(3a)
It will be seen hereafter that there is a limit in the ratio p1/p2 beyond which these expressions cease to be true.
§ 63. Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid is
z1 + p1/G1 + v12/2g = z2 + p2/G2 + v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1(γ−1)/γ},
the expansion being adiabatic, because in the flow of the streams of air through an orifice no sensible amount of heat can be communicated from outside.
Suppose the air flows from a vessel, where the pressure is p1 and the velocity sensibly zero, through an orifice, into a space where the pressure is p2. Let v2 be the velocity of the jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also p2. As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that z1z2 may be neglected. Putting these values in the equation above—
p1/G1 = p2/G2 + v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ;
v22/2g = p1/G1 − p2/G2 + (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ}
= (p1/G1) {γ/(γ − 1) − (p2/p1)γ−1 /γ / (γ − 1)} − p2/G2.