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Source Terms For Accidental Discharge Flow Rates

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



Q
C
A
gc
k

ρ
P
PA
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
=  cp / cv  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 ft-lb / ( lbmol · °R )
=  gas temperature, °R
=  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.

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).

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:

(1)   Qi = C A [ ( 2 gc d 2 H ) + ( 2 gc d ) ( P - PA ) ] 1/2

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

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

Average flow:

(3)   Qavg = ( Qi + Qf ) ÷ 2

where: 
 
 
 
 
 
 
 
 Q
C
A
gc
d
P
PA
H
 =  mass flow rate, lb / s
=  discharge coefficient     (dimensionless, usually about 0.62)
=  discharge hole area, ft 2
=  gravitational conversion factor of  32.17 ft / s 2
=  source liquid density, lb / ft 3
=  absolute source pressure, lb / ft 2
=  absolute ambient pressure, lb / ft 2
=  height of liquid above bottom of discharge opening, ft
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Liquid Discharge From A Non-Pressurized Source Vessel: 1, 2, 6, 7

Initial instantaneous flow through the discharge opening:

(1)   Qi = C A ( 2 gc 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
gc
d
H
 =  mass flow rate, lb / s
=  discharge coefficient     (dimensionless, usually about 0.62)
=  discharge hole area, ft 2
=  gravitational conversion factor of  32.17 ft / s 2
=  source liquid density, lb / ft 3
=  height of liquid above bottom of discharge opening, ft
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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 = ( 4.66 x 10- 6 ) u0.75 TF M ( PS / PH )

where:
 
 
 
 
 
 
 
E
u
TA
TF
TP
M
PS
PH
 =  evaporation flux, ( lb / minute ) / ft 2 of pool surface
=  windspeed just above the liquid surface, miles / hour
=  ambient temperature, °K
=  pool liquid temperature correction factor
=  pool liquid temperature, °F
=  pool liquid molecular weight
=  pool liquid vapor pressure at ambient temperature, mm Hg
=  hydrazine vapor pressure at ambient temperature, mm Hg

(2)   If TP is 32 °F or less, then TF = 1.0
       If TP > 32 °F, then TF = 1.0 + 0.00133 ( TP - 32 ) 2

(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 ).

                 ( 0.284 ) u 0.78 M 0.667A P
(1)     E = —————————————
                                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 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 0.033 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 TA )
(2)   k = 0.00293 u  

where: 
 
  
 
 
  
 
 E 
k
TA
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 Rs which only applies to a specific individual gas.  The relationship between the two constants is Rs = 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).

(1)   E = ( 0.018967 ) ( 0.5322 - 0.001035 B ) ( M ) e - (0.0043 B)

where: 
 
 
 
 E
B
M
e
 =  evaporation flux, ( lb / minute) / ft 2 of pool surface
=  atmospheric boiling point of pool liquid, °F
=  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 = 0.1597 D 2 P [ ln ( P / 14.696 ) ] ( TB / T ) ( T / cp )1/2 ( T - TB ) - 1

where: 
 
 
 
 
 
Q
D
P
T
TB
cp
 =  initial instantaneous mass flow, lb / s
=  discharge hole diameter, in
=  absolute source pressure, lb / in 2
=  source liquid temperature, °R
=  atmospheric boiling point of source liquid, °R
=  source liquid specific heat, Btu / lb / °F
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)  QTotal = ( QS2 + QF2 ) 1/2

where: 
 QTotal
 =  initial instantaneous mass flow, lb / 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, Btu / lb   
=  flashed vapor enthalpy at atmospheric boiling point and pressure, Btu / lb   
=  residual liquid enthalpy at atmospheric boiling point and pressure, Btu / lb   

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, Btu / lb / °F  
=  source liquid temperature, °F  
=  source liquid atmospheric boiling point, °F  
=  source liquid heat of vaporization at atmospheric boiling point, Btu / lb

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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 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://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. ]

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