534 HYDKAULICS AND ITS APPLICATIONS 



while if bz and b 3 are the effective breadths of the wheel passages at 

 entrance and exit, so that 2 TT ?> 2 ''2 and 2 TT b 3 r 3 are the effective passage 



areas, we have, for continuity of flow, / 3 = f z y 2 . Also = ^ 3 , so that 



' * 



2 , r \ tan 



" * tan 



/ tan a \ 



b 3 tan y 



(26) 



Assuming the turbine to be designed, however, so that u' 3 = 0, we get 

 from (2) 



tan 



If the wheel is horizontal, or if in a vertical wheel we neglect the 

 differences of level at the highest and lowest points and also neglect 

 losses at entrance, we have : 



Pz I ^2 2 P3 I r a 2 I work done by , losses per Ib. be- ,.. 

 W h 2^ ~ W 2~g H " water per Ib. tween (2) and (3). 



If then H'=H -^ - = head available for producing flow through 



the wheel, i.e., the total head minus that necessary to overcome pipe 

 line losses, we must have : 



rrt _. P* i V * __ ^3 / 5 \ 



- W + 2 W 



so that, substituting in (4) and putting i' 3 2 = w 3 2 + / 3 2 



( = / 3 2 if w a = 0, 



i TTt /3 I < t; 2 / 1 lttu "I /'fi\ 



we have " H' = ^ H I 1 - , g] 



from which, writing / 3 = / 2 r 2 ^- = ^'2 tan a = - we get : 



OQ ^*Q ^3 ^ 3 



W = ^ ( 2 + ( ?^tan a V - 2 5-? 1 (7) 



W2 A ' ^ 7 \ * tan a 



2 g H> (81 



, 3 /3 / tan 



- ( i _ tan a \ / ^ g H (9 



V L t^~8 )\l < i h r* L , \ 2 tan a 



U* 



- 2 



This neglects losses due to friction, etc., in the wheel. 



