Homogenization Technique Applied to

a Smoldering Combustion Model

Ekeoma Rowland Ijioma

July 2010

Homogenization TechniqueApplied to a Smoldering

Combustion Model

by

Ijioma, Ekeoma Rowland

Supervisors:Dr. Adrian Muntean

Dr. Martijn Anthonissen

July 28, 2010

i

Contents

1 Introduction 11.1 Smoldering combustion . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Choice of microstructure . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Mechanism of smoldering combustion . . . . . . . . . . . . . . . . . 41.4 The objective of this thesis . . . . . . . . . . . . . . . . . . . . . . . 61.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Mathematical model of smoldering combustion 72.0.1 Combustion reaction . . . . . . . . . . . . . . . . . . . . . . 72.0.2 Conservation of energy in the gas and solid phases . . . . . . 72.0.3 Conservation of mass in gas . . . . . . . . . . . . . . . . . . 82.0.4 Solid product formation . . . . . . . . . . . . . . . . . . . . 82.0.5 Boundary and initial conditions . . . . . . . . . . . . . . . . 9

2.1 Reduction to a two-dimensional formulation . . . . . . . . . . . . . 92.1.1 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . 14

3 The homogenization method 183.1 General averaging strategy . . . . . . . . . . . . . . . . . . . . . . . 183.2 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Homogenization of the microscopic problem . . . . . . . . . . . . . 213.4 The macroscopic equations and effective coefficients . . . . . . . . . 29

4 Numerical multiscale homogenization approach 314.1 Computation strategy for the homogenized problem . . . . . . . . . 314.2 Numerical computation of the homogenized coefficients . . . . . . . 33

4.2.1 Weak formulation of the cell problem . . . . . . . . . . . . . 334.2.2 Finite element approximation to the solutions of the cell prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.3 Existence and uniqueness of weak solutions to the cell problems 37

4.3 Further properties of the effective coefficients . . . . . . . . . . . . . 404.3.1 Symmetry and positive definiteness of the effective tensors . 404.3.2 A comparison of computed coefficients with various bounds . 41

4.4 Numerical computation of the macroscopic and microscopic solutions 424.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5.1 Smoldering process in a microstructure . . . . . . . . . . . . 454.5.2 One-dimensional comparison of results . . . . . . . . . . . . 49

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4.5.3 Error estimates in the FE approximations in the numericalsimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusion and Future work 55

A Matlab code for the discrete L2 error estimate 64

iii

Abstract

We study a semi-linear reaction-diffusion (RD) system modeling reverse smolderingcombustion of a thin cellulose material (e.g. paper) under the influence of oxidizingwind.The basic geometry of our material consists of a collection of many small thinsolid cylinders periodically distributed in space that we call microstructures. Themain working assumption is that the microstructure depends on two independentparameters: δ – the height of the cylinder (oriented in Oz direction) and ε – thewidth of a square cell that is copied periodically to cover the x− y plane. After avolume averaging of the RD system along the z coordinate, we obtain the so-calledmicro problem.The main objective is to use periodic homogenization methods to study the asymp-totic behavior of the solutions to the micro problem as the parameter ε goes to zero.This method gives an upscaled RD system (that we refer to as macro problem) to-gether with explicit formulae for the effective coefficients.We use COMSOL Multiphysics to solve numerically the micro problem for decreas-ing values of ε and compare the obtained results with the solution of the macroproblem (ε = 0). We see that our model captures the expected behavior of con-centrations and temperature profiles as observed in smoldering combustion experi-ments.

Keywords: Homogenization, reaction-diffusion system, smoldering combustion, mul-tiscale numerical method MSC 2000: 35B27; 76M50; 35K57; 80A25; 65N99

iv

Acknowledgements

Firstly, I acknowledge the hand of God throughout the period of this thesis, forkeeping me safe and in good health. I will like to express my sincere gratitude tomy supervisors Dr. Adrian Muntean and Dr. Martijn Anthonissen for giving methis project, and especially to Dr. Adrian Muntean, for his guidance, encourage-ment and patience during the period of this thesis.

I am also grateful to all my lecturers, both at the University of Technology, Kaiser-slautern and the University of Technology, Eindhoven. Your lectures and usefuldiscussions have inspired me over the years and have contributed to the successfulcompletion of this work.

Finally, to all my friends and well wishers, especially to those that followed theErasmus Mundus Masters’ program in Industrial and Applied Mathematics, I say ahuge thank you for your encouragements and all the interesting moments we workedtogether. More grease to your elbows in your subsequent careers.

Ijioma, Ekeoma RowlandEindhoven, July 2010.

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Chapter 1

Introduction

1.1 Smoldering combustion

Smoldering is a flameless form of combustion that derives its heat from heteroge-neous reactions occurring on the surface of a solid fuel when heated in the presenceof oxygen [1]. It is of interest for the fundamental understanding of combustionprocesses. It can also be studied as a practical fire hazard, since the smoldering cantransit to flaming and is usually accompanied by the emission of toxic gases.

In porous media, smoldering is a very complex situation that happens in situa-tions where a reactive part of the porous material is oxidized using heat contentwithout flaming. Common examples of smoldering combustion in porous mediainclude the initiation of upholstered furniture fires by weak heat sources and theflaming combustion of biomass occurring in wild land fires behind the flame front.

Many porous materials that can sustain a smoldering reaction namely coal, cotton,tobacco, paper, peat, wood and most charring polymers. We look to smolderingcombustion from the perspective of reaction-diffusion-flow phenomena in porousmedia.

Smoldering initiation requires the supply of heat flux to the solid. The subse-quent temperature increase of the solid triggers its thermal-degradation reactions(endothermic pyrolysis and exothermic oxidation) until the net heat released is highenough to balance the heat required for propagation. This net heat released by thereactions is partially transferred by conduction, convection and radiation ahead ofthe reaction and partially lost to the surrounding environment.

Oxygen is transported to the reaction zone by diffusion and convection, and inturn it feeds the oxidation reactions. Once ignition occurs, the smolder front prop-agates through the material in a creeping fashion [1].

In this thesis, we present a model of the smoldering combustion of a thin cellulosematerial exposed to a flow of oxygen confined in a narrow gap above the material.The experimental setup (see Fig.1.1) under consideration is one having a particu-

1

larly simple geometry (see. [2] for more details) in which the sheet is ignited on oneside rising the local temperature by means of a heat pulse. We assume the flow ofoxygen to be parallel to the cellulose material, and in the opposite direction to thecombustion front.

Figure 1.1: Sketch of the experimental setup [2].

We consider the situation where the combustion front proceeds from the ignitionboundary x = L,(see Fig. 1.1) to the opposite boundary at x = 0. The top andbase of the experimental setup is thermally insulated to prevent heat losses to thesurrounding. This means that the transport mechanism is fully controlled withinthe closed system, where convective/conductive heat transfer takes place betweenthe reacting species.

1.2 Choice of microstructure

The material under consideration in this study is a paper sample like that usedin [3]. We consider the paper material to be a porous medium with periodic ar-rangement of pores distributed around cylindrical structures as shown in Fig. 1.3.The microstructure we bear in mind is depicted in Fig. 1.2 (left). The paper hasthickness of size δ. We choose a reference unit cell, Y δ = (0, 1)n, n ∈ 2, 3 as in

Figure 1.2: Left: Volumetric representative cell Y δ; Right: Planar representativeunit cell Y .

Fig. 1.2 (the microstructure). We generate a lattice of copies of cells εY δ that spans

2

Figure 1.3: 3D micro scale geometry of the paper sheet.

the entire paper occupying the region Ωδ := Ω× (0, δ), with Ω := (0, L)2.

Within the unit cell Y δ, we define a geometrical structure Y δs , as the solid part,

i.e. a closed subset of Y δ and Y δg := Y δ \ Y δ

s , the gas part such that Y δ = Y δs ∪ Y δ

g .Also, we assume that any two neighboring Y δ

s do not touch each other and the twoparts of the unit cell satisfy Y δ

s ∩ Y δg = ∅. Next, we make a periodic repetition of

Y δs all over Rn and set Y k,δ

s := Y δs + k, k ∈ Zn.

Clearly, the obtained set

Es :=⋃k∈Zn

Y k,δs (1.1)

is a closed subset of Rn and Eg := Rn \ Es is an open set in Rn.

Moreover, we assume that the paper consists of connected gas parts. We defineby Es ⊂ Rn, the closed set that is obtained by Y δ-periodic repetition of Y δ

s in theentire space Rn and we denote its gas counterpart by Eg. The following hypotheseshave to be satisfied [4]:

i. Y δg and Y δ

s have strictly positive measures in Y δ with ∂Y δs ∩ ∂Y δ = ∅.

ii. Eg and the interior of Es are open sets with C0,1 boundaries. Furthermore Egis connected.

iii. Y δg is an open set with a local Lipschitz boundary.

The implications of these hypotheses for a paper sheet are as follows. One elemen-tary cell Y δ of a paper consists of both solid and gas parts as depicted in Fig. 1.2.Therefore, assumption (i) is fulfilled.

Furthermore, we assume that a paper sheet consist of periodic repeated cells cov-ering Ωδ. Thus, the solid part Y δ

s , as well as the gas part Y δg , of the repeated cell

must be Y δ-periodic. i.e. Eg, must be connected.

3

Both Eg and Es have a C0,1 boundary, and assumption (ii) is fulfilled. Further-more, we assume that the boundary of Y δ

g is sufficiently regular so that (iii) holds.

Now, we generate a lattice of copies of cells εY δ, for any ε > 0, by letting thedomain, Ωδ ⊂ Rn, be covered by regular mesh of size ε as in Fig. 1.3. We denoteeach cell by Y ε

i = (0, ε)n, where n = 2 or 3, with 1 ≤ i ≤ N(ε), and N(ε) = |Ω|ε−ndenotes the number of cells along the x- and y- directions, while keeping (0, δ) fixed,i.e we have in general

N(ε, δ) :=|Ωδ|εn−1δ

. (1.2)

Each cell is homeomorphic to Y δ, by a linear homeomorphism Πεi , with ratio of

magnification 1/ε, i.e. we re-scale the cell Y δ by ε. Hence,

Y δ,εsi

:= (Πεi )−1(Y δ

s ) (1.3)

Y δ,εgi

:= (Πεi )−1(Y δ

g ) (1.4)

denote respectively the solid and the gas part of the unit cell,Y ε,δ, of the order ε.The gas domain, Ωε

g ⊂ Ω, is obtained by removing the periodically distributed solidparts. That is,

Ωεg := Ωδ \

N(ε)⋃i=1

Y ε,δsi

= Ωδ ∩ εEg (1.5)

with

Ωεs =

N(ε)⋃i=1

Y ε,δsi. (1.6)

Also, we define ∂Ωεg = ∂Ωδ ∪ ∂Ωε

s to be the boundary of the gas part of the porousmedium, where ∂Ωε

s is the solid-gas interface. Subsequently, we will, for simplicityof notation, denote ∂Ωε

s as Γ.

1.3 Mechanism of smoldering combustion

Combustion has been studied extensively over the years both experimentally andnumerically. However, few relevant theoretical studies based on smoldering modelshave been done. Previous studies in this area include the work of Ohlemiller [5],who presented a review of the most significant mechanisms involved in the smol-dering combustion of polymers. Rein [1] presented a 1-D computational study toinvestigate smoldering ignition and propagation in polyurethane foam.

In [6], traveling wave solutions were sought by considering cases of sufficiently largeflow and moderate flow of the air flux. They found that for each Peclet number (Pe)beyond some lower threshold, there exist two solutions: a fast wave and a slow wave,

4

where both have peculiar qualitative differences.The slow waves are unstable whilethe fast waves are stable for large values of Pe. K. Ikeda and M. Mimura (2008) (see[7]) proposed a model of reaction-diffusion system for theoretical understanding ofpattern formation in smoldering combustion, and then proved the existence anduniqueness of strong solution to the system using the standard theory of analyticsemigroups. Their numerical simulation exhibits a good qualitative agreement withthe experimental results of Zik and Moses (see Fig. 1.4).

In their fundamental paper [3], Zik and Moses studied the smoldering combus-tion of a thin cellulose material experimentally and developed a phenomenologicalmodel for the analysis of pattern formation and fingering instability. The primaryuse of their model has been to analyze the global behavior of smoldering combustionusing the flow velocity (see also Peclet number) as a control parameter.

Other control parameters considered in [3] include the vertical gap between theplates and the heat conductivity of the bottom plate of the experimental setup.The lateral boundaries of the experimental setup are made to create a uniform heatconduction and to decrease the supply of oxygen from the sides of the sample. Thisensures uniform propagation along the boundaries.

Figure 1.4: Instability of combustion front when the flow velocity (or Peclet number)decreases from a to e. Oxygen flows downward and the smolder front moves upwardas shown by the arrow in d [6]

The major mechanism of their experiments which is most relevant to our study isthe role of oxygen flow velocity (alternatively the Peclet number) in the observedpattern formation of the smolder front. The results of their experiment show dif-ferent behaviors of the smolder front as shown in Fig. 1.4.

In Fig.1.4, the front is initially smooth due to the high flux of oxygen (a). Next,a sinusoidal pattern develops after the supply of oxygen is limited (b-c). Furtherreducing the oxygen flux separates the peaks and a fingering pattern is formed. Thesame behavior is also possible with a nondimensionalized system. In this case, thenon-dimensional control parameter is the Peclet number Pe, which measures therelative importance of advection and molecular diffusion.

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1.4 The objective of this thesis

The primary objective of this thesis is to investigate the smoldering combustion of athin solid fuel (paper)-a porous solid with periodic microstructure. The theoreticalbackground of our proposed model is partly based on a previous work by L. Kaganand G. Sivashinsky [2], where fragmentation1 of the flame front was excluded. Weexpect that our proposed upscaled model will capture some of the qualitative fea-tures of smoldering combustion as given in [3].

The geometry of the periodic microstructure depends on a small parameter ε (seesection 1.2). This means that any sought solution of the model depends on ε. Itis difficult to obtain an exact solution of this problem. Thus, only an approximatesolution depending on ε is available. In a formal asymptotic analysis we show thatas ε approaches zero, the approximate microscopic solution converges to the macro-scopic solution (ε = 0). This procedure becomes computationally expensive withdecreasing values of ε.

Further, we implement a method that drastically reduces the computational ef-fort involved in solving the microscopic model (see chapter 3 (The homogenizationmethod)). We derive an averaged (homogenized) model. Although the homoge-nized model is only an approximation, we show in this study that it is akin to thephysical behavior of the microscopic model without loosing too much accuracy. Weinvestigate how far the solution of the homogenized model is from the approximatesolution of the microscopic model.

