The prediction of a practical lower bound for ignition delay times and a method of scaling times-to-ignition in large reactant masses from laboratory data

Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 1775-1782

T H E P R E D I C T I O N OF A PRACTICAL L O W E R B O U N D FOR IGNITION DELAY TIMES A N D A M E T H O D OF SCALING TIMES-TO-IGNITION IN LARGE REACTANT MASSES FROM LABORATORY DATA B. F. GRAY School of Chemistry Macquarie University N.S.W. 2109, Australia AND

J. H. MERKIN AND J. F. GRIFFITHS Schools of Mathematics and of Chemistry The University of Leeds Leeds LS2 9JT, U.K.

A new theoretical basis is presented for the practically important problem of predicting ignition delays in systems undergoing spontaneous combustion. An approximate lower bound for the ignition delay is derived in the universally applicable form ~, = u~elu,(1 - e-a/u2,), where the dimensionless terms represent ignition delay time, ~i, ambient temperature, u~, and a defined temperature excess within the reactants, A. The predictions err on the side of safety. The application pertains to temperature ranges above criticality that are of practical importance, but the extent to which the ambient temperature exceeds that at criticality, which is extremely difficult to obtain accurately in marginally supercritical conditions, is not required. The prediction of the (long) ignition delays associated with large amounts of material under supercritical conditions during processing, storage or transport, from measured times to ignition of laboratory-scale samples, by use of the approximate theory, is also discussed. The size of the system is taken into account in a natural way by the choice of u~ (in supercritical conditions) at which the prediction of ti is to be made. Comparisons are made between the ignition delay times and their temperature dependence that are predicted from the approximate theory, calculated from numerical simulations of ignition in a system in which heat transport occurs by thermal conduction, and measured in laboratory-scale experimental studies of ignition in a finely divided, particulate solid. The experimental results are also used to show how ignition delays in large amounts of material being stored or processed at different temperatures may be obtained in a practical case.

Introduction Many substances (such as the cellulosic materials, wood chips, hay, bagasse etc, or others such as coal, wool, animal feedstocks, fertilizers, explosives) that are able to undergo exothermic decomposition or aerobic oxidation are stored in large piles over long periods at ambient conditions (usually around 290 K). Alternatively, they may undergo the final stages of processing, such as curing or drying, at elevated temperatures (400-500 K, say). In either case, internal temperatures may rise as a result of exothermic reaction, and spontaneous ignition may take place. 1 A reasonably well understood scaling

problem in spontaneous combustion is that of predicting the critical size of a large reacting mass at a given ambient temperature from laboratory tests on much smaller samples at much higher ambient temperatures. 2-5 Although problems can arise if wet exothermic reactions are involved and the tests are run above 373 K, so that water first evaporates leaving dry material to undergo thermal runaway, the principles of the extrapolation are soundly based. For a single exothermic reaction from which heat loss occurs by conduction alone, the scaling which takes into account both size and shape has been tested experimentally2 and would be considered to be accurate and well understood. 6

