the Flow of Air through a JPijje. 391 



will also depend directly on its density, while the loss of 

 energy per unit mass will depend on the velocity. If this 

 be granted, 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 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 



BG-w-isr-er-G]"-™ 



Now experiment shows that the resistance to flow is, othei 

 things being equal, directly proportional to the length, so 

 that a = l. 



Inserting this value the equation becomes 



[M] . [L]- 1 . [T]- 2 =£. [M]*+y. [L]*-»-»*+»+i. [T]-<H->. 

 Equating indices of like dimensional quantities we get 



a— i/—o~= — 2 — 71 

 y + z =1 



>j-=2 — n, 



and on solving this we have 



x-=.n— 3; ii—'l—n\ z=n—l. 



Substituting these values in (1), this becomes 



8j) = k.d u -*.fL 3 -» . P "~ ] . /■" . hi ... (2) 



Where p and therefore v is constant, the fluid being in- 

 compressible, on integrating over the whole length of pipe 

 this gives the Reynolds's formula* for the flow of water 

 through a uniform pipe: — 



8p=k.d?-*.p2- n .p*- 1 .4*.l. ■ • . (2') 



In a compressible fluid p= nM> ' . where the volume of 



volume 1 ,» 



unit mass is given by the relation pY=CT 3 so that p= i_P \ 

 Substituting y (where c has the value corresponding t 



* Scientific Papers, Osborne Reynolds, vol. ii. p. 97. 



t In the case of air, if the mass of g-as is unity (32'2 lb. wt. units), 

 andjf p be measured in lbs. wt. per sq. ft., the value of c is 53-18 x 32-2 

 = 1710, while if ^ be measured in lbs. per sq. ins. c becomes 11-9. 



