SF
GIFT OF
MICHAEL REE^E
$ ,
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f
WORKS OF
PROF. HENRY T. BOVEY
PUBLISHED BY
JOHN WILEY & SONS.
A Treatise on Hydraulics.
8vo, cloth, $4.00.
Strength of Materials and Theory of Structures.
830 pages, 8vo, cloth, $7.50.
A TREATISE
ON
HYDRAU LICS.
BY
HENRY T. ^JO
M. INST.C.E., LL.D., F.R.S.C.,
Professor of Civil Engineering and Applied Mechanics,
McGill University, Montreal.
FIRST EDITION.
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
LONDON : CHAPMAN & HALL, LIMITED,
1895.
Copyright, 1895,
BY
HENRY T. BOVEY.
ROBERT DRUMMOND, HLHCTROTYPBR AND PRINTER, NEW YORK.
PREFACE.
THE present treatise is the outcome of lectures delivered in
McGill University during the last ten or twelve years, and
although intended primarily for the use and convenience of
the student of hydraulics, it is hoped that it may also prove
acceptable to the engineer in general practice.
In order to render the treatment of the subject more com-
plete, free reference has been made to standard authors on the
subject. The examples introduced to illustrate the text have
also been selected in part from the works of such well-known
writers as Weisbach, Osborne Reynolds, and Cotterill, but the
greater number are such as have occurred in the course of the
author's own experience. The tables of coefficients of discharge
have been prepared from the results of experiments carried
out in the Hydraulic Laboratory of the University. These
experiments are still being continued and may probably form
the subject of a special paper.
The author desires to acknowledge many suggestions
offered by Professor Bamford, and to express his deep obliga-
tion to Professor Chandler for much labor and time given to
the revision of proof sheets.
HENRY T. BOVEY.
MONTREAL, November, 1895.
CONTENTS.
CHAPTER I.
FLOW THROUGH ORIFICES AND OVER WEIRS.
PAGE
Definitions i
Stream-line Motion 2
Motion in Plain Layers 2
Laminar Motion 2
Density of Water 2
Continuity 2
Bernouilli's Theorem 6
Applications of Bernouilli's Theorem 9
Piezometer 9
Orifice in a Thin Plate 13
Torricelli's Theorem 14
Efflux from Orifice in a Vessel in Motion 16
Flow in a Frictionless Pipe of Gradually Changing Section 18
Hydraulic Resistances 20
Coefficient of Velocity 20
Coefficient of Resistance 21
Coefficient of Contraction 22
Coefficient of Discharge 24
Miner's Inch 26
Energy and Momentum of Jet 27
Inversion of the Jet 27
Time Required to Empty and Fill a Lock 29
General Equations 30
Loss of Energy in Shock 32
Mouthpieces 34
Borda's Mouthpiece 34
Ring Nozzle 37
Cylindrical Mouthpiece 39
Divergent Mouthpiece 42
Convergent Mouthpiece 44
Radiating Current 46
Vortex Motion 47
Free Spiral Vortex 48
Forced Vortex 49
v
vi CONTENTS.
Compound Vortex 50
Large Orifices 50
Rectangular Orifices of Large Size 50
Circular Orifices of Large Size 53
Notches 54
Weirs 54
Triangular Notch 56
Broad-crested Weir 58
Examples 60
CHAPTER II.
FLUID FRICTION.
Definition 70
Laws of Fluid Friction 72
Surface Friction in Pipes 73
Resistance of Ships 76
CHAPTER III.
FLOW IN PIPES.
Assumptions .* 78
Steady Motion .. 78
Influence upon the Flow of the Pipe's Position 83
Transmission of Energy by Hydraulic Pressure 84
Flow in a Uniform Pipe Connecting Two Reservoirs 86
Losses of Head due to Abrupt Changes of Section 89
Remarks on the Law of Resistance to Flow 96
Flow in a Pipe of Varying Diameter 98
Equivalent Uniform Main 100
Branch Main of Uniform Diameter 101
Nozzles 104
Motor Driven by Hydraulic Pressure 107
Siphons 108
Inverted Siphons 109
Air in Pipe no
Three Reservoirs Connected by a Branched Pipe m
Orifice Fed by Two Reservoirs 115
Variation of Velocity in a Transverse Section 119
Examples 122
CHAPTER IV.
FLOW OF WATER IN OPEN CHANNELS.
Flow of Water in Channels t 131
Steady Flow in Channels of Constant Section 132
CONTENTS. Vll
Form of a Channel 135
Flow in Aqueducts 142
River Bends 143
Value of/ 144
Darcy and Bazin's Formulae 145
Ganguillet and Kiitter's Formulas 147
Variation of Velocity in a Transverse Section 148
Bazin's Formula 152
Boileau's Formula 153
Relations between Surface, Mean, and Bottom Velocities 154
Flow of Water in Open Channels of Varying Cross-section 156
Standing Wave 165
Examples 170
CHAPTER V.
METHODS OF GAUGING.
Gauging of Streams and Watercourses 173
Hook Gauge '. 173
Surface-floats 175
Subsurface-floats 175
Twin-floats 176
Velocity Rod 176
Pitot Tube 176
Darcy Gauge 178
Current Meters 180
Hydrometric Pendulum 183
Gauging of Pipes 183
Venturi Meter 183
Piston Meter 184
Inferential Meter 184
CHAPTER VI.
IMPACT.
Impact upon a Flat Vane Oblique to Direction of Jet 186
Impact upon a Flat Vane Normal to Direction of Jet 189
Reaction 190
Jet Propeller 190
Impact upon a Surface of Revolution 192
Impact upon a Flat Vane with Rim 195
Pressure in a Pipe upon a Thin Plate Normal to the Direction of Motion. 196
Pressure in a Pipe upon a Cylindrical Body about Three Diameters in
Length 198
Impact upon a Curved Vane 199
Frictional Effect 205
Resistance to the Motion of a Solid in a Fluid Mass 205
Examples ; 209
Vlll CONTENTS.
CHAPTER VII.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
PAGE
Classification 213
Hydraulic Ram 214
Pressure Engine 215
Accumulator 215
Losses of Energy in Pressure Engines 221
Hydraulic Brakes 223
Water-wheels ^ 225
Undershot Wheels 225
Wheels in a Straight Race 227
Poncelet Wheel 232
Form of Bucket 240
Breast-wheels 242
Sluices 244
Overshot Wheels 254
Effect of Centrifugal Force 255
Weight of Water on Wheel 256
Arc of Discharge 256
Pitch-back Wheel 272
Ventilated Bucket 272
Jet Reaction Wheel •> 272
Barker's Mill 272
Scotch Turbine 276
Reaction Turbines 276-
Impulse Turbines 276
Hurdy-gurdy Wheel 279
Pelton Wheel 280
Radial-, Axial-, and Mixed-flow Turbines 281-284
Limit Turbine 283
Theory of Turbine 284
Remarks on Centrifugal Head in Turbine-flow 298
Practical Values of the Velocities, etc. , in Turbines 299
Theory of the Section-tube 301
Losses of Energy in Turbines 303
Centrifugal Pumps ." 307
Theory of Centrifugal Pumps 309
Examples 315
HYDRAULICS.
CHAPTER I.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
I. Fluid Motion. — The term " hydraulics," as its derivation
(vdoop, water ; avXos, a tube or pipe) indicates, was primarily
applied to the conveyance of water in a tube or pipe, but its
meaning now embraces the experimental theory of the motion
of fluids.
The motion of a fluid is said to be steady or permanent
when the molecules successively arriving at any given point
are animated with the same velocity, are subjected to the
same pressure, and are the same in density. As soon as the
motion of a stream becomes steady a permanent regime is said
to be established, and hydraulic investigations are usually
made on the hypothesis of a permanent regime. With such
an hypothesis any portion of the fluid mass which leaves a
given region is replaced by a like portion under conditions
which are identically the same.
The terms "steady motion" and " permanent regime" are
often considered to be synonymous.
The general problem of flow is the determination of the
relation which exists at any point between the density, press-
2 HYDRA ULICS.
ure, and velocity of the molecules which successively pass that
point.
The actual motion of a fluid is exceedingly complex, and
in order to simplify the investigations various assumptions are
made as to the nature of the flow.
2. (a) Stream-line Motion. — The molecules may be re-
garded as flowing along definite paths, and a succession of such
molecules will form a continuous fluid rope which is termed an
elementary stream or a fluid filament, or, if the motion is steady
and the paths therefore fixed in space, a stream-line.
Experiment shows that the velocity of flow in any cross-
section varies from point to point, and hence it is often assumed
that the section is made up of an infinite number of indefi-
nitely small areas, each area being the section of a fluid
filament.
(b) Motion in Plane Layers. — In this motion it is assumed
that the molecules which at any given moment are found in a
plane layer will remain in a plane layer after they have moved
into any new position.
(c) Laminar Motion.— On this hypothesis the stream is
supposed to consist of an infinite number of indefinitely thin
layers. The variation in velocity from point to point of a
cross-section may then be allowed for by giving the several
layers different velocities based upon the law of fluid resistance
between consecutive layers.
3. Density; Compressibility; Head; Continuity.
The weight of ice per cubic foot at 23° F. is 57.2 Ibs.;
"freshwater" " " " 39.2° F. is 62.425 Ibs.;
" " "salt " " " " " 53° F. is 64 Ibs.;
"fresh " " " " " 53° F. is 62.4 Ibs.,
or 1000 kilog. per cubic metre.
The following table from the article on " Hydromechanics "
in the Encyc. Brit, gives the density of water at different
temperatures:
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
Temperature.
Density.
Weight
in Lbs. per
Cu. Ft.
Temperature.
Density.
Weight
in Lbs. per
Cu. Ft.
Cent.
Fahr.
Cent.
Fahr.
0
32
.999884
62.417
20
68
.998272
62.316
I
33-8
.999941
62.420
22
71-6
.997839
62.289
2
35-6
.999982
62.423
24
75-2
.997380
62.261
3
37-4
I . 000004
62.424
26
78.8
.996879
62.229
4
39-2
1.000013
62.425
28
82.4
.996344
62.196
5
4i
1.000003
62.424
30
86
•995778
62. 161
6
42.8
.999983
62.423
35
95
.994690
62.093
7
44.6
•999946
62.421
40
104
.992360
61.947
8
46.4
.999899
62.418
45
H3
.990380
61.823
9
48.2
•999837
62.414
50
122
.988210
61.688
10
50
.999760
62.409
55
131
.985830
61.540
ii
51.8
.999668
62.403
60
140
.983390
61.387
12
53-6
.999562
62.397
65
I49
.980750
6l.222
13
55-4
.999443
62.389
70
158
•977950
61.048
14
57-2
.999312
62.381
75
167
.974990
60.863
15
59
•999!73
62.373
80
176
.971950
60.674
16
60.8
.999015
62.363
85
185
.968800
60.477
17
62.6
.998854
62.353
90
194
.965570
60.275
18
64.4
.998667
62.341
IOO
212
.958660
59.844
19
66.2
.998473
62.329
Fluids are sensibly compressed under heavy pressures, and
the compression is proportional to the pressure up to about
1000 Ibs. (65 atmospheres) per square inch. Grassi's ex-
periments indicate that the compressibility of water diminishes
as the temperature increases.
TABLE OF ELASTICITY OF VOLUME OF LIQUIDS.
(Reduced from Grasses results.)
Liquid.
Elasticity of Volume.
Temperature.
Mercury . ..
717,000,000
o° C.
Water. ...
j 42,000,000
1 45,900,000
o° C.
18° C.
Sea- water..
52,900,000
Ether .
j 1 6, P. 80,000
o° C.
\ 15,000,000
14° C.
Alcohol. . ..
( 25.470,000
( 23,380,000
7.3°oC.
13-1 C.
Oil
44,090,000
N. B. — The value for mercury is probably erroneous.
If a volume Fof a fluid is compressed by an amount AV
under an increase Ap of the pressure, then
"
4 HYDRA ULICS.
AV
is called the cubical compression, and
V— -- is termed the elasticity of volume. This is sensibly
constant.
The vertical distance between the free surface of a mass of
water and any datum plane is called the head with respect to
that plane. If the water extends down to the level of the
plane, a pressure/ is produced at that level, and the value of pr
so long as the water is at rest, is given by the equation
^ = A+4,
u -fir.J^
w being the. specific weight of the water and /0 the pressure
at the free surface. Thus the pressure may be measured in
terms of the head, and hence the expression "head due to
pressure or pressure head."
The mean value of the atmospheric pressure is 14.7 Ibs. per
square inch.
A , , ( is equivalent to
A head Of a pressure of
2.3 ft. of fresh water I Ib. per sq. in.
2.25 ft. of salt water I Ib. per sq. in.
About 34 ft. of fresh water 14.7 Ibs. per sq. in.
" 33 ft. of salt " 14.7 Ibs. per sq. in.
A head of water is a source of energy. A volume of water
descending from an upper to a lower level may be employed
to drive a machine which receives energy from the water and
utilizes it again in overcoming the resistances of other machines
doing useful work.
Let Q cu. ft. of water per second fall through a vertical
distance of 1i ft. Then the total power of the fall = wQIi
ft. -Ibs. — —=— h. p., w being the weight of the water in
pounds per cubic foot.
Let K be the proportion of the total power which is
absorbed in overcoming frictional and other resistances. Then
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 5
the effective power of the fall = ze/(2^ (r ~ ^0> and the efficiency
is i - AT.
Imagine a bounding surface enclosing a space of invariable
volume in the midst of a moving mass of fluid. The principle
of continuity affirms that in any interval of time the flow into
the space must be equal to the outflow during the same inter-
val. Giving the inflow a positive and the outflow a negative
sign, the principle may be expressed symbolically by
= o.
The continuity of a mass of water will be preserved so long
as the pressure exceeds the tension of the air held in solution.
It is on account of the pressure of this air that pumps cannot
draw water to the full height of the water barometer, or about
34 ft.
Generally speaking, the pressure at every point of a contin-
uous fluid must be positive. A negative pressure is equivalent
to a tension which will tend to break up the continuity pre-
supposed by the formulae ; and should negative pressures result
from the calculations, the inference would be that the latter
.are based upon insufficient hypotheses.
The pressure in water flowing through the air cannot at
any point fall below the atmospheric pressure. There are cases,
however, as in water flowing through a closed pipe (Art. 3,
Chap. Ill), in which the pressure may fall below this limit and
become almost nil. But there is then a danger of the air held
in solution being set free, thus tending to interrupt the
•continuity of the flow, which may be wholly stopped if the air
is present in sufficient volume.
Consider a length of a canal or stream bounded by two
normal sections of areas Alf At, and let vlt v^ be the mean
normal velocities of flow across these sections. Then by the
principle of continuity
and the velocities are inversely as the sectional areas.
Again, assume that a moving mass of fluid consists of an
O HYDRA ULICS.
infinite number of stream-lines, and consider a portion of the
mass bounded by stream-lines and by two planes of areas Alt
AI at right angles to the direction of flow. If v^ , ^2 are the
mean velocities of flow across the planes,
V^AI =• Q = VyA9 if the fluid is incompressible.
Assuming that the fluid is compressible, and that the mean
specific weights at the two planes are wl and w9 , then the
weight of fluid flowing across Al is equal to the weight which
flows across A^ , since the weight of fluid between the two
planes remains constant. Hence
4. Bernouilli's Theorem. — This theorem is based on the
following assumptions :
(1) That the fluid mass under consideration is a steadily
moving stream made up of an infinite number of stream-lines
whose paths in space are necessarily fixed.
(2) That the velocities of consecutive stream-lines are not
widely different, so that viscosity, or the frictional resistance
between the stream-lines, is sufficiently small to be disregarded.
(3) That the fluid is incompressible, so that there can be no
internal zvork due to a change of volume.
In any given stream-line let a portion AB, Fig. I, of the
fluid move into the position A'B' in / seconds.
B B'
i' '
FIG. i.
Let al , pl , vl , zl be the normal sectional area, the intensity
of the pressure, the velocity of flow, and the elevation above
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. /
a datum plane ZZ of the fluid at A. Let tfa,/3, z>2, z^ denote
similar quantities at B.
Since the internal work is nil, the work done by external
forces must be equivalent to the change of kinetic energy.
Now the external work
= the work done by gravity -f- the work done by pressure.
But when the fluid AB passes into the position A'ft ', the
work done by gravity is equivalent to the work done in the
transference of the portion BB' , and therefore, t beng the
time.
the work dw by gr^^ty = wa^AA'-z^ — wa^-BB' '
= wQt (*>-*,),
since A A' = vj, BB' = vj, and alvl = Q = a.^-
Again, the work done by the pressures on the ends A and B
The work done by the pressure on the surface of the stream-
line between A and B is nil, since the pressure is at every point
normal to the direction of motion.
The change of kinetic energy
= kinetic energy of A' B' — kinetic energy of AB
= kinetic energy of BB' — kinetic energy of AA' ,
since the motion is steady, and there is therefore no change in
the kinetic energy of the intermediate portion A' B. Thus,
w v w V
the change of kinetic energy = - a^BB'^- -- a. A A—
w
Hence, equating the external worl{ and the change of kinetic
energy,
«>Qt (*, - *,) + & (A ~ A) = & -- ,
8 HYDRA ULICS,
which may be written in the form
w v? , w v? . .
««,+/>, + - -y = ^,+A + --> ... (i)
But A and B are arbitrarily chosen points, and therefore,
at any point of a stream-line, the motion being steady and
the viscosity nil, the gradual interchange of the energies due
to head, pressure, and velocity is expressed by the equation
w V* fa i
W2 j_ p _L = wH, a constant ; / ... . (3)
~r r \ g 2 VJ/
+/ I Is, ** I • V^ \M
frb*' f - vXm r- J V -i
or n -iVj / /
z being the elevation fef thevpoint above the datum line, / the
pressure at the point, w the specific weight, and v the velocity
of flow. This is Bernouilli's theorem.
Thus the total constant energy of wH ft.-lbs. per cubic foot
of fluid, or H ft.-lbs. per pound of fluid, is distributed uniformly
along a stream-line, wH being made up of wz ft.-lbs. due to
w z?
head,/ ft.-lbs. due to pressure, -- ft.-lbs. due to velocity,
and H being made up of z ft.-lbs. due to head, — ft.-lbs. due
v*
to pressure, and — ft.-lbs. due to velocity.
Assuming that
(a) the motion is steady,
(ft) the frictional resistance may be disregarded,
(c) the fluid is incompressible,
Bernouilli's theorem may be applied to currents of finite size
at any normal section, if the stream-lines across that section
are sensibly rectilinear and parallel. There is then no interior
work due to a change of volume, and the distribution of the
pressure in the section under consideration will be the same as
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 9
if the fluid were at rest, that is, in accordance with the hydro-
static law. This is also true whether the flow takes place under
atmospheric pressure only, or whether the fluid is wholly or
partially confined by solid boundaries, as in pipes and canals,
or whether the flow is through another medium already occu-
pied by a volume of the fluid at rest or moving steadily in a
parallel direction. In the last case there must necessarily be
a lateral connection between the two fluids, but the pressure
over the section must follow the hydrostatic law throughout
the separate fluids, and there can be no sudden change of
pressure at the surface of separation, as this would lead to an
interruption of the continuity.
The hypotheses, however, upon which these results are based
are never exactly realized in actual experience, and the results
can only be regarded as tentative. Further, they can only ap-
ply to an indefinitely short length of the current, as the viscosity,
-which is proportional to the surface of contact, would other-
wise become too great to be disregarded.
5. Applications. — If a glass tube, open at both ends, and
called a piezometer (TrieCeiv, to press ; jterpor, a measure)
is inserted vertically in the cur-
rent, Fig. 2, at a point N, z ft.
above the point O in the datum
line, the water will rise in the
tube to a height MN dependent
upon the pressure at N. The
effect of the eddy motion produced
at N by obstructing the stream-
lines may be diminished by mak-
ing this end of the tube parallel
to the direction of flow. Neglect-
ing altogether the effect of the o
eddies, and taking/ to be the in- FlG- 2-
tensity of the pressure at TV, and/0 the intensity of the atmos-
pheric pressure, then,
w w
10
and therefore
HYDRA ULICS.
w w
= ON + MN + -
1 w
= Q M + -.
1 w
(5)
The locus of all such points as M is often designated " the
line of hydraulic gradient," or the " virtual slope," terms also-
used when friction is taken into account.
Let the two piezometers AB, CD, Fig. 3, be inserted in the
current at any two points B and D, z^ ft., and z% ft. respect-
ively above the points E and F in the datum line.
FIG. 3.
Let /, be the intensity of the pressure at B in pounds per
square foot, /2 that at D, and let the water rise in these tubes
to the heights BA, DC. Then
w
and therefore
= zl+, and l
w w
w
• + - + =^-^=^' . . (6)
the line AG being parallel to the datum line.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
II
Thus
, (zl -|- — J — Ls2 + — J is equal to the fall of the free
surface level between the points B and D.
Let vl , 7'2 be the velocities of flow at B and D. Then by
Bernoulli's theorem
W 2g W 2g
and therefore the fall of free surface level between B and D
(7)
W
2g
Equation (7) may also be written in the form
V? V? , I , Pi\ ( . Pl\ V? . rr SQ\
- = r (Zi H J — l*i H } = r £k» . (8)
2g 2g ^ W' * W' Zg
so that the velocity at D is equal to that acquired by a body
with an initial velocity vl falling freely through the vertical
distance CG.
Froude illustrated Bernouilli's theorem experimentally by
means of a tube of varying section, Fig. 4, conveying' a current
FIG. 4.
between two cisterns. The pressure at different points along
the tube is measured by piezometers, and it is found that the
12 HYDRAULICS.
water stands higher and the pressure is therefore greater, where
the cross-section is larger and the velocity consequently less.
If the section of the throat at A is such that the velocity is
that acquired by a body falling freely through the vertical dis-
tance h between A and the surface level of the water in the
cistern, and if / be the pressure at A, and z the elevation of A
above datum, then, neglecting friction,
W 2g W
But v* = 2gh, and therefore / = pQ , so that the pressure at
A is that due to atmospheric pressure only. Thus, a portion
of the pipe in the neighborhood of A may be removed, as in
the throat of the injector.
Again, let the cross-section in the throat at B be less than
that at A. The pressure at B will be less than the atmospheric
pressure, and a column of water will be lifted up in the curved
piezometer to a height k' .
Let tf , , zl ,pl , vl be the sectional area, elevation above datum,
pressure, and velocity at B.
Let #3 , z^ ,pi , z>a be similar symbols at E.
Then
I J ,, + A + ^+A ^ = , l + A_*' + !i2. (9)
V W 2g W ' 2g W ' 2g
Put //, = #,-[- — -, the height above datum to which the
w
water is observed to rise in the piezometer inserted at E, and
also let #;=*, + A - h'. Then
w
since ap^ = alvl , #a being the sectional area at E. Therefore
ft., — a,
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 1 3
an equation giving the theoretical velocity of flow at the throat
B. Hence the theoretical quantity of flow across the section
at B is
- a
(10)
This is the principle of the Venturi water-meter and also of
the aspirator.
The actual quantity of flow is found by multiplying equa-
tion (10) by a coefficient C whose value is to be determined
by experiment.
If the pressure at E is positive, then //, is merely the
height to which the water is observed to rise in an ordinary
piezometer inserted at E.
Again, Froude also points out that when any number of
combinations of enlargements and contractions occur in a pipe,
the pressures on the converging and diverging portions of the
pipe will balance each other if the sectional areas and directions
of the ends are the same.
6. Orifice in a Thin Plate. — If an opening is made in the
wall or bottom of a tank containing water, the fluid particles
FIG. 6.
FIG. 7.
immediately move towards the opening, and arrive there with
a velocity depending upon its depth below the free surface.
The opening is termed an " orifice in a thin plate " when the
water springs clear from the inner edge, and escapes without
again touching the sides of the orifice. This occurs when the
UNIVERSITY
14 HYDRA ULICS*
bounding surface is changed to a sharp edge, as in Fig. 5, and
also when the ratio of the thickness of the bounding surface
to the least transverse dimension of the orifice does not exceed
a certain amount which is usually fixed at unity, as in Figs.
6 and 7.
Owing to the inertia acquired by the fluid filaments there
will be no sudden change in their direction at the edge of the
orifice, and they will continue to converge to a point a little
in front of the orifice, where the jet is observed to contract to
the smallest section. This portion of the jet is called the vena
contracta or contracted vein, and the fluid filaments flow across
the minimum section in sensibly parallel lines, so that here, if
the motion is steady, Bernouilli's theorem is appli-
c cable.
The dimensions of the contracted section and
F its distance from the orifice depend upon the form
and dimensions of the orifice and upon the head
of water over the orifice.
Let Fig. 8 represent the portion of the jet be-
tween a circular orifice of diameter AB and the
contracted section of diameter CD, EF being the distance
between AB and CD. Then, taking the average results of a
number of observations, it is found that AB, CD and EF are
in the ratios of 100 to 80 to 50.
Thus the areas of the contracted section and of the orifice
are in the ratio of 16 to 25, and, generally speaking, this is
assumed to be the ratio whatever may be the form of the
orifice.
7. Torricelli's Theorem. — Let Fig. 9 represent a jet issu-
ing from a thin-plate orifice -in the side of a vessel containing
water kept at a constant level AB.
Let XXbz the datum line, J/A^the contracted section, and
consider any stream-line mn, m being in a region where the
velocity is sensibly zero, and n in the contracted section. Then
by Bernouilli's theorem, the motion being steady,
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 1 5
/, /, being the pressures at n and ;«, and zt 2l their elevations
above datum. Hence
(2)
FIG. 9.
If the flow is into the -atmosphere,
/ — the atmospheric pressure = /0 , and
p, = w.Om +/.,
O being the point in which the vertical through m intersects
the free surface. Thus,
— = z . — z + Om =
2g
(3)
h being the depth of n below the free surface.
The result given by equation (3) was first deduced by Tor-
ricelli.
The depth below the free surface is very nearly the same
for all points of the contracted vein, and the value of v as
given by (3) is taken to be the theoretical mean velocity of
flow across the contracted section.
Equation (3) is equivalent to the statement that when the
orifice is opened the hydrostatic energy of the water, viz., // ft.-
7'2
Ibs. per pound, is converted into the kinetic energy of —
10 i HYDRA ULICS.
ft.-lbs. per pound. Thus, if the jet is directed vertically upwards,
it will very nearly rise to the level of the free surface, and would
reach that level were it not for air resistance, or for viscosity, or
for friction against the sides of the orifice, or for a combination
of these retarding causes.
If the jet issues in any other direction, it describes a para-
bolic arc of which the directrix lies in the free surface.
Let OTV, Fig. 10, be such a jet, its direction at the orifice
FIG. 10.
at O making an angle a with the vertical. With a properly
formed orifice a greater or less length of the jet will have the
appearance of a glass rod, and if this portion were suddenly
solidified and supported at the ends, it would stand as an arch
without any shearing stress in normal sections.
Again, the horizontal component of the velocity of flow at
any point of the jet is constant (= v sin or), so that, for the
unbroken portion of the jet, equidistant vertical planes will
intercept equal amounts of water, and the height of the C. G.
of the jet above the horizontal line O V will be two thirds of
the height of the jet.
8. Efflux through an Orifice in the Bottom or in the Side
of a Vessel in Motion. — If a vessel containing water zit. deep
ascend or descend vertically with an acceleration /, the press-
ure/ at the bottom is given by the equation
w
± -*/ = p - p0 - wzy
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
being the atmospheric pressure. Therefore
/-A
w
=•(
« ±4
If now an orifice is opened at
the bottom, the velocity of efflux v
is still taken as due to the head of
the pressure /, and therefore by
Torricelli's Theorem
Let W, be the weight of the
vessel and water, and let the vessel be connected with a
counterpoise of weight W^ by meansof a rope passing over a
pulley. Then by Newton's second law of motion, and neglect-
ing pulley friction,
g~ W, W, W.+ Wt'
T being the tension of the rope.
Next let a cylindrical vessel,
Fig. 12, of radius r and containing
water, rotate with an angular veloc-
ity oo about its axis. The surface
of the water assumes the form of a
paraboloid of which the latus
2fT
rectum is — ^. If an orifice is made
GO
at Q in the side of the vessel, the
water will flow out with a velocity
v due to the head of pressure at
FIG. 12.
the orifice. This head is PQ, and
PQ = ON ± z =
CA)
z being the vertical distance OM between the orifice and the
vertex of the paraboloid. Hence by Torricelli's theorem
i8
HYDRA ULICS.
*'
or
9. Application to the Flow through a Frictionless Pipe
of Gradually Changing Section (Fig. 13).— Let the pipe be
supplied from a mass of water of which the free surface is H ft.
above datum.
Let al9 pv vv be the sectional area, pressure, and velocity of
flow at any point A, zl ft. above datum and h^
ft. below the free surface.
Let#a,/3, z/3 be similar symbols for a second point B, ^ ft.
above datum and h^ ft. below the free surface.
FIG. 13.
Then by the condition of continuity
a&i = atvt,
and by Torricelli's theorem
2g
/J
t
IV
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
and
2g ~ *"» W
Hence
so that Bernouilli's theorem, viz.,
— + - + 2 = & +— = a constant,
2£- ' w ' ' w
holds true for the assumed conditions.
10. Hydraulic Resistances — (a) Coefficient of Velocity. —
In reality, the velocity v at the vena contracta is a little less
than V2gh (Art. 7, eq. 3) and the ratio of v to V ' 2gh is called
the coefficient of velocity, and may be denoted by cvt so that
v = cv
Again, the equations for the velocity of discharge in the
case of moving vessels now become
2g
and
A mean value of cv for well-formed simple orifices is .974.
An easy method of determining the value of cv, experi-
mentally, may be indicated by reference to the jet represented
in Fig. 10, p. 1 6.
Measure the vertical and horizontal distances from the
orifice of any two points A, B in the jet.
Let jj>,, x^ denote the co-ordinates of A.
Let yv x^ denote the co-ordinates of B.
Then if tl is the time occupied by a fluid particle in moving
from the orifice to A, and t^ the time from the orifice to B,
2O HYDRA ULICS.
— v sin a . tl ; j^ = v cos <* . /, -- £'/12 ;
^ = v sin a . /3 ; J2 = ^ cos ar . /3 -- gt*.
x
,=^ cot a- , .a ,
2 z;2 sin2 a
P-
= x cot a — -
2
2 v* sin3 a'
By means of the two last equations
and
2 sin a (xl cot a — yj
so that
*t
'
Hence /> ,
and
4^ sin2 a (xl cot a — y^ '
and since the values of x^ yv x» y^ are known, equation I
will give the value of a, and equation 2 the value of cv.
Note. — If the jet issues from the orifice horizontally, a = 90°, and the
last equation becomes
so that the position of one point only relatively to the orifice need be
observed.
(b) Coefficient of Resistance. — Let hv be the head required
to produce the velocity v. Let hr be the head required to
overcome the frictional resistance. Then
h, the total head, — hv-\- hr = hv(i -}- cr),
where hr = trhv.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 21
cr is termed the coefficient of resistance, and is approxi-
mately constant for varying heads with simple sharp-edged
orifices. Again,
Hence
and therefore
so that cr can be found when cv is known, and vice versa.
(c) Coefficient of Contraction. — The ratio of the area a of
the vena contracta to the area A of the orifice is called the co-
efficient of contraction, and may be denoted by cc.
The value of cc must be determined in each case, but in
sharp-edged orifices an average value of cc , as already pointed
out, is — - = .64. C<zteris paribus, cc increases as the orifice area
and the head diminish.
The following are some of the conditions which tend to
modify the value of cc :
(1) The contraction is imperfect and will be suppressed
over the lower edge of a square orifice at the bottom of a ves-
sel, and over a side as well if the orifice is in a corner. In fact,
the contraction is more or less imperfect for any orifice within
three diameters from the side or bottom of the vessel. Thus,
the cross-section of the vena contracta is in-
creased, and experiment shows that the dis-
charge is also increased.
(2) ce is increased or diminished according
as the surface surrounding the orifice is convex
or concave to the interior of the vessel.
(3) The contraction is imperfect and ce is FIG. 14.
increased if the orifice is placed in a confined
part of the vessel or if it approaches the orifice through a chan-
nel, as in Fig. 14, the velocity of the fluid filaments being
thereby considerably increased.
22
HYDRA ULICS.
(4) If the inner edge of an orifice is rounded, as shown by
Figs. 15 and 16, the contraction is more or less imperfect.
FIG. 15.
FIG. 16.
The value of ce varies from .64 for a sharp-edged orifice to very
nearly unity for a perfectly rounded orifice.
(5) The contraction is incomplete when a border or rim is
placed round a part of the edge of the orifice, projecting in-
wards or outwards. According to Bidone,
and
•cc = .62(1 + .152 -) for rectangular orifices,
ce = .62! i -j- .128 -) for circular orifices,
n being the length of the edge of the orifice over which the
border extends, and p the perimeter of the orifice.
(6) If the sides of the orifice are curved so as to form a
bell-mouth of the proportions shown by Fig. 17, and corre-
j* 1-.6
FIG. 17.
^
spending approximately to the shape of the vena contracta,
the whole of the contraction will take place within the bell-
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 2$
mouth, and cc is unity if the area of the orifice is taken to be,
the area of the smaller end.
For such an orifice Weisbach gives the following table of
values of c, :
Head over Orifice in Feet.
.66
i .64 ,
11.48
5577
FIG. 18.
cv.
959
967
975
994
337-93 994
The dimensions of the jet at the contracted section or at
any other point may be directly measured by
means of set-screws of fine pitch, arranged
as in Fig. 18. The screws are adjusted so as
to touch the surface of the jet, and the dis-
tance between the screw-points is then meas-
ured.
(d) Coefficient of Discharge. — If Q is the
quantity of flow per second across the con-
tracted section, then
— cccvA V2gh = cA
where c — cccv is the coefficient of/discharge,j and is to be de-
termined by experiment.
The values of c in thousandths for orifices of different
forms, given in tables A and B, have been deduced by the
author from an extended series of experiments carried out in
the hydraulic laboratory, McGill University.
The experimental tank is about 30 ft. in height and its
horizontal section is square, with an interior area of 25 sq.
ft. The inside faces of the tank are plumb, and there are
no projections to interfere with the stream-lines.
The letters T and S at the head of the columns respec-
tively indicate that the orifice is in a plate of thickness .16 in.,
or is sharp-edged.
HYDRA ULICS.
TABLE A.
OF VALUES OF c FOR ORIFICES OF .197 SQ. IN. IN AREA.
•£ •
S
||
._
Jg
|K
15
'*j D
1 5
•£»
_u
a
o
OJ ^
<D c
u c iC
||
u
ho
>eo
JS ° '
>^:s
>Cc „
H3
'Cc75
>
Q
s^
•S "^
."§-^
•"=-•5
^- *c5
rfj
J3
^ S rC
^ 5 •«
^ 3 a
•^ 5 *+-(
6
"o
9
l.L
1';
||
^"fe
III
i» crd
i^s
Hi
s
3
1*1
rt-o
ca u
5-0 4)
!.•§•-
•S'O §
£
£
C
w
C/3
F
4;C75^
IK<S
cS
•|Kh
Head
T
S
T
S
T
S
S
T
S
s
S
S
in Feet.
I
624
618
627
627
623
628
623
635
640
641
658
659
2
616
6n
620
621
6I3
621
619
626
633
632
646
646
4
610
607
615
615
606
617
614
619
629
629
637
637
6
607
605
6I3
613
604
614
612
616
625
627
634
633
8
606
604
612
612
603
612
612
614
625
625
631
63I
10
606
604
611
611
602
610
611
612
624
623
630
629
12
605
603
611
611
60 1
610
611
611
622
622
627
626
14
604
603
610
610
600
610
609
6n
622
621
624
625
16
606
602
610
610
600
610
609
610
620
621
624
624
18
605
602
610
610
600
610
609
609
620
620
623
623
20
604
60 1
609
609
600
610
609
602
620
620
622
622
TABLE B.
OF VALUES OF c FOR FOUR ORIFICES OF .0625 SQ. IN. IN AREA, AND FOR
ONE TRIANGULAR ORIFICE OF .05 SQ. IN. IN AREA.
Form
of -
Orifice.
Circular.
Equilateral
Triangle with
Horizontal
Base
Square with
Vertical Sides
Rectanc
Vertica
equal u
the W
de with
Sides.
> Twice
r;^»H
Rectangle with
Vertical Sides
equal to
Four Times
uppermost.
the Width.
Head
in Feet.
T
S
T
S
T
S
T
S
T
S
I
678
620
657
631
643
627
662
640
688
67I
2
618
613
646
623
63I
621
643
629
655
657
4
610
605
628
616
620
615
63I
620
642
643
6
607 •
601
628
613
615
612
627
616
634
636
8
606
60 1
621
610
612
609
624
613
631
632
10
fo4
600
6l8
608
6I3
608
621
613
629
629
12
663
598
'6l7
607
6n
606
621
611
626
627
14
602
598
6l7
607
610
606
620
610
623
625
16
602
598
616
606
609
606
619
609
622
625
18
60 1
597
6i5
605
607
605
618
608
622
623
20
60 1
597
615
605
607
604
618
608
621
622
T
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 2$
The jet springs clear from the orifice in all cases repre-
sented in Tables A and B.
The following inferences may be drawn from an inspection
of Tables A and B :
(1) The coefficient of discharge diminishes as the head in-
creases, but at a diminishing rate.
(2) The coefficients for the thick-plate orifices are in all
cases greater than the corresponding coefficients for sharp-
edged orifices, excepting in the case of the longest rectangular
orifice in Table B. Under a head of I ft. the coefficient of
discharge for this orifice still exceeds that of the same orifice
with sharp edge, but for heads exceeding I ft. the coefficient
seems to be a little less, but is practically the same. It may
be noted that the thickness of the plate is 2.56 times the
width of the orifice, and the contraction for the thick-plate
orifice is consequently increased.
(3) The coefficient for rectangular orifices seems to be
practically the same whether the longest side is vertical or
horizontal.
(4) The coefficient increases with the area of the orifice,
excepting when the head is very small. The coefficient for
orifices of small area then rapidly increases, as shown in
Table B.
(5) With rectangular orifices the coefficient increases as
the width of the orifice diminishes, i.e., as the orifice becomes
more elongated.
The two last results are in accordance with similar results
deduced by Weisbach, Buff, and others.
The coefficient of discharge is modified when the edges of
the orifices are not sharp, but have a sensible thickness, and
the formula giving the discharge may be written
Q = cA J^H>
H being the depth of the axis of the orifice below the free
surface.
II. Miner's Inch. — The miner's inch is a term applied to
the flow of water through a standard vertical aperture, one
square inch in section, under an average head of 6£ inches.
26
HYDRA ULICS.
Taking c = .62,
the flow = Q
= .62 A
= -62 x
= i£ cu. ft. per minute, approximately.
The term is more or less indefinite, as the different companies
in disposing of water to their customers do not always use the
FIG. 19.
same head, and the flow is thus found to vary from 1.36 to 1.73
cu. ft. for each square inch of aperture.
The aperture is usually 2 in. deep and may be of any re-
quired width, Fig. 19. The upper and lower edges of the
aperture are formed by ij-in. planks, the lower edge being 2
in. above the bottom of the channel, and the plank forming
the upper edge being 5 to 5^ in. deep, so that the head over
the centre of the aperture is from 6 to 6^ inches.
12. Energy and Momentum of the Jet.
The energy of the jet = wav — ft.-lbs. per second
wav
ft.-lbs. per second
s. « ^
*•<
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 2?
= wavhc* ft.-lbs. per second
wavhc* , N
= - - h. p. (horse-power)
p (= wh} being the hydrostatic pressure due to the head h.
w
The momentum of the jet ~ - av . v — wa - = 2wakc*
<£• o
and this is equal to the pressure in pounds produced by the jet
against a fixed plane perpendicular to its direction. Neglect-
ing cv*, the thrust is double the hydrostatic pressure due to
the head h.
13. Inversion of the Jet. — The phenomenon of the inver-
sion of the jet was first noticed by Bidone, and has been subse-
quently investigated by Poncelet, Lesbros, Magnus, Lord
Rayleigh, the author, and others. When a jet issues from an
orifice in a vertical surface, the sections of the jet at points
along its path assume singular forms dependent upon the
nature of the orifice.
Figs. 20 to 27 are from photographs (taken from the same
point) of jets issuing under the same head, viz., 12 ft., from
orifices of different forms and sizes. The dimensions of these
jets are comparable with the jets shown by Figs. 20 and 21,
which are issuing from circular orifices of I in. and J in.
diameter, respectively.
With a square orifice, Fig. 22 (side = I in.), Fig. 23 (side =
.443 in.), and Fig. 24 (side = .25 in.), the section is a star of
four sheets at right angles to the sides.
With a triangular orifice, Fig. 25 (side = .676 in.), the sec-
tion is a star of three sheets at right angles to the sides.
In general, with a polygonal orifice of n sides the section
will be a star of n sheets at right angles to the sides.
Fig. 26 is a jet from a rectangular orifice (J in. X J in.), its
section near the orifice being a star of four sheets.
Fig. 27 is a jet from a semi-circular orifice (diar. — .388 in.),
FIG. 20.
FIG. 21.
FIG. 22.
FIG. 23.
FIG. 24.
FIG. 25.
Fro. 26.
FIG. 27.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 2Q
the section near the orifice being a rounded boundary and a
single sheet at right angles to the diameter.
The changes in the form of the jet are doubtless due to the
mutual action between the fluid particles. A filament issuing
horizontally and freely at B, Fig. 28, has a velocity 2g . AB, and
FIG. 28.
describes a certain parabola BD. A filament issuing horizon-
tally and freely at a lower level C has a velocity 2g . AC, and
describes a parabola CD of less curvature than BD. Now the
two filaments cannot pass simultaneously through the point of
intersection D, and must necessarily press upon each other-
They are thus deviated out of their natural paths, and the jet
spreads out into sheets, as described above.
If the orifice is small and the head not large, the jet, on
leaving the contracted section at the orifice, spreads out
into sheets, and then diminishes to a contracted section similar
to the first, after which it again spreads out into sheets, bisect-
ing the angles between the first set of sheets, and again dimin-
ishes to a contracted section. This action is repeated so long
as the jet remains unbroken.
14. Emptying and Filling a Canal Lock.— When the
head varies, as in filling or emptying a reservoir or a lock, in
filling a vessel by means of an orifice underwater, or in empty-
ing water out of a vessel through a spout, Torricelli's theorem
is still employed.
If the lock or vessel is to be filled, Fig, 29, let X sq. ft. be
the area of the water-surface when it is x ft. below the surface
of the outside water.
3° HYDRA ULICS.
If the lock or vessel is to be emptied, Fig. 30, then X sq. ft.
is the area of the water-surface when it is x ft. above the orifice.
FIG. 29.
FIG 30.
In each case JIT ft. is the effective head over the orifice, and
is the head under which the flow takes place.
In the time dt the water-surface in the lock or vessel will
rise or fall by an amount dx. Then
— A .dx = quantity which has entered the lock
= cA <J~2gx . dt,
A being the area of the orifice.
Hence
t =
Xdx
cA
an equation giving the time of filling or emptying the lock
between the level x and h. The value of c for submerged
orifices seems to be somewhat less than when the flow occurs
freely, but it is usual to take .6 or .625 as a mean value.
15. General Equations. — Bernoulli's theorem may be
easily deduced from the general equations of fluid motion, as
follows:
Let/ be the pressure and p the density at any point whose
co-ordinates parallel to the axes are x, y, 2.
Let «, v, w be the velocities of flow at the same point
parallel to the axes, and let X, Y, Z be the accelerating forces.
Then three equations result from the principle of the equality
of pressure in all directions, viz. :
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 3!
I dp d(u) du du du du
-p-dX^x--^r = x—dt-uTx-v^-w^ <o
I dp d(v) dv dv dv dv
pdj=Y ^i=Y-7t-uTx-v-dj-w-dz' &
I dp d(w) dw dw dw dw
~j~ = Z j' = Z T~ ~ u 3 v ~i w ~r\ fa}
pdz dt dt dx dy dz' u;
If the motion is steady, so that the velocity at any point is
r ^ r 4.1, v 1 4.U du dv dw
a function of the position only, then ^-— = o = -=- — — and
dt dt dt
u, v, w may be expressed as the differential coefficients of a
function F. Thus,
dF dF dF
u = —j-', v = —j—', w = -7-;
dx dy dz
and therefore
du d*_F_ _ ^
dy ~~ dydx ~~ dx'
du d*F _ dw
dz . dzdx ~~ dx '
dv _ d*F _ dw
dz ~~ dzdy ~~ dy '
Hence equations i, 2, and 3 may be written
I dp du dv dw
~~:r—X—u-j v-j w -7- ; ... (4)
p dx dx dx dx
i dp du dv dw
--^-=Y—u-j v-r — w -7-; . . . (5)
p dy dy dy dy'
i dp du dv ' dw
r- = Z — u —, v —r — w —jr. . . . (6)
p dz dz dz dz ^ }
32 HYDRAULICS.
Multiplying eq. 4 by dx, eq. 5 by dy, and eq. 6 by dz, and
adding, then
r dy ' ' dz
Idv , . dv dv \
— vi-j-dx 4- -j-dy + -rdz\
\dx ' dy J ' dz /
/ , dw dw
which may be written
dp
-— Xdx-\- Ydy + Zdz — (udu + v dv + wdw).
Integrating, and assuming the fluid to be homogeneous,
u* 4- v* 4- wz
- —^- — + a constant.
Hence, if gravity is the only force, and if V is the resultant
velocity at the point, then
and the last equation becomes
P C V*
- — — J gdz --- \- a constant
and therefore
= — gz — — + a constant ;
p Fa
z -\ -I = a constant.
Pg ^g
16. Loss of Energy in Shock. — An abrupt change of sec-
tion at any point in a length of piping destroys the parallelism
of the fluid filaments, breaks up the fluid, and energy is dis-
sipated in the production of eddy and other motions. The
energy thus wasted is termed " energy lost in shock."
FLOW THROUGH ORIFICES, OV.
33
In a short length of piping, where the section suddenly
changes from A'B' to EF, consider the fluid mass between the
two transverse sections AB, where the motion of the fluid fila-
FIG. 31.
ments has been undisturbed and is in parallel lines, and CD*
where the parallelism has been again re-established.
In an indefinitely short interval of time t let the mass move'
forward into the position bounded by the plane sections A ' R"
and CD'.
Let #„ 2/,, /\ be the sectional area, velocity of flow, and mean
intensity of pressure at A'B'.
Let a9, ^,, A be similar symbols for CD'.
Let z, , <8-2 be the elevation above datum of the C. G.s of
the sectional areas at A'B' and CD'.
Let //be the vertical distance between the C. G.s of the areas
Let P be the mean intensity of pressure over the annular
surface between £F and A'B'.
The resultant force acting in the direction of motion upon
the mass of fluid under consideration
= component of weight of mass in this direction
-f- pressure on A'B'
-[-pressure on annular surface between EF and A'B'
— pressure on CD'
34 HYDRAULICS.
~ l
assuming that P = fil, or that the mean intensity of pressure is
unchanged throughout the whole of the section EF.
The normal reaction of the pipe-surface between EF and
CD' has no component in the direction of motion, and fric-
tional resistances are disregarded.
Hence the impulse of the resultant force
(p, - A) /
= change of momentum in the same direction
of the fluid masses CDD'C' and ABB' A',
since the momentum of the mass between
A'B' and CD remains unchanged
w w
= -a,v,.v,t--alv,.vlt
IV
= - a&S - v^t,
o
since by the condition of continuity
alvl = a^.
Dividing throughout by the factor waj, the equation be-
comes
* ,/, A < ^.
z. — z« — n H — — — = — — -- ,
1 w w g g
which may be written in the form
Now the pipes are nearly always axial, and in such case
h = o, so that the last equation becomes
.A , V= - A i ».' , (».-».)'
'~*~W~r2g ' *~f~W~r2g~T 2g
FLO W THROUGH ORIFICES, OVER WEIRS, ETC.
35
If there had been no abrupt change of section, or if the
change between A ' B' and CD had been gradual, then no in-
ternal work would have been done in destroying the parallelism
of the fluid filaments, and no energy wasted. Therefore, by
Bernoulli's theorem, the relation
would have held good.
Thus, ^-2 -- - ft.-lbs. of energy per pound of fluid is the
loss in shock between A' B' and CD.
Experiment justifies the assumption P = pl.
17. Mouthpieces. — (a) Bor das Mouthpiece. — This is merely
a short pipe projecting inwards, as in Fig. 32, representing
FIG. 32.
a jet flowing through a re-entrant mouthpiece of sectional
area A, fixed in the vertical side of a vessel of constant hori-
zontal section and containing water kept at a constant level.
The mouthpiece is as long as will allow of the jet springing
clear from the end EF without adhering to the inside surface.
The velocity of the fluid molecules along AC and DK is
sufficiently small to be disregarded, so that the pressure over
this portion of the vessel is distributed in accordance with
36 HYDRAULICS.
the hydrostatic law. The same may also be said of the pressure
on the remainder of the vessel's surface.
Again, the only unbalanced pressure is that on the surface
HG immediately opposite the mouthpiece, and the resultant
horizontal force in the direction of the axis of the mouthpiece
= (A + wti)A — pQA = whA,
h being the depth of the axis below the water-surface and /a
the intensity of the atmospheric pressure.
The jet converges to a minimum or contracted section MN
of area a.
In a unit of time let the fluid mass between AB and MN
take up the position bounded by A 'B and M' N' . Then
whA = impulse of force in direction of motion
= change of momentum in same direction in a unit
of time.
= difference between the momenta of MNN' Mr
and ABB'A', since the momentum of the mass
between A' B' and MN remains unchanged
= momentum of MNN' M' , since the momentum of
ABB' A' is vertical
w w
= — av . v = — av ,
g g
v being the mean velocity of flow across the contracted section.
Hence
w w
whA = — av = — a. 2gh,
g g
and therefore
A — 2a,
or
1 a
- = -j = coefficient of contraction.
2 A
This result has been very closely verified by experiment, the
coefficient having been found to be .5149 by Borda, .5547 by
Bidone, and .5324 by Weisbach.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 37
Borda's mouthpiece gives a smaller discharge than a sharp-
edged orifice, but a discharge which is much more uniform, and
hence it is generally used in vessels from which water is to be
distributed by measure.
Note. — Let Fig. 33 represent a jet flowing through a re-
entrant mouthpiece of sectional area A fixed in the sloping
side of a reservoir containing water kept at a constant level, and
suppose that the reservoir is of such size that GHKL may
represent a cylindrical fluid mass coaxial with the mouthpiece
and so large that the velocity at its surface is sensibly nil.
Let ti, h be the depths below the water-surface of the C. G.'s
of the areas GH and KL, respectively.
T7\.i
FIG. 33-
Then the resultant force along the axis of the mouthpiece
— pressure on GH — pressure on CK and on DL
— pressure on EF
-f- component of the weight of the fluid
mass GHKL
— (po _[_ whf) area GH — (p0 + wJi) (area CK ' + area DL)
— p, . area EF + w . area GH . GK . -75^-1 very nearly
LrK.
— whA.
38 HYDRA ULICS.
Hence, in a unit of time,
whA = impulse of this force
= change of momentum in direction of axis
w w w
— —av . v = —av = — a . 2gh,
g g g
a being the area of the contracted section, while h is also very
approximately the depth of its C.G. below the water-surface.
Thus, as before,
the coefficient of contraction = — = -.
A 2
(b) Ring-Nozzle. — The ring-nozzle (see Fig. 34) is often
used with a fire-engine jet, and
consists of a re-entrant pipe of
sectional area #, fixed in a pipe
of sectional area av The length
of the re-entrant portion is such
that the water springs clear from
the inner end and, without again
touching the surface of the
. mouthpiece, converges to a mini-
FIG. 34 mum or contracted section of
area a at MN.
Consider the fluid mass between MN and a transverse sec-
tion AB, and in a unit of time let it move into the position
bounded by the planes MN1 and A'B'.
It is assumed that the motion is steady and that there is no-
internal work due to the production of eddies or other motions.
Let />0 , v be the intensity of the atmospheric pressure and
the velocity at MN.
Let p^ , vl be the mean intensity of pressure and the veloc-
ity at AB.
Let P be the mean intensity of the pressure over the annu-
lar surface EF, GH.
Let #0 , 2l be the elevations above datum of the C. G.s of
the sections MN and AB.
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 39
Then
ow.fo - *.) + P& - P(a, - a,) —p. a,
impulse in direction of motion
change of momentum in same direction in a unit of time
difference of the momenta of the fluid masses MNN'M' and
ABB' A'
~(av* - a,*,")-
Assuming that P = piy the last equation becomes
IV
wa,(2l - *.) + a,(p, - p.) = —(av* — a,v*). . . (i)
o
By Bernoulli's theorem,
A *>? p. v*
and therefore
W 2g
Now s, — z0 is very small and may be disregarded without
sensible error, and then by eqs. (i) and (2)
v* — v? _ pl — p0 _ i av* —
2g w ~ g a,
Hence
2
_
a ~ av* — av ~~ aa — a*av* ~~ a
since a9vt = av.
If the sectional area #a of the pipe is very large as compared
with a, so that -- may be disregarded without sensible error,
then — = -, and therefore the coefficient of contraction
a, a
= — = -, as before.
HYDRA ULICS.
(c] Cylindrical Mouthpiece. — Whe-n water issues from a
cylindrical mouthpiece (see Fig. 35) at least two to two and
one half diameters in length, the
jet issues full bore or without
contraction at the point of dis-
charge.
If A be the sectional area of
the mouthpiece, h the depth of
its axis below the water-surface,
and Q the amount of the dis-
charge. Then experiment shows
that
Q = .S2A |/^. . (i)
The coefficient .82 is the pro-
duct of the coefficients of veloc-
ity and contraction, but the co-
efficient of contraction is unity,
FIG 35.
and therefore the coefficient of velocity is .82. Now the
mean coefficient of velocity in the case of a simple sharp-
edged orifice is .947, and the difference between .947 and .82
cannot be wholly accounted for by frictional resistances, but
is in part due to a loss of head. In fact, the water as it clears
the inner edge of the mouthpiece converges to a minimum sec-
tion MN of area a and then swells out until at M' N' it again
fills the mouthpiece.
