472 HYDRAULICS AND ITS APPLICATIONS 



Let Q = volume of water discharged per second in cubic feet. 



h = head of water above the orifice in feet. 

 Then the horizontal reaction of \ . ,, 7 

 the jets, i.e., the momentum I = (v u) Ibs. 

 generated per second 



. * . Work done by this reaction ) TJ Q W , 



\ = U = 5 - ( v u) u ft. Ibs. 

 per second g 



The energy given to the wheel 



, 

 per second 



n,Y> 



Jiimciency = 



. 



Again, the total head at the orifice is the sum of the pressure due to the 

 head h and of the kinetic head due to the velocity of whirl u, so that, 

 neglecting friction, 



Substituting this value of v in the expression for the efficiency, we get 



Differentiating this with respect to u and equating the result to zero, we 



finally get for maximum efficiency ^~ 0, a result which can only be 



u 



true when u is infinitely large. It follows that with a frictionless wheel 

 the efficiency would increase with the speed and would become unity when 

 the speed was infinite. Actually, however, frictional losses, which increase 

 with the speed, cause a maximum efficiency to be obtained at some defi- 

 nite speed with any given wheel. 



Taking frictional resistances inside the wheel into account, and assuming 



r 2 

 these to be proportional to v z and to equal F - we have : 



2 ^ 



1i 



Total head behind orifice = h -|- - . 



*9 



This must equal the kinetic energy at the orifice together with the loss 

 by friction 



.'. v 2 (I + F) u 2 = 2 // h. 



Substituting this value of v in the expression for the efficiency, this 

 now becomes 