1.5 Outline of this thesis

The outline of this thesis is as follows: Chapter 2 introduces the mathematicalmodeling of smoldering combustion, from which the three dimensional microscopicmodel was reduced to a two dimensional formulation. We identified a set of govern-ing parameters by using dimensional analysis. The influence of these parametersis illustrated by direct and detailed numerical simulations on the micro and macroproblems. In Chapter 3, we introduce the homogenization method and approximatethe 2 dimensional microscopic model presented in Chapter 2 by a set of macroscopicequations with effective coefficients. Chapter 4 begins with discussion on the themultiscale numerical homogenization approach. We provide solutions to the cellproblems encountered in Chapter 3 as well as a strategy to compute effective coeffi-cients. Finally, we solve the microscopic and macroscopic problems using COMSOLMultiphysics and give an error estimate in terms of the discrete L2 norm. In chapter5, we discuss the results of the previous chapters with concluding remarks. Also,we present some useful insights into some issues that were not covered in the courseof this study as future work.

1By fragmentation, we mean the splitting of the smolder front as shown in Fig. 1.4 (center)

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Chapter 2

Mathematical model ofsmoldering combustion

In order to remain consistent with a possible real life scenario, we adopt the materi-als and operating parameters, corresponding to the laboratory experiment describedin [3], as a reference case. In this section, we describe the transport and reactionmechanisms of smoldering combustion of a thin sheet of paper using a 3-dimensionalsystem of partial differential equations.

Following the ideas in [8], we consider the conservation of various physical quan-tities, governed by flow equations and combustion equations. The flow equationsexpress conservation of mass and momentum. The combustion equation expressconservation of the species mass fractions and energy. However, in this presentstudy, we consider a minimal formulation where we do not consider the dynamicsof the problem due to the gas pressure (see. [6], [2]).

2.0.1 Combustion reaction

The species involved in the chemical model includes C6H10O5, O2, CO2, H2O andthe inert gases such as N2 (cf.[6], [8]). However, we restrict ourselves to the speciesentering the reaction. Basically, the combustion reaction takes place between cel-lulose and oxygen [6]:

C6H10O5 + 6O2 → 6CO2 + 5H2O, (2.1)

where the constant coefficients in the reaction species in (2.1) are the stoichiometriccoefficients for the reaction. The reaction takes place at the solid-gas interface. Wecan obtain the rate of reaction of the solid at this interface by the application ofthe law of mass action, which states that the rate of a reaction is proportional tothe product of the concentrations of the reactants.

2.0.2 Conservation of energy in the gas and solid phases

The balance in energy in the two phases can be described by two heat transportequations in the gas part, Y δ

g and in the solid part Y δs . The transport equation in

7

Figure 2.1: Reaction/transport mechanisms within the microstructure

the gas part is of convection-diffusion type. The oxygen is first preheated and ispassed from the boundary, x = 0.

Oxygen flows through the porous medium and reacts at the reaction zones pre-sented at the solid-gas interface. Heat is carried by the oxygen across the poresites of the microstructure, and also the oxygen is being diffused in the process (seeFig. 2.1). The heat transfer in the solid part of the microstructure involves onlyconduction. The governing equations are thus:

ρgcg∂Tg∂t

+∇ · (ρgcgu · Tg − λg∇Tg) = 0 in Y δg , (2.2)

ρscs∂Ts∂t−∇ · (λs∇Ts) = 0 in Y δ

s , (2.3)

where T is the temperature (K), λ, the thermal conductivity (J/(msK)), ρ, themass density (kg/m3), c the specific heat capacity (J/(kgK)), u = (u, 0, 0) the flowfield (m/s). The subscripts g and s denotes gas and solid parts, respectively.

2.0.3 Conservation of mass in gas

The oxygen concentration, C (dimension kg/m3), is governed by a convection-diffusion equation.

∂C

∂t+∇ · (uC −D∇C) = 0, in Y δ

g (2.4)

where D is the diffusion coefficient (dimension m2/s) following Fick’s law thatdescribes the flow of oxygen from the region above the paper to the region withinthe porous medium as in Fig. 2.1.

2.0.4 Solid product formation

The transition from paper to char involves a decay due to the chemical reaction(2.1). This follows a mass action type law defining W and is given by

∂R

∂t:= W (T,C) at ∂Y δ

s . (2.5)

8

R is the concentration of solid product (kg/m3) and the reaction rate W (T,C),(kg/(m3s)) is defined by

W (T,C) := AC exp

(−TaT

). (2.6)

(2.6) describes an Arrhenius type kinetics, with Ta being the activation temperature(K). The boundary prescribed in (2.5) is at the gas-solid interface in the volumetricunit cell Y δ.

2.0.5 Boundary and initial conditions

The transport equations described in (2.2) and (2.3) are coupled by the continuityof the temperature across the boundary ∂Y δ

s , and also by the source and sink termsincluded in the surface reaction.

The reaction induces source and sink terms for the oxygen and the heat, that areall proportional to the reaction rate. The sources and the sinks, localized on thesurface ∂Y δ

s of the reactive solid part, lead to the following boundary conditions atthe gas-solid interface:

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

Tg = Ts = Tat ∂Y δ

s (2.7)

where T is the interfacial temperature, n is the outward unit normal vector to theinterface, and Q the heat release ( J/kg)Due to the diffusive-thermal insulation of the system, the boundary conditions atthe walls are:

n · ∇C = n · ∇Tg = n · ∇Ts = 0, at z = δ ∪ z = 0. (2.8)

The system is completed by upstream boundary conditions,i.e., by the gas flowrate, the entry gas temperature and oxygen concentration. Radiative effects arenot taken into account in this formulation.

The initial conditions are as follows:Tg = Ts = T0

C = C0

R = R0

in Ωδ. (2.9)

2.1 Reduction to a two-dimensional formulation

In order to successfully study the surface smoldering of the material, we reduce1

the dimension of the model equations from 3-dimensions to 2-dimensions. To be

1 This reduction also has the effect of suppressing any effects due to gravity. This brings us inline with previous studies on smoldering combustion under micro-gravity.

9

more precise, we do this by averaging along the z-axis of the domain.

Integrating (2.2) over z ∈ (0, δ), we have

ρgcg∂

∂t

∫ δ

0

Tgdz + ρgcgu · ∇∫ δ

0

Tgdz + ρgcg · 0 ·∫ δ

0

∂Tg∂z

dz = λg4∫ δ

0

Tgdz + λg

∫ δ

0

∂2Tg∂z2

dz,

(2.10)

where

u · ∇ 7→ (u, 0, 0) ·(∂

∂x,∂

∂y,∂

∂z

)= u · ∇+ 0 · ∂

∂z, (2.11)

with

u = (u, 0) and ∇ =

(∂

∂x,∂

∂y

).

Similarly, we define

4 7→ ∂2

∂x2+

∂2

∂y2+

∂2

∂z2= 4+

∂2

∂z2, (2.12)

with

4 =

(∂2

∂x2,∂2

∂y2

).

Now, we define an average operator along the z-axis as follows:

〈Mn〉z :=1

δ

∫ δ

0

Mn(x, y, z, t)dz for all (x, y, z) ∈ Ωδ. (2.13)

This is a typical way of working in the averaging technique developed by Whitaker[9]. By (2.13), (2.10) becomes

ρgcg

(∂

∂t〈Tg〉z + u · ∇〈Tg〉z

)= λg4〈Tg〉z +

λgδ

∂Tg∂z

∣∣∣z=δ− λg

δ

∂Tg∂z

∣∣∣z=0

. (2.14)

Applying the boundary condition (2.8) in (2.14), we have

ρgcg

(∂

∂t〈Tg〉z + u · ∇〈Tg〉z

)= λg4〈Tg〉z −

λgδ

∂Tg∂z

∣∣∣z=0

. (2.15)

Similarly, we obtain the following from (2.3), but changing the path of integrationto z ∈ (−δ, 0).

ρscs∂

∂t〈Ts〉z = λs4〈Ts〉z +

λsδ

∂Ts∂z

∣∣∣z=0

(2.16)

Note that we have changed the path of integration to maintain the fact that thesolid is defined in a region situated below the gas.

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Now, adding up (2.15) and (2.16), and using the boundary condition in (2.8), weobtain

ρgcg

(∂

∂t〈Tg〉z + u · ∇〈Tg〉z

)+ ρscs

∂

∂t〈Ts〉z = λg4〈Tg〉z + λs4〈Ts〉z +Q

W (T,C)

δin Y.

(2.17)

In (2.17), we can also denote W (T,C), by

W (T,C) :=W (T,C)

δ. (2.18)

Furthermore, integrating (2.4) and using definitions (2.11) and (2.12) as we did for(2.2), we obtain

∂

∂t

∫ δ

0

Cdz + u · ∇∫ δ

0

Cdz = D4∫ δ

0

Cdz +D

∫ δ

0

∂2C

∂z2dz (2.19)

such that by (2.13), we have

∂

∂t〈C〉z + u · ∇〈C〉z = D4〈C〉z +

D

δ

∂C

∂z

∣∣∣z=δ− D

δ

∂C

∂z

∣∣∣z=0

. (2.20)

Using the boundary conditions (2.7) and (2.8), we obtain the following:

∂

∂t〈C〉z + u · ∇〈C〉z = D4〈C〉z − W (T,C) in Yg. (2.21)

Finally, the set of equations is completed by

∂R

∂t= W (T,C) at ∂Y δ

s . (2.22)

However, in order to be consistent with the model reduction in terms of averages,we must also average the differential equation (2.22), as well as the remainingboundary and initial conditions . In what comes next, we present closure relations(strong assumptions) under which we can average the ordinary differential equation.

Claim: In Y , let the temperature of the solid-gas interface be T . Let also thetemperature-dependent Arrhenius exponential factor be given by A(T ). Assumingthat T is below a critical temperature Tc, then A(T ) reduces to

A(T ) =

0, T < Tc

γ, T > Tc,(2.23)

where γ is a positive constant. In particular, if T Ta in A(T ), then

W (T,C) = AC exp(−TaT

) ∼ AC. (2.24)

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Proof of the claim:For T ≤ 25C and Tc = 450C i.e., the auto-ignition temperature of paper, we seethat A(T ) ∼ 0 and conversely, for T > Tc, we have that A(T ) = γ, for some positiveconstant γ. Let k = Ta

T, such that A(T ) = exp(−k). Taking the Taylor’s expansion

of A around k = 0, we have

A(T ) ≈ 1− k +k2

2− k3

6+O(k4) (2.25)

Hence, T Ta, k → 0 implies that A(T ) ∼ 1 and thus W (T,C) ∼ AC.

Now, assuming that the gas occupies the porous region of the material as shown inFig. 1.3, and applying the result of our claim, (2.22) reduces to

∂R

∂t= AC. (2.26)

Averaging the right hand side of (2.26), leads to

∂〈R〉z∂t

= A〈C〉z. (2.27)

Alternatively, the right hand side of (2.22) can be approximated by the mean valuetheorem as follows:We assume, as before, that the gas occupies the porous region of the paper asdepicted in Fig.1.3. For small variations in the local concentration and temperaturearound z = 0, it makes sense to average along 0 ≤ z ≤ δ such that,

∂〈R〉z∂t

=1

δ

∫ δ

0

W (T,C)dz (2.28)

where the unknowns C and T depend on (x, y, z, t).

Now, applying the mean value theorem (MVT), the right hand side of (2.28) be-comes

1

δ

∫ δ

0

W (T,C)dz =1

δ

∫ δ

0

AC exp

(−TaT

)dz

= C(x, y, z∗, t) exp

(−Ta

T (x, y, z∗, t)

)for some z∗ ∈ (0, δ). By the definition of the average operator, we have

〈C〉z =1

δ

∫ δ

0

C(x, y, z, t)dz ≈ C(x, y, z∗, t) and

〈T 〉z =1

δ

∫ δ

0

T (x, y, z, t)dz ≈ T (x, y, z∗, t) (by MVT),

for some z∗ ∈ (0, δ). Thus,

W (T,C) ≈ W (〈C〉z, 〈T 〉z) at ∂Ys. (2.29)

12

Note that the same averaging idea used in (2.29) also applies to the last termof (2.22). Another possibility of averaging (2.22) will be to consider the Frank-Kamenetskii approximations (see [10] and [11] for example.).

From the derivations so far, we see that the equations are given in quantities andparameters associated with a given part of the representative unit cell (Yg or Ys,see Fig. 1.2). However, we wish to reduce the number of variables appearing in theequations by introducing the following definition:

Definition 2.1.1 Let Y be the representative unit cell as shown in Fig. 1.2 (right).The characteristic function χ

Y`restricted to a part of Y is given by

χY`

(x, y) :=

1, (x, y) ∈ Y`, ` ∈ g, s0, otherwise.

(2.30)

Using (2.30) yields

T := TsχYs+ TgχYg

, (2.31)

ρc := ρscsχYs+ ρgcgχYg

, (2.32)

λ := λsχYs+ λgχYg

. (2.33)

Summarizing, the governing equations and initial/boundary conditions, now formu-lated in 2-dimensions, are:

ρc

(∂

∂t〈T 〉z + u · ∇〈T 〉z

)= λ4〈T 〉z +QW (〈T 〉z, 〈C〉z) in Y, (2.34)

∂

∂t〈C〉z + u · ∇〈C〉z = D4〈C〉z −W (〈T 〉z, 〈C〉z) in Yg, (2.35)

∂〈R〉z∂t

= W (〈T 〉z, 〈C〉z) at ∂Ys. (2.36)

The corresponding boundary conditions are:−Dn · ∇〈C〉z = 0

n · λg∇〈Tg〉z = nλs∇〈Ts〉z = nλ∇〈T 〉〈Ts〉z = 〈Tg〉z = 〈T 〉z

at ∂Ys. (2.37)

The upstream boundary conditions include the following:

∂〈C〉z∂x

= 0 at ∂Ω ∩ x = L (2.38)

〈T 〉z = T0

〈C〉z = C0

〈R〉z = 0

at ∂Ω ∩ x = 0. (2.39)

13

Finally, we supplement the model problem with initial conditions〈T 〉z = T0

〈C〉z = C0

〈R〉z = R0

for 0 < (x, y) < L. (2.40)

Note that the 2-D system of equations in (2.34)-(2.36) needs to be understood inthe distributional sense.

