Homogenization and fingering instability of a ...

Combustion and Flame 162 (2015) 4046–4062

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Homogenization and fingering instability of a microgravity smoldering combustion problem with radiative heat transfer Ekeoma R. Ijioma a,c,∗, Hirofumi Izuhara d, Masayasu Mimura a,b,c, Toshiyuki Ogawa a,b,c a

Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan c Center for Mathematical Modeling and Applications (CMMA), Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan d Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki 889-2192, Japan b

a r t i c l e

i n f o

Article history: Received 5 April 2015 Revised 28 July 2015 Accepted 30 July 2015 Available online 17 August 2015 Keywords: Multiscale modeling Homogenization Smoldering combustion Radiative heat transfer Fingering instability

a b s t r a c t The present study concerns the homogenization and fingering instability of a microgravity smoldering combustion problem with radiative heat transfer. The major premise of the homogenization procedure is the slow exothermic fuel oxidation of a reactive porous medium at the pore level. The porous medium consists of periodically distributed cells, with a suitable scale parameter. A nonlinear reaction rate of Arrhenius type accounts for the relationship between the reactants and the heat that sustains the smoldering process. At the gas–solid interface, the balance of thermal fluxes is given by the heat production rate due to the reaction and the radiative heat losses at the interface. Since the size of the inclusions is small with respect to , we derive a kinetic model for fuel conversion in the region occupied by the solid inclusions and hence complete the description of a single-step chemical kinetics. The derived macroscopic model shows a close correspondence to a previous phenomenological reaction-diffusion model, within a suitable choice of parameters. We perform numerical simulations on the microscopic and homogenized models in order to verify the efficiency of the homogenization process in the slow smoldering regime. We show that the results of the macroscopic model capture the distinct fingering states reminiscent of microgravity smoldering combustion. We also show qualitative results that confirm the close relationship between the radiative heat losses and the characteristic length scales of the instability. © 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction The mathematical modeling of reactive processes in porous media has been a subject of enormous interest, specifically, due to the nature of the chemical processes involved. In addition, there is an inherent difficulty usually encountered when attempting to resolve the details of the pertinent porous medium of interest. A typical example of such a problem concerns the slow exothermic chemical reaction that takes place on the surface of reactive porous materials, in the presence of a gaseous mixture containing an oxidizer. When the chemical process proceeds in a nonflaming mode, it is referred to as smoldering. Distinct configurations of smoldering are possible depending on the direction of flow of the gaseous oxidizer relative to the direction of propagation of the reaction front. These configura-

∗ Corresponding author at: Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan. E-mail addresses: [email protected] (E.R. Ijioma), [email protected] (H. Izuhara), [email protected] (M. Mimura), [email protected] (T. Ogawa).

http://dx.doi.org/10.1016/j.combustflame.2015.07.044 0010-2180/© 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

tions have been a subject of practical interest since they exhibit some important characteristic features. If the gaseous oxidizer flows in a direction opposite to the direction of the reaction front, we refer to the configuration as reverse (upstream) smoldering, whereas if the oxidizer flows in the same direction as the direction of propagation, the configuration is known as forward (downstream) smoldering. For a detailed discussion on the distinct smoldering configurations, we refer to [1–3]. Furthermore, it has been reported in several experimental papers [4–9], that the smoldering process occurring under microgravity results in the self-fragmentation of the smolder front. The phenomenon was believed to arise from a destabilizing effect of oxygen transport in the absence of natural convection, which characterizes a microgravity environment. The smolder front has the form of finger-like patterns, which are distinct depending on the oxygen velocity (or its concentration). In a Hele–Shaw cell experiment, the fingering instability involves two characteristic length scales: the gap between adjacent fingers and the finger width. The velocity of the flow field determined the gap between the fingers whereas the finger width depends on heat losses in the system [5]. The experimental results were

E.R. Ijioma et al. / Combustion and Flame 162 (2015) 4046–4062

Nomenclature Y Ys Yg Y n hr

unit cell solid part of the unit cell gas-filled part of the unit cell -periodicity cell outward unit normal vector ( − ) radiative heat transfer coefficient (W/m2 /K ) a radiative heat transfer coefficient ( − ) L characteristic length of the system (m ) characteristic length of the pore (m ) b = (b1 , b2 )T velocity field, T: vector transpose (m/s ) y = (y1 , y2 ) microscopic variable ( − ) x = (x1 , x2 ) macroscopic variable (m ) T temperature (K ) concentration of gas (kg/m3 ) P concentration of solid fuel (kg/m3 ) W1 = W1 (T , , P ) reaction rate (kg/m2 /s ) W10 = W1 (T 0 , 0 , P 0 ) reaction rate (kg/m3 /s ) Q1 heat release ( J/kg ) D molecular diffusion coefficient (m2 /s ) t time (s ) A1 pre-exponential factor (L m3 /kg/s ) A2 pre-exponential factor (m3 /kg/s ) Ta activation temperature (K ) Cg volumetric heat capacity of gas ( J/m3 /K ) Cs volumetric heat capacity of solid ( J/m3 /K ) N() total number of cells in the -periodic medium ( − ) u effective dimensionless temperature (−) v effective dimensionless oxidizer concentration ( − ) w effective solid fuel concentration ( − ) Le effective Lewis number ( − ) Pe effective Péclet number ( − ) Greek symbols ε emissivity ( − ) σ Stefan–Boltzmann constant (W/m2 /K4 ) scale parameter ( − ) ω vector-valued function, solution of cell problems ( − ) periodic medium interior boundary of Y s ensemble of solid inclusions in matrix of interconnected gas part in g gas–solid interface of φ porosity ( − ) φs tortuosity ( − ) λ thermal conductivity (W/m/K ) α thermal diffusivity (m2 /s ) β dimensionless heat release ( − ) γ dimensionless pre-exponential factor ( − ) τ number of time steps ( − ) θ dimensionless activation temperature ( − ) ratio of gas heat capacity to effective heat capacity (− ) Superscripts ∗ dimensionless eff effective -periodic, varying

s · 0

4047

surface extension homogenized

Subscripts a activation b burned c characteristic g gas phase s solid phase ∞ ambient 0 initial r radiation further validated mathematically in different contexts [9–18]. For a mathematical aided understanding of the smoldering combustion problem, a traveling wave analysis of the pertinent mathematical model is usually adopted [11,13]. The interest is to determine the range of the experimental observation through the analysis of the spread rate of the smoldering front. The derivation of the macroscopic behavior of porous media combustion from pore level has been achieved over the past years using either of two approaches. The first approach involves a volume averaging method [19], which has been applied in the derivation of filtration combustion models [20]. However, until now, there has been no mathematical justification of the validity of the volume averaging approach from the pore level. The second approach uses the homogenization theory [21–24], which has the advantage of making the model derivation mathematically rigorous. In [14,17], a macroscopic model of the smoldering combustion problem was derived using the method of periodic homogenization and its rigorous justification via the two-scale convergence method [25] was discussed in [26]. The model emphasizes the role of conductive heat transfer in the system without heat losses to the surroundings, and hence cannot be used for answering questions concerning the influence of heat losses. The homogenization of heat transfer processes has attracted the attention of researchers over the years; see for instance [27,28] for a treatment of conductive heat transfer scenarios. In [29,30], processes involving conductive and radiative heat transfers in porous media with cavities were studied. We also point out some of the studies on radiative heat transfer in the literature [31,32]. The physical (or rather experimental) setup of the problem is as shown in Fig. 2 (cf., [5,9]). It consists of an upper gas layer and a lower solid layer in which the porous sample is placed. A gaseous mixture is supplied from one end of the setup and ignition is initiated at the other end, thus allowing a reverse smoldering combustion to proceed. The region of interest may be simply taken to be twodimensional along the direction of flow of the gaseous mixture, which occupies a homogeneous gas layer since the thickness of the sample is negligibly small. An alternative point of view for a two-dimensional approximation is the fact that the smoldering combustion process we have in mind takes place in a narrow gap [4,5,9]. The size of the gap ensures the absence of natural convection and hence draws the problem much closer to a variant of the phenomenon observed under microgravity [7]. In the latter report, it is shown that, within a suitable range of oxidizer velocity, it is possible for the smoldering propagation to proceed downstream after the smolder front reaches the upstream end of the sample. The result is the subsequent smoldering of parts of the unburned fuel left behind by the upstream smolder wave. Regarding the description of the microstructure of the porous medium, we consider a situation in which the gaseous oxidizer is free to infiltrate the porous medium through diffusion and convective transport. That is, instead of a perforated domain consisting of a solid matrix (see Fig. 1a), we take the solid fibers to be distributed in a gas matrix (see Fig. 1b). An advantage of this microstructure

4048


g

s

s

g

(a)

(b)

Fig. 1. Schematic of the physical porous medium (a) perforated domain with solid matrix, s and holes g with boundary ; (b) gas–solid domain with gas matrix g and solid inclusions s with boundary .

char propagation

Yg gas layer

solid layer

Ys

Y

Yg : gas part Ys : solid part

g

inflow gaseous mixture

s

N( ) Fig. 2. Schematic diagram of the three-dimensional gas–solid geometry. The direction of the arrows illustrates a reverse combustion configuration.

