LARGE ORIFICES 



131 



Then the area of an elementary strip taken in the vena contracta at a 

 distance y from the centre, as shown, is 2 x 6 //. 

 The head over the element = H y, 



.'. Flow over this element = V 2 cj (77 ?/) X 2 x 8 y 



.-. $Q = 2 V~2~g X V (r 2 \f) (H~~y) 8 y. 



For convenience in integrating, expand V H y by the Binomial 

 Theorem, i.e., put 



since (1 - a;)* = 1 - n x + n (ro 



- etc.) 



Then 



Now put y = r cos 0. 

 (I y 



cT0 = ~ r * me - 



.-. 8y = rsm0S0, 

 and V r 2 f = V r 2 (1 cos 2 0) 

 = r sin 0. 



So that 8 Q = 2 



1 - 



r cos # 



r 2 cos 2 . 



7 2 - H etc. 



ff 8H "7J 



Integrating this expression and giving 

 the limits TT and o, since these are the 

 values of corresponding to the values 

 r of y, we get, on introducing C v , the 

 iotal flow Q over the whole section. 



FIG. 71. 



^5, +-etc.( 



the succeeding terms being negligible. 

 If C be the coefficient of discharge for the orifice, then since C c = 



we may write 



As in the case of a rectangular orifice, the coefficient in / general 

 lecreases as the orifice increases and also as the head increases. / 



K.' 



