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.
I6 S
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equation no longer represents the true condition of flow.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 - 02 3
" 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 o w .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 o w .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.
i6 7
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 cd t 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 [ == im P ulse in same direction.
Hence
w
and therefore
=A 1 y l - A,v v ... (9)
the depths below the surface of the centres of
gravity of the sections ab, cd, respectively.
1 68 H YDRA ULICS.
Now, v l = u l + V r 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
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 H l at
ab and H t 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 H l 2
In Bidone's canal, u 1 = 1^.69, H l = 0^.064. Substituting
these values in equation II, the value of H^ is found to be
o w . 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, , <9 2 . 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 O l signals
the observer at (9 2 , who measures the angle O^O^E, and each
marks the time. On reaching CD the observer at O. t signals
the observer at O l , 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
(v t ) of the combination is approximately the
mean of the surface-velocity (v s ) 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 v d can be determined as soon as the value of v t has
been observed and the value of v s 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 (v m ) is approximately the
mean velocity for the whole depth.
Francis gives the following empirical
formula connecting the mean velocity
(v m ) with the observed velocity (v r ) of
the rod :
...*/),
=zv(i.oi2
d being depth of stream, and d' the depth FlG - I0 5-
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 k f 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 = k v -\- 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
V Q 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
a t , # a being the sectional areas at A and B, respectively, and
ff t H l 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, . . . ( 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* 2wc c c v Ah 2wAh, . . . (i)
o e
disregarding the contraction and putting c v 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 v l 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(v l -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 v l 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 v t 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 v l 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* = (v l uj + u* - 2(z/ 1 *) cos (?r /?),
from which
/?
z/j 3 v* = 2^(^j w)(i cos /?) = 4(z/ u) sin 2 . (i)
Also, -A(v l u) = the quantity of water reaching the
.
Au(v l u) sm
sm^. . (4)
W -V v* 2
A
194 HYDRAULICS.
This is a maximum when
, = 3, > - " ... (5)
and therefore
the maximum efficiency = sin 2 -. . . . (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)sm 9 ~', (8)
Avfa w)sin a ; (9)
u(v l u} . 2 ,/?
4 " V* 2
The efficiency is a maximum when
v.
(ii)
Q
its value then being sin a .
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 sin 8 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
= mv t cos T mv^ cos tf 3= mu cos
= m(v l cos ^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 v 1 = Vf Then
N = m(v^ cos 6 -\- v l cos 6)
w
= 2Av? 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 C l C l>
3 , where the stream-lines are again parallel.
At Ci let A , A l , v l , z l 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 Cf v
Let / 3> A 19 v lt * be corre-
sponding symbols at C 9 C 3 .
Let a be the area of the plate.
Let c c be the coefficient of contraction.
Neglect the skin and fluid friction between C l C l 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 C 3 C 3 .
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 )A l + wA l (z l ^,) 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 = c c (A l a)Vr Therefore
=-*${&&>- >}
v? ( m } a
= wa m \ , r I [ ,
2g \ c c (m - i) j
A
where m = , or
a
R =
r m )
where K in \ , r I >
\ c c (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 C 9 C t . They then converge, flow in
parallel lines, and converge a second time at C 3 C 3 , leaving a
mass of eddies behind the body.
Consider the mass bounded by the planes C^ C t C 4 .
As in the previous article, let
/>,, A lt v l , z l 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 , ? . {^s - *$ ,
! * 4++ " t "
^"" being the loss of head between <7 a , and C,C 3 and
being the loss of head between C 3 C 9 and C t C t .
o
Hence
* i A A _ (^. ^) a I (^ - O a
J " 4-f ~^^ "IF IF"
But A& = ^ 3 e; a = ^ 3 ^,,
and A 3 = 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 c f
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 = m V * ~ V * . . . (4)
w A v?
