MACHINERY.] HYDROMECHANICS .525 If the total area of the orifices is &>, the quantity discharged from the wheel per second is Q = cav = ta V2<7 H + cfr" . While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity ar. The absolute velocity of the water is therefore a 2 /- 2 - or . CO The momentum generated per second is (v- or) , which is nu merically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore, per second, GQ (v - ar}ar foot-pounds. The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is aV 2 - ar} ar Let then 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 or= V^H . Then the efficiency apart from fric tion is (V2oV - or)ar f ~ IT _Ji^4 _o-828 f about 17 per cent, of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60 per cent., so that about 21 per cent, of the energy of the fall in lost in friction, in addition to the energy carried away by the water. 170. General Statement of Hydrodynamical Principles necessary for the Theory of Turbines. 1. "When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation be tween the changes of pressure and velocity is given by Bernoulli s theorem ( 26). Suppose that, at a section A of such a passage, h^ is the pressure measured in feet of water, v l the velocity, and j the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by h z , v 2 , z. 2 . Then If the flow i.s horizontal, * 2 = ^ ; and (1). (la). 2. 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 ( 32). Let r lt v z be the velocities before and after the abrupt change, then a stream of velocity i impinges on a stream at a velocity ?-., and the relative velocity is ?-j - ? 2 . The head lost is 1 HiL . 20 Then equation (la) becomes 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. 3. Equality of Angular Impulse and CJiange of Angular Momen tum. 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 Aj to A a (fig. 184). Let i be the velocity of the body at A 1( v 2 its velocity at A,-, and lot p lt p., be the perpendiculars from C on i and v. 2 . Then AM is termed the angular impulse of the couple, and the quantity is the change of angular momentum relatively to C. Then, from the equality of angular impulse and change of angular momentum W or, if the change of momentum is estimated for one second, "W Let TU r. 2 be the radii drawn from C to A 1( A 2 , and let 7/ j, w? 2 be the components of r : , t 2 , perpendicular to these radii, making angles ;3 and a with v v r 2 . Then V 1 = w 1 sec & ; v. 2 = w 2 sec a; p i = r 1 cos ft ; Pz = r cos a. W (3), Fig. 184. 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 cubic feet enter and leave the wheel per second, and let w lt w 2 be the tangential components of the velocity of the water at the receiv ing and discharging surfaces of the wheel, r it r 2 the radii of those surfaces. By the principle above, _ M = GQ, _ wr If o is the angular velocity of the wheel, the work done by the water on the wheel is GQ = M = (w l r l - W 2 r 2 )o foot-pounds per second . (5). 171. 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 J) p be that loss of head, which varies with the local con ditions in which the turbine is placed. Then H = Hj - l) p is the available head for working the turbine, and on this the cal culations for the turbine should be based. In some cases it is neces sary to place the turbine above the tail-water level, and there is then a t allf) 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 - f), but the efficiency of the turbine for the head H. 172. Gross Efficiency and Hydraulic Efficiency of a Turbine. Let T<j be the useful work done by the turbine, in foot-pounds per second, 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 The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is
- -ffW <6)
And the hydraulic efficiency of the turbine alone is -OOT It is this last efficiency only with which the theory of turbines is concerned. From equations (5) and (7) we get _ This is the fundamental equation in the theory of turbines. In general, 1 u and , the tangential components of the water s motion 1 In general, because when the water leaves the turbine wlieel it ceases to act on the machine. If deflecting vanes or a whirlpool are added to a turbine at the discharging side, then i j may in part depend
on v. 2 , and the statement above is no longer true.
