A Model of Transport of Fuel Gases in a Charring Solid and Its Application to Opposed-ﬂow ﬂame Spread

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Proceedings of the Combustion Institute Proceedings of the Combustion Institute 31 (2007) 2633–2641 www.elsevier.com/locate/proci

A model of transport of fuel gases in a charring solid and its application to opposed-ﬂow ﬂame spread

Howard R. Baum a, Arvind Atreya b,* a Building and Fire Research Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA

Abstract

This paper outlines the development of a mathematical model for the transport of gases through the char matrix of a burning solid. Two basic assumptions are made. First, the gases evolved by the degradation of the virgin material are transported by pressure differences through a network of narrow passage- ways created in the char by the conversion of material from the solid to the gas phase. This process is treated as flow through a porous medium, with the mass flux related to the pressure gradient by Darcy’s law. Second, the gas temperature is the same as the local char temperature. This model is first used to study the time-dependent thermal degradation of a semi-infinite charring solid heated above the charring temperature. Then, the opposed flow flame spread treated by Atreya and Baum [Atreya, A., Baum, H.R., Proc. Combust. Inst., 29 (2002) 227–236] is revisited. It was found that the solution to the condensed phase flame spread problem is identical to the initial transient problem. Weak dependence of the solution on the accumulation parameter ‘b’ validates the assumption made in [Atreya, A., Baum, H.R., Proc. Combust. Inst.,29 (2002) 227–236] and completes the flame spread solution. Fuel mass flux follows the heat flux lines and is normal to the isobars. Calculations using representative values for wood show that the pressure generation at the char–virgin material interface is considerable and equal to 13.27 kPa. Finally, in view of the fact that the nonlinear pressure equation poses considerable numerical difficulties, this analytical solution may help in determining the stability and accuracy of the numerical scheme used for more complicated problems. 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Charring solids; Fuel transport; Pressure generation; Flame spread

1. Introduction It introduces an insulating char layer of unknown thickness between the heat transfer from the gas- An understanding of the burning of charring phase ﬂames and the virgin material that gener- materials is essential for developing models of ﬁre ates fuel gases by solid-phase pyrolysis. Further, growth in buildings. The transient growth of the the combustible gases produced at the char–virgin char layer during fuel production complicates material interface must be expelled by generation both the heat and the mass transfer analysis. of high pressures. Initially, when the interface is at or near the surface of the solid being heated, it is often assumed that the gases are instantly *Corresponding author. Fax: +1 734 647 3170. expelled from the solid material into the adjacent E-mail addresses: [email protected] (H.R. oxidizing atmosphere permitting combustion to Baum), [email protected] (A. Atreya). take place in the gas phase. However, if the

