A mathematical model for the prediction of overpressures generated in totally confined and vented explosions

Nineteenth Symposium(International) on Combustion/The Combustion Institute, 1982/pp. 645-653

A MATHEMATICAL M O D E L FOR THE P R E D I C T I O N OF OVERPRESSURES G E N E R A T E D IN TOTALLY C O N F I N E D A N D V E N T E D EXPLOSIONS M. FAIRWEATHER AND M. W. VASEY British Gas Corporation, Midlands Research Station, Solihull, England A mathematical model for the prediction of overpressures generated in totally confined and vented explosions is described. The model may be applied to explosions of any gaseous fuel/oxidant mixture, and its application to combustion in cubic or cuboid geometries is discussed. Sample solutions of the model are presented, together with comparisons with experimental data and alternative prediction methods. Recommendations as to the correct choice of a turbulent burning velocity for vented explosions are made.

1. Introduction The ability to predict overpressures arising from explosions in industrial plant and in buildings is of value in producing risk assessments, in designing explosion reliefs and in the investigation of incidents. Such predictions are still to a large extent based on empirical formulae, for example those reviewed by Marshall, 1 although the range of application of the expressions available is inevitably restricted. Various authors have therefore derived mathematical models based on simple thermodynamic assumptions, 2 or models which are limited to relatively simple geometries. 3 This paper describes the formulation and application of a mathematieal model developed to handle totally or partially confined gas explosions in cubic or cuboid vessels, the type of geometry most frequently encountered in practice. The model also allows accurately for the thermodynamics of the combustion process, and accordingly may be applied to explosions of any gaseous fuel-oxidant mixture. The precise form of the pressure-time relationship during the course of an explosion depends on the enclosure and properties of the flammable mixture. Previous work has demonstrated (a) the single peak pressure-time structure of totally confined explosions and explosions in enclosures with initially uncovered vent areas, and (b) a double peak structure for the ease of vessels with initially covered vent areas. 4 Recent experimental investigations5 have also revealed the existence of a further, late-time, oscillatory pressure peak. However, dominant oscillatory peaks have only been observed for poorly vented, large scale explosions, where the enclosure is completely filled with high burning velocity mixture. For the accidental release of methane or methane-rich fuels in industrial situations the oc645

currence of such a dominant peak is unlikely. Thus, in many practical situations, modelling of the single or double peak phenomena referred to above is sufficient to enable the accurate prediction of maximum explosion pressures. In the present model, therefore, this rapid unstable combustion has not been considered. 2. Description of the Mathematical Model 2.1 Governing Equations The most important principle used in the model is the first law of thermodynamics, ensuring that energy is conserved during the combustion process. This law may be written in the following form for the case of a vented, confined explosion: Internal Energy of Unburnt Gas within the Enclosure

Internal Energy of + within Burnt Gas the Enclosure

Thermal Initial Energy Lost Chemical + from the = Energy of Burnt the Products Reactants

Chemical Energy of Unburnt + Gas within the Enclosure Enthalpy of Vented Unburnt Gas

Chemical Enthalpy of Energy of - Vented - Vented Burnt Gas Unburnt Gas. (1) This equation holds at any time. Evaluation of the various terms in Eq. (1) is now considered.

DETONATION AND EXPLOSION

646

In the circumstances of a gaseous explosion the unburnt gas, which is substantially air, behaves as a gas which is both ideal and polytropic. For such a gas the adiabatic exponent is constant which enables the internal energy per unit mass of the unburnt gas to be written as

(Tu - To) Cu = R ('r - i) Mu'

(2)

TO is the initial temperature of the reactants, norreally 288 K, and is taken as the base temperature for all internal energies. Because the pressure inside a confining enclosure is equalised at the speed of sound and the rates of pressure rise considered herein are relatively slow, the pressure inside the vessel is assumed to be uniform at any time. Reactants and products will therefore have uniform temperatures throughout their respective volumes, and Eq. (2) may be used to calculate the internal energy of the unburnt gas provided its temperature, which will rise due to isentropic compression, is known. The polytropic assumption cannot, however, be applied to the burnt products because of the existence of dissociation reactions at temperatures above 1600 K. Chemical equilibrium was therefore assumed to prevail throughout the burnt gas mixture, and equilibrium compositions and mixture total enthalpies calculated using the methods outlined by Davies and Toth. 6 The variation of enthalpy with temperature was then supplied to the model in terms of a polynomial fit, and the specific internal energy of the burnt gas calculated from the ideal gas relationship Ub = hb _ R (T1, - To) Mb

