HEAT AND MASS TRANSPORT INSIDE A CANDLE WICK

by

Mandhapati P. Raju

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Dissertation Advisor: Dr. James S. T’ien

Department of Mechanical and Aerospace Engineering

CASE WESTERN RESERVE UNIVERSITY

January 2007

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

______

______

______

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(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. TABLE OF CONTENTS

TABLE OF CONTENTS iii

LIST OF TABLES v

LIST OF FIGURES vi

ACKNOWLEDGEMENTS xii

NOMENCLATURE xiv

ABSTRACT xxii

CHAPTER 1: INTRODUCTION 1 1.1 Candle Basics 1 1.2 Candle Burning 4 1.3 Previous Work 6 1.3.1 Previous Work on Candle Flames 7 1.3.1.1 Experimental Work 7 1.3.1.2 Numerical Work 10 1.3.2 Previous Work on Two-phase Flow in Porous Media 13 1.4 Purpose and Scope of this Dissertation 21 1.5 Dissertation outline 21

CHAPTER 2: AXISYMMETRIC WICK MODELING 24 2.1 Formulation of Transport Process in Porous Media 24 2.1.1 Mathematical Formulation 24 2.1.2 Numerical Formulation 31 2.2 Multifrontal Solvers for Large Sparse Linear Systems 33 2.2.1 Introduction 33 2.2.2 Multifrontal Solution Methods 36 2.23.Benchmark Testing 40 2.3 Analysis of an Externally Heated Axisymmetric Wick 41 2.3.1 Physical Description of the Model 42 2.3.2 Sample Case Results 43 2.3.2.1 Saturation and Temperature Distribution 43 2.3.2.2 Pressure Distribution 41 2.3.2.3 Mass Distribution 44 2.3.2.4 Heat Flux Distribution 45 2.3.2.5 Variation Along the Cylindrical Surface and the Axis of the Wick 46 2.33 Mesh Refinement Studies 47 2.3.4 The Effect of Applied Heat Flux 48 2.3.5 Parametric Studies 49

iii 2.3.5.1 The Effect of Gravity 49 2.3.5.2 The Effect of Absolute Permeability 50

CHAPTER 3: GAS PHASE MODELING INCUDING RADIATION 74 3.1 Theoretical Formulation 74 3.1.1 Continuity Equation 75 3.1.2 Momentum Equations 76 3.1.3 Species Equation 77 3.1.4 Energy Equation 78 3.1.5 Boundary Conditions 79 3.2 Non-Dimensional Parameters 82 3.3 Property Values 85 3.4 Numerical Procedure 86 3.4.1 Grid Generation 87 3.4.2 Numerical Implementation 88 3.5 Gas Radiation Model 89 3.5.1 The Equation of Radiative Transfer 90 3.5.2 Numerical Solution of Discrete Ordinates Method 95 3.5.3 Discrete Ordinates Angular Quadrature 97 3.5.4 Solution of Discrete Ordinates Equation 98 3.5.5 Mean Absorption Coefficient 104 3.6 Solution Procedure 105

CHAPTER 4: RESULTS AND DISCUSSIONS 120 4.1 Candle Flame Coupled to a Porous Wick 120 4.1.1 Detailed Flame Structure at Normal Gravity and 21% O2 122 4.1.1.1 Steady State Candle Flame (Wick Length = 4 mm) 123 4.1.1.2 Self Trimmed Candle Flame 130 4.1.2 Effect of Gravity 133 4.1.3 Effect of Wick Permeability 135 4.1.4 Effect of Wick Diameter 137 4.1.5 Effect of Ambient Oxygen 137 4.1.6 Validation of results 138 CHAPTER 6: CONCLUSION 196

Recommendation for Future Work 198

BIBLIOGRAPHY 201

iv

LIST OF TABLES

Table 2.1 Porous Wick Dimensionless variables 52

Table 2.2 Table 2.2 Porous Wick Numerical Values 53

Table 3.1: Correlating equations of specific heats forO2 , N 2 ,CO2 , H 2O , and fuel. 107

Table 3.2: Gas phase property values. 108

Table 3.3: Nondimensional parameters. 109

Table 3.4: Non-dimensional governing differential equations. 110

Table 3.5: The S4 quadrature sets for axisymmetric cylindrical enclosures. 111

Table 3.6: Least-square fitting equations of Planck mean absorption coefficient for CO2 and H2O. 112

Table 4.1(a) : Effect of wick diameter on candle flame characteristics (5mm candle diameter and 21% O2) . 144

Table 4.1(b) : Effect of wick diameter on candle flame characteristics (5mm candle diameter and 21% O2) . 144

Table 4.2: Effect of wick diameter (Alsairafi, 2003) on candle flame characteristics (5mm wick length, 5mm candle diameter and 21% O2) 145

Table 4.3: Comparison with candle flame experiments in high gravity levels. 145

v

LIST OF FIGURES

Figure 1.1: Schematic of a candle flame. 23

Figure 2.1: Comparison of function residuals vs. CPU times for Newton, modified Newton and Picard’s iterative techniques. 54

Figure 2.2: Physical description of an externally heated axisymmetric wick. 55

Figure 2.3: Computational grid of an externally heated axisymmetric wick. 56

Figure 2.4: Plot of (a) saturation profiles (b) non-dimensional temperature profiles and (c) non-dimensional temperature profiles (expanded in the two-phase region) inside the porous wick for parameters shown in Table 2.2. 57

Figure 2.5: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick for parameters shown in Table 2.2. 58

Figure 2.6: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor mass flux vectors (expanded near the tip of the wick) inside the porous wick for parameters shown in Table 2.2. 59

Figure 2.7: Plot of (a) liquid convective heat flux vectors (b) vapor convective heat flux vectors and (c) conductive heat flux vectors inside the porous wick for parameters shown in Table 2.2. 60

Figure 2.8: Plot of saturation and temperature variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2 61

Figure 2.9: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2 62

Figure 2.10: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2 63

Figure 2.11: Plot of saturation and temperature variation along the axis of the wick for parameters shown in Table 2.2 64

Figure 2.12: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick for parameters shown in Table 2.2 65

vi

Figure 2.13: Comparison of (a) saturation profiles (b) pressure profiles and (c) temperature profiles for three different meshes (80x40, 80x80, 160x40) inside the porous wick for parameters shown in Table 2.2. 66

Figure 2.14 The variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick. 67

Figure 2.15 The variation of total mass of wax evaporated from the wick surface with the total heat supplied to the wick. 68

Figure 2.16 The variation of percentage heat that is lost to the reservoir with the total heat supplied to the wick. 69

Figure 2.17 The effect of gravity on the variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick. 70

Figure 2.18 The effect of gravity on the variation of total mass evaporated from the wick surface with the total heat supplied to the wick. 71

Figure 2.19 The effect of absolute permeability on the variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick. 72

Figure 2.20 The effect of absolute permeability on the variation of total mass evaporated from the wick surface with the total heat supplied to the wick. 73

Figure 3.1 Schematic of a candle 113

Figure 3.2: Variable grid structure for modeling candle flames for 1mm wick diameter and 5mm candle diameter 114

Figure 3.3: Schematic of radiation intensity transfer energy balance on arbitrary control volume in a participating medium. 115

Figure 3.4: Geometry and coordinate system for 2D axisymmetric cylindrical enclosure. 116

Figure 3.5: Projection of an S4 quadrature set on the μ,ξ plane using p,q numbering in r-x geometry. 117

Figure 3.6: Four types of space angle sweep direction for SN scheme. 118

Figure 3.7: Solid angle discretization of the S4 quadrature. 119

Figure 4.1: Plot of (a) pressure profiles (b) saturation profiles and (c) temperature profiles inside the porous wick coupled to a candle flame at normal gravity 146

vii

Figure 4.2: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick coupled to a candle flame at normal gravity 147

Figure 4.3: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor mass flux vectors (expanded near the tip of the wick) inside the porous wick coupled to a candle flame at normal gravity. 148

Figure 4.4: Plot of net heat flux supplied by the candle flame along the cylindrical surface of the wick 149

Figure 4.5: Plot of saturation and temperature variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity. 150

Figure 4.6: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity. 151

Figure 4.7: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity. 152

Figure 4.8: Plot of saturation and temperature variation along the axis of the wick coupled to a candle flame at normal gravity. 153

Figure 4.9: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick coupled to a candle flame at normal gravity. 154

Figure 4.10: Gas temperature contours (non-dimensinalized by T∞ = 300 K ) 155

Figure 4.11: Fuel reaction rate contours (g cm-3 s-1) 155

Figure 4.12: Fuel and oxygen mass fraction contours 156

Figure 4.13: Local fuel/oxygen equivalence ratio contours 156

Figure 4.14: Carbon dioxide mass fraction contours 157

Figure 4.15: Water vapor mass fraction contours 157

Figure 4.16 Oxygen mass flux and flow field around the flame 158

p − p Figure 4.17: Isobar contours; p = ∞ 159 ρ U 2 r r

Figure 4.18: Profiles of temperature and species concentration along the symmetry line 160

viii

Figure 4.19: Effective Mean absorption coefficient distribution (cm-1 atm-1) 161

Figure 4.20: Dimensional net radiative flux vectors (W/cm2) 161

Figure 4.21: Contours of divergence of radiative heat flux (W/cm3) 162

Figure 4.22: Heat fluxes on the candle wick surface per unit radian 163

Figure 4.23: Plot of (a) saturation profiles (b) non-dimensional temperature profiles and (c) non-dimensional temperature profiles (expanded in the two-phase region) inside the porous wick for a self trimmed candle flame at normal gravity 164

Figure 4.24: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick for a self trimmed candle flame at normal gravity 165

Figure 4.25: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors inside the porous wick for a self trimmed candle flame at normal gravity 166

Figure 4.26: Plot of net heat flux supplied by the candle flame along the cylindrical surface of the wick 167

Figure 4.27: Plot of saturation and temperature variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity 168

Figure 4.28: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity 169

Figure 4.29: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity 170

Figure 4.30: Plot of saturation and temperature variation along the axis of the wick for a self trimmed candle flame at normal gravity. 171

Figure 4.31: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick for a self trimmed candle flame at normal gravity. 172

Figure 4.32: Gas temperature contours for a self trimmed candle flame (non- dimensinalized by T∞ = 300 K ) 173

Figure 4.33: Fuel reaction rate contours for a self trimmed candle flame (g cm-3 s-1) 173

Figure 4.34: Fuel and oxygen mass fraction contours for a self trimmed candle flame 174

ix Figure 4.35: Local fuel/oxygen equivalence ratio contours for a self trimmed candle flame 174

Figure 4.36: Carbon dioxide mass fraction contours for a self trimmed candle flame 175

Figure 4.37: Water vapor mass fraction contours for a self trimmed candle flame 175

Figure 4.38: Oxygen mass flux and flow field around the self trimmed candle flame 176

p − p Figure 4.39: Isobar contours for a self trimmed candle flame; p = ∞ 177 ρ U 2 r r

Figure 4.40: Profiles of flame structure at the centerline for a self trimmed candle flame 178

Figure 4.41: Effective mean absorption coefficient distribution for a self trimmed candle flame (cm-1 atm-1) 178

Figure 4.42: Dimensional net radiative flux vectors for a self trimmed candle flame (W/cm2) 179

Figure 4.43: Contours of divergence of radiative heat flux for a self trimmed candle flame (W/cm3) 179

Figure 4.44: Heat fluxes on the candle wick surface per unit radian 180

Figure 4.45: Candle flames at various gravity levels for a self trimmed candle flame 181

Figure 4.46: Candle burning rate and self trimmed length of the candle (on log scale) at various gravity levels for a self trimmed candle flame 182

Figure 4.47: Candle burning rate and self trimmed length of the candle (on normal scale) at various gravity levels for a self trimmed candle flame 183

Figure 4.48: Self trimming length of the candle wick using different wick permeabilities. 184

Figure 4.49: Candle burning rates for different wick permeabilities. 185

Figure 4.50: Candle flame lengths for different wick permeabilities. 186

Figure 4.51: Maximum gas phase temperatures for different wick permeabilities. 187

Figure 4.52: Self trimming length of the candle wick for different wick diameters. 188

Figure 4.53: Candle burning rates for different wick diameters. 189

x

Figure 4.54: Maximum gas phase temperatures for different wick diameters. 190

Figure 4.55: Maximum gas phase temperature at various oxygen molar fractions 191

Figure 4.56: Self trimmed length of the candle wick at various oxygen molar fractions 192

Figure 4.57: Candle burning rate at various oxygen molar fractions 193

Figure 4.58: Fuel reaction rate contours (5.e-5 g/cm3s)at various oxygen molar fractions. 194

Figure 4.59: Comparison of flame length and flame widths at various gravity levels with the experiments (Arai and Amagai, 1993) 195

xi ACKNOWLEDGEMENTS

I would like to express my gratitude to Professor James S. T’ien for the valuable guidance, help and encouragement provided during the course of this work. He has been very patient in guiding me all along the research. His comments and suggestions have been thought provoking. It gave a lot of insight into the problem. His encouraging spirit and friendly disposition has been very inspiring.

I am grateful to Professor Krishnan V. Pagalthivarthi (IIT Delhi) for his selfless motivation in encouraging me to pursue my studies in Case Western Reserve University.

Learning Computational Fluid Dynamics under him was a rich experience for me. The credit to my strength in Fluid Mechanics and in Numerical programming goes to him.

His example has been a motivating force for taking up a career in research.

I wish to acknowledge Dr Ammar, for providing me with his candle flame gas- phase code and the relevant materials. He has been extending to me throughout my research through e-mails. His timely clarifications have helped me to smoothly progress in my work. I am thankful to Dr Amit Kumar for his cooperation and help in practical matters. I also thank my colleagues I. Feier, Sheng-Yen Hsu, Ya-Ting Tseng, K. Tolejko,

G. Mittal, A. Garg, Ravikumar, B. Han, and K. Kumar with whom I had useful discussions. I also thank Prof. Sung, Prof. Edward White, Prof. Saylor Beverly and Dr

Daniel Dietrich for serving as my examination committee members and for their valuable time and suggestions. I would also like to thank staff of the department: J. A. Stiggers, A.