1775

IGNITION

1776

A problem of equal importance is the time required when a material is in a supercritical condition for it to self-heat to an extent which is unacceptable either for reasons of safety (i.e. to ignition) or of cost (i.e. spoilage of material). A prediction of the ignition delay or "time to ignition" could be extremely valuable, not only to define the safe lifetime of a stockpile but also to optimise chemical processing if there were to be economic advantages by intentionally operating in a supercritical condition. In general, the application is likely to be associated with ambient temperatures that do not exceed the critical value by very much (<20 K). A perturbation theory, based on very small perturbations from criticality, exists7-9 which gives the result ti ~ e -1/2, where 9 is the degree of supercriticality. However, this formula is of no practical use because (a) it requires 9 to be unrealistically small, and (b) even then the error of the prediction is not on the side of safety. Even for a single exothermic reaction the literature does not contain any satisfactory theory which enables one to answer the question, "If a small laboratory sample placed ATa above its critical ambient temperature, Ta,cr, takes ti minutes to self heat by 50 K, how long would a much larger body of the same material take to self heat by this amount ff it were ATe', above its critical ambient temperature?" The problem contains both of the ingredients, time dependence and scaling, because the most attractive practical route to the prediction of the behaviour of large amounts of material would be based on comparative, experimental measurements made in the laboratory. A b initio calculations require physical, thermochemical and kinetic data which may not be readily obtainable. We present here a novel approach to the prediction of ignition delay which is based on fundamental principles, yet depends only on the ambient temperature and size parameter of the system. The problem is formulated first, and numerical solutions are then presented for the case of a single exothermic reaction in an infinite slab. From these results we deduce an approximate, geometry independent calculation for the time to ignition which (a) is applicable at temperature ranges above criticality that are of practical importance, (b) errs on the side of safety, and (c) is much more accurate than the perturbation theory calculation. The theory is then tested against experimental results for a particulate, solid material undergoing spontaneous ignition in a cube. Mathematical Formulation The equation for the temperature T of the reacting material, neglecting reactant consumption, is, for slab geometry of thickness 2l

KO2T cOT -+ qAtre -~/RT = - O~z Ot '

(1)

in the region - l < ~ < I. For simplicity, the boundary conditions taken are the Dirichlet conditions T=Ta 0T --=0 0~

on

~=l,

(2a)

on ~=0(symmetry)

(2b)

with initial condition, T=T~

at t = 0 .

(2c)

Here K, q, A, E, R, c and ~r are the thermal conductivity, exothermicity, pre-exponential factor, activation energy., gas constant, specific heat and density of the reactant respectively, and ~ be constants for a given reacting material. The ambient temperature T~ and size I are the only physical parameters which can be varied (i.e. the control variables for a given system). To make Eq. (1) non-dimensional we use a set of dimensionless parameters and variables defined in such a way that (i) changes in one of the control variables (T~ or/) can be identified with changes in just one of the non-dimensional parameters and (ii) the non-dimensionalisation of t is independent of both Ta and 1. Thus, u=--,

RT

qA~rRt ~=

~c ,

E

x=-.

cE

(3)

1

Equation (1) then becomes 02u 0u ~0----z' x + h.e-1/~ = h--'0T

(4a)

subject to the boundary conditions u=ua,

on x = l ,

0u

--=0 0x

on x = 0 ,

(4b)

and initial condition, u=ua

at

~=0.

(4c)

The non-dimensional parameters k and Ua are defined by, qAtrR/z

RTa

KE

E

Note that the ambient temperature appears only in

1777

PREDICTION OF IGNITION DELAY TIMES ua and the length l only in h, while the non-dimensionalisation of t does not involve Ta or l. The problem defined by Eqs. (4) has been discussed by Burnell et al 1~ where it was shown that for all h > h i ( h i ~ 4.618), Eq. (4a) has three branches of stationary state solutions, the lower and upper of which are stable, and the middle branch of which is unstable (and represents the "watershed" above which ignition may take place when material is stockpiled at an initial temperature that is in excess of the ambient temperature. 11 The lower branch, which corresponds to subcritical reaction, exists for 0 < ua <- u~,cr, where the critical value u .... depends on h. For ua > Ua,cr the only stationary state is the upper branch solution, which corresponds to the "ignited" state. The purpose of the present work is to'investigate the behaviour of the solution of Eqs. (4) when the ambient temperature parameter Ua is slightly above criticality, i.e. Ua = Ua,cr + ~ with O < ~ < < 1, and, in particular, to examine the time to ignition in this case. The most useful estimate for *i would (a) involve just those variables which can be measured directly, (b) keep to a minimum the physical constants, the values of which need to be known with some degree of accuracy, and (c) not be too sensitive to imprecision in these values of the physical constants. In most practical situations (and in the experiments described below) the non-dimensional parameter h will be large (typically l0 w or higher) while Ua will be small (typically no larger than 0.05). With h large, Eq. (4a) shows that, initially at least, conduction effects will be small (except in the boundary-layer regions of thickness of 0(h ) at the edges of the slab), and so to a first approximation can be neglected. This leaves the temperature profile in the slab governed by the simple equation, .

