 IV. 
 
 V. 
 
 1 
 
 1-109 
 
 1-290 | 1-830 
 
 2-540 
 
 3-360 
 
 4-170 
 
 2 
 
 1-077 
 
 1-205 
 
 1-587 
 
 2-088 
 
 2-668 
 
 3-241 
 
 3 
 
 1-063 
 
 1-167 
 
 1-479 
 
 1-888 
 
 2-361 
 
 2-829 
 
 4 
 
 1-055 
 
 1-145 
 
 1-415 
 
 1-770 
 
 2-180 
 
 2-585 
 
 5 
 
 1-049 
 
 1-129 
 
 1-371 
 
 1-688 
 
 2-054 
 
 2-416 
 
 6 
 
 1-044 
 
 1-118 
 
 1-338 
 
 1-628 
 
 1-062 
 
 2-293 
 
 . 7 
 
 1-041 
 
 1-110 
 
 1-313 
 
 1-582 
 
 1-892 
 
 2-198 
 
 8 
 
 1-039 
 
 1-102 
 
 1-294 
 
 1-545 
 
 1-835 
 
 2-122 
 
 9 
 
 1-036 
 
 1-097 
 
 1-276 
 
 1-512 
 
 1-785 
 
 2-055 
 
 10 
 
 1-034 
 
 1-092 
 
 1-262 
 
 1-486 
 
 1-745 
 
 2-001 
 
 11 
 
 1-033 
 
 1-087 
 
 1-249 
 
 1-463 
 
 1-710 
 
 1-954 
 
 12 
 
 1-032 
 
 1-084 
 
 1-240 
 
 1-445 
 
 1-682 
 
 1-916 
 
 13 
 
 1-030 
 
 1-080 
 
 1-229 
 
 1-426 
 
 1-653 
 
 1-878 
 
 14 
 
 1-029 
 
 1-077 
 
 1-221 
 
 1-411 
 
 1-630 
 
 1-846 
 
 15 
 
 1-028 
 
 1-075 
 
 1-214 
 
 1-397 
 
 1-608 
 
 1-817 
 
 16 
 
 1-027 
 
 1-073 
 
 1-207 
 
 1-385 
 
 1-590 
 
 1-791 
 
 17 
 
 1-026 
 
 1-071 
 
 1-202 
 
 1-374 
 
 1-573 
 
 1-770 
 
 18 
 
 1-026 
 
 1-068 
 
 1-195 
 
 1-363 
 
 1-556 
 
 1-748 
 
 19 
 
 1-025 
 
 1-066 
 
 1-190 
 
 1-352 
 
 1-540 
 
 1 725 
 
 20 
 
 1-024 
 
 1-065 
 
 1-185 
 
 1-344 
 
 1-528 
 
 1-710 
 
 25 
 
 1-022 
 
 1-058 
 
 1-166 
 
 1-308 
 
 1-472 
 
 1-634 
 
 36 
 
 1-018 
 
 1-048 
 
 1-138 
 
 1-257 
 
 1-393 
 
 1-528 
 
XII 
 
 UNIFOEM FLOW OF WATER 
 
 241 
 
 BAZIN'S VALUES OF f IN EQUATION (10) 
 
 
 Values of f for 7 = 
 
 Hydraulic 
 Mean Depth 
 
 109 
 
 290 
 
 830 
 
 1-54 
 
 2-36 
 
 3-17 
 
 in Feet. 
 
 
 
 
 
 
 
 
 for Categories 
 
 
 I. 
 
 II. 
 
 III. 
 
 III. bis 
 
 IV. 
 
 V. 
 
 1 
 
 00319 
 
 00429 
 
 00867 
 
 01670 
 
 02923 
 
 04506 
 
 2 
 
 00300 
 
 00375 
 
 00652 
 
 01129 
 
 01844 
 
 02719 
 
 3 
 
 00293 
 
 00352 
 
 00567 
 
 00922 
 
 01442 
 
 02072 
 
 4 
 
 00287 
 
 00339 
 
 00518 
 
 00810 
 
 01230 
 
 01730 
 
 5 
 
 00285 
 
 00329 
 
 00486 
 
 00738 
 
 01092 
 
 01512 
 
 6 
 
 00282 
 
 00323 
 
 00463 
 
 00686 
 
 00997 
 
 01362 
 
 7 
 
 00279 
 
 00318 
 
 00445 
 
 00647 
 
 00927 
 
 01250 
 
 8 
 
 00279 
 
 00313 
 
 00432 
 
 00619 
 
 00872 
 
 01170 
 
 9 
 
 00277 
 
 00311 
 
 00422 
 
 00592 
 
 00826 
 
 01092 
 
 10 
 
 00277 
 
 00308 
 
 00411 
 
 00572 
 
 00789 
 
 01036 
 
 11 
 
 00277 
 
 00305 
 
 00404 
 
 00554 
 
 00756 
 
 00989 
 
 12 
 
 00276 
 
 00304 
 
 00398 
 
 00541 
 
 00732 
 
 00950 
 
 13 
 
 00274 
 
 00302 
 
 00391 
 
 00525 
 
 00707 
 
 00914 
 
 14 
 
 00274 
 
 00300 
 
 00385 
 
 00515 
 
 00688 
 
 00883 
 
 15 
 
 00274 
 
 00298 
 
 00380 
 
 00505 
 
 00670 
 
 00855 
 
 16 
 
 00273 
 
 00297 
 
 00377 
 
 00497 
 
 00655 
 
 00831 
 
 17 
 
 00271 
 
 00296 
 
 00373 
 
 00488 
 
 00639 
 
 00810 
 
 18 
 
 00271 
 
 00295 
 
 00370 
 
 00481 
 
 00626 
 
 00791 
 
 19 
 
 00271 
 
 00294 
 
 00366 
 
 00473 
 
 00613 
 
 00771 
 
 20 
 
 00270 
 
 00292 
 
 00363 
 
 00467 
 
 00605 
 
 00756 
 
 25 
 
 00269 
 
 00290 
 
 00352 
 
 00442 
 
 00562 
 
 00691 
 
 36 
 
 00267 
 
 00285 
 
 00336 
 
 00409 
 
 00502 
 
 00606 
 
 16 
 
242 
 
 HYDRAULICS 
 
 CHAP. 
 
 BAZIN'S VALUES OP c IN THE EQUATION v- 
 
 Hydraulic 
 Mean Depth 
 in Feet. 
 
 Values of c for 7 = 
 
 I. 
 
 0-109 
 
 II. 
 
 0-290 
 
 III. 
 0-83 
 
 III. bis 
 1-54 
 
 IV. 
 2-36 
 
 V. 
 
 3-17 
 
 1 
 
 142-0 
 
 122-5 
 
 86-2 
 
 62-1 
 
 47-0 
 
 37-8 
 
 2 
 
 146-4 
 
 131-0 
 
 99-3 
 
 75-5 
 
 59-1 
 
 48-6 
 
 3 
 
 148-2 
 
 135-2 
 
 106-5 
 
 83-6 
 
 66-8 
 
 55-8 
 
 4 
 
 149-8 
 
 137-8 
 
 111-4 
 
 89-1 
 
 72-3 
 
 61-0 
 
 5 
 
 150-2 
 
 139-8 
 
 115-0 
 
 93-4 
 
 76-8 
 
 65-3 
 
 6 
 
 151-0 
 
 141-2 
 
 117-9 
 
 96-8 
 
 80-4 
 
 68-8 
 
 7 
 
 151-8 
 
 142'2 
 
 120-2 
 
 99-7 
 
 83-3 
 
 71-7 
 
 8 
 
 152-1 
 
 143-4 
 
 122-0 
 
 101-9 
 
 85-9 
 
 74-1 
 
 9 
 
 152-4 
 
 143-9 
 
 123-4 
 
 104-3 
 
 88-2 
 
 76-8 
 
 10 
 
 }J 
 
 144-5 
 
 125-1 
 
 106-0 
 
 90-3 
 
 79-0 
 
 11 
 
 tt 
 
 145-3 
 
 126-2 
 
 107-8 
 
 92-3 
 
 80-7 
 
 12 
 
 152-7 
 
 145-5 
 
 127-1 
 
 109-1 
 
 93-8 
 
 82-3 
 
 13 
 
 153-3 
 
 146-0 
 
 128-2 
 
 110-7 
 
 95-4 
 
 83-9 
 
 14 
 
 u 
 
 146-5 
 
 129-3 
 
 111-7 
 
 96-8 
 
 85-4 
 
 15 
 
 
 
 147-0 
 
 130-1 
 
 112-9 
 
 98-0 
 
 86-7 
 
 16 
 
 153-5 
 
 147-2 
 
 130-7 
 
 113-7 
 
 99-2 
 
 88-0 
 
 17 
 
 154-1 
 
 147-4 
 
 131-3 
 
 114-8 
 
 100-3 
 
 89-1 
 
 18 
 
 
 147-7 
 
 131-9 
 
 315-7 
 
 101-4 
 
 90-2 
 
 19 
 
 
 147-9 
 
 132-6 
 
 116-6 
 
 102-4 
 
 91-4 
 
 20 
 
 154-4 
 
 148-4 
 
 133-2 
 
 117-3 
 
 103-1 
 
 92-3 
 
 25 
 
 154-7 
 
 149-0 
 
 135-2 
 
 120-6 
 
 107-0 
 
 96-5 
 
 36 
 
 155-2 
 
 149-4 
 
 138-4 
 
 125-5 
 
 113-2 
 
 103-0 
 
 As examples of the great variation of the coefficients f 
 and c in cases of great variation of roughness some results of 
 gauging the Loch Katrine aqueducts may be given. These 
 aqueducts are largely tunnelled in rock, and are only partly 
 lined with cement mortar. In the case of the older Loch 
 Katrine aqueduct, which was largely unlined and in parts very 
 rough, very low values of c were found. Thus, Mr. Gale 1 states 
 that for the Mugdock tunnel, 1^ miles long and exceptionally 
 rough, c = 56'9. The Loch Katrine tunnel, with about 
 11 per cent lined, gave c=67'8. At the northerly end of 
 the aqueduct, with 21-J- per cent lined, the average value of c 
 was 72. In the case of the newer aqueduct the whole of the 
 
 1 "Loch Katrine Waterworks," Proc. Inst. of Engineers and Shipbuilders in 
 Scotland, 1895. 
 
XII 
 
 UNIFOKM JFLOW OF WATEK 
 
 243 
 
 invert was concreted, and about 50 per cent completely lined. 
 The area when running full was 64*8 sq. feet in the lined 
 part, and 78*3 sq. feet in the unlined part. With water 
 flowing 7 feet deep, m = 3'l in the unlined and 2*87 in the 
 lined part. The gradient is 1 in 5500. The following are 
 some results obtained by Mr. Bruce (Proc. Inst. Civil Engineers, 
 cxxiii.) : 
 
 LOCH KATRINE CONDUIT 
 
 Depth of water 
 
 1-72 
 
 2-42 
 
 2-73 
 
 2'94 
 
 Cross section of water 
 
 14-2 
 
 20-8 
 
 23-7 
 
 25-6 
 
 Discharge cubic feet per sec. Q . 
 
 26-6 
 
 45-8 
 
 53-5 
 
 53-5 
 
 Mean velocity .... 
 
 1-87 
 
 2-21 
 
 2-26 
 
 2-08 
 
 Hydraulic mean depth 
 
 1-23 
 
 1-60 
 
 1-74 
 
 1-81 
 
 Value of c 
 
 125-4 
 
 129-3 
 
 126-9 
 
 135-6 
 
 141. Channels of circular section. Aqueducts and 
 sewers are sometimes of circular section, and concrete open 
 channels have been made of semicircular section. For 
 calculations of the discharge of such channels running partly 
 full the following table is useful. Let r be the radius of the 
 channel and d the depth of water : 
 
 d 
 r 
 
 m 
 r 
 
 ft 
 
 ^ 
 
 /m 
 V r 
 
 ftVm 
 
 rjr 
 
 05 
 
 032 
 
 021 
 
 179 
 
 0037 
 
 10 
 
 052 
 
 060 
 
 229 
 
 0137 
 
 15 
 
 096 
 
 107 
 
 310 
 
 0332 
 
 20 
 
 128 
 
 165 
 
 357 
 
 0589 
 
 30 
 
 185 
 
 294 
 
 430 
 
 1264 
 
 40 
 
 242 
 
 450 
 
 492 
 
 2214 
 
 50 
 
 293 
 
 614 
 
 541 
 
 3321 
 
 60 
 
 343 
 
 795 
 
 586 
 
 4658 
 
 70 
 
 387 
 
 979 
 
 622 
 
 6089 
 
 80 
 
 429 
 
 1-175 
 
 655 
 
 772 
 
 90 
 
 466 
 
 1-371 
 
 683 
 
 935 
 
 1-00 
 
 500 
 
 1-571 
 
 707 
 
 1-109 
 
 1-2 
 
 556 
 
 1-968 
 
 746 
 
 1-469 
 
 1-4 
 
 592 
 
 2-349 
 
 769 
 
 1-807 
 
 1-6 
 
 608 
 
 2-694 
 
 780 
 
 2-098 
 
 1-8 
 
 596 
 
 2-978 
 
 772 
 
 2-300 
 
 2-0 
 
 500 
 
 3-141 
 
 707 
 
 2-219 
 
244 
 
 HYDEAULICS 
 
 CHAP. 
 
 and (H ^/m)/(r ^/r) are the relative velocities 
 and discharges with different depths in the channel. The 
 greatest velocity is when the depth is l'6r, and the greatest 
 discharge when the depth is I'Sr approximately. 
 
 142. Egg-shaped channels or sewers. In sewers for 
 discharging storm water and house drainage the volume of 
 flow is extremely variable ; and there is a great liability for 
 
 deposits to be left 
 
 ^ 4 ->; when the flow is 
 
 small, which are not 
 removed during the 
 short periods when 
 the flow is large. The 
 sewer in consequence 
 becomes choked. To 
 obtain uniform scour- 
 ing action the velocity 
 of flow should be con- 
 stant or nearly so ; a 
 complete uniformity 
 of velocity cannot be 
 
 obtained with any form of section suitable for sewers, but an 
 approximation to uniform velocity is obtained by making the 
 sewers of oval section. Various forms of oval have been sug- 
 gested, the simplest being one in which the radius of the 
 crown is double the radius of the invert, and the greatest 
 width is two-thirds the height. The section of such a sewer 
 is shown in Fig. Ill, the numbers marked on the figure 
 being proportional numbers. 
 
 The following results facilitate calculations on sewers 
 flowing partly filled. Let d be the greatest width (that is 
 four units in the figure). Then 
 
 Fig. 111. 
 
 Depth of 
 
 Area of 
 
 Wetted 
 
 Hydraulic 
 
 Relative 
 
 Stream. 
 
 Section. 
 
 Perimeter. 
 
 Mean Depth. 
 
 Discharge. 
 
 Full . 
 
 l-1485d 2 
 
 3-965d 
 
 0-2897d 
 
 1-000 
 
 f Ml 
 
 0'7558d 2 
 
 2-394d 
 
 0-3157d 
 
 0-687 
 
 l full 
 
 0-2840d 2 
 
 T37 5d 
 
 0-2066d 
 
 0-209 
 
XII 
 
 UNIFOKM, FLOW OF WATEK 
 
 245 
 
 The last column gives the relative discharge neglecting the 
 variation of the coefficient c. 
 
 143. Trapezoidal channels. Artificial channels are 
 commonly trapezoidal in section, the side slopes being deter- 
 mined by the stability of the banks and the kind of protection 
 against degradation adopted. 
 
 Angle of Side Slopes. 
 
 Ratio of Side Slopes. 
 
 Character of Bank. 
 
 90 
 6320 
 45 
 3340 
 2630 
 
 to 1 
 0-5 to 1 
 1 to 1 
 l|to 1 
 2 to 1 
 
 Planks or masonry. 
 Masonry or brick walls. 
 Stone pitching. 
 Firm earth. 
 
 2148 
 1820 
 
 3 to 1 
 
 / Loose earth. 
 
 Let B be the top and b the bottom width, d the depth, 
 </> the slope angle, and n the slope ratio, so that tan <j> = l/n. 
 
 Top width = B = b + 
 Area of section = 12 = (b + nd)d. 
 Wetted perimeter = x = & + 2d vV + 1- 
 
 ncf- --- * - -<5 - - 
 
 144. Trapezoidal channel of minimum section for given 
 side slopes. Various practical considerations determine the 
 general form of the section of a channel. In a navigation 
 canal the depth is fixed by the draught of the boats. In 
 large irrigation canals the depth is limited so as to avoid 
 interference with subsoil drainage, and the canals are of a 
 
246 HYDEAULICS CHAP. 
 
 width equal to ten or twenty times the depth. In valuable 
 ground the width is restricted and a rectangular section is 
 used. The longitudinal slope i is determined by the slope of 
 the country and the limiting velocity which can be permitted 
 consistently with the stability of the canal bed. The side 
 slopes are fixed by the character of the banks. 
 
 If a channel is constructed for a given discharge and given 
 longitudinal and given side slopes, then there is a proportion 
 of breadth to depth which makes the area of cross section, and 
 therefore the amount of excavation, a minimum. The resist- 
 ance to flow depends on the wetted perimeter, and the velocity 
 will be greatest and the section least for that form for which 
 the wetted perimeter is least. 
 
 Differentiating the expressions for fl and ^ given above, 
 and equating to zero, 
 
 db \ , 
 
 . + n )a + o + nd 0, 
 
 dd J 
 
 Eliminating dbjdd, 
 b 
 d 
 
 n = 0-5 1-0 1J 2 2A 3 
 
 5 
 
 ^=2 1-24 0-82 0'60 0'48 0'38 0'32. 
 
 If this value of I is inserted in the expressions for 1 and 
 X, we get a very convenient characteristic of channels of the 
 most economical section 
 
 Tfl == ^ == ~ - -j-/ a \ . . (11). 
 
 That is, in channels of the most economical form, with 
 given side slopes, the hydraulic mean depth is half the actual 
 depth. It will easily be seen that this is a characteristic of 
 the semicircle, the half square, and the half hexagon. A 
 simple geometrical construction shows that for all such channels 
 the sides and bottom are tangents to a semicircle having its 
 centre on the water-surface. 
 
XII 
 
 UNIFORM FLOW OF WATER 
 
 247 
 
 Let Fig. 113 represent a trapezoidal channel of minimum 
 section, for side slopes of n to 1. Let E be the centre of the 
 
 // 
 
 \ 
 
 L 
 
 _..-'-:/ 
 
 B C 
 
 Fig. 113. 
 
 C 
 
 water-surface, and drop perpendiculars EF, EG, EH on the 
 sides. Let AB = CD = a ; BC = 6 ; EF = EH = c ; EG = d. 
 
 12 = AEB + EEC + ECD 
 = ac + -bd, 
 
 X = 
 
 Since the hydraulic mean depth is half the actual depth, 
 
 That is, EF, EG, EH are all equal, and a semicircle with 
 centre at E touches AB, BC, CD. A circle struck from A 
 with radius AE will pass through B. 
 
 PROPORTIONS OF CHANNELS OP THE MOST ECONOMICAL SECTION 
 
 Side 
 Slope 
 Angle. 
 
 Ratio 
 of Side 
 Slopes. 
 
 fl 
 
 m 
 ~d 
 
 b 
 d 
 
 X 
 d 
 
 X 
 
 ViT 
 
 = k 
 
 1 
 
 B 
 
 d 
 
 90 
 
 to 1 
 
 2-000 
 
 0-5 
 
 2-000 
 
 4-00 
 
 2-83 
 
 594 
 
 2-000 
 
 63 20' 
 
 1 2 
 
 1-736 
 
 
 1-236 
 
 3-47 
 
 2-63 
 
 616 
 
 2-236 
 
 45 
 
 1 1 
 
 1-828 
 
 
 0-828 
 
 3-66 
 
 2-71 
 
 607 
 
 2-828 
 
 33 40' 
 
 
 2-106 
 
 
 0-606 
 
 4-21 
 
 2-90 
 
 587 
 
 3-606 
 
 26 30' 
 
 2 1 
 
 2-472 
 
 n 
 
 0-472 
 
 4-94 
 
 3-14 
 
 564 
 
 4-472 
 
 21 48' 
 
 2j M 1 
 
 2-885 
 
 " 
 
 0-385 
 
 5-77 
 
 3-40 
 
 543 
 
 5-385 
 
 18 20' 
 
 3 1 
 
 3-325 
 
 
 0-325 
 
 6-65 
 
 3-64 
 
 524 
 
 6-325 
 
 OF THE 
 
 UNIVERSITY 3 
 
 OF 
 
248 HYDEAULICS 
 
 The velocity in a channel is 
 
 /2gi U 
 V f V k 
 
 (12). 
 
 For a given section of channel the velocity and therefore 
 the discharge will be greatest if !/*/& is greatest, so that 
 this can be taken as a value figure for channels of various 
 forms. 
 
 It is not generally convenient to adopt exactly the form 
 of a channel of minimum section, but the theorem indicates 
 the form towards which actual channel sections should tend 
 if practicable. For other forms of section m>d/2, and the 
 mean velocity for a given longitudinal slope is less. The 
 other limit to the value of ra is d. For in a channel of great 
 width Z>, and small depth d, l = 'bd and % = & nearly, so that 
 m = d nearly. 
 
 The mean velocity varies as V f m. Hence, taking the 
 extreme cases of m = d/2 and m = d, the corresponding mean 
 velocities will have the ratio 
 
 a. I L 
 
 v 2 V 2 
 
 0-709. 
 
 For a given discharge the areas of the channels would be 
 in the inverse proportion. 
 
 145. Discharge of a channel with different depths of 
 water flowing. Consider a rectangular channel with a stream 
 
 of water of width b and depth d. 
 The area is O = Id, the hydraulic 
 mean depth is m = ~bdj(b + 2d). The 
 
 <f 
 
 discharge is 
 
 Q = tiv = ci2 V mi = 
 
 _ / Mi 
 
 V mi = cbd & + 2 ^> 
 
 Fig. 114. 
 
 that is, as i is constant for a given 
 
 channel, and c will only vary a little with the variation 
 of size, 
 
XII 
 
 UNIFOKM FLOW OF WATER 
 
 249 
 
 Q varies as 
 
 2d) 
 
 If Qj is the discharge determined by gauging for a depth 
 
 d l} then the discharge for any other depth is 
 
 Q = QiT 
 
 (13). 
 
