390 Mr. A. H. Gibson on the Resistance to 



also with its diameter, with the mean velocity of flow, with 

 the mean pressure, and with the temperature o£ the air. 

 The fact that this coefficient depends on five factors, of which 

 three may vary in the same pipe, renders this, and any more 

 elaborate formula involving the same law of frictional re- 

 si stance,, useless as a means of determining the probable fall 

 in pressure along a given pipe line, with any degree of 

 accuracy. 



Realizing this, the author attempted to determine some 

 formula which should truly represent the state of affairs in 

 such cases of resisted flow; and the following paper is devoted 

 to a deduction of such a rational formula and to a determi- 

 nation of the constants involved in this, from the results of 

 •experiments by himself and by other experimenters. 



(2) Rational Formula for Pipe-flow. 



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 condition 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 x .fjt?.p*.v n .(8l) a , . . . . (1) 



where Bp = pressure difference in lbs. per sq.ft. at two points 

 SI ft. apart along the pipe. 



„ <:^ = pipe diameter in feet. 



„ jjl = coefficient of viscosity of the fluid under the tem- 

 perature conditions obtaining in the pipe. 



„ $1 = length of element of pipe in leet. 



„ p = density of the fluid, at the mean pressure p, and 

 mean temperature (t° abs.) obtaining in the 

 element. 



,, r = mean velocity of flow in this element in ft. per sec. 



,, Jc is a numerical coefficient. 



Although this 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 roughness 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 



