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The '''choked flow''' (often referred to as '''critical flow''' ) of a flowing [[gas]] is a limiting point which occurs under specific conditions when the gas at a certain [[pressure]] and [[temperature]] flows through a restriction (such as a [[valve]], a convergent-divergent nozzle, the hole in an [[orifice plate]], or a leak in a gas pipeline or other gas container) into a lower pressure environment.   
The '''choked flow''' (often referred to as '''critical flow''') of a flowing [[gas]] is a limiting point which occurs under specific conditions when a gas at a certain [[pressure]] and [[temperature]] flows through a restriction<ref>A [[valve]], a [[convergent-divergent nozzle]] such as a [[de Laval nozzle]], an [[orifice plate]] hole, a leak in a gas pipeline or other gas container, a [[rocket engine]] exhaust nozzle, etc.</ref> into a lower pressure environment.   


As the gas flows through the smaller cross-sectional area of the restriction, its [[velocity]] must increase. The limiting point is reached when the gas velocity increases to the [[speed of sound]] in the gas. At that point, the gas velocity becomes independent of the downstream pressure, meaning that the gas velocity can not be increased any further by further lowering of the downstream pressure. The physical point at which the choking occurs (i.e., the cross-sectional area of the restriction) is sometimes called the ''choke plane''. It is important to note that although the gas velocity becomes choked, the [[mass flow]] of the gas can still be increased by increasing the upstream pressure or by decreasing the upstream temperature.
As the gas flows through the smaller cross-sectional area of the restriction, its linear [[velocity]] must increase. The limiting point is reached when the linear gas velocity increases to the [[speed of sound]] ([[sonic velocity]]) in the gas. At that point, the [[mass]] flow rate (mass per unit of time) of the gas becomes independent of the downstream pressure, meaning that the mass flow rate can not be increased any further by further lowering of the downstream pressure. The physical point at which the choking occurs (i.e., the cross-sectional area of the restriction) is sometimes called the ''choke plane''. It is important to note that although the gas velocity becomes choked, the mass flow rate of the gas can still be increased by increasing the upstream pressure or by decreasing the upstream temperature.


The choked flow of gases is useful in many engineering applications because, under choked conditions, valves and calibrated orifice plates can be used to produce a particular mass flow rate.
The choked flow of gases is useful in many engineering applications because, under choked conditions, valves and calibrated orifice plates can be used to produce a particular mass flow rate. Choked flow in a [[de Laval nozzle]] as used in a [[rocket engine]] can be accelerated to [[supersonic]] linear velocities. 


In the case of liquids, a different type of limiting condition (also known as choked flow) occurs when the [[Venturi effect]] acting on the liquid flow through the restriction decreases the liquid pressure to below that of the liquid [[vapor pressure]] at the prevailing liquid temperature.  At that point, the liquid will partially "flash" into bubbles of vapor and the subsequent collapse of the bubbles causes [[cavitation]]. Cavitation is quite noisy and can be sufficiently violent to physically damage valves, pipes and associated equipment. In effect, the vapor bubble formation in the restriction limits the flow from increasing any further.<ref>[http://www.fisherregulators.com/technical/sizingcalculations/ Scroll to discussion of liquid flashing and cavitation]</ref><ref>[http://www.documentation.emersonprocess.com/groups/public/documents/book/cvh99.pdf Search document for "Choked"]</ref>
In the case of liquids, a different type of limiting condition (also known as choked flow) occurs when the Venturi effect acting on the liquid flow through the restriction decreases the liquid pressure to below that of the liquid [[vapor pressure]] at the prevailing liquid temperature.  At that point, the liquid will partially "flash" into bubbles of vapor and the subsequent collapse of the bubbles causes [[cavitation]]. Cavitation is quite noisy and can be sufficiently violent to physically damage valves, pipes and associated equipment. In effect, the vapor bubble formation in the restriction limits the flow from increasing any further.<ref>[http://www.fisherregulators.com/technical/sizingcalculations/ Scroll to discussion of liquid flashing and cavitation]</ref><ref>[http://www.documentation.emersonprocess.com/groups/public/documents/book/cvh99.pdf Search document for "Choked"]</ref>


==Conditions under which gas flow becomes choked==
==Conditions under which gas flow becomes choked==


All gases flow from upstream higher pressure sources to downstream lower pressure environments. Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure, <math>P_d/P_u</math> is equal to or greater than  
All gases flow from upstream higher pressure sources to downstream lower pressure environments. Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than:


:<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 [[heat capacity ratio]] of the 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, k ranges from about 1.09 to about 1.41, and therefore
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.


