Combustion Theory and Modelling Pattern formation

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Pattern formation in reverse smouldering combustion: a homogenisation approach a

b

Ekeoma Rowland Ijioma , Adrian Muntean & Toshiyuki Ogawa

a

a

Graduate School of Advanced Mathematical Science , Meiji University , 1-1-1 Higashimita, Tamaku Kawasaki , Kanagawa , 214-8571 , Japan b

Center of Analysis, Scientific Computing and Applications, Department of Mathematics and Computer Science , Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology , P.O. Box 513, 5600 , MB , Eindhoven , The Netherlands Published online: 30 Nov 2012.

To cite this article: Ekeoma Rowland Ijioma , Adrian Muntean & Toshiyuki Ogawa (2013) Pattern formation in reverse smouldering combustion: a homogenisation approach, Combustion Theory and Modelling, 17:2, 185-223, DOI: 10.1080/13647830.2012.734860 To link to this article: http://dx.doi.org/10.1080/13647830.2012.734860

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Combustion Theory and Modelling, 2013 Vol. 17, No. 2, 185–223, http://dx.doi.org/13647830.2012.734860

Pattern formation in reverse smouldering combustion: a homogenisation approach


Ekeoma Rowland Ijiomaa∗ , Adrian Munteanb and Toshiyuki Ogawaa a Graduate School of Advanced Mathematical Science, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki, Kanagawa 214-8571, Japan; b Center of Analysis, Scientific Computing and Applications, Department of Mathematics and Computer Science, Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(Received 28 May 2012; final version received 24 September 2012) The development of fingering char patterns on the surface of porous thin materials has been investigated in the framework of reverse combustion. This macroscopic characteristic feature of combustible media has also been studied experimentally and through the use of phenomenological models. However, not much attention has been given to the behaviour of the emerging patterns based on characteristic material properties. Starting from a microscopic description of the combustion process, macroscopic models of reverse combustion that are derived by the application of the homogenisation technique are presented. Using proper scaling by means of a small scale parameter ε, the results of the formal asymptotic procedure are justified by qualitative multiscale numerical simulations at the microscopic and macroscopic levels. We consider two equilibrium models that are based on effective conductivity contrasts, in a simple adiabatic situation, to investigate the formation of unstable fingering patterns on the surface of a charred material. The behaviour of the emerging patterns is analysed using primarily the Péclet and Lewis numbers as control parameters. Keywords: periodic homogenisation; reverse combustion; combustion instability; multiscale modelling; fingering pattern

Nomenclature Y reference periodicity cell Ys solid part of the periodicity cell Yg gas part of the periodicity cell Y ε rescaled periodicity cell n normal Lc characteristic length of the system c characteristic length of the pore y microscopic variable x macroscopic variable M, N vector quantities for cell problems Tg temperature of gas in the gas part Ts temperature of solid in the solid part T effective temperature

∗

Corresponding author. Email: [email protected]

C 2013 Taylor & Francis

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C W R Q D A Ta cg cs Greek ε ε εs εg ∂ε φ φs λ ρ α

oxygen concentration reaction rate surface concentration of the solid product heat release molecular diffusion coefficient pre-exponential factor activation temperature specific heat capacity of gas specific heat capacity of solid

small scale parameter periodic medium internal boundary of the periodic medium ensemble of solid inclusions in ε matrix of interconnected gas part in ε gas/solid interface volume porosity surface porosity thermal conductivity density thermal diffusivity, α = λeff /C eff

Superscripts ∼, ∗ non-dimensionalised variable eff effective s surface Subscripts c characteristic quantities s solid phase g gas phase u unburnt or initial values a activation b burnt o initial values th thermal 1. Introduction Combustion phenomena have been studied extensively over the years, both theoretically and numerically. Due to the vast application of combustion especially for industrial and domestic purposes, understanding of its mechanism and flame propagation has attracted the attention of many scientists ranging from combustion engineers and chemists to theoretical physicists and applied mathematicians. Combustion models (see [1, 2] for details) on the


Combustion Theory and Modelling

187

Figure 1. Combustion front instability. Propagation proceeds from bottom to top [6].

other hand, have been developed to account for various physical combustion processes, for instance in the study of combustion phenomena in premixed gaseous substances, oil shale combustion in fixed beds, combustion processes in porous solid fuels, etc. The latter involves a heterogeneous reaction between interacting gaseous and solid phases and has gained much interest in the framework of filtration combustion. Filtration combustion basically involves two configurations which are based on the direction of flow of the inlet gas oxidiser and the direction of propagation of thermal and reaction fronts. It is classified as forward filtration combustion when the oxidiser flow and the reaction zone propagate in the same direction. However, when the reaction zone propagates in the opposite direction to the direction of propagation of the inlet oxidiser, the process is classified as reverse filtration combustion. Reverse combustion has been studied in detail (see [3, 4] for example) and is known to exhibit certain characteristic features. For instance, in a slow combustion regime referred to as smouldering, the combustion process proceeds in a non-flaming mode [5] in the presence of an oxidiser to produce char, toxic gas fumes and the heat that drives the process, without transiting into gaseous flames. In addition, the surface of the charred solid fuel develops fingering patterns (see Figure 1) resulting from a destabilising effect of reactant transport [5]. For the analysis of fingering instability, such a scenario was investigated by Zik and Moses, who presented experimental results on reverse combustion in a Hele Shaw cell containing a thin filter paper sample laid between two parallel plates. The major mechanism of their experiments most relevant to the present study is the role of oxygen flow velocity (alternatively, Péclet number) in the observed pattern formation. In porous media, smouldering becomes very complex and involves various chemical reactions and transport processes [7] in the presence of multiple characteristic time and space scales. The reaction site is the surface of the solid fuel, where smouldering initiation requires the supply of heat flux. Consequently, the temperature of the solid is increased, giving rise to thermal-degradation reactions (endothermic pyrolysis and exothermic oxidation) until the net heat released is high enough to balance the heat required for


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propagation. The transfer of heat involves transport mechanisms such as conduction, convection and radiation. Common examples of smouldering combustion in porous media include the initiation of upholstered furniture fires by weak heat sources and the flaming combustion of biomass occurring in wild land fires behind the flame front. For details on smouldering, we refer to [8,9], for example. Previous theoretical studies in this area include the work of Ohlemiller [9], who presented the most significant mechanisms involved in the smouldering combustion of polymers. Rein [8] presented a one-dimensional computational study to investigate smouldering ignition and propagation in polyurethane foam. Ikeda and Mimura [10] proposed a coupled reaction–diffusion system for the macroscopic theoretical understanding of pattern formation in smouldering combustion. Their numerical simulations exhibit good qualitative agreement with the experimental results presented by Zik and Moses [5, 11, 12]. Other theoretical studies include [6, 13, 14], where travelling wave solutions are presented. In [7], a pioneering three-dimensional microscale numerical simulation of smouldering in fixed beds of solid fuels is presented. In a set of two- and three-dimensional simulations, it was shown, contrary to the predictions of simple macroscopic models, that in some regimes there exists severe thermal disequilibrium in the constituent phases. This is one of the important aspects of a microscopic numerical simulation of a detailed microscopic model. In this paper, we propose a microscopic model of the smouldering combustion of a thin solid fuel – a porous solid with periodic microstructure – by the mathematical concept of periodic homogenisation. The theoretical background of the proposed model is partly inspired by a previous macro-geometric model by Kagan and Sivashinsky [6], which consists of a gas–solid system. We begin with a microscopic description where the solid fuel is assumed to be microscopically dispersed in a homogeneous gas phase. For an illustration of the macro-geometric (reverse flow) configuration, we refer to [5,6,13], where the smoulder front burns against an incoming gas stream that contains oxygen. In order to reduce the complexity of solving the detailed microscopic model, we consider a macroscopic viewpoint by formally deriving averaged models of reverse combustion via the classical homogenisation method. The results of the upscaled models can be rigorously justified, for instance, by two-scale convergence [15,16], eventually combined with periodic unfolding (very much in the spirit of [17]). On the other hand, macroscopic models [3, 18, 19] derived by the volume-averaging method are harder to justify mathematically. The solutions of the upscaled models are illustrated in two-dimensional numerical simulations and then compared with solutions obtained from idealised microscopic models, in order to justify their validity within an appropriate parameter space. A non-dimensional form of the macroscopic models is further considered in a simple adiabatic case to investigate the development of fingering instability on the surface of the charred solid fuel. 2.