Having obtained the set of model equations in terms of averages, we wish to dropthe notation, 〈 〉z, as used for the averages and introduce the notations Θ,Ψ, andR. Also, we drop the tildes and bars on the parameters. The model equationsformulated in the new variables are:

ρc

(∂Θ

∂t+ u · ∇Θ

)= λ4Θ +QW (Θ,Ψ) in Y, (2.41)

∂Ψ

∂t+ u · ∇Ψ = D4Ψ−W (Θ,Ψ) in Yg, (2.42)

∂R∂t

= W (Θ,Ψ) at ∂Ys, (2.43)

and the corresponding boundary conditions are:−Dn · ∇Ψ = 0

n · (λg∇Θg − λs∇Θs) = 0

Θs = Θg = Θ

at ∂Ys, (2.44)

where Θ,Ψ and R are respectively the temperature, gas concentration, and solidproduct(char).

2.1.1 Non-dimensionalization

In this section, we make the model equations dimensionless following the samestrategy given in [2], [6]. Instead of having a large number of physical parametersand variables, all with dimensional units, we wish to scale all the variables with’characteristic’ values-values of the size we expect to see, or dictated by the ge-ometry, boundary conditions etc-so that we are left with an equation written indimensionless variables.

The idea is to collect all the physical parameters and typical values together into asmaller number of dimensionless parameters (or dimensionless groups) which, whensuitably interpreted, should tell us the relative importance of the various mecha-nisms. This means that we are much interested in obtaining relevant dimensionlessnumbers, which can be used to interpret the physical mechanisms involved in the

14

balanced laws.

We begin now with (2.17) and rescale the mass concentration of the gas and solidproduct with their initial values as follows:

Ψ :=〈C〉zC0

, R :=〈R〉zR0

(2.45)

and the temperature by

Θ` :=〈T`〉z − T0

Tb − T0

, where ` ∈ s, g, (2.46)

with Tb = T0 + fracQcg i.e., the temperature of combustion product. Also, werescale the independent variables (x, y) and t respectively by L and

tD :=L2

D, such that τ :=

t

tD(2.47)

where tD is the diffusive time scale. Rewriting (2.17) in terms of the characteristicscales, we obtain

D(Tb − T0)

L2

∂Θg

∂τ+mD

(Tb − T0)

L2

∂Θs

∂τ+u(Tb − T0)

L∇Θg =

λgρgcg

(Tb − T0)

L24Θg

+λsρgcg

(Tb − T0)

L24Θs

+QAΨC0A(Θ)

ρgcg(2.48)

where

m :=ρscsρgcg

(the weighted mass ratio) and A(Θ) = exp

(− Ta

(T0 + (Tb − T0)Θ)

).

Next, we divide through by the coefficient of the term on the left hand side, sothat we have

∂Θg

∂τ+m

∂Θs

∂τ+uL

D∇Θg =

λgρgcgD

4Θg +λs

ρgcgD4Θs +

QAΨC0L2A(Θ)

D(Tb − T0)ρgcg(2.49)

At this moment, we introduce the following the combustion time scale

tB =

(QAC0

(Tb − T0)ρgcg

)−1

, (2.50)

where in (2.49), we define

Les :=λs

ρscsD, Leg :=

λgρgcgD

(2.51)

15

are respectively the Lewis number of the solid (i.e. the ratio between the equivalentheat diffusivity of the paper and oxygen diffusivity) and gas, respectively.Furthermore,

Pe :=uL

D; the Peclet number. (2.52)

Thus, the equation is given by

∂Θg

∂τ+m

∂Θs

∂τ+ Pe∇Θg = Leg4Θg +mLes4Θs +

tDΨ

tBA(Θ). (2.53)

(2.53) can also be written in compact form by introducing the following definition

Θ := Θgχg +mΘsχs (2.54)

Le := Legχg + Lesχs, (2.55)

so that, we may write (2.53) as

∂Θ

∂τ+ Pe∇Θ = Le4Θ +

tDΨ

tBA(Θ). (2.56)

Similarly, the non-dimensional form of (2.21) and (2.36) are:

∂Ψ

∂τ+ Pe∇Ψ = 4Ψ− tDΨ

tGA(Θ), (2.57)

where the gas combustion time scale tG is defined as

tG := (A)−1 (2.58)

∂R∂τ

=L2Ψ

D

C0

R0

AA(Θ) (2.59)

=tDtR

ΨA(Θ). (2.60)

The time scale of solid product tR is given by

tR :=

(C0

R0

A

)−1

, (2.61)

Considering the boundary condition

n · (λg∇Θg − λs∇Θs) = 0, (2.62)

we simplify (2.62) by introducing ρgcgD and ρscs. (2.62) then reduces to the fol-lowing

λgρgcgD

∇Θg −λs

ρscsD

ρscsρscs∇Θs = 0 (2.63)

n · (Leg∇Θg − Les∇Θs) = 0. (2.64)

16

Hence, (2.64) can then be written as

[−n · Le∇Θ] = 0 on ∂Ys. (2.65)

The dimensionless boundary conditions are:−Dn∇Ψ = 0

[−n · Le∇Θ] = 0

Θs = Θg = Θ

on ∂Ys, (2.66)

∂Ψ

∂x= 0 at ∂Ω ∩ x = L. (2.67)

Θ(L, t) =T ∗ − T0

Tb − T0

H(ε

tD− τ), (2.68)

where T ∗ > Tc, for some critical temperature Tc. ε is some sufficiently small timeand H(·) is the 2Heaviside function.

Θs = Θg = 0

Ψ = 1

R = 0

at ∂Ω ∩ x = 0 (2.69)

and the initial conditions are:Θg = Θs = 0

Ψ = 1R = 1

for 0 < (x, y) < L. (2.70)

2The introduction of the Heaviside function is to mimic a heat pulse. (see [6] for more details.)

17

Chapter 3

The homogenization method

3.1 General averaging strategy

In this chapter, we wish to obtain a macroscopic model of the 2-dimensional for-mulation presented in subsection 2.1.1, which is now considered here to be themicro-problem. We do this by applying the mathematical theory of periodic ho-mogenization, i.e., the averaging of the solution of the microscopic equations forfinding effective equations and coefficients (see [12], [13] for details).

By this averaging procedure, the equations are replaced by similar equations withconstant coefficients, which are easy to solve numerically. Therefore, the idea ofthis homogenization technique is to consider the heterogeneous medium as a peri-odically repeating structure with period ε as pointed out earlier (see section 1.2).Then, a sequence of problems for which the length scale ε, becoming increasinglysmall, goes to zero.

The main target in this chapter is to perform an asymptotic analysis as ε tendsto zero. An effective coefficient (more precisely homogenized (macroscopic) tensor)in the limit problem (ε = 0), describes material properties of the medium such asits conductivity. We will see later that the effective coefficients typically depend onthe microstructure information.

An advantage of this approach, despite its complexity, is that the homogenizedproperty is uniquely defined. As a further step, the approximation made by usingonly effective properties instead of the true microscopic coefficients can be rigor-ously justified by quantifying the resulting error (see [12]). We comment more onthis idea in chapter 4 and chapter 5

3.2 The microscopic problem

Let Ω be a region in R2 with boundary ∂Ω containing two phases, gas and solid,as discussed in section 1.2. The gas and the solid matter possess constant material

18

coefficients1 specific to each of the phases. We denote the coefficient by Lij.

Let the spatial variable be given by x := (x1, x2). We refer to x as the macroscale(global) variables. Also, let the temporal variable be τ . We assume that the quan-tities Θ,Ψ, and R defined in chapter 2 are prescribed in the cylindrical domainQ := Ω× (0, T ), where T is a fixed time.

By the assumption of periodicity given in chapter 2, R2 is then divided into pe-riodically repeated unit cells Y. Each unit cell is divided into two parts, the gas andthe solid parts Yg and Ys. The surface porosity of the medium is defined by

Φ :=meas(Yg)

meas(Y ). (3.1)

In (3.1), meas(Y) is the Lebesgue measure (i.e. the area in R2) of Y . In what comeslater on, we will write |Y | instead of meas(Y).

The periodicity in variable Y implies that the coefficient Lij must be periodic.We describe the Y-cell, with the microscale (local) variable, as y := (y1, y2) withyi := xi/ε for i ∈ 1, 2. The scale factor ε suggests that the quantity Θ togetherwith the coefficients in the different parts of the medium is defined by

Θε := Θεgχg(x/ε)±mΘε

sχs(x/ε) (3.2)

Lεij(x) := Legχg(x/ε) + Lesχs(x/ε) (3.3)

P εj (x) := Peχg(x/ε) for all x ∈ Ω. (3.4)

We assume that the convective term is divergence free i.e. ÷~P = 0. Also, we define

Lεij(x) := Lij(x/ε) = L(y)

P εj (x) := Pj(x/ε) = P (y)

(for y = xε) (3.5)

Y-periodicity means that Lij(a) = Lij(b) whenever a and b are homologous by peri-odicity, i.e. have the same positions in corresponding cells. In addition, if the pointsa and b are close enough with respect to the x variable, we say that the coefficientis locally periodic. For more details on Y-periodicity and locally periodic, see [14],[13], [15].

We note that in a cell Y, Y-periodic functions take on the same boundary val-ues twice, but with opposite outer normals n = (n1, n2). Thus, for a Y-periodicvector field (fi(y)), the following relation holds:∫

Y

∂fi∂yi

dy =

∮∂Y

finidS(y) = 0 (3.6)

by Gauss’ theorem. In the subsequent analysis, we will use (3.6) quite frequently.In (3.6), dS(y) denotes the infinitesimal line element of ∂Ω.

19

Figure 3.1: (a): Deformed cell after time, t; (b) Not structurally deformed cell aftertime t.

Also, we define the following Hilbert space for Y -periodic functions:

W (Y ) := v ∈ H1(Y ) : v is Y-periodic;1

|Y |

∫Y

vdy = 0. (3.7)

Further, we assume that the paper is not structurally deformable, that is, the shapeof the paper from the start of the heating process is not deformed due to changesresulting from the smoldering of the paper after a time, say t. This assumption andits implication is illustrated in Fig. 3.1.

Finally, we assume that each Lij belongs to L∞(Ω) and that Lij is positive def-inite, i.e.

Lijξiξj ≥ αξiξi (3.8)

for some α > 0 and all choices of ξi and ξj.

Now, we let ε vary, thereby obtain a class of initial boundary value problems de-pending on the parameter ε:

∂Θε

∂τ+ ~∇ · (P (y)Θε − L(y)∇Θε) = f(Θε,Ψε) in Ω, (3.9)

∂Ψε

∂τ+ ~∇ · (P (y)Ψε −∇Ψε) = g(Θε,Ψε) in Ωε

g, (3.10)

∂Rε

∂τ= h(Θε,Ψε) on Γε, (3.11)

1Coefficients defined differently in solid and gas parts can be interpreted as a single coefficientdiscontinuous in the space variable. As a result, the corresponding equations can be interpretedin the sense of distributions in Ω.

20

where

f(Ψε,Θε) =tDtB

ΨεA(Θε), (3.12)

g(Ψε,Θε) = −tDtG

ΨεA(Θε), (3.13)

h(Ψε,Θε) =tDtR

ΨεA(Θε.) (3.14)

with

A(Θε) = exp

(− Ta

(T0 + (Tb − T0)Θε)

). (3.15)

The corresponding boundary and initial conditions are as follows:−n · L(y)∇Θε = 0

−n · ∇Ψε = 0

Θεg = Θε

s = Θε

on ∂Ys, (3.16)

Θεs = Θε

g = 0

Ψεg = 1

Rε = 0

at ∂Ω ∩ x = 0, (3.17)

∂Ψε

∂x= 0 at ∂Ω ∩ x = L, (3.18)

Θ(L, t) = Θ∗H(ε

tD− τ), (3.19)

Θε(x, 0) = 0

Ψε(x, 0) = 1

Rε(x, 0) = 1

for 0 < (x, y) < L. (3.20)

We refer to (3.9)-(3.11) as the microscopic problem.

3.3 Homogenization of the microscopic problem

We consider (3.9), (3.10), and (3.11) and assume that the microscopic solutionsΘε,Ψε,Rε together with the right hand side terms f ε, gε and hε can be representedby asymptotic expansions of the form:

Θε(x, τ) = Θ0(x, y, τ) + εΘ1(x, y, τ) + ε2Θ2(x, y, τ) +O(ε3)

Ψε(x, τ) = Ψ0(x, y, τ) + εΨ1(x, y, τ) + ε2Ψ2(x, y, τ) +O(ε3)

Rε(x, τ) = R0(x, y, τ) + εR1(x, y, τ) + ε2R2(x, y, τ) +O(ε3)

f(Θε,Ψε) = f(Θ0,Ψ0) + εΘ1∂1f(Θ0,Ψ0) + εΨ1∂2f(Θ0,Ψ0) +O(ε3)

g(Θε,Ψε) = g(Θ0,Ψ0) + εΘ1∂1g(Θ0,Ψ0) + εΨ1∂2g(Θ0,Ψ0) +O(ε3)

h(Θε,Ψε) = h(Θ0,Ψ0) + εΘ1∂1h(Θ0,Ψ0) + εΨ1∂2h(Θ0,Ψ0) +O(ε3)

for y =x

ε,

(3.21)

21

and the functions Θi(x, y, τ),Ψi(x, y, τ) and Ri(x, y, τ) are all Y−periodic in y forany choice of i ∈ N

The first terms of the series expansions (3.21) will be usually identified with thetypically verified homogenized equations whose unknown effective coefficients canbe exactly computed.

Therefore, numerical computations involving the homogenized equation do not re-quire a fine mesh solution since the heterogenities of size ε have been averaged out(see [12] for example).

Now, following the line of argument of [13], we define the following operators:

Aεφ := ~∇ · (P (y)− L(y)∇)φ, (3.22)

Bεφ := ~∇ · (P (y)−∇)φ. (3.23)

By setting φ(x) = θ(x, y), y = xε, we obtain via the chain rule of differentiation:

∇φ = ∇xθ +1

ε∇yθ. (3.24)

Next, we insert the homogenization ansatz (3.21) into (3.9), (3.10), and (3.11). Inthis way, a geometric series in ε is obtained by application of the chain rule of dif-ferentiation (3.24).

