Chapter XII: Impact and Reaction of Water (2)

§ 171. _Site for Water Motor._--Wherever a stream flows from a higher to a lower level it is possible to erect a water motor. The amount of power obtainable depends on the available head and the supply of water. In choosing a site the engineer will select a portion of the stream where there is an abrupt natural fall, or at least a considerable slope of the bed. He will have regard to the facility of constructing the channels which are to convey the water, and will take advantage of any bend in the river which enables him to shorten them. He will have accurate measurements made of the quantity of water flowing in the stream, and he will endeavour to ascertain the average quantity available throughout the year, the minimum quantity in dry seasons, and the maximum for which bye-wash channels must be provided. In many cases the natural fall can be increased by a dam or weir thrown across the stream. The engineer will also examine to what extent the head will vary in different seasons, and whether it is necessary to sacrifice part of the fall and give a steep slope to the tail race to prevent the motor being drowned by backwater in floods. Streams fed from lakes which form natural reservoirs or fed from glaciers are less variable than streams depending directly on rainfall, and are therefore advantageous for water-power purposes.

§ 172. _Water Power at Holyoke, U.S.A._--About 85 m. from the mouth of
the Connecticut river there was a fall of about 60 ft. in a short
distance, forming what were called the Grand Rapids, below which the
river turned sharply, forming a kind of peninsula on which the city of
Holyoke is built. In 1845 the magnitude of the water-power available
attracted attention, and it was decided to build a dam across the
river. The ordinary flow of the river is 6000 cub. ft. per sec.,
giving a gross power of 30,000 h.p. In dry seasons the power is 20,000
h.p., or occasionally less. From above the dam a system of canals
takes the water to mills on three levels. The first canal starts with
a width of 140 ft. and depth of 22 ft., and supplies the highest
range of mills. A second canal takes the water which has driven
turbines in the highest mills and supplies it to a second series of
mills. There is a third canal on a still lower level supplying the
lowest mills. The water then finds its way back to the river. With the
grant of a mill site is also leased the right to use the water-power.
A mill-power is defined as 38 cub. ft. of water per sec. during 16
hours per day on a fall of 20 ft. This gives about 60 h.p. effective.
The charge for the power water is at the rate of 20s. per h.p. per
annum.

§ 173. _Action of Water in a Water Motor._--Water motors may be divided into water-pressure engines, water-wheels and turbines.

Water-pressure engines are machines with a cylinder and piston or ram, in principle identical with the corresponding part of a steam-engine. The water is alternately admitted to and discharged from the cylinder, causing a reciprocating action of the piston or plunger. It is admitted at a high pressure and discharged at a low one, and consequently work is done on the piston. The water in these machines never acquires a high velocity, and for the most part the kinetic energy of the water is wasted. The useful work is due to the difference of the pressure of admission and discharge, whether that pressure is due to the weight of a column of water of more or less considerable height, or is artificially produced in ways to be described presently.

Water-wheels are large vertical wheels driven by water falling from a higher to a lower level. In most water-wheels, the water acts directly by its weight loading one side of the wheel and so causing rotation. But in all water-wheels a portion, and in some a considerable portion, of the work due to gravity is first employed to generate kinetic energy in the water; during its action on the water-wheel the velocity of the water diminishes, and the wheel is therefore in part driven by the impulse due to the change of the water's momentum. Water-wheels are therefore motors on which the water acts, partly by weight, partly by impulse.

Turbines are wheels, generally of small size compared with water wheels, driven chiefly by the impulse of the water. Before entering the moving part of the turbine, the water is allowed to acquire a considerable velocity; during its action on the turbine this velocity is diminished, and the impulse due to the change of momentum drives the turbine.

In designing or selecting a water motor it is not sufficient to consider only its efficiency in normal conditions of working. It is generally quite as important to know how it will act with a scanty water supply or a diminished head. The greatest difference in water motors is in their adaptability to varying conditions of working.

_Water-pressure Engines._

§ 174. In these the water acts by pressure either due to the height of the column in a supply pipe descending from a high-level reservoir, or created by pumping. Pressure engines were first used in mine-pumping on waterfalls of greater height than could at that time be utilized by water wheels. Usually they were single acting, the water-pressure lifting the heavy pump rods which then made the return or pumping stroke by their own weight. To avoid losses by fluid friction and shock the velocity of the water in the pipes and passages was restricted to from 3 to 10 ft. per second, and the mean speed of plunger to 1 ft. per second. The stroke was long and the number of strokes 3 to 6 per minute. The pumping lift being constant, such engines worked practically always at full load, and the efficiency was high, about 84%. But they were cumbrous machines. They are described in Weisbach's _Mechanics of Engineering_.

The convenience of distributing energy from a central station to scattered working-points by pressure water conveyed in pipes--a system invented by Lord Armstrong--has already been mentioned. This system has led to the development of a great variety of hydraulic pressure engines of very various types. The cost of pumping the pressure water to some extent restricts its use to intermittent operations, such as working lifts and cranes, punching, shearing and riveting machines, forging and flanging presses. To keep down the cost of the distributing mains very high pressures are adopted, generally 700 lb. per sq. in. or 1600 ft. of head or more.