Organisation of the study

The remaining sections of this study are organised as follows. In Section 3, the mathematical description of the microscopic problem that describes the combustion process is given. The governing heat transport models are presented in equations (11a) and (11b), while the equation for the mass transport of oxygen is given in (10). At the microscopic level, these equations are coupled at the gas–solid interface (14a) (see Section 3.2.4 for details), where the equation for the solid product (16) resulting from a one step chemical reaction is prescribed. The microscopic system is further rescaled (see equations 19a–20a and 23–25) so as to allow for a discussion of its upscaling in Section 4. In Section 4, two homogenised



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Figure 2. A sample of Whatman filter paper showing (a) sample at the macroscopic level (b) sample at the microscopic level (available in colour online).

models are derived and are referred to as the comparable conductivity (CC) model (49) and the high conductivity (HC) model (68) (see also Sections 4.2.1 and 4.2.2 for more details). It is necessary to consider solutions to some local boundary value problems, i.e. the cell problems in (40) and (63), respectively, for the CC model and the HC model. In this study, the solutions to the cell problems are useful for the calculation of the effective thermal conductivities in formulae (50) and (69). In addition, a homogenised concentration model (80) with effective diffusivity tensor (81) (see Section 4.2.3) is associated with each of the thermal problems, thus completing the basic equations for the combustion process. Section 5.1 begins with a multiscale numerical procedure for the computation of the effective coefficients (83) and (84), and also to justify the results of the derived macroscopic models in Section 4. In Section 5.2, the macroscopic systems (85) and (86) are analysed to investigate the development of combustion instabilities.

3. Formulation of the problem 3.1. Choice of microstructure We consider a microscopic description of a macroscopically homogeneous material illustrated in Figure 2(a). Based on microscopic evidence, it is observed that the structure of the material consists of a complex perforated arrangement of fibre (see Figure 2(b)). At the scale of the heterogeneity, air permeates the pore scale and interacts with the surface of the fibre. Assuming a typical length at the scale of heterogeneity to be δ, a reference periodicity cell of length is identified, and it is sufficiently large, i.e. δ < , to represent the features at the pore scale. The constituent of the unit volume consists of a spherical structure representative of the solid fibre, which is assumed to be dispersed in a gas matrix (see Figure 3). The heterogeneous medium we have in mind satisfies the following properties: (i) the typical length L at the macroscopic scale is in the x1 direction; (ii) the period of the structure representing the scale of heterogeneity is sufficiently small compared to the macroscopic scale; (iii) the length of the region along the x3 -direction is smaller by far compared to the lengths in the (x1 , x2 )-directions; (iv) x3 has arbitrary periodicity.


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Figure 3. Reference periodicity cell showing Yg and Ys (spherical ball) (available in colour online).

From (i) and (ii), the condition of separation of scales applies, i.e. ε = /L 1, where ε is a small scale parameter. By (iii) and (iv), we assume that the size in the x3 -direction is the same as the period. To be more precise in this construction, we begin with a mathematical description of the intended medium. The (macroscopic) porous domain

:=

3

(0, aj ),

j =1

with aj > 0, j = 1, 2, 3, is described with macroscopic (i.e. global) coordinates x = (x1 , x2 , x3 ), and a reference periodicity cell is chosen in R3 (see Figure 3): Y := {(y1 , y2 , y3 )|0 < yj < 1, j = 1, 2, 3},

(1)

with microscopic (i.e. local) coordinates y = (y1 , y2 , y3 ), within which we consider a spherical domain with smooth boundary. We denote this structure as ⎧ ⎫ 3 ⎨ ⎬ 2 yj − yj0 < R 2 ⊂ Y, Ys := (y1 , y2 , y3 )| ⎩ ⎭

(2)

j =1

where R is the radius and yj0 , j = 1, 2, 3, is the centre of the solid structure – see the solid (pink) sphere shown in Figure 3 – and also the centroid of the reference unit cell, Y. Henceforth, this structure is referred to as the solid part. The solid part is embedded in a pore space Yg := Y \Y s , which is referred to as the gas part, so that the reference cell can be written as Y = Yg ∪ Y s . Let ∂Ys represent the interface separating the gas part and the solid part, and ∂Y represent the boundaries enclosing the domains in Y, then the cell boundary of Yg is represented as Sg := ∂Y. A lattice of copies of Y is thus generated to span a generic parallelepiped domain . With the small scale parameter ε, the periodicity cell as well as its distinct parts and interior boundary are rescaled and translated to cover the macroscopic domain. The structure is thus depicted in Figure 4 and defined as follows.



191

Figure 4. Periodically distributed microgeometry of the idealised gas–solid system (available in colour online).

Definition 3.1 – Rescaled periodicity cell. The rescaled periodicity cell can be represented by

Y ε := (y1 , y2 , y3 )|0 < yj < ε, j = 1, 2, 3 , ⎧ ⎫ 3 ⎨ ⎬ 2 Ysε := (y1 , y2 , y3 )| yj − εyj0 < (εR)2 , ⎩ ⎭

(3)

j =1

ε

Ygε := Y ε \Y s .

Definition 3.2 – Translated subsets. By Definition 3.1, the translation of the rescaled subsets is defined as Ykε := {(y¯1 , y¯2 , y¯3 )|y¯j = yj + εkj , j = 1, 2, 3, (y1 , y2 , y3 ) ∈ Y ε }, k = (k1 , k2 , k3 ) ∈ Z3 . Let the number of cells with respect to the scaling parameter ε, in each coordinate direction, be defined as Nj (ε) := |aj |ε−1 ,

(4)

then the total number of cells in the domain is

N (ε) :=

3

Nj (ε),

(5)

j =1

ε = Ykε \ Definition 3.3 – Interconnected matrix and ensemble of inclusions. Let Yg,k ε ε Y s,k be any translated gas part according to Definition 3.2 with Ys,k , the corresponding

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solid part. Then the ensemble of the disconnected solid parts is given by ε εs := {(y1 , y2 , y3 )|(y1 , y2 , y3 ) ∈ Ys,k , k = (k1 , k2 , k3 ) ∈ Z3 ,

0 ≤ kj ≤ Nj (ε), j = 1, 2, 3}

(6)

such that the matrix of interconnected gas parts can be represented by ε

εg := \s .

(7)


Further, the interior boundaries may also be denoted as follows, based on periodic translation of the boundaries, ∂Ys : ε ε := {(y1 , y2 , y3 )|(y1 , y2 , y3 ) ∈ ∂Ys,k , k = (k1 , k2 , k3 ) ∈ Z3 ,

0 ≤ kj ≤ Nj (ε), j = 1, 2, 3}.

(8)

Remark 1. In the experimental configuration (see [5] for more details) the height of the gap above the paper changes the porosity, but the internal structure of the pores (Figure 2(b)) in the paper does not change. From the point of view of the upscaling procedure, the gap above the material is homogeneous and as such does not require upscaling. We thus focus on the description of the heterogeneous porous solid and its upscaling. Changes in the porosity of the effective medium due to the gap above it can be investigated further by understanding the behaviour of the gas–solid effective system with changing porosity.

3.2. Mathematical model In this section, the mathematical model of the combustion process occurring on the surface of the solid structure is described. The full fluid dynamics of the problem posed in the porous material follows a creeping flow, but, following [13] regarding the complications that may arise while investigating the full fluid dynamics, we introduce a simplifying assumption that the flow in the porous medium is a constant laminar field u of pre-heated air in the pore matrix εg . The key point justifying this approximation is that, in the smouldering regime we are considering, the mass flow rate is much larger than the mass exchange rate due to combustion; u is not so much influenced by the combustion process [13]. In view of the spatially periodic geometry, a spatially periodic flow is also assumed and it is divergence-free over the gas domain, εg . Thus, the flow field is given by u( y) =

3.2.1.

v, 0,

y ∈ εg , y ∈ εs .

(9)

Mass transport of the gaseous species

For a combustion process, the flowing gas constitutes a mixture of gaseous chemical species which are governed by mass conservation equations that are similar in form. In a simplified consideration, however, we do not investigate all the gaseous components in the mixture. In particular, we consider the oxygen flow, which is the major component in the gaseous mixture that takes part in the oxidative combustion process. Thus, we assume that its flow


193

is governed by the following convection–diffusion equation: ∂C + u · ∇C − ∇ · (D∇C) = 0, ∂t

x ∈ εg , t > 0,

(10)


where C is the concentration and D is the molecular diffusion coefficient, assumed here to be constant.

3.2.2. Heat transport in the gas and solid phases The transport of heat in the gas part and the subsequent conduction of heat in the solid part follow from the Fourier law of heat conduction. The gas phase heat equation is governed by a convection–conduction equation, whereas the predominant mode of heat transport in the solid phase is governed by heat conduction. Thus, we have

Cg

∂Tg + Cg u · ∇Tg − ∇ · (λg ∇Tg ) = 0, ∂t ∂Ts Cs − ∇ · (λs ∇Ts ) = 0, ∂t

x ∈ εg , t > 0,

(11a)

x ∈ εs , t > 0,

(11b)

where Ci = ρi ci is the volumetric heat capacity of phase i, λi is the heat conductivity and Ti is the temperature. Since the reaction between the phases is purely heterogeneous and takes place on the surface of the solid, the transport equations are coupled at the pore boundary ε .