 Example. A rectangular channel draining an area of 572,000 acres 
 is 60 feet wide, with a depth of water of 3 feet it is found to discharge 
 400 cubic feet per second. Then equation (13) becomes 
 
 = 442 
 
 (30 + df 
 
 The following table gives the mean monthly depth of water deduced 
 from daily observations and the discharge calculated by tbis formula. 
 From this the total discharge of the stream in each month can be found, 
 and this divided by the drainage area, 24,910 million square feet, gives 
 the depth of rainfall in each month equivalent to the stream discharge. 
 The observed mean rainfall is also given. The ratio of the stream 
 discharge or off-flow from the ground to the rainfall varies with the 
 season, and is an important datum in certain problems of water storage. 
 
 DISCHARGE AND KAINFALL ON A DRAINAGE AREA 
 
 Month. 
 
 Mean 
 Depth 
 of Water 
 in Feet, 
 d. 
 
 Mean 
 Discharge, 
 Cubic Feet 
 per Second, 
 
 Q. 
 
 Total Dis- 
 charge per 
 Month in 
 Million 
 Cubic Feet. 
 
 Equivalent 
 Depth on 
 Drainage 
 Area in 
 Inches. 
 
 Mean Rain- 
 fall in 
 Inches. 
 
 Ratio of 
 Discharge 
 to Rainfall 
 Per Cent. 
 
 January 
 
 4-40 
 
 693 
 
 1856 
 
 894 
 
 2-15 
 
 41'6 
 
 February . 
 
 4-50 
 
 720 
 
 1742 
 
 840 
 
 1-78 
 
 47-2 
 
 March 
 
 4-25 
 
 663 
 
 1775 
 
 855 
 
 170 
 
 50-3 
 
 April . 
 
 3-90 
 
 583 
 
 1511 
 
 728 
 
 1-87 
 
 39-0 
 
 May . . 
 
 3-25 
 
 450 
 
 1205 
 
 581 
 
 1-55 
 
 37-5 
 
 June . 
 
 2-80 
 
 361 
 
 936 
 
 451 
 
 173 
 
 26-1 
 
 July . 
 
 3-30 
 
 459 
 
 1229 
 
 592 
 
 1-48 
 
 40-0 
 
 August 
 
 3-00 
 
 400 
 
 1071 
 
 516 
 
 1-29 
 
 40-0 
 
 September . 
 
 2-85 
 
 371 
 
 962 
 
 463 
 
 1-48 
 
 31-3 
 
 October 
 
 3-05 
 
 410 
 
 1097 
 
 529 
 
 1-44 
 
 36-8 
 
 November . 
 
 3-10 
 
 419 
 
 1086 
 
 523 
 
 2-00 
 
 26-1 
 
 December . 
 
 3-80 
 
 565 
 
 1513 
 
 729 
 
 1-95 
 
 37'4 
 
 The mean depth of water and mean rainfall are the average of five 
 years' observations. The smaller the intervals of time for which the 
 means are taken, tbe more approximate would be the result. 
 
 146. Parabola of discharge. In a rectangular channel 
 
250 
 
 HYDEAULICS 
 
 CHAP. 
 
 of width I and depth d, if I is large compared with d, O = Id 
 and m = d nearly. Then 
 
 Q = cbd \' 'di, 
 
 that is, for a given channel Q varies as d%. In a triangular 
 channel the width & is proportional to d, so that H = /j,d 2 and 
 m = vd, where //, and v are constants depending on the inclina- 
 tion of the sides of the channel. Then 
 
 or Q varies as d?. Ordinary channels are of a form between 
 these two, so that at least for a limited variation of d in a 
 given channel the discharge may be taken to vary approximately 
 as d 2 . In that case, if the depths of water are taken as ordinates 
 and the discharges as abscissae the curve of discharge is a 
 parabola. It often happens that an approximate estimate of 
 the total discharge of a stream is required when the only 
 continuous records available are readings on a gauge of the 
 surface-level of the stream. In such cases it may be assumed 
 that Q varies as (d + S) 2 for the range of variation of level which 
 
 occurs in such cases, where 
 d is the actual depth of 
 water and 8 a quantity to be 
 determined. Suppose that, 
 by gauging, the discharges 
 Q!, Q 2 for two depths d lt d 2 
 of water in the stream have 
 been ascertained. 
 
 Take AB = d l} BC = Q x , 
 
 115). Then C and E are 
 points on the discharge 
 curve, which is assumed to 
 be approximately a para- 
 bola with its vertex at 
 From the properties of the 
 
 O 
 
 Fig. 115. 
 
 some point at 8 below A. 
 parabola 
 
 DE = Q 2 = 4=a(d 2 + S) 2 , 
 where a is the parameter of the parabola. Hence 
 
xii UNIFORM FLOW OF WATER 251 
 
 d-d / Ql ^ 
 S = 
 
 -1 
 
 When 8 and a have been determined, the discharge for 
 any value of d is easily calculated to an approximation sufficient 
 in many cases where comparisons of stream discharge and 
 rainfall have to be made. 
 
 147. General distribution of velocity at different points 
 in the cross section of a channel. Even a cursory observa- 
 tion of flow in an open channel shows that the velocity of 
 translation along the channel is greater towards the centre 
 and surface and less towards the bottom and sides. A more 
 careful investigation indicates some marked peculiarities, and 
 a knowledge of these is of practical importance in considering 
 various methods of gauging the volume of flow in streams. 
 
 By means to be described presently, the mean forward 
 velocity at a number of points in the cross section of a stream 
 can be determined. This was first accomplished in a quite 
 satisfactory way by Darcy, and an example from his work 
 will be taken as an illustration. 
 
 Fig. 116 shows the cross section of a rectangular channel, 
 0'25 metre deep and 0*8 metre wide, in which the velocity 
 was observed at 36 points at the intersection of the verticals 
 ee,ff, . . ., and the transversals aa, &&,... The velocities 
 at each point on a transversal set up from the transversal 
 vertically give points on a transverse velocity curve. Thus 
 aaa is the transverse velocity curve along aa, 111) that along 
 11}, and so on. Similarly, the velocities at each point on a 
 vertical set off from the vertical horizontally give points on a 
 vertical velocity curve. Thus ee is the vertical velocity curve 
 for the vertical ee, ff that for ff, and so on. The vertical 
 curves show that the greatest velocity is not at the surface, 
 but somewhat below it. From the level of greatest velocity 
 at any vertical the velocity decreases upwards and downwards. 
 There is another way of representing the distribution of 
 velocity. If at points on the vertical curves where the 
 velocities are 1'2, Tl, TO, 0'9, and 0'8 metres per second, 
 
252 
 
 HYDRAULICS 
 
 CHAP. XII 
 
 horizontals are drawn to the corresponding verticals, points are 
 found in the section on curves of equal velocity. These curves 
 correspond to the contours of a solid whose base is the cross 
 section of the stream, whose height at any point is the velocity 
 at that point, and whose volume is proportional to the 
 discharge of the stream per second. The maximum velocity 
 is on the centre vertical below the surface, and from that 
 point the velocity decreases in all directions. 
 
 Messrs. Fteley and Stearns made very careful gaugings of 
 the brick conduit at Sudbury with different depths of water 
 flowing. The conduit is 9 feet diameter, with an invert of 
 13'2 feet in radius, the height of the conduit being 7*7 feet 
 (Trans. Amer. Soc. Civil Engineers, 1883). With the greatest 
 flow the velocity was measured at 167 points in the cross 
 section. The following are some of the results obtained : 
 
 SUDBURY CONDUIT 
 
 Depth of water 
 
 4-54 
 
 4-01 
 
 3-00 
 
 2-03 
 
 1-51 
 
 Hydraulic mean depth 
 
 2-33 
 
 2-19 
 
 T84 
 
 1-38 
 
 1-07 
 
 Mean velocity 
 
 2-97 
 
 2-90 
 
 2-62 
 
 2-18 
 
 1-90 
 
 Maximum velocity . 
 
 3-37 
 
 3-32 
 
 3-06 
 
 2-47 
 
 2-14 
 
 Bottom velocity (about) . 
 
 2-20 
 
 2-15 
 
 2-10 
 
 1-75 
 
 1-60 
 
 Ratio mean/maximum 
 
 88 
 
 87 
 
 86 
 
 88 
 
 89 
 
 mean/bottom 
 
 1-35 
 
 1-35 
 
 1-25 
 
 1-25 
 
 1-19 
 
 Discharge per sec. Q 
 
 111-5 
 
 94-4 
 
 62-4 
 
 33-3 
 
 20'1 
 
 Value of c 
 
 140-0 
 
 139-5 
 
 137-5 
 
 133-2 
 
 129-5 
 
 
 I 
 
 
 
 
 148. Depression of the point of greatest velocity. In 
 
 calm weather the maximum velocity is below the surface, and 
 this is not due, as has been sometimes supposed, to a resistance 
 of the air similar to that of the stream bed, for it is the case 
 with a wind down stream which should accelerate the surface 
 layer. In a rectangular channel the velocity is highest at the 
 centre and falls to about half depth at the sides. In channels 
 with sloping sides it rises from the centre outwards, and may 
 be at the surface at the edges of the stream. The cause of 
 the depression has been much discussed. Eddies of water 
 stilled by contact with the bed are thrown off, and wander 
 through all parts of the stream, but accumulate and spread out 
 at the surface. In the Mississippi gaugings it was found that 
 
V 
 
254 
 
 HYDEAULICS 
 
 CHAP. 
 
 the depression of the line of maximum velocity increased with 
 an upstream and decreased with a downstream wind, but this 
 result has not been found in some other cases. Perhaps it 
 depends on the presence or absence of waves or ripples on 
 which the wind can act. 
 
 149. Vertical velocity curve. In purely viscous stream- 
 line motion the vertical velocity curve would be a parabola 
 with a horizontal axis at the free surface. In ordinary 
 turbulent motion in streams the vertical velocity curve 
 
 agrees fairly well with a para- 
 bola having a horizontal axis 
 at the level of maximum vel- 
 ocity. Without assuming this 
 to be more than a convenient 
 approximation, it is a result 
 useful in discussing the rela- 
 tions of the velocities at differ- 
 ent depths in a stream. 
 
 Let AOC (Fig. 117) be a 
 parabolic velocity curve, the 
 axis being a horizontal through 
 0. Let V be the maximum, 
 
 rig. 117. 
 
 v the surface, v b the bed vel- 
 ocity, and v the velocity at any point P. Let Z be the depth 
 of the filament of greatest velocity, z the depth of P, and 
 D the whole depth of the stream. Then from the properties 
 of the parabola 
 
 where K is the parameter of the parabola. 
 
 = v- ( -^ Z)2 
 
 K 
 
 Hence 
 
 (15). 
 
 Mean velocity at a vertical. If a fairly large number 
 of velocities at equal distances on a vertical are observed, the 
 arithmetic mean is very approximately the mean velocity at 
 the vertical. If the number is small the arithmetic mean is 
 less than the true mean velocity. If through observed points 
 a fair vertical velocity curve can be drawn, the mean velocity 
 at the vertical is the area of the curve divided by the depth 
 of the stream. 
 
XII 
 
 UNIFOKM FLOW OF WATEK 255 
 
 Assuming that the vertical velocity curve is a parabola 
 such as is shown in Fig. 117, the mean velocity is the mean 
 ordinate of AOC, that is 
 
 U = g {area EAOCF} 
 
 = ijarea EGHF - 1 (area AGOK + OLCH) j 
 
 But by the equation above, when 
 
 K 
 
 (D - Z) 2 
 ~KT 
 
 _ D 2 DZ_Z 2 
 3K + K K 
 
 DZ ID 2 
 = V + - ' 
 
 D 
 
 If v^ is the velocity at half depth, putting z = ^ in the 
 
 equation above, 
 
 D 2 
 
 so that the half -depth velocity is greater than the mean 
 velocity at the vertical only by the small quantity D 2 /(12K), 
 a result which depends on the assumption of a parabolic curve, 
 but which cannot be much wrong, and this is useful in practical 
 gauging. In Cunningham's Eoorkee gaugings with floats, much 
 attention was paid to this point, and the mid-depth velocity 
 was found a little greater than the mean velocity at the vertical 
 in forty-two cases out of forty-six. The average of a large 
 number of results gave 11/17,= 0*94 to 0'98. 
 
256 HYDEAULICS CHAP. 
 
 If two velocities can be observed on a vertical, then a 
 better approximation to the mean velocity U can be found. 
 Thus the parabolic law shows that if the velocity at the surface 
 and -- depth is observed, 
 
 150. Transverse velocity curves. In a channel sym- 
 metrical about its centre line, the transverse velocity curve at 
 any level shows a maximum velocity at the centre, a slow 
 decrease of velocity towards the sides, more rapid as the banks 
 are approached, and very rapid near the banks. In an 
 
 Fig. 118. 
 
 unsymmetric channel the greatest velocity is over the deepest 
 part of the stream. 
 
 Fig. 118 shows the results of a very careful current-meter 
 gauging of the Eger at Falkenau by Wilhelm Plenkner of 
 Prague. The river is 321*6 feet wide. The vertical scale is 
 exaggerated ten times. The curve 1 passes through the points 
 of maximum velocity, which throughout is somewhat below 
 the surface. Curve 2 passes through the points of mean 
 velocity on each vertical a little below half depth. Curve 3 
 passes through points where the velocity is equal to the 
 mean velocity of the whole section. Curve 4 is the transverse 
 mean velocity curve, that is, its ordinates are the mean 
 
XII 
 
 UNIFORM FLOW OF WATER 
 
 257 
 
 velocities on each vertical. Curve 5 is the transverse surface 
 velocity curve. 
 
 151. Ratio of mean and surface velocities. In a gauging 
 of the Rhine at Basel the velocity at 0*58 of the depth was 
 found to be equal to the mean velocity on the same vertical. 
 The ratio of the mean to the surface velocity on one vertical 
 varied from 0'77 to 0'85, the average being 0'82. The ratio 
 of the mean velocity for the whole cross section to the greatest 
 surface velocity was on the average 0*73. Harlacher found 
 the same ratio in gauging the Elbe. The following table 
 gives some values : 
 
 
 Mean 
 Velocity 
 of Whole 
 Section. 
 v m . 
 
 Mean 
 Surface 
 Velocity. 
 
 u m . 
 
 Greatest 
 Surface 
 Velocity. 
 
 U . 
 
 2 
 
 i 
 
 Elbe (high water) 
 
 3-61 
 
 4-17 
 
 4-66 
 
 86 
 
 77 
 
 (average water) . 
 
 3-12 
 
 3-61 
 
 4-17 
 
 86 
 
 75 
 
 (low water) 
 
 2-49 
 
 2-79 
 
 3-64 
 
 89 
 
 68 
 
 Eger at Warta . 
 
 1-75 
 
 1-75 
 
 3-21 
 
 1-00 
 
 55 
 
 at Falkenau 
 
 2-54 
 
 2-77 
 
 4-43 
 
 92 
 
 57 
 
 
 
 1-31 
 
 1-48 
 
 2-26 
 
 89 
 
 58 
 
 Sazawa at Poric 
 
 1-61 
 
 1-60 
 
 2-72 
 
 1-00 
 
 50 
 
 5) 
 
 82 
 
 84 
 
 1-15 
 
 98 
 
 71 
 
 i> 
 
 1-90 
 
 1-67 
 
 2-61 
 
 1-14 
 
 73 
 
 Moldau at Budweis . 
 
 2-55 
 
 2-67 
 
 3-53 
 
 96 
 
 72 
 
 5) 5J 
 
 5-71 
 
 6'51 
 
 8-02 
 
 88 
 
 71 
 
 
 
 3-07 
 
 3-57 
 
 4-28 
 
 86 
 
 72 
 
 152. Aqueducts. Any work by which water is conveyed 
 may be termed an aqueduct, but the term is usually applied 
 to important works in which water flows by gravitation, and 
 specially to those conveying the water-supply of towns. 
 Where the fall of the country is suitable the water may be 
 conveyed in a channel contoured to the slope of the hydraulic 
 gradient. The channel may be an open channel, such as the 
 conduit which brings water from Staines to London. More 
 commonly it is covered to protect the water from deterioration, 
 but the water flows precisely as in an open channel. Generally, 
 
 17 
 
258 H YDKAULICS CHAP, xn 
 
 such an aqueduct is of a composite character part in tunnel 
 where the ground is above the hydraulic gradient ; part in 
 cut and cover, that is, built in an open trench and then 
 covered in. Across valleys the aqueduct must be carried on 
 piers, or more commonly the water is conveyed in one or more 
 pipes, termed inverted siphons, falling from the hydraulic 
 gradient at one end and rising to it again at the other end. 
 
 Roman aqueducts. Amongst the most striking engineer- 
 ing works of antiquity, of which parts still exist, are the 
 aqueducts constructed for the water-supply of Kome and other 
 cities of the Eoman empire. The Appian Aqueduct at Borne 
 was constructed in 3 1 3 B.C., and conveyed water from springs 
 ten miles distant from the city, in a channel 2^- feet wide by 
 5 feet deep. Others were subsequently constructed, till there 
 were fourteen aqueducts, of lengths varying from 11 to 
 59 miles, and aggregating 359 miles. Of the total length, 
 55 miles were on arches, and the remainder chiefly under- 
 ground. The channels were lined with cement and roofed with 
 slabs, and the gradients varied from perhaps 1 in 500 to 
 1 in 3000. Herschel estimates the total supply to the city 
 of Kome at 5 million gallons daily, with an additional supply 
 to districts outside the city. The water was often distributed 
 by lead pipes, and lead siphons of 12 to 18 inches diameter 
 have been found. 
 
 Types of aqueducts. Fig. 119 shows cross sections of 
 some important aqueducts. A, B, C are sections of the 
 new Loch Katrine aqueduct. 
 
 153. Examples of aqueducts. (1) Loch Katrine 
 aqueduct. This was designed to convey 50 million gallons 
 per day from Loch Katrine to Glasgow, but the roughness of 
 the channel was not fully allowed for, and it probably carries 
 only about 40 million gallons. The top water surface in 
 Loch Katrine is 367 feet above mean sea-level, and the water 
 is delivered into a service reservoir at Mugdock, 26 miles 
 distant, where the top water-level is 317 feet above mean 
 sea-level. Of the 26 miles of aqueduct, 3|- are cast-iron 
 pipes across valleys, llj miles are in tunnel, and 10^- miles 
 are bridges and masonry in cut and cover. The tunnels are 
 8 feet in diameter, with a fall of 10 inches per mile. The 
 channel in cut and cover has the same gradient as the tunnels. 
 
CUT AND COVER IN EMBANKMENT 
 
 BRJCK LINING IN HEAVY GROUND- 
 
 Fig. 119. 
 
260 HYDKAULICS 
 
 CHAP. 
 
 Portions of the pipe line consist of two 4 8 -inch and one 
 36-inch pipe, or of four 36-inch pipes, the general hydraulic 
 gradient being 5 feet per mile. An additional aqueduct has 
 now been constructed following generally the line of the old 
 aqueduct, with the object of ultimately maintaining a supply 
 to the city of 100 million gallons per day. In the new 
 aqueduct, with water flowing 7 feet deep, the area of section 
 is 78'3 square feet. The wetted perimeter 24'9 feet. The 
 hydraulic mean depth 3'1 feet. The slope 1 in 5500. The 
 estimated discharge is nearly 72 million gallons per day 
 (Proc. Inst. Civil Engineers, 1883). 
 
 (2) Thirlmere aqueduct, for the supply of water to 
 Manchester. This is designed to convey 50 million gallons 
 per day from Lake Thirlmere to a service reservoir at 
 Prestwich, a distance of 96 miles. There are 14 miles of 
 tunnel, 37 miles of cut and cover, and 45 miles of cast-iron 
 pipes. The tunnels are 7 feet 1 inch wide, the side walls 
 5 feet high, and the arch rises 2 feet. They are for the most 
 part lined with concrete, but in parts only the floor is lined. 
 The thickness of floor lining is 4^ inches in close rock to 
 18 inches in bad ground. Walls 12 inches to 18 inches 
 thick. Arch ring 15 inches thick. "Where the tunnels are 
 unlined their width is increased to 8 feet 6 inches, to allow 
 for the greater friction due to irregularities of the rough rock 
 surface. The cut -and -cover channels are also of concrete. 
 At full supply the water in the conduit will be 5 feet 6 inches 
 deep. The pipe line was designed to have three parallel 
 4 8 -inch pipes in the first part, and five parallel lines of 
 40 -inch pipe in the later part, the pipes varying in thickness 
 from 1 to 1|- inches, with socket joints run with lead. The 
 second pipe laid has been increased in diameter from 40 to 
 45 inches. The surface of the lake when full is at 584 feet 
 above O.D. The aqueduct starts at 527 feet above O.D. and 
 ends at Prestwich at 353 above O.D. The ruling gradient 
 is 20 inches per mile, but extra fall is given to the pipe line. 
 Along the aqueduct there are manholes at every quarter mile. 
 
 New Croton aqueduct, New York, U.S.A. In this 
 aqueduct there are 30 miles of tunnel, 1 mile of cut and 
 cover, and 2^ miles of pipe. About 7 miles of the tunnel 
 is of circular form 1 2^ feet in diameter, and is under pressure. 
 
XII 
 
 UNIFORM FLOW OF WATER 
 
 261 
 
 amounting at one point to 120 feet of head. The remainder 
 of the tunnel is horseshoe-shaped, 13 feet 7 inches in width 
 and height. For 25 miles the gradient is 0'7 feet per mile. 
 The tunnel is lined with brickwork 12 to 24 inches thick. 
 The discharge is about 300 cubic feet per second. 
 
