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'''Accidental release source terms''' are the mathematical equations that quantify the flow rate at which accidental releases of air  [[pollutant]]s into the ambient [[Natural environment|environment]] can occur at industrial facilities such as [[Petroleum refining processes|petroleum refineries]], [[petrochemical]] plants, [[natural gas processing]] plants, oil and [[gas]] transportation [[Pipeline transport|pipelines]], [[chemical plant]]s, and many other industrial activities. 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.  
'''Accidental release source terms''' are the mathematical equations that quantify the flow rate at which accidental releases of air  [[pollutant]]s into the ambient [[Natural environment|environment]] can occur at industrial facilities such as [[Petroleum refining processes|petroleum refineries]], [[petrochemical]] plants, [[natural gas processing]] plants, oil and [[gas]] transportation [[Pipeline transport|pipelines]], [[chemical plant]]s, and many other industrial activities. 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 calculational 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, [[atmospheric dispersion modeling]] studies then be performed.
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 calculational 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, [[atmospheric dispersion modeling]] studies can then be performed.    


==Accidental release of pressurized gas==
==Accidental release of 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 (i.e., it has attained a maximum) or it may be non-choked.
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 flow|''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 velocity, also referred to as sonic velocity, occurs when the ratio of the absolute source pressure to the absolute downstream pressure is equal to or greater than:
Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than:


(1) &nbsp; &nbsp; <math>\big[(k+1)/2 \big]^{k/(k-1}</math>
:<math>(1)</math> &nbsp; &nbsp; <font style="vertical-align:+15%;"><math>\big[(k+1)/2 \big]^{\,k/(k-1)}</math></font>


where <math>k</math> is the [[specific heat ratio]] of the discharged gas (sometimes called the [[isentropic expansion factor]] and sometimes denoted as <math>\gamma</math>).
where <math>k</math> is the [[specific heat ratio]] of the discharged gas (sometimes called the [[isentropic expansion factor]] and sometimes denoted as <math>\gamma</math> ).


For many gases, <math>k</math> 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 source vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream ambient [[atmospheric pressure]].
For many gases, <math>k</math> 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 metric units is:<ref name=Perry>{{cite book|author=Perry, R.H. and Green, D.W.|title=[[Perry's Chemical Engineers' Handbook]] |edition=6th Edition| publisher=McGraw Hill, Inc.|year=1984|id=ISBN ISBN 0-07-049479-7}}</ref><ref name=Hazards>''Handbook of Chemical Hazard Analysis Procedures'', Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. [http://nepis.epa.gov/Adobe/PDF/10003MK5.PDF Handbook of Chemical Hazard Analysis, Appendix B.] Scroll down to page 391 of 520 PDF pages.  
When the gas velocity is choked, the equation for the mass flow rate in [[SI]] units is:<ref name=Perry>{{cite book|author=Perry, R.H. and Green, D.W.|title=[[Perry's Chemical Engineers' Handbook]] |edition=6th Edition| publisher=McGraw Hill, Inc.|year=1984|id=ISBN ISBN 0-07-049479-7}}</ref><ref name=Hazards>''Handbook of Chemical Hazard Analysis Procedures'', Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. [http://nepis.epa.gov/Adobe/PDF/10003MK5.PDF Handbook of Chemical Hazard Analysis, Appendix B.] Scroll down to page 391 of 520 PDF pages.  
This handbook also provides the references below:
This handbook also provides the references below:
*Clewell, H.J., ''A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids'', Energy Systems Laboratory, ESL-TR-83-03, 1983.
*Clewell, H.J., ''A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids'', Energy Systems Laboratory, ESL-TR-83-03, 1983.
Line 25: Line 25:
*Kahler, J.P., Curry, R.C. and Kandler, R.A.,''Calculating Toxic Corridors'' Air Force Weather Service, AWS TR-80/003, 1980.<br></ref><ref name=Risk>"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. &nbsp;[http://www.epa.gov/emergencies/docs/chem/oca-all.pdf  Guidance for Offsite Consequence Analysis] (Appendix D: Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)</ref><ref name=Netherlands>"Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. [http://vrom.nl/pagina.html?id=20725 PGS2 CPR 14E]</ref><BR><BR>
*Kahler, J.P., Curry, R.C. and Kandler, R.A.,''Calculating Toxic Corridors'' Air Force Weather Service, AWS TR-80/003, 1980.<br></ref><ref name=Risk>"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. &nbsp;[http://www.epa.gov/emergencies/docs/chem/oca-all.pdf  Guidance for Offsite Consequence Analysis] (Appendix D: Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)</ref><ref name=Netherlands>"Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. [http://vrom.nl/pagina.html?id=20725 PGS2 CPR 14E]</ref><BR><BR>