3.2.3.

Chemical reactions

The chemical process in the considered combustion regime [4, 5] assumes the following form: Solid fuel + O2 → solid product + gas product + heat.

(12)

For an oxygen-limited reaction, the reaction rate W is given by a one-step first-order reaction with respect to the deficient gaseous reactant. We also assume that adsorption of the reactant on the solid surface is negligible. The temperature-dependent rate coefficient k is governed by the Arrhenius law, viz. Ta . k(T ) = A exp − T

(13)

Here, A is the pre-exponential factor, Ta is the activation temperature of the reaction, and T is the interfacial temperature at the gas–solid interface ε , satisfying T = Tg = Ts . The micro-heterogeneous reaction, localised on the solid–gas interface, induces coupling terms in form of source/sink terms, respectively, for the oxygen and heat equations where Q is the heat release. The reaction at the surface ε is given by the following boundary conditions.

194 3.2.4.

E.R. Ijioma et al. Interface conditions

(λg ∇Tg − λs ∇Ts ) · n = QW (T , C), Tg = Ts ,

x ∈ ε , t > 0,

(14a)

x ∈ , t > 0,

(14b)

x ∈ ε , t > 0,

(14c)

ε

D∇C · n = −W (T , C),


where n is the outward unit normal vector which points in a direction outside Yg , and W (T , C) is given by Ta . W (T , C) = AC exp − T

(15)

Additionally, the char product R, is given by ∂R = W (T , C), ∂t

x ∈ ε , t > 0.

(16)

So far, the governing equations together with the associated interior coupling conditions have been described. Additionally, the following boundary conditions are prescribed at the exterior boundaries. Diffusive thermal insulation conditions n · ∇Ti = 0,

n · ∇C = 0,

∂\({x = 0} ∪ {x = a1 }),

t > 0.

(17)

Upstream/downstream boundary conditions Ti = Tu , n · ∇ x Ti = 0,

C = Cu , n · ∇ x C = 0,

{x = 0} ∩ ∂, {x = a1 } ∩ ∂,

t > 0,

(18a)

t > 0.

(18b)

Here, u denotes the unburnt or initial values, and i represents phase i = {g, s}. The problem is further supplemented with appropriate initial conditions in order to describe the problem fully. 3.3.

Non-dimensionalisation

We rescale the system of equations derived in the previous section and introduce some dimensionless numbers, which will enable us to consider different problems as well as discuss their upscaling. The rescaling process aids in describing the physics at the microscale and it serves as a major step prior to the formal homogenisation technique, which we will consider in the subsequent section. The spatial-temporal variables are rescaled, viz. x = x ∗ Lc ,

t = t ∗ tc ,

where Lc is the characteristic length scale at the macroscopic level and tc is the characteristic time of the observation. After introducing some characteristic quantities subscripted with


195

c, the dimensionless equations take the following form: Cg

∂Tg∗ ∂t ∗

+ ρg∗ cg∗

Kgc tc uc tc v · ∇Tg∗ − ∇ · (λ∗g ∇Tg∗ ) = 0, Lc L2c Kgc tc ∂Ts∗ − K∇ · (λ∗s ∇Ts∗ ) = 0, ∗ ∂t L2c

mCs


where Cg = ρg∗ cg∗ , and Cs = ρs∗ cs∗ are, respectively, the dimensionless heat capacities of the constituents in the gas-phase and solid-phase. Kgc =

λgc , ρgc cgc

PT =

Kgc tc , L2c

m=

ρsc csc , ρgc cgc

K=

λsc , λgc

where Kgc is the gas phase thermal diffusivity and PT is the ratio of thermal transport to the time scale of the observation. The ratio of heat capacities in the material is denoted as m and the corresponding ratio of heat conductivities is K. Further, we define the temperature of combustion product Tb := Tu + Qc Cc /ρgc cgc and introduce the following global characteristic time scales: tD :=

L2c , Dc

tA :=

Lc , uc

tG :=

Lc , Ac

tR :=

Rc , Cc Ac

where tD is the characteristic global diffusion time scale, tA is the characteristic global advection time scale, tG and tR are, respectively, the characteristic time of gas reaction and the characteristic time of combustion product. We introduce also the following characteristic dimensionless numbers: Lc uc tD = Dc tA Kgc Leg := Dc Lc Ac tD Da := = Dc tG

PeL :=

(Péclet number), (gas-phase Lewis number), (Damköhler number).

We take the time of diffusion in the subdomain εg as the characteristic time of the observation at the macroscopic scale, i.e. tc = tD . Cg

∂Tg∗ ∂t ∗

+ Cg PeL v · ∇Tg∗ − Leg ∇ · (λ∗g ∇Tg∗ ) = 0

(19a)

∂Ts∗ − KLeg ∇ · (λ∗s ∇Ts∗ ) = 0. ∂t ∗

(19b)

mCs

After introducing some simplifications, the corresponding boundary conditions to (19a) and (19b) are n · (Leg λ∗g ∇Tg∗ − KLeg λ∗sg ∇Ts∗ ) = DaQ∗ W ∗ (T ∗ , C ∗ ), Tg∗ = Ts∗ ,

(20a) (20b)

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with W (T , C ) = A C exp − ∗

∗

∗

∗

∗

Ta . T ∗ (Tb − Tu ) + Tu

(21)

Similarly, the mass concentration of oxygen in dimensionless form is L2c ∂C ∗ uc Lc + v · ∇C ∗ − ∇ · (D ∗ ∇C ∗ ) = 0, ∗ Dc tc ∂t Dc

(22)


where PC = L2c /Dc tc is the ratio of global characteristic transport times. With the choice of characteristic time of diffusion tD , we introduce the Péclet number in (22), PC

∂C ∗ + PeL v · ∇C ∗ − ∇ · (D ∗ ∇C ∗ ) = 0. ∂t ∗

(23)

The boundary condition corresponding to (23) is D ∗ ∇C ∗ · n = −DaW ∗ (T ∗ , C ∗ ).

(24)

The solid char product is given in dimensionless form as tc Ac Cc ∗ ∗ ∗ ∂R∗ = W (T , C ), ∂t ∗ Rc tD = W ∗ (T ∗ , C ∗ ). tR

(25)

Finally, the system is completed with the following rescaled initial and boundary conditions:

n·

Ti∗ = 0,

C ∗ = 1,

Ti∗ = 0,

C ∗ = 1,

∇Ti∗

= 0,

∗

R∗ = 0,

n · ∇C = 0,

x ∈ ,

t = 0.

{x = 0} ∩ ∂,

t > 0,

{x = a1 } ∩ ∂,

t > 0.

(26)

4. Homogenisation method 4.1. General averaging strategy There are several approaches in the standard homogenisation or upscaling procedure used in the study of media with heterogeneous properties. In these approaches, a description of a microscale problem governed by some physical law on a heterogeneous medium is given. The upscaling or homogenisation of the problem is to arrive at an equivalent problem with effective properties, which characterise the response on some suitable macroscopic (global) length scale. An example of such methods is renormalisation group (RG) theory, which was first applied to the problem of hydraulic conductivity upscaling. For more details on RG approaches, we refer to [20, 21]. In this section, we formally describe the passage to a macroscopic limit problem whose form is in general different from that of the microscopic



197

model presented in the previous sections. We do this by applying the mathematical theory of periodic homogenisation. For a detailed description of homogenisation techniques, we refer to [22–24]. In this technique, the micro-model is replaced with a particular limit model that is usually referred to as the macro-model. The idea of the method is to consider a sequence of problems in a periodically repeating structure of length ε, this length being increasingly small and going to zero. An advantage of this approach, in spite of its complexity, is that one obtains a much simpler and numerically well-conditioned problem whose coefficients are effectively constant. Additionally, one benefits from a numerical simulation in a coarse grid involving the homogenised model, compared to the complexities involved in the direct numerical simulation of the fine scale micro model with periodic microstructure for which ε is very small. The premise of the micro-model is as follows: we assume periodicity with respect to the physical fields and material coefficients of the micro-model. The majority of the physical situation involving a medium under thermal stress usually leads to deformation of the medium. However, to be consistent with the method at hand, we assume that the structure of the medium is not significantly changed, i.e. the porosity is taken to be a constant. Since our interest is to study a bi-composite medium, we denote the volume porosity by

φ :=

|Yg | , |Y |

(27)

for the gas-phase, and the solid-phase porosity by (1 − φ). The surface porosity is denoted by φ s :=

|∂Ys | . |Y |

(28)

In (27) and (28), |Y | is the Lebesgue measure for the volume of Y , with |Yg | and |Ys | being the corresponding volume fractions in the gas and solid parts, respectively. We also denote the volume average of a generic field ϕi over Yi by 1

ϕi := |Y |

ϕi dY.