1540-7489/$ - see front matter 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2006.08.089 2634 H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 process goes on long enough, the interface will no to study the time-dependent thermal degradation longer be adjacent to the heated surfaces and the of a semi-infinite charring material heated above spatial distribution of gases over the surfaces of the charring temperature. Then, the opposed-flow the solid will depend on the internal pressure flame spread treated by [1] is revisited. The ques- generation and the char porosity. Since these fuel tion how the evolved fuel gas is transported from gases are responsible for fire growth, it is neces- the interface where it is generated to the surface is sary to account for their transport through the addressed. char matrix to any surface exposed to the oxidizer. Figure 1 shows how the fuel gases generated at one location and transported to another can affect 2. Gas transport model fire growth. Significant asymmetry is observed due different permeability across and along the grain Basic assumptions made in the model are: direction. This is discussed in greater detail in (i) The gases evolved by thermal degradation are Refs. [6,7]. transported by pressure differences through the Several models of pyrolysis of charring materi- porous char. This process is treated as flow als have been developed in the past motivated through a porous medium, with the mass flux either by fires or biomass pyrolysis. Di Blasi [2] related to the pressure gradient by Darcy’s law. summarizes the state of the art of transport mod- (ii) The charring material is assumed to be initially els for charring solid degradation. However, the moisture-free and non-porous and the porosity primary focus of many of these models was often develops as a result of charring. This assumption not pressure driven gas transport because the one- eliminates the condensation and re-evaporation dimensional assumption usually made obviates of pyrolysis gases and moisture that are forced the need. Some models [3–5] have considered vol- by high pressure into the virgin interior. (iii) The atile products transport using Darcy’s law with temperature in the gas is the same as the local char isotropic permeability despite being one-dimen- temperature and the convective heat transfer sional. Fredlund [9] has considered a two-dimen- between the gas and the char is ignored. Thus, sional model and even measured the internal the model is reduced to an energy equation for pressure generation. All these models were numer- the virgin material and char, together with mass ically solved to obtain the progress of the char conservation for the transport of gaseous degra- front and gas generation. In this work a physics- dation products through the char. (iv) Thermal based model of fuel gas generation and transport decomposition occurs at a well-defined pyrolysis is developed, analytically solved, and applied to temperature Tp. The corresponding equations the flame spread model of [1]. are supplemented by boundary conditions at the First, basic equations and boundary conditions char–virgin material interface, and at the char–ox- controlling the gas transport are derived and used idizer interface. Let qv denote the density of the virgin material, and let qc be the density of the char, where qc < qv. qv and qc are assumed to be constants with abrupt change occurring at a well-defined pyrolysis temperature Tp. The lower char density is assumed to be caused by the creation of small void spaces through which the evolved gases flow. Shrinkage of char is ignored. The void fraction ‘e’ is then given by the expression

