cos [phi] = V_o/v_(ro) = V_i/v_(ri) = 0.5/(1.22 × 0.45) = cos 24º 23´.
This should be corrected for the vane thickness. Neglecting this, u_o
= v_(ro) sin [phi] = v_(ri) sin [phi] = u_i cosec [theta] sin [phi] =
0.5u_i. The discharging area of the wheel must therefore be greater
than the inlet area in the ratio of at least 2 to 1. In some actual
turbines the ratio is 7 to 3. This greater outlet area is obtained by
splaying the wheel, as shown in the section (fig. 199).
[Illustration: FIG. 203.]
§ 200. _Pelton Wheel._--In the mining district of California about
1860 simple impulse wheels were used, termed hurdy-gurdy wheels. The
wheels rotated in a vertical plane, being supported on a horizontal
axis. Round the circumference were fixed flat vanes which were struck
normally by a jet from a nozzle of size varying with the head and
quantity of water. Such wheels have in fact long been used. They are
not efficient, but they are very simply constructed. Then attempts
were made to improve the efficiency, first by using hemispherical cup
vanes, and then by using a double cup vane with a central dividing
ridge, an arrangement invented by Pelton. In this last form the water
from the nozzle passes half to each side of the wheel, just escaping
clear of the backs of the advancing buckets. Fig. 203 shows a Pelton
vane. Some small modifications have been made by other makers, but
they are not of any great importance. Fig. 204 shows a complete Pelton
wheel with frame and casing, supply pipe and nozzle. Pelton wheels
have been very largely used in America and to some extent in Europe.
They are extremely simple and easy to construct or repair and on falls
of 100 ft. or more are very efficient. The jet strikes tangentially to
the mean radius of the buckets, and the face of the buckets is not
quite radial but at right angles to the direction of the jet at the
point of first impact. For greatest efficiency the peripheral velocity
of the wheel at the mean radius of the buckets should be a little less
than half the velocity of the jet. As the radius of the wheel can be
taken arbitrarily, the number of revolutions per minute can be
accommodated to that of the machinery to be driven. Pelton wheels have
been made as small as 4 in. diameter, for driving sewing machines, and
as large as 24 ft. The efficiency on high falls is about 80%. When
large power is required two or three nozzles are used delivering on
one wheel. The width of the buckets should be not less than seven
times the diameter of the jet.
[Illustration: FIG. 204.]
At the Comstock mines, Nevada, there is a 36-in. Pelton wheel made of
a solid steel disk with phosphor bronze buckets riveted to the rim.
The head is 2100 ft. and the wheel makes 1150 revolutions per minute,
the peripheral velocity being 180 ft. per sec. With a ½-in. nozzle the
wheel uses 32 cub. ft. of water per minute and develops 100 h.p. At
the Chollarshaft, Nevada, there are six Pelton wheels on a fall of
1680 ft. driving electrical generators. With 5/8-in. nozzles each
develops 125 h.p.
[Illustration: FIG. 205]
§ 201. _Theory of the Pelton Wheel._--Suppose a jet with a velocity v
strikes tangentially a curved vane AB (fig. 205) moving in the same
direction with the velocity u. The water will flow over the vane with
the relative velocity v - u and at B will have the tangential
relative velocity v - u making an angle [alpha] with the direction of
the vane's motion. Combining this with the velocity u of the vane, the
absolute velocity of the water leaving the vane will be w = Bc. The
component of w in the direction of motion of the vane is Ba = Bb - ab
= u - (v - u) cos [alpha]. Hence if Q is the quantity of water
reaching the vane per second the change of momentum per second in the
direction of the vane's motion is (GQ/g)[v - {u - (v - u) cos
[alpha]}] = (GQ/g)(v - u)(1 + cos [alpha]). If a = 0°, cos [alpha] =
1, and the change of momentum per second, which is equal to the effort
driving the vane, is P = 2(GQ/g)(v - u). The work done on the vane is
Pu = 2(GQ/g)(v - u)u. If a series of vanes are interposed in
succession, the quantity of water impinging on the vanes per second is
the total discharge of the nozzle, and the energy expended at the
nozzle is GQv²/2g. Hence the efficiency of the arrangement is, when
[alpha] = 0°, neglecting friction,
[eta] = 2Pu/GQv² = 4(v - u)u/v²,
which is a maximum and equal to unity if u = ½v. In that case the
whole energy of the jet is usefully expended in driving the series of
vanes. In practice [alpha] cannot be quite zero or the water leaving
one vane would strike the back of the next advancing vane. Fig. 203
shows a Pelton vane. The water divides each way, and leaves the vane
on each side in a direction nearly parallel to the direction of motion
of the vane. The best velocity of the vane is very approximately half
the velocity of the jet.
