Chapter XII: Impact and Reaction of Water (3)

§ 187. _General Description of a Reaction Turbine._--Professor James Thomson's inward flow or vortex turbine has been selected as the type of reaction turbines. It is one of the best in normal conditions of working, and the mode of regulation introduced is decidedly superior to that in most reaction turbines. Figs. 185 and 186 are external views of the turbine case; figs. 187 and 188 are the corresponding sections; fig. 189 is the turbine wheel. The example chosen for illustration has suction pipes, which permit the turbine to be placed above the tail-water level. The water enters the turbine by cast-iron supply pipes at A, and is discharged through two suction pipes S, S. The water on entering the case distributes itself through a rectangular supply chamber SC, from which it finds its way equally to the four guide-blade passages G, G, G, G. In these passages it acquires a velocity about equal to that due to half the fall, and is directed into the wheel at an angle of about 10° or 12° with the tangent to its circumference. The wheel W receives the water in equal proportions from each guide-blade passage. It consists of a centre plate p (fig. 189) keyed on the shaft aa, which passes through stuffing boxes on the suction pipes. On each side of the centre plate are the curved wheel vanes, on which the pressure of the water acts, and the vanes are bounded on each side by dished or conical cover plates c, c. Joint-rings j, j on the cover plates make a sufficiently water-tight joint with the casing, to prevent leakage from the guide-blade chamber into the suction pipes. The pressure near the joint rings is not very great, probably not one-fourth the total head. The wheel vanes receive the water without shock, and deliver it into central spaces, from which it flows on either side to the suction pipes. The mode of regulating the power of the turbine is very simple. The guide-blades are pivoted to the case at their inner ends, and they are connected by a link-work, so that they all open and close simultaneously and equally. In this way the area of opening through the guide-blades is altered without materially altering the angle or the other conditions of the delivery into the wheel. The guide-blade gear may be variously arranged. In this example four spindles, passing through the case, are linked to the guide-blades inside the case, and connected together by the links l, l, l on the outside of the case. A worm wheel on one of the spindles is rotated by a worm d, the motion being thus slow enough to adjust the guide-blades very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal.

Fig. 190 shows another arrangement of a similar turbine, with some
adjuncts not shown in the other drawings. In this case the turbine
rotates horizontally, and the turbine case is placed entirely below
the tail water. The water is supplied to the turbine by a vertical
pipe, over which is a wooden pentrough, containing a strainer, which
prevents sticks and other solid bodies getting into the turbine. The
turbine rests on three foundation stones, and, the pivot for the
vertical shaft being under water, there is a screw and lever
arrangement for adjusting it as it wears. The vertical shaft gives
motion to the machinery driven by a pair of bevel wheels. On the right
are the worm and wheel for working the guide-blade gear.

§ 188. _Hydraulic Power at Niagara._--The largest development of
hydraulic power is that at Niagara. The Niagara Falls Power Company
have constructed two power houses on the United States side, the first
with 10 turbines of 5000 h.p. each, and the second with 10 turbines of
5500 h.p. The effective fall is 136 to 140 ft. In the first power
house the turbines are twin outward flow reaction turbines with
vertical shafts running at 250 revs. per minute and driving the
dynamos direct. In the second power house the turbines are inward flow
turbines with draft tubes or suction pipes. Fig. 191 shows a section
of one of these turbines. There is a balancing piston keyed on the
shaft, to the under side of which the pressure due to the fall is
admitted, so that the weight of turbine, vertical shaft and part of
the dynamo is water borne. About 70,000 h.p. is daily distributed
electrically from these two power houses. The Canadian Niagara Power
Company are erecting a power house to contain eleven units of 10,250
h.p. each, the turbines being twin inward flow reaction turbines. The
Electrical Development Company of Ontario are erecting a power house
to contain 11 units of 12,500 h.p. each. The Ontario Power Company are
carrying out another scheme for developing 200,000 h.p. by twin inward
flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and
Manufacturing Company on the United States side have a station giving
35,000 h.p. and are constructing another to furnish 100,000 h.p. The
mean flow of the Niagara river is about 222,000 cub. ft. per second
with a fall of 160 ft. The works in progress if completed will utilize
650,000 h.p. and require 48,000 cub. ft. per second or 21½% of the
mean flow of the river (Unwin, "The Niagara Falls Power Stations,"
_Proc. Inst. Mech. Eng._, 1906).

