Accidental Release Of Pressurized GasEdit
When gas stored under pressure flows through an opening, the velocity through that opening may be choked (i.e., it has attained a maximum) or it may be non-choked.^{1, 2, 5, 6} 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 (sometimes referred to as the adiabatic index).
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\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}} $
or this equivalent form:
$ Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T}\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 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\;\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_A}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_A}{P}\bigg)^{(k+1)/k}\;\Bigg]} $
or this equivalent form:
$ Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{2\;M}{Z\;R\;T}\bigg)\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{\;P_A}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_A}{P}\bigg)^{(k+1)/k}\;\Bigg]} $
where: | |
Q | = mass flow rate, kg/s |
---|---|
C | = discharge coefficient, dimensionless (usually about 0.72) |
A | = discharge hole area, m² |
k | = c_{p}/c_{v} of the gas |
c_{p} | = specific heat of the gas at constant pressure |
c_{v} | = specific heat of the gas at constant volume |
$ \rho $ | = real gas density at P and T, kg/m³ |
P | = absolute upstream pressure, Pa |
P_{A} | = 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.com/feature2.html.
When expressed in the customary USA units, the above equations also contain the gravitational conversion factor g_{c} ( 32.17 ft / s² in USA units ). The above equations do not include g_{c} because g_{c} 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 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 d is the ideal gas density.
- kgmol = kilogram mole
Evaporation Of Non-Boiling Liquid PoolEdit
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 MethodEdit
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}
E = ( 4.161 x 10^{-5} ) u^{0.75} T_{F} M ( P_{S} / P_{H} )
where: | |
E | = evaporation flux, ( kg / minute) / m² of pool surface |
---|---|
u | = windspeed just above the liquid surface, m / s |
T_{A} | = absolute ambient temperature, °K |
T_{F} | = pool liquid temperature correction factor, dimensionless |
T_{P} | = pool liquid temperature, °C |
M | = pool liquid molecular weight, dimensionless |
P_{S} | = pool liquid vapor pressure at ambient temperature, mm Hg |
P_{H} | = hydrazine vapor pressure at ambient temperature, mm Hg (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_{P}^{2}
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 the value of the constant e (approximately 2.7183) raised to the power of x.
The U.S. EPA MethodEdit
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 ( U.S. EPA )^{4, 5} 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 = ( 10.40 / RT ) u^{ 0.78} M^{ 0.667} A P
where: | |
E | = evaporation rate, kg / minute |
---|---|
u | = windspeed just above the pool liquid surface, m / s |
M | = pool liquid molecular weight, 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 |
R | = the Universal Gas Law constant = 82.05 ( atm·cm³ ) / ( gmol·°K ) |
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.
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 = ( cubic meters of pool liquid ) / ( 0.01 m )
Notes:
- 1 kPa = 0.0102 kg / cm² = 0.01 bar
- gmol = gram mole
- atm = atmosphere
Stiver and Mackay's MethodEdit
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.^{3}
E = k P M / ( R T_{A} )
k = 0.002 u
where: | |
E | = evaporation flux, ( kg / s ) / m² of pool surface |
---|---|
k | = mass transfer coefficient, m / s |
T_{A} | = absolute ambient temperature, °K |
M | = pool liquid molecular weight, dimensionless |
P | = pool liquid vapor pressure at ambient temperature, Pa |
R | = the Universal Gas Law constant = 8314.5 ( Pa·m³ ) / ( kgmol·°K ) |
u | = windspeed just above the liquid surface, m / s |
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.
Evaporation Of Boiling Cold Liquid PoolEdit
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}
E = ( 0.0001 ) ( 7.7026 − 0.0288 B ) ( M ) e ^{-( 0.0077 B ) - 0.1376}
where: | |
E | = evaporation flux, ( kg / minute ) / m² of pool surface |
---|---|
B | = pool liquid atmospheric boiling point, °C |
M | = pool liquid molecular weight, dimensionless |
e | = the base of the natural logarithm system = 2.7183 |
Adiabatic Flash of Liquified Gas ReleaseEdit
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 |
---|---|
H_{s}^{L} | = source liquid enthalpy at source temperature and pressure, J / kg |
H_{a}^{V} | = flashed vapor enthalpy at atmospheric boiling point and pressure, J / kg |
H_{a}^{L} | = 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 |
---|---|
c_{p} | = source liquid specific heat, J / kg / °C |
T_{s} | = source liquid absolute temperature, °K |
T_{b} | = source liquid absolute atmospheric boiling point, °K |
H | = source liquid heat of vaporization at atmospheric boiling point, J / kg |
ReferencesEdit
- "Perry's Chemical Engineers' Handbook, Sixth Edition, McGraw-Hill Co., 1984.
- "Handbook of Chemical Hazard Analysis Procedures" Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Handbook of Chemical Hazard Procedures Also provides references to (2a), (2b) and (2c) below:
- Clewell, H.J., "A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids", Energy Systems Laboratory, ESL-TR-83-03, 1983.
- Ille, G. and Springer, C., "The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spills", Civil and Environmental Engineering Development Office, CEEDO 712-78-30, 1978.
- Kahler, J.P., Curry, R.C. and Kandler, R.A., "Calculating Toxic Corridors", Air Force Weather Service, AWS TR-80/003, 1980.
- Stiver, W. and Mackay, D., "A Spill Hazard Ranking System For Chemicals", Environment Canada First Technical Spills Seminar, Toronto, Canada, 1993.
- "Technical Guidance For Hazards Analysis", U.S. EPA and U.S. FEMA, December 1987. Technical Guidance for Hazards Analysis
- "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999. Guidance for Offsite Consequence Analysis
- "Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, Section 2.5.2.3, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. PGS2 CPR 14E
- More release source terms are available in the feature articles at www.air-dispersion.com
External linksEdit
- Ramskill's equation 4 is an alternative but equivalent form of the non-choked gas flow equation in this article.
Milton Beychok
January 2006