confined explosions. The neglect of the latter contributions is further justified by the fact that hot products only come in contact with the vessel walls late in the event. To calculate radiative losses the emissive characteristics of the combustion products were modelled by assuming the gas to be a mixture of one gray and one clear gas. The surfaces of the enclosure were further assumed to be gray. The rate of heat loss may then be defined by7 dq = -ag Abo"(T~ - ~ ) --+-% Ae

-1

where the parameter ag is given by

(5) a g - 2 % - ~2b' Gas emissivities were supplied to the model in the form of polynomials in beam length and temperature. The effect of pressure on emissivities was also allowed for. The mass rate of discharge from the confining volume to the surrounding atmosphere via the vent is given by the standard orifice equations, s'3 Two possible expressions exist, depending on whether the discharge velocity is subsonic or sonic. If the pressure ratio P/Po is less than the critical pressure ratio, [('y + 1)/2] ~/(~:-I), venting is subsonic and

(1

(3)

Total enthalpies were again calculated with a base at initial conditions. Enthalpy data was in fact determined at atmospheric pressure, and the suppressing effect of increased pressures on dissociation reactions was therefore neglected. This effect should be fairly small, however, for the relatively low pressure levels for which the model was intended. The internal energy of the burnt gas can be found from Eq. (3) provided its current temperature is known. The chemical energy per unit mass of the reactants may be found from thermodynamic tables for net calorific values. This quantity represents the difference between the energies of formation of the reactants and the products. The term for energy loss is used to allow for heat radiated from the burnt gas to the containing vessel, convective and conductive losses being negligible because of the small time scales involved in

(4)

If P/Po is greater than the critical pressure ratio venting is sonic and dt

CoAv ~/Pp

.

(7)

The mass of unburnt and burnt gas vented at any time may be determined by repeated calculation and summation of Eq. (6) or (7), depending on the discharge velocity, at each time-step in the numerical solution of the descriptive equations. The total current enthalpy of the vented gas and the energy loss due to radiation are similarly found by repeated summation so that the energy conservation equation may now be written in full as R(T,, - To) mu

(y~ - 1)Mu

+ mb

[h b

R (TM~ b _-- To/]

MATHEMATICAL MODEL FOR THE PREDICTION OF OVERPRESSURES

this was achieved using a fourth order Runge-Kutta technique.

dq dt = m,C, + muC. - fo tc -~

2.2 Flame Shape Assumptions

fro R~lu(T. - To) dm.~ - - dt

Jo

(-/.- 1) M.

-

dt

dmb~ fo tc hb dt dt

-

-

muvCn . (8)

Equations must now be derived which enable the mass and volumes of the unburnt and burnt gases to be calculated at any instant in time. For a given burning velocity the rate of production of mass of burnt gas may be written as

dmb dt - as~ us..

(9)

The current mass of unburnt gas can be found from the mass of burnt gas and the mass of gas vented using the equation

mu + mb = mr -- m.~ -- mb~,

(10)

and the volume of burnt gas from V, + Vb = V,.

647

(11)

In hazard analysis it is common practice to identify worst case situations. In dealing with explosions in cubic or cuboid vessels, therefore, ignition central to the enclosure is assumed. This gives rise to the maximum possible flame area and hence rate of production of burnt mass. In the model described it is necessary to calculate the flame area, given the volume of burnt gas, at each time-step in the numerical solution in order to determine the rate of production of mass of burnt gas. A spherically expanding flame front was therefore assumed to arise from ignition central to the enclosure (Fig. l(a)), for which the calculation of surface area from volume is elementary. Once this expanding sphere has reached the enclosure walls, however, the volume of burnt gas begins to take the form of a sphere with truncated spherical caps (Fig. l(b)), the volume of these caps increasing with time. An iterative routine was therefore included in the model to enable the calculation of an effective flame radius, and hence surface area, from the burnt gas volume supplied by the model once the flame diameter had exceeded the smallest dimension of the vessel.

