Mass flow rate of a gas through an orifice during choked conditions

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The velocity of a gas flowing through an orifice or an equipment leak attains a maximum or sonic velocity and becomes "choked" when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than [ ( k + 1 ) / 2 ] ^{k / ( k - 1 )}, where k is the specific heat ratio of the gas. For many gases, k ranges from about 1.09 to 1.41, and therefore [ ( k + 1 ) / 2 ] ^{k / ( k - 1 )} ranges from about 1.7 to about 1.9 ... which means that choked velocity usually occurs when the absolute upstream pressure is at least 1.7 to 1.9 times as high as the absolute downstream pressure.

In SI metric units, when the gas velocity is choked, the equation for the mass flow rate is:

[img http://www.air-dispersion.com/ChokedFlowCA.png]

or this equivalent form:

[img http://www.air-dispersion.com/ChokedFlowCAP.png]

[It is important to note that 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 source pressure is increased.]

Q = mass flow rate, kg/s
C = discharge coefficient (dimensionless, about 0.72)
A = orifice hole area, m²
k = gas c_p/c_v = ratio of specific heats
[ρ] = real gas density, kg/m³, at upstream P and T
P = absolute upstream pressure, Pa
M = gas molecular weight (dimensionless)
R = Universal Gas Law constant, (Pa)(m³ / (kgmol)(¦K)
T = gas temperature, ¦K
Z = the gas compressibility factor at P and T

When dealing with the choked flow of a gas through a leak hole in a pressurized gas system or vessel, it is important to realize that the above equations calculate the initial instantaneous mass flow rate for the pressure and temperature existing in the system or vessel when the release first occurs. The initial instantaneous flow rate from a leak in a pressurized gas system or vessel is much higher than the average flow rate during the overall release period because the pressure and flow rate decrease with time as the system or vessel empties. Calculating the flow rate versus time since the initiation of the leak is much more complicated, but more accurate. To learn how such calculations are performed, go to www.air-dispersion.com/feature2.html.

When expressed in the customary USA units, the equations above also contain the gravitational conversion factor g_c which is 32.17 ft/s² in USA units ... and since the factor g_c is 1 (kg-m) / (N-s²) in the SI metric system of units, the above equations do not include it.

The technical literature can be very 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 / (MW).

Notes:
(1) The above equations are for a real gas.
(2) For an ideal gas, Z = 1 and d is the ideal gas density.
(3) kgmol = kilogram mole

Milton Beychok
(Visit me at www.air-dispersion.com)