Accidental release source terms/Citable Version: Difference between revisions
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[[Image:BP Refinery Fire.jpg|right|thumb|225px|{{#ifexist:Template:BP Refinery Fire.jpg/credit|{{BP Refinery Fire.jpg/credit}}<br/>|}}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 [[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. | |||
==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. | |||
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: | |||
(1) <math>\big[(k+1)/2 \big]^{k/(k-1}</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]]. | |||
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. | |||
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.<br></ref><ref name=Risk>"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. [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> | |||
or this equivalent form: | |||
<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. | |||
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> | |||
<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> | |||
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> | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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|= [[ideal gas|real gas]] [[Density (chemistry)|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/kmole (also known as the molecular weight) | |||
|- | |||
!align=right| ''R'' | |||
|align=left|= the [[Molar gas constant|universal gas law constant]] = 8314.5 Pa·m³/(kmole·K) | |||
|- | |||
!align=right| ''T'' | |||
|align=left|= absolute upstream gas [[temperature]], K | |||
|- | |||
!align=right| ''Z'' | |||
|align=left|= the gas [[compressibility factor]] at ''P'' and ''T'', 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 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''. | |||
Notes:<br> | |||
* The above equations are for a real gas. | |||
* For an ideal gas, '''''Z''''' = 1 and '''''ρ''''' is the ideal gas density. | |||
* kmole = 1000 [[mole (unit)|moles]] | |||
===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> | |||
(1) <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> | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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| <math>\rho_A</math> | |||
|align=left|= ideal gas density at the downstream pressure and temperature | |||
|- | |||
!align=right| <math>\rho</math> | |||
|align=left|= ideal gas density at the upstream pressure and temperature | |||
|- | |||
!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 of the gas at constant pressure | |||
|- | |||
!align=right| ''c<sub>v</sub>'' | |||
|align=left|= specific heat of the gas at constant volume | |||
|- | |||
!align=right| ''P'' | |||
|align=left|= absolute upstream pressure, Pa | |||
|- | |||
!align=right| ''P<sub>A</sub>'' | |||
|align=left|= absolute downstream pressure, Pa | |||
|- | |||
!align=right| ''T'' | |||
|align=left|= upstream absolute temperature, K | |||
|- | |||
!align=right| ''T<sub>A</sub>'' | |||
|align=left|= downstream absolute temperature, K | |||
|- | |||
!align=right| ''M'' | |||
|align=left|= the gas molecular mass, kg/kmole | |||
|- | |||
!align=right| ''R'' | |||
|align=left|= the universal gas law constant = 8314.5 Pa·m³/(kmole·K) | |||
|} | |||
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]]: | |||
(2) <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) <math>\frac{T_A}{T} = \bigg(\frac{P_A}{P}\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>''''': | |||
(4) <math>\rho_A = \frac{M \;P^{\;(k-1)/k}}{R \;T \;P_A^{\ -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. | |||
==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 name=Hazards/> | |||
:'''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> )''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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 mass, 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) | |||
|} | |||
:'''If T<sub>P</sub> = 0 °C or less, then ''T''<sub>F</sub> = 1.0''' | |||
:'''If T<sub>P</sub> > 0 °C, then ''T''<sub>F</sub> = 1.0 + 0.0043 ''T''<sub>P</sub><sup>2</sup>''' | |||
:'''''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>]''' | |||
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.<ref name=Risk/> The non-metric units have been converted to metric units for this presentation. | |||
:'''E = ( 0.1268 ÷ ''T'' ) ''u''<sup> 0.78</sup>' ''M''<sup> 0.667</sup> ''A'' ''P''''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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 mass, 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 | |||
|} | |||
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. <ref name=Stiver>Stiver, W. and Mackay, D., ''A Spill Hazard Ranking System For Chemicals'', Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993.</ref> | |||