§ 189. _Different Forms of Turbine Wheel._--The wheel of a turbine or
part of the machine on which the water acts is an annular space,
furnished with curved vanes dividing it into passages exactly or
roughly rectangular in cross section. For radial flow turbines the
wheel may have the form A or B, fig. 192, A being most usual with
inward, and B with outward flow turbines. In A the wheel vanes are
fixed on each side of a centre plate keyed on the turbine shaft. The
vanes are limited by slightly-coned annular cover plates. In B the
vanes are fixed on one side of a disk, keyed on the shaft, and limited
by a cover plate parallel to the disk. Parallel flow or axial flow
turbines have the wheel as in C. The vanes are limited by two
concentric cylinders.

_Theory of Reaction Turbines._

§ 190. _Velocity of Whirl and Velocity of Flow._--Let acb (fig. 193)
be the path of the particles of water in a turbine wheel. That path
will be in a plane normal to the axis of rotation in radial flow
turbines, and on a cylindrical surface in axial flow turbines. At any
point c of the path the water will have some velocity v, in the
direction of a tangent to the path. That velocity may be resolved into
two components, a whirling velocity w in the direction of the wheel's
rotation at the point c, and a component u at right angles to this,
radial in radial flow, and parallel to the axis in axial flow
turbines. This second component is termed the velocity of flow. Let
v_o, w_o, u_o be the velocity of the water, the whirling velocity and
velocity of flow at the outlet surface of the wheel, and v_i, w_i, u_i
the same quantities at the inlet surface of the wheel. Let [alpha] and
[beta] be the angles which the water's direction of motion makes with
the direction of motion of the wheel at those surfaces. Then

w_o = v_o cos [beta]; u_o = v_o sin [beta]

w_i = v_i cos [alpha]; u_i = v_i sin [alpha]. (10)

The velocities of flow are easily ascertained independently from the
dimensions of the wheel. The velocities of flow at the inlet and
outlet surfaces of the wheel are normal to those surfaces. Let
[Omega]_o, [Omega]_i be the areas of the outlet and inlet surfaces of
the wheel, and Q the volume of water passing through the wheel per
second; then

v_o = Q/[Omega]_o; v_i = Q/[Omega]_i. (11)

Using the notation in fig. 191, we have, for an inward flow turbine
(neglecting the space occupied by the vanes),

[Omega]_o = 2[pi]r0d0; [Omega]_i = 2[pi]r_i d_i. (12a)

Similarly, for an outward flow turbine,

[Omega]_o = 2[pi]r_o d; [Omega]_i = 2[pi]r_i d; (12b)

and, for an axial flow turbine,

[Omega]_o = [Omega]_i = [pi](r2² - r1²). (12c)

_Relative and Common Velocity of the Water and Wheel._--There is
another way of resolving the velocity of the water. Let V be the
velocity of the wheel at the point c, fig. 194. Then the velocity of
the water may be resolved into a component V, which the water has in
common with the wheel, and a component v_r, which is the velocity of
the water relatively to the wheel.

_Velocity of Flow._--It is obvious that the frictional losses of head
in the wheel passages will increase as the velocity of flow is
greater, that is, the smaller the wheel is made. But if the wheel
works under water, the skin friction of the wheel cover increases as
the diameter of the wheel is made greater, and in any case the weight
of the wheel and consequently the journal friction increase as the
wheel is made larger. It is therefore desirable to choose, for the
velocity of flow, as large a value as is consistent with the condition
that the frictional losses in the wheel passages are a small fraction
of the total head.

The values most commonly assumed in practice are these:--

In axial flow turbines, u_o = u_i = 0.15 to 0.2 [root](2gH);

In outward flow turbines, u_i = 0.25 [root]2g(H - [h]),
u_o = 0.21 to 0.17 [root]2g(H - [h]);

In inward flow turbines, u_o = u_i = 0.125 [root](2gH).

§ 191. _Speed of the Wheel._--The best speed of the wheel depends
partly on the frictional losses, which the ordinary theory of turbines
disregards. It is best, therefore, to assume for V_o and V_i values
which experiment has shown to be most advantageous.

In axial flow turbines, the circumferential velocities at the mean
radius of the wheel may be taken

V_o = V_i = 0.6 [root](2gH) to 0.66 [root](2gH).

In a radial outward flow turbine,

V_i = 0.56 [root]{2g(H - [h])}

V_o = V_i r_o/r_i,

where r_o, r_i are the radii of the outlet and inlet surfaces.

