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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 facilites such as [[Oil refinery |petroleum refineries]], [[petrochemical]] plants, [[natural gas]] processing plants, oil and gas  transportation [[Pipeline transport|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.
{{AccountNotLive}}
 
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.<br> <br>   
 
==Accidental release of pressurized gas==
When gas stored under [[pressure]] in a closed vessel is discharged to the [[Earth's atmosphere|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.
 
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 '''[(''k'' + 1) ÷ 2 ]<sup> ''k''÷(''k'' - 1 )</sup>''', where ''k'' is the [[heat capacity ratio|specific heat ratio]] of the discharged gas (sometimes called the [[adiabatic index|isentropic expansion factor]] and sometimes denoted as <math>\gamma</math>).
 
For many gases, ''k'' ranges from about 1.09 to about 1.41, and therefore '''[(''k'' + 1) ÷ 2 ]<sup> ''k''÷(''k'' - 1 )</sup>''' 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.
 
When the gas velocity is choked, the equation for the [[mass flow rate]] in SI metric units is: {{ref|Perrys}} {{ref|Hazards}} {{ref|Risk}} {{ref|NetherlandsCPR}}<br><br>
 
<math>Q\;=\;C\;A\;\sqrt{\;k\;\rho\;P\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}</math>
<br><br>
 
or this equivalent form:<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>
 
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.<br><br>
 
Whenever the ratio of the absolute source pressure to the absolute downstream ambient pressure is less than
'''[ ( ''k'' + 1 ) ÷ 2 ]<sup> ''k'' ÷ ( ''k'' - 1 )</sup>''', then the gas velocity is non-choked (i.e., sub-sonic) 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><br><br>
 
or this equivalent form:<br><br>
 
<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><br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''Q''
|align=left|= [[mass flow rate]], kg/s
|-
!align=right| ''C''
|align=left|= discharge coefficient, dimensionless (usually about 0.72)
|-
!align=right| ''A''
|align=left|= discharge hole area, m²
|-
!align=right| ''k''
|align=left|= c<sub>p</sub>/c<sub>v</sub> of the gas
|-
!align=right| ''c<sub>p</sub>''
|align=left|= [[specific heat capacity|specific heat]] of the gas at constant pressure
|-
!align=right| ''c<sub>v</sub>''
|align=left|= specific heat of the gas at constant volume
|-
!align=right| ''<math>\rho</math>''
|align=left|= real gas density at ''P'' and ''T'', kg/m³
|-
!align=right| ''P''
|align=left|= absolute upstream pressure, Pa
|-
!align=right| ''P<sub>A</sub>''
|align=left|= absolute ambient or downstream pressure, Pa
|-
!align=right| ''M''
|align=left|= the gas [[molecular mass]], kg/kmol&nbsp; &nbsp; (also known as the molecular weight)
|-
!align=right| ''R''
|align=left|= the [[Gas constant|Universal Gas Law Constant]]  =  8314.5 Pa·m³/(kmol·K)
|-
!align=right| ''T''
|align=left|= absolute gas temperature, K
|-
!align=right| ''Z''
|align=left|= the gas compressibility factor at ''P'' and ''T'', dimensionless
|}<br>
 
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. Two equivalent methods for performing such calculations are presented and compared at [http://www.air-dispersion.com/feature2.html www.air-dispersion.com/feature2.html].<br><br>
 
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''.
 
Other forms of the above equations have been published by Ramskill.{{ref|Ramskill}}  When used properly, Ramskill's equations yield results which are identical to the results yielded by the above equations.
 
Notes:<br>
* The above equations are for a real gas.<br>
* For an ideal gas, ''Z'' = 1 and ''ρ'' is the ideal gas density.<br>
* kmol = [[mole (unit)|kilomole]], or kilogram mole.
 
