Accidental release source terms are the mathematical equations that quantify the flow rate at which accidental releases of air pollutants into the ambient environment can occur at industrial facilites such as petroleum refineries, petrochemical plants, naural gas processing plants, oil and gas transportation pipelines, chemical plants, and many other industrial activities. Governmental regulations in many countries require that the probability of such accidental releases be analyzed and their quantitative impact upon the environment and human health be determined so that mitigating steps can be planned and implemented.

There are a number of calculation methods for determining the flow rate at which gaseous and liquid pollutants might be released from various types of accidents. Such calculational methods are referred to as ''source terms'', and this article explains some of the methods used for determining the mass flow rate at which gaseous pollutants may be accidentally released.

Accidental release of pressurized gas:

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 (or sonic velocity) occurs when the ratio of the absolute source pressure to the absolute downstream pressure equals or exceeds [(k + 1) / 2 ]^ k / (k - 1) where k is the specific heat ratio of the discharged gas (sometimes called the isentropic expansion factor).

For many gases, k ranges from 1.09 to 1.41, and thus [(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 is: or this equivalent form: 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 the mass flow rate is: or this equivalent form: Q = mass flow rate, kg/s
C = discharge coefficient, dimensionless (usually about 0.72)
A = discharge hole area, m²
k = cp/cv of the gas
cp = specific heat of the gas at constant pressure
cv = specific heat of the gas at constant volume
ρ = real gas density at P and T, kg/m³
P = absolute upstream pressure, Pa
PA = absolute ambient or downstream pressure, Pa
M = the gas molecular weight, kg/kmol
R = the Universal Gas Law Constant = 8314.5 Pa·m³/(kmol·K)
T = absolute upstream gas temperature, K
Z = the gas compressibility 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.com/feature2.html.

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/M.

Notes:
-- The above equations are for a real gas.
-- For an ideal gas, Z = 1 and ρ is the ideal gas density.
-- kmol = kilomole, or kilogram mole.

References:

(1) Perry's Chemical Engineers' Handbook', Sixth Edition, McGraw-Hill Co., 1984

(2) Handbook of Chemical Hazard Analysis Procedures, Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Chemical Hazard Analysis Handbook

(3) Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. U.S. EPA publication, 1999

(4) Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. PGS2 CPR 14E, Yellow Book

Milt Beychok
(Visit mw at www.air-dispersion.com)