Szakacs, , C. Wilson, S. Campbell and M. Marietta for their prompt help at various stages.

xii Finally I would also like to acknowledge NASA for supporting the work under the technical monitoring of Dr Daniel Dietrich.

xiii NOMENCLATURE

Notation

Gas phase parameters a Coefficient in the discretised equation

A Special function for power-law scheme

A Area (cm2)

2 Ae East surface area of a control volume (cm )

2 An North surface area of a control volume (cm )

2 As South surface area of a control volume (cm )

2 Aw West surface area of a control volume (cm ) b Constant term in the discretised equation

3 -1 -1 Bg Gas phase pre-exponential factor (cm g s ) c Speed of sound (cm s-1)

C Correction factor for the Planck mean absorption coefficient

-1 -1 Cp Dimensional gas phase specific heat (cal g K )

-1 -1 C p,r Reference gas phase specific heat (cal g K ) d geometric dimensionality (1, 2, or 3)

D Flame diameter (cm)

Dw Wick diameter (cm)

2 -1 Di Diffusion coefficient of species i (cm s )

eˆθ Unit vector in the polar direction

eˆψ Unit vector in the azimuthal direction

xiv E Gas phase activation energy (cal gmol-1)

Er Radiant energy flux fi Stoichiometric mass ratio of species i/fuel fr Weighting factor in r-direction fx Weighting factor in x-direction g Gravitational acceleration (cm s-2)

-2 ge Gravitational acceleration on Earth (cm s )

G Incident radiation (cal cm-2 s-1)

-1 hi Enthalpy of species i (cal g )

H Flame height (cm)

I Radiation intensity (cal cm-2 s-1)

J Flux influencing a dependent variable φ

L Latent heat (cal g-1)

Lr Reference length (cm)

Lei Lewis number of species i m Average burning mass fluxes (g cm-2 s-1)

-1 mt Total burning rate (mg s )

M Total number of different directions for SN scheme

-1 MWi Molecular weight of species i (g gmole ) G n Outward normal to the surface

N Order of discrete ordinates scheme p Dynamic pressure (atm)

P Pressure (atm)

xv P∞ Ambient pressure (atm)

-2 qc Conductive heat flux (W cm )

-2 qr Radiative heat flux (W cm )

r -2 qr Radiative heat flux in r-direction (W cm )

x -2 qr Radiative heat flux in x-direction (W cm )

y -2 qr Radiative heat flux in y-direction (W cm )

Q Heat of combustion (cal g-1) r Dimensional r-coordinate (cm) G r Position vector

-1 -1 Ru Universal gas constant (cal gmol K )

S Source terms

SN Discrete ordinates scheme of an order N

T Temperature (K)

Tb Boiling temperature of fuel (K)

Tm Melting temperature of fuel (K)

T∞ Ambient temperature (K)

Tr Reference temperature (K) u Axial velocity (cm s-1)

-1 UB Buoyant velocity (cm s )

-1 UD Diffusion reference velocity (cm s )

-1 Ur Reference velocity (cm s ) v Radial velocity (cm s-1)

V Volume of a control volume

xvi w Quadrature weights

 -3 -1 Wi Sink or source term of species i (g cm s ) x,X Dimensional x-coordinate (cm)

Xi Mole fraction of species i (%) y Dimensional y-coordinate (cm)

Yi Mass fraction of species i (%)

α Absorptivity

2 -1 αr Reference thermal diffusivity (cm s )

β Extinction coefficient (cm-1)

γ Reflectivity

o -1 Δh f ,i Heat of formation of species i (cal g )

ε Solid emissivity

η Direction cosine

φ Dependent variable in governing equations

Φ Scattering phase function

κ Absorption coefficient (cm-1)

Γ General diffusion coefficient in the discretised equations

λ Thermal conductivity (cal cm-1 s-1 K-1)

-1 -1 -1 λr Reference thermal conductivity (cal cm s K )

μ Gas viscosity (g cm-1 s-1)

μ Direction cosine

-1 -1 μr Reference gas viscosity (g cm s )

ν Frequency

xvii

θ Polar angle

ρ Dimensional gas density (g cm-3) -3 ρr Reference gas density (g cm )

-3 ρ∞ Ambient gas density (g cm )

σ Stefan-Boltzmann constant (cal cm-2 s-1 K-4)

σs Scattering coefficient

ω Weighting factor

ξ Direction cosine

ψ Azimuthal angle

ζ Expansion factor

G Ω Solid angle in terms of ordinate direction (ξ,μ,η) (sr)

Subscripts b Black body

B Buoyant

E Neighbor in the positive x-direction on the east side f Flame i Species i max maximum min minimum

N Neighbor in the positive y-direction on the north side

P Central grid point r Reference

xviii S Neighbor in the negative y-direction on the south side w Value at wall

W Neighbor in the negative x-direction on the west side x Axial direction

Superscripts

° Old value of a variable

+ Positive component

- Negative component

Non-dimensional quantity

′ Incoming x x-direction r r-direction p p-level of μ direction cosine q q-level of ξ direction cosine

Special symbol a Absolute value of a

a1 ,a2 ,a3 ,... Largest of a1, a2, a3, …

Dimensionless numbers

Bo Boltzmann number

Da Damköhler number

Gr Grashof number

Pe Peclet number

xix

Pr Prandtl number

Ra Raleigh number

Re Reynolds number

Porous wick parameters c specific heat capacity (J/kg-K) ifg latent heat of evaporation (J/kg)

G Gibb’s phase function h enthalpy k thermal conductivity, (W/m-K)

K permeability, (m2) kr relative permeability

Mw molecular weight, (kg/kg-mole) p pressure, (N/m2)

Pe peclet number

R gas constant, (J/kg-K) s saturation

T temperature, (K) u,v velocity, (m/s) x, X Dimensional x-coordinate

α thermal diffusivity, (m2/s)

ε porosity

μ viscosity, (kg/m-s)

ρ density, (kg/m3)

xx σ surface tension, (N/m)

ζ triplet, ()pl ,,sT

σ surface tension, (N/m)

χ thermodynamic state parameter

η entropy

ζ triplet, ()pl ,,sT

Subscripts l liquid g vapor r relative c capillary eff effective m melting point

0 ambient, standard atmospheric conditions s solid

xxi

Heat and Mass Transport Inside a Candle Wick

Abstract

by

MANDHAPATI PADMANABHA RAJU

The purpose of this study is to investigate the effect of heat and mass transfer inside the porous wick on candle flame combustion. The phenomenon of self trimming that is observed in candle flame, whose wick is made of combustible material, is also analyzed. This is accomplished by modeling two-phase flow inside an axisymmetric wick and coupled with a gas-phase candle flame.

Gas-phase model has been taken from Alsairafi (2003). A finite volume method is used to solve the steady mass, momentum (Navier-Stokes), species, and energy equations, and the radiative transfer equation. The gas phase combustion process is modeled by a single-step, second-order, finite rate Arrhenius reaction. The discrete ordinates method is used to solve the radiative heat transfer with mean absorption coefficients. Flame radiation is only from CO2 and H2O, the products of combustion. The wick is modeled using volume averaged equations for two-phase flow inside an axisymmetric wick. Momentum equations for liquid and vapor flow are governed by

Darcy’s law. The liquid is drawn to the surface by capillary action and the vapor is driven in the two-phase region by vapor pressure gradient governed by thermodynamic equilibrium relations.

Before coupling with the gas-phase flame, a decoupled numerical computation of the wick phase is performed to isolate the effects of porous media parameters. This is

xxii done by applying a prescribed constant heat flux along all the exposed surfaces. The results show that the porous wick parameters like absolute permeability play an important role in determining the saturation distribution inside the wick. For the range of the wick parameters chosen for this study, gravity does not play an important role.

The next part of the study involved coupling the candle wick to the candle flame.

For candles, whose wicks are made of combustible materials, self trimming is being modeled as the burn out of the dry region. The computed results show also that for a self trimmed candle flame, the wick permeability plays an important role in determining the candle flame structure. The wick permeability affects the self trimming length of the wick and this affects the flame structure.

The effect of gravity on the self trimmed candle flame is analyzed. Trimmed wick lengths are computed as a function of gravity. The results show that the burning rate increases continuously from 0ge to 2ge and then decreases. At high gravity levels wake flame is observed with a sudden decrease in burning rate.

Calculations have also been performed for different wick diameters. Maximum flame temperature, burning rate, and flame dimensions are computed. A parametric study using different molar oxygen percentage has been made at different gravity levels.

xxiii 1

CHAPTER 1 INTRODUCTION

Combustion has been the subject of many researchers’ interest because it provides

the majority of useful energy production in residential, commercial and industrial devices. The flame, a thin zone of intense chemical reaction undergoing the process of

combustion, generally consists of a mixture of fuel and oxidizer (i.e. oxygen or air). It is

well known that flames are categorized as being either premixed flames or non-premixed

(diffusion) flames. In a premixed flame, the fuel and the oxidizer are mixed at the molecular level prior to any significant chemical reaction. In a diffusion flame, the reactants are initially separated, and reaction occurs only in the mixing zone between the fuel and the oxidizer. A typical diffusion flame is that of a burning candle, (see figure

1.1).

Candle flame studies contribute to the understanding of many fundamental

aspects like chemical reactions, diffusion processes, porous wick transport as well as their

coupled effects. Although the use of candles is common, not all the physical and chemical aspects of candle burning are sufficiently understood. Before going into the scientific aspects of a candle flame, some basic information on candles is provided.

1.1. CANDLE BASICS

The following material has been taken from the website

http://candles.genwax.com/candle_instructions/___0___how_wick.htm .

The body of a candle is comprised of a solid fuel source, usually paraffin wax. A

wick runs through the center of the body of the candle from the bottom, extending out of 2

the top. The wick, which acts as a fuel pump when the candle is burning, is generally

made of cotton fibers that have been braided together.

Candle Wax

There are two main waxes used in candle making, Paraffin Wax and Beeswax.

Paraffin wax, which is classified as a natural wax, is the most common wax used in candle making, and can be said to ultimately come from plant life.

In order to protect themselves from adverse weather conditions plants produce a

layer of wax on their leaves and stems. Material from dead plants 100-700 million years

ago accumulated in large quantities and eventually became buried beneath the surface of

the earth. After a long period of time, forces of heat and pressure turned the slowly

decaying plant material into crude oil, otherwise known as petroleum. Because of the

nature of waxes, being inert and water repellent, they were unaffected by the

decomposition of the plant material and remained intact, suspended within the crude oil.

Petroleum companies "harvest" the crude oil and process it. They refine the oil,

separating the different properties into gasoline, kerosene, lubrication oil, and many other

products. In many cases, the wax in the petroleum is considered undesirable and is

refined out. The refinery will process the wax into a clean, clear liquid, or as a solid

milky white block, and make it available to companies who may have a use for it. The

refined wax is called paraffin, which comes from the Latin "parum = few or without" and

"affinis = connection or attraction (affinity)". Basically there are few substances that will

chemically react with or bind to this type of wax. 3

A less common but more highly renowned wax for candle making is beeswax.

Classified as a natural wax, it is produced by the honeybee for use in the manufacture of

honeycombs. Beeswax is actually a refinement of honey. A female worker bee eats

honey, and her body converts the sugar in the honey into wax. The wax is expelled from

the bee's body in the form of scales beneath her abdomen. The bee will remove a wax scale and chew it up, mixing it with saliva, to soften it and make it pliable enough to work with, then attach it to the comb which is being constructed. Usually another bee

will take the piece of wax which has just been attached to the comb, chew it some more,

adding more saliva to it, and deposit it on another section of the comb. The combs are

built up, honey is deposited inside, and then the combs are capped with more wax. Since

several worker bees construct the comb at the same time, and the hive is constantly active

with other bees flying around and walking on the combs, depositing foreign matter onto

the combs, the composition of the wax becomes very complex.

Candle Wick

A candle without a wick is just a hunk of wax. The wick is what a candle is all

about. The earliest known candles were basically a wick-like material coated with tallow

or beeswax, not even resembling a candle at all. In taper candles the wick is the structure which supports the first layers of wax that create the candle. In all candles it acts as a

fuel pump, supplying liquefied wax up to the top where all of the action takes place. As a

regulator, different size wicks allow different amounts of wax up into the combustion

area providing different size flames. The wick is pretty much the most important element

of a candle. 4

The word wick comes from Old English "weyke or wicke", Anglo Saxon

"wecca", and Germanic "wieche or wicke". It is a name for a bundle of fibers that when braided or twisted together are used to draw oil or wax up into a flame to be burned in a lamp or candle.

A wick without wax around it is just a piece of string. Because the wick is fibrous and absorbent, melted wax adheres to it easily. Dipping a wick in and out of melted wax several times builds up layers of wax, sufficient enough to make a taper candle. The wick works by a principle called capillary action. Cotton fibers are spun into threads, which are bundled and braided together. The spaces between the cotton fibers, the threads, and braids act as capillaries, which cause liquids to be drawn into them. If you place a drop of water in the center of a paper towel you will see that the drop is absorbed and the wet spot expands. Where the expansion occurs is where capillary action is taking place, the candle wicking absorbs wax the same way.

Candle wicking is available in several types. Probably the most popular is the

Flat Braid, or Regular wick. Different sized wicks cause different sized flames simply because of the number of threads in the bundles. Each thread is considered a plait or ply, and a given number of ply are bundled together.

1.2. CANDLE BURNING

When a candle is lighted, the heat from the ignition source melts the wax, a heavy hydrocarbon, at the wick base. The liquid wax rises up, due to capillary action, and is then vaporized by the heat. This vaporizing wax cools the exposed wick and protects parts of it from burning out. The vaporized wax mixes with oxygen and the mixture reacts and generates heat and the process continues. 5

The heat and mass transfer taking place inside the wick is very complex. Two- phase flow regimes will exist inside the wick. All the three regimes – single phase liquid region, two-phase liquid vapor region and single phase vapor region can possibly exist inside the wick. The heat and mass transfer taking place in the two-phase region of a wick is very involved. It is of practical importance in many applications. Heat pipes work on the principle that the heat transfer coefficient is significantly reduced inside the two-phase flow in a wick. So heat pipes are used as efficient cooling devices.

In the gas-phase, heat is released from the combustion process. The hot combustion products are much less dense than the colder ambient air. In a gravity field, they rise upward and draw the oxygen to the reaction zone. This upward buoyant convective flow is the main reason that makes the flame into the so-called “tear-drop” shape.

In zero gravity environments, on the other hand, natural convection is not present and the fuel and oxidizer need to diffuse towards the reaction zone by the mechanism of molecular diffusion. The primary reason for this process to occur is the existence of concentration gradients around the flame. It is worthy to note that the diffusion transport rates in zero gravity conditions are much slower than the natural convection transport rates in normal gravity because of the absence of buoyancy-driven convection. Zero gravity flames are much less robust (in the sense of smaller reaction rate) than normal gravity flames because of the absence of the buoyant flow.

The mixing of fuel and oxidizer in the presence of a high temperature gradient means that both heat and mass transfer must be considered in addition to chemical kinetics to properly understand combustion. The processes are coupled and the 6

governing equations are nonlinear in nature. Numerical computation has become an

important tool to understand combustion phenomena. For the candle flame, it is

important to understand how the system responds to changes in parameters such as

gravity level. Interest in the candle burning has been revived in recent years. The driving force for this interest is due to the necessity of understanding flames in low gravity environment. The research conducted on the candle flame provides valuable insights on

how flames behave in microgravity which is relevant to spacecraft fire safety.

This research is mainly intended to provide a more complete simulation of a

candle flame coupled to a porous wick. The heat and mass transfer inside the wick can

control the flame shape and structure, and the extinction of flame at certain conditions.

When a candle burns, it slowly consumes the wax from the wax shoulder. As the level of wax comes down, the length of the exposed wick is increased. This process is generally very slow compared with the processes occurring in the gas and porous phases. Therefore in modeling the candle burning, a quasi steady approximation can be assumed with wick length treated as a parameter. As the length of the exposed wick increases, the surface exposed to the heat of candle flame increases and on the other hand, more capillary action is required for the liquid wax to reach the surface of the wick. At some point of time, the tip of the wick dries up. This causes the temperature of the wick to increase sharply at the tip of the wick. If the wick is made of pyrolyzable materials, then the dry portion of the wick will be consumed (burnt out). This phenomenon is referred to as the self trimming of the wick. A self trimming candle regulates its wick length by a balance of

the above two processes. 7

1.3. PREVIOUS WORK

In this section, previous works on the study of candle flame and the study two- phase flows inside porous media are reviewed. The first part will cover the candle flame studies and the second part will cover the porous media studies.

1.3.1 PREVIOUS WORK ON CANDLE FLAMES

1.3.1.1 EXPERIMENTAL WORK

Candles have been a focus of attention of scientific study for hundreds of years.

In the 19th century, English scientist Michael Faraday, who discovered many principles of

electricity, delivered one of the most famous of his Christmas lectures for children called

“The Natural History of a Candle” (Faraday, 1988). His observations have served as the basis for lessons in taking observations in combustion.

Although the processes that occur in a candle flame are complex, the setup of a

candle experiment is very easy. The candle flame is an excellent example of wick stabilized diffusion flame. Many researchers have chosen candle flame to understand a

wide range of combustion phenomena. For example, an early work by Lawton and

Weinberg (1969) examined the effect of magnetic fields on flame deflection using a candle. Chan and T’ien (1978) performed experimental work on commercial candles to study the spontaneous flame oscillation phenomena. Buckmaster and Peters (1986) and

Maxworthy (1999) have both studied stability and flickering phenomena for diffusion flames using a candle for a model diffusion flame.

The influence of hyper-gravity (gravity greater than that on earth) on candle

diffusion flames has been reported by Villermaux and Durox (1992) using a 6 meter

diameter centrifuge. They were able to vary the gravity up to seven times the gravity of 8

earth. Above 7ge, the candle flame becomes extinct. They found that the flame length and the candle burning rate both decreases as the gravity level increases. Another experimental investigation to study the behavior of candle flames in high gravitational field has been reported by Arai and Amagai (1993). They varied the gravity level by using a spin tester in the range from 1 to 14ge. There results showed that both the candle flame length and width were monotonically reduced with increasing gravity. In both the works there is no mention about the wick length of the candle and whether it reached the self trimming length.