<math>Q\;=\;C\;A\;\sqrt{\;k\;\rho\;P\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}</math>
:<math>\dot m\;=\;C\;A\;\sqrt{\;k\;\rho_u\;P_u\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}</math>


or this equivalent form:
where the terms are defined as in listed below. If the upstream gas [[Density (chemistry)|density]], <font style="vertical-align:-15%;"><math>\rho_u</math></font> is not known directly, then it is useful to eliminate it using the [[Ideal gas law]] corrected for the real gas [[Compressibility factor (gases)|compressiblity]]: <br><br>


<math>Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T}\bigg)\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}</math><BR><BR>
:<math>\dot m\;=\;C\;A\;P_u\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T_u}\bigg)\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}</math><BR><BR>


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 source pressure is increased.
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 source pressure to the absolute downstream ambient pressure is less than in expression (1) above,  then the gas velocity is non-choked (i.e., sub-sonic) and the equation for mass flow rate is:<BR><BR>
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:<BR><BR>


<math>Q\;=\;C\;A\;\sqrt{\;2\;\rho\;P\;\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_A}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_A}{P}\bigg)^{(k+1)/k}\;\Bigg]}</math>
:<math>\dot m\;=\;C\;A\;\sqrt{\;2\;\rho_u\;P_u\;\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_d}{P_u}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_d}{P_u}\bigg)^{(k+1)/k}\;\Bigg]}</math>


or this equivalent form:
or this equivalent form:


<math>Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{2\;M}{Z\;R\;T}\bigg)\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_A}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_A}{P}\bigg)^{(k+1)/k}\;\Bigg]}</math>
:<math>\dot m\;=\;C\;A\;P_u\;\sqrt{\bigg(\frac{2\;M}{Z\;R\;T_u}\bigg)\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_d}{P_u}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_d}{P_u}\bigg)^{(k+1)/k}\;\Bigg]}</math>


{| border="0" cellpadding="2"
{| border="0" cellpadding="2"
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|&nbsp;
|&nbsp;
|-
|-
!align=right| ''Q''
!align=right| <math>\dot m</math>
|align=left|= mass flow rate, kg/s
|align=left|= mass flow rate, [[Kilogram|kg]]/[[second|s]]
|-
|-
!align=right| ''C''
!align=right| <math>C</math>
|align=left|= [[discharge coefficient]], dimensionless (usually about 0.72)
|align=left|= discharge coefficient, dimensionless (usually about 0.72)
|-
|-
!align=right| ''A''
!align=right| <math>A</math>
|align=left|= discharge hole area,
|align=left|= discharge hole area, [[metre|m]]<sup>2</sup>
|-
|-
!align=right| ''k''
!align=right| <math>k</math>
|align=left|= c<sub>p</sub>/c<sub>v</sub> of the gas
|align=left|= <math>c_p/c_v</math> = specific heat ratio of the gas
|-
|-
!align=right| ''c<sub>p</sub>''
!align=right| <math>c_p</math>  
|align=left|= [[specific heat capacity|specific heat]] of the gas at constant pressure
|align=left|= [[specific heat capacity]] of the gas at constant pressure
|-
|-
!align=right| ''c<sub>v</sub>''
!align=right| <math>c_v</math>  
|align=left|= specific heat of the gas at constant volume
|align=left|= specific heat capacity of the gas at constant volume
|-
|-
!align=right| ''<math>\rho</math>''
!align=right| <math>\rho_u</math>  
|align=left|= [[ideal gas|real gas]] [[Density (chemistry)|density]] at '''''P''''' and '''''T''''', kg/
|align=left|= real gas upstream [[Density (chemistry)|density]], kg/m<sup>3</sup> = <math>(M P_u)/(Z\, R\, T_u)</math>
|-
|-
!align=right| ''P''
!align=right| <math>P_u</math>
|align=left|= absolute upstream pressure, Pa
|align=left|= absolute upstream pressure, [[Pascal (unit)|Pa]]
|-
|-
!align=right| ''P<sub>A</sub>''
!align=right| <math>P_d</math>
|align=left|= absolute ambient or downstream pressure, Pa
|align=left|= absolute downstream, Pa
|-
|-
!align=right| ''M''
!align=right| <math>M</math>
|align=left|= the gas [[molecular mass]], kg/kmole&nbsp; &nbsp; (also known as the molecular weight)
|align=left|= the gas [[molecular mass]], kg/k[[mole]]&nbsp; &nbsp; (also known as the molecular weight)
|-
|-
!align=right| ''R''
!align=right| <math>R</math>
|align=left|= the [[Molar gas constant|universal gas law constant]] = 8314.5 Pa·m³/(kmole·K)
|align=left|= the [[Molar gas constant|universal gas law constant]] = 8314.5 Pa·m<sup>3</sup>/(kmole·[[kelvin|K]])
|-
|-
!align=right| ''T''
!align=right| <math>T_u</math>
|align=left|= absolute upstream gas [[temperature]], K
|align=left|= absolute upstream gas [[temperature]], K
|-
|-
!align=right| ''Z''
!align=right| <math>Z</math>
|align=left|= the gas [[compressibility factor]] at ''P'' and ''T'', dimensionless
|align=left|= the gas [[compressibility factor]] at <math>P_u</math> and <math>T_u</math>, dimensionless
|}
|}


