the machine. When the machine is at rest the water issues from
the orifices with the velocity / (2 gH) (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-l-a.2r2/2g ft. of water, and the velocity of discharge
through the orifices is 11=~[(2gH-1-a2r2). If the total area of the
orifices is w, the quantity discharged from the wheel per second is
Q =wv=<.»1/ (2gH-l-azrf).

Uhile the water passes through the orifices with the Velocity 11, the orifices are moving in the opposite direction with the velocity ar. The absolute velocity of the water is therefore 11- ar=/ (2gH+a.'r2)-af.

The momentum generated per second is (GQ/g)(11-ar), 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) (11-ar)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 (11- ar) ar H 2gH -1-aff'-arlar

7' gH T gH

H ZH2

Let ~/2gH+a2f2=af+§ - .

then ' n=1-gH/ 2ar+

which increases towards the limit 1 as ar 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 ar =/ (2gH). Then the efhciency apart from friction is 11 = l~/ (2a'1'2)-“flat/gH

=O'4I4G2f2, /gH =o-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 2I % of the energy of the fall is lost in friction, in addition to the energy carried away by the water. § 184. General Stalemenl of Hydrodynamical Principles necessary for the T heory of T urb-ines.

(a) Yhen 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, 111 the velocity, and Z1 the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by hz, 112, zz. Then ll-hz = (1122-1112)/2g+Z2*Z1. (I)

If the flow is horizontal, z2=z1; and

h;-hg = (1122-4212)/2g. (Ill)

(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 111, 112 be the velocities before and after the abrupt change, then a stream of velocity 111 impinges on a stream at a velocity 112, and the relative velocity is 111-112. The head lost is (111-11»1)2/2g. Then equation (Ia) becomes hz-hi = (U1L'l'12) /2g-(1'1»112)2, /2g:W (2/'1-112) /Z~ (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 curva-A1

A 'wi ture without discontinuity.

3 g (ic)C§ qual1tyfofA/lngalar ggnpulse an range 0 ngu ar amen!

1' 2 tum.-Sup ose that a couple, the

gli I/ { moment of)which1 is M, acts on a 5 //Pl lv body of weight W for t seconds, 1, f 1 during, which it moves from A1

3, I A2 tolA2 (figg $3.96 dLet Ukbé the I /, /' ve ocity o the o y at 1, 112 its Q.-'flb velocity at Az, and let p1, pg be C 'i;~~- a' the perperidiculars from C on 111 3 and 7.11. Then Ml is termed the

w agigular impulse of the couple, and

f' 3 the quantity

" (W/g>(v1l>1~v1l>1l

Vg is the change of angular momen-FIG

184 tum relatively to C. Then, from

the equality of angular impulse

and change of angular momentum

NH = <'U2P2"01P1l 1

or, if the change of momentum is estimated for one second, M = gl (Y'2P1“U1P1>ULICS

Let r1, rg be the radii drawn from C to A1, A2, and let 101, 7.02 be the 112, perpendicular to these radii, making angles Then

111=w1 sec;3;111=1112 sec a;

p1=r1 cos 13; p2=r2 cos a.

= <'w2I'2'”ZU1}'1),

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 oints.

p Now the water fiowing 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, we be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r1, rg the radii of those surfaces By the principle above,

~ M = (GQ/2) (wm-wm)~ (4)

If a is the angular velocity of the wheel, the work done by the water on the wheel is

T = Ma= (GQ/g) ('w1r1-w2r2)a foot-pounds per second. (5) § 185. Total and Available Fall.-Let H1 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 Y), be that loss of head, which varies with the local conditions in which the turbine is placed. Then

components of 111,

/3 and a with 111, 111.

H = H1-bp

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 D 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-lj, but the efficiency of the turbine for the head H. § 186. Gross Ejicierzcy and Hydraulic Ejiciency of a Turbine.-Let Ta be the useful work done by the turbine, in foot-pounds per second, T1 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¢+T1.

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

171 =T.1/GQH1. (6)

And the hydraulic efficiency of the turbine alone is 11 = T/ GQH ~ (7)

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

From equations (5) and (7) we get

vGQH = (GQ/g)(w1r1-wmla;

T] = (70171-ZUZV2) G/gH.

This is the fundamental equation in the theory of turbines. In general# 1111 and 1111, 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 11 = wma/gH. (9)

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

71 = w1V1/gH. (ga)

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

§ 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 1 In general, because when the water leaves the turbine wheel it ceases to act on the machine. If deflecting vanes or a whirlpool are added to a turbine at the discharging side, then 111 may in part depend

on 112, and the statement above is no longer true.