Bryant (1995) investigated the effects of gravity level on the heat release rate for a candle flame under an imposed low-speed forced flow. The experiment was performed on board an aircraft flying repeated parabolic trajectories. He reported that the rate of heat release is flow rate dependent, decreases significantly in microgravity, and changes insignificantly under elevated-gravity conditions. Oostra et al. (1996) measured the soot production of a candle during microgravity and normal gravity condition during a parabolic flight. Their measurements predict a lower candle burning rate with higher soot production in microgravity. Amagai et al. (1997) investigated the effect of variable gravity using a gaseous butane diffusion flame issued from a tube in a centrifuge with a rotary arm of radius 0.9 meter. In their experimental work, they found that both the length and the width of butane diffusion flames reduced with an increase of gravity level for a constant feed rate. In addition, oscillations and blow off of the flame were observed at high gravity level.

Recent microgravity works sponsored by NASA have been a major source for fundamentally improving the science of combustion. Microgravity experimental data on 9

candle combustion has been obtained by several different techniques, all of which depend

on the experiment achieving a state of free-fall. A number of studies have used drop

towers where a few seconds of free-fall condition can be generated, i.e. between 2 to 5

seconds. Such facilities make it possible to conduct brief periods of microgravity research on earth. An experimental investigation of candle flame ignition behavior in drop tower has been reported by Ross et al. (1991). The experiments conducted at atmospheric pressure, under 19%-25% O2 concentration, and in nitrogen- or helium-

diluted environments. They found that the visible blue candle flame assumes a

hemisphere shape relatively quickly (about 1 second) after the drop. Although the blue

flame shape and size respond to the ignition or g-variation transient quickly, the color of

the flame continuously changes throughout the entire test duration. Thus it suggests that

more time is required to obtain the true steady state.

Besides the drop towers, microgravity research also took place using parabolic

aircraft maneuver which provide additional time of reduced-gravity environment.

-2 However, the g-jitter in the aircraft is too high (∼10 ge) to obtain a steady flame. The

candle flame fluctuates quite extensively in the airplane experiments. Using Space

Shuttle to obtain extended time in buoyant and weakly buoyant atmospheres, Dietrich et

al. (1994) investigated the candle flames behavior in microgravity. The experimental

results showed that the candle flame was spherical and bright yellow, presumably from

soot, immediately after ignition. After a few seconds, the yellow disappeared and the

flame became blue and nearly hemispherical with a large stand-off distance from the

wick. The flame luminosity decreased continuously with time until extinction. The

extinction typically occurred around one minute. They also reported that the mass of 10

liquefied wax grew continuously without dripping off, contrary to a candle burning in normal gravity. It has been observed, however, while the flame behavior was quasi-

steady during the majority of burning time, axisymmetric flame oscillations developed

near extinction due to oxygen depletion in the finite size combustion chamber. The term

“quasi-steady” needs to be explained. Physically, the flame size response is related to the

diffusion time scale (order of seconds) which is much lower than the depletion time scale of oxygen (order of minutes) in the experimental chamber. This means that the flame size and shape will respond in a quasi-steady manner to the oxygen molar fraction of the ambient even though this oxygen molar concentration is changing with time.

A second series of flight experiment on microgravity candle flame was carried out

in the Mir Station. In the Mir station the oxygen mole fraction is higher than that in the

Shuttle (between 0.23 to 0.25). Also the candle cage has more open area to facilitate oxygen diffusion from the ambient to the flame. Dietrich et al. (2000) showed that the candles onboard Mir with the largest wick diameters had the shortest flame lifetime where as the candles with the smallest wick diameters had the largest lifetimes. All of the candles in the Mir tests burned longer than those on Shuttle. The above examples indicate that candle flames onboard the space shuttle and Mir both oscillated prior to extinction with periodic increases and decreases of flame surface area.

1.3.1.2 NUMERICAL WORK

In addition to experimental effort, limited numerical works to model candle

flames have been conducted, all in zero gravity. To the best knowledge of the author,

there are two different candle flame numerical models. The first model is by Shu (1998),

Shu et al. (1998) and the other model is by Chang (see Dietrich et al., 2001). These two 11

different numerical models predict some candle flame behavior in zero gravity

environments. Both calculations utilize a single-step, finite rate, gas-phase chemical

reaction model in a frictionless flow. The main difference between the two models is the

candle and wick geometry. In Shu’s model, a spherical wick and conical candle body

were used and the equations were formulated using spherical coordinates. The model by

Chang used a more realistic wick and candle geometry and the equations were formulated

using cylindrical coordinates. Both models included a wick surface radiation loss term

and flame radiation is represented as a pure heat loss.

The work by Shu (1998) and Shu et al. (1998) were primarily focused on

computing the flame characteristics, the flammability limits, the effects of flame radiative

loss, and the effect of fuel and oxygen Lewis numbers. The model predicts a stable

steady-state of candle flame in an infinite ambient. They also reported that decreasing the oxygen concentration, prior to extinction, raises the flame base relative to the porous sphere without significant changes in the position of the flame top. Unlike oxygen Lewis

number, the numerical experiments reveal no flame properties dependence on the fuel

Lewis number except for the fuel vapor profile. Their model also predicts near-flame

oscillation similar to those observed in the experiments but there were some differences.

The modeling work by Chang (Dietrich et al., 2001), used the grid generation

technique and a body-fitted coordinate system to fit the more realistic wick and candle

shape. Instead of the explicit scheme used by Shu (1998), an implicit time-marching

scheme was developed and it shortened the computational time needed to reach steady

state and made it possible to examine the transient extinction phenomena near the limit.

There are two shortcomings (or limitations) on the candle modeling work by Shu and 12

Chang. The first is the assumption of potential flow. This assumption simplifies the

momentum equations but has the shortcoming of not being able to satisfy the no-slip

boundary condition on the solid surface. It also cannot handle buoyant induced flow so

gravity effect can not be studied. The second limitation is on the treatment of radiation.

In these models, flame (gas-phase) radiation is treated as a simple heat loss term without

knowing the detailed distribution of radiation heat flux. In previous modeling work on

solid fuel combustion (e.g. Rhatigan et al., 1998), part of the flame radiation becomes the

energy feedback to the solid pyrolysis process. A more rigorous treatment of radiative

heat transfer is thus needed to verify whether the simple heat loss model is adequate or

not.

A more detailed gas-phase modeling has been done by Alsairafi (2003). A

simplified two-dimensional axisymmetric flow around a realistic candle has been

numerically simulated. A finite volume method is used to solve the steady state Navier

stokes equations in conjunction with species and energy equation, and the radiative

transfer equation. The gas phase combustion process is modeled as a single step, second

order, finite rate Arrhenius reaction. The discrete ordinate method is used to solve the radiative transfer equation. The main assumption used in this work is the candle wick is

assumed to be a solid coated with liquid fuel all along its surface. This evades the

necessity to model the heat and mass transport inside the candle wick and its coupling

effect on the candle flame. They reported a significant match between their simulated

results and the experimental results. However, there is a major discrepancy even in the

qualitative trends of the candle burning rate with the imposed gravity level. Experiments

report a monotonic decrease in the burning rate with increasing gravity levels but the 13

numerical results indicate an increase in burning rate upto 3ge and a decrease in burning rate with further increase in gravity level. The studies by Alsairafi (2003) indicate the necessity to model the detailed heat and mass transport inside the wick to achieve a more realistic numerical simulation of candle burning.

1.3.2 PREVIOUS WORK ON TWO-PHASE FLOW IN POROUS MEDIA

Simulation of a porous wick involves modeling two phase flow with phase change inside the porous media. Two phase flow in porous media involves many features which distinguishes it from the single phase flow through a porous medium. The flow can be distinguished as either co-current or countercurrent flow depending on the direction of the flow of the two phases. In two phase flow through porous media, there are three possible saturation regimes that can exist. The porous media may be completely saturated with one phase. This is called the complete saturation regime. The porous media may have the lowest possible saturation with one phase. This is called the pendular regime. In this regime, one of the phases occurs in the form of pendular bodies throughout the porous media. These pendular bodies do not touch each other, so there is no possibility of flow for that phase. The porous media exhibits an intermediate saturation with both phases. This regime is known as funicular regime. Earlier it was thought that in this regime, both liquid and vapor phases flowed simultaneously through channels with gas moving in the inner core and the liquid in the annulus between the gas and the solid channel walls [Scheidegger, 1974]. Flow visualizations have shown that the gas and the liquid flow through their own network of channels [Dullien, 1979].

Two-phase flow in porous media is involved in the following classes of problems

1) Drying of porous materials 14

2) Heat pipe applications

3) Burning over porous wick surfaces

The drying problem is essentially transient in nature. It has been a subject of

interest since 1920’s. Lewis (1921) suggested that drying takes place primarily through a diffusion mechanism. Around this time, the soil scientists and chemists were attempting to explain the movement of moisture in porous media in terms of surface tension forces or by capillary action. Hougen et. al. (1940) made an extensive survey of the importance of capillary action in the drying process. They found out that capillary action can play an important role in the movement of moisture during the drying of a porous solid. Krischer

(1940) was the first to identify the importance of energy transport in a drying process.

Phillip and Devries (1957) included the effects of capillary flow and vapor transport, and

incorporated the thermal energy equation into the set of governing equations that describe

the drying process. Luikov (1975) also published similar equations for the heat and mass

transfer in porous media. Whitaker (1977) made a rigorous formulation of the theory of

drying based on the well known transport equations for a continuous media. These

equations were volume averaged to provide a rational route to a set of equations

describing the transport of heat and mass in porous media. These equations are limited by

the restrictions and assumptions that he had used while deriving the volume averaged

equations.

Even though Luikov (1975) published the full set of governing equations

describing the heat and mass transport in porous media, the equations are nonetheless

difficult to solve because of its complexities. A large number of parameters are required

to solve the equations. There have been efforts to study different aspects of the heat and 15

mass transfer by simplifying the equations using suitable assumptions. Initial efforts

involved consideration of only the vapor transport inside the porous media. The capillary

action on the liquid is neglected. Cross (1979) used the momentum and energy equations

for the vapor flow, neglecting the convection terms in the energy equation. He solved

analytically for the maximum pressure that is build up at the dry-wet interface. As drying

takes place, the evaporation takes place at the interface between the dry and the wet

regions. A high pressure is build up at this interface to generate the necessary driving

force for the transport of vapor in the dry region. Dayan (1981) extended this to include

the convective term in the energy equation. They have analyzed an intensely heated porous space using a transient model to obtain the transient pressure, temperature and moisture distributions. Dayan and Glueker (1982) analyzed the same problem using an explicit time marching technique and incorporating the liquid and vapor transport. They have assumed that the migration of liquid phase is primarily governed by pressure gradients generated by the vapor. They were able to get reasonable predictions for the temperature, pressure and moisture distribution during the drying of cement structure.

During intense heating, a region near the heated surface dries out. Once such a region develops, evaporation takes place exclusively at the dry-wet interface. The evaporation process leads to the pore pressurization and subsequent filtration of all the pore constituents towards both the heated surface and the inner wet zone of the concrete. In the above analysis, the effect of capillary pressure and gravity were neglected.

In drying, three different regions have been observed experimentally (Rahli et. al.,

1997) in the porous media. In the initial stages of drying, the temperature inside the

porous media is below the saturation temperature. Two regions are observed at this stage. 16

A two-phase region is observed near the heating surface and a liquid region is present deep inside the porous media. As drying proceeds, the surface exposed to heating becomes completely dried up and single phase vapor region is formed near the heating surface which penetrates deep into the media as time proceeds. The temperature in the single phase vapor region exceeds the saturation temperature. Three zones have been found to exist. Near the heated surface, a vapor saturated zone is observed. Adjacent to this zone and extending into the medium is a two phase region, which is dominated by capillarity and vapor transport. Ahead of this zone, the medium is saturated with the liquid phase.

In most cases, the capillary pressure and gravity play an important role in the two phase flow inside porous media. In the absence of any forced flow, the liquid is transported primarily due to capillary action and gravity forces. Kaviany and Mittal

(1987) analyzed the two phase flow inside porous media, where liquid is governed by capillary forces. They have analyzed the convective drying of a porous slab in the funicular state initially saturated with liquid. The liquid is driven by capillary pressure gradient and the vapor is driven by the partial pressure of the evaporating species generated by the temperature gradient established inside the porous slab. The transport of non-condensable gas and the effect of gravity are neglected. The calculations are performed only in the funicular regime, i.e. till the appearance of dry patches on the surface. They found good agreement between their experimental results and the predicted results for the drying rate, surface temperature and the average saturation up to the time of first appearance of dry patches on the surface (critical time). Rogers and Kaviany

(1991) extended this work to include the evaporative-penetration front. As the porous 17

slab is dried, the surface saturation decreases as time proceeds. After some period termed

as the critical time, dry patches appear on the surface. When dry patches appear on the

surface, the evaporative front starts penetrating into the porous slab. The effect of gravity

and the surface tension non-uniformities are included. They have also included the

transport of non-condensable gases into the porous slab. The speed of the evaporative

front and the mass transfer rate during this regime were predicted. A significant drop in

the drying rate is observed in the evaporative front regime. This is the result of the high

resistance to heat and mass transport in the dry region.

A heat pipe application is another major area which involves heat and mass

transport in the two-phase region of a porous substance. A heat pipe is a simple device

that can quickly transfer heat from one point to another. Heat pipe application is based on the high heat transfer rate obtained due to the evaporation-condensation mechanism

taking place inside the porous wick involving phase changes. Experimental studies

conducted by Hansen (1970) showed an enhancement of heat transfer over that of

conduction in saturated porous media. The experimental data obtained by Somerton et. al.

(1974) on effective thermal conductivity of steam-water saturated porous materials is

several times larger than those of the same medium saturated only with the liquid phase.

These classes of problems have been treated as steady state systems. In the heat

pipe applications, the porous wick is enclosed and the liquid and the vapor re-circulate

inside the porous wick. There is a counter-current transport of liquid and vapor in the

two-phase region. The liquid and the vapor regions above and below the two-phase

region are essentially stationary. A one-dimensional, steady state analytical model (heat

pipe problem) has been developed by Udell (1985) to study the effects of capillarity, 18

gravity and phase change. Their results predict the increase of heat transfer due to the

combined effect of evaporation, convection and condensation inside the porous media.

The heat pipe effect is also observed in two phase geothermal reservoirs. Bodvarsson

(1994) used a two dimensional porous slab model with a non-uniform heat flux at the bottom. Their results show very efficient heat transfer in the vapor dominated zone consistent with the observations in natural geothermal reservoirs.

Burning over porous wick surfaces also involves heat and mass transport in a

porous wick. The influence of capillarity on the combustion behavior and the effects of

the properties of porous beds on the combustion characteristics make this study unique.

Ignition and transition to the flame spread over the ground soaked with spilled

combustible liquid are of interest to researchers and engineers because they have

important implications in terms of fire safety. Kaviany and Tao (1988) had done some

experiments and numerical calculations on the burning of liquids supplied through a

wick. They have analyzed the burning of a porous slab which is initially saturated with

liquid fuel, during the funicular regime. The liquid is driven by capillarity and the vapor

is driven by the vapor pressure gradient. The effect of gravity and the transport of non-

condensable gases have been neglected. The effect of surface saturation, relative

permeability and vapor flow rate on critical time (time during which the surface become

first dried up) has been studied.

An experimental study was conducted by Kong et. al. (2002) to investigate the

effects of sand size and sand layer depth on the burning characteristics of non-spread

diffusion flames of liquid fuel soaked in porous beds. A porous sand bed initially soaked

with methanol is ignited and the burning characteristics of the flame are studied. The 19 flame temperature profiles, location of vapor/liquid interface, vapor region moving speed, combustion duration time, fuel consumption were studied in the experiments. The results indicate that the fuel consumption rate increased rapidly during the beginning stages. As time proceeded, the formation of vapor region increased the resistance to vapor transport, resulting in a decreasing trend in the fuel consumption rate. Their results confirm that the resistance to vapor transport is the controlling mode when a dry region is formed in the porous bed. There is an abrupt increase in the temperature profile in the dry vapor phase region. In the liquid region, the temperatures are close to the boiling point of the fuel.