Ys,i

s

i=1 N( )

is that it allows simultaneous treatment of coupled heat and mass transfer problem in the heterogeneous porous medium consisting of seemingly distinct reference microstructures for the two transport processes. The next challenge concerns the description of the microstructure for the purpose of homogenization. We assume that the region for homogenization, depicted in Fig. 3, consists of a uniformly periodic distribution of -sized cells, with being the size of the period. The size of the -periodically distributed cells is assumed to be increasingly small. In view of the complexity of the chemical reactions taking place on the distributed -scaled surfaces, a direct numerical simulation of the problem at this level becomes almost unattainable and far too costly in terms of computer resources. Also, it is impossible to capture the qualitative behavior of the phenomena experimentally observed at the macroscopic level while attempting to resolve all the details of the processes occurring at the microstructure level. Thus, the present goal of homogenization in this paper is to derive a functional form of the limit homogenized problem in order to overcome the difficulty of dealing with the intricate structure and chemical processes of the heterogeneous porous medium. Additionally, explicit formulas for calculating the effective transport parameters have to be derived as well. The effective parameters allow us to show the efficiency of the homogenization process [33]. Contrary to the models derived earlier [14,17,26], the novelty of the present study is the incorporation of heat transfer by radiation and a kinetic model for solid fuel mass concentration. The kinetic model involves a second order Arrhenius type nonlinearity in the problem. Based on these improvements, we show that the results of the proposed model are within a suitable range for describing the experimentally observed fingering instability of smoldering combustion under microgravity. In

i

i=1

Fig. 3. Two-dimensional periodic domain for homogenization and the reference cell. The solid inclusions are represented by the gray circles with the boundaries in red. The remaining part of the medium consists of the gas domain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

particular, we numerically study the relation between the radiative heat losses and the characteristic length scales of the fingering instability. Although an attempt was made in [10], to connect the effect of convective heat losses on the fingering instability, from our knowledge of the problem this has not been clearly described in the literature through direct numerical simulation of the pertinent system of equations with radiative heat losses. 2. Pore scale description of the problem In this section, we discuss the derivation of the microscopic model for a smoldering combustion problem with a one-step kinetic scheme. 2.1. Geometry of the porous medium Let = (0, L1 ) × (0, L2 ) be a smooth bounded open set in R2 , which is subdivided into N() periodicity cells Yi , i = 1, . . . , N(),


with each cell being equal and defined up to a translation of a reference cell Y = [0, 1]2 . The number of cells with respect to , in each coordinate direction j, is defined by N j () = (0, L j ) −1 , j = 1, 2, where Lj define the lengths of the sides of the domain . Then, the total number of cells in the domain is N() = N1 () × N2 () = || −2 1 + o(1) . Here, = /L is the size of a single period cell defined as the ratio of the characteristic length of Y to the characteristic length of the domain . The reference cell consists of two distinct parts—a gas-filled part and a solid part. We denote by Ys , the solid part of Y having a smooth boundary denoted by . The gas filled part is denoted by Yg = Y \ Ys . The collection of the periodically dis , i = 1, . . . , N (). Similarly, the coltributed solid parts is given by Ys,i is denoted lection of the periodically distributed boundaries of Ys,i by i , i = 1, . . . , N(). Thus, the ensemble of the periodically translated solid parts, the matrix of gas parts and the ensemble of the solid boundaries can be written as

s =

N()

, = Ys,i g

N() s , = i ,

i=1

(1)

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2.3. Balance equations for heat and mass transport We consider a diffusion-convection problem describing the transport of mass of a gaseous mixture, which is coupled to a heat transport problem. One of the properties of the porous medium of interest is that the diffusion coefficients and volumetric heat capacities are varying in space, i.e. the coefficients take only two different values of the same order of magnitude. Let the thermal conductivity of the constituent parts be defined by the constants λg and λs and the heat capacities by the constants Cg and Cs . Then, the varying coefficients of the -periodic medium can be written as:

λ (x) =

λg , x ∈ g (x) = Cg , x ∈ g , C λs , x ∈ s Cs , x ∈ s .

Similarly, the molecular diffusion D (x) = D(x/) is restricted in g . The gaseous oxidizer occupies the subdomain, g , and reacts with the solid inclusions (i.e. solid fuel) at the surface . Convection takes place only in the gaseous subdomain g . Thus, we introduce a given velocity field

i=1

x

where we denote by | · |, the measure of any of the sets introduced above. Specifically, |Y| is the volume measure of the reference cell Y, with | | the surface measure of . We also note that the measures on the periodic cells can be represented in terms of the measures on the reference cells through the following relation:

b (x) = b

|Yi | =

We assume zero flow at the solid boundary , such that the boundary condition for the velocity field at the solid surface is b(y) = 0 on . We assume the given velocity field to be constant along the direction of front propagation and zero in the transversal direction, i.e., the velocity field has the form b(y) = (b1 (y), b2 (y)), with b1 (y) defined as follows

|Y |, 2

| i | =

| |, for i = 1, . . . , N().

In the -periodic porous medium (see Fig. 3), we assume that the physical parameters and functions are rapidly oscillating. That is, for a given function ψ , one may write ψ (x) = ψ(x/), for all x ∈ with y = x/ representing the microscopic variable. We also assume that the function ψ (y) and physical parameters are extended by Yperiodicity in R2 with a period length .

2.2. Chemistry of the problem The model of heterogeneous chemical reactions for smoldering combustion can be described by using the kinetic schemes given by [34], which has been widely employed in numerical simulations of smolder processes [35–37]. Here, the chemical kinetics is governed by a single-step kinetic scheme for fuel oxidation, i.e. W1

Solid fuel + μ1oO2 −→ μ1c Char + μ1gp gaseous products, Q1

(2)

where W1 is the reaction rate and Q1 , the heat release. The stoichiometric coefficients are denoted by μ1i . The oxidation reaction equation (2) is exothermic (i.e. Q1 > 0). The rate law governing Eq. (2) is given by

W1 = P A1 exp ( − E1 /RTs ),

(3)

where P is the mass concentration of the solid fuel, is the concentration of gaseous oxidizer, which is usually oxygen, Ts is the temperature of the solid, E1 is the activation energy, R is the universal gas constant and A1 , the pre-exponential factor in the Arrhenius law. The rate law given by Eq. (3) is second order with respect to the reactants species. This is one of the main extensions to the models of first order rate law studied (e.g., [9,11,14,17]) for a similar smoldering combustion scenario. The major handicap of previous efforts based on the first-order kinetics is the inability of the models to account for the coexistence of upstream and downstream smolder waves reported in [7]. Thus, the presence of P in Eq. (3) is essential for understanding reflection phenomena of smolder waves. Its significance would be demonstrated later in a subsequent section.

, in g ,

(4)

where the vector field b(y) ∈ Yg is spatially periodic with respect to y and it is divergence-free over the gas-filled domain:

∇y · b(y) = 0, in Yg .

