408.
ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES. [Philosophical Magazine, Vol. xxxn. pp. 177—187, 1916.]
THE problem of the passage of gas through a small aperture or nozzle from one vessel to another in which there is a much lower pressure has had a curious history. It was treated theoretically and experimentally a long while ago by Saint-Tenant and Wantzel* in a remarkable memoir, where they point out the absurd result which follows from the usual formula, when we introduce the supposition that the pressure in the escaping jet is the same as that which prevails generally in the recipient vessel. In Lamb's notation f, if the gas be subject to the adiabatic law (p oc pt\
*__?» ft{! _ (£)Jn=. * w-*),......ID
P y-1 PO ( W j 7- ] where q is the velocity corresponding to pressure p; pa, p0 the pressure and density in the discharging vessel where q = 0; c the velocity of sound in the gas when at pressure p and density p; ca that corresponding to p0, p0. According to (1) the velocity increases as p diminishes, but only up to a maximum, equal to c0 \/{2/(y — 1)}, when p — Q. If 7 = 1'408, this limiting velocity is 2'214c0. It is to be observed, however, that in considering the rate of discharge we are concerned with what the authors cited call the " reduced velocity," that is the result of multiplying q by the corresponding density p. Now p diminishes indefinitely with p, so that the reduced velocity corresponding to an evanescent p is zero. Hence if we identify p with the pressure P! in the recipient vessel, we arrive at the impossible conclusion that the rate of discharge into a vacuum is zero. From this our authors infer that the identification cannot be made; and their experiments showed that from pl = 0 upwards to pl = '4p0 the rate of discharge is sensibly constant. As pl still further increases, the discharge falls off, slowly at first,
* "Mdmoire et experiences sur I'diooulement de 1'air, determine par des differences de pressions considerables," Journ. de I'lbcole Polyt. t. xvi. p. 85 (1839). t Hydrodynamics, §§ 23, 25 (1916).. v. p. 608. [1917. See P.S. to Art. 411 for a reference to the work of Prof. Cisotti.], as for an incompressible fluid, is