The volume of unburnt gas appearing in Eq. (11) may be derived from the following relationship which assumes that compression of the unburnt gas from its original volume and pressure is isentropic:

(

p~l/-~, V~m, Po/ -V-~, m-"~

(12)

In order to form a complete mathematical description of the system it only remains to define equations which enable the temperatures of the unburnt and burnt gases to be calculated from the other thermodynamic variables of the system. These equations are supplied by the ideal equations of state of the two media in the form PV. -

m, RTu M.

mbRTb VVb -

Mb

(a)

(b)

(e)

[d)

(13)

(14)

The model described is now mathematically complete and can be solved numerically for the time variation of all of the thermodynamic variables, including the pressure, in the system. Since only ordinary differential equations are involved,

FIG. 1. Diagrammatic representation of flame shape in cuboid enclosures. (a) Initial spherical combustion phase, (b) flame front in contact with vessel faces, (e) spherical/cylindrical transition, (d) cylindrical combustion phase.

648

DETONATION AND EXPLOSION

Once the flame has reached an enclosure edge where the vessel walls join (Fig. l(c)), the calculation of flame area by the method described becomes increasingly complex. It is necessary to continue calculations beyond this point, however, particularly for cuboid vessels with one side length much larger than the remaining dimensions as a large amount of unburnt gas still remains when the flame has reached the first enclosure edge. Beyond this point the flame was therefore assumed to expand in cylindrical symmetry (Fig. l(d)). Thisga pproach has been used previously by Chappell, although the latter author maintained the flame radius constant at the geometry transition leading to a discontinuity in the pressure profile. This discontinuity does not occur in predictions of the present model since the burnt gas volume, and not the flame radius, is maintained constant at the transition. This does lead to a slight fall in the calculated flame radius at this point, although this decrease must be considered in terms of the change in meaning of this quantity at the transition. Finally, the surface area of burnt products required in the calculation of radiative heat losses must be distinguished from the flame area. Thus when the expanding flame front reaches the sides of the vessel the flame area will reduce as noted above. When this occurs the burnt products are still in contact with the enclosure surfaces, however, and radiative losses from these areas must be allowed for. 2.3 Unburnt and Burnt Gas Venting For vented explosions the model assumes that all vent areas are square and situated in the centre of a specified enclosure face. Venting may occur for unburnt or burnt gas, or for a mixture of the two. Once any vent cover has been relieved at a specflied venting pressure, therefore, unburnt gas is vented through the full area available until the spherical or cylindrical flame surface reaches the face containing the vent. At this point both unburnt and burnt gas are vented, and the model therefore contains procedures to determine the area of vent available to the unburnt and burnt gas at each time-step. This calculation is continued until burnt gas occupies the whole of the vent area, this procedure resulting in a smooth transition from unburnt to burnt gas venting. The restraining effects of gravity must also be allowed for in the case of roof located vents, frequently used in practice. Thus the weight of such a vent and the force exerted by the internal overpressure may be used in the equation of motion to calculate the rate of propagation of the vent cover away from the relief area. The distance the vent has travelled during the time immediately following vent relief may therefore be determined.

Thus an effective vent area, derived from the distance travelled and the perimeter length of the vent, can be calculated. This effective area is used until it has reached the full vent value as the relief is forced away from the enclosure. From this point the full vent area is used. 2.4 Burning Velocity Assumptions The laminar burning velocity of the combustible mixture is used for totally confined explosions of initially quiescent mixtures. In vented explosions, however, venting of unburnt gas causes flow induced turbulence which in turn gives rise to substantial increases in the mixture burning velocity. In practice, therefore, the burning velocity begins to increase above the laminar value as soon as any vent cover has been relieved, or from the start of combustion for an initially uncovered vent area, and steadily increases to an effective turbulent value. To complete the model a turbulent burning velocity needs to be specified. In the absence of reliable experimental data for the turbulent burning velocity of combustible mixtures in the geometries in question, an ad hoc approach of fitting theoretical predictions to data available in the literature was chosen. Comparisons showed that the extent to which the burning velocity is increased is dependent upon the vent breaking pressure, and therefore on the velocity of the vented unburnt gas as would be expected. Extensive comparison between predictions of the model and experimental data, existing empirical correlations and alternative models reported in the literature revealed that results based on a turbulent burning velocity of twice the appropriate laminar value appear to give closest agreement with alternative data for explosions in vessels with initially uncovered vent areas. This conclusion also applies to vents with low breaking pressures of the order of 10 mbar, and for enclosures with roof located vents held in place solely by their weight. For vessels with high breaking pressure reliefs, of the order of 100 mbar, predictions based on a turbulent burning velocity of three times the laminar value appear to be most appropriate. Some of the comparisons used to derive this dependence of turbulence enhancement on breaking pressure are reported in the following section. It should be noted that this ad hoc approach also allows for increases in the flame area caused by distortion of the flame front towards the vent area, after the relief of any cover, in vented explosions. All the data utitised in the comparisons referred to above was obtained for vessels containing no significant turbulence generating structures. For vessels containing significant turbulence generating obstacles a turbulent burning velocity of five times the laminar value has been found to be most ap-