In a radial inward flow turbine,

V_i = 0.66 [root](2gH),

V_o = V_i r_o/r_i.

If the wheel were stationary and the water flowed through it, the
water would follow paths parallel to the wheel vane curves, at least
when the vanes were so close that irregular motion was prevented.
Similarly, when the wheel is in motion, the water follows paths
relatively to the wheel, which are curves parallel to the wheel vanes.
Hence the relative component, v_r, of the water's motion at c is
tangential to a wheel vane curve drawn through the point c. Let v_o,
V_o, v_(ro) be the velocity of the water and its common and relative
components at the outlet surface of the wheel, and v_i, V_i, v_(ri) be
the same quantities at the inlet surface; and let [theta] and [phi] be
the angles the wheel vanes make with the inlet and outlet surfaces;
then

v_o² = [root](v_(ro)² + V_o² - 2V_o v_(ro) cos [phi])

v_i = [root](v_(ri)² + V_o² - 2V_i v_(ri) cos [theta]), (13)

equations which may be used to determine [phi] and [theta].

§ 192. _Condition determining the Angle of the Vanes at the Outlet
Surface of the Wheel._--It has been shown that, when the water leaves
the wheel, it should have no tangential velocity, if the efficiency is
to be as great as possible; that is, w_o = 0. Hence, from (10), cos
[beta] = 0, [beta] = 90°, U_o = V_o, and the direction of the water's
motion is normal to the outlet surface of the wheel, radial in radial
flow, and axial in axial flow turbines.

Drawing v_o or u_o radial or axial as the case may be, and V_o
tangential to the direction of motion, v_(ro) can be found by the
parallelogram of velocities. From fig. 195,

tan [phi] = v_o/V_o = u_o/V_o; (14)

but [phi] is the angle which the wheel vane makes with the outlet
surface of the wheel, which is thus determined when the velocity of
flow u_o and velocity of the wheel V_o are known. When [phi] is thus
determined,

v_(ro) = U_o cosec [phi] = V_o [root](1 + u_o²/V_o²). (14a)

_Correction of the Angle [phi] to allow for Thickness of Vanes._--In
determining [phi], it is most convenient to calculate its value
approximately at first, from a value of u_o obtained by neglecting the
thickness of the vanes. As, however, this angle is the most important
angle in the turbine, the value should be afterwards corrected to
allow for the vane thickness.

Let

[phi]´ = tan^(-1)(u_o/V_o) = tan^(-1)(Q/[Omega]_o V_o)

be the first or approximate value of [phi], and let t be the
thickness, and n the number of wheel vanes which reach the outlet
surface of the wheel. As the vanes cut the outlet surface
approximately at the angle [phi]´, their width measured on that
surface is t cosec [phi]´. Hence the space occupied by the vanes on
the outlet surface is

For

A, fig. 192, ntd_o cosec [phi]
B, fig. 192, ntd cosec [phi] (15)
C, fig. 192, nt(r2 - r1) cosec [phi].

Call this area occupied by the vanes [omega]. Then the true value of
the clear discharging outlet of the wheel is [Omega]_o - [omega], and
the true value of u_o is Q/([Omega]_o - [omega]). The corrected value
of the angle of the vanes will be

[phi] = tan [Q/V_o ([Omega]_o - [omega]) ]. (16)

§ 193. _Head producing Velocity with which the Water enters the
Wheel._--Consider the variation of pressure in a wheel passage, which
satisfies the condition that the sections change so gradually that
there is no loss of head in shock. When the flow is in a horizontal
plane, there is no work done by gravity on the water passing through
the wheel. In the case of an axial flow turbine, in which the flow is
vertical, the fall d between the inlet and outlet surfaces should be
taken into account.

Let

V_i, V_o be the velocities of the wheel at the inlet and outlet
surfaces,
v_i, v_o the velocities of the water,
u_i, u_o the velocities of flow,
v_(ri), v_(ro) the relative velocities,
h_i, h_o the pressures, measured in feet of water,
r_i, r_o the radii of the wheel,
[alpha] the angular velocity of the wheel.

At any point in the path of a portion of water, at radius r, the
velocity v of the water may be resolved into a component V = [alpha]r
equal to the velocity at that point of the wheel, and a relative
component v_r. Hence the motion of the water may be considered to
consist of two parts:--(a) a motion identical with that in a forced
vortex of constant angular velocity [alpha]; (b) a flow along curves
parallel to the wheel vane curves. Taking the latter first, and using
Bernoulli's theorem, the change of pressure due to flow through the
wheel passages is given by the equation

h´_i + v_(ri)²/2g = h´_o + v_(ro)²/2g;

h´_i - h´_o = (v_(ro)² - v_(ri)²)/2g.