(5)

b1 (y) = k in Yg , b1 (y) = 0 on Y s , where k > 0 is a positive constant, Y s = Ys ∪ and b2 (y) = 0, such that the given velocity field satisfies the continuity equation, Eq. (5). We point out that in a generalized treatment of the flow problem, one may need to generate the velocity field by solving any incompressible flow problem. This involves considering the full fluid dynamics of the problem within the interior of the gaseous subdomain and at the boundary of the solid inclusions. However, in a simplified treatment, as we have adopted in Eqs. (4) and (5), we have avoided the intricacy of the fluid dynamical problem by simply assuming that the given velocity field is constant and divergence-free over the gaseous region. Let the temperature in the periodic domain be denoted by T , which can be decomposed as

T (t, x) =

Ts (t, x), Tg (t, x),

x ∈ s , x ∈ g ,

where T is continuous across the interface . Besides the continuity of temperature, we also have the balance of heat fluxes given by the heat produced due to reaction at the interface and radiative heat losses to the surroundings, i.e. the following conditions, for the balance of heat, are satisfied across the interface

−λg ∇ Tg · n = −λs ∇ Ts · n + Q1W1 − q (Ts ), x ∈ , t > 0, Tg = Ts , x ∈ , t > 0,

(6)

where Q1 > 0 is a given constant denoting the heat release and q is the net radiative heat flux1 given by

4 q (Ts ) = εσ ((Ts )4 − T∞ ),

(7)

1 The form of the radiative heat flux given in Eq. (7) follows from Kirchhoff’s law for gray surfaces. It implies that emissivity of the surface and the absorption coefficient are the same.

4050


with ε , σ and T∞ respectively the emissivity of the surface, the Stefan–Boltzmann constant and the temperature of the surroundings. Suppose we assume that the temperature difference T = (Ts − T∞ ) is small, then a simple Taylor series expansion of Eq. (7) around T∞ leads to 3 q (Ts ) = 4εσ T∞ (Ts − T∞ ) + O( T 2 ),

(8)

= hr (Ts − T∞ ),

3 is the radiative heat transfer coefficient. Therewhere hr = 4εσ T∞ fore, the linear form of Eq. (6) with respect to the radiative transfer term is given by

−λg ∇ Tg · n = −λs ∇ Ts · n + Q1W1 − hr (Ts − T∞ ), x ∈ , t > 0, Tg = Ts , x ∈ , t > 0.

by assumption (A2 ). The rate of fuel oxidation is given by

∂ P = −k2 P , ∂t 1i

3

2

first order rate law as:

k2 = A2 exp ( − Ta /Ts ),

(10)

T ∂ P a = −A2 P exp − , x ∈ , t > 0, ∂t Ts

x ∈ , x ∈ , x1 = L1 , t > 0, x1 = 0, t > 0, x2 = 0, x2 = L2 , t > 0.

(12)

2.4. Complete kinetic model for the solid fuel mass balance According to Eq. (2), a single-step kinetic scheme is considered for the fuel consumption, i.e. fuel oxidation, on the surface . However, for a complete description of the mass balance of P , the fuel conversion within the inclusions is described by using the information provided by its consumption rate on the surface. The basic assumption for the chemical conversion in the inclusions invokes the notion of fuel pyrolysis within the inclusions. To model this effect, the following assumptions denoted by A are necessary: Assumptions

(A1 ) P is consumed due to fuel oxidation on the surface, . (A2 ) At the level of the inclusions (assumed relatively small), the is given by the average rate rate of fuel conversion in each Ys,i of fuel oxidation on i , i = 1, . . . , N(). (A3 ) The overall rate of fuel conversion in s is balanced by the first order average rate of consumption on .

We proceed as follows: the rate of change of fuel in s is given by

∂ P = −k1i Pi , x ∈ s , i = 1, . . . , N(), ∂t

(13)

1

| |

k2 dS =

0,

and

k1i =

A2

| i |

, for each i, in Ys,i

in g

, i = 1, . . . , N () exp ( − Ta /Ts )dS, in Ys,i

1

| |

A2 exp ( − Ta /Ts )dS

A2

N()

=

i=1 N()

=

| i |

i

exp ( − Ta /Ts )dS

k1i ,

i=1

, i = 1, . . . , N () due to (A ). where k1 ∈ Ys,i 2

2.5. Parameter estimates and dimensionless equations Prior to the homogenization procedure, it is important to normalize the system of equations as discussed in [14,17], which leads to some dimensionless parameters. We introduce dimensionless variables as follows:

T = T ∗ Tc , = ∗ c , P = P ∗ Pc , x = x∗ L, t = tc t ∗ ,

(18)

where the subscript c denotes some constant characteristic quantity and the asterisk (∗) denotes the corresponding dimensionless variable. We point out that similar characteristic quantities are introduced for the physical quantities in the equations.3 The dimensionless parameters are derived from these characteristic quantities and are estimated in orders of magnitude in [38]. The choices of the parameter estimates usually lead to different forms of the limit problem during the asymptotic procedure as → 0. The dimensionless equations are of the following form:

⎧ ∂ Tg ∗ ⎪ ⎪ PT Cg∗ − Cg∗ PeL b∗ · ∇ Tg ∗ − ∇ · (λ∗g ∇ Tg ∗ ) = 0, ⎪ ⎪ ∂t∗ ⎪ ⎪ ⎪ ⎪ ∂ T ∗ ⎪ PT mCs∗ s∗ − K∇ · (λ∗s ∇ Ts ∗ ) = 0, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∗ ∗ ∗ −1 ∗ ∗ ⎪ ⎪ ⎨ ∂ t ∗ − PeL b · ∇ − Leg ∇ · (D ∇ ) = 0, ∗ −λ∗g ∇ Tg ∗ · n = −Kλ∗s ∇ Ts ∗ · n + DaQ1∗W1 ∗ − Bih∗r (Ts ∗ − T∞ ), ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ Tg = Ts , ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎪ ⎪−D ∇ · n = −Leg DaW1 , ⎪ ⎪ ⎪ T∗ ⎪ ∂ P ∗ tλ ∗ ∗ ∗ ⎪ ⎩ = − A P exp − a ∗ , 2 ∂t∗ tP Ts (19)

according to where we define the fuel content in each Ys,i

1 P dS, Pi = | i | i

(17)

i

(11)

where Ta = E1 /R denotes the activation temperature. The initial and external boundary conditions are given by

⎧ T (0, x) = T0 (x), (0, x) = 0 , ⎪ ⎪P (0, x) = P , ⎨ 0 ∂x1 T (t, x) = 0, (t, x) = 0 , ⎪ ⎪ ⎩∂x1 T (t, x) = 0, ∂x1 (t, x) = 0, ∂x2 T (t, x) = 0, ∂xx2 (t, x) = 0,

x ∈ ,

we average Eq. (17) over and thus obtain the following

(9)

x ∈ , t > 0,

(16)

where k2 = A2 exp ( − Ta /Ts ), with the additional condition that k = k , i = 1, . . . , N(), on , according to (A ), i.e. By defining a

We also assume that the following equations hold respectively for the consumption of the gaseous oxidizer and the solid fuel across the interface2 :

−D∇ · n = −W1 ,

x ∈ ,

where (14)

W1 ∗ = A∗1 ∗ P ∗ exp

−

Ta∗ . Ts ∗

The problem introduces the following global characteristic time scales: (15)

i

2 The kinetic model given by Eq. (10) describes fuel oxidation on . The change in the solid fuel within the inclusions is described in the sequel.

tD =

L2 , Dc

tA =

L , bc

tλ =

Cgc L2

λgc

,

tR =

L , A1c Pc

tP =

1 , A2c c

3 For any generic physical quantity ψ , the normalization scheme is given as ψ = ψ ∗ ψc where ψ c is some characteristic value of interest.


where tD is the characteristic global diffusion time scale, tA is the characteristic global advection time scale, tλ is the characteristic global time of conductive transfer, tR and tP are respectively the characteristic global chemical reaction time scale and the characteristic time of fuel reaction. We introduce as well the following characteristic dimensionless numbers:

PeL =

bc Cgc L

=

bc L

t ´ = λ (P eclet number), tA

λgc α tD α = (Lewis number), Leg =

Dc tλ A1c Pc tλ t ¨ Da = number), = λ (Damkohler L tR hrc L Bi = (Biot number).