:<math>\big[(k+1)/2 \big]^{k/(k-1}</math>
==Mass flow rate equations for the choked flow of gases==
 
ranges from 1.7 to about 1.9 ... which means that choked flow usually occurs when the absolute upstream pressure, <math>P_u</math>, is at least 1.7 to 1.9 times as high as the absolute downstream pressure, <math>P_d</math>.
 
==Mass flow equations for the choked flow of gases==


When the gas velocity is choked, the equation for the mass flow rate in SI metric units is: <ref name=Perry>''[[Perry's Chemical Engineers' Handbook]]'', Sixth Edition, McGraw-Hill Co., 1984.</ref><ref>''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]</ref><ref>"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/$file/oca-all.pdf?OpenElement Guidance for Offsite Consequence Analysis]</ref><ref>"Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. [http://vrom.nl/pagina.html?id=20725 PGS2 CPR 14E] (Scroll down to PGS2, download the PDF and then scroll to PDF pages 84-85)</ref>
When the gas velocity is choked, the equation for the mass flow rate in SI metric units is: <ref name=Perry>''[[Perry's Chemical Engineers' Handbook]]'', Sixth Edition, McGraw-Hill Co., 1984.</ref><ref>''Handbook of Chemical Hazard Analysis Procedures'', Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. [http://hazmat.dot.gov/riskmgmt/tools/archie.pdf Handbook of Chemical Hazard Procedures]</ref><ref>"Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. &nbsp;[http://yosemite.epa.gov/oswer/ceppoweb.nsf/vwResourcesByFilename/oca-all.pdf/$file/oca-all.pdf?OpenElement Guidance for Offsite Consequence Analysis]</ref><ref>"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] (Scroll down to PGS2, download the PDF and then scroll to PDF pages 84-85)</ref>
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:<math>\dot m\;=\;C\;A\;\sqrt{\;k\;\rho_u\;P_u\;\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>


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|compressiblity]]:<br><br>
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|compressibility]]:<br><br>


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


so that the mass flow rate is primarily dependent on the cross-sectional area <font style="vertical-align:+10%;"><math>A</math></font> of the hole and the upstream pressure <math>P_u</math>, and only weakly dependent on the temperature <font style="vertical-align:+10%;"><math>T</math></font>.  The rate does not depend on the downstream pressure at all.  All other terms are constants that depend only on the composition of the gas.
so that the mass flow rate is primarily dependent on the cross-sectional area <font style="vertical-align:+10%;"><math>A</math></font> of the hole and the upstream pressure <math>P_u</math>, and only weakly dependent on the temperature <font style="vertical-align:+10%;"><math>T</math></font>.  


