Source Terms for Accidental Releases
Gas Discharge To The Atmosphere From A Pressure Source: 1, 2
When gas stored under pressure in a closed source vessel is discharged to the ambient atmosphere through a hole or other opening, the gas velocity through that opening may be choked (i.e., has attained a maximum) or nonchoked. Choked velocity, which may also be 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:
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 / ( R T ) ]1/2 [ 2 / ( k + 1 ) ] (k + 1) / (2k  2)
[It is important to note that when 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 nonchoked (i.e., subsonic) and the equation for mass flow rate is:
Q = C A ( 2 d P )1/2 [ k / ( k  1 ) ]1/2 [ ( PA / P ) 2 / k  ( PA / P ) (k + 1) / k ]1/2
or this equivalent form:
Q = C A P [ 2 M / ( R T ) ]1/2 [ k / ( k – 1 ) ]1/2 [ ( PA / P ) 2 / k  ( PA / P ) (k + 1) / k ]1/2
where:

Q
C
A
k
d
M
R
T
P
PA 
= mass flow rate, kg/s
= discharge coefficient (dimensionless, usually about 0.72)
= discharge hole area, m 2
= cp / cv of the gas
= (specific heat at constant pressure) / (specific heat at constant volume)
= gas density, kg/m 3, at source pressure
= gas molecular weight
= the Universal Gas Law constant = 8314.5 ( Pa ) ( m 3 ) / ( kgmol ) ( °K )
= gas temperature, °K
= absolute source or upstream pressure, Pa
= absolute ambient or downstream pressure, Pa 
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 is 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.
The technical literature can be confusing since many authors fail to explain whether they are using the
universal gas law constant R applicable to any ideal gas or whether they are using the gas law constant
Rs applicable only to a specific gas. The relationship between the two constants is Rs = R / (MW).
Notes: kgmol = kilogram mole and Pa = Pascal
Liquid Discharge From A Pressurized Source Vessel: 1, 2
Initial instantaneous flow through the discharge opening:
Q i = C A [ ( 2 g d 2 H ) + ( 2 d ) ( P  PA ) ]1/2
Final flow when the liquid level reaches the bottom of the discharge opening:
Q f = C A [ ( 2 d ) ( P  PA ) ]1/2
Average flow:
Q avg = ( Q i + Q f ) ¸ 2
where:

Q
C
A
g
d
P
PA
H 
= mass flow rate, kg/s
= discharge coefficient (dimensionless, usually about 0.62)
= discharge hole area, m2
= gravitational acceleration of 9.807 m/s2
= source liquid density, kg/m3
= absolute source pressure, Pa
= absolute ambient pressure, Pa
= height of liquid above bottom of discharge opening, m 
Liquid Discharge From A NonPressurized Source Vessel: 1, 2
Initial instantaneous flow through the discharge opening:
Q i = C A ( 2 g d 2 H )1/2
Final flow when the liquid level reaches the bottom of the discharge opening:
Q f = 0
Average flow:
Q avg = Q i ¸ 2
where:

Q
C
A
g
d
H 
= mass flow rate, kg/s
= discharge coefficient (dimensionless, usually about 0.62)
= discharge hole area, m2
= gravitational acceleration of 9.807 m/s2
= source liquid density, kg/m3
= height of liquid above bottom of discharge opening, m 
Evaporation From A NonBoiling Liquid Pool:
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
This method predicts the rate of liquid evaporation from the surface of a pool of liquid that is at or near the ambient temperature, and it was derived from U.S. Air Force tests with pools of liquid hydrazine.
E = ( 4.161 x 10  5 ) u0.75 TF M ( PS / PH )
where:

E
u
TA
TF
TP
M
PS
PH 
= evaporation flux, ( kg/minute ) / m 2 of pool surface
= windspeed just above the liquid surface, m/s
= ambient temperature, °K
= pool liquid temperature correction factor
= pool liquid temperature, °C
= pool liquid molecular weight
= pool liquid vapor pressure at ambient temperature, mm Hg
= hydrazine vapor pressure at ambient temperature, mm Hg 
If TP = 0 °C or less, then TF = 1.0
If TP > 0 °C, then TF = 1.0 + 0.0043 TP2
PH = 760 exp[ 65.3319  ( 7245.2 / TA )  ( 8.22 ln TA ) + ( 6.1557 x 10  3 ) TA ]
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 predict 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 U.S. EPA. using metric and nonmetric units. The nonmetric units have been converted to metric for this presentation.
( 10.40 ) u 0.78 M 0.667 A P
E = —————————————
R T
where:

E
u
M
A
P
T
R 
= evaporation rate, kg/minute
= windspeed just above the pool liquid surface, m/s
= molecular weight of the pool liquid
= surface area of the pool liquid, m 2
= vapor pressure of the pool liquid at the pool temperature, kPa
= pool liquid temperature, °K
= the Universal Gas Law constant = 82.05 ( atm) ( cm 3 ) / ( gmol ) ( °K ) 
The technical literature can be confusing since many authors fail to explain whether they are using the
universal gas law constant R applicable to any ideal gas or whether they are using the gas law constant
Rs applicable only to a specific gas. The relationship between the two constants is Rs = R / (MW).
The U.S. EPA defined the pool depth as 0.01 m ( i.e., 1 cm ) and so the pool surface area is:
A = ( cubic meters of pool liquid ) / ( 0.01 m )
Notes: 1 kPa = 0.0102 kg/cm 2 = 0.01 bar
gmol = gram mole
atm = atmosphere
"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.
E = k P M / ( R TA )
k = 0.002 u
where:

E
k
TA
M
P
R
u 
= evaporation flux, ( kg/s ) / m 2 of pool surface
= mass transfer coefficient, m/s
= ambient temperature, °K
= pool liquid molecular weight
= pool liquid vapor pressure at ambient temperature, Pa
= the Universal Gas Law constant = 8314.5 ( Pa ) ( m 3 ) / ( kgmol ) ( °K )
= windspeed just above the liquid surface, m/s 
The technical literature can be confusing since many authors fail to explain whether they are using the
universal gas law constant R applicable to any ideal gas or whether they are using the gas law constant
Rs applicable only to a specific gas. The relationship between the two constants is Rs = R / (MW).
Note: kgmol = kilogram mole
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).
E = ( 0.0001 ) ( 7.7026  0.0288 B ) ( M ) e  ( 0.0077 B )  0.1376
where:

E
B
M
e 
= evaporation flux, ( kg/minute) / m 2 of pool surface
= atmospheric boiling point of pool liquid, °C
= molecular weight of pool liquid
= 2.7183 ( the number that is the base of the natural logarithm system ) 
Discharge Of Flashing Saturated Liquid: 2, 4
Q = 0.7548 D 2 P [ ln ( P / 101,325 ) ] ( TB / T ) ( T / cp )1/2 ( T  TB )  1
where:

Q
D
P
T
TB
cp 
= initial instantaneous mass flow, kg/s
= discharge hole diameter, m
= absolute source pressure, Pa
= source liquid temperature, °K
= atmospheric boiling point of source liquid, °K
= source liquid specific heat, J/kg/°C 
Notes: ln = natural logarithm (base e)
Discharge of Flashing SubCooled Liquid: 4
 Calculate the singlephase flow component ( QS ) 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.
 Calculate the flashing flow component ( QF ) by using the same equation as for a flashing saturated liquid.
(3) QTotal = ( QS2 + QF2 ) 1/2
where: QTotal = initial instantaneous mass flow, kg/s
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.
X = 100 ( HsL  HaL ) / ( HaV  HaL )
where:

X
HsL
HaV
HaL 
= weight percent vaporized
= source liquid enthalpy at source temperature and pressure, J/kg
= flashed vapor enthalpy at atmospheric boiling point and pressure, J/kg
= residual liquid enthalpy at atmospheric boiling point and pressure, J/kg 
If the enthalpy data required for the above equation is unavailable, then the following equation may be used.
X = 100 [ cp ( Ts  Tb ) ] / H
where:

X
cp
Ts
Tb
H 
= weight percent vaporized
= source liquid specific heat, J/kg/°C
= source liquid temperature, °K
= source liquid atmospheric boiling point, °K
= source liquid heat of vaporization at atmospheric boiling point, J/kg 
References:
(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 www.epa.gov/swercepp/pubs/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 www.epa.gov/ceppo/pubs/oca/ocaall.pdf ]


Milton Beychok
Worked as a process design engineer (oil refineries, petrochemical plants, natural gas processing) with a large , international engineering and construction firm for about 25 years. Then he worked as in independent, consulting engineer for another 25 years.
He is a Fellow of the AIChE. He was also a Diplomate of the American Academy of Environmental Engineers. 




 

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