The variation of pressure due to rotation in a forced vortex is

h´´_i - h´´_o = (V_i² - V_o²)/2g.

Consequently the whole difference of pressure at the inlet and outlet
surfaces of the wheel is

h_i - h_o = h´_i + h´´_i - h´_o - h´´_o
= (V_i² - V_o²)/2g + (v_(ro)² - v_(ri)²)/2g. (17)

_Case 1. Axial Flow Turbines._--V_i = V_o; and the first term on the
right, in equation 17, disappears. Adding, however, the work of
gravity due to a fall of d ft. in passing through the wheel,

h_i - h_o = (v_(ro)² - v_(ri)²)/2g - d. (17a)

_Case 2. Outward Flow Turbines._--The inlet radius is less than the
outlet radius, and (V_i² - V_o²)/2g is negative. The centrifugal head
diminishes the pressure at the inlet surface, and increases the
velocity with which the water enters the wheel. This somewhat
increases the frictional loss of head. Further, if the wheel varies in
velocity from variations in the useful work done, the quantity (V_i² -
V_o²)/2g increases when the turbine speed increases, and vice versa.
Consequently the flow into the turbine increases when the speed
increases, and diminishes when the speed diminishes, and this again
augments the variation of speed. The action of the centrifugal head in
an outward flow turbine is therefore prejudicial to steadiness of
motion. For this reason r_o : r_i is made small, generally about 5 :
4. Even then a governor is sometimes required to regulate the speed of
the turbine.

_Case 3. Inward Flow Turbines._--The inlet radius is greater than
the outlet radius, and the centrifugal head diminishes the velocity of
flow into the turbine. This tends to diminish the frictional losses,
but it has a more important influence in securing steadiness of
motion. Any increase of speed diminishes the flow into the turbine,
and vice versa. Hence the variation of speed is less than the
variation of resistance overcome. In the so-called centre vent wheels
in America, the ratio r_i : r_o is about 5 : 4, and then the influence
of the centrifugal head is not very important. Professor James Thomson
first pointed out the advantage of a much greater difference of radii.
By making r_i : r_o = 2 : 1, the centrifugal head balances about half
the head in the supply chamber. Then the velocity through the
guide-blades does not exceed the velocity due to half the fall, and
the action of the centrifugal head in securing steadiness of speed is
considerable.

Since the total head producing flow through the turbine is H -
[h], of this h_i - h_o is expended in overcoming the pressure
in the wheel, the velocity of flow into the wheel is

v_i = c_v[root]{2g(H - [h] - (V_i² - V_o²/2g + (v{r0}² - v_(ri)²)/2g)}, (18)

where c_v may be taken 0.96.

From (14a),

v{r0} = V_o [root](1 + u_o²/V_o²).

It will be shown immediately that

v_(ri) = u_i cosec [theta];

or, as this is only a small term, and [theta] is on the average 90°,
we may take, for the present purpose, v_(ri) = u_i nearly.

Inserting these values, and remembering that for an axial flow turbine
V_i = V_o, [h] = 0, and the fall d in the wheel is to be
added,
_ _
| / V_i² / u_o² \ u_i² \ |
v_i = c_v[root] | 2g ( H - ---- ( 1 + ---- ) + ---- - d ) |.
|_ \ 2g \ V_o² / 2g / _|

For an outward flow turbine,
_ _
| / V_i² / u_o² \ u_i² \ |
v_i = c_v[root] | 2g ( H - [h] - ---- ( 1 + ---- ) + ---- ) |.
|_ \ 2g \ V_i² / 2g / _|

For an inward flow turbine,
_ _
| { V_i² / u_o² \ u_i² } |
v_i = c_v[root] | 2g { H - ---- ( 1 + ---- ) + ---- } |.
|_ { 2g \ V_i² / 2g } _|

§ 194. _Angle which the Guide-Blades make with the Circumference of
the Wheel._--At the moment the water enters the wheel, the radial
component of the velocity is u_i, and the velocity is v_i. Hence, if
[gamma] is the angle between the guide-blades and a tangent to the
wheel

[gamma] = sin^(-1) (u_i/v_i).

