Influences of sample thickness on the early transient stages of concurrent flame spread and solid burning

ELSEVIER

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S0379-

Fire Safety Journal 25 (1995) 287-304 Copyright © 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0379-7112/95/$09-50 7112196)00010-0

Influences of Sample Thickness on the Early Transient Stages of Concurrent Flame Spread and Solid Burning C o l o m b a Di Blasi Dipartimento di Ingegneria Chimica, Universit~ degli Studi di Napoli Federico II, Piazzale V. Tecchio, 80125 Napoli, Italy (Received 8 August 1995; revised version received 5 January 1996; accepted 18 January 1996)

ABSTRACT The effects of solid thickness on the initial stages of concurrent flame spread are studied through numerical simulation. The two-dimensional mathematical formulation o f the problem is based on the fully elliptic, reactive Navier-Stokes equations coupled to energy and mass conservation equations for a charring solid. For all fuel thicknesses, uniform burn-out, pyrolysis and flame propagation rates are approached after an accelerative stage. As with the opposed-flow problem, three main regimes of spread rate are established based on the dependence on fuel thickness. The first (kinetic) regime, where spread rates increase with the thickness, is established for samples below 0.008 × 10 -2 m. Both flame and pyrolysis lengths are very short. In the second (thermally thin) regime, the spread rates decrease as the solid thickness is increased while the flame and the pyrolysis regions become successively larger. Finally, as the fuel thickness is increased above 0.5 × 10 2 m, the thermally thick regime, signified by constant spread rates, is simulated. As no experimental measurements of spread rate dependency on the thickness of charring materials are available, numerical predictions are compared with a thermal theory to assess its validity limits. Copyright © 1996 Elsevier Science Ltd.

NOTATION C

k L

T h e r m a l capacity or specific heat T h e r m a l conductivity Length 287

C. Di Blasi

288 m

q T T~ap V Y

Pyrolysis mass flux Solid-phase conductive heat flux Temperature Flame temperature Vaporization temperature Maximum value of the inlet parabolic velocity profile Spread rate Species mass fraction

Greek symbols Q r

Density Solid half thickness

Subscripts b F f g O p s 0

Burn-out Fuel Flame Gas Oxygen Pyrolysis Solid Initial conditions

1 INTRODUCTION In concurrent flame spread over solids, three main regions have been identified.~ 3 In the first, near to the ignition source, the flow is laminar and convective heat transfer from the flame to the solid is controlling. A transition zone, where both radiative and convective heat transfer modes are important, is then followed by a highy radiative turbulent flame. Though understanding the laminar concurrent spread process is important in relation to material flammability, detailed experimental and numerical studies of this stage are very few. Particularly interesting are the experiments conducted for thick PMMA, 4-5 thick wood 5 and thin paper. 6 In all cases, the effects of the flow velocity and the oxygen concentration have been investigated. For thick solids, a linear dependence of the spread rate on the free-stream velocity has been measured. However, for thin solids, after a weak dependence on low gas velocity, the spread process has been found to be independent of forced flow conditions. A power-law dependence on the oxygen concentration has also been observed, with exponent values in the range 1-2, depending

Early transient stages of concurrent flame spread and solid burning

289

on material characteristics, fuel thickness and experimental configuration. Analytical and mixed analytical-numerical solutions of thermal, boundary-layer models have been proposed (see the reviews by Fernandez-Pello2"3). In some cases, closed-form solutions for the pyrolysis spread rate have been obtained in the limit of thermally thin or thermally thick solids. Of course, these formulae are not valid for near-limit conditions. The assumptions of boundary layer, infinite rate kinetics and constant vaporization temperature have been removed in some thermo-diffusive (Oseen or parabolic flow) models. 7-9 Interesting information on process dynamics and flame structure has been obtained, but disregarding gas expansion effects may cause large overestimation of the pyrolysis and flame spread rates. ~° These effects are accounted for, through the solution of the Navier-Stokes equations, in the model proposed by Ferkul and T'ien, ~ formulated for steady concurrent microgravity spread over thin paper. However, in this case as in previous numerical models, highly simplified treatments of solid-phase processes have been proposed, again valid only in the limit of thermally thin or thermally thick solids. Thus no analysis is available on the effects of sample thickness on the initial stages of concurrent flame spread. These are investigated in this study through a recently proposed mathematical model of flame spread over charring fuels, ~-~4 including all main chemical and physical processes. The fully elliptic gas-phase momentum, mass and energy equations, including finite-rate combustion kinetics, have been coupled to unsteady solid-phase energy and mass balance equations, including pyrolysis reactions leading to volatile and char formation. Interesting features of the model consist of the description of both two-dimensional solid-phase heat transfer and burn-out processes, which allow the treatment of widely variable thicknesses, and of the quasi-steady formulation of the conservation equations, which allows the prediction of process dynamics.

