WATER HAMMER 237 



The longitudinal stress/, produced in the pipe walls by hammer action, 

 hich equals u y Um J , then becomes, on substitution, 



/ = v - / g . a m / ff [ Ibs. per square foot. 

 I V jBtt^-1 1 ^ V -#r^ J 



Valve Shut Suddenly, but not Instantaneously. 



As the time of closure of a valve becomes less and less, the maximum 

 ise in pressure will evidently tend to the limit given by formula (2) 

 I 236. 



I x 



Now if the time of closure is so short that y^ > T = -^r, the distur- 



' P VP 



|)ance initiated at the valve has travelled a distance x, and has not 

 Irrived at the open end when the latter reaches its seat. 

 I In this case if the retardation is uniform (= a), equation (1), p. 222, 



. iv ax 



Becomes p = - - Ibs. 

 9 



\ But a = ^ = ^*, so that 



i w i w 



P -' = ~ 



I 



= V a V 



his being the value obtained with instantaneous stoppage. It follows 

 hat whatever the law of valve closure, if this is completed in a less time 

 han l-r V p the pressure rise will be the same as with instantaneous 

 losure. 



For values of T between l-^- V p and 21-Z- V p the falling off in pressure 

 vill usually be comparatively small, so that it is in general sufficiently 

 iccurate for all practical purposes to count as " sudden," any stoppage 

 )ccupying a shorter time than this. 



For values of T>4/-J-F P , formula (15), p. 227, may be used without 

 serious error. 



Experimental Results with Sudden Closure of Valve. In the author's 

 experiments I = 550 feet ; K' = 251,000 X 144 ; E = 10 7 X 144 Ibs. per 

 quare foot ; a : a m = T275 ; formula (1) p. 234, gives 

 p' = 7780 v Ibs. per square foot, 

 = 54*0 v Ibs. per square inch. 



For comparison the observed results are shown in the following table, 

 Lgainst those as obtained by using this formula. 