Energy is wasted in eddy motions between MN and M'N',
where the action is similar to that which occurs at an abrupt
change of section.
Let /, v be the intensity of the pressure and the mean
velocity of flow at the point of discharge.
Let />, , v1 be similar symbols for the contracted section MN.
Let />„ be the intensity of the atmospheric pressure.
Remembering that
is the loss of head "due to
shock " between MN and M'N, then by Bernoulli's theorem
• • (2)
, A = A v = /_, v
•" W ~ W ' 2g W*2f
2g
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 4!
Hence
W 2g
and
(I-)'!
where 4. = coefficient of contraction = — = — . Therefore
an equation giving the velocity of flow at the point of discharge.
If the discharge is into the atmosphere,^ =/ and equation
4 becomes
* '
where
/ \.
1--IJ
(5)
^.=-+(7,-.)'
If cc = .62, then cv = .85, while experiment gives .82 as
the value of cv. The small difference between .85 and .82 is
probably due to frictional resistance. The value .82 for cv
makes cc approximately .617.
Again, the discharge from a simple sharp-edged orifice of
same sectional area as the mouthpiece is .62A V2gh, or more
than 24 per cent less than the discharge from the cylindrical
mouthpiece..
42 HYDRA ULICS.
The loss of head between MN and M'N'
(by eqs. 5 and 6)
= h(i — O = h X .3276
= — , approximately.
Thus the effective head is only \h, instead of h.
By eq. 3 the difference between the pressure-heads at
MN and at the point of discharge
= — . h = // - —
w
= £^, very nearly.
Now if one end of a tube is inserted in the mouthpiece at
the contracted section (Fig. 35) and the other end immersed
in a vessel of water, the water will at once rise to a height /z, in
the tube, showing that the
pressure at the contracted
section is less than that due to
the atmosphere. By careful
measurement it is found that
h^ is very nearly equal to \hy
which verifies the theory.
(d] Divergent Mouthpiece.
— Suppose that for the cylin-
drical mouthpiece in (c) there
I is substituted a divergent
FIG- 36. mouthpiece of the exact form
of the issuing jet (see Fig. 36), Then —
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 43
(1) The mouthpiece will run full bore.
(2) There will be no loss of head between the minimum
section MN and the plane of discharge AB, as there is now no
abrupt change of section.
Hence by Bernoulli's theorem, and retaining the same
symbols as in (c),
=+ = + (.)
W W 2g W 2g
If the discharge is into the atmosphere,/ = /0, and therefore
v* = 2gh\ .... '. . . . (2)
or introducing a coefficient cv (= .98, nearly, for a smooth
well-formed mouthpiece),
and the discharge is
. , ;, ._., . (4)
From the last equation it would appear as if the discharge
would increase indefinitely with A, but this is manifestly
impossible.
In fact, by eq. I, the flow being into the air, and taking
, (c)
W W 2g\V*
since av^ = Av. But/, cannot be negative, and therefore
so that
a '\ wk+l (7)
gives a maximum limit for the ratio of A to a.
44
HYDRA ULICS.
—
Now — = 34 feet very nearly, and the last equation may be
written
By eqs. 4 and 7,
(9)
which is also the expression for the discharge through the
minimum section a into a vacuum.
If, however, the sectional areas of the mouthpiece at the
point of discharge and at the throat are in the ratio of A to a,
as given by eq. 7, it is found that the full-bore flow will be in-
terrupted either by the disengagement of air, or by any slight
disturbance, as, for example, a slight blow on the mouthpiece,
and hence, in practice, it is usual to make the ratio of A to a
sensibly less than that given by eq. 7.
(e) Convergent Mouthpiece. — With a convergent mouth-
piece (Fig. 37) two points are to be noted :
(i) There is a contraction within the mouthpiece, followed
by a swelling out of the jet until it again fills the mouthpiece.
FIG. 37.
Thus, as in the case of cylindrical mouthpieces, there is a
" loss of head " between the contracted section and the point
FLO W THROUGH ORIFICES, OVER WEIRS, ETC.
45
of discharge, and also a consequent diminution in the velocity
of discharge.
(2) There is a second contraction outside the mouthpiece
due to the convergence of the fluid filaments. The mean
velocity of flow (V) across the section is
v' = C,'
Cvr being the coefficient of velocity and h the effective head
above the centre of the section.
Also, the area of this section
= CC'X area of mouthpiece at point of discharge
= CC'.A,
Cc being the coefficient of contraction. Hence the discharge
Q is given by
Q = CvfCc'A
= C'A
C'(= Cv'Ccf) being the coefficient of discharge.
The coefficients Cv' and Ce' depend upon the angle of con-
vergence, and Castel found that a convergence of 13° 24' gave
a maximum discharge through a mouthpiece 2-6 diameters in
length, the smallest diameter being .05085 foot.
TABLE GIVING CASTEL'S RESULTS.
Angles of
Convergence.
Cc
<v
C'
Angles of
Convergence.
<v
Cv
C'
o° o'
•999
.830
.829
13 °24'
.983
.962
.946
I 36
i .000
.866
.866
14 28
•979
.966
.941
3 10
1. 001
.894
•895
16 36
.969
.971
.938
4 10
i. 002
.910
.912
19 28
•953
.970
.924
5 26
1.004
.920
.924
21 0
•945
.971
.918
7 52
.998
•931
.929
23 o
•937
•974
•913
8 58
.992
.942
•934
29 58
.919
•975
.896
10 20
.987
•950
.938
40 20
.887
.980
.869
12 4
.986
• 955
.942
48 50
.661
.984
.847
46
HYDRAULICS.
18. Radiating Current. — As an application of Bernouilli's
theorem, consider the steady plane motion of a body of water
flowing radially between two horizontal planes a ft. apart and
symmetrical with respect to a central axis (Fig. 38).
Let v ft. per second be the velocity at the surface of a cyl-
FIG. 38.
FIG. 39-
inder of radius r ft. described about the same axis. Then the
volume Q crossing the second per surface is
Q = 2nr . av,
and therefore
Q
rv = =£- = a constant,
since Q is constant.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 4?
Thus v increases as r diminishes, and becomes infinitely
great at the axis; but it is evident that the current must take
a new course at some finite distance from the axis.
If p is the pressure at any point of the cylindrical surface
3 ft. above datum, then, by Bernoulli's theorem,
z + — + — = a constant = h = y + — ,
denoting the dynamic head z + — by y. Hence
w
, v* Q* a constant
2g Zn'a'r'g r'
and therefore
r*(h — y) = a constant
is an equation giving the free surfaces of the pressure columns
(Fig. 39). These surfaces are thus generated by the revolution
of Barlow's curve.
The surfaces of equal pressure are also given by an equa-
tion of the same form.
19. Vortex Motion. — A vortex is a mass of rotating fluid,
and the vortex is termed free when the motion is produced
naturally and under the action of the forces of weight and
pressure only.
In the radiating current already discussed, assume that the
direction of motion at each point is turned through a right
angle. so that the mass of water will now revolve in circular
layers about the central axis. Also, if there is a slow radial
movement, so that fluid particles travel from one circular stream-
line to another, it is assumed that these particles freely take the
velocities proper to the stream-lines which they join. Such a
motion is termed a free circular vortex.
The motion being steady and horizontal, the equation
z + — -| = a constant = //, . . . . (i)
48 HYDRA ULICS.
holds good at every point of a circular stream of radius r.
Again,
w.d\z-\-— ) = increment of dynamic pressure between two-
consecutive elementary stream-lines
= deviating force
= centrifugal force of an element between the
two stream-lines
Hence
,/ p\ w , wv1 j
w . d\z 4- — I = — —vdv = -- . dry
\ wi g gr
and therefore
so that vr — a constant, and v varies inversely as r, as in the
case of the radiating current. Therefore the curves of equal
pressure will also be the same as in a radiating current.
Free Spiral Vortex. — Suppose that the motion of a mass
of water with respect to an axis O is of such a character that at
any point J/the components of the velocity in the direction of
OM, and perpendicular to OM, are each inversely proportional
to the distance OM from O. The motion is thus equivalent to
the superposition of the motions in a radiating current and in
a free circular vortex ; and if 0 is the angle between <9J/and the
direction of the stream-line at M, v cos 6 and v sin 9 are each
inversely proportional to OM, and therefore 6 must be con-
stant. Hence the stream-lines must be equiangular spirals and
the motion is termed a free spiral vortex.
This result is of value in the discussion of certain turbines
and centrifugal pumps. A steady free surface in the case of a
*FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 49
free spiral vortex is impossible, as the stream-lines cross the
surfaces of equal pressure, which are the same as before.
Also if /„, z/0, r0 are the pressure, radius, and velocity at any
other point at the same elevation 2 above datum, then
W 2g W 2g
and the increase of pressure-head
w 2g 2gr* 2g
Forced Vortex. — A forced vortex is one in which the law
of motion is different from that in a free vortex. The simplest
and most useful case is that in which all the particles have an
equal angular velocity, so that the water will revolve bodily, the
velocity at any point being directly proportional to the distance
from the axis.
As before,
wl g r
But
v oc r = cor,
co being the constant angular velocity of the rotating mass.
Therefore
p\ GO*
--
_7I I r \ ^JtJ J
d U + --1 = — r . dr.
\ wl g
Integrating,
z + — = - - + a constant = - f + a constant.
Hence, if/0, r0, v0 are the pressure, radius, and velocity for
any second point at the same elevation z above datum, then
W 2v 2<> °'
<5 <5
HYDRA ULICS.
If the second point is on the axis of revolution, then ra = o,
and the last equation becomes
W 2P~
1 hus the free surface of the pressure columns is evidently a
paraboloid of revolution with its vertex
at O, as in Fig. 40.
A compound vortex is produced by
the combination of a central forced vor-
tex with a free circular vortex, the free
surface being formed by the revolution
of a Barlow curve and a parabola.
For example, the fan of a centri-
fugal pump draws the water into a forced
FIG. 40. vortex and delivers it as a free spiral vor-
tex into a whirlpool-chamber (Chap. VII.).
In this chamber there is thus a gain of pressure-head, and
the water is therefore enabled to rise to a corresponding addi-
tional height. James Thomson adopted the theory of the corn-
compound vortex as the principle of the action of his vortex
turbine.
20. Large Orifices in Vertical Plane Surfaces. — The
issuing jet is approximately of the same sectional form as
the orifice, and the fluid filaments converge to a minimum
section as in the case of simple sharp-edged orifices.
(a) Rectangular Orifice (Fig. 41). — Let E, F be the upper and
lower edges of a large rectangular orifice of breadth B, and let
H^ , H^ be the depths of E and F, respectively, below the free
surface at A. If u be the velocity with which the water reaches
the orifice, then H = -- is the fall of free surface which must
have been expended in producing the velocity u.
Hence, Hl-\- H and H^ + H are the true depths of the
edges E and F below the surface of still water.
Let A/TV be the minimum or contracted section, and assume
that it is a rectangle of breadth b.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 5 1
Let hl , h^ be the depths of M and N, respectively, below
the free surface at A.
Then hl-\- H, h^ -\- H are the true depths of M and N
below the surface of still water.
First, let the flow be into the air, the orifice being clear
above the tail-water level.
Consider a lamina of the fluid at the section MN of the
1
j
?
;- "'
"==:iEi5=
4
T
1
1
! i-
f :
~— f=£
1
i
J
1 1
f '
r i
E
f
|
i
V|
f1
»
.
2
.
M-1-- ;—
H
»L.-i__
J
L..[-
i
i
1
FIG. 41.
width of the section and between the depths x and ^r + dx
below the surface of still water.
The elementary discharge dq, in this lamina, is
dq = bdx <y/2gx,
and therefore the total discharge Q across the section MNis
/[*h.i + H
dq= b.dxjlgx
J hi -^H
Put*=
Then
t\. . . (i)
HYDRA ULICS.
The coefficient c is by no means constant, but is found to
vary both with the head of water and also with the dimensions
of the orifice, and can only be determined by experiment.
Second, let the orifice be partially (Fig. 42) submerged, and
and let //, be the depth between the
surface of the tail-race water and the
free surface at A.
By what precedes, the discharge Qt
through EG, the portion of the orifice
clear above the tail-race, is
. (2)
Every fluid filament flows through
the portion GF of the orifice under an
effective head Hz -f- H, and therefore
with a velocity equal to
FIG. 42.
Hence the discharge <23 through GF is
... (3)
and the total discharge Q is equal to Q, + Qt.
The coefficients clt c^ are to be determined by experiment,
and if cl = c^ = c,
. (4)
Third, let the orifice be wholly submerged (Fig. 43). Then
the total discharge Q is evidently
Q =
'. + Jf, • • • (5)
c being a coefficient to be determined by experiment.
If the velocity of approach, «, is sufficiently small to be
-- a
FLOW THROUGH ORIFICES, OVER WEIRS, ETC.
53
disregarded without sensible error, then H = o, and equations
i, 4, and 5, respectively, become
(8)
(b) Circular Orifices. — Let Fig. 44
represent the minimum section of
the circular jet issuing from a circu-
lar orifice.
Let 26 be the angle subtended at the centre by the fluid
FIG. 43-
FIG. 44.
lamina between the depths x and x + dx below the surface
of still water.
Let r be the radius of the section so that 2r = h^ — h^ h^
and h^ being, as in (a), the depths of the highest and lowest
points of the orifice below the free surface at A.
H, as before, is the head corresponding to the velocity of
approach.
54 HYDRA ULICS.
Then the area of the lamina under consideration
= 2r sin 9 . dx,
and the elementary discharge, dq, in this lamina, is
dq = 2r sin 6. dx^2gx*
h.+H+k.+H
But*=- -^ -
and therefore
dx
Hence
dq = 2r'sin'fttV' *2 -rcos
and the total discharge Q is
- r cos ff V (9)
21. Notches and Weirs. — When an orifice extends up to
the free-surface level it becomes what is called a notch.
A weir is a structure over which the water flows, the
discharge being in the same conditions as for a notch.
Rectangular Notch or Weir. — The discharge may be found
by putting Hl = o.
Thus equation I becomes
(10)
If the velocity of approach be disregarded, then H •=. o,
and the last equation becomes
(ii)
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 55
and //, is the depth to the bottom of the notch or to the crest
of the weir.
The effective sectional area of the water flowing through a
rectangular notch, or over a weir, is less than BH.t, because of
(a) crest contraction, (b) end contraction, (c) the fall of the free
surface towards the point of discharge.
It is reasonable to assume that the diminution of the actual
sectional area, BH^ , due to crest contraction and to the fall of
the free-surface level is proportional to the width B of the
opening, and that the effect of end contractions is very nearly
the same both for wide and narrow openings.
Francis, in his Lowell weir experiments, found that for
depths H^-\- H over the crest, varying from 3 in. to 24 in.,
and for widths B not less than three times the depth, a per-
fect end contraction had the effect of diminishing the width of
the fluid section by an amount approximately equal to one
f-f —I— J-T
tenth of the depth, or 2 "*" — , so that the effective width
IO
Thus, if there are n end contractions, the effective width
= B — — (//, + H), and the equation giving the discharge
becomes
Q = -c \ B - £(#. +/0 } S&\(Ht + H)*- H*\. (12)
According to Francis, the average value of c in this equa-
tion is .622.
Circtilar Notch. — In equation 9, Art. 20, put h^ = o and
h = 2r. Then
Tsin2
Q = 2r2 ^2g sin2 BH + 2r sin'
56 HYDRAULICS.
and if the velocity of approach be disregarded, so that H = o,
C* 6
Q = 2r* -v/4£- / sin2 6 sin - . dB
*/ o
2
sn -sin sn
(13)
22. Triangular Notch, — Disregard the velocity of ap-
proach and let B be the width of the free surface.
As before, consider a lamina
of fluid between the depths x and
^ ^ B
The area of the lamina = -77-
\ — x}dx, and the discharge in
this lamina is
7}
dq = C(H* — x
Hence the total discharge Q is
r"* ry
"
(14)
c is a coefficient introduced to allow for contraction, etc., and
Professor James Thomson gives .617 as its mean value for a
sharp-edged triangular notch.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. $?
r>
Now the ratio jy is constant in a triangular notch and varies
**\
in a rectangular notch. Hence Thomson inferred and proved
by experiment that the value of c is more constant for trian-
gular than for rectangular notches, so that a triangular notch
is more suitable for accurate measurements.
Example. — A sharp-edged triangular notch is opened in
the side of a reservoir, and the water flows out until the free-
surface level sinks to the bottom of the notch.
The discharge in the short interval dt, when the depth of
water in the notch is x ft.,
= —cmx
mx being the width of the free surface corresponding to the
depth x, and m a coefficient depending upon the angle of the
notch.
Again, S . dx is the quantity of the water which leaves the
reservoir in the same time dt, S being the horizontal sectional
area of the reservoir corresponding to the depth x. Hence
4 ,/—
and therefore
— \/2gcmxldt = — Sdx,
— \f2gcmdt = — Sx~*dx,
so that the time in which the free surface sinks to the required
level
x
15 c
= -- 7^— /
4 V2gcmJQ
X being the initial depth.
58 HYDRAULICS.
If 5 is constant, then
the time =
23. Broad-crested Weir. — Let Fig. 46 represent a stream
flowing over a broad-crested weir. On the up-stream side the
FIG. 46.
free surface falls from A to B. For a distance BD on the crest
the fluid filaments are sensibly rectilinear and parallel; the
inner edge of the crest is rounded so as to prevent crest con-
traction.
Consider a filament ab, the point a being taken in a part of
the stream where the velocity of flow is so small that it may be
disregarded without sensible error.
Let A be the thickness MN of the stream at b.
Let the horizontal plane through N be the datum plane.
Let #„ z be the depths below the free surface of a and b.
Let hl be the elevation of a above datum.
Let/0, /,, p be the atmospheric pressure and the pressures
at a and b.
Let v be the velocity of flow at b.
Then, by Bernoulli's theorem,
W
W
2g
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 59
But
= *i + and = , + ;
W WWW
therefore
and hence
_ = k, + zl - A = H, - A,
//", being the depth of the crest of the weir below the surface
of still water.
Thus, if B be the width of the weir, the discharge Q is
(16)
From this equation it appears that Q is nil both when
A = o and when A = //,. Hence there must be some value
of A between o and //, for which Q is a maximum. This value
may be found by putting
and the expression for the discharge becomes
, = .3855 V^rf, . ' . (17)
which is the maximum discharge for the given conditions.
Experiment shows that the more correct value for the dis-
charge is
. . . (18)
60 HYDRAULICS.
This formula agrees with the ordinary expression for the
discharge over a weir as given by equation u, if c = .525.
It might be inferred that for broad-crested weirs and large
masonry sluice-openings the discharge should be determined
by means of equation 18 rather than by the ordinary weir
formula, viz., equation n.
It must be remembered, however, that in deducing equa-
tion 17 frictional resistances have been disregarded and the
gratuitous assumption has been made that the stream adjusts
itself to a thickness / which will give a maximum discharge.
The theory is therefore incomplete.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 6 1
EXAMPLES.
I. A frictionless pipe gradually contracts from a 6-in. diameter at A
to a 3-in. diameter at B, the rise from A to £ being 2 ft. If the de-
livery is i cubic foot per second, find the difference of pressure between
the two points A and B. Ans. 500 Ibs. per sq. ft.
. 2. In a frictionless horizontal pipe discharging 10 cubic feet.of water
per second, the diameter gradually changes from 4 in. at a point A to i/C
6 in. at a point B. The pressure at the point ^5*is 100 Ibs. per square
inch ; find the pressure at the point A. Ans. 4118 Ibs. per sq. ft.
3. A ^-in. horizontal pipe is gradually reduced in diameter to -J- in.
and then gradually expanded again to its mouth, where it is open to the
atmosphere. Determine the maximum quantity of water which can be
forced through the pipe (a) when the diameter of the mouth is \ in., (b)
when the diameter is f in. Also determine the corresponding velocities
at the throat and the total heads (neglect friction, which, however, is
very considerable). Ans. (a) .24 cub. ft. per min.; 46.7 ft. per. sec.
(b) .239 cub. ft. per. min.; 46.66 ft. per sec.
4. A short horizontal pipe A BC connecting two reservoirs gradually
contracts in diameter from i inch at A to £ inch at B and then enlarges
to i inch again at C. If the height of the water in the reservoir over C
be 12 inches, determine the maximum flow through the pipe and sketch
the curve of pressures. Also obtain an equation for this curve, assum-
ing the rates of contraction and expansion of the pipe to be equal and
uniform. Ans. 3.75 cub. ft. per min.
5. The pipe DE in the figure is gradually contracted in diameter
from D to E, where it is enclosed in
another pipe ABC, expanding from B
towards A and C\ at C it is open to the
atmosphere and at A it is connected with D/
a reservoir R ; the water surface in R
being h' below the horizontal axis of DE.
If the velocity in DE at E be v and the
velocity in AB at B be F, what will be the
common velocity after uniting? Explain
what becomes of the energy lost in im- ^
pact. If the diameters at E, B, arid C
are \ in., f in., and i in., the distance between the outside of E and inside
62 HYDRA ULICS.
of B being T\ inch, find the ratio of the quantity pumped from R to the
flow through DE.
6. A 3-in. pipe gradually expands to a bell-mouth ; if the total head,
//, be 40 ft., find the greatest diameter of the mouth at which it will
run full when open to the atmosphere. Compare the discharge from
this pipe with the discharge when the pipe is not expanded at the mouth.
Ans. 4.8 in.; discharge is 18.63 cub. ft- Per minute with bell-
mouth and 7.337 cub. ft. per minute without bell-mouth.
7. The pressure in a 12- in. pipe at A is 50 Ibs.; the pipe then en- ^
larges to a i5-in. pipe at B, the rise from A to B being 3 ft.; the dis- /
charge is Q cubic feet per minute. Find the pressure at^; also find the
pressure at a point C, the rise from B to C being 6 ft,
(6637.5 + T£
Ans. 6637.5 + lbs' er sc- ft'
8. Two equal pipes lead, one from the steam-space, the other from
the water-space of a boiler at pressure/; Ss is the density of the steam
and Sw that of the water. Assuming Torricelli's theory to hold for rate
of efflux of steam and water, show that
vel. of steam-jet _ */•£» _ quantity of water-jet _ energy of steam-jet
vel. of water-jet ~~ * Ss ~ quantity of steam-jet ~~ energy of water-jet,
and that the momentum of each jet is the same.
9. Find the head required to give i cub. ft. of water per second
through an orifice of 2 square inches area, the coefficient of discharge [/
being .625. (g = 32.) Ans. 206 ft.
10. The area of an orifice in a/fcnin plate was 36.3 square centimetres,
the discharge under a head 0^/^396 metres was found to be .01825 cubic
metre per second, and the velocity of flow at the contracted section, as \^/
determined by measurements of the axis of the jet, was 7.98 metres per
second. Find the coefficients of velocity, contraction, discharge, and re-
sistance. (^ = 9.81.) Ans. .978; .631; .617; .045.
11. The piston of a 12-in. cylinder containing saltwater is pressed
down under a force of 3000 lbs. Find the velocity of efflux and the \J
volume of discharge at the end of the cylinder through a well-rounded
i -in. orifice. Also find the power exerted.
Ans. 60.373 ft. per sec.; .1691 cub. ft. per sec.; 1.166 H. P.
12. In the condenser of a marine engine there is a vacuum of 26^ in.
of mercury ; the injection orifices are 6 ft. below the sea-level. With
what velocity will the injection-water enter the condenser? (Neglect re-
sistance.) Ans. 25.3 ft. per sec.
13. Water in the feed-pipe of a steam-engine stands 12 ft. above the
FLO W THROUGH ORIFICES, OVER WEIRS, ETC. 63
surface of the water in the boiler ; the pressure per sq. in. of the steam is
20 Ibs., of the atmosphere 15 Ibs. Find the velocity with which the
water enters the boiler. Ans. 5.376 ft. per sec.
14. The injection orifice of a jet condenser is 5 ft. below sea-level
and vacuum = 27 in. of mercury. Find velocity of water entering con-
denser, supposing three fourths of the head lost by frictional resistance.
Ans. 23.86 ft. per sec.
15. A vessel containing water is placed on scales and weighed. How
will the weight be affected by opening a small orifice in the bottom of
the vessel ?
1 6. Water is supplied by a scoop to a locomotive tender at 7 ft. above
trough. Find lowest speed of train at which the operation is possible.
Ans. 14.44 miles per hour.
Also find the velocity of delivery when train travels at 40 miles per
hour, assuming half the head lost by frictional resistance.
Ans. 35.68 ft. per. sec.
17. The head in a prismatic vessel at the instant of opening an orifice
was 6 ft. and at closing it had decreased to 5 ft. Determine the mean
constant head h at which, in the same time, the orifice would discharge
the same volume of water. Ans. 5.434 ft.
18. A prismatic vessel 5.747 in. in diameter has an orifice of .2 in.
diam. at the bottom; the surface sinks from 16 in. to 12 in. in 53
seconds. Find the coefficient of discharge. Ans. .6.
19. A prismatic basin with a horizontal sectional area of 9 sq. ft. has
an orifice of .09 sq. ft. at the bottom ; it is filled to a depth of 6 ft. above
the centre of the orifice. Find the time required for the surface to sink
2 ft., 3^ ft., 5 ft. Ans. 260 sec.; 502 sec.; 838 sec.
20. The water in a cylindrical cistern of 144 sq. in. sectional area is
16 ft. deep. Upon opening an orifice of I sq. in. in the bottom the
water fell 7 ft. in i minute. Find the coefficient of discharge. The co-
efficient of contraction being .625, find the coefficients of velocity and
resistance. Ans. .6 ; .96 ; 0.85.
21. How long will it take to fill a paraboloidal vessel up to the level
of the outside surface through a hole in the bottom 2 ft. under water?
(g = 32 and c = .625.)
1 76 |/2 B
Ans. —j, B being the parameter of the parabola and A the
sectional area of the orifice.
22. How long will it take to fill a spherical Vessel of radius r up to
the level of the outside surface through a hole of area A at bottom 2 ft.
under water ? (g = 32 and c = .625.)
Ans'
64 HYDRAULICS.
23. A vessel full of water weighs 350 Ibs. and is raised vertically by
means of a weight of 450 Ibs. Find the velocity of efflux through an
orifice in the bottom, the head being 4 ft. Ans. 17.02 ft. per sec.
24. A vessel full of water makes loo-revols. per min. Find the velocity
of efflux through an orifice 2 ft. below the surface of the water at the
centre. Ans. 33.4 ft. per sec.
What will be the velocity if the vessel is at rest ?
Ans. 1 1. 35 ft. per sec.
25. The jet from a circular sharp-edged orifice, £ in. in diameter, un-
der a head of 18 ft., strikes a point distant 5 ft. horizontally and 4.665
in. vertically from the orifice. The discharge is 98.987 gallons in
569.218 seconds. Find the coefficients of discharge, velocity, contraction,
and resistance. Ans. .6014; .945; .636; .118.
26. A square box 2 ft. in length and i ft. across a diagonal is placed
with a diagonal vertical and filled with water. How long will it take for
the whole of the water to flow out through a hole at the bottom of .02
sq. ft. area ? (c — .625.) Ans. 97.48 sees.
27. A pyramid 2 ft. high, on a square base, is inverted and filled
with water. Find the time in which the water will all run out through
a hole of .02 sq. ft. at the apex. A side of the base is i ft. in length.
(c. — .625.) Ans. 15.08 sec,
28. Find the discharge under a head of 25 ft. through a thin-lipped
square orifice of i sq. in. sectional area, (a) when it has a border on one
side, (b) when it has a border on two sides.
Ans. (a) .3575 cu. ft. per sec.; (b) .3706 cu. ft. per sec.
29. A vessel in the form of a paraboloid of revolution has a depth of
16 in. and a diam. of 12 in. at the top. At the bottom is an orifice of
i sq. in. sectional area. If water flows into the vessel at the rate of 2TV
cubic feet per minute, to what level will the water ultimately rise ? How
long will it take to rise (a) 11 in., (b) 11.9 in., (c) 11.99 m-> (X) I2 in-
above the orifice? If the supply is now stopped, how long (e) will it
take to empty the vessel ?
Ans. 12 inches; (a) 83.095 sec.; (b) 124.2 sec.; (c) 263.9 sec.;
(d) an infinite length of time ; (e) 11.3 sec.
30. If the vessel in Question 29 is a semi-sphere i ft. in diameter, to
what height will the water rise ? How long will it take for the water to
rise (a) 11 in., (b) 12 in. above the orifice ? How long (c) will it take to
empty the vessel ?
Ans. 12 inches ; (a) 67.16 sec. ; (£) 81.46 sec. ; (c) 24.13 sec.
31. In a vortical motion two circular filaments of radii ri , r2 , of ve-
locities Vi,Vt, and of equal weight Ware made to change place. Show
7/2
that a stable vortex is produced if — =const.; and if r2 > r\ , show that
the surfaces of equal pressure are cones.
FLOW THROUGH1 ORIFICES, OVER WEIRS, ETC. 65
32. Prove that for a Borda's mouthpiece running full the coefficient
of discharge is — .
4/2
33. The surface of the water in a tank is kept at the same level;
obtain the discharge at 60 in. below the surface (a) through a circular
orifice i sq. in. in area, (U) through a cylindrical ajutage of the same
sectional aYea fitted to the outside, (c) through the same ajutage fitted
to the inside, and determine the mechanical effect of the efflux in each
case. Ans. (a) 4 36 Ibs. per sec.
(ff) 6.356 " " "
(rf 3.488 •• " "
20.514 ft.-lbs. per sec.
21.369 " " "
1744 " " "
34. Water is discharged under a head of 64 ft. through a short cylin-
drical mouthpiece 12 in. in diameter. Find (a) the loss of head due
to shock, (£) the volume of disdharge in cubic feet per secJnd, (c) the
energy of the issuing jet. (g = 32.)
Ans. (a) 20.96 ft. ; (8) 51.54 cub. ft. ; (c) 393.8 H. P.
35. If a bell-mouth is substituted for the mouthpiece in the preced-
ing question, find the discharge and the mechanical effect of the jet.
Ans. 61.6 cub. ft. per sec. ; 470.6 H. P.
36. Compare the energies of a jet issuing under an effective head of
100 ft. through (i) a 12-in. cylindrical ajutage, (2) a 12-in. divergent aju-
tage, (3) a 12-in. convergent ajutage, the angle of convergence being 21°.
Draw the plane of charge in each case.
Ans. (i) 393.8 H. P.; (2) 672.28 H. P.; (3) 618.23 H. P.
37. Find the discharge through a rectangular opening 36 in. wide
and 10 in. deep in the vertical face of a dam, the upper edge of the
opening being 10 ft. below the water surface.
Ans. 40.2 cub. ft. per sec.
38. Find the discharge in pounds per minute through a Borda's
mouthpiece i in. in diameter, the lip being 12 in. below the water-
surface. Ans. 87.714 Ibs.
39. Sometimes the crest of a dam is raised by floating a stick L into
the position Zi , where it is supported against the
verticals. The stick then falls of itself into position
Li and rests on the crest. Explain the reason
of this.
40. A sluice 3 ft. square and with a head of 12
ft. over the centre has, from the thickness of the
frame, the contraction suppressed on all sides when
fully open ; when partially open, the contraction
exists on the upper edge, i.e., against the bottom of the gate, which is
formed of a thin sheet of metal. Find the discharge in cubic feet when
opened i ft., 2 ft. and also when fully open. Ans. 57.77 ; 114.45 ; '75-9.
66 HYDRA ULICS.
41. What quantity of water flows through the vertical aperture of a
dam, its width being 36 in. and its depth 10 in. ; the upper edge of the
aperture is 16 ft. below the surface. Ans. 50.65 cub. ft. per sec.
42. 264 cubic feet of water are discharged through an orifice of 5 sq.
ins. in 3 min. 10 sec. Find the mean velocity of efflux.
Ans. 64 ft. per sec.
43. One of the locks on the Lachine Canal has a superficial area of
about 12,150 sq. ft., and the difference of level between the surfaces of
the water in the lock and in the upper reach is 9 feet. Each leaf of the
gates is supplied with one sluice, and the water is levelled up in 2 min,
48 sees. Determine the proper area of the sluice-opening. (Centre of
sluice 20 ft. below surface of upper reach.)
Ans. Area of one sluice = 43.73 sq. ft.
44. The horizontal section of a lock-chamber may be assumed a
rectangle, the length being 360 ft. When the chamber is full, the sur-
face width between the side walls, which have each a batter of i in 12,
is 45 ft. How long will it take to empty the lock through two sluices in
the gates, each 8 ft. by 2 ft., the height of the water above the centre of
the sluices being 13 feet in the lock and 4 feet in the canal on the down-
stream side. Ans. 594 sec., c being .625.
45. Water approaches a rectangular opening 2 ft. wide with a velocity
of 4 ft. per second. At the opening the head of water over the lower
edge = 13 ft., and over the surface of the tail-race = 12 ft.; the discharge
through the opening is 70 cub. ft. per second. Find the height of the
opening. Ans. 1.022 ft.
46. The water in a regulating-chamber is 8 ft. below the level of the
water in the canal and 8 ft. above the centre of the discharging-sluice.
Determine the rise in the canal which will increase the discharge by 10
per cent. Ans. 1.68 ft.
The horizontal sectional area of the chamber is constant and equal to
400 sq. ft.; in what time will the water in the chamber rise to the level of
that in the canal, if the discharging-sluice is closed; the sluice between
the canal and chamber being 3 sq. ft. in area? Ans. 150.83 sec.
47. A lock on the Lachine Canal is 270 ft. long by 45 ft. wide and has
a lift of 8£ ft.; there are two sluices in each leaf, each 8f ft. wide by
2£ ft. deep ; the head over the horizontal centre line of the sluices is
19 ft. Find the time required to fill the lock. Ans. 164.6 sec.
48. Show that the energy of a jet issuing through a large rectangular
orifice of breadth B is i2$B(ff£ — Hi*), Hi , H* being the depths below
the water-surface of the upper and lower edges of the orifice, and the
coefficient of discharge being .625.
49. A reservoir at full water has a depth of 40 ft. over the centre of
the discharging-sluice, which is rectangular and 24 in. wide by 18 in.
deep. Find the discharge in cubic feet per second at that depth, and also
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 6?
when the water has fallen to 30, 20, and 10 ft., respectively; find the
mechanical effect of the efflux in each case.
Ans. 94.8 cub. ft.; 82.1 cub. ft.; 67 cub. ft.; 47.4 cub. ft.; 431.2
H.P.; 280 H.P.; 152.5 H.P.; 53.95 H.P.
50. Require the head necessary to give 7.8 cubic feet per second
through an orifice 36 sq. in. in sectional area. Aps. 38.9 ft.
51. The upper and lower edges of a vertical rectangular orifice are
6 and 10 feet below the surface of the water in a cistern, respectively ;
the width of the orifice is i ft. Find the discharge through it.
Ans. 5642 cub. ft. per sec.
52. To find the quantity of water conveyed away by a canal 3 ft.
wide, a board with an orifice 2 ft. wide and i ft. deep is placed across
the canal and dams it back until it attains a height of 2£ ft. above the
bottom and if ft. above the lower edge of the orifice. Find the dis-
charge, (c = .625.) Ans. 17.59 cub- ft- per sec.
53. Six thousand gallons of water per minute are forced through a
line of piping ABC and are discharged into the atmosphere at C, which
is 6 ft. vertically above A. The pipe AB is 12 in. in diameter and 12 ft.
in length ; the pipe J5C is 6 in. in diameter and 12 ft. in length. Disre-
garding friction, find the " loss in shock " and draw the plane of charge.
Ans. Loss of head in shock = 57.9 ft.
54. What should be the height of a drowned weir 400 ft. long, to
deepen the water on the up-stream side by 50 per cent, the section of
the stream being 400 ft. x 8 ft., and the velocity of approach 3 ft. per
second ? Ans. 8.396 ft.
55. The two sluices each 4 ft. wide by 2 ft. deep in a lock-gate are
submerged one half their depth. The constant head of water above the
axis of the sluice is 12 ft. Find the discharge through the sluice,
the velocity of approach being 4 ft. per second.
Ans. 16626.2 cub. ft. per minute.
56. Find the flow through a square opening, one diagonal being ver-
tical and 12 in. in length, and the upper extremity of the diagonal be-
ing in the surface of the water. Ans. 1.727 cub. ft. per sec.
57. The locks on the Montgomeryshire Canal are 81 ft. long and 7f
ft. wide ; at one of the locks the lift is 7 ft.; a 24-in. pipe leads the water
from the upper level and discharges below the surface of the lower level
into the lock-chamber ; the mouth of the pipe is square, 2 ft. in the side,
and gradually changes into a circular pipe 2 ft. in diameter. Find time
of filling the lock, (c = i.) Ans. 130 sees.
58. A canal lock is 115.1 ft. long and 30.44 ft. wide; the vertical
depth from centre of sluice to lower reach is 1.0763 ft., the charge being
6.3945 ft. ; the area of the two sluices is 2 x 6.766 sq. ft. Find the time
of filling up to centre of sluices, (c = .625 for the sluice, but is reduced
68 HYDRA ULICS.
to .548 when both are opened.) Also, find time of filling up to level of
upper reach from centre of sluice-doors. Ans. 25 sec.; 298 sees.
59. A reservoir half an acre in area with sides nearly vertical, so that
it may be considered prismatic, receives a stream yielding 9 cub. ft. per
second, and discharges through a sluice 4 ft. wide, which is raised 2 ft.
Calculate the time required to lower the surface 5 ft., the head over the
centre of the sluice when opened being 10 feet. Ans. 1079 sees.
60. Show that in a channel of V section an increment of 10 per cent
in the depth will produce a corresponding increment of 5 per cent in the
velocity of flow and of 25 per cent in the discharge.
61. The angle of a triangular notch is 90°. How high must the
water rise in the notch so that the discharge may be 1000 gallons per
minute? Ans. 1 2 inches very nearly.
62. Show that upon a weir 10 feet long with 12 inches depth of water
flowing over, an error of i/iooo of a foot in measuring the head will
cause an error of 3 cubic feet per minute in the discharge, and an error
of i/ioo of a foot in measuring the length of the weir will cause an error
of 2 cubic feet in the discharge.
63. In the weir at Killaloe the total length is noo ft., of which 779 ft.
from the east abutment is level, while the remainder slopes i in 214, giving
a total rise at the west abutment of 1.5 ft. Calculate the total discharge
over the weir when the depth of water on the level part is 1.8 ft., which
gives .3 ft. on highest part of weir. (Divide slope into 8 lengths of 40
ft. each, and assume them severally level, with a head equal to the
arithmetic mean of the head at the beginning and end of each length.)
Ans. 7483 cub. ft. per sec.
64. A watercourse is to be augmented by the streams and springs
above its level. The latter are severally dammed up at suitable places
and a narrow board is provided in which an opening 12 in. long by 6
in. deep is cut for an overfall ; it was surmised that this would be suf-
ficient for the largest streams; another piece attached to the former
would reduce the length to 6 in. for smaller streams. Calculate the
delivery by the following streams:
In No. i stream with the 12-in. notch, depth over crest = .37 ft.
" No. 2 " " " 6-in. " " " " = .41 ft. -;
" No. 3 " " " 12-in. " " " " = .29 ft.
" No. 4 " " " 6 in. " " " " = .19 ft.
(Take into account the side contractions.)
Ans. No. i, .695 cub. ft. ; No. 2, .3658 cub. ft. ; No. 3, .4904 cub.
ft.; No. 4, .1275 cub. ft.
65. The horizontal sectional area of a reservoir is constant and =
10,000 square feet. When the reservoir is full, a right-angled notch 2
ft. deep is opened. Find the time in which the level of the water falls-
to the bottom of the notch. Ans. 15.3 min.
FLOW THROUGH ORIFICES, OVER WEIRS, ETC. 69
66. A weir passes 6 cubic feet per second, and the head over the crest
is 8 inches. Find the length of the weir. Ans. 3.3068 ft.
67. A weir 400 ft. long, with a 9-in. depth of water on it, discharges
through a lower weir 500 ft. long. Find the depth of water on the latter.
Ans. .6457 ft.
68. A stream 30 ft. wide, 3 ft. deep, discharges 310 cubic feet per
second ; a weir 2 feet deep is built across the stream. Find increased
depth of latter, (a) neglecting velocity of approach, (b) taking velocity
of approach into account. Ans. (a) 1.26 ft. to 1.265 ft-J
(6) 1. 19 ft.
69. A weir is 545 ft. long; how high will the water rise over it when
it rises .68 ft. upon an upper weir 750 ft. long? Ans. .8413 ft.
70. In a stream 50 ft. wide and 4 ft. deep water flows at the rate of
loo ft. per minute ; find the height of a weir which will increase the
depth to 6 ft., (i) neglecting velocity of approach, (2) taking velocity of
approach into account. Ans. (i) 4.4126 ft; (2) 4.4509 ft.
71. A stream 50 ft. wide and 4 ft. deep has a velocity of 3 ft. per
second ; find the height of the weir which will double the depth, (i)
neglecting velocity of approach, (2) taking velocity of approach into ac-
count. Ans. (i) 5.615 ft.; (2) 5.7688 ft.
72. A stream 80 ft. wide by 4 ft. deep discharges across a vertical
section at the rate of 640 cubic feet per second ; a weir is built in the
stream, increasing its depth to 6 ft. Find the height of the weir.
Ans. 4.233 ft.
73. Salmon-gaps are constructed in a weir ; they are each 10 ft. wide
and their crests are 18 in. below the weir crest. Calculate the discharge
down three of these gaps, the water on the level rjart of the weir being
8 in. deep. Ans. 238.15 cub. ft. per sec.
74. A pond whose area is 12,000 sq. ft. has an overfall outlet 36 in.
wide, which at the commencement of the discharge has a head of 2.8 ft.
Find the time required to lower the surface 12 in. Ans. 354.72 sec.
75. How much water will flow in an hour through a rectangular
notch 24 in. wide, the surface of still water being 8 in. above the crest
of the notch ? (Take into account side contraction.) Ans. 3.386 ft.
76. Show that when the water flowing over has a
depth greater than .3874 ft. it is carried completely
over the longitudinal opening, .83 ft. in width. At
what depth does all the water flow in ? ,
Ans. .221 ft. FIG. 49.
CHAPTER II.
FLUID FRICTION.
I. Fluid Friction. — The term fluid friction is applied to
the resistance to motion which is developed when a fluid flows
over a solid surface, and is due to the viscosity of the fluid.
This resistance is necessarily accompanied by a loss of energy
caused by the production of eddies along the surface, and
similar to the loss which occurs at an abrupt change of sec-
tion, or at an angle in a pipe or channel.
Froude's experiments on the resistance to the edgewise
motion of planks in a fluid mass, the planks being T\ in. thick,
19 in deep, and I to 50 ft. long, each plank having a fine cut-
water and run, are summarized in the following table :
Length of Surface in Feet.
Nature of Surface
2 Feet.
8 Feet.
20 Feet.
50 Feet.
Covering.
A
B
c
A
B
C
A
B
C
A
B
C
Varnish
2.OO
• 41
• 3QO
1.85
.325
.264
1.85
.278
.240
I.83
.250
.226
Paraffine
38
• 37O
1.94
.314
?6o
1.93
.271
.237
Tinfoil
2 16
qo
2QC
I »QQ
.278
.263
I .QO
?6^
•244
r 83
.246
272
Calico . . .
i cn
87
• 72^
I .Q2
.626
.504
I. 80
. C.T.I
•447
T 87
.474
• 42^
Fine sand
2.00
.81
.690
2.00
.583
•450
2.00
.480
• 384
2.06
•405
• 337
Medium sand
2.OO
.90
.7302.00
.625
.488
2.00
•534
• 465
2.OO
.488
•456
Coarse sand
2.00
I. 10
.880
2.OO
.714
.520
2.0O
.588
.490
Columns A give the power of the speed (v) to which the re-
sistance is approximately proportional.
Columns B give the mean resistance in Ibs. per square foot of
the whole surface of a board of the lengths stated in the table.
Columns C give the resistance, in pounds, of a square foot
of surface at the distance sternward from the cutwater stated in
70
FLUID FRICTION. 7 1
the heading, each plank having a standard speed of 10 ft. per
second. The resistance at other speeds can be easily calculated.
An examination of the table shows that the mean resistance
per square foot diminishes as the length of the plank increases.
This may be explained by the supposition that the friction in
the forward portion of the plank develops a force which drags
the water along with the surface, so that the relative velocity
of flow over the rear portion is diminished. Again, the de-
crease of the mean resistance per square foot is .132 Ib. when
the length of a varnished plank is increased from 2 to 20 ft., while
it is only .028 Ib. when the length increases from 20 to 50 ft.
Hence, for greater lengths than 50 ft. the decrease of resistance
may be disregarded without much, if any, practical effect.
Thus, generally speaking, these experiments indicate tha-t
the mean resistance is proportional to the #th power of the
relative velocity, n varying from 1.83 to 2.16, and its average
value being very nearly 2.
Colonel Beaufoy, as a result of experiments at Deptford,
also assumed the mean resistance to be proportional to the nth
power of the relative velocity, the value of n in three series of
observations being 1.66, 1.71 and 1.9.
The frictional resistance is evidently proportional to some
function of the velocity, F(v), which should vanish when v is
nil, as when the surface is level, and should increase with v.
Coulomb assumed the function F(v) to be of the form
av -f- bv* , a and b being coefficients to be determined by experi-
ment. Experiment shows that when v does not exceed 5 ft.
per minute the resistance is directly proportional to the veloc-
ity, but that it is more nearly proportional to the square of the
velocity when the velocity exceeds 30 ft. per minute ; or,
F(v) = av when v < 5 ft. per minute,
and
F(v) = bv^ when v > 30 ft. per minute.
Again, observations on the flow of water in town mains
indicate that no difference of resistance is developed under
72 HYDRA ULICS.
widely varying pressures, and this independence of pressure is
also verified by Coulomb's experiment showing that if a disk
is oscillated in water there is no apparent change in the rate of
decrease of the oscillations, whether the water is under atmos-
pheric pressure or not.
From the preceding and other similar experiments the fol-
lowing general laws of fluid friction have been formulated :
(1) The frictional resistance is independent of the pressure
between the fluid and the surface over which it flows.
(2) The frictional resistance is proportional to the area of
the surface.
(3) The frictional resistance is proportional to some func-
tion, usually the square, of the velocity.
To these three laws may probably be added a fourth, viz.:
(4) The frictional resistance is proportional to the density
of the fluid.
A fifth law, viz., that " the frictional resistance is indepen-
dent of the nature of the surface against which the fluid flows,"
has been sometimes enunciated, and at very low velocities
the law is approximately true. At high velocities, however,
such as are common in engineering practice, the resistance has
been shown by experiment, and especially by the experiments
carried out by Darcy, to be very largely influenced by the
nature of the surface.
Let p be the frictional resistance in pounds per square foot
of surface at a velocity of I ft. per second.
Let A be the area of the surface in square feet.
Let v be the relative velocity of the surface and the water
in which it is immersed.
Let R be the total frictional resistance.
Then from the laws of fluid friction
R = p . AV*.
2j?
Take/ = — p, w being the specific weight of the fluid. Then
R = fwA—.
FLUID FRICTION. 73
The coefficient f is approximately constant for any given
surface, and is termed the coefficient of fluid friction. The
power absorbed by the frictional resistance
v*
= pAv' X v = pAv* = fwA — .
o
TABLE GIVING THE AVERAGE VALUES OF / IN THE CASE OF
LARGE SURFACES MOVING IN AN INDEFINITELY LARGE
MASS OF WATER.
Surface. Coefficient of Friction (/").
New well-painted iron plate .............. 00489
Painted and planed plank ............... 0035
Surface of iron ships ... ................. 00362
Varnished surface ........................ 00258
Fine sand surface ........................ 00418
Coarse sand surface .... ................. 00503
2. Surface Friction of Pipes. — Assuming that the laws of
fluid friction already enunciated hold good when water flows
through a pipe, it has been shown by numerous experiments
that the coefficient of friction /lies between the limits .005 and
.01, its average value under ordinary conditions being about
.0075. No single value of f is applicable to very different
cases. Indeed, /depends not only upon the condition of the
surface, but also upon the diameter of the pipe and the veloc-
ity of the water. Some authorities have expressed its value by
a relation of the form
a and b being constants whose values are to be determined by
experiment.
The following table gives some of the best numerical results
obtained for a and b\
74 HYDRA ULICS.
Authority. a b
Prony ........ .......... 00021230 .00003466
D'Aubuisson ............ 0002090 .000037608
Eytelwein ............... 00017059 .00004441
In pipes of small diameter in which the velocity of flow is
less than 4 in. per second the term a may be disregarded so
that
In ordinary practice and when the pipes have been in use
for some time, the velocity usually exceeds 4 in. per second,
and the term — may then be disregarded, so that
Now Darcy's experiments have shown that it is more cor-
rect to assume that a and b, instead of being constant, are
variable, and Darcy expressed them as functions of the diam-
eter of the pipe.
Thus, for pipes in which the velocity exceeds 4 in. per
second, Darcy took
/ , £
g '' ^d'
d being the diameter of the pipe, and a and ft coefficients.
Darcy also gave the following values for a and ft :
a ' ft
For drawn wrought-iron or smooth
cast-iron pipes 0001545 .000012973
For pipes with surfaces covered by
light incrustations 0003093 .00002598
FLUID FRICTION.
75
These coefficients can be put into the following very simple
form without sensibly altering their values :
For clean pipes ........... f= .005(1 -[- - )
For slightly incrusted pipes / = .01(1 -f- - J
d being the diameter in feet.
Darcy proposed to include all cases by expressing /"more
generally in the form
in which, for new and smooth iron pipes,
a = ,00003959, ft — .00002603125 ;
af = .000064375, ft' = .000000335625.
These values are rarely of any practical use.
TABLE GIVING DARCY'S VALUES OF / FOR VELOCITIES
EXCEEDING 4 IN. PER SECOND.
Diam.
Value of/
Diana.
Value of f.
Diam.
Value of/.
of
of
of
Pipe
Pipe
Pipe
m
New
Incrusted
in
New
Incrusted
in
New
Incrusted
Inches.
Pipes.
Pipes.
Inches.
Pipes.
Pipes.
Inches.
Pipes.
Pipes.
2
.0075
.0150
9
.00556
.OITII
27
.00519
.01037
3
.00667
•01333
12
.00542
.01083
30
.00517
.01033
4
.00625
.0125
15
•00533
.01067
36
.00514
.01028
5
.0060
.OI2
18
.00528
.01056
42
.00512
.OIO24
6
.00583
.01167
2.1
.00524
.01048
48
.00510
.01021
7
.00571
.01143
24
.00521
.01042
54
.00509
.OIOI9
8
.00563
.01125
76 HYDRAULICS.
Again, Weisbach has proposed the formula
-*
Vv
where a = .003598 and b = .004289.
3. Resistance of Ships. — The motion of a ship through
water causes the production of waves and eddies, and the total
resistance to the movement of a ship is made up of a frictional
resistance, a wave-making resistance, and an eddy-making re-
sistance. Although there is no theory by which the resistance
at a given speed of a ship of definite design can be absolutely
determined, Froude's experiments render it possible to make
certain inferences and furnish some useful data.
According to Froude, the frictional resistance is sensibly
the same as that of a rectangular surface moving with the same
speed, of the same length as the ship in the direction of motion,
and of an area equal to the immersed surface of the ship.
Experiments seem to indicate that as the speed increases, the
frictional resistance of well-designed ships with clean bottoms
is from 90 to 60 per cent of the total resistance, and that the
percentage is greater when the bottoms become foul.
The wave-making resistance is especially affected by the
form and proportions of the ship, depending, for a given
length, upon the proportions of the entrance, middle body, and
run. For every ship there is a limit of speed below which the
resistance is approximately proportional to the square, of the
speed, being chiefly due to friction, and beyond which it in-
creases more rapidly than as the square.
The eddy-resistance in the case of well-formed ships should
not exceed about 10 per cent of the total resistance, and is
often much less.
Froude's law of resistance may be enunciated as follows :
Let /„ /, be the lengths of a ship and its model.
Let Alt A^ be the displacements of a ship and its model.
Let /?„ R^ be the resistances of a ship and its model at the
.speeds ivl and vt.
FLUID FRICTION. 7?
Then, if
_i _i_ __ \ _
V " /£ ~~ /f *'
2 *s ^a
the resistances are in the ratio of
Hence, too, the H. P., and therefore also the coal consumption
per hour, is proportional to Rv, that is, to
A1 or /5,
and the coal consumption per mile is proportional to
A or to /3.
Again, R is proportional to /3 ;
that is, to / X /3 1
that is, to v* X ^ ;
and it is sometimes convenient to express the resistance irt
pounds in the form
v being the speed in knots, A the displacement in tons, and k
a coefficient depending upon the type of ship and varying from
.55 to .85 when the bottom is clean.
CHAPTER III.
FLOW OF WATER IN PIPES.
1. Assumptions. — In the ordinary theory of the flow of
water in a pipe it is assumed that the water consists of thin
plane layers perpendicular to the axis of the pipe, that each
layer is driven through the pipe by the action of gravity and by
the difference of pressure on its plane faces, and that the liquid
molecules in any layer at any given moment will also be found
in a plane layer after any interval of time. In such motion the
internal work done in deforming a layer may be generally dis-
regarded.
It is further assumed that there is no variation of velocity
over the surface of a layer, and this is equivalent to saying that
each liquid molecule in a cross-section has the same mean ve-
locity.
The disagreement of these assumptions with the results of
recent experimental researches will be referred to in a subse-
quent article.