(29)

Yi

The oscillatory behaviour of the coefficients in the period Y ε implies for example λε (x) = λ (x/ε) = λ( y), where y = x/ε. We assume that the material coefficients all belong to the space L∞ (), and, in particular, the diffusion coefficients are symmetric and positive definite, i.e. for a positive constant γ , and any choice of vector ξ = (ξ1 , ξ2 , ξ3 ), 1 ≤ i, j ≤ 3, we have, for example, λij = λj i , λij ξi ξj ≥ γ ξi ξi . In addition, the coefficients are also taken to be isotropic tensors, i.e. λ = λo I and D = Do I, where for instance, λo is defined as λg , λo ( y) = λs , with I ∈ R3×3 being the identity matrix.

y ∈ Yg , y ∈ Ys ,

(30)

198 4.2.

E.R. Ijioma et al. Asymptotic homogenisation

The main idea of the (formal) asymptotic homogenisation procedure is to assume a solution to the microscopically derived models, in the form of an asymptotic expansion involving the macro variable x and the micro variable y = x/ε, i.e. we consider an expansion of the form

ϕ ε (x, t) =

∞

εj ϕ j (x, x/ε, t).

(31)


j =0

Next, we introduce equation (31) into, for example, equations (11a) and (11b), identifying problems at the same powers of ε, and finally solving a succession of ε-order local boundary value problems in the Y periodicity cell. The existence and uniqueness of auxiliary solutions can be rigorously justified by the Lax–Milgram lemma. We omit this here and focus on the main aspects of the homogenisation procedure.

4.2.1.

Comparable material coefficients and transport parameters

We analyse the microscopic model described in the previous sections for the heat equations. Here, we consider a problem where the material thermal coefficients as well as the transport parameters are comparable at the macroscopic level. In the context of the discussion here and all subsequent sections, we refer to this model as the comparable conductivity (CC) model. Thus, the following parameter regimes are considered.

Case I.

K = O(1), PeL = O(1), Leg = O(1), Da = O(ε)

The orders of magnitude describe the importance (or the dominance) of a particular physical process (or material coefficient). Specifically, we are interested deriving a macroscopic model that relates the conductivities of the two phases. By choosing the scaling K = O(1), we derive a model where the effect of thermal conductivities of the phases is comparable. Comparability here means that there exists no large contrast (i.e. a low contrast case) of material parameters and the connectivity of the media does not influence the form of the model. Other features of the intended macroscopic limit problem, with respect to the choice of scalings, include a comparable effect of convection to molecular diffusion and a diffusion-controlled regime as expressed by Da. The ratio of heat capacities m, is assumed, for simplicity, to be of order O(1).

Cg

∂Tg + Cg v · ∇Tg − ∇ · (λg ∇Tg ) = 0, ∂t ∂Ts − ∇ · (λs ∇Ts ) = 0, Cs ∂t Tg = T s , n · (λg ∇Tg − λs ∇Ts ) = εQW (T , C),

x ∈ εg , t > 0,

(32a)

x ∈ εs , t > 0,

(32b)

x ∈ ε , t > 0,

(32c)

x ∈ , t > 0.

(32d)

ε


199

As pointed out earlier, the first step is to assume that solutions to the unknown fields in (32a)–(32d) follow a formal asymptotic expansion, viz. (0)

(1)

(2)

Ti (x, t) = Ti (x, y, t) + εTi (x, y, t) + ε2 Ti (x, y, t) + O(ε3 ),

(33a)

C(x, t) = C (0) (x, y, t) + εC (1) (x, y, t) + ε2 C (2) (x, y, t) + O(ε3 ),

(33b)

(n)


where Ti and C (n) , n = 1, 2, . . . , i = {g, s} are Y periodic in y with y = x/ε, and describe, in multiple scales, solutions in × Y . Due to the scale separation, the unknowns in (33a) and (33b) are functions of three variables: x, y and t, and, consequently, we transform the derivatives through the chain rule: 1 ∇x = ∇x + ∇ y . ε

(34)

Applying expansions (33a) and (33b), using (34) and collecting terms in powers of ε, we obtain the following succession of problems. (0)

Boundary value problem for Tg and Ts(0) : (0) −∇ y · λg ∇ y Tg = 0, −∇ y · λs ∇ y Ts(0) = 0, (0)

Tg = Ts(0) , (0) λg ∇ y Tg − λs ∇ y Ts(0) · n = 0, (0)

(0)

Tg and Tg are Y periodic.

⎫ ⎪ ⎪ y ∈ Yg , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ∈ Ys , ⎪ ⎪ ⎪ ⎬ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(35)

By introducing the following form into (35) T (n) (x, y, t) = Tg(n) χYg ( y) + Tg(n) χYs ( y), λ = λg χYg ( y) + λs χYs ( y),

(36)

we can rewrite (35) as ∇ y · λ∇ y T (0) = 0, (0) T ∂Ys = 0, λ∇ y T (0) ∂Ys · n = 0, T (0) is Y periodic,

⎫ ⎪ y ∈ Y, ⎪ ⎪ ⎪ ⎪ ⎪ y ∈ ∂Ys , ⎬ ⎪ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(37)

where [ · ]∂Ys represents the jump Tg − Ts across the interface ∂Ys . Problem (37) involves only derivatives in y, while x and t are present as parameters. The only periodic solution satisfying (37) for the local heat problem is that T (0) (x, y, t) must be a constant, i.e. T (0) (x, y, t) = T 0 (x, t). By using (36), we write the following boundary value problem for T (1) .

(38)

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Boundary value problem for Tg and Ts(1) : ⎫ ⎪ y ∈ Y, ⎪ ⎪ ⎪ ⎪ ⎪ y ∈ ∂Y , ⎬

∇ y · λ ∇ y T (1) + ∇ x T (0) = 0, (1) T ∂Ys = 0, λ ∇ y T (1) + ∇ x T (0) ∂Ys · n = 0,

s

T (1) is Y periodic.


(39)

⎪ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

The linearity of (39) makes it possible to consider the following cell problems for the components of the vector M = (M1 , M2 , M3 ), with each problem corresponding to a unit macroscopic gradient ∇ x T (0) = ej , j = 1, 2, 3, where ej is the canonical orthonormal basis in R3 : ⎫ ⎪ y ∈ Y,⎪ ⎪ ⎪ ⎪ ⎪ y ∈ ∂Y , ⎬

∇ y · λ ∇ y M + I = 0, [M]∂Ys = 0, λ ∇ y M + I ∂Ys · n = 0,

M = 0,

s

(40)

⎪ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ M is Y periodic. ⎭

Therefore, the solution to the initial problem (39) can be written as T (1) (x, y, t) = M( y) · ∇ x T (0) (x, t) + T

(1)

(x, t).

(41)

(1)

In (41), M is the vector satisfying (40) and T (x, t) is basically the mean over T (1) (x, y, t). (2) Again, using the compact form (36), the boundary value problem for Tg and Ts(2) is given as follows. (2)

Boundary value problem for Tg and Ts(2) : (Cg + Cs )∂t T (0) − ∇ y · λ ∇ y T (2) + ∇ x T (1) −∇ x · λ ∇ y T (1) + ∇ x T (0) + Cg v · ∇ x T (0) = 0, (2) T ∂Ys = 0, (2) (1) λ ∇ y T + ∇x T · n = QW T (0) , C (0) , ∂Ys T (2) is Y periodic.

y∈Y y ∈ ∂Ys ,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(42)

The existence of T (2) enforces a compatibility condition that involves taking the mean of (42), applying the divergence theorem and using the prescribed conditions on ∂Ys , i.e.

1 ∇ y · λ ∇ y T (2) + ∇ x T (1) dY = − |Y | Y 1 − λ ∇ y T (2) + ∇ x T (1) · n dS. |Y | ∂Ys

1 − |Y |

λ ∇ y T (2) + ∇ x T (1) · n dS Sg

(43)


201

The first integral at the right-hand side of (43) vanishes due to periodicity in Y and the second integral leads to 1 |Y |

|∂Ys | QW T (0) , C (0) . −QW T (0) , C (0) dS = − |Y | ∂Ys

(44)

Also, we obtain the following averages:


∂T (0) ∂t

1 |Y |

(Cg + Cs ) dY = Ceff Y

v · ∇ x T (0)

1 |Y |

∂T (0) , ∂t

(45)

Cg dY = φCg v · ∇ x T (0) .