2 PROBLEM FORMULATION The problem, schematically represented in Fig. 1, considers a cellulosic fuel slab placed in a channel. Ignition at the right boundary, where a forced flow is imposed, is caused by external radiation. The solid exposed to a heat flux (initially from external devices and then from the flame) degrades to volatiles and char. Volatiles flow through the porous char and mix with air in the gas phase to give a flammable mixture.

290

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Y gas phase spread rate

concurrent flow flame

"

~y'Topyrolysis mass ~ ' ~ ' - " ttttltttttf~J

solid fuel q ~4"~-t~c~ symmetry conditions Fig. I.

:

Schematic representation of the concurrent flame spread problem.

Some of the volatiles are thus burned, and some together with the combustion products flow downstream of the flame to pre-heat the solid. The main transport p h e n o m e n a and chemical processes of concurrent flame spread over charring solids are taken into account through m o m e n t u m , mass and energy conservation equations for the gas phase, and mass and energy conservations for the porous solid fuel. Concerning the gas phase, most of the assumptions previously made in the literature are removed. Since finite-rate combustion effects are important near ignition and extinction conditions, they are included in the present formulation through a one-step, second-order reaction: F + VoO----->vpP. Also, the boundary layer assumption has been removed because the convection may not be large enough to make the gradients along the direction perpendicular to the free stream much larger than those along it. The flow field results from the forced flow imposed at the channel inlet, gas expansion, due to the high combustion temperatures, and mass efflux from the vaporizing surface. Surface radiative heat transfer is also taken into account. As for the solid phase, the assumption of a constant ignition or vaporization temperature has been eliminated and pyrolysis is described as an in-depth process according to: solid ~"> volatiles, solid K,> chars. The global, two-step pyrolysis mechanism allows solid consumption to be taken into account, whereas, to account for non-uniform temperature along the thickness, the variable property enthalpy balance is formulated for a two-dimensional solid. Some assumptions are, however, still made in both the solid- and gas-phase models. Gas properties are constant. Thermal swelling a n d / o r shrinkage and surface regression of the porous solid are neglected. Possible oxidation reaction of the char and the accumulation of volatile mass and energy inside the solid are not described. The volatiles, products of pyrolysis, flow only towards the heated surface

Early transient stages of concurrent flame spread and solid burning

291

with no resistance to mass flow (the gas pressure is constant) and are in local thermal equilibrium with the solid. The model equations and the numerical solution technique, already applied to simulate opposed-flow flame spread, are essentially the same as reported previously ~2-z4 and are not listed here again.

3 RESULTS Simulations of the concurrent flame spread problem have been made for a forced maximum flow velocity at the channel inlet equal to 30 x 10 2 m/s (that is of the same order as that gravitationally induced), by varying the fuel half-thickness, r, from 0.002 x 10 -2 m to 0.5 x 10 -2 m. Kinetic data and properties are those typical of cellulosic fuels and are the same as previously used, '~ ,4 while the surface emissivity is taken as equal to 1. From the qualitative point of view, the concurrent flame spread shows rapid propagation of the flame front followed by a slower advancement of the pyrolysis and burn-out fronts. However, the differences in the three spread rates are sometimes dependent on the definitions applied to get their measurements or evaluations. From the theoretical point of view, burn-out is often assumed to occur when the solid conversion at the sample centerline reaches completion within a factor of 1%. From the experimental point of view, burn-out is generally recorded by observing the sudden decrease in the temperature due to the disappearance of the fuel (the pyrolysis mass flux goes to zero and the concurrent flow starts to cool the inert ash layer). In mathematical and numerical modeling, pyrolysis is assumed to occur as soon as the solid density at the surface, the site where the degradation process starts, is decreased by 1% with respect to the virgin solid value. Experimental analyses generally define the position of the pyrolysis front in terms of a critical surface temperature, associated with some physical processes such as the blackening of cellulosic materials or the melting of thermoplastic polymers. According to thermal theories, the flame location is defined as that where the fuel-to-oxidizer equivalent ratio is equal to 1. This definition has also been used in some finite-rate theories. Alternatively, a critical value of the gas phase combustion rate is used. 7'~ Also, in their analysis of experimental data, Atreya and coworkers ~ introduced a definition of the flame tip location as that corresponding to the peak rate of change in the surface temperature, this peak being the result of a sharp increase in the heat flux from the gas to the solid caused by the arrival of the flame tip. Such a definition