2. Steady Motion in a Pipe of Uniform Section. — Since
the motion is to be steady, the same volume Q cub. ft. of water
will always arrive at any given cross-section of A square feet
with the same mean velocity v ft. per second. Then
Q = Av.
But since the pipe is of constant diameter, A is constant, and
hence also v is constant, so that the mean velocity is the same
throughout the whole length of the pipe.
Consider an elementary mass of the fluid AABB, bounded
by the pipe and by the two cross-sections AA, BB. Let dl
78
FLOW OF WATER IN PIPES.
79
FIG. 50.
be the length AB of the element, the length / ft. of the
pipe being measured along the
axis from any origin O.
Let z, z + dz be the eleva-
tions in feet above a datum
line of the centres of pressure
in the cross-sections A A, BB,
respectively.
Let p, p + dp be the intens-
ities of the pressures on these
cross-sections in pounds per
square foot.
Let P be the perimeter of
the pipe.
Let w be the specific weight of the water in pounds per
cubic foot.
Work Done by Gravity. — In one second wQ Ibs. of water
are transferred from AA to BB, falling through a vertical dis-
tance of dz ft. Thus the work done by gravity per second
= — wQ . dz,
a positive quantity if dz is negative, and vice versa.
Work Done by Pressure. — The total pressure on A A paral-
lel to the axis = pA ; the total pressure on BB parallel to
the axis = (p + dp) A.
Therefore ^the total resultant pressure parallel to the axis
in the direction of motion = — A . dp, and the work done per
second on the volume Q by this pressure = — Q . dp.
Note. — The work done by the pressure at the pipe surface is nil, as
its direction is at right angles to the line of motion.
Work Absorbed by Frictional Resistance. — From the laws of
fluid friction this work per second is evidently
p
— — P . dl . F(v) X v = -r . Q . F(v) . dl,
the sign being negative as the work is done against a resistance.
80 HYDRA ULICS.
Since the motion is steady, the work done by the external
forces must be equivalent to the work absorbed by the fric-
tional resistance, and hence
— wQ . dz — Q . dp - —Q . F(v) . dl — o,
or
, dp P F(v} „
(Jn I * _l V ' /y/ Q
w A " w
Integrating,
^ + -f-j../=a constant = H,
w A w
so that H ft.-lbs. per pound of fluid is the uniformly distributed
total constant energy.
A
— is called the hydraulic mean radius of a pipe and will be
denoted by m.
Take
W 2g
the value adopted in ordinary practice, f being the coefficient
of friction. Then
w in 2g
Let #, , Al ,/, be the elevation above datum, the area of the
cross-section, and the intensity of the pressure
at any point X on the axis of the pipe distant
/x from the origin (Fig. 51).
Let £2, AS, pi be the elevation above datum, the area of the
cross-section, and the intensity of the pressure
at any other point Y on the axis distant 4
from the origin (Fig. 51).
FLO W OF WATER IN PIPES.
Then, from the equation just deduced,
81
,.+* + £*=*=*+>• + £«!.
w m 2g w m 2g
Hence
w
m 2g
L being the length /2 — ^ of the pipe between the two points
K
FIG. 51.
Let vertical tubes (pressure-columns) be inserted in the«.pipe
at X and at Y. The water will rise in these tubes to the levels
C and Z>, and evidently
° being the intensity of the atmospheric pressure.
82 HYDRA ULICS.
Hence, if CX and D Fare produced meet the datum line in
E and F,
I A i/^j^iA ri? j A
#. -+- — = -Sj -h CA -f- — = Czi -t- -JL
ze; w w
and
#a + — = £a + jC>F+ — = Z>F+— .
w w w
Therefore
w wi m 2g
G being the point in which the horizontal through C meets FD
produced.
DG is called the " virtual fall " of the pipe, being the fall of
level in the pressure-columns; and since there would be no fall.
of level if the friction were nil, DG is said te be the head lost
in friction in the distance XY.
Denote this head by h\ then
=
m 2g
and therefore
£_/»•
L m 2g
This ratio - is designated the virtual slope of the pipe, and
JL/
is the head lost in friction per unit of length It will be
denoted by *', so. that
If the section of the pipe is a circle of diameter d, or a
square with a side of length d, then
and
FLOW OF WA7^ER IN PIPES. t _ 83
A d
__ =
L ~ d 2g
3. Influence upon the Flow of the Pipe's Position and
Inclination. — In Fig. 5 1 join CD. Now since the fall of level
(h) is proportional to Ly the free surface in any other column
between X and Y must also be on the line CD. Thus the
pressure/7 at any intermediate point M distant x(==. XM) from
X is given by
w w w
Hence, at every point of a pipe laid below CD, the fluid pres-
sure (pr) exceeds the atmospheric pressure (/0) by an amount
w . MN, so that if holes are made in such a pipe the water will
flow out and there will be no tendency on the part of the air
to flow in. In pipes so placed vertical bends may be intro-
duced, care being taken to provide for the removal of the air
which may collect in the upper parts of the bends.
If the line of the pipe coincides with CD, i.e., with the vir-
tual slope or line of free surface level, MN = o, and the fluid
pressure is equal to that of the atmosphere. If holes are now
made in the pipe it can easily be shown by experiment that
there will be neither any tendency on the part of the water to
flow out nor on the part of the air to flow in.
Next take CC' = DD' = and join CD'.
w J
If the pipe is placed in any position between CD and C ' Dr
MN becomes negative, and the fluid pressure in the pipe is less
than that of the atmosphere. If holes are made in this pipe,
there will be no tendency on the part of the water to flow out,
84 HYDRA ULICS.
but the air will flow in. Thus, if a pipe rises above the line of
virtual slope, there is a danger of air accumulating in the pipe
and impeding, or perhaps wholly stopping, the flow. No verti-
cal bends should be introduced, as the air is easily set free and
would collect in the upper parts of the bends, with the effect
of impeding the flow and of acting detrimentally upon the water
itself, which the liberation of the air renders less wholesome.
If the line of pipe coincide with CD', then the fluid pressure
is nil.
Finally, if the pipe at any point rises above CD', the press-
ure becomes negative, which is impossible. In fact, the con-
tinuity of flow is destroyed, and the pipe will no longer run full
bore. Air will be disengaged and will rise and collect at the
point in question, so that in order to prevent the flow being
wholly impeded, it will be necessary to introduce an air-chamber
at this point from which the air can be removed when required.
Note. — In the preceding it has been assumed that the pipe is straight.
If the pipe is curved, so also is the line of virtual slope. In ordinary
practice, however, the vertical changes of level in a pipe at different
points are small as compared with the length of the pipe, and distances
measured along the pipe are sensibly proportional to distances measured
along the horizontal projection of the pipe. Hence the line of virtual
slope may be assumed to be a straight line without error of practical
importance.
4. Transmission of Energy by Hydraulic Pressure. —
Let Q cub. ft. of water per second be driven through a
pipe of diameter d ft. and length L ft. under a total head of
H ft. Also let n per cent, of the total head be absorbed in
overcoming the frictional resistance in the pipe. Then
the head expended in useful work = H — h
H-h
and the efficiency =
FLOW OF WATER IN PIPES. 8$
Again,
— - h - - -
100 : ~d~ 2g ~''
Since Q = — z/, and g is assumed to be 32, thus,
^ InHd*
- toy T£~'
ancf the work transmitted in foot-pounds per second
14
If ^V= the number of horse-power transmitted, then
jv - _L i;_5 /^£ 1 A^8^
"550 H V /^ "28V "7^~'
and this equation also gives the distance L to which TV horse-
power can be transmitted with a loss of n per cent of the total
head.
Again,
ffi . // 2fL V* 2/Lw v*
the efficiency = I — — = I : -77 T — l — -^—
H gH d g pd>
p( = wH) being the pressure corresponding to the head H.
Thus, the efficiency is constant if — - is constant.
pa
Assuming this to be the case, take v* = c* .pd. Then the
total energy transmitted = wQH ' = w vH
4
86 HYDRA ULICS.
If it be also assumed that the thickness / of the pipe-metal
is so small that the formula
pd = 2ft
holds true, f being the circumferential stress induced in the
metal, then
the energy transmitted = —
F being the volume of the pipe per unit of length.
Hence, for a given volume (V) of metal and a constant
efficiency, the energy transmitted is a maximum when pd is a
maximum.
If / is increased beyond a certain limit, the ratio -^ is no
longer small and the thickness t will have a greater value
than that given by the equation pd = 2. ft. Then the cost of
the pipe will also increase. On the other hand, if d is increased
the ratio -^, and therefore also the pressure/, will remain small,
and thus the cost of the pipe will not increase. Hence it is
more economical to employ large pipes and low pressures than
small pipes and high pressures.
Note. — The efficiency diminishes as v increases, so that, as far as the
efficiency is concerned, it is advantageous to transmit the energy at a
low speed.
5. Flow in a Pipe of Uniform Section and of Length Z,
connecting two Reservoirs at Different Levels. — Let z ft.
be the difference of level between the water-surface in the two
reservoirs.
FLO W OF WATER IN PIPES. 8?
FIG. 52.
The work done per second is evidently equal to the work
done by the fall of wQ pounds of water through the vertical
distance z, and is expended—
(1) In producing the velocity of flow v feet per second
which requires a head of zl feet and an expenditure
of wQzl foot-pounds of work per second ;
(2) In overcoming the resistance at the entrance from the
upper reservoir into the pipe, which requires a head
of sa feet and an expenditure of wQz^ foot-pounds
of work per second.
(3) In overcoming the frictional resistance which requires a
head of z^ feet and an expenditure of wQz^ foot-
pounds of work per second. Thus
wQz = wQzl + wQz^ +
or
z = z * *.
Now #, — - - feet, and the corresponding energy wQz^ is
ultimately wasted in producing eddy motions, etc., in the
lower reservoir.
v*
z^ may be expressed in the form n — feet, n being a coeffi-
cient whose value varies with the nature of the construction of
the entrance into the pipe. If the pipe-entrance is bell-mouth
in form, n = .08, but if it is cylindrical, n = .5. Finally,
88 HYDRA ULICS.
f, = ft.,
,
m w d 2g
F(v) v*
Baking - - =f — , as is usual in practice. Hence
2g\ d
since Q = -- v, and g is assumed to be 32.
4
For given values of Q and z a first approximate value of d
may be obtained from the last equation by neglecting the term
Q*
— rr;(l + «)• Call this value dv and substitute it for the d in
A/L
the term — j- within the brackets. A second approximation
may now be made by deducing d from the formula
and the operation may be again repeated if desired.
Generally speaking, I + n is usually very small as compared
with — , and may be disregarded without error of practical
importance.
The formula then becomes
_
which is known as Chezy's formula for long pipes.
In fact, the term I + n need only be taken into account in
the case of short pipes and high velocities.
FLOW OF WATER IN PIPES.
89
6. Losses of Head due to Abrupt Changes of Section,
Elbows, Valves, etc. — When the velocity or the direction of
motion of a mass of water flowing through a pipe is abruptly
changed, the water is broken up into eddies or irregular mo-
tions which are soon destroyed by viscosity, the corresponding
energy being wasted.
CASE I. Loss due to a sudden contraction. (Art. 16, Chap. I.)
(a) Let water flow from a pipe (Fig. 53), or from a reser-
voir (Fig. 54) into a pipe of sectional area A.
FIG. 53-
FIG. 54.
Let cc be the coefficient of contraction.
Then the area of the contracted section = ccA, and
the loss of head = — (--. «,Y
2 V I
2g V,
where m = ( 2 I .
= m
V2
2?
The value of m has not been determined with any great
degree of accuracy ; but if cc = .64, then m = .316.
HYDRAULICS.
When the water enters a cylindrical (not bell-mouthed) pipe
from a large reservoir, the value of m is
about .505.
(b) Let the water flow across the abrupt
change of section through a central ori-
fice in a diaphragm placed as in Fig. 55.
Let a be the area of the orifice.
Then c,a is the area of the contracted section, and
the loss of head = ( —
W
(A y
where m = I I J •
V^z t
According to Weisbach,
I A Vv* v*
= I — — I J — = m — ,
\cea I 2g 2g'
f- =
•I
.2
•3
•4
•5
cc =
.616
.614
.612
.610
.607
m =
231.7
50.99
19.78
9.612
5.256
•i-
.6
•7
.8
•9
I.OO
Cc —
.605
.603
.601
.598
.596
m =
3.077
1.876
1.169
•734
.48
central
~ =i orifice of area a. nlaced in
a cylin-
-.V I
drical pipe of
7 1
sectional area
A as in
FIG. 56.
Fig. 56.
The " contracted area " of the water = c<a and
the loss of head
i IvA V v* IA \
= — I --- v) = -- -- i)
2sr\cja I 2.g\cfa i
= m-—,
where m = {
FLOW OF WATER IN PIPES. 9 1
Generally m must be determined by experiment, but Weis-
bach gives the following results :
if = -1 -2 .3 .4 .5
ce— .624 .632 .643 .659 .681
m= 225.9 4777 30.83 7.801 3.753
if ^ — .6 .7 .8 .9 i. oo
cc= -712 .755 .813 .892 i. oo
m = 1.796 .797 .29 .06 oo
CASE II. Loss due to a Sudden Enlargement. (Fig. 57.)
Let Al = external area of small pipe.
" A, = " " " large "
FIG. 57-
r , • , i fvA9 V v* (A, V
Then, loss of head = — \—-± — v\ = — -^- — i]
*f\A. I 2jr\A. I
2£-
= m — ,
(A Y
where m — \-~ — i) .
A
Note.— The losses of head in Case I (a) and in Case II may be
avoided by substituting a gradual and regular change of section for the
abrupt changes.
CASE III. Loss of Head due to Elbows. (Fig. 58.)— The
loss of head due to the disturbance caused by an elbow is ex-
v*
pressed by Weisbach in the form m — ,
o
where m =. 9457 sin2 — + 2.047 sin4 — >
0 being the elbow angle.
Weisbach deduced this formula from the results of experi-
ments with pipes 1.2 in. in diameter.
92
HYDRA ULICS.
The velocity vl with which the water flows along the length
AB may be resolved into a component v with which the water
flows along BC and a component u at right angles to the
FIG. 58.
direction of v. The component u and therefore the corre-
sponding head, viz., — , is wasted. The component u evidently
diminishes with the angle 0 and becomes nil when a gradually
and continuously curved bend is substituted for the elbow.
CASE IV. Weisbach gives the following empirical formula
for the loss of head at a bend in a pipe :
hb = mt
, d\k
where m = .131 -f 1.847 —
for a circular pipe of diameter d, p being
the radius of curvature of the bend, and
FIG. 59.
m = .124+ 3-104—
for a pipe of rectangular section, s being the length of a side
of the section parallel to the radius of curvature (p) of the bend.
CASE V. Valves, Cocks, Sluices, etc. — The loss of head in
each of the cases represented by the several figures may be
traced to a contraction of the stream similar to the con-
FLO W OF WATER IN PIPES. 93
traction which occurs in the case of an abrupt change of sec-
v*
tion. The loss may be expressed in the form m — , and the
following tables give the results obtained by Weisbach.
(a) Sluice in Pipe of Rectangular Section. (Fig. 60.) — Area
of pipe = a ; area of sluice = s.
— = i .9
•5 -4
.2 .1
m
= .oo .09 .39 .95 2.08 4.02 8.12 17.8 44.5 193
FIG. 60.
(b) Sluice in Cylindrical Pipe. (Fig. 61). — s =
ratio of height of opening to diameter of pipe.
s= i .875 -75 -625 .5 .375 .25 .125
m = .00 .07 .26 .81 2.06 5.52 17.00 97.8
(c) Cock in Cylindrical Pipe (Fig. 62).
s = ratio of cross-sections;
6 = angle through which cock is turned.
FIG. 61.
FIG. 62.
If 0 = 5°
s = ,926
m = .05
If/= 40°
.85
.29
45°
= •385 .315
= 17.3 31.2
15°
.772
•75
50°
.25
52.6
FIG 63.
20° 25° 30° 35°
.692 .613 .535 .458
1-56 3-1 547
55
106
60° 65°
.137 .091
206 486
82°
00
oo
(d) Throttle-valve in Cylindrical Pipe (Fig. 63)
0 — angle through which valve is turned.
94 HYDRA ULICS.
If 61 =5° 10° 15° 20° 25° 30° 35° 40°
^ = .24 .52 .90 1.54 2.51 3.91 6.22 10.8
If 0=45° 50° 55° 60° 65° 70° 90
#2=18.7 32.6 58.8 118 256
oo
CASE VI. The fall of free surface-level, or loss of head, due
to sudden changes of section, frictional resistance, etc., may be
graphically represented as in Fig. 64.
FIG. 64.
Let a length of piping AE connect two reservoirs, and let
h be the difference of surface-level of the water in the reser-
voirs.
Let Llt rl be length and radius of portion AB of pipe.
« T ~ it « « « « « nr" " <(
/-„ rt ZJCx
" L,, r, " " " " " " CD " •
" L T " " " " " " JD fi 4i "
" ut,u9,ut, ut be the velocities of flow in AB, BC, CD,
DE, respectively.
FLOW OF WATER IN PIPES. 95
The reservoir opens abruptly into the pipe at A.
There is an abrupt change at B from a pipe of radius rv. to
one of radius r^.
There is an abrupt change at C from a pipe of radius ra to
one of radius ra.
At D the water flows through an orifice of area A in a dia-
phragm. At E the velocity of the water as it enters the lower
reservoir is immediately dissipated in eddies or vortices.
Draw the horizontal plane amnop at a distance from the
water-surface in the upper reservoir equal to the head due to
atmospheric pressure.
Draw vertical lines at A, B, C, D, E. Take
ab =loss of head at the entrance A = .49— - ;
= « u tt due to faction from A to B =fci-j ;
r> ^g
r* \a^ a
cd=. " " " due to change of section at B=l-^— il -*-
V i I -%3
re — " " " due to friction from B to C =^-£a ;
= " " " due to change of section at ^=.316- ;
o
— " « « due to friction from C to D =^ . ^-Z
" « " due to change of section at D= (^ - 1) — ;
tk = " " " due to friction from D to E —^- ^L, ;
z/2
kl-= " " " corresponding to u — -— .
HYDRA UL1CS.
Through / draw a horizontal plane Ix. This plane must
evidently be at a distance from the water-surface in the lower
reservoir equal to the pressure-head due to the atmosphere.
Then the total loss of head = Ip
ef + gh + M + 0C + re + sg+ tk,
i
, 2g r r% 2g ' r9 2g
2 3 3
The broken line abcdefghkl is the hydraulic gradient.
7. Remarks on the Law of Resistance. — Poiseuille's ex-
periments on the flow of water through capillary tubes showed
that the loss of head was directly proportional to the ve-
locity.
In the case of pipes used in ordinary practice the loss is
undoubtedly more nearly proportional to the square of the
velocity, and must be mainly due to the formation of eddies.
These eddies, again, are formed more or less readily according
as the water possesses less or greater viscosity.
FLO W OF WATER IN PIPES. 97
The experiments of Unwin and others have shown that the
surface friction is diminished by about i<f> for every rise of 5° F.
in the temperature, and it is also known that the viscosity
diminishes as the temperature rises and vice versa. Reynolds
has propounded a single law of resistance to the flow through
pipes, which embraces the results of Poiseuille and of Darcy,
and takes into account the effects of viscosity, temperature,
etc. This law may be expressed in the form
Bn vn
slope = * = ____
where d is the diameter of the pipe, A = 67,700,000, B = 396,
and P= (i + .0336^ + .00022 1/2), the units being metres and
degrees centigrade (/).
Unwin considers that the index of the diameter d is not
exactly 3 — n, and should be determined independently. For a
rough surface n — 2, for a smooth cast-iron pipe n = 1.9, and
for a lead pipe n = 1.723 ; a limitation which is analogous to
that found by Froude in his experiments upon surface fric-
tion.
Experimenting with glass tubes, Reynolds found for veloc-
ities below a certain critical velocity given by the formula
that the motion of the water is undisturbed, i.e., that it was in
parallel stream-lines. At and above this critical velocity eddies-
are formed, and the parallel stream-line motion is completely
broken up within a very short distance from the mouth of the
tube.
In capillary tubes— = 43.79.
In ordinary pipes — = 278.
98
HYDRA ULICS.
8. Flow of Water in a Pipe of Varying Diameter.—
The variation in the diameter is supposed to be so gradual
that the fluid filaments may still
be assumed to flow in sensible
parallel lines.
Consider a thin slice of the
moving fluid, bounded by the
transverse sections AB, CD, dis-
tant s and s -f- ds, respectively,
from an origin on the axis of the
pipe.
FIG. 65. Let/ be the mean intensity of
pressure, A the water area, P the wetted perimeter for the sec-
tion AB.
Let these symbols become / + dp, A + dA, P + dP, re-
spectively, for the section CD.
Let z be the height of the C. of G. of the section AB
above datum.
Let z -f- dz be the height of the C. of G. of the section CD
above datum.
Let «, u + du be the velocities of flow across the sections
AB, CDy respectively.
Then
The rate of increase of
momentum of the slice
ABCD in the direction of
the axis
f momentum generated by
the effective forces acting
Iupon the slice in the same
direction.
The acceleration in time dt = —Au . dt-j- = — Au . du.
g dt g
The total pressure on AB = p .A, and acts along the axis.
The total pressure on CD = (p + dp) (A + dA\ and acts along
the axis.
The total normal pressure on the surface ACBD of the pipe
= 27t[r-\ J \p + — j A C = 2nrp . A C, very nearly.
FLOW OF WATER IN PIPES. 99
The component of this pressure along the axis
= 2nrpAC .sin 6
= 2 npr . dr, nearly,
6 being the angle between AC and the axis.
Thus the total resultant pressure along the axis
= pA - (p -\-dp\A + dA) + 2npr.dr
= — p.dA — A.dp-}- 27rpr . dr
= -A.dp,
since A — 7tr\ and therefore dA = 27tr . dr.
The component of the weight of the slice along the axis
dA\ I dA\
• w sm i = — \A H )«/• dz= — iv A . dz.
The frictional resistance = P.AC. F(u) = P . ds . F(u), very
nearly. Hence
wAu . du
—~ — - = — A . dp — wA . dz —P. ds . F(u\
o
and therefore
dpu.du. PF(u) y
Integrating,
p ,u' CP F(u} j
z + w + ^+J A ~^ds = a Constant
Then
100
HYDRAULICS.
The integration can be effected as soon as the relation be-
tween r and s is fixed.
Example. — Take r = a -f- bst and assume /and Q to be con-
stant. Then
£_L"C__L — + -r — 3 / — = a constant,
' w~ 2g~ b gn J r"
and therefore
z i <_ _|_ — i _ — _ — — a constant.
' w 2 ' -2 4
9. Equivalent Uniform Main. — A water-main usually con-
sists of a series of lengths of different diameters.
As a first approximation the smaller losses of head due to
changes of section, etc., may be disregarded, and the calcula-
tions may be further simplified by substituting for the several
lengths a single pipe of uniform diameter giving the same fric-
tional loss of head. Such a pipe is called an equivalent main.
FIG. 66.
Let /,,/„, /3 be the successive lengths of the main.
Let dt , d^ , d^ be the diameters of these lengths.
Let z/v, vt, v3 be the velocities of flow in these lengths.
Let //, , h^ , h^ be the frictional losses of head in these lengths.
Let Z,, d, v, h be the corresponding quantities for the
equivalent uniform main.
Then
h = //, + h, + h, + . . . ,
and therefore
r _ , , , ,
~~ ~ 1
FLOW OF WATER IN PIPES. IOI
Hence
where it is assumed that /is the same for the several lengths
of the main and also for the equivalent pipe.
But
nd* nd? nd?
TV = Q = —Vi = —v,= «c.
Hence
L I, /2 /3
an equation giving, the length L of an equivalent pipe having
the same total frictional loss of head.
10. Branch Main of Uniform Diameter. — In a branch main
AB of length L and diameter d, receiving its supply at A. —
Let Qw be the way-service, i.e., the amount of water given
up to the service-pipes on each side.
Let Q be the end-service i.e., the amount of water dis-
charged at the end B.
Then it may be assumed, and it is approximately true, that
the way-service per lineal foot, viz., -JT-, is constant.
Thus the amount of water consumed in way-service in a
length AC of the main, where BC = s, is
while the total amount of water flowing across the section of
the pipe at C
•v being the velocity of flow at C.
I O2 H YDRA ULICS.
Now dh, the frictional loss of head at C for an elementary
length ds of the pipe, is given by the equation
= 32.
Integrating, the total loss of head is
SPECIAL CASES.
CASE I. Let <2/ be the total discharge for the same fric-
lional loss of head, ^, when the whole of the way-service is
stopped. Then
or = &• + Q.Q. + Qjf-
0
and therefore
Hence
and <2/ lies between g« + — and Qe+ — 7=QW, its mean value
V 3
being &
CASE II. If there is no end-service, all the water having
been absorbed in way-service, Qe = o, and therefore Q'e = —r=
V $
and
FLOW OF WATER IN PIPES. 1 03
CASE III. If Qe = o,
fQ™
dh = vTsT^ds == elementary f fictional loss of head.
Integrating between o and s,
and the vertical slope, or line of free pressure, becomes a cubical
parabola.
CASE IV. Let the main receive its supply at A from a
reservoir X in which the surface of the water is hl above datum,
and let it discharge at the end B into a reservoir Fwith its
surface I? above datum.
Since (QeJ = Q? + 0CQW + ^, therefore
If Qw = TsQ.', Qe = o; and if Qw > 3g/, then the res-
ervoir Fwill furnish a portion of the way-service.
Suppose that X gives the supply for the distance AO
(= /,) and that Ksupplies BO (= /a).
Let z be the height above datum of the surface in a press-
ure column inserted at O.
Then, neglecting the loss of head at entrance,
w
i fQ V"
= loss of head between A and O = — r» /a,
3 ^ a L,
and
J fQ 2/S
= loss of head between B and O = rybi
3 n d L
Also A +/, = L.
104
HYDRA ULICS.
II. Nozzles. — Let a pipe AB, of length / and diameter d,
lead from a reservoir h ft. above the end B.
First, let the pipe be open to the atmosphere at B.
FIG. 68.
Then
(v*
= n —
2g
I
-f- head to overcome resistance due to bends, etc. = m—
V 2
•4- head to overcome frictional resistance (= — >)
\ d 2gl
-|- head corresponding to the velocity v in the pipe and at
the outlet f= * J
4/A I ^
d } 2g
Hence the height to which the water is capable of rising
B
v\
or, again, is
=f=A--
h
4/
^-t
d r
£
~d
Second, let a nozzle be fitted on the pipe at B.
Let V be the velocity with which the water leaves the
nozzle.
FLO W OF WATER IN PIPES. 1 05
Let D be the diameter of the nozzle-outlet.
This diameter is very small as compared with the diameter
d of the pipe. But
T7
V = — v,
4 4
and therefore
so that Fis very large as compared with i>.
Also,
h = head to overcome the resistance to entrance at A
-\- head to overcome the resistance due to bends, etc.
-f- head to overcome the frictional resistance in pipe
+ head to overcome the frictional resistance in nozzle
(=*•£)
V 2g )
-f- head corresponding to the velocity V with which the
/ V**\
water leaves the nozzle — —
, 4/A ,F3 . V
— — n -f m + ^- + m'— + — ,
2g\ dl 2g^ 2g
and the height to which the water is now capable of rising at
j5is
v* 7 v*( . . 4/A ,Fa
— = h — — (n-\- m-4- ^—} — m' —
2g 2g\ d I 2g
h
Let — , = //„, be the pressure-head at the entrance to the
w
nozzle. Then the effective head at the same point
Hence
io6
HYDRAULICS.
It will be observed that the delivery from the nozzle is less
than that from the pipe before the nozzle was attached, but
that the velocity-head at the nozzle-outlet is enormously in-
creased. The actual height to which the water rises on leav-
ing a nozzle is less than the calculated height, owing to air-
resistance and to the impact of particles of water as they fall
back.
The force required to hold the nozzle is evidently
g £4
If the water flowing through a pipe, or hose, of length / ft.,
with a velocity of v ft. per second, is quickly and uniformly
shut off by a stop-valve t sec., the pressure in the pipe near the
valve is increased by an amount - - Ibs. per square foot.
<3
Of two forms of nozzle in general use, the one (Fig. 70) is a
FIG. 69. FIG. 70.
surface of revolution with a section which gradually diminishes
to the outlet, while the other (Fig. 69) is a frustum of a cone,
having a diaphragm with a small circular orifice at the outlet.
Denoting the former by A and the latter by £, the following-
table gives the results of Ellis's experiments :
Height of jet from
i-inch Nozzle.
Height of jet from
i£-mch Nozzle.
Height of jet from
iHnch Nozzle.
Pressure in Ibs.
Head in
per sq. in.
feet.
A
B
A
B
A
B
10
23
22
22
22
22
23
22
20
46
43
42
43
43
43
43
30
69
62
61
63
62
63
63
40
92
79
78
81
79
82
80
50
"5
94
92
97
94
99
95
60 ,
138
108
104
112
108
"5
no
70
161
121
"5
125
121
129
123
Sc
184
131
124
137
131
142
135
90
207
140
132
148
141
154
146
• IOO
230
148
136
157
149
164
155
FLOW OF WATER IN PIPES. IO/
Third, if an engine, working against a pressure of pc Ibs. per
square foot, pumps Q cubic feet of water per second through
a nozzle at the end of a hose / feet in length, then
the pumping H.P. of the engine == — — .
The total head at the engine end of the hose = the head
corresponding to the pressure p in the hose -f~ the head re-
quired to produce the velocity of flow v
W 2g
and this head is expended in overcoming the frictional resist-
ance of the hose (all other resistances are disregarded) and in
producing the velocity of flow Fat the outlet. Hence
W W 2g d 2g 2g
and therefore
W d 2g 2g
- _ JL
gn* *
.- Ttd 41.J-S T-
since Q = v = F.
4 4
The pumping H.P.
8wff_(j_ 4/7\
-o7t*\D* d* r
12. Motor Driven by Water from a Pipe. — Let the
nozzle in the preceding article be replaced by a cylinder hav-
ing its piston driven by the water from the pipe.
Let u = the velocity of the piston per second.
Let pm = unit pressure at the end of the pipe, i.e., in the
cylinder.
Let dm — diameter of cylinder.
IO8 HYDRA ULICS.
Then, velocity of flow in pipe = ~jp-u* Hence
, _ d^ ul , 4// dm< u* pm
(other losses of head being disregarded).
13. Siphons. — A siphon is a bent tube, ABCD, Fig. 71, and
>-— r— is often employed to convey
water from one reservoir to
another at a lower level.
Let hv //3, respectively, be
the differences of level be-
tween the top of the siphon
and the entrance A and outlet
D to the siphon. Then, so
long as the height k1 does not
exceed the head of water
(= 32.8 ft.) which measures
the atmospheric pressure, the
FlG* 7I> water will flow along the tube
in the direction of the arrow, with a velocity v given by the
equation
/being the length of the tube ABCD, and all resistances, ex-
.cept that due to frictional resistance, being disregarded.
If //, > 32.8 feet, each of the branches AB and DC becomes
a water-barometer, and the siphon will no longer work.
Even when the siphon does work, an arrangement must
be made for withdrawing the air which will always collect at
the upper part of the siphon,
14. Inverted Siphons. — The existence of a cutting or a
valley sometimes renders it necessary to convey the water
from a course AB to a course DE by means of an inverted
siphon BCD of length.
Let u be the velocity of flow in AB, and h the height of B
above a datum line.
FLOW OF WATER IN PIPES, 1 09
Let v be the velocity of flow in the siphon, and ht the height
of D above datum.
FIG. 72.
Then
h^ — 7za = loss of head at B
-\- frictional loss of head in siphon
loss of head at D
= ,
zg d 2g "•" 2g
4/7 v*
= ZL -- , approximately,
assuming the entrance and outlet to the siphon formed in
u* v*
such a manner as to considerably reduce the losses — and — ,
zg 2g
and to allow of these losses being disregarded without practical
error. Find, by chaining along the ground, the length of the
siphon from B up to a point F not far from D. Call this
length /, , and let \ be the height above datum of F, obtained
with a level. Generally speaking, DF is nearly always of
uniform slope. Call the slope a. Then,
DF = (k^ — h^) cosec a.
But
= hl — h^ — DF. sin #,
an equation from which DF can be found, as /^ — h^ can be
determined by means of a level.
10
HYDXA ULICS.
15. Air in a Pipe. — The effect of an air-bubble in a pipe
ABCD may be discussed as follows:
Let the air occupy the portion BC of a pipe.
Let the surface of the water in the reservoir supplying the
pipe be h^ ft. vertically above E, and hz ft. above D.
FIG. 73.
Also, let h^ be the difference of level between C and D, ht
the difference of level between B and C, and / the thickness of
the water-layer EF.
Let H designate the head equivalent to the elastic resist-
ance of the air in BC. Then, approximately,
and
A
/! being length of portion of pipe from A to E, and /, the length
from E to D.
Adding the two equations,
L IL f - 4/ *>* ,, , ... _ 4// v*
/*,+;,,-/-. __(/l + /,). __,
/ being total length of pipe.
But //! — / + h< = //, — h^ , very nearly. Hence
an equation showing the variation of v with a variation in the
height /*4 of the space occupied by the air.
Note. — H o>{ course varies with the temperature.
FLOW OF WATER IN PIPES.
Ill
16. Three Reservoirs at Different Levels connected by
a Branched Pipe. — Let a pipe DO of length /x ft. and radius
rl ft., leading from a reservoir A in which the water stands hl
ft. above datum, divide at O into two branches, the one, OE,
of length /2 ft. and radius r2 ft., leading to a reservoir B in
which the water stands //2 ft. above datum, the other, OF, of
length /3 ft. and radius r3 ft., leading to a reservoir C in which
the water stands h ft. above datum.
I
-.
FIG. 74.
Let vlt v^ vz be the velocities of flow in DO, OE, OF, re-
spectively.
Let Qlt <2a, Q, be the quantities of flow in DO, OE, OF,
respectively.
Let z be the height above datum to which the water
will rise in a tube inserted at the junction.
Two problems will be considered, and all losses of head
excepting those due to frictional resistance will be disre-
garded.
PROBLEM I. Given h,, h^ hz ; rlf r9, r3 ; to find Qlf <23, Q3 1
•z.', , z>2, i>3, and -S".
fo ^ ^ a
For the pipe DO, -~— = a- . . (i) and Q^Ttrfv,. . . (2)
** ^\
=«V . . (3) " Q.= «r:Vf • (4)
112 HYDRAULICS.
For the pipe OF, Z-^-^=a^*- . . (5) and Q3=nr3*vs. . . (6)
Also, 01= ±<2,+ <2.. • V.;V ; , . (7)
From these seven equations the seven required quantities
can be found.
In equations (3) and (7), the upper or lower signs are to be
taken according as the flow is from O towards £ or from R
towards O.
This may be easily determined as follows :
Assume z — 7za, and then find vl and v^ by means of equa-
tions (i) and (5), and hence Q^ and <23 by means of equations
(2) and (6). If it is found that Ql > Q9, then the flow is from
O to E, and equations (3) and (7) become
s=«^ and =
while if <2, < <23, the flow is from E to O, and the equations
are
-?— — = <*— and
(-9-
. — It is assumed that a — - is the same for each
pipe.
SPECIAL CASE. Fig. 75. — Suppose the pipe OE closed at E.
Also let r^ = ra = ra = r, and let V be the velocity of flow
from A to C.
The " plane of charge " for the reservoir A is a horizontal
plane MQ distant — from the water surface, /0 being the at-
mospheric pressure.
The " plane of charge " for the reservoir C is a horizontal
plane TS distant — from the water-surface.
w
Fa
In the vertical line VTQ, take TN — — and join MN.
2g'
Then, neglecting the loss of head at entrance, MN is the
FLOW OF WATER IN PIPES. 113
" line of charge," or hydraulic gradient, for the pipe DF, and
is approximately a straight line.
Let the " plane of charge " KK for the reservoir B, distant
— from the water-surface, meet MN in G.
If the junction O is vertically below G, there is no head
._£-- — £ £ _<?___
1
I
»
!
Ps,
,
•
• if*
^j._ — :>,
^ ...
7^
>i
vvv
7^^
°x--*
tSJnfa
TIT
\^
f\- -
m.
f 3
KTxv
^^
/
x\
r*4
,
f\
;s
i " /
V m
f
i
FIG. 75-
available for producing flow either from E towards O or from
O towards E, and hydrostatic equilibrium is established.
If the junction O is on the left of G, and a vertical line
OKHL is drawn intersecting KK, MN, and MQ in the points
K, H, and L, there is the head Hf£ available for producing
flow from O towards E.
If the junction O is on the right of G, and the vertical line
OHKL is drawn, the head HK is now available for producing
flow from E towards O.
Let the vertical through G meet MQ in Pt and take
PG = Y. Then, approximately,
I, + / ~~ MN~ QN~ h,-
H4 HYDRA ULICS.
and therefore
y - k* ~ k* j
' A+A ' ''
If HL < F, the flow is from O towards £.
If HL > F, " " " " E " O.
Again,
w 2gl r
and therefore, approximately,
Next assume the junction O to be on the left of G, and
open the valve at E. Then
and Q,= Q,+ Q,,
or z/t = v, + vt.
Thus
«y(A+ O = *,-*, = "(/^,1+ A*,*) = " j A(».+ fJ'+Af ,' } ;
and therefore
^.'(A + A) + 2/w, + A^.1 - (A + A) ^ = o.
Hence, assuming z/a very small as compared with V,
or
where Q = nr* V.
FLOW OF WATER IN PIPES. 115
Thus it appears that if a quantity <23 of water is drawn off
by means of a branch from a main capable of giving a total
end service Q, this end service will be diminished by j-<22, \Q^
\Qv etc. according as the junction O divides the pipe DF into
two portions in the ratio of I to I, I to 2, I to 3, etc.
Note. — The more correct value of v^ is
/,+/
and the maximum value of — — :-LTT-« does not exceed — .
4
Orifice Fed by Two Reservoirs. — Neglect all losses of head
except the losses due to frictional resistance.
FIG. 76.
When the valve at 0 is closed the flow is wholly from A to
and the delivery is
The line of charge (hydraulic gradient) is -M/V, where
. w
Il6 HYDRAULICS.
Open the valve a little : a volume <2a will now flow through
<9, and a volume (23 mto £ where
The " line of charge" becomes the broken line MiN.
As the opening of the valve continues, the pressure-head at
O diminishes, and when it is equal to /z3 + — ° the line of charge
\sM2N, 2 N being horizontal. Hydrostatic equilibrium is now
established between O and C, and the whole of the water from
A passes through O, the delivery being given by
Opening O still further, both reservoirs will serve the ori-
fice, and the line of charge will continue to fall.
When the valve is full open the "line of charge" is
where 3(9 = — , and the discharge is
w
The supply from A is equal to that from C when -1 = — *.
The above investigation shows the advantage of a second
reservoir in emergent cases when an excessive supply is sud-
denly demanded, as, e.g., on the occasion of a fire.
PROBLEM II. Given /z,, //2, h^\ Q2, Qs, and therefore
Qt(= ± a+G3); tofirtdrI,r;,rt,f>l,9t,f'.,jr.
As before, let z be the pressure-head at O. Then
... (i) and <2, = «•>,; ... (2)
••• (3) " e, = *r,'Vt; ... (4)
... (5) " C. = ^>.. . . . (6>
FLOW OF WATER IN PIPES. 1 1/
These six equations contain the seven required quantities,
viz., r1 , ra , r5 , z\ , v9, vt, and z. Thus a seventh equation
must be obtained before their values can be found. This
equation is given by the condition " that the cost of the piping
laid in place should be a minimum/' it being assumed that the
cost of a pipe laid in place is proportional to its diameter.
Hence
llrl + 4ra + 4ra — a minimum (7)
From equations (i) and (2), -L— — — — j-^;
(3) (4), - -^
" (5) " (6), *-^b =
^3 rr,
Differentiating these three equations,
dz _ $aQ* ,
t
But by equation (7)
/X^ -|_ l^dr^ + /3^r3 = O.
Hence
6 „ 6
which is the seventh equation required.
Il8 HYDRAULICS.
This last equation may be written in the forms
and
a = ± a v a
^3 z/,1 ^33"
17. Mains with any Required Number of Branches.
Let there be n junctions and m pipes.
Let hl , //a , . . . hm be the m pressure-heads at the end of
each successive length of pipe.
Let #!,£,,...#„ be the n pressure-heads at the 1st, 2dy
3d, . . . 72th junctions.
Let /!,/,,.../,„ be the lengths of the ;// pipes.
PROBLEM I. Given //, , h^ , . . . hm , rl , r2 , . . . rm , to find
?i;tf9,. *•»»,*, »*,».••**«•
_i. ^ nz £ £>2
There are 772 equations of the type - -—— - = a—.
Also, the quantity flowing through the first portion of the
main is equal to the sum of the quantities flowing through all
the branches at the first junction, and an analogous equation
will hold for each of the remaining n — I junctions. Thus n
additional equations are obtained.
From these m -f- n equations, vl , vz , . . . vm , z^ , ^ , . . . zn
may be found analytically or by the method of repeated ap-
proximation.
PROBLEM II. Given h^ , h^ , . . . hm , Ql , <22 , . . . Qm , to find
There are now only m equations of the type
-[- h If % _ V*
~T~ a~r '
involving m -\- n unknown quantities, and the problem admits
of an infinite number of solutions.
It is therefore assumed that the cost of the piping laid
in place is to be a minimum. Thus n new equations are ob-
FLO W OF WATER IN PIPES.
tained, and the m -\- n equations may be solved analytically or
by repeated trial.
18. Variation of Velocity in a Transverse Section.—
Assumption. — That the water in any portion of a pipe is made
up of an infinite number of hollow concen-
tric cylinders of fluid, each moving parallel
to the axis with a certain definite velocity.
Let u be the velocity of one of these cyl-
inders of radius x and thickness dx. Then
the flow across a transverse section is given
by the equation FlG ?7
dq = 2nx dx . u,
and the total flow
Q—27tl uxdx, (i)
r being the radius of the pipe.
If vm be the mean velocity for the whole transverse section
of the pipe,
nr*
(2)
Again, assuming with Navier that the surface resistance
between two concentric cylinders is of the nature of a viscous
resistance and may be represented by k— per unit of area at
dx
the radius x, k being a coefficient called the coefficient of vis-
cosity, then the total resistance at the radius x for a length ds
of the cylinder
, du du
= — 2nx . ds . k — - = — 2nk . as . x--.
dx dx
The total resistance at the radius x -f- dx
du , <
I2O HYDRAULICS.
Hence the total resultant resistance for the length ds of the
cylinder under consideration
= 2nkds —
The component of the weight of the slice of the cylinder
in the direction of the axis
= w . 2nx . dx . ds . sin 0,
0 being the inclination of the axis to the horizon.
Let — dz be the fall of level in the distance ds. Then
— dz = ds . sin 6.
Therefore, component of weight in direction of axis
= — w . 2nx dx . dz.
The resultant pressure on the slice in the direction of motion
= P — (P ~\~ d£) . znx . dx = 2nx .dx .dp.
Then, since the motion is uniform,
w . 2nk . ds . —r\x—-\dx — w . 2rtx .dx.dz — 2nx .dx .dp — o,
dx\ dxi
and therefore
k . ds d f du\ dp
/t* _ ,/V ty f.
-T-U-H - dz - -£ = o.
x ax\ ax I w
Integrating only for the cylinder under consideration,
ks d f du\ ( p\
-- —\x-r] — (z + — ) = a constant.
x ax\ ax I \ w'
But z + — is evidently independent of x, and is a linear
w
function of s (Art. 2, Chap. III.). Hence
I d I du\
-- (x— = a constant = A, suppose.
x dx\ dx>
FLO W OF WATER IN PIPES. 121
Therefore
d I du\
-(,-J = ^.. . . . . . .. (3)
Integrating,
du x*
x—- = A \- B.
dx 2
Assuming that the central fluid filament is the filament of
maximum velocity, then when x = o, — - is also nil. Therefore
B = o and x^ = Ax\
dx
and therefore
Integrating,
u = A- + C.
4
Let wmax be the velocity of the central filament, i.e., the
value of u when x = o.
Then
(5)
where D = .
4
Again, by equation I,
Q = 27tJ (#max — Dx^x.dx = 7rr*\2£max — J :
and by equation 2,
Dr*
vm = umax (6)
2
If ;/, = surface velocity, then, by equation 5,
U, = «max - DS (7)
Hence, by equations 6 and 7,
us + «inax — zvm (8)
122
i v ,-r y
U
• '$*£'
V-M '• f5 T
: 5:<
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c
(
- ^
ULICS.
'
-1
-^
I
^\:
i^
EXAMPLES.
r sec. /
4 ft. in J
1. A water-main is to be laid with a virtual slope of i in 850, and is to
give a maximum discharge of 35 cubic feet per second. Determine the
requisite diameter of pipe and the maximum velocity, taking/ = .0064. Q V-
Ans. 3.679 ft.; 3.2888 ft. per sec.
2. Find the loss of head due to friction in a pipe : diameter of pipe /
= 12 in., length of pipe = 5280 ft., velocity of flow = 3 ft. per second ; *^
f = .0064.; Also find the discharge.
Ans. 19.008 ft. ; 2.3562 cub. ft. per
3. A pipe has a fall of 10 ft. per mile ; it is 10 miles long and
diameter. Find the discharge, assuming/ = .0064.
Ans. 54.7 cub. ft. per sec.
4. A pipe discharges 250 gallons per minute and the head lost in fric-
tion is 3 ft.. Find approximately the head lost when the discharge is 300
gallons per minute ; also find the work consumed by friction in both
cases. Ans. 4.32(1.; 7500 ft.-lbs. ; 12,960 ft.-lbs.
5. What is the mean hydraulic depth fn1 a circular pipe when the
diameter
water rises to the height -- -=- above the centre ?
2^2 I0
Ans. — x diameter.
6. A 12-inch pipe has a slope of 12 feet per mile; find the discharge.
(/=.oo5.) Ans. 2. ii^ cub. ft. per sec.
7. The mean velocity of flow in a 24-in. pipe is 5 ft. per second ; find
its virtual slope,/ being .0064. Ans. i in 200.
8. Calculate the discharge per minute from a 24-in. pipe of 4000 ft,
length under a head of 80 ft., using a coefficient suitable for a clean iron
pipe. Ans. 34.909 cub. ft. per sec.
9. How long does it take to empty a dock, whose depth is 31 ft, 6
ins. and which has a horizontal sectional area of 550,000 sq. ft., through
two 7-ft. circular pipes 50 ft. long, taking into account resistance at en-
trance ? Ans. 214 min. 6 sec.
10. The virtual slope of a pipe is i in 700; the delivery is 180 cubic
-feet per minute. Find the diameter and velocity of flow.
Ans. 1.26 ft.; 2.401 ft. per sec.
n. Determine the diameter of a clean iron pipe, 100 feet in length,
which is to deliver .5 cub. ft. of water per second under a head of 5 feet..
Assume/ = .006. Ans. .326ft.
FLOW OF WATER IN PIPES. 123
12. A reservoir has a superficial area of 12,000 ft. and a depth of 60
ft. ; it is emptied in 60 minutes through four horizontal circular pipes,
equal in diameter and 50 ft. long. Find the diameter. Ans. 1.75 ft.
Explain how the total head is made up, and draw the plane of charge.
13. A 3-inch pipe is very gradually reduced to i inch. If the press-
ure-head in the pipe is 40 ft., find the greatest velocity with which the
water can flow through. Ans. 1.4 ft. per sec.
14. Water flows through a 24-inch pipe 5000 yards in length. At 1000
yards it yields up 300 cubic feet per minute to a branch. At 2800 yards
it yields up 400 cubic feet per minute to a second branch. At 4000 yards
it yields up 600 cubic feet per minute to a third branch. The delivery at
the end is 500 cubic feet per minute. Find the head absorbed by friction.
(/=.oo75.) Ans. 176.801 ft.
1 5. Find the H. P. required to raise 550 gallons per minute to a height
of 60 feet, through a pipe 100 feet in length and 6 in. in diameter, the
coefficient of friction being .0064. , Ans. 10.74.
1 6. What head of water is required for a $-in. pipe, 150 ft. in length,,
to carry off 25 cub. ft. of water per minute ? Ans. 1.56223 ft.
What head will be required if the pipe contains two rectangular
knees? Ans. 1.84918 ft.
17. Determine the delivery of a 2- in. pipe, 48 ft. long, under a 5-ft.
head. Ans. .1349 cub. ft. per sec.
What will be the delivery if the pipe has five small curves of 90° cur-
vature, the ratio of the radius of the pipe to that of the curves being
1:2? Ans. .1327 cub. ft. per sec.
1 8. The curved buckets of a turbine form channels 12 in. long, 2 in.
wide, and 2 in. deep; the mean radius of curvature of the axis is 8 in.
the water flows along the channel with a velocity of 50 ft. per minute.
What is the head lost through curvature ? Ans. .00138 ft.
19. Find the maximum power transmitted by water in a 36-inch pipe,
the metal being \\ inches thick and the allowable stress 2800 Ibs. per
square inch. If the pipe is \\ miles in length, find the loss of power.
Ans. 576 H. P. ; 720.2 ft. -Ibs.
20. Find the diameter of a pipe \ mile long to deliver 1500 gallons of
water per minute with a loss of 20 feet of head. (/ = .005.)
Ans .1.0135 ft-
21. Water is to be raised 20 ft. through a 3O<ft. pipe of 6 in.
diameter. Find the velocity of flow, assuming that 10 per cent of
additional power is required to overcome friction.
Ans. 8.44 ft. per sec.
22. In a pipe 3280 ft. in length the loss of head in friction is 83 ft.
Taking/ — .0064, find the diameter. Ans. 1.527 ft.
23. A pipe 2000 ft. long and 2 ft. in diameter discharges at the rate
of 1 6 ft. per second. Find the increase in the discharge if for the last
1 24 H V D RA ULICS.
1000 ft. a second pipe of same size be laid by the side of the first and
connected with it so that the water may flow equally well along either
pipe. Ans. 7.24 cub. ft. per sec.
24. A pipe of length / and radius r gives a discharge Q. How will
the discharge be affected (i) by doubling the radius for the whole
length ; (2) by doubling the radius f :>r half the length ; (3) by dividing it
into three sections of equal length, of which the radii are r, —, and — ,
.respectively ? (f = coefficient of friction.)
Ans. i. New discharge =
r + 64/A1
-4
9' + 4//
4228//y •
25. A 24-inch pipe 2000 ft. long gives a discharge of Q cubic feet of
water per minute. Determine the change in Q by the substitution for
the foregoing of either of the following systems : (i) two lengths, each
of looo ft., whose diameters are 24 in. and 48 in. respectively; (2) four
lengths, each of 500 ft., whose diameters are 24 in., 18 in., 16 in., and
24 in.
Draw the " plane of charge " in each case.
Ans. (i) Discharge is increased 33.2 per cent taking loss at
change of section into account;
Discharge is increased 35.7 per cent disregarding loss
at change of section.
(2) Discharge is diminished 45 per cent disregarding
losses at change of section.
26. Q is the discharge from a pipe of length / and radius r \ examine
the effect upon Q of increasing r to nr for a length ml of the pipe.
*
Ans. New discharge = Q
(n* - i)2
«*
27. A reducer, I ft. in length, discharges at the rate of 400 gallons per
minute, and its diameter diminishes from 12 in. to 6 in.; find the total
loss of head due to friction. Ans. .0055297.
28. A reservoir of 10,000 square feet superficial area and 100 feet /
deep discharges through a pipe 24 in. in diameter and 2000 feet long. *
Find the velocity of flow in the pipe.
What should be the diameter of the pipe in order that the reservoir
might be emptied in two hours ? Ans. 15.36 ft. per sec.; 3.67 ft.
29. Eight cubic feet of ore is to be raised at the rate of 900 ft. per
•. FLOW OF WATER IN PIPES. 12$
minute by a water-pressure engine with four single acting cylinders of
6 in. diameter and 18 in. stroke, making 60 revolutions per minute.
Find the diameters of a supply- pipe 230 ft. long for a head of 230 ft.,
disregarding friction of machinery, etc. Ans. 4 in.
30. A 2-inch pipe A suddenly enlarges to a 3-inch pipe B, the quan-
tity of water flowing through being 100 gallons per minute. Find the
loss of head and the difference of pressure in the pipes (i) when the
flow is from A to B ; (2) when the flow is from B to A.
Ans. (i) Loss of head = 8.639 in-
Gain of pressure-head = 13.83 "
(2) Loss of head = 7.428 "
Diminution of pressure-head = 29.88 "
31. A 3-inch horizontal pipe rapidly contracts to a i-inch mouih-
piece, whence the water emerges into the air, the discharge being-
660 Ibs. per minute. Find the pressure in the 3-inch main.
If the 3-inch pipe is 200 ft. in length and receives water from an
open tank, find the height of the tank.
Ans. 1003.5 Ibs. Per sq. ft.; 19.92 ft.
32. The efficiency of an engine is f ; it burns 8 Ibs. of coal per hour
per H.P., and works 8 hours a day for 300 days in the year; the cost of
the engine is $12.00 per H.P., and the cost of the coal is $3.00 per ton ;
4500 gallons of water per minute have to be raised a height of 200 ft.
through a pipe of which the diameter is to be a minimum. Cost of
piping = $£> per lineal foot, D being the diameter. Find the value of D.
Ans. 2.923 ft.
33. A reservoir is to be supplied with water at the rate of 11,000
gallons per minute, through a vertical pipe 30 ft. high; find the
minimum diameter of pipe consistent with economy. Cost of pipe per
foot = &/, d being the diameter; cost of pumping = i cent per H.P.
per hour; original cost of engine per H.P. = $100.00; add 10 per cent
for depreciation. Engine works 12 hours per day for 300 days in the
year. Ans. 4.375 ft.