e ¼ðqv qcÞ=qv: ð1Þ To characterize the thermodynamic state of the gas in terms of the usual density, temperature, and pressure, q must denote the gas density per unit volume of space occupied by the gas. The velocity ~u must be defined such that m~ ¼ q~u is the local mass flux of gas through a macroscopic surface element in the char. Here, char is treated as a continuum, with gas and solid matter coexis- ting in the volume occupied by the char. Also, Fig. 1. Axis-symmetric flame spread on horizontal since the generation of decomposition products surface of Douglas fir ignited at the center in quiescent atmosphere (taken from Ref. [6]). The remaining char occur at a predefined temperature (Tp) that sepa- pattern shows the effect of different permeability across rates char from the virgin material, there are no and along the grain direction. Clearly, transport of fuel gas-generation terms in the porous char matrix. gases generated in one location to another can signifi- Thus the equation expressing conservation of cantly affect the flame spread. mass for the evolved gas takes the form: H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 2635 oðqeÞ mined as part of the solution for the associated þr:ðq~uÞ¼0: ð2Þ ot problem for the evolution of the temperature field. At surfaces exposed to the surrounding atmo- Darcy’s Law is used to relate the local mass sphere, the pressure is ambient, p . The boundary flux and pressure gradient in char. For gas perco- a condition at the interface between the char and lating through char Landau’s [10] approach is the virgin material is as follows: Let V~ be the local used. Accordingly, for an elliptical pipe of cross- velocity of the interface and ~n be the unit normal section with semi-axes a and b, the mass flux vec- to the surface defined with positive ~n pointing into tor is related to the pressure gradient through the the virgin material. Then: formula: 2 prp ~n ðabÞ q~u ~n ¼ eð2=3ÞK m~ ¼ rp: ð3Þ lðT ÞRT 4mða2 þ b2Þ ¼ðq q ÞV~ ~n: ð8Þ Here, m is the kinematic viscosity, related to densi- v c ty and gas viscosity l(T)bym = l/q. The pressure Physically, Eq. (8) states that the rate at which in the gas is denoted as p, while the temperature of evolved gases are created is equal to the product both gas and adjacent char is T. Since the model is of the velocity of the char front normal to itself based on the actual thermodynamic properties in multiplied by the density difference between the the gas, the perfect gas law can be used to relate char and the virgin material. Thus, all the mass density to pressure and temperature. lost by the solid phase is taken up by the gas, with the pressure adjusting accordingly. Finally, at an p ¼ qRT ; ð4Þ impermeable boundary between the char and an where, R is the gas constant for the evolved gases. inert solid material, the gas cannot penetrate into Finally, using the fact that the cross-sectional area the inert solid. Thus, if ~m denotes a unit normal to A of the pipe is given by the expression A = pab, the impermeable boundary: the relation between the mass flux and pressure rp ~m ¼ 0: ð9Þ gradient takes the form: ðabÞA prp m~ ¼ 2 : ð5Þ 4pða2 þ b Þ lðT ÞRT 3. Transient thermal model Equation (5) shows that the local mass flux is composed of two parts; a shape dependent factor Consider an idealized scenario in which the proportional to the cross-sectional area of the temperature at the surface of a semi-infinite solid pipe, multiplied by a term which is proportional is instantaneously raised from an initial ambient to the gradient of pressure squared, and which temperature T1 to a surface temperature Ts > Tp. depends only on the properties of the gas. The Since the pressure field is determined by the tem- shape factor will have to be replaced by an empir- perature distribution in the char and the virgin ical factor, since there is no way of knowing the material, it is considered first. Let Tv (x,t) be the actual shape of the passages created by the gasifi- temperature, kv be the thermal conductivity, and cation processes. Cpv be the heat capacity of the virgin material, The final form of Eq. (5) comes from noting with an analogous notation in the char. Then: that the mass flux averaged over an area that is oT o2T large compared with any individual pore passage q C c ¼ k c 0 6 x 6 X ðtÞ; c pc o c o 2 must account for the fact that only a fraction of t x 2/3 oT o2T the area of order e contains voids. Thus, the q C v ¼ k v X ðtÞ 6 x 6 1: ð10Þ form of Darcy’s law that will actually be used is: v pv ot v ox2 prp Here, x = X(t) is the position of the interface be- m~ ¼ q~u ¼eð2=3ÞK : ð6Þ lðT ÞRT tween materials, with x measured from the heated surface. The permeability K has dimensions of area. It is The boundary conditions are: At the surface an empirical parameter that characterizes the char x =0, Tc = Ts. Far from the surface, the virgin as a porous medium. The mass conservation equa- material is at the ambient temperature. Thus; tion, which controls the transport of gases Tv = T1 as x fi 1. Finally, at the interface through the char, then takes the final form: x = X(t), three conditions must be satisfied. o p eð2=3ÞKp oT oT e ¼r rp : ð7Þ T ¼ T ¼ T ; k v ¼ k c þ q X 0ðtÞQ: ð11Þ ot RT lðT ÞRT v c p v ox c ox v Note that Eq. (7) is parabolic. However, the The first two conditions require that the tempera- domain through which the evolved gases can per- ture at the interface is continuous and equal to the colate is not known in advance, but must be deter- char formation temperature. The final condition 2636 H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 states that the heat flux transmitted from the char must supply an energy Q per unit mass of virgin material to liberate the gas at the char front, with the excess conducted into the interior of the solid. Solutions satisfying the temperature boundary conditions at the surface, interface, and in the interior can be obtained by noting that the ab- sence of any independent length or time scale implies that all physical quantities depend only on a similarity variable g, defined as: pffiffiffiffiffiffiffi g ¼ x= av t ; av ¼ kv=ðqv CpvÞ: ð12Þ This implies that the interface position corresponds to a constant value of g. Denoting this Fig. 2. Interface location parameter C as function of value as g = C, the interface position can be dimensionless endothermic energy Q/(Cpv (Tp T1)). expressed as: The remaining parameters are fixed at the values pffiffiffiffiffiffiffiffiffiffi kc/kv = 1/3, (Ts Tp)/ (Tp T1) = 3, (av/ac) = 3/4. X ðtÞ¼C ðavtÞ: ð13Þ The solutions satisfying the first two of Eq. (11) the char or the virgin material. As this value is can be written in the form: approached, the char layer depth approaches infinity. Beyond this point, there are no solutions T v ¼ T pF ðgÞ to Eq. (17), and thus no solutions to the energy T T erfcðg=2Þ F ðgÞ¼ 1 þ 1 1 ; ð14Þ equations of the postulated form. T p T p erfcðC=2Þ

