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ELEMENTS OF HYDRAULICS 



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McGraw-Hill CookCompaiiy 

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ElGCtrical World TheEiigingerii^aiidMning Journal 
LngiaeGriiig Record Engineering News 

Railway A^ Gazette American Machinist 

Signal Engineer AraericanEngjneer 

Electric Railway Journal Coal Age 

Mptallui-gical anl Chemical Engineering Power 



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ELEMENTS 



OF 



HYDRAULICS 



BY 
S. E. SLOCUM, B. E., Ph. D., 

PROFESSOR OF APPLI10D MATHEMATICS IN THE 
UNIVERSITY OF CINCINNATI 



Second Edition 
Revised and Enlarged 



McGRAW-HILL BOOK COMPANY, Inc. 

239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Ltd, 

6 & 8 BOUVERIE ST., E. C. 

1917 



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copymght, 1916, 1917, by the 
McGraw-Hill Book Company, Inc. 



. • • • .• t T. ; • • • ; : • 



T H K M A I* I. K I» K K K S Y <) K JC !» X 



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PREFACE TO SECOND EDITION 

The present revision is based largely on the experience of those 
who have used the book in the class-room, and is intended to 
make it more efficient as a text as well as a more complete work 
for general reference. To effect this several changes in typog- 
raphy have been made, as well as important alterations in the 
text. For instance, each paragraph has been numbered for ease 
of reference, while the classical terms hydrostatics, hydrokinetics, 
and hydrodynamics have been replaced by the more familiar and 
descriptive names "pressure of water,'^ "flow of water,'' and 
"energy of flow.'' 

The principal additions to the text consist of a more complete 
and up-to-date discussion of the flow of water in pipes, with 
special reference to the exponential formula and its graphical 
solution; a summary of the principal formulas for the strength 
of pipe; a more extended discussion of weir formulas; a fuller 
presentation of the modern use of siphons on a large scale; recent 
developments in the theory of water hammer; and the modern 
solution of penstock and surge tank problems. Numerous minor 
changes have also been made wherever such changes seemed 
indicated for additional clearness or proper emphasis. For in- 
stance, the applicability of Chezy's formula to pipe flow as well 
as to open channels has been pointed out; the turbine constants 
have been revised to date; the principles of draft tube design 
indicated; the mathematical discussion of back-water has been 
replaced by a simple application of Chezy's formula; while special 
care has been taken to correct all misprints as well as inaccuracies 
of statement. 

A complete set of answers to problems has been prepared, and 

is printed separately. 

S. E. Slocum. 
December, 1916. 



357398 n i 

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PREFACE TO FIRST EDITION 

The remarkable impetus recently given to hydraulic develop- 
ment in this country has caused the whole subject to assume a 
new aspect. Not only is this apparent in new and improved con- 
struction details, but in the scientific study which is beginning to 
be given a subject which seemed to have crystallized into a set 
of empirical formulas. 

Such comprehensive plans as those recently undertaken by the 
State of New York and the Dominion of Canada for the system- 
atic development of all their available water power, indicates the 
extent of the field now opening to the hydraulic engineer. The 
extent and cheapness of the natural power obtained not only from 
the development of existing streams but also from the artificial 
pondage of storm water is sufficient to convince even the most 
casual observer that no phase of conservation will have a more 
immediate effect on our industrial development or be more far 
reaching in its consequences. 

The present text is intended to be a modern presentation of the 
fundamental principles of hydraulics, with applications to recent 
important works such as the Catskill aqueduct, the New York 
State barge canal, and the power plants at Niagara Falls and 
Keokuk. Although the text stops short of turbine design, the 
recent work of Zowski and of Baashuus is so presented as to en- 
able the young engineer to make an intelligent choice of the type 
of development and selection of runner. 

In order to make the book of practical working value, a col- 
lection of typical modern problems is added at the end of each 
section, and a set of the most useful hydraulic data has been 
compiled and is tabulated at the end of the volume. 

Cincinnati, Ohio, S. E. Slocum. 

January, 1916. . 



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CONTENTS 



Preface 



Page 
V, vi 



SECTION I 
PRESSURE OF WATER 

Art. 1. Properties op a Perfect Fluid 1-3 

1. Definition of fluid — 2. Distinction between liquid 
and gas — ^3. Elasticity of water — 4. Fluid pressure nor- 
mal to surface — ^5. Viscosity — 6. Density of water — 

7. Specific weight. 

2. Pressttbe of Water 3-8 

8. Equal transmission of pressure — 9. Pressure propor- 
tional to area — 10. Hydraulic press — 11. Frictional re- 
sistance of packing — 12. Efficiency of hydraulic press. 

3. Simple Pressure Machines 8-13 

13. Hydraulic intensifier — 14. Hydraulic accumulator 
— 15. Hydraulic jack — 16. Hydraulic crane — 17. Hy- 
draulic elevator. 

4. Pressure on Submerged Surfaces 13-17 

18. Change of pressure with depth — 19. Pressure on 
submerged area — 20. Center of pressure — ^21. General 
formula for center of pressure — 22. Application. 

5. Strength of Pipes under Internal Pressure . . . 17-19 

23. Thin cylinder — ^24. Lame's formula — 25. Barlow's 
formula — 26. Clavarino's formula — 27. Birnie's 
formula. 

6. Equilibrium of Two Fluids in Contact 19-22 

28. Head inversely proportional to specific weight — ^29. 
Water barometer — ^30. Mercury barometer — ^31. Piezo- 
meter — 32. Mercury pressure gage — ^33. Differential 
gage. 

7. Equilibrium of Floating Bodies 22-26 

34. Buoyancy — ^35. Floating equilibrium — 36. Theo- 
rem of Archimedes — 37. Physical definition of specific 
weight — 38. Determination of specific weight by experi- 
ment — 39. Application to alloy — 40. Zero buoyancy. 

8. Mbtagbnter 26-30 

41. Stability of floating body — 42. Metacenter — 43. 
Codrdinates of metacentei^— 44. Metacentric height — 
45. Period of oscillation— 46. Rolling and pitching. 

Applications 30-46 

xi 



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xu 



CONTENTS 



SECTION II 

FLOW OF WATER 

Page 

Abt. 9. Flow op Watbb fbom Reservoirs and Tanks . . . 47-51 
47. Stream line — 48. Liquid vein — 49. Ideal velocity 
head — 50. Torricelli's theorem — 51. Actual velocity of 
flow — 52. Contraction coefficient — 53. Efflux coeffi- 
cient — 54. Effective head — 55. Discharge from large 
rectangular orifice — 56. Discharge of a rectangular 
notch weir. 

10. Discharge through Sharp-edged Orifice 51-53 

67. Contraction of jet — 58. Complete contraction — 

69. Partial contraction — 60. Velocity of approach. 

11. Rectangular Notch Weirs 53-60 

61. Contracted weir — 62. Suppressed weir — 63. Sub- 
merged weir — 64. Triangular weir — 65. Trapezoidal or 
Cippoletti weir — 66. Formulas for rectangular notch 
weirs. 

12. Standard Weir Measurements 60-63 

67. Construction of weir — 68. Hook gage — 69. Pro- 
portioning weirs. 

13. Time Required for Filling and Emptying Tanks. . 63-67 

70. Change in level under constant head — 71. Varying 
head — 72. Canal lock — 73. Rise and fall in connected 
tanks — 74. Mariotte's flask. 

14. Flow through Short Tubes and Nozzles 67-70 

75. Standard mouthpiece — 76. Stream Kne mouthpiece 
— 77. Borda mouthpiece — ^78. Diverging conical mouth- 
piece — ^79. Venturi adjutage — 80. Converging conical 
mouthpiece — 81. Fire nozzles. 

15. Kinetic Pressure in a Flowing LiQino 71-74 

82. Kinetic pressure — 83. Bernoulli's theorem — 84. 
^ Kinetic pressure head — 85. Application to standard 
mouthpiece. 

16. Venturi Meter 74-78 

86.' Principle of operation — 87. Formula for flow — 88. 
Commercial meter — 89. Catskill Aquaduct meter — 90. 
Rate of flow controller. 

17. Flow op Water in Pipes 78-83 

91. Critical velocity — 92. Viscosity coefficient — ^93. 
Parallel (non-sinuous) flow — ^94. Average velocity of 
flow in small pipes — 95. Loss of head in small pipes — ^96. 
Ordinary pipe flow. 

18. Practical Formxtlas for Loss op Head in Pipe Flow. 83-97 

97. Effective and lost head — 98. Loss at entrance — 99. 
Friction loss — 100. Wood stave pipe — 101. Graphical 



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CONTENTS xiu 

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solution — 102. Cast iron pipe — 103. Deterioration with 
age — 104. Riveted steel pipe — 105. Concrete pipe — 
106. Bends and elbows — 107. Enlargement of section 
— 108. Contraction of section — 109. Gate valve in 
circular pipe — 110. Cock in circular pipe — 111. Throt- 
tle valve in circular pipe — 112. Summary of losses — 

113. Application. 

19. Hydraulic Gradient 97-99 

114. Kinetic pressure head — 116. Slope of hydraulic 
gradient — 116. Peaks above hydraulic gradient. 

20. Hydraulic Radius 99-102 

117. Definition of hydraulic radius — 118. Chezy's 
formula for pipe flow — 119. Kutter's and Bazin's for- 
mulas for pipe flow — 120. Williams and Hazen's expo- 
nential formula. 

21. Divided Flow 102-105 

121. Compound pipes — 122. Branching pipes. 

22. Fire Streams 105-106 

123." Freeman's experiments — 124. Formulas for dis- 
charge — 125. Height of effective fire stream — 126. 
Fleming's experiments. 

23. Experiments on the Flow op Water 107-111 

127. Verification of theory by experiment — 128. Method 
of conducting experiments — 129. Effect of sudden 
contraction or enlargement — 130. Disturbance pro- 
duced by obstacle in current — 131. Stream line motion 
in thin film— 132. Cylinder and flat plate— 133. Ve- 
locity and pressure. 

24. Modern Siphons 111-120 

134. Principle of operation — 135. Siphon spillways — 
136. Siphon lock — 137. Siphon wheel settings. 

25. Flow in Open Channels 120-123 

138. Open and closed conduits — 139. Steady imiform 
flow — 140. Kutter's formula — 141. Limitations to Kut- 
ter's formula — 142. Bazin's formula — 143. Kutter's 
simplified formula. 

26. Channel Cross-section 123-126 

144. Condition for maximum discharge — 145. Maxi- 
mum hydraulic efi&cienxjy — 146. Regular circumscribed 
polygon — 147. Properties of circular an^ oval sections. 

27. Flow in Natural Channels 12^129 

148. Stream gaging — 149. Current meter measure- 
ments — 150. Float measurements — 151. Variation of 
velocity with depth — 152. Calculation of discharge. 

28. The Pitot Tube 129-138 

153. Description of instrument — 154. Darcy's modifi- 



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xiv CONTENTS 

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cation of Pitot's tube — 155. Pitometer — 156. Pitot re- 
corders — 157. Theory of the impact tube — 158. Con- 
struction and calibration of Pitot tubes — 159. DuBuat's 
paradox. 

29. Non-uniform Flow; Backwater . ^ 138-139 

160. Surface elevation. 

Applications 139-156 

4 

SECTION III 
ENERGY OF FLOW 

Art. 30. Pressure op Jet against Stationary Deflecting 

Surface 157-162 

161. Normal impact on plane surface — 162. Relation 
of static to dynamic pressure — 163. Oblique impact on 
plane surface — 164. Axial impact on surface of revolu- 
tion — 165. Complete reversal of jet — 166. Deflection 
of jet — 167. Dynamic pressure in pipe bends and 
elbows. 

31. Pressure Exerted by Jet on Moving Vane. . . . 162-167 

168. Relative velocity of jet and vane — 169. Work 
done on moving vane — 170. Speed at which work be- 
comes a maximum — 171 . Maximum efficiency for single 
vane — 172. Maximum efficiency for continuous succes- 
sion of vanes — 173. Impulse wheel; direction of vanes 
at entrance and exit — 174. Work absorbed by impulse 
wheel. 

32. Reaction op a Jet 167-169 

175. Effect of issuing jet on equilibrium of tank — 176. 
Energy of flow absorbed by work on tank — 177. Prin- 
ciple of reaction turbine — 178. Barker's mill. 

33. Types op Hydraulic Motors 169-171 

179. Current wheels — 180. Impulse wheels — 181. Re- 
action turbines — 182. Classification of reaction tur- 
bines — 183. Classification of hydraulic motors. 

34. Current and Gravity Wheels 171-173 

184. Current wheels — 185. Undershot wheels — 186. 
Poncelet wheels — 187. Breast wheels — 188. Overshot 
wheels. 

35. Impulse Wheels and Turbines 173-185 

189. Pelton wheel— 190. Efficiency of Pelton wheel— 
191. Characteristics of impulse wheels — 192. Girard 
impulse turbine — 193. Power and efficiency of Girard 
turbine. 

36. Reaction Turbines 185-202 

194. Historical development — 195. Mixed flow or 



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CONTENTS XV 

Page 
- American type — 196. Use of draft tube — 197. Draft 
tube design— 198. Profile of draft tube— 199. Time of 
flow through draft tube — 200. Recent practice in tur- 
bine setting. 

37. Characteristics of Rbaction Turbines 202-214 

201. Selection of type — 202. Action and reaction 
wheels — 203. Speed criterion — 204. Capacity criterion 
— ^205. Characteristic speed — 206. Specific discharge — 
207. Specific power — 208. Specific speed — 209. Rela- 
tion between characteristic speed and specific speed — 
210. Classification of reaction turbines — 211. Numer- 
ical application — 212. Normal operating range — 213. 
Selection of stock runner. 

38. Power Transmitted through Pipe Lines 214-217 

214. Economical size of penstock — 215. Numerical 
application. 

39. Effect of Translation and Rotation 217-220 

216. Equilibrium under horizontal linear acceleration 
— 217. Equilibrium under vertical linear acceleration 
— 218. Free surface of liquid in rotation — ^219. Depres- 
sion of cup below original level in open vessel — ^220. 
Depression of cup below original level in closed vessel 
— 221. Practical applications 

40. Water-hammer in Pipes 221-227 

222. Maximum-water hammer — 223. Velocity of com- 
pression wave — 224. Ordinary water-hammer — ^225. 
Joukovsky's formula — 226. Allievi's formula — 227. 
Occurrence of water-hammer in supply systems. 

41. Surge Tanks 228-236 

228. Surge in surge tanks — 229. Differential surge 
tank. 

42. Hydraulic Ram 236-238 

230. Principle of operation — ^231. Efficiency of ram. 

43. Displacement Pumps 238-246 

232. Pump types — ^233. Suction pump — 234. Maxi- 
mum suction lift — 235. Force pump — 236. Stress in 
pump rod — 237. Direct driven steam pump — 238. Cal- 
culation of pump sizes — 239. Power required for opera- 
tion — ^240. Diameter of pump cyUnder — 241. Steam 
pressure required for operation — 242. Numerical 
application. 

44. Centrifugal Pumps 246-255 

243. Historical development — 244. Principle of opera- 
tion — 245. Impeller forms — 246. Conversion of kinetic 
energy into pressure — 247. Volute casing — 248. Vortex 
chamber — 249. Diffusion vanes — 250. Stage pumps. 



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xvi CONTENTS 

Paob 

45. Pressube Developed in Cbntripugal Pump .... 255-258 

251. Pressure developed in impeller — ^252. Pressure de- 
veloped in diffusor — ^253. General expression for pres- 
sure head developed. 

46. Centrifugal Pump Characteristics. 258-265 

254. Effect of impeller design on operation — ^255. Ris- 
ing and drooping characteristics — ^256. Head developed 
by pump— 257. Effect of throttling the discharge — 

258. Numerical illustration. 

47. Eppicibncy and Design op Centrifugal Pumps . . 265-266 

259. Essential features of design — 260. Hydraulic and 
commercial efficiency. 

48. Centrifugal Pump Applications 26^273 

261. Floating dry docks— 262. Deep wells— 263. Mine 
drainage— 264. Fire pumps — 265. Hydraulic dredging 
— ^266. Hydraulic mining. 

Applications 273-289 



SECTION IV 

HYDRAULIC DATA AND TABLES 

Table 1. Properties of water 290 

2. Head and pressure equivalents 291 

3. Discharge equivalents 292 

4. Weigbts and measures 293 

5. Specific weights of various substances 294 

6. Standard dimensions of pipes 295 

7. Capacity of reciprocating pumps 29^297 

8. Circumferences and areas of circles 298-302 

9. Efflux coefficients for circular orifice 303 

10. Efflux coefficients for square orifice 304 

11. Fire streams 305-307 

12. Coefficients of pipe friction 308 

13. Friction bead in pipes 309-312 

14. Bazin's values of Chezy's coefficient 313 

15. Kutter's values of Cbezy's coefficient 314-315 

16. Discbarge from wood-stave pipe 316-317 

17. Discbarge coefficients for rectangular notcb weirs ..... 318 

18. Discbarge per incb of lengtb over rectangular notcb weirs . 319 

19. Discbarge per foot of lengtb over rectangular notcb weirs . 320 

20. Discbarge per foot of lengtb over suppressed weirs .... 321 

21. Principles of mecbanics 322-323 

22. Submerged weir coefficients 324 

Index 325 



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ELEMENTS OF HYDRAULICS 



SECTION 1 
PRESSURE OF WATER 

I. PROPERTIES OF A PERFECT FLUID 

1. Definition of Fluid. — ^A fluid is defined, in general, as a 
substance which offers no resistance to change in form provided 
this deformation is not accompanied by change in volume. 
The fundamental property of a fluid is the perfect mobility of 
all its parts. 

Most of the applications of the mechanics of fluids relate to 
water, and this domain of mechanics is therefore usually called 
hydraulics or hydromechanics. It is customary to subdivide 
the subject into hydrostaiics, relating to water at rest; hydro^ 
kinetics, relating to water in motion; and hydrodynamicSf relating 
to the inertia forces exerted by fluids in motion, and the energy 
available from them. 

2. Distinction between Liquid and Gas. — ^A liquid such as 
water has a certain degree of cohesion, causing it to form in 
drops, whereas a gaseous fluid tends to expand indefinitely. A 
gas is therefore only in equilibrium when it is entirely enclosed. 
In considering elastic fluids such as gas and steam, it is always 
necessary to take account of the relation between volume and 
pressure. For a constant pressure the volume also changes 
greatly with the temperature. For this reason the mechanics of 
gases is concerned chiefly with heat phenomena, and forms a 
separate field called thermodynamics, l3dng outside, and sup- 
plementary to, the domain of ordinary mechanics. 

3. Elasticity of Water. — Water, like other fiuids, is elastic, 
and under heavy pressure its volume is slightly diminished. It 
has been found by experiment that a pressure of one atmosphere, 
or 14.7 lb. per sq. in., exerted on each face of a cube of water at 

1 



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2 ELEMENTS OF HYDRAULICS 

32®F. causes it to diminish about 0.00005 in volume. Conse- 
quently, the bulk modulus of elasticity of water, B, defined as 

„ unit stress 

"" unit volume deformation' 

has as its numerical value 

As the change in volume is so small it is sufficiently accurate for 
most purposes to assume that water is incompressible. An ex- 
ception to this rule will be found in Art. XL. 

4. Fluid Presstire Normal to Surface. — Since a perfect fluid is 
one which offers no resistance to change in form, it follows that 
the pressure on any element of surface of the fluid is everywhere 
normal to the surface. To prove this proposition, consider any 
small portion of a fluid at rest, say a small cube. Since this cube 
is assumed to be at rest, the forces acting on it must be in equi- 
librium. In the case of a fluid, however, the general conditions 
of equilibrium are necessary but not sufficient, since they take no 
account of the fact that the fluid offers no resistance to change 
in form. Suppose, therefore, that the small cube under con- 
sideration undergoes a change in form and position without any 
change in volume. Since the fluid offers no resistance to this 
deformation, the total work done on the elementary cube in 
producing the given change must be zero. 

In particular, suppose that the cube is separated into two parts 
by a plane section, and that the deformation consists in sliding 
one of these parts on the other, or a shear as it is called. Then 
in addition to the forces acting on the outside of each part, it is 
necessary to consider those acting across the plane section. But 
the total work done on each part separately must be zero inde- 
pendently of the other part, and also the total work done on the 
entire cube must be zero. Therefore, by subtraction, the work 
done by the forces acting across the plane section must be zero. 
But when any force is displaced it does work equal in amount to 
the product of this displacement by the component of the force 
in the direction of the displacement. Therefore if the work done 
by the force acting across the plane section of the cube is zero, 
this force can have no component in the plane of the section, and 
must therefore be normal, i.e., perpendicular, to the section. 



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PRESSURE OF WATER 3 

6. Viscosity. — This absence of shear is only rigorously true for 
an ideal fluid. For water there is a certain amount of shear due 
to internal friction, or viscosity, but it is so small as to be practi- 
cally negligible. The greater the shear the more viscous the 
fluid is said to be, and its amount may be taken as a measure of 
viscosity. It is found by experiment that the internal friction 
depends on the difference in velocity between adjacent particles, 
and for a given difference in velocity, on the nature of the fluid. 
The viscosity of fluids is therefore of great importance in consider- 
ing their motion, but does not affect their static equilibrium. 
For any fluid at rest, the pressure is always normal to any element 
of surface. 

6. Density of Water. — In hydraulic calculations the unit of 
weight may be taken as the weight of a cubic foot of water at 
its temperature of greatest density, namely, 39°F. or 4°C. It 
is found by accurate measurement that a cubic foot of water at 
39°F. weighs 62.42 lb. This constant will be denoted in what 
follows by the Greek letter 7. In all numerical calculations it 
must be remembered therefore that 

y = 62.4 lb. per cu. ft. (2) 

The density and volume of water at various temperatures are 
given in Table 1. 

7. Specific Weight. — The weights of all substances, whether 
liquids or solids, may be expressed in terms of •the weight of an 
equal volume of water. This ratio of the weight of a given 
volume of any substance to that of an equal volume of water is 
called the specific weight of the substance, and will be denoted in 
what follows by s. For instance, a cubic foot of mercury weighs 
848.7 lb., and its specific weight is therefore 

848.7 



s = 



62 



^= 13.6 approximately. 



Its exact value^t 0°C. is s = 13.596, as may be found in Table 5. 
The weight of 1 cu. ft. of any substance in terms of its specific 
weight is then given by the relation 

Weight = 7S = 62.4s lb. per cu. ft. 

n. PRESSURE OF WATER 

8. Equal Transmission of Pressure. — The fundamental princi- 
ple of hydrostatics is that when a fluid at rest has pressure applied 



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ELEMENTS OF HYDRAULICS 




Fig. 1. 



to any portion of its surface, this pressure is transmitted equally 

to all parts of the fluid. 
To prove this principle consider any portion of the fluid limited 

by a bounding surface of any form, 
and suppose that a small cylin- 
drical portion is forced in at one 
point and out at another, the rest 
of the boundary remaining un- 
changed (Fig. 1). Then if AA de- 
notes the cross-sectional area of 
one cylinder and An its height, its 
volume is AA-An. Similarly, the 
volume of the other cylinder is 
AA'An', and, since the fluid is as- 
sumed to be incompressible, 
AAAn = AA'-Aw'. 

Now let p denote the unit pressure on the end of the first cylinder, 
i.e., the intensity of pressure, or its amount in pounds per square 
inch. Then the total pressure normal to the end is pAA, and the 
work done by this force in moving the distance An is pAA-An. 
Similarly, the work done on the other cylinder is p'AA'-An'. 
Also, if 7 denotes the heaviness of the fluid per unit volume, the 
work done by gravity in moving this weight yAA-An through the 
distance A, where h denotes the difference in level between 
the two elements considered, is yAA-An-h. Therefore, equating 
the work done on the fluid to that done by it, we have 

pAA'An + yAA'Anh = p'-AA' An'. 
Since AA-An = AA'-An', this reduces to 



p' = p + 7h. 



(3) 



If A = 0, then p' = p. Therefore the pressure at any point in 
a perfect fluid is the same in every direction. Also the pressure 
at the same level is everywhere the same. 

Moreover, if the intensity of pressure p at any point is increased 
by an amount w, so that it becomes p + w, then by Eq. (3) the 
intensity of pressure at any other point at a difference of level 
h becomes 

p" = (p + y>) + yh. 
But since p' = p + yh, we have by subtraction. 



// 



p' + w, 



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PRESSURE OF WATER 5 

that is, the intensity of pressure at any other point is increased 
by the same amount w, A pressiu'e applied at any point is 
therefore transmitted equally to all parts of the fluid. 

For fluids such as gas and steam the term yh is negligible, and 
consequently for such fluids the intensity of pressure may be 
assumed to be everywhere the same. 

9. Pressure Proportional to Area. — To illustrate the applica- 
tion of this principle consider a closed vessel or tank, filled with 
water, having two cylindrical openings at the same level, closed 
by movable pistons (Fig. 2). If a load P is applied to one piston. 




— D— 



Fig. 2. 



Fig. 3. 



then in accordance with the result just proved there is an in- 
crease of pressure throughout the vessel of amount 

P 

where A denotes the area of the piston. The force P' exerted 
on the second piston of area A' is therefore 



whence 



p a' 



The two forces considered are therefore in the same ratio as their 
respective areas. This relation remains true whatever shape the 
ends of the pistons may have, the areas A and A' in any case 
being the crossHsectional areas of the openings. For instance, 
an enlargement of the end of the piston, such as shown in Fig. 3, 
has no effect on the force transmitted, since the upward and 



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ELEMENTS OF HYDRAULICS 



ir(D^ - d^) 



cancel, leaving 



downward pressures on the ring of area 

-7- as the effective area. 
4 

10. Hydraulic Press. — An important practical application of 

the law of hydrostatic pressure is found in the hydraulic press. 

In its essential features this consists of two cylinders, one large 

and one small, each fitted with a piston or plunger, and connected 

by a pipe through which water can pass from one cylinder to the 




Fig. 4. 



other (Fig. 4). Let p denote the intensity of pressure within the 
fluid, Z>, d the diameters of the two plungers, P the load applied 
to one and W the load supported by the other, as indicated in 
the figure. Then 



and consequently 



W = -j-'p, 

W ^ 
P 



d2* 



(4) 



If the small plunger moves inward a distance A, the large one 
will be forced out a distance H such that each will displace the 
same volume, or 

Td% tD^H 



whence 



h = H 



4 
D2 



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PRESSURE OF WATER 



Neglecting friction, the work done by the force P in moving 
the distance h is then 

Ph=p(H^)=H(p^)=HW, 

and is therefore equal to the work done in raising W the dis- 
tance H. 

11. Frictional Resistance of Packing. — Usually, however, 
there is considerable frictional resistance to be overcome, since for 
high pressures, common in hy- 
draulic presses, heavy packing 
is necessary to prevent leakage. 
One form of packing extensively 
used is the U leather packing 
shown in Fig. 5. In this form 
of packing the water leaking 
past the plunger, or ram as it is 
often called, enters the leather 
cup, pressing one side against 
the cylinder and the other 
against the ram, the pressure 
preventing leakage being pro- 
portional to the pressure of the 
water. 

To take into account the fric- 
tional resistance in this case, let 
/i denote the coefficient of friction between leather and ram, 
C, .c the depths of the packing on the large and small rams 
(Fig. 4), and p the intensity of water pressure. Then the area 
of leather in contact with the large ram is tDC and its frictional 
resistance is therefore TtDCpfx. Similarly, the frictional resistance 
for the small ram is irdcpii. Consequently 




W = ^v - fiprDC, 



and 



whence 



P = 



rd2 



p 4" iMpirdc, 




(6) 



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ELEMENTS OF HYDRAULICS 



12. Efficiency of Hydraulic Press. — The efficiency of any ap- 
paratus or machine for transforming energy is defined as 

T?ffi • Useful work or effort 

^ "" Total available work or effort' 

and is therefore always less than unity. In the present case 
if there were no frictional resistance, the relation between W 
and P would be given by Eq. (4). The efficiency for this type 
of press is therefore the ratio of the two equations (5) and (4), or 



Efficiency = 






1+4^ 



(6) 



m. SIMPLE PRESSURE MACHHTES 



m 



HiffhPresrare 
Outlet 



£ 



JDl 



HT 



P. 



m 



^ 






B 



% 



IHL 



nm 



A *—-\ LowPressuia 
Intake 



±^ 



Fig. 6. — Intensifier. 

form another cylinder C, fitted 
fixed to the yoke at the top. 



13. Hydraulic Intensifier. — 
Besides the hydraulic press de- 
scribed in Art. II there are a 
number of simple pressure ma- 
chines based on the principle of 
equal distribution of pressure 
throughout a liquid. Four types 
are here illustrated and de- 
scribed, as well as their combiaa- 
tion in a hydraulic installation. 

When a hydraulic machine 
such as a punch or riveter is 
finishiQg the operation, it is re- 
quired to exert a much greater 
force than at the beginniag of 
the stroke. To provide this in- 
crease in pressure, an intensifier 
is used (Fig. 6). This consists 
of several cylinders telescoped 
one inside another. Thus in 
Fig. 6, which shows a simple 
form of intensifier, the largest 
cylinder A is fitted with a ram B. 
This ram is hollowed out to 
with a smaller ram D, which is 



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PRESSURE OF WATER 



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In operation, water at the ordinary pump pressure enters 
the cylinder A through the intake, thereby forcing the ram B 
upward. This has the effect of forcing the ram D into the 
cylinder C, and the water in C is thereby forced out through D, 
which is hollow, at an increased pressure. Let pi denote the 
pressure of the feed water, pt the intensified pressure in C and 
di, dt the diameters of the cylinders B and C respectively, as 
indicated in the figure. Then 



irdi^ 



whence 



Pi 




A 



^ 



EZ) CD 



to, 



■^ 



Weiflrhts 



■^ 



The intensity of pressure in the 
cylinders is therefore inversely pro- 
portional to the areas of the rams. 

When a greater intensification of 
pressure is required a compound 
intensifier is used, consisting of 
three or four cylinders and rams, 
nested in telescopic form, the 
general arrangement and principle 
of operation being the same as in 
the simple intensifier shown in 
Fig. 6. 

14. Hydraulic Accumulator. — ^A 
hydraulic accumvldlor is a pressure 
regulator or governor, and bears 
somewhat the same relation to a 
hydraulic system that the flywheel 
does to an engine; that is, it stores 
up the excess pump delivery when Fia. 7.-— Accumulator, 
the pumps are delivering more than 

is being used, and delivers it again under pressure when the 
demand is greater than the supply. 

There are two principal types of hydraulic accumulator, in one 
of which the ram is fixed or stationary, and in the other the cylin- 
der. The latter type is shown in Fig. 7. When the delivery of 
the pumps is greater or less than required by the machine, water 
enters or leaves the cylinder of the accumulator through the 



^3 Water 



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ELEMENTS OF HYDRAULICS 



pipe A, The ram is thereby raised or lowered, and with it the 
weights suspended by the yoke from its upper end. The pressure 
in the system is thereby maintained constant and free from the 
pulsations of the pump. 

The capacity of the accumulator is equal to the volume of the 
ram displacement, and should be equal to the delivery from the 
pmnp in five or six revolutions. 

The diameter of the ram should be large enough to prevent a 
high speed in descent, so as to avoid the inertia forces set up by 
sudden changes in speed. 

15. Hydraulic Jack. — The hydraulic jack is a lifting apparatus 
operated by the pressure of a liquid under the action of a force 



'^^^ 




Fig. 8. — Hydraulic jack, 

pump. Thus in Fig. 8 the hand lever operates the pump piston 
B, which forces water from the reservoir A in the top of the ram 
through the valve at C into the pressure chamber D under the 
ram. The force exerted is thereby increased in the direct ratio 
of the areas of the two pistons. Thus if the diameter of the pump 
piston is 1 in. and the diameter of the lifting piston or ram is 4 
in., the area of the ram will be sixteen times that of the pump 
piston. If then a load of, say, 3 tons is applied to the pump 



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PRESSURE OF WATER 



11 



piston by means of the lever, the ram will exert an upward lift- 
ing force of 48 tons. 

16. Hydraulic Crane. — The hydraulic crane, shown in Fig. 9, 
consists essentially of a ram and cylinder, each carrying a set of 
pulleys. A chain or rope is passed continuously over the two 
sets of pulleys as in the case of an ordinary block and tackle, the 




I Cylinder ^ 

i ^1 




Fig. 9. — Hydraulic crane. 

free end passing over guide pulleys to the load to be lifted. 
When water is pumped into the cylinder under pressure, the two 
pulley blocks are forced apart, thereby lifting the load at the 
free end of the chain or rope. 

17. Hydraulic Elevator. — In hydraulic installations two or 
more of these simple pressure machines are often combined, as 
in the hydraulic elevator shown in Fig. 10. In this case the ac- 
cumulator serves to equalize the pump pressure, making the 
operation of the system smooth and uniform. 



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ELEMENTS OF HYDRAULICS 



The main valve for starting and stopping is operated, in the 
type shown, by the discharge pressure, maintained by means of 
an elevated discharge reservoir. A pilot valve, operated from 




Pump to BotQrD Yalvti Dt^chiifee 

Fia. 10. — Hydraulic elevator. 

the elevator cab, admits this low-pressure discharge water to 
opposite sides of the main valve piston as desired, thereby either 
admitting high-pressure water from the pump and accumulator. 



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PRESSURE OF WATER 13 

or opening the outlet valve into the discharge. The other details 
of the installation are indicated on the diagram. 

IV. PRESSURE ON SUBMERGED SURFACES 

18. Change of Pressure with Depth. — For a liquid at rest in 
an open vessel or tank, the free upper surface is perfectly level. 
Let the atmospheric pressure on this surface be denoted by p. 
Then, from Eq. (3), the pressure p' at a depth h below the surface 
is given by 

?>' = P + yh. 

Since the atmospheric pressure is practically constant, the 
free surface of the liquid may be assumed as a surface of zero 
pressure when considering only the pressure due to the weight 
of the liquid. In this case p = 0, and the pressure p' at any depth 
h, due to the weight of the liquid, becomes 

P' = yh. (7) 

Hence the pressure at any point in a liquid due to its own weight 
is directly proportional to the depth of this point below the free 
upper surface. 

Moreover, let AA denote any element of a submerged surface. 
Then the pressure on it is 

p'AA = yhAA. 

Therefore, the pressure on any 
element of area of a submerged 
surface is equal to the weight of 
a column of water of cross-sec- 
tion equal to the element con- 
sidered, and of height equal to 
the depth of this element below 
the surface. 

19. Pressure on Submerged 
Area. — Consider the pressure 
on any finite area A in the side 
of a tank a reservoir contain- 
ing a liquid at rest (Fig. 11). Let AA denote any element of this 
area and x its distance below the surface of the liquid. Then, 
by what precedes, the pressure on this elementary area is 

yxAA, 




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14 



ELEMENTS OF HYDRAULICS 



and consequently the total pressure P on th^ entire area A is 
given by the summation 

P = J^yxAA = yl^xAA. 

But from the ordinary formula for finding the center of gravity 
of an area, the distance Xq of the center of gravity of A below the 
surface is given by 

Axq = J^xAA 
and consequently 

P = tAxo. (8) 

Therefore^ the pressure of a liquid on any submerged plane surface 
is equal ta the weight of a column of the liquid of cross-section equal 
to the given area and of height equal to the depth of the center of 
gravity of this area below the free surface of the liquid. 

20. Center of Pressure. — The point of application of the 
resultant pressure on any submerged area is called the center of 




Fig. 12. 

pressure, and for any plane area which is not horizontal, lies 
deeper than the center of gravity of this area. For instance, 
consider the water pressure against a masonry dam with plane 
vertical face (Fig. 12). By Eq, (7) the pressure at any point A 
is proportional to the depth of A below the water surface. If, 
then, a length AB is laid off perpendicular to the wall and equal 
to the depth of A below the surface, that is, AB = AO, then AB 
will represent to a certain scale the normal pressure at A. If the 
same is done at various other points of the wall, their ends, jB, 
D, etc., will lie in a straight line inclined at 45° to the horizontal. 



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PRESSURE OF WATER 15 

For a portion of the wall of length 6, the pressure acting on it 
will then be equal to the weight of the water prism OEF, namely, 

and the center of pressure will coincide with the center of gravity 
of this prism. It therefore lies at a distance of %h below the 
water surface, which is below the center of gravity of the rectan- 
gular area under pressure since the latter is at a distance of ^ 

from the surface. 

21. General Formula for Center of Pressure. — To obtain a 
general expression for the location of the center of pressure, con- 
sider any plane area inclined at an angle a to the horizontal (or 



Fig. 13. 

water surface) and subjected to a hydrostatic pressure on one 
side. Let 00' denote the line of intersection of the plane in 
which the given area lies with the water surface (Fig. 13). Also, 
let AA denote any element of the given area, h the depth of this 
element below the surface, and x its distance from 00', as indi- 
cated in the figure. Then from Eq. (7), the pressure AP acting 
on this element is 

AP = yhAA, 

and the moment of this force with respect to the line 00' is 

xAP = yhxAA. 

Now let P denote the total resultant pressure on the area A and 
Xe the distance of the point of application of this resultant from 



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16 ELEMENTS OF HYDRAULICS 

00', i.e., Xe represents the x coordinate of the center of pressure. 
Then since the sum of the moments of aJl the elements of pres- 
sure with respect to any axis 00' is equal to the moment of their 
resultant with respect to this axis, we have 

XxAP = Pxc 

But AP = yhAA and P = Xyh^. Consequently this becomes 

XyhxAA = Xc^yhAA. 
Also, since 

h = xmi a, 
this may be written 

y sin aXx^AA = y sin aXe'SxAA 

or, cancelling the common factor 7 sin a, 

Tlx^AA = Xc^xAA. 

The left member of this expression is by definition the moment of 
inertia, I, of the area A with respect to the line 00', that is 

I = Sx«AA, 

while by the formula for the center of gravity of any area A we 
also have 

SxAA = xoA 

where Xo denotes the x coordinate of the center of gravity of A. 
The X coordinate of the center of pressure is therefore determined 
by the general formula 

_ _I_ _ Moment of inertia 

" Axo ~" Statical moment ' 

22. Application. — In applying this formula it is convenient 
to use the familiar relation 

I = la + Ad« 

where 
J. = moment of inertia of A with respect to the axis 00'; 
Ig = moment of inertia of A with respect to a gravity axis 

parallel to 00'; 
d = distance between these two parallel axes. 
For example, in the case of the vertical dam under a hydrostatic 
head h, considered above, we have for a rectangle of breadth b 
and height h, 

12' 



/a = 



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PRESSURE OF WATER 



17 



Consequently the moment of inertia I with .respect to its upper 
edge is 

and therefore the depth of the center of pressure below the sur- 
face is 

bh* 



Xc = 



Axo 



3 _2fc 



HI) 



which, of course, agrees with the result obtained geometrically. 

V. STRENGTH OF PIPES UNDER INTERNAL PRESSURE 

23. ThiA Cylinder. — In the case of a pipe flowing fuH, the 
pressure of the liquid produces stress in the pipe walls. Assum- 
ing the internal pressure of the liquid to be constant at any sec- 
tion, let 

/ = unit fiber stress in pipe 

(hoop stress) in pounds 

per square inch; 
w = imit internal fluid pressure 

in pounds per square inch; 
t = thickness of pipe walls; 
d = inside diameter of pipe; 
D = outside diameter of pipe. 

Now suppose that the pipe is 
divided longitudinally by a plane 
through its axis, and consider a 
section cut out of either half 
by two planes perpendicular to the axis, at a distance apart 
denoted by c (Fig. 14). Then the total internal pressure on the 
strip under consideration is cwd, and the total resisting tension 
in the pipe walls is 2ctf. Consequently cwd = 2ctf, whence 

wd 




f = 



2t 



(10) 



This formula applies primarily to a thin cylinder or pipe, that is, 
one for which -3 ^ 0.025. 
2 



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18 ELEMENTS OF HYDRAULICS 

24. Lamp's Formula. — For thick cylinders, in which the 

thickness of the cylinder is not negligible in comparison with the 

t 
diameter, or in other words cylinders for which ^ > 0.025; the 

formula commonly used is that due to Lam6, namely, 

^ w(D» + d») i, 
^ D« - d« ^ ^ 

26. Barlow's Formula. — ^Another formula which is widely used 
because of its simplicity is that due to Barlow. The derivation 
of this formula is based on the assmnptions that the area of 
cross-section of the tube remains constant under the strain, and 
that the length of the tube also remains unaltered. As neither 
of these assumptions is correct, the resulting formula is only 
approximate. Using the same notation as above. Barlow's 
formula is 

Evidently it is of the same form as the formula derived above for 
the hoop stress in a thin cylinder, except that it is expressed in 
terms of the oviside diameter of the pipe instead of its inside 
diameter. 

From the results of their experience in the manufacture and 
testing of tubes, however, the National Tube Co. asserts that 

for any ratio of ^ < 0.3, Barlow's formula "is best suited for all 

ordinary calculations pertaining to the bursting strength of 
commercial tubes, pipes and cylinders." 

For certain classes of seamless tubes and cylinders, however, 
and for critical examination of welded pipe for which the least 
thickness of wall, yield point of the material, etc., are known 
with accuracy and close results are desired, they recommend 
that the following formulas due to Clavarino and Birnie be used 
rather than Barlow's. 

26. Clavarino's Formula. — In the derivation of Clavarino's 
formula each particle of the tube is assumed to be subjected to 
radial stress, hoop stress and longitudinal stress, due to a uni- 
form internal pressure acting jointly on the walls of the tube and 
its closed ends. The derivation also involves Poisson's ratio 

' For the derivation of this formula see Slocum, "Resistance of Mate- 
rials," p. 123 (Ginn & Co.). 



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PRESSURE OF WATER 



19 



of lateral contraction, and is theoretically correct provided the 
maximum stress does not exceed the elastic limit of the material. 
Assuming the value of Poisson's ratio to be 0.3 and using the 
same notation as above, Clavarino's formula is 



f = 



w(13D« + 4d«) 



whence also 



10(D« - d«) ♦ 



/lQf-13w. 
lOf + 4w 



(13) 
(14) 



27. Bimie's Formula. — The derivation of Birnie's formula is 
based upon the same assumption as Clavarino's except that the 
longitudinal stress is assumed to be zero. Assuming Poisson's 
ratio for steel to be 0.3 and using the same notation as previously, 
Birnie's formula is 

w(13D* + 7d*) 



f = 



whence also 



10(D* - d») 



= D^ 



lOf - 18w 
lOf + 7w* 



(15) 



(16) 



VL EQUILIBRroM OF TWO FLUIDS IN CONTACT 

28. Head Inversely Proportional to Specific Weight— If two 

open vessels containing the same fluid, say water, are connected 














^^,s^ 


-=-— :^j^ 


=r=~^t 


~«fi 









:=-_=-:Lr^ 


'- ^ 



^^^^^^l- 



r 



i-. 



Fig. 16. 



by a tube, the fluid will stand at the same level in both vessels 
(Fig. 15). If the two vessels contain different fluids which are 
of different weights per unit of volume, that is to say, of different 
specific gravities, then since the fluid in the connecting tube 



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ELEMENTS OF HYDRAULICS 



must exert the same pressure in either direction, the surface of 
the lighter fluid will be higher than that of the heavier. 

For instance, let «i and 8% denote the specific gravities of the 
two fluids in the apparatus shown in Fig. 16, and let A denote 
the area of their surface of contact. Then for equilibrium 

ySiAh = ySzAH 
whence 

h Ss 



Si 



(17) 



1 


I 


m 


A 


1 




~ 


7? / 


V i 


- -f 


1 


— 


1 



The ratio of the heights of the two fluids above their surface of 

separation is therefore inversely proportional to the ratio of their 

specific gravities. 
29. Water Barometer. — ^If one of the fluids is air and the other 

water, we have what is called a water 
barometer. For example, suppose 
that a long tube closed at one end is 
filled with water and the open end 
corked. Then if it is placed cork 
downward in a vessel of water and 
the cork removed, the water in the 
tube will fall until it stands at a cer- 
tain height h above the surface of the 
water in the open vessel, thus leaving 
a vacuum in the upper end of the 
tube. The absolute pressure in the 
top of the tube, A (Fig. 17), is there- 
fore zero, and at the surface, B, is 

equal to the pressure of the atmosphere, or approximately 14.7 

lb. per square inch. But from Eq. (3) we have 

Pb =^ Pa + yh 
where in the present case Pb = 14.7 lb. per square inch; Pa = 0; 
y = 62.4 lb. per cubic foot, and by substitution of these values 
we find that 

A = 34 ft. approximately. 

This is the height, therefore, at which a water column may be 
maintained by ordinary atmospheric pressure. It is therefore 
also the theoretical height to which water may be raised by means 
of an ordinary suction pump. As it is impossible in practice to 
secure a perfect vacuum, however, the actual working lift for a 
suction pump does not exceed 20 or 25 ft. 



Fig. 17. 



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PRESSURE OF WATER 



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30. Mercury Barometer. — ^If mercury is used instead of water, 
since the specific gravity of mercury is 8 = 13.5956, we have 



14.7 X 144 
13.6 X 62.4 



= 30 in., approximately. 



which is accordingly the approximate height of an ordinary 
mercury barometer. 

31. Piezometer. — ^When a vessel contains liquid under pressure, 
this pressure is conveniently measured by a simple device called 
a piezometer. In its simplest form this consists merely of a tube 
inserted in the side of the vessel, of suflScient height to prevent 
overflow and large enough in diameter to avoid capillary action, 
say over J^ in. inside diameter. The height of the free surface 
of the liquid in the tube above any point B in the vessel then 
measures the pressure at B (Fig. 18). Since the top of the tube 
is open to the atmosphere, the absolute pressure at jB is that 
due to a head of A + 34 ft. 



i 







Fia. 18.— Piezometer. 




Fia. 19. — Pressure gage. 



32. Mercury Pressure Gage. — In general it is convenient to 
use a mercury column instead of a water column, and change the 
form of the apparatus slightly. Thus Fig. 19 shows a simple 
form of mercury pressure gage, the difference in level, h, of the 
two ends of the mercury column measuring the pressure at B. 
Let 8 denote the specific weight of mercury and s' the specific 
weight of the fluid in the vessel. The pressure at any point C 
in the vessel is then 

p. = 7sh - Ts'h'. (18) 



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ELEMENTS OF HYDRAULICS 



For example, if the fluid in the vessel is water, then y = 62.4, 
«' = 1, « = 13.596, and consequently 

Po = 62.4(13.596A - h'). 




^ 

c = 


= ' ' 


" 


\ ^' 


: 


^_L 


-■ ^ 



Fig. 20. — Vacuum gage. 

In case there is a partial vacuum in the vessel, the gage may be 
of the form shown in Fig. 20. The pressure on the free surface 

AB in the reservoir is then the same 
as at the top of the barometer col- 
umn, C, namely, 

Pc = 14.7 — 7s/i. 

33. Differential Gage.— When 
differences in pressure are to be 
measured, the gage commonly used 
is the U-tube differential gage, one 
form of which is shown in Fig. 21. 
In this form the lower part of the 
U-tube is filled with mercury, or some 
other heavy fluid, and from the differ- 
ence in elevation of the two ends of 
the mercury column the difference in 
pressure in the connecting tubes may 
be calculated. 

Vn. EQUILIBRIUM OF FLOATING 
BODIES 

34. Buoyancy. — ^When a solid body 
floats on the water partially submerged, as in the case of a piece 




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PRESSURE OF WATER 



23 



of timber or the hull of a ship^ each element of the wetted 
surface experiences a unit normal pressure of amount 

p = yh 

where h denotes the depth of the element in question below the 
water surface. Since the body is at rest, the total pressure acting 
on the wetted surface together with the weight of the body, 
which in this case is the only other external force, must then form 
a system in equilibrium. Since the weight of the body acts 
vertically downward, the water must therefore exert an upward 
pressure of the same amount. This resultant upward pressure 
of the water is called the buoyant effort, or buoyancy ^ and the 
point of application of this upward force is called the center of 



T^ W-B 




FiGL 22. 

buoyancy. For equilibrium, therefore, the buoyancy must be 
equal to the weight of the body and act vertically upward along 
the same line, since otherwise these two forces would form a 
couple tending to tip or rotate the body (Fig. 22). 

36. Floating Equilibrium. — To. calculate the buoyancy, sup- 
pose that the solid body is removed and the space it occupied 
below the water line refilled with water. Then since the lateral 
pressure of the water in every direction must be exactly the same 
as before, the buoyancy must be equal to the weight of this vol- 
ume of water. The buoyancy is therefore equal to the weight 
of the volume of water displaced by the floating body, and the 
center of buoyancy coincides with the center of gravity of the 
displacement. For equilibrium, therefore, a solid body must 



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24 ELEMENTS OF HYDRAULICS 

sink until the weight of the water it displaces is equal to the weight 
of the body, and the centers of gravity of the body and its dis- 
placement must lie in the same vertical. 

These conditions also apply to the case when a body is entirely 
submerged. As the density of water increases with the depth, 
if a solid is slightly heavier than the water it displaces, it will 
sink until it reaches a depth at which the density is such that the 
weight of the water it displaces is exactly equal to its own weight. 

36. Theorem of Archimedes. — If a solid is heavier than the 
weight of water it displaces, equilibrium may be maintained by 
suspending the body in water by a cord (Fig. 22), in which case 
the tension, T, in the cord is equal to the difference between the 
weight of the body and its buoyancy, that is, the weight of the 
water it displaces. A solid immersed in a liquid therefore loses 
in weight an amount equal to the weight of the liquid displaced. 
This is known as the Theorem of ArchimedeSf and was discovered 
by him about the year 260 B. C. 

37. Physical Definition of Specific Weight. — Consider a solid 
completely immersed in a liquid, and let V denote the volume of 
the solid, and y the weight of a cubic unit of the liquid, say 1 
cu. ft. Then the buoyancy, J?, of the body is 

y B^yV. 

Also, if 7i demotes the weight of a cubic unit of the solid, regarded 
as uniform and homogeneous, its weight is 

W==yiV. 
The ratio 

!=?=• (") 

is called the specific weight of the solid with respect to the liquid 
in which it is immersed (compare Art. I). In general, the liquid 
to which the specific weight refers is assumed to be water at a 
temperature of 39^F. The specific weight of any substance is 
then that abstract number which expresses how many times 
heavier it is than an equal volume of water at 39®F. The specific 
weight of water is therefore unity; for lighter substances such as 
wood or oil it is less than unity; and for heavier substances like 
lead and mercury it is greater than unity. 

38. Determination of Specific Weight by Experiment. — The 
specific weight of a body may be determined by first weighing it 



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PRESSURE OF WATER 26 

in air and again when immersed in water. The actual weight 
of the body in air is then 

W = yiV =- ysV 

where 8 denotes its specific weight, and its apparent weight T 
when immersed (Fig. 22) is 

T = W -B ^yaV -yV ^ yV(8 - 1), 
that is, 

T = 7V(s - 1). (20) 

Therefore, by division, 

W ^ 8 

T s- 1 
whence 

s = ^. (21) 

The specific weight of a body is therefore equal to its weight in 
air divided by its loss in weight when immersed in water. 

39. Application to Alloy. — If a body is an alloy or mixture of 
two different substances whose specific weights are known, the 
volume of each substance may be determined by weighing the 
body in air and in water. Thus let Vi denote the volmne, and 
«i the specific weight, of one substance, and Vs, 8% of the other. 
Then the weight of the body in air is . 

and its apparent weight T when immersed is, from Eq. (20), 

T =:77i(si - 1) + 772(«2 - 1). 

Solving these two equations simultaneously for V\ and V%, the 
result is 

T-(l^-)w 

Vl = 



T 

72 = - 



(r:-')^ 

-{^-y 



c^')• 



This method of determining relative volumes was invented by 
Archimedes in order to solve a practical problem. Hiero, King 



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26 



ELEMENTS OF HYDRAULICS 



of Syracuse, had furnished a quantity of gold to a goldsmith to 
be made into a crown. When the work was completed the crown 
was found to be of full weight, but it was suspected that the gold- 
smith had kept out a considerable amoimt of gold and substi- 
tuted an equal weight of silver. To test the truth of this sus- 
picion Archimedes first balanced the crown in air against an 
equal weight of gold, and then immersed both in water, when the 
gold was found to outweigh the crown, proving the goldsmith to 
be dishonest. 

40. Zero Buoyancy. — When a body lies flat against the bottom 
of a vessel filled with water, fitting the bottom so closely that no 

water can get under it, its buoy- 
ancy is zero. In this case if W 
denotes the weight of the body, 
A the area of its horizontal cross- 
section, and h the depth of water 
on it, the force T required to 
lift it is (Fig. 23) 

r = TT + yAh. 

That is to say, the force T is the 
same as would be necessary to lift 
the body itself and the entire col- 
FiQ. 23. umn of water vertically over it. 

This same principle underlies 
the action of a leather sucker or vacuum-tipped arrow, the fluid 
in that case being air. 






Vm. METACENTER 

41. Stability of Floating Body. — ^When a floating body is 
shoved to one side it remains in this position and is therefore in 
neutral equilibrium as regards lateral translation. In deter- 
mining the stability of a floating body it is therefore only neces- 
sary to consider its equilibrium as regards rotation. 

After a floating body has been tipped or rotated a small amoimt 
from its position of equilibrium, the buoyancy, in general, no 
longer passes through the center of gravity of the body. Conse- 
quently the weight and buoyancy together form a couple tending 
to produce rotation or tip the body. If this couple tends to 
right the body the equilibrium is stable, whereas if it tends to tip 
it over it is unstable. This evidently depends on the form of the 



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PRESSURE OF WATER 



27 



wetted surface, and abo on the form of the part immersed by the 
rotation. 

42. Metacenter. — For example, consider a floating box of 
rectangular cross-section, injmersed to a depth d below the 
surface A A (Fig. 24), and suppose it is tipped by an external 
couple until the water line becomes A'A\ In this new position 
the displacement is trapezoidal, and the .center of buoyancy B 
is the center of gravity of this trapezoid. But since the buoy- 
ancy is of the same amount as before the box was tipped and the 
triangle of immersion mno is equal to the triangle of emersion 
opq, the lines AA and A'A' intersect on the vertical axis CC. 
The intersection M of the line of action of the buoyancy with 
the vertical axis CC is called the metacenter. Evidently the loca- 
tion of the metacenter depends on the angle of tip and is different 




Fig. 24. 

for each position. It is also apparent that the equilibrimn is 
stable if the metacenter M lies above the center of gravity of 
the body, and unstable if M lies below 0. It is also shown in 
what follows that the metacenter moves higher as the angle of 
tip, a, increases. Its lowest position is called the true metacenter. 
43. Coordinates of Metacenter. — ^For the special case of the 
rectangular cross-section shown in Fig. 24, let x, y denote the 
coordinates of the center of gravity of the trapezoid, and a, 6, c 
the lengths of three sides (Fig. 26). Then from geometry, 
_ c(2a + b) __ a^ + ab + b^ 
^ ~ 8(a + 6) ' ^ ~ 3(a + 6) 

From Fig. 24 the sides a and b of the trapezoid expressed in terms 
of d and a are 

c c 

a ^ d " - tan a; 6 = d + 5 tan a. 



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28 ELEMENTS OF HYDRAULICS 

Inserting these values of a and b in the expressions for x and y, 
the result is 

c(3d - I tan a) Sd^ + ^ tan^ a 

^ " 6d ' ^ " 6d 

Also, from Fig. 24, the total height H of the metacenter above 
the bottom of the vessel or box is 



ff = y + (| - «j cot a. 




Hence by inserting the above values for x and y in this expression 
for H and reducing, we obtain the relation 

H = | + ^(tan*« + 2). (22) 

The height H therefore increases with a; that is, the greater the 

angle of tip, the higher the meta- 
center M. Moreover, by sub- 
stituting a = in Eq. (22) the 
position of the true metacenter, 
or limiting position of M, is found 
to be at a height H' above the 
bottom given by 

H' = | + & W 

To prevent a ship from capsizing, it is necessary to so design and 
load it that the height of its center of gravity above the bottom 
shall be less than H'. 

44. Metacentric Height. — To consider the general ease of 
equiUbrium of a floating body, take a vertical cross-section 
through the center of gravity of the body (Fig. 26), and suppose 
that by the application of an external couple it is slightly tipped 
or rotated about an axis OY, drawn through perpendicular 
to the plane of the paper. Then the volume displaced remains 
unchanged, but the center of buoyancy B is moved to some other 
point B\ To find the metacentric height Ay, or distance from the 
center of gravity G of the body to the metacenter 3f , let 

V = volume of liquid displaced; 

A = cross-sectional area of body in plane of flotation; 

6 = distance from center of gravity G to center of buoyancy B; 

ky = radius of gyration of area of flotation A about the axis OY. 



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PRESSURE OF WATER 29 

Then it can be shown that^ 

h, = ^*-b. (24) 

Similarly, for rotation about the axis OX, the metacentric height 
hs is given by 

h. * ^ - b, (26) 

where fc, denotes the radius of gyration of the area of flotation A 
about the axis OX, 

Evidently the metacentric height is greater for a displacement 
about the shorter principal axis of the section A. For instance, 
it is easier to make a ship roll than to cause it to tip endwise or 
pitch. 




Fig. 26. 

The locus of the centers of buoyancy for all possible displace- 
ments is called the surface of buoyancy, and the two metacenters 
given by Eqs. (24) and (25) are the centers of curvature of its 
principal sections. 

46. Period of Oscillation. — When a floating body is tipped and 
then released, it will oscillate, or roll, with a simple harmonic 
motion. To find the period of the oscillation, the general ex- 
pression for the period of oscillation of a solid body rotating 
about a fixed axis may be applied, namely,^ 



^^ym' 



(26) 



where P = period or time of a complete oscillation; 



1 Webster, " Dynamics of Particles," p. 474 (Teubner). 

* Sloctjm, "Theory and Practice of Mechanics," p. 302 (Holt & Co.). 



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30 ELEMENTS OF HYDRAULICS 

W = weight of the body; 
I = moment of inertia of the body with respect to the 

axis of rotation; 
h = distance from the center of gravity of the body to the 

axis of rotation. 

Since I = MK^, where M denotes the mass of the body and K 
its radius of gyration, and also W = Mg, Eq. (26) for the period 
may be written 

P = 2.JMK^=2^. (27) 

\Mgh Vgh 

46. Rolling and Pitching. — In the present case, consider rota- 
tion about the two principal axes OX and OF of the section A in 
the plane of floatation, and let X,, Ky denote the radii of gyration 
of the solid with respect to these axes, and Px, Py the correspond- 
ing periods, or times of performing a complete oscillation about 
these axes. Then from Eq. (27), 

p — ^Ka ^ p _ 2fjrKy 

VgK' * Vghy 
Substituting in these expressions the values of hx and hy given by 
Eqs. (24) and (26), they become 

1, /"7A M ' I, /"^A bT (28) 

For a body shaped like a ship, K and k increase together, and 
consequently the larger value of k corresponds to the smaller 
period P. A ship therefore pitches more rapidly than it rolls. 

For further applications of the metacenter the student is re- 
ferred to works on naval architecture. 



APPLICATIONS 

1. The ram of a hydraulic press is 10 in. in diameter and the 
plunger is 2 in. in diameter. If the plunger is operated by a 
handle having a leverage of 8 to 1, find the pressure exerted by 
the ram, neglecting friction, when a force of 150 lb. is applied to 
the handle. 

2. In a hydraulic press the diameter of the ram is 15 in. and of 
the plunger is % in. The coefficient of friction may be assumed 
as 0.12 and the width of the packing on ram and plunger is 0.2 



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PRESSURE OF WATER 



31 



of their respective diameters. What pressure will be exerted 
by the ram when a force of 200 lb. is applied to the plunger? 

3. Water in a pipe AB is to be kept at a constant pressure of 
1,200 lb. per square inch by forcmg in a plimger of diameter d 
(Fig. 27). This is operated by a piston of diameter D, whose 
lower surface is subjected to the pressure of a colunm of water 
75 ft. high. Find the ratio of the two diameters d and D. 

4. In a hydraulic pivot bearing, a vertical shaft carrymg a 
total load W is supported by hydraulic pressure (Fig. 28). The 
pivot is of diameter Z>, and is surrounded by a 17 leather packing 



B 
3 



jU 



— o — 




Fig. 27. 

of width c. Show that the frictional moment, or resistance to 
rotation, is given by the relation 

M = 2/icTr 

where /x denotes the coefficient of friction. 

6. For an ordinary flat pivot bearing of the same diameter D 
and for the same coefficient of friction /a as in the preceding prob- 
lem, the frictional moment is given by the relation^ 

M = HWDn. 

Show that the hydraulic pivot bearing is the more efficient of the 
two provided that 



c < 



D 
6" 



Calculate their relative efficiency when c = 0.2D. 

^ Slocttm, ''Theory and Practice of Mechanics," p. 194 (Holt). 



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ELEMENTS OF HYDRAULICS 



6. An instrument for measuring the depth of the sea consists 
of a strong steel flask, divided into two compartments which are 
connected by a valve. The upper compartment is filled with 
920 grams of distilled water and the lower compartment with 
mercury (Fig. 29). When lowered to the bottom, the outside 
pressure forces the sea water through a small opening in the side 
of the flask and thereby forces the mercury through the valve 
into the upper compartment. Assuming that the depth of the 
sea in certain parts of the Pacific ocean is 9,429 meters, and that 
the ratio of the densities of distilled and salt water is 35: 36, find 
how many grams of mercury enter the 
upper compartment.^ The modulus of 
compressibility of water is 0.000047, that 
is, an increase in pressure of one atmos- 
phere produces this decrease in volume. 

Note. — ^Assuming that a pressure of one at- 
mosphere is equal to a fresh-water head of 10J{ 
meters, the corresponding salt water head >- 10^ 
X 8^0 . 10.045 meters. At a depth of 9,429 

9 429 
meters the pressure is therefore = ^'q., « 938.7 




Fig. 29. 



atmospheres. 



7. A hydraulic jack has a 3-in. ram and 
a %-in. plimger. If the leverage of the handle is 10 : 1, find 
what force must be applied to the handle to lift a weight of 5 
tons, assuming the efliciency of the jack to be 76 per cent. 

8. A hydraulic intensifier is required to raise the pressure from 
600 lb. per square inch to 2,600 lb. per square inch with a stroke 
of 3 ft. and a capacity of 4 gal. Find the required diameters of 
the rams. 

9. In a hydraulic intensifier like that shown in Fig. 6, the 
diameters are 2 in., 6 in. and 8 in., respectively. If water is sup- 
plied to the large cylinder at a pressure of 500 lb. per square inch, 
find the pressure at the high-pressure outlet. 

10. How would the results of the preceding problem be modi- 
fied if the frictional resistance of the glands, or packing, is taken 
into account, assuming that the frictional resistance of one stuff- 
ing box is 0.05 pd, where p denotes the water pressure in pounds 
per square inch, and d is the diameter of the ram in inches? 

11. A hydraulic crane has a ram 10 in. in diameter and a 

iWiTTENBAUEB, "Aufgaben ausder Technischen Mechanik,'' Bd. III. 



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PRESSURE OF WATER 



33 



velocity ratio of 1 : 12, that is, the speed of the lift is twelve times 
the speed of the ram. Assuming the efficiency of the crane to be 
SO per cent., find what load it will lift with a water pressure of 
1,500 lb. per square inch. 

12. A hydraulic crane has a velocity ratio of 1:9 and is re- 
quired to lift a load of 4 tons. Find the required size of the 
ram for a pressure in the mains of 750 lb. per square inch, a loss 
of head due to friction of 75 lb. per square inch, and a mechanical 
efficiency of 70 per cent. 

13. How many foot-pounds of work can be stored up in a 
hydraulic accumulator having 



^P 



T 



T 



rr" 
-I i 



T\B 



CL. 



i- 



w 



._l_i— 



a ram 10 in. in diameter and 
a lift of 12 ft., with a water 
pressure of 800 lb. per square 
inch? 

14. Find the energy stored 
in an accumulator which has 
a ram 10 in. in diameter, 
loaded to a pressure of 1,000 
lb. per square inch, and hav- 
ing a stroke of 25 ft. If the 
full stroke is made in 1 min. 
find the horsepower available 
during this time. 

16. The stroke of a hy- 
draulic accumulator is fifteen 
times the diameter of the ram 
and the water pressure is 
1,200 lb. per square inch. 
Find the diameter of the ram for a capacity of 125 hp.-min. 

16. The ram of a hydraulic accumulator is 20 in. in diameter, 
the stroke 25 ft., and the water pressure, 1,050 lb. per square 
inch. If the work during one full downward stroke is utilized 
to operate a hydraulic crane which has an efficiency of 50 per 
cent, and a lift of 35 ft., find the load raised. 

17. An accumulator is balanced by means of a chain of length 
I passing over two pulleys A and B (Fig. 30) and carrying a coun- 
terweight W equal to the total weight of the chain. Find the 
distance apart of the pulleys and the required weight of chain per 
unit of length in order that this arrangement may balance the 
difference in pressure during motion. 

3 



Fig. 30. 



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34 



ELEMENTS OF HYDRAULICS 



Hint, — Let A denote the area of the ram and w the weight of the chain 
per unit of length. Then for the dimensions shown in the figure, we have 
the relations 

wx — wz ^ yACf 
x + y+z^h 
x + k ''b + Cf 
whence 

yAc 

18. A hydraulic accumulator has a ram 15 in. in diameter and 
carries a load of 60 tons. Assuming the total frictional resistance 

to be 3 tons, find the required 
water pressure when the load 
is being raised and when it is 
being lowered. 

19. Show that the depth of 
the center of pressure below 
the surface for a vertical rec- 
tangle of breadth b and depth 
d, with upper edge immersed 
to a depth hi and lower edge 
to a depth ht (Fig. 31) is 

hi* - hi\ 



jj_. 



Fig. 31. 



given by the expression 



-<i^^ 




Fig. 32. 



20. Show that the center of pressure for a vertical plane tri- 
angle with base horizontal and vertex at a distance hi below the^ 
surface (Fig. 32) is given by the expression 



= H( 



3/i2^ -f- 2hih2 + hi\ 



2h2 + hi / 
21. From the results of the preceding problem show that if the 



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PRESSURE OF WAJER 



35 



vertex of the triangle lies in the surface, the depth of the center of 
pressure is 

Xc = %d, 

and if the base of the triangle lies in the surface 

Xc = Hd. 

22. Show that the depth of the center of pressure below the sur- 
face for a vertical circular area of radius r, immersed so that its 
center lies at a depth h below the surface is given by 

23. A circular opening, 2 ft. in diameter in the vertical side of 
a tank is closed by a circular cover held on by two bolts, one 14 
in. above the center of the 
cover and the other 14 in. 
below its center. When 
water stands in the tank at a 
level of 20 ft. above the center 
of the opening, find the stress 
in each bolt. 

24. A pipe of 4 ft. inside 
diameter flows just full, and 
is closed by a valve in the 
form of a flat circular plate 
balanced on a horizontal axis. At what distance from the cen- . 
ter should the axis be placed in order that the valve may bal- 
ance about it? 

26. An automatic movable flood dam, or flashboard, is made of 
timber and pivoted to a back stay at a certain point C, as shown 
in Fig. 33. The point C is so located that the dam is stable pro- 
vided the water does not rise above a certain point A, but when 
it rises above this point the dam automatically tips over. Deter- 
mine where the point C should be located. 

26. An opening in a reservoir wall is closed by a plate 2}4 ft. 
square, hinged at the upper edge, and inclined at 60^ to the hori- 
zontal. The plate wei^ 250 lb., and is raised by a vertical 
chain attached to the middle point of its lower edge. If the 
center of the plate is 15 ft. below the surface, find the pull on the 
chain required to open it. 

27. A rectangular cast-iron sluice gate in the bottom of a 




Fio. 33. 



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36 



ELEMENTS OF HYDRAULICS 



dam is 3 ft. high, 4 ft. wide and 3 in. thick. The head of water 
on the center of the gate is 35 ft. Assuming the coefficient of 
friction of the gate on the slides to be H> ai^d that there is no 
water on the lower side of the gate, find the force required to lift 
it. Weight of cast iron is 450 lb. per cubic foot. 

28. Flow from a reservoir into a pipe is shut off by a flap valve, 
as shown in Fig. 34. The pivot A is so placed that the weight of 
the valve and arm balance about this point. Calculate the pull P 
in the chain required to open the valve for the dimensions given 
in the figure and a head of 16 ft. on the center of the valve. 

29. The waste gate of a power canal is 8 ft. high and 5 ft. wide, 
and when closed there is a head of 10 ft. of water on its center. 

If the gate weighs 1,000 lb. 
and the coefficient of friction 
between gate and seat is 0.4, 
find the force required to 
raise it. 

30. A lock gate is 30 ft. 
wide and the depth of water 
on the two sides is 28 ft. and 
14 ft. respectively. Find the 
total pressure on the gate 
and its point of application. 

31. A lock is 20 ft. wide 
and is closed by two gates, each 10 ft. wide. If the depth of 
water on the two sides is 16 ft. and 4 ft. respectively, find the 
resultant pressure on each gate and its point of application. 

32. A dry dock is 60 ft. wide at water level and 52 ft. wide at 
floor, which is 40 ft. below water level. The side walls have a 
straight batter. Find the total pressure on the gates and its 
point of application when the gates are closed and the dock empty. 

33. A concrete dam is 6 ft. thick at the bottom, 2 ft. thick at 
top and 20 ft. high. The inside face is vertical and the outside 
face has a straight batter. How high may the water rise without 
causing the resultant pressure on the base to pass more than 
6 in. outside the center of the base? 

Note.— A dam may fail either by sliding or by overturning. In general, 
however, if a well-laid masonry dam is stable against overturning, it will 
not fail by sliding on a horizontal joint. This kind of failure could occur 
only when the shearing stress at any joint exceeded the joint friction. 
Ordinarily the resultant pressure on any joint makes only a small angle with 




Fig. 34. 



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PkESSUkE OP WAtER 



37 



a I^^rjieiidiculai' to its plane," and since the angle of repose for masonry is 
largei failure by shear of this kind is not likely to occur. As a criterion 
against failure by shear^ it may be assumed that when the resultant pressure 
on any Joint makes an angle less than 30** with the normal to the joints it 
is safe against sliding at that joint. 

To guard against sUding on the base an anchorage should be provided 
by cutting steps or trenches in the foundation if it is of rock, or in the case 
of clay and similar material, by making the dam so massive that the 
angle which the resultant pressure on the base makes with the vertical is 
well within the angle of friction. Usually if the dam is heavy enough to 
satisfy the condition for stability against overturning, as explained belowi, 
it will also be safe against sliding on the base. 

In order for a dam to fail by overturning, one or more joints must open 
at the face, in which case this edge of the joint must be in tension. Al- 
though a well-laid masonry joint has considerable tensile strength, it is 
customary to disregard this entirely in designing, in which case the condi- 
tion necessary to assure stability against overturning is that every joint 
shall be subjected to compressive stress only. 






Fia. 35. 

Assuming a H^mmx distribution of pressure over the joint, as indicated by 
the trapezoid ABCD in Fig. 35, the resultant pressure R must pass through 
the eeater of gravity of this trapezoid. Consequently when the compres- 
sion at one face, Z>, becomes zero, as indicated in Fig. 35 the trapezoid be- 
comes a trian^^e, and the resultant is applied at a distance » from the 

opposite face C, where h denotes the width of the joint. Moreover, the 
resultant cannot approach any nearer to C without producing tensile stress 
at Z> as indicated in Fig. 35. For stability against overturning, therefore, 
the resultant pressure must ciU the base (or joint) within the middle third. 

If water is allowed to seep under a dam, it will exert a lifting effort equal 
to the weight of a column of water of height equal to the static head at 
fbhts point. To assure stability it is, therefore, essential to prevent seepage 
by means of a cutoff wall, as indicated in Figs. 36 and 37. In investigating 
the stabiUty of a dam, however, the best practice allows for accidental seep- 
age by making allowance for an upward pressure on the base due to a 
hydrostatic head of two-thirds the actual depth of water back of the dam. 

34. Figure 36 shows a typical section of the Kensico Dam, 
forming part of the Catskill Water System of the City of New 



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ELEMENTS OF HYDRAULICS 



amS. 







UM 



//Atf/W f/J^jJ 






WASTE WEIR 

r^-r^"-^ — c— ^^ MAXIMUM SECTJON 












u_ 



J 3^-3"* 



KENSICO DAM 



Fig. 36. 



