Accidental Release Source Terms

Gas Discharge To The Atmosphere From A Pressure Source:^{1, 2}

When gas stored under pressure in a closed vessel is discharged to the atmosphere through a hole or other opening, the gas velocity through that opening may be choked (i.e., it has attained a maximum) or it may be non-choked. Choked velocity, which is also referred to as sonic velocity, occurs when the ratio of the absolute source pressure to the absolute downstream ambient pressure is equal to or greater than [ ( k + 1 ) / 2 ] ^{k / ( k - 1 )} , where k is the specific heat ratio of the discharged gas.

For many gases, the k value ranges from about 1.09 to about 1.41, and therefore [ ( k + 1 ) / 2 ] ^{k / ( k - 1 )} ranges from 1.7 to about 1.9 ... which means that choked velocity usually occurs when the absolute source vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream ambient atmospheric pressure.

When the gas velocity is choked, the equation for the mass flow rate in SI metric units is:

Q = C A [ k d P ]^1/2 [ 2 / ( k + 1 ) ] ^{(k + 1) / (2k - 2)}

  or this equivalent form:

Q = C A P [ k M / ( Z R T ) ]^1/2 [ 2 / ( k + 1 ) ] ^{(k + 1) / (2k - 2)}

For the above equations, 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 source pressure is increased.

Whenever the ratio of the absolute source pressure to the absolute downstream ambient pressure is less than [ ( k + 1 ) / 2 ] ^{k / ( k - 1 )}, then the gas velocity is non-choked (i.e., sub-sonic) and the equation for mass flow rate is:

Q = C A [ 2 d P ]^1/2 [ k / ( k - 1 ) ]^1/2 [ ( P_A / P ) ^{2 / k} - ( P_A / P ) ^{(k + 1) / k} ]^1/2

or this equivalent form:

Q = C A P[ 2 M / (Z R T)]^1/2 [ k / ( k - 1 )]^1/2 [( P_A / P) ^{2 / k} - ( P_A / P )^{(k + 1) / k}]^1/2

where:
Q = mass flow rate, kg/s
C = discharge coefficient, dimensionless (usually about 0.72)
A = discharge hole area, m²
k = c_p/c_v of the gas
c_p = specific heat of the gas at constant pressure
c_v = specific heat of the gas at constant volume
d = real gas density at P and T, kg/m³
P = absolute source or upstream pressure, Pa
P_A = absolute ambient or downstream pressure, Pa
M = gas molecular weight, dimensionless
R = the Universal Gas Law Constant = 8314.5 ( Pa ) ( m³ ) / ( kgmol ) ( °K )
T = absolute gas temperature, °K
Z = the gas compresibility factor at P and T, dimensionless

The above equations calculate the initial instantaneous mass flow rate for the pressure and temperature existing in the source vessel when a 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. Two equivalent methods for performing such calculations are presented and compared at www.air-dispersion/feature2.html.

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

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 R_s which only applies to a specific individual gas. The relationship between the two constants is R_s = 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

This is still a work in progress. Please make no changes as yet.