§ 202. _Regulation of the Pelton Wheel._--At first Pelton wheels were
adjusted to varying loads merely by throttling the supply. This method
involves a total loss of part of the head at the sluice or throttle
valve. In addition as the working head is reduced, the relation
between wheel velocity and jet velocity is no longer that of greatest
efficiency. Next a plan was adopted of deflecting the jet so that only
part of the water reached the wheel when the load was reduced, the
rest going to waste. This involved the use of an equal quantity of
water for large and small loads, but it had, what in some cases is an
advantage, the effect of preventing any water hammer in the supply
pipe due to the action of the regulator. In most cases now regulation
is effected by varying the section of the jet. A conical needle in the
nozzle can be advanced or withdrawn so as to occupy more or less of
the aperture of the nozzle. Such a needle can be controlled by an
ordinary governor.
§ 203. _General Considerations on the Choice of a Type of Turbine._--The
circumferential speed of any turbine is necessarily a fraction of the
initial velocity of the water, and therefore is greater as the head is
greater. In reaction turbines with complete admission the number of
revolutions per minute becomes inconveniently great, for the diameter
cannot be increased beyond certain limits without greatly reducing the
efficiency. In impulse turbines with partial admission the diameter can
be chosen arbitrarily and the number of revolutions kept down on high
falls to any desired amount. Hence broadly reaction turbines are better
and less costly on low falls, and impulse turbines on high falls. For
variable water flow impulse turbines have some advantage, being more
efficiently regulated. On the other hand, impulse turbines lose
efficiency seriously if their speed varies from the normal speed due to
the head. If the head is very variable, as it often is on low falls, and
the turbine must run at the same speed whatever the head, the impulse
turbine is not suitable. Reaction turbines can be constructed so as to
overcome this difficulty to a great extent. Axial flow turbines with
vertical shafts have the disadvantage that in addition to the weight of
the turbine there is an unbalanced water pressure to be carried by the
footstep or collar bearing. In radial flow turbines the hydraulic
pressures are balanced. The application of turbines to drive dynamos
directly has involved some new conditions. The electrical engineer
generally desires a high speed of rotation, and a very constant speed at
all times. The reaction turbine is generally more suitable than the
impulse turbine. As the diameter of the turbine depends on the quantity
of water and cannot be much varied without great inefficiency, a
difficulty arises on low falls. This has been met by constructing four
independent reaction turbines on the same shaft, each having of course
the diameter suitable for one-quarter of the whole discharge, and having
a higher speed of rotation than a larger turbine. The turbines at
Rheinfelden and Chevres are so constructed. To ensure constant speed of
rotation when the head varies considerably without serious inefficiency,
an axial flow turbine is generally used. It is constructed of three or
four concentric rings of vanes, with independent regulating sluices,
forming practically independent turbines of different radii. Any one of
these or any combination can be used according to the state of the
water. With a high fall the turbine of largest radius only is used, and
the speed of rotation is less than with a turbine of smaller radius. On
the other hand, as the fall decreases the inner turbines are used either
singly or together, according to the power required. At the Zürich
waterworks there are turbines of 90 h.p. on a fall varying from 10½ ft.
to 4¾ ft. The power and speed are kept constant. Each turbine has three
concentric rings. The outermost ring gives 90 h.p. with 105 cub. ft. per
second and the maximum fall. The outer and middle compartments give the
same power with 140 cub. ft. per second and a fall of 7 ft. 10 in. All
three compartments working together develop the power with about 250
cub. ft. per second. In some tests the efficiency was 74% with the outer
ring working alone, 75.4% with the outer and middle ring working and a
fall of 7 ft., and 80.7% with all the rings working.
[Illustration: FIG. 206.]