The above equations calculate the '''<u>initial instantaneous</u>''' 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.<ref>[http://www.air-dispersion.com/feature2.html Calculating Accidental Release Rates From Pressurized Gas Systems]</ref>
The above equations calculate the '''<u>initial instantaneous</u>''' 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.<ref>[http://www.air-dispersion.com/feature2.html Calculating Accidental Release Rates From Pressurized Gas Systems]</ref>


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''<sub>s</sub> which only applies to a specific individual gas.  The relationship between the two constants is ''R''<sub>s</sub> = ''R''/''M''.
The technical literature can be confusing because many authors do not explain whether they are using the universal gas law constant <math>R</math> which applies to any [[ideal gas]] or whether they are using <math>R_s</math> which only applies to a specific individual gas.  The relationship between the two constants is <math>R_s</math> = <math>R/M</math>.


Notes:<br>
Notes:<br>
* The above equations are for a real gas.
* The above equations are for a real gas.
* For an ideal gas, '''''Z''''' = 1 and '''''ρ''''' is the ideal gas density.
* For an [[Ideal gas law|ideal gas]], '''''Z''''' = 1 and '''''ρ''''' is the ideal gas density.
* kmole = 1000 [[mole (unit)|moles]]
* kmole = 1000 moles


===Ramskill's equation for non-choked mass flow===
===Ramskill's equation for non-choked mass flow===


P.K. Ramskill's equation <ref>[http://www.che.utexas.edu/cache/newsletters/Spr_99.pdf CACHE Newsletter No.48, Spring 1999] Gierer, C. and Hyatt, N.,''Using Source Term Analysis Software for Calculating Fluid Flow Release Rates'' Dyadem International Ltd.</ref><ref>Ramskill, P.K. (1986), ''Discharge Rate Calculation Methods for Use In Plant Safety Assessments'', Safety and Reliability Directory, United Kingdom Atomic Energy Authority</ref> for the non-choked flow of an ideal gas is:<BR><BR>
P.K. Ramskill's equation <ref>[http://www.che.utexas.edu/cache/newsletters/Spr_99.pdf CACHE Newsletter No.48, Spring 1999] Gierer, C. and Hyatt, N.,''Using Source Term Analysis Software for Calculating Fluid Flow Release Rates'' Dyadem International Ltd.</ref><ref>Ramskill, P.K. (1986), ''Discharge Rate Calculation Methods for Use In Plant Safety Assessments'', Safety and Reliability Directory, United Kingdom Atomic Energy Authority</ref> for the non-choked flow of a gas is somewhat different than the above non-choked flow equation but both versions provide identical results:<BR><BR>


(1) &nbsp; &nbsp; &nbsp; <math>Q = C \;\rho_A\;A\;\sqrt{\frac{\;\,2\;P}{\rho}\cdot\frac{k}{k-1}\cdot{\Bigg[\; 1 - {\bigg(\frac{P_A}{P}\bigg)^{(k-1)/k)}}\Bigg]}}</math>
:(1) &nbsp; &nbsp; &nbsp; <math>\dot m = C \;\rho_d\;A\;\sqrt{\frac{\;\,2\;P_u}{\rho_u}\cdot\frac{k}{k-1}\cdot{\Bigg[\; 1 - {\bigg(\frac{P_d}{P_u}\bigg)^{(k-1)/k)}}\Bigg]}}</math>


