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When determining the consequences of accidental release flow rates from pressurized gas systems, it is important to select the appropriate type of air pollution dispersion model. For released gases which are lighter than or equal to the ambient air density, Gaussian dispersion models as described in Beychok's text^{1} should be used. For released gases which are heavier than air, a dense gas model such as SLAB^{2} or DEGADIS^{3} should be used.
It is also important to determine realistic flow rates for accidental release scenarios selected for dispersion modeling. Most offsite consequence analyses have used accidental releases determined by socalled "sourceterm models" which calculate the initial instantaneous flow rate for the pressure and temperature existing in the source system or 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. Much of the current technical literature on accidental release sourceterm models fails to offer guidance on how to calculate the average flow rate ... and that may explain why so many offsite consequence analyses for pressurized gas releases have been based on initial instantaneous flow rates.
The purpose of this article is to present and explain two published sourceterm models for calculating the timedependent decrease in pressure, temperature and weight of gas in a pressurized gas system or vessel during an accidental release.
It should be emphasized that the sourceterm models discussed in this article are only applicable to systems or vessels containing pressurized gases with very low atmospheric boiling points that therefore will still be in the gas phase when released into the atmosphere. The models are also only applicable to gases at a high pressure such that the release occurs at choked flow conditions during most of the time that the system or vessel is emptying due to an accidental release.The Rasouli and Williams^{4} sourceterm model for choked gas flows from a pressurized gas system was published in 1995. Choked flow is also referred to as sonic flow and it occurs when the ratio of the source gas pressure to the downstream ambient atmospheric pressure is equal to or greater than [ (k + 1) / 2 ]^{ k / (k – 1)}, where k is the specific heat ratio (c_{p} / c_{v }). For many gases, k ranges from about 1.1 to about 1.4, and so choked gas flow usually occurs when the source gas pressure is about 25 to 28 psia or greater (see Table 1). Thus, the large majority of accidental gas releases will usually involve choked flow.
As originally published, the Rasouli and Williams model was in a form specific for methane gas releases and contained a typographical error as well as a minor derivational error. However, based on the original detailed derivation (as kindly provided by Dr. Rasouli), the errors were corrected and the model was generalized to obtain Equation (1) below:
(1) 
where:
^{ }P_{1}
^{ }P_{2}
t_{1}
t_{2}
C
^{ }A
^{ }V
_{ }k
^{ }g_{c}
^{ }R
M
T_{0}
^{}P_{0}

= the gas pressure in the source vessel at t_{1}, in lbs / ft^{2} absolute
= the gas pressure in the source vessel at t_{2}, in lbs / ft^{2} absolute
= any time after leak flow starts, in seconds_{ }
= any time (later than t_{1}) after leak flow starts , in seconds
= coefficient of discharge
= area of the source leak, in ft^{2}
= volume of the source vessel, in ft^{3}
= c_{p} / c_{v}
= gravitational conversion factor of 32.17 ft / s^{2}
= universal gas law constant of 1545 (lbs / ft^{2})(ft^{3}) / (lbmol · °R)
= molecular weight of the gas
= initial gas temperature in the source vessel, in °R_{ }
= initial gas pressure in the source vessel, in lbs / ft^{2}_{ } absolute

The Bird, Stewart and Lightfoot^{5} sourceterm model for choked gas flows from a pressurized gas system was published in 1960 in its generalized form which was rearranged to obtain Equation (2) below:
(2) 
where:
t
F
^{ }V
C
^{ }A
_{ }k
^{ }g_{c}
^{}P_{0}
^{}d_{0}

= any time after leak flow starts, in seconds
= fraction of initial gas weight remaining in source vessel at time t
= volume of the source vessel, in ft^{3}
= coefficient of discharge
= area of the source leak, in ft^{2}
= c_{p} / c_{v}
= gravitational conversion factor of 32.17 ft / s^{2}
= initial gas pressure in the source vessel, in lbs / ft^{2}_{ } absolute
= initial gas density in the source vessel, in lbs / ft^{3}_{ }

Each model was used to obtain a profile of the timedependent decrease in the pressure, the temperature and the weight of gas in a vessel storing methane gas at 60 °F and 3,430 psia when a 0.5 inch diameter leak occurs.
The Rasouli and Williams model becomes specific for this example by substituting these values into equation (1):