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PRESSURE OF WATER 



39 



York. The Kensico Reservoir covers 2,218 acres, with a shore 
line 40 miles in length, and has a storage capacity of 38,000,000,- 
000 gal. The dimensions of the main dam are length 1,843 ft.; 
height 300 ft.; thickness at base 230 ft.; thickness at top 28 ft. 

Investigate the stability of this dam in accordance with the 
conditions stated in the note to Problem 33. 

36, Figure 37 shows a section of the Olive Bridge Dam and 
typical dyke section of the Ashokan Reservoir, which forms part 



a.^ft 



f /tag /tihf-^^ I 




TYPICAL SECTION OF DIKE 

Cats^U aqueduct system. 



i-*:ra 




OLIVE BRIDGE DAM 

MAXIMUM MASONRY SECTION 



Fig. 37. 



of the Catskill Water System of the City of New York. This 
reservoir covers 8,180 acres, with a shore line 40 miles in length 
and a storage capacity of 132,000,000,000 gal. The principal 
dimensions of the main dam are, length 4,650 ft.; height 220 ft.; 
thickness at base 190 ft.; thickness at top 23 ft. 

Investigate the stability of this dam as in the preceding prob- 
lem. 

36. In the $25,000,000 hydraulic-power development on the 
Mississippi River at Keokuk, Iowa, the dam proper is 4,650 ft. 
long, with a spillway length of 4,278 ft. The power plant is 
designed for an ultimate development of 300,000 hp., and consists 



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40 



ELEMENTS OF HYDRAULICS 



of vertical shaft turbines and generators in units of 10,000 hp. 
each. Transmission lines convey the current at 110,000 volts to 
St. Louis, 137 miles distant, and to other points.^ 

A notable feature of the plant is the ship lock which is of un- 
usual size for river navigation, the lock chamber being 400 ft. long 
by 110 ft. wide with a single Uft of from 30 to 40 ft., the total 
water content of the lock when full being about 2,200,000 cu. ft. 
The locks at Panama are the same width but the maximum lift 
on the Isthmus is 32 ft., the average lift being about 28 ft. Find 
the maximmn pressure on the lock gates at Keokuk and its point 
of application. (See frontispiece.) 






Oage L^mp 






^......-.,....„^,_..5^;^?_ 






liMM 




Fig. 38. 

37. The side walls of the Keokuk lock are monoUthic masses 
of concrete, with a base width of 33 ft., a top width of 8 ft., and 
an outside batter of 1: 1.5, as shown in Fig. 38. If the water 
stands 48 ft. above the floor of the lock on the inside and 8 ft. 
on the outside, find the point where the resultant pressure on the 
side walls intersects the base, neglecting the weight of the road- 
way on top and the arches which support it. 

38. The lower lock gates at Keokuk are of the mitering type, 
as shown in Figs. 39 and 40, and are very similar to those in 
the Panama canal locks. The gates are 49 ft. high and each 
leaf consists of 13 horizontal ribs curved to a radius of 66 ft. 4^^ 

1 Eng, News, Sept. 28, 1911. 



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PRESSURE OF WATER 



41 



in. on the center line, framed together at the ends of the quoin 
and miter posts, and also having seven lines of intermediate 
framing. The chord length over the posts is 66 ft. 4^ in. and 
the rise of the curve is 10 ft. 8J^ in.^ 

Each leaf contains a buoyancy chamber to relieve the weight 
on the top hinge. This consists of a tank of about 3,840 cu. ft. 
capacity, placed between the curve of the face and the chord line 
of the bracing. The total weight of each gate in air is about 240 
tons. Find how much the buoyancy chamber reUeves the weight 
on the top hinge. 




Fig. 39; 

39. The upper gates of the Keokuk lock are of a floating type 
never before used, and consist essentially of floating tanks moving 
in vertical guides and sinking below the level of the sill (Fig. 41). 
To close the lock, compressed air is admitted to an open-bottom 
chamber in the gate, which forces out the water and causes the 
gate to rise. To open the lock, the air in this chamber is al- 
lowed to escape, when the weight of the gate sinks it to its lower 
position. 

The flotation of the gate is controlled by two closed displace- 
ment chambers, one at each end, and one open buoyancy cham- 
ber. Each of the former is 42 ft. long, 4 ft. deep and 16 ft. wide. 
The buoyancy chamber is 2}^ ft. high beneath the displacement 
chambers and 63^ ft. high in the 28-ft. space between them, its 
capacity being 6,000 cu. ft. 

» Eng. News, Nov. 13, 1913. 



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42 



ELEMENTS OF HYDRAULICS 



With the gate floating and its bottom just clear of the sill, the 
weight of the part above water is 190 tons, which is increased by 
the ballast in the displacement chambers to 210 tons. The dis- 



^-rl?. 




^^ 






Section of 6fl+fi ^ 
Ci-rs+cr(E-F) 



Sectional Plan C-D 



Fig, 40. — Mitering gate, Keokuk lock. 

placement of the submerged part of the gate is 12 tons so that the 
buoyant efifort required is 198 tons. 
Find the equivalent displacement in cubic feet, from this result 



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PRESSURE OF WATER 



43 




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44 



ELEMENTS OF HYDRAULICS 



subtract the volume of the displacement chambers, and then find 
the required air pressure in the buoyancy chamber. 

In raising the gate it is actually found that this pressure varies 
from 2 lb. per square inch to as high as 12 lb. per square inch 
when the gate is leaving its lower seat. 

40. A gas tank is fitted with a mercury gage as shown in Fig. 
19. The height h of the mercury column is 20 in. Find the 
excess of pressure in the tank above atmospheric. 

41. A piece of lead weighs 20 lb. in air. What will be its ap- 
parent weight when suspended in water, assuming the specific 
weight of lead to be 11.4? 

42. A pail of water is placed on a platform scales and found to 
weigh 12 lb. A 6-lb. iron weight is then suspended by a Ught 
cord from a spring balance and lowered into the water in a pail 




Fig. 42. 

until completely immersed. Find the reading on the spring 
balance and on the platform scales. 

43* A brass casting' (alloy of copper and zinc) weighs 200 lb. 
in air and 175 lb. in water. If the specific weight of copper is 
8.8 and of zinc is 7, how many pounds of each metal does the 
casting contain? 

44. One end of a wooden pole 12 ft. long, floats on the water and 
the other end rests on a wall so that 2 ft. project inward beyond 
the point of support (Fig. 42). If the point of support is 18 in. 
above the water surface, find how much of the pole is immersed. 

46. A floating platform is constructed of two square wooden 
beams each 16 ft. long, one 18 in. square and the other 1 ft. 
square. On these is laid a platform of 2-in. plank, 10 ft. wide. 
Find where a man weighing 160 lb. must stand on the platform 
to make it float level, and how high its surface will then be above 



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PRESSURE OF WATER 



45 



the water (Fig. 43). The weight of timber may be assumed 
as 50 lb. per cubic foot. 

46, A piece of timber 4 ft. long and 4 in. square has a weight 
W attached to its lower end so that it floats in water at an angle 
of45MFig. 44). FindTT. 

47. A rectangular wooden barge is 30 ft. long, 12 ft. wide and 
4 ft. deepi outside measurement, and is sheathed with plank 




Fig. 43. 



3 in. thick, the frame weighing half as much as the planking. 
Find the position of the water line when the barge floats empty, 
and also the load in tons it carries when the water line is 1 ft. 
from the top. Assume the weight of wood as 50 lb. per cubic 
foot. 




Fig. 44 

48. A prismatic wooden beam 10 ft. long, 1 ft. wide and 6 in. 
thick floats flat on the water with 4 in. submerged and 2 in. above 
water. Find its specific weight. 

49. A dipper dredge weighs 1,200 tons and floats on an even 
keel with bucket extended and empty. When the bucket 
carries a load of 3 tons at a distance of 50 ft. from the center 
line of the scow, a plumb line 15 ft. long, suspended from a vertical 
mast, swings out 5 in. Find the metacentric height. 



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46 



ELEMENTS OF HYDRAULICS 



NoTB. — Let W denote the weight of the veBsd and O its center of gravity 

(Fig. 45). When an eccentric load P is added at a distance d from OM^ 

the center of gravity moves to G\ where for a small displacement: 

p 
GG' a i^y approximately. 

If B denotes the angle of heel, then also 
QQ' = GAT tan ^, 



whence 



GM 



GG' Pd 



tan $ W 



cot e. 




50. A steamer is of 14,000 tons displacement. When its life 
boats on one side are filled with water, a plmnb line 20 ft. long 
suspended from a mast is found to swing out 9}4 "!• If the 
total weight of water in the boats is 75 tons and their distance 
from the center line of the vessel is 26 ft., find the metacentric 
height and period with which the ship will roll. 



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SECTION 2 



FLOW OF WATER 

IX. FLOW OF WATER FROM RESERVOIRS AND TANKS 

47. Stream Line, — In the case of a flowing liquid, the path fol- 
lowed by any particle of the liquid in its course is called a stream 
line. In particular, if a reservoir or tank is filled with water and 
a small opening is made in one side at a depth h below the surface, 
the water flows out with a certain velocity depending on the 
depth, or head, h. Since the par- 
ticles of water flowing out converge 
at the opening, the stream lines 
inside the vessel are, in general, 
comparatively far apart, but be- 
come crowded more closely to- 
gether at the orifice. 

48. Liquid Vein. — Under the 
conditions just considered, suppose 
that a closed curve is drawn in any 
horizontal cross-section of the vesr 
sel and through each point of the 
closed curve draw a stream line. 
The totality of all these stream 
lines will then form a tube, called 
a liquid vein (Fig. 46). From the 
definition of a stream line it is 
evident that the flow through such a tube or vein is the same 
as though it were an actual material tube. In particular, the 
same amount of liquid will flow through each cross-section of 
the vein and therefore the velocity of flow will be greatest 
where the cross-section of the vein is least, and vice versa, 

49. Ideal Velocity Head. — ^In any particular vein let v denote 
the velocity of flow at a distance h below the surface, and Q the 
quantity of water per second flowing through a cross-section of 
the vein at this depth. Then the weight of water flowing through 
the cross-section per second is yQ and its potential energy at the 

47 




¥io. 46. 



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48 ELEMENTS OF HYDRAULICS 

height h is yQh. The kinetic energy of this quantity of water 

flowing at the velocity v is -^— . Therefore, by equating the 

potential energy lost to the kinetic energy gained and neglecting 
all frictional and other losses we have 

whence 

V = V^. (29) 

This relation may also be written in the form 

The quantity h is therefore called the ideal velocity head, since 
it is the theoretical head required to produce a velocity of flow v. 

60. TorriceUi's Theorem. — The relation 

V = y/2gh 

is known as TorricelWs Theorem. Expressed in words, it says 
that the ideal velocity of flow under a static head h is the same 
as would be acquired by a solid body faUing in a vacuum from 
a height equal to the depth of the opening below the free surface 
of the liquid. 

61. Actual Velocity of Flow. — The viscosity of the liquid, as 
well as the form and dimensions of the opening, have an important 
effect in modifying the discharge. 

Considering viscosity first, its effect is to reduce the velocity 
of the issuing liquid below the ideal velocity given by the relation 
V = y/2gh. It is therefore necessary to modify this relation so as 
to conform to experiment by introducing an empirical constant 
called a velocity coefficient. Denoting this coefficient by C„, the 
expression for the velocity becomes 

V = C.\/2^. (30) 

For water the value of the velocity (or viscosity) coefficient for an 
orifice or a nozzle is approximately Cv = 0.97. 

62. Contraction Coefficient. — In the case of flow through an 
orifice or over a weir, the obUque pressure of the water approach- 
ing from various directions causes a contraction of the jet or 
stream so that the cross-section of the jet just outside the orifice 



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FLOW OF WATER 49 

is somewhat less than the area of the opening. Consequently 
the discharge is also less than it would be if the jet were the full 
size of the opening. 

If the area of the orifice is denoted by A, the area of the jet at 
the contracted section will be some fraction of this amoimt, say 
CcA, where Ce is an empirical constant called a contraction coeffi- 
cient, which must be determined experimentally for openings of 
various forms and dimensions. 

63. EflSiux CoeflSicient. — ^Taking into account both the viscosity 
of the liquid and the contraction of the jet, the formula for dis- 
charge becomes 

Q = actual velocity X area of jet 
= (C,V2^) X (CcA) 
= CvCcA\/2gh, 

where A denotes the area of the orifice. Since there is no object 
in determining C» and Ce separately, they are usually replaced by 
a single empirical constant K = CvCc, called the coefficient of 
efflux, or discharge. In general, therefore, the formula for the 
actual discharge becomes 

Q = KAV2^. (31) 

64. Effective Head. — The head h may be the actual head of 
water on the orifice; or if the vessel is closed and the pressure is 
produced by steam oi: compressed air, the effective head is the 
height to which the given pressure would sustain a colunm of 
water. 

The height of the equivalent water column corresponding to 
any given pressure may be determined by calculating the weight 
of a colunm of water 1 ft. high and 1 sq. in. in section, from which 
it is found that 

1 ft. head =: 0.434 Ib./in.^ pressure, 

and conversely, 

1 Ib./in.^ pressure = 2.304 ft. head. 

For an orifice in the bottom of a vessel, the head A is of course 
the same at every point of the opening, but if the orifice is in the 
side of the vessel, the head h varies with the depth. However, if 
the depth of the opening is small in comparison with h, as is frer 
quently the case, the head may be assiuned to be constant over 

4 



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50 



ELEMENT S.OF HYDRAULICS 



the entire orifice and equal to the distance of its center of gravity 
from the free surface of the liquid (Fig. 47(a)). 

If an orifice is entirely submerged, as shown in Fig. 47(6), the 
effective head on it is the difference in level between the water 
surfaces on the two sides of the opening. 








V 


f 
k 


\f 


L _. 


y r^j=^z ^ — 


•■S-. -.^ 



Fig. 47. 



(t>l 



66. Discharge from Large Rectangular Orifice. — In the case 
of a comparatively large orifice, the effective head is not the depth 
of itfi center of gravity below the surface, and the discharge must 
be determined in a different manner. 

To illustrate the method of procedure consider the particular 
case of an orifice in the form of a rectangle of breadth &, the upper 




Fig. 48. 



and lower edges being horizontal and at depths of h and H re- 
spectively below the surface, as shown in Fig. 48. Let this rec- 
tangle be divided up into narrow horizontal strips, each of breadth 
h and depth dy. Then the ideal velocity of flow in any one of 
these strips at a distance y below the surface is r = \/2gfj/, and 



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FLOW OF WATER 61 

since its area is bdy, the ideal discharge dQ through this elemen* 
tary area per second is 

dQ = bdyV2gy. 

The total discharge per second, Q, through the entire orifice is 
therefore 



Q = Kb jT^V^dy = %KbV2g(H'^ - h^). 



(32) 



This expression inay also be written in the form 

Q = %KbHV2gH - %KbhV2^ 

which makes it easier to remember from analogy with the weir 
formula which follows. 

66. Discharge of a Rectangular Notch Weir. — ^If the upper 
edge of the rectangular orifice just considered coincides with the 
water surface, the opening is called a rectangular notch weir. In 
this case A = and the preceding formula for discharge becomes 

Q = %KbH\/2gH = %KAV2gH (33) 

where H denotes the head on the crest of the weir, and A is the 
area of that part of the opening 
which lies below the surface. 



X. DISCHARGE THROUGH SHARP- tZT-I 
EDGED ORIFICE 




67. Contraction of Jet. — ^In con- 
sidering the flow of water through 
an orifice it is assumed in what 
follows that a sharp-edged orifice 
is meant, that is, one in which the 
jet is in contact with the wall of - 
the vessel along a Une only (Fig. 
49). When this is not the case, 
the opening is called an adjutage 

or mouthpiece, and the flow is modified, owing to various causes, 
as explained in Par. 75. 

The value of the constant K in Eq. 33 depends on the form 
of the opening and also on the nature of the contraction of the 
jet. The contraction is said to be complete when it takes place 
on all sides of the jet; that is to say, when the size of the opening 
is small in comparison with its distance from the sides and botr 



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52 ELEMENTS OF HYDRAULICS 

torn of the vessel and from the water surface. The contraction 
is called incomplete when one or more of the edges of the orifice is 
continuous with the sides of the vessel. 

68. Complete Contraction. — ^For a sharp-edged orifice with 
complete contraction the mean value of the efflux coefficient K is 

K = 0.62. 

The actual value of this coefficient varies slightly with the size 
of the orifice and effective head on it. The value given, however, 
is sufficiently accurate for all ordinary practical calculations. 
More exact values are given in Tables 9 and 10. 

69. Partial Contraction. — ^In the case of incomplete contraction, 
let P denote the entire perimeter of the orifice, and nP that frac- 
tion of the perimeter which experiences no contraction. Then 
denoting the coefficient of efflux by Ki, its value as determined by 
experiment for sharp-edged orifices is as follows: • 

Rectangular orifice, Ki = K(l + 0.16n) \ (34) 

Circular orifice, Ki = K(l + O.lSn) j 

Assmning K = 0.62, the following table gives the corresponding 
values of Ki as determined from these relations.^ 





n = i 


n = i 


n = i 


Re^rtaninilar orifice 


Ki =0.643 


Ki =0.667 


Ki =0.690 




Circular orifice 


Ki =0.640 


Ki =0.660 


Ki =0.680 



60. Velocity of Approach. — So far it has been assumed that the 
effective head h in the formula for discharge through an orifice, 
namely, 

Q = KAV2gh, 

is simply the static head, measured from the center of the orifice, 
if it is small, to the surface level. If the velocity of approach is 
considerable, however, the velocity head must also be included 
in the effective head. Thus let 

A = area of orifice; 

A' = cross-section of channel of approach; 

V = ideal velocity corresponding to the total head H; 

V = velocity of approach; 

h' = velocity head = ^; 
1 Lauenstoin, " Mechanik," p. 173. 



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FLOW OF WATER 63 

h = static head; 

H = effective head = -s— 
2g 

Sin^e the total flow through the channel of approach must equal 
the discharge through the orifice, we have 

A'v = Q = KAV 
whence 

KAV 

Also the effective head H = h + h', or, expressed in terms of the 
velocities, 

Substituting v = ., in this relation, it becomes] 



whence 



2g "+ 2gA'^ 



^-4: 



2gh 



The expression for the discharge Q is then 



^ = ^^ = ^^/FwAv• (36) 



K« 



m 



From this relation it is evident that if the area A of the orifice is 
small in comparison with the cross-section A' of the channel, say 
A' not less than fifteen times A, the error due to neglecting the 

velocity of approach will be negligible; that is, the term K^ \-j-,) 

in Eq. (35) may be neglected, in which case the formula for 
the discharge simplifies into the original expression given by 

Eq. (33), namely, 

Q = KAV2gh. 

XI. RECTANGULAR NOTCH WEIRS 

61, Contracted Weir. — ^The most common type of weir consists 
of a rectangular notch cut in the upper edge of a vertical wall, 



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54 



ELEMENTS OF HYDRAULICS 



and is called a contracted weir (Fig. 50(a)). In order that the con- 
traction shall be complete, there should be a clearance of not less 
than ih from the sides of the notch to the sides of the channel, 
and from the bottom of the notch (called the crest of the weir) 
to the bottom of the channel. 

62. Suppressed Weir. — ^If the sides of the notch are continuous 
with the sides of the channel, it is called a suppressed weir (Fig. 
50(6)). 

For both types of weir it has been found by experiment that 
the velocity of approach may be neglected when the product bh 
is less than one-sixth Ihe cross-section of the channel. For a 
suppressed weir this is equivalent to saying that the height of 
the weir crest above the bottom should be at least five times the 
head on the weir. 





Fia. 50. 



63. Submerged Weir. — ^When the water level on the down- 
stream side of a weir is higher than the crest of the weir, the latter 
is called a submerged weir (Fig. 51(a)). At present the sub- 
merged weir is seldom used for measuring flow because of the 
lack of reliable data from which to determine the experimental 
constants involved. 

The principal experiments on submerged weirs are six made by 
Francis in 1848; 22 made by Fteley and Steams in 1877; and a 
more extensive set made by Bazin about 1897. 

From the experiments made by Francis and by themselves, 
Fteley and Steams derived the following formula for sharp- 
crested submerged weirs having no end contractions: 



Q = kl(h + |')Vh^=^ 



(36) 



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FLOW OF WATER 



55 



where b = breadth of weir (Fig. 51(a)); 

h = depth of water on upstream side above level of crest; 
ft' = depth of water on downstream side above level of 
crest; 

k = empirical coefficient depending on the ratio -r- 

Values of the coefficient k are given in Table 22. 




Trianatilar Weir 
ib) 

Fia. 51. 

From a later study of the experiments made by Francis and by 
Fteley and Steams, Clemens Herschel proposed the following 
simple formula for sharp-crested submerged weirs: 

I, Q = 3.33b(nh)^, (37) 

where n is an empirical constant depending on the ratio -j-' 

Values of the coefficient n are given in Table 22. 

The formula proposed by Bazin for sharp-crested submerged 
weirs was as follows: 

Q = m (l.05 + 0.21^') (^^^) \hV2gh, (38) 



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56 



ELEMENTS OF HYDRAULICS 



where d denotes the height of the crest of the weir above the bot- 
tom of the channel, and m is the coefficient for a similar sharp- 
crested weir with free overfall and full crest contraction, having 
the same values of h and d. 

64. Triangular Notch Weir. — ^This form of weir is best adapted 
to the calculation of small discharge. To obtain the formula for 
discharge the notation indicated in Fig. 51 (6) will be used. Then 
the element of area may be taken as xdy, the theoretical velo- 
city of flow through this area is >/2gy, and consequently the 
actual total discharge is 

Q = c ( xdyV2gy, 

where c denotes the contraction coefficient to be determined by 
experiment. By similar triangles we have x = jih — y). Sub- 
stituting this value in the expression under the integral sign and 
performing the integration, the result is 

Q = c ^5& V2g ft^ 

Denoting the constant part of this expression by a single letter 
fc, the formula becomes simply 

Q = kh^. (39) 

The following values of the coefficient k for use in this formula 
were obtained experimentally by Professor James Thompson 
of Glasgow in 1860: 

Thompson's Values op Coeppicibnt k in Formula Q = kh!^ for 90° 
Triangular Weir 



Head h in inches 






measured from vertex 


Discharge Q in 


Coefficient k 


of notch to still-water 


cu. ft./min. 


surface of pond 






2 


1.748 


0.3088 


3 


4.780 


0.3067 


4 


9.819 


0.3068 


5 


17.07 


0.3053 


6 


26.87 


0.3048 


7 


39.69 


0.3061 



More recently an accurate series of experiments on triangular 
notch weirs of 54° and 90° angle has been made by James Barr^ 
^London Engineering^ 1910, pp. 435, 473. 



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FLOW OF WATER 



57 



at the Watt Engineering Laboratories of the University of Glas- 
gow, with the following results, among others: 

Barb's Values of Cobfficibnt k in Fobmxjla Q = hl^ for 90® 
Triangular Weir 



Head h in inches 








measured from 








vertex of notch 


Discharge Q 


Discharge Q 


Coefficient Jk 


to still-water 


in cu. ft./min. 


in gal./min. 




surface of 








pond 








2 


1.755 


10.94 


0.3104 


2i 


3.045 


18.97 


0.3084 


3 


4.782 


29.79 


0.3068 


3J 


7.002 


43.63 


0.3057 


4 


9.75 


60.74 


0.3047 


4i 


13.05 


81.29 


0.3038 


5 


16.95 


105.6 


0.3032 


5i 


21.46 


133.7 


0.3026 


6 


26.63 


166.0 


0.3021 


6i 


32.49 


202.4 


0.3017 


7 


39.05 


243.0 


0.3013 


7i 


46.34 


288.7 


0.3009 


8 


54.06 


339.9 


0.3006 


8i 


62.92 


392.0 


0.3003 


9 


72.90 


454.2 


0.3000 


9J 


83.33 


519.2 


0.2998 


10 


94.70 


590.0 


0.2995 



The results obtained by using a 54° triangular weir, as well as 
other results, are summarized in graphical form in the paper 
mentioned. 

65. Trapezoidal or Cippoletti Weir. — A form of weir frequently 
used in irrigation practice is the trapezoidal weir, shown in Fig. 




Fia. 52. — Cippoletti weir. 

52. It is evident that the discharge from a trapezoidal weir 
may be computed by considering it as equivalent to a rectangular 
suppressed weir and a triangular weir. Denoting the slopes of 



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58 



ELEMENTS OF HYDRAULICS 






I 






s 



I 



•s ^ 

i i 



L 

p 

s2 






O w 

1° 

Q 



I 



H 



Us 

+ 

Si 

CO 



CO 
CO 
CO 

II 



I 
' + 

^^ — ^^ 

CO ^ 
CO 



^ 
^ 



CO 



S • ^ 

JH ^ O 

S o "** 

--; I-* «o 






CO 
CO 



s 

d 



I 

CO 
CO 







s 



! 

II 



d 
II 



si 

lo CO ^* 






? 



61 

II u^ 



o 



> 



S co' 






n 



« 



,2 



+ 1 



?■ 



II 



ss 






I 

d 

Is 



O 



a « 



^ 



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FLOW OF WATER 59 

the sides or ends by « so that the top width is b + 2$/i} and 
neglecting end contractions, the discharge from a trapezoidal 
weir will therefore be 

The total discharge is therefore greater, in general, than from a 
rectangular contracted weir of width h. An Italian engineer 
named Cippoletti proposed giving the sides such a slope that this 
increase would just equal the decrease in discharge through a con- 
tracted weir due to end contractions, and found that a slope of 
1:4 would accomplish this result. A trapezoidal weir of side 
slope 1 horizontal to 4 vertical is therefore called a Cippoletti 
weir, and the formula proposed by Cippoletti for calculating its 
discharge is 

= 3.367bh^. (40) 

In using this type of weir the effect of end contractions may thus 
be neglected altogether, which makes it especially convenient 
for use under varying heads. 

66. Formulas for Rectangular Notch Weirs. — ^Numerous ex- 
periments have been made on the flow of water over weirs for 
the purpose of deriving an empirical formula for the discharge. 
The most important of these results, including the formulas in 
common use, are tabulated on page 58. Although these for- 
mulas apparently differ somewhat in form, they are f oimd irf prac- 
tice to give results which agree very closely, 

A rational formula for the discharge over a rectangular weir 
was derived in Par. 56 as expressed by Eq. (33), namely, 

Q = %KA\/2gh. 

For a sharp-edged opening the mean value of the efflux coefficient 
is X = 0.62, as stated in Par. 58. In the present case, therefore, 
KA = 0.626A, and if h and h are expressed in feet, the above 
formula becomes 

Q = %(0.62bh)v^ih; 
" s 3.3bh^ cu. ft per sec. (41) 

It is often convenient to express h and h in inches, and the dis- 
charge Q in cubic feet per minute. Expressed in these units, the 
formula becomes 



Q = %(0.62^)^^X60. 



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60 



ELEMENTS OF HYDRAULICS 



or, reducing and simplifjdng, 

Q = OAhh^ cu. ft per min. (42) 

where b and h are both expressed in inches. These formulas are 
the basis of many of the weir tables used in practical work, such 
as Tables 18 and 19 in this book. 

Xn. STANDARD WEIR MEASUREMENTS 

6T. Constructioa of Weir. — ^From the experiments summarized 
in the preceding article it was found that any empirical weir 
formula could only be relied upon to give accurate results when 




Fia. 53. 

the conditions under which the measurement was made were 
approximately the same as those imder which the formula was 
deduced. To obtain accurate results from weir measurements 
it is therefore customary to construct the weir according to cer- 
tain standard specifications, as follows: 

1. A rectangular notch weir is constructed with its edges 
sharply beveled toward the intake, as shown in Fig. 53. The 
bottom of the notch, called the crest of the weir, must be per- 
fectly level and the sides vertical. 



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FLOW OF WATER 



61 



2. The length, or width, of the weir should be between four 
and eight times the depth of water flowing over the crest of the 
weir. 

3. The channel or pond back of the weir should be at least 
40 per cent, wider than the notch, and of sufficient depth so that 
the velocity of approach shall not be 
over 1 ft. per second. In general it is 
sufficient if the area feft is not over one- 
sixth the area of the channel section 
where fe denotes the width of the 
notch and h the head of water on the 
crest. 

4. To make the end contractions 
complete there must be a clearance 
of from 2ft to 3ft between each side of 
the notch and the corresponding side 
of the channel. 

5. The head ft must be accurately 
measured. This is usually accom- 
plished by means of an instrument 
called B, hook gage (Fig. 54), located as 
explained below. For rough work, 
however, the head may be measured 
by a graduated rod or scale, set back 
of the weir at a distance not less than 
the length of the notch, with its zero 
on a level with the crest of the weir 
(Fig. 53). 

68. Hook Gage. — ^As usually con- 
structed, the hook gage consists of a 
wooden or metal frame carrying in a 
groove a metallic sliding scale gradu- 
ated to feet and hundredths, which is 
raised and lowered by means of a 
milled head nut at the top (Fig. 54). 
By means of a vernier attached to the frame, the scale may be 
read to thousandths of a foot. The lower end of the frame carries 
a sharp-pointed brass hook, from which the instrument gets its 
name. 

In use, the hook gage is set up in the channel above the weir 
and leveled by means of a leveling instrument so that the scale 




Fia. 54. — Hook gage. 



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62 ELEMENTS OF HYDRAULIC^ 

reads zero when the point of the hook is at the exact level of the 
crest of the weir. The hook is then raised until its point just 
reaches the surface, causing a distortion in the reflection of Hght 
from the surface of the water. If slightly lowered the distortion 
disappears, thus indicating the surface level with precision. 
The reading of the vernier on the scale then gives the head on 
the crest to thousandths of a foot. 

To avoid surface oscillations, and thereby obtain more precise 
readings, the hook gage should be set up in a still box communi- 
cating with the channel. The channel end of the opening or 
pipe leading into the still box must be flush with a flat surface 
set parallel to the direction of flow, and the pipe itself must be 
normal to this direction. 

The channel end of the pipe must be set far enough above the 
weir to avoid the slope of the surface curve, but not so far as to 
increase the head by the natural slope of the stream. 

If the formula of any particular experimenter is to be used, his 
location for the still box should be duplicated. 

69. Proportioning Weirs. — ^To illxistrate the method of propor- 
tioning a weir, suppose that the stream to be measured is 5^ ft. 
wide and 13^ ft. deep, and that its average velocity, determined 
by timing a float over a measured distance or by using a current 
meter or a Ktot tube (Pars. 149 and 153), is approximately 4 ft. 
per second. The flow is then approximately 1,980 cu. ft. per 
minute. To determine the size of weir which will flow approxi- 
mately this amount, try first a depth of say 10 in. From Table 
17 it is found that each inch of length for this depth will deliver 
12.64 cu. ft. per minute. The required length of weir would then 

1 980 
be TKgT = 156.6, which is fifteen and tworthirds times the depth 

and therefore too long by Rule 2 of the specifications. 

Since the weir must evidently be deeper, try 18 in. From 'the 

table the discharge per linear inch for this depth is 30.54 cu. ft. 

. per minute, and consequently the required length would be 

1 980 

or^KA = 64.8 in., which is now only 3.6 times the depth and 

therefore too short. 

By further trial it is found that a depth of 15 in. gives a length 

1 980 
of 2323 ~ ^^'^ ^'* which is 5.7 times the depth and therefore 

comes within the limits required by Rule 2. ^ 



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FLOW OF WATER 63 

Suppose then that the notch is made 7 ft. long and say 20 in. 
deep, so that the depth may be increased over the calculated 
amount if necessary. If then the width of the pond back of the 
weir is not 50 per cent, greater than the width of the notch, or 
if the velocity of flow should be in excess of 1 ft. per second, the 
pond should, if possible, be enlarged or deepened to give the 
desired result. With the weir so constructed suppose that the 
depth of water over the stake back of the weir is found to be 
15^ in. From the table the discharge per linear inch corre- 
sponding to this head is found to be 23.52 cu. ft. per minute, and 
this multiplied by 84, the length of the* weir in inches, gives 
1,975.7 cu. ft. per minute for the actual measured discharge. 

XDI. TIME REQUIRED FOR FILLING AND EMPTYING TANKS 

70. Change m Level under Constant Head. — ^To find the time 
required to raise or lower the water level in a tank, reservoir, or 




^^^^^^^^^ 



Fia. 55. 

lock, let A denote the area of the orifice through which the flow 
takes place and K its coefficient of discharge or efflux. Several 
simple cases will be considered. 

The simplest case is that in which the water level in a tank is 
raised, say from AB to CD (Fig. 56), by water flowing in under a 
constant head h. Let V denote the total volume of water flow- 
ing in, represented in cross-section by the area ABCD in the 
figure. Then since the discharge Q through the orifice per second 

is 

Q:^[KAV2^, 



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64 



ELEMENTS OF HYDRAULICS 



the time t in seconds required to raise the surface to the levd CD is 

V 



t ='^ = 

Q KAv^ 



(43) 



71. Varjring Head. — It is often necessary to find the time re- 
quired to empty a tank or reservoir, or raise or lower its level a 
certain amount. A common case is that in which the level is' to 
be raised or lowered from AB to CD (Fig. 56) by flow through 




Level Lowered 
Fig. 56. 

a submerged orifice, the head on one side, EFy of the orifice being 
constant. If the cross-section of the tank is variable, let Y 
denote its area at any section mn. In the time dt the level 
changes from the height y to y — dy, and consequently the vol- 
ume changes by the amoimt 

dV = Ydy. 

But by considering the flow through the orifice, of area A, the 
volume of flow in the time dt is 

dV = KAV2gidt. 



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FLOW OF WATER 



65 



Hence, by equating these values of dV, we obtain the relation 
KAV2gydt = Ydy 



whence 



t = 



zaV: 



1 n 



Ydy 



(44) 



72. Canal Lock. — A practical application of Eq. (44) is in find- 
ing the time required to fill or empty a canal lock. For an 
ordinary rectangular lock of breadth b and length Z, the cross- 
section is constant, namely Y = 6f, and consequently the ex- 
pression for the time integrates into 

rHjdy ^ bl\/2 

^2gji Vy 



t = 



bl 



KA\/2g 



KAVg 



(VH - Vh). 



(46) 



73. Rise and Fall in Connected Tanks. — When one tank dis- 
charges into another without any additional supply from outside, 




Fig. 57. 

the level in one tank falls as that in the other rises. If both tanks 
are of constant cross-section, then when the level in one tank has 
been lowered a distance y, that in the other tank will have been 
raised a distance y' (Fig. 57), such that if M and N denote their 
sectional areas, respectively. 

My = Ny\ 

In the interval of time dt suppose y changes to y + dy. Then 
considering the flow through the orifice of area A, as in the pre- 
ceding case, we have 

Mdy = KA\^2g[H - {y + y')]dt, 



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t = 



(46) 



66 ELEMENTS OF HYDRAULICS 

or, since j/' = -j^, this may be written 

^^ ^ MdyVN 

KAV2g[NH - y(M + N)] 

Simplifying this expression and integrating, the resulting expres- 
sion for the time t is found to be 

MVN r ^ dy ^ 

KAV2gl VNH - y{M + N) 

Substituting the given limits, the time t required to lower the 
level a distance D is 

' = ^^[mTn{^^ - VNH - D(M + N) j]. 

When the level becomes the same in both tanks, since the 
volume discharged by one is received by the other, we have 

MD = N{H - D), 
or 

D = ^^^. 

M + N 

Substituting this value of D in Eq. (46), it becomes 

t = 2 mnVh_ 

KAV2g(M + N)' 

which is therefore the length of time required for the water in the 
tanks to reach a common level. 

74. Mariotte's Flask. — It is sometimes desirable in measuring 
flow to keep the head constant. It is difficult to accomplish this 
by keeping the supply constant, a more convenient method being 
by the arrangement shown in Fig. 58, which is known as Mariotte's 
Flask. This consists of putting an air-tight cover on the tank, 
having a corked orifice holding a vertical pipe open to the atmos- 
phere. Since the pressure at the lower end A of the tube is 
always atmospheric, the flow is the same as though the water 
level was constantly maintained at this height. As water flows 
out, air enters through the tube and takes its place so that the 
effective pressure remains constant, the pressure of air in the 



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FLOW OF WATER 



67 



tank and of the column of water above the lower end of the tube 
together being constantly equal to the pressure of one atmosphere. 
Therefore as long as the water level does not sink below the bot- 
tom of the pipe, the effective head on the orifice is its distance 
h below the bottom of the pipe, and the discharge is given by the 
formula 

Q = KA V2gh. (48) 




Fig. 58.-rMariotte's flask. 

XIV. FLOW THROUGH SHORT TUBES AND NOZZLES 

75. Standard Mouthpiece. — When a short tube (adjutage, 
mouthpiece or nozzle) is added to an orifice, the flow through 
the opening is changed both in velocity and in amount. In 
general the velocity is diminished by the mouthpiece, due to 
increased frictional resistance, whereas the quantity discharged 
may be either increased or diminished, depending on the form of 
the mouthpiece. 

' What is called the standard mouthpiece consists of a circular 
tube projecting outward from a circular orifice, and of length 
equal to two or three diameters of the orifice (Fig. 60). At the 
inner end of the tube the jet is contracted as in the case of a 
standard orifice, but farther out it expands and fills the tube. 
The velocity of the jet is reduced by this form of mouthpiece to 

V = 0.82\/2^ 
which is considerably less than for a standard orifice, but since 
there is no contraction, the quantity discharged is 

Q;= 0.82AV2gh, 



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68 ELEMENTS OF HYDRAULICS 

where A denotes the area of the orifice. The discharge is there- 
fore nearly one-third larger than for a standard orifice of the same 
area with complete contraction (Fig. 59). 

76. Stream-line Mouthpiece. — By rounding the inner edge of 
the mouthpiece so that its contour approximates the form of a 
stream line, the velocity of the jet is greatly increased, its value 
for the relative dimensions shown in Fig. 61 being about 

V = 0M\/2gh; 

and since the jet suffers no contraction, the quantity discharged is 
Q = 0MAV2ghy 

the area A, as before, referring to the area of the orifice. 

77. Borda Mouthpiece. — ^A mouthpiece projecting inward and 
having a length of only half a diameter is called a Borda movih- 
piece (Fig. 62). The velocity is greatly increased by this form of 
mouthpiece, its value being about 

V = 0.99\/2^, 

but the contraction of the jet is more than for a standard orifice, 
so that the discharge is only 

Q = 0.53il\/2^, 

where A denotes the area of the orifice. 

If, however, the length of the mouthpiece is increased to two or 
three diameters (Fig. 63) the discharge is increased nearly 50 per 
cent., becoming 

Q = 0,72AV2gh. 

78. Diverging Conical Mouthpiece. — For a conical diverging 
tube with sharp edge at entrance (Fig. 64) the jet contracts at the 
inner end as for an orifice, but farther on expands so as to fill the 
tube at outlet provided the angle of divergence is not over 8°. 
The discharge is therefore greater than for a standard mouth- 
piece, its amount referred to the area A at the smallest section 
being 

Q = 0,95AV2gh. 

79. Venturi Adjutage. — If the entrance to a diverging conical 
mouthpiece has a stream-line contour, it is called a Venturi 



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FLOW OF WATER 



69 



w a ■»■ gj » j" * *^ g ^ 



4r «=•' 



Sharp Edffed Orifice 
Q=.64 Av «^62 J^sfloh 



^^r^:^ 






i b?j^^ ! j ! j^;^;^^i^!ji^^ 



Standard Mouthpiece 




Area A Measured 
on Section AB 




Fia. 59. 



Fig. 60. 






=!- 



Streamtine Contour 

■^>] ^[Area A Measured 
I Section AB 





Borda'i 
Moatitp iece 

Q-.64 Ay- 
,^AVZgh 



Area A Measamd 

~on Sectlen 

1^-.^^ AB 



FiQ. 61. 



[FiQ. 62. 




itrant Tube 

\Q=Ay^.72^2gh 

Area A Measured 
on Section AB 



. ■g ggJ T j E -j - ie'^ 






Conical Diversing 
Tube 



Area A Measaied on 
Section AB 



Fig. 63. 




Venturi Adjutage 
Anjrle a = G^'to 8** 

Q=1.6 A\/2gh 
Area A Measured on 
A Bestion AB 



Conical Convererinflr 

Tube 
An^le a= 6° to 10* 

Q— .98 A\/2gh 

Area A Measured on 

Section AB i 




Fig. 65. 



Fig. 66. 



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70 



ELEMENTS OF HYDRAULICS 



aSjiUage (Fig. 65). In experiments by Venturi and Eytelwein 
with diverging mouthpieces of the relative dimensions shown in 
Fig. 65; a discharge was obtained nearly two and one-half times 
as great as for a standard orifice of the same diameter as that at 
the smallest section, or about twice that for a standard short tube 
of this diameter, the formula for discharge referred to the area A 
at the smallest section being 

Q = 1.55AV2gh, 

80. Converging Conical Mouthpiece. — ^In the case of a conical 
converging tube with sharp corners at entrance (Fig. 66) the jet 
contracts on entering and then expands again until it fills the 



Fire Hoee; Smooth G>ne Nozzle 
Q=.91AVTgh~ 

Fig. 67. 



Fire Hoee; Smooth Convex Nozzle 

Fig. 68. 




Fire Hose; Square "Ring Nozzle 

^.74AV2^ 

Fig. 69. 



Fire Hose; Undercut Ring Nozzle 

Fig. 70. 



tube, the most contracted section being just beyond the tip, and 
the greatest discharge occurring for an angle of convergence of 
approximately 13®. 

81. Fire Nozzles. — The fire nozzles shown in Figs. 67, 68, 69 
and 70 are practical examples of converging mouthpieces. The 
smooth cone nozzle with gradually tapering bore has been found 
to be the most efficient, the coefficient of discharge for the best 
specimen being 0.977 with an average coefficient for this type 
of 0.97. For a square ring nozzle like that shown in Fig. 69 the 
coefficient of discharge is 0.74; and for the undercut type shown 
in Fig. 70 the coefficient of discharge is 0.71. 



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FLOW OF WATER 71 

XV. KINETIC PRESSURE IN A FLOWING LIQUID 

82. Kinetic Pressure. — For a liquid at rest, the normal pres- 
sure exerted by it on any bounding surface is called the hydro- 
static pressure and is given by the expression deduced in Par. 8, 
namely, 

P = p' + yh. 

If a liquid is in motion, however, the normal pressure it exerts 
on the walls of the vessel containing it, or on the bounding sur- 
face of a liquid vein or filament, follows an entirely different 
law, as shown below. 

To distinguish the hydrostatic pressure from the normal pres- 
sure exerted on any bounding surface by a liquid in motion, the 
latter will be called the kinetic pressure. 

83. Bernoulli's Theorem. — To determine the kinetic pressure 
at any point in a flowing liquid, consider a small tube or vein of 
the liquid bounded by stream 
lines, as explained in Par. 48, 
and follow the motion of the 
liquid through this tube for a 
brief interval of time. 

Let A and A' denote the areas 
of two normal cross-sections of 
the vein (Fig. 71). Then since 
the liquid is assumed to be in- 
compressible, the volume Ad 
displaced at one end of the tube 
must equal the volume A'd' 
displaced at the other end. If 
p denotes the average unit 
pressure on A, and p' on A', the work done by the pressure on 
the upper cap, A, is 

+ pAd, 

and that on the lower cap. A', is 

- p'A'd\ 

the negative sign indicating that the element of work is of 
opposite sign to that at the other end of the vein. 
Also, if h denotes their difference in static head, as indicated in 
Fig. 71, the work done by gravity in the displacement of the 
volume Ad a distance h is 

yAdh. 




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72 ELEMENTS OF HYDRAULICS 

Since the forces acting on the lateral surface of the vein are 
normal to this surface they do no work. Assuming, then, the 
case of steady flow, that is to say, assuming that each particle 
arriving at a given cross-section experiences the same velocity 
and pressure as that experienced by the preceding particle at this 
point, so that the velocities v and t;' through the caps A and A' 
are constant, the change in kinetic energy between these two 
positions is 

Therefore, equating the total work done to the change in energy, 
the result is 

pAd - p'A'd' + yAdh = ^{v'^ - v^), 

or, since Ad = A'd\ this reduces to 

p' + ^* = P + ^' + 7h. (49) 

This result is known as Bernoulli's Theorem, and shows that in 
the case of steady parallel flow of an ideal liquid, an increase in 
velocity at any point is accompanied by a corresponding decrease 
in kinetic pressure, or vice versa, in accordance with the relation 
just obtained. 

84. Kinetic Pressure Head. — If the theoretic heads corre- 
sponding to the velocities v and v' are denoted hy H and H\ 
respectively, then in accordance with Torricelli's theorem (Par. 
50) we have 

J7 — . IJf ^ ^ ' 

"-2-g' " -2^' 
and consequently Eq. (49) may be written in the form 

p' = p -f- 7(h + H - HO, (50) 

which is a convenient form from which to compute the kinetic 
pressure at any given point. 

If this relation is written in the form 

L' + H' = 5 + H + h, (51) 

then since p/.y is the head corresponding to the hydrostatic 
pressure p, each term is a length, and Bernoulli's theorem may be 
expressed by saying that: 

In the case of steady, parallel flow of an ideal liquid, the sum of 



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FLOW OF WATER 



73 



5=4^^:^-=^ 






yyyyy^yyyy^yy^yy^^y^^y^^^^^^ 



mP^^^^^^^' 



the pressure head, velocity head and potential head is a constant 
quantity for any particle throughout its course. 

85. Application to Standard Mouthpiece. — An illustration of 
Bernoulli's theorem is afforded by the flow through a standard 
mouthpiece. At the con- 
tracted section A (Fig. 72) 
the velocity is evidently 
greater than at the outlet 
B. Therefore, by Ber- 
noulli's theorem, the kinetic 
pressure must be less at A 
than at B, Thus if a 
piezometer is inserted in 
the mouthpiece at A, the 
liquid in it will rise, shew- 
ing that the pressure in the 
jet at this point is less than 
atmospheric. It was found 
by Venturi, and can also be 
proved theoretically, that 
for a standard mouthpiece 
the negative pressure head at A is approximately three-fourths of 
the static head on the opening, or, referring to Fig. 72, 

hi = ^h. 

To prove this relation apply Bernoulli's equation between a 
point at the surface and one in the contracted section A. Then 
if p denotes the unit atmospheric pressure at the surface, we have 



■f 
^1 



Fig. 72. 



Va' 



Pa 



+ ^-f-;i = -7^--|-^ + + head lost at entrance. 

Assuming the coefficient of contraction at A as 0.64, we have 

Va X 0.64A = VbXA, 
or 

1 



Va = 



0.64 



Vb. 



0.82 



Since Vs = O.S2\/2gh, we have Va = ^rrwTy/2gh = 1.28\/2gh 
From Art. 98 we have 



head lost at entrance = (0072 "" ^) "y~"~ ^ 



063 



2g 



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74 ELEMENTS OF HYDRAULICS 

Substituting all these values in Bernoulli's equation, it becomes 

^ + fc = (1.28)% + ^ + 0.063 (1.28)2A 
7 7 

whence 

^ = £-0.741fc. 
7 7 

Consequently the negative pressure head at A is 0.74fc, or f^/i 
approximately. 

XVI. VENTURI METER 

86. Principle of Operation. — The Venturi meter, invented by 
Clemens Herschel in 1887 for measuring flow in pipe lines, 
illustrates an important commercial application of Bernoulli's 
theorem. This device consists simply of two frustums of conical 
tubes with their small ends connected by a short cylindrical 
section, inserted in the pipe line through which the flow is to be 
measured (Fig. 73). If a pressure gage is inserted in the pipe 




Fig. 73. 



line at any point A and another at the throat of the meter JB, 
as indicated in the figure, it will be found that the pressure at 
B is less than at A. 

87. Formula for Flow. — ^Let va and Vb denote the velocities at 
A and JB, ahd Pa and pa the. kinetic pressures at these points, 
respectively. Then since both points are under the same static 
head, Bernoulli's theorem, disregarding frictional losses, gives the 
relation 

^ iPa ^Vb^ .Pb 

2g ^ y 2g'^ y' 

If a and b denote the cross-sectional areas at A and B, the dis- 
charge Q is given by 

Q = avA = bvB 
whence 

Q Q 

Va--; VB-y 



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FLOW OF WATER 



75 



n 



Chafftbef drarnbcp 




Pfpet 



H^giitttf-' 




Fig. 74. — Venturi meter and recording gage manufactured by the Builders 

Iron Foundry. 



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76 



ELEMENTS OF HYDRAULICS 




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FLOW OF WATER 77 

Substituting these values of Va and Vb in the preceding equation 
and solving for Q, the result is 

ab 



« - ^^biv/? <- - -)■ 



If Ha and hs denote the static piezometer heads corresponding to 
the kinetic pressures Pa and psj respectively, this formula may 
be written 

^ = :;;;^pV2g(hA - hB)- K^) 

Ordinarily the throat diameter in this type of meter is made 
one-third the diameter of the main pipe, in which case a = 96. 
If, then, h denotes the difference in piezometer head between the 
upstream end and the throat, the formula for discharge, ignoring 
frictional losses, becomes 

Q = 1.062b\/2^. (53) 

By experiment it has been found that ordinarily for all sizes of 
Venturi meters and actual velocities through them, the actual 
discharge through the meter is given by the empirical formula 

Q = (0.97 ± 0.03)b\/2^. (54) 

88. Commercial Meter. — A typical arrangement of meter tube 
and recording apparatus is shown in Fig. 74, the lower dial indi- 
cating the tate of flow, and the upper dial making a continuous 
autographic record of this rate on a circular chart. 

89. Catskill Aqueduct Meter. — The Venturi meter affords the 
most accurate method yet devised for measuring the flow in pipe 
lines. Fig. 75 shows one of the three large Venturi meters built 
on the line of the Catskill Aqueduct, which is part of tlie water 
supply system of the City of New York. Each of these meters is 
410 ft. long and is built entirely of reinforced concrete except for 
the throat castings and piezometer ring, which are of cast bronze. 
Provision is also made in connection with the City aqueduct for 
the installation of a Venturi meter upon each connection between 
the aqueduct and the street distribution pipes. 

' 90. Rate of Flow Controller. — Figure 76 illustrates a rate of flow 
controller operated by the difference in pressure in a Venturi tube. 
This apparatus is designed for use in a water pipe or conduit 
through which a constant discharge must be maintained regard- 



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78 



ELEMENTS OF HYDRAULICS 



less of the head on the valve. It consists of a perfectly balanced 
valve operated by a diaphragm which is actuated by the differ- 
ence in pressure between the full and contracted sections of a 
Venturi tube. The valve and diaphragm are balanced by an 
adjustable counterweight, which when set for any required rate of 
flow will hold the valve discs in the proper position for that flow. 






Fia. 76.- 



-Venturi rate of flow controller manufactured by the Simplex 
Valve and Meter Co. 



XVII. FLOW OF WATER IN PIPES 

91. Critical Velocity. — ^Innumerable experiments and investi- 
gations have been made to determine the laws governing the flow 
of water in pipes, but so far with only partial "success, as no gen- 
eral and universal law has yet been discovered. 

Experiments made by Professor Osborne Reynolds have shown 
that for a pipe of a given diameter there is a certain critical 
velocity, such that if the velocity of flow is less than this critical 
value, the flow proceeds in parallel filaments with true stream- 
line motion; whereas if this critical value is exceeded, the flow 
becomes turbulent, that is, broken by whirls and eddies and 
similar disturbances. The results of Professor Reynolds' experi- 
ments showed that at a temperature of 60°F. this critical 
velocity occurred when 

Dva = 0.02 

where D denotes the diameter of the pipe in feet and Va is the 
average velocity of flow in feet per second. 



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FLOW OF WATER 



79 



For parallel, or non-sinuous, flow it is possible to give a theoret- 
ical explanation of what occurs and deduce the mathematical law 
governing it, as shown below. No one, however, has yet ex- 
plained why the flow suddenly becomes turbulent at the critical 
velocity, or what law governs it subsequently. 

92. Viscosity Coefficient. — ^The loss of energy accompanying 
pipe flow is due to the internal resistance arising from the viscosity 
of the liquid. This shear or drag between adjacent filaments 
is analogous to ordinary friction but follows entirely different 
laws.- Unlike friction between the surfaces of solids, fluid friction 
has been found by experiment to be dependent on the tempera- 
ture and the nature of the liquid; independent of the pressure; 
and, for ordinary velocities at least, approximately proportional 
to the difference in velocity between adjacent filaments. When 
this difference in velocity disappears, the f rictional resistance also 
disappears. 

The constant of proportionahty required to give a definite 
numerical value to fluid friction is called the viscosity coefficient 
and will be denoted by fi. This coefficient /i is an empirical con- 
stant determined by experiment, the values tabulated below being 
the result of experiments made by 0. E. Meyer. 



Temperature in degrees 
Fahr. 


50* 


60* 


65* 


70* 


Viscosity coefficient /i in 

lb. sec. 

ft.* 


32 X 10-« 


28 X 10-« 


26 X 10-« 


24 X 10-« 



The dimensions of /* are, of course, such as to make the equa- 
tion in which it appears homogeneous in the units involved, as 
will appear in what follows. 

93. Parallel (non-sinuous) Flow.^ — Consider non-sinuous flow in 
a straight pipe of uniform circular cross-section, that is, at a 
velocity less than the critical velocity and therefore such that 
the filaments or stream lines are all parallel" to the axis 
of the pipe. By reason of symmetry the velocity of any par- 
ticle depends only on its distance from the axis of the pipe. 
Let V denote the velocity of any particle and x its distance 
from the center (Fig. 77). Then if x changes by an amount 

* The following derivation is substantially that given by FOppl in his 
"Dynamik." 



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80 



ELEMENTS OF HYDRAULICS 



dx, the velocity changes by a corresponding amount dVj and 
since the velocity is least near the pipe walls, v decreases as x 

increases and consequently the rato -r- is negative. 

Since the pipe is assumed to be of constant cross-section and the 
flow uniform and parallel, the forces acting on any element of 
volume must be in equilibrium. Considering therefore a small 
water cyUnder of radius x and length dt/, in order to equiUbrate 
the frictional resistance acting on the convex surface of this cyUn- 
der there must be a difference in pressure on its ends. This ex- 
plains the fall in pressure along a pipe, well known by experiment. 




Fig. 77. 

Let dp denote the difference in pressure in a length dy. Then 
the difference in pressure on the ends of a cyhnder of radius x 

is {Trx^)d'p and the shear on its convex surface is {2jrxdy)fA^' 

dv 
Equating these two forces and remembering that -r- is negative, 

we obtain the relation 

(7rx2)dp = - {2Txdy)tJL^' 

Also, since the difference in pressure on the ends of any cyUnder 
is proportional to its length, we have 

dp _ pi - p2 _ , 
dy - I " ^' 

where pi and p2 denote the unit pressures at two sections at a 
distance I apart, and the constant ratio is denoted by k for 
convenience. 

Substituting dp = kdy from the second equation in the first 
and cancelling common factors, we have finally 

dvM_ k^ 

dx^~ 2/' 



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FWW OF WATER 81 

whence, by integration, 

fee* , 

where c denotes a constant of integration. 

To determine c assume that the frictional resistance between 
the pipe wall and the liquid follows the same law as that between 
adjacent filaments of the liquid. Then it follows that the hquid 
in contact with the pipe must have zero velocity, as otherwise 
it would experience an infinite resistance. This seems also to be 
confirmed by the experiments of Professor Hele-Shaw, who 
showed that in the case of turbulent flow there was always a 
thin fiilm of Uquid adjoining the pipe walls which showed true 
stream-line motion, proving that its velocity was certainly less 
than the critical velocity and therefore small. Furthermore, the 
walls of commercial pipes are comparatively rough and conse- 
quently a thin sldn or layer of liquid must be caught in these 
roughnesses and held practically stationary. 

Assuming then that t; = when x = r, and substituting this 
pair of simultaneous values in the above equation, the value of c 
is found to be 

4m 



c = 



and consequently 



= |;(r»-x»). (56) 



This is the equation of a parabola, and therefore the velocity dia- 
gram is a parabolic arc with its vertex in the axis of the pipe; 
that is, the velocity is a maximum at the center where a; = 0, its 
value being 

^-' = 4^7- 

94. Average Velocity of Blow in Small Pipes. — Let the dis- 
charge through any cross-section of the pipe be denoted by Q. 
Then if the velocity at any radius x is denoted by v, we have 



-X 



or, since 



Q = I 2Trxdxv, 



, = |^(r« - »*), 



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82 ELEMENTS OF HYDRAULICS 

this becomes 

irfcr* 



'>-W."^-''''"t 



But if Va denotes the average velocity of flow we also have 

Q = Vaiirr^) 
whence by substituting the above value for Q, we have 

^'^ -KT^ 7rr2 8^- 

Comparing this expression with that previously obtained for the 
maximum velocity, it is evident that the maximum velocity is 
twice the average velocity of flow. 

96. Loss of Head in Small Pipes. — ^The loss in pressure in a 
length I is given by the relation obtained above, namely, 

or, if the difference in head corresponding to this difference in 
pressure is denoted by 7i, then, since p = yh, we have 

loss in head, h = — — = — 

7 7 

Substituting in this relation the value of k in terms of the average 
velocity of flow, the result is 

For small pipes, therefore, the loss of head is proportional to the 
first power of the average velocity, and inversely proportional 
to the square of the diameter of the pipe. 

This result has been verified experimentally for small pipes by 
the experiments carried out by Poiseuille. 

96. Ordinary Pipe Flow. — Under the conditions usually found 

in practice the velocity of flow exceeds the critical velocity and 

consequently the flow is turbulent and a greater amoimt of energy 

is dissipated in overcoming internal resistance than in the case of 

parallel flow. The result of Professor Rejoiolds' experiments 

indicated that the loss of head in turbulent flow was given by the 

relation 

, Va'-'H 
hoz — z— . 



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FLOW OF WATER 83 

In commercial pipes the degree of roughness is a variable and un- 
certain quantity, so that the exact loss of head cannot be pre- 
dicted with accuracy. Practical experiments have shown, how- 
ever, that ordinarily the loss in head is proportional to the square 
of the average velocity, so that the relation becomes 

a 
Since the theoretical head corresponding to a velocity v is 
A = ^, the expression for the loss in head for a circular pipe 
nmning full may in general be written 

2gd 
or, denoting the constant of proportionality by/, this becomes 

Here /is an empirical constant, depending on the condition of the 
inner surface of the pipe, and is determined by experiment. 
Eq. (57) is identical with Chezy's well-known formula 

as will be shown in Par. 118. 



XVm. PRACTICAL FORMULAS FOR LOSS OF HEAD IN PIPE FLOW 

97. Eflfective and Lost Head. — ^In the case of steady flow 
through long pipes, much of the available pressure head disap- 
pears in frictional and other losses, so that the velocity is greatly 
diminished. Thus if h denotes the static head at the outlet and 
hi the head lost in overcoming frictional and other resistances to 
flow, the velocity v at the outlet is given by the relation 

or its equivalent, 

The lost head h is the sum of a number of terms, which will be 
considered separately. 



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84 ELEMENTS OF HYDRAULICS 

98. Loss at Entrance. — ^A certain amount of head is lost at the 
entrance to the pipe, as in the case of a standard adjutage. If 
V denotes the velocity ilue to the head h with no losses, then 

h - ^' 

whereas if Va denotes the actual velocity of flow the head corre- 
sponding to this velocity is 

h' = ^*. 

• 2g 
The head, hi, lost at entrance, is therefore 

h, = h-h' = '^-'/. 

If Cv denotes the velocity coefficient for the entrance, then 

Va = CvV, 

and consequently the expression for the head lost at entrance 
may be written 

^' 2g 2g 2g 2g 



^2^(c?"^^)' 



2g 

For the standard short tube C„ = 0.82 (Par. 75) and therefore 
^ — 1 = .^ ^o\2 — 1 = 0.5 The head lost at entrance is 
therefore 

hi = 0.6g. (69) 

If the pipe projects into the reservoir, C = 0.72 (Par. 77), and 
the head lost at entrance is thereby increased to 

hi = 0.93 ^. 
^g 

For ordinary service taps on water mains it may be assumed as 

A. = 0.62|. 

99. Friction Loss. — ^In flow through long pipes the greatest 
loss in head is that due to the friction between the liquid and the 



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FLOW OF WATER 86 

walls of the pipe. Until recently this loss in head was assumed to 
be given exactly by the formula 

h, = f|iXg, (60) 

where d = internal diameter of pipe; 
I = length of pipe; 
/ = empirical constant determined by experiment. 

Values of the constant / as determined by experiment for various 
kinds of pipe are given in Table 12. Average values ordinarily 
assumed for cast-iron pipe are: 

for new smooth pipes f - 0.024, \ /g^x 

for old rusty pipes f = 0.03. / ^ ^ 

More recent and accurate experiments have shown, however, 
that the loss in head due to pipe friction does not vary exactly 
as the square of the velocity nor inversely as the first power of 
the diameter, the results of such experiments leading to what is 
known as the exponential formula. This formula is of the form 

h2 = mj, (62) 

in which ^2 denotes the loss in head for a given length of pipe, 
say, 1,000 ft., thereby eUminating the length I, and m is an 

empirical constant which replaces the combination d- in the 

older formula. I'he essential diBFerence between the two formu- 
las consists in the fact that the exponent z is not 2 but varies 
between 1.7 and 2, and the exponent z is not unity but is ap- 
proximately 1.25 for all kinds of pipe. 

100. Wood Stave Pipe. — From accurate and comprehensive 
experiments made by Moritz^ on wood stave pipe, ranging from 
4 in. to 55^ in. in diameter, it was found that the formula in this 
case should read 

h2 = 0.38^ (63) 

where hi = friction head in feet per 1,000 ft. of pipe; 
V = mean velocity of flow in feet per second; 
d = diameter of pipe in feet. 

^ E. A. MoRiTZ, Assoc. M. Am. Soc. C. E., Engineer U. S. Reclamation 
Service, "Experiments on the Flow of Water in Wood Stave Pipe," Trans. 
Am. Soc. C. E., vol. 74, pp. 411-482. 



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86 ELEMENTS OF HYDRAULICS 

Solving this relation for v, we have 

V = 1.72d«ni2^"» (64) 

and consequently the discharge Q in cubic feet per second is 

Q = 1.36d« W "* (66) 

From these formulas, the velocity and discharge have been 
calculated by Moritz^ for pipes ranging from 6 to 120 in. in diame- 
ter and for a large number of hydraulic slopes, and the results 
are summarized in Table 16 at the end of this book. 

If it is necessary to find the discharge or velocity for slopes 
not given in Table 16, it wiU in general be sufficiently accurate 
to interpolate between the next lower and next higher slopes. 
Greater accuracy may be obtained by calculating the value of 
h^o.hii Qj^^ multiplying it by the value of l.SbcP'^ given in column 
4 of the table, which has been inserted especially for this purpose. 

101. Graphical Solution. — ^Any exponential formula is par- 
ticularly adapted to graphical solution by plotting on logarithmic 
paper, as in this case the exponential curves are transformed into 
straight lines and therefore require the plotting of only two 
points to determine each. Such a logarithmic diagram has been 
prepared by Moritz for his formula for wood stave pipe, and is 
shown in Kg. 78. On this diagram all the essential factors, 
namely diameter, area, velocity, discharge and friction head, 
are shown at a glance. As a numerical example, the diagram 
shows that a 16-in. pipe has an area of 1.4 sq. ft., and with a 
velocity of 3 ft. per second will discharge 4.2 cu. ft. per second 
with a loss of head of 1.9 ft. per 1,000 ft. length of pipe. 

As another instance of the use of the diagram, suppose that it 
is required to obtain a discharge of 3 cu. ft. per sec. with an avail- 
able fall of 2 ft. per 1,000. Starting from the bottom scale, 
follow the vertical line for H = 2 to its intersection with the 
horizontal line representing Q = 3. Since the point of inter- 
section lies almost on the 14-in. pipe line, it shows that this size 
pipe is required. Interpolating between the lines representing 
velocities of 2 ft. per second and 3 ft. per second, the velocity of 
flow is found to be 2.8 ft. per second. 

102. Cast-iron Pipe. — Many experiments have been made to 
determine the friction loss in cast-iron pipes, and a number of 

•.formulas have been proposed. The most reliable of these results 

1 Eng, Record, vol. 68, No. 24, Dec. 13, 1913. 



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FLOW OF WATER 



87 



100,0 



I?-^!^W.J'-Vf"^*^ 




Loss of Head in Feet per 1000 

Fia 78. — Graphical solution of exponential formula for flow of water in 
wood stave pipe. {MorUz,) 



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88 ELEMENTS OF HYDRAULICS 

are the experimental coefficients C detennined by Smith for use in 
the Chezy formula 

V = CVrSf 
and the exponential formula proposed by Gardner S. Williams, 
which has the form 

V = eT.Tdo^'V"* (66) 

From a careful investigation of available data and a comparison 
of the results of 16 formulas, including the two just given, 
Moritz^ obtained the exponential formula 

V = 77do-7sO"5' (67) 

whence 

Q = 1.31d2TiO"5 (68) 

This formula differs from that for wood stave pipe only in the 
value of the constant coefficient. The velocity and discharge 
for cast-iron pipe may therefore be obtained from Table 16 for 
wood stave pipe by simply multiplying the results there given 

1.31 
by the ratio of these coefficients, namely, j-^ = 0.97. 

103. Deterioration with Age. — ^The above formula appUes'to 
new cast-iron pipe with smooth alignment and profile. To make 
allowance for deterioration with age, Moritz, adopts the assump- 
tion made by Williams and Hazen in their hydrauUc tables, 
namely, that the friction head increases 3 per cent, per year due 
to tuberculation, and that the diameter decreases 0.01 in. per 
year from the same cause. Applying these assumptions to the 
equation Q = 1.31(P-7Ao,565^ ^nd denoting by K the ratio of 
discharge after n years of service to the discharge when new, we 
have 

[n n2.7 r 1 -10.566 

For example, this equation shows that a 12-in. pipe 10 years old 
will carry only 85 per cent, as much as the same pipe when new. 

104. Riveted Steel Pipe. — Experiments on riveted steel pipes 
are too few to make an accurate formula possible. The formula 
proposed by Moritz for new asphalted pipes having smooth 
alignment and profile is 

Q = 1.18d2 Tio "5 (70) 

which gives results about 10 per cent, less than the formula for 
cast-iron pipe. 
* Eng. Record, Dec. 13, 1913. 



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FLOW OF WATER 89 

The discharge and velocity of flow for riveted steel pipes may 
be obtained from Table 16 for wood-stave pipes by multiplying 
the results there given by the ratio of the constants, namely, 

106. Concrete Pipe. — Concrete pipe is manufactured in various 
ways and by different methods: namely, the dry-mix pipe which 
is built in short sections and laid Uke clay sewer pipe; the wet- 
mix pipe, also built and laid in short sections; and the wet-mix 
pipe built continuously in the trench. In the absence of reliable 
experiments with concrete pipe, Moritz assumes that the dis- 
charge may be closely approximated by classing the dry-mix 
pipe with riveted steel pipe; the wet-mix built in short sections 
with cast-iron pipe; and the continuous wet-mix pipe with wood 
stave pipe. 

106. Bends and Elbows. — ^Bends and elbows in a pipe also 
greatly diminish the effective head. Until recently the formulas 




_>jLgg_^ — x_ 



Fig. 79. 

obtained by Weisbach from experiments on small pipe were 
generally accepted for lack of better authority. According to 
Weisbach the lost head due to a sharp elbow of angle a (Fig. 79) 
is given by the formula 

h3 = mg. (71) 

where w is a function of the angle a, given by the equation 

m = 0.9457 sin^ (|) + 2.047 sin* (|) • 

Values of m, calculated from this formula for various values of 
the angle a, are tabulated as follows: 



a = 


1 20'' 


30*' 1 


40° 


50° 


60° 


70° 1 


80° 


90° 


m = 


1 .046 


.073 


.139 


.234 


.364 


.533 


.740 


.984 



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ELEMENTS OF HYDRAULICS 



For a curved elbow of radius 22 and central angle a (Fig. 80) 
Weisbach's formula for lost head is 



*»="©£ 



(72) 



d\»» 



2g' 

where the coefficient n has the value 

n = 0.131 +0.163(1) 
Values of n calculated from this formula for various values of the 
ratio ^ are tabulated below for convenience in substitution. 



d 


.2 


.8 


.4 


.6 


.6 


.8 


1.0 1.2 


1.26 


1.3 


1.4 


1.6 


1.8 


2.0 


It- |.13li.l33|.138|.145|.168|.206| .294| .440| .487| .639| .66l| .977| 1.40| 1.98| 




Weisbach's formula if it was of general application would imply 
that the greatest loss of head occurs in bends of smallest radius, 
and conversely, as the radius of the bend increases, the loss in 

head diminishes. Experiments 
made by Williams, Hubbell and 
Fenkell^ at Detroit on pipes of 12, 
16 and 30 in. diameter, however, 
indicated that the loss of head is a 
minimum for bends with radii of 
about two and one-half times the 
diameter of the pipe. Further ex- 
periments made by Schoder* at 
Cornell on 6-in. pipe; by Bright- 
more* in England on 3- and 4-in. pipe; and by Davis* and 
Balch* at the University of Wisconsin on 2- and 3-in. pipes 
have shown that the Weisbach formula is not valid for larger 

^Gardner S. Williams, Clarence W. Hubbell and George H. 
Fenkbll, "Experiments at Detroit, Mich., on the Effect of Curvature 
upon the Flow of Water in Pipes," Trans, Am. Soc. C. E., vol. 47. 

•Ernest W. Schodbr, "Curve Resistance in Water Pipes," Trans, Am. 
Soc. C. E., vol. 62. 

»A. W. Brightmore, "Loss of Pressure in Water Flowing Through 
Straight and Curved Pipes," Minutes of Proc. Inst. C. E., vol. 169, p. 323. 

♦George Jacob Davis, Jr., "Investigation of Hydraulic Curve Resist- 
ance. Experiments with 2-in. Pipe," BuU, Univ. of Wis., No. 403, January, 
1911. 

*L. R. Balgh under direction of George Jacob Davis, Jr., "Investiga- 
tion of Hydraulic Curve Resistance. Experiments with 3-in. Pipe, BvU, 
Univ. of Wis., No. 578, 1913. 



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FLOW OF WATER 91 

pipes under ordinary conditions of service. The Wisconsin ex- 
periments, however, did not confirm the Detroit experiments as 
to the minimum loss of head occurring with bends of two and 
one-half pipe diameters. 

The conclusions reached from the Wisconsin experiments were 
as follows: 

(a) The total loss of heads in bends decreases with an increase 
in radius until the radius of the bend equals about four pipe 
diameters. For bends with radii greater than five pipe diame- 
ters, the total loss in head increases with an increase in radius. 

(6) The constant / in the relation 

as applied to curve resistance is independent of the diameter of 
the pipe, but varies inversely as the radius of the bend. 

(c) The net curve resistance, that is, the loss of head due to the 
bend alone and not including pipe friction, decreases to a mini- 
mum for a radius of bend equal to about six pipe diameters, then 
increases until the radius is about 14 pipe diameters, after which 
it again decreases. 

(d) The net curve resistance per unit length of bend is inde- 
pendent of the diameter of the pipe; decreases with an increase 
in the radius of curvature; and varies approximately as the square 
of the velocity of flow. 

It has been pointed out by W. E. Fuller, ^ Consulting Engineer 
of New York City, that in all these experiments it was assumed 
that the loss of head in bends in different sizes of pipes should 
be the same when the radius of the bend in terms of the diameter 
of the pipe were alike, whereas with so many different factors 
contributing to the loss, there seems to be no adequate grounds 
for assuming such a relation to exist. 

From a careful. comparison of all available data. Fuller found 
that the loss is more nearly the same for different sizes of pipes 
with bends of the same actual radius than for bends of the same 
radius in pipe diameters. From this comparison the formula 
for loss of head was found to be 

hs = kv2-26 (73) 

^ W. E. Fuller, "Loss of Head in Bends," Jour. New Eng. Water Works 
Assoc, Vol. 27, No. 4, December, 1913. 



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ELEMENTS OF HYDRAULICS 



where hi — loss of head in excess of the loss for an equal length 
I lot straight pipe; 

Jfc = coefficient depending on the radius of the bend; 
V = velocity of flow in feet per second. 
Values of the coefficient k are given in Fig. 81. 