{| border="0" cellpadding="2"
{| border="0" cellpadding="2"
Line 106: Line 106:
|&nbsp;
|&nbsp;
|-
|-
!align=right| ''Q''
!align=right| <math>\dot m</math>
|align=left|= mass flow rate, kg/s
|align=left|= mass flow rate, kg/s
|-
|-
!align=right| ''C''
!align=right| <math>C</math>
|align=left|= discharge coefficient, dimensionless (usually about 0.72)
|align=left|= discharge coefficient, dimensionless (usually about 0.72)
|-
|-
!align=right| ''A''
!align=right| <math>A</math>
|align=left|= discharge hole area,
|align=left|= discharge hole area, m<sup>2</sup>
|-
|-
!align=right| <math>\rho_A</math>  
!align=right| <math>k</math>
|align=left|= ideal gas density at the downstream pressure and temperature
|align=left|= <math>c_p/c_v</math> = specific heat ratio of the gas
|-
|-
!align=right| <math>\rho</math>  
!align=right| <math>c_p</math>  
|align=left|= ideal gas density at the upstream pressure and temperature
|align=left|= specific heat capacity of the gas at constant pressure
|-
|-
!align=right| ''k''
!align=right| <math>c_v</math>
|align=left|= c<sub>p</sub>/c<sub>v</sub> of the gas
|align=left|= specific heat capacity of the gas at constant volume
|-
|-
!align=right| ''c<sub>p</sub>''
!align=right| <math>\rho_u</math>  
|align=left|= specific heat of the gas at constant pressure
|align=left|= real gas upstream density, kg/m<sup>3</sup> = <math>(M P_u)/(Z\, R\, T_u)</math> 
|-
|-
!align=right| ''c<sub>v</sub>''
!align=right| <math>\rho_d</math>  
|align=left|= specific heat of the gas at constant volume
|align=left|= real gas downstream density, kg/m<sup>3</sup> = <math>(M P_d)/(Z\, R\, T_d)</math>
|-
|-
!align=right| ''P''
!align=right| <math>P_u</math>
|align=left|= absolute upstream pressure, Pa
|align=left|= absolute upstream pressure, Pa
|-
|-
!align=right| ''P<sub>A</sub>''
!align=right| <math>P_d</math>
|align=left|= absolute downstream pressure, Pa
|align=left|= absolute downstream, Pa
|-
!align=right| <math>M</math>
|align=left|= the gas molecular mass, kg/kmole&nbsp; &nbsp; (also known as the molecular weight)
|-
|-
!align=right| ''T''
!align=right| <math>R</math>
|align=left|= upstream absolute temperature, K
|align=left|= the universal gas law constant = 8314.5 Pa·m<sup>3</sup>/(kmole·K)
|-
|-
!align=right| ''T<sub>A</sub>''
!align=right| <math>T_u</math>  
|align=left|= downstream absolute temperature, K
|align=left|= absolute upstream gas temperature, K
|-
|-
!align=right| ''M''
!align=right| <math>T_d</math>
|align=left|= the gas molecular mass, kg/kmole
|align=left|= absolute downstream gas temperature, K
|-
|-
!align=right| ''R''
!align=right| <math>Z</math>
|align=left|= the universal gas law constant  =  8314.5 Pa·m³/(kmole·K)
|align=left|= the gas compressibility factor at the pertinent <math>P</math> and <math>T</math> conditions, dimensionless
|}
|}


Calculation of the downstream ideal gas density '''''<math>\rho_A</math>'''''  requires that the downstream temperature '''''<math>T_A</math>''''' be obtained. Starting with the [[ideal gas law]]:
Since the downstream temperature <math>T_d</math> is unknown and is required to calculate the downstream density <math>\rho_d</math>, the isentropic expansion equation below <ref>[http://www.grc.nasa.gov/WWW/K-12/airplane/compexp.html Isentropic Compression or Expansion]</ref> is used to obtain <math>T_d</math> in terms of the known upstream temperature <math>T_u</math> :  
 
(2) &nbsp; &nbsp; &nbsp; <math>\rho_A = \frac{M\;P_A}{R \;T_A}</math>
 
Since the downstream temperature '''''<math>T_A</math>''''' is not known, the isentropic expansion equation below <ref>[http://www.grc.nasa.gov/WWW/K-12/airplane/compexp.html Isentropic Compression or Expansion]</ref> is used to determine '''''<math>T_A</math>''''' in terms of the known upstream temperature '''''<math>T</math>''''':  