 154. River bends. In rivers flowing in alluvial plains 
 the windings which already exist tend to increase in curvature 
 by the scouring away of material from the outer bank and the 
 deposition of detritus along the inner bank. The sinuosities 
 sometimes increase till a loop is formed with only a narrow 
 strip of land between the two encroaching branches of the 
 river. Finally a "cut off" may occur, a waterway being 
 opened through the strip of land and the loop left separated 
 from the stream, forming a horseshoe-shaped lagoon or marsh. 
 Professor James Thomson has pointed out (Proc. Royal Soc. 
 1877, p. 356; Proc. Inst. of Mech. Engineers, 1879, p. 456) 
 that the usual supposition is that the water, tending to go 
 forwards in a straight line, rushes against the outer bank and 
 scours it, at the same time creating deposits at the inner bank. 
 That view is very far 
 from a complete account 
 of the matter, and Pro- 
 fessor James Thomson 
 has given a much more 
 ingenious account of the 
 action at the bend, which 
 he has completely con- 
 firmed by experiment. 
 
 When water moves 
 round a circular curve 
 under the action of 
 gravity only, it takes a 
 motion like that in a free 
 vortex. Its velocity is 
 greater parallel to the axis 
 
 of the stream at the inner than at the outer side of the bend. 
 Hence the scouring at the outer side and the deposit at the 
 inner side of the bend are not due to mere difference of velocity 
 of flow in the general direction of the stream ; but, in virtue 
 of the centrifugal force, the water passing round the bend 
 
 Fig. 120. 
 
262 HYDKAULICS CHAP. 
 
 presses outwards, and the free surface in a radial cross section 
 has a slope from the inner side upwards to the outer side 
 (Fig. 121). For the greater part of the water flowing in 
 
 curved paths, this 
 
 INNE(U5ANK OUTHMMK. diffel .e n ce of pressure 
 
 produces no tendency 
 to transverse motion. 
 But the water im- 
 mediately in contact 
 
 SECT.ON At MM with ^ rough ^ 
 
 Fig. 121. torn and sides of the 
 
 channel is retarded, 
 
 and its centrifugal force is insufficient to balance the pressure 
 due to the greater depth at the outside of the bend. It 
 therefore flows inwards towards the inner side of the bend, 
 carrying with it detritus which is deposited at the inner bank. 
 Conjointly with this flow inwards along the bottom and sides, 
 the general mass of water must flow outwards to take its place. 
 Fig. 120 shows the directions of flow as observed in a small 
 artificial stream, by means of light seeds and specks of aniline 
 dye. The lines CO show the directions of flow immediately in 
 contact with the sides and bottom. The dotted line AB shows 
 the direction of motion of floating particles on the surface of 
 the stream. 
 
 PROBLEMS. 1 
 
 1. A river has the following section : bottom width, 300 feet ; depth 
 
 of water, 20 feet ; side slopes, 1 to 1 ; fall, 1 foot per mile. 
 
 Find the discharge, using Darcy's coefficient for earth channels. 
 
 Darcy, c = 100 ; Q = 37,340 cubic feet per second. 
 
 2. A canal is to be constructed for a discharge of 2000 cubic feet 
 
 per second. The fall is 1-5 feet per mile; side slopes, 1 to 1 ; 
 bottom width, ten times the depth; c=120. Find the 
 dimensions of the canal. 
 
 Depth, 6-23 feet ; bottom width, 62-3 feet. 
 
 3. Required the dimensions of a trapezoidal channel of the most 
 
 economical section to convey 600 cubic feet per second, with a 
 
 faU of 2 feet per mile, and side slopes l to 1. f = -0035. 
 
 Depth, 7-48 feet; bottom width, 4'49 feet. 
 
 1 When not otherwise stated, Bazin's values .of the coefficients for channels 
 have been used. 
 
XII 
 
 UNIFORM FLOW OF WATER 263 
 
 4. Kecalculate the discharge of the channel determined in (3), taking 
 
 Bazin's coefficient for sides covered with stone pitching. 
 
 388 cubic feet per second. 
 
 5. An irrigation canal in earth with side slopes Ij to 1 conveys 
 
 600 cubic feet per second at a velocity of 2j feet per second. 
 Design a suitable canal section with a depth of 3 feet 
 
 Area of section, 240 square feet ; m = 2-687 ; 
 c = 64-6 ; i = '000557 or 2'94 feet per mile. 
 
 6. A brick culvert, 5 feet 6 inches in diameter and 4000 feet long, 
 
 conveys 150 cubic feet per second when running full Find the 
 fall in feet necessary. 7 '3 feet 
 
 7. An oval brick sewer, flowing two- thirds full, is 4 feet wide and 
 
 6 feet high. Find the fall in feet per mile to give a velocity of 
 3 feet per second, and the discharge. 
 
 2-4 feet ; 36*3 cubic feet per second. 
 
 8. A canal is to be cut in earth with side slopes 2 to 1, and a fall of 
 
 9 inches per mile. The discharge is to be 6000 cubic feet per 
 minute, and the depth 3 feet. Find the dimensions of canaL 
 (Solve by approximation.) 
 
 Assuming ra = 3, 6 = 18*2 feet ; 
 
 then m = 2-29, and 6 = 24 feet. 
 
 9. A semicircular channel of smooth cement is 5 feet deep and slopes 
 
 at 1 in 1000. Find the discharge. 
 
 115-7 cubic feet per second. 
 
 10. A trapezoidal channel of the most economical form, with sides of 
 
 rubble masonry, has a depth of 10 feet and side slopes of 1 to 1. 
 Find the discharge when the fall is 18 inches per mile. 
 
 6 = 8-2; v = 3-28; 12 = 182; Q = 597. 
 
 11. A rectangular ashlar masonry channel is 12 feet wide and 4 feet 
 
 deep, and has a slope of 1 in 5000. Find the velocity and 
 discharge. 2-91 feet per second ; 139-6 cubic feet per second. 
 
 12. The water section in the aqueduct at Dijon is 2 feet wide and 1 foot 
 
 deep, and the sides are smooth cement The slope is 1 in 1000. 
 Find the velocity and discharge. 
 
 3'05 feet per second ; 6-1 cubic feet per second. 
 
 13. Find the equation to the discharge parabola of the Sudbury 
 
 aqueduct from the data in 147, and draw the curve. 
 
 Q = 4(rf + 0-738) 2 . 
 
 14. A channel has an hydraulic mean depth of 5 feet Compare the 
 
 discharges if the sides are of smooth cement, and of rubble 
 masonry. 1*30 to 1. 
 
 15. The top width of an irrigation canal is 200 feet, the depth 10 feet, 
 
 and the side slopes 3 to 1. The slope is 15 inches per mile. 
 Find the discharge. v = 3-86 ; Q = 6567. 
 
CHAPTER XIII 
 
 GAUGING OF STREAMS 
 
 155. FOR various purposes the engineer needs to gauge the 
 flow of streams. For instance, in determining the value of 
 a fall as a source of water power the volume of flow throughout 
 the year must be ascertained. The flood discharge is of little 
 value unless storage reservoirs can be constructed. The 
 ordinary summer flow and the minimum flow are factors of 
 greater importance generally. Then again, the water-supply 
 of many towns is derived from the drainage of large gathering 
 grounds, flowing off by a stream. In considering the sufficiency 
 of the supply, the flow must be determined partly by rainfall 
 observations, partly by gauging the stream so as to establish 
 a relation between the rainfall and flow from the catchment 
 basin. Usually gauging operations are carried on for a 
 considerable period, as accurate statistics are required in the 
 settlement of difficult questions such as the apportionment 
 of compensation water. Lastly, in the management of irriga- 
 tion works it is frequently necessary to gauge the flow in 
 canals and distribution channels. 
 
 156. Water-level gauge. Wherever stream discharge 
 measurements are carried on, water-level gauges should be 
 established, on which readings of the varying water-level can 
 be taken simultaneously with the velocity observations. The 
 zero of the gauge should be connected by levelling with a 
 permanent bench mark, and the zero should be below the 
 lowest water-level to avoid minus readings. The scale of 
 the gauge should be in feet and tenths. The scale may be 
 fixed to a pile driven into the stream bed or fixed to a 
 masonry structure. Sometimes a scale attached to a float 
 
 264 
 
CHAP, xni GAUGING OF STEEAMS 265 
 
 is convenient, the reading being taken against a fixed mark. 
 Automatic gauges are used in important investigations. A 
 cord attached to a float gives motion through reducing-gear 
 to a pencil which records the water-level on a drum driven 
 by clockwork. 
 
 157. Mean velocity calculated from the longitudinal 
 slope. If the longitudinal surface slope of a stream is 
 determined in a part where the channel is of fairly regular 
 section, then the discharge can be ascertained by the formulae 
 of flow, subject, however, to the difficulty of selecting a 
 coefficient suitable to the character of the stream. In most 
 cases, however, the surface slope is an extremely small quantity, 
 generally less than 1 in 5000, and the oscillations of the 
 water surface render its determination difficult. The slope 
 in natural streams often differs to some extent on the two 
 sides as the current sets to one bank or the other. In 
 Cunningham's experiments on the Ganges Canal twelve 
 measurements of slope on symmetrical 2000 and 4000 feet 
 lengths differed by 25 per cent, but the site was probably a 
 specially difficult one. Usually the mean of the slope 
 determined at the two banks is taken as the virtual slope 
 of the stream. 
 
 158. Gauging by observation of the velocity of flow. 
 In streams of moderate size the most accurate method of 
 gauging is by a weir constructed for the purpose across the 
 stream. But often it is impracticable to erect a weir, and 
 the operation of gauging is then effected by determining the 
 cross section H and the mean velocity v m of the stream. The 
 discharge is Q = lv m . For gauging purposes a straight and 
 unobstructed reach of the stream should be selected, where the 
 cross section is fairly uniform in area and form. Then two 
 series of observations are required : (1) a survey of one or 
 more cross sections of the stream: (2) observations of the 
 velocity at one or more points of the cross section. 
 
 159. Measurement of transverse sections. The depth 
 of the stream is ascertained at a series of points, equidistant 
 if possible, along the line of the required cross section. For 
 small streams a wire may be stretched across, with equal 
 distances of about 10 feet or less marked on it by tags. If 
 the wire is first set up on land and stretched with a given 
 
266 
 
 HYDKAULICS 
 
 CHAP. 
 
 weight, the position of the tags can be fixed so that their 
 horizontal distances are equal. The wire is then stretched 
 across the stream with the same tension. The depth at each 
 tag can be taken with a light graduated and loaded rod. Care 
 should be taken that the wire is perpendicular to the thread 
 of the stream. 
 
 For large rivers the position of soundings is fixed by angu- 
 lar measurement. A base line AB (Fig. 122), parallel to the 
 
 stream, is first laid out and 
 _ . _8 - measured. Next staves are 
 set up at CA and D along 
 the line of the required 
 section and at right angles 
 to AB. Observers are 
 placed at C and B ; a boat 
 r ' drops down stream, and at 
 the moment it crosses the 
 section at E the observer C 
 signals, the sounding is 
 taken in the boat, and B 
 Fig. 122. with a box sextant takes 
 
 the angle ABE. This is 
 
 repeated till soundings at a sufficient number of points 
 have been ascertained from which to plot the cross section. 
 The soundings may be taken by a graduated rod if the depth 
 is less than 15 or 18 feet, or by a weighted cord or lead-line 
 or chain. If the velocity of the stream is considerable, the 
 weight should be disc-shaped or lenticular, so as to expose as 
 little surface normal to the current as possible. A simple 
 winch and wire are convenient for lowering the weight, and 
 the winch may have a counter which shows the depth. From 
 the observations the section is plotted, and the area H and 
 wetted perimeter % are calculated. 
 
 The area of a plotted cross section may be obtained by a 
 planimeter, or by dividing the width of the stream into n equal 
 spaces and measuring the n + 1 vertical ordinates at the divid- 
 ing points. Let 6 be the width of a division, and h , k l} . . . h n 
 be the measured ordinates. Then by the trapezoidal rule the 
 area is 
 
xiii GAUGING OF STEEAMS 
 
 If the end ordinates are zero, 
 
 267 
 
 If there are ten spaces, Simpson's rule may be used with 
 somewhat greater accuracy 
 j 
 
 As the level of a stream varies from time to time, a level 
 gauge should be fixed before operations are begun. The water- 
 level should be noted on this gauge when taking the cross 
 sections, and afterwards when the velocity observations are 
 made. 
 
 If velocity observations are to be taken, at least two cross 
 sections should be measured and the average values of ^ and H 
 computed for use in calculations. 
 
 160. Float gauging. The velocity in a stream may be 
 directly observed by taking the time of transit of a float over 
 a measured length of stream. Surface floats are used to 
 determine surface velocities. They may be balls, or discs of 
 wood or cork. A tuft of oily cotton-wool, which does not get 
 wet, is a useful means of rendering them visible. Captain 
 Cunningham at Eoorkee 1 used 
 thin deal discs 3 inches diameter 
 and 1 inch thick. Sub -surface 
 floats. To observe velocities be- 
 low the surface, a large relatively 
 heavy float (Fig. 123), connected by 
 a thin wire (about 0*015 inch thick) 
 to a small, light surface float, has 
 been used. It is assumed that the 
 motion of the combination is prac- 
 tically that of the sub-surface float, 
 the influence of the surface float and 
 connector being negligible. But if 
 the large float is made nearly of the 
 
 density of water, so that the surface float may be small, the 
 eddies prevent the large float from keeping its depth. If the 
 
 1 Eoorkee Hydraulic Experiments, by Captain Allan Cunningham, R.E. 
 (Thomason College Press). 
 
 Fig. 123. 
 
268 
 
 HYDEAULICS 
 
 CHAP. 
 
 lower float is heavy, the upper float must be large, and then 
 its influence on the motion of the combination is not negligible 
 and the velocity observed is not the true sub-surface velocity. 
 Fig. 124 shows the form of sub-surface float used by Captain 
 Cunningham at Eoorkee. It consists of a hollow metal ball 
 connected to a disc of cork. The influence of the connecting 
 wire on the motion increases as the depth of the sub-surface 
 
 K3 diet* 
 
 -t l 
 
 float increases, and the observations become less trustworthy the 
 greater the depth. Twin floats. Fig. 125 shows two equal 
 balls connected by a wire, the lower being loaded so that the 
 combination just floats. The motion of the twin float must 
 be nearly the mean of the surface velocity and the velocity at 
 the depth at which the lower float swims. Thus if v s is the 
 surface velocity, and v d the velocity at the depth d, the velocity 
 
 of the twin float is v = -(v s + v d ). If v s is ascertained by 
 means of a surface float, 
 
 v d = 2v - v s 
 
 (3). 
 
 Captain Cunningham found the twin float more satisfactory 
 than the sub-surface float, but the influence of the connector 
 increases with the depth, and also the uncertainty as to whether 
 the lower float keeps its depth or is tossed about by eddies 
 in the water. 
 
 161. Rod floats. Fig. 126 shows another form of float 
 
XIII 
 
 GAUGING OF STEEAMS 
 
 269 
 
 used in some early researches. Its use has been revived by 
 Captain Cunningham in India. In its simplest form it con- 
 sists of a wooden rod with a cap at the lower end in which 
 shot can be placed, so that the rod floats nearly upright, and 
 with little projection above the 
 water -surface. Wood rods may 
 be made in lengths which can be 
 screwed together. Cunningham 
 used sets consisting of lengths 0*1, 
 0'2, 0-3 ... up to 1 foot, and 1, 
 2, 3 ... up to 12 feet; but tube 
 rods of tinplate about 1 inch in 
 diameter made of graduated lengths, 
 adjusted to float at definite depths 
 in still water and marked, were 
 found more convenient. He found 
 that the velocity of a rod, the 
 immersed length of which was 
 nearly equal to the depth of the 
 stream, is a close approximation 
 to the mean velocity on the verti- 
 cal corresponding to its path, and 
 
 he considered it the most accurate means of float gauging in 
 suitable conditions. At any rate the gaugings showed that 
 though the rod necessarily was shorter than the full depth of 
 the stream, its velocity was very approximately the mean 
 velocity at the vertical corresponding to its path. The rod 
 float is certainly free from the chief objections to the sub- 
 surface or twin float. 
 
 162. Float paths and time of transit. In the part of 
 the stream selected for gauging two cross sections are fixed at 
 a measured distance apart, and the time of transit of the floats 
 between these sections is observed. The floats are thrown in 
 above the upper section at various points in the width of the 
 stream. In careful gauging the exact float paths should be 
 observed. The two end sections may be marked by cords 
 stretched across the stream, and if these have coloured tags at 
 equal distances it is possible to note approximately the 
 distance from the bank at which each float crosses each section. 
 If I is the distance between the cross sections, and t the 
 
 Fig. 126. 
 
270 
 
 HYDKAULICS 
 
 CHAP. 
 
 time of transit, then v = l/t is the velocity of the stream at the 
 position of the float path normal to the cross sections. 
 
 In large streams the float paths must be observed by box 
 sextants or theodolites. A base line AB (Fig. 127) is set out 
 
 parallel to the thread of the stream. 
 fi Ranging rods are set up at A 1? B p 
 on lines at right angles to the base, 
 usually on the lines of surveyed 
 transverse sections. Observers are 
 stationed at A and B with sextants. 
 Floats are dropped into the stream 
 from a boat upstream of AA . 
 As the float crosses AA X at C, the 
 observer at A signals, and B takes 
 
 B / x vlD B, the angle ABC. When the float 
 
 crosses BB X at D, B signals and A 
 takes the angle BAD. An observer 
 Fig. 127. also notes with a chronograph the 
 
 time between the signals. All the 
 
 data are so obtained for calculating the velocity and plotting 
 the float path CD. 
 
 The best length of the float path depends on the velocity 
 and regularity of the stream; lengths of 50 to 250 feet have 
 been used. The longer the base the less the error of the 
 time observation. But, on the other hand, the longer the base 
 the more the floats stray about into regions of differing velocity. 
 In the Ganges Canal researches Captain Cunningham found a 
 run of 50 feet best for the central parts of the stream, but 
 near the banks this had to be shortened to 12j feet. With 
 any longer run the floats strayed to the banks. 
 
 163. The screw current meter. This was termed by 
 early hydraulicians the Woltmann Mill. In improved form 
 it is the most generally useful, and, if properly calibrated, the 
 most accurate apparatus for measuring velocity in streams. 
 A screw propeller, like that in Fig. 128, delicately supported, 
 drives a counter by a worm. The counter can be put in or 
 out of gear by a cord. The meter is fixed on a rod or length 
 of gas-pipe, and held in the water in the desired position. 
 A rudder keeps the propeller facing the stream. The counter 
 is put in gear for one minute or more, and from the difference 
 
XIII 
 
 GAUGING OF STEEAMS 
 
 271 
 
 of the counter readings divided by the duration of run the 
 velocity is calculated. In its ordinary form the meter must 
 be lifted from the water to read the counter, and cannot be 
 conveniently used at greater depths than about seven feet. 
 
 Harlacher screw current meter. This is a current meter 
 with an electrically actuated indicator showing the revolutions. 
 The meter is on a sleeve which slides on a substantial hollow 
 cast-iron rod, and can be moved up and down the rod by a 
 cord passing down inside it. The rod is long enough to be 
 firmly fixed in the bottom of the river. The cord is wound 
 
 Fig. 128. 
 
 on a barrel fixed to the rod, and this has an indicator showing 
 the depth of the meter from the surface. The whole apparatus 
 is fixed on a raft which can be moved across the stream, and 
 anchored at each vertical at which the velocities are to be 
 taken. A current from a small primary battery passes down 
 an insulated wire and back by the rod. A contact-piece on 
 the shaft of the screw closes the circuit every revolution. 
 The current drives a kind of electrical clock with two dials, 
 one showing revolutions and the other hundreds of revolutions. 
 The apparatus being fixed at a vertical in the cross section of 
 the stream, the meter is dropped by the cord to points equi- 
 distant on the vertical, and at each the revolutions in one 
 
272 HYDKAULICS CHAP. 
 
 minute or more are observed. The meter is then moved to 
 the next vertical, and similar observations made. The mean 
 velocity on each vertical is calculated from the observations. 
 Otherwise, the mean velocity on a vertical may be found 
 directly by moving the meter slowly and regularly down the 
 vertical, and noting the revolutions and time of transit. It 
 will be seen that all the observations at each vertical can be 
 made rapidly without removing the apparatus from the water. 
 Harlacher used this meter on the Danube in water 26 feet 
 deep running at 10 feet per second (Proc. Inst. Civil 
 Engineers, Ixvii., 1881). 
 
 Current meter of J. Amsler Laffon (Fig. 128). This can 
 be used on a rod like the primitive meter, and then its chief 
 peculiarity is an improved method of putting the counter in 
 or out of gear. There is a double ratchet, and alternate pulls 
 on a cord throw the counter into gear and out of gear. 
 
 But there is a wholly different way in which this meter 
 can be used, the meter M being hung in gimbals, permitting 
 freedom of motion in all directions, and suspended in the 
 water by a wire (Fig. 129). A conical rudder keeps the 
 meter facing the current. The suspending wire is coiled on 
 a small winch A, and this has an index which can be set to 
 show the precise depth at which the meter is suspended. 
 Below the meter, to keep the suspension wire vertical, is a 
 lenticular weight W, of 85 Ibs., presenting little resistance to 
 the water, so that the wire is practically vertical. For 
 indicating the revolutions of the meter there is an electric 
 circuit formed by an insulated wire from a battery B, and 
 return through the suspension wire. This circuit is closed, 
 by a contact on one of the counting wheels shown in Fig. 128, 
 at every hundred revolutions of the screw, and a bell is rung. 
 It is only necessary, therefore, to note the time by a stop-watch 
 for 100, 200, or 500 revolutions. A subsidiary arrangement 
 is that, when the foot of the lenticular weight touches ground, 
 a contact is made and the circuit closed, so that the bell rings 
 continuously. The meter is then one foot above the ground. 
 This gives warning, and has the further advantage that the 
 apparatus can be used as a satisfactory sounding instrument 
 in any depth of water. 
 
 The suspended meter is generally used thus : The boat 
 
XIII 
 
 GAUGING OF STKEAMS 
 
 273 
 
 is anchored at a vertical, its position being fixed by angular 
 measurement. The meter is then lowered till its axis is at 
 the water surface and the depth index on the winch is set to 
 zero. The meter is then lowered till the foot touches bottom. 
 
 Fig. 129. 
 
 If h t is the reading, the whole depth of the stream is 
 
 H = ^ + 1. Then velocities are observed. 
 
 velocity at h l ; v 2 at h 2 = A x d ; v^ at h 3 = ^ 2d ; . 
 
