CENTRIFUGAL PUMPS 663 



where the head is constant. Here the lowest points A and A' of each 

 curve indicate the minimum volume the pump will lift, and the speed 

 below which pumping will not take place. Obviously the minimum 

 speed is increased and the minimum quantity diminished by any recur- 

 vature of the vanes. 



Pump used for Circulating Purposes. Where a centrifugal pump is used 

 for circulating water through the tubes of a surface condenser or of a cooler, 

 and where the actual height of lift is small, the resistance to flow, and 

 therefore the head against which the pump works, varies approximately 

 as the square of the velocity of flow. In such a case the suction and 

 delivery pipe line is often arranged so as to form a syphon, in which case 

 the whole work of the pump consists in overcoming frictional resistances. 



f L v 2 . 

 Here, putting //' = ' in equation (17), p. 659, we get : 



/ 8 = siny V U .*-J- 



m 

 and since v is proportioned to/ 3 for all speeds, 



.-. / 3 a ?/ 3 = B u 3 for all speeds. 

 Equation (13) now becomes : 



, _ 1 B 2 cosec 2 y 

 ~ 2 (1 - iTcoTyj' 



so that the hydraulic efficiency is independent of the speed of rotation. 

 Since this discussion neglects frictional losses in the wheel which 

 increase with the speed, the actual efficiency will then diminish as the 

 speed increases. 



(2) Pump with Whirlpool Chamber. The same general considerations 

 apply to the case of the pump fitted with vortex chamber or guide vanes, 

 as to the simple pump, though modified to some extent quantitatively. 

 Where a whirlpool chamber is fitted we have, from (9) on equating 

 the gain of pressure head in the pump to the head pumped against, and 

 assuming v = f% : 



K (ir a a + ./ 3 2 ) (1 - c 2 ) + ?/3 2 -/ 3 2 cosec 2 y = 2 g H'. 

 Putting w a = u 3 / 3 cot y, we get 

 A' 0/3 - /g cot y) (1 - c 2 ) + % 2 - /a 2 cosec 2 y + Kff (1 - c- 2 ) 



= 2.</H', (18) 



from which / 3 may be found in terms of the peripheral speed and the head. 

 The efficiency ?/ is thus equal to 



K (I - c 2 ) + 1 1 + /3 2 cosec 2 y { K (1 - c 2 ) - 1 }1 

 - 2 Ku 3 / 3 cot y (1 - c 2 ) _ J 



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