212 



HYDRAULICS AND ITS APPLICATIONS 



revolution (Fig. 95), whose height represents the velocity at any radius, 



is given by 



f a 



I 2 IT xvdx 



Jo 



Ca 



2 TT \ xvdx 



. * . mean height = 2 



TT CL 



Substituting for v from equation (1) we have 



mean velocity, v = v max = k \/ai. 



giving the radius at which 



Velocity in reec per second 

 01234 



a) ~ 7 



x = -689 a 



the velocity is equal to the mean 

 over a cross-section. 



More recent experiments 

 by Bazin l on a cement pipe 

 2'63 feet diameter, and by 

 Messrs. Williams, Hubbell, 

 and Fenkell 2 on cast iron 

 Q asphalted pipes having dia- 

 meters of 12", 16", 30", and 

 42", indicate that the curve 

 of velocities is very nearly 

 an ellipse to which the pipe 

 walls are tangential. Calling 

 u the minimum velocity at 

 the walls, and V the maxi- 

 mum velocity at the centre, 

 the equation to the curve 

 would then be 

 ^ * 2 , 



y-u> 



The volume of the solid of 

 revolution bounded by this 

 curve is equal to 



FIGS. 96 and 97. 



TT a?u x va?(V u) 



= 7ra a 



v = u + .(Vu). 



1 " Mem. de I'Acade'mie des Sciences," xxxii. 1897. 



2 "Proc. Am. Soc. C. E.," vol. xxvii., 1901, p. 3H. 