oqtt

--

&r

= e -1/u,

.

u(O)

Uo = n, + A

(7)

where A will be some (not necessarily small) fraction of Ua. Using Eq. (7) as our basis for the definition of hi, from Eq. (6) we get, tta+A Ti = If |dUa

~Ua,cr +~+A el/Udu =

eX/Udu

(8)

dUa,cr+~

Equation (8) involves only Ua which, from (5), means that we need to have accurate values only for the ambient temperature T a and activation energy E, both of which can be obtained with some reasonable degree of accuracy for a given reacting system. Also, Eq. (8) is independent of the type of boundary conditions employed and is the same whether we assume the Dirichlet boundary conditions (lib) or use more general Robin boundary conditions. Moreover, since the effects of conduction within the reacting mass will be to reduce the rate of increase in the temperature at the centre of the slab, when we integrate Eq. (6) to obtain Eq. (8) for the time to ignition, this will give an underestimate for ~i (at least in the practical cases where Uo is not much larger than ua,cr), and hence will be in error on the side of safety. The integral in Eq. (8) cannot be evaluated analytically, though it is easy to do so numerically, using Simpson's rule for example. However, an approximation to the integral can be obtained on the assumption that Ua < < 1, as, ri = u]el/Ua{1 -- e - a / u ~ .

(9)

-1/2

=

u a

(6)

Use of this equation does n o t mean that we are discussing an adiabatic model. The heat losses are encapsulated in the initial conditions and the exact form of the boundary conditions for the full model (4a), which will be different for a different reacting mass. These conditions affect the calculated times to ignition through the corresponding values of Ua,c~ for each case. For practical purposes (e.g. in the experiments described below) the dimensionless 'time to ignition,' Ti, will be the time when the difference between the dimensionless temperature at the centre (uo) and the dimensionless ambient temperature has risen to some fixed, pre-determined value. In terms of our non-dimensional variables (Eqs. 3, 5) this will be when

A similar expression was derived by Gray and Harper 12 for times to ignition of reactants with spatially uniform internal temperatures. Finally, it remains to check that expression (9) does in fact, give a good approximation to "ri. To do this we integrated Eq. (4a), subject to boundary and initial conditions (4b) and (4c). Values of h that corresponded as nearly as possible to the experimental results described below were used. This meant taking h = 8 • 1011 to represent the 30 mm cube, h = 1.8 • 1012 to represent the 50 mm cube and h = 4.9 • 1012 to represent the 75 mm cube, with corresponding critical values of u . . . . = 0.0289, 0.0282 and 0.0273 respectively (with then Uo,c~ = 0.0300, 0.0292 and 0.0282, where Uo,cr is the critical value of u at the centre of the slab for the given value of Ua,cr). To fix ~i we took A = 0.003. This corresponds to an experimental temperature rise of 50 K. The numerical integrations confirmed our original conjecture that, in the initial development of the temperature profile, conduction effects are very small and can be neglected. This could be sustained for times for which Uo - ua remained rel-