λgc

where α = λgc /Cgc is the thermal diffusivity. Other dimensionless quantities introduced in Eq. (19) include PT = tλ /tc , the ratio of characteristic time of conductive transfer to the characteristic time scale of the observation, m = Csc /Cgc is the ratio of heat capacities, K = λsc /λgc is the ratio of heat conductivities. We take the time of conductive heat transfer in the subdomain, g , as the characteristic time of the observation at the macroscopic scale, i.e. tc = tλ . The characteristic temperature of the combustion product is given by Tc = Q1c c /Cgc so that Ta∗ = Ta /Tc is the dimensionless activation temperature and ∗ = T /T < 1 is the dimensionless temperature of the surroundT∞ ∞ c ings. In order to simplify the problem when passing to the homogenization limit, we assume that the constituents have heat capacities of the same order of magnitude, i.e. m = O(1). Since tc = tλ , it follows that PT = O(1). We study a problem for which the constituent conductivities are of the same order of magnitude, i.e. K = O(1), and the time scale of advection and conductive transfer are comparable at the macroscopic level, i.e. PeL = O(1). For the regime in which the chemical reaction is slow, we assume that Da = O(). To compensate for the slow combustion scenario, we assume that Bi = O(). We estimate the remaining parameters to be of a unit order of magnitude, i.e. Leg = tλ /tP = O(1). For simplicity of notation, we drop the (∗) in the dimensionless system given in Eq. (19). After taking into account the estimates described above, the microscopic problem can be written in the following dimensionless form:

⎧ ∂ Tg ⎪ ⎪ = Cg b· ∇ Tg + ∇ · (λg ∇ Tg ), Cg ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂T ⎪ ⎪ Cs s = ∇ · (λs ∇ Ts ), ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ = b· ∇ + ∇ · (D∇ ), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ N() ∂ P ⎪ ⎪ ⎨ ∂ t = − i=1 k1i Pi , −λg ∇ Tg · n = −λs ∇ Ts · n + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q1W1 + hr (T∞ − Ts ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Tg = Ts , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −D∇ ·n = − W1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ P = −A2 P exp − Ta , ∂t Ts

x ∈ g , t > 0, x ∈ s , t > 0, x ∈ g , t > 0, x ∈ s , t > 0, (20)

x ∈ , t > 0, x ∈ , t > 0, x ∈ , t > 0, x ∈ , t > 0.

2.6. Numerical study of the microscopic model In this subsection, our objective is to show that the essential physics of the phenomenon of interest is well-captured at the pore

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Table 1 Parameter values used in the numerical simulations.

λg 2.38 × 10

λs −4

7.0 × 10

D −4

0.25

Cg 1.57 × 10

−3

Cs

T∞

Q1

0.6858

0.1

2.5

level. Specifically, we show that the radiative heat transfer at the pore surfaces and the effect of a second order reaction rate, which gives rise to upstream and downstream waves, are captured by the pore scale description. To fix ideas, we assume that the lengths of the porous medium in each coordinate direction are L1 = 5 and L2 = 1. We choose the characteristic length of the observation to be L = 5. Then, by using the scaling argument, = /L = 0.2, and choosing = 1, the number of cells in each coordinate direction can be calculated as: N1 () = 25, N2 () = 5, so that the total number of cells in the medium is given by N() = N1 () × N2 () = 125. Here, we have assumed that the R2 space is partitioned with -multiples of the standard periodicity cell. We also assume that the domain size is sufficiently large to suffice our model of radiative heat transfer. However, we have chosen the length, L2 , of the domain in the transversal direction to be relatively small in order to minimize the total number of cells in the system. Thus, periodic boundary conditions are imposed on the lateral boundaries. The numerical method for the discretization of the problem uses the standard Galerkin method with Lagrange P2 finite elements. The time integration of the resulting discrete system of equations was performed with a Backward Differentiation Formula (BDF) as implemented in COMSOL Multiphysics package. The typical scaled parameter values used in the numerical simulations are given in Table 1. These values are normalized according to the scheme introduced in Eq. (18), in which all the physical quantities are normalized with some characteristic values subscripted by c. The numerical tests presented in this section and in the subsequent sections are such that the values of the dimensionless parameters are chosen for ease of numerical calculation. The calculations are meant to capture the qualitative essence of the phenomenon. To be able to achieve quantitative results, more parameter identification study is needed in this direction. We choose the following initial condition for T :

T (0, x) = T0 exp ( − x2 /μ), with T0 , μ > 0 in . In the numerical simulations, μ = 1.0 and T0 = 1.0 unless specified otherwise. and P are chosen as follows:

(0, x) = 0.23, in g , P (0, x) = 1.0, in s . Figure 4 depicts the distribution of temperatures (a) Tg , (b) Ts and the mass concentration of (c) and (d) P at time t = 450. The maximum temperatures of the two subdomains at the given instant of time are at equilibrium. The spatial distribution of the solid fuel depicted in (d) indicates that the material is partially consumed, which is conditioned by the presence of heat losses in the system. The cells in blue shades indicate regions of higher fuel conversion whereas the regions with darker red shades indicate the unburned region. Figure 5 delineates the coexistence of upstream and downstream smolder waves. It illustrates the evolution of the spatial distribution of P at various time instances. The emergence of such waves is conditioned by the second-order reaction rate, contrary to models of the first-order rate in which downstream propagation is not viable. Figure 5a–d shows upstream smolder waves whereas Fig. 5e–g illustrates the downstream waves. The cross-sectional plots illustrated in Fig. 5b–h are one-dimensional plots of the structure of the waves captured at the different time instants across the flame propagation direction, i.e. for a fixed value of y, the wave propagation is considered along the x-direction. The plots also correspond to the different time instances depicted in Fig. 5a, d, e, and g. Ignition is initiated from the left and the oxidizer gas is supplied from the upstream boundary (the right end),

4052


In the next step, we substitute the expansions Eq. (21)–(23), using the transformation Eq. (24), in Eq. (20) and identifying a succession of boundary value problems at the same powers of : • Equation of order −2 :

⎧ 0 ⎪ ⎪−∇y · (λg ∇ Tg ) = 0, ⎪ ⎪ ⎪ ⎪−∇y · (λs ∇ Ts0 ) = 0, ⎪ ⎪ ⎨

in Yg , in Ys ,

T 0 − T 0 = 0,

g s ⎪ ⎪ ⎪ ⎪ ⎪(λs ∇y Ts0 − λg ∇y Tg0 ) ·n = 0, ⎪ ⎪ ⎪ ⎩ 0 0 {Tg , Ts } is Y -periodic.

on ,

(25)

on ,

We deduce that

Tg0 (t, x, y) = Ts0 (t, x, y) = T 0 (t, x).

(26)

• Equation of order −1 : Fig. 4. Spatial distribution of (a) Tg , (b) Ts , (c) , and (d) P . Ignition is initiated from the left and oxidizer gas flows from the right. The circular color pattern indicates regions of unburned fuel (dark red) and regions of burned fuel (remaining blue shades indicate higher conversion of the fuel). The parameter values used in the simulation include: T0 = 4.0, Ta = 5.0, A1 = A2 = 1.5, hr = 1 × 10−3 and b1 = 1 × 10−3 . Other values are taken from Table 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and hence allowing reverse (counterflow) smoldering combustion waves to emanate. The waves propagate initially upstream. On reaching the right edge of the fuel sample, the waves reflect downstream and propagate toward the left. The phenomenon of reflecting smolder waves has been observed in microgravity experiment aboard a spacecraft [7]. We point out that hr essentially controls the intensity of the radiative heat losses to the surroundings. Its introduction in the model results in significant changes in the structure of waves. Specifically, the spread rate of the wave decreases and increases the time for the emergence of downstream waves. Also, we notice the leakage of across the front (Fig. 5d, f, and h), which intensifies as the waves move downstream. The pattern of the solid fuel shows incomplete conversion of the fuel (Fig. 5g and h). 3. Asymptotic expansions

x

(t, x) = 0 t, x,

+ T 1 t, x,

x

x

+ 1 t, x,

x

+ 2 T 2 t, x,

x

+ 2 2 t, x,

+ · · · , (21)

x

+ ··· , (22)

P (t, x) = P 0 t, x,

x

+ P 1 t, x,

x

+ 2 P 2 t, x,

x

+ · · · , (23)

where Ti (t, x, y), i (t, x, y) and Pi (t, x, y) are assumed to be Y-periodic, with y = x/ . Note that the functions at the right hand side of Eqs. (21)–(23) depend on two spatial variables: x = (x1 , x2 ) (the macro scale variable) and y = (y1 , y2 ) (the micro scale variable). The derivative operator is transformed based on the two spatial variables (x, y) introduced in the problem, i.e.

