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The choked 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, the hole in an orifice plate, or a leak in a gas pipeline) 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.

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.

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.^[1]^[2]

1 Conditions under which gas flow becomes choked
2 Minimum pressure ratio required for choked flow to occur
3 See also
4 External links
5 References

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, $P_d/P_u$ is equal to or greater than

$\big[(k+1)/2 \big]^{k/(k-1}$

where $k$ is the heat capacity ratio of the gas (sometimes called the isentropic expansion factor and sometimes denoted as $\gamma$ ).

For many gases, k ranges from about 1.09 to about 1.41, and therefore

$\big[(k+1)/2 \big]^{k/(k-1}$

ranges from 1.7 to about 1.9 ... which means that choked flow usually occurs when the absolute upstream pressure, $P_u$ , is at least 1.7 to 1.9 times as high as the absolute downstream pressure, $P_d$ .

When the gas velocity is choked, the equation for the mass flow rate in SI metric units is: ^[3]^[4]^[5]^[6]

$\dot m\;=\;C\;A\;\sqrt{\;k\;\rho_u\;P_u\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}$

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

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

so that the mass flow rate is primarily dependent on the cross-sectional area $A$ of the hole and the upstream pressure $P_u$ , and only weakly dependent on the temperature $T$ . 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.

where:
Q	= mass flow rate, kg/s
C	= discharge coefficient, dimensionless (usually about 0.72)
A	= discharge hole cross-sectional area, m²
k	= c_p/c_v of the gas
c_p	= specific heat capacity of the gas at constant pressure
c_v	= specific heat capacity of the gas at constant volume
$\rho$	= real gas density at P and T, kg/m³
P	= absolute upstream pressure, Pa
M	= the gas molecular mass, kg/kgmol (also known as the molecular weight)
R	= Universal gas law constant = 8314.5 (N·m) / (kgmol·K)
T	= absolute 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 upstream pressure source when a discharge first occurs.

If the gas is being released from a closed high-pressure vessel, the flow rate will drop during the discharge as the source vessel empties and the pressure drops. Calculating the flow rate versus time since the initiation of the discharge is much more complicated, but more accurate. Two equivalent methods for performing such calculations are compared at www.air-dispersion.com/feature2.html.

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_s which only applies to a specific individual gas. The relationship between the two constants is R_s = R / M.

Notes:

The above equations are for a real gas.
For a monatomic ideal gas, Z = 1 and ρ is the ideal gas density.
kgmol = kilogram mole

Minimum pressure ratio required for choked flow to occur

The minimum pressure ratios required for choked conditions to occur (when some typical industrial gases are flowing) are presented in Table 1. The ratios were obtained using the criteria that choked flow occurs when the ratio of the absolute upstream pressure, P_u, to the absolute downstream pressure, P_d, is equal to or greater than [ ( k + 1 ) / 2 ]^{k / ( k - 1 )} , where k is the heat capacity ratio|specific heat ratio of the gas.

Table 1^[3]^[7]
Gas	k = c_p/c_v	Minimum P_u/P_d 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

External links

References

↑ Scroll to discussion of liquid flashing and cavitation
↑ Search document for "Choked"
↑ ^3.0 ^3.1 Perry's Chemical Engineers' Handbook, Sixth Edition, McGraw-Hill Co., 1984.
↑ 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
↑ "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. Guidance for Offsite Consequence Analysis
↑ "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
↑ Phillips Petroleum Company (1962). Reference Data For Hydrocarbons And Petro-Sulfur Compounds, Second Printing. Phillips Petroleum Company.

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@@ Line 1: / Line 1: @@
-The '''choked 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, the hole in an orifice plate, or a leak in a gas pipeline) into a lower pressure environment.
+The '''choked 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]], the hole in an [[orifice plate]], or a leak in a gas pipeline) 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 [[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.
 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.
-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==
-All gases flow from upstream higher pressure sources to downstream lower pressure sources. Choked flow occurs when the ratio of the absolute upstream 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]] of the gas (sometimes called the [[isentropic expansion factor]] and sometimes denoted as '''''<math>\gamma</math>''''' ).
+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
-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 flow usually occurs when the absolute source 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>
+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>).
+For many gases, k ranges from about 1.09 to about 1.41, and therefore
+:<math>\big[(k+1)/2 \big]^{k/(k-1}</math>
+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>.
 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=22297 PGS2 CPR 14E]</ref><br><br>
-:<math>\dot m\;=\;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>
-where the terms are defined in the table below. If the [[density]] '''''ρ''''' is not known directly, then it is useful to eliminate it using the [[Ideal gas law]] corrected for the real gas compressiblity:<br><br>
+where the terms are defined as in the table below. If the upstream gas [[density]], <math>\rho_u</math> 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>
-<math>\dot m\;=\;C\;A\;P\;\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>
 <br><br>
-so that the mass flow rate is primarily dependent on the cross-sectional area '''''A''''' of the hole and the supply pressure '''''P''''', and only weakly dependent on the temperature '''''T'''''.  The rate does not depend on the downstream pressure at all.  All other terms are constants that depend only on the composition of the material in the flow.
+so that the mass flow rate is primarily dependent on the cross-sectional area <math>A</math> of the hole and the upstream pressure <math>P_u</math>, and only weakly dependent on the temperature <math>T</math>.  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.
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User:Milton Beychok > Sandbox2

From Citizendium, the Citizens' Compendium

Revision as of 03:32, 2 January 2009

Contents

Conditions under which gas flow becomes choked

Minimum pressure ratio required for choked flow to occur

See also

External links

References

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