(3) &nbsp; &nbsp; &nbsp; <math>\frac{T_A}{T} = \bigg(\frac{P_A}{P}\bigg)^{(k-1)/k}</math>
:(2) &nbsp; &nbsp; &nbsp; <math>T_d = T_u \bigg(\frac{P_d}{P_u}\bigg)^{(k-1)/k}</math>


Combining the equations (2) and (3) defines '''''<math>\rho_A</math>''''' in terms of the known upstream temperature '''''<math>T</math>''''':
Combining equation (2) with the definition of <math>\rho_d</math> (in the parameter list just above) provides <math>\rho_d</math> in terms of the known upstream temperature <math>T_u</math> :


(4) &nbsp; &nbsp; &nbsp; <math>\rho_A = \frac{M \;P^{\;(k-1)/k}}{R \;T \;P_A^{\ -1/k}}</math>
:(3) &nbsp; &nbsp; &nbsp; <math>\rho_d = \frac{M \;P_u^{\;(k-1)/k}}{Z R \;T_u \;P_d^{\ -1/k}}</math>


Using equation (4) with Ramskill's equation (1) to determine non-choked mass flow rates for ideal gases gives results identical to those obtained using the non-choked flow equation presented in the previous section above.
Using equation (3) with Ramskill's equation (1) to determine non-choked mass flow rates for gases gives results identical to those obtained using the non-choked flow equation presented in the previous section above.


==Evaporation of non-boiling liquid pool==
==Evaporation of non-boiling liquid pool==
Line 305: Line 304:
==Adiabatic flash of liquified gas release==  
==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 known as [[flash evaporation|"adiabatic flashing"]] and the following equation, derived from a simple heat balance, is used to predict how much of the liquified gas is vaporized.  
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 [[flash evaporation|"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 ( ''H''<sub>s</sub><sup>L</sup> − ''H''<sub>a</sub><sup>L</sup> ) ÷ ( ''H''<sub>a</sub><sup>V</sup> − ''H''<sub>a</sub><sup>L</sup> )'''
:'''''X'' = 100 ( ''H''<sub>s</sub><sup>L</sup> − ''H''<sub>a</sub><sup>L</sup> ) ÷ ( ''H''<sub>a</sub><sup>V</sup> − ''H''<sub>a</sub><sup>L</sup> )'''
Line 349: Line 348:
|-
|-
!align=right| ''H''
!align=right| ''H''
|align=left|= source liquid [[heat of vaporization]] at atmospheric boiling point, J/kg
|align=left|= source liquid [[heat of vaporization]] at atmospheric boiling point, J/kg  
|}
|}


==References==
==References==
{{reflist}}
{{reflist}}

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Fire and explosion damage in BP refinery (Texas) caused by accidental release of hydrocarbon vapors.

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, oil and gas transportation pipelines, chemical plants, and many other industrial activities. 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 calculational 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, atmospheric dispersion modeling studies can then be performed.

Accidental release of 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 is the specific heat ratio of the discharged gas (sometimes called the isentropic expansion factor and sometimes denoted as ).

For many gases, 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:[1][2][3][4]

where the terms are defined as in listed below. If the upstream gas density, is not known directly, then it is useful to eliminate it using the Ideal gas law corrected for the real gas compressiblity:



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:  
= mass flow rate, kg/s
= discharge coefficient, dimensionless (usually about 0.72)
= discharge hole area, m2
= = specific heat ratio of the gas
= specific heat capacity of the gas at constant pressure
= specific heat capacity of the gas at constant volume
= real gas upstream density, kg/m3 =
= absolute upstream pressure, Pa
= absolute downstream, Pa
= the gas molecular mass, kg/kmole    (also known as the molecular weight)
= the universal gas law constant = 8314.5 Pa·m3/(kmole·K)
= absolute upstream gas temperature, K
= the gas compressibility factor at and , 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.[5]

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 using which only applies to a specific individual gas. The relationship between the two constants is = .