1778

IGNITION

atively small, thus allowing us to use Eq. (9) to calculate "ri. Subsequently, the predicted temperature at the centre rose very rapidly and steep gradients developed in the temperature profile, with conduction effects then becoming much more important. It is shown in Fig. 1 that away from criticality, ua - Ua,cr > 1 • 10-3, the approximate solution based on Eq. 9 (solid line) agrees with the exact solutions (solid symbols for A, k = 8 • 1011; B, k = 1.8 • 1012; C, h = 4.9 • 1012). Even at ua - Ua.r ~ 8 • 10 -4, which corresponds to an excess ambient temperature of 14 K above that at criticality in the present case, the agreement is within 20%. The agreement is less good at more marginally supercritical conditions but in each case the discrepancy is always on the side of safety. Finally, we note that the dependence of xi on k in Eq. (8) is quite subtle and comes through the initial condition rather than the equation. The calculation of times to ignition will be a good approximation only when uo is comparable to Ua,~r, i.e. the ambient temperature is just above its critical value, as in the results shown in Fig. 1. Now, for k large, the critical value of h c r is of O(U2a.cr exp (1/Ua, cr), so that, from Eq. (9), xi will be of O(kc~) for k large,

32

28

24-

it

"

'\

i

20-

16 ~2 8 " - , o "11. 4.

~

Ua

FIG. 1. Numerical and analytical solutions for predicted dimensionless ignition delay times, ~, as a function of dimensionless ambient temperature, us. The marked points, equivalent to reaction in the 30 (A), 50 (B) and 75 (C) mm cubes respectively, were obtained from the full numerical solutions of x~ for a first order exothermic reaction in slab geometry, with heat loss by conduction (Eq. 4). The critical values of u~.~ are marked on the abscissa. The analytical solution to the approximate lower bound (Eq. 9) is given as the solid line.

and the dependence of x~ on k (and hence the size of the reacting mass) is seen.

Experimental Methods and Materials Ignition delay times (ti) and their dependence on the control temperature (Ta) were measured in laboratory-scale experiments from thermal ignition following the aerobic oxidation of particulate material packed in cube-shaped, stainless steel, wire mesh baskets, maintained at constant T~ in an oven. The main study was carried out by use of one cube of side dimension 30 mm. Supplementary measurements of the variation of critical ignition temperature (Ta,cr) with cube size were also made in three additional cubes, of side dimension 20, 50 and 75 mm respectively, in order to determine an overall activation energy for the combustion processes taking place. The materials studied was lspaghula, which is a finely-milled cereal crop, rather similar to flour. Its natural packing density in each of the cubes was 600 (-+ 10) kg m -3. Each basket was located in turn in a pre-heated, re-circulating air oven designed to maintain a constant temperature (-+1 K) uniformly over the cube surface for long periods. Successive experiments at different oven temperatures were carried out on each cube, first to bracket the oven temperature at which the critical transition from sub-critical reaction to ignition occurred, 3'5 and secondly, for the 30 mm cube, to determine ti under supercritical conditions. A pair of thermocouples (Tx/T2, 0.1 mm dia wire) was used to monitor the temperature change within the reactant. The thermocouples were linked with opposed polarity on a horizontal axis of each cube. One junction was located at the surface of the cube and the other at its centre. The e.m.f, generated was fed directly to a chart recorder so that difference of temperature between the reactant centre and edge was monitored continuously. Supplementary experiments were also carried out with three thermocouples set on the horizontal axis of the 30 mm cube. These were located at 0, 7 and 15 mm with respect to one edge. Ignition delay times were measured in experiments in which the oven temperature was stabilized at a specific, supercritical temperature before each experiment. Although a disturbance of this temperature was inevitable when the cube of material was put into the oven, the recovery time for the oven was appreciably shorter than the heating period required to take the reactant mass to a uniform internal temperature. The ignition delay was determined from the time at which no temperature difference was detected between the centre and the edge of the cube.

1779

PREDICTION OF IGNITION DELAY TIMES

Experimental Results

to be defined in relation to a particular temperature excess, in practice the experimental results are not sensitive to the magnitude of the chosen excess provided that the experimentally measured delay corresponds to a time at which AT exceeds 50 K.