.016 
.014 


















































•012 


























•:£ .010 

|.006 

> 

.006 

.004 

• .002 







































— 
















,/ 


^ 


















L_ 




/ 





















10 



15 



40 



46 



50 



55 



20 26 80 35 

Radius of Bend in Ft. 
Value of coeflBcient k in formula for curve resistance, hfln**^ 

FlQ. 81. 

The following table shows the loss of head for ordinary 90** 
bends of the New England Water Works Association standard. 

Loss OF Head Due to 90® Bends of the New England Water Works 
" Association Standard 



Size of pipe, 
inches 


Radius of 
bend, feet 


Excess loss over loss in straight pipe 
of length equal to tangents 


» = 3 ft. /sec. 


V = 5 ft./sec. 


»=10ft./sec- 


4 


1.33 


0.021 


0.073 


0.37 


6 


1.33 


0.025 


0.082 


0.40 


8 


1.33 


0.026 


0.086 


0.41 


10 


1.33 


0.027 


0.089 


0.42 


12 


1.33 


0.028 


0.090 


0.43 


16 


2.0 


0.026 


0.085 


0.41 


20 


2.0 


0.027 


0.086 


0.41 


24 


2.5 


0.026 


0.085 


0.41 


30 


3.0 


0.026 


0.083 


0.41 


36 


4.0 


0.026 


0.083 


0.40 



For bends less than 90**, the use of the following values for loss 
of head is recommended: 



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93 



For 45** bends, use three-fourths of that due to 90° bends of 
same radius. 

For 223^** bends, use one-half of that due to 90** bends of same 
radius. 

For a F-branch, use three-fourths of that due to a tee. 

For velocities of 3 to 6 ft. per second the loss of head in bends 
is approximately proportional to the velocity head, and for rough 
approrimations the following rules may be used: 

For 90** bends of radius greater than 1 J^ ft. and less than 10 ft., 

A, = 0.25 ^• 
For tees, that is, bends of zero radius, 

;i. = 1.25|- 
For sharp 90** bends of 6-in. radius 



A, = 0.5 



2g 



107. Enlargement of Section. — ^A sudden enlargement in the 
cross-section of a pipe decreases the velocity of flow and causes a 
loss of head due to eddying in the corners, etc. (Fig. 82). If the 



I 



mmmmmmmmmm/ 






1^^^^^^=-- = - 



^^2 




Fig. 82. 

velocity is decreased by the enlargement from V\ to v^^ it has been 
found by experiment, and can also be proved theoretically, 4;hat 
the head lost in this way is given by the formula 



h4 = 



(Vl - V2)^ 

2g 



(74) 



To prove this relation let A\ denote the smaller area of diameter 
d (Fig. 82) and A^ the larger area of diameter JD, so that from the 
law of continuity of flow we have 

Axvx = A%^)%. 



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94 ELEMENTS OF HYDRAULICS 

Applying Bernoulli's equation to points on opposite sides of 
the change in section^ we have 

I" +?- 1' +?+'""'-<'■ 

whence 

lost head = ^' - ^V ^i-^:^^ 

Since the dijBFerence in pressure is a force of amount P2A2 — P1A2, 
we have by the principle of impulse and momentum 

PzAi - P1A2 = "T ('^i " ^i) 

V 

where W = v^A^j and consequently this reduces to 

Substituting this in the expression for lost head it becomes 
Io8thead = |*-|*-^*(.x-r,) 
_ (»i - fs)' 

-—2r~' 



^»^^<^<^<^^^sg<9g^^t^»t^<<^g^^?^^^^<^^s^^^g^<^^<^t^^^^^^^ 






i ^^"^.^ 



1 __^ -^ C^ ^mvmv^^^^^^^^^^ 



r^^mW^M'M'MMMW^^: 



> ; tl^v^^\vvvvv^vvv\^v^^^^^ 



Fia. 83. 

To obtain a more convenient expression for hi, let a denote 
the area of cross-section of the smaller pipe and A of the larger. 
Then 

Via = V2A, 
whence 

V2A 

and consequently the expression for A4 may be written 

■.-|■(^»)■ m 

108. Contraction of Section. — ^A sudden contraction in section 
also causes a loss in head, similar to that due to a standard orifice 



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FLOW OF WATER 



95 



or adjutage (Fig. 83). The lost head in this case has been found 
by experiment to be given by the equation 



h5 = q^. 



(76) 



where q denotes an empirical constant, determined experimen- 
tally. The following tabulated values of the coefficient q are 
based on experiments by Weisbach, A being the cross-sectional 
area of the larger pipe and a of the smaller. ^^ 



a 
A 


.1 


.2 


.3 


A 


.6 


.6 


.7 


.8 


.9 


1.0 


« 


.362 


.338 


.308 


.267 


.221 


.164 


.105 


.053 


.0151.000 



109. Gate Valve in Circular Pipe.— The loss in 
head due to a partly closed gate valve (Fig. 84) has 
been determined by experiment for different ratios 
of height of opening to diameter of pipe with the 
following results.^ In this Table, x denotes the 
height of the opening, d the diameter of the pipe, 
^6 the loss in head and f the empirical coefficient in 

the formula Ae = f o"' 




Fio. 84. 



X 

d 


i 


i 


\ 


J 


f 


i 


\ 


r 97.8 


17.0 


5.52 


2.06 


0.81 


0.26 


Q.07 




Fig. 85. 



110. Cock in Circular Pipe.— For 
a cock in a cylindrical pipe (Fig* 
85) the coefficient f has been de- 
termined in terms of the angle of 
closure with the following results. 



e 
f 


5° 


|io*» 


15^ 


20'' 


25** 


30*" 


35** 


40** 


45** 


50« 


55« 


eo'' 


65** 


82* 


.05 


.29 


.75 


1.56 


3.1 


5.47 


9.68 


17.3 


31.2 


52.6 


106 


206 


486 


Valve 
closed 



^HosKiNS, "rXext-book^on^Hydraulics/'^p. 74. 

* The coefficient for losses at valves are based on experiments by Weis- 
bach and are given in most standard texts on ''Hydraulics." See for 
example Wittbnbaxjbr, ''Aufgabensammlung/' Bd. Ill, S. 318; Gibson, 
"Hydraulics and its Applications," pp. 249, 250. 



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ELEMENTS OF HYDRAULICS 

/ 111. Throttle Valve in Circular Pipe. 

A — ^The coefficient f in the formula h^ = 

for a throttle valve of the butterfly 

' type (Fig. 86) for various angles of 

Fig. 86. closure with results as follows: 



/ 






1 5» 


1 10° 


20° 1 


30° 


40° 


45° 


50° 


60° 


70° 


f 


1 0.21 


1 0.52 


1.54 


3.91 


10.80 


18.70 


32.6 


118 


751 



112. Summary of Losses. — ^The total head, hi, lost in flow 
through a pipe line is then the sum of the six partial losses in 
head mentioned above, namely, 

hi = hi + h2 + hi + hi + hs + Ae- 

The values of these six terms may be tabulated as follows: 



Loss of head in pipe flow | 


Head lost at entrance 


».-o.»| 


Coefficient modified by 
nature of entrance and 
varies from 0.5 to 0.9 


Friction head 


ht-mj. 


Table 12 


Head lost at bends and el- 
bows 


h, - *»»•«• 




Head lost at sudden enlarge- 
ment 






Head lost at sudden con- 
traction 




For values of coeffi- 
cient, see table, p. 95 


Head lost at partially closed 
valve 


'•-^S 


See tabular values of 
f, pages 95 and 96 



From Eq. (58) we have 

h=\^^+ht = ^^+hi + hi + h + Jn + hi + ft,, 



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For a short straight pipe, where by a short pipe is meant one for 

I _ 
which 2 < 4,000, the loss in head is simply 

h^'^ + h + h 

and inserting in this the values of hj and h^ given, above, we have 



^l + 0.6+f(J) 



Combining this with the relation Q = At; = -j- and solving 

for d, we have as the diameter required to furnish a given dis- 
charge Q, 

d = 0.4789[ (l.6d + fl) ^*]^ (77) 

which is best solved by trial. 

For a long straight pipe, where -j > 4,000, all other losses may 

be neglected in comparison with friction loss, in which case the 
above formula simplifies into 

d = 0.4789 (^')K (78) 

113. Application. — To give a simple illustration of the applica- 
tion of the formula, suppose it is required to find the velocity of 
flow for a straight new cast-iron pipe, 1 ft. in diameter and 5,000 
ft. long, with no valve obstructions, which conducts water from 
a reservoir the surface of which is 150 ft. above the outlet of the 
pipe. 

In this case 



^l+0.5+/y) -^1+0.5 + 0.024 (^^) per sec. 

and the discharge is 

Q = At; = ^ X 8.9 X 60 = 419.4 cu. ft. per min. 

XDL HYDRAULIC GRADIBlf T 

114. Kinetic Pressure Head. — ^In the case of steady flow through 
a long pipe, if open piezometer tubes are inserted at different 
points of its length and at right angles to the pipe, the height 



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ELEMENTS OF HYDRAULICS 



at which the water stands in any tube represents the kinetic 
pressure head at this point. Assuming that the pipe is straight 
and of uniform cross-section, the velocity head is constant through- 
out, and therefore as the frictional head increases the pressure 
head decreases. The head lo^t in friction between any two points 
m and n (Fig. 87) as given by Eq. 60 Par. 99 is 



h 



"■•^U2? 



2g' 



and is therefore proportional to the distance I between these 
points. Consequently, the drop in the piezometer column be- 
tween any two points is proportional to their distance apart, and 
therefore the tops of these columns must lie in a straight line. 




Fig. 87. 



This line is called the hydraulic gradient^ or virtual slope of the 
pipe. Evidently the vertical ordinate between any point in the 
pipe and the hydraulic gradient measures the kinetic pressure 
head at the point in question. 

116. Slope of Hydraulic Gradient. — When a pipe is not straight, 
successive points on the hydraulic gradient may be determined by 
computing the loss of head between these points from the relation 

taking as successive values of I the length of pipe between the 
points considered. 

In water mains the vertical curvature of the pipe line is gen- 
erally small, and its effect on the hydraulic gradient is usually 
neglected. When, however, a valve or other obstruction occurs 
in a pipe there is a sudden drop in the hydraulic gradient at the 
obstruction, due to the loss of head caused by it. 

It should be noted that the upper end of the hydraulic gradient 



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99 



lies below the water level in the reservoir a distance equal to the 

head lost at entrance plus the velocity head. The slope of the 

hydraulic gradient is usually defined, however, as 

oi ^ 1. J T J' X static head 
Slope of hydraulic gradient = length of pipe ' 

which is equivalent to neglecting the velocity head and head lost 
at entrance, thereby making the assumed hydraulic gradient 
slightly steeper than it actually is. 

116. Peaks above Hydraulic Gradient. — When part of the 
pipe line rises above the hydraulic gradient (Fig. 88), the pressure 
in this portion must be less than atmospheric since the pressure 




Fig. 88. 

head V becomes negative. If the pipe is air-tight and filled be- 
fore the flow is started this will not affect the discharge. If the 
pipe is not air-tight, air will collect at the summit above the 
hydraulic gradient, changing the slope of the latter from AB to 
AC as indicated in Fig. 88, thereby reducing the head to V 
with a corresponding diminution of the flow. Before laying 
a long pipe line the hydraulic gradient should therefore be plotted 
on the profile to make sure there are no summits projecting above 
the gradient. In case such summits are imavoidable, provision 
should be made for exhausting the air which may collect at these 
points, so as to maintain full flow. 

XX. HYDRAULIC RADIUS 
117. Definition of Hydraulic Radius. — That part of the 
boundary of the cross-section of a channel or pipe which is in 
contact with the water in it is called the wetted perimeter, and the 
area of the cross-section of the stream divided by the wetted 
perimeter is called the hydraulic radius, or hydraulic mean depth. 
In what follows the hydraulic radius will be denoted by r, defined 
as 

TT J 1. J. Area of flow ,_^v 

Hydraulic radius, r = Netted perimeter <^»> 



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100 ELEMENTS OF HYDRAULICS 

Some writers apply the term hydraulic radius only to circular 
pipes, and use the term hydraulic mean depth for flow in channels. 
For a channel of rectangular cross-section having a breadth b 
and depth of water h, the hydraulic radius is 

^ ' b + 2h 
In a circular pipe of diameter d, running full, the hydraulic 
radius is 

For the same pipe running half full, 

ird^ 

-A ^ 
^ " Td "4' 

~2 

and is therefore the same as when the pipe is full. 

Other examples of the hydraulic radius are shown in Figs. 
108 to 114. 

118. Chezy's Formula for Pipe Flow. — The formula proposed 
by Chezy for the velocity of flow in a long pipe is 

V = CVrs, (80) 

where s denotes the slope of the hydraulic gradient, defined in the 
preceding article; r is the hydraulic radius, defined above; and 
C is an empirical constant which depends on the velocity of flow, 
diameter of pipe, and roughness of its lining. 

For a circular pipe flowing full Chezy's formula is identical 
with the formula for friction loss in a pipe, given by Eq. (60), 
Par. 99, namely. 

To show this identity, substitute in Cheasy's formula the values 

d , h 
r = -7 and s = t* 
4 { 



n 1^ 



Then it becomes 

(dh 

ITI' 
whence, by squaring and solving for A, it takes the form 



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FLOW OF WATER 101 

C!onsequently, if the constant term ^ is denoted by /, that is, 

Chezy's formula assumes the standard form, 

Chezy's formula also applies to flow in open channels, as ex- 
plained in Art. XXV. 

119. Kutter and Bazin Formulas for Pipe Flow. — ^The use of 

I v^ 
Weisbach's formula '^ = /^ ' o^* ^ i^^ shown, is equivalent to 

using Chezy's formula v = Cy/ri under the assumption that the 
coefficient C is constant. It has been found by experiment, 
however, that the coefficient C in Chezy's formula is not strictly 
constant for any particular pipe or channel, nor dependent only 
on the roughness of the pipe or channel lining, but that it also 
varies with the slope and the hydraulic radius. Expressions for 
C in terms of these variables have been proposed by various 
engineers, the two formulas most widely used being those due to 
Kutter and to Bazin, given in Pars. 140 and 142. Although 
Kutter's and Bazin's formulas were intended primarily to apply 
to flow in open channels, they are now also used extensively for 
calculating flow in pipes and conduits. Kutter's and Bazin's 
values of Chezy's coefficient are also given and tabulated in 
Tables 14 and 15. 

120. Williams and Hazen's Exponential Formula. — ^The varia- 
tion in Chezy's coefficient may also be taken into account by 
writing Chezy's formula in the exponential form 

V = Cr*s», (81) 

in which the exponents m and n as well as the coefficient C depend 
on the roughness of the channel lining, and to a certain extent 
on the form of the channel. An exponential formula of this type 
is more flexible as well as simpler than the Kutter and Bazin 
formulas, and is coming to be generally accepted as the standard 
type for calculating flow in pipes as well as in open channels. 

At present the most generally used formula of this type is 
that due to Williams and Hazen, namely, 

V = Cro "s« «H).001-« ?* (82) 



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ELEMENTS OF HYDRAULICS 



where v = velocity of flow in feet per second; 
8 = slope; 

r = hydraulic radius in feet; 
C = 100. 

To obviate the inconvenience of using this formula, Williams and 
Hazen have prepared extensive hydraulic tables, as well as a 
special slide rule. 

A brief table giving friction head in pipes computed by means 
of this formula is given at the end of this volume as Table 13. 

XXI. DIVIDED FLOW 

121, Compound Pipes. — In water works calculations the prob- 
lem often arises of determining the flow through a compound 
system of branching mains. 




WJ?WJWM??WW/^J. 



Profile 



M 



^^^^^^m 



U ^1 B 



V2 



S\ 



h 



•iiii^i^^^i^^^ii^^^^^^ti^ 



HiD 



Vi 



V-i 



N 



Vi 



Plan 



Fig. 89. 



To illustrate the method of finding the discharge through the 
various branches, consider first the simple case of a main tapped 
by a branch pipe which later returns to the main, as indictated 
in Fig. 89. The solution in this case is based on the fimdamental 
relation deduced in Par. 97, namely, 



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FLOW OF WATER r03 

where h denotes the static head, and h the head lost in friction. 
Using the notation indicated on the figure and considering the 
two branches separately, we obtain the following equations: 

For line ABMCD, 

trance + resistance at F-branch. (83) 

For line ABNCD, 

trance + resistance at F-branch. (84) 

By subtraction of these two equations we have 

^(Dl'=4:)l'. <") 

which shows that the frictional head lost in the branch BMC is 
equal to that lost in BNC. 

Since the total discharge through the branches is the same as 
that through the main before dividing and after uniting, we also 
have the two relations 

aiOi = a2V2 + a^vz = 04^4. (86) 

By assuming an average value for the frictional coefficient /, 
the four equations 83, 84, 85 and 86 may then be solved for 
the four unknowns Vi, V2, Vz, Va. Having found approximate 
values of the velocities, corresponding values of / may be sub- 
stituted in these equations and the solution repeated, thus giving 
more accurate values of the velocities. 

Having found the velocities, the discharge through the various 
pipes may be obtained from the relations 

Qi = Q4 = aivi = 04^4; Q2 = a2V2; Qz = asv». 

The solution for more complicated cases is identical with the 
above, except that more equations are involved. 

122. Branching Pipes. — ^Another simple case of divided flow 
which is often met is that in which a pipe AB of diameter d 
divides at some point B into two other pipes, BC and BD, of 
diameters di and da respectively, which discharge into reservoirs 
or into the air (Fig. 90). If any outlet, as C, is higher than the 



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ELEMENTS OF HYDRAULICS 



junction B, then in order for flow to take place in the direction 
BC, the hydraulic gradient must slope in this direction; that is to 
say there must be a drop in pressure between the junction B and 
the level of the outlet reservoir C, or, in the notation of the figure, 
the condition for flow in the direction BC is hi > h. 

Assuming this to be the case, the solution is obtained from the 
same fundamental relation as above, namely. 

h-'^ + h,. 



ga^j^g S i^gigg 



ssss^^ss^^^ 




D ^ 



Fig. 90. 

Using the notation indicated on the figure for length, diameter 
and velocity in the various pipes and considering one line at a 
time, we thus obtain the following equations: 
For line ABC 



For line ABD, 






(87) 



(88) 



Also, from the condition that the discharge through the main 
pipe must equal the sum of the discharges through the branches, 
denoting the crossnsectional areas by a, ai, aa respectively, we 
have 

av = aiVi + a2V2. (89) 

By assimiing an average value for the frictional coefficient /, 
these three equations may then be solved for the thr^e unknowns 
V, vx and v^. Having thus found approximate values of the veloc- 
ities, the exact value of / corresponding to each velocity may be 
substituted in the above equations and the solution repeated. 



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FLOW OF WATER 105 

giving more accurate values of the velocities. Having found the 
velocities, the discharge from each pipe is obtained at once from 
the relations 

The method of solution is the same for any number of branches, 
there being as many equations in any given case as there are un- 
known velocities to be determined. 

Other simple cases of divided flow are illustrated in the numer- 
ical examples at the end of the chapter. 

XXn. FIRE STREAMS 

123. Freeman's Experiments. — ^Extensive and accurate experi- 
ments on discharge through fire hose and nozzles were made by 
John R. Freeman at Lawrence, Mass., in 1888 and 1890.^ 

From these experiments it was found that the smooth cone 
nozzle with simple play pipe is the most efficient for fire streams, 
the coefficient of discharge being nearly constant for the various 
types tried and having an average value of 0.974 for smooth cone 
nozzles and 0.74 for square ring nozzles. 

The friction losses for fire hose were found to be given approxi- 
mately by the empirical formula 

_ P , . 
P ^ Fld^K» "^ ^ 
6472 
with notation as given below. 

For fire hose laid in ordinary smooth curves but not cramped or 
kinked, the friction loss was found to be about 6 per cent, greater 
than in perfectly straight hose. 

124. Eormulas for Discharge. — The following formulas for 
discharge were deduced by Freeman from these experiments. 
Notation: 

j Q = discharge in cubic feet per second; 

' Q = discharge in gallons per minute = 448.83Q; 

h == piezometer reading at base of nozzle in feet of water; 
p == pressure at base of nozzle in lb. per sq. in. = 0.434A; 
P = hydrant pressure in lb. per sq. in.; 
K = coefficient of discharge =- 0.974 for smooth cone nozzles 

and 0.74 for square ring nozzles; 
C« = coefficient of contraction; 
» Tran». Am. Soc. C. E., vol. 21, pp. 303-482; voL 24, pp. 492-527. 



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ELEMENTS OP HYDRAULICS 



d ^ diameter of nozzle orifice in inches; 
D = diam. of channel, where pressure is measured, in in.; 
I = length of hose in feet; 

H = total hydrostatic head in feet = ,,, .• 

F = 30 for unlined linen hose 2)4 ui- in diam.; 

= 26 for inferior rubber-lined hose 2}4 in* in diam.; 
= 13 for best rubber-lined hose 2}4 in. in diam. 
Then 



and 



Q » 0.01374Ed' 



0.0664eKd< 



G = 19.636Kd' 



= 29.83 Kd* 



d\* 



P 



d\« 



P 



H 



H 



(90) 



d\* 



-■^•S) 



H 



91 



126. Height of Effective Fire Stream. — It was also found that 
the height, y, of extreme drops in still air from nozzles ranging 
in size from % in. to 1% in. in diameter was given by the formula 

y = H -0.00136-^' • (92) 

The height of a first-class fire stream will then be a certain frac- 
tion of y as indicated in the following table: 



When y 



Height of first class fire stream = 



50 ft. 



0.82 y 



75 ft. 



0.79 y 



100 ft. 



0.73 1/ 



125 ft. 



0.67 y 



150 ft 



0.63 y 



Table 11 is abridged from a similar table computed by Freeman 
from these and other formulas, not here given, and will be found 
convenient to use in solving fire-stream problems. 

126. Fleming's Experiments. — A series of experiments on fire 
streams from small hose and nozzles was made by Virgil R. Flem- 
ing at the University of Illinois in 1911. The results of these 
experiments are also summarized in Table 11. 



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FLOW OF WATER 107 

XXm. EXPERIMENTS ON THE FLOW OF WATER 

127, Verification of Theory by Experiment. — The subject of 
hydraulics as presented in an elementary text book is necessarily 
limited to simple demonstrations of the fundamental principles. 
It should not be inferred from this that the subject, is largely ex- 
perimental and not susceptible of mathematical analysis. As a 
matter of fact, hydrodynamics is one of the most diflBlcult branches 
of applied mathematics, and its development has absorbed the 
best efforts of such eminent mathematical physicists as Poinsot, 
Kirchhoflf, Helmholz, Maxwell, Kelvin, Stokes and Lamb. Nat- 
urally the results are too technical to be generally appreciated, 
but aflford a rich field for study to those with sufficient mathe- 
matical preparation. 

Some of the results concerning the fiow of liquids derived by 
mathematical analysis have been verified experimentally by the 
English engineers. Professor H. S. Hele-Shaw and Professor 
Osborne Reynolds. The chief importance of these experiments 
is that they serve to visualize difficult theoretical results. 

128. Method of Conducting Experiments. — In Par. 47 a stream 
line was defined as the path followed by a particle of liquid in its 
motion. A set of stream lines distributed through a fiowing 
liquid therefore completely determines the nature of the fiow. 
To make such stream lines visible, so as to make it possible to 
actually trace the motion of the particles of a clear fiuid, both 
experimenters named above allowed small bubbles of air to enter 




Sudden contraction. Sudden'enlargement. 

Fig. 91. 

a fiowing stream. These bubbles do not make the motion 
directly visible to the eye, but by making the pipe or channel of 
glass and projecting a portion of it on a screen by means of a 
lantern, its image on the screen as viewed in this transmitted 
Ught clearly shows certain characteristic features. 

129. Effect of Sudden Contraction or Enlargement. — Figures 
91 and 92, reproduced by permission of Professor Hele-Shaw, 



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108 



ELEMENTS OF HYDRAULICS 



show the effect of a sudden- contraction or enlargement of the 
channel section. It is noteworthy that the disturbance or eddy- 
ing is much greater for a sudden enlargement than for a sudden 




Ck)ntraction. 



Fig. 92. 



Enlargement. 



contraction. This is due to the inertia of the fluid which pre- 
vents it from immediately fiUing the channel after passing through 
the orifice. This also confirms what has already been observed 




in practice, namely, that the loss of energy due to a sudden en- 
largement in a pipe is much greater than that due to a corre- 
sponding contraction. 



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130. Disturbance Produced by Obstacle in Current. — If the 
channel is of considerable extent and a small obstacle is placed in 
it, the stream lines curve around the obstacle, leaving a small 
space behind it, as shown in Fig. 93. If the object is a square 
block or flat plate this effect is greatly magnified, as shown in 
Fig. 94. The water is prevented from closing at once behind the 
obstacle by reason of its inertia. This indicates why the design 
of the stem of a ship is so much more important than that of the 
bow, since if there are eddies in the wake of a ship, the pressure 
of the water at the stern is decreased, thereby increasing by juat 
this much the effective resistance to motion at the bow. 




Fia. 94. 

131. Stream-line Motion in Thin Film. — In these experiments 
it was also observed that there was always a clear film of liquid, 
or border Une, on the sides of the channel and around the obstacle. 
This observed fact was accounted for on the ground that by 
reason of the friction between a viscous liquid and the sides of the 
channel or obstacle, the thin film of liquid affected was not mov- 
ing with turbulent motion but with true stream-line motion, as 
in an ideal fluid. To isolate this film so as to observe its motion, 
water was allowed to flow between two plates of glass in a sheet 
so thin that its depth corresponded to the thickness of the border 
line previously observed. When this was done it immediately 
became apparent that the flow was no longer turbulent but a 
steady stream-line motion. The flow of a viscous fluid Uke glyc- 



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ELEMENTS OF HYDRAULICS 




Fig. 95. 



Fig. 96. 




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FLOW OF WATER 



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erine in a thin film thus not only eliminates turbulent flow, but 
also to a certain extent the inertia effects, thereby resulting in 
true stream-line flow. 

132. Cylinder and Flat Plate. — To make the stream lines vis- 
ible, colored liquid was injected through a series of small openings, 
the result being to produce an equal number of colored bands or 
stream lines in the liquid. Fig. 95 shows these stream lines for 
a cylinder, and Fig. 96 for a flat plate placed directly across the 
current, while Fig. 97 shows a comparison of theory and experi- 
ment for a flat plate inclined to the current. 

133. Velocity and Pressure. — The variation in thickness of the 
bands is due to the difference in velocity in various parts of the 
channel, the bands of course being thinnest where the velocity is 




— ► 



Fig. 98. 

greatest. Since a decrease in velocity is accompanied by a cer- 
tain increase in pressure, the wide bands before and behind the 
obstacle indicate a region of higher pressure. This accounts for 
the standing bow and stern waves of a ship, whereas the narrow- 
ing of the bands at the sides indicates an increase of velocity and 
reduction of pressure, and accounts for the depression of the water 
level at this part of a ship. 

In the case of a sudden contraction or enlargement of the chan- 
nel section, the true stream-line nature of the flow was clearly 
apparent, as shown in Fig. 98, the stream lines following closely 
the form derived by mathematical analysis for a perfect fluid. 

XXIV. MODERN SIPHONS 

134. Principle of Operation. — In its simplest form, a siphon is 
merely an inverted U-shaped tube, with one leg longer than the 
other, which is used for emptying tanks from the top when no 



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ELEMENTS OF HYDRAULICS 



outlet is available below the surface. In use, the tube is filled 
with liquid and the ends corked, or otherwise closed. The short 
end of the tube is then placed in the reservoir to be lowered, so 
that the level of the end outside is lower than the surface of the 
reservoir (Fig. 99). When the ends of the tube are opened, the 
liquid in the reservoir begins to flow through the tube with a head, 
hy equal to the difference in level between the surface of the 
reservoir and the lower, or outer, end of the tube. If the inner 
end of the siphon is placed close to the bottom of the reservoir 




Fig. 99 

it can be practically emptied in this manner. For emptying 
small tanks a siphon can conveniently be made of a piece of ordi- 
nary tubing or hose. 

136. Siphon Spillways. — The siphon principle is now being 
applied on a large scale in the construction of spillways, locks and 
wheel settings. 

In many cases the common overflow spillway requires such a 
great length for proper regulation of the pond or forebay as to 
make its use undesirable. This form of spillway is also ineffi- 
cient because of the low head under which it operates. This 
often makes it necessary to use flashboards and automatic gates 
to increase the head and consequently the velocity of discharge. 
The available head, however, is the total head between the water 
surfaces above and below the dam, and this may be utilized by 
building water passages through the dam and submerging the 
downstream end, thus forming a siphon. 

Such siphon spillways have been in use in Italy for a niunber of 
years. Until recently it was supposed that they could not be 
used in colder countries on account of the impossibility of keep- 



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113 



ing them free from ice. The first siphon spillways to be built 
outside of Italy were in Zurich, Switzerland, where for several 
years a number have been in successful operation on streams 
where ice forms for 2 or 3 months every year. 

In designing siphons there are at least two important prin- 
ciples which must be observed.^ 

, (a) The upper part must be so designed that as soon as the 
water rises above the level to be maintained, the siphon intake 
is sealed and remains sealed until the water level is brought down 
again to normal. The air openings must then be large enough to 
admit sufficient air to break the siphonic action immediately. 
Both of these features may be secured by having long and sharp 
edges on the intake to the siphon at the normal water level. 




CourteBy Eng. Record, 

Fig. 100. — Siphon spillway in use at Seon, Switzerland. 

(6) The lower edge of the siphon must be submerged deep 
enough to secure a constant seal. The upper edge of this open- 
ing must also be as sharp as possible to permit of an easy escape 
of the enclosed air. 

A siphon spillway in use at Seon, Switzerland, is shown in 
Fig. 100. The action of such a siphon is as follows: The pond 
rises until the water seals the upper sharp-edged slots of the in- 
take. As soon as this happens, the water flowing through the 
siphon carries the air with it, which escapes around the sharp 
lower edge, and the siphon is primed. The siphon then con- 
tinues in full action until the pond level is lowered sufficiently 
to admit air under the upstream edge. 

1 HUiLBBRG, Eng. Record, May 3, 1913, p. 488. 
8 



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ELEMENTS OF HYDRAULICS 



It might be supposed that siphons could be used only where 
the difference in elevation of the water surfaces was less than the 
suction head for the siphon, but this condition has been found to 
be not essential. The siphon spillway at Gibswil, Switzerland, 
shown in section in Fig. ^101, operates under a head of 52.48 ft. 
It consists of a riveted steel pipe }^ in. thick, tapering from 
31.5 to 23.6 in., this taper being intended to^keep the water 
column from parting under the high head. At^the upper water 
level the pipe is cut on a- horizontal plane and covered by a 
reinforced-concrete hood, projecting 3.28 ft. below normal water 
level, the purpose of this hood being to prevent ice from clogging 




Courtesy Eng, Record. 
Fig. 101. — High-head siphon spillway in use at Gibfwil, Switzerland. 

the siphon. To prevent the water from being lowered to the 
edge of the hood by siphonic action, long, narrow slots are cut 
through it on three sides at the normal level of the pool, these 
slots closing as the water rises above normal. 

A test of this siphon gave a discharge of 98.9 cu. ft. per second, 
but as the air slots were not all fully closed, it was estimated 
that the marimum discharge would be about 123.6 cu. ft. per 
second. As the end area is 3.03 sq. ft., this would mean a vdocity 
of 40.8 ft. per second. The friction head in the siphon itself 
was 10.2 ft., leaving a net effective head of 62.48 - 10.2 = 42.28 
ft. Since the theoretical velocity of flow due to this head is 

V = 'V2gh = 62.18 ft. per second, the coefficient of discharge, 

40.8 
or efficiency, of the siphon is koTq = 78 per cent. 



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115 



The first siphon spillways to be constructed in this country 
were the three located on the champlain division of the New 
York State Barge Canal. 

At the place shown in Fig. 102 it was necessary to provide for 
a maximum outflow of about 700 cu. ft. per second and to limit 
the fluctuation of water surface to about 1 ft. The ordinary 
waste weir of a capacity sufficient to take care of this flow, with 
a depth of only 1 ft. of water on the crest, would require a spill- 
way 200 ft. long. The siphon spillway measures only 57 ft. 




Fig. 



102. — Siphon spillway, Champlain division, New York State Barge 
Canal. 



between abutments and accomplishes the same results. This 
particular structure consists of four siphons, each having a^ 
cross-sectional area of 7% sq. ft. and working under a lOj^-ft.' 
head. There is also a 20-ft. drift gap to carry off floating debris. 
The main features of construction are shown in Fig. 103. The| 
siphon spillway was designed and patented by Mr. George F. 
Stickney, one of the Barge Canal engineers. 

Another instance is furnished by the second hydro-electric 
development of the Tennessee Power Co. on the Ocoee River, 
Tenn., where a spillway consisting of a battery of eight siphons 
has been constructed. The general features of the design are 
shown in Fig. 104. The entrance area is located 5}4 ft. below 



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ELEMENTS OF HYDRAULICS 





09 

O 



.a 

I 
I 

I 

'a 






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117 



water surface to insure freedom from floating debris, and is 
33^ by 6 ft. in area, protected by ^-in. screen bars spaced 
4 in. apart on centers. The entrance area gradually diminishes 
in the upper leg to 8 by 1 ft. at the top or throat, the larger dimen- 
sion being horizontal. The lower leg is rectangular in cross- 
section and 8 sq. ft. in area throughout, but gradually changes in 
shape to a point 12.8 ft. below the crest, whence its section is 4 
by 2 ft. to the outlet. Four of the siphons operate under a head 
of 27.2 ft., and the other four under a head of 19.2 ft. Hori- 
zontal air inlets 6 by 18 in. in section are provided for each 
siphon unit, extending through the throat casting to the upstream 
face of the dam. 

In a test of these siphons it was found that two of them dis- 
charged 422.8 cu. ft. per second, giving a velocity of flow of 






ewjjfl 













I 



rr 



ffl 




Fig. 104. — Siphon spillway constructed at Ocoee River, Tennessee. 

422.8 

2 y g = 26.425 ft. per second. Since the average head acting 

on the siphon during the test was 26.65 ft., the theoretical veloc- 
ity of flow is t; = '\r2gh = 40.54 ft. per second. The efficiency 



in this case is therefore 



26.425 
40.54 



= 65 per cent. 



136. Siphon Lock. — The siphon lock on the New York State 
Barge Canal is located in the City of Oswego, and is the only 
lock of this type in this country and the largest ever built on 
this principle. It consists of two siphons, as shown in Fig. 105 
the crown of each being connected by a 4-in. pipe to an air tank 
in which a partial vacuum is maintained. To start the flow, the 
air valve is opened, the vacuimi in the tank drawing the air from 
the siphon and thereby starting the flow. When the siphon is 
discharging fully, its draft is such that the air is sucked out of 
the tank, thus restoring the partial vacuum. To stop the flow, 



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ELEMENTS OF HYDRAULICS 




o 






OS 

I 



I 

44 



d 

T 

o 






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FLOW OF WATER 



119 



outer air is admitted to the crest of the siphon by another valve, 
thereby breaking the flow, as indicated in Fig. 106. The operat- 




ing power is thus self-renewing, and, except for air leakage, lock- 
ages can be conducted by merely manipulating the 4-in. air 



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ELEMENTS OF HYDRAULICS 



valves. However, to avoid the necessity of refilling the -tank 
when traffic is infrequent, it is customary to close the 20-in. 
outlet valve, thus holding the water in the tank. Using both 
siphons, the lock chamber can be filled in 4J^ to 5 min., and 
emptied in 5}4 to 6 min. 

137. Siphon Wheel Settings. — The siphon principle has been 
utilized in several instances for waterwheel intakes. Fig. 107 
shows the type of siphonic wheel setting used in the pump house 



r:\ 




Courtesy Eng. Record. 

Fig. 107.— Siphonic wheel setting in the pump house at Geneva, Switzerland. 

at Geneva, Switzerland. The chief advantage of this type is 
that it eliminates the use of headgates, which in design and opera- 
tion are one of the most difficult details of a hydro-electric 
development. The design and operation of such an intake is 
very similar to that for the siphonic locks at Oswego, described 
above. 

XXV. FLOW IN OPEN CHANNELS 

138. Open and Closed Conduits. — Conduits for conveying 
water are usually classified as open and closed. By a closed 



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FLOW OF WATER 121 

conduit is meant one flowing under pressure, as in the case of ordi- 
nary pipe flow discussed in Art. XVII. Water mains, penstocks, 
draft tubes and fire hose are all examples of closed conduits. 

Open channels, or conduits, are those in which the upper sur- 
face of the liquid is exposed to atmospheric pressure only, the 
pressure at any point in the stream depending merely on the 
depth of this point below the free surface. Rivers, canals, 
flumes, aqueducts and sewers are ordinarily open channels. A 
river or canal, however, may temporarily become a closed channel 
when covered with ice, and an aqueduct or sewer may also be- 
come a closed channel if flowing full under pressure. 

139. Steady Unifonn Flow. — The fundamental laws applying 
to flow in open and closed channels are probably identical, and in 
the case of steady, uniform flow the same formulas ajpply to both. 
For steady flow in an open channel the quantity of water passing 
any transverse section W the stream is constant, and for uniform 
flow the mean velocity is also constant. Under these conditions 
the cross-sectional area of the stream is constant throughout its 
length, and the hydrauUc gradient is the slope of the surface of 
the stream. The formula for velocity of flow is then the one 
given in Par. 118 under the name of Chezy's formula, namely 

V = CVSre. (93) 

140. Eutter's Formula. — ^Numerous experiments have been 
made to determine the value of the coefficient C for open channels. 
In 1869, E. Ganguillet and W. R. Kutter, two Swiss engineers, 
made a careful determination of this constant, the result being 
expressed in the following form: 



V = 



41.6. + »=»^ + J:Mi 



+ [u.« + <^]^. 



Vri (94) 



in which 

« = hydraulic gradient, or slope of channel; 

V J ,. J. area of flow 

r = hydraulic radius = — tt— j — —-. — r— ; 
•^ wetted penmeter' 

n = coefficient of roughness. 

The coefficient of roughness, w, depends on the nature of the 
channel lining. Approximate values of n for various surfaces are 
given in the following table: 



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122 



ELEMENTS OF HYDRAULICS 



Nature of Channel Lining 



Planed timber carefully joined, glazed or enameled siirfaces 

Smooth clean cement 

Cement mortar, one-third sand 

Unplaned timber or good new brickwork 

Smooth stonework, vitrified sewer pipe and ordinary bridcwork . 

Rough ashlar and good rubble masonry 

Urm gravel 

Ordinary earth 

Earth with stones, weeds, etc 

Earth or gravel in bad condition 



0.009 
0.010 
0.011 
0.012 
0.013 
0.017 
0.020 
0.025 
0.030 
0.035 



141. Limitations to Eutter's Fonnula. — Kutter's formula, Eq. 
(94), is widely used and is reliable when applied to steady, uni- 
form flow under normal conditions. From a study of the data 
on which this formula is based, its use has been found to be sub- 
ject to the following Umitations: 

It is not rehable for hydrauUc radii greater than 10 ft., or veloci- 
ties greater than 10 ft. per second, or slopes flatter than 1 in 
10,000. Within these Umits a variation of about 5 per cent, may 
be expected between actual results and those computed from the 
formula. 

Table 15 gives numerical values of the coefficient C calculated 
from Eq. (94). 

142. Bazin's Fonnula. — In 1897, H. Bazin also made a careful 
determination of the coefficient C from all the experimental data 
then available, as the result of which he proposed the following 
formula: 

87 



V = 



0.552 + -^ 

V r 



V: 



rs 



(95) 



where r = hydraulic radius; 

m = coefficient of roughness. 

Bazin's formula has the advantage of being simpler than 
Kutter's, and is independent of the slope s. Values of the coeffi- 
cient of roughness, m, for use with this formula are given in the 
following table: 



Nature of Channel Lining 



m 



Planed timber or smooth cement 

Unplaned timber, well-laid brick or concrete 

Ashlar, good rubble masonry or poor brickwork . 

Earth in good condition 

Earth in ordinary condition 

Earth in bad condition 



0.06 
0.16 
0.46 
0.86 
1.30 
1.75 



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FLOW OF WATER 123 

Table 14 gives numerical values of the coefficient C calculated 
from Eq. (95). 

143. Eutter's Simplified Fonnula. — ^A simplified form of Eut- 
ter's formula which is also widely used is the following: 

where 6 is a coefficient of roughness which varies from 0.12 to 
2.44. For ordinary sewer work the value of this coefficient may 
be assumed as 6 = 0.35. 



XXVI. CHANNEL CROSS-SECTION 

144. Condition for Maximum Discharge. — From the Chezy 
formula for fiow in open channels, namely, 

Q = Av = ACVrSy 

it is evident that for a given stream section A and given slope s, 
the maximum discharge wiU be obtained for that form of cross- 
section for which the hydraulic radius r is ja maximum. Since 

area of fiow 
r = 



wetted perimeter 



this condition means that for constant area the radius r, and 
therefore the discharge, is a maximum when the wetted perimeter 
is a minimum. The reason for this is simply that by making the 
area of contact between channel lining and water as small as 
possible, the f rictional resistance is reduced to a minimum, thus 
giving the maximum discharge. 

145. Maximum Hydraulic Efficiency. — ^In consequence of this, 
it follows that the maximum hydraulic efficiency is obtained 
from fiC semicircular cross-section, since for a given area its wetted 
perimeter is less than for any other form (Fig. 108). For rec- 
tangular sections the half square has the least perimeter for a 
given area, and consequently is most efficient (Fig. 109). Simi- 
larly, for a trapezoidal section the half hexagon is the most effi- 
cient (Fig. 110). In each case the hydraulic radius is half the 
water depth, as proved below. 

In the case of unUned open channels it is necessary to use the 
trapezoidal section, the slope of the sides being determined by 
the nature of the soil forming the sides. This angle having been 



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124 



ELEMENTS OF HYDRAULICS 



determined, the best proportions for the section are obtained by 
making the sides and bottom of the channel tangent to a semi- 
circle drawn with center in the water surface (Fig. 111). 

146. Regular Circumscribed Polygon. — ^Any section which 
forms half of a regular polygon of an even number of sides, and 
has each of its faces tangent to a semicircle having its center in 





Semicircle 
Fig. 108. 



4R tan aO** 



Half Square 
Fig. 109. 




Half Octagon 
Fig. 112. 



Triangle 
Fig. 113. 



the water surface, will have its hydraulic radius equal to half the 
radius of this inscribed circle (Figs. 108-113). To prove this, 
draw radii from the center of the inscribed circle to each angle of 
the polygon. Then since the area of each of the triangles so 
formed is equal to one-half its base times its altitude, and since 
the altitude in each case is a radius of the inscribed circle, the 
total area is 

Area = ^ X perimeter. 



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FLOW OF WATER 



125 



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depth 

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Digitized by 



Google 



126 



ELEMENTS OF HYDRAULICS 



Consequently the hydraulic radius r is 



12 



r = 



area of flow 



X perimeter 



wetted perimeter perimeter 



B 
2* 



147. Properties of Circular and Oval Sections. — For circular 
and oval cross-sections, the maximum velocity and maximum dis- 
charge are obtained when the conduit 
"f^ is flowing partly f uU, as apparent from 
the table on page 125, which is a 
collection of the most important data 
for circular and oval sections,^ as shown 
in Figs. 114 and 115. 

Theoretically, the maximum discharge 
for a circular pipe occurs when the pipe 
is filled to a depth of 0.949D, but if 
it is attempted to maintain flow at this 
depth, the waves formed in the pipe 
strike against the top, filhng it at periodic intervals and thus 
producing impact losses. To obtain the maxiipum discharge 
without danger of impact, the actual depth of flow should not 
exceed %D. 




Circular Section 
Fig. 114. 




l_ 

I Standard Oval Section 



Fig. 115. 



XXVn. FLOW IN NATURAL CHANNELS 

148. Stream Gaging. — ^In the case of a stream flowing in a 
natural channel the conditions determining the flow are so vari- 

» Wbyrauch, '^Hydrauliches Rechnen," S. 61. 



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FLOW OF WATER 



127 



able that no formula for computing the discharge has been de- 
vised that can claim to give results even approximately correct. 
To obtain accurate results, direct measurements of cross-sections 
and velocities must be made in the field. 

The two methods of direct measurement in general use are as 
follows: Either 




Fig. 116. — ^Electric current meter. 

1. The construction of a weir across the stream, and the cal- 
culation of the discharge from a weir formula; or, if this is not 
feasible, 

2. The measurement of cross-sections of the stream by means 
of soundings taken at intervals, and the determination of ^ average 
velocities by a current meter or floats. 



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128 



ELEMENTS OF HYDRAULICS 



The first of these methods is explained in Art. XII. 

149. Current Meter Measurements. — ^The current meter, one 
type of which is shown in Fig. 116, consists essentially of a bucket 
wheel with a heavy weight suspended from it to keep its axis 
horizontal, and a vane to keep it directed against the cur- 
rent, together with some form of counter to indicate the speed 
at which the wheel revolves. The meter is first rated by towing 
it through still water at various known velocities and tabulating 
the corresponding wheel speeds. From these results a table, or 
chart, is constructed ^ving the velocity of the current corre- 
sponding to any given speed of the wheel as indicated by the 
counter. This method of calibration, however, is more or less 
inaccurate, as apparent from Du Buat's paradox, explained in 
Par. 159. 

160. Float Measurements. — ^When fioats are used to determine 
the velocity, a uniform stretch of the stream is selected, and two 

Water Surface 






5 



Velocity ^ 



1 



s 



:^ 



;^ 



y 



wmmmmmmmmA 

Fig. 117. 



Bed of > Stream 



cross-sections chosen at a known distance apart. Floats are 
then put into the stream above the upper section and their 
times of transit from one section to the other observed by means 
of a stop watch. A subsurface float is commonly used, so ar- 
ranged that it can be run at any desired depth, its position being 
located by means of a small surface float attached to it. 

If the cross-section of the stream is fairly uniform, rod floats 
may be used. These consist of hollow tubes, so weighted as to 
float upright and extend nearly to the bottom. The velocity 
of the float may then be assumed to be equal to the mean velocity 
of the vertical strip through which it runs. 

161. Variation of Velocity with Depth. — ^The results of such 
measurements show, in general, that the velocity of a stream is 



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FLOW OF WATER 



129 



greatest midway between the banks and just beneath the surface. 
In particular, the velocities at different depths along any vertical 
are found to vary as the ordinates to a parabola, the axis of the 
parabola being vertical and its vertex just beneath the surface, as 
indicated in Fig. 117. From this relation it follows that if a 
float is adjusted to run at about 0.6 of the depth in any vertical 
strip, it will move with approximately the average velocity of all 
the particles in the vertical strip through which it runs. 

162. Calculation of Discharge. — ^In order to calculate the dis- 
charge it is necessary to measure the area of a cross-section as 
well as the average velocities at various points of this section. 
The total crosshsection is therefore subdivided into parts, say 
Ai, A^i Azf etc. (Fig. 118), the area of each being determined by 
measuring the ordinates by means of soundings. The average 




velocity for each division is then measured by one of the methods 
explained above, and finally the discharge is computed from the 
relation 

Q = AiVi + A2V2 + AzVz + 

XXVm. THE PITOT TUBE 

163. Description of Instrument. — ^An important device for 
measuring the velocity of flow is the instrument known as the 
Pilot tvbe. In 1732 Pitot observed that if a small vertical tube, 
open at both ends, with one end bent at a right angile, was dipped 
in a current so that the horizontal arm was directed against the 
current as indicated in Fig. 119^1, the liquid rises in the vertical 
arm to a height proportional to the velocity head. The height 
of the column sustained in this way, or hydrostatic head, is not 
exactly equal to the velocity head on account of the disturbance 
created by the presence of the tube. No matter how small the 
tube may be, its dimensions are never negligible, and its presence 
9 



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130 



ELEMENTS OF HYDRAULICS 



has the effect of causing the filaments of liquid, or stream lines, 
to curve around it, thereby considerably modifying the pressure. 
Since the column of liquid in the tube is sustained by the impact 
of the current, this arrangement is called an impact tvbe. 

If a straight vertical tube is submerged, or a bent tube having 
its horizontal arm directed transversely, that is, perpendicularly, 
to the current, the presence of the tube causes the stream lines to 
turn their concavity toward the orifice, thereby producing a suc- 
tion which is made apparent by a lowering of the water level in 
this tube, as shown in Fig. 119B. In the case of the bent tube, if 



Impact 
Tube 



Suction or 

Pressure ' Trailing 
Tube Tube 



At 


■I-.I.I 1 

^ . 


c 




= 


r 1" 


1= 


z 




-= 1 


■£ 


- 




-E 


= 




- 


r 


_: . - 


Z - 


~ 


= 


— = 


1^ z 


:z 


r 


— 


— 


- 


~ z 


- z 


-'z 


1 


1 'z 


i \ 


.z 


- 


-.-. 


M 


-E ^^^ 


-^ 


7 : 


" 


^^^n- - 



Direction of Flow 

Fig. 119. 



the horizontal arm is directed with the current, as shown in Fig. 
119C, the effect is not so pronounced as when the tube is turned 
at right angles to the current, as for ordinary velocities the suc- 
tion effect due to viscosity predominates over that due to change 
in energy. When the horizontal arm of a bent tube is directed 
with the current, the arrangement is called a suction or trailing 
tvbe. 

It is practically impossible, however, to obtain satisfactory 
numerical results with this simple type of Pitot tube, as in the 
case of flow in open channels the free surface of the liquid is 



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FLOW OF WATER 



131 



i: 






usually disturbed by waves and ripples and other variations in 
level, which are often of the same order of magnitude as the quan- 
tities to be measured; while in the case of pipe flow under pressure 
there are other conditions which strongly aflfect the result, as will 
appear in what follows. 

164. Darcy's Modification of Ktot's Tube —In 1850 Darcy 
modified the Pitot tube so as to adapt it to general current 
measurements. This modification consisted in combining two 
Pitot tubes, as shown in Fig. 120, the 
orifice of the impact tube being directed 
upstream, and the orifice of the suction 
tube transverse to the current. In some 
forms of this apparatus, the suction tube 
is of the trailing type, that is, the hori- 
zontal arm is turned directly downstream. 

To make the readings more accurate, 
the difference in elevation of the water 
in the two tubes is magnified by means 
of a differential gage, as shown in Fig. 
120. Here A denotes the impact tube 
and B the suction tube (often called the 
pressure tube), connected with the tubes 
C and Z>, between which is a graduated 
scale. After placing the apparatus in 
the stream to be gaged, the air in both 
tubes is equally rarified by suction at F, 
thereby causing the water level in both 
to rise proportional amounts. The valve 
at F is then closed, also the valve at E, 
and the apparatus is lifted from the water 
and the reading on the scale taken. 

It was assumed by Pitot and Darcy that the difference in level 

in the tubes was proportional to the velocity head «-, where v 

denotes the velocity of the current. CaUing hi and hi the dif- 
ferences in level, that is, the elevation or depression of the water 
in the impact and suction tubes respectively, and mi, mi the con- 
stants of proportionality, we have therefore 



[ 



T" 



J„, 



D 



>0^ 



B 



Fig. 120. 



WlAl = Off ~ ^2^i* 



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132 ELEMENTS OF HYDRAULICS 

If, then, h denotes the difference in elevation in the two tubes 
(Fig. 120), we hSve 

The velocity v is therefore given in terms of h by the equation 

v = mV^ (97) 

where 



'4, 






ifhi + wis 



The coefficient m depends, like mi and m2, on the form and dimen- 
sions of the apparatus, and when^ properly determined is a con- 
stant for each instrument, provided that the conditions imder 
which the instrument is used are the same as those for which m 
was determined. 
The vMue of m in this formula has been found to vary from 1 

to as low as 0.7; the value m = 1 corresponding to A = «-; and 

the value m = 0.7 to A ?= — . The explanation of this apparent 

discrepancy is given below under the theory of the impact tube. 
In the case of variable velocity of flow it has been shown by 
Rateau^ that the Pitot, or Darcy, tube measures not the mean 
velocity but the mean of the squares of the velocities at the point 
where it is placed during the experiment. To obtain the mean 

velocity it is necessary to multiply ^r by a coefficient which 

^ . . 

varies ietccording to the rate of change of the velocity with respect 

to the time. From Rateau's experiments this coefficient was 
found to vary from 1.012 to 1.37, having a mean value of 1.16. 
This corresponds to a mean value for m of 0.93. 

166. Pitometer. — ^A recent modification of the Pitot tube is an 
instrument called the Pitometer (Fig. 121). The mouthpiece of 
this apparatus consists of two small orifices pointing in oppo- 
site directions and each provided with a cutwater, as shown in the 
figure. When in use, these are set in line parallel to the current, 
so that one points directly against the current and the other with 
it. The differential gage used with this instrument consists of a 
U-tube, one arm of which is connected* with one mouthpiece and 
the other arm with the other mouthpiece, and which is about 

^ Anndlea dea Mines, Mars, 1898. 



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FLOW OF WATF!R 



133 



half filled with a mixture of gasoline and carbon tetrachloride, 
colored dark red. The formula for velocity as measured by this 
instrument is given in the form 

V = k[2gi8 - l)cq^ 



_J. 




Fig. 121. 

where k = empirical constant = 0.84 for the instrument as 

manufactured and calibrated; 
8 = specific weight of the tetrachloride mixture = 1.26; 
d = difference in elevation in feet between the tops of the 

two colunms of tetrachloride. 

Inserting these numerical values, the formula reduces to 

't; = 3M8Vdf 

It is claimed that velocities as low as 3^ ft. per second can be 
measm^ed with this instrument. 



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ELEMENTS OF HYDRAULICS 



166. Pitot Recorders. — ^The Pitot meter is used in power 
houses, pumping stations and other places where a Venturi tube 
cannot be installed, and is invaluable as a water- works instrument 
to determine the pipe flow in any pipe of the system. 




Fia. 122. — Pilot recording meter. Simplex valve and meter Co. 

A recent portable type, especially adapted to this purpose is 
shown in Fig. 122. This instrument is 34 in. high, weighs 75 lb., 
and fiu'nishes charts of the Bristol type which are averaged with 
a special planimeter furnished with the instrument. A 1-in. 
tap in the water main is required for inserting the Pitot 
mouthpiece. 



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FLOW OF WATER 135 

It is claimed that these instruments have a range from }4 ft- 
per second to any desired maximum. 

167. Theory of the Impact Tube. — ^The wide variation in the 
range of coeflBcients recommended by hydraulic engineers for use 
with the Pitot tube can be accounted for only on the ground 
bf a faulty understanding of the hydraulic principles on which 
its action is based. The most important of these are indicated 
below, without presuming to be a complete exposition of its 
action. 

It will be shown in Par. 162 that the force produced by the 
impact of a jet on a flat plate is twice as great as that due to the 
hydrostatic head causing the flow. That is to say, if the theo- 
retical velocity of a jet is that due to a head h, where 

the force exerted on a fixed plate by the impact of this jet is 
equal to that due to a hydrostatic head of A' = 2h, in which case 

The orifice in a Pitot tube is essentially a flat plate subjected to 
the impact of the current. Considering only the impact efifect, 
therefore, the head which it is theoretically possible to attain in a 
Pitot tube is 

9 
which corresponds to a value of m of 0.7 in the formula 

V = niy/2gh' 

There are other considerations, however, which often modify 
this result considerably. The efifect of immersing a circular 
plate in a uniform parallel current has been fully analyzed theo- 
retically and the results confirmed experimentally. The results 
of such an analysis made by Professor Prasil, as presented in a 
paper by Mr. N. W. Akimofif,^ are shown in Mg. 123. The 
diagram here shown represents a vertical section of a current 
flowing vertically downward against a horizontal circular plate. 

The stream lines S, shown by the full lines in the figure, are 
curves of the third degree, possessing the property that the vol- 
umes of the cyUnders inscribed in the surface of revolution gen- 

^ Jour. Amer. Water Works Assoc., May, 1914. 



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136 



ELEMENTS OF HYDRAULICS 



erated by each stream line are equal. For instance, the volume 
of the circular cylinder shown in section by AA'BB' is equal to 
that of the cylinder CC'DD\ etc. It may also be noted that the 
size of the plate does not affect the general shape or properties 
of the curves shown in the diagram. 

The surfaces of equal velocity are ellipsoids of revolution hav- 
ing the center of the plate as center, and are shown in section 
in the figure by the eUipses marked EV. In general, each of 




these ellipses intersects any stream line in two points, such as 
F and G. Therefore somewhere between F and G there must be 
a point of minimum velocity, this being obviously the point of 
contact of the corresponding ellipse with the stream hne. The 
locus of these points of minimum velocity is a straight Une OH 
in section, inclined to the plate at an angle of approximately 20°. 
The surface of minimum velocity is therefore a cone of revolution 
with center at 0, of which OH is an element. 



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FLOW OF WATER 137 

The surfaces of equal pressure are also ellipsoids of revolution 
with common center below 0, and are shown in section by the 
ellipses marked PP in the figure. The surface of maximum pres- 
sure is a hyperboloid of revolution of one sheet, shown in section 
by the hyperbola YOY. 

It should be especially noted that the cone of minimum veloc- 
ity is distinct from the hyperboloid of maximum pressure so that 
in this case minimum velocity does not necessarily imply maxi- 
mum pressure, as might be assumed from a careless appUcation 
of Bernoulli's theorem. 

This analysis shows the reason for the wide variation in the 
results obtained by different experimenters with the Pitot tube, 
and makes it plain that they will continue to differ imtil the 
hydraulic principles imderlying the action of the impact tube 
are generally recognized and taken into account. 

168. Construction and Calibration of Pitot Tubes. — ^The im- 
pact end of a Pitot tube is usuaUy drawn to a fine point with a 
very small orifice, whereas the vertical arm is given a much larger 
diameter in order to avoid the effect of capillarity. The tubes 
used by Darcy had an orifice about 0.06 in. in diameter which was 
enlarged in the vertical arm to an inside diameter of about 0.4 
in. In his well-known experiments for determining the velocity of 
fire streams (Par. 123), Freeman used for the mouthpiece of his 
impact tube the tip of a stylographic pen, having an aperture 
0.006 in. in diameter. With this apparatus and for the high 
velocities used in the tests, the head was found to be almost ex- 

actly equal to s;-, corresponding to a value of m = 1.0 in the 

formula v = m\/2gh. 

It is also important that the impact arm should be long enough 
so that its orifice is clear of the standing wave produced by the 
current flowing against the vertical arm. The cutwater used 
with some forms of apparatus (see Fig. 121) is intended to elimi- 
nate this effect but it is doubtful just how far it accomplishes its 
piu'pose. 

The most proUfic source of error in Pitot-tube measurements 
is in the calibration of the apparatus. The fundamental prin- 
ciple of calibration is that the tube must be caUbrated under the 
same conditions as those for which it is to be used. Thus it has 
been shown in Art. XVII that flow below the critical velocity fol- 
lows an entirely different law from that above this velocity. 



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138 ELEMENTS OF HYDRAULICS 

Flow in a pipe under pressure is also essentially different from 
flow in an open channel. 

169. Du Buat's Paradox. — Furthermore, the method of calibra- 
tion is of especial importance. This is apparent from the weD- 
known hydrauUc principle known as Du BuaVs 'paradox. By ex- 
periment Du Buat has proved that the resistance, or pressure, 
offered by a body moving with a velocity v through a stationary 
liquid is quite different from that due to the Uquid flowing with 
the same velocity v past a stationary object. The pressure of 
the moving liquid on the stationary object was found by him to 
be greater than the resistance experienced by the moving object 
in a stationary liquid in the ratio of 13 to 10. All methods of 
calibration which depend on towing the instrument through a 
liquid at rest therefore necessarily lead to erroneous and mis- 
leading results. 

Since the Pitot tube is so widely used for measuring velocity 
of flow, its construction and caUbration should be standardized, 
so that results obtained by different experimenters may be subject 
to comparison, and utiUzed for a more accurate and scientific 
construction of the instrument. 

XXIX. NON-UNIFORM FLOW; BACKWATER 

160. Surface Elevation. — The case of most practical impor- 
tance is that in which the level of a stream is to be raised by means 
of a dam or weir, and it is required to determine the new surface 
elevation at any given distance back of the dam or weir. As the 
mathematical solution of the problem is somewhat compUcated, 
the method commonly followed in practice is to obtain the hy- 
draulic gradient by a series of approximations. Thus having 
given the discharge and the dimensions of the channel cross- 
section, the velocity of flow, v, and the hydraulic radius, r, become 
known. Then assuming a value for the hydraulic gradient, s, 
the value of C is computed from Chezy^s formula 

V = Cy/rSj 

and also from Kutter's formula 



C = 



,, ^^ , 0.00281 , 1.811 
41.66 -\ H 



+ (41.65 +«:°^)i 



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FLOW OF WATER 



139 



and the two values compared. If these values are not equal, a 
new value of s is assumed and the process repeated until both 
formulas give the same value of C. The corresponding value of 
s is then taken to be the correct hydrauUc gradient, from which 
the actual elevation of the water surface at any point may be 
computed if the slope of the bed of the stream is known. 

The hydraulic gradient, s, may also be computed directly trom 
a formula of the exponential type such as that of Williams and 
Hazen, namely, 

'V = Cr»-6V ^^0.001 - ®-®^ 

provided the engineer's experience warrants him in assuming a 
value for C. As the channel of an ordinary stream varies con- 
siderably, giving rise to non-uniform flow, an exact solution of 
the problem is impossible and the assumption of C is usually 
accurate enough to satisfy all practical considerations. 



APPLICATIONS 

61. A device used by Prony for measuring discharge consists 
of a fixed tank A (Fig. 124) containing water, in which floats a 
cylinder C which carries a second 

tank B, Water flows through the 

opening D from A into B. Show CAB 

that the head on the opening D, and 
consequently the velocity of flow 
through this opening, remains con- 
stant (Wittenbauer). 

62. A cylindrical tank of 6-ft. in- 
side diameter and 10 ft. high con- 
tains 8 ft. of water. An orifice 2 in. 
in diameter is opened in the bottom, Fig. 124. 
and it is found that the water level 

is lowered 21 in. in 3 min. Calculate the coefiicient of discharge. 

63. Water flows through a circular sharp-edged orifice }4 in. 
in diameter in the side of a tank, the head on the center of the 
opening being 6 ft. A ring slightly larger than the jet is held so 
that the jet passes through it, and it is then found that the center 
of the ring is 8.23 ft. distant from the orifice horizontally, and 3 ft. 
below it. In 5 min. the weight of water discharged is 301 lb. 
Calculate the coefficients of velocity, contraction and discharge 
for this orifice. 







c 




A 




r-.- 




-"- 














D 













^ 


N., 




xa 


- 


- ■ 



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140 ELEMENTS OF HYDRAULICS 

Note. — ^This is an interesting method of determining the 
coefficients by experiment but is not very accurate. 

If the velocity of the jet at exit is denoted by v, its abscissa x 
after t sec. will be approximately 

X = vt 

and the ordinate of the same point; considering the water as a 
freely falling body, will be 

Eliminating t between these two relations, the equation of the 
path followed by the jet is foimd to be 

x^ = > 

g 

which represents a parabola with axis vertical and vertex at the 
orifice. Having found the actual velocity v from this equation, 
the velocity coefficient is obtained from the relation 



V2ih 

The effiux coefficient K is then calculated from the measured dis- 
charge Q from the relation 

Q = KAv, 

and the contraction coefficient from 

64. Find^the velocity with which water will flow through a 
hole in a steam boiler shell at a point 2 ft. below the surface of the 
water when the steam pressure gage indicates 70 lb. per square 
inch. 

56. A reservoir having a superficial area of 0.6 sq. mile has an 
outlet through a rectangular notch weir 8 ft. long. If the head 
on the crest when the weir is opened is 2.6 ft., how long will it 
take to lower the level of the reservoir I ft.? 

66. A rectangular notch weir 12 ft. long has a head of 16 in. of 
water on the crest. The cross-sectional area of the approach 
channel is 60 sq. ft. Calculate the flow. 

67. A suppressed weir 6 ft. long has its crest 3 ft. above the 
bottom of the channel, and the head on the crest is 18 in. Com- 
pute the discharge. 



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t = 



FLOW OF WATER 141 

68. A lock chamber 500 ft. long and 110 ft. wide is emptied 
through a submerged opening 6 ft. long by 3 ft. high, having a 
coeflBcient of discharge of 0.68. If the depth of water on the 
center of the opening is initially 30 ft. on the inside and 8 £t. on 
the outside, find how long it will take to lower the water in the 
lock to the outside level. 

69. A hemisphere filled with water has a small orifice of area A 
at its lowest point. Calculate the time required for it to empty. 

Note. — If x denotes the depth of water at any instant, the 
area X of the water surface is X = v{2rx — x*), where r denotes 
the radius of the hemisphere. The time required to empty the 
hemisphere is therefore 

1 r^Xdx ^ 14 m^ 
K^^jo V5 15 KAV2g 

60. A tank 10 ft. square and 12 ft. deep is filled with water. A 
sharp-edged circular orifice 3 in. in diameter is then opened in the 
bottom. How long will it take to empty the tank through this 
opening? 

61. Compute the discharge through a Borda mouthpiece 1.6 
in. in diameter under a head of 12 ft., and determine the loss of 
head in feet. 

62. Compute the discharge through a reentrant short tube 2 
in. in diameter under a head of 20 ft., and determine the loss of 
head in feet. 

63. Compute the discharge through a standard short tube of 
1.76 in. inside diameter under a head of 6 ft., and also find the 
negative pressure head at the most contracted section of the vein. 

64. Find the discharge in gallons per minute through a 1.6-in. 
smooth fire nozzle attached to a 2.6-in. play pipe under a pres- 
sure at base of nozzle of 90 lb. per square inch. 

66. Water flows through a 6-in. horizontal pipe at 200 ft. per 
minute under a pressure of 30 lb. per square inch. If the pipe 
gradually tapers to 4 in. diameter, find the pressure at this point. 

66. A 12-in. horizontal pipe gradually tapers to a diameter of 
6 in. If the flow is 60,000 gal. per hour, calculate the difference 
in pressure at two sections having these diameters. 

67. A Venturi meter in an 18-in. main tapers to 6 in. at the 
throat, and the difference in pressure in main and throat is 
equivalent to 11 in. of mercury. Find the discharge in gallons 
per minute. 



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142 ELEMENTS OF HYDRAULICS 

68^ The Ashokan Venturi meter on the line of the Catskill 
Aqueduct is 7 ft. 9 in. inside diameter at the throat, the diameter 
of the main being 17 ft. 6 in. (see Fig. 76). Find the difference 
in pressure between main and throat for the estimated daily- 
flow of 600,000,000 gal. 

69. The velocity of flow in a water main 4 ft. in diameter is 3.6 
ft. per second. Assuming the coefficient of friction to be 0.0216, 
find the frictional head lost in feet per mile. 

70. Two cylindricar tanks each 8 ft. in diameter are connected 
near the bottom by a 2-in. horizontal pipe 26 ft. long. If the 
water level in one tank is initially 12 ft. and in the other 3 ft. 
above the center line of the pipe, find how long it will take for the 
water to reach the same level in both tanks. 

71. Find the frictional head lost in a pipe 2 ft. in diameter and 
6 miles long which discharges 200,000 gal. per hour, assuming the 
coefiicient of friction to be 0.024. 

72. Find the required diameter for a cast-iron pipe 10 miles 
long to discharge 60,000 gal. per hour under a head of 200 ft. 

73. A house service pipe is required to supply 4,000 gal. per 
hour through a 1.6-in. pipe and a 1-in. tap. The total length of 
the service pipe is 74 ft., including the tap which is 1.6 ft. long. 
Find the total pressure required in the main. 

Solution, — In the solution of water-supply problems of this 
type, it is recommended by W. P. Gerhard^ that the following 
formulas be used. 



Head lost in tap, hi = 0.024 l^j w^ ; 



V 



,2 



Head lost at entrance, hi = 0.62 i, , 

Head lost at stopcock = }4 liead lost in tap; 
Head lost in pipe by Prony's formula. 



G = [M!^]^1.20032) 



where, in this last formula, 

d = diameter of pipe in inches, 
H = head in feet, 
L = length in yards, 
G = discharge in U. S. gallons per minute. 

1 Discharge of Water through Street Taps and House Service Pipes," 
Cassier's Mag., November, 1905. 



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FLOW OF WATER 143 

Using these formulas we obtain in the present case the following 
numerical results: 

Pressure lost in tap = 2.24 lb. per square inch. 

Pressure lost at stopcock = 1.12 lb. per square inch. 

Pressure lost at entrance = 3.08 lb. per square inch. 

Pressure lost in 72.5 ft. of 1-in. pipe = 17.63 lb. per square inch. 
Total pressure required in main = 23.97 lb. per square inch. 

74. A building is to be supplied with 2,600 gal. of water per 
hour through 180 ft. of service pipe at a pressure at the building 
line of 15 lb. per square inch. The pressure in the main is 35 lb. 
per square inch. Find the required size of service pipes and 
taps. 

Solution, — The total drop in pressure in this case is 20 lb. per 
square inch. Therefore, using the formulas given in the pre- 
ceding problem and assuming different sizes of service pipes, the 
results are as follows: 

One 1.26-in. full-size pipe 180 ft. long discharges, 1,715 gal. per 
hour. 

Two 1-in. full-size pipes discharge together, 1,920 gal. per hour. 

Two 1.25-in. pipes with %-in. taps discharge together 2,880 
gal. per hour. 

One 1.5-in. pipe with 1-in. tap discharges 2,619 gal. per hour. 

The last has sujfficient capacity and is cheapest to install, and 
is therefore the one to be chosen. 

76. A pipe 1 ft. in diameter connects two reservoirs 3 miles 
apart and has a slope of 1 per cent. Assuming the coeflBcient 
of friction as 0.024, find the discharge and the slope of the hydrau- 
lic gradient when the water stands 30 ft. above the inlet end and 
10 ft. above the outlet end. 

76. Two reservoirs 5 miles apart are connected by a pipe line 
1 ft. in diameter, the difference in water level of the two reservoirs 
being 40 ft. Assuming the value of Chezy's constant in feet 
and second units to be 125, find the discharge in gallons per hour. 

77. A 12-in. main 5,000 ft. long divides into three other mains, 
one 6 in. in diameter and 6,000 ft. long, one 10 in. in diameter and 
7,000 ft. long, and one 8 in. in diameter and 4,000 ft. long. The- 
total static head lost in each line between reservoir and outlet is 
the same and equal to 100 ft. Find the discharge in gallons per 
24 hr. at each of the three outlets. 



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144 



ELEMENTS OF HYDRAULICS 



SoltUion. — ^The head lost in friction in the length I is given by 
the relation 



»-/(!) 



and the discharge by 









Eliminating v between these relations, we have 



A, = 



whence 



2gdir*d*' 



o-V^vs^ 



Assuming/ = 0.02 and I = 1,000 ft., the discharge Q for pipes of 
various sizes in terms of the head lost per 1,000 ft. is given by 
the following relations: 



Diameter of pipe in 


Discharge in gal. per 24 hours in terms of head 


inches 


lost per 1000 ft. 


4 


= 58,430 V;iz 


6 


= 161,000V;ii 


8 


= 330,500V/n 


10 


= - 577,500 V/u 


12 


= 911,000V^ 


16 


= l,870,000V^z 


20 


= 3,266,000V;ii 


24 


= 5,147,000V/ij 


30 


= 9,002,000V/u 


36 


= 14,200,000V^i 


48 


= 29,150,000V/ii 


56 


= 42,850,000V^ 


60 


= 50,920,000V^ 


66 


- 64,600,000 V^ii 


72 


= 80,320,000V;iz 



In the present case let the flow in gallons per 24 hr. be denoted 
by Q with a subscript indicating the size of pipe. Then 

Qn = Qe + Qs + Qio. 

Also if ft with the proper subscript denotes the head lost in each 
pipe per 1,000 ft., we have from the above relations 

0» = 911,000Vft^ 



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FLOW OF WATER 146 

Q, = 161,000 Vftii^ 

Qb =330,500V^, 

. Qio = 577,500VAi^ 

and since from the conditions of the problem the head lost in 
each line amounts to 100 ft., we also have the relations 

5fti2 + efte = 100, 

5fti2 + 7Aio = 100, 

5/ii2 + 4ft8 = 100. 

