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"^"W^^
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ELEMENTS OF HYDRAULICS
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McGraw-Hill CookCompaiiy
PujStis/iers c^3ooI^/c^
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and hydrodynamics have been replaced by the more familiar and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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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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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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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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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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8
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
9
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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10
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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12
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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20
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
21
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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32
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.
Digitized by
Google
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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38
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
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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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90
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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92
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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FLOW OF WATER
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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96
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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102
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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FLOW OF WATER
109
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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110
ELEMENTS OF HYDRAULICS
Fig. 95.
Fig. 96.
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FLOW OF WATER
111
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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FLOW OF WATER
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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FLOW OF WATER
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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FLOW OF WATER
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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118
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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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
Digitized by vnOOQlC
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.
Digitized by VnOOQlC
FLOW OF WATER
125
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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.
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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
Digitized by LnOOQlC
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
Digitized by VnOOQlC
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*
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
{
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.
Digitized by VnOOQlC
134
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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I
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.
Digitized by VnOOQlC
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.
Digitized by vnOOQlC
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.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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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152
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
173
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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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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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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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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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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Fig. 160.
Fig. 161.
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188
ELEMENTS OF HYDRAULICS
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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
191
Fig. 165. — Gate mechanism
Fig. 166
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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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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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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
Digitized by VnOOQlC
ENERGY OF FLOW
199
FiQ. 170. — Center section of main turbine on transverse axis of power house,
Mississippi River Power Co.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
ENEMY OF FLOW
201
Digitized by VjOOQIC
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
Digitized by VnOOQlC
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.)
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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
JtaiflTA.
r^
z.
! *^
K-K^^^
,^
^
^.^
/
I
4
fP
y
;>
\
:3
Cjv^
r OS
d
C;,i5
lev
y
/
\
/
':
/
-^ '?
^
/
^
/
\
^
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
Digitized by
Google
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
Digitized by VnOOQlC
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.
Digitized by VnOOQlC
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,
Digitized by VnOOQlC
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.
Digitized by LnOOQlC
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
Digitized by vnOOQlC
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).
Digitized by VnOOQlC
260
ELEMENTS OF HYDRAULICS
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CAPACITY
FiQ. 206. — Characteristics and efficiency curves obtained from De Laval
centrifugal pumps.
Digitized by VnOOQlC
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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10 20 30 40 GO 60 TO 80 90 100 110 120 m UQ 1^ 160 170
Percentage of Nonnal Oapacltj
FiQ. 207.
For a high-lift pump under an approximately constant head,
as in the case of elevator work, a pump with radial vanes is most
suitable as the discharge may be varied with a small alteration
in deUvery head. This is also true for a pump working under a
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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.
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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
Digitized by VnOOQlC
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
Digitized by LnOOQlC
272
ELEMENTS OF HYDRAULICS
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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;
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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
-
-
-
-
-
-
-
-
-
^
~
-
'^
-
-
-
-
-
_
=
s
B
=
«
^
^
-
^
■^
^
■^
s
a
^
^^
's
?
70
J
^
^
H
^
^
1
n
fiO
A
/
.^
?
^
A
7
"
f
40
^
A
_J
so
f
1
J
'
70
f
"
\
J
^
10
f
„
>
h
~
I]
_
_
_
_
_
_
.
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,
Digitized by VnOOQlC
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
,0'
•^
'
^
««■
•*•«
^
80
^
^
--
*•
■■
n
hi
80
^
.«*
70
■
^
f^
-*
"
~
~
/"
,^
/^
70
60
/
/
'
/'
/
50
/
/
60
y
i
40
r
r
iO
/
J
30
f,
f
J
/
80
20
/
f
20
7
10
^
10
_
__
_
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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ENERGY OF FLOW
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ELEMENTS OF HYDRAULICS
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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
Digitized by VnOOQlC
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
Digitized by
Google
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
Digitized by VnOOQlC
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
Digitized by
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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
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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
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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
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Digitized by V^aOOQ iC
HYDRAULIC DATA AND TABLES
317
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318
ELEMENTS OF HYDRAULICS
If!
Ill
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o o o o o
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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
Digitized by vnOOQlC
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
Digitized by
GoQgle
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.
Digitized by VnOOQlC
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.
Digitized by
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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.
Digitized by VnOOQlC
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
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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
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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
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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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AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO BO CENTS ON THE FOURTH
DAY AND TO $1.00 ON THE SEVENTH DAY
OVERDUE.
SEP 22 1932
OCT 28 1§3J
A«G 24 t934
OCT 1 1934
m 6 1938
SEP 14 1840
MAR 24 1941 M
LD 21-20m-6,'3a
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