Critical ignition temperatures and temperaturetime profiles leading to ignition: The minimum ignition temperatures (T~.cr - 1 K) for Ispaghula in the 20, 30, 50 and 75 mm cubes were 500, 487.5, 476 and 462 K, respectively. The overall activation energy obtained from the dependence of Ta.cr on cube size3'5 was 140 -+ 10 kJ mo1-1. This parameter enabled ua to be calculated and thus establish a link between theory and experiment. The centre temperature-excess versus time for ignition of Ispaghula in the 30 mm cube at successively higher (supercritical) oven temperatures are shown in Fig. 2. The records begin at the time when the temperature throughout the sample is approximately uniform and equal to the oven temperature (Ta). There is an interval of 45 min preceding this stage (the "'assembly time") during which the material heats up to the temperature of the oven and water is evaporated. 3"5"13 There is a common, initial evolution of the excess centre temperatures shown in Fig. 2, but the runaway to ignition takes place after successively shorter intervals at increased oven temperatures. The transition to high temperature combustion is extremely rapid, which means that although the theoretical foundation requires the time to ignition

500.5 K

120-

497.5 K

The dependence of t~, on the ambient temperature: The variation of the measured ignition delay times with oven temperature for reaction of Ispaghula in the 30 mm cube is plotted in Fig. 3. The sensitivity of the ignition delay varies considerably even when the oven temperature exceeds that a criticality by only a few degrees, and may be regarded to converge to a limit at Ta - 500 K, which is ca 13 K above the critical oven temperature. In the absence of temperature control to within +--0.01 or ---0.001 K it seems most unlikely that the dependence of ignition delay in the limit of marginal supercriticality as required for application of the perturbation theory7-~ could be determined in experiments of the kind reported here, or that the results would ever be of use for the prediction ignition delays in practical cases. When the oven temperature was set at Ta > 500 K, the measured value of ti, as detected from the difference between the centre and edge of the reactants, decreased further. This departure from a limiting value arose because ignition was no longer ini-

493 K

495.5 K

492 K

489 K

100-

80,.s 60-

/

40-

J 20-

0

0

'

1 0/

i

21 0

,

, 30

I

, 40

i

i 50

t / min FIG. 2. The centre temperature-excess measured experimentally under supercritical conditions for reaction of Ispaghula in the 30 mm cube. The zero of the abscissa corresponds to the time at which the temperature at the centre is equal to the surface (ambient) temperature. The ambient temperature is marked on each curve.

IGNITION

1780

60-

50-

40-

r

"C= 30.~..,-

20I I

i

10-

".'O-

Ta, or

0

t

484

I

488

I

I--

492

I

I

496

I

I

500

i

I

i

504

I

508

I

[

512

Ta/K Flc. 3. Ignition delay time-ambient temperature (ti-To) relationships for the reaction of Isphaghula in the 30 mm cube. The experimental results shown as a solid line are consistent with the present theory. The fine, broken line that continues at 3", > 500 K represents the data obtained from asymmetric initiation of ignition (see text). The heavy, broken line represents the approximate lower bound derived from Eq. 9 scaled to the ignition delay measured at 500.5 K.

tiated at the centre of the reactants. The problem is that the "assembly time" is only weakly dependent on the oven temperature, and may be long relative to the ignition delay. Thus at sufficiently supercritical temperatures, the reactants and subsurface layers may have already reached a temperature at which quite vigorous exothermic reaction takes place even before the reactant at the centre has reached the oven temperature, and ignition is then initiated asymmetrically in the reactants. We illustrate this behaviour by reference to the temperature-time records obtained at 7 mm and 15 mm along the horizontal axis of the 30 mm cube (Fig. 4). At Ta = 510.5 K (at which the oven temperature exceeded Ta.cr by 23 K), the reactants at 7 mm below the surface reached Ta at some time during "the assembly," and considerably in advance of the reactants at the centre. This advantage gained over the material at the core was maintained throughout, and ignition was detected first at the 7 mm position. The combustion wave reached the centre 30 s later. Circumstances of this kind are outside the scope of application of the present theory, and they must be avoided if the predictive methods de-

scribed below are to be exploited. The theory of ignition at a surface temperature that greatly exceeds the critical temperature and is initially higher than the internal reactant temperature was addressed by Merzhanov and Averson. 14 This is a sufficiently important hazard to merit further attention. Discussion