1

∇ → ∇x + ∇y .

Tg1 − Ts1 = 0,

⎪ ⎪ ⎪ −(λ (∇ T 1 + ∇x T 0 ) + λs (∇y Ts1 + ∇x T 0 )) ·n = 0, ⎪ ⎩ 1g1 y g {Tg , Ts } is Y -periodic.

in Yg , in Yg , on , on , (27)

The linearity of the problem allows to decompose the solution into elementary solutions ωi (y) associated with unit macroscopic gradients, i.e. ∇x T 0 = ei , i = 1, 2, and satisfying the following socalled cell problems

⎧ ∂ωgi ∂ ⎪ ⎪ + I = 0, λ g ij ⎪ ⎪ ∂yj ∂yj ⎪ ⎪ ⎪

⎪ ⎪ ∂ ∂ωsi ⎪ ⎪ + I = 0, λ ⎪ ij ⎨ ∂yj s ∂yj

⎪ ωgi − ωsi = 0, ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ∂ωgi ∂ωsi ⎪ ⎪ + I − λ + I ·n j = 0, λ s g ⎪ ij ij ⎪ ∂yj ∂yj ⎪ ⎩ ω is Y -periodic.

in Yg , in Ys , on , on , (28)

In this section, we briefly describe the homogenization procedure for deriving an approximate macroscopic description of the coupled system of equations presented in Eq. (20). The starting point of the solution procedure is to assume solutions to system Eq. (20) in the form of asymptotic expansions:

T (t, x) = T 0 t, x,

⎧ −∇y · (λg (Tg1 + ∇x T 0 )) = 0, ⎪ ⎪ ⎪ ⎪−∇y · (λs (Ts1 + ∇x T 0 )) = 0, ⎨

(24)

The solution,

T1 ,

of Eq. (27) is defined by

T 1 (t, x, y) = ω(y) · ∇ T 0 + T

1

(t, x).

(29)

• Equation of order 0 :

⎧ −∇y · (λg (∇y Tg2 + ∇x Tg1 )) − ∇x · (λg (∇y Tg1 + ∇x T 0 )) ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪−Cg b· ∇x T 0 + Cg ∂ T = 0, ⎪ ⎪ ∂ t ⎪ ⎪ ⎪ ⎪ −∇y · (λs (∇y Ts2 + ∇x Ts1 )) − ∇x · (λs (∇y Ts1 + ∇x T 0 )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂T0 = 0, +Cs ∂t ⎪ ⎪ 2 2 ⎪ ⎪Tg − Ts = 0, ⎪ ⎪ ⎪ ⎪ ⎪ (λs (∇y Ts2 + ∇x Ts1 ) − λg (∇y Tg2 + ∇x Tg1 )) ·n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = Q1W10 − hr (T 0 − T∞ ), ⎪ ⎪ ⎪ ⎪ ⎩ 2 2 {Tg , Ts } is Y -periodic.

in Yg ,

in Ys , on ,

on , (30)

The existence of a Y-periodic solution implies a compatibility condition which can be established by divergence theorem over Yg


4053

Fig. 5. Evolution of the spatial profiles of solid fuel concentration P (a) and (c) (respectively (e) and (g)) with their corresponding wave structures; (b) and (d) upstream smolder waves (respectively (f) and (h) downstream smolder waves). The arrows indicate the direction of propagation of the waves. The parameter values used in the simulation include: Ta = 1.0, hr = 8.5 × 10−3 , A1 = A2 = 0.5 and b1 = 1 × 10−2 . Others values are taken from Table 1.

and Ys . First, we consider the first terms in Eq. (30):

Yg

∇y · (λg (∇y Tg2 + ∇x Tg1 ))dY = −

λg (∇y Tg2 + ∇x Tg1 ) ·ndS, (31)

Ys

∇y · (λs (∇

2 y Ts

+∇

1 x Ts

))dY =

the outward normal to Ys . The balance of fluxes on implies that the sum of the integrals is

(λs (∇y Ts2 + ∇x Ts1 ) − λg (∇y Tg2 + ∇x Tg1 )) ·ndS

=

λs (∇

2 y Ts

+∇

1 x Ts

) ·ndS.

(32)

The boundary integrals, Eqs. (31) and (32), vanish on the edges of the unit cell Y due to the periodicity of the fluxes and the sign of the integral on the right-hand side of Eq. (31) changes by taking

(Q1W10 − hr (T 0 − T∞ ))dS

= | |Q1W10 − | |hr (T 0 − T∞ ).

(33)

The second terms of Eq. (30) are evaluated by noting that

∇y T 1 + ∇x T 0 = (∇y ω + I )∇x T0

(34)

4054


according to definition Eq. (29). It follows, by interchanging the integration with respect to y and the derivative with respect to x, that

∇x · (λ(∇y T 1 + ∇x T 0 ))dY = ∇x ·

Y

Y

λ(∇y ω + I )dY ∇x T 0

(35) The averaging of the remaining terms in Eq. (30) is given by

∂T ∂t

0

Yg

Yg

Cg dY +

Ys

Cs dY

= ceff

∂T , ∂t

Description

r

0.4 0.497 2.513 3.9562 × 10−4 0.080523 0.345502

Radius of solid fuel cell Porosity Tortuosity Effective thermal conductivity Effective mass diffusivity Effective heat capacity

φ = 1 − π r2 φ s = 2π r λeff

(36) 4. Results and discussion 4.1. Numerical study of the efficiency of the homogenization procedure

V =

Yg

Cg bdY = φ Cg b, and ceff =

Yg

Cg dY +

Ys

Cs dY.

(38)

Combining the averages of all terms together, we arrive at the following macroscopic description:

∂T0 = ∇x · (λeff ∇x T 0 ) + V · ∇x T 0 + φ s Q1W10 − φ s hr (T 0 − T∞ ), ∂t (39)

where

λ

eff ij

=

Yg

λg

∂ωgj ∂ωsj Ii j + dY + λs Ii j + dY , ∂ yi ∂ yi Ys

i, j = 1, 2.

(40)

For the homogenization of the concentration, (t, x), we follow the lead described in [14]. We also point out that the final homogenized concentration model is functionally identical, except for the presence of the solid fuel mass concentration. The homogenized equation for the solid fuel to first order can be written as

∂ P0 = −W10 + O(). ∂t

(41)

We simply recall the form of the homogenized concentration model,

0 (t, x), [14], which has the form:

∂0 · ∇x 0 − φ s W 0 , = ∇x · (Deff ∇x 0 ) + V 1 ∂t

(42)

where

= V

Values

(37)

where

φ

Parameters

Deff ceff

0

Cg b· ∇x T 0 dY = V · ∇x T 0 ,

ceff

Table 2 Parameter values used in the numerical simulations.

Yg

bdY = φ b

and

Deff ij = D

Yg

Ii j +

∂ω j dY, i, j = 1, 2. ∂ yi

(43)

where, the cell function, ω, given in Eq. (43) satisfies the following cell problem for the concentration model:

⎧

∂ωi ∂ ⎪ ⎪ D +I = 0, ⎪ ⎪ ∂yj ij ⎪ ⎨ ∂yj

∂ωi ⎪ D + I · n j = 0, ⎪ ⎪ ∂yj ij ⎪ ⎪ ⎩ ω is Y -periodic.

in Yg , on ,

(44)

In Eqs. (39) and (42), φ = |Yg | is the system porosity. In a similar notation, we denote the surface porosity given above by φ s = | |. In practice, the nonlinear reaction rate can be approximated by a step function in the spirit of [12,39]. This follows the introduction of an ignition temperature Tign such that the Arrhenius law can be written in the following form

k(T 0 ) =

A1 exp ( − Ta /T 0 ), if T 0 ≥ Tign 0, 0 < T 0 < Tign .