From (3.9), we obtain

∂

∂τ(Θ0 + εΘ1 + · · · ) + (~∇x +

1

ε~∇y) · (P (y)− L(y)((∇x +

1

ε∇y))(Θ

0 + εΘ1 + · · · ))

= f(Θ0,Ψ0) + εΘ1∂1f(Θ0,Ψ0) + εΨ1∂2f(Θ0,Ψ0) + · · ·(3.25)

⇔ (ε−2A0 + ε−1A1 + ε0A2)(Θ0 + εΘ1 + · · · ) = f(Θ0,Ψ0)

+ εΘ1∂1f(Θ0,Ψ0)+

εΨ1∂2f(Θ0,Ψ0) + · · ·(3.26)

where

A0 := − ~∇y · (L(y)∇y), (3.27)

A1 := ~∇y · (P (y)− L(y)∇x)− ~∇x · (L(y)∇y), (3.28)

A2 := ~∇x · (P (y)− L(y)∇x). (3.29)

Now, equating the terms with the same powers of ε leads us to the following threelowest order equations:

A0Θ0 = 0 (ε−2 term), (3.30)

A0Θ1 + A1Θ0 = 0 (ε−1 terms), (3.31)

∂

∂τΘ0 + A0Θ2 + A1Θ1 + A2Θ0 = f(Ψ0,Θ0) (ε0 terms). (3.32)

22

(3.30), (3.31) and (3.32) are posed in Ω. Likewise, we expand the boundary conditioncorresponding to (3.10) and obtain the following expressions:

−n · L(y)(∇x +1

ε∇y)(Θ

0 + εΘ1 + ε2Θ2 +O(ε2)) = 0 (3.33)

from which we derive:

−n · L(y)∇yΘ0 = 0 (ε−1 term), (3.34)

−n · L(y)(∇xΘ0 +∇yΘ

1) = 0 (ε0 term), (3.35)

−n · L(y)(∇xΘ1 +∇yΘ

2) = 0 (ε1 term). (3.36)

(3.30), (3.31), and (3.32) together with (3.34), (3.35) and (3.36) suggest solvinga succession of differential problems PΘ

−j, corresponding to each ε−j term for j ∈0, 1, 2. To this end, we pose the following problems:

PΘ−2

− ~∇y · (L(y)∇yΘ

0) = 0 in Y

− [n · L(y)∇yΘ0] = 0 on ∂Ys,

Θ0 is Y-periodic.

(3.37)

This implies that Θ0 does not depend on y. The only periodic solution satisfying(3.37) is that Θ0 must be a constant with x and τ as parameters, i.e.

Θ0(x, y, τ) = Θ0(x, τ). (3.38)

Taking into account (3.38), we find the corresponding problem for ε−1 :

PΘ−1

− ~∇y · (L(y)(∇xΘ

0 +∇yΘ1)) +∇y · P (y)Θ0 = 0 in Y ,

− [n · L(y)(∇xΘ0 +∇yΘ

1)] = 0 on ∂Ys,

Θ1 is Y-periodic.

(3.39)

Since (3.39) is linear, we expect that Θ1 is a linear function of ∇xΘ0. Thus, we

compute Θ1 in terms of the gradient of Θ0 :

Θ1(x, y, τ) :=2∑j=1

∂Θ0

∂xjwj(y) =

∂Θ0

∂x1

w1(y) +∂Θ0

∂x2

w2(y), (3.40)

where wj(y)j ∈ 1, 2 are the so-called cell functions.

It is clearly pointed out in [12], [14], that Θ1 is defined up to an additive func-tion Θ1(x, τ), depending only on x and τ . This does not matter since only itsgradient w.r.t y arises in the homogenization equation (3.43).

Also, we recall the definition of the gradient of a scalar field:

∇xΘ0 =

2∑j=1

∂Θ0

∂xj~ej =

∂Θ0

∂x1

e1 +∂Θ0

∂x2

e2, (3.41)

23

where (~ej)j=1,2, is the canonical basis in R2.

We apply (3.40) and (3.41) in (3.39) and due to linearity, we solve the follow-ing problem:Find w = (w1, w2) ∈ W (Y ) satisfying

PΘj

− ~∇y · (L(y)(∇ywj(y) + ~ej)) = 0 in Y ,

− [n · L(y)(∇ywj(y) + ~ej)] = 0 on ∂Ys,

wj(y) is Y-periodic,

(3.42)

where the term ∇yP (y)Θ0 in (3.39) vanishes due to the fact that P (y) is divergencefree, i.e. ∇ · P (y) = 0.Note that for all j ∈ 1, 2, wj(y) ∈ W (Y ) = v ∈ H1(Y ) : v is Y-periodic;

1|Y |

∫Yvdy = 0 the function w = (w1, w2) is the unique solution of the so-called

local or cell problems (for more details, see [14], [15] e.g.).

Finally, we solve the problem for ε0 taking into account (3.38), as well as thesolutions of (3.42), and thus (3.39).

PΘ0

∂τΘ

0 − ~∇y · (L(y)(∇xΘ1 +∇yΘ

2)− P (y)Θ1)

− ~∇x · (L(y)(∇xΘ0 +∇yΘ

1)− P (y)Θ0) = f(Ψ0,Θ0) in Y,

− [n · L(y)(∇xΘ1 +∇yΘ

2)] = 0 on ∂Ys,

Θ2 is Y-periodic.

(3.43)

Now, we obtain the homogenized equation from (3.43) by averaging it with respectto y, and inserting the definition of Θ1 given in (3.40) as described below:

∂Θ0

∂τ− 1

|Y |

∫Y

~∇y · L(y)(∇xΘ1 +∇yΘ

2)dy︸ ︷︷ ︸T1

+1

|Y |

∫Y

~∇yP (y)Θ1dy︸ ︷︷ ︸T2

1

|Y |

∫Y

− ~∇x · (L(y)∇xΘ0 +∇yΘ

1)dy︸ ︷︷ ︸T3

+1

|Y |

∫Y

~∇xP (y)Θ0dy︸ ︷︷ ︸T4

=1

|Y |

∫Y

f(Ψ0,Θ0)dy︸ ︷︷ ︸T5

(3.44)

We solve the integrals labeled T1, T2, · · · , T5 as follows:

For T1 :

1

|Y |

∫Y

~∇y · L(y)(∇xΘ1 +∇yΘ

2)dy =1

|Y |

∫∂Ys

−n · L(y)(∇xΘ1 +∇yΘ

2)︸ ︷︷ ︸=0 on ∂Ys

dS(y)

− 1

|Y |

∫∂Y

n · L(y)(∇xΘ1 +∇yΘ

2)dS(y)︸ ︷︷ ︸=0 by periodicity of Y

24

This implies that the integral T1 = 0.

For T2 :

1

|Y |

∫Y

∇yP (y)Θ1dy =1

|Y |

∫Y

2∑j=1

∂Θ0

∂xj∇y(P (y)wj(y))dy

⇒ 1

|Y |

2∑j=1

∂Θ0

∂xj

∫Y

∇yP (y)wj(y)dy =1

|Y |

2∑j=1

∂Θ0

∂xj

∫∂Y

n · (P (y)wj(y))dS(y)

= 0 (by the periodicity of P (y)wj(y)).

For T4 :

1

|Y |

∫Y

∇xP (y)Θ0dy = ∇xΘ0 1

|Y |

∫Y

P (y)dy (since Θ0 is y-independent) (3.45)

= P∇xΘ0, (3.46)

where

P :=1

|Y |

∫Y

P (y)dy.

Similarly, the fact that T5 = f(Ψ0,Θ0) follows from the fact that Ψ0 and Θ0 arey-independent.

Finally, we have for T3 :

− 1

|Y |

∫Y

~∇x · (L(y)(∇xΘ0 +∇yΘ

1))dy =

− 1

|Y |

∫Y

[~∇x ·

(L(y)∇y

(∂Θ0

∂x1

w1 +∂Θ0

∂x2

w2

))+ ~∇x ·

(L(y)

∂Θ0

∂x1

e1 +∂Θ0

∂x2

e2

)]dy.

Collecting conveniently the terms and since that the differential operator ∇ com-mutes with the integral operator

∫, we obtain:

− ~∇x ·[∂Θ0

∂x1

1

|Y |

∫Y

(L(y)∇yw1 + L(y)e1)dy +∂Θ0

∂x2

1

|Y |

∫Y

(L(y)∇yw2 + e2)dy

]= −∇x · (A(w)∇xΘ

0).

(3.47)

where, on expanding the integral in (3.47), we obtain the matrix, A(w) as

A(w) =

1|Y |

∫Y

(L(y)∂w1

∂y1+ L(y))dy 1

|Y |

∫Y

(L(y)∂w2

∂y1)dy

1|Y |

∫Y

(L(y)∂w1

∂y2)dy 1

|Y |

∫Y

(L(y)∂w2

∂y2+ L(y))dy

(3.48)

25

where w = (w1, w2).

Similarly, we follow the same procedure as above to deal with (3.9).

∂

∂τ(Ψ0 + εΨ1 + · · · ) + (~∇x +

1

ε~∇y) · (P (y)− ((∇x +

1

ε∇y))(Ψ

0 + εΨ1 + · · · ))

= g(Θ0,Ψ0) + εΘ1∂1g(Θ0,Ψ0) + εΨ1∂2g(Θ0,Ψ0) + · · ·(3.49)

= (ε−2B0 + ε−1B1 + ε0B2)(Ψ0 + εΨ1 + · · · ) = g(Θ0,Ψ0)

+ εΘ1∂1f(Θ0,Ψ0) + εΨ1∂2g(Θ0,Ψ0) + · · ·(3.50)

where

B0 := − ~∇y · ∇y, (3.51)

B1 := ~∇y · (P (y)−∇x)− ( ~∇x · ∇y), (3.52)

B2 := ~∇x · (P (y)−∇x). (3.53)

Now, equating in powers of ε leads us to the following three lowest order equations:

B0Ψ0 = 0 (ε−2 term), (3.54)

B0Ψ1 +B1Ψ0 = 0 (ε−1 terms), (3.55)

∂

∂τΨ0 +B0Ψ2 +B1Ψ1 +B2Ψ0 = g(Ψ0,Θ0) (ε0 terms). (3.56)

Similarly, we expand the boundary conditions corresponding to (3.9) to obtain:

−n · (∇x +1

ε∇y)(Ψ

0 + εΨ1 + ε2Ψ2 + · · · )− n · ∇yΨ0 = 0 (ε−1 term), (3.57)

−n · (∇xΨ0 +∇yΨ

1) = 0 (ε0 term), (3.58)

−n · (∇xΨ1 +∇yΨ

2) = 0 (ε1 term). (3.59)

From the information provided in (3.54), (3.55), (3.56) and (3.57), we construct asequence of problems as follows:

PΨ−2

− ~∇y · (∇yΨ

0) = 0 in Yg

−n · (∇yΨ0) = 0 on ∂Ys,

Ψ0 is Y-periodic.

(3.60)

(3.60) indicates that Ψ0 does not depend on y. The only periodic solution satisfying(3.60) is that Ψ0 must be a constant in y and having x and τ as parameters, i.e.

Ψ0(x, y, τ) = Ψ0(x, τ) for all x ∈ Ω and τ ≥ 0. (3.61)

26

Taking into account (3.61), we give the corresponding problem for ε−1 :

PΨ−1

− ~∇y · (∇xΨ

0 +∇yΨ1) + ~∇yP (y)Ψ0 = 0 in Yg,

−n · (∇xΨ0 +∇yΨ

1) = 0 on ∂Ys,

Ψ1 is Y−periodic.

(3.62)

Following the same argument as in the case of Θ1, we compute Ψ1 in terms of thegradient of Ψ0 :

Ψ1(x, y, τ) =2∑j=1

∂Ψ0

∂xjωj(y) =

∂Ψ0(x, τ)

∂x1

ω1(y) +∂Ψ0(x, τ)

∂x2

ω2(y)

for all x ∈ Ω, y ∈ Y and τ ≥ 0

(3.63)

Also, we use the following definition of a gradient:

∇xΨ0 =

2∑j=1

∂Ψ0

∂xj~ej =

∂Ψ0

∂x1

e1 +∂Ψ0

∂x2

e2. (3.64)

We apply (3.63) and (3.64) in (3.62) and due to linearity of the problem, we obtainthe following cell problem: Find

ωj(y) = (ω1, ω2) ∈ W (Y )

such that the following BVP is solvable, viz.

PΨj

− ~∇y · (∇yωj(y) + ~ej) = 0 in Yg,

−n · (∇yωj(y) + ~ej) = 0 on ∂Ys,

ωj(y) is Y-periodic.

(3.65)

One easily sees that ωj(y) is the unique solution of cell problems. A simple calcu-lation shows that the term ∇y · (P (y)Ψ0) in (3.62) vanishes due to ∇y · P (y) = 0Finally, we solve the problem obtained for ε0 taking into account (3.61), as well asthe solutions of (3.65), and thus (3.62). We get

PΨ0

∂τΨ

0 − ~∇y · ((∇xΨ1 +∇yΨ

2))− ~∇x · ((∇xΨ0 +∇yΨ

1))

+∇x(P (y)Ψ0) +∇y(P (y)Ψ1) = g(Ψ0,Θ0) in Yg,

−n · (∇xΨ1 +∇yΨ

2) = 0 on ∂Ys,

Ψ2 is Y-periodic.

(3.66)

Averaging (3.66) over Y , we have:

∂Ψ0

∂τ− 1

|Y |

∫Yg

~∇y · (∇xΨ1 +∇yΨ

2)dy︸ ︷︷ ︸S1

+1

|Y |

∫Yg

~∇yP (y)Ψ1dy︸ ︷︷ ︸S2

1

|Y |

∫Yg

− ~∇x · (∇xΨ0 +∇yΨ

1)dy︸ ︷︷ ︸S3

+1

|Y |

∫Yg

~∇xP (y)Ψ0dy︸ ︷︷ ︸S4

=1

|Y |

∫Yg

g(Ψ0,Θ0)dy︸ ︷︷ ︸S5

(3.67)

27

As before, we solve the integrals labeled S1, S2, · · · , S5 as follows:

For S1 :

1

|Y |

∫Yg

~∇y · (∇xΘ1 +∇yΨ

2)dy =1

|Y |

∫∂Ys

−n · (∇xΘ1 +∇yΘ

2)︸ ︷︷ ︸=0 on ∂Ys

dS(y)

− 1

|Y |

∫∂Y

n · (∇xΨ1 +∇yΨ

2)dS(y)︸ ︷︷ ︸=0 by periodicity of Y

(3.68)

This implies that S1 = 0.