They have also studied the effect of sand sizes and the sand layer depths on the burning characteristics of the fuel.

Modeling Two-phase flows in Porous Media

Numerical analysis of multidimensional two-phase flow including phase change in porous media is intrinsically complicated. One reason is the strongly nonlinear and coupled nature of the governing equations for the two-phase flow. Another fundamental difficulty lies in the presence of moving and irregular interfaces between the single and two-phase subregions in a domain of interest. The location of such an interface is not known a priori and must be determined by the coupled flows in adjacent regions.

Primarily three different numerical approaches have been used by researchers – separate flow model, enthalpy model and thermodynamic equilibrium model.

Traditionally separate flow model (Bear, 1972) has been used in which separate equations for the two phases are formulated and the interface between the different regions is explicitly tracked. The explicit tracking of moving interface involves complex coordinate mapping or numerical remeshing (Ramesh and Torrance, 1990). To reduce 20

the complexities of separate flow models, Wang and Beckermann (1993) developed two-

phase mixture model based on enthalpy formulation. The separate equations for the two phases are combined to form a single set of equations for the mixture. In this way, the number of governing differential equations to be solved is reduced by almost half and the rest are replaced by algebraic relations. Based on the enthalpy values of the mixture, the state of the system inside the porous wick is identified. The inherent assumption in this model is that the two-phase region is at a constant temperature (equal to the boiling temperature of the liquid) and hence there is no phase change taking place inside the two- phase region. Phase change is taking place only at the interfaces between the two-phase and the single phase regions. Usually in many physical systems, either condensation or evaporation does occur inside the two-phase region and the temperature of the two-phase region does slightly vary. This can only be accounted by invoking thermodynamic equilibrium relations. Recently, Benard et al. (2005) extended the enthalpy modeling approach of Wang et. al. (1993) by incorporating the equilibrium thermodynamic relations, which determines the thermodynamic state of the system and also accounts for phase change taking place inside the two-phase region.

Raju (2004) developed a steady state one-dimensional stagnation point diffusion

flame stabilized next to a porous media. The detailed one-dimensional heat and mass

transport inside the porous wick has been studied. In this study the liquid in the porous

media is assumed to be driven by capillary action and the vapor is driven by the vapor

pressure gradient induced by the temperature gradient (based on Classius-Clayperon

equilibrium relationship) inside the porous media. These studies reveal that the liquid and 21

the vapor flow counter currently inside the two-phase region. It has also been found out that the steady state solution exists only for stretch rates below a critical value.

1.4. PURPOSE AND SCOPE OF THIS DISSERTATION

The purpose of this thesis is to provide a more complete picture of the candle

flame behavior which is coupled to a porous wick. A more realistic picture is obtained by considering the role of heat and mass transport inside the candle wick. The thesis aims at improving our understanding of wick stabilized candle flames by addressing the following issues

(1) The role of heat and mass transport inside the candle wick

(2) Develop a model for simulating the self trimming of a candle wick

(3) The role of gravity on the shape and size of self trimmed candle flames.

(4) The effect of different wick diameters on burning rate and flame temperature.

The present study is primary computational. The computational results are

generated by solving the full Navier-Stokes equations with a one-step finite rate chemical

reaction rate in the gas phase and two-phase flow equations inside the porous wick.

1.5. DISSERTATION OUTLINE

This dissertation describes numerically the heat and mass transfer taking place

inside the burning candle wick and its effect on the candle flame structure. The first

Chapter of this dissertation describes the background related to candle burning and two-

phase flow inside porous media. A brief review on previous work in this field is

presented. 22

Chapters 2 and 3 include the theoretical and numerical aspects of the current

computational model. In Chapter 2, the governing equations for the fluid flow describing the mass conservation, energy conservation, inert gas species conservation, capillary and thermodynamic relations inside the wick. The numerical method adopted for solving the discretized equations is also included. The description of using multifrontal solvers in this context is presented. Different variants of Newton solvers are also presented.

Preliminary analysis of two-phase flow inside the wick is done by applying a constant heat flux all along the wick surface.

In Chapter 3, the governing equations for the gas phase fluid flow and combustion

reaction are presented using mass, momentum and energy conservation equations. The

pressure-velocity coupling is handled by the SIMPLER algorithm. The numerical method

adopted for solving the discretized equations is also included.

The numerical test results are presented in Chapter 4 of this dissertation. The gas phase model is coupled with the two-phase axisymmetric wick model. The detailed flow structures in the gas phase and in the wick are presented. The phenomenon of self trimming of candle wick is modeled as a burn out of the dry region. Parametric analysis has been done to study the effect of gravity, absolute permeability of the wick, wick diameter and ambient oxygen percentage on the candle flame structure and the burning rate.

Finally, in Chapter 5, a summary will be given and some future work is

suggested.

23

Flow direction

Products of combustion

conduction

Fuel vapor radiation g x

r

oxidizer

Figure 1.1 Schematic of a Candle Flame 24

CHAPTER 2 AXISYMMETRIC WICK MODELING

2.1 FORMULATION OF TWO PHASE FLOW INSIDE POROUS MEDIA

Two phase flow in porous media typically consists of three phases: the solid phase, the liquid phase and the gas phase. In addition to the transport of individual phases, there is a phase change process involved inside the porous media (but no chemical reaction is assumed). Treatment of individual phases from the well known point equations of continuum physics is rather complicated and computationally expensive.

Hence volume averaging technique described by Whitaker (1977) is used to provide a rational route to a set of equations describing the transport of heat and mass in a porous media.

Candle wicks are usually cylindrical in shape although other shapes are also used.

The gas phase and the wick characteristics are assumed to be symmetric around the angular direction and hence the present wick is modeled for an axisymmetric geometry.

2.1.1 MATHEMATICAL FORMULATION

The constitutive equations for the solid, liquid and gaseous phases are volume averaged (Whitaker, 1977). The equations are written in cylindrical coordinates for two phase flow.

In addition to the assumptions of Whitaker (1977), the following additional assumptions are made in this study.

1. The liquid phase is assumed to be continuous. 25

2. The flow is assumed to be laminar. Darcy’s law is assumed to be valid both for

the liquid and gas phases.

3. The transport of non-condensable gases inside the porous wick is neglected. The

term “non-condensable gases”, refers to all the gases other than the fuel vapor

(e.g. ambient air, combustion products like CO2, H2O etc.)

4. Radiative heat transfer inside the porous media is neglected.

5. The surface tension of the liquid is assumed to be constant. It does not vary with

temperature.

6. The vapor is locally in thermal equilibrium with the liquid and the thermodynamic

Gibbs phase equilibrium relations are assumed to be valid in the two-phase

region.

7. Although phase change can occur, there is no chemical reaction in the porous

media.

8. The properties of the wick, like thermal conductivity, permeability etc is assumed

to be isotropic. In principle, the thermal conductivity and permeability can be

different in axial and longitudinal directions.

9. Deformation of the wick material due to thermal stresses or bending of the wick

during burning of the candle is neglected.

10. Ideal gas law is assumed to be valid for the vapor phase

The resultant simplified volume averaged equations are written down for an axisymmetric, steady state, two-phase flow inside a porous wick. 26

Continuity equation:

∂∂11 ∂ ∂ ()ρρlluurvrv++()gg () ρll +() ρgg =0 . (2.1) ∂∂x x rr ∂ rr ∂

The individual terms represent the net mass flux of liquid and vapor at a point in x and r directions respectively.

Momentum equations:

The momentum equations are given by Darcy’s law.

kKrl ⎛⎞∂ Pl ugll=−−⎜⎟ρ , μl ⎝⎠∂x

kKrg ⎛⎞∂ Pg uggg=−−⎜⎟ρ , μg ⎝∂x ⎠ (2.2-5) kKrl ⎛⎞∂ Pl vl =−⎜⎟, μl ⎝⎠∂r

kKrg ⎛⎞∂ Pg vg =−⎜⎟. μg ⎝⎠∂r

The liquid phase is treated as incompressible and the gaseous phase is treated as an ideal gas.

Energy equation:

∂∂11 ∂⎛⎞⎛∂TT∂∂⎞ ()ρρlllhu++ gg h u g () r() ρρlll hv + ggg h v =⎜⎟⎜ keff +rkeff ⎟. (2.6) ∂∂x rr ∂x⎝⎠⎝∂ x rr∂∂ r⎠

Using the relations hcThll== and gg cTi+ fg, equation (2.6) becomes

∂∂11⎛⎞∂∂ρρllurv ll ()ρρlllcu++ gg c u g T() r() ρρlll cv + ggg c v T +−− i fg⎜⎟= ∂∂xrr⎝⎠ ∂ xr∂r (2.7) ∂∂⎛⎞⎛TT1 ∂ ∂ ⎞ ⎜⎟⎜kreff + keff ⎟ ∂∂∂xxrrr⎝⎠⎝ ∂ ⎠

The first two terms on the left hand side of equation 2.7 represent the convective heat transport of liquid and vapor in the x and r direction. The third term represents the heat 27 source term due to phase change taking place between the liquid and the vapor. The right

hand side of this equation represents the conductive heat transfer. keff represents the effective thermal conductivity.

Capillary and permeability relations

The gas pressure is related to the liquid pressure using the capillary relation

Pscg( ) = P− Pl. (2.8)

The capillary pressure is related to the saturation given by Leverett’s function (Leverett,

1941),

σ ⎡ 23⎤ Pssc =−1.42() 1−− 2.12 () 1+ 1.26 () 1−s. (2.9) ()K / ε 1/2 ⎣ ⎦

The relative permeability of the porous media is given by the following approximation

(Bau and Torrence, 1982)

kskrl = , rg =−( 1 s) . (2.10 a-b)

The non-dimensionalization of the porous wick variables are carried out according to the variables indicated in table 2.1. The non-dimensionalized variables are indicated by a

‘hat’ symbol on the top of the variable.

Non-dimensionalized equations:

∂∂11 ∂ ∂ ()uurvrvˆˆˆˆˆˆlg+++()ρˆg() l()ρˆ gg=0 , (2.11) ∂∂xˆˆ x rr ∂ˆˆ rr ∂ˆ 28

⎛⎞∂Pˆ uPekˆˆ=−−l g, lrl⎜⎟ ⎝⎠∂xˆ Pek⎛⎞∂ Pˆ ugˆˆ=−−rg⎜⎟ g ρˆ , ggμˆˆ⎜⎟∂x g ⎝⎠ (2.12-15) ⎛⎞∂Pˆ vPekˆ =−l , lrl⎜⎟ ⎝⎠∂rˆ Pek⎛⎞∂ Pˆ vˆ =−rg⎜⎟ g . g μˆˆ⎜⎟∂r g ⎝⎠

∂∂11⎛⎞∂∂urvˆˆˆ∂⎛⎞⎛∂TTˆˆ1∂∂⎞ ˆˆˆˆˆˆˆ ˆˆˆˆ ˆ ll ˆˆˆ ()ucuTrvcvTil+++ρρ g g g ()()l g g g +−−=+ fg ⎜⎟⎜⎟⎜keff rkeff ⎟ ∂∂xˆˆrrˆ⎝⎠ ∂x ˆ rˆ∂ rˆˆˆ∂ x⎝⎠⎝∂ x rrˆ∂ˆ∂ rˆ⎠ (2.16)

The effective thermal conductivity of the wick is function of saturation given by the expression (Udell and flitch, 1985)

ˆˆ kseff =+(1 − sk) s , (2.17)

ˆ where ks is the thermal conductivity of the solid wick material.

Conditions for Phase Transition:

The equilibrium thermodynamic state of candle wax (single phase liquid, single phase vapor, two-phase) can be determined for given liquid and gaseous pressures and temperature conditions, using the vapor pressure equilibrium data obtained from the thermodynamic Gibbs phase relationships (Benard et. al. 2005).

GhTˆ =−ˆ ˆηˆ , llll (2.18-19) ˆ ˆ ˆ GhTg =−gggηˆ , where G is the Gibbs potential per unit mass of the corresponding phase and symbol

“hat” denotes the non-dimensional value. h and η are respectively the enthalpy and the 29 entropy of the corresponding phase. The dimensional expressions for the enthalpy and entropy are given by

hcTTll=−( 0 ),

higfgg=+ cTT() −0 , (2.20-23) ηll= cTTlog() /0 ,

ηgfgg=+iTc/00 log() TTR / − log() PPg /0 .

The thermodynamic equilibrium relations based on the minimization of Gibbs function

(Saad, 1966) are given as follows

ˆˆ State 1. GGlg< , no vapor phase is present ( s =1 )

ˆˆ State 2. GGlg= , liquid and vapor are in equilibrium ( 01< s < )

ˆˆ State 3. GGl> g, no liquid phase is present ( s = 0 )

(2.24-26)

Equations 2.24-26 simply mean that whichever phase has the least Gibbs phase potential will dominate over the other phase. Equations 2.18-23 in conjuction with the phase equilibrium condition (Eq. 2.25) yield the well known, Classius-Clayperon equation.

Boundary conditions:

(1) Base of the wick

The wick is immersed in the candle wax pool, which can be assumed to be at its melting temperature.

ˆˆ sT=1, = Tm

(2) Cylindrical surface of the wick 30

Depending on the thermodynamic state of the wick on the surface, the boundary conditions will vary. In the case of either pure liquid or pure vapor, all the heat flux imposed on the surface is conducted into the wick. In the case of two-phase region, part of the heat supplied is used for evaporating the liquid on the surface of the wick and part of the heat is conducted into the wick.

ˆ ˆ ⎛⎞∂T Liquid region: sv==1, ˆˆlfe 0, qk =−ff⎜⎟ ⎝⎠∂rˆ

⎛⎞ˆ ˆ ∂T ˆ ˆˆ Two-phase region: 0<

ˆ ˆ ⎛⎞∂T Vapor region: spppqk==−0, ˆˆˆlcfe0 (0), ˆ =−ff⎜⎟ ⎝⎠∂rˆ

(3) Tip of the wick

Similar to that of the boundary conditions on the cylindrical surface.

ˆ ˆ ⎛⎞∂T Liquid region: sv==1, ˆˆlfe 0, qk =−ff⎜⎟ ⎝⎠∂rˆ

⎛⎞ˆ ˆ ∂T ˆ ˆˆ Two-phase region: 0<

ˆ ˆ ⎛⎞∂T Vapor region: spppqk==−0, ˆˆˆlcfe0 (0), ˆ =−ff⎜⎟ ⎝⎠∂rˆ

(4) Symmetry line

Symmetry boundary conditions are imposed along the symmetry line 31

2.1.2. NUMERICAL FORMULATION:

The continuity, momentum and energy equations are discretized using finite difference approximation. This results in a set of non-linear discrete balance equations.

These equations are coupled with the inequalities resulting from the phase equilibrium relationships. The system is solved using Newton’s method. The thermodynamic state of each grid block is updated at each iteration of the method. In this method there is no need to separately track the interface between the single phase and the two-phase regions.

Equations 2.11-2.16 are discretized using finite difference approximation and combined appropriately with the non-dimensionalized forms of equations 2.8-2.9 to form

2 equations for the variables Psl , and T. The equations are closed by using the thermodynamic equilibrium relationships (equations 2.24-2.26). The thermodynamic relationships are incorporated in the iteration scheme as described below.

For each grid node i, we set a thermodynamic state parameter χi ( 1 for pure liquid , 2 for liquid-vapor equilibrium, 3 for pure vapor) which is determined based on

the previous guess value of saturation distribution, ζ ilii= ( p ,,sTi) the triplet of unknowns and

Bs()ζ i ,1=− 1, ˆˆ B()ζ il,2=−GGg , (2.27-29)

Bs()ζ i ,3= .

Note here that B is the thermodynamic equation which depends on the

thermodynamic state χi of the system at any given grid point. The only speciality of this 32 formulation is that the equation B is different at each grid node depending upon the

thermodynamic state at that grid point χi .

