Choked flow

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When a flowing fluid at a certain pressure and temperature flows through a restriction (such as the hole in an orifice plate or a valve in a pipe) into a lower pressure environment, under the conservation of mass the fluid velocity must increase as it flows through the smaller cross-sectional area of the restriction. At the same time, the Venturi effect causes the pressure to decrease.

Choked flow is a limiting condition which, if the flowing fluid is a gas, occurs when the gas velocity traveling through the restriction increases to the speed of sound in the gas. At that point, the gas velocity becomes independent of the downstream environment pressure (i.e, lowering the downstream pressure will not increase the gas velocity any further). 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 linear velocity of the gas becomes choked, the mass flow rate of the gas can still be increased by increasing the upstream pressure.

The choked flow of gases is useful in many engineering applications because the mass flow rate is independent of the downstream pressure, depending only on the temperature and pressure on the upstream side of the restriction. Under choked conditions, valves and calibrated orifice plates can be used to produce a particular mass flow rate.

If the fluid is a liquid, 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]

Contents

[edit] Mass flow rate of a gas at choked conditions

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 ] k / ( k - 1 ) , where k is the specific heat ratio of the gas (sometimes called the isentropic expansion factor and sometimes denoted as γ ).

For many gases, k ranges from about 1.09 to about 1.41, and therefore [ ( k + 1 ) / 2 ] k / ( k - 1 ) 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.

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

Q\;=\;C\;A\;\sqrt{\;k\;\rho\;P\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}

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 compressibility:

Q\;=\;C\;A\;P\;\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 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.

where:  
Q = mass flow rate, kg/s
C = discharge coefficient, dimensionless (usually about 0.72)
A = discharge hole cross-sectional 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
ρ = 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 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 a monatomic ideal gas, Z = 1 and ρ is the ideal gas density.
  • kgmol = kilogram mole

[edit] 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 to the absolute downstream pressure is equal to or greater than [ ( k + 1 ) / 2 ] k / ( k - 1 ) , where k is the specific heat ratio of the gas.

Table 1
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

Notes:

  • Pu = absolute upstream gas pressure
  • Pd = absolute downstream gas pressure
  • k values obtained from:
    1. Perry, Robert H. and Green, Don W. (1984). Perry's Chemical Engineers' Handbook, 6th Edition, McGraw-Hill Company. ISBN 0-07-049479-7.
    2. Phillips Petroleum Company (1962). Reference Data For Hydrocarbons And Petro-Sulfur Compounds, Second Printing, Phillips Petroleum Company.

[edit] See also

  • Accidental release source terms includes mass flow rate equations for non-choked gas flows as well.
  • Orifice plate includes derivation of non-choked gas flow equation.
  • Laval nozzles are Venturi tubes that produce supersonic gas velocities as the tube and the gas are first constricted and then the tube and gas are expanded beyond the choke plane.
  • Rocket engine nozzles discusses how to calculate the exit velocity from nozzles used in rocket engines.

[edit] External links

[edit] References

  1. ^ Scroll to discussion of liquid flashing and cavitation
  2. ^ Search document for "Choked"
  3. ^ Perry's Chemical Engineers' Handbook, Sixth Edition, McGraw-Hill Co., 1984.
  4. ^ 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
  5. ^ "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999.  Guidance for Offsite Consequence Analysis
  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
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