==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. {{ref|Hazards2}}<br><br>
 
'''E = ( 4.161 x 10<sup>-5</sup> ) u<sup>0.75</sup> T<sub>F</sub> M ( P<sub>S</sub> ÷ P<sub>H</sub> )'''<br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''E''
|align=left|= evaporation flux, (kg/min)/m²  of pool surface
|-
!align=right| ''u''
|align=left|= windspeed just above the liquid surface, m/s
|-
!align=right| ''T<sub>A</sub>''
|align=left|= absolute ambient temperature, K
|-
!align=right| ''T<sub>F</sub>''
|align=left|= pool liquid temperature correction factor, dimensionless
|-
!align=right| ''T<sub>P</sub>''
|align=left|= pool liquid temperature, °C
|-
!align=right| ''M''
|align=left|= pool liquid molecular weight, dimensionless
|-
!align=right| ''P<sub>S</sub>''
|align=left|= pool liquid vapor pressure at ambient temperature, [[mmHg]]
|-
!align=right| ''P<sub>H</sub>''
|align=left|= hydrazine vapor pressure at ambient temperature, mmHg (see equation below)
|}<br>
'''If  T<sub>P</sub> = 0 °C or less, then ''T''<sub>F</sub> = 1.0'''<br>
'''If  T<sub>P</sub> > 0 °C, then ''T''<sub>F</sub> = 1.0 + 0.0043 ''T''<sub>P</sub><sup>2</sup>'''<br><br>
 
'''''P''<sub>H</sub> = 760 exp[ 65.3319 − (7245.2 ÷ ''T''<sub>A</sub> ) − (8.22 ln ''T''<sub>A</sub>) + ( 6.1557 x 10<sup>-3</sup>) ''T''<sub>A</sub>]'''<br><br>
 
Note:<br>
* The function "ln ''x''" is the [[natural logarithm]] (base e) of ''x'', and the function "exp ''x''" is the value of the [[exponential function]] 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]] {{ref|Risk2}}  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.<br><br>
 
: '''E = ( 10.40 ÷ ''RT'' ) ''u''<sup> 0.78</sup>' ''M''<sup> 0.667</sup> ''A'' ''P'''''<br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''E''
|align=left|= evaporation rate, kg/min
|-
!align=right| ''u''
|align=left|= windspeed just above the pool liquid surface, m/s
|-
!align=right| ''M''
|align=left|= pool liquid molecular weight, dimensionless
|-
!align=right| ''A''
|align=left|= surface area of the pool liquid, m²
|-
!align=right| ''P''
|align=left|= vapor pressure of the pool liquid at the pool temperature, kPa
|-
!align=right| ''T''
|align=left|= pool liquid absolute temperature, K
|-
!align=right| ''R''
|align=left|= the [[universal gas law]] constant  =  82.05 atm·cm³/(mol·K )
|}<br>
 
 
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:<br><br>
 
:''A'' = ( cubic meters of pool liquid ) ÷ (0.01 m)<br><br>
 
Notes:<br>
* 1 kPa = 0.0102 [kgf]]/cm² = 0.01 bar<br>
* mol  = [[mole (unit)|mole]]<br>
* atm  = atmosphere
 
===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. {{ref|Stiver}}<br><br>
 
:'''''E'' = ''k'' ''P'' ''M'' ÷ (''R'' ''T''<sub>A</sub>)'''<br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''E''
|align=left|= evaporation flux, (kg/s)/m² of pool surface
|-
!align=right| ''k''
|align=left|= mass transfer coefficient, m/s = 0.002 '''''u'''''
|-
!align=right| ''T<sub>A</sub>''
|align=left|= absolute ambient temperature, K
|-
!align=right| ''M''
|align=left|= pool liquid molecular weight, dimensionless
|-
!align=right| ''P''
|align=left|= pool liquid vapor pressure at ambient temperature, Pa
|-
!align=right| ''R''
|align=left|= the universal gas law constant =  8314.5 Pa·m³/(kmol·K)
|-
!align=right| ''u''
|align=left|= windspeed just above the liquid surface, m/s
|}<br>
 
