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 so-called 'source-term 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 source-term 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 source-term models for calculating the time-dependent decrease in pressure, temperature and weight of gas in a pressurized gas system or vessel during an accidental release.

The Rasouli and Williams source model:

The Rasouli and Williams4 source-term 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 atmospheric pressure is equal to or greater than [(k + 1)/2]k/(k - 1), where k is the specific heat ratio (cp/cv). 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 this form:

CD (A / V) (g R / M)1/2 [(k - 1)/(2k)] k3/2 [2/(k + 1)]a (T0 / P0b)1/2 (t2 - t1) = P2c- P1c (1)

where:

CD = coefficient of discharge

A = area of the source leak, in ft2

V = source vessel volume, in ft3

g = gravitational constant of 32.17 ft/s2

R = universal gas constant of 1545 (lbs/ft2)(ft3)/(lb-mol) (°R)

M = molecular weight of the gas

k = cp/cv of the gas

a = (k + 1)/(2k - 2)

T0 = initial gas temperature in the source vessel, in °R

P0 = initial gas pressure in the source vessel, in lbs/ft2 absolute

b = (k - 1)/k

t0 = the time of flow initiation through the leak, in seconds

t1 = any time t0 or later, in seconds

t2 = any time later than t1, in seconds

P1 = the gas pressure in the source vessel at t1, in lbs/ft2 absolute

P2 = the gas pressure in the source vessel at t2, in lbs/ft2 absolute

c = - (k - 1)/(2k)

The Bird, Stewart and Lightfoot source model:

The Bird, Stewart and Lightfoot5 source-term model for choked gas flows from a pressurized gas system was published in 1960 in its generalized form as:

[2/(k - 1)](Fa -1)

t = ——————————————————

CD (A / V){g k (P0 / d0) [2/(k + 1)] b }1/2 (2)

where:

t = the time since the leak flow initiated, in seconds

k = cp/cv of the gas

F = the fraction of initial gas weight remaining in source vessel at any time t

a = (1 - k)/2

CD = coefficient of discharge

A = area of the source leak, in ft2

V = source vessel volume, in ft3

g = gravitational constant of 32.17 ft/s2

P0 = the initial gas pressure in the source vessel, in lbs/ft2 absolute

d0 = the initial gas density in the source vessel, in lbs/ft3

b = (k + 1)/(k - 1)

Comparison of the two models based on an example calculation:

Each model was used to obtain a profile of the time-dependent decrease in the pressure, temperature and 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):

CD = 0.72

A = 0.001363 ft2

V = 51.4 ft3

M = 16.04

k = 1.307

T0 = 520 °R

P0 = 493,920 lbs/ft2 absolute

The resulting expression is:

P2 = [(5.3329 × 10-4 )(t2 - t1) + P1-0.1174 ] -8.5179 (1a)

For the Rasouli and Williams model, equation (1a) was then used to obtain P2 values for each value of (t2 - t1). The corresponding T2 temperature values were obtained from this expression for the isentropic expansion or compression of an ideal gas:

(T2 / T1) = (P2 / P1)(k -1)/k (3)

and the weight of gas (W, in pounds) remaining in the source vessel at the end of each increment of time (t2 - t1) was obtained from the universal gas law expression:

W = P V M / R T (4)

The Bird, Stewart and Lightfoot model becomes specific for this example by substituting these values into equation (2):

CD = 0.72

A = 0.001363 ft2

V = 51.4 ft3

k = 1.307

P0 = 493,920 lbs/ft2 absolute

d0 = 9.861 lbs/ft2

The resulting expression is:

t = 402.1(F-0.1535 - 1) (2a)

which can be re-arranged to obtain:

F = [1 + (0.002487) t ]-6.5147 (2b)

For the Bird, Stewart and Lightfoot model, equation (2b) 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 (3) for the isentropic expansion or compression of an ideal gas can be manipulated and re-arranged to obtain the following expressions:

P = P0 Fk (5)

T = T0 Fk-1 (6)

The corresponding P and T values were calculated, using equations (5) and (6), 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.