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Fig. 2. Burn-out, pyrolysis and flame lenghts as functions of time for r = 0.03 x 10-" m. Burn-out: (la) Q~(x, 0 ) = I%Q,,~; ( l b ) T~(x, r)=650K behind the flame leading edge. Pyrolysis: (2a) Q~(x, r)-99%0,~,: (2b) T~(x, r)=650K; (2c) T,(x, r ) = 6 7 5 K. Flame length: (3a) distance along the x-axis corresponding to a combustion rate equal to 1.72 kg/m ~ s: (3b) distance along the x-axis corresponding to a local fuel-to-oxygen equivalent ratio equal to 1.

was shown to give flame lengths in agreement with those determined through video records. In order to compare different definitions, flame, pyrolysis and burn-out lengths, measured from the right boundary of the sample, where ignition is achieved, are reported in Fig. 2 as functions of time for v = 0 . 0 3 x 10 2 m. Several definitions are used for the burn-out position, pyrolysis length and flame length. 1.

Two definitions are used for the burn-out position: a. b.

2.

Three definitions are compared for the pyrolysis length: a. b. c.

3.

where O~(x, 0 ) = 1%0~,,; and where ~(x, r) = 650 K behind the flame leading edge (here surface temperatures up to about 1000 K are attained).

where O~(x, r ) = 99%0~; where T.(x, r) = 650 K; and where T,(x, r) = 675 K.

As for flame length, two definitions are compared: a.

the distance along the x-axis corresponding to a critical

Early transient stages of concurrent flame spread and solid burning

b.

293

combustion rate (here taken as 1.72 kg/m3s, corresponding to about 1/70 of the m a x i m u m value); and the distance along the x-axis corresponding to a local fuel-to-oxygen equivalent ratio of 1.

The same qualitative dependence of the flame, pyrolysis and burn-out lengths on time is shown for all the definitions listed above. Furthermore, all of them, after an initial accelerative stage, show a linear dependence on time, indicating that constant spread rates are approached. Quantitative differences are, however, significant. As expected, flame lengths defined according to the flame sheet assumption (3b) are shorter than those based on the finite-rate combustion (3a), but almost coincide with the definition based on the peak of surface temperature derivative (not shown). The definition of pyrolysis length (2a), based on the decrease of surface density, is almost the same as that given by the position where the surface temperature is equal to 600 K (not shown). However, larger temperatures, still in the range of those used in the experiments (650-675 K), give much shorter values. Significant differences are also seen in the burn-out lengths. Therefore, the most appropriate choice should be made when theoretical predictions are to be compared with experimental measurements, i.e. the most equivalent definitions for the two cases should be adopted. Since the simulation results are not compared with experiments (as there are no systematic investigations on the effects of the fuel thickness on the spread process), the 'theoretical' definitions (la), (2a) and (3a) are used in this study for burn-out, pyrolysis length and flame length. The dependence of the pyrolysis and burn-out lengths on time is shown in Figs 3 and 4 for several sample thicknesses. For very thin solids ( r - < 0 . 0 0 4 x 10 2 m), the pyrolysis length increases with the sample thickness, then continuously decreases. The dependence on the solid thickness becomes, however, less and less important as thicker solids are considered. Indeed, for l: -> 0.25 x 1 0 2 m, the process becomes independent of this variable. The dependence on time of the burn-out length is shown only for sample half-thicknesses in the range from 0.002 x 10 2 to 0.07 x 10 2 m (thicker solids are characterized by burning times much longer than those associated with the propagation rates of the pyrolysis front). As with the pyrolysis lengths, the burn-out lengths initially increase with the solid thickness and then decrease. The flame lengths (not shown) present a dependence on the solid thickness similar to the pyrolysis lengths, though they are always larger. The spread rates, measured in the last part of the computational

