198 HYDRAULICS AND ITS APPLICATIONS 



Kutter's formula, introducing as it does a roughness coefficient N, has a 

 much wider range of application, and gives much more consistent results 

 than those previously mentioned, from Hagen to Weisbach. This formula 

 however, assumes the resistance to be always proportional to v z , whereas 

 experiment indicates that for smooth surfaces the power of the velocity is 

 always less than the second, and we are led to the conclusion that none 

 of these formulae, while giving good results within their own particular 

 range of application, can be looked upon as representing the general state 

 of affairs in pipe flow. 



ART 64. 



A rational law of resistance to pipe flow, applicable to either steady or 

 unsteady motion, was first evolved by Professor Osborne Reynolds. This 

 is deduced on the assumptions that the resistance to flow along any small 

 element of the pipe depends on the diameter, length, and surface condi- 

 tion of the element ; on the viscosity and density of the fluid ; and on the 

 mean velocity of flow through the element ; and also that it depends on 

 some power of each of these factors. 

 This being so we may write 



8p = k . d* . n v . p* . v n . (Sl) a , (I) 



where Bp = pressure difference in Ibs. per square foot at two points SZ ft. 



apart along the pipe. 

 d = pipe diameter in feet. 

 f> p = coefficient of viscosity of the fluid under the temperature 



conditions obtaining in the pipe. 

 81 = length of element of pipe in feet. 

 ,, p = density of the fluid. 



v = mean velocity of flow in this element in feet per second. 

 k = is a numerical coefficient. 



Although the expression contains no term directly marking the effect 

 of the roughness of the pipe surface, this effect is included in the terms 

 P z and v n . This will be seen if it be granted that the effect of the rough- 

 ness in increasing the resistance to flow is due to loss of available energy 

 in eddy production, the eddies being formed by the sudden deflexion of 

 particles of fluid in close proximity to the walls. 



The mass of fluid thus affected will be greater as the roughness 

 increases and as the velocity of flow increases, and will also depend 

 directly on its density, while the loss of energy per unit mass will depend 

 on the velocity. It follows that the effect of a variation in the roughness 

 will be felt in the factors involving both p and v, and that the values of 