T ¼ T GðgÞ 4. Mass transfer in char c p T S T S erfðg =2Þ GðgÞ¼ 1 ; ð15Þ For the calculation of the mass transport T p T p erfðC =2Þ through the char, it will be assumed that the inter- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi face is endothermic. The mass transport and hence C ¼ ða =a ÞC g ¼ ða =a Þg: ð16Þ v c v c the pressure distribution is confined to ap domainffiffiffiffiffiffiffi bounded by the interface x X t C a t and The parameter C that determines the location of ¼ ð Þ¼ c the char surface x = 0. The coefficients in Eq. (7) the interface is found by requiring that the last depend only on the variable g*, while the char of Eq. (11) be satisfied. The result is identical with interface and surface correspond to the fixed val- that obtained for the interface condition in the ues g* = C* and g* = 0, respectively. Thus, it flame spread problem considered in [1]. In slightly makes sense to assume that the pressure is a func- different notation it can be expressed as follows: tion of g* only. Denoting the pressure at the char Q kc ðT S T pÞ surface by ps, the pressure in the evolved gas is ¼ f1ðC Þf2ðCÞ; CpvðT p T 1Þ kv ðT p T 1Þ written as: ð17Þ pðx; tÞ¼psPðg Þ: ð19Þ where Moreover, since the evolved gas viscosity is a func- 2 tion of temperature only, a reasonable assumption 2 exp C =4 n ffiffiffi ð Þ is that l/l =(T/T ) . Using the solution for the f1ðC Þ¼p ; p p pC erfðC =2Þ temperature field obtained above, Eq. (7) becomes: 2 2 expðC =4Þ ffiffiffi g d f 2ðCÞ¼p : ð18Þ b ðPðg Þ=Gðg ÞÞ pC erfcðC=2Þ 2 dg Figure 2 shows the variation of the interface d dP þ PðgÞ=ðGðgÞÞnþ1 ¼ 0; ð20Þ location parameter C with the dimensionless dg dg endothermic energy parameter Q/Cpv((Tp T1)). The other parameters are fixed at the values indi- b ¼ e1=3a l =ðp KÞ: ð21Þ cated in the figure caption, which are chosen to be c p s consistent with those used in [1]. For these values, The boundary conditions at the interface and the when the interface is no longer endothermic, char surface respectively are: C = 1.15392. In principle, an exothermic interface dP C is possible, but the maximum possible energy P ðg ¼ CÞ¼ bðq T Þ=ðq T ÞM; dg 2 v p s s release corresponds to Q/Cpv ((Tp T1)) = 1 no matter what the thermal properties of either Pð0Þ¼1: ð22Þ H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 2637