From these relations we find 

fts = H^h] hio = ^A«; hn = ^ -> 

and substituting these values in the first equation, the result is 
161,000\/Ai + 330,500\/^ + 577,500\/^ 



= 911,000 



/lOO - 6*6 



5 

whence Ae = 7.52 and consequently 

hs = 12.28; hio = 6.446; hn = 10.976. 

Substituting these values of h in the formulas for discharge, the 
results are 

Qe = 444,360 gal. per 24 hr. 

Qs = 1,110,480 gal. per 24 hr. 

Qio = 1,465,700 gal. per 24 hr. 
Qe + Qs + Qio = 3,020,540 gal. per 24 hr. 

The actual calculated value of Q12 is 

Q12 = 3,015,400 gal. per 24 hr., 

the discrepancy between these results being due to slight inac- 
curacy in extracting the square roots. 

78. A pipe of constant diameter d discharges through a number 
of laterals, each of area A and spaced at equal distances I apart 
(e.g., street main and house service connections). Find the rela- 
tion between the volume of flow in three successive segments of 
the maini (Fig. 125). 

ij. P. Fbizell, Jour FranUin Inst., 1878. 
10 



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146 ELEMENTS OF HYDRAULICS 

Solution. — The discharge at A is 

where vi denotes the velocity of flow at this point. Also the head 
lost in friction in the segment AB is 



^-'(^'^ 



and consequently 
wKere 



2ff 



hi = oQi* 
8fl 



E 



ffjr«d» 
D C 



B 




\ \ \ 



M N R 
FiQ. 125. 

At B the pressure head \&h + hi and the discharge is 

0« - Qi = KA y/2Q V/i + Ai = i£:A\/2^VA+"aQr*. 
Similarly, for the discharge at C and D we obtain the relations 

whence by elimination 

(Q4 - QzY - (Qs - Q2)» = 6U* 



where 



,..i^»x4?. 



The general relation is therefore 



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FLOW OF WATER 147 

The following geometrical construction may be used for deter- 
mining Qn» Determine an angle ^ such that 

b = tan 

and lay off Q».i and Qn-s on a straight line so that OM » 
Q«_2 and ON = Qn-i as shown in Fig. 125. At N erect a per- 
pendicular NP to ON J and then lay off NR = MP. Then OR 

79. A reservoir discharges through a pipe line made up of pipes 
of different sizes, the first section being 4,000 ft. of 24-in. pipe, 
followed by 5,000 ft. of 20-in. pipe, 6,000 ft. of 16-in. pipe and 
7,000 ft. of 12-in. pipe. The outlet is 100 ft. below the level of 
the reservoir. Find the discharge in gallons per 24 hr. 

Sohdion. — Using the same notation as in Problem 77, we have 
in the present case 

4^24 + 5hio + 6Ai6 + 7hi% = 100. 
Also, since 

8/7 



»'-(i^)«'. 



the loss in head per 1,000 ft. varies inversely as the fifth power 
of the diameter, and consequently 

fti. = (^)'*"= 7.594A,«, 
/24\» 



^20 = Q hu = 2.488^,4. 



Solving these three equations simultaneously with the first one, 
the results are 

hu = 0.35; hio = 0.871; hu = 2.658; hit = 11.20. 

As a check on the correctness of these results we have 

4X 0.35 = 1.400 

5 X 0.871 = 4.355 

6 X 2.658 = 15.948 

7 X 11.20 = 78.400 

100.103 



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148 



ELEMENTS OF HYDRAULICS 



The discharge may be found from the formulas given in Problem 
77, the results being as follows: 

Qi2 = Qi6 = Q20 = Q24 = 3,046,000 gal. per 24 hr. 

80. A new cast-iron pipe AB, 1,000 ft. in length, divides at B 
into two branches, BC which is 600 ft. long and BD which is 900 
ft. long. The fall for AB is 26 ft., for BC is 10 ft. and for BD 
is 20 ft., and the velocity of flow in AB is 4 ft. per second. Find 
the required diameters of the three pipes to deliver 500 gal. per 
minute at C and 300 gal. per minute at D. 

81. A reservoir empties through a pipe AB (Fig. 126) which 
branches at B into two pipes BC and BD, one of which discharges 
20,000 gal. per hour at C and the other 30,000 gal. per hour at D. 




FiQ. 126. 

The lengths of the pipes are AB = 1,200 ft., BC = 900 it, BD = 
600 ft., and the depths of the outlets below the surface of the 
reservoir are hi = 25 ft., hi = 60 ft. The pipes are of cast iron, 
and the velocity of flow in AB is to be 3 ft. per second. Calculate 
the diameters of all three, and the velocity of flow in BC and BD. 

82. Two reservoirs empty through pipes which unite at C 
(Fig. 127) into a single pipe which discharges at D. The lengths 
of the pipes are h = 1,500 ft., h = 900 ft. and I = 2,400 ft. The 
diameters of the pipes are di = 6 in., ^2 = 4 in., and d = 9 in., 
and the depths of the outlet below the levels of the reservoirs 
are fti = 75 ft., hi = 100 ft. Find the velocity of flow in each 
pipe and the total discharge in gallons per hour. 

83. A water main 3 ft. in diameter divides into two smaller 
mains of the same diameter and whose combined area equals 



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FLOW OF WATER 



149 



that of the large main. If the velocity of flow is 3 ft. per second, 
compare the heads lost per mile in the large and small mains. 

84. The head on a fire hydrant is 300 ft. Find its discharge in 
gallons per minute through 400 ft. of inferior rubber-lined cotton 
hose 2.5 in. in diameter and a 1.5-in. smooth nozzle. 

86. What head is required at a fire hydrant to discharge 250 
gal.' per minute through a 1.25-in. ring nozzle and 500 ft. of 
2.5-in. best rubber-lined cotton hose? 

86. A fire stream is deUvered through 100 ft. of 2-in. rubber- 
lined cotton hose and a nozzle l}i in. in diameter. The hy- 
drant pressure is 75 lb. per square inch. Find pressure at nozzle, 
discharge in gallons per minute and height of effective fire stream. 




Fig. 127. 

87. Two reservoirs are connected by a siphon 16 in. in diameter 
and 50 ft. long. If the difference in level in the reservoirs is 
25 ft., calculate the discharge, assuming the coefficient of pipe 
friction to be 0.02 and considering only friction losses. 

88. A cast-iron pipe 2 ft. in diameter has a longitudinal slope 
of 1 in 2,500. If the depth of water in the pipe is 18 in., calculate 
the discharge. 

89. A rectangular flume 6 ft. wide, 3 ft. deep and 1 mile long 
is constructed of unplaned lumber and is required to deliver 120 
cu. ft. per second. Determine the necessary gradient and the 
total head lost. 

90. The Aqua Claudia, shown in Fig. 128, was one of the nine 
principal aqueducts in use in first century, A.D., for supply- 
ing Rome with water. The lengths and capacities of these 
nine aqueducts were as follows: 



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150 



ELEMENTS OF HYDRAULICS 













Alti- 






Date of 


Length 


Length 


Discharge 


tude of 
springs 


Level 


Name 


construc- 


m 


in 


per day 


above 


Rome 




tion 


feet 


miles 


in cu. ft. 


sea 












level 


in feet 












in feet 




Aqua Appia 


312 B. C. 


53,950 


. 10.2 


4,072,500 


98 


65 


Anio Vetus 


272-269 


209,000 


39.6 


9,814,200 


918 


157 


Aqua Marcia... 


144-140 


299,960 


56.8 


10,465,800 


1043 


192 


Aqua Tepula. . . 


125 


58,200 


11.0 


993,000 


495 


199 


Aqua Julia 


33 


74,980 


14.2 


2,691,200 


1148 


209 


Aqua Virgo 


19 


67,900 


12.9 


5,587,700 


79 


65 


Aqua Absietina. 


# 


107,775 
225,570 


20.4 


874,800 


685 


54 


Aqua Claudia. . . 


38-52 A. D. 


42.7 


7,390,800 


1050 


221 


Anio Novus 


3a-52A.D. 


285,330 


54.0 


10,572,900 


1312 


231 




CatskiU. 



Croton . 




Old Croton. 



Aqua Claudia. 
Fig. 128. — Comparison of ancient and modem aqueducts. 

The con£i;ruction of the earliest aqueducts was the simplest, most of 
them being underground. In the Aqua Appia only 300 ft. were above 



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FLOW OF WATER 161 

ground, and in the Anio Vetus only 1,100 ft. were above ground. In the 
Aqua Marcia 7.5 miles were supported on arches; in the Aqua Claudia 10 
miles were on arches, and in the Anio Novus 9.5 miles were on arches. 
The construction of the last shows the greatest engineering skill, as it fol- 
lows a winding course, at certain points tunndling through hills and at* 
other crossing ravines 300 ft. deep. 

The cross-flection of the channels {apecua) varied at different points of the 
course, that of the largest, the Anio I^vus, being 3 to 4 ft. wide and 9 ft. 
high to the top, which was of pointed shape. The channels were lined with 
hard cement (opiia signinum) containing fragments of broken brick. The 
water was so hard that it was necessary to clean out the calcareous deposits 
frequently, and for this purpose shafts or openings were constructed at 
intervals of 240 ft. 

Filtering and settling tanks (piaciruB limarioBf or " purgatories '0 were 
constructed on the line of the aqueduct just outside the city, and within the 
city the aqueducts ended in huge distributing reservoirs {CMteUa) from 
which the water was conducted to smaller reservoirs for distribution to the 
various baths and fountains. 

Supposing the population of Rome and suburbs to have then numbered 
one million, there was a daily water supply of nearly 400 gal per capita. 
Modem Rome with a population of half a million has a supply of about 200 
gal. per capita. The volume of water may also be compared with that of the 
Tiber which discharges 342,395,000 gal. per day, whereas in the first cen- 
tury, A.D., the aqueducts carried not less than 392,422,500 gal. per day, 
which by the fourth century had been increased by additional supplies to 
461,628,200 gal. per day. 

Assuming that the Aqua Claudia had an average i/vidth of 3 ft. 
with 6 ft. depth of water, and that the grade was uniform and the 
difference in head lost in friction, calculate from the values tabu- 
lated above the velocity of flow and Chezy's constant C in the 
formula v = Cy/rs. 

91. A channel of trapezoidal section with side slopes of two 
horizontal to one vertical is required to discharge 100 cu. ft. per 
second with a velocity of flow of 3 ft. per second. Assuming 
Chezy's constant as 115, compute the required bottom width of 
channel and its longitudinal slope. 

92. A channel of trapezoidal cross-section has a bottom width 
of 25 ft. and side slopes of 1:1. If the depth of water is 6 ft. 
and the longitudinal inclination of the bed is 1 in 5,000, find the 
discharge, assuming the coefficient of roughness, n, in Kutter's 
formula to be 0.02. 

93. A channel of rectangular section has a bottom width of 
20 ft., depth of water 6 ft. and longitudinal slope of 1 in 1,000. 
Calculate the discharge, assuming the coefficient of roughness, 
n, in Kutter's formula to be 0.01. 



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ELEMENTS OF HYDRAULICS 



94. A reservoir A supplies another reservoir B with 400 cu. ft. 
of water per second through a ditch of trapezoidal section, with 
earth banks, 5 milies long. To avoid erosion, the flow in this 
channel must not exceed 2 ft. per second. 

From reservoir B the water flows to three other reservoirs, C, 
D, E. Prom B to C the channel is to be rectangular in section 
and 4 miles long, constructed of unplaned lumber, with a 10-ft. 
fall and a discharge of 150 cu. ft. per second. 

From B to D the channel is to be 5 miles long, semicircular in 
section and constructed of concrete, with 12-ft. fall and a dis- 
charge of 120 cu. ft. per second. 

From B to E the channel is to be 3 miles long, rectangular in 
section and constructed of rubble masonry, with 15-ft. fall and a 
discharge of 130 cu. ft. per second. 

Find the proper dimensions for each channel section. 



1 


i 





.2 


0. 


4 


0. 


6 


0. 


8 


1 





1. 


lo 


^ 1 








\ 














) 


\ 


08 






\ 




\ 








^ 


y 








5 










\ 




% 


y 






J 


f 


A P 


$ 












^ 


X 








/ 






% 








/ 


/ 










/ 






n 1 


"g 






y 


/^ 


/ 






V 


/ 








U.4 


o 




/ 






/ 




^ 


X 










A A 


S 


/ 






A 


^ 


^ 
















\....._^ 


/ 


^ 


^ 


>^ 






















0.2 0.4 0.6 0.8 1.0 1.2 

Ratio of Q and V to their Values when Pipe is Full 

Fig. 129. 

95. The flow through a circular pipe when completely filled is 
25 cu. ft. per second at a velocity of 9 ft. per second. How much 
would it discharge if filled to 0.8 of its depth, and with what 
velocity? 

Solution. — Fig. 129 shows a convenient diagram for solving a 
problem of this kind graphically.^ The curve marked v (velocity) 
is plotted from Kutter's simplified formula 



V = 



/lOOVr \ 



rjy/Ts 



\b + Vr 
^ Imhoff, "Taschenbuch flir Kanalisations Ingenieure." 



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FLOW OF WATER 



153 



for a value of b of 0.35, and the discharge Q from the formula 
Q =lAVf the ordinates to the curves shown in the figure being the 
ratio of the depth of the stream to the diameter of the pipe, and 
the abscissas the ratios of Q and v respectively to their values 
when the pipe flows full. 

To apply the diagram to the problem under consideration, 
observe that for a depth of 0.8d the abscissa of the discharge 
curve is unity, and consequently the discharge for this depth is 
the same as when the pipe is completely filled. The abscissa of 
the velocity curve corresponding to this depth 0.8d {i.e., with 
abscissa 0.8) is 1.13, and consequently the velocity at this depth 
is 1.13 X 9 = 10.17 ft. per second. 




Similar diagrams have been prepared by Imhoff for a large 
variety of standard cross-sections and are supplemented by 
other diagrams or charts which greatly simplify ordinary sewer 
calculations. 

96. In the Catskill Aqueduct, which forms part of the water 
supply system of the City of New York, there are four distinct 
types of conduit; the cut-and-cover type, grade tunnel, pressure 
tunnel, and steel pipe siphon. The cut-and-cover type, shown in 
section in Fig. 130, is 55 miles in length, and is constructed of 
concrete and covered with an earth embankment. This is the 
least expensive type, and is used wherever the elevation and 
nature of the ground permits. 

The hydraulic data for the standard type in open cut' is as 
follows: 



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154 



ELEMENTS OF HYDRAULICS 



s = 0.00021 


Depth of 


Area of 


Wetted 


Hydraulic 




flow 


flow 


perimeter "^ 


radius 




mfeet 


in sq. ft. 


in feet 


in feet 




17.0 


241.0 


67.4 


4.20 


FuU, 


16.2 


237.7 


50.8 


4.67 


Max. cap. 


15.3 


230.9 


47.7 


4.84 




14.0 


217.6 


44.1 


4.92 




12.0 


192.6 


39.4 


4.88 




10.0 


163.2 


35.0 


4.65 




8.0 


130.7 


30.9 


4.24 




6.0 


97.1 


26.8 


3.61 




4.0 


62.2 


22.8 


2.72 




2.0 


27.0 


18.8 


1.47 





In the preliminary calculations the relative value of Chezy's 
coefficient for this type was assumed to be C = 125. Using this 
value, calculate the maximum daily discharge. 

97. Where hills or mountains cross the line of the Aqueduct, 
tunnels are driven through them at the natural elevation of the 
Aqueduct (Kg. 131). There are 24 of these grade tunnels, aggre- 
gating 14 miles. The hydraulic data for the standard type of 
grade tunnel is as follows: 



8 = 0.00037 1 


Depth of 


Area of 


Wetted 


Hydraulic 




flow 


flow 


perimeter 


radius 




in feet 


in sq. ft. 


in feet 


in feet 




17.0 


198.6 


52.2 


3.80 


FuU, 


16.26 


195.6 


46.0 


4.25 


Max. cap. 


15.3 


188.5 


42.7 


4.41 




14.0 


175.7 


39.3 


4.46 




12.0 


152.4 


35.0 


4.35 




10.0 


126.8 


31.0 


4.10 




8.0 


100.2 


26.9 


3.72 




6.0 


73.8 


22.9 


3.22 




4.0 


47.6 


18.9 


2.51 




2.0 


21.0 


14.9 


1.49 





The relative value of Chezy'a coefficient for this type was assumed 
in the preliminary calculations to be C « 120. Using this value, 
calculate the maximum daily discharge and the corresponding 
velocity of flow. 

98. Where the line of the Aqueduct crosses broad and deep 
valleys and there is suitable rock beneath them, circular tunnels 



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FLOW OF WATER 



155 



are driven deep in the rock and lined with concrete (Fig. 132). 
There are seven of these pressure tunnels, with an aggregate 




FiQ. 131. 



Fig. 132. 



length of 17 miles. The hydraulic data fo/these pressure tunnels 
are as follows: 



Slope 


Diameter 


Area of 
waterway 


Wetted 
perimeter 


Hydraulic 
radius 


0.00059 


14 ft. 6 in. 


165.1 sq. ft. 


45.55 ft. 


3.625 ft. 



Assuming the relative value Chezy's coefficient to be C = 120, 
calculate the velocity of flow and the daily discharge. 

99. In valleys where the rock is not sound, or where for other 
reasons pressure tunnels are impracticable, steel pipe siphons are 



\- Width]- 3 --^ £»T ^MtfM^^f^^^ 




Fig. 133. 

used (Fig. 133). These are made of steel plates riveted together, 
from J4e to % in, in thickness, and are 9 ft. and 11 ft. in diameter 
respectively. These pipes are embedded in concrete and covered 



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156 ELEMENTS OF HYDRAULICS 

with an earth embankment, and are lined with 2 in. of cement 
mortar as a protection to the steel and also for the sake of smooth- 
ness. There are 14 of these siphons aggregating 6 miles in length, 
and three pipes are required for the full capacity of the Aqueduct. 
Assuming three mortar-lined 11-ft. pipes, having a relative coeffi- 
cient of C = 120 and a slope s = 0.00059, calculate the velocity 
of flow through them and the maximum daily discharge. 

100. A broad shallow stream has naturally a depth of 3 ft. 
and a longitudinal slope of 5 ft. per mile. If a dam 8 ft. high is 
erected across the stream, determine the rise in level 1 mile up 
stream assuming the value of the constant C in Chezy's formula 
as 76. 



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SECTION 3 



ENERGY OF FLOW 

PRESSURE OF JET AGAINST STATIONARY DEFLECTING 
SURFACE 



161. Normal Impact on Plane Surface. — When a jet of water 
strikes a stationary flat plate or plane surface at right angles, 
the water spreads out equally in all directions and flows along 
this plane surface, as indicated 
in Fig. 134. The momentum 
of the water after striking the 
surface is equal to the sum of 
the momenta of its separate 
particles, but since these flow 
off in opposite directions their 
algebraic sum is zero. Conse- 
quently the entire momentum 
of a jet is destroyed by normal 
impact against a stationary 
plane surface. 

To find the pressure, P, exerted by the jet on the surface, let 
A denote the cross section of the jet and v its velocity. Then 
the mass of water flowing per unit of time is 




M = 



yAv 



and, consequently, from the principle of impulse and momentum, 

yAv^ 



I 



Pdt=- Mv -- 



g 



For uniform or steady flow, P is constant, and if M denotes the 
quantity flowing per unit of time, then t is unity. In this case 
the above expression for the hydrodynamic pressure P of the 
jet on the surface becomes 

7Av* 



P = 



g 



(98) 



167 



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158 ELEMENTS OF HYDRAULICS 

If h denotes tiie velocity head, then * = s"' ^^^ ^^' ^^^^ 

may be written 

P = 27Ah. (99) 

162. Relation of Static to Dynamic Pressure. — ^If the orifice is 
closed by a cover or stopper, then the hydrostatic pressure P' 
on this cover is approximately equal to the weight of a column 
of water^of height h and cross-section A; and consequently. 

P' = yAh. (100) 

Comparing Eqs. (99) and (100), it is apparent that the normal 
hydrodynamic pressure of a jet on an external plane surface is 
twice as great as the hydrostatic pressure on this surface would 
be if it was shoved up against the opening so as to entirely close 
the orifice. 

In deriving this relation, the coefficient of efflux is assumed to 
be unity; that is, the area A of the jet is assumed to be the same 
as that of the orifice, and the velocity v to be the full value 
corresponding to the head h. Since the coefficient is actually 
less than unity, the hydrodynamic pressure never attains the 
value given by Eq. (99). For instance, in the case of flow from 
a standard orifice, if A denotes the area of the orifice and a the 
crossHsection of the jet, then from Arts. IX and X 

a = 0.62A, and v = 0.97 V2^. 
Therefore the expression for P becomes 

p = 1^ = 27(0.62A)(0.97*A) = l.lQy Ah 

instead of 2y Ah, as given by Eq. (99). Note, however, that this 
apparently large discrepancy is due chiefly to the fact that the 
area A in Eq. (99) denotes the cross-section of the jet, whereas 
in Eq. (100) it denotes the area of the oriflce. If the area A in 
both expressions denotes the cross-section of the jet, Eq. (99) 
is practically true, and the hydrodynamic pressure is approxi- 
mately twice the hydrostatic pressure on an equal area. 

163. ObUque Impact on Plane Surface. — ^If a jet strikes a 
stationary plane surface obliquely, at an angle a (Kg. 135), the 
axial velocity!; of the jet may be resolved into two components, 
v sin a normal to the surface, and v cos a tangential to the surface. 
If the surface is perfectly smooth, the water flowing along the 



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ENERGY OF FLOW 



159 



surface experiences no resistance to motion, and the pressure, 
P) exerted on the surface is that corresponding to the normal 
velocity component w' = i; sin a. The area to be considered, 
however, is no\r a right sec- 
tion, A, of the jet, but a sec- 
tion A' normal to the com- 
ponent t; sin a, as indicated 
in Fig. 135. The total pres- 
sure, P, exerted on the surface, 
is then 




or, since w' = t; sin a and A' = 

-, this may be written in the form 



Fia. 135. 



sma' 



P = - — sm a. 
g 



(101) 



If a = 90**, this reduces to Eq. (98). 

164. Axial Impact on Surface of Revolution. — If the surface 
on which the jet impinges is a surface of revolution, coaxial with 




the jet (Fig. 136), then in this case also the particles spread out 
equally in ^ directions, and consequently the sum of the mo- 
menta of the particles in the direction perpendicular to the axis 
of the jet is zero. The velocity of any particle in a direction 
parallel to the axis of the jet, however, becomes v cos a, where a 
denotes the angle which the final direction taken by the particles 



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160 



ELEMENTS OF HYDRAULICS 



makes with their initial direction, as indicated in Fig. 136. The 
total initial momentum is then 

9 
and the total final momentum is 

Mv cos a = cos a. 

9 

Therefore, equating the impulse to the change in momentum, 
we obtain the relation 



g 



(1 — cos q). 



(102) 



For a jet impinging normally on a plane surface, a = 90**, and 
this expression reduces to Eq. (98). 

166. Complete Reversal of Jet — If a is greater than 90**, then 
cos a becomes negative and the pressure P is correspondingly 




Fig. 137. 

increased. For example, if the direction of flow is completely, 
reversed, as shown in Fig. 137, then a = 180**, cos a = — 1, 
and hence * 

P = 2l|vl (108) 

The hydrodynamic pressure in this case is therefore twice as 
great as the normal pressure on a flat surface, and four times as 
great as the hydrostatic pressure on a cover over an orifice of 
the same area as the cross-section of the jet. 

166. Deflectioa of Jet. — When a jet is deflected in an oblique 
direction, the final velocity v may be resolved into components 
V cos a and v sin a, as indicated in Fig. 138. The component 



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ENERGY OF FLOW 



161 



of the final momentum parallel to the initial direction of the jet 
is then 

Mv(l — cos a) = - — (1 — cos a), 
and the horizontal component, J?, exerted in this direction is 

(104) 



H = ^^'(1 - cos a). 
g 



Similarly, the component of the final momentimi perpendicular 
to the initial direction of the jet is 

Mv sin « = sm a, 

9 



Vcostf 




Fia. 138. 

and the vertical component, F, exerted in this direction is 

V = -^ sin a, (106) 

The total pressure of the jet on the deflecting surface, or reaction 
of the surface on the jet, is, then, 

P = VlP~+Y^ = ^^V(l - cos a)2 + sin2 a 
which simpUfies into 



P = ^^V2(l - cos a). 



(106) 



A more convenient expression for P may be obtained by using 
11 



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162 



ELEMENTS OF HYDRAULICS 



the trigonometric relation ^- 



— cos a . 1 V r 

— = sin -a, by means of 

2 ^ 



which Eq. (106) may be written in the form 

27Av' 



sin ^. 

g 2 



(107) 



167. Dynamic Pressure in Pipe Bends and Elbows. — When a 
bend or elbow occurs in a pipe through which water is flowing, 
the change in direction of flow produces a thrust in the elbow, as 
in the case of the deflection of a jet by a curved vane, considered 
in the preceding paragraph. From Eqs. (106) and (107), the 
amount of this thrust P is 



P =^ V 2(1 — cos a) = — 



sm 



2' 




Fig. 139. 

and the direction of the thrust evidently bisects the angle a, as 
indicated in Fig. 139. 

In the case of jointed pipe lines if the angle of deflection is 
large or the velocity of flow considerable, this thrust may be 
sufficient to disjoint the pipe unless provision is made for taking 
up the thrust by some form of anchorage, as, for example, by 
filling in with concrete on the outside of the elbow. 

XXXI. PRESSURE EXERTED BY JET ON MOVING VANE 

168. Relative Velocity of Jet and Vane. — In the preceding 
article it was assumed that the surface on which the jet impinged 
was fixed or stationary. The results obtained, however, remain 
valid if the surface moves parallel to the jet in the same or op- 



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ENERGY OF FLOW 



163 



posite direction, provided the velocity, v, refers to the relative 
velocity between jet and surface. Thus if the surface moves in an 
opposite direction to the jet with a velocity v\ the relative veloc- 
ity of jet and surface is t; + v' and the pressure is correspondingly 
increased, whereas if they move in the same direction, their rela- 
tive velocity is v— v', and the pressure is diminished. 

169. Work Done on Moving Vane. — Consider, for example, the 
case of a jet striking a deflecting surface and assume first that 
this surface moves in the same direction as the jet with velocity 
v' (Fig. 140). Since the surface, or vane, is in motion, the mass 
of water, ilf ', reaching the vane per second is not the same as the 
mass of water, M, passing a given cross-section of the jet per 
second. That is, the mass, ilf , issuing from the jet per second is 




Fig. 140. 



whereas the mass, M', flowing over the vane per second is 

yA(v - v') 



M' = 



a 



(108) 



Therefore the components of the force acting on the vane, given 
by Eqs. (104) and (105), become in this case 

H = M'(v - vO(l - cos a) = ^ (v - vy{l - cos a), 

y 
yA 
V = M\v — t;') sin a = ^—{v — v'y sin a. 

Since the motion of the vane is assumed to be in the direction of 
the component H, the component V, perpendicular to this direc- 
tion, does no work. The total work, TF, done on the vane by 
the jet is therefore 

W = Hv' = 7AvXv - vV (J _ ^^g ^^^ ^lOg^ 



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164 ELEMENTS OF HYDRAULICS 

170. Speed at which Work Becomes a Maximum. — ^The con- 
dition that the work done shall be a maximum is 



^ = = ^ (1 - cos «)[(. - »')» - 2v'iv -t/)], 



dW _^_Ay 
9 
whence 



V = }. (110) 



Substituting this value of v' in Eq. (109), the maximum amount 
of work that can be realized under the given conditions is found 
to be 

'^3 7 v\^ 4rYAv^ 

TTma. = "7^ («' T gj (1 " COS a) = -^ (1 - COS a). 

171. Maximum Efficiency for Single Vane. — ^The efficiency of 
a motor or machino is defined in general as 

-^ . Useful work .^-^v 

^^"•'^''y = Total energy available ' ^^"^ 

Since, in the present case, the total kinetic energy of the jet is 

the efficiency, E, becomes 

4yAv^ ,- V 

2g 
The maximum efficiency occurs when a = 180**, in which case 

E«a. = 1^ = 69-2 per cent. (112) 

172. Maximum Efficiency for Continuous Succession of Vanes. 
— If there is a series of vanes following each other in succession 
so that each receives only a portion of the water, allowing this 
portion to expend its energy completely on this vane before leav- 
ing it, then the mass M' in Eq. (108) is replaced by M, and the 
component H becomes 

H = ilf(v - i;')(l - cos a) = ^^^(^ - ^') (1 - cos a). 



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MEROY of Flow 166 

The work done on the series of vanes is therefore 

W = Hv' = 'yAw^C^ -^0 (1 _ cos a). (US) 

The condition for a maximum in this case is 

^ = =^(1 - cos«)[(r - t/) - t;'], 



whence 



(114) 



Substituting this value of r' in' Eq. (113), the maximum work 
which can be realized from a series of vanes moving parallel to 
the jet is 

yAv^ ( _v\ 
^ 2 V 2/,, , yAv\, . 

Wfnax = (1 - cos c) = ^Tr-(1 - COS o). 

Hence the efficiency in this case is 

^ — (1 — cos a) ^ 
''-^^-^^ i(l-cosa). 

The maximum efficiency therefore occurs when a = 180®, its 
value being 

E«a« = 2 (2) = 100 per cent (116) 

The actual efficiency of course can never reach this upper limit, 
as the conditions assumed are ideal, and no account is taken of 
frictional and other losses. 

173. Impulse Wheel; Direction of Vanes at Entrance and Exit. 
— In general, it is not practicable to arrange a series of vanes so 
as to move continuously in a direction parallel to the jet. As 
usually constructed, the vanes are attached to the circumference 
of a wheel revolving about a fixed axis (Fig. 141). Let « denote 
the angular velocity of the wheel about its axis, and ri, rt the 
radii of the inner and outer edges of the vanes. Then the tan- 
gential or Unear velocities at these points, say ui and ti2, are 

Ui = rico; U2 = r2W. 

Now let Vi denote the absolute velocity of the jet at entrance to 



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166 



ELEMENTS OF HYDRAULICS 



the vane, and V2 its absolute velocity at exit. Then by forming 
a parallelogram of velocities on ui and Vi as sides, the relative 
velocity, Wi, between jet and vane at entrance is determined, 
as indicated in Fig. 141. Similarly, the parallelogram on U2, W2 
as sides determines the absolute velocity, V2, at exit. In order 
that the water may gUde on the vane without shock, the tip 
of the vane at entrance must coincide in direction with the vec- 
tor tVi. 



Shaft 




Fig. 141. 

174. Work Absorbed by Impulse Wheel. — ^Let M denote the 
mass of water passing over the vane per second. At entrance 
the velocity of this mass in the direction of motion {i.e., its tan- 
gential velocity) is Vi cos a, and at exit is V2 cos j8, where a and 
P are the angles indicated in Fig. 141. The Unear momentum 
of the mass M at entrance is then Mvi cos a, and its angular 
momentum is MviVi cos a. Similarly, its Unear momentum at 
exit is Mv2 cos P and its angular momentum is Mv2r2 cos p. The 
total change in «,ngular momentum per second, that is, the 
amount given up by the water or imparted to the wheel, is then 

MviTi cos a — Mv2r2 cos p. 

For a continuous succession of vanes, as in the case of an ordinary 
impulse wheel, the mass M is the total amount of water supplied 
by the jet per second. Hence, if T denotes the total torque ex- 
erted on the wheel, by the principle of angular impulse and 



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ENERGY OF FLOW 



167 



momentum, remembering that M is the mass of water flowing 
per unit of time, and consequently that the time is unity, 

T = M(viri cos a - V2r2 cos j8). (116) 

The total work imparted to the wheel is 

W = Tw, 

or, since M = -) Ui = r{(a, u% = r2w, the expression for the 

work becomes 

W = Tw = '^^ (uiVi cos o - U2V2 cos j3). (117) 

These relations will be applied in Art. XXXV to calculating 
the power and efficiency of certain types of impulse wheels. 

XXXn. REACTION OF A JET 

176. Effect of Issuing Jet on Equilibrium of Tank. — Consider 
a closed tank containing water or other liquid, and having an 
orifice in one side closed by a cover. When the cover is removed 




Fig. 142. 

the equiUbrium of water and tank will be destroyed. At the 
instant of removal this is due to the disappearance of the pressure 
previously exerted on the cover considered as part of the tank. 
After the jet has formed and a steady flow has been set up, as- 
suming that the depth of water is maintained constant by supply- 
ing an amount equal to that flowing out, as indicated in Fig. 142, 
the pressure within the fluid and on the walls of the tank will not 
regain its original static value, since, in accordance with Ber- 
noulU's theorem, an increase in velocity must be accompanied by 
a corresponding decrease in pressiure. 



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168 ELEMENTS OF HYDRAULICS 

176, Energy of Flow Absorbed by Work on Tank. — To calcu- 
late the effect of the flow on the equilibrium of the tank, suppose 
that the tank is moved in the direction opposite to that of the 
jet and with the same velocity, v, as that of the jet. Then the 
relative velocity of the jet with respect to the tank is still v, but 
the absolute velocity of the jet is zero and consequently its kinetic 
energy is also zero. If h denotes the head of water on the orifice 
(Fig. 142) and Q the quantity of water flowing per second, then 
its loss in potential energy per second is yQh. Moreover, while 
this volume of water Q moved with the tank, it had a velocity v, 

yQv^ 
and therefore possessed kinetic energy of amount 2~' '^^® 

total energy given up by the water in flowing from the tank is 
then 

.0A + ^««'*- 



t;2 

or, since A = h" approximately, these terms are equal, and the 



2g ' 
hese 
total energy lost by the water becomes 

Energy given up = ^*. 

Now let P denote the reaction of the jet, that is, the resultant of 
all the pressure exerted on the tank by the water except that due 
to its weight. Then, since the distance traversed by the force P 
in a unit of time is the .velocity v of the tank, by equating the 
work done by P to the energy givea up by the water, we have 

g 

whence 

g g 

The reaction P is therefore twice the hydrostatic pressure due to 
the head h. 

This is also apparent from the results of Par. 162, since the 
pressure of a jet on a fixed surface close to the orifice must be 
equal to its reaction on the vessel from which the jet issues. The 
actual reaction of the jet is of course somewhat less than its 
theoretical value, as given by the relation P = 2yAh, since there 
are various losses, due to internal friction, etc. 

177, Principle of Reaction Turbine. — In order for the tank to 
retain its uniform velocity, v, a resistance of amount P must 



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ENERGY OF FLOW 



169 



constantly be overcome, for if the resistance is less than this 
amount the motion will be accelerated. It is apparent, there- 
fore, that by a proper choice of the velocity, w, of the tank it is 
possible to utilize almost the entire energy of the jet in over- 
coming a resistance coupled up with the tank. This is the prin- 
ciple on which the reaction turbine is based, as explained in Art. 
XXXVI. 

It should be noted that if the water flowing out is continually 
replaced from above, half of the available energy must be used in 
giving the supply water the same ve- 
locity as the tank. The useful work 
is therefore reduced to one-half the 
previous amount, and the available 
energy is only that due to the velocity 
head h. 

178, Barker's Mill,— The simplest 
practical appUcation of the reaction 
of a jet is the apparatus known as 
Barker's mill (Fig. 143). In this ap- 
paratus water flows from a tank into 
a hollow vertical arm, or spindle, 
pivoted at the lower end, and from 
this into a horizontal tubular arm, 
having two orifices near the ends on 
opposite sides. The jets issue from 
these orifices, and their reactions 
cause the horizontal arm to rotate, 
driving the central spindle from which 
the power is taken oflf by a belt and 
pulley. 

The steam turbine invented by Hero of Alexandria in the first 
century B.C., is an almost identical arrangement, the motive 
power in this case being due to the reaction of a jet of steam 
instead of a jet of water. 



\L 



Y 



1] 



wmmmmmmm. 

SIDE ELEVATION 



a 







PLAN 



Fig. 143. 



XXXm. TYPES OF HYDRAULIC MOTORS 

179* Current Wheels. — ^There are three general types of hy- 
draulic motors, namely: 

1. Current and gravity wheels. 

2. Impulse wheeds and turbines. 



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170 ELEMENTS OF HYDRAULICS 

3. Reaction turbines. 

The current wheel is the oldest type of prime mover, and in 
its primitive form consisted of a large vertical wheel, with a set 
of paddles or buckets attached to its circumference, and so placed 
in a running stream that the current acting on the lower, or im- 
mersed, portion produced revolution of the wheel. A later im- 
provement consisted in placing the wheel at the foot of a waterfall 
and conducting the water by a flume to the top of the wheel, the 
action of the water in this case being due almost entirely to its 
weight (Art. XXXIV). 

180. Impulse Wheels. — The impulse wheel, as its name indi- 
cates, is designed to utilize the impulsive force exerted by a jet 
moving with a high velocity and striking the wheel tangentially. 
The wheel, or runner, in this case carries a series of curved buckets 
or vanes which discharge into the atmosphere. A feature of this 
type is that the runner rotates at a high velocity and can there- 
fore be made of comparatively small diameter. The two prin- 
cipal types of impulse wheel are the Girard impulse turbine, 
which originated in Europe, and the Pelton wheel, which was 
developed in the United States (Art. XXXV). 

181. Reaction Turbines. — The reaction turbine depends chiefly 
on the reaction exerted by a jet on the vessel from which it flows, 
which in this case is the passage between the vanes on the runner. 
In an impulse wheel the energy of the water as it enters the 
wheel is entirely kinetic, and as there is free circulation of air 
between the vanes and they discharge into the atmosphere, the 
velocity of the water is that due to the actual head. In a reac- 
tion turbine the energy of the water as it enters the wheel is 
partly kinetic and partly pressure energy, and as the water 
completely fills the passages between the vanes, its velocity at 
entrance may be either greater or less than that due to the static 
head at that point. A feature of the reaction turbine is that it 
will operate when completely submerged. 

182. Classification of Reaction Turbines. — Reaction turbines 
are subdivided into four classes, according to the direction in 
which the water fiows through the wheel. These are: 

1. Radial outward-fiow turbines. 

2. Radial inward-flow turbines. 

3. Parallel or axial-flow turbines. 

4. Mixed-flow turbines, the direction of flow being partly radial 



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ENERGY OF FLOW 



171 



and partly axial, changing from one to the other in passing over 
the vanes (Art. XXXVI). 

183, Classification of Hydratilic Motors. — The following tabu- 
lated classification is useful as a basis for the description of the 
various types of hydraulic motors given in Arts. XXXIV, XXXV 
and XXXVI. 



Current and Gravity Wheels: 
Utilizes impact of current or weight 
of the water. 

Impuhe Wheels and Turbines: 
Utilizes kinetic energy of jet at 
high velocity. Suitable for limited 
amount of water imder high head. 
Ordinarily used for heads from 300 
ft. to 3000 ft. 

Reaction Turbines: 
Utilizes both kinetic and pressure en- 
ergy of water. Suitable for large 
quantities of water under low or me- 
dium head. Ordinarily used for 
heads from 5 to 600 ft. 



Current wheel, 
Undershot wheel (Poncelet), 
Breast wheel, 
Overshot wheel. 



Girard turbine (European), 
Pelton wheel (American). 



Radial inward flow (Francis type), 
Radial outward flow (Foumeyron 

type), 
Parallel or axial flow (Jonval type), 
Mixed flow (American type). 






XXXIV. CURRENT AND GRAVITY WHEELS 

184. Current Wheels. — The vertical current wheel, mentioned 
in Par. 179, was the earliest type of hydraulic motor, dating from 
prehistoric times, although they 
are still in use in China and 
Syria. 

186. Undershot Wheels.— 
The first improvement consist- 
ed in confining the water in a 
sluice and delivering it directly 
on the vanes. This type was 
known as the Undershot wheel, 
and was in common use until 
about the year 1800 A.D. (Fig. 144). Flat radial vanes were 
used with this type, for which the maximum theoretical effi- 
ciency was 50 per cent., the velocity of the vanes to reaUze this 
efficiency being one-half the velocity of the stream, as explained 
in Art. XXX. The actual efficiency of such wheels was much 
lower, being only from 20 to 30 per cent. 



Fig. 144. — Undershot wheel. 



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ELEMENTS OF HYDRAULICS 



186. Poncelet Wheels. — Undershot wheels were greatly im- 
proved by Poncelet who curved the vanes, so that the water 
entered without shock and was discharged in a nearly vertical 
direction (Fig. 145). The water thus exerted an impulse on the 
vanes during the entire time it remained in the wheel, thereby 
raising the actual efficiency to about 60 per cent. Undershot 




Fig. 145. — Poncelet wheel. 



Fia. 146.— Breast wheel. 



Head Race J 



wheels of the Poncelet type are adapted to low falls, not exceed- 
ing 7 ft. in height. 

187. Breast Wheels. — ^A modification of the undershot wheel 
is the Brectst wheel, the water being deUvered higher up than for 
an ordinary undershot wheel, and retained in the buckets during 
the descent by means of a breast, or casing, which fits the wheel 

as closely as practicable (Fig. 
146). Wheels of this type are 
known as high-breast, breast, and 
low-breast according as the water 
is deUvered to the wheel above, 
at, or below the level of the 
center of the wheel. The high- 
breast wheel operates almost en- 
i Race tirely by gravity, that is, by the 
unbalanced weight of the water 
Fig. 147.— Overshot wheel. in the buckets, its efficiency be- 
ing from 70 to 80 per cent. 

Breast and low-breast wheels operate partly by gravity and 

partly by impulse, the efficiency varying from about 50 per cent. 

for small wheels to 80 per cent, for large wheels. This type 

was in use until about 1850. 

188. Overshot WheeU. — ^A more recent type is the Overshot 
wheel, the characteristic of this type being that the water is 



OT 

M 



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ENERGY OF FLOW 



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delivered at the top of the wheel by a sluice, as indicated in Fig. 
147. For maximum efficiency the diameter of the wheel should 
be nearly equal to the height of the fall, the efficiency for well- 
designed overshot wheels ranging from 70 to 85 per cent., which 
is nearly as high as for a modem turbine. An overshot wheel 
at Troy, N, Y., is 62 ft. in diameter, 22 ft. wide, weighs 230 tons, 
and develops 550 h.p. Another on the Isle of Man is 72 ft. in 
diameter and develops 150 h.p. 

XXXy. IMPT7LSE WHEELS AND TURBINES 

189, Pelton WheeL — ^The intermediate link between the old 
type of waterwheel and the modem impulse wheel was the Hurdy 




Pelton Bucket 




Doble Bucket 
Fio. 148. 

Giwdy, which was introduced into the mining districts of Cali- 
fornia about 1865. This somewhat resembled the old current 
wheel, being vertical with flat radial vanes, but differed from it in 



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174 ELEMENTS OF HYDRAULICS 



L 






^H 


ir IZX4J 






r 






-L^H 


H 


1 


liiJ.---J.^- _ 


^ t^^^l^^^^H 


r 


..-^.JlMH^^Hririfl^^l 




■ 



Fig. 149. 




Fig. 150. 



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ENERGY OFFLOW 



176 



that it was operated by a jet impinging on the vanes at high 
velocity. The maximum theoretical efficiency of the Hurdy 
Gurdy was 50 per cent. (Art. XXX), while its actual efficiency 
varied from 25 to 35 per cent. 

The substitution of curved buckets for the flat radial vanes 
was the great improvement which converted the Hurdy Gurdy 
into the Pelton wheel. The construction of the bucket is shown 
in Fig. 14S, the jet being divided by the central ridge and each 
half deflected through nearly 180°. Evidently the angle of de- 
flection must be slightly less than 180°, so that the discharge 
from one bucket may clear the one following. A later improve- 
ment* is the Doble bucket, also shown in Fig. 148, each .half of 
which is ellipsoidal in form, with part of the outer lip cut away 
so as to clear the jet when coming into action. 




Fig. 151. 

The relation of the jet to the wheel is shown in Fig. 149, the 
type there shown being arranged with a deflecting nozzle for eco- 
nomic regulation. A more recent type of Pelton wheel is shown 
in Fig. 150, the features of this type being the Doble buckets and 
the so-called chain type of attachment of the buckets. 

One of the most important features of construction in this type 
of impulse wheel is the needle valve for regulating the flow. The 
cross-section shown in Fig. 151 indicates the location of the 
needle valve with respect to the nozzle. The methods of oper- 
ating the valve and of elevating and depressing the nozzle are 
shown in Fig. 152. This form of nozzle under the high heads 
ordinarily used gives a very smooth and compact jet, as shown 
by the instantaneous photograph reproduced in Fig. 153. 



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176 ELEMENTS OF HYDRAULICS 




FiQ. 152. — ^Pelton.regiilating-needle Dozzle. 




Fia. 153. 



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ENEBOY OF FLOW 177 

190. Efficiency of Pelton WheeL — ^If the jet was completely 
reversed in direction and the speed of the buckets was one-half 
that of the jet, the theoretical efficiency of the Pelton whed 
would be unity, or 100 per cent., as shown by Par. 172, Eq. (116). 
This is also apparent from other considerations; for if the velocity 

of the jet is v and that of the buckets is ^, then the velocity of the 
water relative to the lowest bucket is v — 2> or g (Fig. 154). 
Therefore, at exit the water is moving with velocity g relative 




Fig. 154. 

to the bucket while the bucket itself is moving in the opposite 
direction with velocity s- Hence the absolute velocity of the 

water at exit ia ^ — ^9 or zero, and therefore, since the total 

kinetic energy of the water has been utilized, the theoretical 
efficiency of the wheel is imity. As a matter of fact there are 
hydraulic friction losses to be taken into account and also the 
direction of flow is not completely reversed. The efficiency of 
the Pelton wheel has been found in a niunber of authentic tests 
to -exceed 86 per cent. The actual efficiency in operation de- 
pends of course on the particular hydrauKc conditions under 
which the wheel operates. A good idea of what may be ex- 
pected in practice, ^however, is given by the following data, ob- 
tained from a imit of 4,000 kw. normal capacity, operating under 
a head of 1,300 ft.: 

12 



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178 



ELEMENTS OF HYDRAULICS 



' Load in KW 


Percentage of nonnal 
capacity 


Wheel efficiency 


5000 
4000 
3000 
2000 


125% 

100% 

76% 

50% 


81% 
83% 
82% 
70% 



Since the Pelton wheel operates by utilizing the kinetic energy 
of the water, it is best adapted to a small discharge under a high 
head. 

191. Characteristics of Impulse Wheels. — The performance of 
an impulse wheel may be judged from the value of a certain 
combination of wheel constants known as the "specific speed/' 
or better, as the "characteristic speed/' The nature of this 
quantity is explained in Art. XXXVII, under the discussion of re- 
action turbines, the form there given applying also to impulse 
wheels. In discussing its application the following notation will 
be used: 

Let h = effective head in feet; 
H.P. = horse power developed by wheel; 

D = diameter of runner at pitch circle in inches; 

n = speed of runner in r.p.m. ; 

d = diameter of jet in inches; 

Q = discharge in cubic feet per minute; 

e = hydraulic efficiency of wheel; 
Ns = characteristic (specific) speed. 



Then as explained in Art. 
Ne is defined as 



XXXVII the characteristic speed 



N. = 



h^ 



(118) 



This quantity, N„ may be used to classify the various types 
of impulse wheels as indicated in the following table: 



Impulse wheels 


N. 


1 


2 


3 


4 


5 


Efficiency at ?i 
Load 


80% 


79% 


78% 


77% 


76% 



The numerical values of N, in this table refer to the maxi- 



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ENERGY OF FLOW 170 

mum power of one nozzle only. In case the characteristic 
speed lies beween 5 and 10 it is therefore necessary to use more 
than one nozzle. 

For example, suppose that an impulse wheel is required 
to develop 1,300 hp. under a 400-ft. head at an efficiency 
of not less than 78 per cent. From the preceding table 
it is apparent that it is necessary to use a wheel having a 
characteristic i^)eed of about 3. If a single wheel and 
nozzle is used, the speed in r.p.m. at which it must run is found 
from Eq. (118) to be 

n = N.hyj^^ = 3 X 400 ^^^ = 150 r.p.m. 

If two nozzles are used, each furnishes half of the power and the 
corresponding speed is 150\/2 = 212 r.p.m. 

With four nozzles acting on two runners the required speed 
would be ISOVi = 300 r.p.m., and for 6 nozzles acting on 3 
runners, n = 150\/6 = 367 r.p.m. Since the value of iV, is the 
same in each case, the efficiency is practically 78 per cent, in 
each case although there is a wide difference in the speed and 
setting. 

A quantity equally valuable in determining the performance of 
an impulse wheel is the ratio obtained by dividing the pitch 
diameter of the wheel by the diameter of the jet.* An expression 
for this ratio in terms of the wheel constants may be obtained 
as follows: 

The peripheral velocity of the wheel on the pitch circle is 
some fraction, say <py of the jet velocity. Since this peripheral 

velocity is of amount To~o an f^- P®r second we have therefore 



vDn 
12 X60 
whence 



= <pV2gh 



j^ ^ 12 X 60 ipV^gh ^ 1 838^. 






Moreover, the horse power developed is 
„_ 62.37Qhe 

^•^' — 66(r~ 

» Carpenter, Eng, Record, June 17, 1916, p, 795. 



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< 



180 ELEMENTS OF HYDRAULICS 

or, assuming the average efficiency as 80 per cent., this becomes 

Since Q = ^^ ..^ \/2gh cu. ft. per second, we have therefore 

whence 

d = p'-^"X^** = l6.9l^^E (120) 



/ .P. X 11 > 



4 

The required ratio, say R, is then 
l,838»>h>* 

R == D ?L_^ ^ 116.6 -^. (121) 

^ 16.91^ "^"^ 

The value of <pm this expression varies from about 0.42 for ordi- 
nary foundry finish to 0.47 for polished buckets. Its average 
value may be taken as ^ = 0.45. 

Since the characteristic speed is given by 

the expression for R may also be written in the form 

R ^ 116.6 ^ 

or its equivalent, 

RNt = 115.5^ = constant. 

The following table gives an idea of the comparative values of 
R and Ntj computed for <p = 0.45 and 6 = 80 per cent. 



Comparative values of Nu and R = 


D 

d 








fi 1 7 1 8 9 1 10 1 11 1 12 1 13 1 14 1 16 1 16 


|17 


18 


19 


20 


i\r.|7.43 6.60 5.78 5.20 4.73|4.33|4.00i3.72|3.47|3.25 3.05|2.89 2.74|2.60| 



The distinction between these two quantities is that the 
characteristic speed N, indicates whether the runner is to be of a 
high- or low-speed type, whereas the ratio R conveys some idea 



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ENERGY OF FLOW 181 

as to the size of the finished wheel and the number of buckets 
required. 

For a single jet acting on a single runner, the characteristic 
speed ranges from to about 5, or possibly 6 as a maximum. 
For an efficient unit, therefore, the value of R should not lie 
below 10. 

In the case of a runner with more than one jet, the comparative 
characteristic speed varies directly as the square root of the 
nimiber of jets, and the equivalent R varies inversely as the square 
root of the nimiber of jets. Thus a runner for ff = 10 but with 

two jets instead of one has an equivalent R of -y= = 7.07 and 

a corresponding characteristic speed of 5.20\/2 = 7.35. 

192. Girard Impulse Turbine. — A type of impulse wheel 
has been developed in Europe, known as the Girard im- 
pulse turbine. In this type the shaft may be mounted 
either vertically or horizontally, and the flow may be either 
radial or axial. The type shown in Fig. 155 is arranged for 
radial flow,- with vertical shaft. The construction is practically 
the same in all cases, the water entering through a pipe J?, as 
shown in Fig. 155, and proceeding through one or more guide 
passages C, which direct the water onto the vanes. The quan- 
tity of water admitted to the vanes is regulated by some kind of 
gate, that indicated in Fig. 155 being a sliding gate operated by 
a rack and pinion not shown in the figure. 

As the vanes are more oblique at exit than at entrance, they 
are necessarily closer tpgether at exit. To prevent choking, 
it is therefore necessary to widen the vanes laterally at exit, as 
shown in elevation in Fig. 155. As the water discharges under 
atmospheric pressure, ventilating holes are made through the 
sides of the vanes at the back to allow free admission of air. 

193. Power and Efficiency of Girard Turbine. — The power and 
efficiency of a Girard turbine may easily be calculated from the 
results of Art. XXXI. Using the same notation, as indicated 
on Fig. 141, from Eq. (117), Par. 174, the work done per 
second on the wheel is given by the relation 

Work per second = {uiVi cos a — u^vz cos fi). 

Since the water is under atmospheric pressure, the absolute 
velocity vi of the water at entrance is calculated from the effective 



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182 



ELEMENTS OF HYDRAULICS 




Fig. 155. 




Fig. 156. 



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ENERGY OF FLOW 183 

head h by means of the relation vi = y/2gh. From Fig. 156 
we have by geometry 

vi cos a = Ui + Wi cos $, 
V2 cos P = Ui — Wi cos ^, 

and from the law of cosines 

Vi^ = Wi* + toi* + 2iiiix;i cos $. 
v^ = ti2^ + W2^ — 2U2W2 cos ^. 

Now the total energy imparted to the wheel per second is the 
difference in kinetic energy at entrance and exit, namely, 

Therefore, equating this to the expression for the work done per 
second, as given above, we have 

}i{vi^ — V2^) = UiVi cos a — U2V2 cos j8. 

Substituting in this equation the values given above for Vi cos a, 
V2 cos j8, vi^ and V2^ and reducing, the result is finally 

Wi« - Wa^ = Ui^ - U2^ (122) 

It is evident that the efficiency will be greater the more nearly 
the jet is reversed in direction, that is, the smaller <p becomes, 
or what amounts to the same thing, the smaller the absolute 
velocity ^2 of the water at exit. However, <p cannot be decreased 
indefinitely as it is necessary to provide a sufficient area at exit 
to carry the discharge. For a given value of <p, V2 will attain 
nearly its minimum value when U2 = lOj. In this case, however, 
by Eq. (122) we have ui = Wi, in consequence of which 



and hence 



e = 2a, and | = 90*" - /5, 



^2 = 2u2 sin ^, and Ui — ^ 



2 ' ""^ ^^ 2 cos a 



Now the peripheral velocity of the inner and outer ends of the 
vanes in terms of the angular velocity w of the runner is given by 
the relations wi= riw, 1^2= r2w, whence 

ti2 r2 Uir2 

— = ~, or ti2 = -^• 
Ui ri' ri 



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184 ELEMENTS OF HYDRAULICS 

Substituting this value of Ut in the expression given above for vt, 
we have 

2uir2 sin ^ ViU sin ^ 

2 ri ri cos a 

Consequently if Q denotes the quantity of water discharged per 
second, the energy utilized per second is 

O O J r2^sin2-|\ 

Energy per second = g (n* - .,*) = ^(^1 - ^:^^^J- 

Substituting y = 62.4"and dividing by 550, the expression for the 
horse power of the wheel is therefore 



H.P. = «2!*QlL* 



2g(660) 






\ri/ cos* a 



(123) 



yQvi^ 
Since the total kinetic energy available is —5 — , the efficiency, 

E, of the wheel, as defined by Eq. (Ill), Par. 171, is 

,sin»| 

E = 1 - (-') — ^. (124) 

\ri/ cos* a ^ ' 

Since E is less than unity, it is evident that the maximum 
theoretical efficiency is always less than 100 per cent., and also 
that the efficiency is greater the smaller the angles a and <p. 

In practice the angle a is usually between 20° and 25°; <p be- 
tween 15° and 20°; and the ratio — between 1,15 and 1.25. 

Assuming as average numerical values 

a = 20°, <p = 15°, ^ = 1.15, 

and substituting these values in Eq. (124), the theoretical effi- 
ciency of the wheel in this case is found to be 



E 



[sin 7 5 n ^ 
1.15 ^ = 97.5 per cent, approximately. 



This efficiency is, of course, merely ideal as it takes no account 
of hydraulic friction losses. 



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ENERGY OF FLOW 185 

The Girard tjrpe of impulse turbine was formerly manufac- 
tured in this country by the Stilwell-Bierce and Smith-Vaile Co. 
(now the Piatt Iron Works Co., of Dayton, Ohio) under the trade 
name of the "Victor High Pressure Turbine." In a test of a 
45-in. wheel of this type installed in the power plant of the 
Quebec Railway Light and Power Co., Montmorency Falls, 
Quebec, with a rated capacity of 1,000 h.p. under a head of 195 
ft. at a speed of 286 r.p.m., a maximimi efficiency of 78.38 per 
cent, was attained. 

In another test of a 25-in. Victor wheel installed in the Napa- 
noch Power Station of the Honk Falls Power Co., Ellensville, 
N. Y., with a rated capacity of 500 h.p. under 145 ft. head at a 
speed of 480 r.p.m., a maximum efficiency of 84.2 per cent, was 
attained. 

The average efficiency of Victor wheels in plants installed is 
said by the manufacturers to vary from 70 to 80 per cent., de- 
pending on the design of unit. 

In this type of unit, no draft tube is used and consequently 
that portion of the head from the center Une of the wheel to the 
level of the tailrace is lost. Various attempts have been made, 
both with this and other types of impulse wheel, to regain at 
least part of this lost head by means of an automatically regu- 
lated draft tube, designed to keep the water at a certain fixed 
level beneath the runner, but this feature has never proved suc- 
cessful in operation. 

XXXVI. REACTION TURBINES 

194. Historical Development — In Art. XXX it was shown that 
a jet flowing from a vessel or tank exerted a pressure or reaction 
on the vessel from which it flows. A simple appUcation of this 
principle was shown in Barker's mill, Fig. 143, in which the reac- 
tions of two jets caused a horizontal arm to revolve. Later this 
device was improved by curving the arms so that the jets issued 
directly from the ends of the arms instead of from orifices in the 
side, and in this form it was known as the Scotch mill. Subse- 
quently the number of arms was increased and the openings en- 
larged, until it finally developed into a complete wheel. 

In 1826 a French engineer, Foiu'neyron, placed stationary guide 
vanes in the center to direct the water onto the runner, or wheel, 
the result being the first reaction tiu'bine, now known as the 



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ELEMENTS OF HYDRAULICS 



Fourneyron or outward-flow type (Fig. 157). This type was 
introduced into the United States in 1843. 




Fig. 157. — Radial outward flow, Fourneyron type. 





^m 



Fig. 158. — Axial flow, Jonval type. 




Fig. 159. — Radial inward flow, Francis type. 

A later modification of design resulted in the axial or parallel- 
flow turbine, known as the Jonval type, which was also of Euro- 
pean origin, and was introduced into the United States about 
1860 (Fig. 168). 



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ENERGY OF FLOW 



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Fig. 160. 




Fig. 161. 



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ELEMENTS OF HYDRAULICS 




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ENEBOY OF FLOW 



189 



A crude form of inward-flow turbine was built in the United 
States as early as 1838. Subsequently the design was greatly 
improved by the noted American hydraulic engineer, J. B. 
Francis, and it has since been known as the Francis type (Fig. 
159). 

Figure 160 shows a runner of the Francis type used in the plant 
of the Ontario Power Co. at Niagara Falls. These are double 
central discharge, or balanced twin turbines, designed to deliver 
13,400 h.p. per unit, imder 180 ft. head. The runners are of 
bronze, 82^ in. in diameter; the shafts 24 in. maximum diameter; 
and the housings of reinforced-steel plate 16 ft. in diameter, 
spiral in elevation, and rectangular in plan, as shown in sectional 




Fig. 163. 

detail in Fig. 161. A cross-section of the power house in which 
these turbines are installed is shown in Fig. 162. 

196. Mixed-flow, or American, Type. — ^The mixed-flow tur- 
bine, or American type, is a modification of the Francis turbine 
resulting from a demand for higher speed and power under low 
heads. Higher speed could only be obtained by using nmners of 
smaller diameter, which meant less power if the design was un- 
altered in other respects. To increase the capacity of a runner 
of given diameter the width of the runner was increased, fewer 
vanes were used, and they were extended further toward the cen- 
ter. As this decreased the discharge area, the vanes were curved 
so as to discharge the water axially (Fig. 163). In a standard 
turbine of this type, the water from the conduit or penstock, 



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ELEMENTS OF HYDRAULICS 



after passing through the shut-off valves, enters a cast-iron or 
cast-steel casing of spiral form encircling the runner, by which 
it is delivered to the whole circumference of the runner at a uni- 
form velocity (Fig. 164). The detail of the gate work for regu- 




lating the admission of water to the runner is shown in Fig. 165, 

and the entire turbine unit is shown in perspective in Fig. 166. 

196. Use of Draft Tube. — In a reaction, or pressure, turbine 

the passages between the vanes are completely filled with water, 



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ENERGY OF FLOW 



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Fig. 165. — Gate mechanism 




Fig. 166 



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192 



ELEMENTS OF HYDRAULICS 



and since this is the case, it will run submerged. By the use of 
a draft tube or suction tube, invented by Jonval in 1843, it is 
possible, however, to set the turbine above the level of the tail- 
water without losing head (Fig. 167). This is due to the fact 




Fia. 167. 

that the pressure at the upper end of the draft tube is enough 
less than atmospheric to compensate for the loss of hydrostatic 
pressure at the point of entrance to the wheel. The chief advan- 
tage of a draft tube, however, is that its use permits of setting the 
turbine in a more accessible position without any sacrifice of head. 



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ENEBOY OF FLOW 



193 



The effect of using a draft tube may be explained mathematic- 
ally as follows: In Fig. 168 let A refer to a point in the free water 
surface of the headrace; B the point at which water enters the 
turbine; C its point of exit into the draft tube; and D a point in 
the free water surface of the tailrace, the level of the latter being 



Head Race 




Fig. 168. 

assumed as datum. Then, neglecting friction and writing out 
Bernoulli's equation between the points A and B, we obtain the 
relation 

2g y 2g 7 

or, since Va = 0, Jia = K and Kb = ^2, this becomes 



V^B 



7 7 ' 2g y 2g 

Similarly, writing Bernoulli's equation between the points C and 
Z>, we have 

2^ + - + Ac-^ + - + A., 

or, since Vd = 0, Ai> = 0, and he = A2, this becomes 

7 7 ^ 2sf' 

13 



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194 ELEMENTS OF HYDRAULICS 

Now the effective head on the runner is equal to the difference 
between the total heads at B and C. Consequently 

Effective head = (I + ^' + ^.) - (^ + ^^ + /ic) 

^^ J- I. -J- J. ^^ 

= — + Ai + A2 - — • 

But since Ai + A2 = A, and Pa = Pdi since both are atmospheric, 
this expression for the effective head reduces to simply 
Effective head = h. 

This, however, includes all frictional losses in the intake and 
draft tubes. Including such losses the expression for the effective 
head becomes 

Effective head = A — friction head. 

Provided the head in the draft tube does not exceed the ordinary 
suction head, say about 25 ft., the use of a draft tube therefore 
causes no loss in the static head h except the small amount due 
to friction. 

197. Draft-tube Design. — The design of the draft tube is an 
important element in any hydraulic installation as a considerable 
percentage of the efficiency of the plant depends on this feature. 
The main object of the draft tube is to gradually reduce the 
velocity of discharge so as to make the final velocity at exit as 
small as possible, thereby wasting as little as possible of the kinetic 
* energy of flow. As the upper end of the draft tube where it 
joins the turbine case is necessarily circular, the ideal form of tube 
would be horn-shaped, that is, having a gradual flare and keeping 
the cross-section circular. For practical reasons, however, the 
outlet must usually be oval or flattened, which leads to certain 
difficulties in calculating the profile and cross-section of the tube. 
The following is perhaps the simplest solution of these difficulties.^ 

Cross-section of Draft Tvbe. — At the upper end of the draft tube, 
its cross-section as well as the discharge, Q, and velocity of flow, 
Vj are known. Assimiing a tentative value for the velocity of 

1 Given by R. Dubs, Zurich Switzerland, in "La Houille Blanche," 
abstract by A. G. Hillbebg, Eng, Record, Aug. 9, 1913. For an elaborate 
analysis of draft-tube design see articles by A. G. Hillberg, Eng. Record^ 
Nov. 13, 1915, p. 604; Nov. 20, 1915, p. 630. 



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ENERGY OF FLOW 



195 



flow at the lower end, the cross-section area, A, at this end may- 
then be determined from the relation 

A ^9. 

V 

If the shape of the cross-section is assumed to be a rectangle 
of length B and depth Z>, with circular corner fillets, as indicated 
in Fig. 169, then if E denotes the area of one fillet, we have 

A = BZ> - iE. 
Since the fillet is a quadrant of a circle, its area is 

E ^r^-"^ = ^(0.8684) 

and consequently 

A = J5D - OMMr\ 




Elementa of Desisn of a Flattened Concrete Draft Tube 
Courtesy of Eng. Record, 

Fig. 169. 

198. Profile of Draft Tube. — The profile or longitudinal sec- 
tion of the draft tube is assumed to be formed of the involutes 
of circles, as indicated in Fig. 169 in which the large and small 



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196 ELEMENTS OF HYDRAULICS 

cirdes are the generating circles of the two involute curves, 
respectively. In order to determine the elements of these 
curves it is necessary to have given the dimensions H and T. 
In applying this method, therefore, it is necessary to begin by 
assimiing H and T tentatively until the other elements of the 
design have been worked out. 

To determine an involute curve it is necessary only to fix the 
center and radius of its generating circle. Since any tangent 
to the involute is perpendicular to the corresponding tangent to 
the circle, it is evident first of all that both generating circles 
must touch the planes of the cross-sections at entrance and exit. 
Let B denote the radius of the large generating circle, and n the 
distance of its center from the tangent to its involute at entrance, 
as indicated in the figure. Then, ilT > H,ia order to make the 

tB 

draft tube as short as possible we may assume w = -s", in which 

case, frdln the properties of the involute, the radius at the lowest 
point will be r + JB = tB, From this relation we have 

B = -^ = 0.467r, 
ir — 1 

and also 

n = ^ = 0.7334r, 

which together determine'the construction of the outer curve. 
For the special case when T = H these relations become 

22 = 0.467H; n = 0.7334iJ. 

To determine the elements of the inner curve, let r denote the 
radius of the small generating circle, m the distance of its center 
from the tangent to its involute at entrance, and a the angle be- 
tween the planes of the cross-sections at entrance and exit. 
Then from the property of involutes and the dimensions given in 
Fig. 169 we have evidently 

Di + m + z = n + y, 
m + (gg^) a + D + {y -x) =z. 
Eliminating m between these two equations the result is 



n + 22/ - 2x + [^) a^z-D + Dx. 



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ENERGY OF FLOW 197 



Inserting in this relation the values of x, y, z and n, namely, 

X = r tan ^; y = /2 tan ^ 
and solving for r, the result is 



X = r tan g; y = B tan 2; « = n + \^^J a;n = -jf 



r^B- ^^-^ 



which gives the radius of the smaller generating circle. The loca- 
tion of its center is then determined from the relation 

m = n + y-x — Di = ^ + 22 tan ^-r tan ^ — Di 
= ^ + (/2-r)tan|-Di. 



For the special case when a = 90**, these relations become 

IT 



^ "^ 0.4292,' 



2 2 

X = r. 
199. Time of Flow through Draft Tube. — By the determina- 
tion of the cross-section and profile of the tube, the velocity of 
flow at entrance and exit, say Vi and V2, have been fixed, and also 
the mean length, L, of the draft tube. To determine the time of 
flow through the draft tube it is assumed that the kinetic energy 
of a particle of water decreases linearly as it flows through the 
tube. That is to say, if M denotes the mass of the particle and 
s its distance from some origin back of the entrance, then }^Mv^ 

varies as -, or, since Af is a constant, this may be written 

8V^ = Jfc, 

where fc is a constant as yet xmdetermined. The law of variation 
in velocity is indicated in Fig. 169. If the distances of the par- 
ticle from the origin at entrance and exit are denoted by Si and 
82 respectively, and the corresponding velocities by vi and 1^2, then 

si^i^ = K; 82^2^ = K. 
and consequently 



«.-ax=L=A;(l,-i) 



whence tl _ Lvih>2^ 



fl* — V2^ 



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Google 



198 ELEMENTS OF HYDRAULICS 

which determines k and therefore also 81 and St- Now from 
the ordinary differential notation for velocity we have 



whence 



ds Ik 



dt = — 7=.Vsds. 



Integrating this between the known limits si and 82 the time of 
flow through the tube is found to be 

200. Recent Practice in Turbine Setting. — A typical illustra- 
tion of recent development in the design and installation of 
reaction turbines in the United States is furnished by the plant 
of the Mississippi River Power Co. at Keokuk, Iowa (see frontis- 
piece). These turbines are of the Francis type, and develop 
10,000 h.p. per unit at a speed of 57.7 r.p.m. under a head of 32 
ft., with an overload capacity of 13,000 h.p. 

The Keokuk power station is designed to accommodate four 
auxiliary and 30 main power units, the sole function of the aux- 
iliaries being to generate power to drive exciter machines. One- 
half of the station has been fully completed and is in operation, 
that is, two auxiliaries and 15 main power units. For the re- 
maining half of the station, all under-water work has been fin- 
ished complete, including the draft tubes. Likewise the forebay 
protection wall, intakes and headgate masonry are entirely com- 
pleted, with sufficient masonry backing the latter to make it 
stable as a dam. 

Seven of the main turbines were built by Wellman-Seaver- 
Morgan Co. and eight by I. P. Morris Co. and all are identical 
in every respect excepting the runners only. The Wellman- 
Seaver-Morgan runners are 147 in. rated diameter and 16 ft. 
6 in. diameter at outside of band. Six of the I. P. Morris runners 
are 1533^ in. rated diameter, two are 1393^ in. rated diameter and 
all eight are 17 ft. 4 in. diameter at outside of band. Both wheel 
builders were required to guarantee an efficiency of 83 per cent. 
When tested at Holyoke, the Wellman-Seaver-Morgan model 
developed 88 per cent, efficiency and the I. P. Morris models 
(type 1533^ in. diameter) 86 per cent, and (type 139}^ in. diame- 
ter) 90 per cent, efficiency. After several months of conunercial 



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ENERGY OF FLOW 



199 




FiQ. 170. — Center section of main turbine on transverse axis of power house, 
Mississippi River Power Co. 



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200 ELEMENTS OF HYDRAULICS 

operation, one of the Wellman-Seaver-Morgan wheels and one of 
the I. P. Morris wheels (1533^ in. diameter) were tested in place 
at the power station and both developed about 90 per cent, effi- 
ciency. These tests were conducted by Mr. B. H. Parsons, of 
New York City. 

Each of the units is of the single-runner vertical-shaft type as 
shown in Fig. 170. The trend of present development of the 
reaction turbine seems to indicate a still wider application of 
this type to aU conditions of head and speed, and that the 
single-nmner vertical-shaft turbine will eventually supersede 
the multi-runner horizontal-shaft type (Fig. 161) and the multi- 
runner vertical-shaft type (Fig. 167). 

In the Keokuk plant the intakes and draft tubes are of con- 
crete moulded in the substructure of the power house. The water 
from the forebay reaches each turbine through four intake open- 
ings, the outer dimensions of which are 22 by 7^ ft., leading 
into a scroll chamber 39 ft. in diameter (Fig. 171). 

The draft tubes are about 60 ft. in length along the center line 
and contain a right angle bend to 24 ft. in. radius. They are 
18 ft. in. diameter at the throat, from whence they flare in 
straight lines to the mouth, where the cross-section is oblong 
22 ft. 8 in. by 40 ft. 2 in., bounded by two semicircles connected 
by straight lines. The velocity of flow is thereby diminished 
from 14 ft. per second at the top of the draft tube to 4 ft. per sec- 
ond at the outlet, the effect being to increase the hydraulic 
efficiency of the plant about 7 per cent. This type of^ construc- 
tion is representative of recent practice, which seems to favor 
the moulding of the volute casing directly in the substructure of 
the power house for all low-head work. Foj heads exceeding 
100 ft., the amount of reinforcement in the concrete becomes 
so great as to warrant the use of cast-iron casings, and for heads 
exceeding 250 ft. the use of cast steel for turbine casings is stand- 
ard practice. 

In the vertical-shaft turbine the weight is carried on a thrust 
bearing, the design of which has been one of the most important 
considerations affecting the adoption of this type. In the 
Keokuk plant the turbine nmner is coupled to the generator 
above by a shaft 25 in. in diameter the total weight of the revolv- 
ing parts, amounting to 550,000 lb., being carried on a single 
thrust bearing 6 ft. in diameter. This bearing is of the oil- 
pressure type, a thin film of oil being maintained at a pressure 



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ENEMY OF FLOW 



201 




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202 ELEMENTS OF HYDRA ULICS 

of 250 lb. per square inch between the faces of the bearings. As a 
momentary failure of the oil supply would result in the immediate 
destruction of the bearing, provision is made for such an emerg- 
ency by introducing an auxiliary roller bearing which is normally 
unloaded. A slight decrease in the oil supply, however, allows 
the weight to settle on this roller bearing, which although not 
intended for permanent use is sufficiently large to carry the 
weight temporarily until the turbine can be shut down. 

