When gas stored under
pressure in a closed vessel is discharged to the
atmosphere through a hole or other opening, the gas
velocity through that opening may be choked (i.e., it has attained a maximum) or it may be non-choked. Choked velocity, also referred to as sonic velocity, occurs when the ratio of the absolute source 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 discharged 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 velocity usually occurs when the absolute source vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream ambient
atmospheric 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\; \left(\frac{2}{k+ 1}\right)^\frac{k + 1}{k - 1}} or this equivalent form: : Q \;=\; C\;A\;P\;\sqrt{\left(\frac{\;\,k\;M}{Z\;R\;T}\right)\left(\frac{2}{k + 1}\right)^\frac{k + 1}{k - 1}} For the above equations,
it is important to note that although the gas velocity reaches a maximum and becomes choked, the mass flow rate is not choked. The mass flow rate can still be increased if the source pressure is increased. Whenever the ratio of the absolute source pressure to the absolute downstream ambient pressure is less than '
[ (k
+ 1) / 2]k
/ (k
− 1)', then the gas velocity is non-choked (i.e., sub-sonic) and the equation for mass flow rate is: : Q \;=\; C\;A\;\sqrt{\;2\;\rho\;P\;\left[\frac{k}{k - 1}\right]\left[\left(\frac{P_A}{P}\right)^\frac{2}{k}-\;\,\left(\frac{\;P_A}{P}\right)^\frac{k + 1}{k}\;\right]} or this equivalent form: : Q \;=\; C\;A\;P\; \sqrt{\left[\frac{2\;M}{Z\;R\;T}\right]\left[\frac{k}{k - 1}\right]\left[\,\left(\frac{P_A}{P}\right)^\frac{2}{k} -\;\,\left(\frac{P_A}{P}\right)^\frac{k + 1}{k}\;\right]} The above equations calculate the
initial instantaneous mass flow rate for the pressure and temperature existing in the source vessel when a release 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 pressure and flow rate decrease with time as the system or vessel empties. Calculating the flow rate versus time since the initiation of the leak is much more complicated, but more accurate. Two equivalent methods for performing such calculations are presented and compared at. 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 an ideal gas,
Z = 1 and
ρ is the ideal gas density. • 1kilomole (kmol) = 1000
moles = 1000 gram-moles = kilogram-mole.
Ramskill's equation for non-choked mass flow P.K. Ramskill's equation for the non-choked flow of an ideal gas is shown below as equation (1): : (1) Q = C \;\rho_A\;A\;\sqrt{\frac{\;\,2\;P}{\rho} \cdot \frac{k}{k-1} \cdot \left[\; 1 - {\left(\frac{P_A}{P}\right)^\frac{k - 1}{k}}\right]} The gas density,
\rhoA, in Ramskill's equation is the ideal gas density at the downstream conditions of temperature and pressure and it is defined in equation (2) using the
ideal gas law: : (2) \rho_A = \frac{M\;P_A}{R \;T_A} Since the downstream temperature
TA is not known, the isentropic expansion equation below is used to determine
TA in terms of the known upstream temperature
T: : (3) \frac{T_A}{T} = \left(\frac{P_A}{P}\right)^\frac{k - 1}{k} Combining equations (2) and (3) results in equation (4) which defines
\rhoA in terms of the known upstream temperature
T: : (4) \rho_A = \frac{M \;P^\frac{k - 1}{k}}{R \;T \;P_A^{-\frac{1}{k}}} Using equation (4) with Ramskill's equation (1) to determine non-choked mass flow rates for ideal gases gives identical results to the results obtained using the non-choked flow equation presented in the previous section above. ==Evaporation of non-boiling liquid pool==