In a large number of hydraulic machines worked by water at high pressure, especially lifting machines, the motor consists of a direct, single acting ram and cylinder. In a few cases double-acting pistons and cylinders are used; but they involve a water-tight packing of the piston not easily accessible. In some cases pressure engines are used to obtain rotative movement, and then two double-acting cylinders or three single-acting cylinders are used, driving a crank shaft. Some double-acting cylinders have a piston rod half the area of the piston. The pressure water acts continuously on the annular area in front of the piston. During the forward stroke the pressure on the front of the piston balances half the pressure on the back. During the return stroke the pressure on the front is unopposed. The water in front of the piston is not exhausted, but returns to the supply pipe. As the frictional losses in a fluid are independent of the pressure, and the work done increases directly as the pressure, the percentage loss decreases for given velocities of flow as the pressure increases. Hence for high-pressure machines somewhat greater velocities are permitted in the passages than for low-pressure machines. In supply mains the velocity is from 3 to 6 ft. per second, in valve passages 5 to 10 ft. per second, or in extreme cases 20 ft. per second, where there is less object in economizing energy. As the water is incompressible, slide valves must have neither lap nor lead, and piston valves are preferable to ordinary slide valves. To prevent injurious compression from exhaust valves closing too soon in rotative engines with a fixed stroke, small self-acting relief valves are fitted to the cylinder ends, opening outwards against the pressure into the valve chest. Imprisoned water can then escape without over-straining the machines.

In direct single-acting lift machines, in which the stroke is fixed, and in rotative machines at constant speed it is obvious that the cylinder must be filled at each stroke irrespective of the amount of work to be done. The same amount of water is used whether much or little work is done, or whether great or small weights are lifted. Hence while pressure engines are very efficient at full load, their efficiency decreases as the load decreases. Various arrangements have been adopted to diminish this defect in engines working with a variable load. In lifting machinery there is sometimes a double ram, a hollow ram enclosing a solid ram. By simple arrangements the solid ram only is used for small loads, but for large loads the hollow ram is locked to the solid ram, and the two act as a ram of larger area. In rotative engines the case is more difficult. In Hastie's and Rigg's engines the stroke is automatically varied with the load, increasing when the load is large and decreasing when it is small. But such engines are complicated and have not achieved much success. Where pressure engines are used simplicity is generally a first consideration, and economy is of less importance.

§ 175. _Efficiency of Pressure Engines._--It is hardly possible to
form a theoretical expression for the efficiency of pressure engines,
but some general considerations are useful. Consider the case of a
long stroke hydraulic ram, which has a fairly constant velocity v
during the stroke, and valves which are fairly wide open during most
of the stroke. Let r be the ratio of area of ram to area of valve
passage, a ratio which may vary in ordinary cases from 4 to 12. Then
the loss in shock of the water entering the cylinder will be (r -
1)²v²/2g in ft. of head. The friction in the supply pipe is also
proportional to v². The energy carried away in exhaust will be
proportional to v². Hence the total hydraulic losses may be taken to
be approximately [zeta]v²/2g ft., where [zeta] is a coefficient
depending on the proportions of the machine. Let f be the friction of
the ram packing and mechanism reckoned in lb. per sq. ft. of ram area.
Then if the supply-pipe pressure driving the machine is p lb. per sq.
ft., the effective working pressure will be

p - G[zeta]v²/2g - f lb. per sq. ft.

Let A be the area of the ram in sq. ft., v its velocity in ft. per
sec. The useful work done will be

(p - G[zeta]v²/2g - f)Av ft. lb. per sec.,

and the efficiency of the machine will be

[eta] = (p - G[zeta]v²/2g - f)/p.

This shows that the efficiency increases with the pressure p, and
diminishes with the speed v, other things being the same. If in
regulating the engine for varying load the pressure is throttled,
part of the available head is destroyed at the throttle valve, and p
in the bracket above is reduced. Direct-acting hydraulic lifts,
without intermediate gearing, may have an efficiency of 95% during the
working stroke. If a hydraulic jigger is used with ropes and sheaves
to change the speed of the ram to the speed of the lift, the
efficiency may be only 50%. E. B. Ellington has given the efficiency
of lifts with hydraulic balance at 85% during the working stroke.
Large pressure engines have an efficiency of 85%, but small rotative
engines probably not more than 50% and that only when fully loaded.

§ 176. _Direct-Acting Hydraulic Lift_ (fig. 171).--This is the simplest of all kinds of hydraulic motor. A cage W is lifted directly by water pressure acting in a cylinder C, the length of which is a little greater than the lift. A ram or plunger R of the same length is attached to the cage. The water-pressure admitted by a cock to the cylinder forces up the ram, and when the supply valve is closed and the discharge valve opened, the ram descends. In this case the ram is 9 in. diameter, with a stroke of 49 ft. It consists of lengths of wrought-iron pipe screwed together perfectly water-tight, the lower end being closed by a cast-iron plug. The ram works in a cylinder 11 in. diameter of 9 ft. lengths of flanged cast-iron pipe. The ram passes water-tight through the cylinder cover, which is provided with double hat leathers to prevent leakage outwards or inwards. As the weight of the ram and cage is much more than sufficient to cause a descent of the cage, part of the weight is balanced. A chain attached to the cage passes over a pulley at the top of the lift, and carries at its free end a balance weight B, working in T iron guides. Water is admitted to the cylinder from a 4-in. supply pipe through a two-way slide, worked by a rack, spindle and endless rope. The lift works under 73 ft. of head, and lifts 1350 lb at 2 ft. per second. The efficiency is from 75 to 80%.