We return to the practical problem discussed in paragraph two of the introduction, i.e. how are we now placed with respect to the prediction of safe estimates of time-to-ignition for low ambient temperatures and large size, from experimentally measured times at small size and high ambient temperatures? It is shown in Fig. 1 that the approximate curve for dimensionless time to ignition (given by Eq. 9) is a safe (lower) bound for the exact times to ignition in cases A, B and C. The exact solutions (and also the experimentally measured results shown in Fig. 3) diverge very significantly from the limiting

PREDICTION OF IGNITION DELAY TIMES

1

580]

('.,~)

560] 540~

l

(is ~,,==~n,)

//

500480460 ! 20

"

3'0

9 t/min

4'0

5'0

FIG. 4. Temperature-time profiles measured during the reaction of Ispaghula in the 30 mm cube at T, = 510.5 K. The thermocouple locations were at 7 mm and 15 mm (centre) with respect to one surface. The cold reactant was put into the pre-heated oven at t = 0. At this markedly supercritical condition, the temperature at 7 mm depth exceeded that at the centre throughout the heating up and the reaction periods. Ignition was initiated in the vicinity of the 7 mm thermocouple. The broken line marks the oven temperature (510.5 K).

envelope curve only at ambient temperatures that are very close to their respective critical temperatures (<2 K, say). The ignition delay time is also very sensitive to ambient temperature in this narrow range above criticality. In a practical situation (such as in a ship's hold, or a large stockpile) we can never guarantee knowledge of ambient temperature to this kind of accuracy, let alone its constancy in time, so it would be extremely imprudent to rely on the longer times to ignition implied by the exact curves (A-C in Fig. 1) at dimensionless temperatures very close to Ua,cr. We therefore propose use of the limiting curve as a working form for safe solution of the identity of the time-to-ignition. It remains to obtain the two scaling factors (Eq. 3) from the dimensionless variables "ri and Ua to the dimensional variables Ta and ti. In this context it is important to stress that the Ti-ua curve is universal, the specific material properties coming into the scaling factors. The temperature scaling factor requires knowledge of the activation energy. In a practical investigation this quantity will already be known if the standard 'critical ambient temperature'-size/scaling has been carried out. 2-5 The time scaling factor c E / q A ~ R , in addition to E/R requires knowledge of the heat production rate the thermal capacity of the material, thus requiring additional measurements if ab initio predictions are

1781

to be made. Heat flow calorimetry and thermal analysis are the natural experimental routes to these data since these techniques are used routinely in many quality and safety assurance laboratories in industry. However, the need may be avoided if the scale factor is determined empirically by fitting (normalising) the shortest measured time to ignition, ti, to the ri value obtained from the "Ti-ua curve at the appropriate value of Ua. It is crucial to use the shortest, acceptable measured time (i.e. the highest ambient temperature used in the tests) for this normalisation in order to retain the lower bound nature of the envelope curve. (This is easily seen from Fig. 1 since, in terms of absolute values, the slope of the envelope curve is always less than that of the exact solutions at a corresponding Ua). We illustrate this method with an example taken from Fig. 3. We use the highest ambient temperature and shortest ignition delay to obtain the time scale-factor. At Ta = 500.5 K the time-to-ignition was observed to be 16.5 min. From Eq. 9, "q = 7.46 x 107 at Ua = 0.0297 (which corresponds to Ta = 500.5 K). Therefore, the scaling factor S = 16.5/7.46 x 107 = 2.21 • 10-7 min. By use of this scaling factor and the experimentally determined activation energy we can derive the lower bound for the experimental measurements. This is shown in Fig. 3. The lower bound is seen to be a very good approximation to the observed ignition delay times within the practically-accessible range of oven temperatures close to criticality. The extrapolation of the lower bound to ambient temperatures for normal stockpiling (<300 K) yields the prediction for the time-to-ignition of Ispaghula as >100,000 years. This implies an inherent stability for this material at normal, ambient temperatures, regardless of the amount. If, however, we consider the possibility of forced drying of the material, the predicted lower bound is less than 12 hours at 450 K and falls to ca. 2 hours at 470 K. This clearly indicates that there is a potential combustion hazard if drying is carried out (either intentionally or unintentionally) at supercritical conditions, unless the residence time of the material in the oven is appreciable shorter than the predicted ignition delay. Such considerations may affect the choice of equipment, such as the use of a fluidized bed type of drying oven in the present case. The temperature dependence of the lower bound for the ignition delay time (Eq. 9) is governed predominantly by the exponential term e I/ua (= eE/RTa). As the application above shows, the predicted lower bound is extremely sensitive to ambient temperature, and thus a very precise value for the activation energy is required for accurate predictions of ignition to be made. However, this requirement is

