88 HYDEAULICS AND ITS APPLICATIONS 



change of velocity per unit length of pipe ( y ) constant, or the loss of 



head per unit length of pipe -4^, constant. The latter pipes were found 



cl x 



to be the more efficient, the saving, over a straight taper pipe of the 

 same length ranging from 20 per cent, to 60 per cent, and being greater 

 the shorter the pipe. 



In a rectangular pipe of length Z, satisfying this condition and having 

 one pair of sides parallel, if the half-breadths at the small and the large 

 ends are respectively y\ and 3/2, the half-breadth y, at a distance x from the 

 small end is given by 



1 1 _*/JL * ^ i 



v5"~"Vy! rvvR v^/' 



while in a circular pipe whose smaller and larger radii are i\ and r 2 , the 

 radius at a distance x from the small end is given by 



1 1_ _ x ( 1 1 \ 



Vr* . Vr? 1 Wf? Vr?/ 



Where the length of pipe is great, or the ratio of areas small, the 

 curves thus formed may, at the smaller end of the pipe, diverge at an 

 angle less than that (6 in a circular pipe and 11 in a rectangular pipe) 

 giving minimum loss, and in such a case the pipe would be made to 



1 The loss in a straight taper pipe whose angle of divergence is is proportional to 8 (r>) 2 and 



(6\ n 

 tan - J where n = 1-40 for a rectangular pipe. Hence in a length 



S x of a trumpet-shaped rectangular pipe, over which the mean angle is 0, the loss is 

 presumably proportional to 8 O) 2 f^-|) or to ~4~ - \j~-j.) ^ s x where y is the half- 



breadth of the pipe and where x measures the distance of the element under consideration 

 from some datum point on the axis of the pipe. 

 But in such a pipe r 2 ^ y - 2 , 



/. loss in length S x -^ (y - 2) (-^ j S x. 

 For this to be constant per unit length 



OT 



~ s 2 ' 4 



= constant, 



If the origin from which sc is measured be taken at the small end of the pipe, where the 



