548 HYDRAULICS AND ITS APPLICATIONS 



(7) Loss due to rejection of kinetic energy in the discharge. 



(8) Loss due to mechanical friction at bearings. 



(1) Of these losses, that due to the friction of the pipe line may be 

 readily estimated, and depends on the length, diameter and velocity of 

 flow through the pipes. 



(2) Loss by Leakage, H L . This loss being proportional to the possible 

 area through which leakage may take place and to the velocity of efflux 

 over this area, is proportional to the product of the periphery of the wheel 

 and the radial clearance at entrance, and to the square root of the head at 

 entrance. Its magnitude is greatest in turbines of the inward flow type, 

 and may amount to from 2 to 4 per cent, of the total energy supplied. 



(8) Loss due to Shock at Entrance, H E . if angle /3 is not correctly 

 proportioned. 



Let a b = v (Fig. 258) represent the velocity at the exit from guides. 



Let c b = u<i represent the velocity of vane tips. 



Then before entering the buckets the relative velocity of water and of 

 wheel in the direction of the tangent at c = c d. 



Also if m c q = /3, the relative velocity in the same direction on 

 entering the buckets = c q. 



.'. Loss of head due to shock at entrance = ^- feet of water. 



* 9 



But c d = w$ u% 



And c q = m q cot ft = / 2 cot /3. 



.-. Loss of head H Fj = <"* ~ " "/ C0t ft} * feet 



_H! l^l'faet (1) 



wz tan ft) 



The actual loss due to shock is in general less than that calculated, 

 since only a portion of the whole supply stream suffers the extreme 

 change of velocity. 



(4) Frictional Losses in the Wheel, H F . These may be divided into the 

 losses caused by (a) Flow over the surfaces of the vanes and crowns ; 

 (&) Disc friction due to rotation of the turbine crown or crowns through 

 the surrounding water. 



(a) If v r = mean relative velocity of flow through turbine, m the 

 hydraulic mean depth of turbine passages, I the effective length of 

 passages, F the coefficient of friction, this loss is given by 



F I v? , 



HF = -?r- - feet of water. 

 2 g m 



