WATER HAMMER 227 



or log, r -l* . l^JLo = j 

 m ( q r r + r j A; a t t 



' 



v Q = . mr feet per second (13) 



where r = J c + ^- -- ; i = Vc + ?- + ^; w = 2 

 v 4 i2 4 4 



This gives the velocity of efflux, from which the velocity v a at any instant 

 within the range of valve opening over which ( J is negligibly small, 



may be readily obtained. 1 



Writing p v for the difference of pressure on the two sides of the valve 

 at any instant during closure, we get, as in equation (5), 



f /7 2 / -fj \ ~\ 



Pv = Ps + -^ I j^r -I ~ l \ ( * + ') } Ibs. per square foot, 



and on substituting for - r - - from (8) this becomes 

 (L t 



p v = Ps + {*- - %- - g h\ Ibs. per sq. ft. (14) 



9 v ^ ^ J 



Evidently this has its maximum value when ?; is a maximum and there- 

 fore, from (13), when t vanishes, i.e., at the instant the valve reaches its 

 seat. At this instant VQ attains the limiting value q, while v a becomes 

 zero, so that we get 



(Mww = Ps + ^ ] ^ 9 h } Ibs. per sq. ft. 

 2 _ f h = - -f - + - 



It follows that the rise in pressure behind the valve at the instant 

 when closure is complete, above that obtaining with no flow through the 

 pipe, is given by 



" = 7 



1 Where ( ) ^ not negligible the treatment follows the lines outlined on p. 28. 



" Water Hammer," ante eit., p. 224. This case is, however, not of great practical importance, 

 since the rise in pressure before the valve gets near to its seat, and hence before the state 

 of affairs hypothecated obtains, is usually very small. 



Q 2 