MATHEMATICAL MODEL FOR THE PREDICTION OF OVERPRESSURES propriate. 1~ This choice is discussed further in the following section. Finally, the length of time over which turbulence build-up occurs, causing the burning velocity to increase to its fully turbulent value, was expressed as a function of vessel volume by optimising predictions and available experimental data.

3. Results and Discussion Due to the restrictions of space, a comprehensive validation of the model cannot be presented. This section therefore reports sample runs and applications of the model, and details a few of the comparisons used in order to derive the turbulent burning velocity-vent relief pressure dependence outlined in the previous section. It is hoped, however, that sufficient information is included to illustrate the use of the model as a predictive tool in the hazard assessment of confined explosions in a wide range of situations. Sample solutions of the model for both totally confined and vented explosions are presented in Fig. 2 in the form of overpressure-time profiles. All predictions were made for a 3 m • 3 m • 6 m vessel completely filled with a stoichiometric methane-air mixture, and for central ignition. Heat loss from the flame was allowed for. Figure 2(a) was derived using a constant laminar burning velocity throughout the course of the explosion, and effectively illustrates the large overpressures which can occur if no relief venting is used. Experimentally observed final pressures can be found in the literature for the totally confined combustion of such a mixture. The works of Bone and Townend, 11 and Sapko et al.,lz report a final absolute pressure of approximately 7.9 bar. Predictions of the model of 7.8 bar for explosions in equivalent vessels are in good agreement. Figure 2(b) shows profiles for the same mixture and enclosure as Fig. 2(a), but fo~: a vent area of

NO VENT

i

O 10 UNCOVEREO VENT AREA

~0o .

' 06

~

o

01

02

03

O~

OS

0S

07

liME AFTER IONITION.S (a)

F1c. 2. Overpressure/time profiles for (a) totally confined explosion, (b) vented, confined explosions.

649

2.25 m 2 situated in a 3 m x 3 m face. The single peak curve was derived for an initially uncovered vent area, and accordingly with a turbulent burning velocity of twice the laminar value. The second curve was obtained for the vent area initially covered and a relief breaking pressure of 100 mbar. A turbulent burning velocity of three times the appropriate laminar value was therefore used in this case once the vent had broken. The large reduction in final overpressure caused by the inclusion of a vent area is immediately apparent from these profiles. In the uncovered vent area case the burning velocity begins to increase to its turbulent value from the onset of combustion, and the rate of pressure rise surpasses that of the initially covered case once the burning velocity has increased sufficiently. In the latter case pressure rises more slowly, with the laminar burning velocity, until the vent is relieved at 100 mbar. Pressure then falls whilst venting predominates, until the flame area and burning velocity have increased sufficiently for the rate of pressure rise to equal, and eventually surpass, the rate of pressure relief. This causes the pressure curve to pass through a minimum and then increase rapidly. The rate of increase of pressure in this second phase is greater than for the uncovered vent area situation since a turbulent burning velocity of three times the laminar value has been employed in the former case, this also leading to a higher maximum pressure. For the geometry in question, both of the maximum pressures shown in Fig. 2(b) occurred as a result of reducing flame area. Once these maxima have been reached pressure begins to fall steadily, and both pressure-time profiles exhibit a much increased rate of pressure drop when burnt gas venting begins. The latter finding is in agreement with the theoretical predictions of Bradley and Mitcheson. 3 It is interesting to note that in the case of the vented explosions described above, radiative losses from the burnt products had a negligibly small effect on the explosion pressures generated. This is also the case for all other results relating to vented explosions described in this paper. It is only for explosions in vessels with little or no vent area, when peak pressures occur at a very late time, that radiative losses have a significant effect on predicted overpressures. Thus in Fig. 2(a), the maximum overpressure is increased by approximately 24% when such losses are excluded from the calculation. A comparison between theoretically determined and experimentally observed 13 pressure-time profiles is shown in Fig. 3 for the case of a vented methane-air explosion in a cubic vessel of 2.55 m 3 volume. An initially covered vent area of 0.56 m z was employed, with a vent breaking pressure of 112 mbar. Agreement between the time of occurrence and magnitude of the maximum pressure at-