(45)

In the previous sections, we started from a pore scale description, which lead to a homogenization problem. Consequently, a homogenized model was derived with formulas for calculating the effective diffusion coefficients. In this section, we discuss the efficiency of the homogenization procedure. Specifically, we compare the numerical results of the microscopic model with the homogenized model. It should be noted that convection, chemical reaction and radiative heat transfer do not play dominant roles at the pore scale (cf., Eq. (28) for the cell problems and the -scaling of Eq. (20)). We are interested in studying a suitable range for a slow smoldering combustion scenario under microgravity. The convergence study is done by varying in the microscopic model. Since the smoldering propagation is essentially one-dimensional, we consider a one-dimensional solution of homogenized equations. However, the microscopic problem is considered in two-dimensions. In order to minimize computational cost, we impose periodic boundary conditions on the lateral boundaries while we increase the total number of cells along the fixed length L1 . We assume that the length L2 = O() for each varying and hence the problem is quasi-one-dimensional. In the numerical experiments, the values of satisfy 0 < f ≤ , i.e., is considered to be moderately small and f is its final value used in the numerical experiments. In each numerical experiment, we run the simulation by varying the characteristic size of the Y-cell. We begin with cells of characteristic size l = 2.5, which corresponds in this case to = 0.5. Then, we decrease l in each subsequent simulation until the last experiment corresponding to f . The time step of the simulations is dt = 10. For the computation of the homogenized model, we follow the general homogenization procedure for periodic structures: 1. Solve the corresponding cell problems, Eqs. (28) and (44) of the field variables T0 and 0 , in each of the canonical directions, e j , j = 1, 2. 2. Calculate the effective thermal conductivity, λeff , and mass diffusivity, Deff , coefficients using the solutions of their corresponding cell problems 3. Solve the coupled system of homogenized equations for T0 , 0 and P0 . For the computations of the cell problems, the homogenized effective coefficients and hence the system of equations, we use the parameter values given in Table 1. Specifically, the effective transport parameters given in Table 2 are calculated from formulas, Eqs. (38), (40) and (43), using the parameter values in Table 1. The error, ERR, in the approximation is evaluated in the L2 (0, τ ; ) norm, i.e.,

ERRi = ψ − ψ 0 L2 (0,τi ;) , i = 1, . . . , Nt , where i represents the error after a given number of time steps τ i and Nt is an integer corresponding to the final time step. This is done to study the evolution of the error for various values of τ due to the transient nature of the problem. One advantage of adopting the homogenized model is that it is easier to solve since the details of the pore scale description have


4055

Fig. 6. Comparison between the microscopic solutions (a) T , (b) , (c) P and the homogenized solutions (d) T0 , (e) 0 , (f) P0 at τ = 450 and for = 0.2. The parameter values considered include: T0 = 4.0, Ta = 5.0, A1 = 1.5, hr = 1 × 10−3 , b1 = 1 × 10−3 and the values given in Tables 1 and 2.

(a)

(b)

(c)

(d)

Fig. 7. Comparison between the homogenized and microscopic solutions for varying in the temperature distribution at various instants of time: (1), (2) upstream waves at τ1 = 50, τ2 = 100; (3), (4) downstream waves at τ3 = 400, τ4 = 500. The parameter values used in the simulation include: hr = 1.0 × 10−2 , Ta = 1.0, A1 = A2 = 1.5, b1 = 1 × 10−3 .

been eliminated in the homogenized limit problem. The problem now involves a coarse mesh that does not consume much of computer resources (e.g., CPU time and memory). Thus, the first step to demonstrating the efficiency of the homogenization procedure is to show that, while the homogenized model captures the qualitative behavior inherent in the microscopic description, it minimizes the computational time. In Fig. 6, the mesh of the microscopic model consists of 29, 500 elements with 124, 842 degrees of freedom. However, the homogenized model consists of 8502 mesh elements with 51, 735 degrees of freedom. The CPU time used in the computation of the microscopic model is about 293 s, while only about 72 s is needed for the homogenized model. It can be concluded that as becomes much smaller the computation of the microscopic model becomes less viable. It can also be seen from Fig. 6 that the solutions of the homogenized problems are consistent with the microscopic solutions.

4.2. Upstream and downstream waves As we illustrated numerically in Fig. 5, the proposed model has the feature of exhibiting both upstream and downstream waves if the parameter values are within a suitable range. Presently, we show that the results of the homogenization procedure are consistent with the predictions of the microscopic model. We run simulations by varying the size of in the microscopic model in each run. In this case, the value of the intensity of the radiative heat loss is fixed, e.g., hr = 1.0 × 10−2 . We consider a final time step τ = 600 when the waves reach an intermediate point in the spatial domain after reflecting downstream. Figures 7 and 8 depict comparisons between the homogenized and microscopic solutions for varying in the distribution of the temperature and solid fuel mass concentration respectively. The pictures consist of various time instants at which the error between the homogenized and microscopic solutions of the upstream and downstream waves are compared. It can be seen that the results

4056


Fig. 8. Comparison between the homogenized and microscopic solutions for varying in the solid fuel concentration distribution at various instants of time: (1), (2) upstream waves at τ1 = 50, τ2 = 100; (3), (4) downstream waves at τ3 = 400, τ4 = 500. The parameter values used in the simulation include: hr = 1.0 × 10−2 , Ta = 1.0, A1 = A2 = 1.5, b1 = 1 × 10−3 .

show good agreement between the homogenized solution and the microscopic solution as decreases; specifically at = 0.05. We also mention that the convergence test for the oxygen concentration follows a similar trend. However, we omit the plots for brevity of presentation.

4.3. Effect of the radiative heat losses One of the features of the models derived in this study is the incorporation of the radiative heat losses. We recall that the parameter estimation of the microscopic model demands that the radiative effects do not play dominant roles at the pore scale. However, we examine the effect of the intensity of radiative heat loss, hr , on the efficiency of the homogenization process. Thus, we run simulations for various values of hr . For brevity of presentation, we fix = 0.05, which serves, for the convergence tests in the previous sections, as the final value of . Figure 9 shows the distributions of temperature, oxygen concentration and solid fuel mass concentration at hr = 0.0 (profile 1), hr = 8 × 10−4 (profile 2), hr = 2 × 10−3 (profile 3), hr = 2 × 10−2 (profile 4). At lower values of hr (hr = 0, 8 × 10−4 , 2 × 10−3 ), the extent of propagation of the waves in the spatial domain, in both upstream and downstream regimes, is larger compared with when hr = 2 × 10−2 (profile 4). Thus, the spread rate of the waves decreases at higher values of hr , which is consistent with previous microgravity experiments (e.g., [40]). Figure 9a (profile 4) indicates an abrupt drop in the temperature distribution behind the front. Increasing hr beyond this value may lead to the extinction of the combustion process. The effects of radiative heat losses in combustion have been notably studied in the literature (e.g., [31,32]) as well as its role in microgravity smoldering combustion experiments (e.g., [5,40]). It is known, at sufficiently high heat losses, that the range of thermaldiffusive instability of the smolder waves is enhanced [32]. This involves the leakage of reactants through the front as the heat loss intensifies. The latter is depicted in Fig. 9c, d, and f (profile 4); the solid fuel and oxygen are not completely consumed by the chemical reaction.

4.4. Two-dimensional fingering instability In order to illustrate the behavior of the macroscopic model for a problem of microgravity reverse smoldering combustion, we perform numerical simulations on the macroscopic model (see Eqs. (39), (41) and (42)). We scale the dimensional form4 of the macroscopic model as detailed in [13] so that the following reference quantities are considered:

u= ut =

T0 , Tb

v=

λeff Deff ceff

0 , 0

u +

w=

P0 , P0

(x∗ , y∗ ) =

(x, y) L

,

t∗ =

t , tD

Cg b 1 L tD Q1 0 φ ux + φ s eff f (u, v, w) ceff Deff c Tb

tD hr (u − u∞ ), ceff b L φvt = v + φ 1eff vx − φ stD f (u, v, w), D wt = −tD Hw f (u, v, w) − φs

and the dimensionless system of equations reduces to

ut = Le u + φ Peux + βγ f (u, v, w) − a(u − u∞ ),

(46)

φvt = v + φ Pevx − γ f (u, v, w), wt = −Hw γ f (u, v, w), where

Le = tD =

λeff Deff ceff L2 , Deff

,

Pe =

b1 L , Deff

=

γ = φ stD A1 P0 , a =

Cg , ceff

φ stD hr ceff

β= ,

Q1 0 , ceff Tb

Hw =

0 . φ s P0

4 It should be pointed out that prior to introducing dimensionless quantities into the system of homogenized equations, it is customary to first transform the homogenized equations to the dimensional quantities.