Definition of terms:
{| border="0" cellpadding="2"
:{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
|-
!align=right| <math>\dot m</math>
!align=right| <math>\dot m</math>
|align=left|= mass flow rate, kg/s
|align=left|= mass flow rate, [[Kilogram|kg]]/[[second|s]]
|-
|-
!align=right| <math>C</math>
!align=right| <math>C</math>
Line 45: Line 43:
|-
|-
!align=right| <math>A</math>  
!align=right| <math>A</math>  
|align=left|= restriction cross-sectional area, m<sup>2</sup>
|align=left|= discharge hole area, [[metre|m]]<sup>2</sup>
|-
|-
!align=right| <math>k</math>
!align=right| <math>k</math>
|align=left|= <math>c_p/c_v</math> = heat capacity ratio of the gas
|align=left|= <math>c_p/c_v</math> = [[specific heat ratio]] of the gas
|-
|-
!align=right| <math>c_p</math>  
!align=right| <math>c_p</math>  
|align=left|= [[specific heat capacity]] of the gas at constant pressure
|align=left|= [[specific heat]] of the gas at constant pressure
|-
|-
!align=right| <math>c_v</math>  
!align=right| <math>c_v</math>  
|align=left|= specific heat capacity of the gas at constant volume
|align=left|= specific heat of the gas at constant volume
|-
|-
!align=right| <math>\rho_u</math>  
!align=right| <math>\rho_u</math>  
|align=left|= real gas density at <math>P_u</math> and <font style="vertical-align:+10%;"><math>T</math></font>, kg/m<sup>3</sup>
|align=left|= real gas upstream [[Density (chemistry)|density]], kg/m<sup>3</sup> = <math>(M P_u)/(Z\, R\, T_u)</math>  
|-
|-
!align=right| <math>P_u</math>  
!align=right| <math>P_u</math>  
Line 63: Line 61:
|-
|-
!align=right| <math>M</math>  
!align=right| <math>M</math>  
|align=left|= the gas [[molecular mass]], kg/kmol &nbsp; &nbsp;(also known as the molecular weight)
|align=left|= the gas [[molecular mass]], kg/k[[mol]]&nbsp; &nbsp; (also known as the molecular weight)
|-
|-
!align=right| <math>R</math>  
!align=right| <math>R</math>  
|align=left|= [[Molar gas constant|Universal gas law constant]] = 8314.5 (N·m) / (kmol·K)
|align=left|= the [[Molar gas constant|universal gas law constant]] = 8314.5 Pa·m<sup>3</sup>/(kmole·[[kelvin|K]])
|-
|-
!align=right| <math>T</math>  
!align=right| <math>T_u</math>  
|align=left|= absolute gas temperature, [[Kelvin (unit)|K]]
|align=left|= absolute upstream gas [[temperature]], K
|-
|-
!align=right| <math>Z</math>  
!align=right| <math>Z</math>  
|align=left|= the gas [[compressibility factor]] at <math>P_u</math> and <font style="vertical-align:+10%;"><math>T</math></font>, dimensionless
|align=left|= the gas [[compressibility factor]] at <math>P_u</math> and <math>T_u</math>, dimensionless
|}
|}


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 <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>.
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>.


If the gas flow restriction is a leak hole in a closed [[pressure vessel]] or a gas pipeline, then the above equations calculate the '''initial instantaneous ''' mass flow rate for the pressure and temperature existing in the source vessel when the leak 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 upstream pressure and the flow rate decrease with time as the vessel or pipeline 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>
If the gas flow restriction is a leak hole in a closed [[pressure vessel]] or a gas pipeline, then the above equations calculate the '''initial instantaneous ''' mass flow rate for the pressure and temperature existing in the source vessel when the leak 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 upstream pressure and the flow rate decrease with time as the vessel or pipeline 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>
Line 81: Line 79:
Notes:
Notes:
* The above equations are for a real gas.
* The above equations are for a real gas.
* For a monatomic ideal gas, <font style="vertical-align:+10%;"><math>Z</math></font> = 1 and <font style="vertical-align:-15%;"><math>\rho_u</math></font> is the ideal gas density.
* For an ideal gas, <font style="vertical-align:+10%;"><math>Z</math></font> = 1 and <font style="vertical-align:-15%;"><math>\rho_u</math></font> is the ideal gas density.
* kmole = 1000 moles


==Minimum pressure ratio required for choked flow to occur==
==Minimum pressure ratio required for choked flow to occur==
Line 108: Line 107:
|}
|}


The above tabulated minimum pressure ratios were calculated by using the criterion that choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure, <math>P_d/P_u</math> is equal to or greater than <math>\big[(k+1)/2 \big]^{k/(k-1}</math>.
The above tabulated minimum pressure ratios were calculated by using the criterion that choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure, <math>P_u/P_d</math> is equal to or greater than <math>\big[(k+1)/2 \big]^{\,k/(k-1}</math>.