The principal prejudicial resistance to the motion of a ram of this
kind is the friction of the cup leathers, which make the joint between
the cylinder and ram. Some experiments by John Hick give for the
friction of these leathers the following formula. Let F = the total
friction in pounds; d = diameter of ram in ft.; p = water-pressure in
pounds per sq. ft.; k a coefficient.

F = k p d

k = 0.00393 if the leathers are new or badly lubricated;
= 0.00262 if the leathers are in good condition and well lubricated.

Since the total pressure on the ram is P = ¼[pi]d²p, the fraction of
the total pressure expended in overcoming the friction of the leathers
is F/P = .005/d to .0033/d, d being in feet.

Let H be the height of the pressure column measured from the free
surface of the supply reservoir to the bottom of the ram in its lowest
position, H_b the height from the discharge reservoir to the same
point, h the height of the ram above its lowest point at any moment, S
the length of stroke, [Omega] the area of the ram, W the weight of
cage, R the weight of ram, B the weight of balance weight, w the
weight of balance chain per foot run, F the friction of the cup
leather and slides. Then, neglecting fluid friction, if the ram is
rising the accelerating force is

P1 = G(H - h)[Omega] - R - W + B - w(S - h) + wh - F,

and if the ram is descending

P2 = G(H_b - h)[Omega] + W + R - B + w(S - h) - wh - F.

If w = ½ G[Omega], P1 and P2 are constant throughout the stroke; and
the moving force in ascending and descending is the same, if

B = W + R + wS - G[Omega](H - H_b)/2.

Using the values just found for w and B,

P1 = P2 = ½G[Omega](H - H_b) - F.

Let W + R + wS + B = U, and let P be the constant accelerating force
acting on the system, then the acceleration is (P/U)g. The velocity at
the end of the stroke is (assuming the friction to be constant)

v = [root](2PgS/U);

and the mean velocity of ascent is ½v.

§ 177. _Armstrong's Hydraulic Jigger._--This is simply a single-acting hydraulic cylinder and ram, provided with sheaves so as to give motion to a wire rope or chain. It is used in various forms of lift and crane. Fig. 172 shows the arrangement. A hydraulic ram or plunger B works in a stationary cylinder A. Ram and cylinder carry sets of sheaves over which passes a chain or rope, fixed at one end to the cylinder, and at the other connected over guide pulleys to a lift or crane. For each pair of pulleys, one on the cylinder and one on the ram, the movement of the free end of the rope is doubled compared with that of the ram. With three pairs of pulleys the free end of the rope has a movement equal to six times the stroke of the ram, the force exerted being in the inverse proportion.

§ 178. _Rotative Hydraulic Engines._--Valve-gear mechanism similar in principle to that of steam engines can be applied to actuate the admission and discharge valves, and the pressure engine is then converted into a continuously-acting motor.

Let H be the available fall to work the engine after deducting the
loss of head in the supply and discharge pipes, Q the supply of water
in cubic feet per second, and [eta] the efficiency of the engine. Then
the horse-power of the engine is

H.P. = [eta]GQH/550.

The efficiency of large slow-moving pressure engines is [eta] = .66 to
.8. In small motors of this kind probably [eta] is not greater than
.5. Let v be the mean velocity of the piston, then its diameter d is
given by the relation

Q = [pi]d²v/4 in double-acting engines,
= [pi]d²v/8 in single-acting engines.

If there are n cylinders put Q/n for Q in these equations.

Small rotative pressure engines form extremely convenient motors for hoists, capstans or winches, and for driving small machinery. The single-acting engine has the advantage that the pressure of the piston on the crank pin is always in one direction; there is then no knocking as the dead centres are passed. Generally three single-acting cylinders are used, so that the engine will readily start in all positions, and the driving effort on the crank pin is very uniform.

_Brotherhood Hydraulic Engine._--Three cylinders at angles of 120°
with each other are formed in one casting with the frame. The
plungers are hollow trunks, and the connecting rods abut in
cylindrical recesses in them and are connected to a common crank pin.
A circular valve disk with concentric segmental ports revolves at the
same rate as the crank over ports in the valve face common to the
three cylinders. Each cylinder is always in communication with either
an admission or exhaust port. The blank parts of the circular valve
close the admission and exhaust ports alternately. The fixed valve
face is of lignum vitae in a metal recess, and the revolving valve of
gun-metal. In the case of a small capstan engine the cylinders are 3½
in. diameter and 3 in. stroke. At 40 revs. per minute, the piston
speed is 31 ft. per minute. The ports are 1 in. diameter or 1/12 of
the piston area, and the mean velocity in the ports 6.4 ft. per sec.
With 700 lb. per sq. in. water pressure and an efficiency of 50%, the
engine is about 3 h.p. A common arrangement is to have three parallel
cylinders acting on a three-throw crank shaft, the cylinders
oscillating on trunnions.

_Hastie's Engine._--Fig. 173 shows a similar engine made by Messrs
Hastie of Greenock. G, G, G are the three plungers which pass out of
the cylinders through cup leathers, and act on the same crank pin. A
is the inlet pipe which communicates with the cock B. This cock
controls the action of the engine, being so constructed that it acts
as a reversing valve when the handle C is in its extreme positions and
as a brake when in its middle position. With the handle in its middle
position, the ports of the cylinders are in communication with the
exhaust. Two passages are formed in the framing leading from the cock
B to the ends of the cylinders, one being in communication with the
supply pipe A, the other with the discharge pipe Q. These passages end
as shown at E. The oscillation of the cylinders puts them alternately
in communication with each of these passages, and thus the water is
alternately admitted and exhausted.

