Choked flow

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, 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.

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.

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, $P_{d}$.

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:

${\dot {m}}\;=\;C\;A\;{\sqrt {\;k\;\rho _{u}\;P_{u}\;{\bigg (}{\frac {2}{k+1}}{\bigg )}^{(k+1)/(k-1)}}}$

where the terms are defined as in listed below. If the upstream gas density, $\rho _{u}$ 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.

Definition of terms:

${\dot {m}}$	= mass flow rate, kg/s
$C$	= discharge coefficient, dimensionless (usually about 0.72)
$A$	= restriction cross-sectional 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} = heat capacity 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 capacity 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 capacity 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 density 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} , 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 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}	= Universal gas law constant = 8314.5 (N·m) / (kmol·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}	= absolute 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} , 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 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.

Notes:

• The above equations are for a real gas.
• For a monatomic 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.

Minimum pressure ratio required for choked flow to occur

The minimum pressure ratios required for choked gas flow conditions to occur are presented in Table 1 for some industrial gases:

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

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_d/P_u} 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}} .