294

C. Di Blasi

2.00 1.80

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Pyrolysis lengths, measured from the right boundary of the sample, as functions of time for the solid half-thicknesses ( x 100 [m]) reported in the figure.

domain, are reported in Fig. 5 where the three different regions previously discussed are again seen. These, similar to opposed-flow flame spread, j2 correspond to the kinetic (or very thin), thermally thin and thermally thick regimes. Furthermore, the differences between the propagation rates of the pyrolysis and the flame front tend to disappear

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t[sJ Fig. 4. Burn-out lengths, measured from the right boundary of the sample, as functions of time for the solid half-thicknesses ( x 100 tin]) reported in the figure.

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a) analytical theory b) numerical solution \ flame

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T x 100 [m] Fig. 5. Flame, pyrolysis and burn-out spread rates as functions of the solid halfthickness. For comparison purposes, the analytical solution by Fernandez-Pello 3 is also reported.

in the kinetic and thermally thick regimes, whereas the propagation rate of the burn-out front is always significantly slower. To assess the validity of simpler thermal theories, the analytical solutions of the pyrolysis spread rates given by F e r n a n d e z - P e l l o 3 are also included in Fig. 5 (given the small flame size, radiative contributions are assumed to be negligible). These predict that the spread rate varies inversely with the sample thickness for thermally thin fuels:

Vp=c(ClkgQgcpV~tp)°5 ~ Tf ~. T,,ap'~ e~c,~r tTvav- To)

(1)

whereas it is independent of the fuel thickness for the case of thermally thick fuels:

Vv=4CC, V.Ogcpkg~ Tf: Tvap~2 tc~c~k~ tTv,v- T,,)

(2)

The gas and solid properties in eqn (1)eqn (2) are those used in the simulations with average values for 08, Tf and Tv,p (Og = 1 × 10 s k g / m 3, Tf= 2800K and Tv.v= 650K). Moreover, it has been assumed that C=4.6, L p = 0 . 5 x 10 2 m and C~ =2.7 to best fit the numerical solution in the limit of thin and thick solids. The trend observed in the second region is in qualitative agreement with eqn (1) which is valid for thermally thin fuels, although the functional dependence of the spread rate on the fuel thickness is not

296

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exactly 1/r. Also, the trend shown by the third region corresponds to the thermally thick regime described by eqn (2) with very close numerically and analytically predicted spread rates. However, the two models predict the transition from the thermally thin to the thermally thick regimes for different values of the solid thickness. Indeed, eqns (1) and (2) give the same spread rate, for r = 0.093 × 10 -2 m, a value about 2.5 times smaller than that numerically predicted (T = 0.25 x 10 2 m). It should be noted that spread rates given by eqns (1) and (2) are affected not only by property values but also by the values chosen for C and C~. Thus, the transition from one regime to a n o t h e r can be predicted for largely different values of r if the a g r e e m e n t with the numerical t h e o r y in the limit of a very thin or very thick solid is not imposed through an ad hoc value for C~. Finally, the first region of spread rate increase with fuel thickness (the kinetic or very thin regime) cannot be predicted by analytical theories based on the assumption of an infinite combustion reaction rate. A n example of the gas-phase dynamics of the early stages of concurrent flame spread is given through the constant contour levels of the combustion reaction rates (Fig. 6), the gas-phase isotherms [Fig. 7 ( A ) - ( C ) ] , the constant contour levels of chemical species mass fractions (Fig. 8) and the vector velocity field (Fig. 9) for r = 0.1 x 10 2 m. The contours of a critical value of the combustion rate (about 1/70 of the m a x i m u m value), generally considered as representative of the flame (shape and size), show that very high values are attained initially at the leading edge of the slab (until burn-out effects are negligible) and then at the burn-out front. The flame is very close to the pyrolyzing solid in this region, where both oxidizer and fuel are at very high levels and give rise to a small premixed region (Fig. 8). Downstrean of this region, the flame assumes a diffusional character. The combustion rates

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2.000 .800 1200 L600 2.000 1.600 2.000 X x 10o [m] Constant contour levels of the gas-phase temperature (values 1000, 1700 and 2300 K) for t = (A) 3 s; (B) 5 s and (C) 8.5 s for r = 0.1xl0 -2 m. I_900

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are still rather high because of the very high fuel concentrations, resulting from solid pyrolysis or gas-phase convective transport, and the high temperatures. The distance of the flame from the surface first increases and then starts to decrease, i.e. the flame tip again bends towards the fuel surface. Indeed, given the high flow velocities, the .700

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Vector velocity field (maximum velocity 0.9 m/s) for r = 0.1 s (the flame, defined as in Fig. 6, is also shown).