(46)

Yg

Further, we see by using (41) that ∇ y T (1) + ∇ x T (0) = (∇ y M + I)∇ x T (0) .

(47)

With (47), it follows from (42) by noting that the integration with respect to y commutes with the derivative with respect to x that

∇ x · λ ∇ y T (1) + ∇ x T (0) dY = ∇ x ·

Y

1 |Y |

λ[∇ y M + I] dY ∇ x T

(0)

. (48)

Y

Finally, the macroscopic description is

Ceff

∂T (0) − ∇ x · λeff ∇ x T (0) + φCg v · ∇ x T (0) = φ s QW T (0) , C (0) , ∂t

(49)

∂Mjs dY λs Iij + dY + ∂yi Ys

(50)

with λeff ij

C

eff

1 = |Y | 1 = |Y |

λg Yg

j

∂Mg Iij + ∂yi

Cg dY +

Yg

Cs dY

= φCg + (1 − φ)Cs .

4.2.2.

(51)

Ys

(52)

Effect of high contrast in thermal conductivities

We discuss the upscaling of the derived microscopic heat model in a framework of high contrast in thermal conductivities of the constituents of the medium. As pointed out in [5,25], an important mode of heat transfer for smoulder propagation is solid-phase conduction. Therefore, we investigate the behaviour of a thermal model with high thermal conductivity by considering the following parameter regime.

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Case II.

K = O(ε−1 ), PeL = O(1), Leg = O(1), Da = O(ε)

In this case, the objective is to derive a macroscopic model based on the contrast K = O(ε−1 ) in the conductivity of the medium. A similar consideration in the context of a double porosity soil model was presented in [26, 27]. In our case, the contrast K describes a situation where the conductivity of the solid inclusion is high compared to the gas phase conductivity. Thus, we refer to the derived model as the high conductivity (HC) model.


Cg

∂Tg + Cg v · ∇Tg − ∇ · (λg ∇Tg ) = 0, ∂t ∂Ts − ∇ · (λs ∇Ts ) = 0, εCs ∂t Tg = Ts , n · (ελg ∇Tg − λs ∇Ts ) = ε2 QW (T , C),

x ∈ εg , t > 0,

(53a)

x ∈ εs , t > 0,

(53b)

x ∈ ε , t > 0,

(53c)

x ∈ ε , t > 0.

(53d)

We now solve the following succession of boundary value problems corresponding to the ε−2 term. (0)

Boundary value problem for Ts(0) and Tg : ∇ y · λs ∇ y Ts(0) = 0, λs ∇ y Ts(0) · n = 0,

y ∈ Ys

⎫ ⎬ (54)

y ∈ ∂Ys , ⎭

of which the solution has the form Ts(0) (x, y) = Ts(0) (x),

(55)

and for the problem posed in Yg (0) = 0, ∇ y · λ g ∇ y Tg (0)

Tg

= Ts(0) (x),

y ∈ Yg

⎫ ⎬ (56)

y ∈ ∂Ys , ⎭

we obtain the following solution: Ts(0) (x, y) = Ts(0) (x) = T (0) (x).

(57)

(0)

Since Ts(0) and Tg are y-independent, all subsequent spatial derivatives in y vanish. The next problem uses the ε−1 term of Ts(0) and the ε0 term of the interface condition. Boundary value problem for Ts(1) : ∇ y · λs ∇ y Ts(1) + ∇ x T (0) = 0, λs ∇ y Ts(1) + ∇ x T (0) · n = 0,

y ∈ Ys

⎫ ⎪ ⎪ ⎬

⎪ ⎭ y ∈ ∂Ys . ⎪

(58)


203

The problem (58) is different from the problems considered earlier. We multiply the terms (1) in (58) by y · ∇ x T (0) + T s , integrating by parts over Ys , and using the prescribed condition on ∂Ys : Ys

(1) y · ∇ x T (0) + T s dY ∇ y · λs ∇ y Ts(1) + ∇ x T (0)

2 λs ∇ y Ts(1) + ∇ x T (0) dY −

= Ys


(1) × y · ∇ x T (0) + T s dS =

Ys

∂Ys

n · λs ∇ y Ts(1) + ∇ x T (0)

2 λs ∇ y Ts(1) + ∇ x T (0) dY = 0.

(59)

The positivity of λs implies that ∇ y T (1) + ∇ x T (0) = 0. Thus, we obtain Ts(1) (x, y) = − y · ∇ x T (0) + T

(1)

(x).

(60)

(1)

Boundary value problem for Tg : (1) ∇ y · λg ∇ y Tg + ∇ x T (0) = 0 , (1)

Tg = Ts(1) = − y · ∇ x T (0) + T where ∇ x T (0) and T in the form

(1)

(1)

y ∈ Yg

⎫ ⎬ (61)

y ∈ ∂Ys , ⎭

(x) ,

(x) are taken as forcing terms. The solution of (61) can be written

Tg(1) (x, y) = N ( y) · ∇ x T (0) + T

(1)

(x),

(62)

where the vector N ( y) is the solution of the following boundary value problem: ∇ y · (λg (∇ y N + I)) = 0, N + y = 0,

⎫ y ∈ Yg ⎬ y ∈ ∂Y . ⎭

(63)

s

(2)

For the macroscopic equations, we look at the ε0 term from Tg and the ε1 term from Ts(3) , i.e. Cg

∂T (0) = ∇ y · λg ∇ y Tg(2) + ∇ x Tg(1) + ∇ x · λg ∇ y Tg(1) + ∇ x T (0) ∂t − Cg v · ∇ x Tg(0) ,

y ∈ Yg ,

∂T (0) = ∇ y · λs ∇ y Ts(3) + ∇ x Ts(2) + ∇ x · λs ∇ y Ts(2) + ∇ x Ts(1) , y ∈ Ys , ∂t λg ∇ y Tg(2) + ∇ x Tg(1) · n = λs ∇ y Ts(3) + ∇ x Ts(2) · n + QW T (0) , C (0) , y ∈ ∂Ys . Cs

(64)

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Averaging (64) over Yg and Ys , and using the prescribed boundary condition: ∂T (0) 1 (0) (1) (0) dY λ g ∇ y Tg + ∇ x T ([1 − φ]Cs + φCg ) + φCg v · ∇ x T = ∇x · ∂t |Y | Yg 1 ∇x · + (65) λs ∇ y Ts(2) + ∇ x Ts(1) dY + φ s QW T (0) , C (0) . |Y | Ys


Since Ts(2) is unknown, we transform the integrals by identifying the heat fluxes in each constituent, viz. [26]: qg = λg ∇ y Tg(1) + ∇ x T (0) , qs = λg ∇ y Tg(2) + ∇ x Ts(1) .

(66a) (66b)

The flux in (66a) satisfies the following dyadic product identity: qik =

∂ (yk qij ), ∂yj

where

qi = ∇ y · ( y ⊗ qi ),

i = {g, s}.

Since the solid inclusions are completely embedded in the matrix, it makes sense to apply the divergence theorem to the integrals in (65), i.e.

qg dY = Yg

∂Ys

( y ⊗ qg ) · n dS +

qs dY = − Ys

( y ⊗ qg ) · n dS

(67a)

Sg

∂Ys

(qg − qs ) · n = 0,

( y ⊗ qs ) · dS,

y ∈ ∂Ys .

(67b) (67c)

Substituting (67a) and (67b) for the integrals in (65) and making use of the condition of flux continuity (i.e. the boundary condition corresponding to the ε1 term) over ∂Ys , we obtain

Ceff

∂T (0) + φCg v · ∇ x = ∇ x · λˆ eff ∇ x T (0) + φ s QW T (0) , C (0) , ∂t

(68)

where the effective conductivity tensor, λˆ eff , is given by the following surface integral: 1 λˆ eff kj = k S g

Sgk

λki g

∂Nj + Iij yk ni dS. ∂yi

(69)

From (69), we see that the effective conductivity is evaluated over the cross-sectional surface of Y , orthogonal to the kth-direction. Remark 2. We point out that the heat models (49) and (68) derived based on the conductivity parameter regime K = λsc /λgc = O(εβ ) can be grouped into three categories, in terms of the value of β. We have treated two cases of β in the present study. A third case applies, for example, when α = 2. In this case, the conductivity of the solid inclusion is very low compared to the gas phase conductivity. This model is similar in form to the


205

distributed microstructure models presented in [28–30]. Generally, such problems exhibit memory effects in the limit problem due to the time retardation introduced by the solid phase conductivity.