Based on the thermodynamic relationships described in the previous section, the following relationships hold

if χζii== 1, then BB( ,1) 0, and ( ζ i ,2) < 0,

⎧⎪B()ζ i ,1< 0 if χζii== 2, then B() ,2 0, and ⎨ , (2.30-32) ⎩⎪B()ζ i ,3> 0

if χζii== 3, then BB() ,3 0, and () ζ i ,2> 0.

The solution procedure is described as follows

Let ζ ()n , χ (n) , be the values at the beginning of (n+1) th iteration and α be the under relaxation parameter. First ζ (n+1) is computed from Newton’s step.

JF()ζχ()nn,, () () δζ() n=− ( ζχ() nn ()), (2.33-34) ζζαδζ(nn+1 )=+ () ()() n.

Next the thermodynamic variable is updated at each grid node by the following relations

()nn()++11() n if χζii=> 1 and if B( ,2) 0, then χ i= 2; ()nn()++11()n if χζii=> 2 and if B() ,1 0, then χ i= 1; ()nn()+1()n+1 if χζii=< 2 and if B() ,3 0, then χ i= 3; ()nn()++11()n if χζii=< 3 and if B() ,2 0, then χ i= 2; ()nn+1 () otherwise χχii= .

The linear system of equations 2.33 is solved using a direct sparse solver

UMFPACK, which is based on a multifrontal technique. This solution procedure is outlined in the next section. Since the system of equations is highly non-linear, very high 33 under relaxation parameter of 0.001 is imposed to ensure smooth convergence of the variables.

2.2 MULTIFRONTAL SOLVERS FOR LARGE SPARSE LINEAR SYSTEMS

This section describes the implementation of sparse direct solvers based on multifrontal techniques for solving the highly non-linear equations resulting from the two-phase flow equations inside the porous wick. The implementation of standard solvers like ADI solvers failed to produce convergence. Implementation of direct solution techniques based on sparse Gaussian elimination (specifically the multifrontal technique) resulted in a stable convergent solution and henceforth developed in this study. A modified Newton’s method is implemented, which when coupled with a multifrontal solver resulted in significant reduction in the computational time. The description of multifrontal technique is beyond the scope of this work but an attempt is made to give a brief introduction of the sparse matrix storage techniques and frontal solution techniques. Finally, the efficiency of the multifrontal solver is tested for a differential cavity benchmark problem.

2.2.1 INTRODUCTION

Solving general sparse linear systems can be accomplished either by direct solution methods or by iterative solution techniques. Earlier direct solution method was often preferred to iterative methods in real applications because of their robustness and predictable behavior. Iterative solvers were often special-purpose in nature and were developed with certain applications in mind. For Finite Volume problems, the ADI method (Peaceman and Rachford, 1955) and implicit Stone (Stone, 1965) algorithms are 34 preferred because of their high computational efficiency and very low memory requirements compared to the direct solvers.

Over the years in 1960’s and 1970’s, there has been significant development in the solutions of large linear systems. Techniques were developed to take advantage of the sparsity to design special direct methods that can be quite economical. The frontal method (Irons, 1970 and Hood, 1976) is a variant of Gaussian elimination and makes use of the sparsity pattern for matrices resulting from the discretization of PDE’s. Later Duff

(1984), and Duff et al. (1986) extended the frontal techniques for solving any general sparse matrices. Over the recent years, there has been significant improvement in the development of multifrontal direct solvers which take advantage of the sparsity pattern and the Level 3 BLAS routines to enhance the computational speed on high power computing architectures.

Before going into the discussion of solution techniques for sparse linear systems, it is important to discuss the different sparse storage schemes for storing the non-zero elements of a sparse matrix.

Sparse Matrix Storage Schemes

In order to take advantage of the large number of zero elements, special schemes are required to store sparse matrices. The main goal is to represent only the nonzero elements and to be able to perform the common matrix operations. Only the most popular schemes (Duff et al., 1986) are presented. The following material is taken from Saad,

(2003) and is presented here for completeness.

Compressed row format 35

A(n*n) with nz nonzero entries is represented using three arrays ax(nz), aj(nz), ai(n+1). The entries are entered row wise.

Example: n=5, nz = 12

[1 2 3 ] [ 4 5 ] [6 7 8 ] [9 10 ] [ 11 12 ]

row pointer array, ai(n+1) = [ 1 4 6 9 11 13 ]

element value array, ax(nz) = [ 1 2 3 4 5 6 7 8 9 10 11 12 ]

element column index array, aj(nz) = [ 1 3 4 2 5 1 3 4 1 4 3 5 ]

Compressed column format

A(n*n) with nz nonzero entries is represented using three arrays ax(nz), ai(nz), aj(n+1). The entries are entered column wise.

Example: n=5, nz = 12

[1 2 3 ] [ 4 5 ] [6 7 8 ] [9 10 ] [ 11 12 ]

Column pointer array, aj(n+1) = [ 1 4 5 9 11 13 ]

element value array, ax(nz) = [ 1 6 9 4 2 7 10 11 3 8 5 12 ]

element row index array, ai(nz) = [ 1 3 4 2 1 3 4 5 1 3 2 5 ]

Triplet format 36

A(n*n) with nz nonzero entries is represented using three arrays ax(nz), ai(nz), aj(nz). Any nonzero entry is identified by a triplet (ax,ai,aj), where ax is the value of the entry, ai is the row entry and aj is the column entry

Example: n=5, nz = 12

[1 2 3 ] [ 4 5 ] [6 7 8 ] [9 10 ] [ 11 12 ]

element value array, ax(nz) = [ 1 2 3 4 5 6 7 8 9 10 11 12 ]

element row index array, ai(nz) = [ 1 1 1 2 2 3 3 3 4 4 5 5 ] element column index array, aj(nz) = [ 1 3 4 2 5 1 3 4 1 4 3 5 ]

2.2.2 MULTIFRONTAL SOLUTION METHODS

Sparse direct solution methods involve band width algorithms, unifrontal and multifrontal techniques. The frontal techniques are much efficient compared to the band width algorithms and are widely used in the solving matrices resulting from finite element formulations.

This section gives a brief introduction to unifrontal and multifrontal techniques which are direct methods for solving the linear equation Ax= B where A can be a symmetric, unsymmetric, definite or indefinite matrix. Unifrontal method is a derivative of the classical gaussian algorithm. Although it is developed for finite element applications, it can be used in other fields also. It uses a small dense sub-matrix called

“frontal matrix’’. Original matrix is read sequentially and frontal matrix is filled with the rows and columns. All-rows must be chosen to be fully-summed i.e. there are no further 37 contributions to come to the rows. When the frontal is filled with such rows, pivot(s) are chosen from the fully-summed columns and basic gaussian elimination start. LU factors are stored in RAM or in disk. Shur complement for the part that cannot be eliminated further is calculated. Pivot row(s) and column(s) are deleted from the frontal to accommodate new elements coming from original matrix. Having built the new frontal, previous steps are repeated. Frontal matrices can be rectangular and doesn't have to be the same size. But important thing is that, there is no more than one frontal at the same time.

Since the same working array is used throughout the factorization, the method is memory-efficient. Partial pivoting among the fully-summed columns may be applied to preserve numerical stability. Unifrontal method is competitive with the best band matrix routines.

Frontal method is first described by Irons (1907) with the article ``A frontal solution program for finite element analysis''. Method is primarily designed for finite element analysis and for symmetric and positive definite matrices. Later, Hood (1976) modified the method for unsymmetric matrices. Again the algorithm was designed to solve finite element problems. In both papers, performance and memory management issues are compared with band matrix routines. Frontal method is found to be efficient especially in the case where the bandwidth is large. Band matrix routines require too much memory for large bandwidths so factorization must be preceded by an ordering phase to obtain smaller bandwidths.

In 1983, ``Multifrontal Method'' was derived by Duff and Reid (1983) in which more than one frontal matrix is used. This method was first designed for symmetric and undefinite matrices but later modified for unsymmetric and definite matrices. If there is 38 more than one frontal matrix at the same time unifrontal method is called Multifrontal method. The description of multifrontal algorithm is beyond the scope of this work, but a brief overview is presented here. Since couple of frontals exists, there are some dependency relations between them. These relations are analyzed to build the assembly tree (factorization tree). This tree roughly says which frontal must be handled first and which one is last. Then from the leaf nodes to the root, factorization starts. First, leaf nodes are factorized and assembled to the parents i.e. Shur complements are calculated and contribution blocks are sent to upper nodes. Since, there is more than one frontal matrix at the same time, management of these sub-matrices, calculating shur complements and assembling them to the upper part of the tree is a difficult task.

However, it gives the chance for doing these calculations in a parallel environment.

Additionally, multifrontal method is more powerful then unifrontal method in the case of the matrices of which bandwith is very large.

A parallel version of multifrontal method is also introduced in by I. S. Duff

(1986). Parallel implementation is based on the elimination tree concept. There is more than one frontal matrix in multifrontal approach and these sub-matrices also called nodes are dependent on each other. If one draw the dependency graph it will be observed that the graph will be a tree. Since leaf nodes of the tree are independent of each other, they can be handled separately. This is the first step of parallelization. At later steps, other nodes are factorized by different processors and so on.

In the present work, a multifrontal solver (Davis et al., 1997, 1999 and 2004) termed as the unsymmetric multifrontal package (UMFPACK v2.2) is used. UMFPACK is a set of routines for solving unsymmetric sparse linear systems, Ax=b, using the 39

Unsymmetric MultiFrontal method and direct sparse LU factorization. UMFPACK v2.2

(Davis et al., 1999) is a set of fortran subroutines. UMFPACK relies on the Level-3 Basic

Linear Algebra Subprograms (dense matrix multiply) for its performance. This code works on Windows and many versions of Unix (Sun Solaris, Red Hat Linux, IBM AIX,

SGI IRIX, and Compaq Alpha). The present code is being run on a windows machine

(Intel Pentium 4 machine) using optimized GOTO BLAS library available for free download from the internet. Using GOTO BLAS has enhanced the performance of

UMFPACK significantly.

Direct solution methods for solving set of non-linear equations usually involve a

Newton’s linearization step to convert the set of non-linear equations into a set of linear equations which is then updated during each iterative step. The disadvantage of using direct solution methods is that the solution of set of linear equations during each iteration takes a huge amount of computational time. Moreover it requires a huge memory. There has been significant improvement in the available of large RAM in the present market. In addition, the onset of 64 bit machines has made it possible to access more than 2GB

RAM. Thus it has been possible to solve even large scale problems using direct solvers.

The development of efficient direct sparse solvers has significantly reduced the computation time for solving a linear system. The most time consuming step in solving the linear system is the factorization step. In the present work, a modified Newton’s method is used which can save a significant amount of computational time.

Newton step:

Jx()nnδ () =− F() n, () (2.35-36) x()nn+1 =+xx ()αδ ()n. 40

Modified Newton step:

Jx()0 δ ()nn=− F(), () (2.37-38) x(nn+1 )=+xx ()αδ ()n, where F is the function residual, J is the Jacobian matrix consisting of the derivatives of the function residual, x and δ x are the variable and the increment in the variables to be solved, n is the iteration number and α is the under relaxation parameter.

In the modified Newton’s step, the Jacobian matrix is not updated. It is calculated only during the first iteration and the same Jacobian matrix is reused. The advantage with

Modified Newton’s method is that the left hand matrix needs to be factorized only during the first iteration. The factors calculated by UMFPACK solver during the first iteration are stored for reuse in the subsequent iterations. Since the factorization is the most expensive step, by skipping the factorization step in the subsequent iterations, the subsequent iterations become very cheap. Although the rate of convergence is reduced compared to the Newton’s step, the savings in computational time per iteration overrules the decrease in convergence and hence significant amount of saving is obtained.

2.2.3 BENCHMARK TESTING

To test the performance of sparse direct solvers over the standard SIMPLE technique used for solving Navier-Stokes equations, a differential cavity problem is chosen as a benchmark problem and the total computational times are compared.

Differential Cavity problem

∂U ∂V + = 0, (2.39) ∂X ∂Y ∂U ∂U ∂P ⎛ ∂2U ∂2U ⎞ U +V −= + Pr⎜ + ⎟ , (2.40) ⎜ 2 2 ⎟ ∂X ∂Y ∂X ⎝ ∂X ∂Y ⎠ 41

∂V ∂V ∂P ⎛ ∂2V ∂2V ⎞ U +V −= + Pr⎜ + ⎟ + PrTRa , (2.41) ⎜ 2 2 ⎟ ∂X ∂Y ∂Y ⎝ ∂X ∂Y ⎠ ∂T ∂T ⎛ ∂2T ∂2T ⎞ U +V = ⎜ + ⎟. (2.42) ⎜ 2 2 ⎟ ∂X ∂Y ⎝ ∂X ∂Y ⎠

The boundary conditions for the driven cavity problem can be expressed as

U = V = ,0 allat other boundaries T = ,1 X = ,0 T = ,0 X = ,1 ∂T = ,0 ,0 and YY == .1 ∂Y

Figure 2.1 compares the function residuals FF= for the Direct solvers ( R ∞ )

(Newton, Picard and Modified Newton) and the SIMPLE method. Figure 2.1 indicates that the direct solvers are more efficient compared to the SIMPLE method. Modified

Newton’s method is found to extremely efficient compared to all the other methods. This is because of the skipping the factorization step after the first iteration which leads to a significant amount of savings in computational time.

2.3 ANALYSIS OF AN EXTERNALLY HEATED AXISYMMETRIC WICK

In a realistic candle, both the candle flame and the wick are coupled to each other.

The heat from the candle flame evaporates the candle wax, providing the driving force for the liquid to rise up through capillary action. The wax evaporated from the surface provides the fuel for the candle flame. In this way the fuel supplied by the wick and the heat supplied by the candle flame are coupled together.

The heat and mass transfer taking place inside the porous wick is very complex due to the presence of both liquid and vapor inside the wick. To gain sufficient insight into the physics of two-phase flow inside the wick, the wick is first analyzed separately 42 by decoupling the wick from the flame. A constant heat flux is imposed along the cylindrical surface and at the tip of the wick. The detailed structure and the flow patterns inside the porous wick are analyzed. Later, the wick is coupled to the candle flame and the detailed study of gas phase flame characteristics and porous flow fields is done in chapter 4.

2.3 1 PHYSICAL DESCRIPTION OF THE MODEL

The wick is now treated as being heated from a constant external heating source.

The heat flux is uniformly distributed along the cylindrical surface of the wick and on the tip surface of the wick (Figure 2.2). Experiments have been conducted by Zhao and Liao

(2000) to study the heat transfer characteristics of a capillary-driven flow in a porous structure heated with a permeable heating source at the top. Their experimental set up is essentially one-dimensional in nature. This present physical situation is different from their experimental set up but their experiments reveal certain essential characteristics of two-phase flow inside a wick.

In this present set up, the wick is dipped in a liquid wax pool. The level of the wax pool is assumed to be constant. The wax pool is at the melting point temperature of the wax (330 K). A constant heat flux is applied along the cylindrical surface and on the wick tip. A wick of length 5mm and diameter 1mm is chosen for the present study. The two-phase flow inside the wick is simulated.

Figure 2.3 shows the computational grid used for the axisymmetric wick model.

The grid size is chosen as 80x40. Grid clustering is used in both the x and r directions. 43

2.3.2 SAMPLE CASE RESULTS

The porous wick parameter chosen are shown in the Table 2.2. A constant heat flux of 810× 4 W/m2 is applied on the wick as described above. The detailed structure inside the porous wick is presented below.

2.3.2.1 Saturation and Temperature Distribution Figure 2.4(a) shows the saturation profiles inside the porous wick. The figure indicates that there are two regions inside the wick – single phase liquid region and two- phase vapor liquid region. The contour line s=1 demarcates the two-phase region and the pure liquid region. The saturation is lowest on the cylindrical corner of the wick reaching a value of 0.825. As the wick is receiving heat on both the cylindrical surface and the tip surface from the external heating source, the evaporation causes a decrease in the saturation at the surface of the wick. Figure 2.4 (b) shows the non-dimensional temperature distribution inside the wick. The base of the wick is at the melting point (323

K) of the liquid wax and on the wick surface, where there is evaporation; the temperature is at its boiling point (620 K). Fig. 2.4 (b) shows the presence of temperature gradients near the base of the wick indicating the heat lost to the wax pool. Figure 2.4 (c) shows the temperature contours inside the two-phase region of the wick. Notice that the temperature variation inside the two-phase region is very small. Still this slight temperature variation causes a significant variation in the vapor pressure distribution inside the two-phase region given by the equilibrium phase relations (Eq. 2.25).