:'''''E'' = ''k'' ''P'' ''M'' ÷ (''R'' ''T''<sub>A</sub>)''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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 mass, 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 | |||
|} | |||
==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 name=Hazards/> | |||
'''''E'' = ( 0.0001 ) ( 7.7026 − 0.0288 ''B'' ) ( ''M'' ) e<sup>-0.0077''B'' - 0.1376</sup>''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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 mass, dimensionless | |||
|- | |||
!align=right| ''e'' | |||
|align=left|= 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 [[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> )''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!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 | |||
|} | |||
If the enthalpy data required for the above equation is unavailable, then the following equation may be used: | |||
:'''''X'' = 100 [ ''c''<sub>p</sub> ( ''T''<sub>s</sub> − ''T''<sub>b</sub> ) ] ÷ ''H''''' | |||
{| border="0" cellpadding="2" | |||
|- | |||
|align=right|where: | |||
| | |||
|- | |||
!align=right| ''X'' | |||
|align=left|= weight percent vaporized | |||
|- | |||
!align=right| ''c<sub>p</sub>'' | |||
|align=left|= source liquid specific heat, J/(kg·K) | |||
|- | |||
!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== | |||
{{reflist}} |
Revision as of 19:16, 27 October 2008
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 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 (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:
(1)
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 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:[1][2][3][4]
or this equivalent form:
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.
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:
or this equivalent form:
where: | |
Q | = mass flow rate, kg/s |
---|---|
C | = discharge coefficient, dimensionless (usually about 0.72) |
A | = discharge hole area, m² |
k | = cp/cv of the gas |
cp | = specific heat of the gas at constant pressure |
cv | = specific heat of the gas at constant volume |
= real gas density at P and T, kg/m³ | |
P | = absolute upstream pressure, Pa |
PA | = absolute ambient or downstream pressure, Pa |
M | = the gas molecular mass, kg/kmole (also known as the molecular weight) |
R | = the universal gas law constant = 8314.5 Pa·m³/(kmole·K) |
T | = absolute upstream gas temperature, K |
Z | = 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. A comparison between two methods for performing such calculations is available online.[5]
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/M.
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 an ideal gas is:
(1)
where: | |
Q | = mass flow rate, kg/s |
---|---|
C | = discharge coefficient, dimensionless (usually about 0.72) |
A | = discharge hole area, m² |
= ideal gas density at the downstream pressure and temperature | |
= ideal gas density at the upstream pressure and temperature | |
k | = cp/cv of the gas |
cp | = specific heat of the gas at constant pressure |
cv | = specific heat of the gas at constant volume |
P | = absolute upstream pressure, Pa |
PA | = absolute downstream pressure, Pa |
T | = upstream absolute temperature, K |
TA | = downstream absolute temperature, K |
M | = the gas molecular mass, kg/kmole |
R | = the universal gas law constant = 8314.5 Pa·m³/(kmole·K) |
Calculation of the downstream ideal gas density requires that the downstream temperature be obtained. Starting with the ideal gas law:
(2)
Since the downstream temperature is not known, the isentropic expansion equation below [8] is used to determine in terms of the known upstream temperature :
(3)
Combining the equations (2) and (3) defines in terms of the known upstream temperature :
(4)
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.
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 ( HsL − HaL ) ÷ ( HaV − HaL )
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 ( Ts − Tb ) ] ÷ 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
- ↑ Perry, R.H. and Green, D.W. (1984). Perry's Chemical Engineers' Handbook, 6th Edition. McGraw Hill, Inc.. ISBN ISBN 0-07-049479-7.
- ↑ 2.0 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.
- ↑ 3.0 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)
- ↑ "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
- ↑ Calculating Accidental Release Rates From Pressurized Gas Systems
- ↑ 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.
- ↑ Ramskill, P.K. (1986), Discharge Rate Calculation Methods for Use In Plant Safety Assessments, Safety and Reliability Directory, United Kingdom Atomic Energy Authority
- ↑ Isentropic Compression or Expansion
- ↑ Stiver, W. and Mackay, D., A Spill Hazard Ranking System For Chemicals, Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993.
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