In any ordinary rotative engine the length of stroke is invariable.
Consequently the consumption of water depends simply on the speed of
the engine, irrespective of the effort overcome. If the power of the
engine must be varied without altering the number of rotations, then
the stroke must be made variable. Messrs Hastie have contrived an
exceedingly ingenious method of varying the stroke automatically, in
proportion to the amount of work to be done (fig. 174). The crank pin
I is carried in a slide H moving in a disk M. In this is a double cam
K acting on two small steel rollers J, L attached to the slide H. If
the cam rotates it moves the slide and increases or decreases the
radius of the circle in which the crank pin I rotates. The disk M is
keyed on a hollow shaft surrounding the driving shaft P, to which the
cams are attached. The hollow shaft N has two snugs to which the
chains RR are attached (fig. 175). The shaft P carries the spring case
SS to which also are attached the other ends of the chains. When the
engine is at rest the springs extend themselves, rotating the hollow
shaft N and the frame M, so as to place the crank pin I at its nearest
position to the axis of rotation. When a resistance has to be
overcome, the shaft N rotates relatively to P, compressing the
springs, till their resistance balances the pressure due to the
resistance to the rotation of P. The engine then commences to work,
the crank pin being in the position in which the turning effort just
overcomes the resistance. If the resistance diminishes, the springs
force out the chains and shorten the stroke of the plungers, and vice
versa. The following experiments, on an engine of this kind working a
hoist, show how the automatic arrangement adjusted the water used to
the work done. The lift was 22 ft. and the water pressure in the
cylinders 80 lb. per sq. in.

Weight lifted, Chain 427 633 745 857 969 1081 1193
in lb. only

Water used, in 7½ 10 14 16 17 20 21 22
gallons

§ 179. _Accumulator Machinery._--It has already been pointed out that it is in some cases convenient to use a steam engine to create an artificial head of water, which is afterwards employed in driving water-pressure machinery. Where power is required intermittently, for short periods, at a number of different points, as, for instance, in moving the cranes, lock gates, &c., of a dockyard, a separate steam engine and boiler at each point is very inconvenient; nor can engines worked from a common boiler be used, because of the great loss of heat and the difficulties which arise out of condensation in the pipes. If a tank, into which water is continuously pumped, can be placed at a great elevation, the water can then be used in hydraulic machinery in a very convenient way. Each hydraulic machine is put in communication with the tank by a pipe, and on opening a valve it commences work, using a quantity of water directly proportional to the work done. No attendance is required when the machine is not working.

A site for such an elevated tank is, however, seldom available, and in place of it a beautiful arrangement termed an accumulator, invented by Lord Armstrong, is used. This consists of a tall vertical cylinder; into this works a solid ram through cup leathers or hemp packing, and the ram is loaded by fixed weights, so that the pressure in the cylinder is 700 lb. or 800 lb. per sq. in. In some cases the ram is fixed and the cylinder moves on it. The pumping engines which supply the energy that is stored in the accumulator should be a pair coupled at right angles, so as to start in any position. The engines pump into the accumulator cylinder till the ram is at the top of its stroke, when by a catch arrangement acting on the engine throttle valve the engines are stopped. If the accumulator ram descends, in consequence of water being taken to work machinery, the engines immediately recommence working. Pipes lead from the accumulator to each of the machines requiring to be driven, and do not require to be of large size, as the pressure is so great.

Fig. 176 shows a diagrammatic way the scheme of a system of
accumulator machinery. A is the accumulator, with its ram carrying a
cylindrical wrought-iron tank W, in which weights are placed to load
the accumulator. At R is one of the pressure engines or jiggers,
worked from the accumulator, discharging the water after use into the
tank T. In this case the pressure engine is shown working a set of
blocks, the fixed block being on the ram cylinder, the running block
on the ram. The chain running over these blocks works a lift cage C,
the speed of which is as many times greater than that of the ram as
there are plies of chain on the block tackle. B is the balance weight
of the cage.

In the use of accumulators on shipboard for working gun gear or
steering gear, the accumulator ram is loaded by springs, or by steam
pressure acting on a piston much larger than the ram.

R. H. Tweddell has used accumulators with a pressure of 2000 lb. per
sq. in. to work hydraulic riveting machinery.

The amount of energy stored in the accumulator, having a ram d in. in
diameter, a stroke of S ft., and delivering at p lb. pressure per sq.
in., is

[pi]
---- p d²S foot-pounds.
4

Thus, if the ram is 9 in., the stroke 20 ft., and the pressure 800 lb.
per sq. in., the work stored in the accumulator when the ram is at the
top of the stroke is 1,017,600 foot-pounds, that is, enough to drive a
machine requiring one horse power for about half an hour. As, however,
the pumping engine replaces water as soon as it is drawn off, the
working capacity of the accumulator is very much greater than this.
Tweddell found that an accumulator charged at 1250 lb. discharged at
1225 lb. per sq. in. Hence the friction was equivalent to 12½ lb. per
sq. in. and the efficiency 98%.