4.2.3.

Homogenisation of the concentration model

As in the case of the temperature models, we derive a macroscopic model for the concentration of the oxidiser based on the following parameter regime:


PeL = O(1),

PC = O(1),

Da = O(ε)

∂C + v · ∇C − ∇ · (D∇C) = 0, ∂t −D∇C · n = εW (T , C),

x ∈ εg , t > 0,

(70a)

x ∈ ε , t > 0.

(70b)

We introduce into (70a) and (70b) the expansions (33a) and (33b), and identify powers of ε. From these expansions, we are led to a few local boundary value problems, as follows. Boundary value problem for C (0) : −∇ y · D∇ y C (0) = 0, −D∇ y C (0) · n = 0, C (0) is Y periodic.

⎫ ⎪ y ∈ Yg , ⎪ ⎪ ⎬ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎭

(71)

As seen earlier, any solution of (71) is unique up to an additive function of x and t, where x and t are parameters. Thus, the solution to (71) is C (0) (x, y, t) = C (0) (x, t).

(72)

Equation (72) implies that the concentration C (0) is a constant over the period. Boundary value problem for C (1) : −∇ y · D ∇ y C (1) + ∇ x C (0) = 0, −D ∇ y C (1) + ∇ x C (0) · n = 0, C (1) is Y periodic.

⎫ ⎪ y ∈ Yg , ⎪ ⎪ ⎬ y ∈ ∂Ys , ⎪ ⎪ ⎪ ⎭

(73)

The gradient,∇ x C (0) , is only present as a source term in (73). Due to linearity and the fact that spatial derivatives are only in y, it makes sense to consider the following local problem

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for the periodic vector N = (N1 , N2 , N3 ), with a mean of zero over Yg , i.e. ∇ y · D ∇ y N + I = 0, D(∇ y N + I) · n = 0,

N = 0,

⎫ ⎪ y ∈ Yg , ⎪ ⎪ ⎬ y ∈ ∂Ys , ⎪ ⎪ ⎪ N is Y periodic, ⎭

(74)

where the solution to (73) can be written in the form


C (1) (x, y, t) = N ( y) · ∇ x C (0) + C

(1)

(x, t),

(75)

(1)

and C (x, t) is only an arbitrary function of x and t. With the results obtained from (72) and (75), we write the boundary value problem for C (2) as follows. Boundary value problem for C (2) : ∂t C (0) − ∇ y · D ∇ y C (2) + ∇ x C (1) − ∇ x · D ∇ y C (1) + ∇ x C (0) + v · ∇ x C (0) = 0, −D ∇ y C (2) + ∇ x C (1) · n = W T (0) , C (0) , C (2) is Y periodic.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ∈ Yg , ⎬ ⎪ y ∈ ∂Ys ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(76)

Applying the compatibility condition for the existence of C (2) , and using the expressions for C (0) and C (1) , respectively in (76), we obtain

1 ∇ y · D ∇ y C (2) + ∇ x C (1) dY = − |Y | Yg 1 − D ∇ y C (2) + ∇ x C (1) · n dS. |Y | ∂Ys

1 − |Y |

D ∇ y C (2) + ∇ x C (1) · n dS

Sg

(77)

The first integral on the right-hand side of (77) vanishes due to the periodicity condition in Y and the second integral yields 1 |Y |

∂Ys

W T (0) , C (0) dS = φ s W T (0) , C (0) .

(78)

The average of the remaining terms in (77) is ∂C (0) + φv · ∇ x C (0) − ∇ x φ ∂t

1 |Y |

D ∇ y N ( y) + I dY

∇ x C (0) .

(79)

∂C (0) + φv · ∇ x C (0) − ∇ x · Deff ∇ x C (0) = −φ s W T (0) , C (0) , ∂t

(80)

Yg

Collecting the terms together, the macro-model can be written as φ



207

Figure 5. Two-dimensional periodicity cell showing Yg and Ys (available in colour online).

where the effective diffusion tensor D eff is defined as 1 ∂N j dY. Dijeff = D Iij + |Y | Yg ∂yi 5.

(81)

Results and discussion

5.1. Multiscale numerical experiments The derived macroscopic models are considered in a two-dimensional x1 , x2 -plane along the direction of the prescribed flow field. The domain of the idealised microscopic problem has the dimensions 0 ≤ x1 ≤ 5 × 10−2 m, 0 ≤ x2 ≤ 3 × 10−2 m. It consists of a uniform arrangement of the periodicity cell depicted in Figure 5 in which the subdomains are spatially homogeneous. The typical geometry in Figure 5 has been chosen for computational simplicity. However, more realistic geometries are also possible. The domain intended for the homogenisation problem also has the same dimensions but the medium is homogenous. The numerical procedure to compute the solution to the homogenisation problem is as follows: first, the boundary value problems for the cell problems (40)–(63) and (74) arising, respectively, from the heat and concentration equations are solved. Then, their solutions are used for the calculation of the effective conductivity and diffusivity tensors through formulae (50)–(69) and (81). Finally, the solution to the homogenised models are computed. 5.1.1.

Computation of the effective coefficients

For the two-dimensional macroscopic flow, the contributions of (40) and (74) to the effective conductivity and diffusivity tensors are dependent only on the canonical e1 and e2 directions, i.e. the local boundary value problems are solved for Mj (y1 , y2 ) and Nj (y1 , y2 ), j = 1, 2, respectively. The problems must be supplemented with periodic boundary conditions, viz. Mj (y1 = 0, y2 ) = Mj (y1 = 1, y2 ),

(82a)

Mj (y1 , y2 = 0) = Mj (y1 , y2 = 1),

(82b)

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E.R. Ijioma et al. Table 1. Physicochemical properties used in numerical simulation. Parameter


cp ρ D λ Ta Tu Q A

Unit

Gas

Solid

Other

J kg−1 K−1 kg m−3 m2 s−1 W m−1 K−1 K K J mol−1 m s−1

1142 [31] 1376 [5] 2.5 ×10−5 [5] 0.0238 – 300 – –

1270 [32] 540 [5] – 0.07 [32] – 300 – –

– – – – 14,432 [4] – 11,200 [4] 2 ×106 [4]

for j = 1, 2. The solutions to (40) are computed on the unit structure as depicted in Figure 5 R . However, the computation of the solution and implemented with COMSOL Multiphysics to (74) is restricted only to the gas part, Yg , of the unit structure (see Figure 5). The coefficient values used in the numerical experiment are given in Table 1. For the implementation, the coefficients must satisfy the scalings implied in the problem. The solutions to the vector valued functions M( y) and N ( y) are illustrated in Figure 6, while the corresponding effective coefficients are computed, respectively, from the formulae

Figure 6. Solutions to the local cell problems. Dimensionless temperatures (a) M1 , (b) M2 . Dimensionless concentrations (c) N1 , (d) N2 .


209

(50) and (81), viz.

λeff =

1.476 × 10−14 0.080523 eff , D = 3.9562 × 10−4 −4.4 × 10−12

3.9562 × 10−4 1.476 × 10−14

−4.4 × 10−12 . 0.080523 (83)

Similarly, after solving the local boundary value problem (63) for the HC model, the solution to the effective conductivity tensor is calculated by the formula (69)

λ


eff

7.39 × 10−4 = 0.00

0.00 . 7.39 × 10−4

(84)

We point out that the implication of the high contrast in the inclusion is such that the same result (84) is obtained, even for λs → ∞. 5.1.2.