For S2 :

1

|Y |

∫Yg

∇yP (y)Ψ1dy =1

|Y |

∫Yg

2∑j=1

∂Ψ0

∂xj∇y(P (y)ωj(y))dy

This leads to1

|Y |

2∑j=1

∂Ψ0

∂xj

∫Yg

∇yP (y)ωj(y)dy =1

|Y |

2∑j=1

∂Ψ0

∂xj

∫∂Y

n · (P (y)ωj(y))dS(y)

= 0 (by the periodicity of P (y)ωj(y)).

For S4 :

1

|Y |

∫Yg

∇xP (y)Ψ0dy = ∇xΨ0 1

|Y |

∫Yg

P (y)dy (since Ψ0 is y-independent)

= P∇xΨ0,

(3.69)

where

P :=1

|Y |

∫Yg

P (y)dy.

Similarly, S5 = |Yg ||Y | g(Ψ0,Θ0) follows from the fact that Ψ0 and Θ0 are y-independent.

The quantity |Yg ||Y | can be interpreted as the medium porosity, i.e. the area of the

gas part of the basic cell to the total area of the cell.

Finally, we have for S3 :

− 1

|Y |

∫Yg

~∇x · (∇xΨ0 +∇yΨ

1)dy =

− 1

|Y |

∫Yg

[~∇x ·

(∇y

(∂Ψ0

∂x1

ω1 +∂Ψ0

∂x2

ω2

))+ ~∇x ·

(∂Ψ0

∂x1

e1 +∂Ψ0

∂x2

e2

)]dy.

−~∇x ·

[∂Ψ0

∂x1

1

|Y |

∫Yg

(∇yω1 + e1)dy +∂Ψ0

∂x2

1

|Y |

∫Yg

(∇yω2 + e2)dy

]= −~∇x · (B(ω)∇xΨ

0).

(3.70)

28

where, on expanding the integral in (3.70), we obtain the matrix B(ω) defined by

B(ω) :=

1|Y |

∫Yg

(∂ω1

∂y1+ 1)dy 1

|Y |

∫Yg

∂ω2

∂y1dy

1|Y |

∫Yg

∂ω1

∂y2dy 1

|Y |

∫Yg

(∂ω2

∂y2+ 1)dy

. (3.71)

3.4 The macroscopic equations and effective co-

efficients

Figure 3.2: Left: Heterogeneous medium; Right: Homogeneous representation ofthe perforated domain.

Summarizing, we obtain the following effective (upscaled) equations for the requiredmacroscopic quantities, namely:

Θ(x, τ) = Θ0(x, τ),

Ψ(x, τ) = Ψ0(x, τ),

R(x, τ) = R0(x, τ),

and

∂Θ

∂τ+ P∇Θ−∇ ·

(A∇Θ

)= f(Ψ,Θ) in Ω× (0, T ), (3.72)

∂Ψ

∂τ+ P∇Ψ−∇ ·

(B∇Ψ

)= g(Ψ,Θ) in Ω× (0, T ), (3.73)

∂R(x, τ)

∂τ= h(Ψ,Θ) in (0,T) and a.e x ∈ Ω, (3.74)

where

A(w) :=

1|Y |

∫Y

(L(y)∂w1

∂y1+ L(y))dy 1

|Y |

∫Y

(L(y)∂w2

∂y1)dy

1|Y |

∫Y

(L(y)∂w1

∂y2)dy 1

|Y |

∫Y

(L(y)∂w2

∂y2+ L(y))dy

, (3.75)

29

B(ω) :=

1|Y |

∫Yg

(∂ω1

∂y1+ 1)dy 1

|Y |

∫Yg

∂ω2

∂y1dy

1|Y |

∫Yg

∂ω1

∂y2dy 1

|Y |

∫Yg

(∂ω2

∂y2+ 1)dy

, (3.76)

and

P :=1

|Y |

∫Yg

P (y)dy, (3.77)

with Θ = 0

Ψ = 1

R = 0

at ∂Ω ∩ x = 0, (3.78)

∂Ψ

∂x= 0 at ∂Ω ∩ x = L, (3.79)

Θ(L, t) = Θ∗H(

ε

tD− τ), (3.80)

Θ(x, 0) = 0

Ψ(x, 0) = 1

R = 1

for 0 < (x, y) < L. (3.81)

It turns out that the effective coefficients describe a homogeneous medium that isindependent of the right-hand side terms f, g and h as well as of the prescribedboundary and initial conditions.

Furthermore, the application of (3.21) in (3.9), (3.10) and (3.11) leads to a rig-orous deductive procedure for obtaining the macroscopic equations (3.72), (3.73)and (3.74). These equations can then be used for predicting the global behavior ofthe heterogeneous medium.

Thus, homogenization process gives the passage from a microscopic descriptionto a macroscopic description of the model problems (3.9), (3.10) and (3.11) [16].Intuitively, we illustrate in Fig. 3.2 (left) the heterogeneous material (left) that ishomogenized, leading to its homogeneous representation in Fig 3.2, by the applica-tion of the homogenization method.

30

Chapter 4

Numerical multiscalehomogenization approach

In this chapter, we wish to explore numerically the solution of the system of equa-tions obtained from the multiscale structure discussed in chapter 2 and chapter 3.

4.1 Computation strategy for the homogenized

problem

In order to compute the solution to the homogenized problem numerically, we con-sider the following three-steps procedure:

Step I: Computation of the cell problems

For the purpose of the discussion in this section, we write the cell problems describedin (3.42) and (3.65), in the following form:

− ∂

∂yi

(Lik(y)

∂wj∂yk

)=

∂

∂yiLij(y), wj − Y periodic, (4.1)

− ∂

∂yi

(∂ωj∂yk

)=

∂

∂yiej = 0, ωj − Y periodic. (4.2)

We use COMSOL Multiphysics [17] to solve (4.1) and (4.2) posed in the unit cellsdepicted in Fig. 4.1 and Fig. 4.2 respectively. We essentially use the Finite Ele-ment Method (FEM), since this method is well suited for problems with complexgeometries.

Looking at the geometry depicted in Fig. 4.1, we see that the unit cell has twodisjoint subdomains Yg and Ys, where the subdomains have different material coef-ficients given by Lij.

31

The coefficient term Lij(y), y ∈ Y , is defined as follows:

Lij(y) :=

Lgij, y ∈ YgLsij, y ∈ Ys.

(4.3)

Consequently, we define (4.1) in both Yg and Ys but with their corresponding ma-terial coefficients. In particular, we define (4.2) in Yg for the geometry in Fig. 4.2

Further, we compute Lij for the two materials and we assume, in the present study,that these coefficients are isotropic in the sense that L`ij = L`ijδij for ` ∈ g, s.

Figure 4.1: Unit cell with two subdomains Yg and Ys.

Figure 4.2: Unit cell with hole: one subdomain.

Step II: Computation of the homogenized (effective) coeffi-cients

We insert the solutions of the cell problems into the following equations of thehomogenized tensors-A and B:

aij :=⟨Lij(y) + Lik(y)

∂wj∂yk

⟩=

1

|Y |

∫Y

(Lij(y) + Lik(y)∂wj∂yk

)dy, (4.4)

32

bij :=⟨δij + δik

∂ωj∂yk

⟩=

1

|Y |

∫Y

(δij + δik∂ωj∂yk

)dy. (4.5)

P :=⟨Pk

⟩=

1

|Y |

∫Yg

Pkdy. (4.6)

The homogenized coefficient are thus calculated by integration over the unit cell(precisely with respect to the subdomain(s)).

Step III: Solution of the homogenized problem

With the homogenized coefficients computed in step II, the upscaled model (3.72)-(3.74) can then be solved numerically provided the numerical approximation of thesolutions to (4.1)-(4.2) are available.

4.2 Numerical computation of the homogenized

coefficients

4.2.1 Weak formulation of the cell problem

The homogenized coefficients are given by (4.4) and (4.5). In order to computethese coefficients, we need to solve the cell problems (4.1) and (4.2) respectively forwj and ωj. We then compute the integrals in (4.4) and (4.5).

To get the weak formulation of (4.1), we note that

− ∂

∂yi

(Lik

∂

∂yk

)yj = − ∂

∂yiLikδkj = − ∂

∂yiLij (4.7)

− ∂

∂yi

(∂

∂yk

)ej = 0 (4.8)

so that (4.1) and (4.2) can be written as:

− ∂

∂yi

(Lik

∂

∂yk(wj + yj)

)= 0, wj is Y − periodic. (4.9)

− ∂

∂yi

(∂

∂yk(wj + ej)

)= 0, ωj is Y − periodic. (4.10)

Now, we recall the space W(Y).

W (Y ) = ψ ∈ H1(Y ) : ψ is Y-periodic. (4.11)

Assume we have solutions ujk := wj + yj and vj := ωj + ej to (4.9) and (4.10)respectively, then (4.9) and (4.10) can be written as

− ∂

∂yi

(Lik

∂u

∂yk

)= 0, (4.12)

− ∂

∂yi

(∂v

∂yk

)= 0. (4.13)

33

Multiplying (4.12) and (4.13) by ψ ∈ W (Y ) and integrating over Y, we have∫Y

− ∂

∂yi

(Lik

∂u

∂yk

)ψdy =

∫Y

Lik∂u

∂yk

∂ψ

∂yidy −

∫∂Y

Lik∂u

∂ykψnidS (4.14)∫

Yg

− ∂

∂yi

(∂v

∂yk

)ψdy =

∫Y

∂v

∂yk

∂ψ

∂yidy −

∫∂Y

Lik∂v

∂ykψnidS (4.15)

by Green’s theorem. The boundary integral terms in (4.14) and (4.15) vanish dueto Gauss’ theorem and periodicity. Hence we have for all ψ ∈ W (Y ), the identities∫

Y

Lik∂u

∂yk

∂ψ

∂yidy = 0 (4.16)∫

Y

∂v

∂yk

∂ψ

∂yidy = 0. (4.17)

We define the following bilinear form for (4.16) and (4.17) respectively:

aY (u, ψ) :=

∫Y

Lik∂u

∂yk

∂ψ

∂yidy, (4.18)

bY (v, ψ) :=

∫Y

∂v

∂yk

∂ψ

∂yidy for all ψ ∈ W (Y ). (4.19)

Thus, the weak formulations of (4.1) and (4.2) are then:Find wj, ωj ∈ W (Y ) such that

aY ((wj + yj), ψ) = 0 for all ψ ∈ W (Y ), (4.20)

aY ((ωj + ej), ψ) = 0 for all ψ ∈ W (Y ), (4.21)

or alternatively,find wj, ωj ∈ W (Y ) such that

aY (wj, ψ) = F jY (ψ), (4.22)

aY (ωj, ψ) = GjY (ψ) for all ψ ∈ W (Y ), (4.23)

where

FY (ψ) :=

∫Y

Lij(y)∂ψ

∂yidy, (4.24)

GY (ψ) :=

∫Yg

ej∂ψ

∂yidy. (4.25)

4.2.2 Finite element approximation to the solutions of thecell problems

In order to obtain a FEM formulation of (4.20) or (4.22), we will use the Galerkinmethod [18] to describe the idea behind our later implementation of the methodin COMSOL Multiphysis solver. We start by making a subdivision of the unit cell

34

Y in N triangles Kn and M nodes Ni according to [18]. Let Wh(Y ) be a finite-dimensional subspace of W (Y ) of dimension π. We introduce the finite elementbasis functions ϕk(y) ∈ Wh(Y ), k = 1, 2, · · · , π, where π denotes the number ofdegrees of freedom.

Generally, π is seldom equal M , because in the case of Y-periodicity, nodes atopposite positions on the boundary of the Y -cell must correspond to the same de-grees of freedom. To relate the nodes to the degrees of freedom, we let Q be amapping from node Ni to the corresponding degree of freedom l, that is l = Q(Ni).We also choose the basis functions to be piecewise linear and define them by

ϕk(Ni) = δkl, where l = Q(Ni) and k = 1, 2, · · · , π. (4.26)

We assume the existence of an approximate solution to (4.22) of the form

whj (y) =π∑k=1

ξjkϕk(y) for all y ∈ Y, (4.27)

where ξjk are constants to be determined. This is the so-called Galerkin ansatz.These constants correspond to the approximation of whj at the nodal points. Letψ ∈ Wh(Y ) be such that ψ has a unique representation

ψ =π∑i=1

ηiϕi. (4.28)

We now formulate the discrete analogue to the problem (4.22):Find whj ∈ Wh(Y ) such that

aY (whj , ψ) = F jY (ψ) for all ψ ∈ Wh(Y ). (4.29)

Because (4.29) must be valid for all ψ ∈ Wh(Y ), it makes sense to use ψ =ϕi, for i = 1, 2, · · · , π as test functions for the finite dimensional approximation.With ψ = ϕi(y), (4.29) becomes

π∑k=1

Aikξjk = bji , i = 1, 2, · · · , π; j = 1, 2, (4.30)

or, in matrix form,

Aξj = bj, (4.31)

where

Aik = aY (ϕk, ϕi) =

∫Y

Lij∂ϕk∂yj

∂ϕi∂yi

dy (4.32)

and

bji = F jY (ϕi) =

∫Y

Lij(y)∂ϕi∂yi

dy for all ϕi ∈ Wh(Y ). (4.33)

35

The finite element approximation of the second problem (4.23) follows in a similarway.

We can now compute the approximate homogenized coefficients by inserting (4.27)in (4.4) to get

aij =1

|Y |

∫Y

(Lij + Lik

π∑l=1

ξjl∂ϕl∂yj

)dy

=1

|Y |

∫Y

Lijdy +1

|Y |

π∑l=1

ξjl

∫Y

∂ϕl∂yj

dy.

(4.34)

We note that

aY (ϕl, yi) =

∫Y

Lmj∂ϕl∂yj

∂yi∂ym

dy (4.35)

because ∂yi

∂ym= δim. Hence, we obtain

aij =1

|Y |

∫Y

Lijdy +1

|Y |

π∑l=1

aY (ϕl, yi)ξjl (4.36)

Suppose Lij is symmetric, (4.24) and (4.33) gives∫Y

aik∂ϕl∂yk

dy =

∫Y

aki∂ϕl∂yk

dy = F iY (ϕl) = bil (4.37)

so that

aij =1

|Y |

∫Y

(Lijdy +1

|Y |

π∑l=1

bilξjl . (4.38)

In order to implement the FEM discretization of subsection 4.2.1 in COMSOLMultiphysics, we use the conservative formulation of the cell problems (3.42) and(3.65). These equations are given in matrix form as:

∇y ·([L11 00 L22

] [∂1w1

∂2w1

]+

[L11

0

])= 0 (4.39)

∇y ·([L11 00 L22

] [∂1w2

∂2w2

]+

[0L11

])= 0 (4.40)

(4.39) and (4.40) corresponds to the following PDE Coefficient form applicationmode of COMSOL:

∇ · (−c∇u− αu+ γ) + β · ∇u+ au = f in Y, (4.41)

n · (c∇u− αu+ γ) = hTµ− qu+ g on ∂Ys (4.42)

with the following periodic boundary conditions u(yi) = u(yi+Y ) on ∂Y for i ∈ 1, 2.The model problems (4.39) and (4.40) are then obtained by setting a, α, f, β, g, q

36

Figure 4.3: Sample geometry of the unit cell

and µ to zero. We then implement the steps outlined in section 4.1 as follows:

Create the geometry and insert equations with corresponding materialcoefficients:The computational geometry is given in Fig. 4.3, where we have inserted, in eachof the subdomains, an equation with its corresponding material coefficient.