When a very great pressure is required a differential accumulator
(fig. 177) is convenient. The ram is fixed and passes through both
ends of the cylinder, but is of different diameters at the two ends, A
and B. Hence if d1, d2 are the diameters of the ram in inches and p
the required pressure in lb. per sq. in., the load required is
¼p[pi](d1² - d2²). An accumulator of this kind used with riveting
machines has d1 = 5½ in., d2 = 4¾ in. The pressure is 2000 lb. per sq.
in. and the load 5.4 tons.

Sometimes an accumulator is loaded by water or steam pressure instead
of by a dead weight. Fig. 178 shows the arrangement. A piston A is
connected to a plunger B of much smaller area. Water pressure, say
from town mains, is admitted below A, and the high pressure water is
pumped into and discharged from the cylinder C in which B works. If r
is the ratio of the areas of A and B, then, neglecting friction, the
pressure in the upper cylinder is r times that under the piston A.
With a variable rate of supply and demand from the upper cylinder, the
piston A rises and falls, maintaining always a constant pressure in
the upper cylinder.

_Water Wheels._

§ 180. _Overshot and High Breast Wheels._--When a water fall ranges between 10 and 70 ft. and the water supply is from 3 to 25 cub. ft. per second, it is possible to construct a bucket wheel on which the water acts chiefly by its weight. If the variation of the head-water level does not exceed 2 ft., an overshot wheel may be used (fig. 179). The water is then projected over the summit of the wheel, and falls in a parabolic path into the buckets. With greater variation of head-water level, a pitch-back or high breast wheel is better. The water falls over the top of a sliding sluice into the wheel, on the same side as the head race channel. By adjusting the height of the sluice, the requisite supply is given to the wheel in all positions of the head-water level.

The wheel consists of a cast-iron or wrought-iron axle C supporting
the weight of the wheel. To this are attached two sets of arms A of
wood or iron, which support circular segmental plates, B, termed
shrouds. A cylindrical sole plate dd extends between the shrouds on
the inner side. The buckets are formed by wood planks or curved
wrought-iron plates extending from shroud to shroud, the back of the
buckets being formed by the sole plate.

The efficiency may be taken at 0.75. Hence, if h.p. is the effective
horse power, H the available fall, and Q the available water supply
per second,

h.p. = 0.75 (GQH/550) = 0.085 QH.

If the peripheral velocity of the water wheel is too great, water is
thrown out of the buckets before reaching the bottom of the fall. In
practice, the circumferential velocity of water wheels of the kind now
described is from 4½ to 10 ft. per second, about 6 ft. being the usual
velocity of good iron wheels not of very small size. In order that the
water may enter the buckets easily, it must have a greater velocity
than the wheel. Usually the velocity of the water at the point where
it enters the wheel is from 9 to 12 ft. per second, and to produce
this it must enter the wheel at a point 16 to 27 in. below the
head-water level. Hence the diameter of an overshot wheel may be

D = H - 1(1/3) to H - 2¼ ft.

Overshot and high breast wheels work badly in backwater, and hence if
the tail-water level varies, it is better to reduce the diameter of
the wheel so that its greatest immersion in flood is not more than 1
ft. The depth d of the shrouds is about 10 to 16 in. The number of
buckets may be about

N = [pi]D/d.

Let v be the peripheral velocity of the wheel. Then the capacity of
that portion of the wheel which passes the sluice in one second is

Q1 = vb(Dd - d²)/D
= v b d nearly,

b being the breadth of the wheel between the shrouds. If, however,
this quantity of water were allowed to pass on to the wheel the
buckets would begin to spill their contents almost at the top of the
fall. To diminish the loss from spilling, it is not only necessary to
give the buckets a suitable form, but to restrict the water supply to
one-fourth or one-third of the gross bucket capacity. Let m be the
value of this ratio; then, Q being the supply of water per second,

Q = mQ1 = mb dv.

This gives the breadth of the wheel if the water supply is known. The
form of the buckets should be determined thus. The outer element of
the bucket should be in the direction of motion of the water entering
relatively to the wheel, so that the water may enter without splashing
or shock. The buckets should retain the water as long as possible, and
the width of opening of the buckets should be 2 or 3 in. greater than
the thickness of the sheet of water entering.

For a wooden bucket (fig. 180, A), take ab = distance between two
buckets on periphery of wheel. Make ed = ½ eb and bc = 6/5 to 5/4
ab. Join cd. For an iron bucket (fig. 180, B), take ed = 1/3 eb; bc =
6/5 ab. Draw cO making an angle of 10° to 15° with the radius at c.
On Oc take a centre giving a circular arc passing near d, and round
the curve into the radial part of the bucket de.

There are two ways in which the power of a water wheel is given off to the machinery driven. In wooden wheels and wheels with rigid arms, a spur or bevil wheel keyed on the axle of the turbine will transmit the power to the shafting. It is obvious that the whole turning moment due to the weight of the water is then transmitted through the arms and axle of the water wheel. When the water wheel is an iron one, it usually has light iron suspension arms incapable of resisting the bending action due to the transmission of the turning effort to the axle. In that case spur segments are bolted to one of the shrouds, and the pinion to which the power is transmitted is placed so that the teeth in gear are, as nearly as may be, on the line of action of the resultant of the weight of the water in the loaded arc of the wheel.

