Instantaneous Flow Rates Overstate Gas Release Determination

Milton R. Beychok
Consultant Irvine, Calif.

• Released gas flow rate caculations [36,353 bytes]

The average flow rate can be determined from two published source-term models that calculate the time-dependent decrease in pressure, temperature, and weight of gas during a release.

Need for calculation

The U.S. Environmental Protection Agency's (EPA) recent federal risk management program¹ has started a focus on off site consequence analyses for facilities handling designated toxic or flammable materials at or above those specified by the EPA.

EPA's draft guidance document² includes look-ups defining the impact of accidental releases. As noted in a recent article,³ these look-up tables may overestimate the impact by as much as a factor of ten.

Therefore, as suggested in that article, many facilities will need refined air-dispersion modeling to determine realistic impacts, and it is important to select the appropriate dispersion model.

Models

Gaussian dispersion models as described in Beychok's text⁴ should be used for released gases with densities lighter than or equal to the ambient air density. A dense gas model, such as SLAB5 or Degadis,⁶ should be used for released gases that are heavier than air.

Also, it is very important to determine realistic flow rates for accidental release scenarios selected for dispersion modeling.

In the past, most off site consequence analyses determined accidental release rates with "source-term models." These models 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 is much higher than the average flow rate during the overall release period because pressure and flow rate decrease with time as the system empties.

Most technical literature on accidental release source-term models fails to offer guidance on how to calculate the average flow rate. This may explain why many off site consequence analyses for pressurized gas releases have been based on initial instantaneous flow rates.

Rasouli and Williams

The Rasouli and Williams⁷ 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, PS/PA, is equal to or greater than [(k+1)/2]k/(k-1), where k is the gas specific heat ratio (cp/cv).

For many gases, k ranges from about 1.1 to about 1.4, and therefore, choked flow usually occurs when the source gas pressure is about 25-28 psia or greater (Table 1) [10456 bytes]. Thus, the large majority of accidental gas releases will usually involve choked flow.

As originally published, the Rasouli and Williams model was 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 provided by Rasouli, the errors were corrected and the model was generalized to obtain Equation 1.

The nomenclature in Equation 1 is as follows:

• CD = Discharge coefficient

• A = Leak source area, sq ft

• V = Source vessel volume, cu ft

• g = Gravitational constant, 32.17 ft/sq sec

• R = Universal gas constant, 1,545 (lb/sq ft)(cu ft)/(lb-mol)(?R.)

• M = Molecular weight of the gas

• k = cp/cv of the gas

• T0 = Initial gas temperature in the source vessel, ?R.

• P0 = Initial gas pressure in the source vessel, lb/sq ft

• t0 = Time of flow initiation through the leak, sec

• t1 = Time t0 or later, sec

• t2 = Time later than t1 , sec

• P1 = Gas pressure in the source vessel at t1 , lb/sq ft

• P2 = Gas pressure in the source vessel at t2 , lb/sq ft.

Bird, Stewart, Lightfoot

The Bird, Stewart, and Lightfoot source model⁸ for choked flow releases from a pressurized gas system was published in 1960 in its generalized form as Equation 2.

The nomenclature is as follows:

• t = Time since flow initiated, sec

• F = Fraction of the initial gas weight remaining in the source vessel at time t

• CD = Discharge coefficient

• A = Leak source area, sq ft

• V = Source vessel volume, cu ft

• g = Gravitational acceleration, 32.17 ft/sq sec

• k = cp/cv of released gas

• P0 = Initial gas pressure in the source vessel, lb/sq ft

• r0 = Initial gas density in the source vessel, lb/cu ft

Model comparison

To compare the two models, each was used to obtain profiles of the time-dependent decrease in pressure, temperature, and weight of gas in a vessel storing methane at 3,430 psia when a 0.5-in. diameter leak occurs.

Parameters needed for the Rasouli and Williams model (Equation 1) are M = 16.04, k = 1.307, CD = 0.72, A = 0.001363 sq ft, V = 51.4 cu ft, T0 = 520 ?R., and P0 = 493,920 lb/sq ft. Equation 3 is the resulting expression.

For the Rasouli and Williams model, Equation 3 was 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 (Equation 4), and the weight of gas, W, remaining in the source vessel at the end of each time increment (t2 - t1) was obtained from the universal gas law expression (Equation 5).

The substitutions in the Bird, Stewart, and Lightfoot model are k = 1.307, CD = 0.72, A = 0.001363 sq ft, V = 51.4 cu ft, P0 = 493,920 lb/sq ft, and r0 = 9.861 lb/ cu ft. Equation 6 expresses the results of the substitution and can be rearranged as Equation 7.

For the Bird, Stewart, and Lightfoot model, Equation 7 was used to obtain F for each t since the initiation of flow through the leak. The corresponding W values for each t were obtained by multiplying the original weight of gas in the source vessel, 507 lb, by the residual weight fraction F at t.

Equation 4 can be manipulated and rearranged to obtain Equations 8 and 9.

The corresponding P and T were calculated with Equations 8 and 9, for F obtained at each t.

Table 2 [44,851 bytes] shows that the profiles from each model are identical. Fig. 1 [22,877 bytes] graphically depicts the time-dependent vessel pressure and gas release rate profiles.

As can be seen in Table 2 and Fig. 1, the initial methane release rate during the first 30 sec is (507 - 317)/30 = 6.3 lb/sec and the rate during the last 30 sec is (18 - 14)/30 = 0.1 lb/sec (after which only 2.65% of the initial 507 lb of methane remains in the vessel).

Also shown is that the overall average release rate is (507 - 14)/300 = 1.6 lb/sec, which is very much less than the 6.3 lb/sec rate during the initial 30 sec.

References

1. Federal Register, Vol. 61, No. 120, June 20, 1996, p. 31668.

2. U.S. EPA, RMP Off site Consequence Analysis Guidance, Research Triangle Park, N.C., 1996.

3. Sung, H., "Refined Modeling for EPA's New Risk Management Program," Environmental Technology, September/October 1996.

4. Beychok, M.R., Fundamentals of Stack Gas Dispersion, published by the author, Irvine, Calif., 1994.

5. Ermak, D.L., User's Manual for Slab-An Atmospheric Dispersion Model for Denser-Than-Air Releases, Lawrence Livermore National Laboratory, Livermore, Calif, 1990.

6. 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, N.C., 1989.

7. Rasouli, F., and Williams, T.A., Journal Air & Waste Management Assoc., March 1995.

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