Equations (20)–(22) reveal some important g ðgÞ¼g½erfðg=2Þ 2 2 physics. Clearly, the parameter b plays a major 4 role in the mass transport. Small values of b þ pffiffiffi erfðg=2Þ exp ðgÞ2=4 p correspond to nearly instantaneous transport rffiffiffi of the evolved gases from the interface to the 2 pffiffiffi 2 erfðg= 2Þ: ð26Þ surface, with little accumulation in the interior p of the char. Thus, models that postulate instant surface emission of gases inherently assume that Figure 3 shows how the pressure distribution of the material in question has small values of b. the evolved gases in the char changes as the mass flux parameter increases, assuming that the accu- Conversely, if the material has a value of b that (0) is not small, then the local accumulation in the mulation parameter b = 0. Note that since P interior cannot be neglected, and the mass flux represents a pressure normalized with respect to at the surface is not the same as that evolved the ambient pressure at the char surface, the pres- in the interior. Moreover, the spatial distribu- sure rise is quite significant. Thus, a linearized tion of the evolved gases on the surface is no treatment of this equation would be very inaccu- longer simply related to the spatial generation rate. This nonlinear pressure equation poses con- pattern in the interior. siderable difficulties in numerical solution. The The magnitude of the pressure rise can be above analytical solution is helpful in determining inferred from Eq. (22). Note that even for small stable and accurate numerical schemes. values of b the pressure rise can be considerable. The above solution, although derived for b =0 The right-hand side of the first of Eq. (22) is and M fixed, has a much wider range of validity. Indeed, if we write M ¼ bM~ and treat M~ as a large proportional to b(qv/qs). Here, qs is the evolved gas density evaluated at the char surface. Even parameter with b fixed, then it is easy to seep thatffiffiffiffiffi ~ if the first factor is small, the density ratio of the same solution holds with an error Oð1= MÞ. virgin solid to gas is quite large. The remaining The accuracy of the analytical solution has been terms in the right hand side of this equation confirmed by testing it against numerical results are typically of order unity. This implies that computed for b =1,M = 10, 100, and 1000. The M is typically of order unity, even if b is small. errors are actually much smaller for the cases Thus, the pressure rise will typically not be small investigated than the error estimate. This is in fact compared with the ambient pressure in the gas at the most realistic case for this particular problem, ~ the surface of the char. Indeed, if the internal since M takes the form: pressure is high enough, it can rupture the char C M~ ¼ ðq T Þ=ðq T Þ: ð27Þ surface producing a crackling sound – a familiar 2 v p s s experience of a wood log in a fire place. A natural starting point for the analysis of In general, C* is a number of order one (see Eqs. (20)–(22) is the observation that when Fig. 2), as is the temperature ratio Tp/Ts. Howev- b = 0, an analytical solution can be found for er, the density ratio of virgin solid to gas at the any value of M. Denoting the resulting pressure char surface will almost always be large. Finally, distribution by P = P(0), the solution takes the as shown below, the solution is also valid for the form: pressure distribution in the opposed flow flame Z spread problem studied in [1]. g 1=2 P ð0ÞðgÞ¼ 1 þ 2M ½GðxÞð1þnÞ dx : ð23Þ 0

The integral in Eq. (23) represents the effect of the temperature dependence of the viscosity of the evolved gases. If it is further assumed that l T so that n = 1, the integral can be evaluated explicitly. Denoting the integral as I (g*), the result is: T s T s T s g1ðg Þ Iðg Þ¼ g 2 1 T p T p T p erfðC =2Þ 2 T s þ 1 =erfðC =2Þ g2ðg Þ; ð24Þ T p Fig. 3. Evolved gas pressure distribution p/p = P(0)(g*) s g1ðg Þ¼g erfðg =2Þ in char for different values of mass flux parameter M. 2 The solutions are valid for values of the transport pffiffiffi 1 exp ðgÞ2=4 ; ð25Þ parameter b = 0. The remaining parameters are * p Ts/Tp = 2, and C = 0.5. 2638 H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641

5. Opposed flow flame spread not mean that there is no preheating of the gas or solid. While the gas phase dynamics is not of Analysis of the flame spread problem requires interest here, it should be noted that both the flow the solution of a coupled problem involving both and temperature distributions depart from their the gas and condensed phases. Most previous ambient values ahead of the flame. The second research has focused on the gas phase. The ther- assumption is that the surface temperature of mal degradation of the condensed-phase material the char is uniform downstream of the flame is typically treated using a surface pyrolysis model front. This was experimentally found to be coupled to a simple heat conduction analysis in approximately constant [8]. This assumption the interior of the solid. There has been almost yields an internally consistent coupled solution no work on charring materials like wood, where to the heat transfer problem in both phases. Spec- the gaseous ‘‘fuel’’ is liberated at an interior sur- ification of a value for this temperature, together face whose location must be found as part of the with all the material properties and upstream flow solution. conditions, then uniquely determines the flame Figure 4 shows the overall geometry of the speed. flame spread problem in a coordinate system mov- Up to this point, all assumptions and simplifi- ing with the flame. While this paper is concerned cations are contained in the solutions described only with the condensed phase gas transport mod- in [1]. However, that analysis required a third el, it is important to explain certain features of the major assumption that relates the spatial distribu- overall problem. To begin, it is assumed that the tion of the mass flux of gas liberated at the problem can be regarded steady in a frame of ref- char–virgin material interface to that at the sur- erence moving with the flame speed V. The objec- face. The analysis presented below removes that tive of the analysis is to determine V as a function assumption, and replaces it with a physics-based of the material properties of the solid fuel, the gas- model for the evolved gas transport. It will be eous oxidizer, and the ambient speed U1 of the demonstrated that the results obtained using this opposed flow. model are consistent with the earlier analysis, Two simplifications are introduced that permit and thus complete the solution for the opposed the gas phase and condensed phase analyses to be flow flame spread over charring materials present- considered separately, with the results of each ed in [1]. analysis combined to produce the desired results. The starting point for the analysis is the repre- First, it is assumed that there is no heat or mass sentation of the steady state version of Eq. (7) in transfer between the gas and condensed phase the parabolic coordinate s*, x* system shown in upstream of the flame front. Note that this does Fig. 4 (which is similar to Fig. 5 of Ref. [1]).