The largest high breast wheels ever constructed were probably the four wheels, each 50 ft. in diameter, and of 125 h.p., erected by Sir W. Fairbairn in 1825 at Catrine in Ayrshire. These wheels are still working.

§ 181. _Poncelet Water Wheel._--When the fall does not exceed 6 ft., the best water motor to adopt in many cases is the Poncelet undershot water wheel. In this the water acts very nearly in the same way as in a turbine, and the Poncelet wheel, although slightly less efficient than the best turbines, in normal conditions of working, is superior to most of them when working with a reduced supply of water. A general notion of the action of the water on a Poncelet wheel has already been given in § 159. Fig. 181 shows its construction. The water penned back between the side walls of the wheel pit is allowed to flow to the wheel under a movable sluice, at a velocity nearly equal to the velocity due to the whole fall. The water is guided down a slope of 1 in 10, or a curved race, and enters the wheel without shock. Gliding up the curved floats it comes to rest, falls back, and acquires at the point of discharge a backward velocity relative to the wheel nearly equal to the forward velocity of the wheel. Consequently it leaves the wheel deprived of nearly the whole of its original kinetic energy.

Taking the efficiency at 0.60, and putting H for the available fall,
h.p. for the horse-power, and Q for the water supply per second,

h.p. = 0.068 QH.

The diameter D of the wheel may be taken arbitrarily. It should not be
less than twice the fall and is more often four times the fall. For
ordinary cases the smallest convenient diameter is 14 ft. with a
straight, or 10 ft. with a curved, approach channel. The radial depth
of bucket should be at least half the fall, and radius of curvature of
buckets about half the radius of the wheel. The shrouds are usually of
cast iron with flanges to receive the buckets. The buckets may be of
iron 1/8 in. thick bolted to the flanges with 5/16 in. bolts.

Let H´ be the fall measured from the free surface of the head-water to
the point F where the mean layer enters the wheel; then the velocity
at which the water enters is v = [root](2gH´), and the best
circumferential velocity of the wheel is V = 0.55f to 0.6v. The number
of rotations of the wheel per second is N = V/[pi]D. The thickness
of the sheet of water entering the wheel is very important. The best
thickness according to experiment is 8 to 10 in. The maximum thickness
should not exceed 12 to 15 in., when there is a surplus water supply.
Let e be the thickness of the sheet of water entering the wheel, and b
its width; then

bev = Q; or b = Q/ev.

Grashof takes e = (1/6)H, and then

b = 6Q/H [root](2gH).

Allowing for the contraction of the stream, the area of opening
through the sluice may be 1.25 be to 1.3 be. The inside width of the
wheel is made about 4 in. greater than b.

Several constructions have been given for the floats of Poncelet
wheels. One of the simplest is that shown in figs. 181, 182.

Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD
making angles of 15° with OA. Then BD may be the length of the close
breasting fitted to the wheel. Draw the bottom of the head face BC at
a slope of 1 in 10. Parallel to this, at distances ½e and e, draw EF
and GH. Then EF is the mean layer and GH the surface layer entering
the wheel. Join OF, and make OFK = 23°. Take FK = 0.5 to 0.7 H. Then K
is the centre from which the bucket curve is struck and KF is the
radius. The depth of the shrouds must be sufficient to prevent the
water from rising over the top of the float. It is ½H to 2/3 H. The
number of buckets is not very important. They are usually 1 ft. apart
on the circumference of the wheel.

The efficiency of a Poncelet wheel has been found in experiments to
reach 0.68. It is better to take it at 0.6 in estimating the power of
the wheel, so as to allow some margin.

In fig. 182 v_i is the initial and v_o the final velocity of the
water, v_r parallel to the vane the relative velocity of the water and
wheel, and V the velocity of the wheel.

_Turbines._

§ 182. The name turbine was originally given in France to any water motor which revolved in a horizontal plane, the axis being vertical. The rapid development of this class of motors dates from 1827, when a prize was offered by the Société d'Encouragement for a motor of this kind, which should be an improvement on certain wheels then in use. The prize was ultimately awarded to Benoît Fourneyron (1802-1867), whose turbine, but little modified, is still constructed.

_Classification of Turbines._--In some turbines the whole available energy of the water is converted into kinetic energy before the water acts on the moving part of the turbine. Such turbines are termed _Impulse or Action Turbines_, and they are distinguished by this that the wheel passages are never entirely filled by the water. To ensure this condition they must be placed a little above the tail water and discharge into free air. Turbines in which part only of the available energy is converted into kinetic energy before the water enters the wheel are termed _Pressure or Reaction Turbines_. In these there is a pressure which in some cases amounts to half the head in the clearance space between the guide vanes and wheel vanes. The velocity with which the water enters the wheel is due to the difference between the pressure due to the head and the pressure in the clearance space. In pressure turbines the wheel passages must be continuously filled with water for good efficiency, and the wheel may be and generally is placed below the tail water level.

Some turbines are designed to act normally as impulse turbines discharging above the tail water level. But the passages are so designed that they are just filled by the water. If the tail water rises and drowns the turbine they become pressure turbines with a small clearance pressure, but the efficiency is not much affected. Such turbines are termed _Limit turbines_.