This angle can, if necessary, be corrected to allow for the thickness
of the guide-blades.

§ 195. _Condition determining the Angle of the Vanes at the Inlet
Surface of the Wheel._--The single condition necessary to be satisfied
at the inlet surface of the wheel is that the water should enter the
wheel without shock. This condition is satisfied if the direction of
relative motion of the water and wheel is parallel to the first
element of the wheel vanes.

Let A (fig. 196) be a point on the inlet surface of the wheel, and let
v_i represent in magnitude and direction the velocity of the water
entering the wheel, and V_i the velocity of the wheel. Completing the
parallelogram, v_(ri) is the direction of relative motion. Hence the
angle between v_(ri) and V_i is the angle [theta] which the vanes
should make with the inlet surface of the wheel.

§ 196. _Example of the Method of designing a Turbine. Professor James
Thomson's Inward Flow Turbine._--

Let

H = the available fall after deducting loss of head in pipes and
channels from the gross fall;
Q = the supply of water in cubic feet per second; and
[eta] = the efficiency of the turbine.

The work done per second is [eta]GQH, and the horse-power of the
turbine is h.p. = [eta]GQH/550. If [eta] is taken at 0.75, an
allowance will be made for the frictional losses in the turbine, the
leakage and the friction of the turbine shaft. Then h.p. = 0.085QH.

The velocity of flow through the turbine (uncorrected for the space
occupied by the vanes and guide-blades) may be taken

u_i = u_i = 0.125 [root](2gH),

in which case about (1/64)th of the energy of the fall is carried away
by the water discharged.

The areas of the outlet and inlet surface of the wheel are then

2[pi]r_o d_o = 2[pi]r_i d_i = Q/0.125 [root](2gH).

If we take r_o, so that the axial velocity of discharge from the
central orifices of the wheel is equal to u_o, we get

r_o = 0.3984 [root](Q/[root]H),

d_o = r_o.

If, to obtain considerable steadying action of the centrifugal head,
r_i = 2r_o, then d_i = ½d_o.

_Speed of the Wheel._--Let V_i = 0.66 [root](2gH), or the speed due to
half the fall nearly. Then the number of rotations of the turbine per
second is

N = V_i/2[pi]r_i = 1.0579 [root](H[root]H/Q);

also

V_o = V_i r_o/r_i = 0.33 [root](2gH).

_Angle of Vanes with Outlet Surface._

Tan[phi] = u_o/V_o = 0.125/0.33 = .3788;

[phi] = 21º nearly.

If this value is revised for the vane thickness it will ordinarily
become about 25º.

_Velocity with which the Water enters the Wheel._--The head producing
the velocity is

H - (V_i²/2g) (1 + u_o²/V_i²) + u_i²/2g
= H {1 - .4356 (1 + 0.0358) + .0156}
= 0.5646H.

Then the velocity is

V_i = .96 [root](2g(.5646H)) = 0.721 [root](2gH).

_Angle of Guide-Blades._

Sin [gamma] = u_i/v_i = 0.125/0.721 = 0.173;

[gamma] = 10° nearly.

_Tangential Velocity of Water entering Wheel._

w_i = v_i cos [gamma] = 0.7101 [root](2gH).

_Angle of Vanes at Inlet Surface._

Cot [theta] = (w_i - V_i)/u_i = (.7101 - .66)/.125 = .4008;

[theta] = 68° nearly.

_Hydraulic Efficiency of Wheel._

[eta] = w_iV_i/gH = .7101 × .66 × 2
= 0.9373.

This, however, neglects the friction of wheel covers and leakage. The
efficiency from experiment has been found to be 0.75 to 0.80.

_Impulse and Partial Admission Turbines._

§ 197. The principal defect of most turbines with complete admission is the imperfection of the arrangements for working with less than the normal supply. With many forms of reaction turbine the efficiency is considerably reduced when the regulating sluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest possible amount of work. The imperfection of the regulating arrangements is therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. B. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial flow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and guide-blades. Fontaine similarly employed two concentric axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but they are not applicable to every form of turbine.

C. Callon, in 1840, patented an arrangement of sluices for axial or outward flow turbines, which were to be closed successively as the water supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The water was thus admitted to opposite but equal arcs of the wheel, and the forces driving the turbine were symmetrically placed. As soon as this arrangement was adopted, a modification of the mode of action of the water in the turbine became necessary. If the turbine wheel passages remain full of water during the whole rotation, the water contained in each passage must be put into motion each time it passes an open portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must be placed above the tail water, and the wheel passages be allowed to clear themselves of water, while passing from one open portion of the sluices to the next.