Notes:

  • The above equations are for a real gas.
  • For an ideal gas, Z = 1 and ρ is the ideal gas density.
  • kmole = 1000 moles

Ramskill's equation for non-choked mass flow

P.K. Ramskill's equation [6][7] for the non-choked flow of a gas is somewhat different than the above non-choked flow equation but both versions provide identical results:

(1)      
where:  
= mass flow rate, kg/s
= discharge coefficient, dimensionless (usually about 0.72)
= discharge hole area, m2
= = specific heat ratio of the gas
= specific heat capacity of the gas at constant pressure
= specific heat capacity of the gas at constant volume
= real gas upstream density, kg/m3 =
= real gas downstream density, kg/m3 =
= absolute upstream pressure, Pa
= absolute downstream, Pa
= the gas molecular mass, kg/kmole    (also known as the molecular weight)
= the universal gas law constant = 8314.5 Pa·m3/(kmole·K)
= absolute upstream gas temperature, K
= absolute downstream gas temperature, K
= the gas compressibility factor at the pertinent and conditions, dimensionless

Since the downstream temperature is unknown and is required to calculate the downstream density , the isentropic expansion equation below [8] is used to obtain in terms of the known upstream temperature  :

(2)      

Combining equation (2) with the definition of (in the parameter list just above) provides in terms of the known upstream temperature  :

(3)      

Using equation (3) with Ramskill's equation (1) to determine non-choked mass flow rates for gases gives results identical to those obtained using the non-choked flow equation presented in the previous section above.

Evaporation of 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. 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. [2]

E = ( 4.161 x 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
TP = pool liquid temperature, °C
M = pool liquid molecular mass, dimensionless
PS = pool liquid vapor pressure at ambient temperature, mmHg
PH = hydrazine vapor pressure at ambient temperature, mmHg (see equation below)
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]

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.

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 United States Environmental Protection Agency using units which were a mixture of metric usage and United States usage.[3] The non-metric units have been converted to metric units for this presentation.

E = ( 0.1268 ÷ T ) u 0.78' M 0.667 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)

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]

E = k P M ÷ (R TA)
where:  
E = evaporation flux, (kg/s)/m² of pool surface
k = mass transfer coefficient, m/s = 0.002 u
TA = absolute ambient temperature, K
M = pool liquid molecular mass, dimensionless
P = pool liquid vapor pressure at ambient temperature, Pa
R = the universal gas law constant = 8314.5 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). [2]

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

where:  
E = evaporation flux, (kg/min)/m² of pool surface
B = pool liquid atmospheric boiling point, °C
M = pool liquid molecular mass, dimensionless
e = the base of the natural logarithm = 2.7183

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 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 ( HsLHaL ) ÷ ( HaVHaL )
where:  
X = weight percent vaporized
HsL = source liquid enthalpy at source temperature and pressure, J/kg
HaV = flashed vapor enthalpy at atmospheric boiling point and pressure, J/kg
HaL = 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 ( TsTb ) ] ÷ H
where:  
X = weight percent vaporized
cp = source liquid specific heat, J/(kg·K)
Ts = source liquid absolute temperature, K
Tb = source liquid absolute atmospheric boiling point, K
H = source liquid heat of vaporization at atmospheric boiling point, J/kg

References

  1. Perry, R.H. and Green, D.W. (1984). Perry's Chemical Engineers' Handbook, 6th Edition. McGraw Hill, Inc.. ISBN ISBN 0-07-049479-7. 
  2. 2.0 2.1 2.2 Handbook of Chemical Hazard Analysis Procedures, Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Handbook of Chemical Hazard Analysis, Appendix B. Scroll down to page 391 of 520 PDF pages. This handbook also provides the references below:
    • Clewell, H.J., A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids, Energy Systems Laboratory, ESL-TR-83-03, 1983.
    • Ille, G. and Springer, C., The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spill, Environmental Engineering Development Office, CEEDO 712-78-30, 1978.
    • Kahler, J.P., Curry, R.C. and Kandler, R.A.,Calculating Toxic Corridors Air Force Weather Service, AWS TR-80/003, 1980.
  3. 3.0 3.1 "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999.  Guidance for Offsite Consequence Analysis (Appendix D: Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)
  4. "Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. PGS2 CPR 14E
  5. Calculating Accidental Release Rates From Pressurized Gas Systems
  6. CACHE Newsletter No.48, Spring 1999 Gierer, C. and Hyatt, N.,Using Source Term Analysis Software for Calculating Fluid Flow Release Rates Dyadem International Ltd.
  7. Ramskill, P.K. (1986), Discharge Rate Calculation Methods for Use In Plant Safety Assessments, Safety and Reliability Directory, United Kingdom Atomic Energy Authority
  8. Isentropic Compression or Expansion
  9. Stiver, W. and Mackay, D., A Spill Hazard Ranking System For Chemicals, Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993.