Accidental release source terms are the mathematical equations that quantify the flow rate at which accidental releases of air pollutants into the ambient environment can occur at industrial facilities such as petroleum refineries, petrochemical plants, natural gas processing plants, chemical plants, oil and gas transportation pipelines, and many other industrial facilities. Governmental regulations in a good many countries require that the probability of such accidental releases be analyzed and their quantitative impact upon the environment and human health be determined so that mitigating steps can be planned and implemented.

There are a number of mathematical calculation methods for determining the flow rate at which gaseous and liquid pollutants might be released from various types of accidents. Such calculation methods are referred to as source terms, and this article on accidental release source terms explains some of the calculation methods used for determining the mass flow rate at which gaseous pollutants may be accidentally released. Given those mass flow rates, air pollution dispersion modeling^{[1]} studies can then be performed.

Accidental release of a pressurized gas

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 or it may be non-choked. Choked flow (also referred to as ''critical flow'') is a limiting or maximum condition at which the gas velocity has attained the speed of sound in the gas.

Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than:

where k is the specific heat ratio of the discharged gas (sometimes called the isentropic expansion factor and sometimes denoted as γ).

For many gases, k ranges from about 1.09 to about 1.41, and therefore the expression in (1) ranges from 1.7 to about 1.9, which means that choked velocity usually occurs when the absolute upstream vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream pressure. In the case of a leak to the ambient atmosphere, the downstream pressure is the atmospheric pressure.

When the gas velocity is choked, the equation for the mass flow rate in SI units is:^{[2]}^{[3]}^{[4]}^{[5]}

where the terms are defined as stated below. If the upstream gas density, ρu is not known directly, then it is useful to eliminate it by using the ideal gas law corrected for the real gas compressibility:

For the above equations, 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 upstream pressure is increased or the temperature is decreased.

Whenever the ratio of the absolute upstream pressure to the absolute downstream pressure is less than in expression (1) above, then the gas velocity is non-choked and the equation for mass flow rate is:

or this equivalent form:

where:

ṁ = mass flow rate, kg/s

C = discharge coefficient, dimensionless (usually about 0.72)

A = discharge hole area, m2

k = cp÷cv = specific heat ratio of the gas

cp = specific heat capacity of the gas at constant pressure

cv = specific heat capacity of the gas at constant volume

ρu = real gas upstream density, kg/m3 = (M Pu)÷(Z R Tu)

Pu = absolute upstream pressure, Pa

Pd = absolute downstream pressure, Pa

M = the gas molecular mass, kg/kmol (also known as the molecular weight)

Z = the gas compressibility factor at Pu and Tu, 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. A comparison between two methods for performing such calculations is available online.^{[6]}

The technical literature can be confusing because many authors do not explain whether they are using the universal gas law constant R which applies to any ideal gas or whether they are using Rs which only applies to a specific individual gas. The relationship between the two is Rs = R/M.

Notes:

The above equations are for a real gas.

For an ideal gas, Z = 1 and ρ is the ideal gas density.

kmol = 1000 mol

Evaporation of 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. The results obtained by the three methods are somewhat different.

The U.S. EPA method

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 U.S. Environmental Protection Agency using units which were a mixture of metric usage and United States usage.^{[7]} The non-metric units have been converted to metric units for this presentation.

E = ( 0.1268 ÷ T) u0.78 M0.667 A P

where:

E = evaporation rate, kg/min

u = windspeed just above the pool liquid surface, m/s

M = pool liquid molecular mass, dimensionless

A = surface area of the pool liquid, m2

P = vapor pressure of the pool liquid at the pool temperature, kPa

T = pool liquid absolute temperature, K

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:

A = (pool volume, in m3) ÷ (0.01)

The U.S. Air Force method

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.^{[8]}

E = (4.161 × 10– 5) u0.75 TF M (PS ÷ PH)

where:

E = evaporation flux, (kg/min) / m2 of pool surface

u = windspeed just above the liquid surface, m/s

TA = absolute ambient temperature, K

TF = pool liquid temperature correction factor, dimensionless (see equations [a] and [b] below)

TP = pool liquid temperature, °C

M = pool liquid molecular weight, g/mol

PS = pool liquid vapor pressure at ambient temperature, mmHg

PH = hydrazine vapor pressure at ambient temperature, mmHg (see equation [c] below)

[a] If TP = 0 °C or less, then TF =1.0

[b] If TP > 0 °C, then TF = 1.0 + 0.0043 TP2

[c] PH = 760 exp[ 65.3319 − ( 7245.2 ÷ TA ) − ( 8.22 ln TA ) + ( 6.1557 × 10– 3 ) TA ]

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

Stiver and Mackay's method

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.^{[9]}^{[10]}

E = k P M ÷ (R TA)

where:

E = evaporation flux, (kg/s)/m2 of pool surface

k = mass transfer coefficient, m/s (which is taken to be 0.002 u)