DETONATION AND EXPLOSION

650

--'--

01C 21

=-

/i

EXPERIMENTAL

I

VENT BREAKING PRESSURE

-#

} O 0 =.

GO lIME

~ , F T E R IGNJ110N .S

FIG. 3. Comparison between theoretical and experimental overpressure/Ume profiles for a vented explosion. rained is very good, although only a single peak exists for the experimental results whereas predicted pressures show the classic double peak form. It was observed experimentally that burnt gas was vented almost immediately after the vent covering was relieved, thus causing the two experimental peaks to merge into a single peak. The second theoretieal peak also occurs when burnt gas is vented. It would therefore appear that flame distortion, due to the rapid venting of unburnt gas once the vent cover is removed, is responsible for the disappearance of the first peak from the experimental results. As noted previously, however, flame distortion is allowed for to some extent in the model by the choice of a turbulent burning velocity, and hence agreement between the maximum pressures attained is good. A comparison with two of the most commonly used empirical correlation equations for the prediction of overpressures in vented confined explosions is shown in Fig. 4, The equations used are due to Cubbage and Marshall, 14 and Rasbash et al., 1~ detailed consideration of these equations, together with their ranges of application, being given in reference 1. These formulae predict the maximum pressure generated irrespective of whether it is a first or second peak, and were applied, together with the present model, to the case of a 3 m side length cube with a vent area of 2.25 m 2 located in the vessel roof. A venting pressure of 100 mbar and a mass per unit area of the vent cover of 3 kg m -z were also used. The variation of first and second peak pressures with gas concentration predicted by the model are shown in this figure. At the extremes of the flammability range, first peak pressures are in good agreement with the formula of Cubbage and Marshall. Some slight underestimation of the predictions of the latter equation is apparent before second peak pressures become

dominant. However, around the stoichiometric concentration the second peaks predicted by the model are seen to be conservative estimates of overpressure. As has been noted previously, 1 the expression of Rasbash et al. tends to overestimate the pressures generated, and hence overestimates the overpressures predicted by the present numerical solutions and the expression of Cubbage and Marshall. Comparisons were also made with the simplified venting theory of Bradley and Mitcheson a who use the dimensionless parameter ,it/S o in order to correlate their results. It is interesting to note that for explosions of methane-air and propane-air mixtures in cubic vessels the present model was in good agreement with the latter theory. For non-cubic vessels, however, agreement was poor because of the spherical geometry used by these authors, correlation of peak overpressures predicted by the present model being more successful in terms of the more commonly used vent coefficient, K, as defined in the nomenclature. Finally, the model may be readily modified in order to predict the overpressures which arise from pocketed, or low energy content, explosions, Comparisons were made with the full scale explosion tests performed by the U.S. Coast Guard 15 in the pump room of a merchant tank vessel. Two particular propane tests, where the flammable mixture occupied approximately 14% of the room volume, were modelled in detail by replacing the pump room volume by an equivalent cuboid in the model. The first test examined was with the volume flooded to a depth of approximately 2 m in order to cover the majority of turbulence generating structures within the room. A measured maximum pressure of 234 mbar compares well with a predicted value of 241 mbar, derived using a turbulent burning velocity of twice the appropriate laminar value as the room contained an open vent area. For the

,

O20

OIS

PREDICTED SECOND PEAK PRES.~J~E CUBBAGE AND

0 10

G&S CONCENTRATION, % ~ v

FIG, 4. Variation of overpressure with mixture gas concentration as predicted by the present model and empirical correlation equations.