 Let tfj be the 
 
 at h n = h l (nl}d. 
 very nearly 
 
 The mean velocity on the vertical is 
 
 18 
 
2*74 
 
 1 (,/v, 
 
 -l 
 
 HYDKAULICS 
 
 v 
 
 . . . +V n _i + 
 
 O 
 
 + + 
 
 CHAP. 
 
 (4). 
 
 Or the vertical velocity curve may be plotted, and its mean 
 ordinate found. The meter can be used with great facility 
 in rivers even in flood. 
 
 EXAMPLE OF CURRENT METER OBSERVATIONS ON A VERTICAL 
 
 Vertical No. 3. 
 
 Depth at vertical, 2-6 feet. 
 
 Distance from zero of transverse section, 32 feet. 
 
 Water-level on gauge, 1-65 feet. 
 
 2 h. 50 m. p.m. 
 
 Depth. 
 
 No. of 
 Revolutions. 
 
 Time. 
 
 Revolutions 
 per Second. 
 
 Mean 
 Revolutions 
 per Second. 
 
 Velocity. 
 Feet 
 per Second. 
 
 0-3 
 0'3 
 
 296 
 240 
 
 75 
 60 
 
 3-946 
 4-000 
 
 } 3-973 
 
 3-325 
 
 0-63 
 0-63 
 
 237 
 238 
 
 60 
 60 
 
 3-950 
 3-966 
 
 } 3-958 
 
 3-309 
 
 0-96 
 
 217 
 
 60 
 
 3-616 
 
 ^1 
 
 
 0-96 
 0-96 
 
 240 
 
 198 
 
 60 
 60 
 
 4-000 
 3-300 
 
 I 3-637 
 
 3-050 
 
 0-96 
 
 218 
 
 60 
 
 3-633 
 
 J 
 
 
 1-29 
 
 234 
 
 65 
 
 3-600 
 
 } 
 
 
 1-29 
 
 211 
 
 65 
 
 3-246 
 
 I 3-419 
 
 2-870 
 
 1-29 
 
 256 
 
 75 
 
 3-413 
 
 J 
 
 
 1-62 
 1-62 
 
 192 
 179 
 
 60 
 60 
 
 3-200 
 2-976 
 
 } 3-088 
 
 2-600 
 
 1'95 
 1-95 
 
 168 
 165 
 
 60 
 61 
 
 2-800 
 2705 
 
 | 2753 
 
 2-325 
 
 2-60 
 
 Bed of 
 
 stream. 
 
 
 
 
 Here the mean velocity on the vertical by eq. (4) is 
 
 
 -87 + 3-05 + 3-309 + 1-662 
 
 -3 x 3-325^ + 1( '65 x 2-325 } 1 =2 -631 feet per second. 
 
 In connection with this it may be mentioned that in 
 gauging the Severn at Worcester, in 1880, a Deacon electric 
 current meter was used, fixed in a frame suspended from a 
 No. 12 steel wire stretched across the river. The river was 
 180 feet wide and about 25 feet deep. Velocity measurements 
 were made at every foot of depth on verticals 10 or 20 feet 
 
xni GAUGING. OF STKEAMS 275 
 
 apart in the cross section. The frame carrying the meter 
 was suspended from a small carriage on two 3 -inch pulleys, 
 and traversed by an endless wire passing over pulleys on the 
 end supports of the carrying wire. Other wires from the 
 frame, carried over a pulley on the carriage, served for raising 
 and lowering the frame. Lastly, a wire with a cast-iron 
 anchor-plate of 70 Ibs. passed through the frame and over 
 the carriage, and served to keep the frame vertically in position 
 during the observations. Insulated wires from the meter, 
 through which a current passed when contact was made at 
 the meter, indicated on shore the revolutions of the meter 
 (Turner, Proc. Inst. Civil Engineers, Ixxx., 1884). In some 
 cases the meter has been used by observers on a travelling 
 platform suspended from a wire rope stretched across the 
 stream. In a gauging of the Khine by Baum (Proc. Inst. 
 Civil Engineers, Ixxi. 456) the current meter was used on a 
 platform between two coupled boats, sliding on a T-iron 
 4" x 2f ". 
 
 164. Calibrating the screw current meter. The 
 accuracy of velocity observations by current meter depends 
 entirely on the care and skill used in determining the constants 
 of the instrument. If the screw propeller were of uniform 
 pitch p, and if it were Motionless, then it would make one 
 revolution for p feet of water passing it. The relation of 
 velocity v and revolutions per second n would be v=pn. In 
 any actual instrument these conditions are not satisfied. At 
 some velocity V Q (about 4 inches per second or less) the meter 
 ceases to revolve, being held by friction. Also the pitch 
 cannot be accurately measured. Hence the relation of v and 
 n must be determined by experiment. It is generally assumed 
 that the form of the relation is linear, so that 
 
 v = an + ft . . . . (5), 
 
 where a and /3 are constants, and {$ is the velocity at which 
 rotation ceases. Exner has shown that the following equation, 
 on theoretical grounds, is more exact and better agrees with 
 experiment : 
 
 t>= < v /(a% 2 + 2 ) . . (6). 
 
 But when the lowest velocity is not less than 1 foot per 
 second, eq. (5) is practically accurate and more convenient. 
 
276 
 
 HYDKAULICS 
 
 CHAP. 
 
 Suppose a current meter towed over a length I feet in 
 still water, and that it makes N revolutions, the time of 
 transit being t seconds. The speed of towing or velocity of 
 the water relatively to the meter is l/t = v feet per second, 
 and the speed of the meter is N/ = n revolutions per second. 
 Let a number of observations be taken in this way at different 
 speeds, and let n l} n 2 , % ... be the meter speeds correspond- 
 ing to the velocities v lt v z , v s . . . Let ra be the number of 
 observations. Then, assuming the relation v = an + ft, the 
 values of a and /? may be found by the method of least 
 squares. 
 
 - 2(n)2(wt;) 
 
 . (7). 
 
 Example. For instance, the following table contains the results of 
 a series of tests on a meter and the summation of the quantities required 
 in determining the constants. The length of run was 336 feet. 
 
 
 Time of 
 
 Velocity, 
 
 No. of 
 
 
 
 No. of 
 
 Transit, 
 
 Feet per 
 
 Revolutions 
 
 
 
 
 Run. 
 
 t 
 
 Second, 
 
 per Second, 
 
 n . 
 
 " 
 
 
 Seconds. 
 
 V. 
 
 71. 
 
 
 
 1 
 
 115 
 
 2-921 
 
 2-043 
 
 4-174 
 
 5-969 
 
 2 
 
 116 
 
 2-896 
 
 2-000 
 
 4-000 
 
 5-792 
 
 3 
 
 113 
 
 2-973 
 
 2-053 
 
 4-215 
 
 6-103 
 
 4 
 
 130 
 
 2-584 
 
 1-776 
 
 3-154 
 
 4-590 
 
 5 
 
 113 
 
 2-973 
 
 2-088 
 
 4-360 
 
 6-209 
 
 6 
 
 121 
 
 2-776 
 
 1-892 
 
 3-580 
 
 5-253 
 
 7 
 
 125 
 
 2-687 
 
 1-824 
 
 3-327 
 
 4-903 
 
 Sums 
 
 19-810 
 
 13-676 
 
 26-810 
 
 38-819 
 
 [S(?i)] 2 =187'03 
 
 7 x 38-819 - 13-676 x 19'810_Q-81 _ 
 7x26-810-187-03 ~0'64~ 
 19-810x26-810 -13-676x38-819 0'22 
 
 7x26-810-187-03 
 
 0'64 
 
 
 Kecalculating the velocities from the revolutions, using these values of 
 the constants, and comparing the results with the observed velocities, the 
 following table is obtained : 
 
 Velocity observed 2-921 2-896 2'973 2-584 2-973 2776 2-687 
 
 calculated 2-931 2'876 2'943 2'592 2'987 2'739 2'653 
 
 + 01 -'02 --03 +-008 +-014 -'037 -'034 
 
GAUGING OF STEEAMS 277 
 
 A different formula of reduction is used by some American 
 engineers. If in the equation v = an + ft observed values of 
 v and n are inserted, then for m observations a series of ra 
 equations can be formed 
 
 an 2 + /3-v 2 = 2 { . . (8), 
 
 where e l} e 2 are small errors of individual observations. Since 
 ft enters in the same way into all the equations, its most 
 probable value is the arithmetical mean. Let n m = (2w)/m be 
 the] mean value of n, and v m = (2v)/ra the mean value of v. 
 Then, as the errors cancel, 
 
 . . . (9). 
 Inserting this value in eq. (8), 
 
 a(l ~ n m) ~ ( V l ~ V m) = f v 
 
 To weight these equations multiply each by the coefficient 
 of a. Then 
 
 -*.)- 
 
 Adding these equations, 
 
 = 0. 
 
 and ft can then be found from eq. (9). 
 
 Example. Taking the data in the table above, the following are the 
 quantities required to determine a and /? : 
 
278 
 
 HYDEAULICS 
 
 CHAP. 
 
 
 t. 
 
 71. 
 
 V. 
 
 n-nm. 
 
 V-Vm. 
 
 (n - 7tm)2. 
 
 (n-nmXy-vm). 
 
 1 
 
 115 
 
 2-043 
 
 2-921 
 
 089 
 
 091 
 
 00792 
 
 00810 
 
 2 
 
 116 
 
 2-000 
 
 2-896 
 
 046 
 
 066 
 
 00211 
 
 00304 
 
 3 
 
 113 
 
 2-053 
 
 2-973 
 
 099 
 
 143 
 
 00980 
 
 01416 
 
 4 
 
 130 
 
 1-776 
 
 2-584 
 
 -178 
 
 - '246 
 
 03168 
 
 04379 
 
 5 
 
 113 
 
 2-088 
 
 2-973 
 
 134 
 
 143 
 
 01796 
 
 01916 
 
 6 
 
 121 
 
 1-892 
 
 2-776 
 
 -062 
 
 - '054 
 
 00384 
 
 00335 
 
 7 
 
 125 
 
 1-824 
 
 2-687 
 
 -130 
 
 -143 
 
 01690 
 
 01859 
 
 Sums 
 
 ... 
 
 13-676 
 
 19-810 
 
 ... 
 
 ... 
 
 09021 
 
 11019 
 
 0-11019 
 
 = 1-2215, 
 
 0-09021 
 = 2-830- (1-2215 x 1-954) = 0'444. 
 
 Recalculating v from the revolutions 
 
 v observed . 
 v calculated . 
 Difference , 
 
 2-921 
 2-940 
 + 019 
 
 2-896 
 2-887 
 -009 
 
 2-973 
 2-952 
 -021 
 
 2-584 
 2-613 
 + 029 
 
 2-973 
 2-995 
 + 022 
 
 2-776 
 2-755 
 -021 
 
 2-687 
 2-672 
 -015 
 
 When the constants of a meter are determined, a diagram 
 (Fig. 130) may be drawn from which the velocity corresponding 
 to any number of revolutions per second can be read off. 
 
 REVOLUTIONS 
 Fig. 130. 
 
 The relation is linear, and the line ab starts from an 
 ordinate ft on the axis of velocities, and has a slope Icjac = a. 
 
 The calibration of a meter by fixing it on a boat towed 
 over a measured base-line at different speeds is an operation 
 
XIII 
 
 GAUGING OF STEEAMS 
 
 279 
 
 requiring a good deal of care. It should be repeated many 
 times to eliminate errors. A better plan is to fix the meter 
 on a truck running on rails alongside a quay wall Slow 
 velocities are best obtained by towing the meter by a winch. 
 Sometimes one current meter can be calibrated by comparing 
 it with another previously calibrated. It is not very satis- 
 factory to obtain the constants by placing the meter in a 
 stream the velocity of which has been determined by floats, 
 but perhaps good results would be obtained if the speed of a 
 stream was determined by a Pitot tube and the current meter 
 used in the same stream at the same place. A check on the 
 calibration of a current meter has sometimes been obtained 
 by using it to measure the volume of flow in a channel the 
 
 V 
 
 B 
 
 Fig. 131. 
 
 discharge of which was also measured by a weir. In a few 
 cases the constants of meters have been ascertained by towing 
 them in the Admiralty tank at Torquay, in which ship models 
 are tested. The means of registering time and speed are so 
 perfect in this case that the results are very trustworthy (see 
 Gordon, Proc. Inst. Mech. Engineers, 1884). 
 
 165. Pitot tube and Darcy gauge. A very early 
 instrument invented by Pitot in 1730, employed in a modified 
 form by Darcy and Bazin in their classical researches, has 
 again come into use in determining the velocity of currents of 
 water and air. Suppose a bent tube, such as that shown in 
 Fig. 131, immersed in a stream of water. When the mouth 
 of the tube points upstream as at A, the impact of the fluid 
 produces a pressure which raises the water in the tube to a 
 height h above the surface outside. If, as at B, the mouth is 
 
280 
 
 HYDKAULICS 
 
 CHAP. 
 
 m 
 
 parallel to the stream, there is no impact, and the water inside 
 and outside are at the same level. If, as at C, the mouth 
 points downstream there is a certain amount of suction, and 
 the level in the tube is depressed by 
 some distance h^ Pi tot used two 
 tubes arranged as at A and B, and 
 found that the difference of level was 
 very nearly v 2 /2g. Hence the special 
 advantage of this instrument is that, 
 if properly constructed, it is almost 
 independent of the need of calibra- 
 tion. 
 
 An objection to the original Pitot 
 gauge was the difficulty of reading the 
 height h when the gauge was in the 
 water. This is overcome in the modified 
 Darcy gauge shown in Fig. 132. The 
 gauge is shown clamped at B on a rod 
 AA resting on the stream bed. The 
 tubes corresponding to A and B in 
 Fig. 131 are at d, being made very 
 small to avoid disturbing the flow. The 
 mouth of the statical tube opens down- 
 wards. The tubes d communicate with 
 the glass tubes I, b, which can be shut 
 off by a two-way cock c actuated by 
 cords. In order to bring the water 
 columns in &, 5 into a convenient posi- 
 tion for reading, a partial vacuum is 
 made above them by sucking out a 
 little air by the tube m and then 
 closing a cock at a. The difference 
 of height of the columns is not altered 
 by raising them. The columns having 
 come to rest, the cock c is closed, and 
 Fig. 132. the readings taken by verniers. For 
 
 a velocity of one foot per second 
 
 &=0'186 inch, which is rather small, but h increases as 
 the square of the velocity, so that at 4 feet per second h = 3 
 inches nearly. 
 
XIII 
 
 GAUGING OF STKEAMS 
 
 281 
 
 If v is the velocity of the stream and h the difference of 
 level of the columns, 
 
 . . (11), 
 
 where k is a constant depending on the form of the mouths of 
 the instrument and the way they are placed. But if the tubes 
 and orifices are small so as not to create eddies, k differs hardly 
 at all from unity. Darcy calibrated his gauge with great care 
 in three ways. Towing the gauge in still water he found 
 k = 1*034 ; observing velocities simultaneously in a stream by 
 floats and by the gauge he found k= 1'006 ; and by taking 
 a number of readings in the cross section of a channel the 
 flow in which was known, he found k = 0*993. He concluded 
 
 Fig. 133. 
 
 Fig. 134. 
 
 that the true value of k did not sensibly differ from unity. 
 White (Journ. Am. Assoc. of Eng. Soc., 1901), and Williams, 
 Hubbell, and Fenkell (Trans. Am. Soc. of Civil Engineers, 
 1902), found that if the tubes were well formed the 
 coefficient was unity. Threlfall (Proc. Inst. MecJi. Engineers), 
 using Pitot tubes in a current of air, found k =0*9 74; and 
 Stanton, in extremely accurate experiments on the flow of 
 air, found k= 1'03 (Proc. Inst. Civil Engineers, 1903). 
 
 The chief cause of variation of the coefficient seems to be 
 the action on the mouth of the statical pressure tube. If this 
 is at all large, the stream lines are bent concave to the mouth 
 (Fig. 133), and there is a slight sucking action which increases 
 h. This may be obviated by a plane disc fitted to the tube, 
 as in Fig. 134. A good arrangement is to form the two tubes 
 
282 
 
 HYDEAULICS 
 
 CHAP. 
 
 concentric, as in Fig. 135, and to place the statical pressure 
 opening on the cylindrical part of the outer tube. 
 
 In the case of air of density G Ibs. per cubic foot, the head 
 
 x & HOLE. 
 
 ,& THICK. 
 
 f fo SLOT. 
 
 Fig. 135. 
 
 corresponding to P Ibs. per square foot is P/G-. Or if the 
 pressure is measured in inches of water h w , the head is 
 5'2A W /G in feet of air. Then 
 
 If the air is at ordinary pressure and temperature, and 
 k = unity, 
 
 . . . (12). 
 
 166. Ratio of different velocities in a stream. Surface 
 and mean velocities. In reducing gauging observations it 
 is necessary to know the relation of the velocities at different 
 parts of a stream. Thus a rough gauging may be made by 
 observing the greatest surface velocity only, if the relation of 
 the mean to the greatest surface velocity is known. 
 
 Let V be the mean velocity of the whole cross section, and 
 v the greatest surface velocity, which may be found by using 
 a surface float or current meter. If O is the area of cross 
 section the discharge is Q = JYV. Darcy and Bazin deduced 
 from their researches on small regular channels that 
 
 V = v - 25*4 ^mi (13)- 
 
 But V = c \frni, where c is a constant for a given type of 
 channel ( 137). Hence 
 
 V = 
 
 c + 25-4 
 
XIII 
 
 GAUGING OF STREAMS 
 
 283 
 
 The following table gives values of V/v for the values of 
 c in 138 : 
 
 Hydraulic 
 Mean Depth 
 m 
 in Feet. 
 
 Values of V/v for Darcy's Classes of Channels. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 0-5 
 
 84 
 
 81 
 
 74 
 
 59 
 
 51 
 
 1-0 
 
 85 
 
 83 
 
 78 
 
 66 
 
 58 
 
 2-0 
 
 85 
 
 83 
 
 80 
 
 71 
 
 65 
 
 5-0 
 
 85 
 
 84 
 
 81 
 
 76 
 
 71 
 
 10-0 
 
 86 
 
 84 
 
 82 
 
 79 
 
 74 
 
 20-0 
 
 86 
 
 84 
 
 82 
 
 80 
 
 76 
 
 50-0 
 
 86 
 
 84 
 
 82 
 
 81 
 
 78 
 
 00 
 
 86 
 
 84 
 
 82 
 
 81 
 
 78 
 
 The ratio decreases as the size of channel decreases, and 
 still more considerably as the roughness of the bed increases. 
 In small wooden channels, probably fairly smooth, Prony found 
 V/-y =0'82. In the smooth brick conduit at Sudbury, with 
 a depth of 3 feet, the mean velocity was 0*85 of the maximum 
 velocity observed, and about 0'9 of the central surface velocity. 
 In the Vyrnwy stream gauged by Mr. Deacon, the bed width 
 was 33 feet, with side slopes 2 to 1, the bed and sides being 
 pitched with stone and the gauging section lined with concrete. 
 Here in extreme cases the ratio varied from 0*78 to 0*94, the 
 mean of all observations being 0'834. 
 
 In rivers with greater roughness and less well-proportioned 
 sections the ratio falls to much lower values. 
 
 [TABLE. 
 
284 
 
 HYDKAULICS 
 
 CHAP. 
 
 RATIO OP MEAX TO GREATEST SURFACE VELOCITY 
 
 
 Surface 
 Width. 
 
 Hydraulic 
 Mean 
 Depth. 
 
 V 
 
 Authority. 
 
 Weser . 
 
 276 
 
 7-5 
 
 69 
 
 Wagner. 
 
 Elbe . 
 
 361 
 
 4-6 
 
 78 
 
 M 
 
 Rhine . 
 
 705 
 
 8-9 
 
 72 
 
 Wagner and Grebenau. 
 
 Elbe . 
 
 397 
 
 6'6 
 
 77 
 
 Harlacher. 
 
 
 
 394 
 
 5-3 
 
 75 
 
 }J 
 
 ,, ... 
 
 344 
 
 3'9 
 
 69 
 
 n 
 
 Rhine . 
 
 718 
 
 5-9 
 
 70 
 
 Grebenau. 
 
 
 
 728 
 
 8-1 
 
 70 
 
 n 
 
 ,, 
 
 741 
 
 9-2 
 
 74 
 
 
 
 
 
 778 
 
 13-0 
 
 72 
 
 
 
 5} 
 
 902 
 
 19-3 
 
 74 
 
 )j 
 
 Eger at Falkenau . 
 
 
 0-62 
 
 57 
 
 Plenkner. 
 
 JJ 5> 
 
 ... 
 
 1-09 
 
 58 
 
 5) 
 
 Sazawa at Poric 
 
 
 1-09 
 
 59 
 
 
 
 )> j? 
 
 
 0-90 
 
 71 
 
 J} 
 
 ) 
 
 
 0-38 
 
 73 
 
 5) 
 
 Moldau at Budweis 
 
 
 1-12 
 
 73 
 
 ,, 
 
 
 
 
 0-91 
 
 71 
 
 5J 
 
 " 
 
 
 1-58 
 
 72 
 
 
 
 Wagner deduces for rivers the relation 
 V = 0*705v + 0'003v 2 . 
 
 which agrees with some other cases, and is useful in rough 
 gauging. 
 
 The central surface velocity is somewhat variable, being 
 affected by wind and other accidents. The mean surface 
 velocity, which can be obtained by a series of surface float 
 observations, has probably a more constant relation to the 
 mean velocity of the cross section. In the gaugings of the 
 Eger and Moldau the ratio (mean velocity) / (mean surface 
 velocity) was 0*90. Wagner found the value 0'88. 
 
 The ratio of the mean velocity for the cross section to the 
 mean velocity on the central vertical was 0'95 to 0'98 in 
 Cunningham's float gaugings on the Ganges Canal, 0'93 
 in Deacon's gaugings of the Vyrnwy stream, and 0'67 on 
 the Eger. 
 
 It seems probable that the ratio of the mean velocity for 
 
XIII 
 
 GAUGING OF STEEAMS 
 
 285 
 
 the cross section to the velocity at the centre of figure of the 
 cross section is a fairly constant ratio. The latter could be 
 easily determined by a current meter. In the Vyrnwy stream 
 this ratio was 0*888. 
 