Computation of the homogenised solution

Having computed the values of the effective tensors, we proceed to the problem of numerically evaluating the homogenised models (49) and (80). We use the streamline diffusion method (see [33] for details) for the computation of the homogenised model. Following the same procedure as described above, we also obtain solutions for the HC model. The normalisation [26,34] of the problem with the orders of magnitude in ε must be reflected in the choice of coefficient values. In the HC model, for example, the contrast in the thermal conductivities is satisfied by considering the following scaling in the inclusion: λs = ε−1 λg . For this case, we have e.g. λg = 0.0238, λs = ε−1 λg , for ε = 0.04. The solutions to the homogenised problems (49) and (80) together with the microscopic problem are studied in their dimensional form. For this study, an inlet velocity of 0.005 m s−1 is utilised with the parameters given in Table 1. The main aspect of this subsection is to carry out a comparative numerical simulation for the upscaled models approximating an idealised microscopic model. The temperature distribution for the model problems is illustrated in Figure 7. The solutions indicate an overall temperature drop in the vicinity of the reaction region for the HC model compared with the CC model. The bump at approximately x = 2 cm on the spatial axis of the temperature profile indicates the location of the reaction region. 5.1.3. Effect of geometric properties The porosity of the periodicity cell is modified and the effective tensors are recomputed to investigate the effect of variation of porosity on the effective tensors. In Figure 9, it is evident that the value of the conductivity tensor decreases with increasing porosity whereas that of the diffusion tensor increases with porosity. These results are consistent with other studies [35, 36] which treat the effect of porosity on material properties. However, the converse is the case when the effect of the solid volume fraction c is considered. 5.2. Analysis of macroscopic system The macroscopic systems are considered in their two-dimensional forms along the plane of propagation. In this subsection, we are particularly interested in investigating interfacial front instabilities exhibited by the effective one-temperature combustion system. For a simple case of isotropic effective material coefficients and geometric properties of the


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Figure 7. Two-dimensional spatial distribution of temperature for the CC model ((a) and (b)) and the HC model ((c) and (d)). Comparison of the microscopic solution (ε = 0.04) (left) with the homogenised solution (right) (available in colour online).

microstructure, the following system of equations is considered:

C eff

∂T − λeff ∇ 2 T + φCg v · ∇T = φ s QW (T , C), ∂t ∂C − D eff ∇ 2 C + φv · ∇C = −φ s W (T , C), φ ∂t ∂R = W (T , C). ∂t

(85)



211

Figure 8. Two-dimensional spatial distribution of concentration. Comparison of the microscopic solution (ε = 0.04) (a) to the homogenised solution (b) (available in colour online).

System (85) is analysed in its non-dimensional form by introducing some characteristic units based on the effective equations and the typical values given in Table 1. C , C˜ = Co N=

˜ = R, R Ro Ta , Tb

T T˜ = , Tb

φ s Aα A˜ = 2 N , U Ne

U2 , α

x˜ =

x , lth

Wα W˜ = 2 , U

lth =

α , U

τ =t

where Co and Ro are, respectively, the initial/unburnt concentrations of the deficient oxidiser and the fuel sample. α is the effective thermal diffusivity defined as α := λeff /C eff and Tb := QCo /C eff is the temperature of combustion product. U is a virtual velocity of a uniformly propagating plane reaction front. N is a non-dimensional activation temperature, W˜ is a non-dimensional reaction rate and the parameter A˜ is introduced to normalise the virtual velocity of normal propagation to one [37]. In terms of the quantities introduced, the non-dimensional system is given by ∂T˜ ˜ − ∇ 2 T˜ + φP e · ∇ T˜ = W (T˜ , C), ∂τ ∂C˜ 1 2˜ ˜ φ + φP e · ∇ C˜ − ∇ C = −W (T˜ , C), ∂τ Le ˜ ∂R ˜ = HR W (T˜ , C), ∂τ where ˜

˜ Ce ˜ N(1−1/T ) . W = AN

(86)

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10

λeff

a

eff

D

−1

−2

10

−3

10

−4

10

0.4

0.5

0.6

0.7

φ

0.8

0.9

1

0

10

eff

λ

b

Deff

−1

10

λeff,Deff


λeff,Deff

10

−2

10

−3

10

−4

10

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c Figure 9. Variation of (a) porosity φ, (b) solid volume fraction c on the effective conductivity λeff and diffusivity D eff tensors.


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In (86), Pe = (vlth /α, 0) and Le = α/D are, respectively, the effective Péclet number and Lewis number. H R = C0 /φ s R0 and = Cg /C eff represent non-dimensional coefficients. System (86) is solved numerically using the Streamline Upstream Petrov–Garlekin (SUPG) scheme. The time integration was performed using an implicit BDF solver. The set of equations (85) and (86) are functionally identical to Equations 25 and 26 of [6] provided that the derived effective properties in (85) can be associated with those in Equations 25 and 26 of [6]. The non-dimensional system (86) was solved using the typical values in Table 1. A feature of the presented numerical results lies in the chosen characteristic length and range of chosen Péclet numbers. The non-dimensional spatial coordinates were expressed in units of lth = α/U, the thermal width of the front, where U is a fixed virtual velocity.

Figure 10. Spatial profiles of the flame structure: (a) temperature (b) char (c) concentration (d) heat release rate. Solution of the HC model for Pe = 10, Le = 0.09. Ignition at bottom, char propagation from bottom to top, gas inlet at top. The spatial axes are in units of the thermal length of the flame, lth (available in colour online).

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The value of the non-dimensional pre-exponential factor, A, was chosen within the range [2.3 × 103 , 104 ]. In the numerical simulations, all other parameters correspond to the typical values given in Table 1. We point out that, in all cases considered, the combustion process is incomplete. In Figure 10, the location in space of the flame can be identified at the reaction region that is delineated within a limited vicinity of tips of the fingers. This region is traced by darker shades in Figure 10(d), where oxygen is fully consumed as the front advances. Behind (or along) the fragmented advancing flame front, there is no reaction but only a trace of the heterogeneous charred region (see Figure 10(b)). The uncharred area of the pattern represents the quenched parts of the fragmented reaction zone and smouldering is only intensified at the tips of the fingers where the oxidant is fully consumed.

5.2.1.

Effect of Péclet number

In this subsection, we focus on a fingering regime that is conditioned by the effective Lewis number represented by Le =

λeff , (Cg φ + Cs [1 − φ])D eff

(87)

where λeff and D eff correspond, respectively, to the values of the effective thermal conductivity and diffusivity tensors derived from the homogenisation procedure. In (87), we see that the Lewis number depends on the porosity of the medium through the heat capacity. Already, we have seen in Figure 9 the dependence of the effective thermal conductivity and mass diffusivity on the porosity. The variation of porosity with the Lewis number is shown in Figure 11. Initially, the Lewis number increases gradually with porosity, but as the porosity increases, the Lewis number changes abruptly within a small interval of the porosity. Also, it can be seen clearly in the figure that the range of values of the Lewis

Figure 11. Variation of the Lewis number with porosity for the CC model and the HC model. The range of Lewis number values is higher in the HC model (available in colour online).

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Figure 12. Spatial profiles of the char pattern for (a) the CC model at Pe = 0.01, Le = 0.014,22, A = 0.27×104 (i.e. effective Lewis number), (b) the HC model at Pe = 0.01, Le = 0.026,563, A = 1×104 . The patterns are elongated fingers with no tip-splitting. (c),(d) contour plots showing hot spots at the vicinity of the tips and regions of higher conversion depths (lighter shade).

number is higher for the HC model compared to the CC model. This is basically due to the higher conductivity of the HC model. The emerging fingering instability is controlled through the Péclet number, Pe. At a relatively low blowing rate, the char pattern emerged as elongated fingers which strictly avoid each other (Figure 12). The patterns show a low contrast in the spatially heterogeneous conversion depth. The conversion depth describes the intensity of the solid fuel conversion to char. Darker shades in Figures 12(a) and 12(b) represent regions of the pattern with high


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Figure 13. Spatial profiles of the char pattern for (a) the CC model at Pe = 17, Le = 0.014,22, A = 0.27×104 , (b) the HC model at Pe = 17, Le = 0.026,563, A = 1×104 . Fingering patterns with tip-splitting and low conversion depth (available in colour online).

conversion intensity or for clarity, this region is shown in lighter shades in Figures 12(c) and 12(d). At the vicinity of the tips which are indicated in lighter shades, the hot spots emerge prior to extinction. These are points of higher temperature values in the numerical simulations. The maximum temperature at an active finger may also alternate from one finger to another due to the localised reaction at the reaction zone. In this adiabatic consideration, it can be seen that the visibility of the patterns is impaired by the inability of the fingers to lose heat, thus the persistence of fingers with weak hot spots. The latter may also contribute to the eventual widening around the tip of fingers closest to the oxygen source, and consequently the fingers shield adjacent fingers, i.e. fingers with weak hot spots, from the oxygen supply. A further increase in the Péclet number, i.e. Pe = 17 increases the contrast in conversion depth and the fingers tip-split (Figure 13).

5.2.2.

Disparity in the models

In this subsection, the behaviour of the developed fingering state is further examined. Figures 14(a) and 14(b) indicate fingering patterns at Pe = 40, respectively, for the CC model and the HC model. In Figure 14(b), the pattern is fully developed with the characteristic features of tip-splitting, increase in conversion depth and bifurcation. Bifurcation of the fingers describes the splitting of a single finger into multiple branches. However, these features are not substantial for the CC model (see Figure 14(a)). A reason for the qualitative difference in the two models can be seen from their sensitivity to the choice of the non-dimensional frequency factor A. For instance, ignition does not occur for some choice of A in the two models. In the CC model, the value of the frequency factor is small compared with the minimum values which are sufficient for ignition to occur in the HC model. In the numerical simulations, the higher value of the frequency factor in the HC model, relative to the CC model, affects the solution, viz.