2.3.2.2 Pressure Distribution Figure 2.5 (a) shows the non-dimensional liquid pressure distribution inside the wick. This drives the liquid from the base of the wick to the surface of the wick. As the 44 wick is receiving heat on both the cylindrical surface and the tip surface from the external heating source, the evaporation causes a decrease in the saturation at the surface of the wick. This causes a decrease in liquid pressure, given by capillary relations, on the surface (both the cylindrical surface and the wick tip surface) of the wick. Pressure gradients are present along the length of the wick and in the radial direction indicating liquid motion in both the directions (refer fig. 2.4 (a)). Figure 2.5 (b) shows the non- dimensional capillary pressure distribution inside the wick. Capillary pressure is a function of the saturation (refer Eq. 2.9). Hence the capillary pressure distribution is qualitatively similar to that of the saturation distribution. At the interface between the liquid and the two-phase region, the capillary pressure is zero. Figure 2.5 (c) shows the non-dimensional vapor pressure distribution inside the wick. The vapor pressure in the two-phase region is a function of the temperature as given by the Gibbs phase relations

(Eq. 2.25). Hence the vapor pressure distribution variation is qualitatively similar to the temperature variation inside the two-phase region. Figure 2.5 (c) shows sharp vapor pressure gradients in the r-direction near the cylindrical surface of the wick.

2.3.2.3 Mass flux Distribution Figure 2.6 (a) shows the liquid mass flux vectors indicating the flow of liquid inside the wick. The liquid is drawn from the base and it comes out of the wick along the cylindrical surface and the tip of the wick. As the liquid is evaporated along the cylindrical surface, the liquid mass flux along the length of the wick decreases. Figure

2.6 (b) shows the vapor mass flux distribution inside the wick. Vapor motion is very intricate and also interesting. Vapor is being driven by the temperature gradients inside the two-phase region. Since the temperature gradients (related to the pressure gradient 45 through the equilibrium relationship Eq. 2.25) are directed into the wick, the vapor moves into the wick. The liquid and the vapor move counter currently to each other. This is also found in one-dimensional wick model [Raju, 2004] and is confirmed by experimental results [Zhao and Liao (2000)]. The vapor eventually condenses at the interface between the liquid and the two-phase region.

2.3.2.4 Heat flux Distribution Figure 2.7 shows the heat flux distribution inside the porous wick. There is convective heat flux due to both liquid and vapor motion and there is also conductive heat flux due to temperature gradient. The heat flux vectors can be expressed as follows

qcTv*****= ρ llll qcTi*****=+ρ v* (2.43-45) g gg( fgg) *** qkTceff=− ∇

Equation 2.43 represents the heat flux vector due to liquid convection. This is depicted in figure 2.7 (a). Equation 2.44 represents the heat flux vector due to vapor convection. This also includes both the sensible heat and the latent heat of evaporation.

This is depicted in Figure 2.7 (b). These vectors are qualitatively similar to the vapor mass flux vectors. Equation 2.45 shows the heat flux vectors due to conduction. This is depicted in Figure 2.7 (c). Since the temperature variation in the two-phase region is very less, the conductive heat flux vectors are small. In the liquid region, there is significant variation in temperature and hence the conduction is prominent in the liquid region.

46

2.3.2.5 Variation along the Cylindrical Surface and the Axis of the Wick Figure 2.8 shows the saturation and the temperature distribution along the cylindrical surface of the wick. The saturation plots shows the presence of liquid region

(s=1) near the base of the wick. The rest of the surface is in two-phase region. The temperature in the liquid region is below the boiling temperature. Figure 2.9 shows the liquid and vapor motion in the r-direction along the cylindrical surface. This figure gives an indication of the evaporation taking place on the surface. Constant heat flux is applied on the surface. In the liquid region near the base of the wick, no liquid is evaporated from the surface. This implies that all the heat supplied on the surface in this region is conducted into the surface and no evaporation takes place. As we move into the two- phase region, a part of the heat is used for evaporating the liquid at the surface and a part of it is conducted into the wick. Therefore the mass flux of liquid evaporated from the surface increases. A part of the liquid that is evaporated at the surface is convected into the wick interior as vapor. Therefore the vapor mass flux shows negative values. The net mass flux of vapor goes out of the wick surface. The vapor mass flux first increases as we traverse along the surface and then it reduce gradually to zero at the tip of the wick.

The reason for this is explained as follows. The vapor motion (Eq. 2.15) is a function of the vapor saturation (1-s) and the vapor pressure gradient (equivalently temperature gradient). The temperature gradient in the r-direction decreases as we traverse to the tip along the cylindrical surface of the wick and the vapor saturation correspondingly increases. The maximum vapor mass flux is achieved in between as a result of the interaction of the two terms. Figure 2.9 shows a maximum vapor mass flux is achieved at x = 2.8 mm. Figure 2.10 shows the variation of the x direction liquid mass flux along the 47 cylindrical surface. There is a continuous decrease in the mass flux due to mass loss by evaporation along the surface of the wick.

Figure 2.11 shows the variation of the saturation and temperature along the axis of the wick. The two-phase region starts at approximately x = 3.8mm. The variation of temperature in the two-phase region (depicted in the sub figure) is very small but it is significant in the liquid region. Figure 2.12 shows the liquid and the vapor mass flux along the axis. The vapor mass flux is negative in the two-phase region indicating that the vapor is moving inwards into the wick. At the interface between the liquid and the two-phase region, the entire vapor condenses. The liquid mass flux continuously decreases because the liquid is continuously drawn along the cylindrical surface of the wick.

2.3.3 MESH REFINEMENT STUDIES

Mesh refinement studies has been performed. The above sample case was run with different mesh sizes to see its effect on the porous wick solution. Mesh sizes 80x40,

80x80, 160x40has been run.

Figure 2.13 shows the comparison of saturation, non-dimensional liquid pressure and non-dimensional temperature contours for all these mesh sizes. The contours give agree very closely for all the three meshes being chosen. The reference grid size is chosen as 80x40. Doubling the grid size either in the x-direction or in the r-direction did not affect the solution much. So the grid size 80x40 is chosen for all the subsequent calculations in this work. 48

2.3.4 THE EFFECT OF APPLIED HEAT FLUX

The effect of heat flux on the heat and mass transport inside the porous wick has been studied. A direct consequence of the variation in applied heat flux is that the amount of wax evaporated from the wick surface varies. The detailed distribution of saturation, pressure and temperature inside the wick also changes with the applied heat flux.

Figure 2.14 shows the variation of saturation at the tip of the cylindrical surface of the wick with the heat supplied to the wick. This is the location where the saturation reaches its minimum value. It gives an overall idea of the saturation distribution inside the wick surface. As the heat supplied to the wick increases, more evaporation takes place from the surface leading to a decrease in saturation distribution on the surface of the wick. This decreased saturation distribution on the wick surface causes the capillary action to increase and hence more wax is drawn from the wax pool. Figure 2.14 shows that the rate of decrease of saturation with the heat supplied increases drastically as the saturation approaches zero. The tip saturation approaches a value of zero for total heat of

1.65X10-5 W. With further increase of heat supplied to the wick surface, numerical difficulties were observed in obtaining a converged solution. This may be due to lack of robustness of the thermodynamic model being used to solve for evaporative front regime.

The code exhibited spurious oscillations. The behavior of the steady state code with the appearance of evaporative front regime seems to indicate that the system might be transient in nature. Figure 2.15 shows the effect of applied heat flux on the total mass of wax evaporated from the wick surface (both from the cylindrical surface and from the wick tip). The mass evaporated from the wick surface varies linearly with the applied 49 heat. The heat that is being supplied to the wick surface is mostly used for evaporating the wax from the surface (supplies both sensible heat to raise the temperature of wax from 330K to 620K and also the latent heat of vaporization at 620K) and a part of it is lost to the wax pool. Figure 2.16 shows the percentage of heat supplied that is lost to the reservoir. At lower heat fluxes, the percentage of heat loss to the reservoir is large compared to that at high heat fluxes. This is also observed in the one-dimensional porous wick analysis (Raju, 2004).

2.3.5 PARAMETRIC STUDY

2.3.5 1 The Effect of Gravity The effect of gravity on the two phase flow in porous wick is studied applying a constant heat flux on the surface of the wick. Gravity acts as an opposing force to the upward motion of the liquid induced by capillary action. This is important to understand the role of gravity on the heat and mass transport inside the wick. When the porous wick is coupled to the candle flame, the effect of gravity on the porous transport cannot be isolated because gravity influences both the porous wick transport and the candle flame through the buoyancy effect. Figure 2.17 shows the effect of gravity on the saturation at the tip of the cylindrical surface of the wick. Although the gravity is varied from 0ge to

2ge, the change in the saturation distribution is small. The capillary driving force seems to dominate the gravity forces for the given set of porous wick parameters and wax properties. The order of estimate of capillary and gravity forces is given below.

∂pl capillary force ps()= 0 = ∂x ∼ c gravity force ρρllwggL capillary force ∼ 122 (for normal gravity) gravity force 50

Hence the effect of gravity on the heat and mass transport in the porous wick seems to be negligible. Figure 2.18 shows the effect of the gravity on the mass of wax evaporated from the wick surface. The mass evaporated from the wick surface remains unchanged with gravity. Since the problem is a steady state problem, this result is expected unless the heat that is lost to the reservoir is affected by gravity.

2.3.5 2 The Effect of Absolute Permeability Absolute permeability is a measure of the ability of porous wick to transport fluids through the medium. It is similar to the concept of thermal conductivity in heat flow. The absolute permeability is a function of the wick material chosen and the way the wick is formed from the fibrous material. Studying the effect of changing the absolute permeability gives us an idea of how the heat and mass transport inside the wick is affected by the wick material or the type of wick being chosen. Figure 2.19 shows the effect of absolute permeability on the saturation at the tip of the cylindrical surface of the wick. Decreasing the absolute permeability decreases the tip saturation. This can be explained as follows. For a given heat flux, the mass evaporated from the wick surface should remain approximately same for a steady state problem. By decreasing the permeability, the resistance to the flow inside the porous media increased. To maintain the flow rate the driving force (capillary action) should increase. This is affected by decreasing the saturation at the surface of the wick. For a wick with high permeability, the opposite effect takes place and hence the saturation at the surface is less. For wicks with low permeability values, the tip saturation reaches zero for less heat flux values and vice versa. 51

Figure 2.20 shows the effect of absolute permeability on the mass of wax evaporated from the surface of the wick. There is a slight decrease in mass evaporated from the wick surface for high permeability values. This is caused due to slight increase in heat lost to the reservoir at high permeability values.

In effect, gravity does not play a major role in the heat and mass transport of the inside the wick, but permeability of the wick plays an important role in determining the phase (saturation) distribution inside the wick.

52

Table 2.1 Porous Wick Dimensionless Variables

* cg cg * cl

** ( ερ ) 21 LKg * g g σ

* i fg i fg * * l Tc 0

K rg (1− s)

K rl s

εσ()K ε 12 Pe αμll

* (Kp ε ) 21 p g g σ

ρε**RT( K )12 R lg0 g σ

ε s l ε

T * T T * b

* uDg u g αl

* uDl ul αl

x* x D 53

* μ g μ g * μl

* ρ g ρ g * ρl

Table 2.2 Porous Wick Numerical Values

* cg − KkgJ 1430

* cl − KkgJ 2452

* 5 i fg kgJ 8.8×10

* e ()sk = 0 − KmW 6.40

* e ()sk = 1 − KmW 6.31

K m 2 1.09×10-11

* Rg − KkgJ 25.2

* Tm K 330

* Tb K 620

ε 0.55

* 2 -6 α l sm 3.4×10

* 3 ρl mkg 770

* -4 μl ⋅ sPa 5×10

* -5 μg ⋅ sPa 1.1×10

σ mN 0.035

54

Differential Cavity (Ra = 104, 60x60 mesh)

100

10-1 NEWTON

-2 PICARD(α=0.6) 10 SIMPLE(α=0.8) MODIFIED NEWTON ( α=0.3) 10-3

10-4 R F 10-5

10-6

10-7

10-8

50 100 150 CPU time (s)

Figure 2.1 Comparison of function residuals vs. CPU times for Newton, modified Newton and Picard’s iterative techniques

55

Dw=1mm q

g

q Wick q Lw=5mm

Liquid wax pool at its melting point

Figure 2.2 Physical description of an externally heated axisymmetric wick.

56

80x40 g 0.5

0.4

0.3

0.2 r(mm)

0.1

0 0 1 2 3 4 5 X (mm) Figure 2.3 Computational grid of an externally heated axisymmetric wick.

57

0.6 Saturation contours 0.5 0 0 0 w 0.4 . .9 0 0 . 84 0.830 5 . . 8 0.3 0 9 8 5 0 7 2 0 0 r/D 0.2 1 5 .0 0.1 0 0 0 1 2 3 4 5 X/Dw (a) 0.6 Non-dimensional temperature contours 0.5 0. 0 0.99 999 w 0.4 0 .9 0 7142 54 0 0 . 80 .9 3 . .8 95 4 9 0.3 6 6 3 68 2 5 3 7 0

r/D 1 0.2 7 13 3 2 8 4 7 3 0.1 4 0 0 1 2 3 4 5 X/Dw (b) Non-dimensional temperature contours (expanded in two-phase region) 0.5 0 0 0. .99 0.4 0.9 0.99 .9 999 99 988 969 99 97 95 w 18 8 9 7 0.3 10 0.2 r/D 0.1 0 3 3.5 4 4.5 5 X/Dw (c)

Figure 2.4: Plot of (a) saturation profiles (b) non-dimensional temperature profiles and (c) non-dimensional temperature profiles (expanded in the two-phase region) inside the porous wick for parameters shown in Table 2.2.

58

0.6 Non-dimensional liquid pressure contours

0.5

0.4

1 12.8742

1

1 1 12.8531

12.811 2 w

1

2

2 2

.

2

.

. . 7

7 0.3 12.8216

7 7

. 2

7

r/D 3

7 5

6

9

7

9 8

9

4 0.2 5 4

0.1

0 0 1 2 3 4 5 X/Dw

0.6 Non-dimensional capillary pressure contours 0.5

0

0 . 1 0.4 0 0 . 1 6 . . 5

w 0 1 4

8 3 0.3 3 9 2 2 0 4

r/D 4 1 0 8

3 3 5 0 7 0.2 8 8 0.1

0 0 1 2 3 4 5 X/Dw 0.6 Non-dimensional vapor pressure contours 0.5 12.8846

0.4 1 2

w 1 . 8 1 2 8 0.3 1 2 . 2 8 4 r/D . 8 . 8 1 0.2 8 7 3 12.827 6 6 3 4 6 0.1

0 0 1 2 X/D 3 4 5 w Figure 2.5: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick for parameters shown in Table 2.2.

59

0.5 0.4 10 kg/m2 s w 0.3 0.2 r/D 0.1 0 0 1 2 3 4 5 X/Dw (a)

0.5 0.4 0.1 kg/m2 s w 0.3 0.2 r/D 0.1 0 2 3 4 5 X/Dw (b) 0.5 0.4

w 0.3 0.003 kg/m2 s 0.2 r/D 0.1 0 4 4.25 4.5 4.75 5 X/Dw (c)

Figure 2.6: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor mass flux vectors (expanded near the tip of the wick) inside the porous wick for parameters shown in Table 2.2.