Fig. 4. A composite flame spread figure similar to Fig. 5 of Ref. [1] showing computed isobars in char and streamlines in the gas phase. The property values are the same as those in [1]. However, instead of isotherms, isobars are shown. Both isotherms and isobars follow lines of constant x*. Fuel gases are driven along lines of constant s*, as shown. Property values are: Q/Cpv (Tp T1)=1; av/ac = 0.75; (Tp T1)/(Ts T1) = 0.275; kc/kv = 1/3; c = 0.5018; 5 7 lp = 2.3 · 10 kg/ms; ac = 6.164 · 10 . Also, char porosity = 0.76; fuel gas molecular weight = 0.08 kg/mol; 13 2 03 permeability of char = 10 m ; b = 1.063 · 10 ; Ts = 850 K; Tp = 451 K; T1 = 300 K. H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 2639 o p o eð2=3ÞKp op axis corresponds to x* = 0. Assuming all physi- V e ¼ ox RT ox lðT ÞRT ox cal quantities to be functions of x* ensures that the gradient of all physical quantities in the sol- o eð2=3ÞKp op þ : ð28Þ id is perpendicular to the surface for x <0. oy lðT ÞRT oy Since the temperature and pressure are constant for x P 0, assigning these values at x* = 0 spec- Equation (28) must be solved subject to the fol- ifies the boundary conditions at the gas–solid lowing boundary conditions at the gas–solid inter- interface. face y =0. To transform the equations to parabolic coor- op dinates, note that the pressure and temperature pðx; 0Þ¼p for x P 0 ðx; 0Þ¼0 for x < 0: depend only on x*. Introducing the dimensionless s oy variables: * n * ð29Þ T = TpG(x ), l = lp(T/Tp) , and p = psP(x ), The first of Eq. (29) states that at the surface of the conservation of mass for the evolved gases the solid, downstream of the flame front, the becomes: pressure must be the ambient pressure. The sec- d ond of these equations states that there is no 2bx ðPðx Þ=Gðx ÞÞ dx mass flux of gasified fuel through the surface d dP upstream of the flame front. If this were not P x = G x nþ1 0 þ ð Þ ð ð ÞÞ ¼ true, the flame would begin further upstream, vio- dx dx 1=3 lating the geometry on which the analysis is based. Where; b ¼ e aclp=ðpsKÞ: ð33Þ Similarly, the temperature Tv in the virgin Here, b is the accumulation parameter identified material and Tc in the char are solutions to the heat conduction equations in each material. in Eq. (21). A similar analysis yields the analogous equations for the char temperature function G(x*) oT o2T o2T and virgin material temperature T = T F(x). V c ¼ a c þ c ; v p ox c ox2 oy2 dG d dG 2 2 2x þ ¼ 0 oT o T o T V v ¼ a v þ v : ð30Þ dx dxdx o v o 2 o 2 x x y dF d dF 2x þ ¼ 0: ð34Þ At the char–gas interface, temperature satisfies dx dx dx boundary conditions analogous to pressure. It is clear from the discussion of the temper- oT ature boundary conditions that by identifying T ðx; 0Þ¼T for x P 0 ðx; 0Þ¼0 for x < 0: c s oy x* = g*/2 and x = g/2, the solutions for the temperature field given in Eqs. (14) and (15) ð31Þ are also the solutions for the flame spread prob- Again, for x P 0, the temperature takes on the lem. Since we also require that P(0) = 1 to constant value Ts, while for x < 0 there is no inter- enforce the surface pressure boundary condition, phase heat transfer. At the char–virgin material if we can demonstrate that the interface condi- interface, continuity of temperature requires that tion for the flame spread problem is identical Tc = Tv = Tp. As in the earlier problem, the loca- to Eq. (22), then the equivalence of the two tion of this interface must be determined as part of problems is complete. the solution. Finally, far from the gas–solid inter- The mass flux m_ of evolved gas at the interface face, the temperature in the virgin material retains is given by the expression: its ambient value; T = T . v 1 ~ Solutions are obtained in parabolic coordi- m_ ¼ðqv qcÞV i ~n; ð35Þ nates defined as (see Fig. 4): where,~i is a unit vector in the flow direction, ~n is a rffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unit normal to the interface pointing into the vir- av V s þ ix ¼ ðs þ ixÞ¼ ðx þ iyÞ; gin material and V is the flame spread velocity. a a c c Since the pressure and temperature are only func- kc tions of x*, the interface is a surface defined by where V is the flame speed and ac ¼ : ð32Þ * * qc Cpc x = C . Using this we obtain: The isotherms shown in Fig. 5 of Ref. [1] are jrxj2 m_ ¼ðq q Þ2a C : ð36Þ similar to isobars shown in Fig. 4. These are v c c jrxj lines of constant x* (or constant x). The transformation is defined so that the branch cut is The mass flux can also be written in the form: taken to be the positive x axis. Thus,pffiffiffiffiffiffiffiffiffiffiffiffis* =0 2=3 prp ~n dp on the negative x axis, and s ¼ Vx=ac on m_ ¼ e K rp ~n ¼ jrx jð37Þ the positive x axis in the char. The positive x lðT ÞRT dx 2640 H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641