==Thin-plate orifices==
==Thin-plate orifices==
Line 116: Line 115:
Cunningham’s work included tests using air and steam with the results and conclusions presented as tables, charts and formulas. Limited information was provided for the tests with steam.<ref name=DKB/>
Cunningham’s work included tests using air and steam with the results and conclusions presented as tables, charts and formulas. Limited information was provided for the tests with steam.<ref name=DKB/>


Critical flow can, however, be expected for thick orifice plates (i.e. plates with a thickness of at least 6 times the orifice hole diameter).<ref name=DKB/>
Critical flow can, however, be expected for thick orifice plates (i.e., plates with a thickness of at least 6 times the orifice hole diameter).<ref name=DKB/>
 
==Choked flow of liquids==
 
In the case of liquids, a different type of limiting condition (also known as choked flow) occurs when the Venturi effect acting on the liquid flow through the restriction decreases the liquid pressure to below that of the liquid [[vapor pressure]] at the prevailing liquid temperature.  At that point, the liquid will partially "flash" into bubbles of vapor and the subsequent collapse of the bubbles causes [[cavitation]]. Cavitation is quite noisy and can be sufficiently violent to physically damage valves, pipes and associated equipment. In effect, the vapor bubble formation in the restriction limits the flow from increasing any further.<ref>[http://www.documentation.emersonprocess.com/groups/public/documents/brochures/d351912x012.pdf Read page 2 of this brochure.][</ref><ref>[http://www.documentation.emersonprocess.com/groups/public/documents/book/cvh99.pdf Search this pdf document for "Choked"]</ref>


==References==
==References==
<references/>
<references/>[[Category:Suggestion Bot Tag]]

Latest revision as of 06:01, 28 July 2024

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The choked flow (often referred to as critical flow) of a flowing gas is a limiting point which occurs under specific conditions when a gas at a certain pressure and temperature flows through a restriction[1] into a lower pressure environment.

As the gas flows through the smaller cross-sectional area of the restriction, its linear velocity must increase. The limiting point is reached when the linear gas velocity increases to the speed of sound (sonic velocity) in the gas. At that point, the mass flow rate (mass per unit of time) of the gas becomes independent of the downstream pressure, meaning that the mass flow rate can not be increased any further by further lowering of the downstream pressure. The physical point at which the choking occurs (i.e., the cross-sectional area of the restriction) is sometimes called the choke plane. It is important to note that although the gas velocity becomes choked, the mass flow rate of the gas can still be increased by increasing the upstream pressure or by decreasing the upstream temperature.

The choked flow of gases is useful in many engineering applications because, under choked conditions, valves and calibrated orifice plates can be used to produce a particular mass flow rate. Choked flow in a de Laval nozzle as used in a rocket engine can be accelerated to supersonic linear velocities.

In the case of liquids, a different type of limiting condition (also known as choked flow) occurs when the Venturi effect acting on the liquid flow through the restriction decreases the liquid pressure to below that of the liquid vapor pressure at the prevailing liquid temperature. At that point, the liquid will partially "flash" into bubbles of vapor and the subsequent collapse of the bubbles causes cavitation. Cavitation is quite noisy and can be sufficiently violent to physically damage valves, pipes and associated equipment. In effect, the vapor bubble formation in the restriction limits the flow from increasing any further.[2][3]

Conditions under which gas flow becomes choked

All gases flow from upstream higher pressure sources to downstream lower pressure environments. 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Mass flow rate equations for the choked flow of gases

When the gas velocity is choked, the equation for the mass flow rate in SI metric units is: [4][5][6][7]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot m\;=\;C\;A\;\sqrt{\;k\;\rho_u\;P_u\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}}

where the terms are defined as in listed below. If the upstream gas density, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_u} is not known directly, then it is useful to eliminate it using the Ideal gas law corrected for the real gas compressibility:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot m\;=\;C\;A\;P_u\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T}\bigg)\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}}

so that the mass flow rate is primarily dependent on the cross-sectional area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of the hole and the upstream pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_u} , and only weakly dependent on the temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} .

where:  
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot m} = mass flow rate, kg/s
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} = discharge coefficient, dimensionless (usually about 0.72)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = discharge hole area, m2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_p/c_v} = specific heat ratio of the gas
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_p} = specific heat of the gas at constant pressure
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_v} = specific heat of the gas at constant volume
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_u} = real gas upstream density, kg/m3 = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M P_u)/(Z\, R\, T_u)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_u} = absolute upstream pressure, Pa
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} = the gas molecular mass, kg/kmol    (also known as the molecular weight)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} = the universal gas law constant = 8314.5 Pa·m3/(kmole·K)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_u} = absolute upstream gas temperature, K
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} = the gas compressibility factor at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_u} , dimensionless

The technical literature can be confusing because many authors do not explain whether they are using the universal gas law constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} which applies to any ideal gas or whether they are using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_s} which only applies to a specific individual gas. The relationship between the two constants is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_s} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/M} .