60

0.5 0.4 30 W/cm2 w 0.3 0.2 r/D 0.1 0 0 1 2 3 4 5 X/Dw (a)

0.5 0.4 0.3 W/cm2 w 0.3 0.2 r/D 0.1 0 2 3 4 5 X/Dw (b)

0.5 0.4 0.1 W/cm2 w 0.3 0.2 r/D 0.1 0 2 3 4 5 X/Dw (c)

Figure 2.7: Plot of (a) liquid convective heat flux vectors (b) vapor convective heat flux vectors and (c) conductive heat flux vectors inside the porous wick for parameters shown in Table 2.2.

61

s 1 1 T 0.95 0.95

0.9 s 0.9

0.85 0.85

0.8 T 0.8

s 0.75 0.75

0.7 0.7

0.65 0.65

0.6 0.6

0.55 0.55 Non-dimensional temperature

0.5 0.5 0 1 2 3 4 5 X/Dw

Figure 2.8: Plot of saturation and temperature variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2

62

1

0.9 ρv 0.8 l

0.7 s)

2 0.6

0.5 ρlvl + ρgvg

0.4

0.3

0.2

0.1 mass flux (kg/m ρv 0 g

-0.1

0 1 2 3 4 5 X/Dw

Figure 2.9: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2

63

8

7

6 s) 2 5

ρul 4

3

2 mass flux (kg/m

1

0 0 1 2 3 4 5 X/Dw

Figure 2.10: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2

64

s T 1 1

0.9 T s 0.9

1 0.8 T 0.8

0.9998 s

0.9996 0.7 0.7

0.9994

0.6 0.9992 0.6 Non-dimensional temperature 0.999 4 4.25 4.5 4.75 5 Non-dimensional temperature X/Dw 0.5 0.5 0 1 2 3 4 5 X/Dw

Figure 2.11: Plot of saturation and temperature variation along the axis of the wick for parameters shown in Table 2.2

65

0 6

-0.0025 5 ρlul ρ u g g -0.005

4 s) s) -0.0075 2 2

3 -0.01 (kg/m (kg/m l g u u l

-0.0125 g ρ 2 ρ -0.015

1 -0.0175

0 -0.02 0 1 2 3 4 5 X/D w Figure 2.12: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick for parameters shown in Table 2.2

66

0.8 80x40 80x80 0.6 160x40

w 0 0 . 0 0 8 0 . 0 . 8 0.4 0 . 5 8 8 . 3 . 8 .9 9 4 4 r/D 7 8 6 3 0 8 4 5 5 7 6 5 8 5 8 6 6 7 2 3 1 1 3 0 1 4 1 8 0.2 7 1 4 9

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 X/Dw (a) 0.8 80x40 80x80 0.6 160x40 w 1

2 9 1

1 12.8162 1 1 . 1 0.4 1 2 2 7 2 9

2 6 .

r/D . 0 7 7 . 3

. 8 7 7

7 . 4

7 8 12.725 1 2 .

9 2 7

0 3 2

3 1 8

6 6 1 0.2 4

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 X/Dw (b)

0.8 80x40 80x80 0.6 160x40 w

0.4 0. 0 9 0 r/D 0 1 .9 . .8 0 7 7 5 18 0. 00 6 0 1 94 6 0 3 0 0 0.2 4 2 121 8 4 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 X/Dw (c)

Figure 2.13: Comparison of (a) saturation profiles (b) pressure profiles and (c) temperature profiles for three different meshes (80x40, 80x80, 160x40) inside the porous wick for parameters shown in Table 2.2.

67

1

0.9

0.8

0.7

0.6

0.5

0.4

Tip Saturation0.3

0.2

0.1

0 0 50 100 150 200 Total Heat Applied on the Wick Surface (W)

Figure 2.14 The variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick.

68

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2 Total mass evaporated (mg/s)

0 0 50 100 150 200 Total Heat Applied on the Wick Surface (W)

Figure 2.15 The variation of total mass of wax evaporated from the wick surface with the total heat supplied to the wick.

69

20 19 18 17 16 15 14 13 12 11 10 9 8 7 % heat lost to the reservoir 6 5 0 50 100 150 200 Total Heat Applied on the Wick Surface (W)

Figure 2.16 The variation of percentage heat that is lost to the reservoir with the total heat supplied to the wick.

70

g=0ge 0.9 g=1ge 0.8 g=2ge

0.7

0.6

0.5

0.4

Tip saturation0.3

0.2

0.1

0 50 100 150 200 Total Heat supplied to the wick (W)

Figure 2.17 The effect of gravity on the variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick.

71

2 g=0ge 1.8 g=1ge g=2g 1.6 e

1.4

1.2

1

0.8

0.6

0.4 Total mass evaporated (mg/s)

0.2

0 50 100 150 200 Total Heat supplied to the wick (W)

Figure 2.18 The effect of gravity on the variation of total mass evaporated from the wick surface with the total heat supplied to the wick.

72

K=5x10-12 m2 1 K=1x10-11 m2 -11 2 0.9 K=2x10 m

0.8

0.7

0.6

0.5

0.4

Tip saturation0.3

0.2

0.1

0 0 50 100 150 200 Total Heat supplied to the wick (W)

Figure 2.19 The effect of absolute permeability on the variation of saturation at the cylindrical tip of the wick surface with the total heat supplied to the wick.

73

2 K=5x10-12 m2 K=1x10-11 m2 1.8 K=2x10-11 m2 1.6

1.4

1.2

1

0.8

0.6

0.4 Total mass evaporated (mg/s)

0.2

0 0 50 100 150 200 Total Heat supplied to the wick (kW)

Figure 2.20 The effect of absolute permeability on the variation of total mass evaporated from the wick surface with the total heat supplied to the wick.

74

CHAPTER 3 GAS PHASE MODEL

To describe the candle burning phenomena, a gas-phase combustion model is

needed to couple with the wick transport model. In this thesis, the axisymmetric gas-

phase model formulated by Alsairafi (2003) is adopted. This chapter is taken directly

from Alsairafi’s thesis and is presented here for the sake of completeness

3.1 THEORETICAL FORMULATION

In this section, the steady state equations for the gas phase are presented. The flow field is solved using the full Navier-Stokes equations. The governing equations for the gas phase model are initially presented in dimensional form. The contribution of each term in the equations and the boundary conditions will be discussed briefly. Next, the non-dimensional procedure will be given.

The geometry of the candle and the wick is shown schematically in figure 3.1. In this model, the coordinate system is attached to the bottom of the exposed wick at the point (x=0, r=0). The x-coordinate, along the symmetry line, is in the axial direction and the other coordinate is along the radial direction. For the reference case to be studied, the wick is 1mm in diameter, the candle body has a diameter of 5mm and is 20mm high. The length of the wick varies from case to case and for a self trimmed candle flame, the self trimmed length of the wick is obtained as a part of the solution. The candle is placed on a solid plate that extends to infinity. The ambient is air with mole fractions of 21% O2 and

79% N2. In the subsequent parametric study, however, the wick diameter and the ambient

oxygen molar fraction will be varied. 75

The following assumptions in the gas phase are used.

• The flow is steady, laminar and axisymmetric in the gas phase.

• The ideal gas law is applicable to the gas mixture.

• This study assumes a single-step reaction, finite rate gas-phase chemical kinetics.

The property values of the fuel are based on a blend of C25 H 52 with stearic acid

(C18 H 36O2 ).

• Viscosity, conductivity, and the product of ρDi have a power law dependence on

temperature.

• Viscous dissipation is negligible.

• Soot formation is neglected.

• The liquid wax pool is assumed to be uniformly at its melting point temperature.

3.1.1 Continuity Equation The mass conservation equation for two-dimensional flow in cylindrical coordinates has the following form:

∂∂1 ()ρρurv+= ( )0, (3.1) ∂∂xrr where u and v are the axial and radial velocity components respectively and ρ is the mass density. The individual terms represent the net flux of mass at a point in the x and r directions respectively. The local density of the mixture depends on the mixture temperature, pressure, and the species mass fractions through the ideal gas relation:

Yi P = ρRuT ∑ , (3.2) i MWi

where MWi is the molecular weight of species i. 76

3.1.2 Momentum Equation Modeling of the buoyant flow requires additional attention. In candle flames, flows between the flame and ambient are driven by natural convection resulting from temperature differences through the flame. Hence, the gravity force needs to be included in the model. The axial momentum equation is

∂∂141 ∂∂∂∂∂Pu u ()ρρuu+=−++ ( r vu ) ( μ ) ( r μ ) + ∂∂xrrxxxrrr ∂∂∂∂∂3 (3.3) 112()∂∂vrv ∂ ∂ ( rμ )−− [μ ] ρg rr∂∂ x rx ∂3 ∂ r and the radial momentum equation has the following form:

∂∂11 ∂∂∂∂∂Pv v ()ρρuv+=−+++ ( r vv ) ( μ ) (2) r μ ∂∂xrrrxxrrr ∂∂∂∂∂ . (3.4) ∂ ∂∂∂∂uv22μ vvu (μμ )−− [ ( ++ )] ∂x ∂∂∂∂rr2 r3 rrx

The two terms on the left hand sides of the x- and r-momentum equations represent the rate of change of the x- and r-momentums respectively. The terms on the right side are pressure, viscous and body forces respectively. P is the pressure that can be

dp decomposed into the hydrostatic pressure, i.e. hydrostatic = −ρ g , and the flow dynamic dx ∞ pressure. Subtracting the hydrostatic pressure from the x-momentum equation, and since there is no gravity force acting in the r-direction, the momentum equations in x- and r- directions take the following forms:

∂∂141 ∂∂∂∂∂pu u ()ρρuu+=−++ ( r vu ) ( μ ) ( r μ ) + ∂∂xrrxxxrrr ∂∂∂∂∂3 ,(3.5) 112()∂∂vrv ∂ ∂ ( rμ )−+− [μρ ] (ρ )g rr∂∂ x rx ∂3 ∂ r ∞ 77

∂∂11 ∂∂∂∂∂pv v ()ρρuv+=−+++ ( r vv ) ( μ ) (2) r μ ∂∂xrrrxxrrr ∂∂∂∂∂ . (3.6) ∂ ∂∂∂∂uv22μ vvu (μμ )−− [ ( ++ )] ∂x ∂∂∂∂rr2 r3 rrx

3.1.3 Species equation The candle composition, as described in Dietrich et al. (2000), is assumed to be a

blend of 80 percent (by weight) n-paraffin wax (C25 H 52 is chosen for the numerical

simulation) and 20 percent stearic acid (C18 H 36O2 ). Assuming one-step chemical reaction, the overall stoichiometric combustion reaction of a candle and air can be described as follow:

C H + 0.31 C H O + 46.06 [O + 3.76 N ] → 25 52 18 36 2 2 2 (3.7) 30.58 CO2 + 31.58 H 2O +173.19 N 2

If f i is the stochiometric mass ratio of species i and fuel, then from the above equation we have:

f = −1, f = −3.3495, f = 3.0577 , and f = 1. 2918 (3.8) F O2 CO2 H 2O

The negative sign in f i indicates that the species is consumed while the positive sign is for produced species. The species equations for fuel, oxygen, carbon dioxide, and water vapor can be written as

∂∂∂11∂YY ∂∂ ()ρρρuY+= ( r vY ) ( Dii ) + ( r ρ D ) + W . (3.9) ∂∂∂∂∂∂xrrxxrrriii ii

The first and second terms represent the convective fluxes of species i. On the right hand side, the first two terms represent mass diffusion rate, the last term is the production rate of species i. It should be noted that Fick’s law has been implemented for the diffusion velocity. 78

3.1.4 Energy equation The energy conservation equation can be written in several different forms. The form using temperature as the dependant variable will be presented here since it is easy to implement in the numerical codes. Then,

∂∂∂∂∂∂TTT1 T ρρuC+= vC() λ + ( r λ ) ++ hW pp∂∂∂∂∂∂xrxxrrr∑ ii i . (3.10) ∂∂YYii∂∂TTG ∑ [ρCDpi i (+ )]−∇⋅ q r i ∂∂xx ∂∂ rr

On the left hand side, the terms represent the convection energy fluxes. The first two terms on the right hand side are the heat diffusion rate expressed by the Fourier’s

Law. The third term represents the rate of heat release from chemical reaction term. The fourth term is associated with the diffusion of species with different enthalpies due to variable specific heat and the last term is the radiation energy flux. The reaction term can be written as follows

T T h W = C dT W + ΔhD W = C dT W + QW . (3.11) ∑ i i ∑ ∫ p,i i ∑ f ,i i ∑ ∫ p,i i F i i TD i i TD Contribution of sensible enthalpy heat of Combustion

It is assumed in this work that a finite rate expression for a one-step, second-order

Arrhenius reaction exists and has the following form:

E W = −B ρ 2Y Y exp(− ) . (3.12) i g F O2 RuT

The energy equation can be rewritten in a different form for numerical purposes. First, the diffusion term can be rewritten as follows: 79

∂ ∂T 1 ∂ ∂T ∂ λ ∂T 1 ∂ λ ∂T (λ ) + (rλ ) = [C p ( )] + [rC p ( )] ∂x ∂x r ∂r ∂r ∂x C p ∂x r ∂r C p ∂r . (3.13) ∂C p λ ∂T 1 ∂(rC p ) λ ∂T ∂ λ ∂T 1 ∂ λ ∂T = [( )( ) + ( )( )] + C p { ( ) + [r( )]} ∂x C p ∂x r ∂r C p ∂r ∂x C p ∂x r ∂r C p ∂r

Using the continuity equation, and dividing by the totalC p , the final form of the energy equation becomes:

∂∂∂∂∂∂11λ TTλ ()ρρuT+=+ ( r vT ) ( ) [()] r + ∂∂∂∂∂∂xrrxCxrrCrpp λ ∂∂CC∂∂TT1 T [pp+ ]+++ [CdTWQW  ] (3.14) 2 ∑ ∫ pi, i F CxxrrCpp∂∂ ∂∂ i TD

ρCDpi, i ∂∂YYii∂∂TT1 G ∑ [ (+−∇⋅ )] qr i CxxrrCpp∂∂ ∂∂

Clearly, the flow field is affected by changes in temperature and density. Hence, all the flow equations must be solved together with the species and energy equations.

The last term on the right hand side of the energy equation, the radiation energy flux, can be rewritten in the following form (Modest, 1993)

G 4 ∇ ⋅ qr = κ[4σT − G], (3.15) where κ is the absorption coefficient of the mixture and G is the incident radiation. It is obtained from the computed radiation intensity. The development of radiation transfer equation will be described in the next section.

3.1.5 Boundary conditions The bottom of the computational domain is bounded by the candle and the solid plate that is placed on as shown in figure 3.1. The other boundaries are theoretically at infinity, however, in the numerical computation the boundary conditions are applied at a large but finite distance away from the candle (i.e. rmax and xmax). Except for the wick, the 80

remaining boundary conditions are similar to that used by Alsairafi (2003). If qr,in and qr,out represent the incident radiation into the wick surface and the outward radiation from the surface, respectively, then the boundary conditions can be written as follows:

On the wick surface:

∂Y − ρD F = ρv(1− Y ) F ∂r F

∂Yi ρD = ρvY (i=O2, CO2, and H2O) i ∂r i

∑Yi =1 (i=F,O2, CO2, H2O and N2) where ρv is the mass flux of fuel supplied by the wick to the gas phase, which is given by the following expression

ρρρvvv=+ ()ll gg wick

The RHS term indicates the net mass flux of fuel (liquid+vapor) supplied by the wick at this surface.

The energy balance equation in the gas phase is given by

∂T λ +=qqq + ∂r rin,, wick rout

where qwick is the heat provided by the gas phase to the candle wick. This expression gives the amount of heat flux that is supplied to the candle wick for evaporating the wax on the surface.

The temperature is determined as a part of the solution of the wick governing equations.

On the wick-tip surface:

∂Y − ρD F = ρu(1− Y ) F ∂x F 81

∂Yi ρD = ρuY (i=O2, CO2, and H2O) i ∂x i

∑Yi =1 (i=F,O2, CO2, H2O and N2) where ρu is the mass flux of fuel supplied by the wick to the gas phase, which is given by the following expression

ρρρuuu=+ ( ll gg)wick surface

The RHS term indicates the net mass flux of fuel (liquid+vapor) supplied by the wick at this surface.