The total available work = - A -- (5)
,, ~ . mu. LM v? v*
The efficiency -- = img -- r- ...... (6)
w A v? * 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
F 3 = CK* = CB* = AB* + AC* - 2AB . AC cos A
= v* -\-u* 2v ji cos A .......... (8)
Hence
L = u \ v t cos (A + 6) v l 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 sin 3 - = 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
v l ~~ 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. u y m = A(v. 11) ;
g
P = Y= A(v t u)\i cos a) = 2 A(v^ u) sin 2 -;
5 o
W &
useful work = Pu = 2 Ati(v, uY sin 2 ;
g 2
U(V. U}" OL
efficiency = 4 sin .
IMP A CT. 20$
This is a maximum and equal to sin 2 when v l = $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.; T V
At what angle should the jet strike the plane so that the efficiency
might be a maximum? Find the maximum efficiency.
Ans. sin 1 ; -..
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.; ().6 3 ; (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 = 2347'.
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 = i43'.
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 2348'; (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) 2btu(y u}* sin 5 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 h l 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 h l 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.
" W l 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 h f be the head consumed in frictional and other hy-
draulic resistances.
Then
W,(h' + h f } = 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, r t 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 __ 5r t
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 p l 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
F 2
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/ be the unit pressure at the accumulator-ram.
Let/ be the unit pressure in the working cylinder.
Then
/ & 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
v 1
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
F 2
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
F a
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 P t and
let v 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 V Q 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 L a 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 L 4 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 L 6 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 L t due to the Velocity with which the Water
leaves the Discharge-pipe.
A =
Hence
the effective head ==-- (L^ + A - A + A + L 6 + ),
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
P l = wA(m i) 2
*g
Let F be the friction of the slide. Then
o
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 22$
and P l -}- 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 v l 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(v l 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 = kwAv lt k being some coefficient (< i) to be
oletermined'by experiment. Then
^
the useful work per second = kAw l (y l 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 = -v l ,
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 h l 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 Q l 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 <2 2 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, s l = BC, when a float AB is in its lowest
position, Fig. 140, to a maximum, B l C l = CD=^C t , when
FIG. 140.
two floats A l B l , A^Bs are equidistant from the lowest position,
Fig. 140. Thus the mean clearance
= J(25, + BD) = 5, +-, nearly,
r l 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 = S l -\ --- *.
Again, the difference of head on the up-stream and down
stream sides
and the velocity of discharge, v d , 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 J a 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 , _ f 7
\ 2 " 2 gl ( 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 > v r 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 = v l 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 V 1 u and with an
absolute velocity v l 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 u l 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 # = v l V u = 2u(v l cos Y u\
2
Therefore the useful work per second
= ^ 2U fa cos y - u ) (s>
wQ v? cos 8 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,
V R 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 = v t 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 mbdu l ,
m being a fraction whose value may be taken to be J or f
Hence
mbdu l btv^ ,
and therefore
u. md u.
t = md = cos y
V l 2 r U
md R
= cos v .
2 r r,
According to Morin,
r, = 2d to $d.
The mean velocity at entrance = c v < 2g(H /), an aver-
age value of c v 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 = Aq f +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 (V 2 ). From Fig. 142,
ao (= \ag) _ sin y __ Q a )
ad sin aod u
Hence for any given values of u and y, v z 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,
i7 1 = 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 aofx. 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.
" m fa = the angle maf (180 fmd)
= "+-*- 45- '
Let r lt r^ be the radii of the outer and inner peripheries of
the wheel. Then
sin (f!L _ 45)
r l 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 k l 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 b l 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
v l 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 = 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, sin 2 -.
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 T L.
The head required to produce the velocity at entrance, v l9
(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
F 2 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.
= -(v l 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 . dv l = 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 v t = 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 = v l cos y,
and therefore
V = v l 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
F a v? . v> i if i
head = = sin 2 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 h l = 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 s 1 be the play between the ends of the buckets and the
channel.
Let s^ be the play at the sides. (^, = J a , approximately.)
Let z l , # 2 , . . . z n 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 , ... l n 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 w l = weight of water on the wheel.