If the gas flow restriction is a leak hole in a closed pressure vessel or a gas pipeline, then the above equations calculate the initial instantaneous mass flow rate for the pressure and temperature existing in the source vessel when the leak 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 upstream pressure and the flow rate decrease with time as the vessel or pipeline 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.[8]

Notes:

  • The above equations are for a real gas.
  • For an ideal gas, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} = 1 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_u} is the ideal gas density.
  • kmole = 1000 moles

Minimum pressure ratio required for choked flow to occur

The minimum pressure ratios (also referred to as the critical pressure ratio) required for choked gas flow conditions to occur are presented in Table 1 for some industrial gases:

Table 1[4][9]
Gas  k = cp/cv  Minimum
Pu/Pd
required for
choked flow
Hydrogen 1.410 1.899
Methane 1.307 1.837
Propane 1.131 1.729
Butane 1.096 1.708
Ammonia 1.310 1.838
Chlorine 1.355 1.866
Sulfur dioxide 1.290 1.826
Carbon monoxide 1.404 1.895

The above tabulated minimum pressure ratios were calculated by using the criterion that choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_u/P_d} is equal to or greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big[(k+1)/2 \big]^{\,k/(k-1}} .

Thin-plate orifices

Experiments carried out by RG Cunningham (1951)[10][11] demonstrated that the assumption that choked flow could occur for gases flowing through thin, square-edged orifice plates is not correct. The flow continued to increase as the downstream pressure was reduced to as low as 0.2 of the upstream pressure.[12]

Cunningham’s work included tests using air and steam with the results and conclusions presented as tables, charts and formulas. Limited information was provided for the tests with steam.[12]

Critical flow can, however, be expected for thick orifice plates (i.e., plates with a thickness of at least 6 times the orifice hole diameter).[12]

Choked flow of liquids

In the case of liquids, a different type of limiting condition (also known as choked flow) occurs when the Venturi effect acting on the liquid flow through the restriction decreases the liquid pressure to below that of the liquid vapor pressure at the prevailing liquid temperature. At that point, the liquid will partially "flash" into bubbles of vapor and the subsequent collapse of the bubbles causes cavitation. Cavitation is quite noisy and can be sufficiently violent to physically damage valves, pipes and associated equipment. In effect, the vapor bubble formation in the restriction limits the flow from increasing any further.[13][14]

References

  1. ↑ A valve, a convergent-divergent nozzle such as a de Laval nozzle, an orifice plate hole, a leak in a gas pipeline or other gas container, a rocket engine exhaust nozzle, etc.
  2. ↑ Scroll to discussion of liquid flashing and cavitation
  3. ↑ Search document for "Choked"
  4. ↑ 4.0 4.1 Perry's Chemical Engineers' Handbook, Sixth Edition, McGraw-Hill Co., 1984.
  5. ↑ 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 Procedures
  6. ↑ "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999.  Guidance for Offsite Consequence Analysis
  7. ↑ "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 (Scroll down to PGS2, download the PDF and then scroll to PDF pages 84-85)
  8. ↑ Calculating Accidental Release Rates From Pressurized Gas Systems
  9. ↑ Phillips Petroleum Company (1962). Reference Data For Hydrocarbons And Petro-Sulfur Compounds, Second Printing. Phillips Petroleum Company. 
  10. ↑ R.G. Cunningham (1951). "Orifice Meters With Supercritical Flow". Trans. ASME 73: 625-630.
  11. ↑ Richard W. Miller (1996). Flow Measurement Engineering Handbook, Third Edition. McGraw Hill. ISBN 0-07-042366-0. 
  12. ↑ 12.0 12.1 12.2 Flow through Orifice Plates in Compressible Fluid Service at High Pressure Drop
  13. ↑ Read page 2 of this brochure.[
  14. ↑ Search this pdf document for "Choked"
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