28 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II 



Efflux of Gases. 



26. We consider next the efflux of a gas, supposed to flow 

 through a small orifice from a vessel in which the pressure is 

 />o and density p into a space where the pressure is p lt We assume 

 that the motion has become steady, and that the expansion takes 

 place according to the adiabatic law. 



If the ratio PQ/PI of the pressures inside and outside the vessel do not 

 exceed a certain limit, to be indicated presently, the flow will take place in 

 much the same manner as in the case of a liquid, and the rate of discharge 

 may be found by putting p=p in Art. 24 (9), and multiplying the resulting 

 value of q by the area S' of the vena contracta. This gives for the rate of 

 discharge of mass 



2 y+I 



It is plain, however, that there must be a limit to the applicability of this 

 result; for otherwise we should be led to the paradoxical conclusion that 

 when Pi = 0, i.e. the discharge is into a vacuum, the flow of matter is nil. 

 The elucidation of this point is due to Prof. Osborne Reynolds f. It is easily 

 found by means of Art. 24 (8), that qp is a maximum, i.e. the section of an 

 elementary stream is a minimum, when q 2 dp/dp, that is, the velocity of the 

 stream is equal to the velocity of sound in gas of the pressure and density 

 which prevail there. On the adiabatic hypothesis this gives, by Art. 24 (10), 



c 2 \i 



and therefore, since c 2 oc p 7 " 1 , 



1 _y_ 



P_( 2 Ny- 1 P_f 2 VR 



or, if y=l'408, 



p = '634p , jt? = '527j0 (iv). 



If jo x be less than this value, the stream after passing the point in question, 

 widens out again, until it is lost at a distance in the eddies due to viscosity. 

 The minimum sections of the elementary streams will be situate in the 

 neighbourhood of the orifice, and their sum S may be called the virtual 

 area of the latter. The velocity of efflux, as found from (ii), is 



The rate of discharge is then =qpS, where q and p have the values just 



* A result equivalent to this was given by de Saint Venant and Wantzel, 

 Journ. de VEcole Polyt., t. xvi., p. 92 (1839). 



t " On the Flow of Gases," Proc. Nanch. Lit. and Phil. Soc., Nov. 17, 1885 ; 

 Phil. Mag., March, 1876. A similar explanation was given by Hugoniot, Comptes 

 Rendus, June 28, July 26, and Dec. 13, 1886. I have attempted, above, to condense 

 the reasoning of these writers. 