 If v c is the velocity at the centre of figure and V the mean 
 velocity for the cross section, Wagner found in rivers 
 
 . . (15). 
 
 This ratio does not differ much from the ratio of the mean 
 velocity of cross section to central mid-depth velocity, which 
 was 0*8 7 6 at the Vyrnwy stream. 
 
 167. Velocities on one vertical. The following table 
 contains averages from the large mass of float gaugings made 
 by Captain Cunningham on the Ganges Canal. 1 The aqueducts 
 were 85 feet wide with 10 feet depth and less. The main 
 embankment site was 170 feet wide with 11 feet depth and 
 less. The averages are fairly consistent. The individual 
 results vary a good deal. 
 
 
 Central 
 Surface 
 Velocity, 
 
 V Q . 
 
 Maximum 
 Velocity, 
 
 t'mo*. 
 
 Bed 
 
 Velocity, 
 ^ 
 
 o 
 
 Vmax.' 
 
 Vb 
 
 Vmaz" 
 
 Solani left aqueduct 
 
 3-97 
 
 4-01 
 
 3-35 
 
 0-99 
 
 0-84 
 
 Solani right aqueduct . 
 
 4-06 
 
 4-18 
 
 3-70 
 
 0-97 
 
 0-88 
 
 Embankment main site 
 
 3-56 
 
 3-65 
 
 3-08 
 
 0-98 
 
 0-84 
 
 The following averages are from the same large mass of 
 observations : 
 
 
 Mean 
 
 
 
 
 
 
 
 Velocity 
 
 Half- 
 
 Rod 
 
 
 
 
 
 at 
 
 depth 
 
 Float 
 
 Vd/2 
 
 V r 
 
 tfd/2 
 
 
 Central 
 
 Velocity, 
 
 Velocity, 
 
 v. 
 
 V, 
 
 v r 
 
 
 Vertical, 
 
 "d/2- 
 
 v r . 
 
 
 
 
 
 V,. 
 
 
 
 
 
 
 Solani left aqueduct 
 
 3-65 
 
 3-69 
 
 3-55 
 
 1-010 
 
 0-972 
 
 1-039 
 
 Solani right aqueduct . 
 
 4-02 
 
 4-07 
 
 3-85 
 
 1-012 
 
 0-958 
 
 1-057 
 
 Embankment main site 
 
 3-44 
 
 3-48 
 
 3-31 
 
 1-011 
 
 0-963 
 
 1-051 
 
 Eoorkee Hydraulic Experiments, Cunningham. Roorkee, 1881. 
 
286 
 
 HYDKAULICS 
 
 CHAP. 
 
 The half-depth velocity was 1 per cent greater than the 
 mean velocity at a vertical. The rod float velocity was about 
 4 per cent less than the mean. The mean velocity was com- 
 puted from double float observations. 
 
 Wagner found the mean velocity at a vertical to be 0'8 of 
 the surface velocity at the vertical when the surface velocity 
 was not greater than 2 feet per second. The ratio was 0'85 
 for velocities from 2 to 4 feet per second, and 0*9 for velocities 
 from 4 to 10 feet per second. 
 
 The depth at which the maximum velocity is found at 
 
 Section/ 
 
 SecUon.lI 3 
 
 Fig. 136. 
 
 the central vertical is from to 0*3 of the whole depth. On 
 other verticals it varies a good deal according to the form of 
 the channel section. The position on a vertical at which the 
 velocity is equal to the mean velocity is fairly constant, and 
 equal to 0'58 to 0'6 of the whole depth. The mid-depth 
 velocity is very slightly greater than the mean velocity. 
 
 168. Surface or rod float gauging. Fig. 136 shows 
 a gauging of the Thames by surface floats. Two sections, 
 I. and II., were surveyed at the ends of a 200-foot base-line. 
 These sections are divided into ten compartments of equal 
 width. Between the sections the float paths are plotted. A 
 base-line AB is taken midway between the sections, and at 
 
XIII 
 
 GAUGING OF STEEAMS 
 
 287 
 
 the points where the float paths cross the line AB the 
 observed velocities are set up as ordinates. Through the 
 points so found the surface velocity curve is drawn. The 
 curve of mean velocities on verticals can be found from this 
 by taking ordinates 0'85 to 0'95 of those of the surface 
 velocity curve, according to the character of the stream. Let 
 Hj, H 2 . . . be the mean areas of the ten pairs of compartments 
 in the two end sections in square feet, and v lt v 2 . . . the mean 
 ordinate of the curve of mean velocities corresponding to each 
 compartment in feet per second. Then the discharge of the 
 stream is 
 
 Q = ft.^ + Q 2 v 2 + . . . + fl 10 %) cubic feet per second . (16). 
 
 The mean velocities might have been observed directly 
 by using rod floats or sub-surface mid-depth floats. In that 
 case the uncertainty due to the selection of the ratio of surface 
 to mean velocity is obviated. The following table gives the 
 results of the gauging shown in Fig. 136. The mean velocities 
 on the verticals are taken at 0*93 of the surface velocities. 
 
 RIVER GAUGING, OCTOBER 1877 
 
 Compartment 
 
 Mean Area 
 of Section, 
 Square Feet. 
 
 Mean Surface 
 Velocity, 
 Feet* 
 per Second. 
 
 Mean Velocity, 
 Feet 
 per Second. 
 
 Discharge, 
 Cubic Feet 
 per Second. 
 
 I. . 
 
 59-2 
 
 409 
 
 380 
 
 22-5 
 
 II. . 
 
 93-5 
 
 659 
 
 613 
 
 57-0 
 
 IIL . 
 
 111-8 
 
 905 
 
 842 
 
 93-9 
 
 IV. . 
 
 128-1 
 
 1-206 
 
 1-120 
 
 143-5 
 
 V. . 
 
 138-2 
 
 1-710 
 
 1-590 
 
 219-7 
 
 VI. . 
 
 153-3 
 
 1-798 
 
 1-670 
 
 256-1 
 
 VII. . 
 
 157-3 
 
 1-631 
 
 1-520 
 
 239-1 
 
 VIIL . 
 
 144-1 
 
 1-421 
 
 1-339 
 
 190-2 
 
 IX. . 
 
 116-4 
 
 1-115 
 
 1-037 
 
 121-0 
 
 X. . 
 
 44-2 
 
 579 
 
 538 
 
 25-7 
 
 Total . . 1368-7 
 
 169. Discharge curve. A very convenient method of 
 deducing the discharge from a curve of mean velocities on 
 verticals is to construct a curve with the stream width as base, 
 
288 
 
 HYDEAULICS 
 
 CHAP. 
 
 and ordinates proportional at each point to the discharge at 
 that point. 
 
 Let aeE (Fig. 137) be the stream section, A/B the curve 
 of mean velocities on verticals. Take ab = af=v ; ac = k = 
 any convenient unit. Join ce, and draw bd parallel to it. 
 Then d is a point on the discharge curve. 
 
 Fig. 137. 
 
 If D = ae is the depth, and v = af=ab is the mean 
 velocity at a, the discharge for any small portion dx of the 
 width of the stream at a is Dvdx, and the whole discharge of 
 the stream is 
 
 Q = f 
 
 But ad = (ae x ab}j(ac), that is ad = (Dv)/k. Let y = ad, 
 then . 
 
 . . (17); 
 
 that is, the whole discharge is proportional to the area of the 
 curve A^B. 
 
 If the area of the curve is measured in square inches, and 
 the scales are ra feet per second, and n feet to one inch, and k 
 is set off in inches, then the area of the curve must be 
 multiplied by mn 2 k to give cubic feet per second. 
 
 170. Calculation of discharge from the vertical velocity 
 curves. If the vertical velocity curves have been drawn from 
 current meter observations at different depths, the discharge 
 
GAUGING OF STKEAMS 
 
 289 
 
 between each pair of verticals can be regarded as the volume 
 of a truncated pyramid having the velocity curves as bases. 
 Let 1,1... (Fig. 138) be the distances between the 
 
 Fig. 138. 
 
 verticals ; a v a z . . . the areas of the vertical velocity curves. 
 Then the discharge between the verticals ra 1 and ra is 
 
 The discharge of the two end sections may be taken as the 
 volumes of pyramids on the bases ^ and a n . Hence the whole 
 discharge is 
 
 Q = 2 
 
 (18). 
 
 If the vertical velocity curve is plotted so that m feet per 
 second = one inch, and n feet of depth = one inch, then 
 one square inch of area represents mn square feet of water 
 passing the vertical per second. The areas of the curves 
 measured in square inches should be multiplied by mn, and 
 the widths taken in feet in the equation, to get the result in 
 cubic feet per second. 
 
 171. Calculation of discharge from contours of equal 
 velocity. If contours of equal velocity have been plotted, 
 as in Fig. 116, 147, a method due to Culmann may be 
 used. Let H be the area of cross section of the stream, and 
 Op I1 2 . . . the areas included in the successive contours; 
 these should be reckoned in square feet, so that if the scale is 
 m feet to an inch the areas measured in square inches must 
 be multiplied by ra 2 . Let d be the intervals of velocity for 
 which the contours are plotted in feet per second. Then the 
 discharge of any one layer of thickness d is %(&* 
 
 19 
 
290 HYDKAULICS CHAP, xm 
 
 The top layer of small volume will usually have a thickness 8 
 less than d, and its volume may be reckoned with accuracy 
 enough as -f l^S. Hence the whole discharge is 
 
 . (19). 
 
 172. Gauging streams by chemical means. Mr. C. E. 
 
 Stromeyer has experimented with a chemical gauging method 
 (Proc. Inst. Civil Engineers, clx. 349). A fairly concentrated 
 solution of a chemical for which a sensitive reagent is known 
 is discharged at a uniform rate into the stream to be gauged. 
 Analyses are made of the water before the chemical is added, 
 and after it has become well mixed with the stream. Let x 
 be the percentage of chemical in the solution, y the percentage 
 found in the water, a the volume of solution added per second, 
 and' Q the discharge of the stream. 
 
 Chloride of calcium, of magnesium, or of sodium and other 
 chemicals may be used. 
 
CHAPTEK XIV 
 
 IMPACT AND KEACTION OF FLUIDS 
 
 
 
 173. WHEN a stream of fluid impinges on a solid surface, it 
 exerts a pressure on the surface which is equal and opposite 
 to the force exerted by the surface on the fluid in changing 
 its momentum. 
 
 If a fluid glides over a solid also moving, the motion of 
 the former can be resolved into two components one a motion 
 which the fluid and solid have in common, the other a motion 
 of the fluid relatively to the solid. The motion which the 
 fluid has in common with the solid cannot be affected by their 
 contact. The relative component can be altered in direction, 
 but not in magnitude, for the relative motion must be 
 tangential to the surface, while the pressure between the 
 fluid and solid (friction being neglected) must be normal to 
 the surface. The pressure can deviate the fluid, but cannot 
 alter the magnitude of the relative motion. The absolute 
 velocity of the fluid, after contact with the surface, is found 
 by combining the deviated but otherwise unchanged relative 
 motion, tangential to the solid at the point where the fluid 
 leaves it, with the common velocity of fluid and solid. 
 
 The principle of the conservation of momentum has 
 already been explained in 35. The impulse of the mass 
 of fluid impinging in a given time is equal to the change of 
 momentum, the impulse and change of momentum being 
 estimated in the same direction. If Q cubic feet or GQ/<? 
 units of mass impinge in one second with a velocity v l in a 
 given direction, and v 2 is the velocity in the same direction 
 after impact, then the pressure exerted, also in the same 
 direction, is 
 
 291 
 
292 
 
 HYDKAULICS 
 
 CHAP. 
 
 (I)' 
 
 174. Jet deviated wholly in one direction. Let a jet 
 
 of water (Fig. 139) impinge on a curved trough-shaped vane 
 
 ae, so that it is deviated in the plane of the figure. Let ab 
 represent in magnitude and direction the velocity v of the jet, 
 and ac = u that of the vane. Completing the parallelogram, 
 ab = v may be resolved into two components a velocity in 
 common with the vane ac = u, and a velocity relative to the 
 
xiv IMPACT AND BEACTION OF FLUIDS 293 
 
 vane ad = v r In order that there may be no shock or 
 disturbance of the water at a, the tangent to the lip of the 
 vane must be parallel to ad. The water glides up the vane 
 with the velocity v r , and leaves it tangentially with this 
 relative velocity unchanged. Take ef tangential to the vane 
 and equal to v r) and eg equal and parallel to the common 
 velocity ac = u. Completing the parallelogram, eh is the 
 absolute velocity and direction of motion of the water leaving 
 the vane. Take ok equal and parallel to eh, and join kb, kc. 
 Then the initial velocity and direction of motion db are 
 changed during impact to ok, and kb = w is the change of 
 motion. If Q cubic feet of water impinge per second the 
 pressure on the vane is in the direction kb and equal to 
 
 Since ak is equal and parallel to eh and ac to eg, kc is equal 
 and parallel to hg, and therefore to ef. Hence ck, cb are each 
 equal to v r and parallel to the initial and final directions of 
 relative motion. It is unnecessary to consider the common 
 velocity in treating the problem. The change of motion kb 
 is represented in magnitude and direction by the third side of 
 an isosceles triangle ckb, the other sides of which are equal to 
 the relative velocity and parallel to the initial and final direc- 
 tions of relative motion. 
 
 175. A jet of water impinges axially on a solid of 
 revolution, which is moving in the same direction. 
 
 The section of the jet (Fig. 140) is supposed much smaller 
 than the solid. The water is deviated symmetrically in all 
 directions and flows away at an angle 6 with the axis, each 
 elementary stream being deviated through the same angle. 
 From the symmetry of the conditions the resultant pressure 
 on the solid will be axial. Let v be the velocity of the water, 
 u that of the solid. Since the common velocity is the same 
 before and after impact, it may be disregarded. Parallel to 
 the axis the relative velocity is v u before impact, and after 
 impact its component in the same direction is (v u) cos 0. 
 If a) is the section of the jet, the quantity of water impinging 
 per second is o)(v u\ and its mass is Ga)(v u)/g. The 
 resultant pressure on the surface, which is equal to the 
 
294 
 
 HYDKAULICS 
 
 CHAP. 
 
 change of momentum per second, estimated in the same 
 direction, is 
 
 P = o>(# - u){(v - u) - (v - u) cos 0} 
 
 Ibs. 
 
 . (2). 
 
 Fig. 140. 
 
 The work done by the water in driving the solid is 
 
 Pw = a*u(v - u) 2 (l - cos 6) ft.-lbs. per second. . (3). 
 
 If the solid is at rest, u = o, and then 
 
 P = *o 
 
 g 
 
 and no work is done. The work done will also be zero if 
 u = v. Hence there must be an intermediate ratio of u to v, 
 
 -COS0), 
 
xiv IMPACT AND KEACTION OF FLUIDS 295 
 
 for which the work is a maximum. The total energy issuing 
 from a fixed nozzle would be 
 
 G tf G 
 
 and the efficiency of the arrangement, considered as a means 
 of utilising the energy of the jet, is 
 
 *? = 
 
 Differentiating and equating to zero, 
 
 du 
 whence rj is a maximum if u = v/3. Inserting this value, 
 
 (l~H49 - - (5). 
 
 In a number of hydraulic machines, a jet acts on a series 
 of vanes which succeed one another in the same position at 
 very short intervals of time. Such vanes are attached to a wheel 
 and therefore have a circular path. But the path of each 
 during the action of the jet is very short, and if the radius 
 of the wheel is large, the curvature of the path may be 
 neglected. Then the quantity of water per second which acts 
 on the series of vanes is a)V, and the equations become 
 
 P = -wt<;-i*Xl-cos0)lbs. . . (6), 
 
 i/ 
 
 Pu = <avu(v u)(l cos 6) ft.-lbs. per second (7), 
 2u(v-u)(l -cos0) 
 
 The efficiency is greatest if u = v/2, and then 
 
 *7ma* = i(l-cos0) . . (8). 
 
 176. Special Cases. Case I. A jet impinges normally 
 on a plane moving in the same direction. Let v (Fig. 141) 
 
296 
 
 HYDRAULICS 
 
 CHAP. 
 
 be the velocity of the jet, and u that of the plane. The 
 
 relative velocity is v u. If o> is 
 the section of the jet, the quantity 
 of water which reaches the plane is 
 (t)(v u) cubic feet per second. In 
 the direction of the jet the initial 
 velocity of the water is v, and its 
 final velocity after impact is u. The 
 pressure on the plane, which is equal 
 to the change of momentum per 
 second, is 
 
 p 
 
 P = u(v - u)(v u) 
 
 Fig. 141. 
 
 and the work done in driving the plane is 
 
 = w(v u) 2 Ibs., 
 y 
 
 ~Pu = G>(0 - ufu ft.-lbs. per second. 
 This is a maximum for u = v/3, and then 
 
 Pu = <t)V B ft.-lbs. per second. 
 
 These results can be obtained by putting = 90 in eqs. (2) 
 and (3). If the plane is atjrest, u = 0, and then 
 
 P 
 
 P = W fl 2 Ibs. 
 9 
 
 It appears that if the area of the plane is less than 16 
 times the area of the jet, the effective deviation is less than 
 90, and the pressure is less. 
 
 Case II. A series of plane vanes are interposed in 
 front of the jet in succession. The other conditions are 
 supposed the same as in the last case. This arrangement is 
 roughly identical with that of an undershot wheel with plane 
 floats which enter in succession in front of a stream issuing 
 with the velocity due to the head driving the wheel. The 
 quantity of water acting per second on the vanes is cov cubic 
 feet. The pressure on the series of vanes is 
 
XIV 
 
 IMPACT AND KEACTION OF FLUIDS 297 
 
 /-^ 
 
 P = (jiv(v - u) Ibs. 
 
 The work done in driving the vanes is 
 
 /-i 
 Pu = (avu(v - u) ft.-lbs. per second. 
 
 This is a maximum if u = v/2, and then 
 
 These results can be obtained by putting 6= 90 in eqs. (6) 
 and (7). 
 
 Case III. A jet of water impinges on a series of hemi- 
 spherical cups moving in 
 the same direction (Fig. 
 142). Here the water is 
 deviated through 180. The 
 initial relative velocity is 
 v u, and the final (v u) 
 = u v, both parallel to the 
 direction of the jet. The 
 quantity of water impinging 
 per second is cov cubic feet. 
 
 P = (0v{(v u)-(u v)} 
 
 p 
 
 = 2 a>v(v - u) Ibs. 
 
 9 Fig. 142. 
 
 The work done is 
 
 Pu = 2 o>w(v - u) ft.-lbs. per second. 
 
 This is greatest when u = v/2, so that 2u - v = 0, and then 
 
 Pu max = wv 3 ft.-lbs. per second, 
 20 
 
 or equal to the whole kinetic energy of the jet. This roughly 
 corresponds to the case of the Pelton wheel, which on high 
 falls reaches an efficiency of 0'8 or more, the loss being due 
 to friction and imperfect deviation of the water as the buckets 
 pass in front of and away from the jet. 
 
298 
 
 HYDRAULICS 
 
 CHAP. 
 
 177. Pressure of a steady stream of limited section on 
 a plane normal to the direction of motion. Let CD (Fig. 
 
 143) be a thin plate normal to the axis of a pipe through 
 which water is flowing, which for simplicity is taken horizontal. 
 The elementary streams, parallel afc A , are deviated in front 
 of the plate, form a contraction at A lf and then converge, 
 leaving a mass of eddies at the back of the plate, and at 
 some section A 2 become parallel again. It may be inferred 
 from the convexity of the stream lines in front and the 
 concavity behind the plate that there is an excess pressure 
 
 Fig. 143. 
 
 in front and a negative pressure behind the plate, the sum 
 of which forms the reaction E causing changes of momentum 
 in the water, and which is equal and opposite to the total 
 pressure of the water on the plate. Since the same amount 
 of water at the same velocity passes the sections A , A 2 in a 
 given time, the kinetic energy flowing in and out is the same, 
 and the external forces acting on the mass between A and A 2 
 must be balanced. Let H be the section of the stream at 
 AO or A 2 , and &> the area of the plate CD. The area of the 
 contracted section of the stream at A! is c c (H -co), where c c is 
 a coefficient of contraction. For simplicity let l/o> = p and 
 Il/{c c (ft )} =r. Then r = p/{c c (p - 1)}. Let v be the 
 velocity at A and A 2 , and v t the velocity at A x . 
 
 v 12 = c12 - w' 
 
 -co) 
 
 = rr. 
 
xiv IMPACT AND KEACTION OF FLUIDS 299 
 
 Let p Q , p lt p 2 be the pressures at AQ, A x , A 2 respectively. 
 Applying Bernoulli's theorem to A and A I} 
 
 &+*.&*. 
 
 G + 2g a V 
 
 and similarly for A l and A 2 , allowing for the loss in shock 
 due to the relative velocity v l v ( 36), 
 
 ^i + V_5 + ^ + (A^_ 2 
 G 2g G 2g 2g 
 
 Pi = P- 2 v(Vi-v) 
 G G g 
 
 or replacing v 1 by its value above, 
 
 The external horizontal forces acting on the mass between 
 A and A 2 are the difference of the pressures on the sections 
 A and A 2 and the reaction of the plate CD, and these are in 
 equilibrium, there being no resultant change of momentum. 
 Hence 
 
 and the total resultant pressure on CD is 
 
 where K is a coefficient depending only on p and c c . Thus if 
 c c =0-85, 
 
 P= K = 
 
 2 3-6 
 
 3 1-8 
 
 4 1-3 
 10 -9 
 50 2-0 
 
 As p increases, K diminishes to a minimum and then increases. 
 This is not intelligible, and therefore c c cannot have a constant 
 
300 HYDEAULICS CHAP. 
 
 value, or, what probably is the same thing, the influence of the 
 plate in deviating the stream lines extends only to a limited 
 distance. 
 
 From the equation above, 
 
 fl 
 P 1 =p 1 --(v 1 -v)v 
 
 Now in the eddying mass behind the plate the pressure 
 must be practically identical with p lt and hence the defect of 
 pressure forming part of the reaction E is 
 
 Consequently the front pressure must be 
 
 P /= R - P 6 = {GHr - I) 2 - 2Go>(r - 1)} J 
 = G{p(r - I) 2 - 2(r - l)} 
 
 The following values have been calculated, using values 
 of c c selected by Zeuner on the basis of some experiments of 
 Weisbach. 
 