1782

IGNITION

no more stringent than the appropriate to accurate scaling of the critical temperature of large reactant masses from laboratory measurements. 1-6A3 In summary, the strengths of the theoretical treatment presented in this paper lie in four main features: (i) the application pertains to temperature ranges above criticality that are of practical importance; (ii) the predictions of the approximate theory err on the side of safety; (iii) the size of the system is taken into account in a natural way by the choice of ua (in supercritical conditions) at which the prediction of xi is to be made; (iv) the extent to which the ambient temperature exceeds that at criticality, which is extremely difficult to obtain accurately in marginally supercritical conditions, is not required.

REFERENCES 1. BOWES, P. C.: Self-heating: evaluating and controlling the hazards, H.M.S.O., 1984. 2. EGEIBAN, O. M., GRIFFITHS, J. F., MULLINS, J. R. AND SCOTr, S. K.: Nineteenth Symposium (International) on Combustion, p. 825, The Combustion Institute, 1983.

3. GRAY, B. F., GRIFFITHS, J. F. AND HASKO, S. M.: J. Chem. Tech. Biotechnol. 34A, 453 (1984). 4. TOGNOTI3, L., PETARCA, L. AND ZANELLI, S.: Twenty-second Symposium (International) on Combustion, p. 201, The Combustion Institute, 1989. 5. BEEVER, P. F. AND GRIFFITHS, J. F.: International Symposium on Runaway Reactions, p 1, AIChE, 1989. 6. BODDINCTON,T., GRAY, P. AND HAaVEY, I.: Phil. Trans. R. Soc. Lond. A270, 467 (1971). 7. BODDINGTON,T., FENG, C. G., KAY, S. R. AND GRAY, P.: R. Soc. Chem. Farad. Trans. 2 81, 1795 (1985). 8. GRAY, P. AND KORDYLEWSKI, W.: Chem. Eng. Sci. 40, 1703 (1985). 9. GRAY, B. F. AND MERKIN, J. H.: R. Soc. Chem. Farad. Trans. 2, 86, 597 (1990). 10. BURNELL, J. G., GRAHAM-EAGLE, J. G., GRAY, B. F. AND WAKE, G. C.: I.M.A. Journ. Appl. Maths 42, 147 (1989). 11. GRAY, B. F. AND SCOTr, S. K.: Comb. Flame 61, 227 (1985). 12. GRAY, P. AND HARPER, n . J.: Seventh Symposium (International) on Combustion, p. 425, The Combustion Institute, 1959. 13. GRIFFITHS,J. F., HASKO, S. M. AND TONG, W. A.: Comb. Flame 59, 1 (1985). 14. MERZHANOV,A. G. AND AVERSON,A. E.: Comb. Flame 16, 89 (1971).