Let r l = radius of wheel's outer periphery.
Let r 1 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=b Sl ( 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 W Q
\wQ c V2g 1 ! v l 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 v , 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 T Vths 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 h l 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 u l 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 ed t 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 be the inclination of Od to the horizon.
Let be the inclination of ad to the horizon.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 26 1
Let r l be the radius of the outer periphery.
Let R be the radius of the division circle.
Let r a be the radius of the inner periphery.
Then
od^ __ r l _ sin _ sin
oe ""^"sin j 90 0+0} ~ cos (0+0)'
and therefore
Again,
Therefore
sin
af = fd tan (0 -|- 0), approximately.
the area dfa=< tan (0 + 0) = tan (8 + 0),
2 2
where d = r l r 2 . Hence
the area abed = area cod area bof area ^/iz
Equations (i) and (2) give 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 and 0, and let
262 HYDRA ULICS.
y t 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., r l (sin l 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 dd r = 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
mbdu l 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
= / >
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=v l 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' = s 1 = ee r .
Let arc bM = s t .
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 , V i U 27Cr i V i U \
(Note.ef eb = eb- - = -^T. - - , nearly.)
\ J u u N u J i
Thus
and by equations 2 and 3,
( v i + v *' 2U \ _ 27tr i !!L
S \ 2u I ~ N u>
268
HYDRA ULICS.
an equation giving approximately the distance s l 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 h lt 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 -\- r l 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(h l + kh^ k being a frac-
tion < I and approximately = .5.
Let A 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 /z 2 , insert an even number s of
buckets, and let their water-areas A l , A 9 , A 3 , . . . A s be care-
fully calculated.
Let Q m be the mean amount of water per bucket in the
discharging arc.
Let A m be the mean water-area per bucket in the discharg-
ing arc.
Then
The value of k can now be easily found, since
Q m _A m
~-~"
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 =
useful work _ v V
The reaction = linear ve i oci t y 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,
= j t | 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-
J 73. 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
2 7 8
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,, r a be the radii of the wheel inlet and outlet surfaces
or an I. F. or O. F.
Let r lt r t 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 A lf 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 d lt d t be the depths of the same in an I. F. or O. F.
Let d lt d^ be the widths of the same in an A. F.
Let h be the thickness of the wheel in an A. F.
Let H l 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 H 9 is negative.
Let v lt v t be the absolute velocities of the water at the
inlet- and outlet-surfaces.
Let u lt #, 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 ac t making
an angle y with the tangent ad. Take ac to represent v l and
ad to represent u lt Complete the parallelogram bd. The side
ab represents V lt 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, v r ' at the inlet-surface ;
v' i v r " 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 ^ ;# = V l sin or ; (2)
and at the outlet-surface
vj' = z> a cos 6 =fn =.fk kn = u 9 V^ cos /? ; . (3)
v r " = ^ 2 sin 6 = /* F 2 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 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 - I8 9- 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,
A l -fad l \2nr l nt cosec y n v t l cosec a\ . . (6)
and
^. = TWi 2 ^.- *i** cosec ft\> ...... (7)
T 9 being a fraction depending on practical considerations.
288
HYDRAULICS.
In an axial-flow turbine R is to be substituted for r l ind r y
in the values of A l and A 9 .
n l 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 - v w "r t )
since
since
Ui 2
r*= = > - - (9)
'1 * 2
and the hydraulic efficiency
T v w 'ui - z>"w a
and the hydraulic
r/ f
\Viv
efficiency
-^")i / I0 x
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* 2v l u l cos y, . . . . (ii)
and
sn y
sin of
(12)
From the triangle ./M,
, 1 = , 1 +F, 1 -2,r,COS/ (I 3 )
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 k l
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.