 P= | 4 9 
 
 c c = -824 -852 -873 '892 
 
 r= 2-19 1-56 1-36 1-26 
 
 K= 3-18 1-26 -81 -68 
 
 K 6 = 2-38 1-12 -72 -52 
 
 K /= -80 -13 '09 -09 
 
 178. Distribution of pressure on a plane struck 
 normally by a jet. Mr. J. S. Beresford made some experi- 
 ments on the distribution of pressure on a plane struck by a 
 jet. A small hole in the plane communicated by a flexible 
 tube with a pressure column. This aperture was moved across 
 the area struck by the jet. In the following abstract, columns 
 A give the ratio (distance from axis of jet) /(diameter of jet) 
 and the columns B the ratio (pressure head) / (velocity head 
 of jet). 
 
XIV 
 
 IMPACT AND.KEACTION OF FLUIDS 301 
 
 Jet 0'475 Inch Diameter. 
 Velocity Head 43 Inches. 
 
 Jet 0-988 Inch Diameter. 
 Velocity Head 42 Inches. 
 
 Jet 1 '95 Inches Diameter. 
 Velocity Head 27 Inches. 
 
 A 
 
 B 
 
 A 
 
 B 
 
 A 
 
 B 
 
 
 
 965 
 
 
 
 998 
 
 
 
 993 
 
 21 
 
 917 
 
 10 
 
 988 
 
 17 
 
 967 
 
 42 
 
 839 
 
 20 
 
 958 
 
 37 
 
 808 
 
 63 
 
 672 
 
 30 
 
 891 
 
 48 
 
 669 
 
 84 
 
 334 
 
 51 
 
 546 
 
 61 
 
 463 
 
 1-05 
 
 083 
 
 71 
 
 118 
 
 71 
 
 261 
 
 1-26 
 
 012 
 
 86 
 
 038 
 
 97 
 
 074 
 
 179. Pressure of an unlimited stream of water on a 
 solid at rest. The theorem in 177, although it elucidates 
 the general action of a stream on a solid immersed in it, does 
 not furnish a numerical solution for the case of a very large 
 stream acting on a small solid. But the general expressions 
 
 where co is the projected area of the solid normal to the direc- 
 tion of motion, and K^ K b , K are experimental coefficients, 
 have been generally adopted, and appear to agree with the 
 results of experiments so far as they have been carried for 
 unshipshape bodies, the resistance of which is due to the 
 creation of eddies at sharp changes of section in the stream. 
 For quite shipshape bodies, in which the surfaces over which 
 the water slides are of gradual and continuous curvature, and 
 which are wholly immersed, the resistance is due to skin 
 friction, and depends on the total surface of the body, not its 
 projected area. 
 
 From some experiments by Dubuat and Duchemin on 
 prisms of cross section ax a and length I, immersed in a 
 stream of water, the following results were deduced : 
 
 J-O-03 
 
 1^=1-19 
 
 K 6 = -67 
 
 K = l-86 
 
 1-19 
 
 27 
 
 1-46 
 
 2 
 
 1-19 
 
 16 
 
 1-35 
 
 1-19 
 
 14 
 
 1-33 
 
 6 
 
 1-19 
 
 27 
 
 1-46 
 
 It is difficult to believe that K^ can be greater than unity. 
 The shortest prism corresponds with a thin plate. In the 
 
302 
 
 HYDKAULICS 
 
 CHAP. 
 
 case of the longest prism it would seem that the increase of 
 resistance is due to skin friction. For a plane one foot 
 square moved in still water Dubuat found K / = 1, K 5 = 0*433, 
 K= 1-433. Morin, Piobert, and Didion found K = T36 for 
 planes moved normally through air, and Thibault obtained a 
 mean value K = T83. 
 
 180. Stanton's experiments. 1 A very careful research 
 has been carried out by Dr. Stan ton at the National Physical 
 Laboratory. The solids were placed in a cylindrical trunk 
 2 feet in diameter and 4 feet 6 inches long, through which 
 a steady current of air was drawn by a fan. It was found 
 that if the area of a plane placed in this trunk was more 
 than 1-1 44th of the cross section of the trunk, there was a 
 perceptible increase of resistance due to the action of the sides 
 of the trunk which caused an increase of the negative back 
 pressure. Hence the experiments were limited to very small 
 planes. The maximum intensity of front pressure at the 
 centre of a circular or square plane, normal to the current, 
 was always very approximately 
 
 2 
 
 G Ibs. per square foot, 
 
 and the intensity of pressure diminished towards the edges. 
 At the back of the plate there was a negative pressure nearly 
 
 Front 
 
 Back 
 
 ids per sa. in. 
 
 Fig. 144. 
 
 uniformly distributed. Fig. 144 shows the distribution of 
 pressure on a square plate and some lines of equal pressure. 
 
 1 Proc. Inst. Civil Engineers, clvi., 1903-4. 
 
XIV 
 
 IMPACT AND REACTION OF FLUIDS 303 
 
 The average value of K 6 was 0'48 for a circular and 0'67 for 
 a square plate. So far as the tests went, the total resistance 
 of similar plates when normal to the stream was directly 
 proportional to the area. The total resistance of square or 
 circular plates, normal to the stream, the velocity of which 
 was v feet per second or V miles per hour, was 
 
 P = 0-00126^ = 0-0027V 2 Ibs. per square foot, 
 
 which is nearly in agreement with the result obtained by 
 Mr. Dines, namely, 
 
 P = 0-0029V 2 . 
 
 If the weight of a cubic foot of air at 60 and 1 atm. is 
 taken at 0*0764 lb., Stanton's result can be put in the form 
 
 v 2 
 P=1-061G Ibs. per square foot, 
 
 and using the result as to negative pressure stated above, this 
 gives 
 
 
 Coefficient of 
 
 Front Pressure, 
 K,- 
 
 Back Pressure, 
 K* 
 
 Total Pressure, 
 K. 
 
 Circular Plate . 
 Square Plate . 
 
 0-581 
 0-391 
 
 0-48 
 0-67 
 
 1-061 
 1-061 
 
 Stanton's results give somewhat lower pressures than those 
 obtained by earlier observers. He has since carried out 
 experiments on larger planes and solids acted on by wind 
 pressure, and has found that almost uniformly the pressures in 
 these conditions are 18 per cent greater than in the previous 
 experiments on small planes and solids tested in the air trunk. 
 It would appear, therefore, that for planes in an indefinitely 
 large stream 
 
 v 2 
 P= 1'252G-- Ibs. per square foot. 
 
 For rectangular plates, the total resistance was found to 
 increase with the ratio of length to width of plate. The 
 
304 
 
 HYDEAULICS 
 
 CHAP. 
 
 following are some examples deduced from Dr. Stanton's 
 results : 
 
 Dimensions. 
 Inches. 
 
 Katio of Length 
 to Width. 
 
 Total Pressure in Lbs. 
 
 3 xl 
 
 3 
 
 P=-00134tf 2 
 
 3-75 x -75 
 
 5 
 
 00135v 2 
 
 5-0 x -5 
 
 10 
 
 00151v 2 
 
 7-5 x -15 
 
 50 
 
 00201v 2 
 
 181. Pressure on solids of various forms. When a 
 solid body is presented to a stream the front pressure is 
 modified if the face of the body is not plane, and the back 
 pressure if the form of the body interferes with or facilitates 
 the convergence in the wake. If k is the ratio of the total 
 pressure on the solid to the pressure on a thin plate normal 
 to the stream and of area equal to the projected area of the 
 solid normal to the stream, then 
 
 Sphere 
 Cube 
 
 
 
 Cylinder (height = diameter) 
 Cone (height = diameter of base) 
 
 k = 
 0-31 
 0-80 
 0-66 
 0-47 
 0-38 
 
 Direction of stream. 
 
 Normal to face. 
 
 Parallel to diagonal of face. 
 
 Normal to axis. 
 
 Parallel to base. 
 
 182. Pressure on planes oblique to the direction of 
 
 the stream. Let Fig. 145 re- 
 present a plane moving in a 
 fluid at rest in the direction E, 
 making an angle 6 with the 
 normal to the plane, or con- 
 versely a plane at rest in a 
 stream moving in the direction 
 E. The resultant pressure on 
 the plane will be a normal pres- 
 sure N, with a component E in 
 
 v the direction of motion and a 
 lateral component L resisted by 
 
 Obviously 
 
 the supports of the plane. 
 
 E = N cos <9, 
 L = N sin 0. 
 
XIV 
 
 IMPACT AND.KEACTION OF FLUIDS 305 
 
 The simplest expression for the pressure on the plane in 
 the direction of motion is that of Duchemin, 
 
 = p 
 
 - - - 
 1 + cos 2 
 
 Ibs. per square foot, 
 
 where P is the pressure per square foot on a plane in similar 
 conditions normal to the direction of the stream. Conse- 
 quently the normal pressure on the plane is 
 
 N _ p 2 cos 2P 
 
 1 + cos 2 sec + cos 6* 
 
 The following table contains some results calculated by 
 this rule. Dr. Stanton experimented on a small plane 3 inches 
 by 1 inch, with a velocity of stream of 21 feet per second. 
 He found the remarkable result that the normal pressure was 
 different according as the short or the long axis of the rectangle 
 was normal to the current. Further, in the case of the long 
 axis normal to the current, the normal pressure for an inclina- 
 tion of about 45 was considerably greater than when the 
 plane was normal to the stream. 
 
 NORMAL PRESSURE ON THIN PLANES 
 
 
 Values of N/P. 
 
 Angle 6. 
 
 By Duchemin's 
 
 Stanton. 
 
 
 Rule. 
 
 Long Axis Normal. 
 
 Short Axis Normal. 
 
 
 
 1-00 
 
 1-00 
 
 1-00 
 
 15 
 
 TOO 
 
 I'OO 
 
 97 
 
 30 
 
 99 
 
 1-01 
 
 87 
 
 45 
 
 94 
 
 1-11 
 
 79 
 
 60 
 
 80 
 
 88 
 
 71 
 
 75 
 
 49 
 
 30 
 
 64 
 
 80 
 
 34 
 
 16 
 
 56 
 
 85 
 
 17 
 
 08 
 
 34 
 
 90 
 
 
 
 
 
 
 
 In 1872 some experiments were made for the Aeronautical 
 Society on the pressure of air on oblique planes. These plates, 
 of 1 to 2 feet square, were balanced by ingenious mechanism 
 designed by Mr. Wenham and Mr. Spencer Browning, in such 
 
 20 
 
306 
 
 HYDKAULICS 
 
 CHAP. 
 
 a manner that both the pressure in the direction of the air 
 current and the lateral force were separately measured. These 
 planes were placed opposite a blast from a fan issuing from a 
 wooden pipe 18 inches square. The pressure of the blast 
 varied from -f^ to 1 inch of water pressure. The following 
 are the results given in pounds per square foot of the plane, 
 and a comparison of the experimental results with the pressures 
 given by Duchemin's rule. These last values are obtained by 
 taking P = 3'31, the observed pressure on a normal surface: 
 
 
 
 e. 
 
 = 
 
 
 
 75 
 
 70. 
 
 30. 
 
 0. 
 
 Horizontal pressure R 
 Lateral pressure L . 
 
 0-4 
 1-6 
 
 0-61 
 1-96 
 
 2-73 
 1-26 
 
 3-31 
 
 
 
 Normal pressure \/L 2 + R 2 
 Normal pressure by Duchemin's rule . 
 
 1-65 
 1-605 
 
 2-05 
 2-027 
 
 3-01 
 3-276 
 
 3-31 
 3-31 
 
 Lord Eayleigh obtained theoretically the expression 
 
 N _ P ( 4 + ff ) sin 
 4 + 7rsin0' 
 
 but this gives the normal component of the front pressure only. 
 
 Dr. Stanton found the variation of total normal pressure with 
 
 inclination to be very different in the case of rectangular plates 
 according as the longer or shorter side 
 was perpendicular to the stream. 
 
 183. Distribution of pressure on 
 an inclined plane. In the case of a 
 plane inclined to a stream there is an 
 excess of pressure at the forward part 
 and less pressure sternwards. Fig. 146, 
 from Dr. Stanton's results, shows gener- 
 ally the distribution of positive pressure 
 on the windward and negative pressure 
 on the leeward side of a plane at 45 
 to the direction of an air current. 
 Clearly the resultant pressure does not 
 act through the centre of the plane. 
 Conversely, if a plane is pivoted about 
 
 an axis eccentric to its centre line and placed in a stream, 
 
 Fig. 146. 
 
XIV 
 
 IMPACT AND EEACTION OF FLUIDS 307 
 
 it will assume a position inclined to the stream such that 
 the resultant normal pressure passes through the axis 
 about which it can turn. If, therefore, planes pivoted BO 
 
 that the ratio T (Fig. 147) is varied are placed in water, and 
 the angle they make with the direction of the stream is 
 
 f 
 
 Fig. 147. 
 
 observed, the position of the resultant of the pressures on the 
 plane is determined for different angular positions. Experi- 
 ments of this kind have been made by Hagen. Some of his 
 results are given in the following table : 
 
 a 
 b' 
 
 a 
 
 Values of <f>. 
 
 a + b' 
 
 Larger Plane. 
 
 Smaller Plane. 
 
 Calculation. 
 
 I'O 
 
 500 
 
 
 90 
 
 90 
 
 0-9 
 
 474 
 
 75 
 
 72j 
 
 66i 
 
 0'8 
 
 445 
 
 60 
 
 57 
 
 55 
 
 0-7 
 
 412 
 
 48 
 
 43 
 
 45 
 
 0-6 
 
 375 
 
 25 
 
 29 
 
 35A 
 
 0-5 
 
 333 
 
 13 
 
 13 
 
 26 
 
 0'4 
 0-3 
 
 286 
 231 
 
 8 
 
 $ 
 
 16J 
 
 6 
 
 0-2 
 
 167 
 
 4 
 
 ... 
 
 ... 
 
 Joessel has given the formula 
 a 
 
 = 0-2 + 0-3 sin <f>. 
 
 The last column in the table above gives angles calculated by 
 this rule. 
 
308 
 
 HYDEAULICS 
 
 CHAP. 
 
 184. Wind Pressure. One of the most important cases to 
 the engineer, in which the pressure of a fluid stream on bodies 
 immersed in it has to be considered, is that of the pressure of 
 wind on structures. Unfortunately the action of the wind is 
 so complex and variable that there is not general agreement 
 as to the allowance to be made for it. 
 
 Storm winds are generally rotating eddies generated 
 between two oppositely flowing air currents not of themselves 
 of violent character. Once put in motion, the energy of such 
 
 LOUIS STORM 
 
 Fig. 148. 
 
 an eddy accumulates and the distribution of the energy is a 
 purely mechanical problem. Conditions of dynamical stability 
 involve this, that the pressure diminishes and the velocity 
 increases from the circumference to the centre of the eddy 
 ( 33). Fig. 148 is a diagram of the St. Louis storm of 1896, 
 which shows that the isobars formed closed curves round the 
 storm centre, the barometric pressure decreasing from 3 inches 
 at the outside to 29*4 inches at the centre. On the other 
 hand, the velocity and violence of the wind increase towards 
 the centre. A storm of this kind is not fixed in position. 
 
xiv IMPACT AND KEACTION OF FLUIDS 309 
 
 Its centre travels along a track generally in the northern 
 hemisphere eastwards or north-eastwards. At any given place, 
 as the storm passes, the wind veers round contrary to the hands 
 of a watch. The storm centre may travel 20 or 30 miles per 
 hour, but the wind velocity near the centre of the storm may 
 be 80 or 100 miles per hour. The area of a storm is 
 extremely variable. It may be 600 or 1200 miles in 
 diameter. In other cases the width of the track over 
 which the wind is violent enough to cause destruction may 
 be only 60 to 1000 feet. Some whirlwinds cut down the 
 trees in a forest along a track as narrow as a road, leaving 
 trees on either side undamaged. 
 
 Wind pressures are measured on anemometers of two types, 
 pressure and velocity anemometers. In the former the pressure 
 is measured on a thin vertical plate exposed normally to the 
 wind. It is rare for pressures on such a plate to exceed 
 30 Ibs. per square foot. But at Bidston Observatory near 
 Liverpool pressures of 50 to 80 Ibs. per square foot have been 
 registered. There the anemometer is 56 feet above the 
 ground and 251 feet above sea-level. The exposure of the 
 anemometer is complete and severe, but the Board of Trade 
 Committee on the Tay Bridge disaster found no reason to 
 doubt the records. Baier came to the conclusion, after 
 examining some cases of destruction, that the wind pressure 
 in the tornado at St. Louis in 1896 must have ranged from 
 45 to 90 Ibs. per square foot. 
 
 A large number of records have been obtained with 
 velocity anemometers of the Eobison type, in which hemi- 
 spherical cups are rotated by the wind, the velocity of 
 the cups being about one-third that of the wind. These 
 records give the average velocity over a more or less consider- 
 able period of time. The Board of Trade Committee found 
 that if v m is the mean velocity during an hour, then the 
 highest pressure during the hour would be approximately 
 
 P = O'Olv 2 Ibs. per square foot. 
 
 Now observations at Aberdeen show a wind travel of 
 69 miles an hour, corresponding to a maximum pressure of 
 48 Ibs. per square foot; at Falmouth a travel of 71 miles per 
 hour, corresponding to 5 Ibs. per square foot ; at Holyhead a 
 
310 
 
 HYDRAULICS 
 
 CHAP. 
 
 travel of 80 miles an hour, corresponding to 64 Ibs. per 
 square foot. The velocity anemometer is free of inertia errors, 
 and its indications are not consistent with the supposition 
 that gusts during which the pressure is excessive are necessarily 
 of short duration. 
 
 185. Increase of pressure with elevation. Numerous 
 experiments show that the wind velocity and pressure is greater 
 
 the greater the height 
 from the ground. In 
 some experiments by 
 Mr. Thomas Stevenson 
 in 1878, six velocity 
 anemometers were fixed 
 on a vertical pole 50 
 feet in height, and ob- 
 servations were taken 
 at various dates when 
 strong winds were blow- 
 ing. For a height of 
 1 5 feet from the ground 
 the velocities were low 
 and irregular even when 
 strong winds were blow- 
 ing. For heights above 
 20 feet the velocities 
 increased in a fairly 
 Fig. 149. regular way with in- 
 
 crease of elevation. 
 
 Plotted horizontally the wind velocities gave the irregular 
 curves in Fig. 149. For heights above 20 feet the velocity 
 curves agreed fairly with parabolas having their vertices 72 
 feet below ground-level. If V and v are velocities, and P and 
 p pressures at heights of H and h feet, 
 
 / 
 
 = i>. / 
 
 V 
 
 H + 72 
 
 1772' 
 
 H + 72 
 
 Suppose that at 25 feet above ground the mean hourly 
 velocity is 30 miles per hour, corresponding to a maximum 
 
XIV 
 
 IMPACT AND KEACTION OF FLUIDS 311 
 
 pressure during the hour of 9 Ibs. per square foot. Then at 
 higher elevations the velocities and pressures by Stevenson's 
 rule would be as follows : 
 
 Elevation. 
 
 Mean Velocity. 
 Miles per hour. 
 
 Maximum Pressure. 
 Lbs. per sq. ft. 
 
 25 
 
 30-0 
 
 9'0 
 
 50 
 
 33-6 
 
 11-3 
 
 100 
 
 39-9 
 
 16-0 
 
 200 
 
 50-4 
 
 25-3 
 
 300 
 
 58-8 
 
 34-6 
 
 These results apply only to the case of a flat and nearly 
 unobstructed ground surface. 
 
 186. Evidence of high wind pressure in storms. It 
 may be shown that a pressure of 2 5 to 35 Ibs. per square foot 
 distributed over the area of a railway carriage is necessary to 
 overturn it, and this must be chiefly front pressure, as in the 
 case of such a body it cannot be supposed that the negative 
 pressure due to a wake is as completely established as in the 
 case of a thin plate. Now Mr. Seyrig has described the over- 
 turning of five carriages of a passenger train at Salces, in 
 France, in 1860. On the same day five waggons of a freight 
 train were overturned at Kivesaltes, and three others thrown 
 off the track. On the same railway, in 1867", a passenger 
 train was almost completely overturned. In 1867 a brake- 
 van and post-office tender were blown over between Chester 
 and Holy head. In 1864 carriages in two trains on the 
 Eastern Bengal Bail way were overturned by wind. In 1870 
 two spans of a bridge at Decatur, U.S.A., were blown over ; 
 and in 1880, one 150-foot span of a bridge at Meredocia. 
 On September 10, 1897, in Paris, a cyclone uprooted every 
 tree from the Quai St. Michel to the Pont Neuf, some barges 
 were sunk, an omnibus overturned, and at the Palais de Justice 
 not a pane was left in the windows. 
 
 On the other hand, those who have carefully examined 
 cases of damage by wind have found that structures such as 
 windows, chimneys, roofs, etc., of weak construction, and 
 incapable of standing any considerable lateral pressure, have 
 
312 HYDKAULICS CHAP. 
 
 stood for long periods unharmed. Whether any adequate 
 explanation of the paradox thus presented can be given is 
 doubtful, but certain considerations may be noted : (a) At any 
 one place the occurrence of high wind pressure must be very 
 exceptional ; (&) A structure must be still more rarely struck 
 normally ; (c) Its form may prevent the creation of a negative 
 pressure ; (d) Neighbouring obstructions may have the effect 
 of shielding a structure. In this connection the great decrease 
 of wind velocity near the ground is instructive. 
 
 187. The Forth Bridge experiments. During the con- 
 struction of the Forth Bridge some important experiments 
 were carried out by Sir B. Baker. A very large pressure- 
 plate anemometer was erected on Inchgarvie, 20 feet long 
 by 15 feet high, facing east and west. Beside it were erected 
 two small pressure plates, one facing east and west, the other 
 revolving to face the wind. Between 1883 and 1890, on 
 fourteen occasions of storm, pressures ranging from 25 to 65 
 Ibs. per square foot were registered by the revolving pressure 
 plate. In the same period the pressure on the small fixed 
 pressure plate ranged from 16 to 41 Ibs. per square foot. Also, 
 during the same period, pressures were registered by the large 
 plate of 300 square feet area ranging from 7 to 35 Ibs. per 
 square foot. 
 