Prescribe the interior and periodic conditions and solve the problem:For the cell problems, we prescribe periodic boundary conditions on the outerboundaries of Y and use the Neumann condition on the interior boundary of ∂Ys.We solve the problems and obtain the solutions for the Y periodic cell functionsas depicted in Fig. 4.5 and Fig. 4.7. Fig. 4.5 shows the periodic solution of thetemperature in a unit cell, while in Fig. 4.7, we obtain the periodic solution forconcentration of oxygen in the part of the unit cell (see Fig. 4.2) Similar qualita-

(a) (b)

Figure 4.4: Solution of cell problems, (a): w1(xε) and (b): w2(x

ε) with x ∈ Ω.

tive results as depicted in Fig. 4.5–Fig. 4.7 can be seen, for example, in [19], [20],and [21].

4.2.3 Existence and uniqueness of weak solutions to the cellproblems

In this section, we introduce some important definitions and lemmas. These will beused in the subsequent section to show the existence and uniqueness of weak solu-tions of the cell problems. Finally, some properties of the homogenized coefficientse.g. like symmetricity are treated in subsequent sections.

37

(c) (d)

Figure 4.5: Solution of cell problems, (c): w1(xε) and (d): w2(x

ε) with x ∈ Ω.

3D visualization of the solution to the cell problems for temperature.

(e) (f)

Figure 4.6: Solution of cell problem (e): ω1(xε) and (f): ω2(x

ε) with x ∈ Ω.

(g) (h)

Figure 4.7: Solution of cell problem (g): ω1(xε) and (h): ω2(x

ε) with x ∈ Ω.

3D visualization of the solution to the cell problems for concentration.

Definition 4.2.1 Let X be a real Hilbert space. A mapping

a(·, ·) : X ×X → R

is called a bilinear form on X if

i. a(x1 + x2, y) = a(x1, y) + a(x2, y) for all x1, x2, y ∈ X ,

ii. a(αx, y) = αa(x, y) for all x, y ∈ X and α ∈ R,

38

iii. a(x, y1 + y2) = a(x, y1) + a(x, y2) for all x, y1, y2 ∈ X ,

iv. a(x, βy) = βa(x, y) for all x, y ∈ X and β ∈ R.

Definition 4.2.2 Let a(·, ·) be a bilinear form on the real Hilbert space X. We saythat a(·, ·) is

i. bounded if there exists a constant γ > 0 such that |a(x, y)| ≤ γ‖x‖X‖y‖X for all x, y ∈ X,

ii. symmetric if a(x, y) = a(y, x) for all x, y ∈ X,

iii. coercive if there exists a constant α > 0 such that a(x, x) ≥ α‖x‖2X for all x ∈ X.

Lemma 4.2.3 (Poincare inequality): Let Ω be an open and bounded set in Rn

which has the Lipschitz property. Then

‖u‖2H1(Ω)/R ≤

∫Ω

∂u

∂xi

∂u

∂xi, (4.43)

for each u ∈ H1(Ω)/R and some constant C = C(Ω).Proof. See e.g. [22].

Lemma 4.2.4 (Lax-Milgram): Let X be a real Hilbert space. If the bilinear forma(·, ·) : X×X → R is bounded and coercive and if, in addition, the linear functionalf : X → R is bounded, then there exists a unique element x ∈ X such that a(x, y) =(f, y) for all y ∈ X.

Proof. See e.g. [23], p.118-119.

Lemma 4.2.5 Let F be square integrable over Y and consider the boundary valueproblem

A0Φ = F in a unit Y-cell, Φ is Y-periodic. (4.44)

where A0 is similar to the operator (3.30). The following holds true:

i. There exists a weak Y-periodic solution of (4.2.5) if and only if 〈F 〉 = 0.

ii. If there exists a weak Y-periodic solution of (4.2.5), then it is unique up to anadditive constant.

Proof. We recall the Hilbert space W (Y ) and define on it the bilinear form

aY (Φ, ψ) =

∫Y

Lij∂Φ

∂yj

∂ψ

∂yidy for all ψ ∈ W (Y )

and the linear functional

(F, ψ)Y =

∫Y

Fψdy. (4.45)

39

Then the weak formulation of (4.44) is:Find Φ ∈ W (Y ) such that

aY (Φ, ψ) = (F, ψ)Y for all ψ ∈ W (Y ). (4.46)

Assume that Φ is a solution and choose ψ ≡ 1. Then

0 = aY (Φ, 1) = (F, 1)Y = |Y |〈F 〉, (4.47)

which shows that 〈F 〉 = 0. Conversely, assume that 〈F 〉 = 0. It is clear that thebilinear form aY (Φ, ψ) is not coercive in W (Y ), since if ψ is a nonzero constantfunction, then aY (·, ·) = 0 but ‖ψ‖W (Y ) > 0. However, according to Lemma 4.2.3,it is coercive in W (Y ). Moreover, since Lij is bounded, it follows that aY (·, ·) isbounded in W (Y ). (F, ψ)Y is a well-bounded continuous linear functional on W (Y ),since if c is a constant, ∫

Y

Fcdy = c

∫Y

Fdy = c|Y |〈F 〉 = 0, (4.48)

and therefore

(F, ψ)Y =

∫Y

Fψdy =

∫Y

F (ψ − c)dy. (4.49)

Hence

|(F, ψ)Y | ≤ ‖F‖L2(Y ) infc∈R‖ψ − c‖L2(Y ) ≤ ‖F‖L2(Y )‖ψ‖W (Y ). (4.50)

Now, by Lemma 4.2.4, there exists a unique solution in W (Y ) to (4.46).

4.3 Further properties of the effective coefficients

This section is very much inspired by [13].

4.3.1 Symmetry and positive definiteness of the effectivetensors

Theorem 4.3.1 Let Lij be a symmetric and positively definite tensor. Then theassociated effective tensor aij, defined as in (3.75), is symmetric and positive defi-nite.

Proof. We consider the cell problem (3.42) and recall the weak formulation (4.20):Find wj ∈ W (Y ) such that

aY ((wj + yj), ψ) = 0 for all ψ ∈ W (Y ) (4.51)

40

where W (Y ) and aY (·, ·) have their usual meaning. According to (4.4), we have

aij =1

|Y |

∫Y

(Lij(y) + Lik(y)∂wj∂yk

)dy =1

|Y |

∫Y

(Lmkδkjδim + Lmk∂wj∂yk

)dy =

=1

|Y |

∫Y

(Lmk∂yj∂yk

∂yi∂ym

+ Lmk∂wj∂yk

)dy

=1

|Y |

∫Y

Lmk(∂wj∂yk

+∂yj∂yk

)∂yi∂ym

dy =1

|Y |aY (wj + yj, yi),

(4.52)

which, by using (4.51) with ψ = wi, gives

aij = aij + 0 =1

|Y |aY (wj + yj, yi) +

1

|Y |aY (wj + yj, wi) =

=1

|Y |aY (wj + yj, wi + yi).

(4.53)

If Lik = Lki, then aY (u, v) = aY (v, u) and hence by (4.53), aij = aji, i.e., aij issymmetric.Now we consider aijξiξj for an arbitrary ξ ∈ Rn. Let ϑ := ξi(wi + yi). The positivedefiniteness of aij and (4.53) yield

1

|Y |aY (ϑ, ϑ) ≥ α

⟨ ∂ϑ∂yi

∂ϑ

∂yi

⟩≥ 0. (4.54)

Thus, aijξiξj ≥ 0 with equality if and only if ∂ϑ∂yi

= 0, that is, if and only if ϑ ≡ c

(constant), or equivalently, ξiyi = ξiwi − c. Here wi is Y -periodic so that equalityin (4.54) holds if and only if ξi = 0, thus the theorem is proved.

4.3.2 A comparison of computed coefficients with variousbounds

With the solutions of the cell problems computed, we compute the effective co-efficients from section 3.4(3.75)-(3.77) by evaluating the integrals. We compareafterwards the effective coefficients aij obtained with the different bounds that ex-ist. Let L1 and L2 denote the specific thermal coefficients of the gas and solid partsof the medium. Moreover, let m1 and m2 be the corresponding surface fractions ofthe gas and solid respectively.

Having obtained the effective coefficients a11, a22 and b11, b22, in two specified di-rections, from (3.75)-(3.77), we show that our estimates of the effective coefficientssatisfy the following inequalities between the harmonic and arithmetic means. Wefollow now the strategy presented in [13] Chapter 2, p. 16-18 and [20] p. 29-31.The inequality is given by

Mh ≤ aii ≤Ma, i ∈ 1, 2, (4.55)

where Mh denotes the harmonic mean

Mh :=1

m1

L1+ m2

L2

(4.56)

41

and Ma denotes the arithmetic mean

Ma := m1L1 +m2L2. (4.57)

However, there are stronger bounds, the so-called Hashin-Shtrikman bounds (seeWendt et al.[24]), which can be obtained for L1 < L2. The Hashin-Shtrikmanbounds are thus defined through the following relation:

a− := L1 +m2

1L2−L1

+ m1

2L1

, (4.58)

a+ := L2 +m1

1L1−L2

+ m2

2L2

(4.59)

such that

a− ≤ aii ≤ a+, i ∈ 1, 2. (4.60)

In Table 4.1, aii for i ∈ 1, 2 represents the diagonal entries of the homogenized

L1 0.01 0.721777L2 0.3 1.349831Mh 0.019451 0.94211Ma 0.15577 1.037475a− 0.027752 0.981312a+ 0.109482 0.996104aii 0.02954 0.9855

Table 4.1: Comparison of the homogenized coefficients with known bounds

matrix for the temperature field. The last two columns of Table 4.1 contain valuesobtained for different choices of the material coefficients L1 and L2. We compare theharmonic and arithmetic means with the effective coefficients given by aii, i ∈ 1, 2.We do the same using the lower a− and upper a+, Hashin-Shtrikman bounds. Inall cases, we see that the computed values satisfy the inequalities given in (4.55)and (4.60). Also, within the limits of discretization errors, the difference betweenthe computed effective coefficients are relatively small if compared with the meansor the Hashin-Shtrikman bounds.

4.4 Numerical computation of the macroscopic

and microscopic solutions

Having computed the homogenized coefficients as in subsection 4.3.2, we turn nowto the aspect of numerically evaluating the macroscopic model. Since the systemof equations given in (3.72)-(3.74) is convection-dominated for P 1, i.e. theconvection term, it is not efficient to use the standard finite element method for(3.72)-(3.74) due to stability issues that may arise. We use the streamline diffu-sion method, (see [25],[18] for details). In [18], the discretization of the transient

42

convection-diffusion equation using streamline diffusion is explained in detail.

Consequently, we adopt the theory explained in [18] and implement this method inCOMSOL Multiphysics (see [17] for more details) with the domain of interest asΩ = (0, 1)2. In particular, we use the Streamline-upwind Petrov/Galerkin method(SUPG). The homogenized matrix are given as follows:

(aij) =

[a11 a12

a21 a22

], (bij) =

[b11 b12

b21 b22

], (4.61)

and

Pj = (P1, 0), (4.62)

where a11 = a22 as in Table 4.1 and b11 = b22 = 0.349884. Also, we have aij = bij = 0for i 6= j. For the microscopic problem of section 1.2, we see that it is difficult tofind the exact solution of the original problem, hence we use the standard finiteelement solution, implemented in COMSOL, to replace the exact solutions Θε andΨε so that we can compare our the solution of the macroscopic problem with the1ideal exact solution. In COMSOL Multiphysics, we walk through the followingsteps:

Step I: Choose a representative physicsSince our model equations are of transient convection-diffusion (i.e. for the trans-port of oxygen) and conduction-convection (i.e. for heat transport) types, we choosein COMSOL, representative physics based on (3.72) and (3.73). Similarly, we dothe same for the microscopic equations (3.9) and (3.10)We customize the chosen multiphysics models by modifying its coefficients suitably.The reaction rate (3.74) for the macro models is implemented using the subdomainweak form PDE module of COMSOL. Finally, we select the SUPG from the sta-bilization tab in the subdomain dialog. We see that for the micro model, 2eachequation is prescribed appropriately on each of the periodic cells (see Fig. 4.8 (a)).

Step II: Create a geometryThe geometry of the macroscopic problem in consideration is relatively simple. Wecreate the domain Ω = (0, 1)2 as a unit square in COMSOL (see Fig. 4.8).We create the geometry of the microscopic problem is as follows: We begin with atypical unit cell as shown in Fig. 4.3. For any ε > 0, we rescale the unit cell (Fig.4.3) and generate a lattice of copies of cells that span the entire domain Ω = (0, 1)2

as described in section 1.2. For example, a typical microscopic domain consistingof 100 periodic cells is depicted in Fig. 4.8 (a).

Step III: Set the material properties i.e. setting all the constants that

1A solution that can be used in place of the actual solution.2The microscopic domain has two subdomains in each of its periodic cells. we prescribe equa-

tions related to either subdomain of each periodic cell.

43

(a) Geometry of the microscopic problem for 10x10 cells

(b) Geometry of the macroscopic problem

Figure 4.8: 2D Geometry of the model problems

appear in the PDEWith the chosen physics in step I, we customize the settings of the parameters ofthe model appropriately. First, we declare all constants and expressions that arepart of the model respectively in the constant and expression settings dialog boxes.We also enter their names correctly in the subdomain settings dialog box underthe physics menu. The reaction rate (3.11) for the micro model is implemented

44

using the boundary weak form PDE module of COMSOL. Here, we prescribe theequations at each of the interior circular boundaries of the geometry in Fig. 4.8 (a).Most importantly, we ensure continuity of fluxes across the interior boundaries ofthe micro model geometry Fig. 4.8 (a). Table 5.2 contains information of the listof parameters used.

