
Source Terms for Accidental
Discharge Flow Rates 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

where: ^{ } _{ } _{ } ^{ } ^{ } _{ } 
Q C A^{ } g_{c}^{ } k_{ } Rho P^{ } P_{A}^{ } M R T Z 
=
mass flow rate, lb / s = discharge coefficient (dimensionless, usually about 0.72) = discharge hole area, ft^{ 2} = gravitational conversion factor of 32.17 ft / s^{ 2}_{ } = c_{p} / c_{v} of the gas = (specific heat at constant pressure) / (specific heat at constant volume) = real gas density, lb / ft^{ 3} at P and T = absolute source or upstream pressure, lb / ft^{ 2} = absolute ambient or downstream pressure, lb / ft^{ 2}_{ } = gas molecular weight = the Universal Gas Law Constant = 1545.3 ftlb / ( lbmol · °R ) = gas temperature, °R = the gas compressibility factor at P and T (dimensionless) 
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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Initial instantaneous flow through the discharge opening:

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

Average flow:

where:  Q  = mass flow rate, lb/s 
C  = discharge coefficient (usually about 0.62)  
A  = discharge hole area, ft^{2}  
g  = gravitational constant of 32.17 ft/s^{2}  
d  = source liquid density, lb/ft^{3}  
P  = absolute source pressure, lb/ft^{2}  
P_{A}  = absolute ambient pressure, lb/ft^{2}  
H  = height of liquid above bottom of discharge opening, ft 
Initial instantaneous flow through the discharge opening:

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

Average flow:

where:  Q  = mass flow rate, lb/s 
C  = discharge coefficient (usually about 0.62)  
A  = discharge hole area, ft^{2}  
g  = gravitational constant of 32.17 ft/s^{2}  
d  = source liquid density, lb/ft^{3}  
H  = height of liquid above bottom of discharge opening, ft 
Three different methods of calculating the rate of evaporation from a nonboiling 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.

where:  E  = evaporation flux, lb/minute/ft^{2} of pool surface 
u  = windspeed, miles/hour  
T_{A}  = ambient temperature, °K  
T_{F}  = pool liquid temperature correction factor  
T_{P}  = pool liquid temperature, °F  
M  = pool liquid molecular weight  
P_{S}  = pool liquid vapor pressure at ambient temperature, mm Hg  
P_{H}  = hydrazine vapor pressure at ambient temperature, mm Hg 


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 ).
(1) 
E =  ( 0.284 ) u^{ 0.78} M^{
0.667}A P ————————————— R T 
where: ^{ } ^{ } 
E u M A^{ } P T R^{ } 
= evaporation rate, lb / minute = windspeed just above the pool liquid surface, m / second = molecular weight of the pool liquid = surface area of the pool liquid, ft^{ 2} = vapor pressure of the pool liquid at the pool temperature, mm Hg = pool liquid temperature, °K = the Universal Gas Law constant = 82.05 ( atm · cm^{ 3} ) / ( gmol · °K ) 
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:
(2) A = ( cubic feet of pool liquid ) / ( 0.033 ft ) 
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.
(1) E = k P M / ( R T_{A
}) (2) k = 0.00293 u ^{ } 
where:^{ } _{ } ^{ } 
E^{ } k T_{A} M P R^{ } u 
= evaporation flux, ( lb / s ) / ft^{
2} of pool surface = mass transfer coefficient, ft / s = ambient temperature, °R _{ } = pool liquid molecular weight = pool liquid vapor pressure at ambient temperature, mm Hg = the Universal Gas Law constant = 555 ( mm Hg · ft^{ 3} ) / ( lbmol · °R ) = windspeed just above the liquid surface, miles / hour 
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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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).

where:  E  = evaporation flux, lb/minute/ft^{2} of pool surface 
B  = atmospheric boiling point of pool liquid, °F  
M  = molecular weight of pool liquid  
e  = 2.7183 

where:  Q  = initial instantaneous mass flow, lb/minute 
D  = discharge hole diameter, in  
P  = absolute source pressure, lb/in^{2}  
T  = source liquid temperature, °R  
T_{B}  = atmospheric boiling point of source liquid, °R  
C_{p}  = source liquid specific heat, Btu/lb/°R 
Notes: ln = natural logarithm (base e); in = inch; ° = ° = 460 + °
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(1)  Calculate the singlephase flow component (Q_{S}) for the source liquid by using the same equation as for a liquid discharge from a pressurized source, except substitute the source pressure minus the source liquid vapor pressure for the source pressure. 
(2)  Calculate the flashing flow component (Q_{F}) by using the same equation as for a flashing saturated liquid. 

where: 
Q 
= initial instantaneous mass flow, lb/minute 
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.

where:  X  = weight percent vaporized 
H_{s}^{L}  = source liquid enthalpy at source temperature and pressure, Btu/lb  
H_{a}^{V}  = flashed vapor enthalpy at atmos. boiling point and pressure, Btu/lb_{ }  
H_{a}^{L}  = residual liquid enthalpy at atmos. boiling point and pressure, Btu/lb_{ } ^{ } 
If the enthalpy data required for the above equation is unavailable, then the following
equation may be used.

where:  X  = weight percent vaporized 
C_{p}  = source liquid specific heat, Btu/lb/°F _{ }  
T_{s}  = source liquid temperature, °F _{ }  
T_{b}  = source liquid atmos. boiling point, °F _{ }  
H  = source liquid heat of vaporization at atmos. boiling point, Btu/lb 
Note: atmos. = atmospheric
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(1)
"Perry's Chemical Engineers' Handbook, Sixth Edition, McGrawHill 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, ESLTR8303, 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 7127830, 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 TR80/003, 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 G2, Appendix G. Available at
http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/tech.pdf/$File/tech.pdf ]
(6) "Risk Management Program Guidance For Offsite Consequence
Analysis", U.S. EPA publication EPA550B99009, April 1999. [ Equation (D1),
Section D.2.3, Appendix D. Available at
http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/ocaall.PDF/$File/ocaall.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@airdispersion.com