 For experiments on bodies of complex form, Sir B. Baker 
 adopted a very ingenious device. Experiments in wind storms 
 would have been difficult and inconvenient. Instead of this 
 a light wooden rod was suspended by a cord. At one end, 
 the complex form the resistance of which was required was 
 fixed; at the other, a small cardboard plane. Setting the 
 apparatus swinging, it was obvious at once at which end of 
 the rod the resistance was greatest. Then the area of the 
 cardboard plane was altered until its resistance just balanced 
 that of the body to be tested. In this way the areas of plane 
 having resistance equivalent to that of various bodies of 
 complex form was determined. 
 
 For bodies of comparatively simple form, such as cubes 
 and cylinders, the relative resistances were found to be the 
 same as those directly determined by earlier observers. The 
 most interesting point to determine next was the influence of 
 one surface in sheltering another. With discs placed at from 
 
xiv IMPACT AND KE ACTION OF FLUIDS 313 
 
 one to four diameters apart, there was complete shelter when 
 the distance was one diameter, the resistance being the same 
 as for a simple disc. The resistance was increased by 25 per 
 cent when the discs were 1|- diameters apart; by 40 per cent 
 at 2 diameters ; by 6 per cent for 3 diameters ; and by 
 80 per cent for 4 diameters. Intermediate discs did not 
 much increase the resistance. Four discs in series behind 
 each other, with a total distance between first and fourth of 
 3^- diameters, had no more resistance than two discs at 
 4 diameters. 
 
 Perforated discs were then tried to imitate the effect of 
 shelter of one lattice girder on another. With openings in 
 the discs equal to one-fourth the whole area, the discs being 
 1 diameter apart, the resistance of the sheltered disc was only 
 8 per cent of that of the front disc. But with openings half 
 the whole area, the resistance of the sheltered disc was 30 per 
 cent of that of the front disc. At 2 diameters apart, the 
 resistances of the sheltered disc were 40 per cent to 66 per 
 cent of that of the front disc, and at 4 diameters apart, with 
 openings half the total area, the resistance of the sheltered 
 disc was 94 per cent of that of the front disc. 
 
 The top members of the Forth Bridge consist each of a 
 pair of box-lattice girders, that is, they are nearly equivalent 
 to four single lattice girders in series. Models of single-web 
 girders made to imitate these were tested in pairs. With 
 distances apart equal to once, twice, and three times the 
 depth of the girders, the resistance of the sheltered girder was 
 20 per cent, 50 per cent, and 70 per cent of the resistance of 
 the front girder. With additional girders placed between 
 the others the increase of resistance was small With a 
 complete model of a bay of one top member of the bridge, 
 that is, with the equivalent of two single-lattice girders, the 
 total resistance was 1*75 times the resistance of a plate equal 
 in area to the projection of one lattice girder, that is, to the 
 projection of the solid surfaces excluding the openings. 
 
 The bottom member of the Forth Bridge consists of two 
 tubes of circular section braced together by lattice girders. A 
 complete model of one bay was tested. It had a resistance 
 10 per cent greater than the resistance of a plane surface of 
 the projected area of one tube. 
 
314 HYDEAULICS CHAP, xiv 
 
 EXAMPLES 
 
 1. A jet 3 inches in diameter under a head of 400 feet strikes normally 
 
 a plane at rest Find the pressure on the plane. 2452 Ibs. 
 
 2. A jet of water delivers 160 cubic feet per minute at a velocity of 
 
 20 feet per second, and strikes a plane normally. Find the 
 pressure on the plane : (1) when the plane is at rest ; (2) when 
 it is moving at 7 feet per second in the direction of the jet. 
 In the latter case find the rate at which work is done in 
 driving the plane. 
 
 103'4 Ibs. ; 43-7 Ibs. ; 305-8 ft.-lbs. per second. 
 
 3. Water impinges on a Poncelet float at 10 with the tangent to the 
 
 circumference of the wheel. The velocity of the water is double 
 that of the float. Find by construction the angle of the float 
 to receive the water without shock. A slope of 10 is nearly 
 1 in 6. 
 
 4. A cylindrical chimney shaft 100 feet high and 75 feet in diameter 
 
 is exposed to a wind pressure of 30 Ibs. per square foot. Find 
 the overturning moment. 105,750 Ibs. 
 
 5. A fixed curved vane has a receiving edge making an angle of 45 
 
 and a delivering edge an angle of 20 with a line AB. A jet 
 delivers 10 cubic feet per second at a velocity of 30 feet per 
 second, without shock, so that it is deviated along the vane. 
 Find the resultant pressure on the vane, the angle it makes 
 with AB, and the components of the pressure along and at 
 right angles to AB. 970 Ibs. ; 12| ; 946 Ibs. ; 210 Ibs. 
 
 6. Suppose the vane in the previous question is moving in the direc- 
 
 tion AB at 10 feet per second, and the jet at 45 with AB at 
 30 feet per second. Find the angle the receiving edge of the 
 vane must make with AB that there may be no shock. Also 
 the relative velocity. 63 ; 24 ft. per second. 
 
APPENDIX 
 
APPENDIX 
 
 317 
 
 TABLE I. FUNCTIONS OF NUMBERS FROM 0-1 TO 10*0 
 
 n. 
 
 ^ tfa 
 
 */3. 
 
 Natural 
 log. n. 
 
 n. 
 
 x^. tfi. 
 
 ^ 
 
 Natural 
 log. n. 
 
 1 
 
 3162 '4642 
 
 0316 
 
 
 5-1 
 
 2-258 1721 
 
 11-52 
 
 1-6292 
 
 2 
 
 4472 '5848 
 
 0894 
 
 ... 
 
 5-2 
 
 2-280 1732 
 
 11-86 
 
 6487 
 
 3 
 
 5477 '6694 
 
 164 
 
 
 5-3 
 
 2-302 1744 
 
 12-20 
 
 6677 
 
 4 
 
 6325 7368 
 
 253 
 
 ... 
 
 5-4 
 
 2-324 1754 
 
 12-55 
 
 6864 
 
 5 
 
 7071 7937 
 
 354 
 
 ... 
 
 5-5 
 
 2-345 1765 
 
 12-90 
 
 17047 
 
 6 
 
 7746 '8434 
 
 465 
 
 
 5-6 
 
 2-366 1776 
 
 13-25 
 
 7228 
 
 7 
 
 8367 '8879 
 
 586 
 
 
 57 
 
 2-387 1786 
 
 13-61 
 
 7405 
 
 8 
 
 8944 '9283 
 
 716 
 
 
 5-8 
 
 2-408 1797 
 
 13-97 
 
 7579 
 
 9 
 
 9487 '9655 
 
 854 
 
 ... 
 
 5'9 
 
 2-429 1-807 
 
 14-33 
 
 7750 
 
 I'O 
 
 1-0000 I'OOOO 
 
 1-000 
 
 o-oooo 
 
 6-0 
 
 2-449 1-817 
 
 1470 
 
 17918 
 
 1-1 
 
 1-0488 1-0323 
 
 1-153 
 
 0953 
 
 6-1 
 
 2-470 1-827 
 
 15-07 
 
 8083 
 
 1-2 
 
 1-0954 1-0627 
 
 1-315 
 
 1823 
 
 6-2 
 
 2-490 1-837 
 
 15*44 
 
 8245 
 
 1-3 
 
 1-1402 1-0914 
 
 1-483 
 
 2624 
 
 6-3 
 
 2-510 1-847 
 
 15-81 
 
 8405 
 
 1-4 
 
 1-1832 1-1187 
 
 1-655 
 
 3365 
 
 6-4 
 
 2-530 1-857 
 
 16-19 
 
 8563 
 
 1-5 
 
 1-2247 1-1447 
 
 1-837 
 
 0-4055 
 
 6-5 
 
 2-550 1-866 
 
 16-57 
 
 1-8718 
 
 1-6 
 
 1-2649 1-1696 
 
 2-024 
 
 4700 
 
 6-6 
 
 2-569 -876 
 
 16-96 
 
 8871 
 
 17 
 
 1-3038 1-1935 
 
 2-216 
 
 5306 
 
 67 
 
 2-588 -885 
 
 17-34 
 
 9021 
 
 1-8 
 
 1-3416 1-2164 
 
 2-414 
 
 5878 
 
 6-8 
 
 2-608 -895 
 
 1773 
 
 9169 
 
 1-9 
 
 1-3784 1-2386 
 
 2-62 
 
 6419 
 
 6-9 
 
 2-627 -904 
 
 18-13 
 
 9315 
 
 2-0 
 
 1-4142 1-2599 
 
 2-83 
 
 0-6931 
 
 7-0 
 
 2-646 -913 
 
 18-52 
 
 1-9459 
 
 2-1 
 
 1-4491 1-2806 
 
 3-04 
 
 7419 
 
 7-1 
 
 2-665 -922 
 
 18-92 
 
 9601 
 
 2-2 
 
 1-4832 1-3006 
 
 3-26 
 
 7885 
 
 7-2 
 
 2-683 -931 
 
 19-32 
 
 9741 
 
 2'3 
 
 1-5166 1-3200 
 
 3-49 
 
 8329 
 
 7-3 
 
 2702 -940 
 
 1972 
 
 9879 
 
 2-4 
 
 1-5492 1-3389 
 
 372 
 
 8755 
 
 7-4 
 
 2720 '949 
 
 20-13 
 
 2-0015 
 
 2-5 
 
 1-5811 1-3572 
 
 3-95 
 
 0-9163 
 
 7-5 
 
 2739 1-957 
 
 20-54 
 
 2-0149 
 
 2'6 
 
 1-6125 1-3751 
 
 4-19 
 
 9555 
 
 7-6 
 
 2757 1-966 
 
 20-95 
 
 0281 
 
 27 
 
 1-6432 1-3925 
 
 4-44 
 
 9933 
 
 77 
 
 2775 1-975 
 
 21-36 
 
 0412 
 
 2-8 
 
 1-6733 1-4095 
 
 4*69 
 
 1-0296 
 
 7'8 
 
 2793 1-983 
 
 2179 
 
 0541 
 
 2-9 
 
 17029 1-4260 
 
 4-94 
 
 0647 
 
 7-9 
 
 2-811 1-992 
 
 22-20 
 
 0669 
 
 3-0 
 
 17321 1-4422 
 
 5-20 
 
 1-0986 
 
 8-0 
 
 2-828 2-000 
 
 22-63 
 
 2-0794 
 
 3-1 
 
 17607 1-4581 
 
 5-46 
 
 1314 
 
 8-1 
 
 2-846 2-008 
 
 23-05 
 
 0919 
 
 3-2 
 
 17889 1-4736 
 
 573 
 
 1632 
 
 8-2 
 
 2-864 2-017 
 
 23-48 
 
 1041 
 
 3'3 
 
 1-8166 1-4888 
 
 6-00 
 
 1939 
 
 8-3 
 
 2-881 2-025 
 
 23-91 
 
 1163 
 
 3-4 
 
 1-8439 1-5037 
 
 6-27 
 
 2238 
 
 8-4 
 
 2-898 2-033 
 
 24-34 
 
 1282 
 
 3-5 
 
 1-871 1-518 
 
 6-55 
 
 1-2528 
 
 8-5 
 
 2-915 2-041 
 
 2478 
 
 2-1401 
 
 3-6 
 
 1-897 1-533 
 
 6-83 
 
 2809 
 
 8-6 
 
 2-933 2-049 
 
 25-22 
 
 1518 
 
 37 
 
 1-924 1-547 
 
 7-12 
 
 3083 
 
 87 
 
 2-950 2-057 
 
 25-66 
 
 1633 
 
 3'8 
 
 1-949 1-560 
 
 7'41 
 
 3350 
 
 8-8 
 
 2-966 2-065 
 
 26-10 
 
 1748 
 
 3-9 
 
 1-975 1-574 
 
 770 
 
 3610 
 
 8-9 
 
 2-983 2-072 
 
 26-55 
 
 1861 
 
 4'0 
 
 2-000 1-587 
 
 8-00 
 
 1-3863 
 
 9-0 
 
 3-000 2-080 
 
 27-00 
 
 2-1972 
 
 4-1 
 
 2-025 1-601 
 
 8*30 
 
 4110 
 
 9'1 
 
 3-017 2-088 
 
 27-45 
 
 2083 
 
 4-2 
 
 2-049 1-613 
 
 8-61 
 
 4351 
 
 9'2 
 
 3-033 2-095 
 
 27-91 
 
 2192 
 
 4-3 
 
 2-074 1-626 
 
 8-92 
 
 4586 
 
 9-3 
 
 3-050 2-103 
 
 28-36 
 
 2300 
 
 4-4 
 
 2-098 1-639 
 
 9-23 
 
 4816 
 
 9'4 
 
 3-066 2-110 
 
 28-82 
 
 2407 
 
 4-5 
 
 2-121 1-651 
 
 9-55 
 
 1-5041 
 
 9-5 
 
 3-082 2-118 
 
 29-28 
 
 2-2513 
 
 4'6 
 
 2-145 1-663 
 
 9-87 
 
 5261 
 
 9'6 
 
 3-098 2-125 
 
 2974 
 
 2618 
 
 47 
 
 2-168 1-675 
 
 10-19 
 
 5476 
 
 97 
 
 3'114 2-133 
 
 30-21 
 
 2721 
 
 4'8 
 
 2-191 1-687 
 
 10-51 
 
 5686 
 
 9'8 
 
 3-130 2-140 
 
 30-68 
 
 2824 
 
 4-9 
 
 2-214 1-698 
 
 10-85 
 
 5892 
 
 9'9 
 
 3-146 2-147 
 
 31-15 
 
 2925 
 
 5-0 
 
 2-236 1710 
 
 11-18 
 
 1-6094 
 
 10-0 
 
 3-162 2-154 
 
 31-62 
 
 2-3026 
 
318 
 
 HYDEAULICS 
 
 TABLE II. VELOCITY AND HEAD 
 
 n. 
 
 Height due to 
 Velocity, 
 
 %2 
 
 20' 
 
 Velocity due to 
 Height, 
 
 ^/2gn. 
 
 n. 
 
 Height due to 
 Velocity, 
 3 
 
 2/ 
 
 Velocity due 
 to Height, 
 
 N/2071. 
 
 
 Metres. Feet. 
 
 Metres. Feet. 
 
 
 Metres. Feet. 
 
 Metres. Feet. 
 
 1 
 
 00051 '000155 
 
 1-401 2-537 
 
 5-1 
 
 1-326 -4041 
 
 10-00 18-12 
 
 2 
 
 00203 -000622 
 
 1-981 3-588 
 
 5-2 
 
 378 '4201 
 
 10 '29 
 
 3 
 
 00459 -001398 
 
 2-426 4-394 
 
 5-3 
 
 432 -4365 
 
 20 '47 
 
 4 
 
 00816 -002486 
 
 2-801 5-074 
 
 5-4 
 
 486 '4531 
 
 29 '64 
 
 5 
 
 01274 -003885 
 
 3-132 5-673 
 
 5-5 
 
 1-542 -4700 
 
 10-39 18-81 
 
 6 
 
 01835 -005593 
 
 3-431 6-214 
 
 5-6 
 
 599 -4873 
 
 48 '98 
 
 7 
 
 02498 -007613 
 
 3706 6712 
 
 57 
 
 656 '5048 
 
 57 19-15 
 
 8 
 
 03262 -009943 
 
 3-962 7-176 
 
 5-8 
 
 715 '5227 
 
 67 '32 
 
 9 
 
 04129 '01259 
 
 4-202 7'611 
 
 5-9 
 
 774 -5408 
 
 76 -48 
 
 1-0 
 
 0-0510 -01554 
 
 4-429 8-022 
 
 6-0 
 
 1-835 -5593 
 
 10-85 19-65 
 
 1-1 
 
 0617 '01880 
 
 4-645 8-414 
 
 6-1 
 
 897 -5782 
 
 94 -81 
 
 1-2 
 
 0734 -02237 
 
 4-852 8788 
 
 6-2 
 
 959 -5973 
 
 11-03 -97 
 
 1-8 
 
 0861 -02626 
 
 5-050 9-147 
 
 6-3 
 
 2-023 -6167 
 
 12 20-13 
 
 1*4 
 
 0999 -03045 
 
 5-241 9-492 
 
 6-4 
 
 088 -6364 
 
 21 -29 
 
 1-5 
 
 0-1147 '03496 
 
 5-425 9-826 
 
 6-5 
 
 2-154 -6564 
 
 11-29 20-45 
 
 1-6 
 
 1305 -03978 
 
 603 -148 
 
 6-6 
 
 220 -6768 
 
 38 -61 
 
 17 
 
 1473 -04490 
 
 775 '460 
 
 67 
 
 288 '6975 
 
 46 -76 
 
 1-8 
 
 1652 -05034 
 
 942 764 
 
 6-8 
 
 357 7185 
 
 55 -92 
 
 1-9 
 
 1840 -05609 
 
 6-105 11-059 
 
 6-9 
 
 427 7397 
 
 63 21-07 
 
 2-0 
 
 0-2039 -06215 
 
 6-264 11-346 
 
 7-0 
 
 2-498 7613 
 
 1172 21-23 
 
 2-1 
 
 2248 -06852 
 
 418 -626 
 
 7-1 
 
 570 7832 
 
 80 -38 
 
 2-2 
 
 2467 '07520 
 
 570 -899 
 
 7'2 
 
 643 '8055 
 
 "88 '53 
 
 2'3 
 
 2697 '08219 
 
 717 12-167 
 
 7-3 
 
 716 '8280 
 
 97 -67 
 
 2-4 
 
 2936 '08950 
 
 862 '429 
 
 7-4 
 
 791 -8508 
 
 12-05 -82 
 
 2-5 
 
 0-3186 '09711 
 
 7-003 12-685 
 
 7-5 
 
 2-867 -8740 
 
 12-13 21-97 
 
 2-6 
 
 3446 '10503 
 
 142 '936 
 
 7-6 
 
 944 -8974 
 
 21 22-11 
 
 27 
 
 3716 -11326 
 
 278 13-182 
 
 77 
 
 3-022 -9212 
 
 29 -26 
 
 2-8 
 
 3996 '12182 
 
 411 '424 
 
 7-8 
 
 101 -9475 
 
 37 -40 
 
 2-9 
 
 4287 '13067 
 
 543 -662 
 
 7-9 
 
 181 -9697 
 
 45 '55 
 
 3-0 
 
 0-4588 -1398 
 
 7-672 13-90 
 
 8-0 
 
 3-262 -9944 
 
 12-53 22-69 
 
 3'1 
 
 4899 '1493 
 
 798 14-13 
 
 8-1 
 
 344 1-0194 
 
 61 -83 
 
 3-2 
 
 5220 -1591 
 
 923 '35 
 
 8-2 
 
 428 1-0447 
 
 68 -97 
 
 3'3 
 
 5551 '1692 
 
 8-046 '57 
 
 8-3 
 
 512 1-0704 
 
 76 23-11 
 
 3-4 
 
 5893 '1796 
 
 167 79 
 
 8-4 
 
 597 1-0963 
 
 84 -25 
 
 3-5 
 
 0-6244 -1904 
 
 8-286 15-01 
 
 8-5 
 
 3-683 1-1226 
 
 12-91 23-39 
 
 3-6 
 
 6606 '2014 
 
 404 '21 
 
 8-6 
 
 770 1-1492 
 
 99 '53 
 
 37 
 
 6978 '2127 
 
 520 '42 
 
 87 
 
 858 1-1761 
 
 13-06 -66 
 
 3-8 
 
 7361 '2244 
 
 634 '63 
 
 8-8 
 
 947 1-2032 
 
 14 -80 
 
 3'9 
 
 7753 '2363 
 
 747 '84 
 
 8-9 
 
 4-038 1-2307 
 
 21 -93 
 
 4-0 
 
 0-8156 '2486 
 
 8-858 16-05 
 
 9-0 
 
 4-129 1-259 
 
 13-29 24-07 
 
 4'1 
 
 8569 '2612 
 
 968 -24 
 
 9-1 
 
 221 1-287 
 
 36 -20 
 
 4-2 
 
 8992 '2741 
 
 9-077 -44 
 
 9-2 
 
 314 1-315 
 
 43 '33 
 
 4-3 
 
 9425 '2873 
 
 184 '63 
 
 9-3 
 
 409 1-344 
 
 51 -47 
 
 4-4 
 
 9869 '3008 
 
 291 '83 
 
 9-4 
 
 504 1-373 
 
 58 '60 
 
 4-5 
 
 1-0322 '3146 
 
 9-396 17-02 
 
 9-5 
 
 4-600 1-402 
 
 13'65 2473 
 
 4-6 
 
 0786 -3288 
 
 500 ,'20 
 
 9-6 
 
 698 1-432 
 
 72 -86 
 
 47 
 
 1260 '3432 
 
 602 -39 
 
 97 
 
 796 1-462 
 
 79 '99 
 
 4-8 
 
 1745 '3580 
 
 704 -57 
 
 9-8 
 
 896 1-492 
 
 87 25-11 
 
 4-9 
 
 2239 '3731 
 
 804 76 
 
 9-9 
 
 996 1-523 
 
 94 -24 
 
 5-0 
 
 1-2744 -3884 
 
 9-904 17'94 
 
 10-0 
 
 5-097 1-554 
 
 14-01 25-37 
 
APPENDIX 
 
 319 
 
 TABLE III. SLOPE TABLE 
 
 Fall in Feet 
 per Mile. 
 
 Slope 1 in n. 
 n= 
 
 Slope 
 Foot per Foot 
 
 Slope 1 in n. 
 
 71 = 
 
 Slope 
 Foot per Foot. 
 
 Fall in Feet 
 per Mile. 
 