4057

Fig. 9. Comparison between the homogenized solution and microscopic solution ( = 0.05) for varying hr in the distribution of (a), (b) temperature; (c), (d) oxygen concentration; (e), (f) solid fuel concentration, at two instants of time corresponding to (a), (c) and (e) upstream waves at τ = 100; (b), (d) and (f) downstream waves at τ = 600. The parameter values used in the simulation include: Ta = 1.0, A1 = A2 = 1.5, b1 = 1 × 10−3 and hr = 0.0 (1), hr = 8.0 × 10−4 (2), hr = 2.0 × 10−3 (3), hr = 2.0 × 10−2 (4).

The consumption rate of the reactants is given by the following Arrhenius law:

f (u, v, w) =

vw exp ( − θ /u), if u ≥ u∗ 0,

otherwise.

(47)

where Tb = T∞ + Q1 0 /ceff is the adiabatic temperature of combustion products; θ = Ta /Tb is the dimensionless activation temperature; u∞ = T∞ /Tb < 1 is the dimensionless tempera-

ture of the surroundings and u∗ is the dimensionless ignition temperature. It should be pointed out that for most phenomenological models (cf., [12]) that neglect the contribution of the presence of obstacles, the system porosity, φ , in Eq. (46) may be set equal to unity. However, for such an approximation, the essential physics of the phenomenon of interest is still viable. We consider the problem Eq. (46) with Eq. (47) in a bounded rectangular domain 0 < x < Lx , 0 < y < Ly (Lx = 400, Ly = 200), in which

4058


Fig. 10. Spatial profiles of char patterns on the solid fuel (w) as a function of Pe. In all cases, a = 0.25 and θ = 0.5. The parameters used in the simulation are taken from Table 3. In the pictures, ignition is initiated from the bottom and the gaseous oxidizer is passed from the top. The combustion fronts propagate from the ignition line at the bottom toward the upstream end at the top. (a) and (b) correspond to patterns in the connected state; (c) and (d) patterns in the developed fingering regime.

the lateral walls are subjected to Neumann conditions:

where N (y) is an appropriately chosen pseudo-random perturbation5 and

uy = vy = 0

(v, w)(0, x, y) = (v0 = 0.1, w0 = 1.0), 0 < x < Lx , 0 < y < Ly .

at y = 0, Ly .

At the inlet end (i.e., x = Lx ) and the outlet end (i.e., x = 0), the following conditions are prescribed for the field variables u and v:

(ux , vx )(t, 0, y) = (0, 0), (ux , v)(t, Lx , y) = (0, v0 ), t > 0.

(48)

The boundary descriptions, Eq. (48), imply that ignition is initiated at the left boundary and the combustion fronts propagate from the left to the right boundary, thus setting up a reverse combustion. This configuration is known to exert a destabilizing effect on the emerging combustion fronts [5,7]. Finally, the system Eq. (46) is closed by the following initial conditions:

u(0, x, y) =

2 + N (y), u∞ ,

if 0 ≤ x ≤ 5 + N (y), 0 ≤ y ≤ Ly , otherwise,

(49)

For the numerical solution of the problem, we use an explicit finite difference method with a central difference discretization of the diffusion and advection terms. The spatial step sizes of the discrete grid are dx = dy = 0.2 and the time step size is dt = 0.008, which satisfies the stability condition for the explicit scheme. The parameter values used in the numerical simulations are given in Table 3. 4.5. Distinct fingering states In the numerical simulations, the instability of the front is controlled by using the Péclet number as a free parameter in system Eq. (46) while other parameters assume the values given in 5 The forced ignition to initiate the combustion process is accomplished through a uniformly distributed pseudo-random perturbation of the initial condition of the temperature.


4059

Fig. 11. Spatial profiles of char patterns on the solid fuel (w) as a function of Pe. In all cases, a = 0.25 and θ = 0.0. The parameters used in the simulation are taken from Table 3. (a) and (b) correspond to patterns in the connected state; (c) and (d) patterns in the developed fingering regime.

Table 3 Parameter values used in all the numerical simulation.

θ

β

γ

φ

Le

u∞

Hw

u∗

0.5

20.

5.0

1.0

0.3

0.25

0.0

1.0

1.25

Table 3. Specifically, we fix the radiative heat transfer coefficient a = 0.25 in the simulations discussed in this subsection. Figures 10 and 11 show distinct fingering states and the transition from the regime of stable planar front to the unstable regimes (fingering with and without tip splitting). We consider two different forms of the combustion model based on the nature of the Arrhenius kinetics, which is conditioned by the parameter, θ . The first form is related to when the dimensionless activation energy is greater than zero, i.e., θ > 0 and the second form, when we assumed it to be zero, i.e. θ = 0. In the latter form, we point out that a fingering pattern can be obtained provided that the reaction is suppressed for u < u∗ .

4.5.1. Model with Arrhenius kinetics (θ > 0) We consider the behavior of the fingering instability when θ = 0.5 in the Arrhenius law, Eq. (47). In this case, the Arrhenius kinetics has a temperature dependent exponential factor, which enhances the nonlinear dynamics of the problem. Figure 10 shows the evolution of fingering instability patterns on the solid fuel as a function of Péclet number. We see that, at a relatively large Pe value (Pe = 1.2), the smoldering pattern has the form of a stable planar front (Fig. 10a). Decreasing the value of Pe slightly (Pe = 0.5) results in a cellular structure, which marks the onset of the instability (Fig. 10b). It should be noted that combustion occurs only at the vicinity of the tip of the front. The pattern behavior as depicted in Fig. 10 manifests as an imprint of the propagating thermal front of u, of which the location is clearly visible within the vicinity of the reaction zone. A corresponding pattern is also cast on the spatial distribution of oxygen due to the chemical reaction. For an illustrative picture of the pattern behavior on the temperature and concentration profiles, we refer the interested reader to [14]. As Pe is further decreased, the patterns

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Fig. 12. Effect of heat losses on the developed fingering instability. In all cases, Pe = 0.3 and θ = 0.5. The parameters used in the simulation are taken from Table 3.

separate into distinct fingers with tip splitting and having the characteristic feature of screening neighboring fingers from the supply of oxygen (cf., [5]). At Pe = 0.25, the emerging fingering patterns propagate as distinctly spaced fingers without tip splitting. In this regime, the thermal front propagates in the form of isolated hot spots toward the upstream boundary. We point out that the mechanism of the pattern formation demonstrated presently can also be found in the paper [10], in which a pore network simulator was employed. 4.5.2. Model with simplified Arrhenius kinetics (θ = 0) In order to ascertain the behavior of fingering patterns resulting from the system Eq. (46) with the simplified kinetics, i.e., θ = 0, we examine the problem as we did earlier on. Figure 11 shows the spatial distribution of the fingering patterns on the solid fuel in this case. The parameter values used in the numerical simulation are similar to those used in Fig. 10. Based on the pattern-forming dynamics, the results depicted in Fig. 11 show no significant difference with the results demonstrated in Fig. 10. However, an obvious difference between the numerical results can be seen in the range of Pe values, which exhibit the distinct patterns in the two models. In connection with the phenomena of interest, we see that the model with the simplified kinetics captures similar qualitative fingering patterns. The patterns show good agreement with the

dynamics of the fingering instability described in the experiments [5,7,41].

