Home     How To Buy     Contents     Site Map


Calculating Accidental Release Flow Rates
From Pressurized Gas Systems

Milton R. Beychok, Consulting Engineer
Newport Beach, California, USA

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 text1 should be used. For released gases which are heavier than air, a dense gas model such as SLAB2 or DEGADIS3 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 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.

It should be emphasized that the source-term 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 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 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 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:
 P1
 P2
t1
t2
C
 A
 V
 k
 gc
 R
M
T0
P0
 
= the gas pressure in the source vessel at t1, in lbs / ft2 absolute
= the gas pressure in the source vessel at t2, in lbs / ft2 absolute
= 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

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 which was re-arranged to obtain Equation (2) below:

(2)
 
where:
t
F
 V
C
 A
 k
 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 ft3
= 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 

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 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 ft2
V   = 51.4 ft3
M  = 16.04 lb/lbmol
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 (3)

For the Rasouli and Williams model, equation (3) 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 (4)

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 (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 ft2
V   = 51.4 ft3
k   = 1.307
P0 = 493,920 lbs/ft2 absolute
d0 = 9.861 lbs/ft3

The resulting expression is:

t = 402.1(F – 0.1535 – 1) (6)

which can be re-arranged 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 re-arranged to obtain the following expressions:

P = P0 F k (8)
T = T0 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.

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



Stored gas


cp / cv
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


Table 2

  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





>> Was this article useful? The book "Fundamentals of Stack Gas Dispersion" is better yet. <<
>> Click on "How To Buy" (see below) for price and method of payment. <<


Click 
  HERE  
 to see the front cover of the book or
click any of these links to visit other parts of this site:
    Home     How To Buy     Contents     Site Map

Or click on one of these Feature Technical Articles:
    Error Propagation in Air Dispersion Modeling
    Air Dispersion Modeling Conversions and Formulas
    Source Terms For Accidental Discharge Flow Rates
    Air pollution dispersion modeling
    Petroleum Refining Processes
    Natural Gas Processing