But if the wheel passages are free of water when they arrive at the open guide passages, then there can be no pressure other than atmospheric pressure in the clearance space between guides and wheel. The water must issue from the sluices with the whole velocity due to the head; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a small final velocity only, precisely in the same way as when a single jet strikes a curved vane in the free air. Turbines of this kind are therefore termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action on the wheel, and the whole energy of the jet is imparted to the wheel, simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efficiency of the wheel. The diameter of the wheel may be made as large as convenient, and the water admitted to a small fraction of the circumference only. Then the number of revolutions is independent of the water velocity, and may be kept down to a manageable value.

§ 198. _General Description of an Impulse Turbine or Turbine with Free
Deviation._--Fig. 197 shows a general sectional elevation of a Girard
turbine, in which the flow is axial. The water, admitted above a
horizontal floor, passes down through the annular wheel containing the
guide-blades G, G, and thence into the revolving wheel WW. The
revolving wheel is fixed to a hollow shaft suspended from the pivot p.
The solid internal shaft ss is merely a fixed column supporting the
pivot. The advantage of this is that the pivot is accessible for
lubrication and adjustment. B is the mortise bevel wheel by which the
power of the turbine is given off. The sluices are worked by the hand
wheel h, which raises them successively, in a way to be described
presently. d, d are the sluice rods. Figs. 198, 199 show the sectional
form of the guide-blade chamber and wheel and the curves of the wheel
vanes and guide-blades, when drawn on a plane development of the
cylindrical section of the wheel; a, a, a are the sluices for cutting
off the water; b, b, b are apertures by which the entrance or exit of
air is facilitated as the buckets empty and fill. Figs. 200, 201 show
the guide-blade gear. a, a, a are the sluice rods as before. At the
top of each sluice rod is a small block c, having a projecting tongue,
which slides in the groove of the circular cam plate d, d. This
circular plate is supported on the frame e, and revolves on it by
means of the flanged rollers f. Inside, at the top, the cam plate is
toothed, and gears into a spur pinion connected with the hand wheel h.
At gg is an inclined groove or shunt. When the tongues of the blocks
c, c arrive at g, they slide up to a second groove, or the reverse,
according as the cam plate is revolved in one direction or in the
other. As this operation takes place with each sluice successively,
any number of sluices can be opened or closed as desired. The turbine
is of 48 horse power on 5.12 ft. fall, and the supply of water varies
from 35 to 112 cub. ft. per second. The efficiency in normal working
is given as 73%. The mean diameter of the wheel is 6 ft., and the
speed 27.4 revolutions per minute.

As an example of a partial admission radial flow impulse turbine, a
100 h.p. turbine at Immenstadt may be taken. The fall varies from 538
to 570 ft. The external diameter of the wheel is 4½ ft., and its
internal diameter 3 ft. 10 in. Normal speed 400 revs. per minute.
Water is discharged into the wheel by a single nozzle, shown in fig.
202 with its regulating apparatus and some of the vanes. The water
enters the wheel at an angle of 22° with the direction of motion, and
the final angle of the wheel vanes is 20°. The efficiency on trial was
from 75 to 78%.

§ 199. _Theory of the Impulse Turbine._--The theory of the impulse
turbine does not essentially differ from that of the reaction turbine,
except that there is no pressure in the wheel opposing the discharge
from the guide-blades. Hence the velocity with which the water enters
the wheel is simply

v_i = 0.96 [root]{2g(H - [h])},

where [heta] is the height of the top of the wheel above the tail
water. If the hydropneumatic system is used, then [h] = 0. Let
Q_m be the maximum supply of water, r1, r2 the internal and external
radii of the wheel at the inlet surface; then

u_i = Q_m/{[pi](r2² - r1²)}.

The value of u_i may be about 0.45 [root]{2g(H - [eta][h])},
whence r1, r2 can be determined.

The guide-blade angle is then given by the equation

sin [gamma] = u_i/v_i = 0.45/0.94 = .48;

[gamma] = 29°.

The value of u_i should, however, be corrected for the space occupied
by the guide-blades.

The tangential velocity of the entering water is

w_i = v_i cos [gamma] = 0.82 [root]{2g(H - [h])}.

