User:Milton Beychok/Sandbox2

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.

The specific 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 ]k / ( k - 1 ), where k is the specific heat 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 [ ( 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:

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

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 compressiblity:

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

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.

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

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

Notes:
 * k values obtained from: