Accidental release source terms/Citable Version

(PD) Photo: U.S. Dept. of Energy
Fire and explosion damage in BP refinery (Texas) caused by accidental release of hydrocarbon vapors.

Accidental release source terms are the mathematical equations that quantify the flow rate at which accidental releases of air pollutants into the ambient environment can occur at industrial facilities such as petroleum refineries, petrochemical plants, natural gas processing plants, oil and gas transportation pipelines, chemical plants, and many other industrial activities. Governmental regulations in a good many countries require that the probability of such accidental releases be analyzed and their quantitative impact upon the environment and human health be determined so that mitigating steps can be planned and implemented.

There are a number of mathematical calculation methods for determining the flow rate at which gaseous and liquid pollutants might be released from various types of accidents. Such calculational methods are referred to as source terms, and this article on accidental release source terms explains some of the calculation methods used for determining the mass flow rate at which gaseous pollutants may be accidentally released. Given those mass flow rates, atmospheric dispersion modeling studies then be performed.

Accidental release of pressurized gas

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 or it may be non-choked. Choked flow (also referred to as critical flow) is a limiting or maximum condition at which the gas velocity has attained the speed of sound in the gas.

Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure 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 (1)}     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)}}

where 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} is the specific heat ratio of the discharged gas (sometimes called the isentropic expansion factor and sometimes denoted as 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 \gamma} ).

For many gases, 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} ranges from about 1.09 to about 1.41, and therefore the expression in (1) ranges from 1.7 to about 1.9, which means that choked velocity usually occurs when the absolute upstream vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream pressure. In the case of a leak to the ambient atmosphere, the downstream pressure is the atmospheric pressure.

When the gas velocity is choked, the equation for the mass flow rate in SI units 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 \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, 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 not known directly, then it is useful to eliminate it using the Ideal gas law corrected for the real gas compressiblity:

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 \dot m\;=\;C\;A\;P_u\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T_u}\bigg)\bigg(\frac{2}{k+1}\bigg)^{(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 upstream pressure is increased or the temperature is decreased.

Whenever the ratio of the absolute upstream pressure to the absolute downstream pressure is less than in expression (1) above, then the gas velocity is non-choked and the equation for mass flow rate 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 \dot m\;=\;C\;A\;\sqrt{\;2\;\rho_u\;P_u\;\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_d}{P_u}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_d}{P_u}\bigg)^{(k+1)/k}\;\Bigg]}}

or this equivalent form:

${\dot {m}}\;=\;C\;A\;P_{u}\;{\sqrt {{\bigg (}{\frac {2\;M}{Z\;R\;T_{u}}}{\bigg )}{\bigg (}{\frac {k}{k-1}}{\bigg )}{\Bigg [}\,{\bigg (}{\frac {\;P_{d}}{P_{u}}}{\bigg )}^{2/k}-\;\,{\bigg (}{\frac {\;P_{d}}{P_{u}}}{\bigg )}^{(k+1)/k}\;{\Bigg ]}}}$

where:
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 \dot m}	= mass flow rate, kg/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 C}	= discharge coefficient, dimensionless (usually about 0.72)
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 A}	= discharge hole 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} = specific heat 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 upstream density, 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 (M P_u)/(Z\, R\, T_u)}
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 P_d}	= absolute downstream, 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/kmole    (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}	= the universal gas law constant = 8314.5 Pa·m3/(kmole·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_u}	= absolute upstream 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_u} , 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. A comparison between two methods for performing such calculations is available online.

The technical literature can be confusing because many authors do not explain whether they are using the universal 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} which applies to any ideal gas or whether they are using 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} .

Notes:

• The above equations are for a real gas.
• For an ideal gas, Z = 1 and ρ is the ideal gas density.
• kmole = 1000 moles

Ramskill's equation for non-choked mass flow

P.K. Ramskill's equation for the non-choked flow of a gas is somewhat different than the above non-choked flow equation but both versions provide identical results:

(1)       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 \dot m = C \;\rho_d\;A\;\sqrt{\frac{\;\,2\;P_u}{\rho_u}\cdot\frac{k}{k-1}\cdot{\Bigg[\; 1 - {\bigg(\frac{P_d}{P_u}\bigg)^{(k-1)/k)}}\Bigg]}}}

where:
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 \dot m}	= mass flow rate, kg/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 C}	= discharge coefficient, dimensionless (usually about 0.72)
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 A}	= discharge hole 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} = specific heat 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 upstream density, 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 (M P_u)/(Z\, R\, T_u)}
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_d}	= real gas downstream density, 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 (M P_d)/(Z\, R\, T_d)}
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 P_d}	= absolute downstream, 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/kmole    (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}	= the universal gas law constant = 8314.5 Pa·m3/(kmole·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_u}	= absolute upstream 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 T_d}	= absolute downstream 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 the pertinent $P$ and $T$ conditions, dimensionless

Since the downstream temperature $T_{d}$ is unknown and is required to calculate the downstream density $\rho _{d}$, the isentropic expansion equation below is used to obtain $T_{d}$ in terms of the known upstream temperature $T_{u}$ :

(2)       $T_{d}=T_{u}{\bigg (}{\frac {P_{d}}{P_{u}}}{\bigg )}^{(k-1)/k}$

Combining equation (2) with the definition of $\rho _{d}$ (in the parameter list just above) provides $\rho _{d}$ in terms of the known upstream temperature $T_{u}$ :

(3)       $\rho _{d}={\frac {M\;P_{u}^{\;(k-1)/k}}{ZR\;T_{u}\;P_{d}^{\ -1/k}}}$

Using equation (3) with Ramskill's equation (1) to determine non-choked mass flow rates for gases gives results identical to those obtained using the non-choked flow equation presented in the previous section above.

Evaporation of non-boiling liquid pool

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.

E = ( 4.161 x 10^-5 ) u^0.75 T_F M ( P_S ÷ P_H )

where:
E	= evaporation flux, (kg/min)/m² 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 mass, 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

P_H = 760 exp[ 65.3319 − (7245.2 ÷ T_A ) − (8.22 ln T_A) + ( 6.1557 x 10^-3) T_A]

Note: The function "ln x" is the natural logarithm (base e) of x, and the function "exp x" is e (approximately 2.7183) raised to the power of x.

The U.S. EPA 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 developed by the United States Environmental Protection Agency using units which were a mixture of metric usage and United States usage. The non-metric units have been converted to metric units for this presentation.

E = ( 0.1268 ÷ T ) u ^0.78' M ^0.667 A P

where:
E	= evaporation rate, kg/min
u	= windspeed just above the pool liquid surface, m/s
M	= pool liquid molecular mass, dimensionless
A	= surface area of the pool liquid, m²
P	= vapor pressure of the pool liquid at the pool temperature, kPa
T	= pool liquid absolute temperature, K

The U.S. EPA also defined the pool depth as 0.01 m (i.e., 1 cm) so that the surface area of the pool liquid could be calculated as:

A = ( pool volume, in m³ ) / (0.01)

Stiver and Mackay's 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 developed by Warren Stiver and Dennis Mackay of the Chemical Engineering Department at the University of Toronto.

E = k P M ÷ (R T_A)

where:
E	= evaporation flux, (kg/s)/m² of pool surface
k	= mass transfer coefficient, m/s = 0.002 u
TA	= absolute ambient temperature, K
M	= pool liquid molecular mass, dimensionless
P	= pool liquid vapor pressure at ambient temperature, Pa
R	= the universal gas law constant = 8314.5 Pa·m³/(kmol·K)
u	= windspeed just above the liquid surface, m/s

Evaporation of boiling cold liquid pool

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).

E = ( 0.0001 ) ( 7.7026 − 0.0288 B ) ( M ) e^{-0.0077B - 0.1376}

where:
E	= evaporation flux, (kg/min)/m² of pool surface
B	= pool liquid atmospheric boiling point, °C
M	= pool liquid molecular mass, dimensionless
e	= the base of the natural logarithm = 2.7183

Adiabatic flash of liquified gas release

Liquified 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 liquified gas is released into the ambient atmosphere, the resultant reduction of pressure causes some of the liquified 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 liquified gas is vaporized.

X = 100 ( H_s^L − H_a^L ) ÷ ( H_a^V − H_a^L )

where:
X	= weight percent vaporized
HsL	= source liquid enthalpy at source temperature and pressure, J/kg
HaV	= flashed vapor enthalpy at atmospheric boiling point and pressure, J/kg
HaL	= residual liquid enthalpy at atmospheric boiling point and pressure, J/kg

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

X = 100 [ c_p ( T_s − T_b ) ] ÷ H

where:
X	= weight percent vaporized
cp	= source liquid specific heat, J/(kg·K)
Ts	= source liquid absolute temperature, K
Tb	= source liquid absolute atmospheric boiling point, K
H	= source liquid heat of vaporization at atmospheric boiling point, J/kg