34. A horizontal pipe 4 in. in diameter suddenly enlarges to a
diameter of 6 in.; find the force required to cause a flow of 300
gallons of water per minute through the sudden enlargement.
Ans. .06 H.P.
35. 1000 gallons per minute is to be forced through a system of
pipes AB, BC, CD, of which the lengths are 100 ft., 50 ft., 120 ft., and
the radii 4 in., 6 in., and 3 in., respectively. Draw the plane of
charge.
Ans. Loss in friction from A to B = 111.96 ft.; loss at B — 4.499 ft.;
" " " " B to C — 7.372 " " " C — 14.56 "
" " " " C to D = 566.17 "
126 HYDRA ULICS.
36. A pipe 4 in. in diameter suddenly contracts to one 3 in. in
diameter; find the power necessary to force 250 gallons per minute
through the sudden contraction. Ans. 1.23997 H.P.
37. If a pipe whose diameter is 8 in. suddenly enlarges to one whose
diameter is 12 ins., find the power required to force 1000 gallons per
minute through the enlargement, and draw to scale the plane of charge.
Ans. Energy expended = .1377 H.P.
38. 1000 gallons per minute are forced through a system of pipes
AB, BC, CD, of which the lengths are 100 ft., 50 ft., and 120 ft., and the
radii 6 in., 3 in., and 4 in., respectively. Draw to scale the plane of
charge.
Ans. Loss in friction from A to B = 14.744 ft.; loss at B = 14.56 ft.
" " " " B to c = 235.9 " ; " " c= 8.819"
" " " " CtoZ>= 134.36 "
39. Water flows from a 3-inch pipe through a i^-inch orifice in
a diaphragm into a 2-inch pipe. What head is required if the delivery
is to be 8 cubic feet of water per minute ? Ans. 2.826 ft.
40. 500 gallons of waiter per minute are forced through a continuous
line of pipes AB, BC, CD, of which the radii are 3 in., 4 in., 2 in., and
the lengths 100 ft., 150 ft., and 80 ft., respectively. Find the total loss
of head (a) due to the sudden changes of form at B and C, (b) due to
friction. Find (c) the diameter of an equivalent uniform pipe of the
same total length.
Ans. (a) .1378 ft.; 1.152 ft.
(b) 3.688 ft. in AB; 1.313 ft. in BC\ 22.393 ft- in CD.
(c) .4212 ft.
41. AB, BC, CD is a system of three pipes of which the lengths are
looo ft., 50 ft., and 800 ft., and the diameters 24 in., 12 in., and 24 in.,
respectively; the water flows from CD through a i-inch orifice in a
thin diaphragm, and the velocity of flow in AB is 2 ft. per second.
Draw the plane of charge and find the mechanical effect of the
efflux.
Ans. Loss at B = -& ft.; at C = -/fa ft.; in friction from A to
B = .8 ft. ; from B to C = 1.28 ft.; from C to D = .64 ft. ;
energy of jet = 14,81 if H.P.
42. looo gallons per minute flows through a sudden contraction from
12 inches to 8 inches at A, then through a sudden enlargement from 8
inches to 12 inches at B, the intermediate pipe AB being 100 ft. long.
Draw the plane of charge.
Ans. Loss at A = .288 ft. ; at B = .281 ft. ; in friction from A
to B = 3.499 ft.
43. Water flows from one tube into another of twice the diameter;
the velocity in the latter is 10 ft. Find the head corresponding to the
resistance. Ans. 14.0625 ft.
FLOW OF WATER IN PIPES. 12 7
44. In a given length / of a circular pipe whose inner radius is r
and thickness <?, a column of water flowing with a velocity v is sud-
denly checked by the shutting off of cocks, etc. Show that
\e
in which ^ = head due to the velocity vt E = coefficient of elasticity,
E\ = coefficient of compressibility of water, A = extension of pipe cir-
cumference due to E.
45. A loo-gallon tank, 100 feet above the ground, is filled by a i^-in.
pipe connected with an accumulator consisting of a 3-ft. cylinder with a
piston loaded with 50 tons. How long will it take to fill the tank,
assuming that frictional resistances absorb nine tenths of the head and
that the mean height of the piston above the ground is 10 feet?
Ans. 13.9 sees.
46. Determine the discharge from a pipe of 12 in. radius and 3280 fty
in length which connects two reservoirs having a difference of level of
128 ft. Take into account resistance at entrance. Draw the plane of
charge. Ans. 48.571 cub. ft. per sec.
47. Determine the diameter of a clean iron pipe 5000 ft. in length
which connects two reservoirs having a total head of 40 ft. and dis-
charges into the lower at the rate of 20 cub. ft. per second. Draw to
scale the line of charge. Ans. 1.9219 ft.
48. The difference of level between the two reservoirs is 100 ft., and
they are connected by a pipe 10,000 ft. long. Find the diameter of the
pipe so as to give a discharge of 2000 cubic feet per minute (a) by
Darcy's formula, (b} assuming / = .0064. (Allow for loss of head at
entrance.) Ans. (a) 2.266 ft.; (b) 2.360 ft.
49. Two reservoirs are connected by a 1 2-inch pipe ij miles long.
For the first 500 yards it has a slope of i in 30, for the next half mile a
slope of i in 100, and for the remainder of its length it is level. The
head of water over the inlet is 55 ft. and that over the outlet is 15 ft.
Determine the discharge in gallons per minute. (Take/ = .0064.)
Ans. 1950.66.
50. Two reservoirs are connected by a 6-inch pipe in three sections,
each section being three quarters of a mile in length. The head over
the inlet is 20 ft., that over the outlet 9 ft. The virtual slope of the first
section is i in 50, of the second i in 100, and the third section is level.
Find the velocity of flow, and the delivery.
Ans. 4.5 ft. per sec. ; 332 gallons per minute.
51. A pipe 5 miles long, of uniform diameter equal to 12 in., conveys
water from a reservoir in which the water stands at a height of 300 ft.
above Trinity high-water mark, to a reservoir in which the water stands
128 HYDRA ULICSl
at a height of 1 50 ft. above the same datum. To what height will water
rise in a supply-pipe taken one mile from the lower end? For what
pressure would you design the main at this point, if it lies 20 ft. above
the level of the lower reservoir ? Ans. 179.93 ft.; 19 Ibs. per sq. in.
52. The water surface in one reservoir is 500 ft. above datum, and is
100 ft. above the surface of the water in a second reservoir 20,000 ft.
away, and connected with the first by an i8-in. main. Find the delivery
per second, taking into account the loss of head at the upper entrance.
53. Water surface of a reservoir is 300 ft. above datum, and a 4- in.
pipe 600 ft. long leads from reservoir to a point 200 ft. above datum.
Find the height to which the water would rise (a) if end of pipe is open
to atmosphere, (b) if it terminates in a i-inch nozzle. In latter case find
longitudinal force on nozzle. Ans. (a) 2f ft.; (b} 87.52 ft.; 59.693 Ibs.
54. The surface of the water in a tank is 388 ft. above datum and is
connected by a 4-in. pipe 200 ft. long with a turbine 146 ft. above
datum. Determine the velocity of the water in the pipe at which the
power obtained from the turbine will be a maximum. Assuming the
efficiency of the turbine to be 85 per cent, determine the power.
Ans. 19.928 ft. per sec.; 31.895 H. P.
55. A pipe 12 in. in diameter and 900 ft. long is used as an inverted
siphon to cross a valley. Water is led to it and away from it by an
aqueduct of rectangular section 3 ft. broad and running full to a depth
of 2 ft. with an inclination of i in 1000. What should be the difference
of level between the end of one aqueduct and the beginning of the
other ? Ans. 575.8 ft.
56. Water flows through a pipe 20 ft. long with a velocity of 10 ft.
per second. If the flow is stopped in -^ sec. and if retardation during
the stoppage is uniform, find the increase in the pressure produced.
(g = 32 and the density of the water = 62.5 Ibs. per cub. ft.)
Ans. 62$ cu. ft. of water.
57. An hydraulic motor is driven by means of an accumulator giving
750 Ibs. per square inch. The supply-pipe is 900 ft. long and 4 in. in
diameter. Find the maximum power attainable, and velocity in pipe.
(/= .0075.) Ans. 242.4 H. P.; 21.203 ft- Per sec-
58. A 2-inch hose conveys 2 gallons of water per second. Find the
longitudinal tension in the hose. Ans. 9.18 Ibs.
59. Find the pumping H. P. to deliver i cub. ft. of water per second
through a i-inch nozzle at end of a 3-inch hose 200 ft. long,/ being .016,
Ans. 97.335 H. P.
60. A volume of water 50 ft. in length flowing through a pipe with a
velocity of 24 ft. per second is quickly and uniformly stopped in one
tenth ol a second by closing a stop-valve. Find the increase of pressure
per square inch in the pipe near the valve. Ans. 162.5 Ibs.
61. The surface of the water in a tank is 286 ft. above datum. The
I/
FLOW OF WATER IN PIPES. 12$
tank is connected by a 4-in. pipe 500 ft. long with a 36-in. cylinder
170 ft. above datum. Find (a) the velocity of flow in the pipe for which
the available power will be a maximum ; (b) the power. If the piston
moves at the rate of i ft. per minute, find (c} the pressure on the piston.
Also find the height to which the water would rise if (d) the cylinder
/end of the pipe were open to the atmosphere and if (e) the pipe termi-
nated in a nozzle I inch in diameter, neglecting the frictional resistance
of the nozzle. Finally, find (/) the power required to hold the nozzle.
(Coeff. of friction = .005.)
Ans. (a) 8.93 ft. per sec. ; (ff) 6.85 H. P.; (c) 22.8 tons per sq. ft.;
(d) 3.74 ft.; (e) 103.8 ft.; (/) 70,8 Ibs.
62. The conduit-pipe for a fountain is 250 ft. long and 2 in. in diam-
eter ; the coefficient of resistance for the mouthpiece is .32 ; the entrance
orifice is sufficiently rounded, and the bends have sufficiently long radii
of curvature to allow of our neglecting the corresponding coefficient of
resistance. How high will a ^-in. jet rise uuder a head of 30 ft. ?
Ans. 1 9. 14 ft.
63. The difference in level of two reservoirs is 250 ft. and they are
connected by a 24-inch pipe AB, 6000 ft. long. If f= .0064, draw the
plane of charge. A third reservoir is so placed that the difference
between its level and that of the first (or highest) is 100 ft., and is con-
nected to the main at a point O by a branch OC, 3000 ft. long and 12 in.
in diameter. Examine the distribution.
Ans. Upper reservoir will supply the two lower reservoirs if
AO < %BO.
The two upper reservoirs will both discharge into the
, lower reservoir if AO > %BO.
(x If AO = 2000 ft'., the pressure-head at O =161 ft.; 2/1 = 14.9
ft. ; 2/2 = 3.02 ft.; v* = 14.18 ft.
If AO=4ooo ft., the pressure-head at 0=96 ft.; 2/1=13.8 ft.;
2/2 = 6.7 ft.; 2/3 = 15.4 ft.
64. A pipe 24 in. in diameter and 2000 ft. long leads from a reservoir
in which the level of the water is 400 ft. above .datum to a point B, at
which it divides into two branches, viz., a 12-in. pipe J3C, 1000 ft. long,
leading to a reservoir in which the surface of the water is 250 ft. above
datum, and a branch BD, 1500 ft. long, leading to a reservoir in which
the surface of the water is 50 ft. above datum. Determine the diameter
of BD when the free surface-level at B is (a) 300 ft., (b) 250 ft., and (c)
200 above datum. Ans. (a) 1.454 ft.; (&) 1.783 ft.; (c) 2.096 ft.
65. Two reservoirs A and B are connected by a line of piping MON,
2000 ft. in length. From the middle point O of this pipe a branch OP,
looo feet in length, leads to a reservoir C. The reservoirs A and (Tare
200 feet and 100 feet, respectively, above the level of C. The deliveries
in MO, OP, ON, in cubic feet per second, are ^-it, ^-TT, and TC, respec-
1 30 HYDRA ULICS.
lively. Find (a) the velocities of flow in MO, OP, ON\ (b] the radii of
these lengths; (c) the height of the free surface-level at O above C.
Ans. (a) 1 1. 121 ft. per sec. in MO; 10.158 ft. per sec. in OP;
14.145 ft. per sec. in OAr.
(b) .49976^.; .41831 ft.; .26588 ft.
(c) 150.5 ft., very nearly.
66. A main, 1000 ft. long and with a fall of 5 ft. discharges into two
branches, the one 750 ft. long with a fall of 3 ft., the other 250 ft. long
with a fall of i ft. The longer branch passes twice as much water as
the other and the total delivery is 47^ cu. ft. per minute. The velocity
of flow in the main is i\ ft. per second. Find the diameters of the main
and branches. Ans. .63245 ft.; .288ft.; 488ft.
67. How far can 100 H.P. be transmitted by a 3^ in. pipe with a loss
of head not exceeding 25 per cent under an effective head of 750 Ibs. per
square inch ? Ans. 5426.3 ft.
68. A city is supplied with water by means of an aqueduct of rect-
angular section, 24 ft. wide, running 4 ft. deep, and sloping i in 2400.
One-fourth of the supply is pumped into a reservoir through a pipe 3000
ft. long, rising 25 ft. in the first 1500 ft., and 75 ft. in the second 1500 ft.
The pumping is effected by an engine burning 2| Ibs. or coal per H.P.
per hour, and working constantly through the year. A percentage is to
be allowed for repairs and maintenance; the cost of the coal per ton of
20oolbs. is $4 ; the prime cost of the engine is $100 per H.P. ; the effi-
ciency of the engine is f ; the coefficient of pipe friction is .0064, the
cost of the piping is $30 per ton. Determine the most economical diam-
eter of pipe, and the H.P. of the engine. Ans. 4.84 ft. ; 456.455 H.P.
CHAPTER IV.
FLOW OF WATER IN OPEN CHANNELS.
i. Flow of Water in Open Channels. — A transverse sec-
tion of the water flowing in an open channel may be supposed
to consist of an infinite number of elementary areas represent-
ing the sectional areas of fluid filaments or stream-lines. The
velocities of these stream-lines are very different at different
points of the same transverse section, and the distribution of
the pressure is also of a complicated character. Generally
speaking, the side and bed of a channel exert the greatest
retarding influence on the flow, and therefore along these
surfaces are to be found the stream-lines of minimum velocity.
The stream-lines of maximum velocity are those farthest
removed from retarding influences. If the stream-line velo-
cities for any given section are plotted, a series of equal
velocity-curves may be obtained. In a channel of symmetrical
FIG. 78.
section, the depth of the stream-line of maximum velocity
below the water-surface is less than one fourth of the depth of
the water, while the mean velocity-curve cuts the central
vertical line at a point below the surface about three fourths of
the depth of the water.
In the ordinary theory of flow in open channels, the
variation of velocity from point to point in a transverse section
is disregarded, and it is assumed that all the stream-lines are
sensibly parallel and move normally to the section with a
common velocity equal to the mean velocity of the stream.
With this assumption, it also necessarily follows that the
131
132
HYDRA ULICS.
distribution o4" pressure over the section is in accordance with
the hydrostatic law.
Again, it is assumed that the laws of fluid friction already
enunciated are applicable to the flow of water in open chan-
nels. Thus, the resistance to flow is proportional to some
function of the velocity (F(v)}, to the area (S) of the wetted
surface, is independent of the pressure, and may be expressed
by the term S.F(v). An obvious error in this assumption is
that v is the mean velocity of the stream and not the velocity
of the stream-lines along the bed and sides of the channel. In
practice, however, the errors in the formulae based upon these
imperfect hypotheses are largely neutralized by giving suitable
values to the coefficient of friction (/).
When a constant volume (Q) of water feeds a channel of
given form, the water assumes a definite depth. A permanent
regime is said to be established and the flow is steady. If the
transverse sectional area (A) is also constant, then, since
Q = vA, the velocity v is constant from section to section and
the flow is said to be uniform. Usually the sectional area A is
variable and therefore the velocity v also varies, so that the
motion is steady with a varying velocity. Any convenient
short stretch of a channel, free from obstructions, may be
selected, and treated without error of practical importance, as
being of a uniform sectional area equal to that of the mean
section for the whole length under consideration.
2. Steady Flow in Channels of Constant Section (A). —
The flow is evidently uniform ; and since A is constant, the
depth of the water is also con-
stant, so that the water-surface
is parallel to the channel-bed.
JH Consider a portion of the
stream, of length /, between the
two transverse sections aa, bb.
Let i be the inclination of
the bed (or water-surface) to
the horizon.
Iff
FIG. 79.
Let Pbe the length of the wetted perimeter of a cross-section.
FLOW OF WATER IN OPEN CHANNELS. 133
Then, since the motion is uniform, the external forces
acting upon the mass between aa and bb in the direction of
motion must be in equilibrium.
These forces are :
(1) The component of the weight of the mass, viz.,
wAl sin i = wAli = wAl— = wAk,
h being the fall of level in the length /.
Note. — When i is small, as is usually the case in streams,
-j = tan / = sin / = /, approximately.
(2) The pressures upon the areas aa and bb, which evi-
dently neutralize each other.
(3) The frictional resistance developed by the sides and
bed, viz.,
Hence
wAh - PlF(v) = o,
or
FM Ah
-^ = w=m*>
m being the hydraulic mean depth.
It now remains to determine the form of the function F(v).
In ordinary English practice it is usual to take
W 2g
f being the coefficient of friction. Then
or
jig
v = \i ~~f y mi = cy mi.
1 34 H YDRA UL1CS.
c being a coefficient whose value depends upon the roughness
of the channel surface and upon the form of*its transverse
section.
Prony and Eytelwein adopted the formula
F(v]
-±— = av -j- bv* = mi,
w
and carried out different experiments to determine the values
of a and b.
According to Prony, - = 22472.5 and -r — 10607.02,
" "Eytelwein, - = 41688.02 " -= 8975.43.
For a velocity of about 70 ft. per minute Prony's and
Eytelwein's results give the same value for mi. For other
velocities, Prony's values of mi are greater or less than those
of Eytelwein, according as the velocity v is greater or less
than 70 ft. If v, however, does not differ very widely from
70 ft., the change of value is small and of no practical
importance.
For values of v exceeding 20 ft. per minute the term av
may be disregarded without practical error, and the formula
then becomes
mi = bv*t
or
Hence
v = 105 \/mif according to Prony,
and
v = 95 ^/mi, according to Eytelwein,
giving as a mean
v = loo^mz, which is Beardmore's formula.
The total head H in a stream is made up of two parts, the
FLOW OF WATER IN OPEN CHANNELS. 13$
one required to produce the velocity of flow, and the other
absorbed by the frictional resistance. Thus,
2g ?;/ w
In long canals, and in rivers with slopes not exceeding 3 ft.
v*
per mile, the term — is very small as compared with the term
/ Mv)
— , and may be disregarded without sensible error.
m w
Note. — The retarding effect of the air upon the free surface
of a stream or river has yet to be determined by careful
observation and experiment. It may, however, be assumed
that the resistance offered by calm air per unit of free surface
is approximately one tenth of the resistance offered by similar
units at the bottom and sides of smooth channels. Thus, in
smooth channels, if X is the width of the free surface, the
Y
wetted perimeter is more correctly P -\-
In general, the wetted perimeter may be expressed in the
form P -f- -ip ft being 10 for smooth channels and greater
than 10 for rough channels. The value of ft is obviously
diminished by opposing winds and increased by following
winds.
3. On the Form of a Channel.— In the formula
F(v)
mt'= *T'
= — J and ilss -yl are similarly related in the deter-
mination of v, the mean velocity of flow. If v is constant, the
product mi must also be constant, so that if m increases i must
diminish, and vice versa. Thus, in a very flat country the flow
may be maintained by making m sufficiently large, while again
if the channel-bed is steep m is small.
136 HYDRAULICS.
The erosion caused by a watercourse increases with the
rapidity of flow. At the same time the sectional area (A)
of the waterway also increases, so that the velocity of flow v
diminishes. Thus there is a tendency to approximate to a
" permanent regime " when the resistance to erosion balances
the tendency to scour.
Hence, throughout any long stretch of a river, passing
through a specific soil, the mean velocity of flow will be very
nearly constant if the amount of flow (Q) does not vary. Gen-
erally speaking, the volume conveyed by a river increases from
source to mouth on account of the additions received from
tributaries, etc. Since Q increases, A must also increase ; and
if mi or v is to remain constant, i must diminish. It is also
observed that the surface slopes of large rivers diminish gradu-
ally from source to mouth.
Again, various problems relating to the proper sectional
form of a channel may be discussed by means of the formulae
'A .
v = c \mi =
and
Suppose the slope to be constant. Then
A
v* is proportional to 75
and
A9
Q* is proportional to —p. '
PROBLEM I. The section of the waterway being a rectangle
of width x and depth y, and of given area (A — xy\ it is
required to find the ratio of x to y for which the velocity of
flow (v) will be a maximum. Then dv = o, and therefore
P.dA—A.dP
P* '
FLO W OF WATER IN OPEN CHANNELS.
137
Hence
PdA-AdP=o.
But dA — o = xdy -\-ydx, and
therefore also ^
dP — o = dx + 2dy, J
since
P=x + 2y.
Hence,
FIG. 80.
and the mean hydraulic depth
_ A _ xy _y
~P~x-\-2y~2
= one half of the depth of the water.
The same results follow if the discharge Q instead of v
is to be a maximum. In such case
dQ
M'\
- o - d\-p) =
.dA - A*.dP
and therefore $PdA — AdP — o.
But dA = o, and therefore dP = o. Hence, etc.
Note. — The same results also follow if, instead of A being
given, the wetted perimeter P is to be a minimum, since then
dP = o, and therefore also dA = o.
PROBLEM II. The waterway being trapezoidal in section,
FIG. 81.
of bottom width x, depth y, and sides sloping at a given angle
Q to the horizontal, it is required to find the ratio of x to y
which, for a given wetted perimeter (P^ or area (A), will make
the velocity of flow or the discharge a maximum.
HYDRAULICS.
As in Problem I,
dA = o and dP = o.
But A = (x +y cot 6)y and P = x + 2y cosec 6.
Hence
^£4=0= ydx + <^/O + 27 cot 0)
and
</P = o — dx + */j/ . 2 cosec 0.
Therefore
x -4- 2y cot # dx
— ! — - --- = — -j- = 2 cosec 0.
dy
y
Hence
x := 2i/(cosec 0 — cot 0) = 2i/
and therefore
- cos 0
^ == 2v tan — t
sin 0 2
x 0
— = 2 tan — .
/ 2
Then mean hydraulic depth
A (*-\-y cot 0)y
j/2 — cos 6K y
0 = 2(2 - cos 0} ~ 2
P x -|- 27 cosec
= one half of the depth of the water.
The section may be easily sketched as in Figs. 82 and 83.
G
FIG. 82.
From the middle point C of AB, the bottom width, draw
CF at right angles to AB and equal in length to the depth of
the water. Then
AB _ 0
- 2 tan ,
0 being the given slope of the sides.
FLOW OF WATER IN OPEN CHANNELS. 139
With F as centre and FC as radius describe a circle. From
the points A and B draw tang.ents to touch this circle at D
and E. FA evidently bisects the angle CAD. Therefore
CAD CF CF 6
tan - = tan CAP = = - = = cot 2'
Hence TT — CAD = #, and ^/?, /?£ have the slope required.
PROBLEM III. To find the proper sectional form of a
channel of bottom width 2a so that the mean velocity of flow
may be constant for all depths of water.
Let x, y, Fig. 84, be the co-ordinates of any point P in the
profile referred to the middle point O of AB, the bottom width,.
as origin and let s be the length of A P.
FIG. 84.
Since v is to be constant m must also be constant, and
therefore
• A=,[y^_=
which may be written
/ ydx = m(s + a).
Differentiating,
ydx = mds — m^dx*
and therefore
dx dy
m ~~ (/ _ w?
Integrating,
x
m~ *>e \s
c being a constant of integration.
1 4° H YDRA ULICS.
When x = o, y = a, and therefore
o = log, (a + *V - M2) + c = loge& + c,
where b = a -f- j/#a — ^2. Hence
£ = te
or
is the equation to the required profile, which, as may be easily
shown, is a curve which flattens very rapidly.
PROBLEM IV. If water flows through a circular aqueduct,
find the angle 6 subtended at the centre by the wetted perim-
eter, for which the velocity of flow is a
maximum.
Let r = radius of aqueduct.
Area of waterway = — (6 — sin 0).
Wetted perimeter = r6.
FIG. 85. Then
r 0 — sin 0 r I sin
m = — -ft —
2 C^
sin 0
Now v is to be* a maximum and therefore — -^— must be a
minimum. Hence
0 cos 0 - sin 0 ,
/sin 0\ 0COS
4—)=°=-
and therefore 0 cos — sin 0 = o.
Hence 6= tan 0, and the angle 0 in degrees is about 77° 27'.
FLOW OF WATER IN OPEN CHANNELS.
141
Also, the mean hydraulic depth = — ^i --- -r— )
= - (i _ cos d)
= rsin— = . 39 X r.
PROBLEM V. A channel of given slope has a given surface-
width AC, vertical sides AB (=yl) and CD (=7,) of given
depths, and a curved bed BD (= L) of given length.
FIG. 86.
The amount and velocity of flow in the channel will be a
maximum when the form of the bed BD is a circular arc. This
can be easily proved as follows :
Since the slope is constant, v oc
/~A
a \f -p.
But P (= L -\- y^ + >0 is a constant quantity, and therefore
v and also Q will be a maximum when ^4 is a maximum.
Hence, too, the area between the chord BD and the curve
must be a maximum, and therefore the curve must be a circu-
lar arc. The proof of this by the Calculus of Variations is as
follows :
Take O in CA produced as the origin, OC as the axis of x,
and the vertical through O as the axis of y. Then
ydx is to be a maximum.
1 42 H YD RA ULICS.
Also,
dy
is a given quantity, OA being = JT, , OC = x^ , and Hr
Let V =• y -f- a Vi -\- p\ a being some constant.
Then
/*;
/ F. dx is to be a maximum,
*X ^i
and therefore
that is,
and thus
^ + ~^=r^ = ^ •
Therefore
^ / ' Va* - (c, - <y)* '
Integrating,
the equation to a circle of radius a.
Hence the profile BD is a circular arc.
The maximum depth of the channel is cl — a.
The constants c1 , c^ , a can be found from the three con-
ditions that the arc is of given length and has to pass through
the two fixed points B and D.
4. Flow in Aqueducts. — The velocity v depends upon m
A
(m = - and therefore upon the depth of the water in the
FLOW OF WATER IN OPEN CHANNELS. 143
aqueduct. For some definite depth the velocity will be a
maximum. If the water fills the aqueduct, the aqueduct be-
comes a pipe, and the formula for channel-flow ought to change
suddenly so as to agree with that for pipe-flow. The theory is
thus imperfect.
5. River-bends. — The following explanation is due to Pro-
fessor James Thomson (Inst. Mechl. Engs., 1879 5 Pi'oc. Royal
Soc. 1877). In rivers flowing in alluvial plains, the curvature
of the windings which already exist tends to increase owing to
the scouring away of material from the outer bank and to the
deposition of detritus along the inner bank. The sinuosities
often increase until a loop is formed, with only a narrow isth-
mus of land between two encroaching banks of a river. Finally
a cut-off occurs, a short passage for the water is opened
through the isthmus, and the loop is separated from the river-
course, taking the form of a horseshoe-shaped lagoon or swamp.
The ordinary supposition that the water always tends to move
forward in a straight line, rushing against the outer bank and
wearing it away, and at the same time causing deposits at the
inner bank, is correct, but it is very far from being a complete
explanation of what takes place.
When water flows round a circular curve under the action
of gravity only, it takes a motion like that in a free vortex.
Its velocity parallel to the axis of the stream is greater at the
inner than at the outer side of the curve.
Thus, too, the water in a river-*bank flows more quickly
along courses adjacent to the inner bank of the bend than
FIG. 87.
along courses adjacent to the outer. The water, in virtue of
centrifugal force, presses outwards so that the water-surface of
a transverse section (Fig. 87) has a slope rising upwards from
144
HYDRA ULICS.
the inner to the outer bank. Hence the free level for any
particle of the water near the outer bank is higher than the
free level for any particle in the same transverse section near
the inner bank, but the tendency to flow from the higher to
the lower level is counteracted by centrifugal action. Now
the water immediately in contact with the bottom and sides
of the course is retarded, and its centrifugal force is not suf-
ficient to balance the pressure due to the greater depth at the
outside of the bend. This water therefore tends to flow from
FIG. 88.
the outer bank towards the inner (Fig. 88), carrying with it
detritus which is deposited at the inner bank. Simultaneously
with the flow of water inwards, the mass of the water must
necessarily flow outwards to take its place.
6. Value of/. — The value of /depends upon
(a) the roughness of the sides and bed ;
(b] the velocity of flow ;
(c) the dimensions of the transverse section ;
(d] the slope of the channel-bed.
An average mean value of /is .00757.
FLO W OF WATER IN OPEN CHANNELS. 145
Weisbach has proposed to take
the values of a and /?, obtained as the results of 255 experi-
ments, being a = .007409, ft = .192, so that
, .0014225
/=. 007409+ .
Darcy and Bazin assume f to be given by an expression of
the form
ft
giving the following values of a and ft as the results of their
experiments :
In very smooth channels, with sides of planed timber or
rendered in cement,
.000316
a = .00316, /3 = .1 ; .'. /= .00316 + - — .
In smooth channels with sides of planks, brick-work, or
ashlar
, .0009223
a = .00401, ft = .23 ; /. /= .00401 + - --.
In rough channels with sides of rubble masonry or pitched
with stone
<*=:. 00507, 0 = .82; .-./ = .00507 + : °^574.
In very rough channels in earth
or = . 00592, 0 = 4.i; .-./=. 00592+ '°21272.
146 HYDRA ULICS.
In torrential streams encumbered with detritus
a = .00846, /? = 8.2 ; .-./= .00846 + '—.
Ganguillet and Kutter, taking the formula
v = c m =
kave endeavored to obtain a more correct value of c by a care-
ful investigation of :
(a) The experimental results of Darcy and Bazin. These
results show that the value of c depends upon the roughness of
the channel and also upon its dimensions. The values given
for a and ft are different for different classes of channel even
when the dimensions are infinite. But while in small channels
the influence of differences of roughness upon the flow must
be very great, it is certainly more than probable that this in-
fluence diminishes as the section of the channel increases, and
that it will be nil in the case of an indefinitely large channel.
(b) The measurements of Humphreys and Abbott on the
Mississippi, a stream of very large section and of very low
slope.
(c) Their own gaugings in the regulated channels of certain
Swiss torrents with exceptionally steep slopes and running
through extremely rough channels.
(d) The effect of the slope.
From the Mississippi data it was found that
c — 256 for a slope of .0034 per looo
and
c = 154 " " " " .02 " "
Thus c, and therefore also the discharge, will be subject
to considerable variations in the case of large streams with low
slopes. The value of c does not vary much with the slope in
FLOW OF WATER IN OPEN CHANNELS. 147
small rivers. Proceeding in a purely empirical manner, Gan-
guillet and Kutter arrived at the formula
C =
where n is a coefficient depending only on the roughness of
the channel sides and bed, while A and / are new coefficients
whose values remain to be determined.
Now c depends upon the slope i and decreases as i in-
creases. This may be allowed for by taking
so that
c =
the form finally adopted by Ganguillet and Kutter.
The values given for the constants, the unit being a foot,
are
0 = 41.6; /=i.8n; p •=• .00281 ; n = .008 to .05.
The following table gives the values of n which will be
found of most use in practice :
In a channel with sides of well-planed timber n = .009
" " " " rendered with cement n = .01
In a channel with sides rendered with a mixture of
3 of cement to I of sand n = .01 1
In a channel with sides of unplaned planks n = .012
" " ashlar or brickwork n = .013
" " " " " " canvas on frames w = .oi5
" " rubble masonry n = .017
1 48 H YDRA ULICS.
In rivers and canals in very firm gravel n = .02
In rivers and canals in perfect order and free from
detritus (stones and weeds) n = .025
In rivers and canals in moderately good order, not
quite free from stones and weeds n — .03
In rivers and canals in bad order, with weeds and
detritus n = .035
In torrential streams encumbered with detritus n = .05
To the above Jackson adds the following classification for
artificial canals :
In canals in very firm gravel in perfect order n = .02
u u " earth above the average order n = .0225
" " " " in fair order n = .025
" " " " below the average order # = .0275
In canals in earth in rather bad order, partially over-
grown with weeds and obstructed with detritus, n = .03
The difficulty of properly selecting the value of n is due to
the fact that there is no absolute measure of the roughness of
channel-beds.
In Cunningham's experiments on the Ganges c varied
from 48 to 130.
In Humphreys and Abbott's experiments on the Missis-
sippi c varied from 53 to 167, the units in each case being a
foot and a second.
7. Variation of Velocity in different parts of the trans-
verse section of a stream.
Assumptions. — (a) That the stream is of uniform depth h
and of indefinite width.
(b) That the fluid filaments flow across the section in sen-
sibly parallel lines.
(c) That a permanent regime has been established, and that
the flow is uniform. The pressure in the section is therefore
distributed in accordance with the hydrostatic law.
(d) That the resistance to the relative sliding of consecutive
filaments is of the nature of viscous resistance.
FLOW OF WATER IN OPEN CHANNELS.
149
Let Fig. 89 represent a portion of a vertical longitudinal
section of the stream intersected by two transverse sections
AB, CD, I being the distance between them.
FIG 89.
Consider a thin layer abed of thickness dy and width b,
bounded by the sections AB, CD, and by the planes ad, be, at
depths y and/ -j- dy, respectively, below the free surface.
The forces acting upon the layer in the direction of motion
are :
(1) The pressures on the ends ab, cd, which evidently neu-
tralize each other.
(2) The component of the weight— wbl. dy . sin i = wbli . dy ;
i being the slope of the bed.
(3) The viscous resistances on the lateral faces of the layer
under consideration. These are nil, since in a stream of indefi-
nite width there will be no relative sliding between abed and
the vertical faces on each side.
(4) The viscous resistances along the planes ad and be.
The frictional resistance to distortion, i.e., to shearing,
along such planes is found to be proportional to the shear per
unit of time, and is measured by the shear per unit of area at
the actual rate of shearing. The coefficient of viscosity, or
shear per unit of area
simply the viscosity, is the quotient -. —
shear per unit of time
and defines that quality of the fluid in virtue of which it resists
a change of shape.
Adopting Navier's hypothesis,
di}
the viscous resistance along ad = — kbl--.
150
HYDRAULICS.
k being the coefficient of viscosity. The sign is negative as,
dv .
since v increases withjj/, -y- is positive, and, at the same time,
the- action of the layers above ad is of the character of a re-
tardation.
dv
The viscous resistance along be = kbl-j-
dy
Then, as the motion is uniform,
dv
kbl . d-j-
y
;
dy
wbli . dy - kbl^- + kbl~ + kbl^dy =
dy dy dy "
Hence
w
Integrating twice,
(I)
a and vs being constants of integration.
It is evident that vs is the surface-velocity, i.e., the value
of v when y — o.
The equation may be written in the form
ka* wi f ka\*
FIG. 90.
dv
= °)
and
Thus the velocity - curve is a parabola
ka
having a horizontal axis at a depth Y '= — r
wi
below the free surface. This is also the
depth of the filament of maximum velocity
(3)
FLOW OP WATER IN OPEN CHANNELS. !$!
Hence, by equations I and 3,
wi
v = v^--k(y- Y}\ . .. , . . (4)
Let vm be the " mean " velocity for the whole depth h.
Let v\ be the mid-depth velocity. Then
f
and
with V
(6)
Hence
witf
a result upon which Humphreys and Abbott have based a rapid
method of gauging rivers.
Let vb be the bottom velocity, i.e., the value of v when
y = h. Then by equation 4,
wi
and therefore
^^ f 1 T/"\2 /O\
^max — Vb = — 7-(/2 — Yy (8)
2k ^
When the filament of maximum velocity was below the free
surface Bazin found the value of the difference z>max — vb to be
constant. Take
IllUX O sy 7y \ /
IS2 HYDRA ULICS.
Then the general equation (4) of the velocity-curve becomes
• . .... (9)
Now if Y'— o, i.e., if the filament of maximum velocity is
in the free surface,
H P=tw-A^.
But in such case Bazin's experiments led to the relation
Hence
^=36.3
and the general equation of the velocity-curve becomes
^iv- FV
..... (10)
This is Bazin's formula, and it agrees well with his experi-
ments on artificial channels and also with the results of
experiments on the Saone, Seine, Garonne, and Rhine. It
was found that
*7)
- — 1.17 in the Rhine at Basle and ranged from i.i to 1.13
^»«
in the others;
36.3^ i/ .
(h _ KY" y between r3 and 2o;
Y
— = .33 in some artificial channels and ranged from O to 0.2
in the other cases ;
*W — vb ranged from Jz/max to i^max.
These results are not in agreement with the Mississippi
measurements.
FLOW OF WATER IN OPEN CHANNELS. 153
Note. — When the filament of maximum velocity is in the
free surface, Y = o, and therefore, by equation 5,
wih*
«TI — 191
um — ^max ~~" z- 7 >
and by equation 8,
wit?
Hence, combining these two equations,
Boileau assumes that the velocity-curve is given by the
equation
.. ..... (12)
below the filament of maximum velocity, being MMl in Fig. 91,
and by the equation
v = a-Bf + Cy (13)
above the filament of maximum velocity, being MM9 in Fig. 92.
Let vs be the surface-velocity, i.e., the
value of v when y — o. Then, by equa-
tion 13,
vs = a.
Also, the two equations (12) and (13)
must each give the same value for the
maximum velocity (zw), and therefore
A - BY* = vmax = a — BY* + CY, FlG' 9I>
from which
A — a A —vs
Again, taking A = z/max + ^ Boileau deduced experimen-
tally that d is sensibly constant for different streams.
1 54 HYDRA ULICS.
But A = ?w + d = A - B Y* + d, and therefore B
Hence Boileau's equation becomes
for points below the filament of maximum velocity, and
V = V, - *' + (ZW + d- ».
for points above the filament of maximum velocity.
8. Relations between Surface, Mean, and Bottom Ve-
locities. — Bazin deduced from his experiments on canals the
relation
, — vm
vm = vs — 25.4 Vmt = vs — 25.4—,
where c — V -~. Therefore
cvs
vm =
- c+ 25.4
Darcy and Bazin give the relation
10.87 ^wt = vb + 10.87—.
Therefore
v =
~
C — I0.8/
A mean value of c is 45.7, which makes
vm = 1.312. zv ib
Dubuat gives the following table of maximum bottom
velocities consistent with stability :
FLOW OF WATER IN OPEN CHANNELS.
155
Nature of Canal Bed. Vj,.
Soft earth 0.25
Loam 0.50
Sand i.oo
Gravel 2.00
Pebbles 3.40
Broken stone, flint 4.00
Chalk, soft shale 5.00
Rock in beds 6.00
Hard rock , 10.00
TABLE OF MAXIMUM VELOCITIES FROM INGENIEURS
TASCHENBUCH.
Nature of Canal-bed. vs vm vb
Slimy earth or brown clay 49 .36 .26
Clay 98 .75 .52
Firm sand 1.97 1.51 1.02
Pebbly bed 4.00 3.15 2.30
Boulder bed 5.00 4.03 3.08
Conglomerate of slaty fragments 7.28 6.10 4.90
Stratified rocks 8.00 7.45 6.00
Hard rocks 14.00 12.15 10.36
TABLE OF VISCOSITY OF WATER AND MERCURY.
(From Everett's System of Units.)
WATER.
MERCURY.
Temp.
(Cent.)
o
5
10
15
20
25
30
Viscosity.
.Ol8l
.0154
•0133
.0116
.OIO2
.OOQI
.0081
Temp.
(Cent.)
35
40
45
50
60
80
90
Viscosity.
.0073
.0067
.0061
.0056
.0047
.0036
.0032
Temp.
(Cent.)
Ou
IO
18
99
154
197
249
Viscosity.
.0169
.0162
.0156
.0123
.0109
.OIO2
. 00964
Temp.
(Cent.)
315
340
Viscosity.
.00918
.00897
156 HYDRA ULICS.
The viscosity is given by
_.oi83
and by
.0369^
, according to Meyer ;
j| .00131, according to Slotte;
/ being the temperature centigrade.
9. Flow of Water in Open Channels of Varying Cross-
section and Slope,
Assumptions. — (a) That the motion is steady.
Thus the mean velocity is constant for any given cross-
section, but varies gradually from section to section.
(b) That the change of cross-section is also gradual.
(c) That, as in cases of uniform motion, the work absorbed
by the frictional resistance of the channel-bed and sides is the
only internal work which need be taken into consideration.
Xy
FIG. Q2.
Let Fig. 92 represent a longitudinal section of the stream.
The fluid molecules which are found in any plane section db
at the commencement of an interval will be found in a curved
surface dc at the end of the interval, on account of the differ-
ent velocities of the fluid filaments.
Suppose that the mass of water bounded by the two trans-
verse sections ab, ef, comes into the position cdhg in a unit of
time. Then the change of kinetic energy in this mass is equal
to the algebraic sum of the work done by gravity, of the work
done by pressure, and of the work done against the frictional
resistance.
Change of Kinetic Energy. — This is evidently the difference
FLOW OF WATER IN OPEN CHANNELS. I 57
between the kinetic energies of the masses efgh and abed,
since, as the motion is steady, the kinetic energy of the mass
between cd and ef remains constant.
Let A1 be the area of the cross-section ab.
" «j " " mean velocity across this section.
*' v " " velocity at this section of any given fluid,
filament of sectional area a.
Let v = ul± V.
Then
Aji, = 2(av) and 2(aV) = O.
The kinetic energy of the mass abed
Since S(aV) — o and 3«x ± V — 2«, + v.
Now 2«j + v is evidently positive. Hence the kinetic en-
ergy of the mass abed
a' being a coefficient of correction whose value depends upon
the law of the distribution of the velocity throughout the sec-
tion ab. It is positive and greater than unity. Assume that
a has the same value for the sections ab and ef. Then if A^
158 H YDRA ULICS.
#a, are the area and mean velocity at the transverse section ef,
the kinetic energy of the mass efgh
= <x~A^'
Hence the change of kinetic energy in the mass under con-
sideration
g 2
since A^ut = Q = A.u^
Work done by Gravity. — Consider any fluid filament mn,
the depth of m below the surface being y^ and of n, y^.
Let z be the fall in the surface-level from a to e.
Then the fall from m to n
and the work done by gravity on the elementary volume dQ
in a unit of time
Work done by Pressure.
The pressure per unit of area at m — wyl -\-p0 ;
0 being the atmospheric pressure.
Hence the work due to these pressures per unit of time
= dQ(wy, + A) - dQ(wy, +/.),
= w .
Thus the total work done by gravity and by pressure
= 2(w .dQ.z) = wQz,
for the mass under consideration.
FLO W OF WATER IN OPEN CHANNELS. 159
Work absorbed by Friction. — Consider a thin lamina of water
of thickness ds, bounded by the transverse planes xx, yy, the
distance of xx from ab being s.
Since the change of velocity is gradual, the mean velocity
from xx to yy may be assumed to be constant.
Let u be this mean velocity.
" Pbe the wetted perimeter at the section xx.
" A be the area of the waterway at the section xx.
Then the work absorbed by friction per second from xx
to yy
= P.ds.u.F(u\
and the total work absorbed between ab and ef
= <2
*.
/ being the distance between ab and ef. Hence
a
and therefore z = a""* ~ "' + / -
2g J0Aw
~ . F(u) -u* , A
Take — ^ = /— and — = m. Then
w 2g P
2g
If the two planes ab and ef are indefinitely near one an-
other (Fig. 93), the last equation evidently gives,
j a j f u* j / \
dz = —u . du -4- - -- as. ..... (2)
g m2g
160 HYDRA ULICS.
which is the fundamental differential equation of steady varied
motion, dz being the fall of surface level in
the distance ds.
In the figure aa' is drawn parallel to the
bed and aa" is horizontal. The distance
a" e may, without sensible error, be assumed
equal to dz.
Also a" a' = i . aa' — i . ds, very nearly.
ids — a' a" = a'e + a" e = dh + dz. . . . (3)
Substituting the value of dz from this equation in equa-
tion 2,
i . ds — dh = -u . du + — — . ds. . (4)
g
Also, since Au = Q, a constant,
^4 . du + » . dk4 = o,
and dA = x . dk, very nearly, if x is the width of the stream.
Therefore
Adu + ux . dh = o,
and hence, by equation 4,
. , ,, &a x .. . f if .
i .ds — dh — —a --- - . dh + - --- ds.
g A m 2g
Therefore
i-L± i-t*-.
dh m 2g m 2gi
~3s= ~ur^=l~ u'x ..... (^
I — a- I — a—
gA gA
Let the position of any point a in the surface be defined by
its perpendicular distance h from the bed and by the distance s
of the transverse section at a from an origin in the bed. Then
r is the tangent of the angle which the tangent to the surface
ds
FLOW OF WATER IN OPEN CHANNELS. l6l
at a makes with the bed. It is positive or negative according
as the depth increases or diminishes in the direction of flow,
thus defining two states of steady varied motion.
Between these there is an intermediate state defined by
dh f u*
_- = o = * «- ^- — ,
as m2g
f u*
and i = — — is the equation for steady flow with uniform
m2g
motion.
Let £/", M, H be the corresponding values of #, m, h in the
case of uniform motion. Then
and equation 5 becomes
__
dh m U*
I — a
EXAMPLE. — Consider a stream of rectangular section and
of a width x which is very great as compared with the depth.
Then
A = xh ; P = x very nearly ; m = -- =- h ; M = - - = H.
Hence
*-TW '-(T)'
/* £7* \ h I
dh
I — a—f I — a--
gh gh
since xhu = xHU and therefore — -. = -r-.
£/ h
Note. — In each of the following cases the line PQ drawn
1 62 H YD RA ULICS.
parallel to the bed, represents the surface of uniform motion,
H being the distance between PQ and the bed.
CASE I. au* < gh and H < h.
— is positive, and therefore h increases in the direction of
as
flow. Thus the actual surface MN of the stream is wholly
above the line PQ.
FIG. 94.
Proceeding up stream, h becomes more and more nearly
equal to H, so that the numerator of equation 8, and therefore
also — -, approximates more and more closely to zero,
as
Again, proceeding down-stream, h increases and u dimin-
ishes, so that the numerator and denominator in equation 8
approximate each more and more closely to the value unity,
and therefore — becomes more and more nearly equal to i,
as
the slope corresponding to uniform motion.
Hence up-stream, MN is asymptotic to PQ, and down-
stream MN is asymptotic to a horizontal line. This form of
water-surface is produced when a weir is built across a channel
in which the water had previously flowed with a uniform
motion.
CASE II. au* < gh and H > h.
— is now negative, and the depth diminishes in the direc-
ds
tion of flow.
Up-stream, h increases and approaches H in value, so that
MN is asymptotic to PQ.
FLOW OF WATER IN OPEN CHANNELS.
163
Down-stream, h diminishes, u increases, and therefore the
value of — is more and more nearly equal to unity,
gh
Thus, in the limit, the denominator in equation 8 becomes
zero, and therefore — = 00. Hence theory indicates that at a
as
certain point down-stream the surface line MN takes a direc-
tion which is at right angles to the general direction of flow.
This is contrary to the fundamental hypothesis that the fluid
filaments flow in sensibly parallel lines. In fact, before the
FIG. 95.
limit could be reached this hypothesis would cease to be even
approximately true, and the general equation would cease to
be applicable. This form of water-surface is produced when
there is an abrupt depression in the bed of the stream.
Fig. 96 shows one of the abrupt falls in the Ganges canal
as at first constructed. The surface of the water flowing freely
FIG. 96.
over the crest of the fall took a form similar to MN below the
line PQ.oi uniform motion. The diminution of depth in the
approach to the fall caused an increase in the velocity of flow,
with the result that for several miles above the fall a serious
164
HYDRA ULICS.
erosion of the bed and sides took place. In order to remedy
this, temporary weirs were constructed so as to raise the level
of the water until the surface-line assumed a form MN' cor-
responding approximately to PQ. In some cases the water
was raised above its normal height and a backwater produced*
CASE III. au* > gh and H < h.
—- is negative and the surface-line of the stream is wholly
above PQ.
FIG. 97.
dk
If h gradually increases, u diminishes and —j- approximates
to — i in value.
If h gradually diminishes it approximates to H in value,
dk
and in the limit -T~= o.
ds
Between these two extremes there is a value of h for which
the denominator of equation 8 becomes nil, viz.,
and the corresponding value of -y- is infinity.
Thus one part of the surface line is asymptotic to PQ, the
line of uniform motion, another part is asymptotic to a hori-
zontal line, while at a certain point at which the depth is
the surface of the stream is normal to the bed.
FLOW OF WATER IN OPEN CHANNELS.
I6S
This is contrary to the fundamental hypothesis that the
fluid filaments flow in sensibly parallel lines, and the general
equation no longer represents the true condition of flow.
In cases such as this, there has been an abrupt rise of the
surface of the stream, and what is called a " standing wave "
has been produced.
In a stream of depth H flowing with a uniform velocity
tgr
•depth to h^ which is >
U which is > \ / — — , construct a weir so as to increase the
all*
Then in one portion of the stream near the weir the depth
aU* aU*
is > , while further up the stream the depth is < — — .
o o
U*
Thus at some intermediate point the depth = a , the cor-
o
dh
responding value of -r- being oo , so that at this point a stand-
ds
ing wave is produced.
Now
flT
= Mi=-Hi\
and since
1 66
HYDRAULICS.
and therefore
which condition must be fulfilled for a standing wave.
Bazin gives the following table of values of/:
Nature of Bed.
Slope (A = /)
below which stand-
ing wave is im-
possible. In
Metres per Metre.
Standing Wave Produced.
Slope in Metres
per Metre (or
Feet per Foot).
Least Depth
in Metres.
Very smooth cemented surface
.00147
.00186
.00235
•00275
{.002
.003
.004
( -003
4 .004
( .006
.004
• .006
.010
.006
• .010
.015
.08
•03
.02
.12
.06
•03
.36'
.16
.08
I. O6
• 47
.28
Earth
A standing wave rarely occurs in channels with earthen
beds, as their slope is almost always less than the limit, .00275.
The formation of a standing wave was first observed by
Bidone in a small masonry canal of rectangular section.
The width of the canal = 0^.325 = x •
" slope f= -j) of the canal — -023 »
" uniform velocity of flow = 1^.69 = U\
" depth corresponding to U = 0^.064 = H.
A weir built across the canal increased the depth of the
water near the weir to ow.287 = h^
It was found that the " uniform regime " was maintained
up to a point within 4^.5 of the weir. At this point the
depth suddenly increased from 0^.064 to about ow.i7O, and
between the point and the weir the surface of the stream was
slightly convex in form (Fig. 98).
FLOW OF WATER IN OPEN CHANNELS.
i67
With the preceding data and taking a = i.i,
is therefore > I at a section ab, Fig. 99.
At the section cd,
=q
H_
h
.064
^87
X 1.69 = 0^.377,
and
= .055 and is therefore < i.
au
FIG. 99*
Thus the expression I -- —is negative for a section ao
and positive for a section cdt so
that z must change sign between
these sections, and — will then
as
become infinite.
Consider a portion of a
stream bounded by two- trans-
verse sections ab, cd, in which a standing wave occurs, Fig. 99.
Assume that the fluid filaments flow across the sections in
sensibly parallel lines.
Let the velocities and area at section ab be distinguished
by the suffix i, and those at cd\sy the suffix 2. Then
Change of momentum in di- )
rection of flow [ == imPulse in same direction.
Hence
w
—
and therefore
=A1yl - A,vv ... (9)
the depths below the surface of the centres of
gravity of the sections ab, cd, respectively.
1 68 H YDRA ULICS.
Now, vl = ul + Vr Therefore
Also, as already shown,
a,A,U: = 2av? = AM'
and, neglecting F, as compared with 3«, ,
**#• =-Arf +
Thus
and hence
ufA-i,
= ~^-L(a + 2) = aA*u»
0
a 4- 2
where a' = — ! — , and is 1.033 « * — !•!•
Similarly it may be shown that
Thus equation 9 becomes
~(A^ - Ap?) = ^^ - Aj,. . . . (10)
Let the section of the canal be a rectangle of depth Hl at
ab and Ht at ^. Then
ff H
ufr = u,H, ; -± = >, ; -y-= ^.
FLOW OF WATER IN OPEN CHANNELS. 169
Therefore, by equation 10,
which reduces to
//, = H^ satisfies the equation and corresponds to a condition
of uniform motion.
Also
a'u? ^ff.ff. + ff,
g Hl 2
In Bidone's canal, u1 = 1^.69, Hl = 0^.064. Substituting
these values in equation II, the value of H^ is found to be
ow. 16, which agrees somewhat closely with the actual measure-
ments.
N.B. — The coefficients a and a' have not been very accu-
rately determined, but their exact values are not of great
importance. They are often taken equal to unity.
H YD RA ULICS.
EXAMPLES.
1. What fall must be given to a canal 2600 ft. long, 7 ft. wide at the
top, 3 ft. wide at the bottom, \\ ft. deep, and conveying 40 cubic ft. of
water per second ? /=^¥. Ans. i in 135.
2. Determine the fall of a canal 1500 ft. long, of 2 ft. lower, 8 ft.
upper breadth, and 4 ft. deep, which is to convey 70 cubic feet of water
per second. Ans. i in 1365.4.
3. For a distance of 300 ft. a brook with a mean water perimeter of
40 ft. has a fall of 9.6 in.; the area of the upper transverse profile is 70
sq. ft., that of the lower 60 sq. ft. Find the discharge.
Ans. 662.87 cub. ft. per sec.
4. In a horizontal trench 5 ft. broad and 800 ft. long it is desired to
carry off 20 cub. ft. discharge and to let it flow in at a depth of 2 ft. ;
what must be the depth at the end of the canal ? (/ = .008.)