Step IV: Set the boundary conditions and initial conditionsHere, we set the boundary conditions. For (3.72) and (3.73), the boundary condi-tions at the top/bottom are of thermal/diffusive insulation type, as stated earlierin chapter 2, while the side boundaries are either of Dirichlet or Neumann (i.e. forpurely convective outflow) type. The heat pulse maintained at one of the ends,specifically at x = 1, is to imitate the impulsive onset of combustion [6]. In COM-SOL, we implement this using the smoothed heaviside function, flc2hs (see [17] andTable 5.2). The boundary and initial conditions are respectively specified under theboundary settings and the subdomain settings in COMSOL.

Step V: Choose an element type and mesh the geometryFor the element type, we use the quadratic Lagrange elements with a triangularmesh as shown in Fig. 4.13. The mesh is created with a click on the mesh button.

Step VI: Choose a solver and solve for the unknownsFor the implementation here, we choose the direct linear solver (e.g. UMFPACK)and click the solve button.

Step VII: Post-process the results to find the information that is re-quiredTo postprocess the solution, we export the data to Matlab. The idea is to be ableto compute the error estimate between the microscopic problem with decreasingvalue of ε (i.e the scale parameter) and the macroscopic solution. We can also plotdifferent types of graph of the computed solutions.

4.5 Results and discussion

In this section, we discuss the results of our numerical simulations in 4.5.

4.5.1 Smoldering process in a microstructure

Smoldering phenomenon has been studied extensively and interpreted from a macro-scopic standpoint [3], [6]. It can as well be studied at a microscale level, wheremicroscopic descriptions are given to smoldering combustion process [26]. Here, weillustrate the numerical results of our simulation, giving insight into the microscopicbehavior of smoldering process of a paper sample.

For the plots depicted in Fig. 4.16 and Fig. 4.17, we have used the Peclet number inthe numerical simulation as a control parameter. At Pe = 10 (see Fig. 4.16 (left)),the smolder front spreads uniformly on each cell from the ignition line at x = 1,

45

(a) Mesh consists of 14261 elements.

(b) Mesh consists of 24042 elements.

Figure 4.9: 2D mesh of the model poblems.

and proceeding away from this line. We then cut the flux of oxygen to Pe = 8 andPe = 5 respectively as illustrated in Fig. 4.16 (right) and Fig. 4.17 (left). It isobserved that the front spreads towards the ignition line, and gradually generatesconcentration gradients and lateral diffusion currents across adjacent cells. (see [3]p.2817, Fig. 4.18 (right) and Fig.4.18 (left)) This observation is expected since itdescribes a gradual drop in the advective dominance over diffusion.

46

Figure 4.10: Qualitative comparison of the temperature distribution of a smolderedpaper. Left: Macroscopic solution Θh

0(Pe = 2); Right: Microscopic solution Θε(Pe =2, ε = 1

20).

Figure 4.11: 3D plot of the temperature distribution of a smoldered paper. Left: Macro-scopic solution Θh

0(Pe = 2); Right: Microscopic solution Θε(Pe = 2, ε = 120).

Figure 4.12: One dimensional plot of the temperature distribution of a smoldered paper.Left: Averaged solution Θh

0(Pe = 2) without oscillations; Right: Microscopic solutionΘε(Pe = 2, ε = 1

20) with oscillations.

47

Figure 4.13: Qualitative comparison of the concentration distribution on a smolderedpaper. Left: Macroscopic solution Ψh

0(Pe = 2); Right: Microscopic solution Ψε(Pe =2, ε = 1

20).

Figure 4.14: 3D plot of the concentration distribution on a smoldered paper. Left:Macroscopic solution Ψh

0(Pe = 2); Right: Microscopic solution Ψε(Pe = 2, ε = 120).

Figure 4.15: One dimensional plot of the concentration distribution on a smolderedpaper. Left: Averaged solution Ψh

0(Pe = 2) without oscillations; Right: Microscopicsolution Ψε(Pe = 2, ε = 1

20) with oscillations.

As the Peclet number is reduced further to Pe = 2, we notice an increase in lateral

48

Figure 4.16: Smoldering process of a paper. Left: numerical solution for Pe = 10; Right:numerical solution for Pe = 8.

Figure 4.17: Smoldering process of a paper. Left: numerical solution for Pe = 5; Right:numerical solution for Pe = 2.

diffusion currents that eventually spreads across the entire 3 reaction region with asinusoidal pattern ahead of the front.

4.5.2 One-dimensional comparison of results

We illustrate here a one-dimensional analysis of the results obtained from the mi-croscopic and macroscopic problems. These results were achieved by considering aset of points on the two-dimensional domain and computing the solutions at thesepoints.In Fig. 4.19, we see that the reaction term, which depends on the oxygen flux, is

a decreasing function. it decreases fastest at R1 close to the reaction region whichsuggest that the oxygen is continually consumed at this region. At R5, we observean almost steady flow with no much effect on the oxygen flux.

3The reaction region is a region where the transport and reaction mechanisms have the mostcritical influence. [26]

49

Diffusive flux (arrow) of oxygen on a temperature field at Pe = 2 and 49 cells.

Concentration gradient (arrow) of oxygen on a temperature field at Pe = 2 and 49 cells.

Figure 4.18: Numerical solution for Pe = 2 and 49 cells.

4.5.3 Error estimates in the FE approximations in the nu-merical simulations

In this subsection, we present the relative error estimates describing the quantitativebehavior of solutions between the FE approximations in the multiscale numericalhomogenization approach introduced in 4.5 and the direct computation of the microproblem. The direct computation provides an ideal exact solution. This is done byconsecutively increasing the number of periodic cells or decreasing the value of ε.

50

Macroscopic solution of reaction term arising in (3.74).

Microscopic solution of the reaction term arising in (3.11).

Figure 4.19: Numerical computation of the reaction term.

We will see later that as we make the value of ε smaller and smaller in the directcomputational approach, the computational time of the 4CPU increases.

The error estimate we wish to adopt here is called the discrete L2 norm and is

4An acronym for Central Processing Unit that represents the part of a computer (a micropro-cessor chip) that does most of the data processing

51

presented, for example, in [27]. It is defined as follows:

‖uh‖h =

(hd∑i

(uhi )2

) 12

(4.63)

where d is the dimension of the domain. We adapt this to our problem to approxi-mate the continuous L2 error estimate as follows:

‖Θε −Θh‖h =

(hd∑i

(Θε −Θhi )

2

) 12

(4.64)

where Θε is the FE approximation of the micro solution of the temperature fieldand Θh is the FE approximation of the homogenization solution of the tempera-ture. We obtain similar estimates for Ψ, the concentration. Obviously, we see thatthe estimate depends on any chosen ε. As mentioned in 4.4, step VII, we exportfrom COMSOL the solutions of the macro problem and of the micro problem fordecreasing ε.

However, looking back at the solutions of the macro and the micro problems, we seethat these solutions are obtained in an 5unstructured mesh. Therefore, to enable uscompare the solutions in the two problems, we map the solutions by linear interpo-lation, to a structured mesh of 500× 500 data points on the (x,y)-plane. We thenimplement (4.64) in Matlab for the relative error estimates. The error estimatesare summarized in table 4.2 and depicted in Fig. 4.20.

ε L2 error estimates in Θ1 3.672712

0.788614

0.272017

0.1492110

0.1138114

0.0940120

0.0790

ε L2 error estimates in Ψ1 0.315312

0.269114

0.212617

0.2053110

0.2045114

0.2031120

0.2061

Table 4.2: Left: Error estimates in the FE approximation of ‖Θε−Θh‖h for variousε; Right: Error estimates in the FE approximation of ‖Ψε −Ψh‖h for various ε. Inall cases, Pe = 10.

In Fig. 4.20 (left), we see that the error estimates for the temperature drops expo-nentially with decreasing ε. This shows that the solution gets better with decreasingε. However, the estimate for the concentration behaves poorly. This is due to thewrong implementation of the boundary reactive terms in the microscopic geome-try. The results is also shown in Table 4.2. From Fig. 4.21 and Table 4.3, itis now evident that the computational time of the direct implementation of the

5By unstructured mesh, we mean the values of the solutions are obtained in a randomly gen-erated space variables

52

Error estimates in the FE approximation of ‖Θε −Θh‖h for various ε

Error estimates in the FE approximation of ‖Ψε −Ψh‖h for various ε

Figure 4.20: Error estimates in the temperature and concentration. In all cases,Pe = 10. The plots correspond to values in Table 4.2.

ε No. of Mesh No. of solved d.o.f Computational time (s)1 7577 31196 39.31212

7838 32578 152.14414

8560 35778 198.04717

11879 50412 306.801110

24042 101610 597.398114

43120 182674 1399.9120

52000 221602 2020.342

Table 4.3: Left: Comparing the computational time with decreasing ε. In all cases,Pe = 10

53

Figure 4.21: Comparing the computational time with decreasing ε. In all cases,Pe = 10. The plots correspond to values in table 4.3.

micro problem increases quadratically with decreasing ε. This is one importantreason why we consider the numerical multiscale homogenization method, since thecomputational time is drastically reduced. For example, the computation time ofour homogenization numerical simulation is 64.466 s for Pe = 10 and 59.105s forPe = 2.

54

Chapter 5

Conclusion and Future work

In this thesis, we modeled the smoldering combustion of a paper assuming a peri-odic distribution of microstructures. The realized model is a system of semi-linearreaction-diffusion equations. Due to the usual difficulties involved in solving systemsof this nature with periodic microstructures, we applied the homogenization tech-nique that yields a macroscopic description of the problem. This is one of the mainachievements of this thesis. The numerical implementation of the homogenizationtechnique results in a model that drastically reduces the CPU time compared tothe direct computation of the micro model with the periodic structure. The resultsof our analysis have shown that the macroscopic solutions are in agreement withthe predictions of the microscopic ones. Furthermore, we point out that the qual-itative description of our numerical simulation is not a direct consequence of theexperimental observations in [3]. A similar observation, however, can be accountedfor in a microscale smoldering simulations like those conducted in [26]. These werethe subject of subsection 4.5.2.

For the purpose of further research in this subject, we present other areas of interestwhich were not covered in the limited time frame: In the construction of the mi-crostructure, we assumed, for simplicity, a structure consisting of cylindrical solidsdistributed periodically over the volume of paper. However, it will be of interest,in subsequent analysis of the techniques implemented in the course of this thesis,to consider the effect of other geometries on the solution of the homogenized model(see [20] more details and related questions). Another possibility is to consider adesign where the pore space is ”disconnected” and the solid matrix ”connected” asshown in Fig. 5.1.

Figure 5.1: An alternative geometry with connected solid matrix

55

The proposed model can be modified to address other issues such as modelingthe velocity of the smolder front and the effect of heat losses due to radiation [6].Stokes model of flows in porous media can also be adopted in computing the microvelocity field of the gas. We have used a one-temperature model in this thesis. Afurther implementation could be to consider a two-temperature model accountingfor the temperature fields in the solid and in the oxidizer. This model (i.e the two-temperature model) can be used to address the questions:(Q1) : Under which conditions will the two-temperature model be in diffusive-thermal equilibrium?(Q2) : Under which conditions will the instantaneous oxidation of the gas beachieved?

On the other hand, the reaction rate (3.11) can be reformulated to include a charredor uncharred part of the material (see [6], for instance). Also, we see that the righthand side of (3.11) has some regularity issue that was not immediately addressed inthe present study. We have approximated this term by considering (2.23). Finally,in the homogenization technique implemented in this study, a formal asymptoticexpansion (3.21) was considered. The approximate solutions obtained in this casewere of order zero i.e. Θ0,Ψ0 and R0. These gives rise to an amount of error in ournumerical solutions, since the method only approximate the solutions with the firstterms of their corresponding asymptotic expansions.

The question here is two fold:(Q3) : Can we show the convergence of Θε,Ψε,Rε to Θ0,Ψ0,R0?(Q4) : How large is the error when replacing Θε,Ψε,Rε by Θ0,Ψ0,R0?

A possible answer to (Q4) is to introduce an asymptotic expansion of order one(see [25], [19] for example). This expansion has the following form:

Θε1 = Θ0 + ε

2∑l=1

wl(y)∂Θ0

∂xl, (5.1)

where wl are the solutions of the cell problems e.g. (3.42) and (3.65). It was shownin [25] for example, that Θε

1 is an approximation of Θε. The enhancement of theerror estimate of the two-scale finite element solution will then be to compute theterms of correction εΘh

1(x, xε, τ) and εΨh

1(x, xε, τ). These terms will improve the

approximation of Θε and Ψε. Thus, the relative error estimate can that can beachieved in the L2 norm is :‖Θε − (Θh

0 + εΘh1)‖L2(Ωε) ≤ Cε, with C independent of

ε.

In addition, a detailed procedure on how to get this error estimate is given forinstance in [25]. Obviously, the enhanced approximation (5.1) recovers the smalloscillation which is not present in Θh

0 .

56

Nomenclature

Quantity Description (units).δ paper thickness [m]Ωδ 3D outer domain∂Ωδ boundary of outer domainΩ planar outer domain∂Ω boundary of planar outer domainY δ volumetric representative unit cellY planar representative unit cells solid partg gas partYs, Y

δs solid parts of representative unit cell

Yg, Yδg gas parts of representative unit cell

∂Y δs inner boundary of 3D representative unit cell

∂Ys inner boundary of planar representative unit cellΩgε pore space

Ωsε solid space

T temperature[K]C volumetric mass fraction in gas part [kg/m3]R volumetric mass fraction of the solid product [kg/m3]cg,s specific heat capacities [Jkg−1K−1]ρg,s densities[kg/m3]λg,s thermal conductivities[J/m · s]D molecular diffusivity[m2/s]W reaction rate with Arrhenius temperature dependence[kg/m3 · s]u = (u, 0, 0) prescribed flow-field [m/s]Q heat release [J/kg]Ta activation temperature[K]A pre-exponential factor[s−1]Ψ non-dimensional gas concentrationΘ non-dimensional temperatureT0 initial temperature[K]W (Y ) Hilbert space of Y-periodic functions

Table 5.1: Notation.