 0-5 
 
 10560 
 
 000095 
 
 6000 
 
 000088 
 
 88 
 
 075 
 
 7040 
 
 000142 
 
 5000 
 
 0002 
 
 1-06 
 
 i-o 
 
 5280 
 
 000189 
 
 4500 
 
 000222 
 
 1-17 
 
 1-25 
 
 4224 
 
 000237 
 
 4000 
 
 00025 
 
 1-32 
 
 1-5 
 
 3520 
 
 000284 
 
 3500 
 
 000286 
 
 1-51 
 
 175 
 
 3017 
 
 000331 
 
 3000 
 
 000333 
 
 176 
 
 2-0 
 
 2640 
 
 000379 
 
 2500 
 
 0004 
 
 2-11 
 
 3-0 
 
 1760 
 
 000568 
 
 2000 
 
 0005 
 
 2-64 
 
 4*0 
 
 1320 
 
 000758 
 
 1500 
 
 000667 
 
 3'52 
 
 5-0 
 
 1056 
 
 000947 
 
 1250 
 
 0008 
 
 4-23 
 
 6-0 
 
 880 
 
 001136 
 
 1000 
 
 001 
 
 5-28 
 
 7-0 
 
 754 
 
 001326 
 
 750 
 
 00133 
 
 7-03 
 
 8-0 
 
 660 
 
 001515 
 
 500 
 
 002 
 
 10-56 
 
 9-0 
 
 587 
 
 001704 
 
 400 
 
 0025 
 
 13-2 
 
 10-0 
 
 528 
 
 001894 
 
 300 
 
 00333 
 
 17-6 
 
 11-0 
 
 444 
 
 002083 
 
 250 
 
 004 
 
 21-1 
 
 12-0 
 
 440 
 
 002273 
 
 200 
 
 005 
 
 26-4 
 
 13-0 
 
 406 
 
 002462 
 
 175 
 
 00571 
 
 30-2 
 
 14-0 
 
 377 
 
 002651 
 
 150 
 
 00667 
 
 35-2 
 
 15-0 
 
 352 
 
 002841 
 
 125 
 
 008 
 
 42-3 
 
 17'5 
 
 302 
 
 003311 
 
 100 
 
 01 
 
 52-8 
 
 20-0 
 
 264 
 
 003788 
 
 75 
 
 0133 
 
 70-3 
 
 22-5 
 
 235 
 
 004255 
 
 50 
 
 02 
 
 105-6 
 
 25-0 
 
 211 
 
 004735 
 
 40 
 
 025 
 
 132 
 
 30-0 
 
 176 
 
 005682 
 
 30 
 
 0333 
 
 176 
 
 35-0 
 
 151 
 
 006629 
 
 25 
 
 04 
 
 211 
 
 40'0 
 
 132 
 
 007576 
 
 20 
 
 05 
 
 264 
 
 45-0 
 
 117 
 
 008523 
 
 15 
 
 0667 
 
 352 
 
 50-0 
 
 105-6 
 
 009470 
 
 10 
 
 1 
 
 528 
 
 60-0 
 
 88-0 
 
 011364 
 
 
 
 
 70-0 
 
 75-4 
 
 01326 
 
 
 
 
 80-0 
 
 66-0 
 
 01515 
 
 
 
 
 90-0 
 
 587 
 
 01705 
 
 
 
 
 100-0 
 
 52-8 
 
 01894 
 
 
 
 
 120-0 
 
 44-0 
 
 02273 
 
 
 
 
 140-0 
 
 377 
 
 02652 
 
 
 
 
 160-0 
 
 33-0 
 
 03030 
 
 
 
 
 180-0 
 
 29-3 
 
 03409 
 
 
 
 
 200-0 
 
 26-4 
 
 03788 
 
 
 
 
 300-0 
 
 17-6 
 
 05682 
 
 
 
 
 400-0 
 
 13-2 
 
 07576 
 
 
 
 
 500-0 
 
 10-6 
 
 09470 
 
 
 
 
320 
 
 HYDKAULICS 
 
 TABLE IV. TABLE TO FACILITATE CALCULATIONS ON PIPES 
 
 Diameter. 
 
 Area of Section 
 in Square Feet. 
 
 n. 
 
 Hydraulic 
 Mean Radius 
 in Feet. 
 
 d 
 m= r 
 
 \/w.. 
 
 Os/m. 
 
 Inches. 
 
 Feet. 
 d, 
 
 3 
 
 0-250 
 
 0-0491 
 
 0-0625 
 
 250 
 
 0122 
 
 4 
 
 0-333 
 
 0-0873 
 
 0833 
 
 289 
 
 0252 
 
 5 
 
 0-417 
 
 0-136 
 
 104 
 
 322 
 
 0437 
 
 6 
 
 0-500 
 
 0-196 
 
 125 
 
 354 
 
 0693 
 
 7 
 
 0-583 
 
 0-267 
 
 146 
 
 382 
 
 1019 
 
 8 
 
 0-666 
 
 0-349 
 
 166 
 
 407 
 
 1420 
 
 9 
 
 0-750 
 
 0-442 
 
 188 
 
 434 
 
 1918 
 
 10 
 
 0-833 
 
 0-545 
 
 208 
 
 456 
 
 2485 
 
 12 
 
 1-000 
 
 0-785 
 
 250 
 
 500 
 
 3925 
 
 14 
 
 1-167 
 
 1-069 
 
 292 
 
 540 
 
 577 
 
 15 
 
 1-250 
 
 1-227 
 
 312 
 
 559 
 
 687 
 
 16 
 
 1-333 
 
 1-396 
 
 333 
 
 577 
 
 807 
 
 18 
 
 1-500 
 
 1-767 
 
 375 
 
 612 
 
 1-083 
 
 20 
 
 1-666 
 
 2-182 
 
 417 
 
 646 
 
 1-408 
 
 21 
 
 1-750 
 
 2-405 
 
 438 
 
 662 
 
 1-588 
 
 24 
 
 2-000 
 
 3-142 
 
 500 
 
 707 
 
 2-219 
 
 27 
 
 2-250 
 
 3-976 
 
 563 
 
 750 
 
 2-985 
 
 30 
 
 2-500 
 
 4-909 
 
 625 
 
 791 
 
 3'88 
 
 33 
 
 2-750 
 
 5-939 
 
 688 
 
 829 
 
 4'92 
 
 36 
 
 3-000 
 
 7-068 
 
 750 
 
 866 
 
 6-14 
 
 40 
 
 3-333 
 
 8-725 
 
 833 
 
 913 
 
 7-96 
 
 42 
 
 3-500 
 
 9-621 
 
 875 
 
 935 
 
 8-99 
 
 45 
 
 3-750 
 
 11-050 
 
 938 
 
 968 
 
 10-65 
 
 48 
 
 4-000 
 
 12-566 
 
 1-000 
 
 1-000 
 
 12-56 
 
APPENDIX 
 
 321 
 
 cooo^oooo 
 
 OO O CO 
 
 0000000000000 
 l>-COTj<OO-3<OOOOOOi5O 
 
 <NCOO3r- ICOOJOOOOOOOOOO 
 
 cooo<Mc<ic<ioo-r}<i i 
 
 OO 
 
 10 o 
 
 oooooooooooooo 
 
 *O i5 C^ 1^* r-H OO O O O O O O O O O O 
 C^COOCOOOCOOOi lOOCOr iOC<l^*IOl 
 
 ooooooooooooo 
 
 oooooooooooooooo 
 
 O>CO(NrDCOCOOOOOOOOOOOO 
 rH(NOlO^ OSCOi 1 ItOOilQ-^COO* 
 
 vaOirtO 
 
 OOCOOOOOOOOOOOOOOO 
 
 i I i i O3 C^ CO "* VQ 
 COOOt^OOOOOOOOOOOOO 
 
 i-lO^H-J-^CO 
 ^OOOC^COO 
 
 i-l i-H rH <N IM CO -^ 
 
 l-l rH ^H (N 
 
 O CO 
 
 T-H OO 
 
 21 
 
322 
 
 HYDEAULICS 
 
 co eo oo o 
 
 OO OO CO ^ C^l CO r-l 
 i-Ht^OOil-i-lO 
 
 OOOOOOOOOO 
 
 C<liOO5CO 
 rH 1-t I-t Cfl 
 
 COOOJi ( 
 
 C<> r-i O5 
 
 i-t 1-1 r-l (N (N CO 
 
 O Oi (M 
 
 00 O CO 
 
 SO CO O l>* rH 
 
 OOOC^i *l^^lOI>**-H<OCOrHOCOCO 
 
 CO rH -rt* 
 l^-OiiO 
 
 CO 
 CO 
 
 *> CO 
 
 O CO 
 OOCOCO 
 
APPENDIX 
 
 323 
 
 * 
 
 OO r- 1 -3< CO T I O t 1 
 
 " X 
 
 pop pppppppoppppp 
 
 i 1 CO CO T 1 Oi Ci O OO t 1 
 OOOO-^OS-^ 
 
 mot>-^T"'- i 
 C<IC^i i i i i l 
 
 CiC^rHr-Hr-tOOOOOOOOOOO 
 
 pppppppopppppppp 
 
 oooooooooooooooo 
 
 gppgggpg 
 
 OOOOOOO 
 
 ooooopo 
 
 t^^ic^i iOioot--?ours 
 
 i li 1 I i lOO^OO 
 OOOOOO 
 
 pooopo 
 
 lO^t<COtOS<lr- 1 i ii I 
 
 t^ ?O O O Tf CO 
 
 _ ^oooooo 
 
 oooooooooooooooo 
 o o o o o o o o o o o o o o o o 
 
 OiiC v ?O5OOt^iOO 
 <M!Mi-HT-iOOOOOOOOOO 
 
 OJ r-( ,-H r-l O 
 
 o o o o o 
 99999 
 
 
 OCDOOO 
 
 O 
 ^-Ti li ' i IT I^OOO 
 
 oooooooooooooooo 
 oooooooooooooooo 
 
 I * 
 
 s 
 
 12 
 
 o 
 
 II 
 
 -< I--* 
 
 =H 
 
 tfi 
 1 I 
 
 .8 
 
 o 
 .2 
 
 1 
 1 
 
 , 
 
 'PH 
 
 I 
 -2 
 'I 
 
 1 
 
324 
 
 HYDEAULICS 
 
 CO CO O CO IO CO OO 
 
 pppppppppppppppp 
 
 ? !C<ll>.OCOt"-O5O5-<tlO ; 5Oi i * !>. OO J 1 
 t-^COOOCO*nO*O<Nr 1 O5 OO t-~ CO O ^*l ** 
 i IT- lOOOOOOO 
 
 
 CO CO C^ C<l TI r i i IOOOOOOOOC 
 
 pppppppppppppppp 
 
 Orf<Ot--COOOOCODTfl-^COCOC'OC<IC^ 
 COfMtNi-Hi-Hi-HOOOOOO-OOOO 
 
 oooooooooooooooo 
 
 rHi-Hr-iOOOOOOOOOOOOO 
 
 opoppopppooppppo 
 
 r-i rjt O HO 
 
 T I i IT IT IOOOOO 
 
 pppppppppppppppp 
 
 T 'GOGOi IOCOOrHi^>OOOO 
 OOCOOOCDOOOOCOt^CO<^)O 
 OiiTii (O5OOCOOO-^-rtlCOCO 
 
 
 
 t^ CO t^. CO 
 O CO CO CO 
 T 1 OOCOCO 
 
INDEX 
 
 Acceleration, numerical values of, 7 
 
 Air, flow through orifices, 123 ; discharge 
 
 from orifices, 130 ; flow in mains, 
 
 226 
 
 Air-valves, 191 
 Alexander, 171 
 Amsler Laffon, 272 
 Anemometers, 309 
 Appold, 189 
 
 Approach, velocity of, 98, 109 
 Aqueduct, pipe, 190 ; Vyrnwy, 192 ; East 
 
 Jersey, 193 ; Coolgardie, 193 
 Aqueduct, Loch Katrine, 242, 258; Roman, 
 
 258 ; Thirlmere, 260 ; New Croton, 260 
 Atmospheric pressure, 6 
 
 Baker, Sir B., 312 
 
 Barometer, 121 ; measurement of heights 
 by, 122 
 
 Batcheller, 230 
 
 Baum, 275 
 
 Bazin, 109, 113, 218, 235, 236, 279, 
 282 
 
 Bellmouth, 70 
 
 Bends of pipes, thrust at, 17 ; resistance 
 at, 170 ; in rivers, 261 
 
 Beresford, 300 
 
 Bernoulli's theorem, 42 ; illustrations of, 
 45 ; application to orifices, 81 ; modi- 
 fication for compressible fluids, 127 
 
 Bidone, 81 
 
 Black well, 103 
 
 Borda, 81 
 
 Bossut, 205 
 
 Boyden, 70 
 
 Boyle's law, 119 
 
 Break-pressure reservoirs, 179 
 
 Bruce, 243 
 
 Buoyancy, 32 
 
 Calibration of current meters, 275 
 Canals, 230 ; Chezy formula for, 232 ; 
 Darcy's research, 233 ; Ganguillet and 
 Kutter's formula, 234 ; Bazin's investi- 
 gation, 236 ; of circular section, 243 ; 
 egg-shaped, 244 ; trapezoidal, 245 ; 
 minimum section, 245 ; parabola of 
 discharge, 249 ; distribution of velocity 
 
 in, 251 ; velocity curves, 254 ; mean 
 and surface velocities, 257 
 
 Capillary tubes, 146 
 
 Carpenter, 196 
 
 Centre of pressure, 26 
 
 Channels, see Canals 
 
 Charles's law, 120 
 
 Chezy formula for pipes, 150, 199, 201, 
 217 ; for canals, 232 
 
 Church, 38 
 
 Coefficients of velocity and resistance, 63 ; 
 of contraction, 64 ; of discharge, 65 ; 
 for weirs, 98 ; of friction, 133, 157, 
 161 ; of friction in gas mains, 224 ; of 
 air in mains, 229 
 
 Coker, 147 
 
 Compressibility of liquids, 10 
 
 Conduits, 230. See Canals 
 
 Conservation of energy, 42 ; of momen- 
 tum, 56 
 
 Contraction, 64, 77 ; minimum, 79 ; at 
 weirs, 97, 110 
 
 Conversion of English and metric meas- 
 ures, 2 
 
 Corrosion, 189 
 
 Cost of mains, 188 
 
 Cotterill, 48 
 
 Critical velocity, 147 
 
 Cunningham, 255, 284, 285 
 
 Current, radiating, 50 
 
 Current meters, 270 
 
 Dalton's law, 120 
 
 Darcy, 156, 160, 167, 201, 204, 233, 279, 
 
 282 
 
 Darcy gauge, 279 
 Deacon, 190, 283, 284 
 Density of water, 4 
 Discharge curve, 287 
 Distribution of velocity in pipes, 219 ; in 
 
 air mains, 230 ; in streams, 282 
 Duchemin, 301, 305 
 Durley, 124 
 
 Eddying motion, 38 
 
 Elbows, 170 
 
 Exner, 275 
 
 Expansion of compressible fluids, 125 
 
 325 
 
326 
 
 HYDKAULICS 
 
 Fire-hose pipes, 165 
 
 Fire nozzle, 83 
 
 Float gauging of streams, 267 
 
 Floating bodies, equilibrium of, 33 
 
 Fluid pressure, 13 ; on planes, 14 ; on 
 curved surfaces, 15 ; varying as the 
 depth, 22 ; on a wall, 24 ; on valve, 
 25 ; graphic determination of, 31 
 
 Fluids, properties of, 9 
 
 Forth Bridge experiments on wind pres- 
 sure, 312 
 
 Francis, 91, 107, 108 
 
 Free surface, 11, 20 
 
 Freeman, 83 
 
 Friction, fluid, 132 ; of discs in water, 136 
 
 Froude, 45, 133 
 
 Fteley, 98, 100, 107 
 
 Functions of numbers, table of, 317 
 
 Gale, 160 
 
 Gas, flow in mains, 222 
 
 Gaseous laws, 119 
 
 Gauging by weirs, 114 ; of streams, 264, 
 
 286 ; by chemical means, 290 
 Gordon, 279 
 
 Gravity, acceleration due to, 6 
 Gutermuth, 229 
 
 Hagen, 202, 206 
 
 Hamilton Smith, 72, 75, 98, 100, 205 
 
 Hardness of water, 4 
 
 Harlacher, 271 
 
 Head, meaning of term, 44 ; measurement 
 
 of, 68, 114 
 
 Heaviness of water, 3 ; of gases, 118 
 Herschel, 55, 68, 162, 192 
 Hook gauge, 70, 114 
 Hydraulic gradient, 152, 156, 187 
 Hydraulic press, 18 
 
 Impact of fluids, 291 ; jet deviated in one 
 direction, 292 ; impact of jet on solid 
 of revolution, 293 ; impact of jet on 
 planes and cups, 295 ; impact of limited 
 stream on plane, 298 ; of unlimited 
 stream, 301 ; of air on solids, 302 ; dis- 
 tribution of pressure on surface, 306 
 
 Inversion of jets, 78 
 
 Jet impinging on curved surface, 292 
 Jet pump, 86 
 Joukowsky, 198 
 
 Kelvin, Lord, 11 
 Kuichling, 174 
 
 Kutter's formula for pipes, 200 ; for canals, 
 234 
 
 Labyrinth piston, 59 
 Lampe, 205 
 Lesbros, 98 
 Levy, Maurice, 159 
 Lock, of canal, 84 
 
 Mains, water, 178 ; of varying diameter, 
 180 ; branched, 185 ; compound, 187 
 
 Mair, 73, 92, 202, 219 
 
 Marx, 164 
 
 Mean velocity, 41 
 
 Metacentre, 34 
 
 Module, 68, 75 
 
 Mouthpiece, cylindrical, 85 ; convergent, 
 88 ; divergent, 89 
 
 Napier, 130 
 
 Non-sinuous motion, 146 
 
 Notches, 95 ; triangular, 105. Se.e Weirs 
 
 Orifices, 61 ; use in measuring water, 67 ; 
 conoidal, 70 ; sharp-edged, 71 ; rect- 
 angular, 74 ; submerged, 75 ; self- 
 adjusting, 75 ; flow of fluids other than 
 water, 77 ; application of Bernoulli's 
 theorem to, 81 ; head varying with 
 time, 84 ; large rectangular vertical, 
 95 ; flow of air through, 123, 130 
 
 Pascal's law, 13 
 
 Pelton wheel, 297 
 
 Pipe aqueducts, 190 
 
 Pipe scrapers, 189 
 
 Pipes, non-sinuous and turbulent condition 
 of flow, 146 ; permissible velocities, 
 149 ; Chezy formula, 150 ; inlet re- 
 sistance, 155 ; pressure in, 156 ; 
 Darcy's investigation, 156 ; Maurice 
 Levy's formula, 159 ; later investiga- 
 tions, 160 ; riveted pipes, 162 ; timber 
 stave pipes, 164 ; practical calculations, 
 166 ; tables of flow in, 167 ; secondary 
 losses of head, 168 ; later investigation 
 of flow in, 199 ; Kutter's formula, 200 ; 
 general formula and constants, 216 : 
 flow of compressible fluids in, 221 ; 
 tables of discharge of, 321, 322 
 
 Pitot tube, 279 
 
 Poisseuille, 146 
 
 Poncelet, 74 
 
 Pressure column, 20 
 
 Pressure variation along stream lines, 43 ; 
 across stream lines, 48 ; at abrupt 
 changes of section 57 
 
 Prony, 199, 283 
 
 Rafter, 105 
 
 Rankine, 5 
 
 Reaction of fluids, 291 
 
 Reynolds, Osborne, 37, 146, 203 
 
 Riedler, 229 
 
 River gauging, 265, 286 
 
 River weirs, 96 
 
 Rivers, velocity curves, 256 ; ratios of 
 
 mean and surface velocity, 257 
 Riveted pipe, 162 
 Rod floats, 268 
 Rosenhain, 131 
 
INDEX 
 
 327 
 
 Scour valves, 191 
 
 Screw current meter, 270 
 
 Sewers, 244 
 
 Shock, 57 ; in pipes, 196 
 
 Siphon gauge, 21 
 
 Slope table, 319 
 
 Sluices, 173, 191 
 
 Specific heat of gases, 119 
 
 Stanton, 302, 305 
 
 Stave pipes, 164 
 
 Steady and unsteady motion, 40, 150 
 
 Stearns, 98, 100, 107, 160, 205 
 
 Stockalper, 230 
 
 Stream lines, 37, 48, 146 
 
 Strohmeyer, 290 
 
 Suction pipe of pumps, 195 
 
 Temperature, influence on flow from 
 orifices, 91 ; on friction, 143 ; on flow 
 in pipes, 202, 219 ; correction for, 
 213 
 
 Thomson, James, 106, 261 
 
 Town supply, 177 
 
 Transverse sections, measurement of, 
 265 
 
 Turner, 275 
 
 Units of measurement, 1 ; of intensity of 
 pressure, 6 
 
 Valves, 173 ; scour, 191 ; reflux, 191 ; 
 momentum, 191 ; sluice, 191 ; resist- 
 ance to flow at, 173 
 
 Velocity and head, table of, 318 
 
 Velocity curves, 251 
 
 Venant, de St., 202 
 
 Venturi meter, 52 
 
 Viscosity. lp T Iffi 
 
 Volume of flow, 40 
 
 Vortex, free, 51 ; forced, 52 
 
 Wagner, 284, 285 
 
 Water hammer, 196 
 
 Water inch, 68 
 
 Water-level gauge, 264 
 
 Watt's hydrometer, 21 
 
 Weirs, 96 ; drowned, 101, 113 ; broad- 
 crested, 102 ; with no end contractions, 
 106 ; Francis's formula, 108 ; Baziu's 
 researches, 109 ; separating, 115 
 
 Weisbach, 81, 88, 171 
 
 Wenham, 305 
 
 Williams, 172, 218, 281 
 
 Wind pressure, 308 
 
 THE END 
 
 Printed by R. & R. CLARK, LIMITED, Edinburgh. 
 
THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 AN INITIAL FINE OF 25 CENTS 
 
 WILL BE ASSESSED FOR FAILURE TO RETURN 
 THIS BOOK ON THE DATE DUE. THE PENALTY 
 WILL INCREASE TO SO CENTS ON THE FOURTH 
 DAY AND TO S1.OO ON THE SEVENTH DAY 
 OVERDUE. 
 
 OCT 5 1932 
 16 1933 
 
 c 
 
 JAM 31 1935 
 FEB 191935 
 
 DEC 19 I** 6 
 SEP 141940 
 SP 30 1940 
 
 OCT 23 
 
 FEB -4 1943 
 
 940M 
 
 LD 21-50?n-8,'32 
 
YG 13479