Ans. 1.64 ft.
5. Water flows along an open channel 12 ft. wide and 4 ft. deep, at
the rate of 2 ft. per second. What is the fall? A dam 12 ft. by 3 ft.
high is formed across the channel; how high will the water rise over the
crest of the dam ? Ans. i in 48o,/ being .08 ; .899 ft.
6. A stream is rectangular in section, 12 ft. wide, 4 ft. deep, and falls
i in 100. Determine the discharge (i) with an air-perimeter; (2) without
air-perimeter. Ans. (i) 645.398 cub. ft. per sec.
(2) 665.088 cub. ft. per sec.
7. A canal 20 ft. wide at the bottom and having side slopes of i£ to
i has 8 ft. of water in it; find the hydraulic mean depth. Ans. 5.24 ft.
8. The water in a semicircular channel of 10 ft. 'radius, when full
flows with a velocity of 2 ft. per second ; the fall is i in 400. Find the co-
efficient of friction. Ans. .2.
9. Calculate the flow per minute across a given section of a rectarw-
gular canal 20 ft. deep, 45 ft. wide, the slope of the bed being 22 in. per
mile and the coefficient of friction per square foot = .008.
Ans. 279,229 cub. ft.
10. Why does the water of the St. Lawrence rise on the formation
of the ice ?
11. Find the depth and width of a rectangular stream of 900 sq. ft.
sectional area, so that the flow might be a maximum ; also find the flow,
f being .008 and the slope 22 in. per mile.
Ans. 21.21 ft.; 42.42 ft.; 4885 cub. ft. per second.
FLOW OF WATER IN OPEN CHANNELS. \7\
12. Water flows along a symmetrical channel, 20 ft. wide at top and
8 ft. wide at bottom ; the friction at the sides varies as the square of the
velocity, and is i Ib. per square foot for a velocity of 16 ft. per second.
Find the proper slope, so that the water may flow at the rate of 2 ft. per
second when its depth is 6 ft. Arts, i in 3445.
13. Calculate the flow across the vertical section of a stream 4 ft.
deep, 1 8 ft. wide at top, 6 ft. wide at bottom, the slope of the surface
being 18 in. per mile. (/= .008.) Ans. 110.9376 cub. ft. per second.
14. The sewers in Vancouver are square in section and are laid with
one diagonal vertical. To what height should the water rise so that
(a) the velocity of flow may be a maximum ; (b) the discharge may be a
maximum ? (A side of the square = 12 in.)'
Ans. (a) .292 ft. above horizontal diameter.
(b) .5797 ft. "
15. The sides of an open channel of given inclination slope at 45*
and the bottom width is 20 ft. Find the depth of water which will make
the velocity of flow across a vertical section a maximum.
Ans. 6.73 ft.
17. The banks of a channel slope at 45° ; the flow across a transverse
section is to be at the rate of 100 cubic feet at a maximum velocity of 5
ft. per second. Determine the dimensions of the transverse profile.
Ans. 11.05 ft. wide at bottom ; 2.28 ft. deep.
1 8. What dimensions must be given to the transverse profile of a
canal whose banks slope at 40°, and which has to conduct away 75 cubic
feet with a mean velocity of 3 ft. per second ?
Ans. Depth = 3.6 ft. ; width at bottom = 2.62 ft.
19. The section of a canal is a regular trapezoid ; its slope is i in
500 ; its width at the bottom is 8 ft.; the sides are inclined at 30° to the
vertical. On one occasion when the water was 4 ft. deep a wind was
blowing up the canal, causing an air-resistance for each unit of free sur-
face equal to one fifth of that for like units at the bottom and sides,
where the coefficient of friction may be taken to be .08.
Determine the discharge. How will the discharge be affected when
the canal is frozen over? Ans. 75.34 cub. ft. per sec.
20. The section of a channel is a rhombus with diagonal vertical.
How high must the water rise in the channel (a) to give a maximum of
flow, and (b) to give a maximum discharge?
Ans. If D is the length of the horizontal diameter, and if &
is the inclination of a side to the vertical, the water
must rise above the horizontal diameter to the height
Z)cot0 x .207 in (a) and to the height Z>cotfl x .4099
in (b).
21. In the transverse section ABCD of an open channel with a verti-
cal slope of i in 300, the bottom width is 20 ft., the angle ABC — 90*
1 72 H YDRA ULICS.
and the angle BCD = 45°. Find the height to which the water will
rise so that the velocity of flow may be a maximum ; also find the dis-
charge across the section,/ being .008.
Ans. 11.715 ft.; 1584 cub. ft. per second.
22. A canal is 20 ft. wide at the bottom, its side slopes are i| to i, its
longitudinal slope is i in 360; calculate H.M.D. and the flow per minute
across any given vertical section when there is a depth of 8 ft. of water
in the canal. (Coeff. of friction = .008.)
Ans. 5.24 ft.; 2762.7776 cub. ft. per second.
23. If a weir 2 ft. high were built across the canal in the preceding
question, what would be the increase in the depth of the water?
Ans. 2.79 ft.
24. For a small tachometer the velocities are .163, .205, .298, .366,
,61 metre; the number of revolutions per second are .6, .835, 1.467,
1.805, 3.142. Find the constants corresponding to the wheel.
Ans. ,162; .202; .309; .367; .595.
25. If the head of water in a channel increase by one tenth, show
that the velocity and discharge, respectively, increase by -£$ and ^.
approximately.
If the depth diminish by 8$, show that the velocity and discharge,
respectively, diminish by 4% and 12%, approximately.
26. Assuming (i) that a river flows over a bed of uniform resistance
to source ; (2) that to maintain stability the velocity is constant from
source to mouth ; (3) that the river sections at all points are similar ;
(4) that the discharge increases uniformly in consequence of the supply
from affluents — determine the longitudinal section of such a river.
Ans. A parabola.
CHAPTER V.
METHODS OF GAUGING.
I. Gauging of Streams and Watercourses. — The
amount of flow Q in cubic feet per second across a transverse
section of A sq. ft. in area is given by the expression
Q - Au,
u being the mean velocity of flow in the section in
feet per second. Various methods are employed for
the determination of u.
METHOD I. The most convenient method for
gauging small streams, canals, etc., is by means of
a temporarily constructed weir, which usually takes
the form of a rectangular notch. The greatest
care should be exercised to ensure that the crest
of the weir is truly level and properly formed and
that the sides are truly vertical. The difference of
level between the crest of the weir and the surface
of the water at a point where it has not begun to
slope down towards the weir is best es-
timated by means of Boyden's hook gauge,
Fig. 100.
This gauge consists of a carefully grad-
uated rod, or of a rod with a scale attached,
having at the lower end a hook with a thin
flat body and a fine point. The rod slides
in vertical supports, and a slow vertical
movement is given by means of a screw of
fine pitch. In an experiment, the hook
FIG. TOO.
point is set truly level with the crest of the weir, and a read-
ing is taken. The gauge is then moved away from the weir,
HYDRA ULICS.
about 2 to 4 ft. for small weirs and about 6 to 8 ft. for large
weirs. The hook is then slowly raised, until a capillary eleva-
tion of the surface is produced over the point. The hook is
now lowered until this elevation is barely perceptible, and a
second reading is taken. The difference between the two
readings is the difference of level required.
In ordinary light, differences of level as small as the one-
thousandth of a foot, can be easily detected by the hook
gauge, while with a favourable light it is said that an experi-
enced observer can detect a difference of two ten-thousandths
of a foot.
METHOD II. A portion of the stream which is tolerably
straight and of approximately uniform section is defined by
two transverse lines O^B, OfD, at any distance 5 ft. apart.
FIG. 101.
The base-line O,O^ is parallel to the thread EF of the
stream, and observers with chronometers and theodolites (or
sextants) are stationed at (9, , <92. The time T and path EF
taken by a float between AB and CD can now be determined.
At the moment the float leaves A B the observer at Ol signals
the observer at (92, who measures the angle O^O^E, and each
marks the time. On reaching CD the observer at O.t signals
the observer at Ol , who measures the angle O^Of, and each
again marks the time.
Experience alone can guide the observer in fixing the dis-
METHODS OF GAUGING.
175
tance 5 between the points of observation. It should be
remembered that although the errors of time observations are
diminished by increasing S, the errors due to a deviation from
lines parallel to the thread of the stream are increased.
A number of floats may be sent along the same path, and
their velocities UsJ are often found to vary as much as 20 per
cent and even more.
Having thus found the velocities along any required num-
ber of paths in the width of the stream, the mean velocity for
the whole width can be at once determined.
Surface-floats are small pieces of wood, cork, or balls of
wax, hollow metal and wood, colored so as to be clearly seen,
and ballasted so as to float nearly flush with the water-surface
and to be little affected by the wind.
Subsurface-floats. — A subsurface-float consists of a heavy
float with a comparatively large intercepting area, maintained
at any required depth by means of a very fine and nearly
vertical cord attached to a suitable surface-float of minimum
immersion and resistance. Fig. 102 shows such a combina-
tion, the lower float consisting of two pieces of galvanized iron
soldered together at right angles, the upper float being merely
a wooden ball.
FIG. 102.
FIG. 103.
Another combination of a hollow metal ball with a piece
of cork is shown by Fig. 103.
The motion of the combination is sensibly the same as that
HYDRA ULICS.
of the submerged float, and gives the velocity at the depth to
which the heavy float is submerged.
Twin-floats. — Two equal and similar floats (Fig. 104), one
denser and the other less dense than water,
1 are connected by a fine cord. The velocity
(vt) of the combination is approximately the
mean of the surface-velocity (vs) and of the
velocity (v^) at the depth to which the heavier
float is submerged. Thus
FIG. 104. and therefore
d> ~~"~ ^ t */s 9
so that vd can be determined as soon as the value of vt has
been observed and the value of vs found by surface-floats.
Velocity-rod. — This is a hollow cylindrical rod of ad-
justable leiigth and ballasted so as to float nearly vertical. It
sinks almost to the bed of the stream,
and its velocity (vm) is approximately the
mean velocity for the whole depth.
Francis gives the following empirical
formula connecting the mean velocity
(vm) with the observed velocity (vr) of
the rod :
...*/£),
=zv(i.oi2
d being depth of stream, and d' the depth FlG- I05-
of water below bottom of rod ; but d' should not exceed about
one fourth of d.
METHOD III. Pitot Tube and Darcy Gauge.— A Pitot
tube (Figs. 106 to 108) in its simplest form is a glass tube with
a right-angled bend. When the tube is plunged vertically into
the stream to any required depth z below the free surface, with
its mouth pointing up-stream and normal to the direction of
METHODS OF GAUGING.
177
flow, the water rises in the tube to a height h above the out-
side surface, and the weight of the column of water z -f- h
FIG. i 06.
FIG. 107.
FIG. 108.
high, is balanced by the impact of the stream on the mouth.
Hence, (Chap. VI.),
wA(z -f- k) = wAz -f- kwA — ,
and therefore
A being the sectional area of the tube, u the velocity of flow
at the given depth, and k a coefficient to be determined by
experiment.
A mean value of k is 1.19. With a funnel-mouth or a bell-
mouth, Pitot found k to be 1.5. This form of mouth, however,
interferes with the stream-lines, and the velocity in front of
the mouth is probably a little different from that in the unob-
structed stream.
The advantages of tubes of small section are that the dis-
turbance of the stream-lines is diminished and the oscillations
of the column of water are checked. Darcy found by careful
measurement that the difference of level between the surfaces
of the water-column in a tube of small section placed as in
Fig. 106, and of the water-column placed as in Fig. 107 with
HYDRA ULICS.
FIG. 109.
its mouth parallel to the
direction of flow, is almost
exactly equal to — -.
When the tube is placed
as in Fig. 108 with its
mouth pointing down-
stream and normal to the
direction of flow, the level
of the surface of the water
in the tube is at a depth ti
below the outside surface,
and
where kf is a coefficient to
be determined by experi-
ment and a little less than
unity.
In this case the tube
again obstructs the stream-
lines. Pitot's tube does
not give measurable indi-
cations of very low veloc-
ities. A serious objection
to the simple Pitot tube is
the difficulty of obtaining
accurate readings near the
surface of the stream. This
objection is removed in
the case of Darcy's gauge,
shown in the accompany-
ing sketch, Fig. 109.
A and B are the water-
inlets; C and D are two
double tubes ; E is a brass
METHODS OF GAUGING.
tube containing two glass pipes which communicate at the
bottom with the water-inlets and at the top with each other,
and with a pump F by which the air can be drawn out of
the glass, pipes thus allowing the water to rise in them to any
convenient height.
Thus Darcy's gauge really consists of two Pitot tubes con-
nected by a bent tube at the top and having their mouths at
right angles or pointing in opposite directions. If h is the
difference of level between the water-surfaces in the tubes
when the mouths are at right angles, then
and Darcy's experiments showed that k does not sensibly
differ from unity.
When the mouths point in opposite directions, let h^ h^ be
the differences of level between the stream-surface and the
surfaces of the water in the tube pointing up-stream and the
tube pointing downstream, respectively. Then
u*
** = k{2g'>
U*
and therefore
u*
h j. h - (k , + k )—
*>2g
where k = kv -\- k^
k having been determined experimentally once for all, the
difference of level (= h^ -\- h^) between the columns for any
given case can be measured on the gauge and then u can be
at once found.
1 80 H YDRA ULICS.
A cock may be inserted in the bend connecting the two
tubes, and through this cock air may be exhausted and a
partial vacuum created in the upper portion of the gauge.
The water-columns will thus rise to higher levels, but the dif-
ference between them will remain constant. Thus the surface
of the column in the down-stream tube may be brought above
the level of the outside surface, and the reading is then easily
made.
Sometimes the gauge is furnished with cocks at the lower
parts of the tubes, and if these cocks are closed when the
measurement is to be made, the gauge may be removed from
the stream for the readings to be taken.
METHOD IV. Current-meters. — The velocity of flow in
large streams and rivers is most conveniently and most ac-
curately ascertained by means of the current-meter. The
earliest form of meter, the Woltmann mill, is merely a water-
mill with flat vanes, similar in theory and action to the .wind-
mill. When the Woltmann is plunged into a current, a counter
registers the number of revolutions made in a given interval
of time, and the corresponding velocity can then be deter-
mined. This form of meter has gone out of use and has been
replaced by a variety of meters of greater accuracy, of finer
construction, and much better suited to the work. In its sim-
plest form the present meter consists of a screw-propeller
wheel (Fig. 1 10), or a wheel with three or more vanes mounted
on a spindle and connected by a screw-gearing with a counter
which registers the number of revolutions. The meter is put
' in or out of gear by means of a string or wire. When a cur-
rent velocity at any given point is to be found, the reading of
the counter is noted, the meter is sunk to the required position,
and is then set and kept in gear for any specified interval of
time. At the end of the interval the meter is put out of gear
and is raised to the surface when the reading of the counter is
again noted. The difference between the readings gives the
number of revolutions made during the interval, and the veloc-
ity is given by an empirical formula connecting the velocity
and the number of revolutions in a unit of time.
METHODS OF GAUGING.
The vane Fis introduced to compel the meter to take its
proper direction.
In order to prevent the mechanism of the meter from being
FIG. 1 10.
FIG. in.
injuriously affected by floating particles of detritus, Revy en-
closed vthe counter in a brass box, Fig. ill, with a glass face,
FIG. 112.
FIG. ri3.
and filled the box with pure water so as to ensure a constant
coefficient of friction for the parts which rub against each
other. In the best meters, however, the record of the number
1 82 HYDRAULICS.
of revolutions is kept by means of an electric circuit, Fig. 112,
which is made and broken once, or more frequently, each
revolution, and which actuates the recording apparatus. The
time at which an experiment begins and ends is noted, and the
revolutions made in the interval are read on the counter, which
may be kept in a boat or on the shore, as the circumstances of
the case may require. The meter is usually attached to a suit-
ably graduated pole, so that the depth of the meter below the
water-surface can be directly read. The mean velocity for the
whole depth at any point of a stream may be found by moving
the meter vertically down and then up, at a uniform rate.
The mean of the readings at the two surface positions and at
the bottom position will be the number of revolutions corre-
sponding to the mean velocity required. The mean velocity
for the whole cross-section may also be determined by moving
the meter uniformly over all parts of the section.
Before the meter can be used it must be rated. This is
done by driving the meter at different uniform speeds through
still water. Experiment shows that the velocity v and the
number of revolutions n are approximately connected by the
formula
v = an + b,
where a and b are coefficients to be determined by the method
of least squares or otherwise.
Exner gives the formula
VQ being the velocity at which the meter just ceases to re-
volve.
OTHER METHODS. — Many other pieces of apparatus for
the measurement of current velocities have been designed.
PerrodiTs hydrodynamometer, for example, gives the ve-
locity directly in terms of the angle through which a vertical
torsion-rod is twisted, and in this respect is superior to the
current-meter.
METHODS OF GAUGING.
183
FIG. 114.
The hydrometric pendulum (Fig. 114), again, connects the
velocity with the angular devia-
tion from the vertical of a heavy
ball suspended by a string in the
current.
Hydrometric and torsion bal-
ances have also been devised,
but they must be regarded
rather as curiosities than as
being of any real practical use.
2. Gauging of Pipe Flow. — A variety of meters have
been designed to register the quantity of water delivered by a
pipe. The principal requisites of such a meter are :
1. That it should register with accuracy the quantity of
water delivered under different pressures.
2. That it should not appreciably diminish the effective
pressure of the water.
3. That it should be compact and adaptable to every
situation.
4. That it should be simple and durable.
The Venturi Meter (Fig. 115) is so called from Venturi, who
first pointed out the relation between the pressures and veloci-
ties of flow in converging and diverging tubes.
FIG. 115.
As shown by the longitudinal section, Fig. 116, this meter
consists of two truncated cones joined at the smallest sections
by a short throat-piece. At A and B there are air-chambers
with holes for the insertion of piezometers, by which the fluid
1 84
HYDRA ULICS.
pressure may be measured. By Art. 5, Chap. I, the theoretical
quantity Q of flow through the throat at A is
at, #a being the sectional areas at A and B, respectively, and
fft — Hl the difference of head in the piezometers, or the
"head on Venturi," as it is called.
FIG. 116.
Introducing a coefficient of discharge C, the actual delivery
through 'A is
An elaborate series of experiments by Herschel gave C
values varying between .94 and 1-04, but the great majority of
the values lay between .96 and .99.
The piezometers may be connected with a recorder, and
thus a continuous register of the quantity of water passing
through the meter may be obtained at any convenient position
within a radius of 1000 ft. This distance may be extended to
several miles by means of an electric device.
Other meters may be generally classified as Piston or Re-
ciprocating Meters and Inferential or Rotary Meters. They
are all provided with recorders which register the delivery with
a greater or less degree of accuracy.
The piston meter (Fig. 1 1 8) is the more accurate and gives
a positive measurement of the actual delivery of water as
METHODS OF GAUGING.
185
recorded by the strokes of the piston in a cylinder which is
filled from each end alternately. Thus an additional advan-
l_ ..:
PIG. 118. — SCHONHEYDER'S POSITIVE
METER.
FIG. 119. — THE UNIVERSAL
METER.
FIG. 120. — THE BUFFALO METER. FIG. 121. — THE UNION ROTARY PIS-
TON METER.
tage possessed by a water-engine is that the working cylinder
will also serve as a meter.
In inferential meters, a drum or turbine is actuated by the
force of the current passing through the pipe, but it often
happens that when the flow is small the force is insufficient to
cause the turbine to revolve, and there is consequently no
register of the corresponding quantity of water passing through
the meter.
CHAPTER VI.
IMPACT.
Note. — The following symbols are used :
z/, = the velocity of the jet before impact ;
z>2 = " " " " " after leaving the vane ;
u — " " " " vane ;
V — " " " water relatively to the vane ;
A = sectional area of the impinging jet ;
m = mass of the water reaching the vane per second.
i. Impact of a Jet upon a Flat Vane oblique to the
Direction of the Jet.— Let 6 be the angle between the nor-
mal to the vane and the direction of the impinging jet, <p the
angle between the nor-
mal to vane and the
direction of the vane's
motion, and a the an-
gle between the vane
and the vertical.
The jet moving with
its stream-lines paral-
lel, swells out near the
vane, over which it
spreads-and with which
it travels along in the
direction of the vane's
motion, and finally again flows along with its stream-lines sen-
sibly parallel to the vane.
The problem is still further complicated by the production
of eddies and vortices for which allowance can only be made in
a purely empirical manner.
Let N be the normal pressure on the vane due to the impact.
Let N' be the total normal pressure on the vane.
Let W be the weight of water on the vane.
1 86
IMPACT. IS/
Then
N = N' — W sin a = change of momentum in direction of the
normal
= mv^ cos 6 — mu cos 0.
or
N = m(vl cos 6 — u cos 0). i ., . . (i)
(N. B. — The sign in front of u cos 0 will be plus if the jet
and vane move in opposite directions.)
The term W sin a may be designated the static pressure
and the term m(vl cos 6 — u cos 0) the dynamic pressure
which causes the deviation of the stream-lines.
Note. — The pressure when a jet first strikes the plane is
greater than when the flow has become steady, or permanent
regime is established.
This is made evident by the following consideration :
At any moment let MN, PQ, RS be the bounding planes
across which the water is flowing with its stream-lines sensibly
parallel.
In a unit of time let the bounding planes of the mass be
M'N', P'Q, R'S'.
Then, initially, the reaction of the plane must destroy the
motion of the mass of the fluid bounded by M'N', PfQf,
and RfSf.
Take OC to represent vl in direction and magnitude.
In one second the vane AB moves parallel to itself into
the position A'B'. Let A' B' intersect OC in D.
Then
m = -A . DC = -A(v, - OD)
g g
W A( COS 0\ f N
= —A(vl— u -- 4-J ........ (2)
g \ cos Ql
Thus equation i becomes
7JJ A
N= --- s(^i cos 6 — u cos 0)a. ••• (3)
g COS C7
1 88 HYDRAULICS.
Let P be the pressure in the direction of the vane's
motion, then
, . (4)
and the useful work done on the vane per second
= Pu = — A - ^u(v cos 6 — u cos 0)2. (c)
g cos 6
<2jn <7j
The total available work = — A%- ........ (6)
g 2 ^
W . COS 0 ,
~^A ^rtu(v* cose - u cos 0)a
Hence, the efficiency =
cose-ucos^ (7)
This is a maximum when
z/t cos 6 = $u cos 0, (8)
and therefore
o
the maximum efficiency = — cos9 0. . . . (9)
If the vane is of small sectional area a portion of the water
will escape over the boundary and the pressure must neces-
sarily be less than that given by equation 3.
Instead of one vane moving before the jet, let a series of
vanes be introduced at short intervals at the same point in the
path of the jet.
The quantity of water now reaching the vane per second is
evidently
m = -Avl9 ....... (10)
o
IMPACT. 189
and, by equation I, the normal pressure
fl = ^N—Avfa costi — u cos 0). . ./ . (11)
o
Also, the pressure in the direction of the motion of
the vane
= P = N cos 0 = - — A cos 0 v1(vl cos 0 — u cos 0). (12)
o
The useful work done per second
— Pu= ^A cos 0 ^X^i cos 6 — u cos 0), . . (13)
o
and the efficiency
IV
— A cos 0 v^u(vl cos 6 — u cos 0)
2 COS 0 «(Z/ COS 6 — U COS 0)
~~
This is a maximum when vl cos 8 = 2u cos 0, . . (15)
and therefore
the maximum efficiency = --- ..... (16)
SPECIAL CASE I. Let a single vane be at right angles to,
and move in the line of, the jet's motion, Fig. 123.
Then 6 = o = 0.
Hence
the pressure = P= N == — A(VI — u)9; . (17)
"' ~~ -r- —
FIG. 123. the useful work = Pu = — Au(v^ — #)3; . (18)
o
19° HYDRA ULICS.
the efficiency = 2—(vl — u) ; . . (19)
o
the maximum efficiency^ — (20)
Again, if u = o, i.e., if the vane be fixed, and if H be the
head corresponding to the velocity vlt then, by equation 17,
P = Av? = 2wAH
= twice the weight of a column of water
of height H and sectional area A.
SPECIAL CASE 2. Let each of a series of vanes be at right
angles to and move in the line of the jet's motion at the
instant of impact.
Then 6 = o = 0.
w $
The pressure = N = P = — Av\(v^ — u). . (21)
<5
IV
The useful work = Pu = — A^lu(i'1 — u^ . (22)
o
The efficiency = 2*fo " *>. . . (23)
The maximum efficiency= - (24)
2. Reaction — Jet Propeller. — The term reaction is em-
ployed to denote the pressure upon a surface due to the di-
rection and velocity with which the water leaves the surface.
Water, for example, issues under the head h and with the
IMPACT. IQI
velocity v (at contracted section) from an orifice of sectional
area A in the vertical side of a vessel,
Fig. 124.
Let R be the reaction on the opposite
vertical side of the vessel, and let Q be
the quantity of water which flows through
the orifice per second. Then
R = horizontal change of momentum
wQ w
= v — — CcAv* — 2wcccvAh — 2wAh, . . . (i)
o e»
disregarding the contraction and putting cv — I.
Thus the reaction is double the corresponding pressure
when the orifice is closed (Special Case I, Art. i).
Again, let the vessel be propelled in the opposite direction
with a velocity u relatively to the earth.
Then vl — u is the velocity of the jet at the contracted
section relatively to the earth and
R = horizontal change of momentum
= ^Q(Vl-u) . . (2)
o
The useful work done by the jet
IV
= Ru = —Qu(vl-u) (3)
o
The energy carried away by the issuing water
Hence
w w (v. — uY
the total energy = —Qu(v, -u) + —Q —
(5)
IQ2 HYDRAULICS. ,
and
w
g 2U
the efficiency = — —5 r = . — . . . . (6)
w v, — u v, -4- u
Thus the more nearly vl is equal to u, and therefore the
larger the area A of the orifice, the greater is the efficiency.
If the vessel is driven in the same direction as the jet, then
77, -f- u is the relative velocity of the jet with respect to the
earth, and the reaction is
R — horizontal change of momentum
-G& + u) = c^Av^v, + u)
(7)
disregarding the contraction and putting c, = I.
3. A Jet of Water impinging upon a Surface of Rev-
olution moving in the Direction of its Axis and also in
the Line of the Jet's Motion. — Disregarding friction, the
water flows over the surface without any change in the magni-
tude of the relative velocity vt — u, but the stream-lines are
deviated from their original direction through an angle /?.
(N.B. — The sign before u is plus if the surface and jet are
moving in opposite directions.)
Let the water leave the surface at D, and in the direction
of the tangent at D take DE to represent vl — u in direction
and magnitude. Also draw DF parallel to the axis of the sur-
face and take DF to represent //,
Complete the parallelogram EF.
The diagonal DG evidently represents in direction and
magnitude the absolute velocity v^ with which the water leaves
IMPACT. 193
the surface. Hence, from the triangle DFG, since the angle
DFG = n — ft,
v* = (vl — uj + u* - 2(z/1 — *)« cos (?r — /?),
from which
/?
z/j3 — v* = 2^(^j — w)(i — cos /?) = 4«(z/ — u) sin2 — . (i)
Also, — -A(vl — u) = the quantity of water reaching the
<b
surface per second.
Hence, if P is the pressure in the direction of motion, the
useful work done per second
FIG. 125
-1 = 2—Au(vl — u)* sina — ; . (2)
and the pressure
0" 2 \«J/
The efficiency
A f \a - •>.
— Au(vl — u) sm —
sm^. . (4)
W -V v* 2
A
194 HYDRAULICS.
This is a maximum when
», = 3«, • • > - •" ... (5)
and therefore
the maximum efficiency = — sin2 -. . . . (6)
If, instead of one surface, a series of surfaces are succes-
sively introduced at short intervals at the same point in the
path of the jet, the quantity of water reaching each surface
per second becomes
w
m= " (7)
and hence the useful work, pressure, and efficiency also respec-
tively become
w ft
2~A^u(^-u)sm9~', (8)
—Avfa — w)sina— ; (9)
u(vl — u} . 2,/?
4" V* 2
The efficiency is a maximum when
v.
(ii)
Q
its value then being sina — .
2
It will be observed that the results given by equations 2
to II are identical with those given by equations 17 to 20 and
21 to 24, Art. I, except that in each case there is an additional
ft
factor 2 sin8 or I — cos ft. This factor is greater than unity,
and therefore the pressure, useful work, and efficiency are each
IMPACT.
195
increased, if ft > 90°, i.e., in the case of a concave vane ; while
in the case of a convex vane, ft being < 90°, the factor is also
less than unity and they are each diminished.
SPECIAL CASE. Let fi = 180°, i.e., let the vane be of the
cup type and in the form of a hemisphere.
1 80°
The maximum efficiency is sin" = unity, and is per-
fect. The water should therefore leave the surface without
velocity; and this is the case ; for, by equation I,
Hence
v* — v* = ^ii(v^ — u), and u = — .
2
v* — v* = v*, and therefore ^a = o.
4. Impact of a Jet of Water upon a Vane with Borders.
— Let the vane in Art. i be provided with borders, Figs.
126, 127, so as to produce a further deviation of the stream-
lines, and let the water finally flow off with a velocity v* in a
•direction making an angle 0' with the normal to the vane.
FIG. 126.
FIG. 127.
Then
the normal pressure = N
= mvt cos 0 T mv^ cos tf 3= mu cos 0
= m(vl cos 0 ^F z>a cos 0' =F u cos 0),
the sign of the second term being plus or minus according to
the direction in which the stream-lines are finally deviated.
196 HYDRAULICS.
The effect of the borders is therefore to increase or diminish
the normal pressure, and hence also the useful work and the
efficiency.
SPECIAL CASE. Let the vane be at rest, i.e., let u = o, and
let the final and initial directions of the jet be parallel.
Also, let v1 = Vf Then
N = m(v^ cos 6 -\- vl cos 6)
w
= 2—Av? cos 6
o
= 4wAH cos 6.
Hence, if fl— o, the normal pressure N= qwAH •=• four
times the weight of a column of water of height H and sec-
tional area A.
5. Pressure of a Steady Stream in a Uniform Pipe
against a Thin Plate AB Normal to the Direction of
Motion. — The stream-lines in front of the plate are deviated
and a contraction is formed at Cf^ They then converge,
leaving a mass of eddies behind the plate.
Consider the mass bounded by the transverse planes ClCl>
3 , where the stream-lines are again parallel.
At C£i let A , Al , vl , zl be the mean intensity of the press-
ure, the sectional area of the
waterway, the velocity of flow, and
the elevation of the C. of G. of
the section above datum.
Let /2, AS, z>3, z^ be corre-
sponding symbols at Cfv
Let /3> A19 vlt *„ be corre-
sponding symbols at C9C3.
Let a be the area of the plate.
Let cc be the coefficient of contraction.
Neglect the skin and fluid friction between ClCl and
Then by Bernoulli's theorem,
+ +
' ' ' '
W 2g W 2g W 2g 2g
IMPACT. 197
(v — v\
the term — - — representing the loss of head due to the
bending of the stream-lines between Cf^ and C3C3.
Hence
• A -A (v* - v>Y
Again, let R be the total pressure on the plane. Then
x . x A ( fluid pressure in the direction
A -M, = (A - AK = | of the axis_
— 2*
= component of the weight in the direction
of the axis.
Thus
^ __ j>s)Al + wAl(zl — ^,) — R = change of motion in direction
of axis
= 0,
since the motion is steady.
Hence
R A -A (",-".)'
wA l w 2g
But A^, = AM = cc(Al — a)Vr Therefore
=-*${&&>- >}
v? ( m } a
= wa — m \ —, r — I [• ,
2g \ cc(m - i) j
A
where m = — , or
a
R =
r m )
where K — in \ —, — — r — I >
\ cc(m - i) f
198 HYDRAULICS'.
6. Pressure of a Steady Stream in a Uniform Pipe on
a Cylindrical Body about Three Diameters in Length.—
The stream-lines in front of the body are deviated and a
contraction is formed at C9Ct. They then converge, flow in
parallel lines, and converge a second time at C3C3, leaving a
mass of eddies behind the body.
Consider the mass bounded by the planes C£^ CtC4.
As in the previous article, let
/>,, Alt vl, zl be the intensity of pressure, sectional area of
the waterway, velocity of flow, and elevation
of C. of G. above datum at
r A, ^2, z>2, <sra be similar symbols
V3 <?4 <•
for
/3, ^48, ^3, #3 be similar symbols
for Cfy
& &* 3- p^Al,vl,Zi be similar symbols
FIG- I29' for C&.
Neglect the skin and fluid friction between ClCl and C4Ct.
Then, by Bernouilli's theorem,
, .
W 2g W 2g w ' 2g
, A , *>? . {^s - *$ ,
!*4++"t "
^""« being the loss of head between <7a£, and C,C3 and
— being the loss of head between C3C9 and CtCt.
o
Hence
* i A— A _ (^. — ^)a I (^ - Oa
J" 4-f ~^^ "IF IF"
But A& = ^3e;a = ^3^,,
and A3 = A, — a.
IMP A CT. 199
Therefore
, y t A_ _A^n
a I \7JJ^-a) A,- a] J
where m = — *.
Also, as in the preceding article,
(A-
Hence
f
2g (m— i)2 (m - i)a V,
where m = — -, and
a
This value of K is always less than the value of K for the
plate in the preceding article for the same values of m, a,
and cf
Hence the pressure on the cylinder is also less than the
corresponding pressure on the plate.
In every case K should be determined by experiment.
7. Jet impinging upon a Curved Vane and deviated
wholly in one Direction — Best Form of Vane. — Let the
jet, of sectional area A, moving in the direction AB with a
velocity v^ , drive the vane AD in the direction AC with a
velocity u.
200
HYDRAULICS.
Take AB to represent v^ in direction and magnitude.
" AC " " u " " ".
Join CB.
Then CB evidently represents F, the velocity of the water
relatively to the vane, in direction and magnitude. If CB is
parallel to the tangent to the vane at A, there will be no sud-
FIG. 130.
den change in the direction of the water as it strikes the vane,
and, disregarding friction, the water will flow along the vane
from A to D without any change in the magnitude of the rela-
tive velocity V (= CB). The vane is then said to "receive the
water without shock."*
Again, from the triangle ABC, denoting the angles BA C,
ABC, ACB, byA,£, C, respectively.
sin B
u _ AC _ sin B __
^ =" ~AB ~ sin C ~ sin (A + B)' ' '
. . (I)
and therefore
cot B = — cosec A — cot A, .... (2)
IMPACT. 201
a formula giving the angle between the lip and the direction
of the impinging jet, which will ensure the water being received
" without shock."
In the direction of the tangent to the vane at D, take
DE = CB (= V).
Draw DF parallel and equal to AC(= u).
Complete the parallelogram EF.
Then the diagonal DG evidently represents in direction and
magnitude the absolute velocity v^ with which the water leaves
the vane.
Draw AK equal and parallel to DG (= z/a).
Join BK. Then BK represents the total change of velocity
between A and D in direction and magnitude.
Thus, if R is the resultant pressure on the vane, then
R = m. BK.
Let ML be the projection of BK upon AC.
Then ML represents the total change of velocity in the
direction of the vane's motion.
Let P be the pressure upon the vane in this direction.
Then
P=m. LM. (3)
The useful work = Pu = mu . LM = mV* ~ V* . . . (4)
w A v?
The total available work = - A -- (5)
„,, ~ . mu. LM v? — v*
The efficiency -- — = img— -- r- ...... (6)
w Av? * wAv?
Again, join CK.-
Then, since A C is equal and parallel to DF, and AK to DG,
the line CK is equal and parallel to DE, and is therefore equal
to CB.
Thus in the isosceles triangle CBK, CB is equal and parallel
to the relative velocity Fat A, CK is equal and parallel to the
2O2 HYDRA ULICS.
relative velocity Fat D, and the base B K represents the total
change of motion.
Let 8 be the angle through which the direction of the water
is deviated, i.e., the angle between AB and AK. Then
= V* -\- U* — 2V Ji COS (A + #), ...... (7)
and also
F3 = CK* = CB* = AB* + AC* - 2AB . AC cos A
= v* -\-u* — 2v ji cos A .......... (8)
Hence
— L = u \ vt cos (A + 6) — vl cos A } . . . (9)
If BH is drawn parallel to the tangent at D, BK evidently
bisects the angle between BC and BH, and this angle is equal
to the angle between the tangents to the vane at A and D.
Let a be the sttpplcmcnt-^f the angle between the normals
at A and D. Then the angle KCB — a, and
the angle CBK = -(180° - «) = 90° - £
2 2
Therefore
BK = 2CB (cos 00° — - ] = 2Fsin -.
\ 21 2
Hence
;in- (10)
IMP A CT. 2O3
Let X, Fbe the components of R in the direction of the
normal at A and at right angles to this direction. Then
Y=R cos- = mVsm or; .... (n)
X = R sin — = 2m V sin3 - = m V( i — cos a). ( 1 2}
2 2
The efficiency is a maximum when
dP
The efficiency is nil when
Pu = o, i.e., when u = o or P = o. . . . (14)
In the latter case, since P — m. LM, the projection LM
must be nil, and therefore BK must be at right angles to A C,
as in Fig. 131.
FIG. 131.
FIG. 132.
204 HYDRA ULICS.
The angle ACB is now = 180° -- , and therefore
u_ sin ABC
vl ~~ sin A CB
sn
in (180° — -^
(IS)
sm —
2
If BK is parallel to AC (Fig. 132), then
the angle ACB = -(180° -«) + « = 90° + -
2 2
.and therefore
„ sin (90° + - + A\ cost- ~4- A]
u_ _ sin ABC V r 2 ) _ \2 1
sm I Qcr + - 1 cos -
SPECIAL CASE. — Let the direction of the impinging jet be
tangential to the vane at A, and let the jet and vane move in
the same direction. Then
V— v. — uy m = — A(v. — 11) ;
g
P = Y= — A(vt — u)\i — cos a) = 2 — A(v^ — u) sin2 -;
«5 o
W &
useful work = Pu = 2 — Ati(v, — uY sin2 — ;
g 2
U(V. — U}" OL
efficiency = 4 — sin — .
IMP A CT. 20$
This is a maximum and equal to — sin2 — when vl = $u.
27 2
These results are identical with those for a concave cup
when a = 180°.
Instead of one vane let a series of vanes be successively
introduced at short intervals at the same point in the path of
the jet. Then
w
m = —Av^
and hence the pressure P, useful work, and efficiency respec-
tively become
—A
o
w A
— Av, .
S
and
8. Friction. — The effect of friction has been disregarded,
and nothing definite is known as to its action or law of distri-
bution. It has been suggested to assume that the loss of head
due to friction is a fraction of the head due to the velocity of
the jet relatively to the surface over which it spreads. Thus
in Art. 7
V*
the loss of head due to friction =/ —
V*
and the corresponding loss of energy = wQ*f — •
9. Resistance to the Motion of Solids in a Fluid Mass.
— The preceding results indicate that the pressure due to
2O6 HYDRA ULICS.
the impact of a jet upon a surface may be expressed in the
form
A being the sectional area of the jet, V the velocity of the jet
relatively to the surface, and K a coefficient depending on the
position and form of the surface.
Again, the normal pressure (N) on each side of a thin
plate, completely submerged in an indefinitely large mass of
still water, is the same. If the plate is made to move hori-
zontally with a velocity F, a forward momentum is developed
in the water immediately in front of the plate, while the plate
tends to leave behind the water at the back. A portion of the
water carried on by the plate escapes laterally at the edges
and is absorbed in the neighboring mass, while the region it
originally occupied is filled up with other particles of water.
Thus the normal pressure N, in front of the plate, is increased
by an amount n, while at the back eddies and vortices are pro-
duced, and the normal pressure N at the back is diminished
by an amount n' . The total resultant normal pressure, or the
normal resistance to motion, is n-\- n', and this increases with
the speed. In fact, as the speed increases, n' approximates
more and more closely to N, and in the limit the pressure
at the back would be nil, so that a vacuum might be main-
tained.
Confining the attention to a plate moving in a direction
normal to its surface, the resistance is of the same character as
if the plate is imagined to be at rest and the fluid moving
in the opposite direction with a velocity V. So, if both the
water and the plate are in motion, imagine that a velocity
equal and opposite to that of the water is impressed upon
every particle of the plate and of the water. The resistance is
then of the same character as that of a plate rrioving in still
water, the velocity of the plate being the velocity relatively
to the water. Thus, in general, the resistance to the motion
of such a plane moving in the direction of the normal to its
IMPACT. 207
surface, with a velocity V relatively to the water, may be ex-
pressed in the form
R - KwA - ,
A being the area of the plate, and K a coefficient depending
upon the form of the plate and also upon the relative sectional
areas of the plate and of the water in which it is submerged.
According to the experiments of Dubuat, Morin, Piobert,
Didion, Mariotte, and Thibault, the value of K may be taken
at 1.3 for a plate moving in still water, and at 1.8 for a current
moving on a fixed plate. Unwin points out the unlikelihood
of such a difference between the two values, and suggests that
it might possibly be due to errors of measurement.
Again, reasoning from analogy, the resistance to the motion
of a solid body in a mass of water, whether the body is wholly
or only partially immersed, has been expressed by the
formula
R = KwA—,
V being the relative velocity of the body and water, A the
greatest sectional area of the immersed portion of the body at
right angles to the direction -of motion, and K a coefficient de-
pending upon the form of the body, its position, the relative
sectional areas of the body and of the mass of water in which
it is immersed, and also upon the surface wave-motion.
The following values have been given for K\
K = i.i for a prism with plane ends and a length from 3 to 6
times the least transverse dimension ;
K = i.o for a prism, plane .in front, but tapering towards the
stern, the curvature of the surface changing gradu^
ally so that the stream-lines can flow past without
any production of eddy motion, etc.;
208 HYDRA ULICS.
K — .5 for a prism with tapering stern and a cut-water or
semi-circular prow ;
K = .33 for a prism with a tapering stern and a prow with a
plane front inclined at 30° to the horizon ;
K = .16 for a well-formed ship.
Froude's experiments, however, show that the resistance to
the motion of a ship, or of a body tapering in front and in
the rear, so that there is no abrupt change of curvature lead-
ing to the production of an eddy motion, is almost entirely
due to skin-friction (see Art. i, Chap. II).
IMPACT. 209
EXAMPLES,
1. A stream with a transverse section of 24 square inches delivers y
10 cubic feet of water per second against a flat vane in a normal direc- ^
tion. Find the pressure on the vane. Am. 1171! Ibs.
2. If the vane in question i moves in the same direction as the im- ./
pinging jet with a velocity of 24 ft. per second, find (a) the pressure on
the vane ; (b) the useful work done ; (c) the efficiency.
Am. (a) 4211 Ibs.; (ff) 10,125 ft.-lbs.; (c) .288.
3. What must be the speed of the vane in question 2, so that the J
efficiency of the arrangement may be a maximum ? Find the maximum ^
efficiency. Ans. 20 ft. per sec.; ^V %
4. Find (a) the pressure, (b) the useful work done, (c) the efficiency,
when, instead of the single vane in question 2, a series of vanes are intro-
duced at the same point in the path of the jet at short intervals.
Ans. (a) 703^ Ibs.; (b} 16,875 ft.-lbs.; (c) .48.
What must be the speed of the vane to give a maximum efficiency ?
What will be the maximum efficiency? Ans. 30 ft. per sec.; .5.
5. A stream of water delivers 7,500 gallons per minute at a velocity of
15 ft. per second and strikes an indefinite plane. Find the normal pres-
sure on the vane when the stream strikes the vane (a) normally; (d) at
an angle of 60° to the normal. Ans. (a) 585.9 Ibs.; 292.9 Ibs.
6. A railway truck, full of water, moving at the rate of 10 miles an
hour, is retarded by a jet flowing freely from an orifice 2 in. square in
the front, 2 ft. below the surface. Find the retarding force.
Ans. 7.97 Ibs.
7. A jet of water of 48 sq. in. sectional area delivers 100 gallons per Q%
second against an indefinite plane inclined at 30° to the direction of the- (
jet ; find the total pressure on the plane, neglecting friction. How will
the result be affected by friction ? Ans. 750 Ibs. '
8. If the plane in question 7 move at the rate of 24 ft. per second in
a direction inclined at 60° to the normal to the plane, find the useful
work done and the efficiency. Ans. 2250 ft.-lbs.; TV
At what angle should the jet strike the plane so that the efficiency
might be a maximum? Find the maximum efficiency.
Ans. sin1 £ ; -£..
9. A stream of 32 square inches sectional area delivers 32 cub. feet
of water per second. At short intervals a series of flat vanes are intro-
210 HYDRA ULICS.
duced at the same point in the path of the stream. At the instant of
impact the direction of the jet is at right angles to the vane, and the
vane itself moves in a direction inclined at 45° to the normal to the
vane. Find the speed of the vane which will make the efficiency a
maximum. Also find the maximum efficiency and the useful work
done. Ans. 15.08 ft. per sec.; /T; 2io6f|-f ft.-lbs.
10. In a railway truck, full of water, an opening 2 in. in diameter
is made in one of the ends of the truck, 9 ft. below the surface of the
water. Find the reaction (a) when the truck is standing; (b) when the
truck is moving at the rate of 10 ft. per second in the same direction as
the jet ; (c) when the truck is moving at the rate of 10 ft. per second in
a direction opposite to that of the jet. If this movement of the truck
is produced by the reaction of the jet, find the efficiency.
Ans. (a) 24.55 Ibs. per sq. in.; (b) 34.78 Ibs. per sq. in.; (c) 14.3
Ibs. per sq. in.; .588.
11. From a ship moving forward at 6 miles an hour a jet of water is
sent astern with a velocity relative to the ship of 30 feet per second from
a nozzle having an area of 16 square inches; find the propelling force
and the efficiency of the jet as a propeller without reference to the man-
ner in which the supply of water may be obtained.
Ans. i
12. A stream of 64 sq. in. section strikes with a 40- ft. velocity against
a fixed cone having an angle of convergence = 100° ; find the hydraulic
pressure. Ans. 492.1 Ibs.
13. A jet of 9 sq. in. sectional area, moving at the rate of 48 ft. per
second, impinges upon the convex surface of a paraboloid in the direc-
tion of the axis and drives it in the same direction at the rate of 16 ft.
per second. Find the force in the direction of motion, the useful work
done, and the efficiency. The base of the paraboloid is 2 ft. in diameter
and its length is 8 inches. Ans. 25 Ibs.; 400 ft.-lbs.; r£y.
14. A stream of water of 16 sq. in. sectional area delivers 12 cubic feet
of water per second against a vane in the form of a surface of revolu-
tion, and drives in the same direction, which is that of the axis of the
vane. The water is turned through an angle of 120° from its original
direction before it leaves the vane. Neglecting friction, find the
speed of vane which will give a maximum effect. Also find impulse
on vane, the work on vane, and the velocity with which the water
leaves the vane.
Ans. 36 ft. per sec.; 562^ Ibs.; 20,250 ft.-lbs.; 95.24 ft. per sec.
15. At 8 knots an hour the resistance of the Water-witch was 5500
Ibs.; the two orifices of her jet propeller were each 18 in. by 24 in.
Find (a) the velocity of efflux; (b) the delivery of the centrifugal pump;
IMPACT. 211
(V) the useful work done ; (d) the efficiency; (<?) the propelling H.P., as-
suming the efficiency of the pump and engine to be .4.
Ans. (a) 29.4 ft. per sec.; (b) 1 104.6 gallons per sec.; (c) 74,393 ft.-
Ibs.; (</).63; (e) 532.
1 6. If feathering-paddles are substituted for the jet propeller in
question 15, what would be the area of stream driven back for a slip of
25$ ? Find the efficiency and the water acted on in gallons per minute.
Ans. 34 sq.ft.; .75; 236,000.
17. A vane moves in the direction ABC with a velocity of 10 ft. per
second, and a jet of water impinges upon it at B in the direction BD
with a velocity of 20 ft. per second ; the angle between BC and BD is
30°. Determine the direction of the receiving-lip of the vane, so that
there may be no shock.
Ans. The angle between lip and BC = 23°47'.
18. A jet moves in a direction AltCwith a velocity Fand impinges
upon a vane which it drives in the direction BD with a velocity \ V.
The angle ABD is 165°. Determine the direction of the lip of the vane
at B, so that there may be no shock at entrance.
Ans. The angle between lip and direction of stream = i4°3'.
19. A jet issues through a thin-lipped orifice i sq. in. in sectional
area in the vertical side of a vessel under a pressure equivalent to a
head of 900 ft. and impinges on a curved vane, driving it in the direc-
tion of the axis of the jet. The water enters without shock and turns
through an angle of 60° before it leaves the vane. Find (a) the speed
of the vane which will give a maximum effect ; (If) the pressure on the
vane ; (c) the work done ; (d) the absolute velocity with which the water
leaves the vane ; (e) the reaction on the vessel, disregarding contraction.
Ans. (a) 80 ft. per sec. ; (d) 320.9 IDS.; (c) 46.68 H. P.; (d} 184 ft.
per sec.; (e) 781.25 Ibs.
20. A stream moving with a velocity v impinges without shock
upon a curved vane and drives it in a direction inclined at an angle to
the direction of the stream. The angle between the lip of the vane and
the direction of the stream is x, and V is the relative velocity of the
water with respect to the vane. If the speed of the vane is changed by
a small amount, say n per cent, show that the corresponding change in
the direction of the lip, in order that the water might still strike the
v
vane without shock, is n — sin x.
21. A jet of water under a head of 20 feet, issuing from a vertical
thin-lipped orifice i in. in diameter, impinges upon the centre of a vane
3 ft. from the orifice. Determine the position of the vane and the force
of the impact (a) when the vane is a plane surface ; (b) when the vane is
6 in. in diameter and in the form of a portion of a sphere of 6 in. radius.
2 1 2 HYDRA ULICS.
22. A stream of water of 36 sq. in. section moves in a direction ABC
and delivers 4 cub. ft. of water per second upon a vane moving in a
direction BD with a velocity of 8 ft. per second, the angle between BC
and BD being 30°. Find (a) the best form to give to the vane ; (b) the
velocity of the water as it leaves the vane ; (c) the mechanical effect of
the impinging jet ; and (d) the efficiency, the angle turned through by
the jet being 90°.
Ans. (a) The angle between lip and BC — 23°48'; (b) 2.946 ft,
per sec. ; (c) 966.098 ft. per sec.; (d) .966.
23. A stream of thickness / and moving with the velocity v im-
pinges without shock upon the concave surface of a cylindrical vane of
a length subtending an angle 20. at the centre. Determine the total
pressure upon the vane (a) if it is fixed ; (b) if it is moving in the same
direction as the stream with the velocity u. In case (b) also find (c) the
work done on the vane.
iu w yu
Ans. (a) 2— bin* sin a; (b) 2-bt(v — U)* sin a ; (c) 2—btu(y — u}* sin5 a.
O £> £
24. Two cubic feet of water are discharged per second under a press-
ure of loo Ibs. per sq. in. through a thin-lipped orifice in the vertical
side of a vessel, and strike against a vertical plate. Find the pressure
on the plate and the reaction on the vessel. Ans. 475.82 Ibs.
25. A stream moving with a velocity of 16 ft. per second in the direc-
tion ABC, strikes obliquely against a flat vane and drives it with a
velocity of 8 ft. per second in the direction BD, the angle CBD being 30°.
Find {a) the angle between ABC and the normal to the plane for which
the efficiency is a maximum ; (b) the maximum efficiency ; (c) the velocity
with which the water leaves the vane; (d} the useful work per cubic
foot of water.
Ans. (a) 21° 44'; (b) .25664; (c) 12.6 ft. per sec.; (d) 256.64 ft.-lbs.
CHAPTER VII.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
I. Hydraulic Motors are machines designed to utilize the
•energy possessed by a moving mass of water in virtue of its
position, pressure, and velocity.
The motors may be classified as follows :
(1) Bucket Engines. — In this now antiquated form of motor
weights are raised and resistances overcome by allowing water
to flow into suspended buckets, thus causing them to descend
vertically.
(2) Rams and Jet-pumps, in which the impulsive effect of
one mass of water is utilized to drive a second mass of water.
(3) Water-pressure Engines are especially adapted for high
pressures and low speeds, and necessarily have very heavy
moving parts. With low pressures the engine becomes un-
wieldy and costly.
Pressure-engines are either reciprocal or rotative. The
latter are very convenient with moderately high pressures and
-especially when they are to drive machinery which is to be used
intermittently. They also give an exact measurement of the
water used.
Direct-acting pressure-engines are of great advantage where
a slow and steady motion is required, as, for example, in work-
ing cranes, lifts, etc.
(4) Vertical Wat er-iv heels, in which the water acts almost
wholly by weight, or partly by weight and partly by impulse,
or wholly by impulse.
(5) Turbines, in which the water acts wholly by pressure
or wholly by impulse.
214
HYDRAULICS.
2. Hydraulic Rams. — By means of the hydraulic ram a
quantity of water falling through a vertical distance hl is made
to force a smaller weight of water to a higher level.
The water is brought from a reservoir through a supply-
pipe 5. At the end B of this pipe there is a check- or clack-
valve opening into an air-chamber A, which is connected with
a discharge-pipe D. At C there is a weighted check- or clack-
valve opening inwards, and the length of its stem (or the stroke)
is regulated by means of a nut or cottar at E. When the waste-
valve at C is open the water begins to escape with a velocity due
to the head hl and suddenly closes the valve. The momentum.
FIG. 133.
of the water in the pipe opens the valve at B, and a portion of
the water is discharged into the air-vessel. From this vessel it
passes into the discharge-pipe in consequence of the reaction
of the compressed air. At the end of a very short interval of
time the momentum of the water has been destroyed, the valve
at B closes, the waste-valve again opens, and the action com-
mences as before. It is found that the efficiency of the ram is
increased by introducing a small air-vessel at F, supplied with
a check- or clack-valve opening inwards at G. The wave-motion
started up in the supply-pipe by the opening and closing of the
valve at B has been utilized in driving a piston so as to pump
up water from some independent source.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 21$
Let v be the velocity of flow in the supply-pipe at the mo-
ment when the valve at C is closed.