§ 204. _Speed Governing._--When turbines are used to drive dynamos
direct, the question of speed regulation is of great importance. Steam
engines using a light elastic fluid can be easily regulated by governors
acting on throttle or expansion valves. It is different with water
turbines using a fluid of great inertia. In one of the Niagara penstocks
there are 400 tons of water flowing at 10 ft. per second, opposing
enormous resistance to rapid change of speed of flow. The sluices of
water turbines also are necessarily large and heavy. Hence relay
governors must be used, and the tendency of relay governors to
hunt must be overcome. In the Niagara Falls Power House No. 1, each
turbine has a very sensitive centrifugal governor acting on a ratchet
relay. The governor puts into gear one or other of two ratchets driven
by the turbine itself. According as one or the other ratchet is in gear
the sluices are raised or lowered. By a subsidiary arrangement the
ratchets are gradually put out of gear unless the governor puts them in
gear again, and this prevents the over correction of the speed from the
lag in the action of the governor. In the Niagara Power House No. 2, the
relay is an hydraulic relay similar in principle, but rather more
complicated in arrangement, to that shown in fig. 206, which is a
governor used for the 1250 h.p. turbines at Lyons. The sensitive
governor G opens a valve and puts into action a plunger driven by oil
pressure from an oil reservoir. As the plunger moves forward it
gradually closes the oil admission valve by lowering the fulcrum end f
of the valve lever which rests on a wedge w attached to the plunger. If
the speed is still too high, the governor reopens the valve. In the case
of the Niagara turbines the oil pressure is 1200 lb. per sq. in. One
millimetre of movement of the governor sleeve completely opens the relay
valve, and the relay plunger exerts a force of 50 tons. The sluices can
be completely opened or shut in twelve seconds. The ordinary variation
of speed of the turbine with varying load does not exceed 1%. If all the
load is thrown off, the momentary variation of speed is not more than
5%. To prevent hydraulic shock in the supply pipes, a relief valve is
provided which opens if the pressure is in excess of that due to the
head.
[Illustration: FIG. 207.]
§ 205. _The Hydraulic Ram._--The hydraulic ram is an arrangement by
which a quantity of water falling a distance h forces a portion of the
water to rise to a height h1, greater than h. It consists of a supply
reservoir (A, fig. 207), into which the water enters from some natural
stream. A pipe s of considerable length conducts the water to a lower
level, where it is discharged intermittently through a self-acting
pulsating valve at d. The supply pipe s may be fitted with a flap valve
for stopping the ram, and this is attached in some cases to a float, so
that the ram starts and stops itself automatically, according as the
supply cistern fills or empties. The lower float is just sufficient to
keep open the flap after it has been raised by the action of the upper
float. The length of chain is adjusted so that the upper float opens the
flap when the level in the cistern is at the desired height. If the
water-level falls below the lower float the flap closes. The pipe s
should be as long and as straight as possible, and as it is subjected to
considerable pressure from the sudden arrest of the motion of the water,
it must be strong and strongly jointed. a is an air vessel, and e the
delivery pipe leading to the reservoir at a higher level than A, into
which water is to be pumped. Fig. 208 shows in section the construction
of the ram itself. d is the pulsating discharge valve already mentioned,
which opens inwards and downwards. The stroke of the valve is regulated
by the cotter through the spindle, under which are washers by which the
amount of fall can be regulated. At o is a delivery valve, opening
outwards, which is often a ball-valve but sometimes a flap-valve. The
water which is pumped passes through this valve into the air vessel a,
from which it flows by the delivery pipe in a regular stream into the
cistern to which the water is to be raised. In the vertical chamber
behind the outer valve a small air vessel is formed, and into this
opens an aperture ¼ in. in diameter, made in a brass screw plug b. The
hole is reduced to 1/16 in. in diameter at the outer end of the plug
and is closed by a small valve opening inwards. Through this, during the
rebound after each stroke of the ram, a small quantity of air is sucked
in which keeps the air vessel supplied with its elastic cushion of air.
[Illustration: FIG. 208.]
During the recoil after a sudden closing of the valve d, the pressure
below it is diminished and the valve opens, permitting outflow. In
consequence of the flow through this valve, the water in the supply pipe
acquires a gradually increasing velocity. The upward flow of the water,
towards the valve d, increases the pressure tending to lift the valve,
and at last, if the valve is not too heavy, lifts and closes it. The
forward momentum of the column in the supply pipe being destroyed by the
stoppage of the flow, the water exerts a pressure at the end of the pipe
sufficient to open the delivery valve o, and to cause a portion of the
water to flow into the air vessel. As the water in the supply pipe comes
to rest and recoils, the valve d opens again and the operation is
repeated. Part of the energy of the descending column is employed in
compressing the air at the end of the supply pipe and expanding the pipe
itself. This causes a recoil of the water which momentarily diminishes
the pressure in the pipe below the pressure due to the statical head.
This assists in opening the valve d. The recoil of the water is
sufficiently great to enable a pump to be attached to the ram body
instead of the direct rising pipe. With this arrangement a ram working
with muddy water may be employed to raise clear spring water. Instead of
lifting the delivery valve as in the ordinary ram, the momentum of the
column drives a sliding or elastic piston, and the recoil brings it
back. This piston lifts and forces alternately the clear water through
ordinary pump valves.