Introducing the dimensionless pressure P and initial ambient temperature T1 to Ts > Tp under using the equation of state for the evolved gas, the transformation g* =2x*. Weak dependence ps = qsRTs, the two expressions for m_ can be com- of the solution on the accumulation parameter b bined to yield the following condition at the validates the assumption made in Ref. [1] and interface: completes the flame spread solution. The fuel dP q T mass flux follows the heat flux lines and is normal P 2C v p b: 38 to the isobars. Calculations using representative ¼ ð Þ dx qsT s values for wood [1,5] show that the pressure gen- This result is identical with that obtained in Eq. eration at the char–virgin material interface is (22) if we again make the identification x* = g*/2 considerable and equal to 13.27 kPa. While prob- and note that the value of C* obtained here must lems with more complicated geometry will have to satisfy the same condition. Thus, the solution of be numerically solved, the analytical solution pre- the condensed phase portion of the opposed flow sented here helps in determining the stability and flame spread problem is identical to the solution accuracy of potential numerical schemes. of the one-dimensional impulsively heated problem. The flame spread solution obtained in Ref. [1] assumed that the mass flux of gaseous fuel Acknowledgment emerging at the interface moves along the curves of constant s* without change until it reaches the This research was funded by NIST – GRANT # surface of the char layer where it is oxidized by 3D1086. air to produce the flame. The present analysis shows that it is equivalent to assuming that b 1. Thus, the analytical solution for P obtained References in Eq. (23) completes the solution in [1], and pro- vides a self-consistent physics-based model of [1] A. Atreya, H.R. Baum, Proc. Combust. Inst. 29 opposed-flow flame spread over charring materials. (2002) 227–236. Figure 4 shows the isobars (lines of constant [2] C. Di Blasi, Polymer Int. 49 (2000) 1133–1146. dimensionless pressure P) during flame spread [3] J.E.J. Staggs, Polymer Degrad. Stab. 82 (2003) 297– over a charring material. It is similar to Ref. [1] 307. [4] E.J. Kansa, H.E. Perlee, R.F. Chaiken, Combust. where isotherms were shown. It is computed with Flame 29 (1977) 311–332. the same property values (see Fig. 4 caption). [5] A. Atreya, Pyrolysis, ignition and fire spread on Computed streamlines in the gas phase from horizontal surfaces of wood, Ph.D. thesis, Harvard Ref. [1] are also shown for completeness. Both iso- University, Cambridge, MA, 1983. therms and isobars follow lines of constant x* and [6] A. Atreya, Combust. Sci. Technol. 39 (1984) 163– fuel gases are driven along lines of constant s*. 194. [7] A. Atreya, AIChE Symposium Series 246 81 (1985) 46–54. 6. Summary and conclusions [8] K. Mekki, A. Atreya, S. Agrawal, I.S. Wichman, Proc. Combust. Inst. 23 (1990) 1701–1709. [9] Fredlund, B. A model for heat and mass transfer in The solution to the condensed phase flame timber structure during fire, Ph.D. dissertation, spread problem is identical to the initial transient Lund University, 1988. problem where the surface temperature of a semi- [10] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, infinite solid is instantaneously raised from an Addison-Wesley, Reading, MA, 1959, 57–59.

