WATER HAMMER 235 



Putting K = 300,000 X 144 Ibs. per square foot ; w = 62'4 ; g = 32'2 ; 



ihis becomes 



p = 9160 v Ibs. per square foot. 



= 63'7 v Ibs. per square inch. 



A closer approximation to the actual rise in pressure may be obtained 

 >y assuming that while the pipe line is rigid in that the motion is 

 Instantaneously stopped, yet the elasticity is felt in its effect on the value 



f A', this adopting the value K' where -^ =-^> + ^ approx. 1 



I In the author's experimental pipe line this makes K' = 251,000 X 

 .44 Ibs. per square foot, and makes p' = 58'4 v Ibs. per square inch. 



The following demonstration shows how the elasticity of the pipe line 

 Imd water column may be taken fully into account. 

 I Let K' and E have the meaning already attached to them, and let w 



1 Suppose the pipe to be of radius r feet, and of comparatively small thickness t feet, and 

 !2t the material of which it is composed have a modulus of elasticity E Ibs. per square foot, 



( nd a Poisson's ratio -. 



I Then if at any section of the pipe the increase in pressure due to retardation is p' Ibs. per 

 juare foot, the increase in the circumferential stress in the pipe walls is ^^- and in the 



i)ngitudinal stress is 2LL Ibs. per square foot. 



! It, then, 8 x is the change in length of an element of the pipe at this point, whose original 

 !?ngth was x, we have 



8_a? ]>' r _ p r = v' r i 1 2 



x It E <rt E 2t E 

 Sr 



. , 6 r p r 



Also = V - 



r t E 2 a- 1 E' 



The change in the volume of this element is therefore given by 

 * + 8r2,7' + 8 a -*, 



_ - 



' V ~ 2 t E ~ a- j ' 

 But the actual new volume of the liquid = TT r z x ( 1 - ^ J while its apparent new 



)lume = IT r z x \ 1 - ^. - ^L ^ (5 - - J j , S o that the effective value of , which will 

 e. denoted by A'', is given by the relation 



or iron pipes a may be taken as 3'6 approximately. 



