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108
[PUMPS
HYDRAULICS

Fro. 210. same as for a turbine. If Q is the quantity pumped, and H the t, u¢=0'25/2 frf. ' (1) 21rr;d¢ = Also in practice de = I'2fi . . Hence, (2) fi ='257I/ (Q/V H). Usually ro = 271, and d., =d; or § d; . according as the disk is parallel-sided or coned. The water enters the wheel radially with the velocity ur, and ua = Q/21rr, , d., . QS? Fig. 21 I shows the notation adopted for the velocities. Suppose the water enters the wheel with the velocity vg, while

FIG. 211 the velocity of the wheel is Va. Completing »the parallelogram, va is the relative velocity of the water and wheel, and is the proper direction of the wheel vanes. Also, by resolving, us and ws are the component velocities of flow and velocities of whir of the velocity vt bf the water. At the outlet surface, v., is the final; velocity of discharge, and the rest of the notation is similar to that for the inlet surface. Usually the water flows equally in a ll directions in the eye of the wheel, in that case va is radial. Then, in normal conditions of working, at the inlet surface, 11; =1t; wg =0 tan 8=us/Vt (4) v, ¢ = ug cosec 6 = 4 iu¢2 +V, ”, If the ump is raising less or more than its proper quantity, 0 will not satis y the last condition, and there is then some loss of head in F shock. At the outer circumference of the wheel or outlet surface, 11,0 = u., coscc ¢ w, ,=V..-u. cot q> va 5 J iun2'i' (Vu “ua C()t Variation of Pressure in the Pum Disk.+Precisely as in the case 1 Y (5) of turbines, it can be shown that tllie variation of pressure between the inlet and outlet surfaces of the pump is ho"hi = (Vo2*Vi2)!2g (vroz vr'i2) Izg- ' i, Inserting the values of vm. v, ; in (4) and (5), we get for normal conditions of working ll., -ll; = (V, ,'-VF)/2g -11,2 cosec2¢/2g+ (uf -I-VF)/2g =V.,2/2g -uv” coscc 24>/2g-5-11,2/2g. (6) Hydraulic Ejiciency of the Pump.—Neglecting disk friction, lournal friction, and leakage, the efficiency of the pump can be found ln the same way as that of turbines (§ 186). Let M be the moment of the couple 'rotating the ump, and o. its angular velocity; wa, 1, the tangential velocity ofp the water and radius at the outlet surface; wr, rt the same quantities at the inlet surface. Q being the discharge per second, the change of angular momentum per second is (GQ/K) (wir. -wirel-Hence M = (GQ/g) (won, -win). In normal working, wg =o. Also, multiplying by the angular velocity, the work done per second is Ma = (GQ/g)w, , r., a.. But the useful work done in pumping is GQH. Therefore the efficiency is 1| =GQH/Ma. =gH/w, , r, , a =gH/'w0V, ,. (7) § 209. Case I. Centrifugal Pump with no Whirlpool Chamber.-When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a. chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity vo, and impinges on a mass flowing to the discharge pipe at the much slower velocity 11.. The radial component of 'vs is almost necessarily wasted. From the tangential component there is a gain of pressure (w.»='-v?)/2g - (wa-°vS>“/2g =vs(wa°“vs)/grwhich will be small, if v, is small compared with wo. Its greatest value, if -v. =$w¢, , is éwaf/2g, which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure in the pump disk then balances the lift and the head 14,42/2g necessary to give the initial velocity of flow in the eye of the wheel. =vv2/2g uu2 COSQC 2¢/2g+ui2/2g1 H =V, ,2/2g-u, ,2 coscc 2¢»/2gQ (8) or V0 = 4 (2gH +1402 coscc '4> 5 and the efficiency of the pump is, from (7), n=gH/V..w..=gH/{V(V.»-mcot 4>)}. ', = (V02-us” cosec ”¢>), /{2V., (V0 - up cot ¢}, (9). For ¢=90°, 1z=('¢F-1192)/2/V, . which is necessarily less than é. That is, half the work expended in driving the pump is wasted. By re curving the vanes, a plan introduced by Appold, the efficiency is increased, because the velocity v., of discharge from the pump is diminished. If ¢ is very small, cosec ¢ = cot ¢; and then n = (Vo-4-ua Cosec ¢>)/2V0, T which may approach the value I, as qb tends towards 0. Equation (8) shows that. us cosec 4>-cannot be greater than V, ,. Putting u, , =0'254 (2gH) we get the following numerical values of the

efficiency and the circumferential velocity of the pump:ff