C = 0.72 A = 0.001363 ft^{2} V = 51.4 ft^{3} M = 16.04 lb/lbmol k = 1.307 T_{0} = 520 °R P_{0} = 493,920 lbs/ft^{2} absolute 
The resulting expression is:
P_{2} = [(5.3329 × 10^{ – 4} )(t_{2} – t_{1}) + P_{1}^{– 0.1174} ]^{ – 8.5179}  (3) 
For the Rasouli and Williams model, equation (3) was then used to obtain P_{2} values for each value of (t_{2} – t_{1}). The corresponding T_{2} temperature values were obtained from this expression for the isentropic expansion or compression of an ideal gas:
(T_{2 } / T_{1}) = (P_{2} / P_{1})^{(k – 1) / k}  (4) 
and the weight of gas (W, in pounds) remaining in the source vessel at the end of each increment of time (t_{2} – t_{1}) was obtained from the universal gas law expression:
W = P V M / R T  (5) 
The Bird, Stewart and Lightfoot model becomes specific for this example by substituting these values into equation (2):

C = 0.72 A = 0.001363 ft^{2} V = 51.4 ft^{3} k = 1.307 P_{0} = 493,920 lbs/ft^{2} absolute d_{0} = 9.861 lbs/ft^{3} 
The resulting expression is:
t = 402.1(F^{ – 0.1535} – 1)  (6) 
which can be rearranged to obtain:
F = [1 + (0.002487) t ]^{ – 6.5147}  (7) 
For the Bird, Stewart and Lightfoot model, equation (7) was used to obtain F values for each value of time t since the initiation of flow through the leak. The corresponding values of W for each value of time t were obtained by multiplying the original weight of gas in the source vessel (i.e., 507 pounds) by the residual weight fraction F at time t.
Equation (4) for the isentropic expansion or compression of an ideal gas can be manipulated and rearranged to obtain the following expressions:
P = P_{0} F^{ k}  (8)  
T = T_{0} F^{ (k – 1)}  (9) 
The corresponding P and T values were calculated, using equations (8) and (9), for the F value obtained at each value of time t.
The comparative profiles yielded by the two models are tabulated in Table 2 and it is obvious that the two models produced identical results. As can be seen in Table 2, the initial methane release rate during the first 30 seconds is (507  317) / 30 = 6.3 lbs/second and the rate during the last 30 seconds is (18  14) / 30 = 0.1 lbs/second ... after which only 2.65 percent of the initial 507 lbs of methane remains in the vessel.
It can also be seen that the overall average release rate is (507  14) / 300 = 1.6 lb/second, which is very much slower than the rate of 6.3 lbs/second during the initial 30 seconds.
Figure 1 graphically presents the profile of time versus the source vessel pressure as well as the profile of time versus the gas release rate. It is quite obvious that the decay of source vessel pressure and of the gas release rate is not linear.
Stored gas 
c_{p }/ c_{v} 
Storage pressure at which gas flow through a leak would be choked flow 
Butane Propane Sulfur Dioxide Methane Ammonia Chlorine Carbon Monoxide Hydrogen 
1.096 1.131 1.290 1.307 1.310 1.355 1.404 1.410 
25.1 psia or greater 25.4 psia or greater 26.8 psia or greater 27.0 psia or greater 27.0 psia or greater 27.4 psia or greater 27.9 psia or greater 27.9 psia or greater 
Equation 1 (Rasouli and Williams Model) 
Equation 2 (Bird, Stewart and Lightfoot Model)  
t (sec) 
P (psia) 
T (°R) 
W (lbs) 
P (psia) 
T (°R) 
W (lbs) 
F 
0 30 60 90 120 150 180 210 240 270 300 
3,430 1,859 1,050 615 372 231 147 96 64 43 30 
520 450 394 347 309 276 248 224 204 186 170 
507 317 205 136 93 64 46 33 24 18 14 
3,430 1,859 1,050 614 371 231 147 96 64 43 30 
520 450 394 347 308 276 248 224 204 186 171 
507 317 205 136 92 64 46 33 24 18 13 
1.0000 0.6258 0.4041 0.2682 0.1824 0.1268 0.0898 0.0647 0.0474 0.0352 0.0265 
t = P = T = W = F = 
time since initiation of flow, sec gas pressure within source vessel, psia gas temperature within source vessel, °R weight of gas within source vessel, lbs weight fraction (of initial gas weight) remaining in source vessel 
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