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Calculating Accidental Release Flow RatesFrom Pressurized Gas Systems

Milton R. Beychok, Consulting EngineerNewport Beach, California, USA

When determining the consequences of accidental release flow rates from pressurized gassystems, it is important to select the appropriate type of air pollution dispersion model. Forreleased gases which are lighter than or equal to the ambient air density, Gaussian dispersion

models as described in Beychok's text1should be used. For released gases which are heavier

than air, a dense gas model such as SLAB2or DEGADIS3should be used.

It is also important to determine realistic flow rates for accidental release scenarios selected fordispersion modeling. Most offsite consequence analyses have used accidental releasesdetermined 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 releasefirst occurs. The initial instantaneous flow rate from a leak in a pressurized gas system orvessel is much higher than the average flow rate during the overall release period because thepressure and flow rate decrease with time as the system or vessel empties. Much of the currenttechnical literature on accidental release source-term models fails to offer guidance on how tocalculate the average flow rate ... and that may explain why so many offsite consequenceanalyses 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 forcalculating the time-dependent decrease in pressure, temperature and weight of gas in apressurized gas system or vessel during an accidental release.

It should be emphasized that the source-term models discussed in this article are onlyapplicable to systems or vessels containing pressurized gases with very low atmosphericboiling 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 atchoked flow conditions during most of the time that the system or vessel is emptying due to anaccidental release.

The Rasouli and Williams source model:

The Rasouli and Williams4source-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 whenthe 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 (cp/ cv ). For many

gases, k ranges from about 1.1 to about 1.4, and so choked gas flow usually occurs when thesource gas pressure is about 25 to 28 psia or greater (see Table 1). Thus, the large majority ofaccidental gas releases will usually involve choked flow.

As originally published, the Rasouli and Williams model was in a form specific for methane gasreleases 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:

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(1)

where:


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The Bird, Stewart and Lightfoot source model:

The Bird, Stewart and Lightfoot5source-term model for choked gas flows from a pressurizedgas system was published in 1960 in its generalized form which was re-arranged to obtainEquation (2) below:

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, the

temperature and the weight of gas in a vessel storing methane gas at 60 F and 3,430 psia whena 0.5 inch diameter leak occurs.

The Rasouli and Williams modelbecomes specific for this example by substituting these valuesinto equation (1):

P1

P2t1

t2C

A

V

kgc

RMT0

P0

= the gas pressure in the source vessel at t1, in lbs / ft2absolute

= the gas pressure in the source vessel at t2, in lbs / ft2absolute

= any time after leak flow starts, in seconds

= any time (later than t1) after leak flow starts , in seconds

= coefficient of discharge

= area of the source leak, in ft2

= volume of the source vessel, in ft3

= cp/ cv= gravitational conversion factor of 32.17 ft / s2

= universal gas law constant of 1545 (lbs / ft2)(ft3) / (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 / ft2 absolute

(2)

where:t

F

VC

Ak

gc

P0

d0

= 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 ft2

= cp/ cv

= gravitational conversion factor of 32.17 ft / s2

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

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

C = 0.72

A = 0.001363 ft2

V = 51.4 ft3

M = 16.04 lb/lbmolk = 1.307

T0= 520 R

P0= 493,920 lbs/ft2absolute


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The resulting expression is:

For the Rasouli and Williams model, equation (3) was then used to obtain P2values for each

value of (t2 t1). The corresponding T2temperature values were obtained from this expression

for the isentropic expansion or compression of an ideal gas:

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

The Bird, Stewart and Lightfoot modelbecomes specific for this example by substituting thesevalues into equation (2):

The resulting expression is:

which can be re-arranged to obtain:

For the Bird, Stewart and Lightfoot model, equation (7) was used to obtain F values for eachvalue of time t since the initiation of flow through the leak. The corresponding values of W foreach 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 manipulatedand re-arranged to obtain the following expressions:

The corresponding P and T values were calculated, using equations (8) and (9), for the F valueobtained at each value of time t.

The comparative profiles yielded by the two modelsare tabulated in Table 2 and it is obviousthat the two models produced identical results. As can be seen in Table 2, the initial methanerelease rate during the first 30 seconds is (507 - 317) / 30 = 6.3 lbs/second and the rate duringthe last 30 seconds is (18 - 14) / 30 = 0.1 lbs/second ... after which only 2.65 percent of the initial507 lbs of methane remains in the vessel.

P2= [(5.3329 10 4)(t2 t1) + P1

0.1174] 8.5179 (3)

(T2/ T1) = (P2/ P1)(k 1) / k (4)

W = P V M / R T (5)

C = 0.72

A = 0.001363 ft2

V = 51.4 ft3

k = 1.307

P0= 493,920 lbs/ft2absolute

d0= 9.861 lbs/ft3

t = 402.1(F 0.1535 1) (6)

F = [1 + (0.002487) t ] 6.5147 (7)

P = P0Fk (8)

T = T0F(k 1) (9)


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It can also be seen that the overall average release rate is (507 - 14) / 300 = 1.6 lb/second, whichis 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 asthe profile of time versus the gas release rate. It is quite obvious that the decay of source vesselpressure and of the gas release rate is not linear.

References:

(1) Beychok, M.R., Fundamentals of Stack Gas Dispersion, published by the author, Irvine,California, USA, Fourth Edition, 2005

(2) Ermak, D.L., User's Manual for Slab - An Atmospheric Dispersion Model for Denser-Than-Air Releases, Lawrence Livermore Nat'l. Laboratory, Livermore, Calif., USA, 1990

(3) Spicer, T. and Havens, J., User's Guide for the Degadis 2.1 Dense Gas Dispersion Model,EPA-450/4-89-019, Research Triangle Park, North Carolina, USA, 1989

(4) Rasouli, F. and Williams, T.A., J. Air & Waste Management Association, March 1995

(5) Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons,New York, New York, USA, 1960

Table 1

Table 2

Stored gas cp/ cv

Storage pressure at whichgas flow through a leakwould be choked flow

ButanePropaneSulfur DioxideMethaneAmmoniaChlorineCarbon MonoxideHydrogen

1.0961.1311.2901.3071.3101.3551.4041.410

25.1 psia or greater25.4 psia or greater26.8 psia or greater27.0 psia or greater27.0 psia or greater27.4 psia or greater27.9 psia or greater27.9 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

03060

90120150180210240270

3,4301,8591,050

615372231147966443

520450394

347309276248224204186

507317205

136936446332418

3,4301,8591,050

614371231147966443

520450394

347308276248224204186

507317205

136926446332418

1.00000.62580.4041

0.26820.18240.12680.08980.06470.04740.0352


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