222 HYDRAULICS AND ITS APPLICATIONS 



Professor Unwin gives coefficients for formulae deduced by Hagen, 

 which for clean cast iron pipes make 



C' having the value 41 '9. 



In experiments carried out by M. Vallot, Q was found to be proportional 

 to d 2 - 66 



ART. 70. INITIATION AND STOPPAGE OF MOTION IN A PARALLEL PIPE. 



(a). Gradual Stoppage. Neglecting the effect of the elasticity of the 

 water, and thus assuming that during acceleration or retardation in a 

 pipe line, the velocity throughout the pipe is uniform at any instant, if a 



d v 

 retardation j-y~be produced in any way, as by the gradual closing of 



a valve at the outlet to the pipe, or by the retardation of a plunger giving 

 motion to the water column, the rise in pressure behind the valve or 

 piston due to this retardation will be given by 



w 7 d v 



p a = . I . a . -j- 



9 dt 



wl d v 



/. p' = . - Ibs. per square foot . (1) 



9 at 

 where " I " is the length, and "a" the sectional area of the column in 



square feet. The minus sign, is used since -=- being an acceleration, is 



itself negative if the motion is being retarded. This pressure difference 

 at the two ends of the pipe is superposed on that due to steady flow with 

 the velocity obtaining at the given instant. Thus if the loss of pressure 

 from entrance to exit, due to steady flow with velocity v feet per second, 



is -0- ( 1 -j- J ) Ibs. per square foot, and if v and ^ are respectively 



&9 \ In / (t t 



the velocity and the acceleration at a given instant, the pressure at the 

 valve will be less than the statical pressure under conditions of no flow, 

 by an amount 



w ( dv . v* ( - ft \ } ., 

 "alJJIFv/1 per S( l uare foofc * 



A general expression for the pressure changes, accompanying changes 

 of velocity in a pipe line may be obtained from Bernoulli's equation. If 