W t 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 V 9 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., v t 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 (v m ") at the
outlet-surface equal to nil, or to make the relative (F 2 ) and
circumferential (u 9 ) velocities at the outlet-surface, equal and
opposite. In each case v 9 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 = V 9 COS ft. (2O)
\v 2 -v' r V 2
' /
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
H l = 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 - v w ' = o, . . . (27)
and therefore
a = 90
and
Vi Vr
tan y = ,.
Also, the efficiency
and thus
W Q\H 1 + h =-J= effective work
wQ , wQ . .
=v-wUi = UM cos y, (24)
S
and therefore
^i + h - r 1 = ^~ 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 (v w f )
relatively to the wheel at the
inlet-surface, then
Ul - v w ' = o, . . (27)
and therefore
a = 90
and
Also, the efficiency
an thus
(28) uS = g(Hi 4-
. (28)
2 9 2
HYDRA ULICS.
if the efficiency is perfect.
Usually the efficiency of
good turbines is about .85.
Velocity of Efflux.
Z'a 2 = z/a 5 tan 2 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/ 2 2 = 2 2 tan 2 ft
2g(ffi 4- A) tan ft
tan ft -j- 2 cot y
Useful Work
2-^- cot y
-wQfft li . ( 3 o)
tan /?-(- 2-^- cot^
2 cot ^
^>/ TT \ r\ ** / V
tan /3-}- 2-f- cot y
Efficiency
2 ^ co. r
-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
- 27rr 2 2 A / ~ ^ (33)
y tan ^ -(- 2-^ cot y
7^^ pressure-head at the in-
Jff^urfnr.f
27/ft?i . / . G3>
y tan ft-{-2co\.y
The pressure-head at the in-
let-surface
2g
,.) r a V,
Hl< 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
'* sin 2 ;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 p l >
At Le -' if
ffi -- h h ,
w w
that is, if
tan ft
IN IMPULSE TURBINES.
In an I. F. or O. F.:
Speed of Turbine.
Since
V? = 2gff 1 , . . (35)
by equation 22,
riVi 8 sin 2 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 sin 2 y _ uf_ _ uf_
d tan 2 ft ~ z/i 2 ~" z/x a '
and therefore
2 2 = Wj 2 = 2gffi \ ^"a^- (3 6 )
Velocity of Efflux.
, 2 = 7/ 2 2 tan 2 ft
' (37)
294
HYDRA ULICS.
Useful Work
H v '\
~^j
Efficiency
- r ^- s{D ' r= '>- (39)
Work
-/r,g-.in'r). (38;
Efficiency
\ -ij- sin a 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/? r a 2. (43)
27T
In an A. F.
= i si
= F 3 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 _ w 2 _ ^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 = UiV w '. (46)
Also,
Wl sin (a + y}
Hence,
sin a
+
cot a tan X)
- tan ^ cosec ^. (47)
. If the velocity of
whirl (^ w r ) 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 - v w ' = o, . . (48)
and then
f A). (49)
HYDRAULICS.
Velocity of Efflux.
By equations 42 and 47
. ft
8 2 sin 2 -
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
=. 2itridiv r " = 27Tr a a F a sin ft
= 2itr / 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
sin 2 -
(50)
Useful Work
= Q(ffi + h){ i - ~ tan ^ tan y \ (51)
Efficiency
=I * 2 i<&) =I -| tan ^ an7 '- (52)
Amount Q of Water passing
through Turbine
= inRdiVr" = 2itRd/,
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, - h 1 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 2 2 sin 2 --
2.
*-$# ft - .
cos 2
2
Efficiency
. (59)
In an A. F. :
Speed of Turbine.
Since
(55)
^ ^. . . (56)
Velocity of Efflux.
. (57)
Useful Work
-^ cog2 r Ms8)
( 3S 2 )
Efficiency
H\ d^ sin 2 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 d l , however, must not be too small, as there might
be a loss of energy due to a contracted section at entrance,
while if d z 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 F 2 ,
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 = F 2 , or vj r = 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. w a < u, , and the term L is negative.