TA = absolute ambient temperature, K

M = pool liquid molecular weight, g/mol

P = pool liquid vapor pressure at ambient temperature, Pa

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 0 °C or less).^{[11]}

E = ( 0.0001 ) ( 7.7026 − 0.0288 B ) ( M ) e– 0.0077B – 0.1376

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 commonly referred to 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 ( HuL – HdL ) ÷ ( HdV – HdL )

where:

X = weight percent vaporized

HuL = upstream liquid enthalpy at upstream temperature and pressure, J/kg

HdV = flashed vapor enthalpy at downstream pressure and corresponding saturation temperature, J/kg

HdL = residual liquid enthalpy at downstream pressure and corresponding saturation temperature, J/kg

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

X = 100 × cp (Tu – Td) ÷ Hv

where:

X = weight percent vaporized

cp = liquid specific heat at upstream temperature and pressure, J/(kg · °C)

Tu = upstream liquid temperature, °C

Td = liquid saturation temperature corresponding to the downstream pressure, °C

Hv = liquid heat of vaporization at downstream pressure and corresponding saturation temperature, J/kg

Note: The words "upstream" and "downstream" refer to before and after the liquid passes through the release opening.

• J.H. Clewell (1983), A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids, Energy Systems Laboratory, ESL-TR-83-03.

• G. Ille and C. Springer (1978), The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spill, Environmental Engineering Development Office, CEEDO 712-78-30.

• J.P. Kahler, R.C. Curry and R.A. Kandler (1980), Calculating Toxic Corridors, Air Force Weather Service, AWS TR-80/003.

^Risk Management Program Guidance For Offsite Consequence Analysis, U.S. EPA publication EPA-550-B-99-009, April 1999. Available online at Guidance for Offsite Consequence Analysis (Appendix D: Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)

^Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases), CPR 14E (Yellow Book), Chapter 2, pp. 2.67 - 2.68, The Netherlands Organization Of Applied Scientific Research (TNO), 2005. Available for free registration and download here

^ W. Stiver and D. Mackay (1983), A Spill Hazard Ranking System For Chemicals, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 261-266 .

^ W. Stiver and D. Mackay (1983), Evaporation Rates of Chemical Spills, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 1-8.

Accidental release source termsare the mathematical equations that quantify the flow rate at which accidental releases of air pollutants into the ambient environment can occur at industrial facilities such as petroleum refineries, petrochemical plants, natural gas processing plants, chemical plants, oil and gas transportation pipelines, and many other industrial facilities. Governmental regulations in a good many countries require that the probability of such accidental releases be analyzed and their quantitative impact upon the environment and human health be determined so that mitigating steps can be planned and implemented.There are a number of mathematical calculation methods for determining the flow rate at which gaseous and liquid pollutants might be released from various types of accidents. Such calculation methods are referred to as

source terms, and this article on accidental release source terms explains some of the calculation methods used for determining the mass flow rate at which gaseous pollutants may be accidentally released. Given those mass flow rates, air pollution dispersion modeling^{[1]}studies can then be performed.## Accidental release of a pressurized gas

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 bechokedor it may benon-choked. Choked flow (also referred to as ''critical flow'') is a limiting or maximum condition at which the gas velocity has attained the speed of sound in the gas.Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than:

where

is the specific heat ratio of the discharged gas (sometimes called thekisentropic expansion factorand sometimes denoted as).γFor many gases,

ranges from about 1.09 to about 1.41, and therefore the expression ink(1)ranges from 1.7 to about 1.9, which means that choked velocity usually occurs when the absolute upstream vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream pressure. In the case of a leak to the ambient atmosphere, the downstream pressure is the atmospheric pressure.When the gas velocity is choked, the equation for the mass flow rate in SI units is:

^{[2]}^{[3]}^{[4]}^{[5]}where the terms are defined as stated below. If the upstream gas density,

is not known directly, then it is useful to eliminate it by using the ideal gas law corrected for the real gas compressibility:ρuFor the above equations,

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 upstream pressure is increased or the temperature is decreased.Whenever the ratio of the absolute upstream pressure to the absolute downstream pressure is less than in expression

(1)above, then the gas velocity is non-choked and the equation for mass flow rate is:or this equivalent form:

mass flow rate, kg/sṁ == discharge coefficient, dimensionless (usually about 0.72)C= discharge hole area, m2A=kcp÷= specific heat ratio of the gascv= specific heat capacity of the gas at constant pressurecp= specific heat capacity of the gas at constant volumecv= real gas upstream density, kg/m3 =ρu(M Pu)÷(Z R Tu)= absolute upstream pressure, PaPu= absolute downstream pressure, PaPd= the gas molecular mass, kg/kmol (also known as the molecular weight)M= the universal gas law constant = 8314.472 Pa·m3R÷(kmol·K)= absolute upstream gas temperature, KTu= the gas compressibility factor atZandPu, dimensionlessTuThe above equations calculate the

