Accidental release source terms

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 ${\displaystyle \gamma }$).

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:[1][2][3][4]

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

or this equivalent form:

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

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:

${\displaystyle Q\;=\;C\;A\;{\sqrt {\;2\;\rho \;P\;\left[{\frac {k}{k-1}}\right]\left[\left({\frac {P_{A}}{P}}\right)^{\frac {2}{k}}-\;\,\left({\frac {\;P_{A}}{P}}\right)^{\frac {k+1}{k}}\;\right]}}}$

or this equivalent form:

${\displaystyle Q\;=\;C\;A\;P\;{\sqrt {\left[{\frac {2\;M}{Z\;R\;T}}\right]\left[{\frac {k}{k-1}}\right]\left[\,\left({\frac {P_{A}}{P}}\right)^{\frac {2}{k}}-\;\,\left({\frac {P_{A}}{P}}\right)^{\frac {k+1}{k}}\;\right]}}}$

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
${\displaystyle \rho }$	= real gas density at P and T, kg/m3
P	= absolute upstream pressure, Pa
PA	= absolute ambient or downstream pressure, Pa
M	= the gas molecular mass, kg/kmol    (also known as the molecular weight)
R	= the Universal Gas Law Constant = 8314.5 Pa·m3/(kmol·K)
T	= absolute upstream 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 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.[5]

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 R_s which only applies to a specific individual gas. The relationship between the two constants is R_s = R/M.

Notes:

• The above equations are for a real gas.
• For an ideal gas, Z = 1 and ρ is the ideal gas density.
• 1 kilomole (kmol) = 1000 moles = 1000 gram-moles = kilogram-mole.

Three different methods of calculating the rate of evaporation from a non-boiling liquid pool are presented in this section. The results obtained by the three methods are somewhat different.

The U.S. Air Force method

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.[2]

${\displaystyle E=\left(4.161\times 10^{-5}\right)\cdot u^{0.75}\cdot T_{F}\cdot M\cdot {\frac {P_{S}}{P_{H}}}}$

where:
E	= evaporation flux, kg/m2·min of pool surface
u	= windspeed just above the liquid surface, m/s
TA	= absolute ambient temperature, K
TF	= pool liquid temperature correction factor, dimensionless
TP	= pool liquid temperature, °C
M	= pool liquid molecular weight, dimensionless
PS	= pool liquid vapor pressure at ambient temperature, mmHg
PH	= hydrazine vapor pressure at ambient temperature, mmHg (see equation below)

If T_P = 0 °C or less, then T_F = 1.0

If T_P > 0 °C, then T_F = 1.0 + 0.0043 T_P²

${\displaystyle P_{H}=760\cdot e^{65.3319-{\frac {7245.2}{T_{A}}}-8.22\;\ln \left(T_{a}\right)+0.0061557\;T_{A}}}$

where:
${\displaystyle e}$	= 2.7183, the base of the natural logarithm system
${\displaystyle \ln }$	= natural logarithm

The following equation is for predicting the rate at which liquid evaporates from the surface of a pool of cold liquid (i.e., at a liquid temperature of about 0 °C or less).[2]

${\displaystyle E=0.0001\;M(7.7026-0.0288\;B)\;e^{-(0.0077\;B)-0.1376}}$

Liquefied gases such as ammonia or chlorine are often stored in cylinders or vessels at ambient temperatures and pressures well above atmospheric pressure. When such a liquefied gas is released into the ambient atmosphere, the resultant reduction of pressure causes some of the liquefied gas to vaporize immediately. This is known as "adiabatic flashing" and the following equation, derived from a simple heat balance, is used to predict how much of the liquefied gas is vaporized.

${\displaystyle X=100\;{\frac {H_{s}^{L}-H_{a}^{L}}{H_{a}^{V}-H_{a}^{L}}}}$

If the enthalpy data required for the above equation is unavailable, then the following equation may be used.

${\displaystyle X=100\cdot c_{p}\cdot {\frac {T_{s}-T_{b}}{H}}}$

• Choked flow
• Orifice plate
• Flash evaporation

• Ramskill's equations are presented and cited in this pdf file (use search function to find "Ramskill").
• Choked flow of gases
• Development of source emission models