" Wl be the weight of the mass of water in motion.
W v 2
Then - - is the energy of the mass, and this energy is
expended in opening the valve at B, forcing the water into the
air-chamber, compressing the air, and finally causing the eleva-
tion of a weight W^ of the water through a vertical distance k '.
Let hf be the head consumed in frictional and other hy-
draulic resistances.
Then
W,(h' + hf} = the actual work done = — ' -.
This equation shows that, however great h' may be, W^ has
a definite and positive value, and therefore water may be raised
to any required height by the hydraulic ram.
WJt'
The efficiency of the machine = 2 , and may be as much
11
as 66 per cent if the machine is well made.
3. Pressure-engines.— The energy required to drive a press-
ure-engine is usually supplied by means of steam-pumps, but
an accumulator is often interposed between the pumps and the
motor in order to store up the pressure energy of the water.
Indeed, it is perhaps to the introduction of the accumulator
that the success of hydraulic transmission is especially due.
Its cost, however, only allows of its use in cases where the
demand for energy is for short intervals of time.
In its simplest form the accumulator is merely a vertical
cylinder into which the water is pumped and from which it is
then discharged by the descent of a heavily loaded piston.
The water-pressure thus developed in ordinary hydraulic ma-
chinery is from 700 to 800 Ibs. per square inch, but in riveting
and other similar machinery pressures of 1500 Ibs. per square
inch and upwards are often employed.
Fig. 134 represents an accumulator designed by Tweddell
for these higher pressures.
216
HYDRAULICS.
The loaded cylinder A slides upon a fixed spindle B.
The water enters near the base, passes up the hollow spindle,
and fills the annular space surrounding the spindle. Thus
FIG. 134-
the whole of the weight is lifted by the pressure of the water
upon a shoulder C. The water section being small, any large
demand for water will cause the loaded cylinder to fall rapidly,
so that when it is brought to rest there will be a considerable
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2 1/
increase of pressure which is of advantage in punching, rivet-
ing, etc.
Let Wbe the weight of the loaded cylinder.
Let /'"be the friction of each of the two cup-leathers.
Let T-J be the radius of the cylinder, rt the radius of the
spindle.
Let h be the height of the column of water above the pipe D.
Let w be the specific weight of the water.
Then/j, the intensity of the pressure in D when the cylinder
is rising,
W+2F
= Wk -f- ( a __ 5rt
and /, , the intensity of the pressure in D when the cylinder is
falling,
W-2F
Hence an approximate measure of the variation of the
pressure is pl —p^ — —, — ^ r. , which ordinarily varies from
about ifo of the pressure for a i6-in. ram to 4$ for a 4-in. ram.
In a direct-acting pressure-engine let A be the sectional
area of the working cylinder (Fig, 135).
Let a be the sectional area of the supply-
pipe.
Let A = na.
Let IV be the weight of the water, piston, FJG- '35-
and other reciprocating parts in the working cylii.der.
Let / be the length of the supply pipe.
Let f be the acceleration of the piston. Then nf is the
acceleration of the water in the supply-pipe.
The force required to accelerate the piston
218 HYDRAULICS.
and the corresponding pressure in feet of water
W f
~~wAg'
The force required to accelerate the water in the supply
pipe
wal
• : = ^nf'
and the corresponding pressure in feet of water
A.
Similarly, if /' is the length of the discharge-pipe and —
its sectional area, the pressure-head due to the inertia of the
discharge-water
Hence the total pressure in feet of water required to over-
come inertia in the supply-pipe and cylinder
W
The quantity -— ;-)-#/ has been designated the length of
working cylinder equivalent to the inertia of the moving parts.
Let the engine drive a crank of radius r, and assume that the
velocity V of the crank-pin is approximately constant. Then
the acceleration of the piston when it is at a distance x from
its central position
F2
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 21 9
and the pressure due to inertia
wA^
Let v be the velocity of the piston in the working cylinder.
Let u be the velocity of the water in the supply-pipe.
Let h be the vertical distance between the accumulator-
ram and the motor.
Let/0 be the unit pressure at the accumulator-ram.
Let/ be the unit pressure in the working cylinder.
Then
/0 &a _ / V* ( losses due to friction, sudden changes
w 2g ~~ w 2g \ of section, etc.
Thus
A — t — v- -11 + losses.
W 2g
V — U
The term 1- losses may be approximately expressed
o
v1
in the form K— , AT being the coefficient of hydraulic resistance.
Hence
w 2g
the term h being disregarded as it is usually very small as
compared with — .
w
Thus the total pressure-head in feet required to overcome
inertia and the hydraulic resistances
and is represented by the ordinate between the parabola ced
220
HYDRA ULICS.
and the line ab in Fig. 136, in which afgb is a rectangle, ab
representing the stroke 2r,
ac = oa —
the pressure due to inertia at the end of the stroke, and
F2
the pressure required to overcome the hydraulic resistances at
the centre of the stroke.
9
FIG. 136.
The ordinate between the parabola fmg and the line fg
represents the back pressure, which is necessarily proportional
Fa
to the square of the piston-velocity, i.e., to —(r* — x*}. Hence
the effective pressure-head on the piston, transmitted to the
crank-pin, is represented by the ordinate between the curves
amg and ced. The diagram shows that the pressure at the
end of the stroke is very large and may become excessive. It
is therefore usual to introduce relief-valves or air-vessels to
prevent violent shocks. In certain cases, however, as, e.g., in
a riveting-machine, a heavy pressure at the end of the stroke,
just where it is most needed to close the rivet, is of great
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 221
advantage, and therefore the inertia effect is increased by the
use of a supply-pipe of small diameter and an accumulator
with a small water section (Fig. 134).
The effective pressure should be as great as possible, and
therefore the pressures due to inertia and frictional resistance,
and the back pressure, which are each proportional to v*, should
be as small as possible, and hence it is of importance to fix a
low value for the speed of the piston, which in practice rarely
exceeds 80 ft. per minute. The exhaust port should also be
made of large area, as the back pressure diminishes as the area
of the port increases.
By equation I,
(3)
This speed v can be regulated at will by the turning of a
cock, as in this manner the hydraulic resistances may be in-
definitely increased.
Let the engine be working steadily under a pressure Pt and
let v0 be the speed of steady motion. Then
and
_ j useful resistance overcome by the piston
( + friction between piston and accumulator-cylinder.
If P is diminished, the speed VQ will be slightly increased,
but in no case can it exceed,
4. Losses of Energy. — The losses may be enumerated as
follows :
(a) The Loss L^ due to Piston-friction. — It may be assumed
that piston-friction consumes from 10 to 20 per cent of the
total available work.
222 HYDRA ULICS.
(b) The Loss Z, due to Pipe-friction. — The loss of head in
the supply-pipe of diameter </,
The loss of head in the discharge-pipe of diameter d^
Hence the total loss of head in pipe-friction is
Ml (nJ
L'-4f-- —
The loss in the relatively short working cylinder is very
small and may be disregarded.
(c) The Loss La due to Inertia. — The work expended in
moving the water in the supply-pipe
wA v*
gn ~2~'
and in moving the water in the discharge-pipe
_ wA ,, i?_
~ 1 ~
The total work thus expended
/,// l'\v*
= wA(--\- — } — ,
\n ' ril 2g>
and it may be assumed that nearly the whole of this is wasted.
Hence the corresponding loss of head is
~"
_/ /'W _W_ll_ ^_\^__ X
n ' ri)~2 ~~ ^2r\n "• ~n'} ^~~ ~2%
A2r \n ' ri2g ~~ 2rn • n' g~~ 2g
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 22$
(d) The Loss L4 due to Curves and Elbows. — The losses due
curves and elbows may be expressed in the form
A =/4— (Chap. Ill, Art. 6).
(e) The loss L6 due to sudden Changes of Section. — The loss
of head in the passage of the water through the ports may
be expressed in the form/' .
The loss occasioned by valves may also be expressed by
/f/
.
Thus the total loss is
The coefficient/" may be given any desired value between
O and oo by turning a valve, so that any excess of pressure
may be destroyed and the speed regulated at will.
(/) The Loss Lt due to the Velocity with which the Water
leaves the Discharge-pipe.
A =
Hence
the effective head ==£-•- (L^ + A - A + A + L6 + £„),
and the efficiency = I - — (L, + A + A + L< + L>).
The volume of water used per stroke is a constant quan-
tity, and the efficiency, which may be as great as eighty per
cent when the engine is working under a full load, may fall
below forty per cent when the load is light.
5. Brakes. — Hydraulic resistances absorb energy which is
proportional to the square of the speed. This property has
224 H YDRA ULICS.
been taken advantage of in the design of hydraulic brakes
for arresting the motion of a rapidly moving mass, as a gun
or a train, of weight W. In Fig. 137 the fluid is allowed
to pass from one side of the piston to the other through
orifices in the piston.
Let m be the ratio of the area of the piston to the effective
area of the orifices.
Let v be the velocity of the -piston when moving under a
force P.
Let A be the sectional area of the cylinder.
FIG. 137.
Then
the work done per second = Pv
= the kinetic energy produced
and therefore
P= wA(m— i)2 — ,
and is the force required to overcome the hydraulic resistance
at the speed v.
Let V be the initial value of v, and P, the maximum value
of P. Then
Pl = wA(m — i)2—
*g
Let F be the friction of the slide. Then
—
o
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 22$
and Pl -}- F is the maximum retarding force. It would cer-
tainly be an advantage if the retarding force could be constant.
In order that this might be the case (m — i)v must be con-
stant, and therefore as v diminishes m should increase and con-
sequently the orifice area diminish. Various devices have been
adopted to produce this result.
Assuming the retarding force to be constant, let x be the
piston's distance from the end of the stroke when its velocity
is v. Then
and therefore ^2 is proportional to x.
But (m — i)v is constant.
Therefore (m — i) is inversely proportional to
6. Water-wheels. — Water-wheels are large vertical wheels
which are made to turn on a horizontal axis by water falling
from a higher to a lower level. These wheels may be divided
into three classes :
(a) Undershot Wheels, in which the water is received near
the bottom and acts by impulse.
(b) Breast Wheels, in which the water is received a little
below the axis of rotation and acts partly by impulse and partly
by its weight.
(c) Overshot Wheels, in which the water is delivered nearly
at the top and acts chiefly by its weight.
7. Undershot Wheels. — Wheels of this class, with plane
floats or buckets, are simple in construction, are easily kept in
repair, and were in much greater use formerly than they are
now. They are still found in remote districts where there is
an abundance of water-power, and are also employed to work
floating mills, for which purpose they are suspended in an open
current by means of piles or suitably moored barges. They
are made from 10 to 25 feet in diameter, and the floats, which
are from 24 to 28 in. deep, are fixed either normally to the
periphery of the wheel, or with a slight slope towards the
supply-sluice, the angle between the float and radius being
226 HYDRA ULICS.
from 1 5° to 30°. Generally from one half to one third of the
total depth of float is acted upon by the water.
Let Fig. 138 represent a wheel with plane floats working in
an open current.
FIG. 138.
Let vl be the velocity of the current.
Let u be the velocity of the wheel's periphery.
Let Q be the delivery of water in cubic feet per second.
The water impinges upon a float, is reduced to relative rest,
and is carried along with the velocity u. Thus
the impulse = (#, — u),
o
and
wQ
the useful work per second = - u(vl — u).
o
Hence
wQ .
—u(y. — u) , x
^ /*= • £ 2u(v. — U)
the efficiency = — — ^-—^ - = v * a - '-,
which is a maximum and equal to — when u = — v..
^ l
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 22/
Theoretically, therefore, the wheel works to the best advan-
tage when the velocity of its periphery is one half of the cur-
rent velocity. Even then its maximum theoretic effect is only
50$, and in practice this is greatly reduced by frictional and
other losses, so that the useful effect rarely exceeds 30$.
Undershot wheels with plane floats are cumbrous, have little
efficiency, and should not be used for falls of more than 5 feet.
Again, let A be the water-area of a float, and w be the
specific weight of the water.
wQ is somewhat less than wAv^ , as there will be an escape
of water on both sides of the float.
Let wQ = kwAvlt k being some coefficient (< i) to be
oletermined'by experiment. Then
^
the useful work per second = kAw— l— (yl — u),
o
kA
and its maximum value = - v.w.
According to Bossut's and Poncelet's experiments a mean
A *y
value of k is — , and the best effect is obtained when u = -vl ,
the corresponding useful work being — — - - - and the effi-
48
ciency — ,
125
8. Wheels in Straight Race.— Generally the water is let
on to the wheel through a channel made for the purpose, and
closely fitting the wheel, so as to prevent the water escaping
without doing work. For this reason also, the space between
the ends of the floats in their lowest positions and the channel
is made as small as is practicable and should not exceed 2 in.
Hence /&, and therefore also the efficiency, will be increased.
Assume the channel to be of a uniform rectangular section and
to have a bed of so slight a slope that it may be regarded as
horizontal without sensible error.
228
HYDRA ULICS.
The wheel is usually from 24 to 48 ft. in diameter, with 24
to 48 floats, either radial or inclined. The floats are 12 to 20
inches deep, or about 2\ to 3 times the depth of the approach-
ing stream. The fall should not exceed 4 ft. Let the floats
be radial, Fig. 139.
FIG. 139.
Let hl be the depth of the water on the up-stream side of
the wheel.
Let //, be the depth of the water on the down-stream side
of the wheel.
Let £, be the width of the race.
The impulse = impulse due to change of velocity
-|- impulse due to change of pressure
g 2
and the useful work per second
= impulse X u = ^u(v, - u) + ^ - *•)«,
g 2 Vtf, -Ul
The second term is negative, since h^ > /i, , and tne maxi-
mum theoretic efficiency may be easily shown to be <.5.
Three losses have been disregarded, viz. :
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 229
(i) The loss of Ql cubic feet of the deeper fluid elements
which do not impinge upon some of the foremost floats.
According to Gerstner,
o --=c-2(
cQ( *' V
*,'U - u) '
.72, being the number of the floats immersed, and c being -J or
v according as the bottom of the race is straight or falls
.abruptly at the lowest point of the wheel.
(2) The loss of <22 cubic feet of water which escape between
the wheel and the race-bottom.
Approximately, the play at the bottom may be said to vary
from a minimum, sl = BC, when a float AB is in its lowest
position, Fig. 140, to a maximum, BlCl = CD=£^Ct, when
FIG. 140.
two floats AlBl , A^Bs are equidistant from the lowest position,
Fig. 140. Thus the mean clearance
= J(25, + BD) = 5, +-, nearly,
•rl being the wheel's radius.
230 HYDRA ULICS.
But - - = distance between two consecutive floats
ft
= 2 . B^D, very nearly,
n being the total number of floats. Hence
a
and therefore the mean clearance = Sl -\ --- — *.
Again, the difference of head on the up-stream and down
stream sides
and the velocity of discharge, vd, through the clearance is
given by the equation
Hence
Introducing .7 as a coefficient of hydraulic resistance,
^ . / I TrVA
a =.7,+--^
If the depth of the stream is the same on both sides of the
wheel, i.e., if h, = &t, then
(3) The loss of 03 cubic feet of water which escape between
the wheel and the race-sides.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
Let Ja be the clearance on each side. Then
.7 being a coefficient of hydraulic resistance.
Finally, if f^lbs. is the weight on the wheel-journals, the
loss due to journal friction
/* being the journal coefficient of friction, and p the journal
radius.
Thus the actual delivery of the wheel in foot-pounds
These wheels are most defective in principle, as they utilize
only about one third of the total available energy. They may
be made to work to somewhat better advantage by introducing
the following modifications:
(a) The supply may be so regulated by means of a sluice-
board, that the mean thickness of the impinging stream is about
6 or 8 inches. If the thickness is too small, the relative loss of
water along the channel will be very great. If the thickness is
too great, the floats, as they emerge, will have to raise a heavy
weight of water. The sluice-board is inclined at an angle of
30° to 40° to the vertical, so that the sluice-opening may be as
near the wheel as possible, thus diminishing the loss of head
due to channel friction, and is rounded at the bottom to pre-
vent a contraction of the issuing fluid. Neglecting frictional
losses, etc.,
f i re /->/rr . v? u*\ ( loss of energy
the useful effect = wQ[H-\--^ -- — J , _ f7
\ 2£" 2gl ( due to shock
g
232 HYDRA ULICS.
H being the difference of level between the point at which the
water enters the wheel and the surface of the water in the tail-
race, i.e., the fall. H is usually very small and may be negative.
If the vanes are inclined, the resistance to emergence is not
so great, and the frictional bed resistance between the sluice
and float is practically reduced to nil. With a straight bed and
small slope (i in 10) the minimum convenient diameter of
wheel is about 14 ft.
(b) The bed of the channel for a distance at least equal to
the interval between two consecutive vanes may be curved to the
form of a circular arc concentric with the wheel, with the view
of preventing the escape of the water until it has exerted its
full effect upon the wheel. When the bed is curved, the mini-
mum convenient diameter of wheel is about 10 ft.
An undershot wheel with a curb is in reality a low breast-
wheel, and its theory is the same as that described in Arts. 13
and 14.
(c) The down-stream channel may be deepened so that the
velocity of the water as it flows away becomes > vr The im-
pulse due to pressure is then positive, which increases the useful
work and therefore also the efficiency.
(d) The down-stream channel may be widened and a slight
counter-inclination given to the bed. What is known as a
standing-wave is then produced, in virtue of which there is a
sudden rise of surface-level on the down-stream side above that
on the up-stream side. This allows of the wheel being lowered
by an amount equal to the difference of level between the sur-
faces of the standing-wave and of the water-layer as it leaves
the wheel, thus giving a corresponding gain of head.
(e) The introduction of a sudden fall has been advocated
in order to free the wheel from back-water, but it must be
borne in mind that all such falls diminish the available head.
Thus undershot wheels with plane floats have little effect
because of loss of energy by shock at entrance and the loss of
energy carried away by the water on leaving the floats. These
losses have been considerably modified in Poncelet's wheel,
which is often the best motor to adopt when the fall does
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 233
not exceed 6 ft., and which, in its design, is governed by two
principles which should govern every perfect water-motor, viz. :
(1) That the loss of energy by shock at entrance should be
a minimum.
(2) That the velocity of the water as it leaves the wheel
should be a minimum.
The vanes are curved and are comprised between two
crowns, at a slightly greater distance apart than the vane-
width ; the inner ends of the vanes are radial, and the water
acts in nearly the same manner as in an impulse turbine.
First. Assume that the outer end of a vane is tangential
to the wheel's periphery, that the impinging layer is infinitely
thin, and that it strikes a float tangentially.
Let #/(Fig. 141) be a float, and aq the tangent at a.
The velocity of the water relatively
to the float = vl — u.
The water, in virtue of this velocity?
ascends on the bucket to a height
(" - V"
pq — , then falls back and FlG I4I
<§
leaves the float with the relative velocity V1 — u and with an
absolute velocity vl — 2u. This absolute velocity is nil when
the speed of the wheel is such that u = %i\, and the theoreti-
i v 3
cal height of a float is/0 = -. The total available head is
42£-
thus changed into useful work, and the efficiency is unity, or
perfect.
Taking R as the mean radius of the crown and ul as the
corresponding linear velocity, the mean centrifugal force on
•each unit of fluid mass is -~ and acts very nearly at the direc-
tion of gravity, so that the height pq of a float may be
approximately expressed in the form
'R
234
HYDRA ULICS.
V being the velocity with which the water commences to rise
on the float.
Practically, however, the float is not tangential to the pe-
riphery at a, as the water could not then enter the wheel. Also
the impinging water is of sensible thickness, strikes the periph-
ery at some appreciable angle, and in rising and falling on the
floats loses energy in shocks, eddies, etc.
Let the water impinge in the direction ac, Fig. 142, and
take ac = v^
Take ad in the direction of and equal to «, the velocity of
the wheel's periphery.
Complete the parallelogram bd.
Then cd = ab = V is the velocity of the water relatively to
the float.
That there may be no shock at entrance, ab must be a tan-
gent to the vane at a.
FIG. 142.
Again, the water leaves the vane in the direction of ba pro-
duced, and with a relative velocity ae — ab = V.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 23 $
Complete the parallelogram de. Then ag(=. v^ is the
absolute velocity of the water leaving the wheel.
Evidently cdg is a straight line.
Let the angle cad = y, and the angle bad = n — a.
From the triangle adc,
V* = v* -f u* — 2v ji cos Y I • • • • (i)
v? = V* -j~ u* — 2 Vu cos a ; .... (2)
V sin Y
v, sin OL **/
From the triangle adg,
By equations I, 2, and 4,
^,8 ^ rr rra /
— — = — 2 Vu cos # = vl — V — u = 2u(vl cos Y — u\
2
Therefore the useful work per second
= ^2U fa cos y - u) (s>
wQ v? cos8 Y
This is a maximum and equal to when
V. COS Y rr •
u -, and the maximum emciency is cos y, Hence^
too, by equations I and 3,
tan (n — a) = 2 tan y (6)
Also,
VR sin
, by equation 6.
u sin (a -\- y} cos (n — a]
The efficiency is perfect if y is nil, and therefore a = 1 80°.
Practically this is an impossible value, but the preceding cal-
culations indicate that ; should not be too large (usually
< 30°), and that the speed of the wheel should be a little less
than one half of the velocity of the inflowing stream.
236
HYDRA ULICS.
Take y = 15° as a mean value. Then
u = vt X .484, and the efficiency = .993.
Actually the efficiency does not exceed 68 per cent. In-
deed it must be borne in mind that the theory applies to one
elementary layer only, say the mean layer, and that all the
other layers enter the wheel at angles differing from 15°, thus
giving rise to " losses of energy in shock." The losses of
energy in frictional resistance, eddy motion, etc., in the vane
passages, have also been disregarded. The layers of water,
flowing to the wheel under an adjustable sluice and with a
velocity very nearly equal to that due to the total head, may
be all made to enter at angles approximately equal to 15°, and
the corresponding losses in shock reduced to a minimum by
forming the course as follows :
The first part of the course FG, Fig. 143, is curved in such
a manner that the normal pqr at any point/ makes an angle
of 15° with the radius^. The water moves sensibly parallel
to the bottom FG, and therefore in a direction at right angles
FIG. 143.
to/r. Hence at q the direction of motion makes an angle of
15° with the tangent to the wheel's periphery. If or is drawn
perpendicular to/r, then or = oq sin 15° = a constant.
Thus the normal pqr touches at r a circle concentric with
the wheel and of a certain constant diameter.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2$?
The initial point F of the curve FG is the point in which
the tangent to this circle, passing through the upper edge of
the sluice-opening, cuts the bed of the supply-channel.
If t is the thickness (or depth of sluice-opening) and b the
breadth of the layer of water as it leaves the sluice, then
Q = btv, ,
and according to Grashof
H being the available fall.
The thickness should not exceed 12 to 15 inches, and is
generally from 8 to 10 inches.
Neglecting float thickness, the capacity of the portion of
the wheel passing in front of the entering stream per second
= bdu^ , very nearly.
Only a portion of this space can be occupied by the water,
so that
Q — mbdul ,
m being a fraction whose value may be taken to be J or f „
Hence
mbdul — btv^ ,
and therefore
u. md u.
t = md— = — cos y —
Vl 2 r U
md R
= — cos v— .
2 rr,
According to Morin,
r, = 2d to $d.
The mean velocity at entrance = cv< 2g(H — £/), an aver-
age value of cv being .9.
Thus \it = ,
HYDRAULICS.
The diameter of the wheel is often taken to be
The area of the sluice-opening is usually from \\bt to i.^bt.
The inside width of the wheel is about (b + J) ft.
The water should not rise over the top of the buckets, and
in order to prevent this the depth of the shrouding is from J//
to \H.
If A is the angle subtended at the centre O of the wheel by
the water-arc between the point of entrance A and the lowest
point €, Fig. 144, of the wheel, and if Aq' is drawn horizontally,
then Aq' is approximately the height of the float, and the
theoretic depth d of the crown is given by
' + OC - Oq'
= AC = Aqf +Cq' =
In practice it is usual to increase this depth by /, the thick-
ness of the impinging water-layer.
Again,
2 V"1
d — s -f r,(i — cos A) -f- a few inches, approximately.
The buckets are usually placed about I ft. apart, measured
along the circumference, but the number of the buckets is not
a matter of great importance. There are generally 36 buckets
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 239
in wheels of 10 to 14 ft. diameter, and 48 buckets in wheels of
20 to 23 ft. diameter.
It may be assumed that the water-arc is equally divided by
the lowest point C of the wheel, so that
the length of the water-arc = 2\r = 2uT,
T being the time of the ascent or descent of the water in the
bucket.
In the middle position, the upper end of the bucket should
be vertical, and if the float is in the form of a circular arc, its
radius r' = d sec (it — a\ a being the angle between the
bucket's lip and the wheel's periphery.
The time of ascent or descent is also given by
where sin fy = I/cos (it — a).
9. Efficiency corresponding to a Minimum Velocity of
Discharge (V2). — From Fig. 142,
ao (= \ag) _ sin y __ £Qa)
ad sin aod u
Hence for any given values of u and y, vz is a minimum
when sin aod is greatest, that is, when aod = 90°, or ag is at
right angles to de. Then also ad = ae = ab, or u = V, and ac
bisects the angle bad. Thus,
i71 = 2u cos y and v^ — 2u sin y.
The useful work
W v? — v? W WV/cos 2y
= — . -' - '- = —2u* cos 2y = -- 5- - £,
g 2 g g 2 COS' Y
The total available work
240 H YDRA ULICS.
Therefore the efficiency
cos 2v
-
Ex. — If y = 15°, the efficiency = .928 and u = .
In practice the best value of u is found to lie between.
and .60^.
The horse-power of the wheel
rf being the efficiency with an average value of 60$.
Although, under normal conditions of working, the effi-
ciency of a Poncelet wheel is a little less than that of the best
turbines, the advantage is with the former when working with
a reduced supply.
10. Form of Bucket — The form of the bucket is arbitrary,
and may be assumed to be a circular arc. In practice there
are various methods of tracing its form.
METHOD I (Fig. 145), The tangent am to the bucket at a
FIG. 145.
makes a given angle a with the tangent at a to the wheel's
outer periphery. The radius of\s also a tangent to the bucket
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 24!
at/! If the angle aof\s known the position of f on the inner
periphery is at once fixed, and the form of the bucket can be
easily traced.
Let the angle aof—x. Join af and let the tangents to the
bucket at a and /meet in m. Then
the angle oam = a — 90°.
" oma — 1 80° — oam — aom = 270° — a — x.
" mfa = the angle maf — £(180° — fmd)
= "+-*- 45- '
Let rlt r^ be the radii of the outer and inner peripheries of
the wheel. Then
sin (f!L±£ _ 45°)
rl oa sin of a sin mfa \ 2 /
of sin oaf sin oaf
sin (^£-45*)
since the angle oaf ' =• oam — maf '= - 45°.
Hence
r. —
X
tan -
2
tan -
an equation giving ;tr.
The point o' in which the perpendicular o'f to 0/" meets
the perpendicular o'a to am is the centre of the circular arc
required and o'f(^o'd) is the radius.
METHOD II (Fig. 146). Take mad = 150°, and in ma pro-
duced take ak = of. With k as centre and a radius equal to
242
HYDRAULICS.
ao describe the arc of a circle intersecting the inner periphery
in the point f. Join kf, of, and af. The two triangles aof
and akf are evidently equal in every respect, and therefore
the angle kaf is equal to the angle of a. Drawing ao' at right
angles to ak and fo' tangential to the periphery at f, the angle
0'af(= kaf — 90°) is equal to the angle o'f a (= of a — 90°), and
therefore o'a = o'f. Thus o' is the centre of the circular arc
required and o'a (= o'f) is the radius.
FIG. 146.
9-
METHOD III (Fig. 147). Let the bed with a slope of, say,
i in 10 extend to the point C, and then be made concentric
with the wheel for a distance CC subtending an angle of 30°
at the centre of the wheel. Let the mean layer, half way
between the sloping bed and the surface of the advancing
water, strike the outer periphery at the point /. Draw fk
making an angle of 23° with of, and take fk equal to one half
or seven tenths of the available fall, k is the centre of the
circular arc required and £/is its radius.
II. Breast-wheels. — These wheels are usually adopted for
falls of from 5 to 15 feet, and for a delivery of from 5 to 80
cubic feet per second.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 243
The diameter should be at least 1 1 ft. 6 in., and rarely ex-
ceeds 24 ft. The velocity (u) of the wheel's periphery is gen-
erally from 3^ ft. to 5 ft. per second, the most useful average
velocity being about 4^ ft. per second.
The width of the wheel should not exceed from 8 to 10 ft.
It is of great importance to retain the water in the wheel
as long as possible, and this is effected by surrounding the
water-arc with an apron, or a curb, or a breast, which may be
constructed of timber, iron, or stone. Hence, too, the buckets
may be plane floats, but they should be set at an angle to the
periphery of the wheel, so as to rise out of the water with the
least resistance (Art. 8).
The depth of a float should not be less than 2.3 ft., and the
space between two consecutive floats should be filled to at
least one half, and even to two thirds, of its capacity. The
head (measured from still water) over the sill or lip should be
about 9 in.
The play between the outer edge of the floats and the
curb varies from £ in. in the best constructed wheels to
2 inches.
The distances between the floats is from i^ to if times the
head over the sill.
244
HYDRA ULICS.
Breast-wheels are among the best of hydraulic motors,
giving a practical efficiency which may be as large as 80
per cent.
12. Sluices. — The water is rarely admitted to the wheel
without some sluice arrangement, which may take the form of
an overfall sluice (Fig. 148),
an underflow sluice (Fig. 149),
or a bucket or pipe sluice
(Fig. 150).
The pipe sluice is espe-
cially adapted for a varying
supply, being provided, for a
certain vertical distance, with
a series of short tubes, so in-
clined as to ensure that the
water enters the wheel in the
right direction. Taking .85
as the mean coefficient of
hydraulic resistance for these
tubes, the head kl required
to produce the velocity of
entrance z> is
and if H is the total available
fall,
= remainder of fall available for pressure-work.
The profile AB in an overfall and an underflow sluice,
should coincide with the parabolic path of the lowest stream-
lines of the jet. The crest of the overfall should be properly
curved, and the inner edges of the underflow opening should
be carefully rounded so as to eliminate losses due to con-
traction
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 24$
The underflow sluice-opening should also be normal to
the axis of the jet.
Let h^ be the head above the crest of an overfall sluice.
Then
2 T. •' *
Q = -cb,
b^ being the width of the crest and c the coefficient of dis-
charge. The width bl is usually 3 or 4 inches less than the
width b of the wheel.
From this equation
and the depth of water over the crest or lip is usually about
9 inches.
Again, the head h^= CD) required to produce the velocity
vl at the point of entrance B is
10
10 per cent being allowed for loss due to friction.
Thus the height of the crest A above B, the point of
entrance,
= AD = CD - CA = h, -
ii *;/ 36 V
10 2g \2cb^2g)'
But BA is a parabola with its vertex at A, and therefore,
if B is the angle between the horizontal BD and the tangent
the parabola at B,
n • « f\ A
V, sm u 1 1 v*
2g ~ 10 2g
y
)
246 HYDRA ULICS.
Also
v. sin 26
The head available for pressure work
= DE = FG = H - h,.
Let a be the angle between BT and the tangent to the
wheel's periphery at B. Then
a _f 0 = the angle EOF,
BO being the radius to the centre of the wheel and OFG'
vertical.
% If the lowest point G' of the wheel just clears the tail-
race, the head available for pressure work
= H - h, = FG' — OG' - OF
= rfr _ cos BOF) = 2r, si
r, being the radius to the outer periphery of the wheel.
If, again, the water enters the wheel tangentially,
a = o, and the angle BOF = B,
so that
H - h, = 2r, sin2 -.
If the sluice-opening is not at the vertex of the parabola,
the axis of the opening should be tangential to the parabola.
13. Speed of Wheel. — The water leaves the buckets and
flows away in the race with a velocity not sensibly different
from the velocity u of the wheel's periphery.
Let b be the breadth of the wheel (Fig. 151).
Let x be the depth of the water in the lowest bucket.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 247
FIG. 151.
Allowing for the thickness of the buckets, the play between
the wheel and curb, etc.,
Q = cbxu,
c being an empirical coefficient whose average value is about
.0. Hence
10 Q
u = jr.
9 ox
In practice b is often taken to be — to — . It is impor-
tant that b should be as small as possible and hence x should
be as large as possible, its value being usually ij ft. to 2 ft.
It must be borne in mind, however, that any increase i-n
the value of x will cause an increase in the weight of water
lifted by the buckets as they emerge from the race, and will
therefore tend to diminish the efficiency.
14. Mechanical Effect.— Theoretically, the total mechan-
ical effect
248
HYDRA ULTCS.
H being the fall from the surface of still water in the supply-
channel to the surface of the water in the tail-race.
This, however, is reduced by the following losses:
(a) Owing to frictional resistance, etc., there is a loss of
v 3
head in the supply-channel which may be measured by ^-7-
v being approximately JL to TL.
The head required to produce the velocity at entrance, vl9
(b) Let af, Fig. 152, represent in direction and magnitude
v, the velocity of the water entering the bucket.
FIG. 152.
Let ad, in the direction of the tangent to the wheel's
periphery, represent the velocity u of the periphery in direction
and magnitude.
Complete the parallelogram bd. Then ab evidently repre-
sents the velocity V of the water relatively to the wheel.
This velocity V is rapidly destroyed, the corresponding loss of
head being
F2 U*-\-V? — 2UV^ COS y
being the angle daf.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 249
Assuming that the water enters the race with the velocity
u of the wheel, the theoretical useful work per pound per
second due to impact
u.
= -(vl cos y — u).
g
V^
If the loss — is to be a minimum for a given speed of
o
wheel,
v,dv^ — u cos y . dvl = o, or ^ — u cos y. . . (2)
Hence, by equation I, V = u sin 7, and therefore
V df
tan y = - = 2
v, af
so that for a velocity of entrance vt = u cos y the angle afd
should be 90°. But this value is inadmissible, as the water
would arrive tangentially and consequently would not enter the
buckets. In Order that the loss in shock at entrance may be as
small as possible, ab, the direction of the relative velocity F,
should be parallel to the arm xy of the bucket, and should
therefore be approximately normal to the wheel's periphery.
This is equivalent to the assumption that the water arrives in
a given direction (y) with a given velocity (^), and that the
speed (?/) of the wheel is to be such as will make V a mini-
mum. Thus, by equation I,
o — udu — v^ cos y . du, or u = vl cos y,
and therefore
V = vl sin y.
Hence tan y = — — -£, and therefore the angle adf = 90°.
u ad
250
HYDRAULICS.
In practice y is generally 30°, and the corresponding loss of
Fa v? . v> i if i
head = — = — sin2 y = — •-. - = — . -
At point of entrance x falls below y, the water flows up the
inclined plane xy, and F, instead of being wholly destroyed in
eddy motion, is partially destroyed by gravity. This velocity,
destroyed by gravity, is again restored to the water on its
return, and thus adds to the efficiency
of the wheel.
It will be found advantageous to
use curved or polygonal buckets and
not plane floats. A bucket, for ex-
ample, may consist of three straight
portions, ab, be, cd, Fig. 153. Of these
the inner portion cd shoud be radial ;
the outer portion ab is nearly normal to the periphery of the
wheel, and the central portion be may make angles of about
135° with ab and cd.
Disregarding all other losses, the theoretical delivery of the
wheel in foot-pounds
where h^ = total fall — fall (h^ required to produce the veloc-
ity v,.
If 77 be the efficiency, then, according to the results of
Morin's experiments,
rf = .40 to .45 if h^ = -//";
4
rf = .42 to .49 if hl = —H\
rj = .47 if h, = -H;
3
if h, = ff.
4
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2$ I
(c) There is a loss of head due to frictional resistance along
the channel in which the wheel works.
Let / = length of the channel (or curb).
Let t — thickness of water-layer leaving the wheel.
Let b = breadth of wheel.
The mean velocity of flow in this curb channel is approxi-
mately -u, and the loss of head due to channel friction
bt 2g
where/ = coefficiency of friction, b -f- 2t = wetted perimeter,
bt = water area, and y being 30°.
(d] There is a loss of head due to the escape of water over
the ends and sides of the buckets.
Let s1 be the play between the ends of the buckets and the
channel.
Let s^ be the play at the sides. (^, = Ja , approximately.)
Let zl , #2 , . . . zn be the depths of water in a bucket corre-
sponding to n successive positions in its descent
from the receiving to the lowest points.
Let /a , /a, ... ln be the corresponding water-arcs measured
along the wheel's periphery.
The orifice of discharge at end of a bucket = bs^
The mean amount of water escaping from a bucket over
its end
c being the coefficient of discharge.
The water escapes at the sides as over a series of weirs,
and the mean amount of water escaping from a bucket over
the sides
252 HYDRAULICS.
Hence the total loss of effect from escape of water
per sec., ^ being the vertical distance between the point of
entrance and the surface of the water in the tail-race
__.
(e) There is a loss of head due to journal friction.
Let W = weight of wheel.
Let wl = weight of water on the wheel.
Let rl = radius of wheel's outer periphery.
Let r1 — radius of axle.
Loss per second of mechanical effect due to journal friction
r being the coefficient of journal friction.
There is a loss of mechanical effect due to the resistance of
the air to the motion of the floats (buckets), but this is prac-
tically very small, and may be disregarded without sensible
error. A deepening of the tail-race produces a further loss of
effect, and should only be adopted when back-water is feared.
Hence the total actual mechanical effect,
putting
Z=bSl( V^
cos
,s =
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 253
=wQ ff- (i + v) + fa cos r - «)
--*(", cos y-u)
Hence, for a given value of z/,, the mechanical effect (omit-
ting the last term) is a maximum when
« = ^ C°S Y (= -433 X ^ , if r = 30°).
In practice the speed of the wheel is made about one half
of the velocity with which the water enters the wheel.
For a given speed of wheel, and disregarding the loss of
effect due to curb friction, which is always small, the mechani-
cal effect is a maximum for a value of z/, given by
I ^ t/—w'Z\l + v i WQ
— \wQ — c V2g — 1 — ! — vl H -u cos Y = o,
or
U COS Y
The loss by escape of water, viz., c V2g—, varies, on an
average, from 10 to 15 per cent of the whole supply, so that
c V2g- varies from — to 2s,
d n 10 20
254 JfYDRA ULICS.
15. Sagebien Wheels have plane floats inclined to the
radius at from 40° to 45° in the direction of the wheel's rota-
tion. The floats are near together and sink slowly into the
fluid mass. The level of the water in the float-passages grad-
FIG. 154.
ually varies and the volume discharged in a given time may
be very greatly changed. The efficiency of these wheels is
over 80 per cent, and has reached even 90 per cent. The
action is almost the same as if the water were transferred from
upper to lower race, without agitation, frictional resistance,
etc., flowing away without obstruction, into the tail-race.
16. Overshot Wheels. — These wheels are among the best
of hydraulic motors for falls of 8 to 70 ft. and for a delivery of
3 to 25 cub. ft. per second. They are especially useful for falls
of 12 to 20 ft. The efficiency of overshot wheels of the best
construction is from .70 to .85.
If the level of the head-water is liable to a greater variation
than 2 ft., it is most advantageous to employ a pitch-back or
high breast-wheel, which receives the water on the same side
as the channel of approach.
17. Wheel-velocity. — This evidently depends upon the
work to be done, and upon the velocity with which the water
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 255
arrives on the wheel. Overshot wheels should have a low
circumferential speed, varying from 10 ft. per sec. for large
wheels to 3 ft. per sec. for small wheels, and should not be less
than 2-J ft. per sec.
In order that the water may enter the buckets easily, its
velocity should be greater than the peripheral velocity of the
wheel.
18. Effect Of Centrifugal Force. — Consider a molecule
of weight W in the " unknown" surface of the water in a
FIG. 155.
bucket (Fig. 155). At each moment there is a dynamical
equilibrium between the " forces" acting on m, viz.: (i) its
256 HYDRA ULICS.
IV
weight w\ (2) the centrifugal force — coV; (3) the resultant T
o
of the neighboring reactions.
2V
Take MF = w, MG = — coV, and complete parallelogram
o
FG. Then MH = T. The direction of T is, of course, normal
to the surface of the water in the bucket.
Let HM produced meet the vertical through the axis O of
the wheel in E. Then
w_ a
MG z**r FH OM r
MF~ w ~MF~OE"OE'
and therefore
OB =*, =
GO
taking g = 32 ft. and n being the number of revolutions per
minute.
Thus the position of E is independent of r and of the
position of the bucket, so that all the normals to the water-
surface in a bucket meet in E, and the surface is the arc of a
circle having its centre at E, or, rather, a cylindrical surface
with axis through E parallel to the axis of rotation.
19. Weight of Water on Wheel and Arc of Discharge.—
Let Q = volume supplied per sec., and N = number of buckets.
Noo
Then - - = number of buckets fed per sec.,
27T
and — — = volume of water received by each bucket per sec.
Hence the area occupied by the water until spilling com-
mences = , ., , b being the bucket's width (= width of wheel
between the shroudings).
The water flows on to the wheel through a channel (Fig.
156), usually of the same width b as the wheel, and the
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
supply is regulated by means of an adjustable sluice, which
may be either vertical, inclined, or horizontal.
When the water springs clear from the sluice, as in Fig. 156,
the axis of the sluice should be tangential to the axis of the
FIG. 156.
jet, and the inner edges of the sluice-opening should be rounded
so as to eliminate contraction.
Let y, z be the horizontal and vertical distances between
the sluice and the point of entrance.
Let T be the time of flow between the sluice and entrance.
Let v0 , 2\ be the velocities of flow on leaving the sluice and
on entering the bucket.
Then
258 H YDRA ULICS.
and
V? = V* + 2gZ,
d being angular deviation of point of entrance from summit,
and y the angle between the direction of motion of the water
and the wheel at the point of entrance.
Assume the bed of the channel to be horizontal, and the
sluice vertical and of the same, width b as the wheel. The
sluice is also supposed to open upwards from the bed. Then
x being the depth of sluice-opening and h^ the effective head
over the sluice. This effective head is about TVths of the actual
head.
Thus, taking g=. 32, •— = %xh$ gives the delivery per foot
width of wheel.
Taking .6 ft. and 3.6 ft. as the extreme limits between
which hl should lie, and .2 ft. and .33 ft. as the extreme limits
between which x should lie, then ~ must lie between the
o
limits 1.24 and 5, and an average value of ^ is 3. Thus the
width of the wheel should be on the average ^ — .
Again, neglecting the thickness of the buckets, the capacity
of the portion of the wheel passing in front of the water-sup-
ply per second
= b<*> \ — — — - - !• = Mfafr, -- J = bdrja, approximately,
, , Lj
= bdu. — bd
30
r, being the radius and ul the velocity of the outer circumfer-
ence of the wheel, d the depth of the shrouding, and n the
number of revolutions per minute.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 259
Only a portion, however, of the space can be occupied by
the water, so that the capacity of a bucket is mubd, m being
a fraction less than unity and usually -J or J. For very high
wheels m may be \. Hence
, , 27tQ
mbdu. = ~=.
NGO
Again, since the thickness of the buckets is disregarded,
Nu —
Therefore mdu. = ^.
b
The delivery \^j per foot of width must not exceed a
certain limit, otherwise either d or u will be too great. In the
former case the water would acquire too great a velocity on
entering the buckets, which would lead to an excessive loss in
eddy motion and a corresponding loss of efficiency ; while if
the speed u of the wheel is too great the efficiency is again
diminished and might fall even below 40$.
The depth of a bucket or of the shrouding varies from 10
to 1 6 in., being usually from 10 to 12 in., and the buckets are
spread along the outer circumference at intervals of 12 to
14 inches. The number of the buckets is approximately $r or
6r, r being the radius of the wheel in feet.
The efficiency of the wheel necessarily increases with the
number of the buckets, but the number is limited by certain
considerations, viz. : (a) the bucket thickness must not take up
too much of the wheel's periphery ; (b) the number of the
buckets must not be so great as to obstruct the free entrance
of the water; (c) the form of the bucket essentially affects the
number.
Let the bucket, Fig. 157, consist of two portions, an inner
portion be, which is radial, and an outer portion cd\ c being a
point on what is called the division circle. The length be is
usually one half or two thirds of the depth d of the shrouding.
260
HYDRA ULICS.
Take be = \d.
It may also be assumed without much error that the water-
surface ad is approximately perpendicular to the line edt so
that the angle edc is approximately a right angle.
The spilling evidently commences when the cylindrical sur-
face, having its axis at e and cutting off from the bucket a
water-area equal to -~, passes through the outer edge d of
Noo
the bucket.
FIG. 157.
Let /3 be the bucket angle cOd.
Let 0 be the inclination of Od to the horizon.
Let 0 be the inclination of ad to the horizon.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 26 1
Let rl be the radius of the outer periphery.
Let R be the radius of the division circle.
Let ra be the radius of the inner periphery.
Then
od^ __ rl _ sin 0 _ sin 0
oe ""^"sin j 90°— 0+0} ~ cos (0+0)'
and therefore
Again,
Therefore
sin 0
af = fd tan (0 -|- 0), approximately.
the area dfa=<— tan (0 + 0) = — tan (8 + 0),
2 2
where d = rl — r2. Hence
the area abed = area cod — area bof — area ^/iz
Equations (i) and (2) give 0 and 0, and therefore the posi-
tion of the bucket when spilling commences.
The bucket will be completely emptied when it has reached
a position in which cd is perpendicular to a line from e to
middle point of cd, or, approximately, when edc is a right
angle.
Let 0,, 0, be the corresponding values of 0 and 0, and let
262 HYDRA ULICS.
yt be the angle between cd and the tangent at d to the wheel's
periphery. Then
and
= 90° -
sn r, ._. g
sin r
two equations giving 0, and 0^
Also, if ^ is drawn perpendicular to od,
de r — R cos
tan y = cot <:# <? = — =
ce R sin fi
The vertical distance between the points where spilling be-
gins and ends, viz., rl (sin 0l — sin 0) can now be determined.
The pitch-angle(= rp) is the angle between two consecutive
buckets so that ^ = . In order to obtain a small angle
(=: y^ between the lip of the bucket and the wheel's periphery,
it is usual to make the bucket angle ft greater than if}.
For example,
5 5 360° 450°
The interval between the buckets should be at least suf-
ficient to prevent any bucket dipping into the one below at the
moment the latter begins to spill.
Let coo'. Fig. 158, be the division angle and t the thickness
of the bucket.
Then
approximately, and therefore
(3)
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 263
Also, by equation 2,
, 2nQ
„ r> „. 9 Jt
^Sb*.^^
Z, £ £
.. (4)
W
These two last equations give N and 0.
The number of buckets may also be approximately found
from the formula
In practice the bucket may be delineated as follows :
Let ddr = distance between two buckets.
56 d
Take dd" = ~ dd' to - dd'\ also take fo = -, and join dc.
This gives the form of a suitable wooden bucket.
FIG. 158.
If the bucket is of iron, a circular arc is substituted for the
portions be, cd.
Again, let/w, Fig. 159, be the thickness of the stream just
before entering the bucket.
Let dn be the thickness of the stream just after entering
the bucket.
Let \ be the angle between the bucket's lip and the wheel's
periphery.
264 HYDRA ULICS.
Then
mbdul — capacity of bucket = bv^ . pm = bV. dn
= bv^dp sin y = b V. dp . sin A,
and therefore
~ v.smr" FsinA'
Now overshot wheels cannot be ventilated, and it is conse-
FIG. 159.
quently necessary to leave ample space above the entering
stream for the free exit of air. Thus, neglecting float thick-
ness,
' = distance between consecutive floats
= <W'(Fig. 158) > <// >
and N, the number of buckets,
2 Try, F sin \
mdu,
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 26$
For efficient action the number of the buckets is much less
than the limit given by this relation, often not exceeding one
half of such limit.
If y is very small, V=vl — u^ approximately, and therefore
The capacity of a bucket depends upon its form ; and the
bucket must be so designed that the water can enter freely
and without shock, is retained to the lowest possible point, and
is finally discharged without let or hindrance. Hence flat
buckets, Fig. 160, are not so efficient as the curved iron bucket
in Fig. 163 and as the compound bucket made of three or two
FIG. 1 60.
FIG. 161.
FIG. 162.
FIG. 163.
FIG. 164.
pieces in Figs. 161, 162, 164. The resistance to entrance is
least in the curved bucket, as there are no abrupt changes of
direction due to angles. The capacity of a compound bucket
may be increased, without diminishing the ease of entrance, by
making the inner portion strike the inner periphery at an
266
HYDRA ULICS.
acute angle, Fig. 164. The objection to this construction,
especially if the relative velocity V is large, is that the water
tends to return in the opposite direction and escape from the
bucket.
Let bed, efg, Fig. 165, represent two consecutive buckets of
an overshot wheel turning in the direction shown by the arrow.
FIG. 165.
Water will cease to enter the bucket-space between
efg, and impact will therefore cease, when the upper parabolic
boundary of the supply-stream intersects the edge b. The last
fluid elements will then strike the water already in the bucket
at a point M, whose vertical distance below b may be desig-
nated by z. The velocity v' with which the entering particles
reach M is given by the equation
(0
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 267
Again, while the fluid particles move from b to M let the
buckets move into the positions b'c'd' , e'f'g'.
Let arc bb' = s1 = eer.
Let arc bM = st.
Let T be the time of movement from b to b' (or b to M\
Then
s. • = uT
and
assuming that the mean velocity from b to M is an arithmetic
mean between the initial and final velocity of entrance. Thus
l -f- ^i
Also, since the angle between bM and the wheel's periphery
is small, it may be assumed that
the arc bM ' = be -\- ef-\- ee'y approximately,
27tr,
N N u
,
+**'
/,T r^7 ,Vi— U 27Cri Vi — U \
(Note.—ef— eb — = eb- - = -^T. - - , nearly.)
\ J u u N u J i
Thus
and by equations 2 and 3,
(vi + v*' — 2U\ _ 27tri !!L
S\ 2u I ~ N u>
268
HYDRA ULICS.
an equation giving approximately the distance sl passed
through by a float during impact. The buckets can now be
plotted in the positions they occupy at the end of the impact.
The amount of water in each bucket being also known, the
water-surface can be delineated, and hence the vertical distance
x can be at once found.
20. Useful Effect — (a) Effect of Weight. — The wheel
should hang freely, or just clear the tail-water surface, and
the total fall is measured from the surface of the water in the
tail-race to the water-surface just in front of the sluices through
which the water is brought on to the wheel.
FIG. 1 66.
Let hlt Fig. 166, be the vertical distance between the cen-
tres of gravity of the water-areas of the first and last buckets
before spilling commences. Then
//, = R cos d -\- rl sin 0, very nearly.
Let h^ be the vertical distance between the centres of
gravity of the water-area of the bucket which first begins to
spill, and the point at which the spilling is completed. Then
h^ — r,(sin 0, — sin 0), very nearly.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 269
The useful work per sec. = '^Q(hl + kh^ k being a frac-
tion < I and approximately = .5.
Let A0 be the water-area in the bucket which first begins
to spill.
Between this bucket and the one which is first emptied,
i.e., in the vertical distance /z2 , insert an even number s of
buckets, and let their water-areas Al , A9, A3 , . . . As be care-
fully calculated.
Let Qm be the mean amount of water per bucket in the
discharging arc.
Let Am be the mean water-area per bucket in the discharg-
ing arc.
Then
The value of k can now be easily found, since
Qm_Am
~-~"
Let q be the varying amount of water in a bucket frorrr
which spilling is taking place, and at any moment let y be the
vertical distance between the outer edge of the bucket and the
surface of the water in the tail-race.
q is a function of y and depends upon the contour of the
water in the bucket.
Let Y be the mean value of y between the points where
spilling begins and ends, i.e., for values^, and j/a of y. Then
y\
since
Jy .dq=yq — Jq . dy.
2/O HYDRA ULICS.
Again, the elementary quantity of water, dq, having an
initial velocity equal to that of the wheel, viz., &, falls a dis-
tance y and acquires a velocity = <J u' -\- 2gy.
Thus it flows away in the tail-race causing a loss of
w .dq ( if
energy = -"(* + 2^7) = w •
Hence the total loss of energy between the points where
spilling begins and ends
Overshot and pitch-back wheels do not work well in back-
water, as they lift a greater or less weight of water in rising
above the surface.
If the water-level in the race is liable to variation it is better
to diminish the diameter of the wheel and design it so that it
may never be immersed to a greater depth than 12 inches.
(b) Effect of Impact. — The head h' required to produce the
velocity v with which the water reaches the wheel is theoret-
v*
ically — — ; but as there is a loss of at least 5 per cent in the
o
most perfect delivery, it is usual to take h' = v-^, an average
o
value of v being I.I.
Let the water enter the bucket in the direction ac, Fig.
167. Take ac = vr The water now moves round with a
velocity u (assumed the same as that of the division circle),
and leaves the wheel with the same velocity. Take ab in the
direction of the tangent to the division circle at the point of
entrance = u. The component be represents the relative
velocity V of the water with respect to the bucket, and this
velocity is wholly destroyed, ab must necessarily be parallel to
the outer arm of the bucket, so that there may be no loss of
shock at entrance. Then the impulsive effect
g
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2? I
But
V* = v? -\- if — 2v \u cos y,
y being the angle through which the water is deviated from
its original direction at the point of entrance.
FIG. 167.
Hence the impulsive effect
wQ
= u(v^ cos y — u),
o
and the TOTAL USEFUL EFFECT
i+^2)+ ^M7'i cos K— &)—'loss due to journal friction,
o
The loss due to journal friction
p being the radius of the axle and Wthe weight of the wheel.
2/2 HYDRA ULICS.
21. A pitch-back or high breast wheel is to be preferred
to an overshot wheel when the surface-levels of the head- and
tail-water are liable to very considerable variation.