MATHEMATICAL MODEL FOR THE PREDICTION OF OVERPRESSURES case of an unflooded pump room, where significant turbulence generating structures were present, measured and predicted overpressures were 717 mbar and 714 mbar respectively. The latter pressure was derived using a turbulent burning velocity of five times the laminar value in order to allow for the enhancement in burning velocity caused by turbulence generated by venting and flow over structures within the room. This figure of five times the laminar value is in agreement with the recommendation of Rasbash et al. 10

Greek Symbols ~/

Adiabatic Index Gas emissivity Gas emissivity evaluated at twice the mean beam length Density Stefan--Boltzmann constant

r p ~r

Subscripts b

Burnt gas Vented burnt gas Enclosure Flame Ambient conditions Refers to side of enclosure in which vent is located Unburnt gas Vented unburnt gas Vent Pre-ignition state

by 4. Conclusions The model described has been validated by an extensive comparison with experimental data and alternative prediction methods available in the literature, some of these comparisons being reported herein. This has enabled recommendations to be made for the correct choice of a turbulent burning velocity in various types of enclosure. The generality of the model will also allow its extension to cover the flame distortion which occurs along the venting streamlines, and hence enable accurate prediction of the time of onset of burnt gas venting. Any further improvements may then be coneerned with the modelling of pressure instabilities which are of importance in poorly vented explosions of high burning velocity mixtures. In its present form, however, the model should provide an extremely valuable tool in the hazard assessment of confined explosions of any concentration of a gaseous fuel-oxidant mixture in vessels of a very general geometry.

Nomendature A A CD Cn h K M m P q R So S, T t tc U V

Area Vent area ratio = CDAv/Ae Vent discharge coefficient Calorific value of the mixture Total enthalpy of the mixture per unit mass Vent coefficient = As/Av Molecular weight mass Pressure Heat loss Gas constant Ratio of gas velocity ahead of flame front to acoustic velocity in unburnt gas just after ignition Burning velocity, the speed of the flame relative to the unburnt mixture Temperature Time current time Internal energy of the mixture per unit mass Volume

651

e f o s u

uv V x

Acknowledgment This paper is published by permission of British

Gas. REFERENCES 1. MARSHALL, M. R.: I. Chem. E. Symposium Series No. 49, p. 21, 1977. 2. YAO, C.: A. L C h . E . , 8th Loss Prevention Symposium, p. 1, 1973. 3. BRADLEY, D., MITCHESON, A.: Combustion and Flame 32, 221 (1978) and 32, 237 (1978). 4. CUBBAGE,P. A., SIMMONDS,W. A.: Gas Council Research Communication No. GC 23 (1955) and No. GC 43 (1957). 5. SOLBERG, D. M., PAPPAS,J. A., SKRAMSTAD, E.: Loss Prevention and Safety Promotion in the Process Industries, 3rd International Symposium, p. 1295, 1980. 6. DAVIES, R. M., TOTH, H. E.: Proc. Fourth

Symposium on Therm. Prop., p. 350, 1968. 7. HoTrEL, H. C., SAROFIM,A. F.: Radiative Heat Transfer, McGraw-Hill, 1967. 8. BRITISH STANDARDSINSTITUTE: British Standards

1042, Part 1, 1964. 9. CHAPPELL, W. G.: A. I. Ch. E, 8th Loss Prevention Symposium, p. 76, 1973.

10. RASBASH,D. J., DRYSDALE, D. D., KEMP, N.: I. Chem. E. Symposium Series No. 47, p. 145, 1976. 11. BONE, W. A., TOWNENO, D. T. A.: Flame and Combustion in Gases, Longmans, 1927. 12. SAPKO, M. J., FURNO, A. L., KUCHTA, J. M.: Flame and Pressure Development of LargeScale CH4-AIR-N~ Explosions. Buoyancy Ef-

652

DETONATION AND EXPLOSION

fects and Venting Requirements, U.S. Bureau of Mines, Report of Investigations 8176, 1976. 13. MARSHALL, M. R.: Private Communication. 14. CUBBAGE, P. A., MARSHALL, M. R.: I. Chem. E. Symposium Series No. 39a, p. 196, 1974.

15. RICHARDS, R. C.: Development of Explosion Suppression System Requirements for Shipboard Pump Rooms, U.S. Coast Guard, Report No. CG-D-79-76, 1976.

