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Source Terms For Accidental Discharge Flow Rates

Click in table below on desired source term (metric version):
Gas Discharge From
Pressure Source
Liquid Discharge From
Pressurized Vessel
Liquid Discharge From
Non-Pressurized Vessel
Evaporation From
Non-Boiling Liquid Pool
Evaporation From a Pool
of Cold Boiling Liquid
Discharge Of A Flashing
Saturated Liquid
Discharge of a Flashing
Sub-Cooled Liquid
Adiabatic Flash of a Liquified Gas Release References
and Appendices
To return to opening page and select United States version, click 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 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:

  or this equivalent form:

where:

 
 

 

 

 


Q
C
A 
k 

ρ 
P
PA
M
R 
T
Z
=  mass flow rate, kg / s
=  discharge coefficient     (dimensionless, usually about 0.72)
=  discharge hole area, m 2
=  cp / cv  of the gas  =  the isentropic expansion coefficient
=  (specific heat at constant pressure) / (specific heat at constant volume)
=  real gas density, kg / m 3 at P and T
=  absolute source or upstream pressure, Pa
=  absolute ambient or downstream pressure, Pa  
=  gas molecular weight
=  the Universal Gas Law Constant  =  8314.5 ( Pa · m 3) / ( kgmol · °K )
=  gas temperature, °K
=  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. Click here to learn how such calculations are performed.

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

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 / (MW).

Other forms of the above equations have been published by Ramskill.  When used properly, Ramskill's equations yield results which are identical to the results yielded by the above equations.  Appendix 1 at the end of this article discusses Ramskill's equation for gas flow from a pressurized source.

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

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Liquid Discharge From A Pressurized Source Vessel: 1, 2

Initial instantaneous flow through the discharge opening:

(1)   Qi = 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:

(2)   Qf = C A [ ( 2 d ) ( P - PA ) ] 1/2

Average flow:

(3)   Qavg = ( Qi + Qf ) ÷ 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, m 2
=  local gravitational acceleration of  9.807 m / s 2
=  source liquid density, kg / m 3
=  absolute source pressure, Pa  
=  absolute ambient pressure, Pa   
=  height of liquid above bottom of discharge opening, m

When expressed in the customary USA units, the equations (1) and (2) above also contain the gravitational conversion factor gc ( 32.17 ft / s2 in USA units ) associated with the ( P - PA ) term.  Since gc is 1 ( kg-m ) / ( N-s2 ) in the SI metric system of units, equations (1) and (2) above do not include it.

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

Initial instantaneous flow through the discharge opening:

(1)   Qi = C A ( 2 g d 2 H )1/2

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

(2)   Qf = 0

Average flow:

(3)   Qavg = Qi ÷ 2

where: 
 
 
 
   
 
Q
C
A
g
d
H
=  mass flow rate, kg / s
=  discharge coefficient     (dimensionless, usually about 0.62)
=  discharge hole area, m 2
=  local gravitational acceleration of  9.807 m / s 2
=  source liquid density, kg / m 3
=  height of liquid above bottom of discharge opening, m
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Evaporation From A Non-Boiling Liquid Pool:

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.

(1)   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

(2)   If  TP = 0 °C or less, then TF = 1.0
       If  TP > 0 °C, then TF = 1.0 + 0.0043 TP2

(3)   PH = 760 exp[ 65.3319 - ( 7245.2 / TA ) - ( 8.22 ln TA ) + ( 6.1557 x 10 - 3 ) TA ]

Note:  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 ) using units which were a mixture of metric usage and United States usage. The non-metric units have been converted to metric units for this presentation.

                  (10.40 ) u 0.78 M 0.667 A P
(1)     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 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 / (MW).

The U.S. EPA also defined the pool depth as 0.01 m ( i.e., 1 cm ) so that the surface area of the pool liquid could be calculated as:

(2)     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.

(1)   E = k P M / ( R TA )
(2)   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 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 / (MW).

Note:  kgmol = kilogram 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., at a liquid temperature of about zero degrees Centigrade or less ).

(1)   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 )
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Discharge Of Flashing Saturated Liquid: 2, 4

(1)   Q = 0.7584 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 
Note:  ln = natural logarithm (base e)

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Discharge of Flashing Sub-Cooled Liquid: 4

(1) 
 
 
 
(2) 
 
Calculate the single-phase 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)   Q Total = ( QS2 + QF2 ) 1/2

where: 
 Q Total
=  initial instantaneous mass flow, kg / s
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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.

(1)   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.

(2)   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

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References:

(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 (available at http://nepis.epa.gov/Adobe/PDF/10003MK5.PDF) 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, 1983

(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://www.epa.gov/OEM/docs/chem/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, and Equation (D-7), Section D.6, Appendix D.  Available at http://www.epa.gov/emergencies/docs/chem/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. ]


APPENDIX 1: THE RAMSKILL EQUATION FOR THE NON-CHOKED FLOW OF GAS:

The first section in this article, entitled "Gas Discharge To The Atmosphere From A Pressure Source", includes two equivalent forms of an equation for the non-choked flow of a gas from a pressurized source.  Another form of the equation has been published by Ramskill.  When used properly, Ramskill's equation yields results which are identical to the two forms in the above first section.  Ramskill's equation is shown below as Equation (1):



The ideal gas density at the downstream conditions of temperature and pressure in Ramskill's equation is ρA and it is defined in Equation (2) using the ideal Gas Law:



Since the downstream temperature TA is not known, the adiabatic expansion Equation (3) below is used to determine TA in terms of the known upstream temperature T:



Combining Equations (2) and (3) results in Equation (4) which defines ρA in terms of the known upstream temperature T:



Using Equations (1) and (4) to determine non-choked mass flow rates for ideal gases gives identical results to the results obtained using equation (5) below (which is one of the two equivalent equations in the "Gas Discharge To The Atmosphere From A Pressure Source" section of this article):



where:

 
 

 

 

 

Q
C
A 
k 

ρ 
P
PA
M
R 
T
=  mass flow rate, kg / s
=  discharge coefficient     (dimensionless, usually about 0.72)
=  discharge hole area, m 2
=  cp / cv  of the gas  =  the isentropic expansion coefficient
=  (specific heat at constant pressure) / (specific heat at constant volume)
=  gas density, kg / m 3 at upstream source conditions P and T
=  absolute source or upstream pressure, Pa
=  absolute ambient or downstream pressure, Pa  
=  gas molecular weight
=  the Universal Gas Law Constant  =  8314.5 ( Pa · m 3) / ( kgmol · °K )
=  upstream source gas temperature, °K

References for Equation (1):

Gierer, Conrad and Hyatt, Nigel,"Using Source Term Analysis Software for Calculating Fluid Flow Release Rates", Dyadem International Ltd., CACHE Newsletter No.48, Spring 1999,Austin, Texas (www.che.utexas.edu/cache/newsletters/Spr_99.pdf)

Ramskill, P.K., "Discharge Rate Calculation Methods for Use In Plant Safety Assessments", Safety and Reliability Directory, 1986, United Kingdom Atomic Energy Authority

References for Equation (5):

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

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

"Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, Section 2.5.2.3, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005.

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