57

L 1× 10−2[m] characteristic length scaleD 2.5× 10−5[m2/s] diffusivity of oxygen [3]v −[m/s] velocity of gasP (1− 0.42π)vL/D homogenized parameterPe vL/D Peclet numberTa 723.15[K] activation temperature (in this study)A 1.545[1/s] pre-exponential factor (in this study)T0 297.15[K] ambient temperature (in this study)Q 13.37[J/kg] heat release (in this study)C0 0.23[kg/m3] initial conc. of oxygen (in this study)ρg 1376.3[kg/m3] mass density of oxygen[3]ρs 540[kg/m3] mass density of paper[3]cg 923[J/kg/K] specific heat of gascs 1270[J/kg/K] specific heat of paper[28]R0 1.4[kg/m3] initial conc. of unburnt solid (in this study)Tc 656.1[K] critical temperature (in this study)ε 0.5[s] small timehs 0.01 scale for resolution of step functionm (csρs)/(cgρg) density ratiotD (L2)/D diffusive time scaletR R0/(C0A) solid product time scaletG 1/A gas time scaleTb T0 + (Q/cg) temperature combustion producttB 1/((QAC0)/((Tb − T0)ρgcg)) combustion time scale

A 0.86 Arrhenius exponential factor

f tDAΨ/tB temperature source term

g tDΨA/tG gas source term

h tDAΨ/tR reaction termTΓ (460− T0)H(ε/tD − t, hs)/(Tb − T0) ignition sourceλs 0.07[W/m/K] thermal conductivity of solid [28]λg 0.0238[W/m/K] thermal conductivity of gasLeg λg/(ρgcgD) Lewis number for gasLes λs/(ρscsD) Lewis number for solid

Table 5.2: Physical parameters

58

List of Figures

1.1 Sketch of the experimental setup [2]. . . . . . . . . . . . . . . . . . 21.2 Left: Volumetric representative cell Y δ; Right: Planar representative

unit cell Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 3D micro scale geometry of the paper sheet. . . . . . . . . . . . . . 31.4 Instability of combustion front when the flow velocity (or Peclet num-

ber) decreases from a to e. Oxygen flows downward and the smolderfront moves upward as shown by the arrow in d [6] . . . . . . . . . 5

2.1 Reaction/transport mechanisms within the microstructure . . . . . 8

3.1 (a): Deformed cell after time, t; (b) Not structurally deformed cellafter time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Left: Heterogeneous medium; Right: Homogeneous representation ofthe perforated domain. . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Unit cell with two subdomains Yg and Ys. . . . . . . . . . . . . . . . 324.2 Unit cell with hole: one subdomain. . . . . . . . . . . . . . . . . . . 324.3 Sample geometry of the unit cell . . . . . . . . . . . . . . . . . . . . 374.4 Solution of cell problems, (a): w1(x

ε) and (b): w2(x

ε) with x ∈ Ω. . . 37

4.5 Solution of cell problems, (c): w1(xε) and (d): w2(x

ε) with x ∈ Ω. . . 38

4.6 Solution of cell problem (e): ω1(xε) and (f): ω2(x

ε) with x ∈ Ω. . . . 38

4.7 Solution of cell problem (g): ω1(xε) and (h): ω2(x

ε) with x ∈ Ω. . . . 38

4.8 2D Geometry of the model problems . . . . . . . . . . . . . . . . . 444.9 2D mesh of the model poblems. . . . . . . . . . . . . . . . . . . . . 464.10 Qualitative comparison of the temperature distribution of a smoldered pa-

per. Left: Macroscopic solution Θh0(Pe = 2); Right: Microscopic solution

Θε(Pe = 2, ε = 120). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.11 3D plot of the temperature distribution of a smoldered paper. Left:Macroscopic solution Θh

0(Pe = 2); Right: Microscopic solution Θε(Pe =2, ε = 1

20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.12 One dimensional plot of the temperature distribution of a smoldered pa-

per. Left: Averaged solution Θh0(Pe = 2) without oscillations; Right:

Microscopic solution Θε(Pe = 2, ε = 120) with oscillations. . . . . . . . . 47

4.13 Qualitative comparison of the concentration distribution on a smolderedpaper. Left: Macroscopic solution Ψh

0(Pe = 2); Right: Microscopic solu-tion Ψε(Pe = 2, ε = 1

20). . . . . . . . . . . . . . . . . . . . . . . . . . 48

59

4.14 3D plot of the concentration distribution on a smoldered paper. Left:Macroscopic solution Ψh

0(Pe = 2); Right: Microscopic solution Ψε(Pe =2, ε = 1

20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.15 One dimensional plot of the concentration distribution on a smoldered

paper. Left: Averaged solution Ψh0(Pe = 2) without oscillations; Right:

Microscopic solution Ψε(Pe = 2, ε = 120) with oscillations. . . . . . . . . 48

4.16 Smoldering process of a paper. Left: numerical solution for Pe = 10;Right: numerical solution for Pe = 8. . . . . . . . . . . . . . . . . . . 49

4.17 Smoldering process of a paper. Left: numerical solution for Pe = 5; Right:numerical solution for Pe = 2. . . . . . . . . . . . . . . . . . . . . . . 49

4.18 Numerical solution for Pe = 2 and 49 cells. . . . . . . . . . . . . . . 504.19 Numerical computation of the reaction term. . . . . . . . . . . . . . 514.20 Error estimates in the temperature and concentration. In all cases,

Pe = 10. The plots correspond to values in Table 4.2. . . . . . . . . 534.21 Comparing the computational time with decreasing ε. In all cases,

Pe = 10. The plots correspond to values in table 4.3. . . . . . . . . 54

5.1 An alternative geometry with connected solid matrix . . . . . . . . . . 55

60

List of Tables

4.1 Comparison of the homogenized coefficients with known bounds . . 424.2 Left: Error estimates in the FE approximation of ‖Θε − Θh‖h for

various ε; Right: Error estimates in the FE approximation of ‖Ψε −Ψh‖h for various ε. In all cases, Pe = 10. . . . . . . . . . . . . . . . 52

4.3 Left: Comparing the computational time with decreasing ε. In allcases, Pe = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Physical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

61

Bibliography

[1] G. Rein, Computational Model of Forward and Opposed Smoldering Com-bustion with Improved Chemical Kinetics, Ph.D. thesis, Dept of MechanicalEngineering, Univ. of California of Berkeley (2005).

[2] L. Kagan, G. Sivashinsky, Pattern formation in flame spread over thin solidfuels, Combustion Theory and Modelling. 12 (2) (2008) 269 –281.

[3] O. Zik, E. Moses, Fingering instability in combustion, Physical Review E.60 (1) (1999) 1–14.

[4] M. Espedal, A. Fasano, A. Mikelic, Filtration in Porous Media and IndustrialApplication, Springer, Berlin, 2000.

[5] T. Ohlemiller, Modeling of smoldering combustion propagation, Progress inEnergy and Combustion Science. 11 (1985) 277 –310.

[6] A. Fasano, M. Mimura, M. Primicerio, Modelling a slow smoldering combustionprocess, Mathematical Methods in the Applied Sciences. (2009) 1–11.

[7] K. Ikeda, M. Mimura, Mathematical treatment of a model for smoldering com-bustion, Hiroshima Math. J. 38 (2008) 349–361.

[8] M. J. H. Anthonissen, Local Defect Correction Techniques:Analysis and Ap-plication to Combustion, Ph.D. thesis, Eindhoven University of Technology(2001).

[9] S. Whitaker, The Method of Volume Averaging, Springer-Verlag New York,1999.

[10] W. Gill, A. B. Donaldson, A. R. Shouman, The frank-kamenetskii problemrevisited. part i. boundary conditions of first kind, Combustion and Flame. 36(1979) 217–232.

[11] W. Gill, A. B. Donaldson, A. R. Shouman, The frank-kamenetskii problemrevisited, part ii: Gradient boundary conditions, Combustion and Flame. 41(1981) 99–105.

[12] G. Allaire, Shape Optimization by the Homogenization Method, Springer,2002.

62

[13] L. E. Persson, L. Persson, N. Svanstedt, J. Wyller, The HomogenizationMethod: An Introduction, Chartwell-Bratt, 1993.

[14] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Vol. 127of Lecture Notes in Physics, 1980.

[15] A. Bensoussan, J. Lions, G. Papanicolaou, Asymptotic Analysis for Peri-odic Structures, Vol. 5 of Studies in mathematics and its application, North-Holland, 1978.

[16] E. Sanchez-Palencia, A. Zaoui, Homogenization Techniques for Composite Me-dia, Vol. 272 of Lecture Notes in Physics, Springer, 1985.

[17] http://www.comsol.com.

[18] C. Johnson, Numerical Solution of Partial Differential Equations by the FiniteElement Method, Studentlitteratur, Lund, 1987.

[19] G. Allaire, K. E. Ganaoui, Homogenization of a conductive and radiative heattransfer problem, simulation with cast3m, Proceedings of 2005 ASME SummerHeat Transfer Conference. (2005) 1–6.

[20] J. Bystrom, The Homogenization Mathod Applied to the Computation of ef-fective Thermal Conductivities of Composite Materials, Lulea University ofTechnology,S-971 87 Lulea, Sweden.

[21] J. Bystrom, Some mathematical and engineering aspects of the homogeniza-tion theory, Ph.D. thesis, Department of Mathematics, Lulea University ofTechnology, SE-97187, Lulea, Sweden (2002).

[22] J. Necas, Les methodes directes dans la theorie des equations elliptiques, Ed.de l’Acad. Tech. des Sciences, Prague, 1967.

[23] H. Alt, Lineare Funktionalanalysis, Springer, Berlin, 1992.

[24] F. Wendt, H. Liebowitz, N. Perrone, Mechanics of Composite Materials, Perg-amon Press, Oxford, 1970.

[25] W. Zhihua, Y. Ningning, Numerical simulation for convection-diffusion prob-lem with periodic micro-structure, Acta Mathematica Scientia 28B (2) (2008)236–252.

[26] G. Debenest, V. Mourzenko, J. Thovert, Smouldering in fixed beds of oil shalegrains. a three-dimensional microscale numerical model, Combustion Theoryand Modelling 9 (1) (2005) 113–135.

[27] W. L. Briggs, V. E. Henson, S. F. McCormick, A multigrid tutorial, Societyfor Industrial and Applied Mathematics, 2000.

[28] R. Dinwiddie, M. A. White, D. L. McElroy, Thermal Conductivity 28, DEStechPublications. Inc., 2006.

63

Appendix A

Matlab code for the discrete L2

error estimate

% Computation of the Discrete L^2 error of the solution of the homogenized

% problem

% Load data

clear all

% load into Matlab the set of solutions for temperature and concentration

% (macro)

load ’U.txt’;

load ’C.txt’;

%***********************************************************************

% load set of solutions computed from COMSOL for the temperatures (micro)

load ’Ue2D20.txt’;

load ’Ue2D10.txt’;

load ’Ue2D14.txt’;

load ’Ue2D7.txt’;

load ’Ue2D4.txt’;

load ’Ue2D2.txt’;

load ’Ue2D1.txt’;

% load set of solutions computed from COMSOL for the concentration (micro)

load ’Ce2D20.txt’;

load ’Ce2D14.txt’;

load ’Ce2D10.txt’;

load ’Ce2D7.txt’;

load ’Ce2D4.txt’;

load ’Ce2D2.txt’;

load ’Ce2D1.txt’;

64

%***********************************************************************

N=length(U_h);

h=(0.002)^2; % mesh size

%***********************************************************************

% variable assignment

U_h=U(:,3);

C_h=C(:,3);

C_eps20=Ce2D20(:,3);

C_eps14=Ce2D14(:,3);

C_eps10=Ce2D10(:,3);

C_eps7=Ce2D7(:,3);

C_eps4=Ce2D4(:,3);

C_eps2=Ce2D2(:,3);

C_eps1=Ce2D1(:,3);

U_eps20=Ue2D20(:,3);

U_eps14=Ue2D14(:,3);

U_eps10=Ue2D10(:,3);

U_eps7=Ue2D7(:,3);

U_eps4=Ue2D4(:,3);

U_eps2=Ue2D2(:,3);

U_eps1=Ue2D1(:,3);

%***********************************************************************

% compute the difference Ue - Uh

Cerr20=C_eps20-C_h;

Cerr14=C_eps14-C_h;

Cerr10=C_eps10-C_h;

Cerr7=C_eps7-C_h;

Cerr4=C_eps4-C_h;

Cerr2=C_eps2-C_h;

Cerr1=C_eps1-C_h;

err20=U_eps20-U_h;

err14=U_eps14-U_h;

err10=U_eps10-U_h;

err7=U_eps7-U_h;

err4=U_eps4-U_h;

err2=U_eps2-U_h;

err1=U_eps1-U_h;

65

%***********************************************************************

% computation of the discrete L^2 Error

Cerror20=sqrt(h*(sum(Cerr20.^2)));

Cerror14=sqrt(h*(sum(Cerr14.^2)));

Cerror10=sqrt(h*(sum(Cerr10.^2)));

Cerror7=sqrt(h*(sum(Cerr7.^2)));

Cerror4=sqrt(h*(sum(Cerr4.^2)));

Cerror2=sqrt(h*(sum(Cerr2.^2)));

Cerror1=sqrt(h*(sum(Cerr1.^2)));

error20=sqrt(h*(sum(err20.^2)));

error14=sqrt(h*(sum(err14.^2)));

error10=sqrt(h*(sum(err10.^2)));

error7=sqrt(h*(sum(err7.^2)));

error4=sqrt(h*(sum(err4.^2)));

error2=sqrt(h*(sum(err2.^2)));

error1=sqrt(h*(sum(err1.^2)));

%***********************************************************************

% Output results.

figure(1),hold on

H=bar([error1,error2,error4,error7,error10, error14 ,error20]);

plot([error1,error2,error4,error7,error10, error14 ,error20],’r’,...

’LineWidth’,2);

set(get(H(1),’BaseLine’),’LineWidth’,4)

colormap summer % Change the color scheme

xlabel(’increasing no of cells’); ylabel(’value of error’);

title(’Error estimates in temperature \Theta for increasing cell sizes’);

hold off;

%**********************************************************************

figure(2), hold on

G=bar([Cerror1, Cerror2,Cerror4,Cerror7,Cerror10,Cerror14,Cerror20]);

plot([Cerror1, Cerror2,Cerror4,Cerror7,Cerror10,Cerror14,Cerror20],’r’,...

’LineWidth’,2);

set(get(G(1),’BaseLine’),’LineWidth’,4)

colormap summer % Change the color scheme

xlabel(’increasing no of cells’); ylabel(’value of error’);

title(’Error estimates in concentration \Psi for increasing cell sizes’);

hold off;

%***********************************************************************

66