436 HYDRAULICS AND ITS APPLICATIONS 



Let t? = initial velocity of water. 



v f = final 



,, v r = relative ,, ,, and wheel. 



u = velocity of buckets. 

 On entering the buckets the relative velocity is v r , and in virtue of this 



v 2 

 the water will rise to a height approximately equal to ~- feet above its 



normal level. 



It then falls through this height relative to the wheel under the action 

 of gravity, so that on leaving the wheel its relative velocity will again be 

 approximately equal to v r . This assumption neglects the effect of 

 friction and of eddy formation in the buckets, both of which tend to make 

 the final less than the initial relative velocity. Assuming, however, for 

 simplicity that v r is the same at inlet and at discharge, we have 



AC = C'A' = v r 

 CB = B'C' = u. 



For v f to be as small as possible with a given value of v r , evidently 

 A'B' should be perpendicular to B'C'. 



The two diagrams may now be combined graphically so as to give the 

 most suitable angles a and /? by making A'C' coincide with AC. 



Thus, draw c b (Fig. 192 b) = u and produce b c to b r , making 

 b 1 c = c b. From b' draw a perpendicular to b b', and with b as centre 

 describe an arc of a circle with v as radius to cut this at a. Join a c. 



Then ac = v r ; a b c = a ; a c b' = /?. 



From the figure it is evident that for minimum loss of kinetic energy at 

 discharge we must have v cos a = 2 u. 



v 

 u = . cos a. 



In practice a is usually made about 15. 



.-. cos a = -966. /. u = -488 v. 

 The theoretical efficiency is given by 



WQ 



But from the figure we have 