×

10 2 m and t = 5

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C. Di Blasi

shape of the flame is mainly dictated by the fuel availability. Also, together with the increase in the flame length, an increase in the flame stand-off distance is observed. The flow structure at the leading edge shows the existence of an elevated pressure zone. Flow velocities significantly larger than those assigned at the channel inlet are simulated in the burning region, as a consequence of gas expansion effects (up to a factor of 3). Thus, the rates of convective transport of mass and heat are quantitatively very different from those predicted by thermo-diffusive models. The description of gas expansion effects is also very important in terms of gas-phase densities which directly affect the combustion rate and thus the dynamics of the whole process. Gas-phase processes do not show any qualitative dependence on the solid fuel thickness, except for variations in the values of the combustion rate and the flame temperature. These attain their maxima for sample thicknesses belonging to the first part of the thermally thin regime. On the contrary, significant differences, even from the qualitative point of view, are shown in the distribution of the solid-phase variables, as the solid fuel thickness is varied. An example of the time evolution of the solid phase variables is given, for r = 0.07 x 10 2 m, through the profiles of surface temperature and solid density [Fig. 10(A)] and pyrolysis mass and heat fluxes [Fig. 10(B)]. These profiles refer to times longer than those associated with flame lengthening (definition 3a) over the whole sample length. The flame leading edge (maximum heat flux and temperature) propagates along the sample while the surface temperature increase becomes small for times longer than 10 s, when about 80% of the surface has undergone degradation. On the other hand, the forced flow cools the burn-out region. The m a x i m u m temperature and heat flux tend to constant values, whereas the m a x i m u m pyrolysis flux, located downstream of the flame, slightly decreases because of fuel consumption effects. Finally, the heat flux profile is highly unsteady, being the result of the gas-phase temperature and position of the flame. The latter, for instance, for a fixed position, first increases with time and then suddenly decreases because of the passage of the flame leading edge and thus of the burn-out front. A comparison of the distributions of solid-phase variables (temperature, solid density and vector pyrolysis mass flux), for different solid fuel thicknesses and about the same pyrolysis length, is shown through Fig. l l ( A ) - ( D ) . As expected, solid-phase temperature and density gradients along the y-direction are successively reduced as successively thinner solids are considered. Though the depth of the reaction layer is quite narrow, the thermal wave penetrates significantly across thick

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Fig. 10. Surface temperature and solid density profiles for (A) t = 5, 10, 15 and 20 s; and (B) pyrolysis and heat flux profiles for t = 10 and 20 s for r = 0.07 x 10 2 m. solids [Fig. 11(A)]. Thus, solid-phase heat conduction in the direction of flame propagation is a n o t h e r mechanism which contributes to the e n h a n c e m e n t of the combustion process (this m e c h a n i s m is, however, m u c h slower than the convective gas-phase transport of heat). The m a x i m u m surface t e m p e r a t u r e increases as the solid thickness is decreased, because of the r e d u c e d dispersion of heat through the sample (the condition of zero normal gradient at the sample centerline corresponds to a two-sided burning sample). A slight decrease is seen only for very thin samples ( r - < 0 . 0 0 4 × 10 2 m) given the r e d u c e d combustion t e m p e r a t u r e s and heat fluxes resulting from the lower fuel concentrations.

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Solid phase isotherms [K] (400 and then a step of ]00) (solid lines), solid

density [kg/m ~] (from 65 and then with a step of 65) (dashed lines) and vector pyrolysis mass flux for about the same pyrolysis length and different sample half-thicknesses. The mark indicates the position of the flame leading edge.