COMMENTS H. Pasman, Prins Maurits Labs, Netherlands. In line with various comments I would like to emphasize that on the basis of our experiments acoustic wave-flame coupling, in particular from the time of vent opening, seems to be able to disturb the venting process in a 'number of cases, and to give rise to sharp secondary or tertiary pressure pulses. Therefore interpolation to larger volumes on the basis of the type of model you presented could be risky. Could you comment on this?

into an enclosure is most likely to form a pocket of explosive mixture which does not completely fill the vessel. In this case, the time necessary to establish acoustic wave-flame coupling after burnt gas venting begins will not be achieved in most situations due to the completion of combustion within the vessel.

Author's Reply. The mathematical model was formulated in order to describe the pressure-time history of totally confined and vented explosions in empty vessels, and to thereby allow further understanding of the physical mechanisms which give rise to the various pressure peaks observed in such explosions. The dependence of the rate of production of burnt mass on flame distortion and increased burning velocity effects, caused by venting, was derived by comparing predictions with available experimental data. This data was obtained in both large and small scale experiments. We therefore believe that the model in its present form is directly applicable to the prediction of maximum explosion overpressures over a wide range of enclosure sizes. As noted in the question, however, the model does at present neglect the occurrence of late-time acoustically driven pressure peaks that have been observed by some authors, l'z Available experimental evidence suggests that large scale vessels, and hence large time-scales, are necessary for the acoustic wave-flame coupling mechanism to occur. 3 However, the pressure peaks caused by this mechanism have, to the authors' knowledge, only been observed in poorly vented explosions in empty vessels completely filled with flammable mixture. In practical situations where the enclosure is filled with process equipment, the presence of obstacles in the flow field may well disrupt the coupling of pressure waves with the acoustic modes of the enclosure. Also, as noted in tests performed at the Prins Maurits Laboratory, z inclusion of absorbent material within the vessel effectively removes dominant acoustically driven pressure peaks. Finally, in practice the accidental release of a flammable gas

1. SOLBEaG, D. M., PAPPAS, J. A., SKRAMSTAD, E.: 18th Symposium (International) on Combustion, p. 1607, 1981. 2. ZEEUWEN, J. P., International Specialist Meeting on Fuel Air Explosion, November 4-6, 1981, Montreal, Canada. 3. LEE, J. H. S., GUIRAO, C. M., Plant/Operations Progress, 1, p. 75, 1982.

REFERENCES

J. Lee, McGill University, Canada. The weakest link in our ability to predict the overpressure time development in closed or vented vessels is our ignorance of the burning rate. Since the turbulent burning velocity and flame area are strong functions of the gasdynamic flow field what we need is a flame law that gives the flame velocity as a function of the thermodynamic and fluid dynamic variables in the unbounded mixtures. For that we would have to rely on experiments. Hence at present I see that all models must eventually have to use some experimental input which is only as good as the particular scale and geometry in which t h e experiments are actually performed. I see no way of getting around doing a lot of large scale experiments at tlle present to get valid empirical laws. Author's Reply. We agree with Professor Lee's comments, and a consideration of the points mentioned leads on directly from our reply to the previous question. If acoustically driven pressure peaks do not occur in practical situations, as for example in the U.S. Coast Guard tests ~ referenced in the paper (where combustion was of a flammable pocket of gas and

MATHEMATICAL M O D E L FOR THE PREDICTION OF OVERPRESSURES obstacles were present in the flow field), the model may be applied to predict the maximum explosion overpressure. In applying the model to such situations however, it is necessary to allow for the increase in the combustion rate due to flame distortion and turbulence effects caused by flow over the obstacles. In this case it is necessary to rely on experimental data from tests in similar scale enclosures, with similarly distributed internal structures, to that under consideration in order to derive a factor with which to enhance the combustion rate. It is our experience in applying the model that if a

653

realistic maximum flame speed can be chosen, and an enhancement factor derived therefrom, then the model will yield reliable predictions of maximum overpressure.

REFERENCE 1. RICHARDS, R. C.: D e v e l o p m e n t of Explosion Suppression System Requirements for Shipboard Pump Rooms, U.S. Coast Guard, Report No. CG-D-79-76, 1976.