4.6. Effect of heat losses on the fingering instability In this subsection, we examine the influence of the radiative heat losses on the fingering instability. In particular, we qualitatively justify the relation between the characteristic lengths (the gap between the fingers and the finger width) of the instability and the radiative heat transfer coefficient, a. The features exhibited by the incorporation of heat losses in the experiments [5] include changes in finger width, changes in the gap between fingers, and suppression of tip splitting. Also, at larger values of the gap between the plates of the Hele–Shaw cell there is no fingering behavior. While our mathematical model is quite distinct from the system studied experimentally, we attempt to ascertain whether the incorporation of the radiative heat losses can capture similar characteristic features observed in the experiment. Figure 12 illustrates the evolution of the fingering patterns as a function of the radiative heat transfer coefficient, a, for the model problem with Arrhenius temperature dependence. At a relatively low value of a (Fig. 12a), close to the adiabatic problem (a = 0), there are no fingers. This is because the oxidizer is completely consumed at


the vicinity of the reaction zone, hence no substantial leakage of the oxidizer across the front. Increasing a reveals fingering instability with the characteristic features of a minimal gap between adjacent fingers, increased thickness of fingers, and decreased ability to tip split (Fig. 12b). A further increase of a above this value shows fingers with an enhanced ability to tip split (see Fig. 12c). The gap between the fingers increases, but it is not pronounced. This implies that increasing the value of a enhances the range of the fingering instability. As the value of a increases from a = 0.26 to a = 0.28 (Fig. 12d and e), we have a transition from the tip splitting state to fingering without tip splitting. In Fig. 12e, there is substantial leakage of the oxidizer across the fingers due to the elimination of weak hot spots by the increase in a. At a = 0.32 (Fig. 12f), only two propagating fingers remain with a large gap between them. At the same time, the leakage of oxidizer increases even further. It is also evident that the width of the fingers is narrower with the increasing value of a. Thus, we conclude that the width of the individual fingers can be determined by the heat losses. This observation is in good agreement with the experimentally observed fingering instability in the presence of heat losses [5]. We also point out that at large values of a, we have the extinction of the combustion process. We observed similar trends by varying a in the model with the simplified kinetics. That is, starting from the developed tip splitting regime of the instability, two distinct limits are possible: no fingering behavior at a → 0 and the extinction of the combustion process at a → ∞. However, only at moderate values of a in between the two limits does a developed fingering state emerges. 5. Conclusion In this study, we have derived a functional form for the limit homogenized problem, which we used to establish a link to a previous phenomenological reaction-diffusion model [12]. The proposed model has the following improvements: • Kinetic model for the solid fuel mass balance that uses a second order Arrhenius type kinetics as a consumption rate for the reactants and a production rate for the heat release mechanism. The local solid fuel conversion within the inclusions reveals distinct patterns, perceptive of the global behavior of fuel conversion in the homogenized model. It also extended the mechanism of the smoldering combustion process; specifically, the model exhibits coexistence of upstream and downstream smolder waves. • Radiative heat transfer mechanism that accounts for the effect of heat losses from the surface of the solid and to the surroundings. It extended the range of validity of the previously proposed adiabatic model [14]; thus, the visibility of the finger-like patterns is not impaired. Also, it brings the model to a more viable range for the emergence of a steady state of fingers without tip splitting. By incorporating the radiative heat losses, we showed the influence of the heat transfer coefficient on the characteristic length scales of the instability: the gap between fingers and the finger width. Specifically, increasing the value of a may result in the narrowing of the fingers, thereby confirming the dependence of the finger width on heat losses as reported in [5]. We also obtained formulas for calculating the effective transport parameters in the homogenized model. The effective parameters allowed us to show the efficiency of the homogenization process in the slow smoldering regime. The results of the numerical simulations show close agreement between the homogenized solution and the microscopic solutions. We conclude that the homogenization procedure demonstrated in the present study is efficient provided that the regimes of convection and chemical reactions are constrained within the range for a slow smoldering combustion process. Furthermore, we studied the behavior of the macroscopic system of equations numerically for the problem of fingering instability in microgravity

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smoldering combustion. We showed that the proposed model captures the distinct fingering states reminiscent of the experimentally observed finger-like char patterns, thus confirming the close resemblance of the pattern-forming dynamics of our model to the mechanism of diffusion instability observed in the experiment. For example, at a relatively low Péclet number, the proposed model predicts distinctly spaced sparse fingers without tip splitting. The latter result is conditioned by the presence of the radiative heat losses, contrary to the predictions of the previously studied adiabatic model [14]. We showed that for a fixed Pe value, two distinct limits are obtainable; the extinction of the combustion process at large values of a and the vanishing state of no fingering behavior at values of a close to the adiabatic regime. The fact that the regime of the instability is well within these two limits shows that incorporation of the radiative heat losses can expand the range of the thermal-diffusive instability of the phenomenon. We also analyzed the problem of the fingering instability using two forms of the Arrhenius kinetics, which depend on the dimensionless activation energy, θ . It was shown that the patterns observed for θ = 0.5 and θ = 0 are qualitatively similar though the Péclet values as well as the values of the radiative heat transfer coefficient which exhibit the patterns are slightly different. These results imply that, instead of the Arrhenius law with a temperature dependent exponential factor, one can use the model with the simplified kinetics (θ = 0) since it describes the essential feature of the combustion phenomena. For example, the transition of the patterns from planar front to fingering without tip splitting. We also mentioned that the viability of the latter model depends on the suppression of the reaction for u < u∗ . A potential advantage of using the simplified kinetics is the ease of theoretical analysis; particularly, it is easier to treat the smoldering combustion problem from the perspective of traveling wave analysis using the simplified kinetics. Acknowledgments This work was supported by the Mathematical Institute for Advanced Study of Mathematical Sciences (MIMS), Center for Mathematical Modeling and Applications (CMMA), Meiji University. References [1] T. Ohlemiller, D. Lucca, Combust. Flame 54 (1983) 131–147. [2] C. Wahle, B. Matkowsky, A. Aldushin, Combust. Sci. Technol. 175 (2003) 1389– 1499. [3] D. Schult, B. Matkowsky, V. Volpert, A. Fernandez-Pello, Combust. Flame 101 (1995) 471–490. [4] O. Zik, Z. Olami, E. Moses, Phys. Rev. Lett. 81 (1998) 3868–3871. [5] O. Zik, E. Moses, Phys. Rev. E 60 (1999) 518–531. [6] O. Zik, E. Moses, Proc. Combust. Inst. 27 (1998) 2815–2820. [7] S. Olson, H. Baum, T. Kashiwagi, Proc. Combust. Inst. 27 (1998) 2525–2533. [8] S.L. Olson, F.J. Miller, I.S. Wichman, Combust. Theor. Model. 10 (2) (2006) 323– 347. [9] K. Kuwana, G. Kushida, Y. Uchida, Combust. Sci. Technol. 186 (4–5) (2014) 466– 474. [10] C. Lu, Y. Yortsos, Phys. Rev. E 72 (2005) 036201(1–16). [11] L. Kagan, G. Sivashinsky, Combust. Theor. Model. 12 (2008) 269–281. [12] K. Ikeda, M. Mimura, Hiroshima Math. J. 38 (2008) 349–361. [13] A. Fasano, M. Mimura, M. Primicerio, Math. Method. Appl. Sci. (2009) 1–11. [14] E.R. Ijioma, A. Muntean, T. Ogawa, Combust. Theor. and Model. 17 (2) (2013) 185– 223. [15] F.-p. Yuan, Z.-b. Lu, Appl. Math. Mech. Engl. (2013) 1–12. [16] L. Hu, C.-M. Brauner, J. Shen, G.I. Sivashinsky, Math. Models Methods Appl. Sci. 25 (04) (2015) 685–720, doi:10.1142/S0218202515500165. [17] E.R. Ijioma, A. Muntean, T. Ogawa, Int. J. Heat Mass Transfer 81 (2015) 924–938. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.021. [18] Y. Uchida, K. Kuwana, G. Kushida, Combust. Flame 162 (5) (2015) 1957–1963. http://dx.doi.org/10.1016/j.combustflame.2014.12.014. [19] S. Whitaker, Theory and Applications of Transport in Porous Media: The Method of Volume averaging, vol. 13, Kluwer Academic Publishers, The Netherlands, 1998. [20] A. Oliveira, M. Kaviany, Prog. Energ. Combust. 27 (2001) 523–545. [21] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Application, vol. 5, North-Holland, 1978. [22] E. Sanchez-Palencia, A. Zaoui, Homogenization Techniques for Composite Media, Lecture Notes in Physics, vol. 272, Springer, 1985.

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