All gases flow from higher pressure to lower pressure. Choked flow can occur at the change of the cross section in a
de Laval nozzle or through an
orifice plate. The choked velocity is observed upstream of an orifice or nozzle. The upstream
volumetric flow rate is lower than the downstream condition because of the higher upstream density. The choked velocity is a function of the upstream pressure but not the downstream. Although the velocity is constant, the mass flow rate is dependent on the density of the upstream gas, which is a function of the upstream pressure.
Flow velocity reaches the speed of sound in the orifice, and it may be termed a ''''.
Choking in change of cross section flow Assuming ideal gas behavior, steady-state choked flow occurs when the downstream pressure falls below a critical value p^{*}. That critical value can be calculated from the dimensionless critical pressure ratio equation :\frac{p^*}{p_0} = \left(\frac{2}{\gamma + 1}\right)^\frac{\gamma}{\gamma - 1}, where \gamma is the
heat capacity ratio c_p/c_v of the gas and where p_0 is the total (stagnation) upstream pressure. For air with a heat capacity ratio \gamma = 1.4, then p^* = 0.528 p_0; other gases have \gamma in the range 1.09 (e.g. butane) to 1.67 (monatomic gases), so the critical pressure ratio varies in the range 0.487 , which means that, depending on the gas, choked flow usually occurs when the downstream static pressure drops to below 0.487 to 0.587 times the absolute pressure in stagnant upstream source vessel. When the gas velocity is choked, one can obtain the mass flowrate as a function of the upstream pressure. For isentropic flow Bernoulli's equation should hold: h + \frac{v^2}{2} = \frac{C_P T}{\mu} + \frac{v^2}{2} = const, where h - is the
enthalpy of gas, C_P = \frac{\gamma}{\gamma - 1}R - molar specific heat at constant pressure, with R being the universal gas constant, T - absolute temperature. If we neglect the initial gas velocity upstream, we can obtain the ultimate gas velocity as follows: v_{max} = \sqrt{\frac{2}{\mu}C_P T} In a choked flow this velocity happens to coincide exactly with the sonic velocity c_s^* at the critical cross-section: c_s^* = \sqrt{\gamma\frac{p}{\rho^*}} = \sqrt{\frac{2}{\mu}C_P T} = v_{max}, where \rho^* is the density at the critical cross-section. We can now obtain the pressure p as: p = \frac{\rho^* {c_s^*}^2}{\gamma} = \frac{\dot{m}}{A^*} \frac{c_s^*}{\gamma}, taking in account that \rho^* c_s^*\, A^* = \dot{m}. Now remember that we have neglected gas velocity upstream, that is pressure at the critical section must be essentially the same or close to the stagnation pressure upstream P_0 \approx p, and A^* \approx A. Finally we obtain: \dot{m} = \gamma\, P_0\, A\, \left(\frac{2}{\mu} C_P\,T_0\right)^{-1/2} as an approximate equation for the mass flowrate. The more precise equation for the choked
mass flow rate is: : \dot{m} = C_d A \sqrt{\gamma \rho_0 P_0 \left(\frac{2}{\gamma + 1}\right)^\frac{\gamma + 1}{\gamma - 1}} The mass flow rate is primarily dependent on the cross-sectional area A of the nozzle throat and the upstream pressure P, and only weakly dependent on the temperature T. The rate does not depend on the downstream pressure at all. All other terms are constants that depend only on the composition of the material in the flow.
Although the gas velocity reaches a maximum and becomes choked, the mass flow rate is not choked. The mass flow rate can still be increased if the upstream pressure is increased as this increases the density of the gas entering the orifice. The value of C_d can be calculated using the below expression: : C_d = \frac{\dot{m}}{A \sqrt{2\rho\Delta P}} The above equations calculate the steady state mass flow rate for the pressure and temperature existing in the upstream pressure source. If the gas is being released from a closed high-pressure vessel, the above steady state equations may be used to approximate the
initial mass flow rate. Subsequently, the mass flow rate decreases during the discharge as the source vessel empties and the pressure in the vessel decreases. Calculating the flow rate versus time since the initiation of the discharge is much more complicated, but more accurate. The technical literature can be confusing because many authors fail to explain whether they are using the universal gas law constant R, which applies to any
ideal gas or whether they are using the gas law constant Rs, which only applies to a specific individual gas. The relationship between the two constants is Rs = R / M where M is the molecular weight of the gas. ==Real gas effects==