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. A comparison between two methods for performing such calculations is available online.initial instantaneous^{[6]}The technical literature can be confusing because many authors do not explain whether they are using the universal gas law constant

which applies to any ideal gas or whether they are usingRwhich only applies to a specific individual gas. The relationship between the two isRs=Rs.R/MNotes:

= 1 andZis the ideal gas density.ρ## Evaporation of 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. The results obtained by the three methods are somewhat different.## The U.S. EPA method

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 U.S. Environmental Protection Agency using units which were a mixture of metric usage and United States usage.^{[7]}The non-metric units have been converted to metric units for this presentation.E = ( 0.1268 ÷ T) u0.78 M0.667 A PE= evaporation rate, kg/minu= windspeed just above the pool liquid surface, m/sM= pool liquid molecular mass, dimensionlessA= surface area of the pool liquid, m2P= vapor pressure of the pool liquid at the pool temperature, kPaT= pool liquid absolute temperature, KThe 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:

A= (pool volume, in m3) ÷ (0.01)## The U.S. Air Force method

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.^{[8]}E = (4.161 × 10– 5) u0.75 TF M (PS ÷ PH)E= evaporation flux, (kg/min) / m2 of pool surfaceu= windspeed just above the liquid surface, m/sTA= absolute ambient temperature, KTF= pool liquid temperature correction factor, dimensionless (see equations [a] and [b] below)TP= pool liquid temperature, °CM= pool liquid molecular weight, g/molPS= pool liquid vapor pressure at ambient temperature, mmHgPH= hydrazine vapor pressure at ambient temperature, mmHg (see equation [c] below)If TP = 0 °C or less, then TF =1.0If TP > 0 °C, then TF = 1.0 + 0.0043 TP2PH = 760 exp[ 65.3319 − ( 7245.2 ÷ TA ) − ( 8.22 ln TA ) + ( 6.1557 × 10– 3 ) TA ]Note:

The function

ln xis the natural logarithm (base e) ofx, and the functionexp xise(approximately 2.7183) raised to the power ofx.## Stiver and Mackay's method

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.^{[9]}^{[10]}E = k P M ÷ (R TA)E= evaporation flux, (kg/s)/m2 of pool surfacek= mass transfer coefficient, m/s (which is taken to be 0.002u)TA= absolute ambient temperature, KM= pool liquid molecular weight, g/molP= pool liquid vapor pressure at ambient temperature, PaR= the universal gas law constant of 8314.472 Pa·m3÷(kmol·K)u= windspeed just above the liquid surface, m/s## Evaporation of boiling, cold liquid pool

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 0 °C or less).^{[11]}

where:E = ( 0.0001 ) ( 7.7026 − 0.0288 B ) ( M ) e– 0.0077B – 0.1376E= evaporation flux, (kg/min)/m2 of pool surfaceB= pool liquid atmospheric boiling point, °CM= pool liquid molecular weight, g/mole= 2.7183, the base of the natural logarithm## Adiabatic flash of liquified gas release

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 commonly referred to 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 ( HuL – HdL ) ÷ ( HdV – HdL )= weight percent vaporizedX= upstream liquid enthalpy at upstream temperature and pressure, J/kgHuL= flashed vapor enthalpy at downstream pressure and corresponding saturation temperature, J/kgHdV= residual liquid enthalpy at downstream pressure and corresponding saturation temperature, J/kgHdLIf the enthalpy data required for the above equation is unavailable, then the following equation may be used:

X = 100 × cp (Tu – Td) ÷ Hv= weight percent vaporizedX= liquid specific heat at upstream temperature and pressure, J/(kg · °C)cp= upstream liquid temperature, °CTu= liquid saturation temperature corresponding to the downstream pressure, °CTd= liquid heat of vaporization at downstream pressure and corresponding saturation temperature, J/kgHvNote: The words "upstream" and "downstream" refer to before and after the liquid passes through the release opening.

## References

Fundamentals of Stack Gas Dispersion, 4th Edition, ISBN 0-07-049479-7Perry's Chemical Engineers' Handbook, 6th Edition, McGraw Hill, ISBN 0-07-049479-7Handbook of Chemical Hazard Analysis Procedures (Appendix B), Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Available online atHandbook of Chemical Hazard Analysis, Appendix B.Scroll down to page 391 of 520 PDF pages. This handbook also provides the references below:A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids, Energy Systems Laboratory, ESL-TR-83-03.The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spill, Environmental Engineering Development Office, CEEDO 712-78-30.Calculating Toxic Corridors, Air Force Weather Service, AWS TR-80/003.Risk Management Program Guidance For Offsite Consequence Analysis, U.S. EPA publication EPA-550-B-99-009, April 1999. Available online atGuidance for Offsite Consequence Analysis(Appendix D: Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases), CPR 14E (Yellow Book), Chapter 2, pp. 2.67 - 2.68, The Netherlands Organization Of Applied Scientific Research (TNO), 2005. Available for free registration and download hereA Spill Hazard Ranking System For Chemicals, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 261-266 .Evaporation Rates of Chemical Spills, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 1-8.