217



Figure 14. Spatial profiles of the char pattern for (a) the CC model at Pe = 40, Le = 0.014,22, A = 0.27 × 104 , decrease in the spatial extent of the front, (b) the HC model at Pe = 40, Le = 0.026,563. Patterns with characteristic features of tip-splitting and bifurcation. (c) fingering state at A = 0.23×104 for the CC model, (d) fingering state at A = 0.85 × 104 for the HC model. In all cases computational time decreases with an increase in A.

(i) decreases the time required for ignition and faster front velocities, (ii) improves the strength of the convective transport, thus enhancing the visibility of the patterns, (iii) increases the heat release rate, and hence the sensitivity of fingers to tip-splitting, especially in the tip-splitting regime.


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Figure 15. Spatial profiles of char patterns for the HC model, showing (a) regime of constant fingers with no tip-splitting at Pe = 0.01, Le = 0.01 (b) limit of large flows at Le = 1, Pe = 10 and Le = 1 (available in colour online).

From (iii), the higher heat release rate in the HC model leads to an increased number of localised hot spots due to the inability of the front to lose heat. The excess heat at the reaction zone promotes the combustion process and the ability of single fingers to tip split and develop into branches. This behaviour is not significant in the CC model that shows a higher tendency to tip split and an increase in the spatial extent of the reaction zone at a frequency factor of A = 0.23 × 104 (see Figure 14(c)). Figure 14(d) indicates the pattern behaviour at a frequency factor A = 0.85 × 104 for the HC model. (i) is consistent with other studies in combustion processes [38,39]. (ii) follows from the fact that a robust reverse combustion and convective transport may enhance the visibility of patterns [4].

5.2.3.

Effect of Lewis number

The macroscopic models describe, in an effective sense, a one-temperature gas–solid system for the coupled two-temperature microscopic model. The effects arising from the gas phase and solid phase Lewis numbers can be investigated via the non-dimensional Lewis number, Le. Unlike the results presented in the previous subsection, the behaviour of the system (86) is analysed by considering the Lewis number as a free parameter. According to [25], the Lewis number is not the controlling transport mechanism for the fingering behaviour. However, in the models discussed in this study, we point out that it introduces some characteristic features to the fingering behaviour. It relates heat transport to mass transport, and as such, can characterise different flow regimes in a system with simultaneous convective mass and heat transport. The focus of this subsection is to show the close resemblance of the mechanism of the diffusive instability [5,25] to the well-studied thermaldiffusive instability occurring in low Lewis number premixed gas flames. At a very low blowing rate, the effective oxygen mass transport is dominant, and hence the Lewis number is close to a value that is much smaller than unity (Le 1). Figure 15(a) illustrates the resulting fingering pattern for the HC model at Pe = 0.01. In this regime, convective heat



219

Figure 16. Collage of fingering patterns for the CC model in the tip-splitting regime showing the variation of the distance between fingers at (a) Le = 0.1, (b) Le = 0.25, (c) Le = 0.7. In all cases, Pe = 10 (available in colour online).

transport is suppressed by an order of 10−6 . Specifically, φPe ∼ 10−6 . The fingering pattern shows a steady state of elongated fingers with no tip-splitting, but sparse fingering in the sense of d/w 1 [5] did not emerge. Here, d represents the distance between fingers and w the finger width. The emergence of such a steady state regime of sparse fingers with d/w 1 is presented in [4, 10], but in a non-adiabatic situation where the effect of heat losses conditions the state of the emerging fingers. In Figures 16(a) and 16(c) and Figures 17(a) and 17(c), the flow rate is moderate and the development of fingering patterns with characteristic tip-splitting features is substantial. The spacing between the fingers can be controlled by varying the Lewis number, i.e. an increase in the strength of the convective flux decreases the spacing between fingers. On the other hand, the inlet velocity or non-dimensional Péclet number affects the development of the pattern. In Figures 16 and 17, it is evident that the Pe changes the relative spatial extents of the pattern. In the high Pe limit, however, the topology of the fingers is very dense and the process may become extinct. It is also clear that some regions of the pattern also

Figure 17. Collage of fingering patterns for the HC model in the tip-splitting regime showing the variation of the distance between fingers at (a) Le = 0.1, (b) Le = 0.25, (c) Le = 0.7. In all cases, Pe = 10 (available in colour online).

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exhibit a shielding effect [5]. Such shielding of fingers by adjacent fingers that are close to the oxygen source causes the shielded fingers to stop growing and the tips of the shielding fingers to split. The behaviour of the patterns at large flow rates is shown in Figure 15(b) for the HC model. This regime corresponds to a Lewis number close to unity (i.e. Le = 1). The result in Figure 15(b) indicates a stable planar front that advances without fragmentation of the reaction zone. The finger patterns combine together into an unbroken reaction front and no visible pattern is observed. This connected front regime is also valid for Lewis number ranges which are much greater than unity. For example, in the analysis of [40], the results predict stable patterns with an infinite effective Lewis number. 6.

Concluding remarks

We have described, in a simple adiabatic case, the smouldering combustion model of a thin solid fuel that assumes a periodic distribution of its microstructure. We considered also in the present study two models of smouldering combustion which are particularly based on the conductivities in the constituent phases. The form of the homogenised models in both cases resemble the one-temperature model of filtration combustion [6, 41, 42] and are indeed similar to most other models obtained by the volume averaging method [3, 19]. The derivation of the macroscopic models does not require the use of the assumption of thermal-diffusive equilibrium. This requirement follows naturally from the rigorous mathematical treatment by the homogenisation method. However, there are situations where the assumption may not be applicable. For instance, a situation where heat diffusion in the system is not instantaneous because of a time retardation or memory effect introduced into the system by a component with a very low diffusion property. Homogenisation approaches have been widely used in providing rigorous mathematical answers to such physical situations. Moreover, the numerical simulations show that the results of the upscaled models are comparable to the predictions of the microscopic models, thereby justifying the classical homogenisation technique. However, one difficulty that was encountered in the comparative tests is that the presence of (strong) chemical reaction terms leads to a contradiction of the classical homogenisation approach. A possible remedy to this problem would be to seek for solutions to the microscopic problem in a moving frame. This approach was recently presented in [43]. The analysis of the macroscopic system based on the derived effective parameters contributed to the partitioning of the fingering state into different regimes. Specifically, the use the effective Lewis number that has an order of 10−2 demonstrated a fingering state of relatively spaced out fingers. This regime of fingers exhibits tip-splitting at considerably higher flow rates compared with other regimes, which are conditioned by some values of the Lewis number. The dependence of the Lewis number on porosity shows that, in some confined parameter space of the porosity, the Lewis number changes abruptly. In particular, it exhibited a broader range of values for the HC model compared with the CC model. Within the limit of theoretical assumptions in the problem formulation, we demonstrated, in a set of macroscopic numerical simulations, that the proposed models of reverse combustion capture the basic physics underlying the experimental observations of Zik and Moses [5, 11, 12] and Olson et al. [25]. Whereas the approach explored in this paper is quite different from those reported in other literature [4,10,13], the results of the numerical simulations have shown a close resemblance of the dynamics of pattern formation presented in this study with the mechanism of the diffusion driven instability [5, 25].



221

Also, the results obtained from the two equilibrium models show that high or moderate conductivity of fuels is not responsible for the emerging fingering states. The two models exhibit fingering instabilities in spite of the variations in their heat conducting capabilities. This explains the fact that similar fingering behaviour can be obtained in a vast number of combustible media [5]. However, the disparity arising from the two models indicates the need for a detailed understanding of the effects of kinetic and fuel property interactions in the emerging fingering instabilities. While the effect of heat losses has not been considered in the present paper, the results of the numerical simulations presented in this study suggest that heat transport mechanisms play an important role in the structure of the patterns. It was shown that variations in the heat release rates influence the morphology of the patterns. Thus, a better understanding of the interplay between various heat transport mechanism is still required. In addition, the experiments of [5] show that the gap above the material has a strong influence on the width of the fingers. This was also interpreted in terms of a heat loss effect, which is omitted in the present study. The models developed are intended in general for the qualitative treatment of the development of patterns in reverse combustion in porous media. Nevertheless, the results of the homogenisation technique present an effective way of integrating material microstructural properties into a macroscopic system describing real life processes. Thus, it can serve as a paradigm for further investigation of combustion processes in a rigorous mathematical treatment.

Acknowledgements This work is jointly supported by the Meiji University Global Center of Excellence (GCOE) Program and the Japanese Government (Monbukagakusho:MEXT). A.M. thanks Professor Willi Jäeger (Heidelberg) for having drawn his attention to the application of homogenisation methods to combustion in porous media.

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