Next there is a difference of constructive arrangement of turbines, which does not very essentially alter the mode of action of the water. In axial flow or so-called parallel flow turbines, the water enters and leaves the turbine in a direction parallel to the axis of rotation, and the paths of the molecules lie on cylindrical surfaces concentric with that axis. In radial outward and inward flow turbines, the water enters and leaves the turbine in directions normal to the axis of rotation, and the paths of the molecules lie exactly or nearly in planes normal to the axis of rotation. In outward flow turbines the general direction of flow is away from the axis, and in inward flow turbines towards the axis. There are also mixed flow turbines in which the water enters normally and is discharged parallel to the axis of rotation.

Another difference of construction is this, that the water may be admitted equally to every part of the circumference of the turbine wheel or to a portion of the circumference only. In the former case, the condition of the wheel passages is always the same; they receive water equally in all positions during rotation. In the latter case, they receive water during a part of the rotation only. The former may be termed turbines with complete admission, the latter turbines with partial admission. A reaction turbine should always have complete admission. An impulse turbine may have complete or partial admission.

When two turbine wheels similarly constructed are placed on the same axis, in order to balance the pressures and diminish journal friction, the arrangement may be termed a twin turbine.

If the water, having acted on one turbine wheel, is then passed through a second on the same axis, the arrangement may be termed a compound turbine. The object of such an arrangement would be to diminish the speed of rotation.

Many forms of reaction turbine may be placed at any height not exceeding 30 ft. above the tail water. They then discharge into an air-tight suction pipe. The weight of the column of water in this pipe balances part of the atmospheric pressure, and the difference of pressure, producing the flow through the turbine, is the same as if the turbine were placed at the bottom of the fall.

I. Impulse Turbines. | II. Reaction Turbines.
|
(Wheel passages not filled, and | (Wheel passages filled, discha-
discharging above the tail | rging above or below the tail
water.) | water or into a suction-pipe.)
(a) Complete admission. (Rare.) | Always with complete admission.
(b) Partial admission. (Usual.) |
\_________________________________\/_______________________________/
Axial flow, outward flow, inward flow, or mixed flow.
\_________________________________\/_______________________________/
Simple turbines; twin turbines; compound turbines.

§ 183. _The Simple Reaction Wheel._--It has been shown, in § 162,
that, when water issues from a vessel, there is a reaction on the
vessel tending to cause motion in a direction opposite to that of the
jet. This principle was applied in a rotating water motor at a very
early period, and the Scotch turbine, at one time much used, differs
in no essential respect from the older form of reaction wheel.

The old reaction wheel consisted of a vertical pipe balanced on a
vertical axis, and supplied with water (fig. 183). From the bottom of
the vertical pipe two or more hollow horizontal arms extended, at the
ends of which were orifices from which the water was discharged. The
reaction of the jets caused the rotation of the machine.

Let H be the available fall measured from the level of the water in
the vertical pipe to the centres cf the orifices, r the radius from
the axis of rotation to the centres of the orifices, v the velocity of
discharge through the jets, [alpha] the angular velocity of the
machine. When the machine is at rest the water issues from the
orifices with the velocity [root](2gH) (friction being neglected). But
when the machine rotates the water in the arms rotates also, and is in
the condition of a forced vortex, all the particles having the same
angular velocity. Consequently the pressure in the arms at the
orifices is H + [alpha]²r²/2g ft. of water, and the velocity of
discharge through the orifices is v = [root](2gH + [alpha]²r²). If the
total area of the orifices is [omega], the quantity discharged from
the wheel per second is

Q = [omega]v = [omega] [root](2gH + [alpha]²r²).

While the water passes through the orifices with the velocity v, the
orifices are moving in the opposite direction with the velocity
[alpha]r. The absolute velocity of the water is therefore

v - [alpha]r = [root](2gH + [alpha]²r²) - [alpha]r.

The momentum generated per second is (GQ/g)(v - [alpha]r), which is
numerically equal to the force driving the motor at the radius r. The
work done by the water in rotating the wheel is therefore

(GQ/g) (v - [alpha]r) ar foot-pounds per sec.

The work expended by the water fall is GQH foot-pounds per second.
Consequently the efficiency of the motor is

(v - [alpha]r) [alpha]r {[root]{2gH + [alpha]²r²]} - [alpha]r} [alpha]r
[eta] = ----------------------- = -----------------------------------------------.
gH gH

Let

gH g²H²
[root]{2gH + [alpha]²r²} = [alpha]r + -------- - ----------- ...
[alpha]r 2[alpha]³r³

then

[eta] = 1 - gH/2[alpha]r + ...

which increases towards the limit 1 as [alpha]r increases towards
infinity. Neglecting friction, therefore, the maximum efficiency is
reached when the wheel has an infinitely great velocity of rotation.
But this condition is impracticable to realize, and even, at
practicable but high velocities of rotation, the friction would
considerably reduce the efficiency. Experiment seems to show that the
best efficiency is reached when [alpha]r = [root](2gH). Then the
efficiency apart from friction is

[eta] = {[root](2[alpha]²r²) - [alpha]r} [alpha]r/gH
= 0.414 [alpha]²r²/gH = 0.828,

about 17% of the energy of the fall being carried away by the water
discharged. The actual efficiency realized appears to be about 60%, so
that about 21% of the energy of the fall is lost in friction, in
addition to the energy carried away by the water.