B o t h t h e e x t e n s i o n o f t h e p y r o l y s i s z o n e a n d t h e v a l u e s o f the p y r o l y s i s m a s s flux a r e i m p o r t a n t to e v a l u a t e t h e t o t a l m a s s flow o f fuel v a p o r s e n t e r i n g the gas p h a s e a n d t h u s the f l a m m a b i l i t y b e h a v i o r o f the s a m p l e . I n d e e d , the t o t a l m a s s flow of fuel v a p o r s , F [kg/s], can b e e x p r e s s e d as: F =

['q; -h

m dy

)

dx

(3)

Early transient stages of concurrent flame spread and solid burning

301

where m [kg/m 2 s] is the pyrolysis mass flux. A comparison of the solutions obtained for different thicknesses shows that pyrolysis is always an in-depth process with a m a x i m u m mass flux increasing as the solid thickness is decreased, except for very thin solids (~ < 0.004 x 10 2 m), when solid consumption is a very fast process and low quantites of fuel vapors are released in the gas phase. The pyrolysis mass flux attains its m a x i m u m behind the position of the flame leading edge for thick and intermediate solids [Fig. l l ( A ) - ( C ) ] because, apart from a thin layer close to the surface exposed to the flame, solid degradation is rather slow. For thin solids [Fig. II(D)], burn-out effects soon become important and the m a x i m u m pyrolysis mass flux is located close to the flame leading edge. The pyrolysis length always increases for sample half-thicknesses larger than 0.004 × 10 -2 m, whereas for thinner solids, a reduction in the size of the pyrolysis zone is seen again because of the reduced flame size associated with the reduced generation of fuel vapors. Therefore, the whole process is slowed. The combination of the optimal conditions of pyrolysis mass flux (the highest values) and pyrolysis length (again rather large values) gives rise to the highest spread rates and thus to the most hazardous conditions from the point of view of fire safety, for the samples whose behavior can be classified as thermally thin. In other words, in the kinetic regime, m, (Lv - Lb) and v are small so that the process is slow (the low fuel concentrations also give rise to reduced combustion temperatures). In the thermally thin regime, both m and (Lp - Lb) reach high values, whereas v values are high enough to avoid limitations in the instantaneous fuel release rate. In the thermally thick regime, the length (Lp- Lb) reaches the highest values because burn-out characteristic times are much longer than those associated with flame and pyrolysis. However, notwithstanding the large pyrolysis length and the contribution from in-depth degradation, the high heat dispersion through the sample makes the rates of solid devolatilization and reaction front propagation slower. For thick solids, the existence of an extinction zone is always observed at the leading edge of the slab, as a consequence of the reduced vapor fuel production associated with the formation of a thick, low-thermal-conductivity char layer. In fact, the formation of a char layer with the consequent reduction in the heat transfer rate towards the virgin solid reduces the vapor fuel production, making flame stabilization difficult. Therefore, for thick solids, instead of the propagation of a burn-out front, the propagation of an extinction front can be observed, depending on flow conditions, solid degradation rate, etc. The size of the extinction region is significantly reduced for r -> 0.07 x 1

302

C. Di Blasi

0 2 m, but the pyrolysis front enlarges to the whole solid thickness and burn-out effects start to become significant.

4 CONCLUSIONS A fully elliptic, two-dimensional, quasi-steady model has been applied to simulate the initial stages of concurrent flame spread over charring materials and their dependence on sample thickness. These issues have not been previously investigated either experimentally or theoretically by means of numerical simulation. The detailed thermal structure of the flame and its dynamics are qualitatively similar to those obtained by means of thermo-diffusive models. However, variations in the flow field structure and density, due to temperature variations, are noticeable and affect both the spread rates and the convective structure of the flame through the existence of a zone of slow flow at the flame leading edge and a very fast one in the combustion region. The convective transport and combustion rates are thus significantly different. Similar to opposed-flow flame spread, ~2 the simulations have allowed the existence of three regimes of concurrent flame spread to be determined on dependence of the sample thickness: 1.

2.

3.

a kinetic regime ( v - < 0 . 0 0 4 x 10 2 m), where the reduced pyrolysis mass flux associated with very thin samples and the consequent reduced length of the flame cause an enhancement of flammability characteristics with sample thickness: a thermally thin regime, where negligible space gradients along the sample thickness are simulated and the burn-out, pyrolysis and flame spread rates continuously decrease as the sample is made thicker; and a thermally thick regime where the process becomes indepedent of the sample thickness (~'->0.25 × 10 2 m).