Comments

Jose Torero, The University of Edinburgh, UK. c = 0.45. For the correlation only av for poplar was need- What properties were used for comparing with the ed (=0.1 mm2/s, obtained from the literature). experiments? The value of the charring constant ‘c’ can also be determined from wood and char properties, as shown Reply. This apparently simple question is very impor- in Fig. 2, and the model can be used in a predictive tant because it inquires about the fundamental usefulness manner. Using the literature values for poplar and its of the model. The experimental data on measured ﬂame char,: spread rates and char depths for wood correlated well according to the parabolic char-material interface, (Vy/ 2 2 av) = c +2c(Vx/av), given by the model. The data from kc=kv ¼ 1:4; av=ac ¼ 0:248; ðT p T 1Þ=ðT s T 1Þ four experiments on poplar collapsed onto one curve and the parabola that passed through this data had a value of ¼ 0:618; Q=ðCpvðT p T 1ÞÞ ¼ 1 H.R. Baum, A. Atreya / Proceedings of the Combustion Institute 31 (2007) 2633–2641 2641

we get c = 0.502. This is not too different from that However, within the measurement accuracy, the char obtained by the correlation, given the large uncertainty temperature was roughly constant for a given experi- in the value of the heat of pyrolysis of wood and prop- ment but depending on the experimental condition, the erties of char. value of the constant char temperature changed from 550C to 750C with the average being 650C. While it d may seem high, it is not unusual. Similar measurements are available in the literature. The constant char temper- Michael Delichatsios, University of Ulster, UK. Great ature used for numerical calculations and data correla- effort, needing some comments: tion was 850 K. This was also measured during the (1). The surface temperature is constant along the experiment. Fortunately, the constant surface tempera- fuel bed following ignition. This is surpassing and its val- ture boundary condition (see Fig. 4) is actually applied ue at the leading edge of 650C seems too high. (2). some distance behind the flame foot as required by the Wood can form cracks so that pressure build up may parabolic coordinates. This leaves some room for the not occur for all species of wood. temperature to rise from the pyrolysis temperature to the char surface temperature. Reply. (1). The measured char surface temperature (2). Yes indeed, if the wood cracks there is no pressure behind the flame foot was found to be approximately build up. Interestingly, a long time ago, Professor Em- constant for various experiments under different external mons mentioned a method of making uncracked wood radiation conditions. The data showed considerable fluc- charcoal. The secret was to pyrolyze wood in its own vol- tuations in the measurements. This is unavoidable atile atmosphere without oxygen. Thus, it is possible that because there is no way of assuring that the ther- the char underneath the flame does not crack. We will mocouple will stay on the char surface after pyrolysis. have to look at the experiments more carefully next time.