The circumferential velocity of the wheel may be (at mean radius)

V_i = 0.5 [root]{2g(H - [h])}.

Hence the vane angle at inlet surface is given by the equation

cot [theta] = (w_i - V_i)/u_i = (0.82 - 0.5)/0.45 = .71;

[theta] = 55°.

The relative velocity of the water striking the vane at the inlet edge
is v_(ri) = u_i cosec[theta] = 1.22 u_i. This relative velocity remains
unchanged during the passage of the water over the vane; consequently
the relative velocity at the point of discharge is v_(ro) = 1.22 u_i.
Also in an axial flow turbine V_o = V_i.

If the final velocity of the water is axial, then

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).

§ 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.

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.

§ 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.

§ 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.

§ 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.

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.

PUMPS

§ 206. The different classes of pumps correspond almost exactly to the different classes of water motors, although the mechanical details of the construction are somewhat different. They are properly reversed water motors. Ordinary reciprocating pumps correspond to water-pressure engines. Chain and bucket pumps are in principle similar to water wheels in which the water acts by weight. Scoop wheels are similar to undershot water wheels, and centrifugal pumps to turbines.

_Reciprocating Pumps_ are single or double acting, and differ from water-pressure engines in that the valves are moved by the water instead of by automatic machinery. They may be classed thus:--

1. _Lift Pumps._--The water drawn through a foot valve on the ascent of the pump bucket is forced through the bucket valve when it descends, and lifted by the bucket when it reascends. Such pumps give an intermittent discharge.

2. _Plunger or Force Pumps_, in which the water drawn through the foot valve is displaced by the descent of a solid plunger, and forced through a delivery valve. They have the advantage that the friction is less than that of lift pumps, and the packing round the plunger is easily accessible, whilst that round a lift pump bucket is not. The flow is intermittent.

3. _The Double-acting Force Pump_ is in principle a double plunger pump. The discharge fluctuates from zero to a maximum and back to zero each stroke, but is not arrested for any appreciable time.

4. _Bucket and Plunger Pumps_ consist of a lift pump bucket combined with a plunger of half its area. The flow varies as in a double-acting pump.

5. _Diaphragm Pumps_ have been used, in which the solid plunger is replaced by an elastic diaphragm, alternately depressed into and raised out of a cylinder.

As single-acting pumps give an intermittent discharge three are generally used on cranks at 120°. But with all pumps the variation of velocity of discharge would cause great waste of work in the delivery pipes when they are long, and even danger from the hydraulic ramming action of the long column of water. An air vessel is interposed between the pump and the delivery pipes, of a volume from 5 to 100 times the space described by the plunger per stroke. The air in this must be replenished from time to time, or continuously, by a special air-pump. At low speeds not exceeding 30 ft. per minute the delivery of a pump is about 90 to 95% of the volume described by the plunger or bucket, from 5 to 10% of the discharge being lost by leakage. At high speeds the quantity pumped occasionally exceeds the volume described by the plunger, the momentum of the water keeping the valves open after the turn of the stroke.

The velocity of large mining pumps is about 140 ft. per minute, the indoor or suction stroke being sometimes made at 250 ft. per minute. Rotative pumping engines of large size have a plunger speed of 90 ft. per minute. Small rotative pumps are run faster, but at some loss of efficiency. Fire-engine pumps have a speed of 180 to 220 ft. per minute.

The efficiency of reciprocating pumps varies very greatly. Small reciprocating pumps, with metal valves on lifts of 15 ft., were found by Morin to have an efficiency of 16 to 40%, or on the average 25%. When used to pump water at considerable pressure, through hose pipes, the efficiency rose to from 28 to 57%, or on the average, with 50 to 100 ft. of lift, about 50%. A large pump with barrels 18 in. diameter, at speeds under 60 ft. per minute, gave the following results:--

Lift in feet 14½ 34 47
Efficiency .46 .66 .70

The very large steam-pumps employed for waterworks, with 150 ft. or more of lift, appear to reach an efficiency of 90%, not including the friction of the discharge pipes. Reckoned on the indicated work of the steam-engine the efficiency may be 80%.

Many small pumps are now driven electrically and are usually three-throw single-acting pumps driven from the electric motor by gearing. It is not convenient to vary the speed of the motor to accommodate it to the varying rate of pumping usually required. Messrs Hayward Tyler have introduced a mechanism for varying the stroke of the pumps (Sinclair's patent) from full stroke to nil, without stopping the pumps.