Regimes 2 and 3 are also predicted by a thermal (analytical) theory, 3 though the prediction of the transition from one regime to another is d e p e n d e n t on property and parameter values. The process dynamics and the values of the spread rates, for fixed flow conditions, are largely dictated by the total mass flow of fuel vapors entering the gas phase. This is determined by the pyrolysis mass flux and the length of the pyrolysis region. These two variables lead to the highest flammability characteristics for samples in the thermally thin regime.

Early transient stages of concurrent flame spread and solid burning

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F u r t h e r investigations are still n e e d e d to assess the validity limits of thermal theories, u n d e r different conditions, since their closed-form solutions are easily applicable. As for numerical approaches, models allowing the simulation of larger flames and the transition from the l a m i n a r - c o n v e c t i v e regime to the t u r b u l e n t - r a d i a t i v e regime should be developed. Two main lines can be pursued in this direction. The first deals with the coupling of finite-rate theories, such as the one here applied, with thermal theories. The second is c o n c e r n e d with the d e v e l o p m e n t of new numerical treatments which combine fully elliptic formulations of the flame leading edge with parabolic formulations of the d o w n s t r e a m region. Such an approach, which is expected to be m o r e feasible in terms of computational costs c o m p a r e d with fully elliptic treatments, has been succesfully applied for thin paper 11 but further study is n e e d e d for intermediate and thick solids.

REFERENCES 1. Markstein, G. H., & De Ris, J., Upward fire spread over textiles, Fourtheenth Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, 1973, pp. 1085-97. 2. Fernandez-Pello, A. C., Flame spread modelling. Combustion Science and Technol., 39 (1984) 119-134. 3. Fernandez-Pello, A. C., The solid phase, in Combustion Fundamentals of Fire, Ch. 2, G. Cox (Ed.). Academic Press, New York, 1995, pp. 32-100. 4. Loh H. T. & Fernandez-Pello A. C., A study of the controlling mechanisms of flow assisted flame spread. Twentieth Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, 1984, pp. 1575-82. 5. Mekki, K., Atreya, A., Agrawal, S. & Wichman, I. S., Wind-aided flame spread over charring and non-charring solid: an experimental investigation. Twenty-third Symp. (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, 1990, pp. 1701-7. 6. Loh H. T. & Fernandez-Pello A. C., Flow assisted flame spread over thermally thin fuels. Proc. of the First Int. Symposium on Fire Safety Science, Hemisphere, 1985, pp. 65-74. 7. Di Blasi, C., Crescitelli, S. & Russo, G., Near limit flame spread over thick fuels in a concurrent forced flow. Combustion and Flame, 72 (1988) 205-215. 8. Di Blasi, C., Crescitelli, S., Russo, G. & Fernandez-Pello, A. C., Model of the flow assisted spread of flames over a thin charring combustible. Twenty-second Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, 1988, pp. 1205-12. 9. Di Blasi, C., Crescitelli, S. & Russo, G., Numerical modelling of flow assisted flame spread. Computer Methods in appl. Mech. Engng, 5 (1989) 481-492.

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10. Di Blasi, C., Momentum, mass and heat transfer processes during wind-aided flame spread over combustible surfaces, in Numerical Methods in Thermal Problems, Vol. VIII, Part 2, Proc. of the 8th Int. Conf. on Numerical Methods for Thermal Problems, R. W. Lewis (Ed.). Pineridge Press, Swansea, 1993, pp. 1358-69. 11. Ferkul, P. V. & T'ien, J. S., A model of low-speed concurrent flow flame spread over a thin fuel. Combustion Science and Technol., 99 (1994) 345 -370. 12. Di Blasi, C., Processes of flames spreading over the surface of charring solid fuels: effects of fuel thickness. Combustion and Flame, 97 (1994) 225-239. 13. Di Blasi, C., Predictions of unsteady flame spread and burning processes by the vorticity-stream function formulation of the compressible NavierStokes equations. Int. J. Nurner. Meth. for Heat and Fluid Flow, 5 (1995) 511-530. 14. Di Blasi, C., Predictions of wind-opposed flame spread rates and energy feed back analysis for charring solids in a microgravity environment. Combustion and Flame, 100 (1995) 332-340.