Consider a vessel containing ideal gas under a pressure of p which leaks through an orifice of known properties CdA. If the pressure is high enough, the flow through the orifice is choked, and the velocity is the sonic velocity. The flow through the orifice is given by \rho CdA v where all the conditions are at the throat. The total enthalpy and the static enthalpy of a gas are related by h0 = h + 1/2 v^2. Inside the vessel, the value will be the total velocity, and taking the ideal gas relation h = CpT where T is the absolute temperature, we have T0 = T + 1/2v^2/Cp. At the throat, where sonic conditions prevail, v = a, where a, the speed of sound is given by \sqrt(kRT). Thus, the temperature at the throat is given by T = 2/(k+1) T0. If the ideal gas is assumed to under go an adiabatic process, the total pressure can be similarly related to the pressure at the throat. i.e., p0/p = (T0/T)^(k/(k-1)) = (2/(k+1))^(k/(k-1)). Going back to the mass flow equation mdot = Cd A \rho v, we have \rho = p/RT. The throat pressure can be written as p = p0*(p/p0). Similarly, the throat temperature can also be stated in terms of the vessel pressure. Finally, what we have is the mass flow rate as a function of the instantaneous pressure and temperature of the vessel.
Now we can write the mass flow rate in terms of the rate of change of density of the gas in the vessel. The right hand side contains the pressures and temperature of the gas in the vessel, and these can also be written in terms of the density using the ideal gas equation. This will give the variation of density as a function of time. Since we are assuming an adiabatic process, we will also get the variation of pressure and temperature with time.
Some of the applications of the above equations are the following: 1. find the time for the pressure in a leaking vessel to fall. 2. find the amount of gas that will leak from a vessel in a given time.
We can also give the reverse question: what is the nature of the leak if the variation in the pressure with time is known? If we know the gas, we can find the value of CdA.
We can see that the pressure ratio between the vessel and the throat has to be above a certain ratio for choked flow to occur. Now let us consider the case where this pressure ratio is lower, so that the flow is subsonic. In this case, the flow usually occurs between two pressures p1 and p2, which are the upstream and downstream values respectively. Now, even if we assume a value of CdA for the orifice, the conditions at the throat will not be same as the downstream, i.e., the pressure will not be p2. However, the conventional analysis is carried out by assuming the throat value to be same as that of the downstream. Once again, we assume the upstream to be static, so that the velocity can be neglected. We can write the expression for the static enthalpy as in the previous section. However, this time the velocity will not be the sonic velocity. Once again, adiabatic expansion is assumed between the two points, and we have v^2 = 2CpT1(1-T2/T1). Now, T2/T1 = (p2/p1)^((k-1)/k). The density at the throat, \rho2 can be written as \rho2 = \rho1 * (\rho2/\rho1). Using the adiabatic relations, p \propto \rho^k. Thus, we have the subsonic mass flow equation, where it is function of the upstream values and the pressure ratio.
Having talked a lot about choked flow, let us ask the question, does the flow really choke? That is to say, if we continuously increase the pressure ratio, will we find the value of flow not increasing beyond a certain value? Experiments on orifices show that this is not really the case. The effect is explained on the basis of the shifting of the minimum area position as well as the size. As the outlet pressure is decreased to zero, the flow increases, though the increase is not as rapid beyond a certain pressure ratio.
The flow through a given orifice is a strong function of the manufacturing process, in particular, the various radii that are achieved for the edges of the orifice. Since there is a wide variation in these geometric parameters even with products of a single machine, hardly any attempt is made to find the theoretical flow patterns and the like in the real world.