142 HYDRAULICS AND ITS APPLICATIONS 



This end contraction evidently diminishes the discharge by an amount 

 which is constant for a given depth, and it is reasonable to assume that it 

 therefore diminishes the effective breadth of the notch by an amount 

 which is also constant for a given depth, and which depends only on the 

 depth. The law of its variation with the depth has not been accurately 

 determined, but it was assumed, as appears to be approximately true, that 

 the diminution in effective length is directly proportional to the depth 

 and that the effective length is therefore equal to (b n k H) where n is 

 the number of contractions and A; is a constant to be determined 

 experimentally. Francis then assumed (though not on theoretical 

 grounds) that the discharge from notches of equal effective length varies 

 according to a constant power of the depth, so that the formula 

 became 1 



Q = K {b nkH} H x . 

 Though neither of the assumptions made is quite correct, yet if x be 



given the value 1*5, K the value 3*33, and k the value , they lead to 



results closely in accord with experiment. 



In general the Francis formula is suitable for use for all heads above 

 6 inches where the bottom contraction and end contractions, if any, are 

 free, and where the length of crest is greater than three times the head. 



Even where the ratio of length to head is less than three, the formula 

 gives fairly accurate results, the discharge as calculated being always 

 slightly less than observed. Eecent experiments at Cornell University 2 

 on a suppressed weir 6'56 feet wide and with heads ranging from 2'0 to 

 4*85 feet give mean results 98'4 per cent, of those calculated by its use, 

 and indicate its reliability for heads up to 5 feet. 



For perfect contractions Francis specifies a distance to side of approach 

 channel = 2 H , and a depth below the weir crest 3 H, and states that 

 a reduction of the bottom clearance to 2 H and of the side clearance to 

 H increases the flow by about 1 per cent. 



Effect of the Velocity of Approach. As in the case of a submerged 

 orifice, the fact that the stream has kinetic energy in virtue of its velocity 

 of approach to the weir may be taken into account, and correction made 

 for this by adding to the measured head H a supplementary head h, 



A 



where h = ^r. Here v is the velocity of approach, and is necessarily the 



1 A formula of this form was suggested by Boyden in 1846. 



2 " Trans. Am. Soc. C.E.," vol. 44, p. 397. The maximum variation from Francis's for- 

 mula, viz., 4 per cent., occurred vyith 2-64 feet head. 