The oil pressure bearing, when taken in connection with the 
necessary pumps and auxiliary apparatus, is expensive to in- 
stall and maintain, and requires constant inspection. For this 
reason the roller bearing and the Kingsbury bearing are now 
being appUed to large hydro-electric imits. One of the first 
installations in which the roller bearing was appUed to large 
hydraulic units was at the McCall Ferry Plant of the Pennsyl- 
vania Water and Power Co., where both the roller and the Kings- 
bury type of bearing are now in satisfactory use. 

XXXVn.^CHARACT£RISTlCS OF REACTION TURBINES 

201. Selection of Type. — The design of hydrauUc turbines is a 
highly speciaUzed branch of engineering, employing a relatively 
small number of men, and is therefore outside the domain of 
this book. On account of the rapid increase in hydrauUc devel- 
opment, however, every engineer should have a general knowl- 
edge of turbine construction and type characteristics so as to be 
able to make an inteUigent selection of type and size of turbine 
to fit any given set of conditions. For this reason the following 
explanation is. given of the use and significance of commercial 
turbine constants, such, for instance, as those given in the runner 
table in Par. 213. 

202. Action and Reaction Wheels. — The two systems of 
hydrauUc-power development now in use in this country are the 
impulse wheel and the radial inward-flow pressure turbine. 
When an impulse wheel is used, the total effective head on the 
runner is converted into speed at entrance and this type is there- 
fore sometimes called an action wheel. In the case of a pressure 
turbine, however, the effective head on the runner is not all con- 
verted into speed at entrance, the entrance speed being smaller 
than the spouting velocity, so that the water flows through the 
runner under pressure, the effect of which is to accelerate the 



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ENERGY OF FLOW 203 

stream as it passes over the runner. A pressure turbine is there- 
fore called a reaction wheel. 

Reaction turbines are generally used for heads between 5 
and 500 ft., and impulse wheels for heads between about 300 
and 3,000 ft.^ While there is no doubt as to the system proper 
for very low or very high heads, there is a certain intermediate 
range, say from 300 to 500 ft., for which it is not directly ap- 
parent which system is most suitable. To determine the proper 
system within' this range, the criterion called the characteristic 
speed has been introduced, as explained in what follows. 

203. Speed Criterion. — In determining the various criteria 
for speed, capacity, etc., the following notation will be used: 

Let h = net head in feet at turbine casing, 

= gross head minus all losses in headrace, conduit and 
tailrace; 
d = mean entrance diameter of nmner in feet; 
6 = height of guide casing in feet; 
n = runner speed in r.p.m.; 
V = spouting velocity in feet per second; 
Ui — peripheral velocity of runner in feet per second; 

Ui = y = ratio of peripheral speed of runner to spouting 
velocity of jet. 

From Eq. (30), Par. 51, the spouting velocity in terms of the 
head is given by the relation 

where the constant C = 0.96 to 0.97. 

For maximum efficiency the peripheral velocity of the runner 
is some definite fraction of the ideal velocity of the jet, \/2gh, 
that is, 

Ui = (pV^ (126) 

where (p denotes a proper fraction. For tangential or impulse 
wheels the average value of ^ is from 0.45 to 0.51, whereas for 

1 At the hydro-electric plant of the Georgia Railway and Power Co. at 
Tallulah Falls, Ga., the hydraulic head is 680 ft., which is probably the 
highest head that has been developed east of the Rocky Mountains, and the 
highest in this country for which the reaction type of turbine has been 
employed. See Fig. 181. (General Electric Review, June, 1914, pp. 608- 
621.) 



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204 ELEMENTS OF HYDRAULICS 

reaction turbines its value ranges from 0.49 to 0.96, with an 
average range from about 0.57 to 0.87. 

The ratio Ui of peripheral speed of runner to spouting velocity- 
is therefore given by the expression 

V C 

and consequently Ui is about 3 per cent, more than (p. 

Since in Eq. (125) the factor (py/2g is a constant, this equa- 
tion may be written in the form 

til = KVh, i 

where the coefficient K may be called the speed constant. For a 
given runner for which d, h and n are known, this speed constant 
may be calculated from the relation 

k. = -*^ = -^- (126) 

Vh eovh 

By the use of the speed constant fc«, different types of runners 
may be compared as regards speed. In the case of reaction tur- 
bines if the speed constant is much in excess of 7, either the speed 
is too high for maximum efficiency or the nominal diameter of the 
runner is larger than its mean diameter. 

204. Capacity Criterion. — The entrance area A of the runner 
is given by the relation 

A = Ciirdb 

where Ci denotes a proper fraction, since the open circumference 
is somewhat less than the total circumference by reason of the 
space occupied by the ends of the vanes or buckets. The ve- 
locity of the stream normal to this entrance area is the radial 
component of the actual velocity at entrance, say Ur, and like 
this velocity is a multiple of \/A, say 

Ur = C2\/A. 

Since the discharge Q is the passage area multiplied by the speed 
component normal to this area, we have 

Q = Aur = {ciTdb)ct'\/h. 

It is customary, however, to express the height of a runner in 
terms of its diameter as 

6 = Czd, 

where the coefficient c$ is a constant for homologous runners of a 
given type. For American reaction turbines Cz varies from about 



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ENERGY OF FLOW 205 

0.10 to 0.30. Substituting this value for 6 in the expression for 
the discharge, it becomes 

Therefore, if the constant part of this expression is denoted by 
fcg, it may be written 

Q = kgd^Vh. (127) 

The coefficient k^ may be called the capacity constant of the run- 
ner, and for any given runner may be computed from the relation 

For American reaction turbines the capacity constant ranges in 
value from about 2 to 4. Since kg has approximately the same 
value for all runners of a given type, it serves as a criterion for 
comparing the capacity of runners of different types. 

206. Characteristic Speed. — The speed constant and capacity 
constant taken separately are not sufficient to fix the require- 
ments of combined speed and capacity. That is to say, two 
runners may have diiGferent values of K and kg and yet be 
equivalent in operation. To fix the type, therefore, anothei 
criterion must be introduced which shall include both k^ and 
kg. The most convenient combination of these constants is 
that introduced by Professor Camerer of Munich and the well- 
known turbine designer, Mr. N. Baashuus of Toronto, Ontario. 
This criterion may be obtained as follows.^ 

The horse power of a turbine is given by the expression 

where e denotes the hydraulic efficiency of the turbine. If the 
horse power, discharge Q and head h are given, the efficiency 
may be calculated from this relation by writing it in the form 

^ 550H.P. 
^ 62.37 QA' 

If the efficiency is known, the constants in the above expression 
for the horse power may be combined into a single constant fc, 
and the equation written in the form 

H.P. = kQh. (129) 

^S. G. ZowsKi, "A Comparison of American High-speed Runners for 
Water Turbines, Eng. News, Jan. 28, 1909, pp. 99-102. 



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206 ELEMENTS OF HYDRAULICS 

When the efficiency is not known it is usually assumed as 80 per 
cent., in which case fc = jr* 

From Eq. (126) the speed in r.p.m. is given by the relation 

n = ?^ (130) 

and from Eq. (128) the nominal diameter of the runner is given as 



-i. 



Q 



Eliminating d between these two relations, we have therefore 

eoKVKVhVh 

n = j=z • 

tVq 

Moreover, from Eq. (129) we have 

and substituting this value of Q in the preceding expression for n, 
we have finally 

^ )vww: ^"^^ 

The expression in parenthesis is a constant for any given type 
and may be denoted by Ng, in which case we have 

hVh 



^ / eokVi 

\ TT 



For any given type of turbine this constant iV, may be calculated 
from the relation 

Various names have been proposed for this constant N^ such 
as "type constant" and "type characteristic." In Germany, 
where its importance as determining the type and perfonnance 
of a turbine seems to have first been recognized, it is called the 
specific speed (spezifische Geschwindigkeit, or spezifische Um- 
laufzahl). This term, however, is not entirely satisfactory to 
American practice, as it seems desirable to use the term specific 
speed in another connection, as explained in what follows. The 
term for the constant Ns favored by the best authorities as more 



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ENERGY OF FLOW 207 

fully describing its meaning is characteristic speed, which is 
therefore the name adopted in this book.^ 

For impulse wheels the characteristic speed ranges in value 
from about 1 to 5, while for radial, inward-flow turbines its 
value Ues between 10 and 100. 

206. Specific Discharge. — It is convenient to express the 
discharge, power, speed, etc., in terms of their values under a 1-ft. 
head. 

The discharge under a 1-ft. head is called the specific die- 
charge, and its value is found by substituting A = 1 in. Eq. (127). 
Consequently if the specific discharge is denoted by Qi, its value 
is 

and therefore 

Q = QiVh. (133) 

For reaction turbines the specific discharge ranges in value from 
0.302(P for ahe slowest speeds, to 2.866d^ for the highest speeds, 
the diameter d being expressed in feet. 

207. Specific Power. — Similarly, the power developed under 
1-ft. head is called the specific power, and will be denoted in what 
follows by H.P.i. From Eq (129) we have 

H.P. = kQh 
and since from Eq. (127) 

Q = k^d^Vh, 

by eliminating Q between these two relations we have 

H.P. = kk^d^hVh. 

Substituting A = 1 in this equation, the specific power is there- 
fore given as 

H.P.1 = kk^d^, 
and consequently 

H.P. = H.P.ih\/h. (134) 

208. Specific Speed. — ^By a nalogy with what precedes, the speed 
under 1-ft. head will be called the specific speed and denoted 

^ The use of the term characteristic speed has been recommended to the 
author by the well-known hydraulic engineer Mr, W. M. White, who is 
using this term in preparing the American edition of the German handbook 
"de Htitte," and strongly advocates its general adoption in American 
practice. 



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208 ELEMENTS OF HYDRAULICS 

by ni. Substituting ft = 1 in Eq. (130), we have therefore 

60fc, 

and consequently 



^^ = Tr 



n = niVh. (136) 

For reaction turbines the specific speed ranges in value from 

78 147 

-^ for the lowest speeds, to — r- for the highest speeds,-the diame- 
ter d being expi:essed in feet. 

209. Relation between Characteristic Speed and Specific 
Speed. — From the relation 

. nVHT. 

the characteristic speed N, may be defined in terms of the quan- 
tities defined above as specific. Thus, assmning ft = 1 and 
HJP. = 1, we have N, = n, expressed in r.p.m. Therefore, the 
characteristic speed is the speed in r.p.m. of a turbine diminished 
in all its dimensions to such an extent as to develop 1 h.p. when 
working under a head of 1 ft. 

Since it is apparent from Eq. (131) that N, stands for the 
combination 

,, QOK\%Vk 

iV, = 9 

T 

where fc is a function of the efficiency e, the characteristic speed 
Ns is an absolute criterion of turbine performance as regards 
speed, capacity and efficiency. From Eq. (132), however, it 
is evident ' that Ns may be calculated directly from the speed, 
power and head without knowing the actual dimensions of the 
runner, its discharge, or its efficiency. 

210. Classification of Reaction Turbines. — The character- 
istic speed Ng may be used as a means of classifying not only the 
various types of impulse wheels (Par. 191) but also of reaction, or 
pressure, turbines. 

In the following table practically all the different kinds of 
pressure turbines of the radial inward-flow type are classified 
by their characteristic speeds, the corresponding efficiencies 
being also given in each case. 



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ENERGY OF FLOW 



209 



Type of pres- 
sure turbine 


Charac- 
teristic 
speed, Na 


Efficiency 


Maximum 


At half power 


Low-speed 

Medium-speed.. . 

High-speed 

Very high-speed. 


10- 20 
30- 60 
60- 80 
90 -100 


88-92 per cent, at ?i power 
88-92 per cent, at ?i power 
87-91 per cent, at 0.8 power 
86-90 per cent, at 0.9 power 


80-85 per cent. 
78-82 per cent. 
75-80 per cent. 
73-76 per cent. 











The values of the constant iV, in this table refer to the maxi- 
mum power of one runner only. In case the characteristic speed 
is higher than 100, it is necessary to use a multiple unit. At 
maximum power, the efficiencies are slightly lower than the 
maximum efficiencies given above. 

From this table it is apparent that low-speed turbines show a 
favorable efficiency over a wide range of loads but are prac- 
tically limited to high heads, whereas high-speed turbines are 
efficient at about 0.8 load but show a notable decrease in effi- 
ciency at half load. The use of the latter is therefore indicated 
for low heads where the water supply is ample at all seasons. 

211. Numerical Application. — ^To illustrate the use of the pre- 
ceding numerical data, suppose that it is required to determine 
the proper system of hydraulic development for a power site, 
with an available flow of 310 cu. ft. per second under an effective 
head of 324 ft. 

The power capacity in this case is 

H.P. = «?i^ = 9,100. 

Of this amoimt about 100 h.p. will be required for exciter and 
Ughting purposes. There would therefore be installed two 
exciter units running at 550 r.p.m., one of which would be a 
reserve unit. The characteristic speed for these units would then 
be 



^ 650 flOO .^- 



Since this lies between 1 and 5, an impulse wheel would be used 
for driving the exciter generators. 

The main development of 9,000 h.p. would be divided into three 
units of 3,000 h.p. each, running at 500 r.p.m., with a fourth 
unit as a reserve. The characteristic speed for these main units 
would then be 



^ _ 600 /3,000 ^ 

^' "■ 3ioVi7:6 " ^^• 



14 



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210 



ELEMENTS OF HYDRAULICS 



As N, lies between 10 and 100, a pressure turbine would be used 
for driving the main generators. 

212. Normal Operating Range. — Having determined the 
proper type of development, it is necessary, in case a reaction 





















































































































^periatink i 














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r^ 




z. 




! *^ 












K-K^^^ 






















,^ 






^ 
































^.^ 


/ 










I 


























4 


fP 


y 




































;> 














\ 


:3 




Cjv^ 


r OS 


d 










C;,i5 


lev 


y 














/ 






















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-^ '? 


^ 














/ 




























^ 












/ 
























\ 




^ 










y 


/ 
























I 




1 










L 


































1 











10 



20 



40 50 60 70 

Per Cent Turbine Load 

Fia. 172. \ 



90 



10^ 



turbine is used, to determine the required size and type of runner 

to develop maximum efficiency under the given conditions of 

operation. 

For a turbine direct-connected to a generator, the capacity of 

the turbine, in general, should be such as 

to permit the full overload capacity of 

the generator to be developed and at the 

same time place the normal operating 

^^^ "?^ range of the unit at the point of maxi- 

^^5W mum efficiency of the turbine, as indi- 

^^-"""'^ ^ cated in Fig. 172. The normal horse 

\^^ power, or full-load, here means the 

power at which the maximum efficiency 

is attained, any excess" power being 

regarded as an overload. 

When the supply of water is ample but the head is low, effi- 
ciency may to a certain extent be sacrificed to speed and capacity 



J.' V/X C» M IJ.X ILf AXX^ VI 



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ENERGY OF FLOW 



211 



in order that the greatest power may be developed from each 
runner, thereby reducing the investment per horse power of 
the installation. On the other hand, when the flow of water is 
insufficient to meet all power requirements, an increase in effi- 




FiG. 174. 




Fig. 175. 

ciency shows a direct financial return in the increased output of 
the plant. 

213. Selection of Stock Runner. — Ordinarily it is required to 
select a stock runner which will operate most favorably under the 



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212 ELEMENTS OF HYDRAULICS 

given conditions. To explain how an intelligent selection of 
size and type of runner may be made from the commercial con- 
stants given by manufacturers, the following runner table of a 
standard make of turbine is introduced^ (page 213). 

The cut accompanjdng each of the six types given in the table 
shows the outline of runner vane for this t3rpe. To indicate its 
relation to the runner and to the turbine unit as a whole, Fig. 
173 shows a typical cross-section of runner; Fig. 174 shows how 
this is related to the casing; and Fig. 175 shows a cross-section 
of the entire turbine unit. The runner is also shown in per- 
spective in Fig. 176. 




Fig. 176. 

From Eq. (132) it is evident that, other things being equal, 
the characteristic speed for high heads will be relatively small 
whereas for low heads it will be large. Thus in the runner 
table above, type "A," with a characteristic speed of 13.55 is 
adapted to high heads, running up to 600 ft., while at the other 
end of the series, type "F," with a characteristic speed of about 
75, is adapted to effective heads as low as 10 ft. 

To give a numerical illustration of the use of the runner table, 
suppose it is required to determine the tjrpe of runner and the 
speed in r.p.m. to develop 750 h.p. under a head of 49 ft. 

In this case hy/h = 49\/49 = 343, and consequently 

^■^" - AVi " 343 - ^•^' 
^ The Allis-Chalmers Co., Milwaukee. 



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ENERGY OF FLOW 



213 



TVPE"li"^RimilEII 


TYPE^'e'RIUliaR 


TYPE 
t 


Vrunjier 


ife^ 


\ 


1^ 


\ 


K 




ol -^ 




i ^ 


2 


^^ 




11 






N. - 18.56 


N, - 20.3 


Nm m 29.4 


Ui - 0.585 


Ui - 0.625 


Ut m 0.668 


Diam. 


R.P.M.1 


H.P.1 


Qi 


R.P.M.1 


1 H.P.1 


Ql 


R.P.M.1 1 H.P., 1 Qi 


16 


71.7 


0.0358 


0.394 


76.6 


0.0705 


0.776 


81.4 


0.130 


1.43 


18 


50.8 


0.0514 


0.565 


63.8 


0.105 


1.155 


67.8 


0.187 


2.06 


21 


51.2 


0.0705 


0.776 


54.7 


0.138 


1.523 


58.2 


0.225 


2.48 


24 


44.8 


0.0915 


1.007 


47.8 


0.182 


2.00 


51.0 


0.333 


3.66 


27 


39.8 


0.116 


1.276 


42.5 


0.229 


2.52 


45.2 


0.423 


4.65 


90 


35.8 


0.142 


1.562 


38.3 


0.284 


3.12 


40.7 


0.520 


5.72 


34 


31.6 


0.184 


2.024 


33.8 


0.363 


3.99 


35.9 


0.668 


7.35 


38 


28.3 


0.230 


2.53 


30.2 


0.453 


4.98 


32.2 


0.835 


9.19 


42 


25.6 


0.280 


3.08 


27.4 


0.551 


6.06 


29.1 


1.016 


11.18 


46 


23.4 


0.336 


3.69 


25.0 


0.665 


7.32 


26.6 


1.225 


13.48 


50 


21.5 


0.398 


4.38 


23.0 


0.79 


8.69 


24.4 


1.450 


15.95 


55 


19.5 


0.480 


5.28 


20.9 


0.95 


10.45 


22.2 


1.745 


19.20 


60 


17.9 


0.573 


6.30 


19.1 


1.13 


12.43 


20.4 


2.08 


22.88 


66 


16.5 


0.672 


7.39 


17.7 


1.33 


14.63 


18.8 


2.44 


26.84 


70 


15.4 


0.785 


8.64 


16.4 


1.53 


16.83 


17.5 


2.82 


31.00 




TYPE^D^RUffltER 


TYPE'E'RUNJIER 

1 


type"f"rojuier 




S 

< 

S 


w 






1 


p 




^ 


< 


,) 












k 




N. - 40.7 


N, - 61.7—60.5 


i^. -71.4— 79 


Ui - 0.70 


Ui - 0.76 


Ui - 0.85 


Diam. 


1 R.P.M.I 


iHJP.i 


Qi 


Diam. 


R.P.M.1 


H.P.1 Qi 


R.P.M.1 H.P.1 


1 Qi 










14 


98.4 


0.277 


3.05 


111.5 


0.410 


4.51 










16 


86.1 


0.367 


4.04 


97.7 


0.541 


5.95 










18 


76.5 


0.471 


5.18 


86.8 


0.704 


7.74 


15 


85.7 


0.226 


2.49 


20 


69.0 


0.597 


6.57 


78.1 


0.912 


10.03 


18 


71.4 


0.324 


3.56 


22 


62.6 


0.731 


8.04 


71.0 


1.133 


12.46 


21 


61.3 


0.442 


4.86 


24 


57.4 


0.883 


9.70 


65.1 


1.375 


15.13 


24 


53.6 


0.577 


6.35 


26 


53.0 


1.055 


11.60 


60.1 


1.62 


17.85 


27 


47.6 


0.731 


8.04 


28 


49.2 


1.243 


13.67 


55.8 


1.93 


21.25 


30 


42.8 


0.902 


9.92 


30 


46.0 


1.436 


15.80 


52.1 


2.20 


24.20 


34 


37.8 


1.158 


12.74 


32 


43.0 


1.65 


18.15 


48.8 


2.55 


28.10 


38 


33.9 


1.444 


15.88 


34 


40.5 


1.89 


20.80 


46.0 


2.82 


31.10 


42 


30.6 


1.765 


19.4 


36 


38.3 


2.15 


23.65 


43.5 


3.14 


34.55 


46 


28.0 


2.12 


23.3 


38 


36.3 


2.42 


26.60 


41.1 


3.52 


38.70 


50 


25.7^ 


2.50 


27.5 


40 


34.4 


2.75 


30.25 


39.1 


3.93 


43.20 


55 


23.4 


3.04 


33.4 


42^ 


32.4 


3.09 


34.0 


36.8 


4.33 


47.60 


60 


21.4 


3.61 


39.7 


45 


30.6 


3.53 


38.8 


34.7 


4.92 


54.10 


65 


19.8 


4.22 


46.4 


47J4 


29.0 


4.01 


44.1 


32.9 


5.66 


62.25 


70 


18.4 


4.90 


53.9 


50 


27.6 


4.45 


49.0 


31.2 


6.13 


67.40 










5aH 


26.3 


4.95 


54.5 


29.8 


6.75 


74.25 










55 


25.1 


5.52 


60.7 


28.4 


7.60 


82.50 










57H 


24.0 


6.10 


67.1 


27.2 


8.16 


89.75 










60 


23.0 


6.80 ' 


74.8 


26.0 


8.94 


98.30 



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214 ELEMENTS OF HYDRAULICS 

which corresponds to a 30-in. type "F" runner. Referring to 
the table for this type and size, we have ni = 52.1, from which 
the required speed is found to be 

n = 52.lVh = 364.7, say 360 r.p.m. 

If twin turbines were used, we would have 

which corresponds to a 22-in. type "F" runner, having a speed 
of 

n = 71.0y/h = 497, say 500 r.p.m. 

As a second illustration, let it be required to find from the 
table the t3rpe of runner and speed to develop, 4,000 h.p.under an 
eflfective head of 300 ft. 

In this case h\/h = 300\/300 = 5,190, and consequently the 
specific power is 

H.P.. = ^ = i^ = 0.77. 
hVh 5,190 

which corresponds to a 50-in. type "B" runner. Referring to 
the table for this type and size we have 

H.P.i = 0.79, and rii = 23, 
and consequently the power and speed for this type and size is 

H.P. = 0.79hVh = 4,100, 
and 

n = 23 Va = 395, say 400 r.p.m. 

XXXVm. POWER TRANSMITTED THROUGH PIPE LINES 

214. Economical Size of Penstock. — In hydraulic develop- 
ments involving the construction of long pipe lines or pipes under 
a high head, the cost of the pipe line is often a considerable part 
of the total investment. In such cases the determination of the 
most economical size of penstock is of special importance. 

In discussing this problem the following notation will be used: 

a = cost of metal pipe per pound; 

b = cost of wood pipe per foot of diameter, per foot of length; 

c — constant in Chezy's formula v = cy/rs; 

d = inside diameter of pipe; 



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ENERGY OF FLOW 215 

/ = allowable fiber stress in metal pipe in pounds per square 

inch; 
h = friction head per unit of length; 
i = income in dollars per year per foot of head; 

. , irWd 

k = numencal coefficient = -s— ; 

{ = length of pipe; 
p = total percentage return on investment, including operat 

ing expenses, depreciation and profit; 
Q = volume of flow through pipe in cubic feet per second; 

r = hydraulic radius = j for pipe flowing full; 

8 = hydraulic gradient = y-; 

t = thickness of metal pipe; 
V = velocity of pipe flow; 
w = internal water pressure in pipe in pounds per square inch; 

In the Chezy formula, v = c\/rs, if we put ^ = 7> « = r ^^d 
solve for the loss in head, h, the result is 

, 4fo2 

or, since v = -tj, this may be written 

, _ 64ZQ« 

Consequently the loss in annual income due to this loss in head 
is 

QilQH 



Income lost = 



cVd^ 



Another factor which reduces the income is the annual fixed 
charge on the pipe. To calculate this for metal pipe, note that 
the thickness of pipe wall is given by Eq. (10) Par. 23, namely, 

and therefore the weight of pipe per foot of length is irdi. Conse- 

ir'uyd^fL 
quently the cost per foot of length for metal pipe is wdta = ~~o7~"» 

and hence finally 



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216 ELEMENTS Of HYDRAULICS 

Total cost of pipe = — sT" ~ ^h where for metal pipe 

and for wood stove pipe k may be determined from the relation 
Cost of pipe per foot of diameter per foot of length = kd. 
Adding the terms representing these two losses we have then 

Total annual loss = ^7: , + ^^^ . 
c^d^T^ 100 

The most economical diameter of penstock is then the value of 
d which makes this expression a minimuni. The condition for 
such a minimum is obtained by equating to zero the first deriva- 
tive of the expression with respect to d. Performing this differ- 
entiation^ the condition for a minimum is then 

S2QIQH 2kdlp _ 
c«dV2 "^ 100 ~ "' * 

whence, solving for d, the formula for economical diameter of 
penstock is determined as 

c^kp 

216. Numerical Application. — The following numerical exam- 
ple illustrates the use of the formula as applied to wood stave 
pipe.^ 

A stream is to be developed by building a large reservoir to 
equalize the flow, and conducting the water to the surge tank and 
penstocks by a single wood stave pipe 13,000 ft. long. The avail- 
able head is 440 ft., and the average flow through the pipe while 
the plant is in operation is 112 cu. ft. per second, developing about 
3,000 kw. As the plant is to be used as a base- load plant and the 
load is to be nearly constant, the value of Q is assumed as 112 
cu. ft. per second. 

The plant develops annuaUy about 26,000,000 kw.-hr., which 
sold at 1 ct. per kilowatt-hour gives a gross income of $260,000 or 
$590 per foot of fall. 

To determine the most economical value of d, a velocity of 6 ft. 
per second is first assumed, which would require a 5-ft. pipe, the 
cost of which in place is estimated at $10 per foot, or $2 per foot 

» Wabrkn, Trans, Am. Soc. C. E., vol. kxix, p. 270. 



=# 



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ENERGY OF FLOW 



217 



of diameter per foot of length, making h — % — 0.4. Sub- 
stituting these numerical values in the formula, namely, 

= 112; i = 590; C = 113 (assumed); p = 15; fc == 0.4, 
the resulting value of d is 






625(112)2590 



= 5.52 ft. 



: (113)45 

A 5K-ft. penstock would therefore be assumed as most eco- 
nomical. 

XXXIX. EFFECT OF TRANSLATION AND ROTATION 

216. Equilibrium under Horizontal Linear Acceleration. — 
Consider the equiUbrium of a body of water having a motion of 
translation as a whole but with its particles at rest relatively to 
one another, such, for example, 
as the water in the tank of a 
locomotive tender when in mo- 
tion on a straight level track. 
If the speed is constant, the 
forjses acting on any particle of 
the liquid are in equiUbrium, 
and conditions are the same as 
when the tank is at rest. If the 
motion is accelerated, however, 
every particle of the liquid must 
experience an inertia force pro- 
portional to the acceleration. 
Thus, if the acceleration is de- 
noted by a, the inertia force F acting on any particle of mass m^ 
according to Newton's law of motion, is given by the relation 

F = ma. 
For a particle on the free surface of the Uquid (Fig. 177), the 
inertia force F acting on this particle must combine with its 
weight W into a resultant R having a direction normal to the free 
surface of the liquid. From the vector triangle shown in the 
figure we have 

and by Newton's law 




F = TT tan a, 



whence by division 



F = 



tan< 



W 



r«; 



S 



(137) 



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218 



ELEMENTS OF HYDRAULICS 



217. Equilibrium under Vertical Linear Acceleration. — ^If the 
tank is moving vertically upward or downward, the surface of the 
liquid will remain horizontal. If the motion is uniform, that 
is, with constant velocity, the conditions will be the same as 
though the tank was at rest. If it is moving upward with 
acceleration a, the surface will still remain horizontal but the 
pressure on the bottom of the tank will be increased by the 

W 
amoimt ma = — a, where W denotes the weight of a column of 

9 
water of imit cross-section and height equal to the depth of 

water in the tank. Thus if p denotes the pressure on the bottom 

W 
of the tank in pounds per square inch, then since ma = — a = 

— a, we have 

p = 7li + ma = 7li(^^)- (138) 

If the acceleration is vertically downward, the pressure on the 
bottom of the tank is diminished by the amount 7A(~)> its 
value being 

p = ^h(^). (139) 

218. Free Surface of Liquid 
in Rotation. — If the tank is in 
the form of a circular cylinder 
of radius r,' and revolves with 
angular velocity o) about its 
vertical axis YY (Fig. 178), the 
free surface of the Uquid will 
become curved or dished. To 
find the form assumed by the 
surface, let P denote any parti- 
cle on the free surface at a dis- 
tance X from the axis of rota- 
tion. Then if m denotes the 
mass of this particle, the centrifugal force C acting on it is 




mx(a^ = 



W 



-zca'' 



From the vectof triangle shown in the figure we have 

C^ _x<a^ 



tan ^ = = = 



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ENERGY OF FLOW 



219 



dy 
and since the slope of the surface curve is ^^ = tan^, we have 

as its differential equation 

3^ = tan 6 = — I 
dx g 

whence, by integration, its algebraic equation is found to be 



y = 



2g 



(140) 



The surface curve cut out by a diametral section is therefore 
a parabola with vertex in the axis of rotation, and the free surface 
is a paraboloid of revolution. 

219. Depression of Cup below Original Level in Open Vessel. 
— Since the volume of a paraboloid is half the volume of the cir- 
cumscribing cylinder, the volume of liquid above the level OX of 
the vertex (Fig. 179) is 

2 ' 



Vol. OCDEF = 



E 





Y 








. 




D 




J 


\ 




/ 


B 

F 


\ 


-1 


/ \ 


i 







X 














Fig. 179. 



Fig. 180. 



7rr2fc== 



But if AB is the level of the liquid when at rest, then 

2 

where k denotes the depth of the cup below the original level, 
and therefore 



2,.,2 



r^w 



k = - = ■ 
^ 2 4g 



(141) 



Consequently the depth of the cup below the original level is 
proportional to the square of the angular velocity. 

220. Depression of Cup below Original Level in Closed Vessel. 
—If the top of the tank is closed and the angular velocity 



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hr*-- 


= HR* 


R* = 


hr* 
H 



220 ELEMENTS OF HYDRAULICS 

increased until the liquid presses against the top, as shown in 
Fig. 180, the surface will still remain a paraboloid. If the total 
depth of the cup is denoted by H and its greatest radius by R, 
then since its volume must be the same as that of the cup of 
depth h, we have the relation 



whence 



03 X 

But from the equation of the surface curve, y = , by sub- 
stituting the simultaneous values y = H, x =^ R,we have 

„ _ o>'R' 
Jti = —^ — i 

and substituting in this relation the value of fi* from the previous 
equation, the result is 

whence 



H 



= «r^|. (142) 



Therefore after the liquid touches the top cover of the tank, the 
total depth of the cup is proportional to the first power of the 
angular velocity. 

221. Practical Applications. — An important physical applica- 
tion of these results consists in the formation of a true parabolic 
mirror by placing mercury in a circular vessel which is then 
rotated with uniform angular velocity, the focus of the mirror 
depending on the speed of rotation. 

Another practical application has been found in the con- 
struction of a speed indicator. A glass cylinder containing a 
colored liquid is mounted on a vertical spindle which is geared 
to the shaft whose speed is required. The required speed is 
then obtained by noting the position of the vertex of the para- 
boloid on a vertical scale. From the level AB to the level CD 
(Fig. 180) the graduations on the scale are at unequal distances 
apart, as apparent from Eq. (141), but below this point they are 
equidistant, as shown by Eq. (142). 



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ENERGY OF FLOW 221 

XL. WATER-HAMMER IN PIPES 

222. Mazimum Water-hammer. — If water is flowing in a 
pipe with uniform velocity and the flow is suddenly stopped, as by 
the closing of a valve, the pressure in the pipe is greatly increased, 
producing what is known as water-hanuner. The maximum 
pressure occurs of course when the flow is stopped instantane- 
ously. In this case all the kinetic energy of flow is expended in 
compressing the water and distending the pipe. The increase in 
pressure due to instantaneous closing may therefore be deter- 
mined by simply calculating the work done in compressing the 
water and in distending the pipe, and equating their sum to the 
kinetic energy of the water flowing in the pipe when shut off. 
This derivation assumes that all the energy of the water in any 
section of given length is expended in compressing the water and 
distending the pipe in this particular section. This assumption, 
however, can be shown to be true theoretically and is also 
verified by experiment.* 

As a notation for use in the derivation which follows, let: 

W = weight of water in the pipe in pounds; 

V = velocity of flow in feet per second; 

7 = 62.4 = weight of water in pounds per cubic feet; 

A = area of cross-section of pipe in square feet; 

d ^ diameter of pipe in feet; 

I = length of pipe in feet; 

h = head in feet due to water-hammer alone in excess of 
static head; 

B = bulk modulus of elasticity of water = 294,000 lb. per 
square inch (Par. 3) ; 

E = Young'smodulusof elasticity of pipe material; 

h = thickness of pipe wall in feet. 
Then the kinetic energy of the water flowing in the pipe with 
velocity V is 

kinetic energy of flow -^ — = —k — ft.-lb. 

1 All the formulas for maximum watei^hammer are the same however 
derived, although they appear in different forms. The derivation here 
given is probably the simplest and is a modification of that given by Minton 
M. Wabben in his paper on "Penstock and Surge Tank Problems," pub- 
lished in Trans, Am. Soc. C. E., vol. Ixxix, p. 238, 1915. The correct for- 
mula for maximum water-hammer seems to have been first obtained by 

JOTJKOWSKT. 



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222 ELEMENTS OF HYDRAULICS 

To calculate the work expended in compressing the water in the 
pipe, note that the total pressure on any crossnsection of the pipe 
is Ahy lb., and consequently the unit pressure is hy lb. per 
square foot. Since the bulk modulus of elasticity of water B 
is defined as (Par. 3) 



we have 



D _ unit pressure 
"" unit deformation 



hy 
imit deformation = -^« 



Therefore the total compression of a column of water { ft. in 

hyl 
length will be -^. Now the total pressure on any cross-section is * 

Ahy, but since this abnormal pressure starts at zero and increases 
imiformly to its full value, the average pressure on the cross-sec- 
tion is y^Ahy, Consequently the total work expended in com- 
pressing the water is 

work done on water = }4Ahy X -^ = ^J ft.-lb. 

To determine the amount of work expended in distending the 
pipe, note that the total tension on any longitudinal seam in 

the pipe is lb., and consequently the imit stress in the pipe 

is — rr or ~r^- Since by definition of Young's modulus 
jjf unit stress 



unit deformation 
we have 

unit deformation = ^r^> 

and consequently the increase in the length of the circumference 
is 

''^^2&^" 2bE~' 

Since this tension starts at zero and increases uniformly to its 
full value, the work done in distending the pipe is 

, , . 1 ^dlhy^ 7rd%y Adlh^y\^ „ 

work done on pipe = ^ X -— X -^j^ = 2&g ^'^^' 



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ENERGY OF FLOW 223 

Equating the kinetic energy of flow to the work of deformation 
expended on water and pipe wall jointly, we have 
AlyV* Ah*yH Adlh*y* 
2g 2B "•" 2bE 

whence, by reducing and solving for h, we have 

,_ W^ 



» = 



V 



'+§f 



Inserting in this expression the numerical values of the constants, 

namely, 

B = 294,000 lb. per sq. in. = 42,336,000 lb. per sq. ft., 

E = 30,000,000 lb. per sq. in. = 4,320,000,000 lb. per sq. ft., 

7 = 62.4 lb. per cu. ft. 

it becomes 

, 145.27 

^= / =y 

Jl + 0.0098,? 

or, with suflBicient exactness, ^. 

h = — =F=- / (143) 



^1>P.(»| 



223. Velocity of Compression Wave. — ^As mentioned above, 
what happens, when the flow in a pipe is suddenly cut off, is an 
increase in pressure, first exerted at the valve, which compresses 
the water and distends the pipe at this point. Beginning at the 
valve, this effect travels back toward the reservoir or supply, 
producing a wave of compression in the water and a wave of 
distortion in the pipe. When all the water in the pipe has been 
brought to rest, the total kinetic energy originally possessed by 
the flowing water is stored up in the elastic deformation of the 
water and pipe walls. Since this condition cannot be maintained 
under the actual head in the reservoir, the pipe then begins to 
contract and the water to expand, thereby forcing the water 
back into the reservoir until it acquires a velocity approximately 
equal to its original velocity but in the opposite direction, that 
is, back toward the reservoir. After this wave has traversed 
the pipe the water again comes to rest, but the kinetic energy 
acquired by the flow toward the reservoir will have reduced the 



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224 ELEMENTS OF HYDRAULICS 

pressure below normal. Consequently water again enters the 
pipe from the reservoir and flows toward the valve, beginning a 
new cycle of operations. 
Now, in addition to the notation previously adopted, let: 

Vc = velocity of compression wave; 
t =» time required for wave starting at the valve to reach 

upper end of pipe; 
F = total pressure on any crossnsectidH of pipe. 

Then, from the principle of impulse and momentum, we have 

W 

g ' 

or since F =» Ahy; W = Aly; < = -, by substituting these values 
we have 

which reduces to - \ n r^ •--? — '^ 

Since the first expression for h derived in the preceding paragraph 
may also be written in the form 



^V'+i-f 



iyg 

by substituting this value of F in the formula just derived for 
Vc we have 

P 

hg \y 



Vc = 



" ^P"■ 



E b 

Inserting in this relation the numerical values of g, B, E and 7, 
as above, it reduces to the final form 

4,674 



V.= /j . nn^d (1«) 



/I + 0.01^ 



224. Ordinary Water-hammer. — The rise in pressure due to 
the gradual shutting off of the flow, as by the closing of a gate 



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ENERGY OF FLOW 225 

or valve, is more difficult to determine than the maximum water- 
hammer due to instantaneous closing. It is evident that if the 
reflected waves from the reservoir return to the gate before it is 
entirely closed, the rise in pressure will be less than the maximum 
given by Eq. (143). Since the period of the compression wave, 
that is, the time required to make a round trip from the gate to 

21 
the reservoir and back again, is — > the statement just made is 

equivalent to saying that if the time of closure is greater than 

21 

—9 the rise in pressure will not reach the maximum given by 

Eq. (143). 

Several formulas have been proposed for the rise in pressure 
due to ordinary closure of a gate or valve. The one here derived 
is due to Minton M. Warren,^ and is the simplest and also seems 
to agree best with experiment. The assumption on which the 
formula is based is that the pressure head rises linearly from zero 

21 
to A in the time — and then remains constant until the gate is 

closed, after which it falls again. The derivation of the formula 
is based on the principle of impulse and momentum, namely, that 
the momentum destroyed must equal the product of the pressure 
produced by the length of time it acts. 

In applying this principle it is assumed to be sufficiently exact 
to use the momentum of the entire mass of water filling the pipe. 
This assumption would make no appreciable difference for ordi- 
nary closing, but might lead to errors in the case of very high 
velocities or extremely slow closing. 

Derivation. — Since the total pressure in the pipe is Ahy, the 

21 
impulse received in the time — > during which interval the pres- 

Ve 

sure head is assumed to be increasing linearly from zero to h, is 

HAhyxf' 

21 
During the remaining time of closure, namely, t — —» the impulse 

received is 

2l\ 



Vc 



M'-^ 



Aly 
Since the mass of the water filling the pipe is -— , the total mo- 



^ Trans. Am. Soc. C. E., voL Ixxix, p. 250. 
15 



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226 ELEMENTS OF HYDRAULICS 

Aly 
mentum destroyed is V. Equating the impulse received to 

the momentum destroyed we have therefore 

21 , ^^ /, 2l\ _ Aly 

g 



HAhy.?^ + Ahy(t^^)=^V 
Vc \ Vcl g 

whence 



h = --!^-Tr (i«) 



g 



i'-v) 



This formula is subject to the limitations that it is liable to be in 
error for very slow closing or for very high velocities; also, that 
it does not give any value for the fall in pressure following the 
rise, and does not apply to opening the gates. However, it is 
probably the most accurate formula yet derived, and its simplic- 
ity commends it for ordinary use.* 

226. Joukovsky's Formula. — The formula for maximum water- 
hammer derived above, namely, 

9 ' 
was first derived by Joukovsky.^ From this result he obtained 
a formula for ordinary water-hammer by assuming that if the 
time of closure is greater than the period T of the compression 
wave, the excess pressure is inversely proportional to the time 
of closure, that is, 

actual pressu re _ T. 
maximum pressure t 
Substituting the proper values in this relation we have therefore 

21 
h _ Vc 
Vvc^ t 

whence 

21V 
h = ^. (147) 

This formula is now generally regarded as inaccurate, and for 
slow closure may be as much as 100 per cent, in error. 

1 Joukovsky's experiments were made in Moscow in 1897-8, and the re- 
sults published in the Memoirs of the Imperidl Academy of Sciences, St. 
Petersburg, vol. ix, 1897. This account was translated from the original 
Russian into English in 1904 by Miss Olqa Simin, and the results discussed 
and amplified by Mr. Boris Simin, Jour. Amer WcUer Works Assoc, 1904, 
p. 335. 



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ENERGY OF FLOW 227 

226. Allievi's Formula. — Another formula for ordinary water- 
hammer has been derived more recently by L. Allievi.^ The 
derivation of this formula is based on the assumptions that the 
area of the gate decreases linearly, and that the velocity of flow 
decreases at a constant rate throughout the interval of closure. 

21 
It also assumes that the time of closure, L is not less than — , and 

is good for such values of t only. 

The derivation is somewhat complicated, involving the solu- 
tion of a differential equation and other mathematical difficulties, 
and is not given here. The complete derivation may be found 
in Mr. Warren's article, cited above. 

The result of AUievi's discussion is embodied in the formula 



h = ^±'H-^' + N (148) 



2 

where 

N = (-7^) , H = normal pressure head. 

The minus sign in the formula is supposed to apply in opening 
the gates, and therefore, according to the formula, the drop in 
pressiffe can never exceed H but approaches it as a limit. In 

21 
this formula the time t >—, and it has been found by experiment 

Ve 

that the formula becomes inaccurate when t approaches the value 

Ve 

227. Occurrence of Water-hammer in Supply Systems. — The 

question of water-hammer is not restricted to the closure of gates, 
and surge in penstocks, but is also of frequent occurrence in the 
water-supply systems of large manufacturing cities where a large 
number of factories shut down at about the same time, say 5 :30 
or 6 p.ni. 

It has also been found that when a small branch pipe having 
a dead end leads ofif from a larger pipe in which water-hammer 
takes place, the small pipe may be subjected to double the pres- 
sure set up in the large pipe. This should be taken into account 
in designing distributing systems, and also in taking readings 
on water-hammer from a pressure gage set in the end of a small 
pipe leading from a penstock or wheel case. 

1 Annali della Societa degli Ingegneri, Rome, vol. xvii, 1902. 



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228 



ELEMENTS OF HYDRAULICS 



UL. SURGE TANKS 

228. Surge in Surge Tanks. — To relieve the effects of water- 
hammer in penstocks, produced by suddenly starting up or shut- 
ting down the plant, it is customary to provide surge tanks so that 



Surge Tank 
— JffUTtinPbn 




Concrete surge tank, penstocks, 
turbines and draft tune for 680- 
foot developmeut, Tallulah Falls, 
Georgia. 



I4?ntun' Meter-^ 



45 hy&otJicQi^ ^ , . 
operated Gate \kive '^ 



. -^*^.-^^^^ ' 

,,,^_,__„._ . — .i'^of^,U»ivt .^-^-,-j, 



Surge Reservot -Area 2IX'*' 
El. 1529 







Cowrteay Eng. Record, 

Fig. 181. — Changes in hydraulic gradient due toVater hammer in penstock. 

the shock is cushioned by the simple rise or fall of the water level 
in the tank. To make the calculations requisite for surge-tank 
design, it is necessary to know the height of the surge up or down 
in the tank. In discussing this problem the following notation 
will be used: 

A = cross-sectional area of surge tank in square feet; 
a = cross-sectional area of conduit in square feet; 



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ENERGY OF FLOW 



229 



V = velocity of flow in main conduit at time / in feet per 

second; 
V = normal velocity of flow in main conduit in feet per second 

during period of steady flow; 
F == friction head in feet corresponding to velocity of flow V; 
f = friction head in feet corresponding to velocity of flow v, 

and consequently/ = Fy-^] 

q = volume of discharge through penstock at time t in cubic 

feet per second; 
Q == volume of discharge through conduit at time t in cubic 

feet per second, and consequently g = Q when t^ = F; 
H = total head in feet lost between forebay and surge tank 

corresponding to velocity of flow V; 
8 = distance of surge-tank level at any time t from its initial 

level, i.e., below level of forebay when starting up, and 

above a point H ft. below level of forebay when shutting 

down; 
S = maximum surge up or down in feet, measured from the 

initial level the same as for «• 

l«vel at time t 



Forebay 



Head lost in conduit^JSrT 



SUtic Level 



X Initial level when 
/ startinir up 



Draw down when startinar up^^S 




8-= Surge when 
Ishuttinflr down 



=^ Initial level when 
shutting down 



Fig. 182. — Changes in level in surge tank. 



To begin the demonstration assume that the forebay has a 
spillway such that the elevation of the water surface in the fore- 
bay remains constant during the period under discussion. 

Before starting up the plant, the water is at rest in the entire 
system, the water in the surge tank being at the same level as 
in the forebay. When the gate is suddenly opened, full-load flow 
is immediately drawn from the surge tank and the level suddenly 



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230 ELEMENTS OF HYDRAULICS 

drops by an amount called the "draw-down," and then oscillates 
until constant normal flow is established in the conduit. Under 
normal operating conditions the level of the water in the surge 
tank is then at a distance H below the level of the forebay, where 
H denotes the total head lost in the conduit for normal flow. In 
shutting down the plant, a sudden closing of the gate causes an 
upward surge in the surge tank, and the water level again oscil- 
lates iip and down until the motion dies away and the water 
once more comes to rest. 

The problem is, then, to find the maximum draw-down when 
starting the plant, and the upward surge when shutting down. 
The expression for this change in level, or surge, will be derived 
by appljdng the principle of work and energy. 

Consider first the case when the plant is starting up, and calcu- 
late the changes in work and energy which take place when the 
water in the conduit flows a distance dx. The kinetic energy of 
flow in the conduit at velocity v is 

kinetic energy = }4 ''^^^ = — o — ' ^^• 

The rate of change of this kinetic energy with the time is, then, 

^(K.E.) = ^?^.«*. 
at g dt 

Consequently the total change in kinetic energy in the time dt 
during which the particles move a distance dx = vdt, is 

change m K.E. = ^ (K.E.)d« = 

The work done on the water entering the conduit during a motion 

dx is equal to the kinetic energy this water acquires, namely, 

,/ , 62.4adx „ 

y^rnv^ = — ^ • v^, 

2g 

The work expended in overcoming friction is 

work expended in friction = 62Aafdx, 

but since the friction head is proportional to the square of the 
velocity, we have 

f = F — 

where F denotes the friction head at the velocity F, and conse- 
quently the above expression may be written 

work expended in friction = 62.4 adx - Fy^ 



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ENERGY OF FLOW 231 

Finally, remembering that the plant is assumed to be starting up, 
the potential energy given up by the water in the surge tank is 

potential energy used = 62.4 asdx. 

Therefore, by equating the total work done to the kinetic energy 

acquired, we have 

^^ . , 62AaLvdv , 62Aadxv^ , ^cy a ^ i? ^^ 

62Aasdx = ^ h 62AadxF ,7^ 

g 2g 72 

Substituting dx = vdt and then dropping the common factors 
62.4 av, this reduces to 

gdi~^ ^ V 2g 

To simplify this expression, note that in the case of steady flow 

V becomes ^> jT = 0? a^^d s takes the value ff , where H denotes 

the total head lost in the conduit. Substituting these simul- 
taneous values in the differential equation, it becomes 

whence 

72 

"-I' + W 

Using this relation, the differential equation simplifies into 

^S -»-=;. (i4„ 

A second fundamental equation may be obtained from the 
condition for continuous flow, namely, that the volume flowing 
through the conduit in any small interval of time dt must equal 
the sum of the change in volume in the surge tank and the vol- 
ume flowing through the penstock in the same interval. 

Since the rate at which the level in the surge tank is rising or 

falling is -r* its change of volume in a unit of time is A -r. In 

starting up, assuming that the full-load flow is immediately 

drawn from the surge tank, the amount flowing out through the 

penstock per second is 5, while the amount coming in through 

the conduit per second is av, and consequently the amount drawn 

from the surge tank per second \^ q — av] whence 

A ds 
A^ = q-av. 



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232 ELEMENTS OF HYDRAULICS 

For full-load flow we have ff = Q, and therefore this relation may 
also be written 

Differentiating this equation with respect to the time t, we have 

. (Pa do 

and eliminating ^ between this relation and the original differ- 
ential eqtiation, the result is 



dt* 



-M'-^vil- 



Substituting in this relation the value of v obtained from the 
condition for continuous flow, namely, 



V = > 



the result is finally 



d^s _ Hg /^ A ^s\' , ags 
dt^ ALV^a 



(^-^sT+S-'- ("») 



This differential equation embodies all the conditions of the 
problem, and its integral is the required solution. So far, how- 
ever, no one has succeeded in obtaining its integral by any direct 
method.* The simplest solution, and in fact the only one ob- 
tained so far which does not involve an erroneous assumption, 
is that due to Professor I. P. Church of Cornell University, 

^ This differential equation has been obtained independently by Db. 
D. L. Webster of the Physics Department of Harvard University, and 
Pbof. Fbanz Prasil, but neither of them succeeded in integrating it in 
its general form. To effect its integration both made the assumption that 
the friction head is proportional to the first power of the velocity instead 
of to its square. Under this assumption, the differential equation reduces 
to that for a damped harmonic oscillation, the solution of which is well 
known. 

For Webbtbb'b derivation see "Penstock and Surge Tank Problems," 
by M. M. Wabrbn, Trans. Am. Soc. C. E., vol. badx, 1916. For Pbasil's 
derivation see "Surge Tank Problems," authorized translation by E. R. 
Weinmann and D. R. Cooper, Canadian Engineer^ vol. xxvii, 1914. Also 
reprint in pamphlet form of this series of articles. 



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ENERGY OF FLOW 



233 



which consists in plotting the integral curve from its differential 
equation by means of successive tangents and points of tangency.^ 
To carry out this method^ eliminate dt between the two rela^ 
tions 



and 



A^ = Q - at; = a{V - v) 
Ldv _ _ „ j^ 



Instead of the above differential equation we then have the 
equivalent relation 




Fia. 183. 

which contains only two variables, s and v. Integrating this 
between the limits zero and S for s and the corresponding limits 
zero and V for v, we have 



H r-s La 

— ^ I v^ds = — 



{V - v)dv, 



which becomes 



S' 



2^ r 2^ 



LaV^ 
Ag 



In this relation the quadrature J]fv^ds can be evaluated only 
when the relation between v and s is known. 
To obtain such a relation between v and s let AC, Fig. 183, 

*See discussion of Warren's article on "Penstock and Surge Tank 
Problems," by I. P. Church, frans. Am. Soc. C. E., vol. Ixxix, 1915, p. 
272. 



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234 ELEMENTS OF HYDRAULICS 

denote the initial level in the surge tank at the beginning of 
the surge, and BD the level at the end of the first downward 
surge; in which case the distance CD = S. At any instant 
during the surge let v denote the velocity of flow in the conduit, 
and 8 the distance of the surface level in the surge tank below its 
initial position AC. Then by putting the differential equation 
in the form 

ds La V — V 



dv Ag H 

^ y2 



2 



for any pair of values of v and s, the value of the slope -r at this 

point is determined. In particular, at A, which represents the 
beginning of the surge, we have v = and s = and therefore 
tan 1^ = oc ; whereas at D, which represents the end of the first 
downward surge, we have s = S, v = V and tan t> = 0. Using 
numerical data, it was found by Professor Church that the en- 
velope of the tangents so determined was very approximately 
the quadrant of an ellipse. 

The equation of an ellipse referred to the given axes with 
origin A is 

Sv = V(S - \/S' - s2). 

Substituting this relation in the quadrature and integrating, 
its value is found to be 



i: 



Subsituting this value in the differential equation, its complete 
integral is 

S' - 0.1917HS = ^^y 

Ag 

whence 



S = 0m58H + >fe^ + 0.00918/^2. 



Ag 



As an approximation sufficiently accurate for all practical pur- 
poses we may therefore write 



s = roH + >^' (151) 



which is the required formula. 

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ENERGY OF FLOW 



23S 



itial Ports 



This formula gives the maximum draw-doWn at starting up 
due to a sudden full opening of the penstock gates. For a sudden 
shutdown the sign of s is reversed, and the upward surge attains 
the same value S as the maximum draw-down, but is measured 
from the initial level which in this case is at a distance H below 
the static level in the forebay. Referring both the upward and 
downward surge to the static level in the forebay, we may there- 
fore say that the maximmn rise in the surge tank above this level 
is less than the maximum draw- 
down below it by the amount H of 
the friction head lost in the con- 
duit at the normal velocity of 
flow V. 

229. Differential Surge Tank.— 
The differential surge tank is a 
modification of the ordinary surge 
tank, its function being to throttle 
the surges.^ Fig. 184 shows in out- 
line a surge tank of this type 
erected by the Salmon River Power 
Co. near Altmar, N. Y. Its func- 
tion is to absorb the energy of a 
column of water 9,625 ft. long and 
from 11 to 12 ft. in diameter. It 
consists of an elevated steel tank, 
50 ft. in diameter and 80 ft. high, 
with a hemispherical bottom which 
adds 25 ft. to its height, connected 
with the distributor for the power-house penstocks by a 12-ft. 
riser. Inside of the tank is an interior riser 10 ft. in diameter 
flaring to a diameter of 10 ft. 8 in. at bottom and 15 ft. at top. 
As the riser to the tank is 12 ft. in diameter, this leaves an annulai 
opening 8 in. wide between the risers at the bottom of the tank. 
This opening is divided into 12 spaces forming the differential 
ports. The action of the surge tank is therefore as follows: 
When part of the load is thrown oflf the power station and the 
upward wave begins in the riser to the tank, a part of the volume 
is deflected through these ports into the main tank, thereby 

^ Invented by R. D. Johnson, Hydraulic Engineer of Ontario Power Co. 
See A. S. M. E. Pamphlet No. 1,204, 1908; also Eng, Recardy July, 18, 1914, 
p 82. 




Fig. 



184. — Differential surge 
tank. 



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236 



ELEMENTS OF HYDRAULICS 



reducing the ultimate height of the level in the riser. When the 
wave recedes, water flows through the ports into the riser thereby- 
decreasing the depth of the surge. The greater the difference 
in elevation between the levels in the tank and the riser, the larger 
will be the volume of flow through the ports. The action there- 
fore tends to throttle the surges. 

The best precaution against hydraulic shock of this nature has 
been foimd to be the use of slow-closing valves. Air chambers 
placed near the valves have also been found effective if kept filled 
with air, and safety valves of course reduce the shock to a pres- 
sure corresponding to the strength of spring used. 

XLH. HYDRAULIC RAM 

230. Principle of Operation. — ^A useful application of water- 
hammer is made in the hydraulic ram. In principle, a hydraulic 




Drive Pipe Connection A Discharge Pipe F Air Chamber E 

Escape Valve C Delivery Valve D Air Feeder H 

Fig. 185. — Rife hydraulic ram. 

ram is an automatic pimip by which the water-hanuner produced 
by suddenly checking a stream of running water is used to force 
a portion of that water to a higher elevation. 

To illustrate the method of operation, a cross-section of a ram 
is shown in Fig. 185. The ram is located below the level of the 



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ENERGY OF FLOW 



237 



supply water in order to obtain a flow in the drive pipe. If 
located some distance from the supply, the water is first con- 
ducted to a short standpipe, as shown in Fig. 186, and from here 
a drive pipe of smaller diameter than the supply pipe conducts 
the water to the ram. The object of this arrangement is to 
utilize the full head of water available without making the drive 
pipe too long for the capacity of the ram. 




Fig. 186. 

Referring to Kg. 185, the water flowing in the drive pipe A at 
first escapes around the valve C, which is open, or down. This 
permits the velocity of flow to increase imtil the pressure against 
C becomes sufficient to raise it against its seat B. Since the 
water can then no longer escape through the valve C, it enters the 
air chamber E through the valve D, thereby increasing the pres- 
sure precisely as in the case of water-hammer discussed in the 
preceding article. When the pressure in E attains a certain maxi- 
mimi value, the flow is checked and the valve D falls back into 




place, closing the opening and trapping the water which has 
already entered the chamber E. The pressure in E then forces 
this water into the supply pipe F, which delivers it at an elevation 
proportional to the pressure in E. 

Hydraulic rams are also so built that they can be operated 
from one source of supply and pump water from a different 
source (Fig. 187). Muddy or impure water from a creek or 
stream may thuig be used to drive a ram, and the water pumped 
from a pure spring to the delivery tank. 



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238 ELEMENTS OF HYDRAULICS 

231. Efficiency of Ram. — The mechanical efficiency of a ram 
depends on the ratio of fall to pumping head, ranging from 20 per 
cent, for a ratio of 1 to 30, up to 75 per cent, for a ratio of 1 to 4. 
Its efficiency as a pump is of course very small, as only a small 
fraction of the water flowing in the drive pipe reaches the delivery 
pipe. The advantages of the hydraulic ram are its small first 
cost, simplicity of operation, and continuous service day and 
night without any attention. 

To obtain an expression for the mechanical efficiency of a ram, 
let: 
.H: = supply head; 
h = effective delivery head including friction; 
q = quantity delivered; 
Q = quantity wasted at valve. 

Then the total input of energy to the ram is (Q + q)H, and the 
total output is qh. Consequently the mechanical efficiency is 
given by the ratio 

W - g^ 

(Q + q)H 

This is known as d'Aubuisson's efficiency ratio. 

The hydraulic efficiency, however, is the ratio of the energy 
required for delivery to the energy of the supply. Consequently 
its value is 

^^ QH ' 
The latter expression is known as Rankine's formula. 

XLin. DISPLACEMENT PUMPS 

232. Pump Tjrpes. — There are two types of pumps in general 
use; the displacement, or reciprocating type, and the centrifugal 
type. In the displacement pump the liquid is raised by means 
of a bucket, piston, or plunger, which reciprocates backward and 
forward inside a cylindrical tube called the pump barrel or cylinr 
der. In the centrifugal pump, as its name indicates, the opera- 
tion depends on the centrifugal force produced by rotation of the 
liquid. 

233. Suction Pump. — One of the simplest forms of displace- 
ment pump is the ordinary suction pump shown in Fig. 188. 
Here the essential parts are a cylinder or barrel C containing a 



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ENERGY OF FLOW 



239 



bucket B, which is simply a piston provided with a movable 
valve, permitting the water to pass through in one direction only. 
This bucket is made to reciprocate up and down inside the barrel 
'by means of a rod E. A suction pipe S leads from the lower 
end of the barrel to the liquid to be raised, and a delivery pipe 
D discharges the liquid at the desired elevation. 

In operation, the bucket starts from its lowest position, and 
as it rises, the valve m closes of its own weight. The closing of 
this valve prevents the air from en- 
tering the space below the bucket, 
and consequently as the bucket rises 
the increase in volume below it causes 
the air confined in this space to ex- 
pand and thereby lose in pressure. 
As the pressure inside the suction pipe 
S thus becomes less than atmospheric, 
the pressure outside forces some of 
the liquid up into the lower end of the 
pipe. 

When the bucket reaches the top 
of its stroke and starts to descend, 
the valve n closes, trapping the liquid 
already in the suction pipe S and also 
that in the barrel, thereby lifting the 
valve m as the bucket descends. 
When the bucket reaches its lowest 
position, it again rises, repeating the 
whole cycle of operations. At each repetition the water rises 
higher as it replaces the air, until finally it fills the pump and a 
continuous flow is set up through the delivery pipe. 

234. Maximum Suction Lift. — Since atmospheric pressure at 
sea level is 14.7 lb. per square inch, a pump operating by suc- 
tion alone cannot raise water to a height greater than the head 
corresponding to this pressure. Since a cubic foot of water 
weighs 62.4 lb., the head corresponding to a pressure of one atmos- 
phere is 

14 7 14 7 




Fig. 188. — Suction pump. 



h = 



62.4 
144 



0.434 



which is therefore the maximum theoretical height to which water 
can be lifted by suction alone. As there are frictional and other 



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240 



ELEMENTS OF HYDRAULICS 



losses to be considered, the actual suction lift of pumps is only 
about two-thirds of this amount, the practical lift for different 
attitudes and pressures being as given in the following table. 



Altitude 


Barometric 


Equivalent head 


Practical suction 


pressure 


of water 


lift of pumps 


Sea level 


14.701b. per sq. in. 


33.95 ft. 


22 ft. 


1/4 mUe... 


14.021b. per sq. in. 


32.38 ft. 


21ft. 


1/2 mUe... 


13.331b. per sq. in. 


30.79 ft. 


20 ft. 


3/4 mUe... 


12.661b. per sq. in. 


29.24 ft. 


18 ft. 


1 mile 


12.02 1b. persq, in. 


27.76 ft. 


17 ft. 


1-1/4 miles. 


11.421b. persq. in. 


26.38 ft. 


16 ft. 


1-1/2 miles. 


10.881b. persq. in. 


25.13 ft. 


15 ft. 


2 miles 


9.881b. persq. in. 


22.82 ft. 


14 ft. 



235. Force Pump. — When it is necessary to pump a liquid to a 
height greater than the suction lift, or when it is desired to equal- 
ize the work between the up and down strokes, a combination 

suction and force pump may 
be used, as shown in Fig. 189. 
In the simple type here illus- 
trated, the bucket is replaced 
by a solid piston, tlie movable 
valves being at m and n as 
shown. On the up stroke of 
the piston the valve m closes 
and the pump operates like a 
simple suction pump, filling 
the barrel with liquid. When 
the piston starts to descend, 
the valve n closes, and the 
liquid in the bairel is there- 
fore forced out through the 
valve m into the delivery 
pipe D. 

By making the suction and 
pressure heads equal, the 
piston ca'n therefore be made to do the same amoimt of work on 
the down as on the up stroke; or the entire suction head may be 
utilized and the pressure head made whatever may be necessary. 

236. Stress in Pump Rod. — To find the pull P on the pump rod 
E for the type shown in Fig. 188, let A denote the area of the 
bucket and hi, A2, the heads above and below the bucket, as 




FiQ. 189. — Combined suction and 
force pump. 



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ENERGY OF FLOW 



241 



indicated in the figure. Then the downward pressure Pi on top 
of the bucket is 

Pi = 14.7A + 62AhiA, 

and the upward pressure P2 on the bottom of the bucket is 

P2 = U.7A - 62AhiA. 
Therefore the total pull P in the rod is 

P = Pi - P2 = 62.4ii(Ai + hi) = 62.4Ah. 
If I denotes the length of the stroke, the work done per stroke is 
then 

work per stroke = PI = 62.4Ahl. 
For the combined suction and pressure type shown in Fig. 189, 
the pressure in the rod on the down stroke is 

P = 62.4iift2, 
and the tension in the rod on the up stroke is 

P = 62.4iifti. 




Fig. 190. — Direct acting steam pump. 

237. Direct-driven Steam Pump. — The modern form of 
reciprocating power pump of the suction and pressure type is the 
direct-driven steam pump, illustrated in Figs. 190 and 191. In 
this type the steam and water pistons are on opposite ends of 
the same piston rod and therefore both have the same stroke, 
Id 



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242 



ELEMENTS OF HYDRAULICS 




Fio. 191. 



\^=J 



Steam lulet 




Fig. 192, 



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ENERGY OF FLOW 243 

although their diameters are usually different. Until recently 
this was the standard type of general service pump, being used 
for all pressures and capacities, from boiler-feed pumps to muni- 
cipal pumping plants. Although the centrifugal type is rapidly 
taking its place for all classes of service, the displacement pump 
is the most efficient where conditions demand small capacity 
at a high pressure, as in the operation of hydraulic machinery. 
Fig. 192 illustrates the use of a displacement pimip in connection 
with a hydraulic press. The best layout in this case would be 
to use a high-pressure pump and place an accumulator (Par. 14) 
in the discharge line between pump and press. The press cylin- 
der can then be filled immediately at the maximum pressure and 
the ram raised at its greatest speed, the pmnp running meanwhile 
at a normal speed and storing excess p>ower in the accumulator. 

238. Calculation of Pump Sizes. — To illustrate the calculation 
of pump sizes, suppose it is required to find the proper size for 
a duplex (i.e., two cylinder, Fig. 191) boiler-feed pump to supply 
a 100-h.p. boiler. 

For large boilers the required capacity may be figured as 34 J^ 
lb. of water evaporated per hour per horse power. For small 
boilers it is customary to take a larger figiu-e, a safe practical 
rule being to assume Ko gal. per minute per boiler horse power. 

In the present case, therefore, a 100-h.p. boiler would require a 
supply^ of 10 gal. per minute. Assuming 40 strokes per minute 
as the limit for boiler feed pumps, the required capacity is 

j^ = 0.25 gal. per stroke. 

Therefore, assuming the efficiency of the pump as 50 per cent., 
the total capacity of the pump per stroke should be 

^-^ = 0.5 gal. per stroke. 

Since we are figuring on a duplex, or two-cylinder, pmnp, the 
required capacity per cylinder is 

-^ = 0.25 gal. per stroke per cylinder, 

and consequently the displacement per stroke on each side of the 

piston must be 

25 

-^ = 0.125 gal. piston displacement. 

Referring to Table 7 it is found that a pump having a water cylin- 



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244 ELEMENTS OF HYDRAULICS 

der 2^ in. in diameter, with a 6-in. stroke, will have the required 
capacity. 

239. Power Reqtiired for Operation. — ^To find expressions for 
the horse power and steam pressure required to operate a dis- 
placement pump, let: 

Q = discharge of pump in gallons per minute; 

h = total pumping head in feet (including friction and suction 
head if any) ; 

D ^ diameter of steam piston in inches; 

d = diameter of water piston in inches; 

p = steam pressure in pounds per square inch; 

w = water pressure in pounds per square inch; 

n = number of full strokes (i.e., roimd trips) per minute; 

c = number of pump cylinders {e.g., for duplex pump, Fig. 
191, c = 2); 

I = length of stroke in inches; 

E == efficiency of pump. 

Since a gallon of water weighs 8.328 lb., the total work per 
minute required to raise the given amount Q to the height h is 

work = 8.328QA ft.-lb. per minute. 
Taking into account the efficiency of the pump, the actual 
horse power required is therefore 

240. Diameter of Pump Cylinder. — If the pump makes n full 
strokes per minute, the piston displacement per minute for each 
cylinder is 

and the actual effective displacement of the pump per minute is 

(/?2 \ 
-T-lj cu. in. per minute. 

Equating this to the required discharge Q, expressed in cubic 
inches per minute, we have 

2ncE(^)^231Q, 

whence the required diameter of the pump cylinder in terms of the 
speed is found to be 



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whence 



ENERGY OF FLOW 245 

241. Steam Pressure Required for Operation. — Since the total 
pressure on the steam piston cannot be less than that on the 
water piston, the minimum required steam pressure, p, is given by 
the relation 

p(f).0.43«(f), 

p = 0.434h(^)*. (164) 

242. Numerical Application. — ^To illustrate the application of 
these results, suppose it is required to determine the indicated 
horsepowerlio operate a fire engine which delivers two streams of 
250 gal. per minute each, to an effective height of 60 ft. 

Since the height of an effective fire stream is approximately 
four-fifths that of the highest drops in stiU air, the required head 
at the nozzle is 

jh = Jx 60 = 75 ft. 
4 4 

To this must be added the friction head h/ lost in the hose 
between pump and nozzle, which is given by the relation (Par. 99) 

where I is the length of the hose and d its diameter, both expressed 
in inches, v is the velocity of fiow through the hose, and / is an 
empirical constant. For the best rubber-lined hose, / = 0.02 
for the first 100 ft. of hose and 0.0025 for each additional 100 ft. 
whereas for unlined hose / = 0.04 for the first 100 ft. and 0.005 
for each additional 100 ft. In the present case, assmning 100 ft. 
of the best 2J^-in. rubber-lined hose, we have/ = 0.02, and since 
the quantity of water delivered is 

Q = 250 X Y72Q ^^* ^^- P®^ minute, 

and the area of the hose is 

. xd* ir(2.5)2 ' rt^Q . 
A = -^ = ^. ^ = 4.908 sq.m., 

the velocity of flow in the hose is 



V = 



144X60 



= 16.3 ft. per second. 



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S46 ELEMENTS OF HYbRAULtCS 

Consequently the friction head h/ is^ 

f A, = 0.02ig?-^ = 39.6,8ay40ft., 

and therefore the total pumping head H is 

H = 76 + 40 = 115 ft. 

From Eq. (103) the total horse power required, assuming a pimip 
efficiency of 60 per cent., is then found to be 

t; H.P. = 0.00025 ^^^— = 28.76. 

Assiuning the efficiency of the engine to be 60 per cent., the total 
indicated horse power required would be 

XLIV. CENTRIFUGAL PUMPS 

243. Historical Development. — The centrifugal pump in its 
modern form is a development of the last 16 years although as a 
type it is by no means new. The inventor of the centrifugal 
pimip was the celebrated French engineer Denis Papin, who 
brought out the first pump of this type in Hesse, Germany, in 
1703. Another was designed by Euler in 1754. These were 
regarded as curiosities rather than practical machines until the 
type known as the Massachusetts pump was produced in the 
United States in 1818. From this time on, gradual improve- 
ments were made in the centrifugal pump, the most important 
being due to Andrews in 1839, Bessemer in 1845, Appold in 1848, 
and John and Henry Gynne in England in 1851. Experiments 
' seemed to show that the best efficiency obtainable from pumps of 
this type ranged from 46 to 64 per cent, under heads varjdng 
from 4^ to 15 ft., and 40 ft. was considered the maximum head 
for practical operation. 

About the year 1901 it was shown that the centrifugal pump 
was simply a water turbine reversed, and when designed on simi- 
lar lines was capable of handling heads as large, with an efficiency 
as high, as can be obtained from the turbines themselves. Since 
this date, great progress has been made in both design and con- 
struction, the efficiency of centrifugal pumps now ranging from 
55 to over 90 per cent., and it being possible to handle heads as 



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ENERGY OF FLOW 247 

high as 300 ft. with a single-stage turbine pump and practically 
any head with a multi-stage type.^ 

The advantages of the centrifugal over the displacement 
type are its greater smoothness of operation, freedom from water- 
hammer or shock, absence of valves, simplicity and compactness, 
and its adaptability for driving by belt or by direct connection to 
modern high-speed prime movers, such as steam turbines, gas 
engines and electric motors. Under favorable conditions the 
first cost of a high-lift centrifugal pimip may be as low as one* 
third that of a displacement pump, and the floor space occupied 
one-fourth that required by the latter. However, for small 
quantities of water discharged under a high liead the displace- 
ment pmnp is preferable to the centrifugal type, as the latter 
requires too much compounding imder such conditions. 

244. Principle of Operation. — The principle on which the 
original centrifugal pumps of Papin and Euler operated was 
simply that when water is set in rotation by a paddle wheel, the 
centrifugal force created, forces the water outward from the cen- 
ter of rotation. Appold discovered that the eflSciency depended 
chiefly on the form of the blade of the rotary paddle wheel, or 
impeller, and the shape of the enveloping case, and that the best 
form for the blade was a curved surface opening in the opposite 
direction to that in which the impeller revolved, and for the case 
was a spiral form or volute. The first engineer to discover the 
value of compounding, that is, leading the discharge of one cen- 
trifugal pimip into the suction of another similar pump, was the 
Swiss engineer Sulzer of Winterthur, who was closely followed by 
A. C. E. Rateau of Paris, France, and John Richards and Byron 
Jackson of San Francisco, Cal. 