§ 184. _General Statement of Hydrodynamical Principles necessary for
the Theory of Turbines._

(a) When water flows through any pipe-shaped passage, such as the
passage between the vanes of a turbine wheel, the relation between the
changes of pressure and velocity is given by Bernoulli's theorem (§
29). Suppose that, at a section A of such a passage, h1 is the
pressure measured in feet of water, v1 the velocity, and z1 the
elevation above any horizontal datum plane, and that at a section B
the same quantities are denoted by h2, v2, z2. Then

h1 - h2 = (v2² - v1²)/2g + z2 - z1. (1)

If the flow is horizontal, z2 = z1; and

h1 - h2 = (v2² - v1²)/2g. (la)

(b) When there is an abrupt change of section of the passage, or an
abrupt change of section of the stream due to a contraction, then, in
applying Bernoulli's equation allowance must be made for the loss of
head in shock (§ 36). Let v1, v2 be the velocities before and after
the abrupt change, then a stream of velocity v1 impinges on a stream
at a velocity v2, and the relative velocity is v1 - v2. The head lost
is (v1 - v2)²/2g. Then equation (1a) becomes

h1 - h2 = (v1² - v2²)/2g - (v1 - v2)²/2g = v2(v1 - v2)/g (2)

To diminish as much as possible the loss of energy from irregular
eddying motions, the change of section in the turbine passages must be
very gradual, and the curvature without discontinuity.

(c) _Equality of Angular Impulse and Change of Angular
Momentum._--Suppose that a couple, the moment of which is M, acts on a
body of weight W for t seconds, during which it moves from A1 to A2
(fig. 184). Let v1 be the velocity of the body at A1, v2 its velocity
at A2, and let p1, p2 be the perpendiculars from C on v1 and v2. Then
Mt is termed the angular impulse of the couple, and the quantity

(W/g)(v2p2 - v1p1)

is the change of angular momentum relatively to C. Then, from the
equality of angular impulse and change of angular momentum

Mt = (W/g)(v2p2 - v1p1),

or, if the change of momentum is estimated for one second,

M = (W/g)(v2p2 - v1p1).

Let r1, r2 be the radii drawn from C to A1, A2, and let w1, w2 be the
components of v1, v2, perpendicular to these radii, making angles
[beta] and [alpha] with v1, v2. Then

v1 = w1 sec [beta]; v2 = w2 sec [alpha]

p1 = r1 cos [beta]; p2 = r2 cos [alpha],

.: M = (W/g) (w2r2 - w1r1), (3)

where the moment of the couple is expressed in terms of the radii
drawn to the positions of the body at the beginning and end of a
second, and the tangential components of its velocity at those points.

Now the water flowing through a turbine enters at the admission
surface and leaves at the discharge surface of the wheel, with its
angular momentum relatively to the axis of the wheel changed. It
therefore exerts a couple -M tending to rotate the wheel, equal and
opposite to the couple M which the wheel exerts on the water. Let Q
cub. ft. enter and leave the wheel per second, and let w1, w2 be the
tangential components of the velocity of the water at the receiving
and discharging surfaces of the wheel, r1, r2 the radii of those
surfaces. By the principle above,

-M = (GQ/g)(w2r2 - w1r1). (4)

If [alpha] is the angular velocity of the wheel, the work done by the
water on the wheel is

T = Ma = (GQ/g)(w1r1 - w2r2) [alpha] foot-pounds per second. (5)

§ 185. _Total and Available Fall._--Let H_t be the total difference of
level from the head-water to the tail-water surface. Of this total
head a portion is expended in overcoming the resistances of the head
race, tail race, supply pipe, or other channel conveying the water.
Let [h]_p be that loss of head, which varies with the local
conditions in which the turbine is placed. Then

H = H_t - [h]_p

is the available head for working the turbine, and on this the
calculations for the turbine should be based. In some cases it is
necessary to place the turbine above the tail-water level, and there
is then a fall [h] from the centre of the outlet surface of
the turbine to the tail-water level which is wasted, but which is
properly one of the losses belonging to the turbine itself. In that
case the velocities of the water in the turbine should be calculated
for a head H - [h], but the efficiency of the turbine for the
head H.

§ 186. _Gross Efficiency and Hydraulic Efficiency of a Turbine._--Let
T_d be the useful work done by the turbine, in foot-pounds per second,
T_t the work expended in friction of the turbine shaft, gearing, &c.,
a quantity which varies with the local conditions in which the turbine
is placed. Then the effective work done by the water in the turbine is

T = T_d + T_t.

The gross efficiency of the whole arrangement of turbine, races, and
transmissive machinery is

[eta]_t = T_d/CQH_t. (6)

And the hydraulic efficiency of the turbine alone is

[eta] = T/GQH. (7)

It is this last efficiency only with which the theory of turbines is
concerned.

From equations (5) and (7) we get

[eta]GQH = (GQ/g)(w1r1 - w2r2)a;

[eta] = (w1r1 - w2r2)a/gH. (8)

This is the fundamental equation in the theory of turbines. In
general,[7] w1 and w2, the tangential components of the water's motion
on entering and leaving the wheel, are completely independent. That
the efficiency may be as great as possible, it is obviously necessary
that w2 = 0. In that case

[eta] = w1r1a/gH. (9)

ar1 is the circumferential velocity of the wheel at the inlet surface.
Calling this V1, the equation becomes

[eta] = w1V1/gH. (9a)

This remarkably simple equation is the fundamental equation in the
theory of turbines. It was first given by Reiche (_Turbinenbaues_,
1877).