Accidental Releases From Pressurized Gas Systems:1, 2
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, which is also referred to as sonic velocity, occurs when the ratio of the absolute source pressure to the absolute downstream ambient pressure is equal to or greater than [ ( k + 1 ) / 2 ] k / ( k - 1 ) , where k is the specific heat ratio of the discharged gas.
For many gases, the k value 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 [ k d P ]1/2 [ 2 / ( k + 1 ) ] (k + 1) / (2k - 2)
or this equivalent form:
Q = C A P [ k M / ( Z R T ) ]1/2 [ 2 / ( k + 1 ) ] (k + 1) / (2k - 2)
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 [ 2 d P ]1/2 [ k / ( k - 1 ) ]1/2 [ ( PA / P )
2 / k - ( PA / P ) (k + 1) / k ]1/2
or this equivalent form:
Q = C A P[ 2 M / (Z R T)]1/2 [ k / ( k - 1 )]1/2 [( PA / P) 2 / k - ( PA / P )(k + 1) / k]1/2
where:
Q = mass flow rate, kg/s
C = discharge coefficient, dimensionless (usually about 0.72)
A = discharge hole 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
d = real gas density at P and T, kg/m3
P = absolute source or upstream pressure, Pa
PA = absolute ambient or downstream pressure, Pa
M = gas molecular weight, dimensionless
R = the Universal Gas Law Constant = 8314.5 ( Pa ) ( m3 ) / ( kgmol ) ( °K )
T = absolute gas temperature, °K
Z = the gas compresibility factor at P and T, dimensionless
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 www.air-dispersion/feature2.html.
When expressed in the customary USA units, the above equations also contain the gravitational conversion factor gc ( 32.17 ft / s2 in USA units ). The above equations do not include gc because gc is 1 ( kg-m ) / ( N·s2 ) in the SI metric system of units,.
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 / (MW).
Notes:
(1) The above equations are for a real gas.
(2) For an ideal gas, Z = 1 and d is the ideal gas density.
(3) kgmol = kilogram mole
Evaporation From A Non-Boiling Liquid Pool:
Three different methods of calculating the rate of evaporation from a non-boiling liquid pool are presented in this section.
Method developed by the U.S. Air Force: 2
The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were derived from field tests performed by the U.S. Air Force with pools of liquid hydrazine.
E = ( 4.161 x 10-5 ) u0.75 TF M ( PS / PH )
This is still a work in progress. Please make no changes as yet.