In its modern form, the power appUed to the shaft of a cen- 
trifugal pump by the prime mover is transmitted to the water by 
means of a series of curved vanes radiating outward from the 
center and mounted together so as to form a single member called 
the impeller (Fig. 193). The water is picked up at the inner edges 
of the impeller vanes and rapidly accelerated as it flows between 
them, until when it reaches the outer circumference of the impeller 
it has absorbed practically all the energy applied to the shaft. 

'Rateau found by experiment that with a single impeller 3.15 in. in 
diameter, rotating at a speed of 18,000 r.p.m., it was possible to attain a 
head of 863 ft. with an efficiency of approximately 60 per cent. Engineer^ 
Mar. 7, 1902. 



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248 



ELEMENTS OF HYDRAULICS 





Hollow arm impeller. 



Concave arm impeller. 




Sand pump impeller. Open impeller used in sewage pumps. 




Enclosed side suction impeller. Enclosed double suction impeller. 

Fig. 193. — Impeller types. {Courtesy Morris Machivs Works,) 



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ENERGY OF FLOW 



249 



246. Impeller Forms. — There are two general forms of impel- 
ler, the open and the closed types. In the former the vanes are 
attached to a central hub but are open at the sides, revolving 
between two stationary side plates. In the closed type, the 
vanes are formed between two circular disks forming part of 
the impeller, thus forming closed passages between the vanes, 
extending from the inlet opening to the outer periphery of the 
impeller. The friction loss with an open impeller is considerably 
more than with one of the closed type, and consequently the 
design of pumps of high efficiency is Umited to the latter. 

246. Conversion of Kinetic Energy into Pressure. — ^As the 
water leaves the impeller with a high velocity, its kinetic energy 




Fia. 194. 

forms a considerable part of the total energy and the efficiency 
of the pimip therefore depends largely on the extent to which this 
kinetic energy is converted into pressure in the pump casing. 

In some forms of pump no attempt is made to utilize this 
kinetic energy, the water simply discharging into a concentric 
chamber surrounding the impeller, from which it flows into a 
discharge pipe. The result of such an arrangement is that only 
the pressure generated in the impeller is utilized and all the kinetic 
energy of the discharge is dissipated in shock and eddy formation. 

247, Volute Casing. — ^This loss of kinetic energy may be par* 



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250 



ELEMENTS OF HYDRAULICS 



tially avoided by making the casing spiral in section, so that the 
sectional area of the discharge passage increases uniformly, 
making the velocity of flow constant (Fig. 194). This type of 
casing is called a volute chamber (Fig. 195). 

When the volute is properly designed, a high efficiency may be 
obtained with this type of casing.^ 




Fig. 195. — Double-suction volute pump, Piatt Iron Works Co. 

248. Vortex Chamber* — ^An improvement on the simple volute 
chamber is that known as the whirlpool chamber, or vortex 
chamber, suggested by Professor James Thomson. In this type 
the impeller discharges into a concentric chamber considerably 
larger than the impeller, outside of and encircling which is a 
volute chamber. In its original form this necessitated exces- 
sively large dimensions, but in a modified form it is now very 
generally used (Figs. 196 and 197). 

The effectiveness of this arrangement depends on the principle 
of the conservation of angular momentum. Thus, after the 
water leaves the impeller no turning moment is exerted on it 
(neglecting frictional resistance) and consequently as a given 

* With the De Laval volute type of centrifugal pump shown in Fig. 196> 
efficiencies as high as 85 per cent, have been obtained under favorable 
conditions. 



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ENERGY OF FLOW 



261 




Fig. 196. — Longitudinal section of De Laval single-Btage double -suction 

volute pump. 




Fio. 197.— Longitudinal section of Alberger volute pump. 



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252 



ELEMENTS OF HYDRAULICS 



mass of water moves outward, its speed decreases to such an 
extent as to keep its angular momentum constant. For a well- 
designed vortex chamber, the velocity of the water at the outside 
of the diffusion space is less than the velocity of the water as it 
leaves the impeller in the inverse ratio of the radii of these points, 
and if this ratio is large, a large part of the kinetic energy of the 
discharge may therefore be converted into pressure head in 
this manner. This method of diffusion is therefore well adapted 
to the small impellers of high-speed pumps, since the ratio of the 
outer radius of the diffusion chamber to the outer radius of the 
impeller may be made large without unduly increasing the size of 
the casing. 




Fig. 198.— Diffusion ring. 

249. Diffusion Vanes. — ^Another method for converting the 
kinetic energy of discharge into pressure head consists in an 
application of Bernoulli's law as illustrated in the Venturi tube; 
namely, that if a stream flows through a diverging pipe the initial 
velocity head is gradually converted into pressure head without 
appreciable loss. To apply this principle to a centrifugal pump, 
the impeller is surrounded by stationary guide vanes, or diffusion 
vanes (Fig. 198), so designed as to receive the water without 



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ENERGY OF FLOW 



253 



shock on leaving the impeller and conduct it by gradually diverg- 
ing passages into a vortex chamber or volute casing. This type of 
construction is therefore essentially a reversed turbine, and is 
commonly known as a turbine pump (Fig. 199). 




END SECTIONAL VIEW 




S^DE SECTIONAL VIEW 

FiQ. 199. — Alberger two-stage turbine pump. 

The angle which the inner tips of the diffusion vanes make 
with the tangents to the discharge circle is calculated exactly 
as in the case of the inlet vanes of a turbine, that is, so that 'they 
shall be parallel to the path of the water as it leaves the impeller. 



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254 



ELEMENTS OF HYDRAULICS 



As this angle changes with the speed, the angle which is correct 
for one speed is incorrect for any other and may actually obstruct 
the discharge. A turbine pump must therefore be designed for 
a particular speed and discharge, and when required to work 
under variable conditions loses considerably in efficiency. If 
the conditions are very variable, the vortex chamber type is 
preferable, both by reason of its greater average efficiency under 
such conditions and also on account of its greater simplicity and 
cheapness of construction. 

260. Stage Pumps. — Single impellers can operate efficiently 
against heads of several hundred feet, but for practical reasons it 




Fia. 200. — Worthington two-stage turbine pump. 

is desirable that the head generated by a single impeller should 
not exceed about 200 ft. When high heads are to be handled, 
therefore, it is customary to mount two or more impellers on the 
same shaft within a casing so constructed that the water flows 
successively from the discharge of one impeller into the suction 
of the next. Such an arrangement is called a stage pump j 
and each impeller, or stage, raises the pressure an equal amount. 
Fig. 200 shows a multi-stage pump of the turbine type and 
Fig. 201 one of the volute type. 

A single impeller pump may be either of the side-suction or 
double-suction type. In the latter, half of the flow is received on 



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ENERGY OF FLOW 265 

each side of the impeller which is therefore perfectly balanced 
against end thrust (Fig. 196). A side-suction pump, however, is 
simpler in construction, and it is also possible to balance them 
hydraulically against end thrust (Fig. 197). In stage pumps the 
device sometimes used for balancing is to arrange the impellers 
in pairs so that the end thrust of one impeller is balanced by 
the equal and opposite end thrust of its mate. 




Fig. 201. — De Laval three^stage volute pump. 

XLV. PRESSURE DEVELOPED IN CENTRIFUGAL PUMP 

251. Pressure Developed in Impeller. — The pressure produced 
in a centrifugal pump must be sufficient to balance the static and 
frictional heads. When there is no volute, vortex chamber or 
diffusor, the kinetic energy of the discharge is all dissipated and 
the entire change in pressmre is produced in the impeller. If, 
however, the velocity of discharge is gradually reduced by means 
of one of these devices, a further increase in pressmre is produced 
in the casing or diffusion space, and if a diverging discharge pipe 
is used the pressure is still further increased. 

The change in pressure which is produced in passing through 
the impeller may be deduced by applying Bernoulli's theorem. 
For this purpose it is convenient to separate the total difference 
in pressure between the inlet and discharge circles into two com- 
ponents; one due to the rotation of the water in a forced vortex 
with angular velocity w, and the other due to the outward flow, 
i.e., the relative motion of the water with respect to the vanes of 
the impeller. Let the subscripts 1 and 2 refer to points on the 
inlet and discharge circles respectively. Then the radii of 



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256 



ELEMENTS OF HYDRAULICS 



these circles will be denoted by ri, r»; the pressure at any point 
on these circles by pi, p2, etc. Also let « denote the angular 




Fig. 202. — Detail of labyrinth rings in piunp showD in Fig. 201. 




Fig. 203. 



velocity of rotation of the impeller, and ui, u^ the tangential 
velocities of the vanes at their inner and outer ends (Fig. 203), 
in which case u\ = riw and u% = rtw. 



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ENERGY OF FLOW 267 

Applying Bemoull's theorem to the change in pressure pro- 
duced by rotation alone, we have therefore 

7 ^ " V ^' 
Consequently the total change in pressure due to rotation, say 
Pr where Pr = Pi — Pu ^ given by the relation 

7 " 7 " 2/^^* "^^ ^ 2(7 

This expression is often called the centrifugal head. 
By similar reasoning the change in pressure produced by the 
outward flow is given by the relation 



y 


+ 


tot* 
2g 


y 


+ 


2g' 


P't 


— 


P'l 


Wi' 


— 


w»* 



whence _ 

T 2(7 

If the water enters radially, ^1 = 50° and consequently it?!* = 
Vi* + til*. In this case, denoting the difference in pressure at 
inlet and exit due to the flow by p/, where p/ = p'2 — p'l, we 
have 

Pf ^ p'i - p\ ^ vi^ + ^i' - t^2*^ 

7 T 2flf * 

The total increase in pressure in the impeller between the inlet 

and discharge ends of the vanes is therefore given by the relation 

Pr + Pf _ Vl^ + Ul* -• W%^. + U2* -^ til* _ Pi* + ti2* — t£?2* 

7 " 2flf " 2g ' 

262. Pressure Developed in Difihisor. — ^Besides the increase in 

pressure produced in the impeller, the use of a suitable diffusion 

chamber permits part of the kinetic energy at exit, due to the 

absolute velocity V2 of the discharge from the impeller, to be 

converted into pressure. Thus if k denotes the fraction of this 

kinetic energy which is converted into pressure in the diffusor, 

va* 
the expression derived above is increased by the term k^r-. When 

^g 

diffusion vanes are used, as in a turbine pump, the value of k may 
be as high as 0.75, and for a vortex chamber it may reach 0.60. 

263. General Expression for Pressure Head Developed. — 
Combining the terms derived above, the total pressure head H 
developed by the pump is given by the simple expression 

^^kv,* + Vx* + «,»-w.»^ (166) 

17 



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258 



ELEMENTS OF HYDRAULICS 



In applying this formula it is convenient to note that the total 
head H developed in the pump consists of three terms, as follows: 



U2 



2 _ 



W2^ 



2g 



kv£_ 

2g 



= head at eye (entrance) of impeller; 

= head developed in impeller; 

= head developed in casing or diffusor. 



XLVI. CENTRIFUGAL PUMP CHARACTERISTICS 

264. Effect of Impeller Design on Operation. — The greatest 
source of loss in a centrifugal pump is that due to the loss of the 






Velocity jelabVe^ 
*o Blade .Wa 




Fig. 204. 

kinetic energy of the discharge. As only part of this kinetic 
energy can be recovered at most, it is desirable to reduce the 
velocity of discharge to as low a value as is compatible with effi- 
ciency in other directions. This may be accomplished by curving 
the outer tips of the impeller vanes backward so as to make the 



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ENERGY OF FLOW 259 

discharge angle less than 90**. The relative velocity of water and 
vane at exit has then a tangential component acting in the oppo- 
site direction to the peripheral velocity of the impeller, which 
therefore reduces the absolute velocity of discharge. This is 
apparent from Fig. 204 in which the parallelogram of velocities 
in each of the three cases is drawn for the same peripheral velocity 
U2 and radial velocity at exit W2 sin 62. A comparison of these 
diagrams indicates how the absolute velocity at exit V2 increases 
as the angle ^2 increases. The backward curvature of the vanes 
also gives the passages a more imiform cross-section, which is 
favorable to efficiency. The average value of ^2 at exit is about 
30^ 

The effect which the design of the impeller has on the operation 
of the pump is most easily illustrated and imderstood by plotting 
curves showing the relations between the variables under con- 
sideration. Assuming the speed to be constant, which is the 






usual condition of operation, three curves are necessary to com- 
pletely illustrate the operation of the pump; one showing the 
relation between capacity and head, one between capacity and 
power, and one between capacity and efficiency. The first of 
these curves is usually termed the charctcteristic. 

266. Rising and Drooping Characteristics. — The principal fac- 
tor influencing the shape of the characteristic is the direction of 
the tips of the impeller blades at exit, although there are other 
factors which affect this somewhat. If the tips are curved for- 
ward in the direction of rotation the characteristic tends to be of 
the rising type, whereas if they curve backward the characteristic 
tends to be of the drooping type (Figs. 205 and 206). For a ris- 
ing characteristic the head increases as the delivery increases 3,nd 
consequently the power curve also rises, since a greater discharge 
against a higher head necessarily requires more power (Fig. 207). 



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260 



ELEMENTS OF HYDRAULICS 



































































-u 


A 
























































































































u 



































































■^ 


^ 


































































"^ 




^ 


^ 












£fA«Lniv 












































- 


^^ 


::s 


^ 








^_ 




" 


























T 










■ 




( 


A 














^ 










^ 
/" 


V** 












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^ 


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s 


^ 












i 




n 




// 


^ 


























**7 




^V 


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\ 
















/ 
































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V 


s, 






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\ 










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\ 


















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_4 


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( 









80 






lf)0 






120 


N 




LO 





CAPAcnr 




CAPACITY 



1AA— 




















































































































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- 




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T 


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ou 




















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i 


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to 






Jj 


iO 





CAPACITY 



FiQ. 206. — Characteristics and efficiency curves obtained from De Laval 
centrifugal pumps. 



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ENEROY OF. FLOW 



261 



A drooping power curve may be obtained by throttling at the 
eye of the impeller, -but a greater efficiency results from designing 
the impeller so as to give this form of curve normally. 





— 


— 




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140 


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Percentage of Nonnal Oapacltj 

FiQ. 207. 



For a high-lift pump under an approximately constant head, 
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Percentage Of Nonnal OapaoLty 

Fig. 208. 

falling head, as in the case of emptying a lock or dry dock, as it 
makes it possible to obtain a large increase in the discharge as 
the head diminishes, thereby saving time although at a loss of 
efficiency. 



Digitized by VnOOQlC 



262 ELEMENTS. OF HYDRAULICS 

One of the most important advantages of a drooping char- 
acteristic (Fig. 208) is that it is favorable to^ a drooping power- 
delivery curve, making it impossible for the pump to overload 
the driving motor. For an electrically driven pump, in which 
the overload is limited to 20 per cent., or at most 25 per cent., of 
the normal power, backward-curved vanes are therefore essential. 
Moreover, with a pump designed initially to work against a 
certain head, if the vanes at exit are radial, or curved forward, 
the possible diminution in speed is very small, the discharge 
ceasing altogether when the speed falls slightly below normal. 
As the backward curvature of the vanes increases the range of 
speed also increases, and consequently when the actual working 
head is not constant, as in irrigation at different levels, or in 
delivering cooUng water to jet condensers in low-head work, 
where the level of the intake varies considerably, a pump with 
drooping characteristic is much better adapted to meet varying 
conditions without serious loss of efficiency. 

266. Head Developed by Pump. — These facts may be made 
more apparent by the use of the expression for the head developed 
by the pump, derived in the preceding article. Considering only . 
the head developed in the impeller and casing, and omitting that 
due to the velocity of flow at entrance, Vi, which does not depend 
on the design of the pump, the expression for the head developed 
is 

a. = zz • 

Since v^ is the geometric resultant of Ui and w^, we have by the 
law of cosines, 

V2^ = t*2^ + t02* — 2t*2t^2 cos ^2. 

V2^ 

For an ideal pump, that is, one in which all the velocity head ^ - 

is converted into pressure head in the diffusor, k is unity. As- 
suming A; = 1 and substituting the expression for v^^ in the equa- 
tion for Hj the result is 

Tj ^2* — t*2t^2 cos 62 

Q 
For constant speed of rotation, U2 is constant. 

For forward-curved vanes 62 is greater than 90® and therefore 
cos 02 is negative. In this case as W2 increases H also increases; 
i.e., the greater the delivery the greater the head developed. 



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mSR^Y OF FLOW 26S 

For radial-tipped vanes, 62 = 90** and cos 62 = 0. In this case 

H = — , which is constant for all deliveries. 


For backward-curved vanes dt is less than 90® and cos 62 is 
positive. Consequently in this case as the delivery increases the 
head diminishes. 

Although these relations are based on the assumption of 
a perfect pump, they serve to approximately indicate actual 
conditions, as is evident by inspection of the three types of 
characteristic. 

267. Effect of Throttling the Discharge. — It is always neces- 
sary to inake sure that the maximum static head is less than the 
head developed by the jpiunp' ^,t no discharge. This is self-evi- 
dent for the drooping characteristic, but the rising characteristic 
is misleading in this respect as the head rises above that at shut- 
off. Since for a certain range of head two different outputs are 
possible, it might seem that the operation of the pimip under 
such conditions would be imstable. This instability, however, 
is coimteracted by the frictional resistance in the suction and 
delivery pipes, which usually amoimts to a considerable part of 
the total head. Any centrifugal pump with rising characteristic 
will therefore work satisfactorily if the maximum static head is 
less than the head produced at shut-off. If the frictional resist- 
ance is small it may be increased by throttling the discharge, so 
that by adjusting the tlirottle it is possible to operate the pump 
at any point of the curve with absolute stability. 

258. Numerical Illustration. — ^The particular curves shown in 
Fig. 209 were plotted for an 8-in., three-stage turbine fire piunp 
built by the Alberger Co., New York, and designed to deliver 760 
gal. per minute against an effective head of 290 ft., the pump 
being direct connected to a 75-h.p. 60-cycle induction motor op- 
erating at a synchronous speed of 1,200 r.p.m. 

The head curve shows that this pump would deliver two fire 
streams of 260 gal. per minute each, at a pressure of 143 lb. per 
square inch; three streams of 260 gal. per minute each, at a pres- 
sure of 125 lb. per square inch; four streams of 25Crgal. per minute 
each, at a pressure of nearly 100 lb. per square inch; or even five 
fairly good streams at a pressure of 80 lb. per square inch. With 
the discharge valve closed the pump delivers no water but pro- 
duces a pressmre equivalent to a head of 308 ft. If the head 
against which the pump operates exceeds this amount, it is of 



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264 



ELEMENTS OF HYDRAULICS 



course impossible to start the discharge. The head for which 
this particular piunp was designed was 290 ft., which corresponds 
to the point of maximum efficiency. It is therefore appar- 
ent that the operating head must be carefully ascertained in 
advance, for if it is higher than that for which the pump was 
designed, both the efficiency and the capacity are diminished, 
whereas if it is lower, the capacity is increased but the efficiency 
is diminished. 




500 600 700 800 900 1000 1100 1200 
Gallons-per Minute 

Fig. 209. 



1800 



The power curve shows that under low heads the power rises. 
Also that the overload in the present case is confined to about 
12 per cent, of the normal power. Consequently the motor could 
only be overloaded 12 per cent, if all the hose lines should burst, 
whereas the head curve shows that if all the nozzles were shut oflF 
no injurious pressure would result. 

The efficiency curve always starts at zero with zero capacity, 
as the pump does no useful work until it begins to discharge. 
The desirable features of an efficiency curve are steepness at the 



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ENERGY OF FLOW 265 

two ends, a flat top and a large area. Steepness at the beginning 
shows that the efficiency rises rapidly as the capacity increases, 
whereas a flat top and a steep ending show that it is maintained 
at a high value over a wide range. Since the average efficiency 
is obtained by dividing the area enclosed by the length of the 
base, it is apparent that the greater the area for a given length, the 
greater will be the average efficiency. 

XLVn. EFFICIENCY AND DESIGN OF CENTRIFUGAL PUMPS 

269. Essential Features of Design. — The design of centrifugal 
pumps like that of hydraulic turbines requires practical ex- 
perience as well as detailed mathematical analysis. The general 
principles of design, however, are simple and readily understood, 
as will be apparent from what follows: 

Three quantities are predetermined at the outset. The inner 
radius of the impeller, fi, is ordinarily the same as the radius of 
the suction pipe or slightly less; the outer radius, r2, is usually 
made twice n; and the angular speed o) at which the impeller is 
designed to rim is fixed by the particular type of prime mover by 
which the pump is to be operated. 

The chief requirement of the design is to avoid impact losses. 
In order therefore that the water shall glide on the blades of the 
impeller without shock, the relative velocity of water at entrance 
must be tangential to the tips of the vanes. 

Assiuning the direction of flow at entrance to be radial, which 
is the assiunption usually made although only approximately 
realized in practice, the necessary condition for entrance without 
shock is (Fig. 203) 

Vi = Wi tan ^1, 

which determines the angle ^i. The relative velocity of water 
and vane at entrance is then 

Wi = Vwi* + Vi^. 

The direction of the outer tips of the vanes, or angle 6%, Fig. 
203, is determined in practice by the purpose for which the pump 
is designed, as indicated in Art XL VI. For an assigned value 
of $2, the absolute velocity of the water at exit is 
Vi^ = tia* + t»2* — 2uk%wt cos $2 

and consequently as 0% increases, the absolute velocity at exit, 
vt, also increases. 



Digitized by VnOOQlC 



266 ELEMENTS OF HYDRAULICS 

Let sif 82, Fig. 203| denote the radial velocity of flow at en- 
trance and exit, respectively, and Ai, A2 the circumferential 
areas of the impeller at these points. Then for continuous 
flow 81A1 = «ailj. Usually Si = 82, in which case Ai = A2. If 
6] and bs denote the breadth of the impeller at inlet and outlet 
respectively, then Ai = 2Tri&i and A2 — 2irr2b2, and conse- 
quently for ill = A 2 we have biri = ftgrj. Assuming the radial 
velocity of flow throughout the impeller to be constant, the 
breadth b at any radius r is given by the relation br = biVi. 

260. Hydraulic and Commercial Eflkiency. — ^Let H' denote the 
total effective head against which the piunp operates, including 
suction, friction, delivery and velocity heads. Then if w denotes 
the velocity of the water as it leaves the delivery pipe, h the total 
lift including suction and delivery heads, and h/ the friction head, 
we have 

H' = A + A, + |. 

The total theoretical head H developed by the pump, as derived 
in ArtXLV, is 

Vi^ + kV2^ + W2* - W2^ 



H = 



2g 



Consequently the hydravlic efficiency of the pump is the ratio of 

these two quantities, that is, 

H' 
Hydraulic efficiency = =-• (156) 

The commercial efficiency of the pump is the ratio of the work 
actually done in lifting the water through the height h to the 
total work expended in driving the impeller shaft, and is of course 
less than the hydraulic efficiency. 

XLVm. CENTRIFUGAL PUMP APPLICATIONS 

261. Floating Dry Docks. — To illustrate the wide range of ap- 
plications to which centrifugal pumps are adapted, a few typical 
examples of their use will be given. 

The rapid extension of the world's commerce in recent years has 
created a demand for docking facilities in comparatively isolated 
ports, which has given rise to the modern floating dry dock (Fig. 
210). In docks of this type the various compartments into 
which they are divided are provided with separate pumps so that 



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ENERGY OF FLOW 



267 



they may be emptied in accordance with the distribution of 
weight on the dock. Provision is usually made for handling 
one short vessel, two short vessels, or one extremely long ship, 
the balancing of the dock on an even keel being accomplished by 
emptying the various compartments in proportion to the weight 
sustained. The number of pumps in docks of this type varies 
from 6 to 20, depending on the number of compartments. The 
centrifugal pump is widely used and particularly suitable for this 
class of work, where a large quantity of water has to be discharged 
in a short time against a changing head which varies from zero. 




FIG. 210. 

when the piunping begins, to 30 or 40 ft. when the dock is nearly 
dry (Fig. 211).i 

262« Deep Wells. — In obtaining a water supply from deep 
wells, the problem is to secure a pump which will handle a large 
quantity of water efficiently in a drilled well of moderate diame- 
ter, the standard diameters of such wells being 12 to 15 in. To 
meet this demand, centrifugal pumps are now built which will de- 
liver from 300 to 800 gal. per minute from a 12-in. well, and from 
800 to 1,500 gal. per minute from a 15-in. well, with efficiencies 
ranging from 55 to 75 per cent. The depth from which the 
water is pumped may be 300 ft. or more, the pumps being built in 
several stages according to the depth (Fig. 212). 

^ Figs. 201-214 are reproduced by permission of the Piatt Iron Works 
Co., Dayton, Ohio. 



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ELEMENTS OF HYDRAULICS 




z\ 



jm 



fc-* 



t^ 



mrm^ 



Fig. 211. 



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ENERGY OF FLOW 



269 




,--^^h 



Fio. 212. 



Fio. 213. 
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270 



ELEMENTS OF HYDRAULICS 



263. Mine Drainage. — The extensive use of electric power for 
operating mining machinery has led to the employment of cen- 
trifugal pumps for mine drainage. The advantages of this type 
of pump when direct-connected to a high-speed motor are its 
compactness, simplicity and low first cost. Fig. 213 illustrates a 
mine-sinking turbine pump which operates against a 1,250-ft. 
head in a single lift. Pumps of similar design are in operation in 
nearly all the important mining regions of the United States and 
Mexico. The turbine pump is used to best advantage where it 
is required to unwater a flooded mine shaft. For actual sinking 
work a displacement pump is preferable unless an ample smnp is 
provided in order to keep the turbine pump well supplied with 
water so that it will not take air. 

264. Fire Pumps. — The use of centrifugal pumps for fire pro- 
tection has been formally approved by the Fire Insurance Under- 
writers, who have issued specifications covering the essential 




Fia. 214. 

features of a pump of this type to comply with their requirements. 
In the case of fire boats the centrifugal pump has been foimd to 
fully meet all demands. The New York fire boats "James 
Duane" and "Thomas Willett" are equipped with turbine 
pumps, each of which has a capacity of 4,500 gal. per minute 
against 150 lb. per square inch pressure. For automobile fire en- 
gines, the great range of speed for gas engines gives the cen- 
trifugal pump a great advantage, making it possible to throw 
streams to a great height by merely increasing the speed of the 
motor. This type can also be readily moimted on a light chassis 
and driven from the driving shaft of the machine, making a light, 
compact, flexible and efficient unit (Fig. 214). 



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ENERGY OF FLOW 271 

266. Hydraulic Dredging. — The rapid development and im- 
provement of internal waterways in the United States has demon- 
strated the efficiency of the hydraulic or suction dredge. The 
advantage of the hydraulic dredge over the dipper and ladder 
types is that it not only dredges the material but also delivers it 
at the desired point with one operation. Its cost for a given 
capacity is also less than for any other type of dredge, while its 
capacity is enormous, some of the Government dredges on the 
Mississippi handling over 3,000 cu. yd. of material per hour. 

In operation the dredging pump creates a partial vacuum in the 
suction pipe, sufficient to draw in the material and keep it 
moving, and also produces the pressure necessary to force the 
discharge to the required height and distance. Hundreds of 
such pumps, ranging from 6 to 20 in. in diameter, are used 
on Western rivers for dredging sand and gravel for building and 
other purposes. The dredge for this class of service is very sim- 
ple, consisting principally of the dredging pump with its driving 
equipment moimted on a scow, the suction pipe being of suffi- 
cient length to reach to the bottom, and the material being 
delivered into a flat deck scow with raised sides, so that the sand 
is retained and the water flows overboard. 

For general dredging service where hard material is handled, 
it is necessary to use an agitator or cutter to loosen the material 
so that it can be drawn into the suction pipe. In this case the 
suction pipe is mounted within a structural steel ladder of 
heavy proportions to stand the strain of dredging in hard 
material, and of sufficient length to reach to the depth required. 
The cutter is provided with a series of cutting blades, and is 
moimted on a heavy shaft supported on the ladder, and driven 
through gearing by a separate engine (Fig. 215). 

Usually two spuds are arranged in the stern of the dredge to act 
as anchors and hold the dredge in position. The dredge is 
then swimg from side to side on the spuds as pivots by means of 
lines on each side controlled by a hoisting engine, thus controlling 
the operation of the dredge. 

Suction dredges are usually equipped with either 12-, 15-, 18- 
or 20-in. dredging piunps, the last-named being the standard 
size. For most economical operation as regards power, the ve- 
locity through the pipe line should not be greater than just 
sufficient to carry the material satisfactorily. 

With easily handled material the delivery pipe may be a mile 



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272 



ELEMENTS OF HYDRAULICS 




Digitized by VjOOQIC 



ENERGY OF FLOW 273 

or more in length, but with heavy material requiring high velocity 
the length should not exceed 4,000 ft. The practical maximum 
discharge pressure is about 50 lb. per square inch. For long 
pipe lines it therefore becomes necessary to use relay pumps, the 
dredging pump delivering through a certain length of pipe 
into the suction of the relay pxunp, and the latter delivering it 
through the remainder of the line. For high elevations or very 
long lines, several relay pumps may have to be used. 

The efficiency of a dredging pump is usually only 40 or 50 
per cent., a high efficiency in this case not being so important 
as the ability to keep going. 

266. Hydraulic Mining. — The centrifugal pump is also suc- 
cessfully used in hydraulic mining, where a high-pressure jet is 
used to wash down a hill. A number of centrifugal pumps are 
used for this purpose in the phosphate mines of Florida. Other 
uses for centrifugal pumps besides those described above are 
found in municipal water-works, sewage and drainage plants, 
sugar refineries, paper mills and irrigation works. 

APPLICATIONS 

101. A jet 2 in. in diameter discharges 5 cu. ft. of water per 
second which impinges on a flat vane moving in the same direc- 
tion as the jet with a velocity of 12 ft. per second. Find the 
horsepower expended on the vane. 

102. A fireman holds a hose from which a jet of water 1 in. in 
diameter issues at a velocity of 80 ft. per second. What force 
will the fireman have to exert to support the jet? 

103. A small vessel is propelled by two jets each 9 in. in diame- 
ter. The water is taken from the sea through a vertical inlet 
pipe with scoop facing forward, and driven astern by a centrifugal 
pump 2 ft. 6 in. in diameter running at 428 r.p.m. and delivering 
approximately 2,250 cu. ft. of water per second. If the speed of 
the boat is 12.6 knots (1 knot = &,080 ft. per hour), calculate 
the hydraulic efficiency of the jet. 

104. In the preceding problem, the efficiency of the pump was 
48 per cent, and efficiency of engine and shafting may be assumed 
as 80 per cent. Using these values, calculate the total hydraulic 
efficiency of this system of propulsion. 

Note. — The jet propeller is more efficient than the screw pro- 
peller, the obstacle preventing the adoption of this system in 

18 



Digitized by VnOOQlC 



274 ELEMENTS OF HYDRAULICS 

the past being the low efficiency obtainable from centrifugal 
pumps. 

106. A locomotive moving at 60 miles per hour scoops up 
water from a trough between the rails by means of an L-shaped 
pipe with the horizontal arm projecting forward. If the trough 
is 2,000 ft. long, the pipe 10 in. in diameter, the opening into the 
tank 8 ft. above the mouth of the scoop, and half the available 
head is lost at entrance, find how many gallons of water are 
lifted into the tank in going a distance of 1,600 ft. Also find the 
slowest speed at which water will be delivered into the tank. 

106. A tangential wheel is driven by two jets each 2 in. in 
diameter and having a velocity of 75 ft. per second. Assuming 
the wheel efficiency to be 85 per cent, and generator efficiency 90 
per cent., find the power of the motor in kilowatts (1 hp. = 746 
watts = 0.746 kilowatt). 

107. In a commercial test of a Pelton wheel the diameter of 
the jet was found to be 1.89 in., static head on runner 386.5 
ft., head lost in pipe friction 1.8 ft., and discharge 2.819 cu. ft. 
per second. The power developed was found by measurement to 
be 107.4 hp. Calculate the efficiency of the wheel. 

108. A nozzle having an efflux coefficient of 0.8 delivers a jet 
13^-in. in diameter. Find the amount and velocity of the dis- 
charge if the jet exerts a pressure of 200 lb. on a flat surface nor- 
mal to the flow. 

109. A jet 2 in. in diameter is deflected through 120^ by 
striking a stationary vane. Find the pressure exerted on the 
vane when the nozzle is discharging 10 cu. ft. per second. 

110. A power canal is 50 ft. wide and 9 ft. deep, with a velocity 
of flow of 13^ ft. per second. It supplies water to the turbines 
under a head of 30 ft. If the efficiency of the turbines is 80 per 
cent., find the horsepower available 

111. It is proposed to supply 1,200 electrical hp. to a city 25 
miles from a hydraulic plant. The various losses are estimated 
as follows: 

Generating machinery, 10 per cent. ; line, 8 per cent. ; trans- 
formers at load end, 9 per cent. ; turbine efficiency, 80 per cent. 

The average velocity of the stream is 3 ft. per second, available 
width 90 ft., and depth 6 ft. Find the net fall required at the 
dam. 

112. The head race of a vertical water wheel is 6 ft. wide and 
the water 9 in. deep, flowing with a velocity of 5 ft. per second. 



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ENERGY OF FLOW 275 

If the total fall is 20 ft. and the efficiency of the wheel is 70 per 
cent., calculate the horse power available from it. 

113. A stream is 150 ft. wide with an average depth of 4 ft. 
and a velocity of flow of 1 ft. per second. If the net fall at the 
dam is 20 ft. and the efficiency of the wheel is 75 per cent., find 
the horsepower available. 

114. Eighty gallons of water per minute are to be pumped 
from a well 12 ft. deep by a pump situated 50 ft. from the well, 
and delivered to a tank 400 ft. from the pump and at 80 ft. 
elevation. The suction pipe is 3 in. in diameter and has two 
3-in. elbows. The discharge pipe is 2}4 in. in diameter and has 
three 23^-in. elbows. Find the size of engine required. 

Note. — The lift is 92 ft. and the friction head in pipe and 
elbows amounts to about 25 ft., giving a total pumping head of 
117 ft. The pump friction varies greatly, but for a maximum 
may be assumed as 50 per cent, of the total head, or, in the pres- 
ent case, 583^ ft. 

115. A single-acting displacement pump raises water 60 ft. 
through a pipe line 1 mile long. The inside diameter of the pump 
barrel is 18 in., the stroke is 4 ft., and the piston is driven 
by a connecting rod coupled to a crank which makes 30 r.p.m. 
The velocity of flow in the pipe line is 3 ft. per second. Assum- 
ing the mechanical efficiency of the pump to be 75 per cent., and 
the slip 5 per cent., find the horse power required to drive the 
pump and the quantity of water delivered. 

116. A 6-in. centrifugal pump deUvers 1,050 gal. per minute, 
elevating 20 ft. The suction and discharge pipes are each 6 in. 
in diameter and have a combined length of 100 ft. Find the 
friction head, total horse power required, and speed of pump for 
50 per cent, efficiency. 

Note. — The velocity of flow in this case is 12 ft. per second and 
the corresponding friction head for 100 ft. of 6-in. pipe is 8.8 ft. 
The total effective head is therefore 28.8 ft., requiring 15.26 h.p. 
at a speed of 410 r.p.m. 

117. In the preceding problem show that if an 8-in. pipe is 
used instead of 6-in. there will be a saving in power of over 22 
per cent. 

118. A hydraulic ram uses 1,000 gal. of water per minute under 
a 4-ft. head to pump 40 gal. per minute through 300 ft. of 2-in. 
pipe into a reservoir at an elevation of 50 ft. above the ram. 



Digitized by VnOOQlC 



276 ELEMENTS OF HYDRAULICS 

Calculate the mechanical and hydraulic efficiencies of the ram, 
assuming the coefficient of pipe friction as 0.024. 

119. An automobile booster fire pump, used for making a 
quick initial attack on a fire, is required to deliver two streams 
through ^-in. nozzles and 250 ft. of 1-in. hose. The pump is of 
the centrifugal type and is geared up to a speed of 3,500 r.p.m. 
from the gas engine which drives the machine. Calculate the 
discharge in gallons per minute and the horsepower required to 
drive the pump, assuming 50 per cent, efficiency. 

Note. — For this size nozzle, the maximum discharge is reached 
with a nozzle pressure of about 68 lb. per square inch correspond- 
ing to a velocity of about 100 ft. per second. 

120. Feed water is pumped into a boiler from a round vertical 
tank 23^ ft. in diameter. Before starting the pimap the water 
level in the boiler is 38 in., and in the tank 22 in., above the floor 
level, and when the pump is stopped these levels are 40 in. and 
15 in. respectively. If the steam pressure in the boiler while the 
pump is at work is 100 lb. per square inch, find the number of 
foot-pounds of work done by the pump. 

121. A fire pump delivers three fire streams, each discharging 
250 gal. per minute under 80 lb. per square inch pressure. Find 
the horse power of the engine driving the pump if the efficiency 
of the engine is 70 per cent, and of the piunp is 60 per cent. 

122. A mine shaft 580 ft. deep and 8 ft. in diameter is full of 
water. How long will it take a 6-h.p. engine tounwater the shaft 
if the efficiencies of pump and engine are each 75 per cent.? 

123. A fire engine pumps at the rate of 500 gal. per minute 
against a pressure of 100 lb. per square inch. Assuming the 
overall efficiency to be 50 per cent., calculate the indicated horse 
power of the engine. 

124. A water-power plant is equipped with tangential wheels 
having an efficiency of 80 per cent. The water is delivered to 
the wheels through a cylindrical riveted-steel penstock 5 miles 
long with a total fall of 900 ft., practically the entire penstock 
being under this head. 

The cost of power house and equipment is estimated at $50,000, 
penstock 6 cts. per pound, operating expenses $5,000 per annum, 
and interest on total investment 4 per cent, per annum. The 
income is to be derived from the sale of power at $12 per horse 
power per annum. A constant supply of water of 100 cu. ft. 



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ENERGY OF FLOW 277 

per minute is available. Find the diameter of penstock for which 
the net income is a maximum. 

126. A hydraulic pipe line is required to transmit 150 h.p. with 
a velocity of flow not greater than 3 ft. per second and a delivery 
pressure of 900 lb. per square inch. Assuming that the most 
economical size of pipe is one which allows a pressure drop of 
about 10 lb. per square inch per mile, determine the required 
size of pipe. 

126. Find the maximum horse power which can be transmitted 
through a 6-in. pipe 4 miles long assuming the inlet pressure to 
be 800 lb. per square inch and the coefficient of pipe friction to 
be 0.024. Also determine the velocity of flow and outlet pressure. 

127. A 6-in. pipe half a mile long leads from a reservoir to a 
nozzle located 350 ft. below the level of the reservoir and dis- 
charging into the air. Assuming the coefficient of friction to be 
0.03, determine the diameter of nozzle for maximum power. 

Solution. — The discharge is 

IT 7)2 
Q = ^ 62.4r 

where D = inside diameter of pipe and V = velocity of flow 
through pipe. Also the horse power delivered at the nozzle is 

H.P. at nozzle = -^^^^ ((h-f-^ ^) 

where h ^ static head at nozzle and I = length of pipe. The 
value of V for which the horse power is a maximum is found from 
the calculus condition. 

dV "" 
whence, solving for F, we find 



V 



Now let A = area of cross-section of pipe; 

a = area of cross-section of nozzle; 

V = velocity of flow through nozzle; 

p = pressure before entering nozzle. 
Then AV = av, and therefore from Bernoulli's theorem 

62.4 ■^2fif 2g \a) 2g' 

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278 



ELEMENTS OF HYDRAULICS 



and also 



62.4 ^2(7 ^^ D2g "' 



whence, by subtraction and reduction, 



a^ AV^ 



l2ghD-flV^ 

Substituting in this relation the value of V obtained above for 
maximum power, the result is 

** = ^ W 

For a circular pipe and nozzle this becomes 

which gives the required size of nozzle for maximum output of 
power. 

128. A 10-in. water main 900 ft. long is discharging 1,000 gal. 
of water per minute. If water is shut oflF in 2 sec. by closing a 
valve, how much is the pressure in the pipe increased? 

129. In a series of experiments made by Joukowsky on cast- 
iron pipes, the time of valve closure in each case being 0.03 sec, 
the following rises in pressures were observed.^ Show that these 
results give the straight line formula, p = 57v. 

Cast-iron Pipe, Diameter 4 In., Length 1,050 Ft. 



Vel. in ft./sec 


0.6 


2.0 


3.0 


4.0 


9.0 




Observed pressure in Ib./in.*... . 


31 


119 


172 


228 


511 


Cast-iron pipe, diameter 6 in., length 1,066 ft. 


Vel. in ft./sec 


0.6 


2.0 


3.0 


7.5 




Observed rise in pressure in lb. /in.* 


43 


113 


173 


426 



130. It is customary in practice to make allowance for possible 
water-hammer by designing pipes to withstand a pressure of 
100 lb. per square inch in excess of that due to the static head. 
Show that this virtually allows for an instantaneous stoppage at 
a velocity of 1.6 ft. per second. 

131. A bowl in the form of a hemisphere, with horizontal rim, 
is filled with liquid and then given an angular velocity w about 
its vertical axis. How much liquid flows over the rim (Fig. 216) ? 

^ Gibson, "Hydraulics and Its Applications," p. 239. 



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ENERGY OF FLOW 



279 



132. A closed cylindrical vessel of height H is three-fourths full 
of water. With what angular velocity co must it revolve around 
its vertical axis in order that the surface paraboloid shall just 
touch the bottom of the vessel (Fig. 217). 





Fig. 216. 



Fig. 217. 



133. A closed cylindrical vessel of diameter 3 ft. and height 
6 in. contains water to a depth of 2 in. Find the speed in r.p.m. 
at which it must revolve about its vertical axis in order that the 
water shall assume the form of a hollow truncated paraboloid 
for which the radius of the.upper base is 1 per cent, greater than 
the radius of the lower base; or, 
referring to Fig. 218, such that 
ri = l.Olrz. 

134. The test data for a 19-in. 
New American turbine runner 
are as follows: 

Head 25 ft.; speed 339 r.p.m.; 
discharge 2,128 cu. ft. per minute; 
power developed 80 h.p. 

Calculate the turbine constants including the characteristic 
speed. 

Solution, — In this case, from Art. XXXVII, 

^19. 

Tdn 



-r^r- 



FiG. 218. 



Ivy """ 



<P = 



Ky 

VTg 



6OV25 
5.62 

2,128 



= 5.62; 



= 0.7; 



O - ^ - 



60 

V25 



= 7.0933; 



Digitized by VjOOQIC 



280 ELEMENTS OF HYDRAULICS 

_Qi_7_J0m_ 

\i2/ 

i,. = !^^ = ??^0 = 54.24. 

136. Two types of turbine runner, A and S, are to be compared. 
From tests it is known that runner A will develop a maximum 
of 2,080 h.p. at 500 r.p.m. under 100-ft. head, and runner B will 
develop 4,590 h.p. at 580 r.p.m. under 150-ft. head. Determine 
which of these types is the higher speed. 

Solution.-Type A, N. = ^OOV^jSO ^ ^^.n, 

loov^lob 

Type 5, iV, = ^?5^P? = 74.86. 
150\/l50 

136. Show that to transform the characteristic speed iV, from 
the EngHsh to the metric system it is necessary to multiply by 
the coefficient 4.46; that is to say, if the horse power and head 
are expressed in foot-pound units, andiV, in the metric system, 
we have the relation 

Vhjp. 



Ns = 4.46n- 



/i^ 



137. Five two-runner Frajicis turbines installed in the power 
house of the Pennsylvania Water and Power Co. at McCalPs 
Ferry on the Susquehanna River are rated at 13,500 h.p. each 
under a head of 53 ft. at a speed of 94 r.p.m. The quantity of 
water required per turbine is 2,800 cu. ft. per second. Calculate 
from this rating the characteristic speed, efficiency, and other 
turbine constants. 

138. Four two-runner Francis turbines operating in the Little 
Falls plant of the Washington Water Power Co. have a nominal 
power capacity of 9,000 h.p. each under a head of 66 ft. at a 
speed of 150 r.p.m. The quantity of water required per tiu-bine 
is 1,500 cu. ft. per second. From this rating calculate the char- 
acteristic speed, efficiency, and specific constants for these units. 

139. The upper curve shown in Fig. 219 is the official efficiency 
test curve of the 9,000-h.p. turbines, built by the I. P. Morris Co. 
for the Washington Water Power Co. These wheels are of the 



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ENERGY OF FLOW 



281 



horizontal shaft, two-runner, central discharge type, with volute 
casings. Head 66 ft., speed 150 r.p.m., and rated runner 
diameter 6 ft. 2 in. 

The lower curve shown in the figure is derived from a test at 
Holyoke of a homologous experimental runner having a rated 
diameter of 2 ft. 8 i%4 in. These curves are almost identical 
in shape, the eflSciency of the large units exceeding by a small 
margin that of the experimental runner. 

Calculate the discharge and characteristic speed at maximum 
efficiency, and from these results compute the specific constants. 



m 


- 


- 


- 


- 


- 


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so 








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70 






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~ 





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SO 



00 
50& 



30^ 



6u^M)ooJ60oaM^ffia^aal035oo*0(*4500£«)06«0(KM^ 1500 tjooo aGQO 9uaj seoo 

Fig. 219. 



140. In testing a hydraulic turbine it was found by measure- 
ment that the amount of water entering the turbine was 8,000 
cu. ft. per minute with a net fall of 10.6 ft. The power devel- 
oped was measured by a friction brake clamped to a pulley. 
The length of brake arm was 12 ft., reading on scales 4001b., and 
speed of pulley 100 r.p.m. Calculate the efficiency of the turbine. 

141. One of a series of 65 tests of a 31-in. Wellman-Seaver- 
Mopgan turbine runner gave the following data:^ 

Gate opening 75 per cent.; head on runner 17.25 ft.; speed 
186.25 r.p.m.; discharge 63.12 cu. ft. per second; power developed 
111.66 h.p. 

Calculate the efficiency and the various turbine constants. 

142. One of a series of 82 tests of a 30-in. Wellman-Seaver- 
Morgan turbine runner gave the following data:^ 

^ "Characteristics of Modem Hyd. Turbines," C. W. Larnbb, Trans, 
Am. Soc. C. E., vol. Ixvi (1910), pp. 306-386. 
« Ibid, 



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282 



ELEMENTS OF HYDRAULICS 



Gate opening 80.8 per cent.; head on runner 17.19 ft.; speed 
206 r.p.m.; discharge 85.73 cu. ft. per second; power developed 
146.05 h.p. 

Calculate the efficiency and the other turbine constants. 

143. Four of the turbines of the Toronto Power Co. at Niagara 
Falls are of the two-runner Francis type, with a nominal develop- 
ment of 13,000 h.p. each under a head of 133 ft. at a speed of 
250 r.p.m. The quantity of water required per turbine is 1,060 
cu. ft. per second. 

Calculate the efficiency, characteristic speed and specific 
turbine coefficients for these units. 

144. The upper curve shown in Fig. 220 is the official test 
curve of the 6,000-h.p. turbines designed by the I. P. Morris Co. 



90 








































J 












m^ 








=^ 


^ 


a 


90 














































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80 
































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^ 






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hi 


80 
































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70 














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70 


60 














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r 
























































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20 




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L. 






























400 800 12(X) 1600 2000 2400 280082009G00400044004800&200Se00600064W 
Hone Power 

Fig. 220. 

for the Appalachian Power Co. The rated runner diameter 
is 7 ft. 6>i in., head 49 ft., and speed 116 r.p.m. These turbines 
are of the single-runner, vertical-shaft type. 

The lower curve is derived from'a test at Holyoke of the small, 
homologous, experimental runner, having a rated diameter of 
27^ in. The curves are identical in shape, but owing to the 
better arrangement of water passages in the large plant, its 
efficiency considerably exceeds that of the experimental runner. 
It may also be noted that the efficiency shown on this diagram is 
the highest ever recorded in a well-authenticated test. 

Calculate the discharge and characteristic speed at maximum 
efficiency, and from these results compute the specific turbine 
constants. 



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ENERGY OF FLOW 



283 



146. The following data, taken from the official Holyoke test 
reports, give the results of tests made on a 35-in. vertical 
Samson turbine built by the James Leflfel Co. of Springfield, 
Ohio. Calculate the turbine constants and characteristic speeds. 

Tbsts of 35-in. Vertical Samson Tubbinb 





Head on 


Speed 


Discharge 


Horse 


Efficiency 


Gate opening 


wheel in 


in rev. 


in. cu. ft. 


power 


in per 




feet 


per min. 


per sec. . 


developed 


cent. 


Full gate 


16.57 


187 


120.61 


188.27 


83.06 


0.9 gate 


16.69 


191 


114.35 


188.88 


87.26 


0.8 gate 


16.78 


189 


105.10 


179.87 


89.93 


0.75 gate 


16.86 


187 


100.29 


172.57 


89.99 


0.7 gate 


17.08 


188 


92.83 


160.03 


88.99 


0.6 gate 


17.23 


185 


77.15 


128.22 


85.05 


0.5 gate 


17.47 


188 


66.89 


108.72 


82.03 



146. The speed and water consumption of a turbine vary as 
the square root of the head (\/^), and the power varies as the 
square root of the cube of the head ( V^)- Thus if the head on a 
wheel is multiplied by 4, the speed and discharge will be multi- 
plied by 2 and the power by 8. 

Given that a 12-in. turbine under 12-ft. head develops 14 h.p. 
at 480 r.p.m. using 762 cu. ft. of water per minute, find the power, 
speed and discharge for the same tiu'bine under 48-ft. head. 

147. On page 284 is given a rating table of turbines manu- 
factured by the S. Morgan Smith Co. of York, Pa., computed 
from actual tests of each size turbine under the dynamometer at 
the Holyoke testing flume. 

Calculate the nominal efficiency and characteristic speed for 
each size runner, and determine whether it is of the low-, medium- 
or high-speed type. 

Note. — ^^Data of this kind may be used by the instructor as 
problem material for an entire class without duplicating results, 
the final results being collected and tabulated, thus serving as 
a check on the calculations and also showing the range gf the 
constants involved. 

148. On pages 285, 286, and 287 is given a rating table of 
Victbr Turbines manufactured by the Piatt Iron Works Co., 
Dayton, Ohio. 

Calculate the nominal efficiency, characteristic speed, and 
speed and capacity constants for each diameter and head. 



Digitized by VnOOQlC 



284 



ELEMENTS OF HYDRAULICS 



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Digitized by VnOOQlC 



ENERGY OF FLOW 



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cq cq ""^ 



CO 



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lO CO 



00 -^ »o 

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^ CO ^ 



O i-< 
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CO 



'<*' »0 CO 
lib CO Tf 
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CO 



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CO 



CO -^ CO 



Ol t^ Ol 

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^ Q ^ 



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OO CO Ol 
CO OS ^ 



d 1-1 i> 

t^ CO iH 

eo QQ ^ 



lO S 



, OS i-H 
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Ol i 

CO ' 
CO ' 



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CI r* OS 

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1-H A p 

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lO OS CO 



»o -^ OS 
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lO CO CO 



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r* »o CO 
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O 00 d 

t^ iH t* 

^ ^ CO 



OS t* 

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s 



t^ t^ CO 

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tH t^ t^ 

Ol '^ »o 

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Tf o CO 



OS Q 00 

00 O Tf 

CO OS CO 

CO 



CO »0 Tf 
t^ CO -4 
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ELEMENTS OF HYDRAULICS 



T 

QD 

z 

l-l 

S 

H 



^ 



9 

o 






00 



CO 



5! 



CO 



s 



CO 



^ 



CO 



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(N ^H cq 

CO Od CD 
t^ 00 CO 



O 0> 00 
iH 00 lO 
b- t^ CO 



^ 53 Q 

OS 



CO 



-^ t^ CD 

Tt< r^ '<t< 

CO "^ CO 

o» 



CO cq N 
c^ r^ Tf 

CO CO CO 



fH »0 O 

o CO CO 

CO C^ CO 

oa 



1-1 (© Tj< 

00 »0 CO 
lO iH CO 

o» 



O t^ iH 
CO ^ CO 
»0 O CO 



O O 00 

"^ CO w 

lO 05 CO 

00 



O lO c^ 
c^ 53 N 

»0 00 CO 



8»0 Oi 

O 1-1 

»0 t« CO 
OO 



»0 00 

OO 



1-1 »o o 

CO t- «H 
■^ 'ij CO 



CO OS M 
iH 00 CO 
05 p CO 



Od 
CO 



»o ^ N 

00 00 CO 



c^ o c^ 

CO (N C<l 
OO t^ CO 



Tt< 00 

»0 CO 



Q 0> "* 
00 CO iH 
t^ "^ CO 



»o S S 

l^ CO CO 



00 1-H 00 

N 1-H O 

l>. N CO 



CO 

O ■ 



i 



OO r^ o 
t^ ^ o 

CO Oi CO 

o 



S(N 00 

1-1 o 

CO 00 w 

o 



o r^ »o 

C^ t^ Oi 

CD CO cq 

o 



lO CO ^ 

O CO o> 

CO »o cq 

o 



»H Oi CO 
00 0» 00 

»o CO cq 

o 



00 CO c^ 
lO »0 00 
"3 C^ W 



O CO 



S5§ 

O (N N 



c^ t^ t^ 

8a> Od 
p w 



o »o OS 
O OS 01 

CO 



OS 

"5 ^ - 

OS 60 04 
CO 



s 



00 Q t^ 

01 »0 00 
OS CO (N 

CO 



t^ 00 -^ 
OS OS oO 
00 "«*< N 



^ fe f^ 

00 t* 

CO 



§8 



OS OS Ti« 

p 01 t^ 

So o w 

CO 



00 l^ 1-1 

t^ CD t> 

t^ 00 01 

cq 



00 00 ' 
CO o 

Ol 



l> -^ 

01 CO CO 

t« »o Ol 

01 



*-< »0 01 

OS t^ <^ 

CD CO Ol 

04 



Tf lO 00 
CO O "5 
CO 01 01 



t^ lO t^ 

CO *-< t^ 

01 r^ 01 

1-1 CO 



8 CO IQ 

01 »0 01 

tH CO 



01 OS i-H 

CO CO t^ 
^ g 01 



01 OS 



OS 00 »o 

00 iH CO 



CO OS 01 



t^ 00 OS 

iH lO »0 

O CO 01 

iH iC 



d Tj< CO 

00 t^ W5 
Os;g01 



Tj< S lO 

OS 01 01 



CO P 1-H 

1-1 o ^o 



00 "«!< t^ 

r^ th Tt< 

00 OS N 



00 l^ 01 



i-H »0 01 
i-H 01 Tf 

00 »2 01 



OS lO l> 
t^ 01 CO 
t« CO 01 



si** 



kO CO 00 

CO 00 "5 

Tf CO Ol 

1-1 OS 



01 00 lO 

OS 00 "5 

CO iH 01 

iH OS 



OS CO CO 

-^ 00 lO 

CO OS oi 

iH 00 



CO tt 01 

1-i 00 



CO CO t^ 
CO t^ -**< 
01 lO 01 

1-i 00 



1-1 iH -^ 

Ol l^ '^ 

01 CO 01 



00 CO 5S 
1-1 iH 01 
1-1 00 



OS O 00 

CO »0 CO 

iH OS 01 

1-1 l^ 



00 CO iO 
OS CO CO 
O b* 04 

iH t^ 



OS CO CO 

»0 iH CO 

O iC 01 

1-1 r^ 



OS t^ fH 

^ OS CO 

O 01 01 

1-1 t^ 



gTt< CO 
CO 01 

OS O 01 



1-1 lO Tt< 

CO ^ 04 
OS (» 01 



SCO 1-1 
1-1 01 
OS CO 04 
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00 00 04 

OS t^ Tt< 

CO "^ 04 
1-i 04 
01 



r* 01 00 

SiO CO 
01 04 
iH 04 
01 



00 CO ^ 

^ OS CO 

»0 t^ 04 



o o o 

lO 01 CO 
"^ CO 01 



t^ CO CO 

OS 00 04 
CO O 01 



s 



a 



r* 1-1 04 

CO S 01 



tH CD O 
CD »0 04 
04 CO 04 



04 CO 00 

T-l O 1-< 

01 iH 04 

1-1 o 

04 



O CO Tt< 

r* »o ^ 

1-1 op 01 
vi O 



»0 1-1 04 

01 OS ^ 

fh »0 01 

1-1 OS 



1-1 00 OS 
1-1 OS 



t^ 04 t^ 

CO i> o 

S8" 




u 



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1 .5 I 



o 

CO 



O jj & 



CO 
CO 



S3 
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d 
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Digitized by VnOOQlC 



ENERGY OF FLOW 



287 



s 



G ^ ^ 
CO t^ N 

oi lo cq 



w »o o 

00 t^ »-• 

^ 00 w 

C^ 00 



CO c) 00 

Tt< t^ o> 

Tf CO iH 
CO 



O CD Oi 

N CO r* 

CO 



W t^ 01 

r^ 01 (N 

00 CO c^ 






CO t* CO 
t^ ^ Oi 
CO O iH 

CO 



CO r* fh 
cq lo 

CO 



00 



»0 CD O 

iH lO 01 
00 O N 



N N iC 

»o o o 

s S <^ 

C<4 00 



O 00 Tf 

o »-• o> 

CO b» »-• 



CO M< 00 
»0 CO *-< 

CO 



Oi Tf 00 
iC o> *-< 



00 »0 CO 

00 Oi o 
a> Oi c^ 



O l> 01 

CO 00 o 

C^ CO 1-1 

CO 



"^ iH CO 

00 t^ 00 

•^ iSi ^ 

CO 



CO Oi '^ 

O 01 1-H 
t> LO 04 

tH ^ 

Ol 



W5 r^ ^ 
01 o> o 

05 CO w 
04 



»o »o S 

^ O iH 

CSi »-t 

CO 



»o r^ CO 

O O 00 
'^ »o ^ 
cq "^ 

CO 



W 00 Tt< 

»o "^ t^ 

r^ t« fh 

CO 



CO CO I"* 
CO 



^ 



00 CO 04 
^ lO fH 
CO 04 04 



^ 



•CO CO o> 

CO o o 

00 CO iH 
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O t^ 1-1 



b» CO 1-H 
04 »-t 00 
CO 04 1-1 



00 ^ 05 

t^ iH CO 

lO 0> f-H 

CO 



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04 Q O 
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lO O 04 
1-1 CO 
04 



0> 00 CO 
1-1 CO 00 
O CO i-< 



O t* OS 
»0 CO b- 
04 00 1-1 
04 CO 
CO 



s ^ s 

0> O CO 

Tt< Tf fH 

04 t^ 

CO 



CO 



O CO 00 

CO CO O 

»o r^ 04 

1-* CO 
04 



O 00 Tf 

Tt< t^ 0> 

r^ r* iH 

tH CO 

04 



CO fH t^ 

t^ »o t^ 



c^ £2 

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CO 00 ^ 

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»0 0> Tt< 

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CO CO 1-1 

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CO 



CO 00 CO 

CO »o o 

Tt< 1-H ^ 
fH CO 
04 



O 00 o 
04 -^ O 

O 1-1 fH 
tH CO 
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CO lO o> 

1-t 1-4 t>. 
00 CO fH 
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CO CO 04 

04 CO t*. 

O CO tH 

CO 



Q CO OS 
lO »0 CO 
OS 04 1-1 
1-1 04 
CO 



5 



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CO 04 00 

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04 



o <5 00 
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t^ OS 1-t 

1-1 00 

04 



iH fH O 

CD '^t* CO 

fH t* fH 
CO 



FH ^ S 

CO OS OS 
CO lO fH 
tH 04 
04 



t^ 00 CO 
t^ lO CO 

00 00 fH 



00 
CO 



Q 00 l> 
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04 N 1-i 
fH 04 
04 



»0 04 CO 

1-1 lO 
04 



04 O l> 

CO 04 »-• 



lO 04 lO 

O CO CO 
00 "««*< 1-i 



CO b» fH 
lO '^ t^ 
»0 00 iH 

1-1 t^ 

04 



s 



04 t^ lO 

2|- 



CO 
CO 



O OS O 
oO OS OS 



04 t^ 00 

CO OS t« 

CO Tf iH 



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Digitized by VnOOQlC 



288 



ELEMENTS OF HYDRAULICS 



149. Fig. 221 shows a vertical section of the 10,800-h.p. tur- 
bines designed by the I. P. Morris Co. for the Cedar Rapids 
Mfg. and Power Co. The rated diameter of these turbines is 
11 ft. lOJ^ in., head 30 ft., and speed 65.6 r.p.m. 

These turbines are at present the largest in the world, and it 
may be noted that all the latest features have been incorporated 
in the design, namely, volute casings and draft tubes molded in 
the concrete; cast-iron speed rings supporting the concrete, gen- 




FiG. 221. 

erator and thrust-bearing loads from above; lignum vitae turbine 
guide bearing; thrust-bearing support located above the genera- 
tor; Kingsbury thrust bearing with roller auxiliary; and pneu- 
matic brakes acting on the rotor of the generator. 

Calculate the characteristic speed from the rating given above, 
and from the table on page 209 determine to which speed type 
it belongs. 

150. The following table gives the results of 20 tests out of a 
total of 66 made Sept. 3 and 4, 1912, at the testing flmne of the 



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ENERGY OF FLOW 



289 



Holyoke Water Power Co. on a 24-in. Morris turbine type "0" 
runner. Calculate the characteristic speed for each test, and 
note its high value in test number 10. 

Note. — Two of the wheels for the Keokuk installation and 
nine wheels for the Cedar Rapids plant are built with this type 
of runner, the large wheels being geometrically similar to the 
experimental wheel tested at Holyoke. 

Report of Tests op a 24-in. Morris Turbine, Type "O" Runner made 
IN THE Testing Flume op the Holyoke Water Power Co. 



Number 
of experi- 
ment 


Open- 
ing of 
speed 
gate in 
inches 


Per cent, 
of full 

dis- 
charge of 
wheel 


Head 

on 

wheel in 

feet 


Speed in 

rev. per 

min. 


Dis- 
charge 
in cu. 
ft per 

sec. 


Horse 
power 
devel- 
oped 


Effi. 
ciency 
in per 

cent. 


1 


3.0 


0.792 


17.39 


344.25 


84.66 


103.98 


62.31 


2 


3.0 


0.779 


17.40 


298.00 


83.35 


126.01 


76.66 


3 


3.0 


0.775 


17.39 


275.75 


82.91 


133.26 


81.55 


4 


3.0 


0.778 


17.36 


257.50 


83.13 


139.99 


85.59 


5 


3.0 


0.779 


17.83 


249.20 


83.13 


143.01 


87.58 


6 


3.0 


0.782 


17.30 


241.25 


83.35 


145.73 


89.17 


7 


3.0 


0.779 


17.30 


235.50 


83.13 


146.53 


89.89 


8 


3.0 


0.781 


17.28 


238.25 


83.24 


146.80 


90.05 


9 


3.0 


0.783 


17.27 


«40.20 


83.46 


146.55 


89.71 


10 


3.0 


0.781 


17.23 


236.80 


83.13 


146.62 


90.32 


11 


3.0 


0.762 


17.31 


214.00 


81.28 


138.97 


87.15 


12 


3.0 


0.782 


17.24 


237.20 


83.24 


146.51 


90.08 


13 


3.0 


0.744 


17.34 


185.50 


79.44 


128.86 


82.54 


14 


3.5 


0.902 


17.15 


357.20 


95.73 


107.89 


57.98 


15 


3.5 


0.900 


17.10 


320.50 


95.38 


135.52 


73.31 


16 


3.5 


0.903 


17.06 


287.75 


95.61 


156.44 


84.62 


17 


3.5 


0.892 


17.14 


265.75 


94.70 


160.53 


87.26 


.18 


3.5 


0.871 


17.17 


243.25 


92.55 


154.29 


85.67 


19 


3.5 


0.903 


17.15 


280.20 


95.84 


160.80 


86.32 


20 


3.5 


0.904 


17.19 


278.60 


96.07 


164.09 


87.67 



19 



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SECTION 4 



HYDRAULIC DATA AND TABLES 

Table 1. — ^Pbopbbtibs of Wateb 
Density and Volume of Water 



Temp, in 




Volume of 


Temp, in 




Volume of 


degrees 


Density 


1 gram in 


degrees 


Density 


1 gram in 


Centigrade 




cu. cm. 


Centigrade 




cu. cm. 





0.999874 


1.00013 


24 


0.997349 


1.00266 


1 


0.999930 


1.00007 


26 


0.996837 


1.00317 


2 


0.999970 


1.00003 


28 


0.996288 


1.00373 


2 


0.999993 


l.OOOOl 


30 


0.995705 


1.00381 


4 


1.000000 


1.00000 


32 


0.995087 


1.00394 


5 


0.999992 


1.00001 


35 


0.995098 


1.00394 


6 


0.999970 


1.00003 


40 


0.99233 


1.00773 


•7 


0.999932 


1.00007 


45 


0.99035 


1.00974 


8 


0.999881 


1.00012 1 


50 


0.98813 


1.01201 


9 


0.999815 


1.00018-! 


55 


0.98579 


1.01442 


10 


0.999736 


1.00026 j 


60 


0.98331 


1.01697 


11 


0.999643 


1.00036 1 


65 


0.98067 


1.01971 


12 


0.999537 


1.00046 ' 


70 


0.97790 


1.02260 


13 


0.999418 


1.00058 ! 


75 


0.97495 


1.02569 


14 


0.999287 


1.00071 


80 


0.97191 


1.02890 


16 


0.998988 


1.00101 


85 


0.96876 


1.03224 


18 


0.998642 


1.00136 


90 


0.96550 


1.03574 


20 


0.998252 


1.00175 


95 


0.96212 


1.03938 


22 


0.997821 


1.00218 ; 


100 


0.95863 


1.04315 



Weight of Water 



Temp, in 


Weight in 


Temp, in 


Weight in 


Temp, in 


Weight in 


degrees 


pounds per 


degrees 


pounds per 


degrees 


pounds per 


Fahrenheit 


cu. ft. 


Fahrenheit 


cu. ft. 


Fahrenheit 


cu. ft. 


32 62.42 1 


100 


62.02 


170 


60.77 


40 62.42 


110 


61.89 


180 


60.65 


50 


62.41 


120 


61.74 


190 


60.32 


60 


62.37 


130 


61.56 


200 


60.07 


70 


62.31 


140 


61.37 


210 


59.82 


80 


62.23 


150 


61.18 


212 


59.56 


90 


62.13 


160 


60.98 







290 



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HYDRAULIC DATA AND TABLES 



291 



Table 2. — ^Head and I'bessukb Eqthvalbnts 



Head of Water in Feet and Equivalent Pressure 


in Pounds per Sq. In. 


Feet 


Pounds per 


Feet 


Pounds per 


Feet 


Pounds per 


head 


sq. in. 


head 


sq. in. 


head 


sq. in. 


1 


0.43 


55 


23.82 


190 


82.29 


2 


0.87 


60 


25.99 


200 


86.62 


3 


1.30 


65 


28.15 


225 


97.45 


4 


1.73 


■ 70 


30.32 


250 


108.27 


5 


2.17 


75 


32.48 


275 


119.10 


6 


2.60 


80 


34.65 


300 


129.93 


7 


3.03 


85 


36.81 


325 


140.75 


8 


3.40 


90 


38.98 


350 


151.58 


9 


3.90 


95 


41.14 


375 


162.41 


10 


4.33 


100 


43.31 


400 


173.24 


15 


6.50 


110 


47.64 


500 


216.55 


20 


8.66 


120 


61.97 


600 


259.85 


25 


10.83 


130 


66.30 


700 


303.16 


30 


12.99 


140 


60.63 


800 


346.47 


35 


15.16 


150 


64.96 


900 


389.78 


40 


17.32 


160 


69.29 


1000 


433.09 


45 
50 


19.49 
21.65 


170 
180 


73.63 
77.96 











Pressure in Pounds per Sq. In. and Equivalent Head of Water in Feet 



Pounds per 
sq. in. 


Feet 
head 


Pounds per 
sq. in. 


Feet 
head 


Pounds per 
sq. in. 


Feet 
head 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

15 

20 

25 

30 

35 

40 

45 

50 


2.31 

4.62 

6.93 

9.24 

11.54 

13.85 

16.16 

18.47 

20.78 

23.09 

34.63 

46.18 

57.72 

69.27 

80.81 

92.36 

103.90 

115.45 


55 

60 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

125 

130 

140 

150 

160 

170 


126.99 
138.54 
150.08 
161.63 
173.17 
184.72 
196.26 
207.81 
219.35 
230.90 
253.98 
277.07 
288.62 
300.16 
323.25 
346.34 
369.43 
392.52 


180 
190 
200 
225 
250 
275 
300 
325 
350 
375 
400 
500 


415.61 
438.90 
461.78 
519.51 
577.24 
643.03 
692.69 
750.41 
808.13 
865.89 
922.58 
1154.48 























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292 



ELEMENTS OF HYDRAULICS 



Table 3. — Dischabqb Equivalbnts 



Gallons 
per 
min. 


Cubic 

feet per 

sec. 


Cubic feet 
per 
min. 


Gallons 
per 
hour 


Gallons 
per 24 
hours 


Bbls. per 

minute, 

42 gal. 

bbl. 


Bbls. per 
hour, 42 
gal. bbl. 


Bbls. per 

24 hours, 42 

gal. bbl. 


10 

12 

15 

18 

20 

25 

27 

30 

35 

36 

40 

45 

50 

60 

70 

75 

80 

90 

100 

125 

135 

150 

175 

180 

200 

225 

250 

270 

300 

315 

360 

400 

450 

500 

540 

600 

630 

675 

720 

800 

900 

1,000 

1,125 

1,200 

1,350 

1,500 

1,575 

1,800 

2,000 

2,025 

2,250 

2,500 

2,700 

3,000 




1.3368 
1.6042 
2.0052 
2.4063 
2.6733 
3.342 
3.609 
4.001 
4.678 
4.812 
5.348 
6.015 
6.684 
8.021 
9.357 
10.026 
10.694 
12.031 
13.368 
16.710 
18.046 
20.052 
23.394 
24.062 
26.736 
30.079 
33.421 
36.093 
40.104 
42.109 
48.125 
53.472 
60. 158 
66.842 
72.186 
80.208 
84.218 
90.234 
96.25 
106.94 
120.31 
133.68 
150.39 
160.42 
180.46 
200.52 
210.54 
240.62 
267.36 
270.70 
300.78 
334.21 
360.93 
401.04 


600 

720 

900 

1,080 

1,200 

1,500 

1,620 

1,800 

2.100 

2,160 

2,400 

2,700 

3,000 

3,600 

4,200 

4,500 

4,800 

5,400 

6,000 

7,500 

8,100 

9,000 

10,500 

10,800 

12,000 

13,500 

15,000 

16,200 

18,000 

18,900 

21,600 

24,000 

27,000 

30,000 

32.400 

36,000 

37,800 

40.500. 

43.200 

48,000 

64.000 

60.000 

67,500 

72,000 

81.000 

90.000 

94,500 

108,000 

120,000 

121.500 

135,000 

150,000 

162.000 

180.000 


14,400 

17,280 

21,600 

25,920 

28,800 

36.000 

38,880 

43,200 

50,400 

51.840 

57,600 

64,800 

72,000 

86,400 

100,800 

108.000 

115,200 

129.600 

144.000 

180,000 

194.400 

216,000 

252,000 

259,200 

288,000 

324,000 

360,000 

388.800 

432,000 

453.600, 

518.400 

576.000 

648.000 

720,000 

777,600 

864,000 

907.200 

972.000 

1.036,800 

1,152,000 

1,296,000 

1,440,000 

1,620,000 

1.728,000 

1,944,000 

2,160,000 

2,268,000 

2,692,000 

2,880,000 

2,916,000 

3.240.000 

3,600,000 

3.880.000 

4,320,000 


0.24 

0.29 

0.36 

0.43 

0.48 

0.59 

0.64 

0.71 

0.83 

0.86 

0.96 

1.07 

1.19 

1.43 

1.66 

1.78 

1.90 

2.14 

2.39 

2.98 

3.21 

3.57 

4.16 

4.28 

4.76 

6.36 

5.96 

6.43 

7.14 

7.6 

8.57 

9.52 

10.7 

11.9 

12.8 

14.3 

15.0 

16.0 

17.0 

19.05 

21.43 

23.8 

26.78 

28.57 

32.14 

35.71 

37.5 

42.85 

47.64 

48.21 

53.67 

59.52 

64.3 

71.43 


14.28 
17.14 
21.43 
25.71 
28.57 
36.71 
38.57 
42.85 
50.0 
51.43 
57.14 
64.28 
71.43 
85.71 
100.0 
107.14 
114.28 
128.5 
142.8 
178.6 
192.8 
214.3 
250.0 
257.0 
285.7 
321.4 
357.1 
386.7 
428.6 
450.0 
514.3 
671.8 
642.8 
714.3 
771.3 
857.1 
900.0 
964.0 
1,028.0 
1,142.0 
1.285.0 
1,428.0 
1,607.0 
1,714.0 
1,928.0 
2,142.0 
2,250.0 
2,571.0 
2,867.0 
2,892.0 
3,214.0 
3,671.0 
3,857.0 
4,285.0 


342.8 

411.4 

514.3 

617.1 

685.7 

857.0 

925.0 

1,028.0 

1,200.0 

1.234.0 

1.371.0 

1,543.0 

1,714.0 

2,057.0 

2,400.0 

2,570.0 

2,742.0 

3.085.0 

3.428.0 

4,286.0 

4,628.0 

6,143.0 

6,000.0 

6,171.0 

6,857.0 

7.714.0 

8,670.0 

9.267.0 

10.284.0 

10,800.0 

12,342.0 

13.723.0 

16,428.0 

17,143.0 

18.512.0 

20.570.0 

21,600.0 

23,143.0 

24,686.0 

27,387.0 

30,867.0 

34,284.0 

38,671.0 

41.143.0 

46,086.0 

51,427.0 

54,000.0 

61,710.0 

68,568.0 

69,426.0 

77.143.0 

85,704.0 

92,572.0 

102,840.0 






















0.1 










0.2 




0.3 




0.4 


0.5 


0.6 




0.8 


1.0 


1.2 . 


1.4 
1.5 
1.6 


2.0 


2.5 


3.0 


3.5 
4.0 


4.5 




6.0 



Digitized by 



Google 



HYDRAULIC DATA AND TABLES 



293 






H 
.O 

M 



S 8 S 






00 
.1 






adard 
ht 

4536 kil 
8024 kil 
0483 kil 




1 


1 fc fc 








s 


.^ CO «o 




i t^ s s 


|-?<=^S2 




3 


§2SI 




s « s * 


> 2 




P 


^-^^ 




•1 5;-^ « 


t:; B » n 




ti n H 




•5 1. .1 « 


^11 






III 




iii 


1H 1-H T-4 






ft O"^ 

.-1 f-l 1-4 




ft O" M 

T-l ^ 1-4 














m5 






III 




CO 

.'SI 


CO « :S 

N 2j -; 






ft 3 S 




. a&S 


i 58S 




i 


00 W t« 




Sz:>;is 


■g ^ »o csi Tjl 




^^^ 




00 CO O tJ 


'c ^ R n H 




>» 


d d (N 

II II y 




1. , » 






Q 


S .1 
.=5 -3 




ll 1 


am 

ogra 

tinea 






§11 




•*J fc« *» 










b3 2 






^ ^ ^ 




^ ^ ;S 


ri iH 1-H 






















1 


2 2 


« 




fe 




g 


mdard 

gth 

9 centimete 

4 centimete 

3 meters 

3 kilometer 




l22 

§ 3 3 
& & a 


s 
s 

o 


3 M 


II II n 


II 


' 


2^5 

lO g? CO 
-* 6 00 

o o d 


d 


.2 o 

II 


ja ^ -g 


o 




II II 11 


il 




HE 


a 




.B^t 


« 


1 


»H ^ T-4 


^ 




S'S'S' 


1 








^ »H ^ 


,-4 










.Sift 


2 


d 


ches 
ches 
ches 


J 




g'g'S' 


1 




.a .3 .3 


3 




3SS 


1-( 


«o 


S © CO O 


M* 




lO O 00 


1^ 


CO 








-* 




00* *& d d d 


d 


« 


o o 2 


csi 


0) CO 

1 " 

3 


•2 S „ „ 7 




^ 


n n II 


n 


•shJ n " « 


II 




h 




> 


Me 

millimeter 
centimeter 
meter 
kilometer 






o 


3 

1 


--- 


-H 




s'g'S^ 


o 


1 






T-t 1-1 .-1 


•^ 





Xi 


in 


3 


H 




p:; 


N 


p 




-< 


^ 


Ed 


n 


s 


s 


p 


'^ 


15 


00 


<i 


3 




X> 


<£ 




H 


UJ 


W 





o 



o 

I 

fa 
o 

DQ 

p 

a 



C) h 


gs 


.-co 






O 3 


d d 


d 




1-4 

* o 8 o 








, 


^ d ^ 




CO 








t* «o iC 






fe 


CO 04 to 

« u5 o CO 










o o o 




CO ^' d 00 






;:q' 


<N 




1 "^l 






' '"' •"• 




C4 CO ■ 






b. <<|4 










00 




^ 


§ o 


Sc 


=!oSo 


H 


d d 


d c 


i-d- 




o 








lO <N 










•^ 05 


r^ 








t^ 00 


»o 


lO 






o o 


»o o 


o o 




o o 


O N 


^ o 








'"' 








t» 










-d 


CO o >o 










3 


CO O g CO C 
00 d • (N iH 










(2 










rH O O 
















o 






o 




CO 




CO o «c 


»o 


o 




CO 1-1 


Xi 


-4^500 


00 O) O CO 


3 


d d d •^ o 


.-4 lO O »0 


O 






CO 


CO 


A 


-* 










C3 










a 


O <N O O b- 






o o 


o 


•H b. »H 00 b- 






-4 oo 


•s 


00 t* (>♦• c< 








^ 


<N (N b. 






o 


3 










o 








o 




^ 










'C ^ 


w 






CO O CO C 




9* "flS 


« O O (N S 


MONO 


i-t 


o ^ o « o 




g 


g 










Ul 


o e^ o * c 


'(J* OO M O 


QQ 


»-4 ^ O t- O 


CO 00 O ^ 


D 




S N 






c 




















CI 
















11- 


t 










1 




a|f,-2. 








g 








ol 






ilsSI 




OH JO 



Digitized by VnOOQlC 



294 



ELEMENTS OF HYDRAULICS 



Table 6. — Spbcifig Wbighto of Various Substancbs 



Air, press. 76 cm. 'Hg., 

0»C 

Alcohol 

Aluminium, pure 

commercial 

Basalt 

Bismuth 

Brass 

Brick 

Cadmium 

Carbon, charcoal 

diamond 

graphite 

Coal, hard 

Copper, cast 

electrolytic 

wire 

Cork 

Earth 

Gold 

Glass 

Granite 

Hydrogen, press. 76 cm. 

Hg.,0»C 

Ice 



0.001293 
0.79 
2.583 

2.7 - 2.8 
2.4 - 3.3 
9.76- 9.93 

7.8 - 8.7 
1.4 - 2.3 
8.54- 8.69 
1.45- 1.70 
3.4^ 3.53 
2.17- 2.32 

1.2 - 1.8 

8.3 - 8.92 
8.88- 8.95 
8.93- 8.95 

0.24 

1.4 - 2.8 
19.30-19.34 

2.5 - 3.8 
2.5 - 3.0 

0.0000894 
0.926 



Iron, cast 

pure 

steel 

wrought 

Lead 

Lime mortar 

Limestone 

Magnesium ..... 

Marble 

Mercury at 0*C. 

Nickel 

Oil 

Platinum, cast . . 

wire and foil . . 

Quartz 

Rubber 

Sand 

Sandstone 

Seawater 

Silver 

Timber, oak 

fir 

poplar 

Tin 

Zinc 



.03- 7.73 
.85- 7.88 
.60- 7.80 
.79- 7.85 
.21-11.45 
.6 - 1.8 
.4 - 2.8 
.6^ 1.75 
.5 - 2.9 
13.596 
.57- 8.93 
.91- 0.94 
48^21.50 

2 -21.7 

3 - 2.7 
0.93 

2 - 1.9 
9 - 2.7 
02- 1.03 
42-10.57 
62- 1.17 
5 - 0.9 
35- 1.02 
97- 7.37 
85- 7.24 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



295 



nil 



1^ o 5 « 



i-« a» t» U3 

■^ N lo CO i-« 

C4 •« lO 00 r-l 



«C ^ lO M 



« ^ b- ^ us 

CO t^ t^ O CO 

r« M iH t^ o 



o o o o -^ 



•-I N N CO "3 



O Oft 



S 



I 



5 



QQ 



OQ 



§ 
S 

I 

o 

g 

o 

i 



9 



O W M Tf O 



CO CO ^ N O 
^ q6 U5 b- I* 

lO CO t^ <^ CI 



Oi '^ ^ Oi t* 



"* CO N M ^ 



11 



Is 



«* o t» <o -^ 



S;0 t^ ^ -^ 
I* CO 00 «o 

CO csl ci fh »H 



U) h- Od 00 t» 
^ t>. ^ ^ U3 
C4 O O 00 h- 



eo 5 S N S 
CO ub Tf Tf CO 



iH iH O O O 



o o o o o 



08 o 

fi OS 

II 



lO b- t* t* 

■* o « s s 



OS I* •« -* CO 



5 CO o < 

N N Csl 1 



•H U3 O) <«i b. 

§>0 ^ « 00 

OS OO b> CO 

^ o d d d 



t* r-l CO b- »5 

t« Q ^ OS u5 

lO 25 Tf CO CO 

d d d d d 



6 6 



b- oj CO N I* 

O ii -• M CO 



iQ 00 t^ ^ 00 
Oft O Oft t« o 
•^ to t^ O h- 



CO OS •* ■«* «0 
^ t» t» CO •-) 
e5| CO ^ I* CO 



I to CO -^ 
> C4 00 CO CI 

I OS CO O OS 



o o o o o 



N CI CO CO •* 



iQ CO 00 O ^ 



CO 1-t b- 00 ( 
I* ^ 1-1 ^ ( 

§22gl 



00 t^ r-l 

00 00 CO CO ( 
CO QO t^ OS ( 



o o o o o 



O 1-1 M CO Tf 



t^ OS CI kO OS 



^ ^ ^ CI CO I 






r^ d CO I 



00 -^ lO CI 

IQ (O CO CO OS 
CO »H 00 ^ ^ 



b* d M CO CO 

■* ® "<1< I* I* 



o o o o o 



T-4 CI CI -* to 



> CI to OS <^ Tti I 

*H r^ i-l d CO ■ 



Si: 

-* CO 



^5«5»ooo aacoSo8»« 
00i-<>OOSiO cicoo-^t* 



> CO 00 CI OS 

i;ss2s 



> to CO t* 

> b» b- b» 

> o o -* 



CO "^ US CO b» 



o -^ c« -^ >o 



I CO CO < 



e« » 1-1 o) < 
I* OS w CO < 

CI CO *H (O < 



I 1-1 CI 

1 to CO 

1 -* o 



it^oob- eoiocoooci Ti<»o 
coOb» i-4Kdaeot« ^»0 

•"Ib--^ OOOSOCIb- OS© 



li-iCICICO -^lOiOl^OS 



O CI •<«< lO b- 



l?5 



O000i-<«CO ^ lO-Tjt^ l^COl^COOS ^r^CI"^* 

coooaoi-" co^^kOQ THCico-^ua oopci4'< 
ooo^^ i-i^^^55 cicicicici cic5co«< 



o o o o o 



o o o o o 



o o o o o 



o o o o o 



< ± 

ll 

^ 8 



S-^j^co-^ 00 ^t*oo i*ootooo»o kOcocit«os 
oscici "^oOfhcoco ©-^ciq^ tociooeo^ 
ClcO<^tOO0 OCOCOO^ O^OubO ooososo 



o o o o o 
66661^ 



00 •* 00 N •<«< 

»H fH CO -^ CO 



^i-i^CIC« COCO"^-**© 



I kQ >0 

I CO OS CO 00 
i ^ f-I CI ci 



10 o >o o >o 
CO -^ -^ »o "O 



I CO Tf -* I 



CO b- t^ 00 o 



Mi Hi Hi %ei 

CI CI CI CI »o 

CO CO CO CO b* 

CO b» 00 OS o 



CO to OD 01 e 



o o 

»H CI 



Digitized by VnOOQlC 



296 



ELEMENTS OF HYDRAULICS 



Table 7. — Capacttt of Reciprocating Pumps 

Capacity, or piston displacement, of reciprocating pumps in gallons per single stroke 



Diameter 

of 

cylindpr, 

inches 


Length of stroke in inches 


2 


3 


4 


5 6 

1 


7 


8 


9 


10 


1-1/4 


0.0106 


0.0159 


0.0212 


0.0266 


0.0319 


0.0372 


0.0425 


0.0478 


0.0531 


1-3/8 


0.0128 


0.0192 


0.0256 


0.0321 


0.0385 


0.0419 


0.0513 


0.0578 


0.0642 


1-1/2 


0.0153 


0.0229 


0.0306 


0.0382 


0.0459 


0.0535 


0.0612 


0.0688 


0.0765 


1-3/4 


0.0208 


0.0312 


0.0416 


0.0521 


0.0625 


0.0729 


0.0833 


0.0937 


0.1041 


2 


0.0272 


0.0408 


0.0544 


0.068 


0.0816 


0.0952 


0.1088 


0.1224 


0.136 


2-1/4 


0.0344 


0.0516 


0.0688 


0.086 


0.1033 


0.1205 


0.1377 


0.1548 


0.1721 


2-1/2 


0.0425 


0.0637 


0.0850 


0.1062 


0.1275 


0.1487 


0.17 


0.1912 


0.2125 


2-3/4 


0.0514 


0.0771 


0.1028 


D.1285 


0.1543 


0.1799 


0.2057 


0.2313 


0.2571 


3 


0.0612 


0.0918 


0.1224 


0.1530 


0.1836 


0.2142 


0.2448 


0.2754 


0.306 


3-1/4 


0.0718 


0.1077 


0.1436 


0.1795 


.0.2154 


0.2513 


0.2872 


0.3231 


0.3594 


3-1/2 


0.0833 


0.1249 


0.1666 


0.2082 


0.2499 


0.2915 


0.3332 


0.3748 


0.4165 


3-3/4 


0.0956 


0.1434 


0.1912 


0.239 


0.2868 


0.3346 


0.3824 


0.4302 


0.478 


4 


0.1088 


0.1632 


0.2176 


0.272 


0.3264 


0.3808 


0.4352 


0.4896 


0.544 


4-1/4 


0.1228 


0.1842 


0.2456 


0.307 


0.3684 


0.4298 


0.4912 


0.5526 


0.6141 


4-1/2 


0.1377 


0.2065 


0.2754 


0.3442 


0.4131 


0.4819 


0.5508 


0.6196 


0.6885 


4-3/4 


0.1534 


0.2301 


0.3068 


0.3835 


0.4602 


0.5369 


0.6136 


0.6903 


0.7671 


5 


0.17 


0.2550 


0.34 


0.425 


0.51 


0.595 


0.68 


0.765 


0.85 


5-1/4 


0.1874 


0.2811 


0.3748 


0.4685 


0.5622 


0.6559 


0.7496 


0.8433 


0.9371 


5-1/2 


0.2057 


0.3085 


0.4114 


0.5142 


0.6171 


0.7199 


0.8228 


0.9256 


1.0285 


5-3/4 


0.2248 


0.3372 


0.4496 


0.562 


0.6744 


0.7868 


0.8992 


1.011 


1.124 


6 


0.2448 


0.3672 


0.4896 


0.612 


0.7344 


0.8568 


0.9792 


1.1016 


1.2240 


6-1/4 


0.2656 


0.3984 


0.5312 


0.6640 


0.7968 


0.9296 


1.062 


1.195 


1.328 


6-1/2 


0.2872 


0.4308 


0.5744 


0.7182 


0.8610 


1.0052 


1.1488 


1.2926 


1.4364 


6-3/4 


0.3098 


0.4647 


0.6196 


0.7745 


0.9294 


1.084 


1.239 


1.394 


1.549 


7 


0.3332 


0.4998 


0.6664 


0.833 


0.9996 


1 . 1662 


1.3328 


1.4994 


1.666 


7-3/4 


0.4084 


0.6126 


0.8168 


1.021 


1.225 


1.429 


1.633 


1.837 


2.042 


- 8 


0.4352 


0.6528 


0.8704 


1.088 


1.3056 


1.5232 


1.7408 


1.9584 


2.176 


9 


0.5508 


0.8262 


1.1010 


1.377 


1.6524 


1.9278 


2.2032 


2.4786 


2.754 


10 


0.68 


1.02 


1.36 


1.7 


2.04 


2.38 


2.72 


3.06 


3.4 


11 


0.8227 


1.2341 


1.6451 


2.057 


2.464 


2.879 


3.2911 


3.7258 


4.1139 


12 


0.9792 


1.468 


1.9584 


2.448 


2.9376 


3.4222 


3.9168 


4.4064 


4.896 


13 


1.149 


1.723 


2.297 


2.872 


3.445 


4.022 


4.596 


5.170 


5.745 


14 


1.332 


1.998 


2.665 


3.331 


3.997 


4.664 


5.33 


5.996 


6.663 


15 


1.529 


2.294 


3.059 


3.824 


4.589 


5.354 


6.119 


6.884 


7.649 


16 


1.74 


2.61 


3.48 


4.35 


5.22 


6.09 


6.96 


7.83 


8.703 


18 


2.202 


3.303 


4.404 


5.505 


6.606 


7.707 


8.808 


9.909 


11.01 


20 


2.720 


4.08 


5.440 


6.8 


8.16 


9.52 


10.88 


12.24 


13.6 



Digitized by vnOOQlC 



HYDRAULIC DATA AND TABLES 



m 



Tisva 7. — Capacitt of REapROCATiNO Pumps. — {Continued^ 



Diameter 

of 

cylinder, 

inches 


Length of stroke in inchea 


12 


14 


15 


16 


18 


20 


22 


24 


1-1/4 


0.0637 


0.0743 


0.0797 


0.0848 


0.0955 


0.1062 


0.1168 


0.1274 


1-3/8 


0.077 


0.089 


0.0963 


0.1027 


0.1156 


0.1280 


0.1408 


0.1541 


1-1/2 


0.0918 


0.1071 


0.1147 


0.1224 


0.1377 


0.1530 


0.1683 


0.1836 


1-3/4 


0.1249 


0.1457 


0.1562 


0.1666 


0.1874 


0.2082 


0.2290 


0.2499 


2 


0.1632 


0.1904 


0.204 


0.2176 


0.2448 


0.2720 


0.2992 


0.3264 


2-1/4 


0.2063 


0.241 


0.258 


0.2754 


0.3096 


0.344 


0.3784 


0.4128 


2-1/2 


0.255 


0.2975 


0.3187 


0.34 


0.3825 


0.4252 


0.4677 


0.61 


2-3/4 


0.3085 


0.3598 


0.3855 


0.4114 


0.4626 


0.5142 


0.5666 


0.617 


3 


0.3672 


0.4284 


0.459 


0.4896 


0.6608 


0.612 


0.6732 


0.7344 


3-1/4 


0.4312 


0.503 


0.6385 


0.5748 


0.6466 


0.7182 


0.79 


0.8624 


3-1/2 


0.4998 


0.5831 


0.6247 


0.6664 


0.7497 


0.833 


0.9163 


0.9996 


3-3/4 


0.5736 


0.6692 


0.687 


0.7648 


0.8605 


0.9661 


1.0517 


1.147 


4 


0.6528 


0.7616 


0.816 


0.8904 


0.9792 


1.088 


1.1968 


1.3056 


4-1/4 


0.7368 


0.8596 


0.921 


0.9824 


1.105 


1.228 


1.3508 


1.473 


4r-l/2 


0.8262 


0.9639 


1.0327 


1.1016 


1.2393 


1.377 


1.6147 


1.6524 


4r-3/4 


0.9204 


1.073 


1.15 


1.227 


1.380^ 


1.534 


1.6874 


1.84 


5 


1.02 


1.19 


1.275 ' 


1.36 


1.53 


1.7 


1.87 


2.04 


5-1/4 


1.124 


1.311 


1.405 


1.499 


1.686 


1.874 


2.0614 


2.248 


5-1/2 


1.2342 


1.4399 


1.5427 


1.6456 


1.8513 


2.067 


2.2627 


2.4684 


5-3/4 


1.348 


1.573 


1.686 


1.789 


2.022 


2.248 


2.4728 


2.696 


6 


1.4688 


1.7136 


1.8362 


1.9584 


2.2032 


2.448 


2.6928 


2.9376 


6-1/4 


1.593 


1.859 


1.992 


2.124 


2.39 


2.666 


2.9216 


3.186 


6-1/2 


1.7955 


2.0109 


2.1546 


2.2982 


2.6885 


2.8728 


3.16 


3.4473 


6-3/4 


1.858 


2.168 


2.323 


2.479 


2.788 


3.098 


3.4078 


3.716 


7 


1.9992 


2.3324 


2.499 


2.6656 


2.9988 


3.332 


3.6662 


3.9984 


7-3/4 


2.45 


2.858 


3.063 


3.266 


3.674 


4.084 


4.4924 


4.9 


8 


2.6112 


3.0464 


3.264 


3.4816 


3.9168 


4.352 


4.7872 


5.2224 


9 


3.3048 


3.8556 


4.131 


4.4064 


6.0572 


5.608 


6.0588 


6.6096 


10 


4.08 


4.76 


5.1 


5.44 


6.12 


6.8 


7.48 


8.16 


11 


4.9367 


5.7595 


6.1709 


6.5823 


7.4051 


8.2279 


9.0506 


9.8735 


12 


5.8752 


6.8544 


7.344 


7.833 


8.8128 


9.792 


10.7712 


11.7504 


13 


6.894 


8.042 


8.616 


9.192 


10.34 


11.49 


12.639 


13.78 


14 


7.994 


9.328 


9.993 


10.66 


11.99 


13.32 


14.652 


15.98 


15 


9.178 


10.70 


11.47 


12.23 


13.76 


16.29 


16.819 


18.36 


16 


10.44 


12.18 


13.05 


13.92 


15.66 


17.40 


19.14 


20.88 


18 


13.21 


15.41 


16.51 


17.61 


19.81 


22.02 


24.22 


26.42 


20 


16.32 


19.04 


20.4 


21.76 


24.48 


27.2 


29.92 


32.6 



Digitized by VnOOQlC 



298 



ELEMENTS OF HYDRAULICS 



Table 8. — Gibcumfbbbnceb and Abeas of Cibclbs 

DUmeters, 1/16 in. up to and including 120 in. Advancing, 1/16 to 1 ; 1/8 to 60; 1/4 to 80, 

and 1/2 to 120 



Diam- 


Circum- 


Area, 


Diame- 


Circum- 


Area, 


Diame- 


Circum- 


Area. 


eter, 


ference, 


square 


ter, 


ference, 


square 


ter. 


ference. 


square 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


1/16 


0.19635 


0.00307 


4r-l/2 


14.137 


15.904 


9-5/8 


30.237 


72.759 


1/8 


0.3927 


0.01227 


4-5/8 


14.529 


16.800 


9-3/4 


30.630 


74.662 


3/16 


0.6890 


0.02761 


4-3/4 


14.922 


17.720 


9-7/8 


31.023 


76.588 


1/4 


0.7854 


0.04909 


4-7/8 


15.315 


18.665 








5/16 


0.9817 


0.07670 








10 


31.416 


78.540 


3/8 


1.1781 


0.1104 


5 


15.708 


19.635 


10-1/8 


31.808 


80.615 


7/16 


1.3744 


0.1503 


5-1/ 8 


16.100 


20.629 


10-1/4 


32.201 


82.516 


1/2 


1.5708 


0.1963 


5-1/4 


16.493 


21.647 


10-3/8 


32.594 


84.540 


9/16 


1.7771 


0.2485 


5-3/8 


16.886 


22.690 


10-1/2 


32.986 


86.590 


5/8 


1.9635 


0.3068 


5-1/2 


17.278 


23.758 


10-5/8 


33.379 


88.664 


11/16 


2.1598 


0.3712 


5-5/8 


17.671 


24.850 


10-3/4 


33.772 


90.762 


3/4 


2.3562 


0.4417 


5-3/4 


18.064 


25.967 


10-7/8 


34.164 


92.885 


13/16 
7/8 


2.5525 
2.7489 


0.5185 
0.6013 


5-7/8 


18.457 


27.108 


11 


34.558 


95.033 


15/16 


2.9452 


0.6903 


6 


18.849 


28.274 


11-1/8 
11-1/4 


34.960 
35.343 


97.205 
99.402 


1 


3.1416 


0.7854 


6-1/8 


19.242 


29.464 


11-3/8 


35.736 


101.623 


1-1/8 
1-1/4 
1-3/8 
1-1/2 


3.5343 
3.9270 
4.3197 


0.9940 
1.2271 
1.4848 


6-1/4 
6-3/8 
6-1/2 


19.635 
20.027 
20.420 


30.679 
31.919 
33.183 


11-1/2 
11-5/8 
11-3/4 


36.128 
36.621 
36.913 


103.869 
106.139 
108.434 


4.7124 


1 . 7671 


6-5/8 
6-3/4 


20.813 
21.205 


34.471 
35.784 


11-7/8 


37.306 


110.753 


1-5/8 


5.1051 


2.0739 


6-7/8 


21.598 


37.122 


12 


37.699 


113.097 


1-3/4 
1-7/8 


5.4978 
5.8905 


2.4052 
2.7621 


7 


21.991 


38.484 


12-1/8 
12-1/4 


38.091 
38.484 


115.466 
117.869 


2 


6.2832 


3.1416 


7-1/8 


22.383 


39.871 


12-3/8 


38.877 


120.276 


2-1/8 


6.6759 


3.5465 


7-1/4 


22.776 


41.282 


12-1/2 


39.270 


122.718 


2-1/4 


7.0686 


3.9760 


7-3/8 


23.169 


42.718 


12-5/8 


39.662 


126.184 


2-3/8 


7.4613 


4.4302 


7-1/2 


23.562 


44.178 


12-3/4 


40.055 


127.676 


2-1/2 


7.8540 


4.9087 


7-5/8 


23.954 


45.663 


12-7/8 


40.448 


130.192 


2-5/8 
2-3/4 


8.2467 
8.6394 


5.4119 
5.9395 


7-3/4 

7-7/8 


24.347 
24.740 


47.173 
48.707 


13 
13-1/8 


40.840 
41.233 


132.732 
136.297 


2-7/8 


9.0321 


6.4918 








13-1/4 


41.626 


137.886 








8 


25.132 


50.265 


13-3/8 


42.018 


140.500 


3 


9.4248 


7.0686 


8-1/8 


25.515 


51.848 


13-1/2 


42.411 


143.139 


3-1/8 


9.8175 


7.6699 


8-1/4 


25.91$ 


53.456 


13-6/8 


42.804 


145.802 


3-1/4 


10.210 


8.2957 


8-3/8 


26.310 


56.088 


13-3/4 


43.197 


148.489 


3-3/8 


10.602 


8.9462 


8-1/2 


26.703 


66.745 


13-7/8 


43.689 


161.201 


a- 1/2 


10.995 


9.6211 


8-5/8 


27.096 


58.426 








3-5/8 


11.388 


10.320 


8-3/4 


27.489 


60.132 


14 


43.982 


163.938 


3-3/4 


11.781 


11.044 


8-7/8 


27.881 


61.862 


14-1/8 


44.376 


166.699 


3-7/8 


12.173 


11.793 








14-1/4 


44.767 


169.486 








9 


28.274 


63.617 


14-3/8 


46.160 


162.295 


4 


12.566 


12.566 


9-1/8 


28.667 


65.396 


14r-l/2 


45.663 


165.130 


4-1/8 


12.959 


13.364 


9-1/4 


29.059 


67.200 


14-6/8 


46.945 


167.989 


4r-l/4 


13.351 


14.186 


9-3/8 


29.452 


69.029| 


14-3/4 


46.338 


170.873 


4-3/8 


13.744 


15.033 


9-1/2 


29.845 


70.882 


14-7/8 


46.731 


173.782 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



299 



Table 8.— -Cibcumferbncbs and Abbas of Circles— 


{Continued) 


Diame> 


Ciroum- 


Area 


Diame- 


Circum- 


Area 


Diame- 


Circum- 


Area 


ter 


ferenoe 


square 


ter 


ference 


square 


ter 


ference 


square 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


15 


47.124 


176.715 


21 


65.973 


346.361 


27 


84.823 


672.556 


15-1/8 


47.616 


179.672 


21-1/8 


66.366 


360.497 


27-1/8 


86.215 


577.870 


15-1/4 


47.909 


182.654 


21-1/4 


66.769 


354.657 


27-1/4 


86.608 


583.208 


15-8/8 


48.302 


185.661 


21-3/8 


67.161 


358.841 


27-3/8 


86.001 


588.571 


15-1/2 


48.694 


188.692 


21-1/2 


67.544 


363.061 


27-1/2 


86.394 


593.968 


15-6/8 


49.087 


191.748 


21-6/8 


67.937 


367.284 


27-5/8 


86.786 


699.370 


1&-3/4 


49.480 


194.828 


21-3/4 


68.329 


371.643 


27-3/4 


87.179 


604.807 


15-7/8 


49.872 


197.933 


21-7/8 


68.722 


376.826 


27-7/8 


87.572 


610.268 


16 


60.265 


201.062 


22 


69.115 


380.133 


28 


87.964 


616.763 


16-1/8 


60.668 


204.216 


22-1/8 


69.507 


384.466 


28-1/8 


88.357 


621.263 


16-1/4 


61.061 


207.394 


22-1/4 


69.900 


388.822 


28-1/4 


88.750 


626.798 


16-3/8 


51.443 


210.697 


22-3/8 


70.293 


393.203 


28-3/8 


89.142 


632.357 


16-1/2 


61.836 


213.826 


22-1/2 


70.686 


397.608 


28-1/2 


89.535 


637.941 


16-6/8 


62.229 


217.077 


22-6/8 


71.078 


402.038 


28-5/8 


89.928 


643.594 


16-3/4 


62.621 


220.363 


22-3/4 


71.471 


406.493 


28-3/4 


90.321 


649.182 


16-7/8 


63.014 


223.664 


22-7/8 


71.864 


410.972 


28-7/8 


90.713 


654.837 


17 


63.407 


226.980 


23 


72.266 


415.476 


29 


91.106 


660.621 


17-1/8 


53.799 


230.330 


2^1/8 


72.649 


420.004 


29-1/8 


91.499 


666.277 


17-1/4 


64.192 


233.705 


2^1/4 


73.042 


424.557 


29-1/4 


91.891 


671.958 


17-3/8 


64.686 


237.104 


2^3/8 


73.434 


429.136 


2^3/8 


92.284 


677.714 


17-1/2 


64.978 


240.628 


23-1/2 


73.827 


433.731 


29-1/2 


92.677 


683.494 


17-6/8 


66.370 


243.977 


23-6/8 


74.220 


438.363 


29-5/8 


93.069 


689.298 


17-3/4 


66.763 


247.450 


23-3/4 


74.613 


443.014 


29-3/4 


93.462 


695.128 


17-7/8 


66.166 


260.947 


23-7/8 


75.005 


447.699 


29-7/8 


93.856 


700.981 


18 


66.648 


254.469 


24 


75.398 


452.390 


30 


94.248 


706.860 


18-1/8 


66.941 


258.016 


24-1/8 


76.791 


457.116 


30-1/8 


94.640 


712.762 


18-1/4 


57.334 


261.686 


24-1/4 


76.183 


461.864 


30-1/4 


95.033 


718.690 


18-3/8 


67.726 


266.182 


24-3/8 


76.576 


466.638 


30-3/8 


95.426 


724.641 


18-1/2 


58.119 


268.803 


24r-l/2 


76.969 


471.436 


30-1/2 


95.818 


730.618 


18-5/8 


68.612 


272.447 


24-5/8 


77.361 


476.259 


30-5/8 


96.211 


736.619 


18-3/4 


68.905 


276.117 


24-3/4 


77.754 


481.106 


30-3/4 


96.604 


742.644 


18-7/8 


69.297 


279.811 


24-7/8 


78.147 


485.978 


30-7/8 


96.996 


748.694 


19 


69.690 


283.629 


26 


78.640 


490.875 


31 


97.389 


764.769 


19-1/8 


60.083 


287.272 


26-1/8 


78.932 


495.796 


31-1/8 


97.782 


760.868 


19-1/4 


60.476 


291.039 


25-1/4 


79.325 


500.741 


31-1/4 


98.175 


766.992 


19-3/8 


60.868 


294.831 


26-3/8 


79.718 


505.711 


31-3/8 


98.567 


773.140 


19-1/2 


61.261 


298.648 


26-1/2 


80.110 


510.706 


31-1/2 


98.968 


779.313 


19-6/8 


61.663 


302.489 


26-6/8 


80.503 


515.725 


31-5/8 


99.353 


785.510 


19-3/4 


62.046 


306.366 


26-3/4 


80.896 


520.769 


31-3/4 


99.745 


791.732 


19r7/8 


62.439 


310.246 


25-7/8 


81.288 


525.837 


31-7/8 


100.138 


797.978 


20 


62.832 


314.160 


26 


81.681 


530.930 


32 


100.531 


804.249 


20-1/8 


63.224 


318.099 


26-1/8 


82.074 


636.047 


32-1/8 


100.924 


810.546 


20-1/4 


63.617 


322.063 


26-1/4 


82.467 


541.189 


32-1/4 


101.316 


816.866 


20-3/8 


64.010 


326.061 


26-3/8 


82.869 
83.262 


546.366 


32-3/8 


101.709 


823.209 


20-1/2 


64.402 


330.064 


26-1/2 


551.647 


32-1/2 


102.102 


829.578 


20-5/8 


64.795 


334.101 


26-6/8 


83.645 


656.762 


32-6/8 


102.494 


835.972 


20-3/4 


65.188 


338.163 


26-3/4 


84.037 


562.002 


32-3/4 


102.887 


842.390 


20-7/8 


66.680 


342.250 


26-7/8 


84.430 


567.267 


32-7/8 


103.280 


848.833 



Digitized by VnOOQlC 



300 



ELEMENTS OF' HYDRAULICS 



CiRCUMPBRBNCBS AND ArEAS OP 


Circles — (Cont 


inued) 




Diame- 


Circum- 


Area 


Diame- 


Circum- 


Area 


Diame- 


Circum- 


Area 


ter ^ 


ference 


square 


ter 


ference 


square 


ter 


ference 


square 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


33 


103.672 


855.30 


39 


122.622 


1194.69 


45 


141.372 


1590.43 


33^1/8 


104.055 


861.79 


39-1/8 


122.915 


1202.26 


46-1/8 


141.764 


1599.28 


33-1/4 


104.458 


868.30, 


39-1/4 


123.307 


1209.95 


46-1/4 


142.157 


1608.15 


33-3/8 


104.850 


874.84 


39-3/8 


123.700 


1217.67 


46-3/8 


142.660 


1617.04 


33-1/2 


106.243 


881.41; 


39-1/2 


124.093 


1225.42 


46-1/2 


142.942 


1625.97 


33-6/8 


105.636 


888.00 


39-5/8 


124.485 


1233.18 


46-6/8 


143.336 


1634.92 


33-3/4 


106.029 


894.61 


39-3/4 


124.878 


1240.98 


46-3/4 


143.728 


1643.89 


33-7/8 


106.421 


901.25 


39-7/8 


125.271 


1248.79 


46-7/8 


144.120 


1662.88 


34 


106.814 


907.92 


40 


126.664 


1266.64 


46 


144.613 


1661.90 


34-1/8 


107.207 


914.61 


40-1/8 


126.066 


1264.50 


46-1/8 


144.906 


1670.95 


34-1/4 


107.699 


921.32 


40-1/4 


126.449 


1272.39 


46-1/4 


145.299 


1680.01 


34-3/8 


107.992 


928.06 


40-3/8 


126.842 


1280.31 


46-3/8 


145.691 


1689.10 


34-1/2 


108.386 


934.82 


40-1/2 


127.234 


1288.25 


46-1/2 


146.084 


1698.23 


34-5/8 


108.777 


941.60 


40-6/8 


127.627 


1296.21 


46-6/8 


146.477 


1707.37 


34-3/4 


109.170 


948. 4i; 


40-3/4 


128.020 


1304.20 


46-3/4 


146.869 


1716.54 


34-7/8 


109.663 


966.25 


40-7/8 


128.412 


1312.21 


46-7/8 


147.262 


1725.73 


35 


109.966 


962.11 


41 


128.805 


1320.25 


47 


147.666 


1734.94 


35-1/8 


110.348 


968.99 


41-1/8 


129.198 


1328.32 


47-1/8 


148.047 


1744.18 


35-1/4 


110.741 


975.90 


41-1/4 


129.691 


1336.40 


47-1/4 


148.440 


1753.45 


35-3/8 


111.134 


982.84 


41-3/8 


129.983 


1344.61 


47-3/8 


148.833 


1762.73 


35-1/2 


111.526 


989.80 


41-1/2 


130.376 


1362.65 


47-1/2 


149.226 


1772.05 


35-5/8 


111.919 


996.78 


41-6/8 


130.769 


1360.81 


47-5/8 


149.618 


1781.39 


35-3/4 


112.312 


1003.78 


41-3/4 


131.161 


1369.00 


47-3/4 


150.011 


1790.76 


35-7/8 


112.704 


1010.82 


41-7/8 


131.554 


1377.21 


47-7/8 


150.401 


1800.14 


36 


113.097 


1017.88 


42 


131.947 


1386.44 


48 


150.796 


1809.66 


36-1/8 


113.490 


1024.95 


42-1/8 


132.339 


1393.70! 


48-1/8 


161.189 


1818.99 


36-1/4 


113.883 


1032.06! 


42-1/4 


132.732 


1401.98' 


48-1/4 


151.682 


1828.46 


. 36-3/8 


114.275 


1039.19' 


42-3/8 


133.125 


1410.29, 


48-3/8 


151.974 


1837.93 


36-1/2 


114.668 


1046.35' 


42-1/2 


133.518 


1418.62' 


48-1/2 


162.367 


1847.45 


36-5/8 


116.061 


1053.52 


42-6/8 


133.910 


1426.98 


48-6/8 


152.760 


1866.99 


36-3/4 


116.463 


1060.73 


42-3/4 


134.303 


1436.36 


48-3/4 


153.153 


1866.65 


36-7/8 


115.846 


1067.95 


42-7/8 


134.696 


1443.77 


48-7/8 


153.545 


1876.13 


37 


116.239 


1076.21 


43 


135.088 


1462.20 


49 


153.938 


1886.74 


37-1/8 


116.631 


1082.48 


43-1/8 


135.481 


1460.65 


49-1/8 


154.331 


1895.37 


37- 1/4 


117.024 


1089.79 


43-1/4 


135.874 


1469.13 


49-1/4 


154.723 


1905.03 


37-3/8 


117.417 


1097.11 


43-3/8 


136.266 


1477.63 


49-3/8 


155.116 


1914.70 


37-1/2 


117.810 


1104.46 


43-1/2 


136.669 


1486.17 


49-1/2 


155.509 


1924.42 


37-5/8 


118.202 


1111.84 


43-5/8 


137.062 


1494.72 


1 49-5/8 


166.901 


1934.16 


37-3/4 


118.696 


1119.24 


43-3/4 


137.445 


1603.30 


49-3/4 


166.294 


1943.91 


37-7/8 


118.988 


1126.66 


43-7/8 


137.837 


isii.oo 


49-7/8 


156.687 


1953.69 


38 


119.380 


1134.11 


44 


138.230 


1520.53 


50 


167.080 


1963.50 


38-1/8 


119.773 


1141. 69| 


44-1/8 


138.623 


1629.18 


50-1/4 


157.865 


1983.18 


38-1/4 


120.166 


1149.08 


44-1/4 


139.016 


1537.86 


50-1/2 


158.660 


2002.96 


38-3/8 


120.658 


1156.61 


44-3/8 


139.408 


1546.56 


60-3/4 


159.436 


2022.84 


38-1/2 


120.961 


1164.15 


1 44-1/2 


139.801 


1665.28 


51 


160.221 


2042.82 


38-6/8 


121.344 


1171.73 


44-5/8 


140.193 


1664.03 


51-1/4 


161.007 


2062.90 


38-3/4 


121.737 


1179.32 


44-3/4 


140.586 


1672.81 


i 51-1/2 


161.792 


2083.07 


38-7/8 


122.129 


1186.94 


44-7/8 


140.979 '1581.61 


1 51-3/4 


162.577 


2103.35 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



301 



Circumferences 


AND Areas op 


Circles — {Continued) 




Diame- 


Ciroum- 


Area 


Diame- 


Circum- 


Area 


Diame- 


Circum- 


Area 


ter 


ference 


square 


ter 


ference 


square 


ter 


ference 


square 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


inches 


52 


163.363 


2123.72 


63 


197.920 


3117.25 


74 


232.478 


4300.84 


52-1/4 


164.148 


2144.19 


63-1/4 


198.706 


3142.04 


74-1/4 


233.263 


4329.96 


52-1/2 


164.934 


2164.75 


6»-l/2 


199.491 


3166.92 


74-1/2 


234.049 


4359.16 


52-3/4 


165.719 


2185.42 


63-3/4 


200.277 


3191.91 


74r-3/4 


234.834 


4388.47 


53 


166.504 


2206.18 


64 


201.062 


3216.99 


76 


235.620 


4417.86 


53-1/4 


167.490 


2227.05 


64-1/4 


201.847 


3242.17 


76-1/4 


236.405 


4447.37 


6a-l/2 


168.075 


2248.01 


64-1/2 


202.633 


3267.46 


76-1/2 


237.190 


4476.97 


53-3/4 


168.861 


2269.06 


64-3/4 


203.418 


3292.83 


76-3/4 


237.976 


4606.67 


54 


169.646 


2290.22 


65 


204.204 


3318.31 


76 


238.761 


4636.46 


64-1/4 


170.431 


2311.48 


: 65-1/4 


204.989 


3343.88 


76-1/4 


239.547 


4666.36 


54-1/2 


171.217 


2332.83 


65-1/2 


205.774 


3369.56 


76-1/2 


240.332 


4606.35 


54-3/4 


172.002 


2354.28 


64-3/4 


206.560 13395.33 


76-3/4 


241.117 


4626.44 


55 


172.788 


2375.83 


66 


207.345 


3421 . 19 


77 


241.903 


4666.63 


55-1/4 


173.573 


2397.48 


66^1/4 


208.131 


3447.16 


77-1/4 


242.688 


4686.92 


55^1/2 


174.358 


2419.22 


66-1/2 


208.916 


3473.33 


77-1/2 


243.474 


4717.30 


55^3/4 


175.144 


2441.07 


66-3/4 


209.701 


3499.39 


77-3/4 


244.269 


4747.79 


56 


175.929 


2463.01' 


• 67 


210.487 


3525.66 


78 


246.044 


4778.36 


56-1/4 


176.715 


2485.05, 


67-1/4 


211.272 


3552.01 


78-1/4 


245.830 


4809.06 


56-1/2 


177.500 


2507.19 


67-1/2 


212.058 


3578.47 


78-1/2 


246.616 


4839.83 


56-3/4 


178.285 


2529.42 


67-3/4 


212.843 


3605.03 


78-3/4 


247.401 


4870.70 


57 


179.071 


2551.76 


68 


213.628 


3631.68 


79 


248.186 


4901.68 


57-1/4 


179.856 


2574.19 


68-1/4 


214.414 13658. 44| 


7^1/4 


248.971 


4932.76 


57-1/2 


180.642 


2596.72 


1 68-1/2 


215.199 


3686.29 


7^1/2 


249.767 


4963.92 


57-3/4 


181.427 


2619.35 


68-3/4 


215.985 


3712.24 


7^3/4 


260.642 


4996.19 


58 


182.212 


2642.08 


69 


216.770 


3739.28 


80 


261.328 


5026.56 


58-1/4 


182.998 


2664.91 


69-1/4 


217.565 


3766.43 


80-1/2 


262.898 


6089.58 


58-1/2 
58-3/4 


183.783 
184.569 


2687.83 
2710.85 


69-1/2 
69-3/4 

1 


218.341 
219.126 


3793.67 
3821.02 


'81 
81-1/2 


264.469 
266.040 


5163.00 
5216.82 


59 


185.354 


2733.97 


70 


219.912 


3848.45 


82 

82-1/2 


267.611 
269.182 


5281.02 
6346.62 


59-1/4 
59-1/2 


186.139 
186.925 


2757.19 
2780.51 


70-1/4 
70-1/2 


220.697 
221.482 


3875.99 
3903.63 


5^3/4 


187.710 


2803.92 


70-3/4 


222.268 


3931.36 


83 


260.752 


6410.61 


60 


188.496 


2827.43 


71 


223.053 


3959.19 


83-1/2 


262.323 


5476.00 


60-1/4 


189.281 


2851.05 


71-1/4 


223.839 


3987.13 


84 


263.894 


6641.77 


60-1/2 


190.066 


2874.76 


71-1/2 


224.624 


4015.16 


84-1/2 


266.466 


6607.96 


60-3/4 


190.852 


2898.56 


71-3/4 


225.409 


4043.28 


86 


267.036 


6674.61 


61 


191.637 


2922.47 


72 


226.195 


4071.50 


86-1/2 


268.606 


5741.47 


61-1/4 


192.423 


2946.47 


72-1/4 


226.980 


4099.83 


86 
86-1/2 


270.177 
271.748 


6808.80 
6876.66 


61-1/2 
61-3/4 


193.208 
193.993 


2970.57 
2994.77 


72-1/2 
72-3/4 


227.766 
228.551 


4128.25 
4156.77 


62 
62-1/4 


194.779 
195.564 


3019.07 
3043.47 


73 

73-1/4 


229.336 
230.122 


4185.39 
4214.11 


87 
87-1/2 


273.319 
274.890 


6944.68 
6013.21 


62-1/2 


196.350 


3067.96 


73-1/2 


230.907 


4242.92 


88 


276.460 


6082.12 


62-3/4 


197.135 


3092.56 


73-3/4 


231.693 


4271.83 


88-1/2 


278.031 


6161.44 



Digitized by VnOOQlC 



302 



ELEMENTS OF HYDRAULICS 



OiRCUHFEBBNCES AND Arbab OF CiBCiJis — (Continued) 



Diame- 
ter 
inches 


Circum- 
ference 
inches 


Area 
square 
inches 


Diame- 
ter 
inches 


Circum- 
ference 
inches 


Area 
square 
inches 


Diame- 
ter 
inches 


Circum- 
ference 
inches 


Area 
square 
inches 


89 
89-1/2 

90 
90-1/2 

91 
91-1/2 

92 
92-1/2 

93 
93-1/2 

94 
94-1/2 

95 
95-1/2 

96 
96-1/2 

97 
97-1/2 

98 
98-1/2 

99 
99-1/2 


279.602 
281.173 

282.744 
284.314 

285.885 
287.466 

289.027 
290.598 

292.168 
293.739 

295.310 
296.881 

298.452 
300.022 

301.593 
302.164 

304.734 
306.306 

307.876 
309.446 

311.018 
312.588 


6221 . 14 
6291.25 

6361.73 
6432.62 

6503.88 
6573.56 

6647.61 
6720.07 

6792.91 
6866.16 

6939.78 
7013.81 

7088.22 
7163.04 

7238.23 
7313.84 

7389.81 
7474.20 

7542.96 
7620.12 

7697.69 
7775.64 


100 
100-1/2 

101 
101-1/2 

102 
102-1/2 

103 
103-1/2 

104 
104-1/2 

105 
105-1/2 

106 
106-1/2 

107 
107-1/2 

108 
108-1/2 

109 
109-1/2 

110 
110-1/2 


314.159 
315.730 

317.301 
318.872 

320.442 
322.014 

323.584 
325.154 

326.726 
328.296 

329.867 
331.438 

333.009 
334.580 

336.150 
337.722 

339.292 
340.862 

342.434 
344.004 

345.575 
347.146 


7863.98 
7938.72 

8011.86 
8091.36 

8171.28 
8251.60 

8332.29 
8413.40 

8494.87 
8676.76 

8659.01 
8741.68 

8824.73 
8908.20 

8992.02 
9076.24 

9160.88 
9245.92 

9331.32 
9417.12 

9603.32 
9589.92 


111 
111-1/2 

112 
112-1/2 

113 
113-1/2 

114 

114r-l/2 

115 
115-1/2 

116 
116-1/2 

117 
117-1/2 

118 
118-1/2 

119 
119-1/2 

120^ 


348.717 
350.288 

351.868 
363.430 

366.000 
366.670 

368.142 
359.712 

361.283 
362.854 

364.425 
365.996 

367.666 
369.138 

370.708 
372,278 

373.849 
375.420 

376.991 


9766.89 
9674.28 

9852.03 
9940.20 

10028.76 
10117.68 

10207.03 
10296.76 

10386.89 
10477.40 

10668.32 
10669.64 

10751.32 
10843.40 

10936.88 
11028.76 

11122.02 
11215.68 

11309.73 















Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



303 



Tablb 9.' — ^Epfltjx Cobfficibnts fob Circular Obificb 

Values of efflux coeffioient JC in Eq. (32). Par. 55. Q - 2/ZKby/2giHf^* - h*^*), for circular. 

▼ertioal orifices, with sharp edges, full contraction and free discharge in air. 

For heads over 100 ft., use JT >■ 0.592. 



Head 


Diameter of orifice in feet 


on cen- 
ter of 






























orifice 


0.02 


0.03 


0.04 


0.05 


0.07 


0.10 


0.12 


0.15 


0.20 


0.40 


0.60 


0.80 


1.0 


in feet 




























0.3 
0.4 
0.6 
0.6 








0.637 
0.631 
0.627 
0.624 


0.628 
0.624 
0.621 
0.618 


0.621 
0.618 
0.616 
0.613 


0.613 
0.612 
0.610 
0.609 


0.608 
0.606 
0.606 
0.606 
















0.637 
0.633 
0.630 












0.655 


0.643 
0.640 


0.600 
0.601 


0.696 
0.696 


0.592 
0.593 






0.690 




0.7 


0.651 


0.637 


0.628 


0.622 


0.616 


0.611 


0.607 


0.604 


0.601 


0.697 


0.594 


0.691 


0.690 


0.8 


0.648 


0.634 


0.626 


0.620 


0.615 


0.610 


0.606 


0.603 


0.601 


0.697 


0.594 


0.592 


0.591 


0.9 


0.646 


0.632 


0.624 


0.618 


0.613 


0.609 


0.605 


0.603 


0.601 


0.598 


0.595 


0.593 


0.591 


1.0 


0.644 


0.631 


0.623 


0.617 


0.612 


0.608 


0.605 


0.603 


0.600 


0.598 


0.695 


0.693 


0.591 


1.2 


0.641 


0.628 


0.620 


0.615 


0.610 


0.606 


0.604 


0.602 


0.600 


0.598 


0.596 


0.694 


0.592 


1.4 


0.638 


0.625 


0.618 


0.613 


0.609 


0.605 


0.603 


0.601 


0.600 


0.599 


0.696 


0.694 


0.693 


1.6 


0.636 


0.624 


0.617 


0.612 


0.608 


0.605 


0.602 


0.601 


0.600 


0.699 


0.697 


0.696 


0.594 


1.8 


0.634 


0.622 


0.615 


0.611 


0.607 


0.604 


0.602 


0.601 


0.509 


0.699 


0.697 


0.696 


0.595 


2.0 


0.632 


0.621 


0.614 


0.610 


0.607 


0.604 


0.601 


0.600 


0.509 


0.699 


0.597 


0.596 


0.595 


2.5 


0.620 


0.619 


0.612 


0.608 


0.605 


0.603 


0.601 


0.600 


0.599 


0.599 


0.598 


0.697 


0.596 


3.0 


0.627 


0.617 


0.611 


0.606 


0.604 


0.603 


0.601 


0.600 


0.599 


0.599 


0.698 


0.597 


0.697 


3.5 


0.625 


0.616 


0.610 


0.606 


0.604 


0.602 


0.601 


0.600 


0.599 


0.599 


0.698 


0.697 


0.696 


4.0 


0.623 


0.614 


0.609 


0.605 


0.603 


0.602 


0.600 


0.599 


0.599 


0.698 


0.597 


0.697 


0.596 


5.0 


0.621 


0.613 


0.608 


0.605 


0.603 


0.601 


0.599 


0.599 


0.698 


0.698 


0.697 


0.696 


0.696 


6.0 


0.618 


0.611 


0.607 


0.604 


0.602 


0.600 


0.599 


0.599 


0.598 


0.598 


0.597 


0.696 


0.596 


7.0 


0.616 


0.609 


0.606 


0.603 


0.601 


0.600 


0.699 


0.599 


0.598 


0.958 


0.597 


0.696 


0.696 


8.0 


0.614 


0.608 


0.606 


0.603 


0.601 


0.600 


0.599 


0.598 


0.698 


0.697 


0.696 


0.596 


0.696 


9.0 


0.613 


0.607 


0.604 


0.602 


0.600 


0.699 


0.699 


0.698 


0.697 


0.697 


0.596 


0.696 


0.596 


10.0 


0.611 


0.606 


0.603 


0.601 


0.599 


0.598 


0.598 


0.697 


0.597 


0.697 


0.596 


0.596 


0.696 


20.0 


0.601 


0.600 


0.599 


0.698 


0.597 


0.596 


0.596 


0.696 


0.696 


0.596 


0.596 


0.595|0.694 


50.0 


0.596 


0.696 


0.595 


0.595 


0.594 


0.694 


0.594 


0.594 


0.594 


0.594 


0.594 


0.693 0.593 


100.0 


0.593 


0.593 


0.592 


0.592 


0.592 


0.592 


[0.592 


0.592 


0.692 


0.592 


0.502 


0.592 0.692 



1 From Hamilton Smith's "Hydraulics." 



Digitized by LnOOQlC 



304 



ELEMENTS OF HYDRAULICS 



Table 10. ^ — ^Effujx Cobfficibnts pob Square Orifice 

Values of eflBlux coefficient it in Eq. (32), Par. 56. Q - 2/^Khy/2o{H^'^ - h^'\ for square, 

vertical cnrifices, with sharp edges, full contraction, and free discharge in air. 

For heads over 100 ft. use IC - 0.598 



Head 
on cen- 
ter of 


Side of square in feet 




























orifice 
in feet 


0.02 


0.03 


0.04 


0.60. 


0.07 


0.10 


0.12 


0.15 


0.20 


0.40 


0.60 


0.80 


1.0 


0.3 
0.4 
0.5 
0.6 








0.642 
0.637 
0.633 
0.630 


0.632)0 624 


0.617 
0.616 
0.614 
0.613 


0.612 
0.611 
0.610 
0.610 

















0.643 
0.639 
0.636 


0.628 
0.626 
0.623 


0.621 
0.619 
0.617 












0.660 


0.648 
0.645 


0.605 
0.605 


0.601 


0.597 






0.6010.598 


0.596 




0.7 


0.656 


0.642 


0.633 


0.628 


0.621 


0.616 


0.612 


0.609 


0.605 


0.602 0.599 0.598 


0.596 


0.8 


0.652 


0.639 


0.631 


0.626 


0.620 


0.615 


0.611 


0.608 


0.605 


0.602 0.6000.698 


0.697 


0.9 


0.650 


0.637 


0.629 


0.623 


0.619 


0.614 


0.610 


0.608 


0.605 


0.603 0.60110.599 


0.598 


1.0 


0.648 


0.636 


0.628 


0.622 


0.618 


0.613 


0.610 


0.608 


0.606 


0.603 


0.60l|0.600 


0.599 


1.2 


0.644 


0.623 


0.625 


0.620|0.616 


0.611 


0.609 


0.607 


0.605 


0.604 


0.602 0.601:0.600 


1.4 


0.642 


0.630 


0.623 


0.618 


0.614 


0.610 


0.608 


0.606 


0.605 


0.604 


0.602 0.6010.601 


1.6 


0.640 


0.628 


0.621 


0.Q17 


0.613 


0.609 


0.607 


0.606 


0.605 


0.605 


0.603 0.602 0.601 


1.8 


0.638 


0.627 


0.620 


0.616 


0.612 


0.609 


0.607 


0.606 


0.605 


0.605 


0.603 0.602 0.602 


2.0 


0.637 


0.626 


0.619 


0.615 


0.612 


0.608 


0.606 


0.606 


0.605 


0.606 


0.604 


0.602 0.602 


2.5 


0.634 


0.624 


0.617 


0.613 


0.610 


0.607 


0.606 


0.606 


0.605 


0.605 


0.604 


0.603 0.602 


3.0* 


0.632 


0.622 


0.616 


0.612 


0.609 


0.607 


0.606 


0.606 


0.605 


0.605 


0.604 


0.603 0.603 


3.5 


0.630 


0.621 


0.615 


0.611 


0.609 


0.607 


0.606 


0.606 


0.605 


0.605 


0.604 


0.603 0.602 


4.0 


0.628 


0.619 


0.614 


0.610 


0.608 


0.606 


0.606 


0.605 


0.605 


0.605 


0.603 


0.603 0.602 


5.0 


0.626 


0.617 


0.613 


0.610 


0.607 


0.606 


0.605 


0.605 


0.604 


0.604 


0.603 


0.602 0.602 


6.0 


0.623 


0.616|0.612 


0.609 


0.607 


0.605 


0.605 


0.605 


0.604 


0.604 


0.603 


0.602 0.602 


7.0 


0.621 


0.615 


0.611 


0.608 


0.607 


0.605 


0.606 


0.604 


0.604 


0.604 


0.603 


0.602 


0.602 


8.0 


0.619 


0.613 


0.610 


0.608 


0.606 


0.605 


0.604 


0.604 


0.604 


0.603 


0.603 


0.602 


0.602 


9.0 


0.618 


0.612 


0.609 


0.607 


0.606 


0.604 


0.604 


0.604 


0.603 


0.603 


0.602 


0.602 


0.601 


10.0 


0.616 


0.611 


0.608 


0.606 


0.605 


0.604 


0.604 


0.603 


0.603 


0.603 


0.602 


0.602 


0.601 


20.0 


0.606 


0.605 


0.604 


0.603 


0.602 


0.602 


0.602 


0.602 


0.602 


0.601 


0.601 


0.601 


0.600 


50.0 


0.602 


0.601 


0.601 


0.601 


0.601 


0.600 


0.600 


0.600 


0.600 


0.600 


0.699 


0.599 


0.699 


100.0 


0.599 


0.598 


0.598 


0.598 


0.598 


0.598 


0.698 0.59810.598 


0.598 


0.598 


0.598 


0.698 



« From Hamilton Smith's "Hydraulics." 



Digitized by 



Google 



HYDRAULIC DATA AND TABLES 



305 



Table 11. — Fibb Streams 

From Tablet Published by John R. Freeman 







3/4-in 


. Smooth Nossle 










Pressure in pounds per sq. in. re- 


Pressure at 

nossle in 

pounds per 

sq.in. 


Discharge in 

gallons per 

min. 


Height of 

effective 

fire stream 


• 

Horiaontal 
distance 
of stream 


quired at hydrant or pump to main- 
tain pressure at noaxle through vari- 
ous lengths of 2-1/2-in. smooth, 
rubber-lined hose. 


50 


100 


200 


300 400 


500 600 


800 


1000 










ft. 


ft. 


ft. 


ft. 


ft. 


ft. ft. 


ft. 


ft. 


35 


97 


65 


41 


37 


38 


40 


42 44 


46 


48 


53 


57 


40 


104 


60 


44 


42 


43 


46 


48, 60 


53 


55 


60 


65 


46 


110 


64 


47 


47 


48 


51 


54 


67 


69 


62 


68 


73 


50 


116 


67 


50 


52 


54 


57 


60 


63 


66 


69 


75 


81 


55 


122 


70 


52 


58 


69 


63 


66 


69 


73 


76 


83 


89 


60 


127 


72 


54 


63 


65 


68 


72 


76 


79 


83 


90 


97 


65 


132 


74 


56 


68 


70 


74 


78 


82 


86 


90 


98 


106 


70 


137 


76 


58 


73 


75 


80 


84 


88 


92 


97 


105 


114 


75 


142 


78 


60 


79 


81 


85 


90 


94 


99 


104 


113 


122 


80 


147 


79 


62 


84 


86 


91 


96 


101 


106 


111 


120 


130 


• 85 


151 


80 


64 


89 


92 


97 


102 


107 


112 


117 


128 


138 


90 


156 


81 


65 


94 


97 


102 


108 


113 


119 


124 


135 


146 


95 


160 


82 


66 


99 


102 


108 


114 


120 


125 


131 


143 


154 


100 


164 


83 


68 


105 


108 


114 


120 


126 


132 


138 


150 


163 






7/8-i] 


a. Smooth N( 


)axle 


35 


133 


66 


46 


38 


40 


44 


48 


52 


56 


60 


68 


76 


40 


142 


62 


49 


43 


46 


50 


55 


59 


64 


68 


78 


87 


45 


150 


67 


52 


49 


51 


57 


62 


67 


72 


77 


87 


97 


60 


159 


71 


55 


54 


57 


63 


69 


74 


80 


86 


97 


108 


55 


166 


74 


58 


60 


63 


69 


76 


82 


88 


94 


107 


119 


60 


174 


77 


61 


65 


69 


75 


82 


89 


96 


103 


116 


130 


65 


181 


79 


64 


71 


74 


82 


89 


96 


104 


111 


126 


141 


70 


188 


81 


66 


76 


80 


88 


96 


104 


112 


120 


136 


152 


75 


194 


83 


68 


82 


86 


94 


103 


111 


120 


128 


145 


162 


80 


201 


85 


70 


87 


91 


101 


110 


119 


128 


137 


155 


173 


85 


207 


87 


72 


92 


97 


107 


116 


126 


136 


145 


165 


184 


90 


2ia 


88 


74 


98 


103 


113 


123 


134 


144 


154 


174 


195 


95 


219 


89 


75 


103 


109 


119 


130 


141 


152 


163 


184 


206 


100 


224 


90 


76 


109 


114 


126 


137 


148 


160 


171 


194 


216 






1-in 


Smooth Noj 


isle 


35 


174 


68 


51 


40 


44 


51 


57 


64 


71 


78 


92 


105 


40 


186 


64 


65 


46 


50 


58 


66 


73 


81 


89 


105 


120 


45 


198 


69 


58 


62 


56 


65 


74 


83 


91 


100 


118 


135 


50 


208 


73 


61 


57 


62 


72 


82 


92 


102 


111 


131 


151 


55 


218 


76 


64 


63 


69 


79 


90 


101 


112 


122 


144 


166 


60 


228 


79 


67 


67 


75 


87 


98 


110 


122 


134 


157 


181 


65 


237 


82 


70 


75 


81 


94 


107 


119 


132 


145 


170 


196 


70 


246 


85 


72 


80 


87 


101 


115 


128 


142 


156 


183 


211 


75 


255 


87 


74 


86 


94 


110 


123 


138 


152 


167 


196 


226 


80 


263 


89 


76 


92 


100 


115 


131 


147 


162 


178 


209 


241 


85 


274 


91 


78 


98 


106 


123 


139 


156 


173 


189 


222 




90 


279 


92 


80 


103 


112 


130 


147 


165 


183 


200 


236 




95 


287 


94 


' 82 


109 


118 


137 


156 


174 


193 


211 


249 




100 


295 


96 


83 


115 


125 


144 


164 


183 


203 


223 







20 



Digitized by VnOOQlC 



306 



ELEMENTS OF HYDRAULICS 



Fire Streams — (Continued) 



l'l/8-inoh Smooth Nosale 



Pressure at 

nosale in 

pounds per 

sq. in. 



■6& 
40 
45 

50 
£5 

m 

65 
TO 
75 
SO 
&5 
90 
95 
100 



m 

95 
IQO 



Discharge in 

gallons per 

min. 



Height of 

effective 

fire stream 



222 
238 
252 
206 
27D 
29t 

ao3 

314 
325 
336 
340 
350 
S66 
376 



65 
70 

75 
SO 
83 
80 

90 
02 
94 
03 

90 



Horisontal 
distanog 
of stream 



Pressure in pounds per sq. in. required 
at hydrant or pump to maintain pres- 
sure at nosale through various lengths 
of 2-1/2-in. smooth, rubber-lined hoee 



200 
ft. 



300 

ft 



54 



63 



72 
75 
77 
79 
81 
&3 
S5 
S7 
SO 



49 eo 



400 

ft. 



102:118 



500 
ft. 



600 

ft 



800 
ft. 



aooo 
ft. 



105J127 
120 145 
1^5 163 



94 
107 

120 

134 150'l81 



95,ll2jl30| 147 105 200 



lOa, 122; 141 160 



1S012I8 
195 '236 
200,254 
224 ... 
239 

1061 1 191 146 173' 200|237 254 

112,126 155:133 2121241 



112|]32|t53,174 
120,143*165 187 
93jl05il29 153 177.201 

09'll2[l38 163ll88 2l4 

, — ,1 



lis 133^163,194 
124!l40 172:204 



224 254 

230 . . . 



Ii49 
171 

192 
213 
235 
256 



l-l/4-ineh Smooth Noiile 



35 


277 


40 


2^6 


45 


314 


SO 


331 


55 


347 


80 


36^ 


65 


377 


70 


392 


75 


405 


80 


419 


85 


432 



444 
468 



67 
72 
77 
81 
85 
88 
91 
93 
y.T 
97 
99 
100 
101 



59 
63 
67 
70 
73 
76 
79 
81 
83 
85 
SS 
90 
92 
93 



48 
55 

62| 
6S 
75 
82 
89 
96 
103 
110 



57 

65 

73 

81 

89 

07 

105 

113 

121. 

129 



116 137 



74 
84 
05 
106 
116 
127 
137 
148 
158 

im\ 

179| 
190 
210 
211 1 



01 109 
104 
117 
130 
143 
156 



124 
140 
155 
170 198 



186 
169 201 



182 
'195 
208 
221 
234 
247 
261 



217 
232 
248 



]20|l^ll78 

1441164 203 

229 

204 

225 

245 



162' 184 
180^ 



216 
234 
252 



254 



212 
243 



35 
40 
45 
50 
55 
60 
65 
70 
75 
80 
85 
90 
95 
100 



340 
363 
3g5 
406 
426 
445 
463 
480 
497 
514 
529 
545 
560 
574 



l-S/S-inch Bmooth lioute 



62 
69 
74 
79 

m 

87 

90 

92 

95 

97 

09 

100 

101 

103 



62 
66 
70 
73 
76 
79 
82 
84 
8i\ 
88 
00 
92 
94 
96 



54 
62 
70 
78 


67 
77 
87 
06 
106 
116 


04 
107 
120 
134 
147 


120 146 
137 166 
154 187 
171 208 
I88l22ft 


172 
196 
221 
245 
270 


198 
226 
454 


250 




86 








93 


160 205 


250 








101 


\n 


174 222 










109 


135 
145 
154 
164 


187 339 












117 


201 2-^ 












lt>4 


214 
5>fl7 














13S* 














140 


17:1940 














148'l83 254 














156 193 


... 















Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



307 



FiBB SmBAMa— (Continue*!) 

From ezperimenU made by Tirgil R, Heming 



6/16-lN. NOMM 


Preasure 
base of 
nossle 
lb. per 
sq. in. 


Discharge 

gallons 

per 


Loss of head in 
100 ft. of hose 


Vertical 
height of 

jet for 
good fire 


Horisontal distance of | 


Jet for 

good fire 

stream 

feet 


Extreme 

drops at 

level 


Rubber 
lined 


UnUned 
linen 


minute 


lb. per 
sq. in. 


lb. per 
sq. in. 


stream 
feet 


of nossle 
feet 


20 


12 


0.7 


1.3 


28 


15 


53 


30 


15 


1.1 


1 9 


32 


18 


63 


40 


17 


1.5 


2.6 


34 


21 


71 


50 


19 


1.8 


3.2 


35 


23 


78 


60 


21 


2.2 


3 9 


36 


26 


84 


70 


23 


2.6 


4.5 


37 


28 


90 


80 


24 


2.9 


5.2 38 


29 


96 


90 


26 


3.3 


5.9 39 


30 


102 


100 


28 


3.7 


6.5 40 


31 


107 


7/16-In. Nossls 


20 


25 


2.8 


6.1 


23 


10 


46 


3a 


30 


4.2 


7.7 


27 


13 


54 


40 


35 


5.6 


10.2 


30 


16 


63 


60 


39 


7.0 


12.8 


32 


18 


70 


60 


43 


8.5 


15.3 


33 


20 


77 


70 


47 


9.8 


17.8 


34 


21 


84 


80 


50 


11.1 


20.3 


35 


23 


94 


90 


53 


12.7 


22.9 


36 


24 


99 


100 


56 


14.1 


25.5 


37 


25 


lOtf 


1/2-lN. NOMIJl 


20 


33 


6.2 


9.5 


34 


15 


63 


30 


40 


7.7 


14.4 


87 


20 


79 


40 


46 


10.2 


18.8 


38 


25 


91 


50 


52 


12.8 


23.8 


39 


30 


102 


60 


57 


15.4 


28.5 


40 


33 


111 


70 


61 


18.0 


32.7 


41 


37 . 


120 


80 


65 


20.5 


38.4 


42 


40 


127 


90 


69 


23.0 


42.0 


43 


43 


134 


100 


73 


25.6 


47.0 


44 


46 


140 



Digitized by VnOOQlC 



308 



ELEMENTS OF HYDRAULICS 



Table 12." — CosFnciiiNTB of Pipe Fwction 

Value of the frietion coeffioient /, in the formula 
Computed from the exponential formulas of Thrupp, Tutton and Unwin 



Material 


Diameter 
in inches 


Velocity of flow in feet 


per second ] 


2 


4 


6 


8 


10 




1 


0.032 


0.026 


0.024 


0.022 


0.021 


Lead pipe 


2 


0.030 


0.026 


0.023 


0.021 


0.020 




3 


0.029 


0.024 


0.022 


0.020 


0.019 




4 


0.028 


0.023 


0.021 


0.020 


0.019 


Wood pipe 


6 
12 
18 
24 


0.034 
0.027 
0.024 
0.022 


0.033 
0.027 
0.024 
0.022 


0.032 
0.026 
0.023 
0.021 


0.032 
0.026 
0.023 
0.021 












36 


0.020 


0.019 


0.019 


0.019 






48 


0.018 


0.018 


0.017 


0.017 








6 


0.026 


0.023 


0.022 


0.021 


0.020 




9 


0.025 


0.022 


0.021 


0.020 


0.019 




12 


0.024 


0.021 


0.020 


0.019 


0.019 


Asphalted pipe 


18 


0.023 


0.020 


0.019 


0.018 


0.018 




24 


0.022 


0.020 


0.018 


0.017 


0.017 




36 


0.021 


0.019 


0.017 


0.017 


0.016 




48 


0.020 


0.018 


0.017 


0.016 


0.015 




3 


0.024 


0.021 


0.019 


0.018 


0.017 




6 


0.022 


0.019 


0.017 


0.016 


0.016 




12 


0.019 


0.017 


0.015 


0.014 


0.014 


Bare wrought iron 


24 


0.017 


0.015 


0.014 


0.013 


0.012 


pipe 


36 


0.016 
0.015 


0.014 


. 0.013 


0.012 


0.011 




48 


0.013 


0.012 


0.011 


0.011 




60 


0.015 


0.013 


0.012 


0.011 


0.010 




12 


0.025 


0.022 


0.021 


0.020 


0.019 




24 


0.020 


0.018 


0.017 


0.016 


0.016 


Riveted wrought iron 


36 


0.017 


0.016 


0.015 


0.014 


0.014 


or steel pipe 


48 


0.016 


0.014 


0.014 


0.013 


0.013 




60 


0.015 


0.013 


0.013 


0.012 


0.012 




72 


0.014 


0.013 


0.012 


0.011 


0.011 




3 
6 


0.028 
0.024 


0.026 
0.022 


0.025 
0.022 


0.025 
0.021 






New cast-iron pipe 


9 
12 


0.021 
0.020 


0.020 

o.ai9 


0.020 
0.018 


0.019 
0.018 






18 


0.018 


0.017 


0.017 


0.016 






24 


0.017 


0.016 


0.016 


0.015 






36 


0.015 


0.015 


0.014 


0.014 








3 
6 


0.059 
0.050 


0.058 
0.050 


0.058 
0.050 


0.058 
0.049 






Old cast iron pipe 


9 
12 


0.046 
0.043 


0.045 
0.042 


0.045 
0.042 


0.044 
0.042 






18 


0.039 


0.039 


0.038 


0.038 






24 


0.037 


0.036 


0.036 


0.036 






36 


0.033 


0.033 


0.033 


0.032 





I Compiled from data in Gibson's **Hydraulics." 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



309 



Table 13. — Friction Head in Pipes According to Exponential For- 
mula OF Williams and Hazen 

Friction head in feet for each 100 ft. of straight, clean, cast-iron pipe. For old pipes the 
tabular values of the friction head should be doubled. Computed from Williams and . 
Hasen's formula, v »• C7r»«»w»« (o.OOl) -oo«: v = velocity in feet per sec., « ■» slope; 
9 » hydraulic radius in feet, C ■> 100. 