==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). {{ref|Hazards3}}<br><br>
:'''''E'' = ( 0.0001 ) ( 7.7026 − 0.0288 ''B'' ) ( ''M'' ) e<sup>-0.0077''B'' - 0.1376</sup>''' <br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''E''
|align=left|= evaporation flux, (kg/min)/m² of pool surface
|-
!align=right| ''B''
|align=left|= pool liquid atmospheric boiling point, °C
|-
!align=right| ''M''
|align=left|= pool liquid molecular weight, dimensionless
|-
!align=right| ''e''
|align=left|= the base of the natural logarithm system = 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 [[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.<br><br>
 
:'''''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> )'''<br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''X''
|align=left|= weight percent vaporized
|-
!align=right| ''H<sub>s</sub><sup>L</sup>''
|align=left|= source liquid enthalpy at source temperature and pressure, J/kg<br>
|-
!align=right| ''H<sub>a</sub><sup>V</sup>''
|align=left|= flashed vapor enthalpy at atmospheric boiling point and pressure, J/kg
|-
!align=right| ''H<sub>a</sub><sup>L</sup>''
|align=left|= residual liquid enthalpy at atmospheric boiling point and pressure, J/kg
|}<br>
 
If the enthalpy data required for the above equation is unavailable, then the following equation may be used.<br><br>
:'''''X'' = 100 [ ''c''<sub>p</sub> ( ''T''<sub>s</sub> − ''T''<sub>b</sub> ) ] ÷ ''H'''''<br><br>
 
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right| ''X''
|align=left|= weight percent vaporized
|-
!align=right| ''c<sub>p</sub>''
|align=left|= source liquid specific heat, J/(kg&nbsp;°C)
|-
!align=right| ''T<sub>s</sub>''
|align=left|= source liquid absolute temperature, K
|-
!align=right| ''T<sub>b</sub>''
|align=left|= source liquid absolute atmospheric boiling point, K
|-
!align=right| ''H''
|align=left|= source liquid heat of vaporization at atmospheric boiling point, J/kg
|}
 
==References==
#{{note|Perrys}}''Perry's Chemical Engineers' Handbook'', Sixth Edition, McGraw-Hill Co., 1984.
#{{note|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://hazmat.dot.gov/riskmgmt/tools/archie.pdf Handbook of Chemical Hazard Procedures] Also provides the references below:<br>&#149;&nbsp;&nbsp;Clewell, H.J., "A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids", Energy Systems Laboratory, ESL-TR-83-03, 1983.<br>&#149;&nbsp;&nbsp;Ille, G. and Springer, C., "The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spills", Environmental Engineering Development Office, CEEDO 712-78-30, 1978.<br>&#149;&nbsp;&nbsp;Kahler, J.P., Curry, R.C. and Kandler, R.A., "Calculating Toxic Corridors", Air Force Weather Service, AWS TR-80/003, 1980.
#{{note|Risk}}"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. &nbsp;[http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/oca-all.pdf/$file/oca-all.pdf?OpenElement Guidance for Offsite Consequence Analysis]
#{{note|NetherlandsCPR}}"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=22297 PGS2 CPR 14E]
#{{note|Ramskill}} Ramskill, P.K., "Discharge Rate Calculation Methods for Use In Plant Safety Assessments", Safety and Reliability Directory, 1986, United Kingdom Atomic Energy Authority
#{{note|Hazards2}}''Handbook of Chemical Hazard Analysis Procedures'', Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989.
#{{note|Risk2}}"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999.
#{{note|Stiver}}Stiver, W. and Mackay, D., "A Spill Hazard Ranking System For Chemicals", Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993.
#{{note|Hazards3}}''Handbook of Chemical Hazard Analysis Procedures'', Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989.
 
==External links==
*[http://www.che.utexas.edu/cache/newsletters/Spr_99.pdf Ramskill's equations] are presented and cited this pdf file (use search function to find "Ramskill").
*More release source terms are available in the feature articles at [http://www.air-dispersion.com www.air-dispersion.com]
*[http://www.okcc.com/PDF/Choked%20Flow%20of%20Gases%20pg.48.pdf Choked flow of gases]
*[http://www.qub.ac.uk/qc/webpages/whatwedo/researchgroups/environmentalmodelling/ia/documents/chapter5.pdf Development of source emission models]

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