In the pitch-back wheel the water is admitted by an adjust-
able sluice into the buckets on the same side as the supply-
channel, Fig. 168. Thus the wheel revolves in the direction
FIG. 168.
in which the water leaves, and the drowning of the wheel is
prevented. Further, the buckets may be now ventilated, Fig.
169, and may therefore be placed closer together than in the
unventilated overshot wheel.
The efficiency of the pitch-back is at least equal to that of
the overshot.
22. The Jet Reaction Wheel (Scotch Turbine). — In this
form of motor the water enters the centre of the wheel, spreads
out radially in tubular passages, and issues from openings at
the ends tangentially to the direction of rotation.
Fig. 170 represent the simplest wheel of this class. In Eng-
land it is known as Barker's mill, and in Germany it is called
Segner's water-wheel.
A reaction wheel may have several tubular passages, as in
Fig. 172, and the vertical chamber XY may be cylindrical,
rectangular, or conical.
Let r be the horizontal distance between the axis of aa
orifice and the axis of the vertical chamber.
Let h be the head of water over the orifices when closed.
Let v be the velocity of erflux relatively to the tube when
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2?$
the orifices are open, and let Fbe the corresponding linear
velocity of rotation at the centre of an orifice. Then
cv being the coefficient of velocity.
FIG. 1 70.
FIG. 171.
2 74 HYD RA ULICS.
The absolute velocity of efflux '= v — V.
v-V
The angular momentum of each pound of water = r.
o
The useful work of each pound of water
v-V V V.
= r— = —(v — V\
g r g
The total work of each pound of water = h.
FIG. 172.
The efficiency
useful work _
Total work
- V}
= ^ suppose>
useful work _ v — V
The reaction = linear veiocity of rotation = g
For a maximum efficiency
= o =
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2?$
Hence
«f — 2Z/F+ *.•?" = o,
and therefore
v = V(i + Vi - c,') ....... (4)
Experience indicates that the greatest efficiency corresponds
to a speed of rotation equal to the velocity due to a head h,
i.e., to a value of V given by
. - , ..... (5)
By equations (i) and (5)
«f = 4V** ....... (6)
and therefore, by equations (4), (5), and (6),
o
c* = ~» or c, = .94 ....... (7)
Hence, by equations (3), (5), (6), and (7),
the maximum efficiency = — .
o
Thus one third of the head is lost, and of this amount the
(v— F)2/ h\
portion --- — — ^= -j is carried away by the effluent water.
The portion - -- (= -kj is lost in frictional resistance, etc.
Again,
= —jt | cjj* + -yT— terms cont'g higher powers of -~\ — i | .
276 HYDRA ULICS.
The efficiency therefore increases with F, but the value of
V is limited by the practical consideration that, even at
moderately high speeds, so much of
the head is absorbed by friction as
to sensibly diminish the efficiency.
The serious practical defects of
this wheel are that its speed is most
unstable and that it admits of no
efficient system of regulation for a
varying supply of water.
The Scotch or Whitelaw's tur-
J73. bine, Fig. 173, excepting in the
curved arms, does not differ essentially from the reaction
wheel just considered.
23. Reaction and Impulse Turbines.— All turbines be-
long to one of two classes, viz., Reaction Turbines and Impulse
Turbines, and are designed to utilize more or less of the avail-
able energy of a moving mass of water.
In a reaction turbine a portion of the available energy is
converted into kinetic energy at the inlet surface of the wheel.
The water enters the wheel-passages formed by suitably
curved vanes, and acts upon these vanes by pressure, causing
the wheel to rotate. The proportions of the turbine are such
that there is a particular pressure (hence the term pressure-
turbine) at the inlet surface corresponding to the best normal
condition of working. Any variation from this pressure,
caused, e.g., by the partial closure of the passages through
which the water passes to the wheel, changes the working con-
ditions and diminishes the efficiency. In order to avoid such
a variation of pressure, it is essential that there should be a
continuity of flow in every part of the turbine ; the wheel-
passages should be kept completely filled with water, and
therefore must receive the water simultaneously; Such
turbines are said to have complete admission. The admission
is partial when the water is received over a portion of the inlet
surface only.
In an impulse (Girard) turbine, Figs. 174, 175, the energy
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
of the water is wholly converted into kinetic energy at the
inlet surface. Thus the water enters the wheel with a velocity
due to the total available head and therefore without pressure,
is received upon the curved vanes, and imparts to the wheel
the whole of its energy by means of the impulse due to the
FIG. 174. FIG. 175.
Girard Turbine for Low Falls. Girard Turbine for High Falls.
gradual change of momentum. Care must be taken to ensure
that the water may be freely deviated on the curved vanes,
and hence such turbines are sometimes called turbines with free
deviation. For this reason the water-passages should never be
completely filled, and the water should flow through under a
pressure which remains constant. In order to ensure an un-
broken flow through the wheel-passages and that no eddies
are formed at the backs of the vanes, ventilating holes are
arranged in the wheel sides, Fig. 177. Figs. 176 and 177 also
show the relative path AB and the absolute path CD traversed
by the water in an inward-flow and a downward-flow turbine.
If there is a sufficient head, the wheel may be placed clear
278
HYDRAULICS.
above the tail-water, when the stream will be at all times under
atmospheric pressure. With low falls the wheel may be placed
in a casing supplied with air from
an air-pump by which the surface
of the water may be kept at an
invariable level below the outlet
orifices, which is essential for per-
fectly free deviation. While the
wheel-passages of a reaction tur-
bine should be kept completely
'filled with water, no such restric-
tion is necessary with an impulse
turbine. The supply may be par-
tially checked and the water may be received by one or
more vanes without affecting the efficiency. ' Thus the dimen-
sions of an impulse turbine may vary between very wide
TAIL WATER
FIG. 177.
limits, so that for high falls with a small supply, a compara-
tively large wheel with low speed may be employed. The
speed of a reaction turbine under similar conditions would be
disadvantageously great, and any considerable increase of the
diameter would largely increase the fluid friction and would
also render the proper proportioning of the vane-angles
almost impracticable. Impulse turbines may have complete
or partial admission, while in reaction turbines the admission
should be always complete, as in Fig. 178, which shows the
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
relative path AB and absolute path CD traversed by the water.
When there is an ample supply of water the reaction turbine
is usually to be preferred, but on very high falls its speed
FIG. 178.
becomes inconveniently great and it is then better to adopt a
turbine of the impulse type. The diameter of the wheel can
then be increased and the speed proportionately diminished.
The Hurdy-gurdy is the name popularly given to an
impulse wheel which was introduced into the mining districts
of California about the year 1865. Around the periphery of
the wheel is arranged a series of flat iron buckets, about
4 to 6 in. in width, which are struck normally by a jet of
water often not more than three eighths of an inch in
diameter. Theoretically, the efficiency of such an arrange-
ment cannot exceed 50 per cent (Art. 7), while in prac-
tice it rarely reaches 40 per cent. The best speed of the
wheel, in accordance with both theory and practice, is one
half of that of the jet. Although the efficiency is so
low, the wheel found great favor for many reasons. Any
required speed could be obtained by a suitable choice of
diameter ; the plane of the wheel could be placed in any
convenient position ; the wheel could be cheaply constructed
and was largely free from liability to accident. Hence it was
of the utmost importance to increase, if possible, the efficiency
of a wheel possessing such advantages. Obviously a first step
was to substitute cups for the flat buckets, the immediate
result necessarily being a very large increase in the efficiency.
This was increased still further by the adoption of double
2 80 H YDRA ULICS.
buckets, Fig. 179, that is, curved buckets divided in the middle
so that the water is equally deflected on both sides.
Thus developed, the wheel is widely and most favorably
known as the Pelton wheel, Fig. 179. Its efficiency is at least
80 per cent, and it is claimed that it often rises above 90 per
cent. The power of the wheel does not depend upon its
diameter, but upon the available quantity and head of water.
The water passes to the wheel through one or more nozzles,
FIG. 179.
having tips bored to suit any required delivery. These tips
are screwed into the nozzles and can be easily and rapidly
replaced by others of larger or smaller size, so that the Pelton
is especially well adapted for a varying supply of water. It is
claimed that in this manner the power may be varied from a
maximum down to 25 per cent of the same without appreci-
able loss of efficiency.
The character of the construction of turbines has led to
their being classified as (i) Radial-flow turbines; (2) Axial-
flow turbines ; (3) Mixed-flow turbines.
In Radial-flow turbines the water flows through the wheel
in a direction at right angles to the axis of rotation and
approximately radial. The two special types of this class are
the Outward-flow turbine, invented by Fourneyron, and the
Inward-flow or Vortex turbine, invented by James Thomson.
In the former, Figs. 180 and 181, the water enters a cylindrical
chamber and is led by means of fixed guide-blades outwards
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 28 1
from the axis. It is distributed over the inlet-surface, passes
through the curved passages of an annular wheel closely sur-
FIG. 1 80.
FIG. 181.
rounding the chamber, and is finally discharged at the outer
surface. The wheel works best when it is placed clear above
282 HYDRA ULICS.
the tail-water. A serious practical defect is the difficulty of
constructing a suitable sluice for regulating the supply over
the inlet-surface. Fourneyron was led to the design of this
turbine by observing the excessive loss of energy in the ordi-
nary Scotch turbine, or reaction wheel, and introduced guide-
blades in order to give the water an initial forward velocity
and thus cause a diminution of the velocity of the water leav-
ing the outlet-surface.
In the Inward-flow or Vortex turbine, Figs. 182 and 183,
the wheel is enclosed in an annular space, into which the
water flows through one or more pipes, and is usually dis-
tributed over the inlet-surface of the wheel by means of four
guide-blades. The water enters the wheel, flows towards the
space around the axis, and is there discharged. This turbine
possesses the great advantage that there is ample space outside
the wheel for a perfect system of regulating-sluices.
Axial- flow turbines, Figs. 184, are also known as Parallel
and Downward-flow turbines and are sometimes called by the
names of the inventors, Jonval and Fontaine. In these the
water passes downward through an annular casing in a direction
parallel to the axis of rotation, and is distributed by means of
guide-blades over the inlet-surface of an adjacent wheel. It
enters the wheel-passages and is finally discharged vertically, or
nearly so, at the outlet-surface. The sluice regulations are
worse even than in the case of an outward-flow turbine, but
there is this advantage, that the turbine may be placed either
below the tail-water, or, if supplied with a suction-pipe, at any
point not exceeding 30 ft. above the tail-water.
If a turbine is designed so that the pressure at the clear-
ance between the casing and the wheel is nil, and with curved
passages in the form of a freely deviated stream, it becomes
what is called a Limit turbine. In its normal condition of
working it is an Impulse turbine, but when drowned, it is a
Reaction turbine, with a small pressure at the clearance. For
moderate falls with a varying supply its average efficiency is
higher than that of a pressure turbine.
The Mixed- or Combined-flow (Schiele) turbine is a combi-
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 283
nation of the radial and axial types. The water enters in a
nearly radial direction and leaves in a direction approximately
FIG. 182.
X —
///////////////^^^^^
1
--T
FIG. 183.
parallel to the axis of rotation. This type of turbine admits
of a good mode of regulation and is cheap to construct.
24. Theory of Turbines (Figs. 185 to 188).— Denote in-
284
HYDRA ULICS.
ward-flow, outward-flow, and axial-flow turbines by I. F., O. F.,
and A. F., respectively.
FIG. 184.
Let r,, ra be the radii of the wheel inlet and outlet surfaces
or an I. F. or O. F.
Let rlt rt be the outer and inner radii of the wheel inlet-
surface of an A. F.
Let R be the mean radius \== r* "^ r*J of an A. F., assumed
constant throughout.
FIG 185. — Section of an inward-flow turbine.
Let Alf A, be the areas of the wheel inlet and outlet
orifices.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 285
FIG. 186. — Enlarged portion of the section through XY, Fig. 185.
FIG. 187. — Enlarged portion of a section through XY, Fig. 180, of an outward-
flow turbine.
FIG. 188. — Enlarged portion of a cylindrical section XY, Fig. 184, of a down-
ward-flow turbine developed in the plane of the paper.
286 HYDRAULICS.
Let dlt dt be the depths of the same in an I. F. or O. F.
Let dlt d^ be the widths of the same in an A. F.
Let h be the thickness of the wheel in an A. F.
Let Hl be the effective head over the inlet-surface of the
wheel. This is the total head over the inlet-
surface diminished by the head consumed in
frictional resistance in the supply-channel, and
by the head lost in bends, sudden changes of
section, etc.
Let HI be the fall from the outlet-surface to the surface of
the water in the tail-race. If the turbine is
submerged, then H9 is negative.
Let vlt vt be the absolute velocities of the water at the
inlet- and outlet-surfaces.
Let ult #, be the absolute velocities of the inlet- and outlet-
surfaces.
Let V^ Vi be the velocities of the water relatively to the
wheel, at the inlet- and outlet-surfaces.
Let GO be the angular velocity of the wheel.
Let the water enter the wheel in the direction act making
an angle y with the tangent ad. Take ac to represent vl and
ad to represent ult Complete the parallelogram bd. The side
ab represents Vlt and in order that there may be no shock at
entrance, ab must be tangential to the vane at a. Again, at/
drawy^-, a tangent to the vane, and//£, a tangent to the wheel's
periphery.
Take fg and fk to represent V^ and u^ respectively. Com-
plete the parallelogram gk. The diagonal /$ must represent
in direction and magnitude the absolute velocity v^ with which
the water leaves the wheel. Let the angle hfk = d.
The tangential component of the velocity of the water as
it enters or leaves the wheel is termed the velocity of whirl,
and the radial component the velocity of flow. Denote these
components respectively by
vj, vr' at the inlet-surface ;
v»' i vr" at the outlet-surface.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 28/
Let the angle bad = 1 80° — a,
Let the angle gfk = 180° — ft.
Draw cm perpendicular to ad, and hn to fk.
Then at the inlet-surface,
vj=. v^ cos y — ac cos y = #;# = ad^dm = #, — P, cos a ; (i)
?>/ = z/, sin ^ — £;# = Vl sin or ; (2)
and at the outlet-surface
vj' = z>a cos 6 =fn =.fk ±kn = u9— V^ cos /? ; . (3)
vr" = ^2 sin 6 = /*« — F2 sin /? (4)
Let g be the volume of water passed per second. Then •
in an I. F. or O. F.
Vr'Ai = VrZTtridi •= Q
(5)
in an A. F.
i = Q
(5)
In equations (5) the thickness of the vanes has been disre-
garded. If 0 is the angle between the vane, of thickness BC,
A/ and the wheel's periphery AB, then the space
^f^j occupied by the vane along the wheel's periph-
/ / ery is AB = BC cosec 0.
/ Let n be the number of the guide-vanes and /
FlG- I89- their thickness.
Let #, be the number of the wheel-vanes and /, , /2 their
thickness at the inlet- and outlet-surfaces, respect-
ively.
Then, in a radial-flow turbine,
Al — -fadl\2nrl — nt cosec y — nvtl cosec a\ . . (6)
and
^. = TWi2^.- *i** cosec ft\> ...... (7)
T9¥ being a fraction depending on practical considerations.
288
HYDRAULICS.
In an axial-flow turbine R is to be substituted for rl ind ry
in the values of Al and A9.
nl may be made equal to n -f- I or n -f- 2.
Again, as the water flows through the wheel its angular
momentum relatively to the axis of rotation is changed from
— rjsj at the inlet- to rj)J' at the outlet-surface.
o o
Hence, if T is the effective work done by the water on the
turbine, and GO the angular velocity of the turbine,
in an I. F. or O. F.
in an A. F.
T - ^(vv'n - vw"rt)<o
g
wQ
g
s
g
a/')«i» • • (8>
since
since
Ui «2
r*= — = °°> - - (9)
'1 * 2
and the hydraulic efficiency
T vw'ui - z>«"wa
and the hydraulic
r/ f
\Viv
efficiency
-«^")«i /I0x
wQH, gff, ' (
wQ(H, + A) g(i
yi + A) '
Equation 10 is the fundamental equation upon which the
whole design of turbines depends.
From the triangle abc,
V* = v* + u* — 2vlul cos y, . . . . (ii)
and
sn y
sin of
(12)
From the triangle ./M,
»,1 = «,1+F,1-2«,r,COS/» (I3)
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 289
Again, if — , -- are the pressure-heads at the inlet- and outet-
w w
surfaces of the wheel of a REACTION TURBINE,
A -A
— =ff. -
w
(14)
In an IMPULSE TURBINE the water is under atmospheric
pressure only, and therefore
05)
To make allowance for hydraulic resistances k^— may be
o
v*
substituted for — in equations 14 and 15, a mean value of kl
o
^ ' I0
being — .
' 9
Applying Bernoulli's theorem to the filament from a to/,
% * _ u »
and taking account of the head - - due to centrifugal
force —
In a reaction I. F. or O. F.
Wt 2g W
and therefore
VJ- V? _/i
In words, the change of en-
ergy from atof = work due
to pressure -|- work due to
centrifugal force.
In an impulse I. F. or O. F.
V^~ r'* = ***~"1*. (18)
In a reaction A. F.
2g
and therefore
In words, the change of en-
ergy from a to f = work due
to pressure -f- work due to
gravity. The work due to A
centrifugal force is evidently
nil.
In an impulse A. F.
^ ~ V^ = h- - • (I8>
290
HYDRA ULICS.
To make allowance for hydraulic resistances £, F,2 may be
substituted for V9 in equations 17 and 18, a mean value of £a
being i.i.
For a maximum effect the water should leave the wheel
without velocity, i.e., vt should be nil. But this value of v^ is
impracticable, as no water could then pass through the wheel.
It is usual either to make the velocity of whirl (vm") at the
outlet-surface equal to nil, or to make the relative (F2) and
circumferential (u9) velocities at the outlet-surface, equal and
opposite. In each case v9 is small. First let
»-" = <>, d9)
so that the water leaves the wheel with a much-reduced ve-
locity in a direction normal to the out-
let-surface. Thus (Fig. 194), &\*)** fy
£ = 90°; *.=?*/',
and
Aj(j ^ = Z>2 COt /3 = V9 COS ft. (2O)
\v2-v'rV2
' /
Also, by equations 2, 4, 5, and 20 —
FIG. 189.
In an I. F. or O. F.
~= vi sinyridi = V* sin/J>v/2
211
• (21)
In an A. F.
= v\ sin ydi = V* sin fid*
= «2 tan fidi. (21)
The following results are now easily obtained :
In an I. F. or O. F. :
Relation between the Vane-
angles.
By equations 9 and 21, and
from the triangle acd,
r\di sin y «3 r* u\
tan
sin a.
In an A. F. :
Relation between the Vane-
angles.
By equations 9 and 21, and
from the triangle acd,
d\ sin y «2
</2 tan @ ~ v
sin (a -\- y}
sin a
(22)
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2QI
and therefore
^-^ cot ft = cot a 4- cot y. (23)
TVfla
and therefore
— cot ft = cot a -f- cot y. (23)
0a
In an I. F. or O. F.:
Speed of Turbine.
By equations I, 10, and 19,
IN REACTION TURBINES.
In an A. F.:
Speed of Turbine.
By equations I, 10, and 19,
WQ(HI — ^}- effective work
\ ?
wO wQ
~ g S
and therefore
cosy, (24)
Z/2 tt\i>l , x
Hl — = cos y. . (25)
Hence, by equations 20, 22,
and 25,
COt /J
tan ft 4- 2— cot^
Note. — If the water is to
have no velocity of whirl (vj)
relatively to the wheel at the
inlet-surface, then
«i - vw' = o, . . . (27)
and therefore
a = 90°
and
Vi Vr
tan y — — = — ,.
Also, the efficiency
and thus
WQ\H1 + h =-J= effective work
wQ , wQ . .
=—v-wUi = — —UM cos y, (24)
£• S
and therefore
^i + h - r1 = ^~ cos r-
Hence, by equations 20, 22,
and 25,
4- A) cot
-.. (26)
tan ft -\- i-j- cot
. — If the water is to
have no velocity of whirl (vwf)
relatively to the wheel at the
inlet-surface, then
Ul - vw' = o, . . (27)
and therefore
a = 90°
and
Also, the efficiency
an thus
(28) uS = g(Hi 4-
. (28)
292
HYDRA ULICS.
if the efficiency is perfect.
Usually the efficiency of
good turbines is about .85.
Velocity of Efflux.
Z'a2 = z/a5 tan2 ft
2,07/1 tan ft
(20)
if the efficiency is perfect.
Usually the efficiency of
good turbines is about .85.
Velocity of Efflux.
z/22 = «22 tan2 ft
2g(ffi 4- A) tan ft
tan ft -j- 2— cot y
Useful Work
2-^- cot y
-wQfft li . (3o)
tan /?-(- 2-^- cot^
2— cot ^
^>/ TT \ r\ ** / V
tan /3-}- 2-f- cot y
Efficiency
2^ co.r
</l , x
= wQ(Hi + /^) ^ . (30)
tan ft -J- 2 — cot ^
«i
Efficiency
2|cotr
— d ' ' '31)
tan /> -J- 2~r cot X
Amount Q of water passing
through turbine
tan y#-f- 2-^ cot v
«i
Amount Q of water passing
through turbine
1 zgVi tan ft .
, /ig(Hi -\- h} tan /5
- 27rr2</2 A / ~ ^ • (33)
y tan ^ -(- 2-^ cot y
7^^ pressure-head at the in-
Jff^urfnr.f
27/ft?i . / . G3>
y tan ft-{-2—co\.y
The pressure-head at the in-
let-surface
2g
,„.) raV,«
•H»l< I~ „ OV 9
tan^
2g
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 293
When the turbine is work-
ing freely in space above the
surface of the tail-water, there
will be no inflow of air if p^ >
A» f-e-> if
I > — o— ,
tan ft
'* sin2;r(tan/?-]-2-2-cot;K)
d\
If the turbine is drowned
with a head h' of water over
the outlet-surface, there will
be no back-flow of water if
that is, if
7i ^ o
tan ft
When tne turbine is work-
ing freely in space above the
surface of the tail-water, there
will be no inflow of air if pl >
At Le-' if
ffi <tf_ _ tan ft _
ffi-4-A aV2 . , ,, . d*
sin2 ^(tan p-\- 2— cot y)
d\
If the turbine is drowned
with a head h' of water over
the outlet-surface there will
be no back-flow of water if
$l -^ ^2 i j,'
— > -- h h ,
w w
that is, if
tan ft
IN IMPULSE TURBINES.
In an I. F. or O. F.:
Speed of Turbine.
Since
V? = 2gff1 , . . (35)
by equation 22,
riVi8 sin2 y _ uj _ rj u^
and therefore
Velocity of Efflux.
= «2 tan p
~nW
tan ft -f- 2-^ cot y
d\
In an A. F. :
Speed of Turbine.
Since
, - • (35)
by equation 22,
dS sin2 y _ uf_ _ uf_
d<? tan2 ft ~ z/i2 ~" z/xa '
and therefore
«22 = Wj2 = 2gffi \ ^"a^- (36)
Velocity of Efflux.
,2 = 7/22 tan2 ft
'• • (37)
294
HYDRA ULICS.
Useful Work
H v'\
~^j
Efficiency
-r^-s{D'r='>- (39)
Work
-/r,g-.in'r). (38;
Efficiency
\— -ij- sina y = n. (39)
Second, let
so that the water again leaves the wheel with a much-reduced
velocity. Evidently also
J-
a z= 2&2 cos = 22/a sn —
2
sn — .
2
. (42)
Also, by eqs. 2, 4, 5, and 42 —
In an I. F. or O. F.
Q_
zit
= «a sin/? ra</2. (43)
27T
In an A. F.
= »i si
= F3 sin
. (43)'
The following results are now easily obtained :
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 2$$
In an I. F. or O. F. :
Relation between the Vane-
angles.
By eqs. 9 and 43 and from
the triangle acd
sin y _ w2 _ ^a «i_
sin /^ z/2 r\ Vi
r-i sin (a -f* y)
ri sin a
and therefore
cosec ^ = cot
(44)
(45)
Relation between the Vane-
angles.
By eqs. 9 and 43 and from
the triangle acd
d\ sin y _u<i _u\
di sin ft v\ v\
sin (a 4
sm
(44)
and therefore
-i cosec ft = cot or -f cot ^. (45)
IN REACTION TURBINES.
In an I. F. or O. F.:
Speed of Turbine.
By eqs. 14, 17, and 40
UiVi COS y = £-/A = UiVw'. (46)
Also,
Wl sin (a + y}
Hence,
sin a
+
cot a tan X)
- tan ^ cosec ^. (47)
. — If the velocity of
whirl (^wr) relatively to the
wheel at the inlet-surface is to
be nil,
Ul — Vm = O, . . (48)
and then
In an A. F.:
Speed of Turbine.
By eqs. 14, 17, and 40
v\ cos x ~=-S(H\ ~T"^) == WiZ'w'- (46)
Also,
«i _ sin (a -}-X)
sin a
Hence
sin a cos
cot cr tan y)
+ h~ tan ^ cosec /?. (47)
TVi?^. — If the velocity of
whirl (vj) relatively to . the
wheel at the inlet-surface is to
be nil,
Ul - vw' = o, . . (48)
and then
f A). (49)
HYDRAULICS.
Velocity of Efflux.
By equations 42 and 47
0 . 0 ft
82 sin2 -
i— tan L tan
Useful Work
(50)
f i - - tan 6. tan A (51)
^ d* 2 J
Efficiency
Amount Q of Water passing
through Turbine
=. 2itridivr" = 27Tra</a Fa sin ft
= 2itr<id<iU<i sin ft
= 27Tra
tan y sin /?. (53)
Pressure-head at Inlet-surface
by equations 44 and 47.
When the turbine is work-
ing freely in space there will
be no inflow of air if /, > /2 ,
i.e., if
When the turbine is
drowned, with a head h' of
water over the outlet-surface,
Velocity of Efflux.
By equations 42 and 47
ft
sin2 -
(50)
Useful Work
= Q(ffi + h){ i - ~ tan ^ tan y \ (51)
Efficiency
=I*2i<&)=I-|tan^an7'-(52)
Amount Q of Water passing
through Turbine
= inRdiVr" = 2itRd<i Vi sin ft
= 2itRd<iUi sin ft
— 2TtR
n/?. (53)
Pressure-head at Inlet-surface
2g
sin
(54)
by equations 44 and 47.
When the turbine is work-
ing freely in space there will
be no inflow of air if pl >/„,
i.e., if
Hi d± sin ft
H\ -\-k d\ sin 2y'
When the turbine is
drowned, with a head h' of
water over the outlet-surface,
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
there will be no back-flow of
water if
* ^ ^ _u h'
— > — r "• »
IV IV '
that is, if
Ti — W_ r-^d-j sin ft
r^di Sin 2*y'
HI
there will be no back-flow of
water if
A . A , ,,
— > -- h h ,
'
that is, if
H, - h1 d* sin ft
Hi -j- A' di sin 2y
IN IMPULSE TURBINES.
In an I. F. or O. F. :
Speed of Turbine.
Since
t . . (55)
Velocity of Efflux.
jn
•U-? = 4«22 sin2--
2.
*-$# ft- .
cos2 —
2
Efficiency
. (59)
In an A. F. :
Speed of Turbine.
Since
(55)
^ ^. . . (56)
Velocity of Efflux.
. (57)
Useful Work
-^ cog2r Ms8)
( 3S 2 )
Efficiency
H\ d^ sin2 v
= I ~ LJ- i /. T^ *' (59)
298 H YDRA UL ICS.
The great advantages possessed by turbines over vertical
wheels on horizontal axes are shown by a consideration of the
expressions for the useful work and efficiency. The former
involves the available head only, while the latter is independent
even of that. Thus a turbine will work equally well under
water or above water, while its efficiency remains the same,
whatever the available head may be.
The efficiency, also, increases as the ratio — • diminishes.
a,
The value of dl , however, must not be too small, as there might
be a loss of energy due to a contracted section at entrance,
while if dz is made too large, the vane-passages will no longer
run full bore.
Finally, the efficiency -increases as the angles /? and y
diminish.
In practice y usually ranges from 10° to 30° in an I. F.,
and from 20° to 50° in an O. F. and A. F., an average value being
20° for an I. F., and 25° for an O. F. and A. F.
In an I. F. ft generally ranges from 135° to 150° if ?/2 — F2 ,
or from 30° to 45° if vj' — o, and in an O. F. and P. F. from 20°
to 30°, an average value being 145° or 35° for an I. F., accord-
ing as #2 = F2 , or vjr = o, and 25° for an O. F. and A. F.
25. Remarks on the Centrifugal Head
From equations 14 and 17
In an I. F. wa < u, , and the term — L is negative.
Hence the velocity vl diminishes as the speed of the tur-
bine increases and vice versa. The centrifugal head -J— -
therefore tends to secure a steady motion in the case of an I. F.,
and also to diminish the frictional loss of head. For this rea-
son it should be made as large as possible consistent with
practical requirements, and — is usually made equal to 2.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 299
In an O. F., on the other hand, u^ > ul and the centrifugal
head is positive. The velocity vl will, therefore increase and
diminish with the speed of the turbine («). Thus the cen-
trifugal head is adverse to a steady motion, and tends both to
augment a variation from the normal speed and to increase
u * _ u a
the frictional loss of head. It follows that — — — - should be
tg
as small as possible consistent with practical requirements, and
a common value of — is 1.25.
^i
Again, eq. 5 shows that the velocity of flow vr (and there-
fore also £»,) increases as the size of the wheel diminishes, and
is accompanied by a corresponding increase in the frictional loss
of head. Hence it would seem advisable to employ large
wheels ; but if the size of a wheel is increased, it must be
borne in mind that the skin-friction (if the turbine works under
water), the weight, and consequently the journal friction, will
all increase. Belanger has suggested that the efficiency of an
A. F. may be increased by so forming the vane-passages that
the path of a fluid particle gradually approaches the axis of
rotation.
26. Practical Values of the Velocities, etc.— Let v be
the theoretical velocity due to the head H\ i.e., let v* = 2gH.
Experience indicates that the following values will give
good results in reaction turbines :
Inl.R, Vr' = Vr" = ;
In O. F., vr' = - ; vr" = .2iv to .172; ; u, = -u^ = .$6v.
4 ri
In A. F., vrr = vr" = .i$v to .2v ; u, = u9 = -v to -v.
Again, in reaction and impulse turbines the thickness of.
the vanes varies from -J inch to f inch if of wrought iron, and
3OO HYDRAULICS.
from \ inch to f inch if of cast iron. In the latter case the
vanes are usually tapered at the ends.
In axial-flow turbines the mean radius R is often made to
vary
o . _ . _
from - yAJ sin y to 2 yAt sin y if A^ sin y < 2 square feet ;
from --'\fA1 sin y to -\A4,sin y\i A1s\ny > 2sq. ft.< l6sq. ft.;
4 2
from \/ ' Al sin ;/ to —^\/A1 sin ^ if ^4, sin y > 16 square feet.
4
In axial-impulse turbines the mean radius R is often made
to vary from --v/^sin ;/ to 2<\fA1s'my.
4
Also, the depth h of the wheel varies from - r to - - but
o II
must be determined by experience.
Again,
For a delivery of 30 to 60 cubic feet and a fall of 25 ft. to
40 ft. y should be 15° to 18°, and (3 should be 13° to 16°.
For a delivery of 40 to 200 cubic feet, and a fall of 5 ft. to
30 ft. y should be 1 8° to 24°, and fi should be 16° to 24°.
For a delivery of more than 200 cubic feet, and lower falls,
y should be 24° to 30°, and 0 24° to 28°.
In axial-impulse turbines it may also be assumed as a first
approximation that
. ?A vju.
work per pound = — - = _^L_J
2T g
and therefore
Vl = 2#, cos y = 2 Vi cos y.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 30 1
27. Theory of the Suction (or Draught) Tube. — Vortex
and axial-flow turbines sometimes have their outlet orifices
opening into a suction (or draft) tube which extends down-
wards and discharges below the surface of the tail-water. By
such an arrangement the turbine can be placed at any conven-
ient height above the tail-water and thus becomes easily acces-
sible, while at the same time a shorter length of shafting will
suffice. The suction tube is usually cylindrical and of constant
diameter, so that there is an abrupt change of section at the
outlet surface of the turbine, producing a corresponding loss of
energy by eddies, etc. This loss may be prevented by so form-
ing the tube at the upper end that there is no abrupt change
of section, and by gradually increasing the diameter downwards.
The cost of construction is greater, but the action of the tube
is much improved.
Let h' be the head above the inlet orifices of the wheel.
Let h" be the head between the inlet orifices and the sur-
face of the tail-water.
Let Ll be the loss of head up to the inlet surface.
Let L^ be the loss of head between the wheel and the tube
outlet.
Let v^ be the velocity of discharge from the outlet at
bottom of tube.
Let P be the atmospheric pressure.
Then, assuming that there is no sudden change of section
at the outlet surface,
h' ~~= L'
and therefore
w 2g
v*
- — 2gi — K + J**
302 HYDRA ULICS.
where H = h' + h" = total head above tail-water surface ; and
-^aa,_^42, Z-j-, Za are expressed in the forms
2 l ' 4 1 ' *2g' *2g*
*3> /*4> A*6» A<6 being empirical coefficients.
Again, the effective head
and is entirely independent of the position of the turbine in
the tube.
Also, if A i is the area of the outlet from the suction-tube,
A^VI = Q = Alvl sin y,
so that v. can be expressed in terms of z/4, and hence **1 ~ ^ is
w
also independent of the position of the turbine in the tube.
Suppose the velocity of flow to be so small that ^4, v» L9
may be each taken equal to nil. Then
W
and since the minimum value of /, is also nil, the maximum
theoretical height of the wheel above the tail-water surface is
equal to the head due to one atmosphere. Again,
V 3
= vl cos yul — u^u, — F, cos ft) + — L-
But
Alvl sin y = Q = A^ sin d = A^ sin ft = Apt ;
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 303
and hence, taking
gH = z/,(«i cos Y + ****• u* cos ft) — «•" + ^-^
and therefore
- w a _j_ ?7, «i cos r + ^ - «, cos /?
= v? + 2v ^ . — cos y + i/^, . cos
— — (— cos ^ + V/*8
( \ ^
where B — — — cos V*8 cos
Hence it follows that z/, increases with «a, i.e., with the
speed of the turbine, if
A suction-tube is not used with an outward-flow turbine,
but a similar result is obtained by adding a surrounding sta-
tionary casing with bell-mouth outlet. A similar diffusor might
be added with effect to a Jonval working without a suction-tube
below the tail-water. The theory of the diffusor is similar to
that of the suction-tube.
28. Losses and Mechanical Effect. — The losses may be
enumerated as follows:
I. The loss (Z,) of head in the channel by which the water
is taken to the turbine.
L -/-^
*' "7l m 2g>
fi being the coefficient of friction with an average value of
304 HYDRA ULICS.
.0067, / the length of the channel of approach tn its mean
hydraulic depth, and v0 the mean velocity in the channel.
Ll is generally inappreciable in the case of turbines of the
inward- and axial-flow types, as they are usually supplied with
water from a large reservoir in which VQ is sensibly nil.
If AQ is the sectional area of the supply-channel, then
A0v0 = Q = A1v1 sin yy
and
£, = /, -
A,
II. The loss (Za) of head in the guide-passages.
This loss is made up of :
(a) The loss due to resistance at the entrance into the
passages ;
(b) The loss due to the friction between the fluid and the
fixed blades;
(c) The loss due to the curvature of the blades ;
(d) The loss of head on leaving the guide-passages.
These four losses may be included in the expression
/a being a coefficient which has been found to vary from .025
to .2 and upwards. An average value of f9 is .125, but this is
somewhat high for good turbines.
Note. — In Impulse turbines /a has been found to vary from
.11 to .17.
III. The loss (Z,3) due to shock at entrance into the wheel.
In order that there may be no shock at entrance, the relative
velocity ( F,) must be tangential to the lip of the vane. For
any other velocity (z// = ac'} and direc-
tion (dad = yf) of the water at en-
trance, evidently
L3 = the loss of head
FIG. 191.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 3O$
(v' sin y' — ^ sin y)3 (v' cos y' — vl cos y)9
_ (vf sin ;/ — Vl sin <*)a (z/ cos y' — z\ — Vl cos a)8
Generally a? is small, and L3 is always nil when the turbine
is working at full pressure and at the normal speed.
This loss of head in shock caused by abrupt changes of sec-
tion, and also at an angle, may be avoided by causing the sec-
tion to vary gradually, and by substituting a continuous curve
for the angle.
IV. The loss (Z,4) of head due to friction, etc., in passing
through the wheel-passages, including the loss due to leakage
in the space between the guides and the inlet-surface. This
loss is expressed in the form
V:
sn
ftl
where f^ varies from .10 to .20.
Note. — The loss of head due to skin-friction often governs
the dimensions of a turbine, and renders it advisable, in the case
of high falls, to employ small high-speed turbines.
V. The loss of head (Lb) due to the abrupt change of sec-
tion between the outlet-surface and the suction-tube.
As in III, v9 (=ffy is suddenly changed into vt' (•= fh'\
and loss of head is
2g 2g 2g
since h ' x is very small and may be disre-
garded. Thus,
(FiG. 192.
4 =
#/ being the component of vj (fhf) in the direction of the
axis of the suction-tube.
3O6 HYDRA ULICS.
If there is no abrupt change of section between the outlet-
surface and the tube, Z& is nil.
VI. The loss of head (L6) due to friction the in suction-tube.
Assume that the velocity v^ of flow in the tube is equal to v^
the velocity with which the water leaves the turbine. Also let
A be the sectional area of the tube. Then
/ f- f
6 ~~/6 m' 2g ~ /6 m' \ A, I 2g '
/6( =/t) being the coefficient of friction with an average value
of .0067, I' the length of the tube, and m' its mean hydraulic
depth.
VII. The loss (Z7) of head due to entrance to sluice at base
of tube. This loss may be expressed in the form
A
the average value of/7 being about .03.
VIII. The loss (Z8) of head due to the energy carried away
by the water on leaving the suction-tube.
and z>4 usually varies from | V2gH to f V2gH.
In good turbines the loss should not exceed 6#. It might
be reduced to 3$, or even to i$, but this would largely increase
the skin-friction.
IX. The loss of head (L9) produced by the friction of the
bearings.
being the coefficient of journal friction, Wthe weight of the
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
turbine and of the water it contains, and p the radius of the
journal.
Hence the total loss of head
and the total mechanical effect
Note. — If there is no suction-tube, £6 = O = L6 = L, =
and the total loss becomes
fall from outlet-surface
tail-water surface.
to
29. Centrifugal Pumps. — If an hydraulic motor is driven
in the reverse direction, and supplied with water at the point
from which the water originally proceeded, the motor becomes
a pump. All turbines are reversible, and may, therefore, be
converted into pumps, but no pump has yet been constructed
of an inward-flow type. The ordinary centrifugal pump, Fig.
193, is an outward-flow machine.
It is more economical and less
costly for low falls than a recip-
rocating pump, and has been
known to give good and eco-
nomic results for falls as great
as 40 feet.
With compound centrifugal
pumps very much greater lifts
are economically possible.
There are three main differ-
ences between centrifugal pumps
and turbines:
ist. The gross lift with a pump is greater, on account
FIG. 193.
308
HYDRAULICS.
of frictional resistances, than the fall in the case of a tur-
bine.
2d. The water enters the pump-fan without any velocity
of whirl (vj — o) and leaves the fan with a velocity of whirl
(vw") which should be reduced to a minimum in the act of
lifting, but which is by no means small. In a turbine, on the
other hand, the water has a considerable velocity of whirl (vw'}
at entrance, while at exit the velocity of whirl (vw") is reduced
to a minimum, and is generally nil.
3d. In a turbine the direction of the water as it flows
into the wheel is controlled by guide-blades ; whereas in the
case of a pump, the direction of the water, as it flows towards
the discharge-pipe, is controlled by a single guide-blade, which
forms the outer surface of the volute, or chamber, into which
the water flows on leaving the fan.
FIG. 194. — Experimental Centrifugal Pump in the Hydraulic Laboratory,
McGill University.
Before the pump can be put into action it must be filled,
and this can be effected through an opening (closed by a plug)
in the casing when the pump is under water, or, if the pump
is above water, by creating a vacuum in the pump-case by
means of an air pump or a steam-jet pump, when the water
must necessarily rise in the suction-tube.
At first the water rotates as a solid mass, and delivery com-
mences when the speed is such that the head due to centrifugal
force r» —u*\ exceeds the lift. This speed may be after-
\ 2g I
wards reduced, providing a portion of the energy is utilized
at exit.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 309
As soon as the pump, which is keyed on to a shaft driven
by a belt or by gearing, commences to work, the water rises
in the suction-tube and divides so as to enter the eye of the
pump-disc on both sides. As in turbines, the revolving pump-
disc is provided with vanes curved so as to receive the water
at the inlet-surface, for a given normal condition of working,
without shock. Experiment has also tended to show that the
angle between the tangents to a vane and the disc circumfer-
ence at the outlet-surface, may be advantageously made as
small even as 15°, but manufacturers hold different opinions on
this point. The water leaves the disc with a more or less con-
siderable velocity, and impinges upon the fluid mass flowing
round the volute, or spiral casing surrounding the disc, towards
the discharge-pipe. This volute should have a section gradu-
ally increasing to the point of discharge, in order that the
delivery across any transverse section of the volute may be
uniform. This volute is also so designed as to compel rotation
in one direction only, with a velocity corresponding to the
velocity of whirl (vwff) on leaving the fan. There are exam-
ples of pumps in which the delivery is effected in all direc-
tions, and the water is guided to the outlet by a number of
spiral blades.
In these pumps an important advantage is gained by the
addition of a vortex or whirlpool chamber surrounding the
pump-disc. The water discharged from the disc then contin-
ues to rotate in this chamber, and a portion of the kinetic
energy is thus converted into pressure energy, which would
otherwise be largely wasted in eddies in the volute or discharge-
pipe. The water leaves the vortex chamber with a diminished
whirling velocity which cannot be very different in direction
and magnitude from the velocity of the mass of water in the
volute. The vortex chamber is provided with guide-blades
following the direction of free vortex stream-lines (equiangular
spirals) so as to prevent irregular motion. A conical suction-
pipe is advantageous, as it allows of a gradual increase of
velocity, and a still greater advantage is to be found in the
use of a conical discharge-pipe. The velocity in the dis-
charge-pipe should not be too great, as it leads to a waste of
310 HYDRAULICS.
energy. A velocity of 3 to 6 feet is found to give the best
results.
Pumps work under different conditions from turbines, and
hence there are corresponding differences necessary in their
design. They work best for the particular lift for which they
are designed, and any variation from this lift causes a rapid
reduction in the efficiency.
30. Theory of Centrifugal Pump.—
Denote the velocities at the inlet- and out-
let-surfaces of the pump-fan by the same
symbols as in turbines.
Let Q be the delivery of the pump.
Let Hs be the gross lift, including the
actual lift (ffa), the head due to the velocity
FIG. 195. Of delivery, the heads due to the frictional
resistances in the ascending main, in the suction-pipe and in
the wheel-passages, and the head corresponding to the losses
" in shock " at entrance and exit.
Let Ha be the actual lift.
The total work done on the wheel
The useful work done by the pump —
Hence
the efficiency (rf) = g g
At the inlet-surface the flow is usually radial, so that y = 90°,
and the velocity of whirl vj is nil.
Thus,
the efficiency = fr* = 77,
and the equation
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 311
is the fundamental equation governing the design of centrifu-
gal pumps.
Again,
H:
the efficiency n = — jh- = i
"
For a given speeed («3) this is a maximum when
j (VJJ+(U,-V "y
y v y
-77 = a minimum — --
vw w w
Hence, differentiating, -
u* tan2 ft
- -<^- + sec- /> = Q,
and therefore
vjf = u^ sin ft,
and is the velocity of whirl at exit which, for a given speed (wa),
will give a maximum efficiency.
Note.—li u^ = vw", then
and the water leaves the fan with a velocity equal to that due
to at least one half of the gross lift. The efficiency must
therefore be necessarily less than .5.
Again, since vr" cot ft = u^ — vw", ft must be 90° if u^ = vj'-,
but ft is generally much less than 90°, and therefore vw" is
generally less than uy Let vw" = ku^> k being an empirical
coefficient less than unity.
Then kit? — gHe and the efficiency = -~>
KU^
Consider two cases.
CASE I. Pump without a vortex-chamber.
When the water is discharged into the volute, the velocity
of flow (vr") is wasted and the velocity of whirl (vw'f) is sud-
denly changed to the velocity vs of the mass of water in the
312 HYDRAULICS.
volute assumed to be moving in a direction tangential to the
pump-disc. Thus,
(yjy - far (vjf - vy
the gam of pressure-head = -
i fe/')9 ^ "
which is a maximum and equal to— when vs = -^— .
4 g
This gain of head is always very small and may be dis-
regarded as being almost inappreciable. Neglecting also the
losses due to frictional resistances, etc., then, precisely as in
the case of turbines,
v±_ , TT __ f variation of pressure-head between
2g ( outlet and inlet surfaces.
*.*-«.• FV-F?
But V? = u? + T^2, since y = 90°, and therefore
_ u* ~ ~_¥JL. _ u* ~ (ui ~ v™'}* sec2 ft
and
u 2 — (u — v //)2 sec2 ft
the efficiency — - w..
2«^w"
which is a maximum for a given speed &a and equal to
; — j- — : — -5 when vwff = u^ sin /?.
Thus the efficiency increases as ft diminishes.
When ft = 90°, or ^wr/ — &2, the maximum efficiency is £,
and therefore one half of the work done in driving the pump
is wasted.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 313
Note. — Loss of head
= loss due to hydraulic friction
-f- loss due to abrupt change from vw" to v,
-\- loss due to dissipation of vr"
-f- loss due to vs carried away
= loss due to friction (hydraulic)
+ *,(*„" - ^ (*l_ _ (VT
2g 2g 2g
= loss due to friction (hydraulic)
, (*.")• ,
~~
when vs = \vj'.
CASE II. Pump ivith a vortex-chamber (Fig. 199).
The diameter (— 2r3) of the outer surface of this chamber
should be at least twice that of the outlet-surface of the pump-
disc.
Assuming that the motion in the
•chamber is a free vortex, then
the gain of ) _ v^_ I r?\
pressure-head ) 2g \ r32/
and hence
the efficiency =
T,, . . FIG. 106.
This, again, is a maximum for
a given speed, when vj = u^ sin fiy its value being
I +'(l - Sj) sin ft
I + sin ft
3 1 4 HYDRA ULICS.
This expression increases as ft diminishes, but the value of
ft is not of so much importance as in Case I, and it is very
common to make ft equal to 30° or 40°.
When ft = 90° the maximum efficiency = - ( 2 - -M = —
if ra = 2r,.
31. Practical Values. — The following values are often
adopted :
3 = d^ when faces of pump-disc are parallel ;
^ = \d^ when pump-disk is coned.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 31$
EXAMPLES.
1. An accumulator ram is 9 inches diameter and 21 feet stroke. Find
the store of energy in foot-pounds when the ram is at the top of its
stroke, and is loaded till the pressure is 750 Ibs. per square inch.
Ans. 958,000 ft.-lbs.
2. In a differential accumulator the diameters of the spindle are 7
inches and 5 inches ; the stroke is 10 feet. Find the store of energy when
full and loaded to 2000 Ibs. per square inch. Ans. 377,000 ft.-lbs.
3. A direct-acting lift has a ram 9 inches diameter, and works under
a constant head of 73 feet, of which 13 per cent, is required by ram-fric-
tion and friction of mechanism. The supply-pipe is 100 feet long and 4
inches diameter. Find the speed of steady motion when raising a load
of 1350 Ibs., and also the load it would raise at double that speed.
If a valve in the supply-pipe is partially closed so as to increase the
coefficient of resistance by 5!, what would the speed be ?
Ans. Speed = 2 ft. per second ; load = 150 Ibs.
4. Eight cwt. of ore is to be raised from a mine at the rate of 900 feet
per minute by a water-pressure engine, which has four single-acting
cylinders, 6 inches diameter, 18 inches stroke, making 60 revolutions per
minute. Find the diameter of a supply-pipe 230 feet long for a head
of 230 feet, not including friction of mechanism.
Ans. Diameter =• 4 inches.
5. If A. be the length equivalent to the inertia of a water-pressure
engine, F the coefficient of hydraulic resistance, both reduced to the
ram, -z/o the speed of steady motion, find the velocity of ram after
moving from rest through a space x against a constant useful resistance.
Also find the time occupied.
Ans. v* =
—V
6. An hydraulic motor is driven from an accumulator, the pressure
in which is 750 Ibs. per square inch, by means of a supply-pipe 900 feet
long, 4 inches diameter; what would be the maximum power theoreti-
cally attainable, and what would be the velocity in the pipe correspond-
ing to that power? Find approximately the efficiency of transmission at
half power. Ans. H.P. = 240 ; v = 22 ft. ; efficiency = .96 nearly.
7. A gun recoils with a maximum velocity of 10 feet per second.
The area of the orifices in the compressor, after allowing for contraction,
may be taken as one twentieth the area of the piston. Find the initial
pressure in the compressor in feet of liquid.
HYDRAULICS.
Assuming the weight of the gun to be 12 tons, friction of sUde 3
tons, diameter of compressor 6 inches, fluid in compressor, water, find
the recoil.
Find the mean resistance to recoil. Compare the maximum and
mean resistances, each exclusive of friction of slide.
Ans. 621; 4ft. 2^ in. ; total mean resistance = 4.4 tons;
ratio = 2.5.
8. A reaction wheel is inverted and worked as a pump. Find the
speed of maximum efficiency and the maximum efficiency, the coeffi-
cient of hydraulic resistance referred to the orifices being .125.
Ans. Speed = twice that due to lift ; .758.
9. A reaction wheel with orifices 2 in. in diameter makes 80 revolu-
tions per minute under a head of 5 ft. The distance between the centre
of an orifice and the axis of rotation is 12 inches. Find the H.P. and
the efficiency. Ans. .146; .596.
10. In a reaction wheel the speed of maximum efficiency is that due
to the head. In what ratio must the resistance be diminished to work
at | this speed, and what will then be the efficiency? Obtain similar
results when the speed is diminished to three fourths its original
amount. Ans. .949; .8896; 1.071; .753.
11. In a reaction wheel, determine the per cent of available effect
lost, (i) if i? = 2gH\ (2) if tt* = ^gH; (3) if u1 = ZgH.
What conclusion may be drawn from the results?
Efficiencies are respectively .828, .9, .945.
12. An undershot water-wheel with straight floats works in a straight
rectangular channel of the same width as the wheel, viz., 4 ft.; the
stream delivers 28 cub. ft. of water per second, and the efficiency is £.
Find the relation between the up-stream and down-stream velocities.
If the velocity of the inflowing water is 2 ft. per second, find the velocity
on the down-stream side and determine the mechanical effect of the
wheel, its diameter being 20 ft., the diameter of the gudgeons being 4
in., and the coefficient of friction .008.
13. A vane rotates about an axis with an angular velocity A, and
and water moves freely along the vane. Show that the work per unit of
weight of water, due to centrifugal force, in moving from a point distant
A1(<y 2 *• 2\
r\ ft. from the axis to a point distant r-i ft. from the axis is — '— .
14. Determine the effect of a low breast or undershot wheel 15 ft. in
diameter and making 8 revols. per minute; the fall is 4 ft. and the
delivery 20 cub. ft. per second; the velocity of the stream before com-
ing on the wheel is double that of the wheel. Ans. 1490 ft.-lbs.
15. The efficiency of an undershot water-wheel working in a straight
rectangular channel with horizontal bed is \. Find the relation between
the up- and down-stream velocities of flow.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 31 /
16. Determine the mechanical effect of an undershot wheel of 12 ft.
diam. making 10 revols. per minute, the fall being 3 ft. and the quantity
of water passed per second 15 cub. ft.
17. Ascertain the general proportions of a Poncelet wheel, being
given: height of fall = 4^ ft.; delivery of water = 40 cub. ft. per
second; radius of exterior circumference = 9 ft. ; thickness of the
stream = 9 in.
18. Design a Poncelet wheel for a fall of 4.5 ft. and 24 cub. ft. of
water per second, using the formulae on pages 237-239, taking y = 20°,
and also A = 20° as a first approximation.
Ans. a = 143° 57' ; depth of crown = 1.73 ft. ; depth of stream
= .386 ft. ; b = 4.14 ft.; radius of bucket = 2.35 ft. ; ^ =
128° 6' ; A = 18.69°; number of buckets = 44; mechanical
effect = 8.5 H.P. ; efficiency = .69.
19. 15 cub. ft. of water per second with a fall of 8g ft. are brought on
a breast-wheel revolving with a linear velocity of 5 ft.; depth of shroud-
ing = 12 in.; the buckets are half filled, and vi = 2tt ; also r\ — 12 ft.
Find the theoretical mechanical effect. Ans. 7240 ft.-lbs.
20. A wheel is to be constructed for a 3O-ft. fall having an 8-ft.
velocity at circumference and taking on the water at 12° from the sum-
mit with a velocity of 16 ft. Determine the radius of the wheel and the
number of revols. Ans. 12. 8ft.; 5.94.
21. If for the wheel in example 20 the number of revols. is 5, and
Vi = 2u, the water being again taken on at 12°, find the radius and u.
Ans. 13.98 ft. ; 7.3 ft. per sec.
22. A breast wheel passes 12 cub. ft. of water per second* and for the
speed = 3-z/i = 4 ft. per second the loss of mechanical effect due to the
relative velocity V being destroyed is a minimum. Find this effect.
23. In a breast-wheel Q = 10 cub. ft. per second ; H = 10 ft. ; v\ =
-£# ; u = 4i ft. per second ; y = 30° ; diam. of gudgeon = 6 in. ; diam. of
wheel = 30 ft. ; /* =; .08 ; weight of wheel and water = 20,000 Ibs. Find
the mechanical effect of the wheel. (Neglect loss of effect due to escape
of water from buckets and to frictional resistance along the curb.)