Inside diameter of pipe | 


1/2 


in. 


3/4 in. 


lin. 


1-1/2 in. 


2 in. 1 


1 
li 


.a« 
li 




Friction head 
in feet per 
100 ft. 


1 

.9 . 

ll 






Friction head 
in feet per 
100 ft. 




Friction head 
in feet per 
100 ft. 


1 

2 
3 

4 
5 


1.05 
2.10 
3.16 
4.21 
5.26 


2.1 

7.4 

15.8 

27.0 

41.0 


















1.20 
1.80 
2.41 
3.01 


1.9 

4.1 

7.0 

10.5 














1.12 
1.49 
1.86 


1.26 
2.14 
3.26 










q.63 
0.79 


0.26 
0.40 














6 

8 

10 

12 

15 


6.31 

8.42 

10.52 


67.0 

98.0 

147.0 


3.61 
4.81 
6.02 
7.22 
9.02 


14.7 
25.0 
38.0 
53.0 
80.0 


2.23 
2.98 
3.72 
4.46 
5.57 


4.55 

7.8 

11.7 

16.4 

25.0 


0.94 

1.26 
1.57 
1.89 
2.36 


0.56 
0.95 
1.43 
2.01 
3.05 


0.61 

0.82 

1.02- 

1.23 

1.53 


0.20 
0.33 
0.60 
0.70 
1.07 










20 
25 
30 
35 
40 

60 
60 
70 
80 
90 






12:03 


136.0 


7.44 

9.30 

11.15 

13.02 

14.88 


42.0 

64.0 

89.0 

119.0 

152.0 


3.15 
3.93 
4.72 
5.51 
6.30 


5.2 

7.8 

11.0 

14.7 

18.8 


2.04 
2.55 
3.06 
3.57 
4.08 


1.82 

2.73 

3.84 

-5.1 

6.6 






1 i! 






1 






















,' 








7.87 

9.44 

11.02 

12.59 

14.17 


28.4 
39.6 
53.0 
68.0 
84.0 


5.11 
6.13 
7.15 
8.17 
9.19 


9.9 
13.9 
18.4 
23.7 
29.4 


























































1 


100 
120 
140 
160 
180 














15.74 
18.89 
22.04 


102.0 
143.0 
190.0 


10.21 
12.25 
14.30 
16.34 
18.38 


35.8 
50.0 
67.0 
86.0 
107.0 


































































1 


200 
250 





1 




1 


j 






20.42 
25.53 


129.0 
196.0 




II 





1 








Digitized by VnOOQlC 



310 



ELEMENTS OF HYDRAULICS 







Fbiction Head in Pipe»— (C(m«nue(i) 






i 

b 

a a 


Inside diameter of pipe | 


2-1/2 in. 1 


8 in. 1 


4 


in. 


5 in. 


6 in. 1 


u 


it 

|5 


1 

> 


|.9§ 


'J 






11 






10 
15 
20 
25 
30 


0.65 
0.98 
1.31 
1.63 
1.96 


0.17 
0.37 
0.61 
0.92 
1.29 


0.45 
0.68 
0.91 
1.13 
1.36 


0.07 
0.15 
0.25 
0.38 
0.54 


























0.51 
0.64 
0.77 


0.06 
0.09 
0.13 


















0.49 


0.04 










35 
40 
50 
60 
70 


2.29 
2.61 
3.27 
3.92 

4.58 


1.72 
2.20 
3.32 
4.65 
6.2 


1.59 

1.82 

2.27' 

2.72 

3.18 


0.71 
0.91 
1.38 
1.92 
2.57 


0.89 
1.02 
1.28 
1.53 
1.79 


0.17 
0.22 
0.34 
0.47 
0.63 


0.57 
0.65 
0.82 
0.98 
1.14 


0.06 
0.08 
0.11 
0.16 
0.21 






0.45 
0.57 
0.68 
0.79 


0.03 
0.05 
0.07 
0.09 


80 

90 

100 

120 

140 


5.23 
5.88 
6.54 
7.84 
9.15 


7.9 

9.8 

12.0 

16.8 

22.3 


3.63 
4.09 
4.54 
5.45 
6.35 


3.28 

4.08 

4.96 

7.0 

9.2 


2.04 
2.30 
2.55 
3.06 
3.57 


0.81 

1.0 

1.22 

1.71 

2.28 


1.31 
1.47 
1.63 
1.96 
2.29 


0.27 
0.34 
0.41 
0.58 
0.76 


0.91 
1.02 
1.13 
1.36 
1.58 


0.11 
0.14 
0.17 
0.24 
0.31 


160 
180 
200 
250 
300 


10.46 
11.76 
13.07 
16.34 
19.61 


29.0 
35.7 
43.1 
65.5 
92.0 


7.26 

8.17 

9.08 

11.35 

13.62 


11.8 
14.8 
1,7.8 
27.1 
38^0 


4.08 
4.60 
5.11 
6.38 
7.66 


2.91 

3.61 

4.4 

6.7 

9.3 


2.61 
2.94 
3.27 
4.08 
4.90 


0.98 
1.22 
1.48 
2.24 
3.14 


1.82 
2.05 
2.27 
2.84 
3.40 


0.41 
0.53 
0.61 
0.93 
1.29 


350 
400 
450 
500 
550 


22.87 
26.14 
29.41 


122.0 
156.0 
196.0 


15.89 
18.16 
20.43 
22.70 
24.96 


50.5 
65.0 
81.0 
98.0 
117.0 


8.93 
10.21 
11.49 
12:77 
14.04 


12.4 
16.0 
19.8 
24.0 
28.7 


5.72 
6.54 
7.35 
8.17 
8.99 


4.19 

5.4 

6.7 

8.1 

9.6 


3.98 
4.54 
5.11 
5.68 
6.24 


1.73 
2.21 
2.75 
3.35 
3.98 










600 
700 
800 
900 
1000 






27.23 


137.0 


15.32 
17.87 
20.42 
22.98 


33.7 
44.9 
57.0 
71.0 


9.80 
11.44 
13.07 
14.71 
16.34 


11.3 
15.1 
19.4 
24.0 
29.2 


6.81 

7.95 

9.08 

10.22 

11.35 


4.68 
6.24 
7.98 
9.93 
12.04 










































1100 
1200 
1300 
1400 
1500 














17.97 
19.61 


34.9 
40.9 


12.49 
13.62 
14.76 
15.89 
17.03 


14.4 
16.9 
19.6 
22.5 
25.6 






































1 














1 ' * * ' 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



311 







Friction Head in Fifeu— (Continued) i 






H 


p 


Inside diameter of pipe { 


Sin. 1 


10 in. 


12 in. 


16 in. 


20 in'. 1 


1 

.a 

h 


.3 . 


1 

.3 




1 

.a 
I' 




1 

.9 
1^ 




.a 

n 


Ii 

1! 
II 


200 
250 
300 
350 
400 


0.309 
0.386 
0.464 
0.641 
0.619 


0.89 
1.11 
1.33 
1.56 
1.77 


0.08 
0.12 
0.14 
0.22 
0.28 


0.57 
0.71 
0.85 
0.9SL 
1.13 


0.04 
0.05 
0.06 
0.07 
0.09 


0.39 
0.49 
0.59 
0.69 
0.79 


0.01 
0.02 
0.02 
0.03 
0.04 


0.22 
0.28 
0.33 
0.39 
0.44 


0.003 
0.004 
0.006 
0.008 
0.010 


















0.28 


0.003 


450 
500 
550 
600 
700 


0.696 
0.774 
0.851 
0.928 
1.083 


1.99 
2.22 
2.44 
2.66 
3.10 


0.34 
0.42 
0.50 
0.59 
0.78 


1.28 
1.42 
1.56 
1.70 
1.99 


0.12 
0.14 
0.17 
0.20 
0.26 


0.89 
0.99 
1.09 
1.18 
1.38 


0.05 
0.06 
0.07 
0.08 
0.11 


0.50 
0.55 
0.61 
0.66 
0.77 


0.012 

0.016 

0.017 

0.02 

0.03 


0.31 
0.35 
0.39 
0.43 
0.50 


0.004 
0.005 
0.006 
0.007 
0.009 


800 

900 

1,000 

1.100 

1,200 


1.238 
1.392 
1.547 
1.702 
1.867 


3.55 
3.99 
4.43 
4.88 
5.37 


0.99 
1.24 
1.51 
1.80 
2.11 


2.27 
2.55 
2.84 
3.12 
3.40 


0.34 
0.42 
0.51 
0.61 
0.71 


1.58 
1.77 
1.97 
2.17 
2.36 


0.14 
0.17 
0.21 
0.25 
0.29 


0.89 
1.00 
1.11 
1.22 

1.33 

1 


0.03 
0.04 
0.05 
0.06 
0.07 


0.57 
0.64 
0.71 
0.78 
0.85 


0.012 
0.014 
0.017 
0.020 
0.024 


1.500 
2.000 
2.500 
3.000 
3,600 


2.321 
3.094 
3.868 
4.642 
6.41 


6.65 
8.86 
11.08 
13.30 


3.18 
5.4 
8.4 
11.6 


4.26 
5.67 
7.10 
8.51 
9.93 


1.08 
1.84 
2.78 
3.86 
5.19 


2.96 
3.94 
4.92 
5.91 
6.89 


0.44 
0.76 
1.15 
1.60 
2.13 


1.66 
2.22 
2.77 
3.32 
3.88 


0.11 
0.19 
0.28 
0.40 
0.53 


1.06 
1.42 
1.77 
2.13 
2.48 


0.04 
0.06 
0.09 
0.13 
0.18 






4.000 
. 6.000 
6.000 
7,000 
8,000 


6.19 
7.74 
9.28 
10.83 
12.38 




11.35 
14.19 
17.03 


6.65 
10.05 
14.09 


7.88 

9.85 

11.82 

13.79 

15.76 


2.70 

4.10. 

5.8 

7.7 

9.9 


4.43 
5.54 
6.65 
7.76 
8.86 


0.68 
1.02 
1.43 
1.90 
2.42 


2.84 
3.55 
4.26 
4.96 
6.67 


0.23 
0.34 
0.48 
0.64 
0.82 




1 




















9,000 
10,000 
11.000 
12.000 
15,000 


13.92 
15.47 
17.02 
18.67 
23.21 








17.73 
19.70 


12.2 
15.0 


9.97 
11. Q8 
12.19 
13.30 
16.62 


3.02 

3.68 

4.40 

6.2 

7.8 


6.38 
7.09 
7.80 
8.51 
10.64 


1.02 
1.24 
1.48 
1.74 
2.62 


,. 


I 




1 


























1 










16.000 
17.000 
18.000 
19.000 
20.000 


24.76 
26.30 
27.85 
29.40 
30.94 


1 














11.35 
12.06 
12.77 
13.47 
14.18 


2.96 
3.31 
3.68 
4.07 

4.48 


' * : 1 














t 










........... 










1 ' 






:--• •-;. 


1 1 



Digitized by VnOOQlC 



312 



ELEMENTS OF HYDRAULICS 







Friction Head in Pipes — (Continued) 






ij 

is 


i 
g 

H 
1^ 


Inside diameter of pipe | 


24 in. 1 


30 in. 1 


36 in. 1 


42 


in. 


48 


in. 


1 

.9 

h 


.9^ 


1! 


.9;g 

31 


1 

.9 . 
|i 


ss 

•a ^ 

.n 


1 

.9 


Is 

|i 

gl 


1 

II 


1 ^ 


1.0 
1.6 
2.0 
2.5 
3.0 


1.547 
2.321 
3.094 
3.868 
4.642 


0.49 
0.74 
0.98 
1.23 
1.48 


0.007 
0.015 
0.026 
0.039 
0.065 


0.32 
0.47 
0.63 
0.79 
0.95 


0.002 
0.005 
0.009 
0.013 
0.018 


0.22 
0.33 
0.44 
0.55 
0.66 


0.002 
0.003 
0.004 
0.005 
0.008 






1 
















1 












0.48 


0.004 










3.6 
4.0 
4.5 
5.0 
6.0 


5.41 
6.19 
6.96 
7.74 
9.28 


1.72 
1.97 
2.22 
2.46 
2.96 


0.07 
0.09 
0.12 
0.14 
0.20 


1.10 
1.26 
1.42 
1.58 
1.89 


0.025 
0.032 
0.039 
0.048 
0.067 


0.77 
0.88 
0.99 
1.09 
1.31 


0.010 
0.013 

0.020 
0.027 


0.56 
0.64 
0.72 
0.80 
0.96 


0.005 
0.006 
0.007 
0.009 
0.013 






0.49 
0.55 
0.62 
0.74 


0.003 
0.004 
0.005 
0.007 


7.0 
8.0 
9.0 
10.0 
12.0 


10.83 
12.38 
13.92 
15.47 
18.57 


3.45 
3.94 
4.43 
4.92 
5.91 


0.26 
0.34 
0.42 
0.51 
0.71 


2.21 
2.52 
2.84 
3.15 
3.78 


0.09 
0.11 
0.14 
0.17 
0.24 


1.53 
1.75 
1.97 
2.19 
2.63 


0.036 
0.047 
0.058 
0.071 
0.099 


1.13 
1.29 
1.45 
1.61 
1.93 

2.25 
2.57 
2.89 
3.22 
3.53 


0.017 
0.022 
0.027 
0.033 
0.047 


0.86 
0.98 
1.10 
1.23 
1.48 


0.009 
0.012 
0.014 
0.017 
0.024 


14.0 
16.0 
18.0 
20.0 
22.0 


21.66 
24.76 
27.85 
30.94 
34.04 


6.89 
7.88 
8.86 
9.85 
10.83 


0.95 
1.22 
1.52 
1.83 
2.19 


4.41 
5.04 
5.67 
6.30 
6.93 


0.32 
0.41 
0.51 
0.62 
0.74 


3.06 
3.50 
3.94 
4.38 
4.82 


0.13 
0.17 
0.21 
0.25 
0.30 


0.06 
0.08 
0.10 
0.12 
0.14 


1.72 
1.97 
2.22 
2.46 
2.71 


0.032 
0.042 
0.052 
0.063 
0.075 


24.0 
26.0 
28.0 
30.0 
32.0 


37.13 
40.23 
43.32 
46.42 
49.51 


11.82 
12.80 
13.79 
14.77 


2.59 
2.99 
3.42 
3.90 


7.56 
8.20 
8.83 
9.46 
10.09 


0.87 
1.01 
1.16 
1.32 
1.48 


5.25 
5.69 
6.13 
6.57 
7.00 


0.36 
0.41 
0.48 
0.54 
0.61 


3.86 
4.18 
4.50 
4.82 
5.16 


0.17 
0.20 
0.22 
0.26 
0.29 


2.96 
3.20 
3.45 
3.69 
3.94 


0.09 
0.10 
0.12 
0.13 
0.15 






34.0 
36.0 
38.0 
40.0 
50.0 


52.6 
55.7 
58.8 
61.9 
77.4 


1 


10.72 
11.35 
11.98 


1.66 
1.84 
2.04 


7.44 
7.88 
8.32 
8.76 
10.95 


0.68 
0.76 
0.84 
0.92 
1.39 


5.47 
5.79 
6.11 
6.45 
8.04 


0.32 
0.36 
0.40 
0.44 
0.66 


4.19 
4.43 
4.68 
4.92 
6.16 


0.17 
0.19 
0.21 
0.23 
0.34 


' 1 


1 1 






I 


1 


1 


60.0 
70.0 
80.0 
90.0 
100.0 


92.8 
108.3 
123.8 
139.2 
154.7 






13.13! 1.96 


9.65 
11.26 
12.86 


0.92 
1.22 
1.57 


7.39 
8.62 
9.86 
11.08 


0.48 
0.64 
0.82 
1.02 
1.24 








1 


1 




' 1 




1 


1 1 




' 





112.31 



Digitized by vnOOQlC 



HYDRAULIC DATA AND TABLES 



313 



Table 14. — Bazin's Values of Chezy's Coefficient 

Values of the coefficient C in Chesy's formmla t - CVri according to Basin's formula 
(Par. 142): 

^87 



0.552 + 









V" 


r 






Hydraulic 

radius 

r, in 

feet 


Coefficient of roughness, m | 


Planed tim- 
ber or 
smooth 
cement 


Unplaned 
timber, well 
laid brick, 
or concrete 


rubble mas- 
onry, or 
poor brick- 
work 


Earth in 

good 
condition 


Earth in 
ordinary 
condition 


Earth in 

bad 
condition 


m -0.06 


m - 0.16 


m - 0.46 


m - 0.85 


m - 1.30 


m - 1.75 


0.1 


117 


82 


43 


27 


19 


14 


0.2 


127 


96 


55 


35 


25 


19 


0.3 


131 


103 


63 


41 


30 


23 


0.4 


135 


108 


68 


46 


33 


26 


0.5 


136 


112 


71 


50 


36 


29 


0.6 


138 


115 


76 


53 


39 


31 


0.7 


139 


117 


79 


55 


41 


33 


0.8 


141 


119 


82 


58 


43 


35 


0.9 


141 


121 


84 


«r-* 


45 


36 


1.0 


142 


122 


86 


62 


47 


38 


1.25 


143 


125 


90 


66 


51 


41 


1.50 


145 


127 


94 


70 


54 


44 


1.75 


145 


129 


97 


73 


57 


47 


2.00 


146 


131 


99 


75 


59 


49 


2.5 


147 


133 


104 


80 


63 


53 


3.0 


148 


135 


106 


83 


67 


57 


4.0 


150 


a|s\ 


111 


89 


72 


61 


5.0 


160 


140 


115 


93 


77 


65 


6.0 


151 


141 


118 


97 


80 


69 


7.0 


152 


142 


120 


100 


83 


72 


8.0 


152 


143 


122 


102 


86 


74 


9.0 


152 


144 


123 


104 


88 


77 


10.0 


152 


145 


^ 125 


106 


90 


79 


12.0 


153 


145 


127 


109 


94 


82 


15.0 


153 


147 


130 


113 


98 


86 


20.0 


154 


148 


133 


117 


103 


92 


30.0 


155 


150 


137 


123 


110 


100 


40.0 ' 


155 


151 


139 


127 


115 


105 


50.0 


155 


151 


141 


129 


118 


109 



Digitized by VnOOQlC 



314 



ELEMENTS OF HYDRAULICS 



TABIiB 15. — ^KijTTBB'S VikLTJES OF ChEZT'S CoEFFICIBNT 

Value* of the eoefflcient C in Cheiy's formula « — Cs/ri ao«<»rdin( -to Kutten formula 
(Eq. (94) Par. 140): 

41.65 + -?^5o??L + JL«i 



0.00281 \ 









A -r 


V 


.oo 


T 


« 


J 


V 


r 














Slope. 

9 


Coefficient of 
rouBhnesfl, n 


Hydraulic radius r, in feet • | 


0.1 


0.2 


0.4 


0.6 


0.8| 1 


1.5 


2 


3 1 4 1 6 1 8 1 10 


15 


20 




0.009 


65 


87 


111 


127 


138|148 


166 


179 


197209 


226 


238 


246 


262 


271 


^ 


0.010 


57 


75 


97 


112 


122 131 


148 


160 


177 188 


206 


216 


225 


240 


249 


S 


0.011 


50 


67 


87 


100 


109!ll8 


133 


144 


160 172 


188 


199 


207 


222 


231 


0.012 


44 


59 


78 


90 


99|l06 


121 


131 


147 158 


174 


184 


192 


206 


216 


0.013 


40 


53 


70 


81 


90 


97 


111 


121 


135 146 


161 


171 


179 


193 


202 


ih 






















/-,» 












0.017 


28 


38 


51 


60 


66 


72 


83 


91 


103 


113 


126 


135 


142 


155 


164 


5 fl S 


0.020 


23 


31 


42 


49 


55 


60 


69 


77 


88 


96 


108 


117 


124 


136 


144 


6^6 


0.026 ' 


17 


24 


32 


38 


43 


47 


55 


61 


70 


78 


88 


96 


102 


11* 


121 


III 


0.030 


14 


19 


26 


31 


35 


38 


45 


50 


59 


65 


74 


82 


87 


98 


106 




0.035 


12 


16 


22 


26 


30 


32 


38 


43 


50 


56 


64 


71 


76 


86 


94 




0.009 


78 


100 


124 


139 


150 


158 


173 


184 


198 


207 


220 


228 


234 


244 


250 


± 


0.010 ' 


67 


87 


109 


122 


133 


140 


154 


164 


178 


187 


199 


206 


212 


220;228| 


1 


0.011 


59 


77 


97 


109 


119 


126 


139 


148 


161 


170 


182 


189 


195 


205 


211 


Pi 


0.012 


52 


68 


88 


98 


107 


114 


126 


135 


148 


156 


168 


176 


181 


189 


196 


0.013 


47 


62 


79 


90 


98 


104 


116 


124 


136 


146 


156 


163 


169 


179 


184 


0.017 


33 


44 


57 


65 


71 


77 


87 


94 


104 


111 


122 


129 


134 


142 


149 


8.sS 


0.020 


26 


35 


46 


53 


59 


64 


72 


79 


88 


95 


105 


111 


116 


126 


131 


6 r^ O 


0.026 


20 


26 


35 


41 


46 


49 


57 


62 


71 


77 


85 


91 


96 


104 


110 


1 H R 


0.030 


16 


21 


28 


33 


37 


40 


47 


51 


59 


64 


72 


78 


82 


90 


96 




0.035 


13 


18 


24 


28 


31 


34 


40 


44 


50 


56 


63 


68 


72 


79 


85 


V 


0.009 


90 


112 


136 


149 


158 


166 


178 


187 


198 


206 


215 


221 


226 


233 




237 


1 . 


0.010 


78 


98 


119 


131 140 


147 159 


168 


178 


186 


195 


201 


205 


212216 


0.011 


68 


86 


106 


118 126 


132 144 


151 


162 


169 


178 


184 


188 


196 200 


^i 


0.012 


60 


76 


95 


105 


114 


120 130 


138 


149 


156 


164 


170 


174 


181 


185 


0001 
in lO.OG 
628 ft. 


0.013 


54 


69 


86 


96 


103 


109 


120 


127 


137 


143 


152 


158 


162 


169 


173 


0.017 


37 


48 


62 


70 


76 


81 


89 


96 


104 


111 


119 


124 


128 


136 


139 


o ^ 6 


0.020 


30 


39 


50 


57 


63 


67 


75 


81 


89 


94 


102 


107 


111 


118 


122 


H R R 


0.025 


22 


29 


38 


44 


48 


52 


59 


64 


71 


76 


84 


88 


92 


98 


102 


«0 


0.030 


17 


23 


31 


35 


39 


42 


48 


53 


59 


64 


71 


76 


78 


86 


89 




0.035 


14 


19 


25 


30 


33 


35 


41 


45 


51 


65 


61 


66 


69 


76 


79 



Digitized by VnOOQlC 



HYDRAULIC DATA AND TABLES 



315 



Kutteb'b Valdb op Chbzy's 


Coefficient — {Continued) 




Slope. 

9 


Coefficient of 
roughnesa, n 


Hydraulic radius r in feet \ 


0.1 


0.2 


0.3|0.4 


0.6 


0.8 


1.0 


1.6 


2 


3 


4 


6 


10 


15 


20 




0.000 


99 


121 


133 


143 


166 


164 


170 


181 


188 


200 


205 


213 


222 


228 


231 


1 


0.010 


86 


105 


116 


126 


138 


145 


151 


162 


170 


179 


186 


193 


201 


207 


210 


0.011 


74 


93 


103 


112 


122 


131 


136 


146 


164 


163 


168 


176 


185 


190 


194 


).0002 
Lin 5000 
1.056 ft. per 


0.012 


66 


83 


92 


100 


111 


118 


123 


133 


140 


149 


155 


162 


170 


176 


180 


0.013 


69 


74 


83 


91 


100 


107 


113 


122 


129 


137 


143 


150 


158 


164 


168 


0.017 


41 


52 


69 


66 


73 


79 


83 


91 


97 


106 


111 


117 


126 


131 


134 


0.020 


32 


42 


48 


63 


60 


66 


69 


77 


82 


89 


94 


100 


108 


113 


117 


1 1 R 

«0 


0.026 


24 


31 


36 


40 


46 


60 


54 


60 


64 


72 


76 


82 


89 


96 


98 


0.030 


18 


25 


29 


32 


37 


41 


44 


49 


64 


69 


63 


69 


76 


82 


86 




0.036 


16 


21 


24 


27 


31 


34 


37 


42 


45 


51 


66 


60 


67 


72 


76 




0.009 


104 


126 


138 


148 


167 


166 


172 


183 


190 


199 


^ 


211 


219 


224 


227 


1 


0.010 


89 


110 


120 


129 


140 


148 


164 


164 


170 


179 


184 


191 


199 


203 


207 


0.011 


78 


97 


107 


116 


126 


133 


138 


148 


164 


162 


168 


175 


183 


187 


190 


1 

O »H N 

1 1 H 


0.012 


69 


87 


96 


104 


113 


121 


126 


135 


141 


149 


164 


161 


168 


172 


176 


0.013 


^ 


78 


87 


94 


103 


110 


116 


124 


130 


138 


z 


149 


157 


162 


164 


0.017 


43 


64 


62 


68 


76 


81 


86 


93 


98 


106 


110 


116 


123 


128 


131 


0.020 


34 


44 


60 


55 


62 


67 


70 


78 


83 


89 


94 


99 


107 


110 


115 


0.025 


26 


32 


37 


42 


47 


61 


66 


61 


66 


71 


76 


81 


88 


92 


96 


«0 


0.030 


19 


25 


30 


33 


38 


42 


45 


60 


64 


69 


63 


69 


76 


80 


83 




0.035 


16 


21 


24 


27 


31 


36 


37 


42 


45 
191 


51 
199 


65 
204 


60 
211 


66 
218 


70 
222 


73 
225 




0.000 


IIO 


129 


141 


150 


161 


169 


175 


184 


A 


0.010 


94 


113 


124 


1-31 


142 


160 


156 


166 


171 


179 


184 


190 


197 


202 


206 


- 0.001 

- 1 in 1000 

- 5.28 ft. per mil( 


0.011 


83 


99 


109 


117 


127 


134 


139 


149 


156 


163 


168 


174 


181 


186 


188 


0.012 


73 


89 


98 


105 


115 


122 


127 


136 


142 


149 


164 


160 


167 


171 


176 


0.013 


66 


81 


89 


96 


104 


111 


116 


124 


130 


138 


142 


149 


155 


160 


163 


0.017 


45 


57 


63 


69 


76 


82 


86 


93 


98 


105 


110 


116 


122 


127 


129 


0.020 


36 


45 


61 


66 


63 


68 


71 


78 


83 


89 


93 


99 


105 


110 


113 


0.025 


27 


34 


39 


43 


48 


62 


56 


62 


66 


71 


75 


81 


87 


91 


94 


w 


0.03d 


21 


27 


30 


34 


39 


42 


45 


50 


64 


69 


63 


68 


74 


78 


81 




0.035 


17 


22 


25 


28 


32 


36 


38 


43 


46 


61 


64 


69 


65 


68 


72 




0.009 


110 


130 


143 


161 


162 


170 


175 


185 


191 


199 


204 


210 


217 


222 


225 




0.010 


95 


114 


126 


133 


143 


161 


166 


165 


171 


179 


184 


190 


196 200 


204 


1 


0.011 


83 


100 


111 


119 


129 


135 


141 


149 


155 


162 


167 


173 


180 


184 


187 


0.012 


74 


90 


100 


107 


116 


123 


128 


136 


142 


149 


154 


160 


166 


170 


173 


y 100 
8 ft. pe 


0.013 


66 


81 


90 


98 


106 


112 


117 


125 


130 


138 


142 




148 


154 


169 


161 


0.017 


46 


57 


64 


70 


77 


82 


87 


94 


99 


105 


109 


115 


121 


126 


128 


III 


0.020 


36 


46 


62 


57 


64 


68 


72 


79 


83 


89 


93 


99 


105 


108 


112 


0.025 


27 


34 


39 


44 


49 


53 


66 


62 


66 


71 


76 


81 


86 


90 


93 


■> « 


0.030 


21 


27 


31 


35 


39 


43 


46 


61 


56 


69 


63 


68 


74 


77 


80 




0.035 


17 


22 


26 


29 


33 


35 


38 


43 


46 


61 


65 


Iki 


66 


68 


71 



Digitized by VnOOQlC 



316 



ELEMENTS OF HYDRAULICS 



& 



n 



H 
H 
H 



I) 



e4e4eQeo'4t'4t^ioio««« 



CQ <o o CO <o o» ei >o t« ^ '^t <o o» ei '^t e« 



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coaoio^t«<«<«aooo^Oio 

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vH^oe^e^co^^toot^QOoOfHco^aot-ab^^oor* 

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coo 



^<^t<.^XiO>ot<»^t<»coooc4a>ooc4oo«^ooa»<«e4ioe«-<ia>oo 
ddd'-)THC4co^dt^d>-<^ddeodddd^'ddc^ddcoi-<c 






o p r^ rlr' rl.'-' "-i '-'P<c <c^e^e^e^e^e^popoc o' 



^lOcoaSoTHNco^topt^SoiOJc^iSooow 



3P40) 



C0^>O<Ot*( 

coeocoeo'oocococo^^^tQiOtQiOcocbcb 



i^o>«o»c4a)ocoa»a>cop^io<« 
6oo»^»^c<co^>o«Doooe<'*'ddNdd"*ooNi^codd^'oOf 



<oKe^co>o«OooooxoOOOh>a 

COt^OiO'HNCO'^JnOCOt.OOC 



cot^0i-«co«0eocot^o0»«0»' 



iio<<«<coc4oa»to>ocooxco-^i-iaft'-i 
• 0'Hcsco^'^»o«ot^aooojO'H.H'^ 

I e< 01 cq' 01 N (N esi esi (N c^' e^' n co co' co' co' 



lit 



CO 00 



"^-^-^ 



82 



00400)>HCOCOCOtOrHO^COO 



oooo<-if-ic4co^>ocoooa)i-ico>oo 



:Jt^04t^C4C( 

dO>-<THC0^ 



H»-<e<e<eoco-i 



oe¥u5ooo« 

OO^N-^K 



O OO O op O ^ .H »H .H .-L- 

0> » ^ lO ^ O « O -^ 05 CO T}* h 

o»-icoioooNcoc^obiOT*<T*<if 



• •* CO CO -^ 00 CO »H O tH ^ o» CO ^ t- lO 

d d d d d "H .H « « d "^ d d b»* d d N -^ d 00 d CO d d esi lo 00 N >o d d 



e<e^e^eio<c<e^g<wcococococo 



ooN^oooeoTf»ONN 

o«oc^»o^eocococot»060 

e^^xcooobocoo^r^ooco 



OO^NCO 



COiOOOOOCO-^-^iOQ 
OOt^OXOcOt^CO-^iO'- 



it-cO'^coN'^eoo 



poo-Hcie^-^iocoaococt 



ocoio^^Oi-iao^ococ 
*oooboco»^»-«ioi-ii*.co< 

.H ^ >H W C< CO CO -^ II 



MCO 

in 



22 

IP 



«0)iou3a»<ot^^oe^bo<oa>tc>oa)coh>^ 
O»"^"^00«OO6CO»"^f T*<oo»*c 

.-lCO»Ot^OCOt^i-*CO»-lCONG 



<ia6tOCOOOOh>COiOU 



dou3'^cot»«ao««o-<ie9«^ 
dfHcocDb>c<4fH^p^e<9r«coaoio 



O O O O ^ ^ ^ N N CO CO ■* •* •« CO t^ t« 00 Oi O tH 01 CO •* lO b 



9 






3t^C0Ob*C0Ot^C0Ot^C0C 
PCDCOOCOCOQCOCOOCOCOC 
OcOXOi-icOiOCOXOi-icOu 



DCOQCOCOC 
9«O*HC0u 



(bjCOOt-COC 
ICOCOOCO COs 
tCOXOfHCOU 



oo>ooioo>eoiooioo 



1 « N N C« N N CO CO CO CO CO CO ■* -* •* •* •* •* lO lO CO CO t* t- 00 00 » o» O 






oo4'^<o«Qe>4'<ficoxoe>9-<icooooe9'^coooQeq-<fcoooQcoc4«^pcoe4oO'4tQ 
,^^^,HfHcleic<ie4o«cococococo<«<«'^<«<«SSaoiOioioS«t»b-ooSa>oo«Hei 



Digitized by V^aOOQ iC 



HYDRAULIC DATA AND TABLES 



317 





















Mt»050e«io 



to CO t^ 0)0 '-I e^ 



•-•Cl^t«.0>0t-i 



•^ lO ©t- 00 0> O •H rt N 



co^ie«Dcot^oooooioo<-«^c4 






diocoo6>OiHb.ioio 
■1 1-1 >-i c< CO CO Ty le <o 



co^ioiocot^b<.xooa»ooi-iiMC4 



o«HM-i*«(oa» 



t^c4tob»a»oe>4oo 

eOGOcooicDio-^'^o 

*H rH N N CO -^ ub CO t* 



eo-^-^iococot-t^ooooaoiOOi-^^N 



CO^COC4^COOOOOOOiOC4aO>OC4C4C4 
rl ^ N N CO '^ U3 S h- 00 S 



iHOO' 

eo'eo*' 



POOr-l 
|J>OiOO 



csi«^cocsid«oe>i ^ 

t*c)t*c4t*rH«oou3a»coxe4 

C0b>t«XX0)a6OOOiH*-iC4 



CDCOt 



|OJO»ojoO'^(<«.HNe«e«i-tM^ 
icocox^ 



SSI 

c«co\ 



0i-uOOi0«06b«.«0«0t*06C0t* 
•H^NC^COCO-^USOt^OOOrH 

fHi-< 

1* »o o lo oj ^ «r^r« -- 

3 '^ «0 'H »0 05 TlJ 00 N CO O -^ « 0« U3 G 
:5COCOt»t»b«.0000oJo»ddc>i-^THi- 



ioe«e 
d^*e 



I6a6>ot»b>^ococot»o>oc^ 
^o-^xcoacocof-iddcsi^'h^Nb 



N CO CO •'il •** lo «o CO CO CO t-b. t- 00 00 o> d d d d 



;-^!!3S^ooo«OT-icoe«io 
0^X04 CO a)cot^o<9h.^<4i 



00 c«' 



HOOt- 

>C0O^ 

«eoioV 



d^'rt*,-^ci 



5 r-l 0* -^ CO »0 ^ CO 0> ^ N ^ ,H 

ococOrHcoe^o^co^^ioiotsIo'^ 
.H,HC«e«coco-*io«oi^ooS»Hc 



<o«co 

«i? «s ol lO ^ -^ 

N'^ooeco 
ddd»-Icsi« 



2SS2l!*222395*ON»ocoiooooocO'»i«T 

O CO OJ t« 00 CO O O "* « CO « t- O 00 5 S li 5 

^>ocoooococoa»c<idd^diodt^coda 



coo»>otoa)coh>*-ioc4t^coa»to>oa)Ot^«-i 

0)^^Q0C0$C000^^00h*OXOCDb»MNtt^h««HlA(-> 

^ eo Id i>. o CO t- ^ CO »H CO N o lo CO o 00 b. 3 § S 5 S i2 § 
doooiHiHi-ie^c^coco-^^iodt^t^ooddiM 



Oi-ic^co^io 

r-l >H i-l ,H i-< iH 



ocoeoocoeoOcocoScocoScocooSeoOocoS^mS 
35coo65^wiocooo5^eoi5<oooo^«SS8SScoS 
d d d ^* ^ ^* -H .H ^ N ei m' N M* M* ec' eo' eo' CO* eo* eo' «*!« ^* ^* ^* 



•«SSS228aSSS5SSS§SS§l?§58SS 



Digitized by VjOOQIC 



318 



ELEMENTS OF HYDRAULICS 



If! 
Ill 



> O <0 CO CO 



o o o o o 






O O O Q.O 






<6 d CO CO 



a» lo 00 ^ 1 
CO S CO CO ( 



o o o o o 






o o o o o 



' CO (b CO < 



o o o o o 



t^ ^ CO *-• I 



o o o o o 



t* CO I 

SSI 



o o o o o 



o o o o o 



o o o o d 



liiis 



o o o o o 



n 



i;nn 



o o o o o 



CO CO 



to CO CO 

odd 



I e3 S « J 



o o o o o 



tssg 

• CO CO CO 



o o o o o 



CO CO 



X 00 00 

r-l *^ VH 

CO CO CO 
d d d 



isiii 



o o o o o 



i Sa S^ ^ ( 



o o o o o 



o o o o o 



o o o o o 



11^ 

CO CO 






o o o o o 



d d I 



III 



(O CD 

d d 






^ CO t* 






>o to CO CO 

SOI C< 04 
CO CO CO 



CO CO CO CO 

d d d d 






iH ^ N M eo 
d d d d d 



"^ U3 CO t« 00 

d o 6 G o 



a> o ^ CI CO 

O <^ vH >^ >^ 






§ eo 3 



ISI 



St: 

CO CO 



»0 "xl* CO 



w ^ o o < 

IH tH IH IH < 

CO CO CD CO < 



o o o o o 



o o o o o 



O O O O Q 



§111 

d d d d 



«0 N ■ 

CD CO < 






O) CO 

CO to 



<* CO M 

VH «H IH 

«0 CO CO 



ISSSI 

I CO V V < 



o o o o o 



ooooo ooooo 



s^gss; 



,3 

d d 



d d d 



CO CO 



CO N ^ 



ooooo 



CO S 

d d 



« t* CO 

« ei M 



r« CO 

CO CO 

d d 



CO w V 
C> O G 



if 

to CO 



to 
d d 



lH CO FH 



CO CD < 



ooooo 



;22l 

O CO < 



ooooo 



ooooo 



1^2 

I CD CO 



I >o CO < 



ooooo 



ooooo 



> A 00 00 

> Q 53 !i 
) CO CO CO 



S o < 



ooooo 



ooooo 



CO ^ CO 1 
S CO CO < 



f-4 00 lO 



o o 
o >o 

d d 



o o o 



ooooo 



« oi 00 

IH VH Q 

CO CO V 
odd 



^ CD ( 
O lO I 



q5 CD O V o 



^ ^ s s 

S S o s 



ooooo 



00 CO lO ]f 



ooooo 



CO W OB t> CO 



"* M F^ 

SS3 



ooooo 



ooooo 



X lo CO *-i a» 

s s s s s 



ooooo 



CO eo «H o) < 



ooooo 



^ 01 Q 

d d d 



C« O t« I 



ooooo 



CI o 

eg "H 

d d 



lH lO ^ 

« s s 



lO O I 

SSI 



Digitized by vnOOQlC 



HYDRAULIC DATA AND TABLES 



319 



Table 18. — Dischabge per Inch of Length oyer Rectangular Notch 

Weirs 

Discharge over sharp-crested, vertical, rectangular notch weirs in cubic feet per minute 

per inch of length 

Computed from Eq. (42), Par. 66; Q - 0.46*'^' f or 6 >i 1 in. 



Depth on crest in 
inches 





1/8 


1/4 


3/8 


1/2 


6/8 


3/4 


7/8 





0.00 


0.01 


0.06 


0.09 


0.14 


0.19 


0.26 


0.32 


1 • 


0.40 


0.47 


0.65 


0.64 


0.73 


0.82 


0.92 


1.02 


2 


1.13 


1.23 


1.35 


1.46 


1.68 


1.70 


1.82 


1.96 


3 


2.07 


2.21 


2.34 


2.48 


2.61 


2.76 


2.90 


3.05 


4 


3.20 


3.35 


3.50 


3.66 


3.81 


3.97 


4.14 


4.30 


5 


4.47 


4.64 


4.81 


4.98 


5.16 


5.33 


6.61 


5.69 


6 


5.87 


6.06 


6.26 


6.44 


6.62 


6.82 


7.01 


7.21 


7 


7.40 


7.60 


7.80 


8.01 


8.21 


8.42 


8.63 


8.83 


8 


9.05 


9.26 


9.47 


9.69 


9.91 


10.13 


10.36 


10.57 


9 


10.80 


11.02 


11.26 


11.48 


11.71 


11.94 


12.17 


12.41 


10 


12.64 


12.88 


13.12 


13.36 


13.60 


13.86 


14.09 


14.34 


11 


14.59 


14.84 


16.09 


16.34 


16.59 


16.86 


16.11 


16.36 


12 


16.62 


16.88 


17.15 


17.41 


17.67 


17.94 


18.21 


18.47 


13 


18.74 


19.01 


19.29 


19.56 


19.84 


20.11 


20.39 


20.67 


14 


20.95 


21.23 


21.51 


21.80 


22.08 


22.37 


22.65 


22.94 


15 


23.23 


23.52 


23.82 


24.11 


24.40 


24.70 


26.00 


26.30 


16 


25.60 


25.90 


26.20 


26.60 


26.80 


27. U 


27.42 


27.72 


17 


28.03 


28.34 


28.65 


28.97 


29.28 


29.69 


29.91 


30.22 


18 


30.54 


30.86 


31.18 


31.50 


31.82 


32.15 


32.47 


32.80 


19 


33.12 


33.45 


33.78 


34.11 


34.44 


34.77 


35.10 


36.44 


20 


35.77 


36.11 


36.46 


36.78 


37.12 


37.46 


37.80 


38.16 


21 


38.49 


38.84 


39.18 


39.63 


39.87 


40.24 


40.60 


40.96 


22 


41.28 


41.64 


41.98 


42.36 


42.68 


43.04 


43.44 


43.76 


' 23 


44.12 


44.48 


44.84 


46.20 


46.66 


45.96 


46.32 


46.68 


24 


47.04 


47.40 


47.76 


48.12 


48.52 


48.88 


49.28 


49.64 


25 


50.00 


50.40 


50.76 


51.08 


51.52 


61.88 


62.28 


52.64 


26 


53.04 


53.40 


53.80 


64.16 


54.66 


54.96 


55.36 


65.72 


27 


56.12 


66.52 


56.92 


67.32 


67.68 


58.08 


68.48 


68.88 


28 


59.28 


69.68 


60.08 


60.48 


60.84 


61.28 


61.68 


62.08 


29 


62.48 


62.88 


63.28 


63.68 


64.08 


64.62 


64.92 


65.32 


30 


66.72 


66.16 


66.56 


66.96 


67.36 


67.80 


68.20 


68.64 



Digitized by VnOOQ IC 



320 



ELEMENTS OF HYDRAULICS 



TaBLB 19. — DiBCHABOE PER FoOT OP LENGTH OVEB RECTANQULi^R NOTCH 

Weirs 

Disoharge over sharp crested, vertical, rectangular notch weirs in cubic feet per second 
per foot of length. Computed from Eq. (41), Far. 66: 









Q- 


S.Zbh*/* for b 


- 1ft. 










Depth on crest 
in feet 


0.00 


0.01 


0.02 


0.03 


0.04 


0.06 


0.06 


0.07 


0.08 


0.09 


0.0 


0.000 


0.003 


0.009 


0.017 


0.026 


0.037 


0.049 


0.061 


0.075 


0.089 


0.1 


0.104 


0.120 


0.137 


0.155 


0.173 


0.192 


0.211 


0.231 


0.252 


0.273 


0.2 


0.295 


0.317 


0.341 


0.364 


0.388 


0.413 


0.438 


0.463 


0.489 


0.515 


0.3 


0.642 


0.570 


0.697 


0.626 


0.654 


0.683 


0.713 


0.743 


0.773 


0.804 


0.4 


0.835 


0.866 


0.898 


0.931 


0.963 


0.996 


1.030 


1.063 


1.098 


1.132 


0.5 


1.167 


1.202 


1.238 


1.273 


1.309 


1.346 


1.383 


1.420 


1.468 


1.496 


0.6 


1.538 


1.572 


1.611 


1.650 


1.690 


1.729 


1.769 


1.810 


1.850 


1.892 


0.7 


1.933 


1.974 


2.016 


2.068 


2.101 


2.143 


2.187 


2.230 


2.273 


2.317 


0.8 


2.361 


2.406 


2.450 


2.495 


2.541 


2.586 


2.632 


2.678 


2.724 


2.768 


0.9 


2.818 


2.865 


2.912 


2.960 


3.008 


3.065 


3.104 


3.152 


3.202 


3.251 


1.0 


3.300 


3,350 


3.399 


3.449 


3.501 


3.561 


3.600 


3.663 


3.703 


3.755 


1.1 


3.808 


3.858 


3.911 


3.963 


4.016 


4.069 


4.122 


4.178 


4.231 


4.283 


1.2 


4.340 


4.392 


4.448 


4.501 


4.657 


4.613 


4.666 


4.722 


4.778 


4.834 


1.3 


4.891 


4.947 


5.006 


6.062 


6.118 


6.178 


6.^ 


5.293 


6.349 


5.409 


1.4 


5.468 


5.524 


6.684 


6.643 


6.702 


6.762 


6.821 


5.881 


6.940 


6.003 


1.5 


6.062 


6.126 


6.184 


6.247 


6.306 


6.369 


6.428 


6.491 


6.664 


6.617 


1.6 


6.679 


6.742 


6.805 


6.867 


6.930 


6.993 


7.059 


7.121 


7.187 


7.250 


1.7 


7.316 


7.379 


7.446 


7.608 


7.573 


7.639 


7.706 


7.772 


7.838 


7.904 


1.8 


7.970 


8.036 


8.102 


8.171 


8.237 


8.303 


8.372 


8.438 


8.607 


8.673 


1.9 


8.643 


8.712 


8.778 


8.847 


8.917 


8.986 


0.066 


9.126 


9.194 


9.263 


2.0 


9.332 


9.405 


9.474 


9.544 


9.616 


9.686 


9.768 


9.827 


9.900 


9.969 


2.1 


10.042 


10.115 


10.187 


10.260 


10.332 


10.406 


10.478 


10.660 


10.623 


10.695 


2.2 


10.768 


10.841 


10.916 


10.989 


11.065 


11.138 


11.213 


11.286 


11.362 


11.435 


2.3 


11.510 


11.586 


11.662 


11.738 


11.814 


11.887 


11.966 


12.042 


12.118 


12.194 


2.4 


12.269 


12.345 


12.425 


12.500 


12.576 


12.656 


12.731 


12.811 


12.890 


12.966 


2.5 


12.935 


13.124 


13.200 


13.279 


13.368 


13.438 


13.617 


13.696 


13.675 


13.764 


2.6 


13.834 


13.916 


13.995 


14.075 


14.154 


14.236 


14.316 


14.-398 


14.477 


14.660 


2.7 


14.642 


14.721 


14.804 


14.886 


14.969 


15.048 


15.131 


16.213 


15.296 


16.378 


2.8 


15.461 


15.543 


15.629 


16.711 


16.794 


15.876 


16.962 


16.045 


16. 130 


16.213 


2.9 


16.299 


16.381 


16.467 


16.650 


16.636 


16.721 


16.807 


16.889 


16.975 


17.061 


3.0 


17.147 


17.233 


17.318 


17.404 


17.490 


17.679 


17.665 


17.761 


17.837 


17.926 


3.1 


18.011 


18.101 


18.186 


18.275 


18.361 


18.460 


18.636 


18.625 


18.714 


18.803 


3.2 


18.889 


18.978 


19.067 


19.157 


19.246 


19.335 


19.424 


19.613 


19.602 


19.694 


3.3 


19.784 


19.873 


19.962 


20.064 


20.143 


20.236 


20.326 


20.414 


20.606 


20.699 


3.4 


20.688 


20.780 


20.873 


20.962 


21.054 


21.146 


21.239 


21.331 


21.424 


21.616 
22.447 


3.5 


21.608 


21.701 


21.793 


21.886 


21.978 


22.074 


22.166 


22.269 


22.354 


3.6 


22.542 


22.635 


22.730 


22.823 


22.919 


23.011 


23.107 


23.202 


23.296 


23.390 


3.7 


23.486 


23.682 


23.678 


23.773 


23.869 


23.965 


24.060 


24.156 


24.252 


24.347 


3.8 


24.446 


24.642 


24.638 


24.734 


24.833 


24.928 


26.027 


25.123 


25.222 


25.318 


3.9 


25.417 


25.616 


25.611 


25.710 


25.809 


26.905 


26.004 


26.103 


26.202 


26.301 


4.0 


26.400 


26.499 


26.698 


26.697 


26.796 


26.895 


26.997 


27.096 


27.196 


27.298 



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HYDRAULIC DATA AND TABLES 



321 



Table 20.' — Discharqb per Foot of Length over Suppressed Weirs 

Difloharge ovev 8harp<ere«ted, vertical, suporesaed weirs in cubio feet per second per foot 
of length. Computed by Basin's formula (Art. XI) 

Q - (0.405 + 5:5^ ) (l + 0.55(^)*) bhVm. for 6 - 1 ft. 



— 


Head on crest, h, 
in feet 


Height of weir, d, in feet | 


2 


4 


6 


8 


10 


20 


30 


0.1 


0.13 


0.13 


0.13 


0.13 


0.13 


0.13 


0.13 




0.2 


0.33 


0.33 


0.33 


0.33 


0.33 


0.33 


0.33 




0.3 


0.58 


0.58 


0.58 


0.58 


-0.58 


0.58 


0.58 




0.4 


0.88 


0.88 


0.87 


0.87 


0.87 


0.87 


0.87 




0.5 


1.23 


1.21 


1.21 


1.21 


1.21 


1.20 


1.20 




0.6 


1.62 


1.59 


1.58 


1.58 


1.57 


1.57 


1.57 




0.7 


2.04 


1.99 


1.98 


1.98 


1.97 


1.97 


1.97 




0.8 


2.50 


2.43 


2.41 


2.41 


2.40 


2.40 


2.40 




0.9 


3.00 


2.90 


2.88 


2.86 


2.86 


2.85 


2.85 




1.0 


3.53 


3.40 


3.36 


3.35 


3.34 


3.33 


3.33 




1.1 


4.09 


3.92 


3.87 


3.86 


3.85 


3.84 


3.84 




1.2 


4.68 


4.48 


4.42 


4.40 


4.38 


4.36 


4.36 




1.3 


5.31 


5.07 


4.99 


4.96 


4.94 


4.91 


4.91 




1.4 


5.99 


5.68 


5.58 


5.54 


5.52 


5.49 


5.48 




1.5 


6.68 


6.30 


6.20 


6.16 


6.13 


6.10 


6.09 




1.6 


7.40 


6.97 


6.84 


6.78 


6.74 


6.69 


6.69 




1.7 


8.14 


7.66 


7.49 


7.42 


7.39 


7.33 


7.32 




1.8 


8.93 


8.37 


8.18 


8.09 


8.05 


7.98 


7.96 




1.9 


9.75 


9.11 


8.89 


8.79 


8.74 


8.65 


8.63 




2.0 


10.58 


9.87 


9.62 


9.51 


9.44 


9.34 


9.32 




2.1 


11.45 


10.65 


10.37 


10.25 


10.17 


10.05 


10.02 




2.2 


12.34 


11.46 


11.14 


10.99 


10.91 


10.78 


10.75 




2.3 


13.24 


12.29 


11.93 


11.77 


11.66 


11.52 


11.48 




2.4 


14.20 


13.15 


12.75 


12.56 


12.45 


12.28 


12.24 




2.5 


15.17 


14.03 


13.59 


13.38 


13.26 


13.06 


13.01 




2.6 


16.16 


14.92 


14.44 


14.20 


14.07 


13.85 


13.80 




2.7 


17.18 


15.83 


15.31 


15.04 


14.92 


14.65 


14.60 




2.8 


18.23 


16.79 


16.21 


15.92 


15.76 


15.48 


15.42 




2.9 


19.29 


17.77 


17.11 


16.79 


16.63 


16.33 


16.25 




3.0 


20.39 


18.74 


18.06 


17.71 


17.52 


17.18 


17.10 




3.1 


21.50 


19.74 


19.02 


18.64 


18.42 


18.04 


17.96 




3.2 


22.64 


20.77 


19.98 


19.58 


19.34 


18.93 


18.83 




3.3 


23.81 


21.80 


20.98 


20.55 


20.27 


19.82 


19.73 




3.4 


24.98 


22.89 


21.99 


21.52 


21.24 


20.76 


20.63 




3.5 


26.20 


24.00 


23.01 


22.48 


22.22 


21.69 


21.60 




3.6 


27.41 


25.09 


24.06 


23.52 


23.20 


22.62 


22.48 




3.7 


28.64 


26.22 


25.14 


24.56 


24.20 


23.59 


23.43 




3.8 


29.94 


27.38 


26.22 


25.60 


25.23 


24.56 


24.39 




3.9 


31.21 


28.53 


27.33 


26.65 


26.26 


25.53 


25.34 




4.0 


32.54 


29.74 


28.45 


27.74 


27.32 


26.55 


26.35 




4.1 


33.85 


30.95 


29.59 


28.83 


28.36 


27.55 


27.33 




4.2 


35.22 


32.18 


30.75 


29.96 


29.48 


28.59 


28.36 




4.3 


36.59 


33.43 


31.93 


31.10 


30.58 


29.62 


29.37 




4.4 


37.99 


34.70 


33.12 


32.24 


31.70 


30.66 


30.42 




4.6 


39.40 


35.98 


34.33 


33.39 


32.83 


31.74 


31.47 




4.6 


40.83 


37.29 


35.56 


34.58 


33.98 


32.84 


32.53 




4.7 


42.29 


38.62 


36.82 


35.75 


35.13 


33.93 


33.61 




4.8 


43.75 


39.96 


38.07 


37.00 


36.33 


35.05 


34.70 




4.9 


45.22 


41.30 


39.35 


38.20 


37.49 


36.15 


35.77 




5.0 


46.71 


42.67 


40.62 


.39.44_ 


38.70 


37.28 


36.88 



1 Compiled from extensive hydraulic tables by Williams and Hazek. 



21 



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322 



ELEMENTS OF HYDRAULICS 



Table 21. — Principles 



Kinematics (motion) 



Linear motion 



Angular motion 





« " displacement 




9 B displacement 




V - velocity 




<a - velocity 




a « acceleration 




a B acceleration 




vo - initial velocity 




wo - initial velocity 




F - force 




M « torque about fixed axis 


Nototion 


IF - F« - work 




W " Me " work 




m = mass 




J - 2mr« - moment of inertia 




' v B my a. weight 




t - time 




1 t - time 




3ff - impulse 




! Ft - impulse 




Ibi « momentum 




1 mv " momentum 






Definitions 


ds dv d*8 
" "df" ' dt - dt* 




de do» d*e 
" " dt' *" dt " d<8 


Uniformly 


V " vo •\- at 


w * wo + «< 


accelerated motion 


« - ro< + ia/« 


<? - «o< + W^ 


acoel. « const. 


V* = i;o« + 2o« 


\ 
wo 


«« - «o« + 2ae 


Derivation of 
above formulas 


If « = 0, w - uo .'.Ci - 


d/« • «• dt " «' + ^' 

« - ia<« + Cit -f Ca 

If < =» 0, « - «o .'.Ci - uo 




If < - 0. « - .-.Ci = 





If < - 0, « - .-.c* - 


Relation between 


V " ru 




linear and 


at " ra 


at - tang. comp. of accel. 


angular motion 


On » v^/r - r«2 


a* » normal comp. of accel. 




vB 


Arc AB » ds '^ rde u - rw 


Derivation of first 
two formulas 


.rY 


d« d(? dv do 
dt ~' "" dt dt '^ dt 




/^^ u 


u — r« at "■ ra 




^4;'' 


If body at A were free, it would proceed 




in direction of tangent AB and in 




f /^ 


time t would reach B where AB « vf. 


Derivation of 

normal accel. 

for uniform 

circular motion 


lyt> J Since it is found at C instead of B it 
jyy^^^^ must have experienced a central 
acceleration. 
Let an denote this central acceleration. Then BC - \ant'^' 


By geometry BC X BD 


■ .^B> and in the limit BD approaches 2r. 




Hence ian«« X 2r - v2««, 


from which Un « v^/r « wV. 



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HYDRAULIC DATA AND TABLES 



323 



OF Mbchanics 



Dynsmicg (force) 



Fundamental law ; 



Linear motion 



I 



F '^ ma 



Angular motion 



la 



DiBcussion and j 
derivation I 



Principle of work 
and energy 



Derivation 



By experiment it is found that 
F oz a (Newton's 2nd Law) 
.'. F/a >- const., say m, whence 
F — ma. m ■■ intrinsic prop- 
erty of body called its mass. 
Mass " measured inertia. 
i 

If F » then a '^ and hence I 
V s or constant, which ex- | 
presses Newton's 1st Law. 

F » impressed force, ma » kin- 
etic reaction or inertia force, i 

Equality F -■ ma is dynamical 
expression of Newton's 3rd Law. 



Fa 



ir -F« - 



mv* 



mvo* 
2 



Consider rotation 
of rigid body 
about a fixed 
axis. - 
Then for a parti- 
cle of mass m at 
distance r from .azis of rotation, 
law F '^ ma becomes Fr -> mar^ 
or since a — ra, Fr ■■ mr*a. 
By summation 

2 Fr ■> — mr*a 
But 

X Fr " Mt and X mr*' 
:.M - la 



Tf - Af a - 



J«i 



W 
2 



Principle of im- 
pulse Sc momentum 



mat «* ■■ so* + 2a« 
at ■■ — n — , and 

mwo* 



2 
Tf - F« - ma9 - 



2 



Ft . 



Derivation 



Power 



F 


- mo. 


V 


■■ VO 


+ a< 




•. (rt - 


V 


- wo 


and 


Ft 


- merf 


■■ 


mw - 


- mwi 



- Ja, tr» — wi* + 2ao 
. . oo — s • ftnd 



W - Af - Jao ' 



I»« - !«.« 



Aft - J«, Jo>» 



Plower — Fw, h.p. 



Fy_ 
650 



Af — ~ Ja, tp -» tri + al 

.*. a< « w — wi, and 

Aft — lal " luf — etri 



Power — Mw, h.p. — gg^ 



Centrifugal force I 

Derivation ' 

D*Alembert's 
principle 



F. = 



w V* 



— - w*r 




dU 
F - «5^ - 



I F oi ma where F >- external impressed force and a » accel. produced. 

I Introduced another force P, given by P ■> — ma. Then by addition, 

I F + F "■ 0; i.e., the body is in equilibrium under the action of F and 

; P. P is called the kinetic reaction, or reversed effective force, tinee 

[P » — F. By introducing this idea of the kinetic reactions equiU- 

Explanation and i brating the impressed forces, all problems in dynamics are reduced 

to statical problems. This is called d*Alembert's Principle, and is 

usually expressed in the form 

d*a 
F - m^^ - 0. 



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324 



ELEMENTS OF HYDRAULICS 



Tablb 22. — Submerged Weir Coefficients.^ 
Values of the coefficient n in Herschel's submerged weir formula 



h = depth of water measured to crest 
h' = depth of water measured to crest 



level on upstream side; 
level on downstream side. 



Hi 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


1.000 


1.004 


1.006 


1.006 


1.007 


1.007 


1.007 


1.006 


1.006 


1.005 


0.1 


1.005 


•1.003 


1.002 


i.oop 


0.998 


0.996 


0.994 0.992 


0.989 


0.987 


0.2 


0.985 


0.982 


0.980 


0,977 


0.975 


0.972 


0.970 0.967 


0.964 


0.961 


0.3 


0.959 


0.956 


0.953 


0.950 


0.947 


0.944 


0.941 


0.938 


0.935 


0.932 


0.4 


0.929 


0.926 


0.922 


0.919 


0.915 


0.912 


.908 


0.904 


0.900 


0.896 


0.5 


0.892 


0.888 


0.884 


0.880 


0.875 


0.871 


0.866 


0.861 


0.856 


0.851 


0.6 


0.846 


0.841 


0.836 


0.830 


0.824 


0.818 


0.813 


0.806 


0.800 


0.794 


0.7 


0.787 


0.780 


0.773 


0.766 


0.758 


0.750 


0.742 


0.732 


0.723 


0.714 


0.8 


0.703 


0.692 


0.681 


0.669 


0.656 


0.644 


0.631 


0.618 


0.604 


0.590 


0.9 


0.574 


0.557 


0.539 


0.520 


0.498 


0.471 


0.441 


0.402 


0.352 


0.275 



Values of the coefficient k in Pteley and Steams submerged weir 
formula Q = Kl(k + ^) y/h-W 



h 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 




3.33 


3.33 


3.34 


3.34 


3.36 


3.37 


3.37 


3.37 


3.37 


0.1 


3.37 


3.36 


3.35 


3.34 


3.34 


3.33 


3.32 


3.31 


3.30 


3.29 


0.2 


3.29 


3.28 


3.27 


3.26 


3.26 


3.25 


3.24 


3.23 


3.23 


3.22 


0.3 


3.21 


3.21 


3.20 


3.19 


3.19 


3.18 


3.18 


3.17 


3.17 


3.16 


0.4 


3.16 


3.15 


3.15 


3.14 


3.14 


3.13 


3.13 


3.12 


3.12 


3.12 


0.5 


3.11 


3.11 


3.11 


3.10 


3.10 


3.10 


3.10 


3.10 


3.10 


3.09 


0.6 


3.09 


3.09 


3.09 


3.09 


3.09 


3.09 


3.09 


3.09 


3.09 


3.09 


0.7 1 3.09 


3.09 


3.10 


3.10 


3.10 


3.10 


3.11 


•3.11 


3.11 


3.12 


0.8 3.12 


3.13 


3.13 


3.14 


3.14 


3.15 


3.16 


3.16 


3.17 


3.18 


0.9 3.19 


3.20 


3.21 


3.22 


3.23 


3.25 


3.26 


3.28 


3.30 


3.33 



1 "Hydraulics," Hughes and Safford, pp. 228, 229. 



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INDEX 



Accumulator, hydraulic, 9 
Adjutage, 67 

Venturi, 69 
Age, deterioration with, 88 
AUievi's formula for water hammer, 

227 
American type of reaction turbine, 

189 
Appalachian Power Co. turbines, 282 
Aqueduct, Catskill, 153-156 

Roman, 150 
Archimedes, theorem of, 24 



B 



Backwater, 138 
Barge canal, N. Y. State, 115 
Barker's mill, 169 
Barlow's formula, 18 
Barometer, mercury, 21 

water, 20 
Bazin's formula, 122 

for pipe flow, 101 

values of Chezy's coefficient, 
313 
Bends and elb6ws, resistance of, 89 
Bernoulli's theorem, 71 
Bimie's formula, 19 
Borda mouthpiece, 68 
Branching pipes, 103 
Breast wheel, 172 
Bulk modulus of water, 1 
Buoyancy, 22 

zero, 26 



Capacity criterion, 204 
Cast-iron pipe, flOw in, 86 
Catskill aqueduct, 153-156 
meter, 77 



Cedar Rapids turbines, 288 
Center of pressure, 14 
Centrifugal pumps, 246 
characteristics, 258 
design of, 265 
Characteristics of centrifugal pumps, 
258 
of impulse wheels, 178 
Characteristic speed, 205 
Chezy's coefficient, Bazin's values of, 
313 
Kutter's values of, 314 
formula for open channels, 121 
for pipe flow, 100 
Cippoletti weir, 57 
Circles, properties of, 298 
Clavarino's formula, 18 
Cock, head lost at, 95 
Coefficient of pipe friction, 85 
Complete contraction, 52 
Compound pipes, 102 
Concrete pipe, 89 
Conduits, flow in, 120 
Conical mouthpiece, 69-70 
Contracted weir, 53 
Contraction coefficient, 48 
of jet, 51 
of section, 94 
partial and complete, 52 
Crane, hydrauHc, 11 
Critical velocity of flow, 78 
Current meter measurements, 128 
wheels, 169, 171 



D 



Dams, Catskill aqueduct system, 39 

Keokuk, 40 

stability of, 37 
Darcy's mod. of Pitot tube, 131 
Deep well cent, pump, 267 
Deflection of jet, 160 
Density of water, 3, 290 



325 



Digitized by VnOOQlC 



326 



INDEX 



Design of centrifugal pumps, 265 
Differential gage, 22 

surge tank, 235 
Diffusion vanes, 252 
Diffusor, pressure developed in, 257 
Discharge equivalents, 292 
Displacement pumps, 238 
Divided flow, 102 
Doble bucket, 173 
Draft tube design, 194 

profile, 195 

theory of, 193 

time of flow through, 197 

use of, 190 
Draw down in surge tanks, 230 
Dry dock, floating, 266 
DuBuat's paradox, 138 
Dynamic pressure, 158 



Fourneyron type of reaction tur- 
bine, 186 
Francis type of reaction turbine, 180 
Free surface of liquid in rotation, 218 
Freeman's experiments, 105 
Friction head in pipes, 309 

loss, 84 
Fteley and Steams formula, 324 
Fuller, W. E., 91 

G 

Gage, differential, 22 

pressure, 21 
Gate valve, head lost at, 95 
Girard turbine, efficiency of, 181 
Graph of exponential formula, 87 



E 



Effective head, 49 
Efficiency, hydraulic, 123 

of centrifugal pump, 266 

of hydraulic press, 8 
ram, 238 

for moving vanes, 164 
Efflux coefficient, 49 

for circular orifice, 303 

for square orifice, 304 
Elasticity of water, 1 
Elevator, hydraulic, 11 
Energy of flow, 157 
Enlargement of section, 93 
Entrance, loss of head at, 84 
Equilibrium of fluids in contact, 19 
Exponential (ormula, 85, 88 

friction head from, 309 

Williams and Hazen, 101 



Head and pressure equivalents, 291 

developed by cent, pumps, 262 

loss of, in pipes, 83 

lost at bends, 90 

varying, 64 
Hele-Shaw's experiments, 107 
Herschel, Clemens, 74 
Hook gage 61 
Hydraulic dredging, 271 

efficiency, 123 

gradient, 97 

slope of, 98 

mining, 273 

motors, classification of, 171 
types of, 169 

press, 6 

efficiency of, 8 

radius, 99 

ram, 236 



Fire nozzles, 70 

pimip, centrifugal, 270 

streams, 105 

table, 305 
Fleming's experiments, 106 
Fluid, properties of, 1 
Force pump, 240 



Impact on plane surface, 158 

surface of revolution, 159 

tube, theory of, 135 
Impellers, centrifugal pump, 249 
Impulse wheels, 170 

characteristics of, 178 

vane angles, 165 



Digitized by VnOOQlC 



INDEX 



327 



Impulse wheels, work absorbed by, 

166 
Intensificr, 8 



Jack, hydraulic, 10 

Jet, pressure of, on surface, 157 

reaction of, 167 
Jonval type of reaction turbine, 186 
Joukowsky's formula, 226 

K 

Keokuk lock and dam, 39-43 

turbines, 200 
Kinetic pressure, 71 
Kutter's formula, 121 

for pipe flow, 101 
simplified formula, 123 
values of Chezy's coefficient, 
314 



Lamp's formula, 18 
Liquid, definition of, 1 

vein, 47 
Loss of head in pipe flow, 83 

M 

Mariotte's flask, 66 
Mechanics, principles of, 322 
Metacenter, 26 

coordinates of, 27 
Metacentric height, 28 
Mine drainage, cent, pump for, 270 
Mixed flow type of turbine, 189 
Modulus of elasticity of water, 1 
Moritz, E. A., 85 

formula, table from, 316 
Mouthpiece, standard, 67 

N 

Natural channels, flow in, 126 
Needle nozzle, 175 
Non-sinuous flow, 79 
Non-uniform flow, backwater, 138 
Normal pressure of water, 2 



Open channels, flow in, 120 
Operating range, normal. 210 
Orifice, circular, efllux coefficients 

for, 303 
square, efflux coefficients for, 

304 
Oscillation, period of, 29 
Overshot wheel, 172 



Packing, frictional resistance* of , 7 
Parallel flow, 79 
Partial contraction, 52 
Pelton wheel, 173 

efficiency of, 177 
Penstock, economical size of, 214 
Piezometer, 21 
Pipe flow, 78-83 

friction, coefficient of, 85 

lines, power transmitted 
through, 214 

strength of, 17 
Pipe friction, coefficient of, 308 
Pipes, dimensions of, 295 
Pitometer, 132 
Pitot recorders, 134 

tube, 129 

calibration of, 137 
Poncelet wheel, 172 
Pressure, center of, 14 

change with depth, 13 

developed in cent, pump, 255 

equal transmission of, 3 

gage, mercury, 21 

head developed in cent, pumps, 
257 
kinetic, 72 

kinetic, 71, 97 

normal to surface, 2 

of jet on surface, 157 

proportional to area, 5 
Principles of mechanics, 322 
Pump cylinders, diameter of, 244 
Pumps, capacity of reciprocating, 
296 

centrifugal, 246 



Digitized by VnOOQlC 



328 



INDEX 



Pumps, displacement, 238 
Pump sizes, calculation of, 243 



Ram, efficiency of, 238 

hydraulic, 236 
Rate of flow controller, 77 
Reaction of jet, 167 
turbines, 170 

classification of, 170, 208 
principle of, 168 
Rectangular notch weirs, discharge 
* coefficients for, 318-320 
discharge from, 51 
orifice, discharge from, 50 
Riveted steel pipe, 88 
Rolling and pitching, 30 
Roman aqueducts, 150 
Rotation, liquid in, 218 



S 



Selection of stock runner, 211 

Service pipes, house, 143 

Sharp edged orifice, discharge from, 

51 
Siphon lock, 117 
modem, 111 
spillways, 112 
wheel settings, 120 
Slope of hydraulic gradient, 98 
Specific discharge. 207 
power, 207 
speed, 206-208 
weight, 3 

determination by experiment, 

24 
of various substances, 294 
physical definition of, 24 
Speed criterion, 203 
Stage pumps, 254 
Static pressure, 158 
Steam pump, 241 
Stock runner, selection of, 211 
Stream gaging, 126 
line, 47 

mouthpiece, 68 
Strength of pipe, 17 



Submerged weir, 54 

coefficients, 324 
Suction lift, maximum, 239 

pump, 238 
Suppressed weir, 54 

discharge from, 321 
Surge in surge tanks, 228 
tanks, 228 

differential, 235 



Tanks, filling and emptying, 63-65 
Throttling discharge of cent, pump, 

263 
Throttle valve, head lost at, 96 
Torricelli's theorem, 48 
Translation, horizontal linear, 217 

vertical linear, 218 
Trapezoidal weir, 57 
Triangular notch weir, 56 
Turbine pumps, 253 

setting, recent practice in, 198 

U 

Undershot wheel, 171 

V 

Vane, pressure of jet on, 162 

work done on moving, 163 
Varying head, 64 
Velocity, critical, 78 

head, ideal, 47 

of approach, 52 
Venturi adjutage, 69 

meter, 74 
Viscosity coefficient, 79 

of water, 3 
Volute casing, 249 
Vortex chamber, 250 

W y- 

Water hammer in pipes, 221 
ordinary, 224 
Warren, Minton M., 221 
Water, properties of, 290 



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INDEX 



329 



Wave, compression, in pipe, 223 
Weight of water, 290 
Weights and measures, 293 
Weir, Cippoletti, 57 

construction of, 60 

contracted, 53 

formulas, empirical, 58-69 

measurements, 60 
Weirs, proportioning, 62 

rectangular notch, discharge 
coefficients for, 318-320 



Weirs, submerged, 64 

coefficients, 324 
suppressed, 64 

discharge from, 321 
trapezoidal, 57 
triangular notch, 56 
Williams and Hazen's exponential 
formula, 101 
tables from, 309 
Wood stave pipe, 85 
discharge from, 316 



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m 6 1938 




SEP 14 1840 




MAR 24 1941 M 






LD 21-20m-6,'3a 



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