Hence the velocity v l 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^ > u l and the centrifugal
head is positive. The velocity v l 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 v r (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., v r ' = - ; v r " = .2iv to .172; ; u, = -u^ = .$6v.
4 r i
In A. F., v r r = v r " = .i$v to .2v ; u, = u 9 = -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 - yA J sin y to 2 yA t sin y if A^ sin y < 2 square feet ;
from --'\fA 1 sin y to -\A4,sin y\i A 1 s\ny > 2sq. ft.< l6sq. ft.;
4 2
from \/ ' A l sin ;/ to ^\/A 1 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<\fA 1 s'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 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
V l = 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 L l 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*
- 2 gi K + J**
302 HYDRA ULICS.
where H = h' + h" = total head above tail-water surface ; and
-^ a a ,_^ 4 2 , Z-j-, Z a 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 = A l v l 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 L 9
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
= v l cos yu l u^u, F, cos ft) + L -
But
A l v l 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 v the mean velocity in the channel.
L l 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 V Q is sensibly nil.
If A Q is the sectional area of the supply-channel, then
A v = Q = A 1 v 1 sin y y
and
, = /, -
A,
II. The loss (Z a ) 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 f 9 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 = y f ) of the water at en-
trance, evidently
L 3 = the loss of head
FIG. 191.
HYDRAULIC MOTORS AND CENTRIFUGAL PUMPS. 3O$
(v' sin y' ^ sin y) 3 (v' cos y' v l cos y) 9
_ (v f sin ;/ V l sin <*) a (z/ cos y' z\ V l cos a) 8
Generally a? is small, and L 3 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 (L b ) due to the abrupt change of sec-
tion between the outlet-surface and the suction-tube.
As in III, v 9 (=ffy is suddenly changed into v t ' (= 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 (fh f ) 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 (L 6 ) 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 (Z 7 ) 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 (Z 8 ) 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 (L 9 ) 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 = L 6 = 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
(v w ") 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 (v w '}
at entrance, while at exit the velocity of whirl (v w ") 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 (v w ff ) 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 H s be the gross lift, including the
actual lift (ff a ), the head due to the velocity
FIG. 195. O f 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 H a 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 --
v w w w
Hence, differentiating, -
u* tan 2 ft
- -<^- + sec- /> = Q,
and therefore
vj f = u^ sin ft,
and is the velocity of whirl at exit which, for a given speed (w a ),
will give a maximum efficiency.
Note.li u^ = v w ", 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 v r " cot ft = u^ v w ", ft must be 90 if u^ = vj'-,
but ft is generally much less than 90, and therefore v w " is
generally less than u y Let v w " = ku^> k being an empirical
coefficient less than unity.
Then kit? gH e 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 (v r ") is wasted and the velocity of whirl (v w ' f ) is sud-
denly changed to the velocity v s 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 (vj f - vy
the gam of pressure-head = -
i fe/') 9 ^ "
which is a maximum and equal to when v s = -^ .
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 * ~ ( u i ~ v '}* sec2 ft
and
u 2 (u v // ) 2 sec 2 ft
the efficiency - w ..
2^ w "
which is a maximum for a given speed & a and equal to
; j- : -5 when v w ff = u^ sin /?.
Thus the efficiency increases as ft diminishes.
When ft = 90, or ^ w r/ & 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 v w " to v,
-\- loss due to dissipation of v r "
-f- loss due to v s carried away
= loss due to friction (hydraulic)
+ *,(*" - ^ (*l_ _ (VT
2g 2g 2g
= loss due to friction (hydraulic)
, (*.") ,
~~
when v s = \vj'.
CASE II. Pump ivith a vortex-chamber (Fig. 199).
The diameter ( 2r 3 ) 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 \ r 3 2 /
and hence
the efficiency =
T,, . . FIG. 106.
This, again, is a maximum for
a given speed, when vj = u^ sin fi y 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 r a = 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 u 1 = 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(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) = 7 26', = 28 33' ;
(c) e = 73 15'. = 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, ( 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 C v , 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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