PIPE FLOW 199 



the indices z and n of these expressions as determined for equation (1) 

 will implicitly involve the effect of the roughness. 



If [M] , [L] , and [T] be the fundamental units of mass, length, and 

 time, (1) may be expressed dimensionally as 



' 



and since experiment shows that the resistance to flow is, other things 

 being equal, directly proportional to the length, a = I. 

 Inserting this value the equation becomes 



Equating indices of like quantities we get 



x y 3 z = 2 n\ y + z = l; y = 2 n 

 from which we have 



' x = n 3 



z = n 1 



V 



The formula then becomes 



8 p = g h p = K r n ~ B /x 2 " n p n ~ 1 v n . L (2) 1 



If i represents the loss of head in units of length of a water column 



per unit length of pipe, so that i = -=, i is called the hydraulic gradient 

 of the pipe, and gives the slope at which this would need to be laid in 



1 A similar rational law may be deduced for the flow of compressible fluids by writing 



-^ for p in equation (1). Here r is the absolute temperature of the gas, while c is the 

 CT 



constant obtained from the relationship p V = c T. We then get 



n- i 



An examination, by the author, of the results of experiments on the flow of air through 

 pipes, by several experimenters, and with diameters ranging from -125 in. to '98 feet 

 velocities from 10 to 40 feet per second, confirms the assumptions made in deducing this 

 formula (" Phil. Mag.," March, 1909). From these experiments the following values of n Lave 

 been deduced : 



n. 

 Small lead pipe, -125 in. in diameter ......... 1-28 



Lead pipe, 2-16 ins. ......... 1-77 



( 5-9 ......... 1-81 



Cast-iron pipe j 7-87 ......... 1-78 



(ll-8 ,. ......... 1-77 



while A = 125 x 10 ~ 8 ; B = 6 '60. ; if Up is measured in Ibs. per sq. ins. 



Below the C. V. n = 1, and the formula becomes Sp = , a m * indicating that the pressure 



drop is now independent of the absolute pressure in the pipe a result verified by the 

 experiments. 



