90 HYDRAULICS AND ITS APPLICATIONS 



divergence of the first portion of such a tube, the sudden enlarge- 

 ment of section and the accompanying loss is made greater, but 

 the loss in the diverging portion is reduced in a double degree, since not 

 only is the numerical coefficient expressing such loss as a percentage of 

 (t/'i #a) 2 /20 diminished, but A 3 is diminished at the same time, and thus 

 the factor v v 3 is also diminished. A diminution in the angle of 

 divergence therefore causes a rapid diminution in this portion of the 

 loss, which may, or may not, be counter-balanced by the loss at the 

 sudden enlargement of section. Owing to the comparatively low 

 velocities at the large end of the pipe, however, except in pipes whose 

 length is comparatively very short, and whose ratio of enlargement 

 is small, this latter loss is comparatively small and the total loss is a 

 minimum with a pipe straight or curved whose angle of divergence 

 actual or effective is little, if any, greater than that giving minimum 

 loss In the diverging portion of the pipe alone. 



It thus becomes possible to design a pipe often with a considerable 

 reduction in length in which the boundaries are straight, and in which 

 the loss is still appreciably less than in a straight taper pipe giving the 

 full enlargement of section with the best possible value of 0. 



The total loss of head in such a pipe (shown in Fig. 43) is theoretically 

 equal to 



+ - feet 



20 



3 



where K is obtained from the curves of Figs. 41 and 42. As 3 = ~ 



Vi A 3 



while = -A where A represents the corresponding area this becomes 



or 



/, *._\2r/ j, \2f / 4,\ a /A. A. \v\~\ 



feet. 



*}' \K(l - i'V+ (4-' - 

 Aj ( V ^3/ \Ai A 



20 



In a rectangular pipe whose breadth increases uniformly from bi to 6 3 in 

 a length L, 



= bi -f- L (approximately) where is in angular measure, 

 so that A 3 =Ai-\-L 0, and the above expression becomes 

 loss = 



