226 HYDRAULICS AND ITS APPLICATION S 



Writing^ + *.+ f l S. -g 9 h feet, where h is the difference 



WaQ W 



of statical head on the two sides of the valve with no flow taking place, 

 equation (7) becomes 



<'o 2 _ 7i __ I f dv a /r a 2 ) 



Tg "Jilt TSJ 



On substituting for ^- a , its value- | a -^- + t'o ^ ' this becomes 



and on dividing throughout by the coefficient of r 2 , we get 



, Cl TQ n i 



k o -y = r 2 b v c, 

 ct t 



r \ . rf . r - + -P = r[4- t = iT : -f4-* (10) 



J r w r c K j O,Q K ai j t 



%l d a 



~ a ' dt 

 where 6 = ^ ? TQ 5 c 



m \ a / 7?i \ a 



2 



If a /a is small, so that the term ( * ) i s small in comparison with 



unity, and may therefore be neglected, b, c, and k become constants 

 having the values 



k =2J.*S;,^a>*-5* = L'. 



d t a 



This will alwa} 7 s be the case as the valve gets close to its seat, and when 

 in consequence the hammer effect is most noticeable. In such a case 

 both sides of (10) become integrable. Writing this in the form 



-r- { - dV - + D = ,- lOg, * (11) 



J / ~ rr a "\ 2 / 2 ka * 



and determining D from the consideration that when t = T, r has a 

 known value > we have 



^ 