24. The quantity of water laid on a breast-wheel by an overfall sluice
= 6 cub. ft. per second, the total fall being 4 ft. 6 in., and the velocity
of the periphery 5 ft. per second ; also 52/1 = Su, and if d be the depth of
the shrouding 2bdu — $Q (in the present case d = 12 in.). Find the effec-
tive fall, the height of the lip of the guide, the angle of inclination at
the end of the guide-curve, the breadth of the lip of the guide-curve, and
the radius of the wheel that the water may enter tangentially. If the
radius is limited to 12 ft. 6 in., find the deviation of the direction of
motion of the water from that of the wheel at the point of entrance.
Ans. 6.9 ft. ; .325 ft. ; 34° 46' ; 2f ft. ; 38.6 ft. ; 28° 36'.
25. In an overshot wheel r\ = 15 ft., d — 10 in., ft = f^. If the
3l8 HYDRAULICS.
division circle is at one half of the depth of the crown, find the angle
(yi) between the bucket-lip and the wheel's periphery.
Ans. yi = 18° i'.
26. An overshot wheel in which r\ — 18 ft. makes 4 revolutions per
minute, and the velocity of the water on entering the buckets is twice
that of the wheel's periphery. If yi = 20°, find a, and also find the rela-
tive velocity ( V) of the entering water.
Ans. a. = 10° 9' ; V = 7.78 ft. per second.
27. If one fourth of the theoretic capacity of a bucket is filled by the
water, find the greatest number of buckets theoretically possible, the
depth of the crown being i ft., the radius (ri) to the outer periphery 12
ft., the angle yi 20°, and the velocity of the entering water twice that of
the wheel's periphery.
Ans. 103.1. Making allowance for exit of air, the number of
buckets might be about two thirds of this amount, or, say, 69.
28. A wheel of 3o-ft. diam. with 72 buckets makes 7 revolutions per
minute, Q being 5 cub. ft. per second. The division circle is halfway
between the outer and inner peripheries. If d = i ft. and vi = 2u, find
the effect due to impact.
29. A 3O-ft. wheel weighs 24,000 Ibs. and makes 6 revolutions per
minute; its gudgeons are 6 in. in diameter and the coefficient of friction
is .08. The water enters the wheel with a velocity of 1 5 ft. per second,
and in a direction making an angle of 10° with the direction of motion
of the wheel at the point of entrance. The deviation from the summit
of the point of entrance is 12°, of the point where spilling begins is 150°,
of the point where all is spilt is 160°, and 5 cub. ft. of water enter the
wheel per second, of which the partially filled buckets contain one half.
Determine the total mechanical effect. Ans. 9305.6 ft.-lbs.
30. The velocity of the pitch circle is 9! ft.; the angle between the
directions of motion of stream and wheel is 15°. Find impulsive action
of wheel. Ans. 91 ft.-lbs. per cub. ft. of water.
31. An overshot wheel 40 ft. in diameter makes 4 revolutions per
minute and passes 300 cub. ft. of water per minute. Show how to deter-
mine the mechanical effect of the wheel. (Neglect friction of gudgeons.)
If the gudgeons are 6 in. in diameter and the wheel weighs 30,000 Ibs.,
by how much will the mechanical effect be diminished (/= .008) ?
Ans. 25 ft.-lbs. per sec.
32. The diameter of an overshot wheel = 30 ft. ; Vi = 15 ft. ; u = 9$
ft., deviation of impinging water from direction of motion of wheel
(y) = 8^° ; deviation of point of entrance from summit = 12° ; deviation
of point where spilling begins from the centre = 58^° ; deviation of point
where spilling ends = 70^° ; Q = 5 cub. ft. Find total effect of impact
and weight. Ans. 17 H. P.
33. An overshot wheel with a radius of 15 ft. and a i2-in. crown
takes 10 cub. ft. of water per second and makes 5 revolutions per
H YDRA ULIC MO TORS A ND CEN TRIFUGA L P UMPS. 3 1 9
minute. If m = i, find the width of the wheel and the number of the
buckets. Ans. 5TV ft. ; 75 or 90.
34. An overshot wheel of 32 ft. diameter makes 5 revolutions per
minute. Find the angle between the water-surface in a bucket and the
horizontal when the lip is 140° from the summit. Ans. 4° 33'.
35. An overshot wheel of 10 ft. diameter makes 20 revolutions per
minute. Find the angle between the water-surface and the horizontal
when the lip is (i) 90° from the summit, (2) 45° 26' from the summit.
Ans. (i) 34° 16'; (2)45° 26'.
36. The water enters an overshot wheel at 12° from the summit with
a velocity of 16 ft. per second and the linear velocity of the wheel's pe-
riphery is 8 ft. per second. The fall is 30 ft. Find the diameter of the
wheel and the number of revolutions per minute.
Ans. 25.68 ft. ; 5.94.
37. An overshot wheel of 36 ft. diameter and with 96 buckets has a
peripheral velocity of 7^ ft. per second. The water enters with a velocity
of 15 ft. per second and acquires in the wheel a velocity of 16.49 ft- Per
second Find the distance through which the float moves during impact.
Ans. 2.15 ft.
38. The sluice for a lo-ft. overshot wheel is vertically above the centre
and inclined at 45° to the vertical. The water enters the buckets at a
point 2 ft. vertically below the sluice and 10° from the summit of the
wheel. Find the angle between the directions of motion of the entering
water and of the wheel's circumference. Also find the velocity of the
water as it enters the wheel.
39. In an overshot wheel z>i = 17 ft. ; u = n ft. per second; elbow-
angle = 70°; division-angle = 5°; water enters the first bucket at 12°
from summit of wheel. Find (a) the relative velocity Fso that water
may enter unimpeded; (£) the direction of the entering water; (c)
the diameter of the wheel, which makes 5 revolutions per minute ; (d)
the position and direction of the sluice, which is 2 ft., measured hori-
zontally from the point of entrance.
40. In an overshot wheel the deviation of the impinging water from
the direction of motion of the wheel is 10° ; the velocity (vi) of the im-
pinging stream = 15 ft. per second; of the circumference of the wheel
(«) = 15 cos 10°. What proportion of the head is sacrificed?
41. A 3o-ft. water-wheel with 72 buckets and a 12-in. shrouding makes
5 revolutions and receives 240 cub. ft. of water per minute. Find the
width and sectional area of a bucket. The fall is 30 ft. ; at what point
does the water enter the wheel, the inflowing velocity being i| times
that of the wheel's periphery? Also find the deviation of the water-
surface from the horizontal at the point at which discharging com-
mences, i.e., 140° from the summit.
42. What number of buckets should be given to an overshot wheel of
3 2O H YDRA UL ICS.
40 ft. diameter and 12 in. width in wheel, pitch-angle = 4°, thickness of
bucket lip = i in., water area = 24^ sq. in. ?
43. A wheel makes 5 revolutions per minute, the radius is 16 ft., and
the discharging angle 50°. Find deviation of water-surface from the
horizon. Ans. 4° .29.
44. A wheel makes 20 revolutions per minute; radius = 5 ft., angle
of discharge = o°. Find deviation of water-surface from horizon. Also
find deviation at 44° 35' above centre. Ans. 4° 33' ; 44° 34'.
45. The water in a head-race stands 4.66 ft. above the sole and leaves
the race under a gate which is raised 6 in. above the sole, the coefficient
of velocity (v*) being .95. The water enters a breast wheel in a direction
making an angle of 30° with the tangent to the wheel's periphery at the
point of entrance. The speed (u) of the periphery is 10 ft. per second,
the breadth of the wheel is 5 ft., the depth of the water beneath the
axle is 8 in., and the length of the flume is 8.2 ft. Find the loss of
head (a) due to the destruction of the relative velocity (V) at entrance;
(b) due to the velocity of flow in the tail-race ; (c) in the circular
flume. Ans. (a) i.u ft.; (£) 1.57 ft. ; (c] .44 ft.
46. In the preceding example, find how the losses of head would be
modified if the flume were lowered 1.03 ft., and if the point of entrance
were raised so as to make u = v\ cos 30°.
47. A water-wheel has an internal diameter of 4 ft. and an external
diameter of 8 ft.; the direction of the entering water makes an angle of
15° with the tangent to the circumference. Find the angle subtended
at the centre of the wheel by the bucket, which is in the form of a cir-
cular arc, and also find the radius of the bucket.
48. An overshot wheel 5 ft. wide, 30 ft. in diameter, having a 12-in.
crown and 72 buckets, receives 10 cub. ft. of water per second and
makes 5 revolutions per minute. Determine the deviation from the
horizontal at which the water begins to spill, and also the corresponding
depression of the water-surface.
49. An overshot wheel makes — revolutions per minute ; its mean
iTt
diameter is 32 ft. ; the water enters the buckets with a velocity of 8 ft.
per second at a point 12° 30' from the summit of the wheel. At the
point of entrance the path of the inflowing water makes an angle of 30°
with the horizontal. Show that the path is horizontal vertically above
the centre. The sluice-board is placed at a point whose horizontal
distance from the centre is one half that of the point of entrance.
Find its position relatively to the centre and its inclination to the hori-
zon. (Sin 12° 30' = .2165).
50. The water enters the buckets of the wheel in the preceding
example without shock. Find the elbow-angle. Also, if the buckets
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. $21
begin to spill at 150° from the summit, find where the bucket is empty
and the number of buckets. (Depth of crown = 12 in.; thickness of
bucket = \\ in.)
51. Given 7/1 = 15 ft. per second, and S — 2oJ°. Find the position of
the centre of the sluice, which is 4 in. above the point of entrance.
Ans. .097 ft. vertically below and 1.114 ft- horizontally from the
summit. The axis of the sluice is inclined at 9° 58' to the
horizontal.
52. In an overshot water-wheel 2/1 = 15 ft.; u = 10 ft. ; elbow-angle
= 70^° ; division-angle = 4^° ; deviation from summit of point of en-
trance = 12°. Find the deviation of the layer from that of the arm, so
that the water might enter unimpeded; also find the inclination of the
layer to the horizon, and the value of V . If the centre of the sluice-
aperture is to be 4 in. above point of entrance, find its vertical and
horizontal distance Trom the vertex of the stream's parabolic path which
is vertically above the centre of the wheel, and also find inclination oi
sluice-board to horizon.
Ans. 15!° ; 2oJ° ; 5.3 ft. per sec. ; .42 ft. ; 1.04 ft. ; 9° 34',
53. In an overshot wheel Q= 18 cub. ft.; r\ =6 ft. ; d— i ft. ; b =
4 ft. ; N — 24. At the moment spilling commences the area afd = 1.025
sq. ft.; between this point and the point where the spilling is com-
pleted three buckets are interposed, the sectional areas of the water
being .591, .409, and .195 sq. ft., respectively. Find (a) the sectional area
of bucket, (b) the point where the spilling commences, (c] the point where
the spilling is completed, (d) the height of the arc of discharge, (<?) the
mechanical effect due to the fall of the water through the arc, of discharge.
' Ans. (a) .662 sq. ft ; (b) 0 = 7° 26', 0 = 28° 33' ;
(c) e = 73° 15'. 0 = 5° 59' ; (d) 449 ft. I (') 4-93 H.P.
54. In the preceding example, if the water enters with a velocity of
20 ft. per second at 20° below the summit, and if the direction of the
inflowing stream makes an angle of 25° with the wheel's periphery at
the point of entrance, find the mechanical effect (a) due to impulse,.
(b) due to the fall to the point where spilling commences.
Ans. (d) 5.34 H.P.; (b) 12.15 H.P.
55. 300 cub. ft. of water per minute enter the buckets of a 4o-ft.
overshot wheel with a 12-in. crown and making four revolutions per
minute. The wheel has 136 buckets. At the moment when spilling
commences the area afd (Fig. 156) = 102 sq. in., and the area abed ' =
24.5 sq. in. The spilling is completed when the angle between the hori-
zontal and the radius to the lip of the bucket = 62° 30'. Between these
two positions three buckets are interposed, the sectional areas of the
water in the buckets being 24.5, 14.48, and 6.6 sq. in., respectively. The
vertical distance between the water-surface in the first bucket and the
centre is 18 ft. Find (a) the width of the wheel, (b) the cross-section of
3 2 2 HYDRA ULICS.
a bucket, (c) the angle between the horizontal and the radius to the lip
of the bucket when spilling commences, (d) the height of the discharg-
ing arc, (<?) the mechanical effect due to weight.
Ans. (a) 2.4 ft.; (b) 33.09 sq. ft.; (c) 6 = 52° 2*'; (d) 1.9 ft.;
(<?) 19.48 H.P.
56. As the bucket arm cd moves downwards from the horizontal
position, show that while the wheel moves through an angle the last
particle of water at c will move, through a distance approximately equal
?*(/r^* ~{~ u^ }
to - — ^ (6 — sin 6), r being the distance (assumed constant) of the
particle of water from the axis, and u being the linear velocity of the
wheel at the radius.
57. If the last particle of water leaves the buckets just as the lip d
reaches the lowest point of the wheel, and if the arm is i ft. in length,
find the angle between the lip" and the wheel's periphery (i) for a wheel
of 20 ft. diameter, the peripheral velocity being 5 ft. per second ; (2) for
a wheel of 40 ft. diameter, the peripheral velocity being 10 ft. per second ;
(3) for a wheel of 10 ft. diameter, the peripheral velocity being 8 ft. per
second. Ans. (i) 20^° ; (2) 20° ; (3) 40°.
58. In an overshot wheel of 30 ft. diameter, 5 cub. ft. of water per
second enter the buckets with a velocity of 16 ft. per second and the
wheel's velocity at the division circle is 7 ft. per second. The point of
entrance is 18° from the summit, and the angle between the directions
of the inflowing water and the wheel's periphery at the point of entrance
is 12°. The water begins to spill at 148^° from the summit and the
spilling is complete at i6o£° from the summit. Find the total mechani-
cal effect due to impulse and weight. What is the tangential force at
the outer periphery? Ans. 16.28 H.P. ; 1194 Ibs.
59. 20 cub. ft. of water per second enter an undershot wheel of 30 ft.
diameter, making 8 revolutions per minute through an underflow sluice.
The velocity of the entering water is twice that of the wheel's periphery.
Find (a) the head of water behind the sluice, (b) the fall, (c} the theo-
retical mechanical effect, (d) the actual mechanical effect, disregarding
axle friction.
Ans. (a) 2.779 ft.; (b) 1.221 ft.; (c} 5.57 H.P.; (d) 2.62 H.P.
60. 20 cub. ft. of water per second enter a breast wheel of 32 ft. diam
eter and having a peripheral velocity of 8 ft. per second, at an angle of
25^ with the circumference. The depth of the crown is ij ft.; the buc-
kets are half filled, and the fall is 9 ft. The velocity of the entering
water is 12 ft. per second. The centre of the sluice-opening is -54ft.
above the point of entrance, and the width of the sluice is 3! ft. The
wheel has 48 buckets. The distance between the wheel and breast is %
inch. The bucket passes through .9 ft. while receiving water, and the
depth of the water-surface in the bucket below the point of entrance is
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS.
1.25 ft. Find (a) the angular distance of the point of entrance from the
horizontal, (b) the fall in the breast, (<r) the head of water over the
sluice, (d) the velocity of the water in the bucket the moment entrance
ceases, (e) the total mechanical effect, disregarding axle friction.
Ans. (a) 53° 53'; (b) 6.525 ft.; (c) 1.935 ft.; (d) 14.9 ft.; (e) 15.59 H.P.
61. In the preceding question, if the energy absorbed by axle fric-
tion, etc., is 743 ft.-lbs., find the efficiency of the wheel. Ans. f.
62. 20 cub. ft. of water per second enter an undershot wheel of 20 ft.
diameter in a straight race, the fall being 3 ft. The depth of the enter-
ing stream is £ ft. The width of the wheel is 4f ft., and the clearance is
f inch. The number of the floats, of which four are immersed, is 48,
and each is i ft. long. The weight of the wheel is 7200 Ibs., the radius
of the axle is if in., and the coefficient of friction is .1. Find (a) the
best speed for the wheel, (b} the corresponding mechanical effect, (c) the
efficiency.
Ans. (a) 6 ft. per second ; (ff) 2.32 H.P., assuming the speed of
wheel reduced to 5.74 ft. per second by axle friction ; (c} .34.
63. A downward-flow turbine of 24 in. internal diameter passes 10
cub. ft. of water per second under a head of 31 ft ; the depth of the
wheel is i ft. and its width 6 in. Find the efficiency, assuming the
whirling velocity at outlet to be nil. Ans. .997.
64. A downward-flow turbine of 5 ft. external diameter passes 20
cub. ft. of water per second under a head of 4 ft., the depth of the
wheel being 5 ft. The water enters the wheel at an angle of 60° with
the vertical, the receiving-lip of the wheel-vanes is vertical, and the ve-
locity of whirl at outlet is nil. Find the internal diameter and the
speed in revolutions per minute. Ans. 4.68 ft.; 46.53.
65. A downward-flow turbine has an internal diameter of 24 in.; the
breadth of the wheel is 6 in.; the turbine passes 33 cub. ft. per second
under an effective head of 16 ft. Assuming the whirling velocity at out-
let to be nil, find the efficiency and power of the turbine. If the vane-
lip at inlet is radial, finci the direction of the vane at outlet, and the
speed of the turbine in revolutions per minute.
Ans. .931 ; 55.865 H.P.; ft = y = 21° 2' ; 166.7.
66. Discuss the preceding question on the assumption that the
peripheral speed at outlet (»9) is equal to the speed of the water at that
point relatively to the wheel ( F»).
Ans. .928 ; 55.715 H.P.; /3 = 21° 47' and y = 20° 21'.
67. An axial-flow impulse turbine of 5 ft. mean diameter passes 170
cub. ft. of water per second under an effective head of 8.6 ft.; the depth
of the wheel is .9 ft. At what angle should the water enter the wheel to
give an efficiency of 81 per cent, the width of the wheel being constant
and disregarding hydraulic resistances ? Ans. =27 16'.
68. In example 67, find (a) the velocity with which the water enters
324 HYDRA ULICS.
the wheel (b) the speed of the turbine in revolutions per minute, (c) the
directions of the vane edges at inlet and outlet, (d) the velocity of the
water as it leaves the wheel, (e) the power of the turbine.
Ans. (a) 2346ft. per sec.; (b) 45.08 ; (c) a = 130° 05', ft = 42° 19';
(d} 10.748 ft. per sec.; (e) 148.65 H.P.
69. In example 67, if instead of assuming that the whirling velocity
at exit is nil, it is assumed that the peripheral speed (u*) of the wheel at
the mean radius is equal to the relative velocity ( F2) of the water at exit,
show how the several results are affected.
Ans. y = 25° 6'; (a) 23.46 ft. per second ; (b) 54.638 ;
(c) a= 124° 54', 0 = 44° 7' ; (d) 10.748 ft. per second ;
(e) 148.65 H.P.
70. In examples 68 and 69, assuming that the hydraulic resistances
necessitate an increase of i2£ per cent in the head equivalent to the ve-
locity with which the water enters the wheel, and an increase of 10 per
cent in the head equivalent to the relative velocity ( VJ) at outlet, show
how the several results are affected.
Ans. Question 68. (a) 22.12 ft. per sec.; (b) 47.82;
(c) a — 121° 30', ft = 40° 9'; (d) 10.748 ft. per sec.;
(e) 148.65 H.P.
Question 69. (a) 22.119 ft. per sec.; (b} 50.97;
(c) a= 124° 91', ft =47° 28'; (d} 10.748 ft. per sec.;
(<?) 148.65 H.P.
71. The efficiency of an axial -flow turbine is 90 per cent, and it passes
12 cub. ft. per second under an effective head of 40 ft. At the mean
radius the water enters at an angle of 30° with the wheel's face, and the
whirling velocity at outlet is nil. Find (a) the velocity with which the
water enters and leaves the wheel, (b) the directions of the vane at inlet
and outlet, (c) the sectional areas of the inlet- and outlet-orifices, (d} the
speed of the wheel in revolutions per minute, (<?) the power of the tur-
bine. Ans. (a) 42% ft. per second ; 16 ft. per second ;
(b) a = 49° 6', 0 = 2i° 3';
(c) .75 sq.ft.;
W). I9&39; (<f)49& H.P.
72. An axial-flow turbine of 5 ft. mean radius passes 212 cub. ft. of
water per second under a total effective head of 12.1 ft. At the mean
radius, the direction of the inflowing water makes an angle of 70° with
the vertical, and the vane-lip at the outlet makes an angle of 17° with
the wheel's periphery. If the whirling velocity at the outlet-surface is
nil, find (a) the velocity with which the water must enter the wheel to
give an efficiency of .953 per cent. Also find (b) the direction of the
vane-lip at outlet, (c) the speed of the wheel in revolutions per minute,
(d} the widths and areas of the inlet- and outlet-orifices, (e) the power
of the turbine.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. $2$
Ans. (a) 19.9 ft. per sec.;
(6) «=8i°22'; (*) 37.67;
(d) .991 ft.; 31.148 sq.ft.; 1.181 ft; 35.14 sq.ft.;
(e) 277- 799 H. P.
73. An axial-flow impulse turbine passes 170 cub. ft. of water per
second under an effective head of 9.5 ft., the depth of the wheel being .9
ft. and its mean radius 4.2 ft. The vane-lip at the outlet makes an angle
of 72° with the vertical. Assuming that the whole of the effective head
is transformed into useful work, and that the whirling velocity at the
outlet-surface is nil, find (a) the inclination to the vertical of the outlet-
lip of the guide-vane, (b) the direction of the inlet-lip of the wheel-vane,
(c) the efficiency ; first neglecting hydraulic resistances, and second taking
these resistances into account.
Ans. First. (a) 59° 52' ; (b) 60° 16' ; (c) .905;
Second, (a) 52° 52'; (b) 74° 16' ; (c) .804.
74. In the preceding example find the inlet- and outlet-orifice areas
in the two cases. Ans. First. 8.12 sq. ft. ; 22.4 sq. ft. ;
Second. 9.64 sq. ft.; 28.56 sq. ft.
75. An axial-flow turbine passes 200 cub. ft. of water per second
under a head of 14 ft., the depth of the wheel being i ft. The mean
radius of the wheel is 3 ft. ; the areas of the inlet- and outlet-surfaces are
in the ratio of 7 to 8 ; the water enters the wheel at an angle of 21° to
the wheel face, and the outlet edge of the vane makes an angle of 16°
with the face. Find the speed, efficiency, and power of the turbine, and
also the direction of the inlet-lip of the vanes.
Ans. 73.69 revolutions per minute; .954; 325.24311.?.;
a =65° 57'.
76. In question 11, if there are 62 wheel, and 66 guide-vanes, the
thickness of the latter being .2 in. and of the former .4 in., find the
width of the inlet-orifices.
77. Water is delivered to an O. F. turbine at a radius of 24 in. with a
whirling velocity of 20 ft. per second, and leaves in a reverse direction
at a radius of 4 ft. with a whirling velocity of 10 ft. per second. If the
linear velocity of the inlet-surface is 20 ft. per second, find the head
equivalent to the work done in driving the wheel. Ans. 24.8 ft.
78. An outward-flow turbine of 9.5 in. external diameter works under
an effective head of 270 ft. Find the speed in revolutions per minute,
assuming that the whirling velocity at the inlet-surface relatively to the
wheel is nil and that the efficiency is unity. Ans. 2242.
79. An outward-flow turbine, whose external and internal diameters
are 8 ft. and 5^ ft. respectively, makes 26 revolutions per minute under
an effective head of 4 ft. The water enters the wheel in a direction
making an angle of 36° (y) with the direction of motion at the point of
entrance. Determine the angles of the moving vane at ingress and egress,
HYDRAULICS.
the efficiency being .85. Also find the energy per pound of water carried
away by the water as it leaves the turbine.
Ans. a= 129° 59', fi= 29° 38'; .6 ft.-lbs.
80. Construct an outward-flow turbine from the following data : the
fall = 5 ft. ; internal diameter = 1.8 ft.; external diameter = 2.45 ft.;
quantity of water passed per second = 30 cub. it.,y = 30° ; efficiency = .9.
Ans. a = 108° 15'; fi = 21° 3'; Ai =3.897 sq. ft. ; A* = 5.303;
d\ = d* = .688 ft., neglecting thickness of vanes.
Si Assuming that the intensities of the pressure at the receiving
and discharging edges of the moving vanes of a Fourneyron turbine are
equal, and also that the rim velocity and the velocity of the water
relatively to the wheel at the discharging-surface are equal, show that
the direction of the impinging stream must bisect the angle between
the direction of motion and the tangent to the vane at the receiving
r? sin /?
edge. Also show that — „ = -. .
r-i sin ?.y
82. If the areas of the inlet- and outlet-orifices of an inward- or
outward-flow impulse turbine are equal, show that the efficiency of the
turbine is cos2 y , y being the angle which the direction of the entering
water makes with the wheel's periphery.
83. A radial impulse turbine of 4.5 ft. and 4 ft. external and internal
radii passes 8^ cub. ft. of water per second under an effective head
of 560 ft. The direction of the entering water is inclined at 17° to the
wheel's periphery, and the wheel has the same depth at the inlet- and
outlet-surfaces. If the peripheral speed at the outlet-surfe (w2) is equal
to the relative velocity of the water (F2) with respect to the wheel, find
(a) the efficiency, (b] the speed of the turbine in revolutions per minute,
(c) the sectional areas of the stream at inlet and outlet, (d) the direction
of the vane-outlet edge, (e) the velocity of the water as it leaves the
wheel, (/") the power of the turbine.
Ans. (a) .873; (£) 209.94; (c) .15357 sq. ft.; .13651 sq. ft.;
(d) ft = 45° 2' ; (e) 67.39 ft. per sec.; (f) 472.33 H.P.
84. In the preceding question examine how the results will be
affected when hydraulic resistances are taken into account, allowing .94
as a coefficient of velocity for the water on entering the wheel, and
assuming that the head equivalent to the relative velocity ( Fa) on
leaving the wheel is increased by 10 per cent.
Ans. (a) .886; (£) 193.76 revols. per minute;
(c) .163 sq. ft.; .145 sq. ft.;
(d) ft = 46° 1 8'; (*) 63.842 ft. per sec.;
(/) 479-39 H.P.
85. A radial outward-flow turbine of the impulse type passes 8^ cub.
ft. of water per minute under an effective head of 560 ft.; the width of
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 327
the wheel is 7^ in.; the radius to the outlet-surface is 1.15 times the
radius to the inlet-surface ; the linear velocity of the inlet-surface is 87
ft. per second ; the direction of the water at entrance makes an angle of
17° with the wheel's periphery. Find (a) the efficiency, (b} the lip-
angles, (c) the areas of the inlet- and outlet-orifices, neglecting first
hydraulic resistances, and second taking these resistances into account.
Ans. First, (a) .878; (b) a = 149° 31' and ft = 33° 20';
(c) .1535 sq. ft. and .1283 sq. ft.;
Second, (a) .826; (b) a — i48°and ft = 17° 19' ;
(c) .183 sq. ft. and .306 sq. ft.
86. Construct a Fourneyron turbine for a fall of 5 ft. with 30 cub. ft.
of water per second, a. = 80°, y = 30°, — = 1.35. Assume u% = Fa,
and neglect hydraulic resistances.
Ans. ft = 16° 42' ; Ai = 4.29 sq. ft.; A* = 5.8189 ft. ; rj = .915 ;
if ri — 1.8 ft., then di = d* = .38 ft.
87. In an inward-flow turbine passing 400 gallons of water per
minute, the slope of the guide-vane lips is i in 5, the radii to the inlet-
and outlet-surfaces are i ft. and 6 ins., respectively; the breadth of the
inlet-orifices is 1.25 ft. Find the efficiency. Ans. .98.
88. An I. F. turbine, of 4 ft. external diameter, works under an
effective head of 250 ft. Find the speed of the wheel in revolutions per
minute. Ans. 427.
89. An I. F. turbine of 4 ft. external and 3 ft. internal diameter,
makes 360 revolutions per minute. The sectional area of flow is 3 sq.
ft. and is the same in every part of the turbine. The direction of the
inflowing water makes an angle of 30° with the wheel's periphery.
Assuming that the whirling velocity at the outlet-surface is nil, find
(a) the efficiency, (b) the H.P., and (c) the delivery in cubic feet per
minute. The total head is 200 ft. Ans. (a) .86 ; (<£) 2476.8 ; (c) 7593.
90. An inward-flow turbine being required for an available head of
20 ft. and a discharge of 800 cub. ft. per minute, determine (a) the size
and (b) the speed of the wheel, (c) the inclinations of the guide and
wheel-vanes, and (d} the efficiency of the turbine, assuming r2 = \r\ =•
depth of wheel ; vr' = ^ \/2g-ff; v^," = o and a. = 90°.
Ans. (a) r2 = .487 ft., ri = .974 ft.;
(b) 240 revolutions per minute;
(c) y = 10° 21', ft = 36° 8' ; (d) 93! per cent.
91. A vortex turbine passes Q cub. ft. of water per second under an
effective head of H ft. The inlet-lip of the vanes is radial and the
direction of the entering water makes an angle of 30° with the wheel's
periphery. The areas of the inlet- and outlet-orifices are — — and -
o C
328 HYDRA ULICS.
respectively, and the width of the wheel is — , D being the diameter of
the inlet-surface. If the whirling velocity at the outlet-surface is nil,
find (a) the efficiency, (b) the direction of the outlet edge of the vane,
(c) the velocity with which the water enters and leaves the wheel ;
(d) the speed of the wheel in revolutions per minute, (<?) the diameters
of the inlet- and outlet-surfaces.
Ans. (a) .938; (b) /3=24° 17'; (V) 6.3291^ ; 1.977 HI \
(d) "6.7^; (*) -896 j|; .yijrfj.
92. A vortex turbine passes 1 1 cub. ft. of water per second under
a head of 35 -ft.; the diameter of the outlet-surface is 2 ft. and its
breadth 6 in. Find the power of the turbine, disregarding friction and
assuming that the whirling velocity at the outlet-surface is nil.
Ans. 43.5 H.P.
93. An inward-flow turbine has an internal radius of 12 in. and an
external radius of 24 in.; the water enters at 15° with the tangent to the
circumference, and is discharged radially; the velocity of outer periph-
ery of wheel is 16 ft. per second. Find the angles of the vanes at the
inner and outer circumferences, and the useful work done per pound of
fluid. Ans. ft = 32°, a = 118° I'; 9.33 ft.-lbs.
94. A radial impulse turbine passes S^ cub. ft. of water under an
effective head of 560 ft. The direction of the entering water is inclined
at 17° to the wheel's periphery. The linear speed of the inlet-surface is
87 ft. per second. Assuming that the velocity of whirl at the outlet is
nil, and disregarding hydraulic resistances, find (a) the efficiency, (b) the
velocity with which the water enters the wheel, (c) the velocity of the
water as it leaves the wheel, (d) the sectional areas of the inflowing and
outflowing stream, (<?) the direction of the vane-lip at inlet, (/) the
power of the turbine.
The radii of the inlet- and outlet-surfaces are 4^| ft. and 4$ ft. respect-
ively. Find (g) the direction of the vane edge at outlet.
Ans. (a) .879; (b) 189.31 ft. per sec. ; (c) 65.86 ft. per sec.;
(d} .15356 sq. ft. ; .129 sq. ft.; (e) a = 149° 34';
(7)47543 H.P. ; (g) ft = 33° 21'.
95. In the preceding example show how the results are affected by
taking .94 as the coefficient of velocity in calculating the velocity with
which the water enters the wheel.
Ans. (a) .828; (b) 178.49 ft. per sec.; (c) 78.36 ft. per sec. ;
(d) .163 sq. ft. ; .1085 sq. ft.; (e} a = 148° 3' ;
(/) .441.25 H.P.; (£•)/* = 38" 4'*
96. In an I. F. turbine the radius to the inlet-surface is twice that to
the outlet-surface ; the linear velocity of the inlet-surface is one half
that due to the head ; the water enters the wheel with a velocity of flow
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 329
{vr') equal to one eighth that due to the head, and the sectional area of
the water-way is constant from inlet to outlet. Find the angle between
the discharging-lip of the vane and the wheel's periphery, the whirling
velocity at the outlet-surface being nil. Ans. Cot"1 4/8.
97. In a vortex turbine the depth of the inlet-orifices is one eighth of
the diameter of the wheel (= ~] and — of the depth of the outlet-
\ 8 / 32
orifices. The width of the wheel is one tenth of the diameter ( = — -).
\ io/
The inlet-lip of the vanes is radial, and the water enters at an angle of
30° with the inlet periphery. Find the size, speed, and efficiency of the
turbine in terms of the supply of water Q and the effective head H.
Also find the direction of the outlet edge of the vanes.
Q^
Ans. I. Assume vw" = o. Then r\ = .448 — -, ;
H*
No. of revolutions per minute = 1 16.7 — r ;
r; = .938; /3 = 24° 17'.
Qk
II. Assume w2 = F2. Then r\ — .45 — - ;
H 4
H*
No. of revolutions per minute == 116. -3 — r
Qk
77 = .935; $ = 26° 49'.
98. A vortex turbine with a wheel of 2 ft. diameter and 6 in. breadth
passes io cubic ft. of water per second under a head of 32 feet. Find
the inclination of the guides and the power of the turbine. Assume
as = V?. and a = 90°. Ans. 5° 41', 36T4T H.P.
99. An inward-flow turbine has an internal radius of 12 in. and an
external radius of 24 in. ; the water enters at 15° with the tangent to the
circumference, and is discharged radially; the velocity of radial flow is
5 ft. at botn circumferences; the velocity of outer periphery of wheel is
16 ft. per second. Find the angles of the vanes at the inner and outer
circumferences, and the useful work done per pound of fluid.
Ans. ft = 32° ; d = 1 18° i' ; 9.33 ft.-lbs.
100. A radial I. F. reaction turbine, with or without draught-pipe,
passes 113 cub. ft. of water under an effective head of 13 ft. The radius
to the inlet-surface is 1.169 times the radius to the outlet- surface, and the
ratio of the outlet to the inlet area is .92. The vane-lip at outlet makes
an angle of 15° with the wheel's periphery, and the water enters at an
angle of 1 2° with the wheel's periphery. The sectional area of the draught-
tube (if there is one) at the point of discharge is 1.035 times the sectional
area of the outlet-orifice. Show that the effective work per pound of
33° HYDRA ULICS.
water is 13 ft.-lbs., and that the work consumed in hydraulic resistance
(Art. 28, page 303) is nearly 1.96 ft.-lbs. ; also find Ai, A^ z/a, and the
efficiency.
Ans. (a) 28.26 sq. ft. ; 26.12 sq. ft. ; (b) 4.4 ft. per sec. ; .977.
101. In the preceding example, if the radius to the inlet-surface is 4
ft., find (a) the speed of the wheel in revolutions per minute. Also find
(b) the depth of the wheel at inlet and outlet, the guide-vanes being 40
and the wheel-vanes 41 in number, and the thickness of the former being
T\ inch and of the latter J inch. Ans. (a) 38.95 ; (b) 1.23 ft. ; 1.32 ft.
102. In example 38 find the efficiency if the diameter of the draught-
tube is made the same as the diameter of the outlet-surface, the lower
edge of the tube being rounded. What will be the "loss in shock" in
the tube per pound of water? Ans. .861 ; .071 ft.-lbs.
103. An axial-flow turbine is to be used for raising water. Explain
how the vanes should be arranged, write down the resulting equations,
and determine the efficiency.
104. Write down the equations for a Jonval modification of Euler's
turbine.
105. An inward-flow turbine has an external diameter of 3 ft. and an
internal diameter of 2 ft. It passes 12 cub. ft. of water per second
under an effective head of 40 ft. The water enters the wheel at an
angle of 30° with the wheel's periphery, and the depth of the outlet-
orifices is twice the depth of the inlet-orifices. The efficiency of the
turbine is .9. Disregarding friction, find (a) the vane-angles at inlet and
outlet, (b) the velocity with which the water leaves the wheel, (c) the
speed of the turbine in revolutions per minute, (d) the velocity with
which the water enters the wheel, (e) the areas of the outlet- and inlet-
orifices, (/) the power of the turbine.
Ans. (d) a = 105.09', ft = 35° 35' (b) 16 ft. per sec. ; (c) 198.39;
(d) 42! ft. per sec. ; (e) .5625 sq. ft. ; .75 sq. ft. ; (/) 49^ H.P.
106. A centrifugal pump with a 12-in. fan delivers 1000 gallons per
minute, the actual lift being 20 ft. and the gross lift (allowing for fric-
tion, etc.) 30 ft. The velocity of whirl at the outlet-surface is reduced
one half. Find the revolutions of the pump per minute. Ans. 623.1.
107. In a centrifugal pump the external diameter of the fan is 2 ft., the
internal i ft., and the width 6 in. Determine the speed and efficiency
of the pump when delivering 2000 cub. ft. per minute against a pressure
head of 64 ft., the inclination of the wheel-vanes at outlet-surface
being 90°. Ans. 643.36 revols. per min. ; .656.
108. A centrifugal pump delivers 1500 gallons per minute. Fan, 16 in.
diameter; lift, 25 ft. ; inclination of vanes at outer periphery to the tan-
gent, 30°. Find the breadth at the outer periphery that the velocity of
whirl may be reduced one half, and also the rev6lutions per minute, as-
suming iht gross lift to be i£ times the actual lift.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 331
Also find the proper sectional area of the chamber surrounding the fan
for the proposed delivery and lift. Examine the working of the pump at
a lift of 15 ft. Ans. Breadth, f in.; revolutions, 700; 24 sq. in.
109. For a given discharge (0 and head (77), and considering only the
losses of head due to flow and to the resistance in the wheel, show that
the maximum efficiency of a centrifugal pump of chamber D is
A being a constant depending on the size of the wheel.
no. A centrifugal pump lifts 35 cub. ft. of water per second a height
of 20 ft. At the outer periphery the vane-angle (ft) is 15° and the radial
velocity is 5 ft. per second. If the wheel makes 140 revolutions per
minute, find (a} its diameter. If the diameter of the outer periphery of
the wheel is three times that of the inner periphery and if the radial
velocity at the latter is 8 ft. per second, find (b) the vane-angle at the in-
ner periphery and (c) the depths of the wheel at the inner and outer
peripheries. . Ans. (a) 5^ ft. ; (b) 30° 58'; (c) n.i in.; 5.76 in.
in. The pump in the preceding example is supplied with a vortex
chamber of 6£ ft. diameter. Show that the "gain of head" is a maxi-
mum when the velocity of flow in the volute is 8.46 ft. per second. Also
show that the frictional loss of head is 4.1785 ft.
112. In a centrifugal pump the diameter of the fan = 16 in. , the depth
= 2 in., the lift = 25 ft., and the delivery = 300 cub. ft. per minute.
Determine (a) the speed, (b) the efficiency, and (c) the power expended
when the vane-angle (ft) at the outer periphery is (i) 90°, (2) 45°, and (3)
30°. Ans. (i) (a) 785 revols. per min. ; (b) .47 ; (c) 30 H.P. ;
(2) (a) 805.8 •• " " (b) .58 ; (f) 24.4 H.P. ;
(3) (a) 846.1 " «« " (b) .68 ; (c) 22.9 H.P.
113. An Appold pump delivers 10,000 gallons per minute. The gross
lift is 50 ft. The radial velocity at the outlet-surface is one eighth of
that due to the. gross lift, and the velocity of whirl and the peripheral
velocity are reduced one half. Find (a) the radius of the wheel, (b) the
vane- angles, (c} the speed of the wheel, (d) the efficiency.
Take the breadth of the wheel at outlet equal to one sixth of the ra-
dius, and^- = 32.
Ans. (a) 1.9 ft. ; (b) 56° 16' ; 23° 16; (c) 331 revols per min. ; (d) .74.
114. The internal and external diameters of the fan of a centrifugal
pump are 9 in. and 18 in., respectively ; the depth is 6 in., and it passes
400 cub. ft. per minute against a pressure head of 16 ft. The inclination
(ft) of the discharging-lips of the fan being 30°, determine (a) the speed,
(b) the efficiency, (c) the power expended, and (d) the inclination of the
receiving-lips of the fan.
Ans. (0)413.58 revols. per min. ; (b) .571 ; (c) 21.23 H.P. ; (d) 19° 48'.
3 3 2 HYDRA ULICS.
Find the efficiency when a vortex chamber 36 in. in diameter Sur-
rounds the fan. Ans. .581.
115. A centrifugal pump with a gross lift of 17 ft. delivers 25 cub. ft.
of water per second. At the outer periphery the vane-angle is 80° and
the radial velocity is 5 ft. per second. The diameters of the outer and
inner peripheries of the disc are 54 in. and 18 in., respectively, and the
hydraulic efficiency is .75. Find (a) the speed of the fan, (b) the vane
angle at the outlet periphery, (c) the velocity of flow in the volute,
(d) the diameter of the volute, (<?) the diameter of the suction-pipe.
If there are six £-in. vanes, find (/) the width of the disc at the outer
and inner peripheries.
Assuming the discharge-pipe to be 4 ft. per second, show that there is
a loss of 5.026 ft. of head due to hydraulic friction.
Ans. (a) 116 revols. per min. ; (b) 41° 14' ; (c) 26.0 ft. per second ;
(d) 14.7 in. ; (e) 33.8 in. ; (/) 9.64 in. ; 4.8 in.
116. The vane of a centrifugal pump or turbine is the involute of a
circle concentric with the pump circumference. Show that V\ = V* in an
I. F. or O. F., and '£ = - in a D. F.
Vt rz
117. A race is straight and close fitting so that the loss of effect due
to escape of water may be disregarded. A single undershot wheel with
plane floats is replaced by four similar tandem wheels. If the delivery
of each of the four wheels is the same, and if it is assumed that the
water reaches each wheel with the same velocity with which it leaves the
preceding wheel, find the total maximum velocity due to impact.
Ans. i£ times the delivery of the single wheel.
118. Discuss the preceding example, assuming that the delivery of
each wheel is not the same, but that the total delivery is a. maximum.
Ans. 1.6 times the delivery of the single wheel.
119. If n wheels of the same type are substituted for the single wheel
in example 117, and if the assumptions are the same as those in example
1 1 8, show that the total delivery of the n wheels is to the delivery of the
single, wheel in the ratio of 2« to in + i, and that, theoretically, if the
number is made very large, they will approximately give the entire work
of the fall.
INDEX.
Abbott, 148, 151
Abrupt changes of section, loss of
head due to, 89
Accumulator, 215
Air in a pipe, no
Aqueducts, flow in, 142
Arc of discharge in overshot wheel,
256
Axial-flow turbine, 282
Barker's mill, 272
Barlow curve, 47, 50
Barometer, water, 5
Bazin, 145, 152, 154, 166
Bazin's velocity curve and formula, 154
Beard more, 134
Beaufoy, 7
Bends, loss of head due to, 92
Bernouilli's theorem, 6
" applications of, 9
Bidone, 23, 36, 166
Boileau, 153
Boileau's velocity curve and formula,
154
Borda, 36
Borda's mouthpiece, 34
Bossut, 227
Bovey's tables of coefficients of dis-
charge, 24, 25
Boyden's hook gauge, 173
Branched pipe connecting three reser-
voirs, in
Branch main of uniform diameter, 101
Breast-wheel, 225, 242
efficiency of, 250
losses of effect in, 248
mechanical effect of, 247
252
speed of, 246
Broad-crested weir, 58
Bucket, forms of, 240, 250, 265
Buckets, number of, 264
Buff, 26
Canal-lock, time of emptying and fill-
ing a, 29
Capillary tubes, flow in, 97
Castel's table of coefficients of dis-
charge, 45
Centrifugal force, effect of> 255
head in turbine, 298
pumps, 307
" theory of, 309
" vortex-chamber in,
309, 313
Chamber, whirlpool, 50
Channel, bottom velocity of flow in a,
154
flow in an open, 131
form of, 135, 136
" maximum velocity of flow in
a, 150, 153
mean velocity of flow in a,
151, 154
mid-depth velocity of flow in
a, 1ST
" steady flow in a, 132
" surface velocity of flow in a,
150, 15-1
value of yin a, 144
variation of velocity in a sec-
tion of a, 148
Channels, differential equation of flow
in, 159
examples of, 162
" of constant section, steady
flow in, 132
" of varying section, flow in,
156
surface slope in, 160, 161
Chezy's formula, 88
Cock in cylindrical pipe, 93
Cocks, loss of head due to, 93
Coefficient of contraction, 22, 89
" discharge, 24
" friction, 73, 144
" " resistance, 21
333
334
INDEX.
Coefficient of velocity, 20
Combined-flow turbines, 284
Compressibility, 2
Continuity, 2, 5
Contraction, imperfect, 22
incomplete, 23
loss of head due to ab-
rupt, 89
Coulomb, 72
Critical velocity, 97
Cunningham, 148
Current-meters, 180
Darcy, 72, 74, 75, 97, 148, 154
gauge, 176, 178
D'Aubuisson, 74
Density, 2
Downward-flow turbine, 282
Draught-tube, theory of, 301
Dubuat, 154
Elasticity of volume, 3, 4
Elbows, loss of head due to, 91
Ellis, 106
Energy lost in shock, 32
" of fall of water, 4
" jet of water, 27
" transmission of, 84
Enlargement of section, loss of head
due to, 32, 91
Equations, general, 30
Equivalent uniform main, TOO
Erosion caused by watercourses, 136
Examples, 60, 122, 170, 209, 315
Exner, 183
Eytelwein, 74, 134
Floats, sub-surface, 175
" surface, 175
" twin, 175
Flow from vessel in motion, 16
in a frictionless pipe, 18
" in aqueducts, 142
" influence of pipe's inclination
and position upon the, 83
" in pipes, 78
" in pipe of uniform section, 86
" " " of varying diameter, 98.
Fluid friction, 70
" motion, I
Fourneyron's turbine, 281
Francis, 176
Friction, coefficients of, 70, 73, 74, 75
in pipes, surface, 73, 97
laws of fluid, 72
Froude, u, 13, 70, 76, 97
Froude's table of frictional resistances,
70
Ganguillet, 147
Gauge, Darcy, 176, 178
Gauges, experiments on, 148
Gauging, method of, 173
Gaugings on the Ganges, 148
" " Mississippi, 146
General equations, 30
Gerstner's formula, 229
Graphical representation of losses of
head, 94
Grassi, 3
Head, 2, 3
Herschel, 184
Hook gauge, Boy den's, 173
Humphreys, 148, 151
Hurdy-gurdy, 279
Hydraulic gradient, 10
mean depth, 133
" radius, 80
" resistances, 20
Hydraulics, definition of, i
Hydrodynamometer, Perrodil's, 183
Hydrometric pendulum, 183
Impact, 1 86
on a curved vane, 199
on a surface of revolution, 192
on a vane with borders, 195
Inclination, influence of pipe's, 83
Injector, 12
Inward-flow turbine, 282
Jackson, 148
Jet, energy of, 27
inversion ot, 27
momentum of, 27
propeller, 191
reaction wheel, 272
efficiency of, 274
" useful effect of,
274
Kutter, 147
Laminar motion, 2
Lesbros, 27
Limit turbine, 283
Loss of energy in shock, 32
Loss of head due to abrupt change of
section, 89
" " " " bends, 92
" *' " " cocks, 93
" " " " contraction of sec-
tion, 89
" " elbows, 91
" " " enlargement of
section, 91
IAD EX.
335
Loss of head due to orifice in dia-
phragm, 90
" " " " " sluices, 93
" " »*••"« valves, 93
Losses of head, graphical representa-
tion of, 93
Magnus, 27
Main of uniform diameter, branch, 101
" with several branches, 118
Meters, 180
" inferential, 184
" piston, 184
" rotary, 184
Meyer, 156
Miner's inch, 26
Mississippi, experiments on, 148
Mixed-flow turbines, 284
Motion, fluid, I
" in plane layers, 2
" in stream-lines, 2
" laminar, 2
" permanent, I
" steady, I
Motor driven by water flowing along a
pipe, 107
Mouthpiece, Borda's, 34
convergent, 44
cylindrical, 39
divergent, 42
ring-nozzle, 37
Navier, 149
Notch, 54
" circular, 55
" rectangular, 54
triangular, 56
Nozzles, 104
" Ellis's experiments on, 106
Open channels, 131
Orifice fed by two reservoirs, 115
" flow through an, 16
:-•;*' in a diaphragm, loss of head
due to, 98
in a thin plate, 13
" in vertical plane surfaces, 50
with a sharp edge, 14
Orifices, circular, 53
large, 50
" rectangular, 50
Outward-flow turbine, 281
Overshot wheel, 225, 254
" arc of discharge in,
256
bucket angle of, 262
" " division angle in, 262
Overshot wheel, effect of centrifugal
force in, 255
" effect of impact on,
270
" " weight on,
268
" number of buckets in,
262, 264
" pitch-angle in, 262
" " speed of. 254
" useful effect of, 268,
271
weight of water on,
256
Parabolic path of jet, 16
Pelton wheel, 280
Permanent regime, i
Piezometer, 9
Pipe connecting three reservoirs,
branched, in
two reservoirs, 86
" of rectangular section, sluice in,
93
" uniform section, flow in, 78
" " varying section, 18, 98
Pitch-back wheel, 272
Pilot tube, 176
Plane layers, motion in, 2
Poiseuille, 96, 97
Poncelet, 27, 227
Poncelet's wheel, 232
Position, influence of pipe's, 83
Pressure-head, 4
Prony, 74, 134
Pumps, centrifugal, 307
" " theory of, 309
'•* vortex - chamber
in, 309, 313
Radiating current, 46
Rayleigh, Lord, 27
Reaction, 190
Reaction wheel, efficiency of, 274
" " jet, 272
Regime, permanent, I
Reservoirs, Branched pipe connecting
three, in
orifice fed by two, 115
" pipe connecting two, 86
Resistance of ships, 76
" to flow, law of, 96
Revy's meter, 181
Reynolds, 97
Ring-nozzle, 37
River-bends, 143
Sagebien wheels, 254
336
INDEX.
Schiele turbine, 284
Ships, resistance of, 76
Siphon, 108
" inverted, 109
Slotte, 156
Sluice in cylindrical pipe, 93
" in rectangular pipe, 93
loss of head due to a, 93
Sluices, 244
Standing wave, 165, 232
Steady flow in channels of constant
section, 132
Steady motion, i, 132
" in pipe of uniform
section, 78
Stream-line, 2
Suction-tube, theory of, 301
Surface-floats, 175
Surface-friction in pipes, 73
•' slope in channels, 160, 161
Table of bottom velocities, 155
" Castel's results, 45
" " coefficients of discharge, 24,
25, 45
" friction, 73, 75
" " velocity, 23
" " density of water, 3
" " discharge through nozzles,
1 06
" " elasticity of volume of water,
4
" " Ellis's experiments on nozzles,
106
" " frictional resistances, 70, 73,
.74, 75
" maximum velocities, 155
" " values of f, 147
" " " " " Bazin's, 166
" '' viscosity of water and mer-
cury, 155
Table of Weisbach's values of Cv, 33
Theory of suction or draught tube, 301
" " turbines, 284
Thomson, James, 50, 143
Thomson's turbine, 282
Throttle valve, loss of head due to, 83
Time of emptying and filling a canal
lock, 29
Torricelli's theorem, 14
Torricelli's theorem applied to the
flow through a frictionless pipe of
gradually changing section, 18
Transmission of energy by hydraulic
pressure, 84
Turbine, axial-flow, 282
" centrifugal head in, 297
" combined, 284
Turbine, efficiency of, 288, 291, 292,
296, 297
Fontaine's, 282
impulse or Girard, 276
inward-flow, 282
Jouval, 282
limit, 283
losses of effect in, 303
mixed-flow, 284
outward-flow, 281
parallel-flow, 282
practical values of velocities
in, 299
radial flow, 281
Schiele, 284
Scotch, 276
theory of, 284
Thomson, 282
useful work of, 292, 296, 297
ventilated, 278
vortex, 50
Undershot wheel, 225
" actual delivery in ft. -
Ibs. of, 231
depth of crown of,
238
efficiency of, 227,
235, 239
form of course of,
236,
in a straight race, 227
" losses of effect with,
228
" modifications to in-
crease efficiency
of, 231
" number of buckets
in, 238
" Poncelet's, ^32
" Poncelet's efficiency
of, 235, 239
useful work of, 228,
235
with flat vanes, 227
Uniform main, equivalent, 100
Unwin, 97
Valve, loss of head due to a, 93
Vane, best form of, 199
" cup, 195
Velocity, bottom, 151, 154, 155
" critical, 97
curve in a channel, 152, 154,
" formulae, 150, 152, 154
Bazin's, 152
Boileau's, 153
" maximum, 151, 155
INDEX.
337
Velocity, mean, 151, 154
" mid-depth, 151
" of flow, 286
of whirl, 286
rod, 176
" surface, 150, 154
" variations of, 119, 131, 148
Velocities in turbines, practical values
of, 299
Vena contracta, 14
Ventilated buckets, 272
Venturi water-meter, 13, 183
Virtual fall, 82
" slope, 10, 82, 84
Viscosity, 96, 97, 119, 149
Meyer's formula for, 156
Slotte's " "156
Vortex-chamber in centrifugal pump,
309, 313
Vortex, circular, 47
" compound, 50
free, 47
" free-spiral, 48
" forced, 49
Vortex, motion, 47
" turbine, 50, 282
Water-barometer, 5
Water-meter, 13
Weight of fresh water, 2
" " ice, 2
" " salt water, 2
Weir, 54
" broad-crested, 58
" rectangular, 54
Weisbach, 23, 26, 36, 76, 90, 91, 92, 93,
145
Wheel, breast, 242
hurdy-gurdy, 279
jet reaction, 272
overshot, 254
Pelton, 280
pitch-back, 272
Poncelet's, 232
Sagebien, 254
undershot, 225
Whirlpool-chamber, 50
Whirl, velocity of, 286
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