Source Terms for Accidental Discharge Flow Rates
(Special Shared Content from www.air-dispersion.com)

Click in table below on desired source term (United States version):

           SI version of this article can be found here.

Gas Discharge To The Atmosphere
From A Pressure Source: ^{1, 2, 6, 7}

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., has attained a maximum) or 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, k ranges from about 1.09 to about 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 ambient atmospheric pressure.

When the gas velocity is choked, the equation for the mass flow rate is:

or this equivalent form:

[ It is important to note that although the gas velocity reaches a maximum and becomed 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:

The above equations calculate the initial instantaneous 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. Click HERE to learn how such calculations are performed.

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) lbmol = pound mole

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Liquid Discharge From A Pressurized Source Vessel: ^{1, 2}

Initial instantaneous flow through the discharge opening:

Final flow when the liquid level reaches the bottom of the discharge opening:

Average flow:

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Liquid Discharge From A Non-Pressurized Source Vessel: ^{1, 2}

Initial instantaneous flow through the discharge opening:

Final flow when the liquid level reaches the bottom of the discharge opening:

Average flow:

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Evaporation From A Non-Boiling Liquid Pool: ²

Three different methods of calculating the rate of evaporation from a non-boiling liquid pool are presented in this section.

Method developed by the U.S. Air Force: ²

The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were derived from field tests performed by the U.S. Air Force with pools of liquid hydrazine.

Notes:  The function "ln x" is the natural logarithm (base e) of x, and the function "exp x" is the value of the constant e (approximately 2.7183) raised to the power x.

Method developed by U.S. EPA: ^{5, 6}

The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were developed by the United States Environmental Protection Agency ( U.S. EPA ).

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).

The U.S. EPA also defined the pool depth as 0.033 ft ( i.e., 1 cm ) so that the surface area of the pool liquid could be calculated as:

All of the units in the above Equation (1) and Equation (2) are a mixture of metric usage and United States usage. However, they are the units developed by the U.S. EPA and thus they were retained here.

Note:  gmol = gram mole.

Method developed by Stiver and Mackay: ³

The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were developed by Warren Stiver and Dennis Mackay of the Chemical Engineering Department at the University of Toronto.

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).

Note:  lbmol = pound mole

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Evaporation From A Boiling Pool Of Cold Liquid: ²

The following equation is for predicting the rate at which liquid evaporates from the surface of a pool of cold liquid (i.e., liquid temperature of about zero degrees Centigrade or less).

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Discharge Of Flashing Saturated Liquid: ^{2, 3}

Notes:  ln = natural logarithm (base e);  in = inch;  ° = ° = 460 + °


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Discharge of Flashing Sub-Cooled Liquid: ³

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Adiabatic Flash of a Liquified Gas Release Into Atmosphere:

Liquified gases such as ammonia or chlorine are often stored in cylinders or vessels at ambient temperatures and pressures well above atmospheric pressure. When such a liquified gas is released into the ambient atmosphere, the resultant reduction of pressure causes some of the liquified gas to vaporize immediately. This is known as "adiabatic flashing" and the following equation, derived from a simple heat balance, is used to predict how much of the liquified gas is vaporized.

If the enthalpy data required for the above equation is unavailable, then the following equation may be used.

Note:  atmos. = atmospheric

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References: ^{2, 3}

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

(2)   "Handbook of Chemical Hazard Analysis Procedures", Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989 provides references to (2a), (2b) and (2c) below

(2a)   Clewell, H.J., "A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids", Energy Systems Laboratory, ESL-TR-83-03, 1983 (Available at Air Force Weather Technical Library, Asheville, North Carolina)

(2b)   Ille, G. and Springer, C., "The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spills", Civil and Environmental Engineering Development Office, CEEDO 712-78-30, 1978 (Available at Air Force Weather Technical Library, Asheville, North Carolina)

(2c)   Kahler, J.P., Curry, R.C. and Kandler, R.A., "Calculating Toxic Corridors", Air Force Weather Service, AWS TR-80/003, 1980 (Available at Air Force Weather Technical Library, Asheville, North Carolina)

(3)   Stiver, W. and Mackay, D., "A Spill Hazard Ranking System For Chemicals", Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993

(4)   Fauske, Hans K., "Flashing Flows: Some Guidelines For Emergency Releases", Plant/Operations Progress, July 1985

(5)   "Technical Guidance For Hazards Analysis", U.S, EPA and U.S. FEMA, December 1987 [ Equation (7), Section G-2, Appendix G.  Available at http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/tech.pdf/$File/tech.pdf ]

(6)   "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. [ Equation (D-1), Section D.2.3, Appendix D.  Available at http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/oca-all.PDF/$File/oca-all.PDF ]

(7) "Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", CPR 14E, Third Edition Second Revised Print, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. [ Equations (2.22) and (2.25) on page 2.68. ]

By: Milton Beychok, Guest Author, mbeychok@air-dispersion.com