The energy balance equation in the gas phase is given by

∂T λ +=qqq + ∂x rin,, wick rout

where qwick is the heat provided by the gas phase to the candle wick. This expression gives the amount of heat flux that is supplied to the candle wick for evaporating the wax on the surface.

The temperature is determined as a part of the solution of the wick governing equations.

From point A to B in figure 3.1 (candle wax):

The temperature is assumed to be at the melting point of the wax.

TT= m

The other dependent variables are as follows:

∂Yi = 0 ; u = 0; v = 0 ( i=F, O2, CO2, and H2O) ∂x

∑Yi =1 (i=F,O2, CO2, H2O and N2)

82

From point B to C in figure 3.1:

The temperature is fixed at the ambient temperature of 300K and the other dependent variables are as follows:

∂Yi = 0 ; u = 0; v = 0 (i=F, O2, CO2, and H2O) ∂r

∑Yi =1 (i=F,O2, CO2, H2O and N2)

On the solid plate:

∂Yi T=T∞; = 0 ; u = 0; v = 0 (i=F, O2, CO2, and H2O) ∂x

∑Yi =1 (i=F,O2, CO2, H2O and N2)

At r=rmax:

∂v T = T ; Y = Y ; u = 0; = 0 (i=F, O2, CO2, and H2O) ∞ i i,∞ ∂r

∑Yi =1 (i=F,O2, CO2, H2O and N2)

At x=xmax:

∂T ∂Yi ∂u = 0 ; = 0 ; = 0; v = 0 (i=F, O2, CO2, and H2O) ∂x ∂x ∂x

∑Yi =1 (i=F,O2, CO2, H2O and N2)

Symmetry line:

Because of symmetry, computation is performed only on one half of the domain and the boundary conditions along the symmetry line (r=0) are:

∂T ∂Yi ∂u ∂v = 0 ; = 0 ; = = 0 (i=F, O2, CO2, and H2O) ∂r ∂r ∂r ∂r

∑Yi =1 (i=F,O2, CO2, H2O and N2) 83

3.2 NON-DIMENSIONAL PARAMETERS

The basic governing equations are put into non-dimensional forms in order to characterize the importance of the physical parameters as well as to facilitate the numerical computation. The reference velocity used in the non-dimensionalization, Ur, is the velocity seen by the flame near the stabilization zone. It consists of two parts, i.e.

Ur=UB+UD, where UB is the buoyant velocity assumed to take the form:

(ρ∞ − ρ f )α r 1/ 3 U B = [ g] (3.16) ρ r and UD is the molecular diffusion velocity. The buoyant velocity is estimated by a balance between buoyant forces and inertia forces (see T’ien et al., 2001). The diffusion velocity is introduced here to avoid the singularity at zero gravity (UB→0). Diffusion velocity in one atmosphere is of the order of 0.5-5cm/s and it is temperature dependent.

In this work we take UD =2cm/s.

A reference length scale, the smallest scale of importance in the physical problem, needs to be chosen carefully in order to non-dimensionalizing the spatial coordinates in

the governing equations. The thermal length,LUrrr= α / , is chosen for scaling and can be obtained by considering the balance of convection and conduction in the gas-phase flame. This thermal length is a good measure of the flame standoff distance, i.e. distance between flame and candle wick near the flame base stabilization zone. The non- dimensional variables and parameters are: 84

u v α r x r ρ μ λ C p u = ; v = ; Lr = ; x = ; r = ; ρ = ; μ = ; λ = ; C p = ; U r U r U r Lr Lr ρr μr λr C p,r

T Δho + C dT f ,i ∫ pi C o D T E q C = p,i ; h = T ; D = i ; T = ; E = ; q = ; p ,i i i C p,r C p,rT∞ Di,r T∞ RuT∞ C p,rT∞

 α ρ B μ C  Wi r r g α r r p,r ρ rU r Lr ρ rα r Wi = ; Da = 2 ; Lei = ; Pr = ; Re = = ; (ρ rU r / Lr ) U r Di λr μ r μ r

g(ρ − ρ )L3 ρ C U U r Lr ∞ f r G G Lr r p,r r p − p∞ Pe = ;Gr = 2 ; ∇ ⋅ qr = ∇ ⋅ qr ( 4 ) ; Bo = 3 ; p = 2 ; α r ρ rα r σT∞ σT∞ ρrU r

κ = κ ⋅ Lr ; β = β ⋅ Lr

The last two non-dimensional parameters (i.e. κ and β ) are needed for the radiative transfer equation. The non-dimensional governing equations take the form

∂∂1 ()ρρurv+ ( )0= , (3.17) ∂∂xrr

∂∂1141 ∂∂∂∂∂puu ()ρρuu+=−++ ( r vu ) {( μ ) ( r μ ) + ∂∂xrrxxxrrr ∂∂∂∂∂Re 3 , (3.18) 112()∂∂vrv ∂ ∂ ρ − ρ (rμ )−+ [μ ]}Gr (∞ ) rr∂∂ x rx ∂3 ∂ r ρ∞ −ρ f

∂∂111 ∂∂∂∂∂p vv ()ρρuv+=−++ ( r vv ) {() μ (2) r μ + ∂∂x rr ∂∂∂∂∂ rRe x x rr r , (3.19) ∂∂uv22μ ∂ ∂ vvu ∂ (μμ )−− [ ( ++ )]} ∂∂xrr2 ∂ r3 ∂ rrx ∂

∂∂11 ∂∂YYii 1 ∂∂  ()ρρuYii+= ( r vY ) [( ρ D i ) + ( r ρ D ii )] + W , (3.20) ∂∂xrrLexxrrri ∂∂∂∂ 85

∂∂∂∂∂∂11λλTT ()ρρuT+=+ ( r vT ) [()] [()] r + ∂∂∂∂∂∂xrrxCxrrCrpp 1 T λ ∂∂CC∂∂TT [CdTWQW + ]+++ [pp ] , (3.21) ∑ ∫ pi, i F 2 CCxxrrppi TD ∂∂ ∂∂

ρC pi, ∂∂YYii∂∂TT1 G ∑ [ [+−∇⋅ ] qr i CLepi∂∂ x x ∂∂ r r CBo p

W = − f Daρ 2Y Y exp(−E /T ) . (3.22) i i F O2

The physical meaning of each of these dimensionless numbers is as follows: The

Damköhler number Da is the ratio of a characteristic flow time to a characteristic chemical reaction time in one thermal length, The Lewis number Le represents the ratio of energy to mass transport rates. It should be noted that Lewis number is assumed to be constant but different for each species. The Prandtl number Pr compares the momentum and the energy transport rates, The Reynolds number Re indicates the ratio of inertia to viscous forces and it is based on the thermal length and the reference velocity. The Peclet number Pe is the ratio of the bulk heat transfer to the conductive heat transfer which is unity in this work, and the Grashof number Gr indicates the importance of buoyancy force acting on the fluid. The Gr in the system is the consequence of choosing

Ur=UB+UD. If UD is negligible (as the case when g is large), Gr→1. Finally, Bo is called the Boltzmann number and it is a measure of the relative importance of convective energy flux to radiative energy flux. The radiative term is

G 4 ∇ ⋅ qr = κ (4T − G ) . (3.23)

3.3 PROPERTY VALUES

In the analysis of this work, heat capacity, enthalpy, and heat conductivity are functions of temperature. The values of the thermal properties for oxygen, nitrogen, 86 carbon dioxide, and water vapor are well known. The thermal and transport properties used in the model are the same reported by Smooke and Giovangigli (1991), where they

have found that viscosity μ , λ / C p , and ρDi have a power law dependence on temperature as follows:

μ ∝ T 0.7 (3.24)

0.7 λ / C p ∝ T (3.25)

0.7 ρDi ∝ T , i = F, O2 , CO2 , H 2O, N 2 (3.26) The values of the diffusion coefficients for each different species i are not needed since the Lewis numbers are introduced in the non-dimensionalizing procedure. The specific heat of the mixture is composition and temperature dependant. It can be calculated from the relation:

C p = ∑YiC p,i , i = F, O2 , CO2 , H 2O, N 2 , (3.27) i

whereC p,i is a function of temperature for each different species and can be represented in a polynomial form. For the candle fuel vapor, the value of constant pressure specific heat is taken from Middha and Wang (2002). The thermodynamic data of Cp’s were estimated using Ritter and Bozzelli's group additivity as implemented in the THERM code (Ritter and Bozzelli, 1991). The data are listed in Table 3.1. Gas properties values used in this work are listed in Table 3.2 while the values of the non-dimensional parameters are summarized in Table 3.3.

3.4 NUMERICAL PROCEDURE

The strongly coupled and highly nonlinear nature of the model equations derived for the candle flames exclude the possibility of obtaining analytical solutions. Our main 87 task is to develop a suitable numerical procedure of solving this coupled system of equations.

The system of governing equations and boundary conditions introduced in the preceding sections has been solved numerically by Jiang (1995) and Kumar et al. (2003) for two-dimensional flame spread problems over a planar solid fuel in rectangular coordinates. Alsairafi (2004), solved for the candle flame problem using axisymmetric formulation for both the combustion and the radiation parts.

For the combustion equations, a fully implicit control-volume based finite difference method of Patankar (1980) is used. The method is described in details in many

CFD books (Patankar, 1980; Ferziger and Perić, 1996; Versteeg and Malalasekera, 1995).

The procedure for the calculation of the flow field is performed by SIMPLER algorithm, which stands for Semi-Implicit Method for Pressure-Linked Equations Revised. The

SIMPLER scheme is adopted for the treatment of the pressure-velocity coupling.

Irregular geometries are modeled using a blocked-off region procedure.

3.4.1 Grid generation The grid point location has to be chosen properly when solving the system of partial differential equations in order to avoid instability and divergence. Hence, one has to pay a great deal of attention to grid generation before seeking a numerical solution.

The main role of grid generation is the specification of the boundary point distribution as well as the determination of the interior point distribution. Two dimensional grid generations is considerably more complicated than the one-dimensional case. Accurate and economic resolution of the solution requires that grid points be clustered in regions of large gradients and be spread out in regions of small gradients. 88

Many general methods of grid generation exist. Algebraic method would be our method of choice. In this method, algebraic equations are used to generate the algebraic transformation. As an example, in geometric progressions, each spatial increment is a fixed multiple of the previous spatial increment, i.e.

Δxi = ζ Δxi−1 , (3.28) where ζ is the ratio of successive increments. This ratio of successive increments varies with different candle dimensions. Several numerical experiments have been performed to study the grid independence for the problem in this work. The minimum axial spatial

increment, Δxmin , has been chosen so that the wick has 30 control volumes in the axial direction, which is equivalent to the thermal length scale at 300ge. The minimum radial

spatial increment, Δrmin , on the other hand, has been selected so that the candle inert shoulder (from point A to point B in figure 3.1) has 20 control volumes, which is equivalent to the thermal length scale at 70ge. Alsairafi (2003) found that when these two minimum axial and radial spatial were decreased by half, the solutions of dependent variables, i.e. T, do not change significantly (ΔT∼8K which is less than 1% change). The grid generation for the problem at hand is shown in figure 3.2.

3.4.2 Numerical implementation The governing equations (3.17-3.21) can be re-written in the general form for a dependent variableφ :

∂∂∂∂φ 1 φ ()()ρφurvrS−Γ + ρφ − Γ = . (3.29) ∂∂∂∂xxrrrφ φφ

This equation is often called the transport equation for property φ. From this equation, one may easily highlight the different transport processes: the convective and 89

diffusive terms, and the source term. The expressions for Γφ and Sφ for its corresponding dependent variable φ are given in Table 3.4.

The next step is to integrate the transport equation over a control volume and then to develop a suitable numerical method (an approximation) based on this integration scheme. These approximation techniques are needed to obtain the discretized equations.

The differential equations have been discretized using the conventional finite volume differencing techniques for non-uniform mesh spacing to the physical domain. The power-law scheme of Patankar (1981) has been used to approximate the convection- diffusion fluxes. It is found that this scheme represents the exponential behavior very well. In addition, this scheme behaves likes the central differencing scheme at a low

(local) Peclet number and as the upwind scheme at a high (local) Peclet number.

The general equation for a variable φ is integrated over the control volume. The control volumes for the x- and r- momentum equations are different from those discussed for the scalar equations. The control volumes are called staggered control volumes and these staggered grids are used only for the velocity components. The control volume for the x-momentum is staggered in the x-direction only, and the r-momentum control volume is staggered in the r-direction only. As a result, the velocities are stored between the grid points as opposed to at the grid points like temperature and pressure.

3.5 GAS RADIATION MODEL

In the system of equations developed in previous section, radiative flux term appears in the energy equation (3.21). To obtain these quantities, the radiation field needs to be solved. The governing equation for radiative heat transfer in a participating 90 medium is the radiative transfer equation (RTE). This equation is coupled to the energy equation and the boundary conditions and needs to be solved simultaneously. In recent years (T’ien et al., 2001), radiation has been identified as a key mechanism in determining the flame behavior in microgravity.

This chapter describes mathematical and computational considerations pertaining to radiative transfer processes and radiative transfer models in combustion systems. Our approach is to present a detailed derivation of the tools of radiative transfer needed to predict the radiative quantities (irradiation, intensity, and heating fluxes). We begin with discussion of the intensity field. Then, the RTE for a general cylindrical geometry is presented. The discrete ordinates method to solve the RTE is introduced next. Finally, a general numerical method to solve the coupled RTE and energy equation is developed.

3.5.1 The Equation of Radiative Transfer The fundamental quantity defining the radiative energy transfer in a medium is the specific intensity of radiation. Specific intensity measures the flux of radiant energy transported in a given direction per unit cross sectional area orthogonal to the beam per unit time per unit solid angle per unit frequency (or wavelength).

G G Consider the light beam traveling in the direction Ω through the point r and G G construct an infinitesimal element of surface area ds intersecting r and orthogonal to Ω , G the radiant energy flux dE crossing ds in time dt in the solid angle dΩ in the frequency G G range [ν ,ν + dν ] is related to Iν (r,Ω) by

G G G dE = Iν (r,Ω) ds dt dΩ dν (3.30) 91

The radiation field is, generally, a multi-dimensional quantity, depending upon three coordinates in space, one in time, two in angle, and one in frequency. This makes it very difficult to solve for exactly or numerically. Therefore, some assumptions and approximations are introduced to simplify the solution procedure. The main assumptions of the mathematical model are as follows:

• The medium is assumed to be absorbing-emitting and non-scattering with mean

absorption coefficients that varying from location to location.

• The radiative participating gases are CO2 and H2O.

• Surfaces bounding the medium and the solid plate are assumed to be black.

• Candle wax and wick surfaces are assumed to be radiatively gray and diffused.

We reduce the number of spatial dimensions from three to two by assuming the symmetry which is azimuthally homogeneous and in which physical quantities may vary G only in the axial and radial dimensions. Thus we replace r by (r, x). The use of the mean absorption coefficient omits the frequency dependence. With these assumptions, the intensity is a function only of radial and axial positions, time, and of direction, G [i.e.I(t,r, x,Ω ) ]. The angular direction of the radiation is specified in terms of the polar G angle θ and the azimuthal angle ψ, thus Ω = θeˆθ +ψeˆψ . The specific intensity is also referred to as intensity.

The mechanism of radiation in absorbing, emitting, and scattering media can be described mathematically by the radiative transfer equation (Siegel and Howell, 1992;

Modest, 1993). The RTE is an integro-differential equation in terms of the radiative intensity. By solving the RTE and energy conservation equation simultaneously, the 92 temperature distribution as well as heat flux in both the medium and on the enclosure surfaces can be obtained.

The key idea is to make an energy balance on the radiative energy, see figure 3.3, which undergoes absorption, emission, and scattering during traveling an infinitesimal G length along a line of sight in the direction Ω .

G G G G G G G G G G 1 ∂I(r,Ω) G G G G G σ s (r ) G + (Ω ⋅∇)I(r,Ω) = κ (r)I b (r ) − β (r)I(r,Ω) + Φ(Ω′,Ω)I(r,Ω′)dΩ′ ∫G c ∂t 4π Ω′

(3.31) where I is called the radiation intensity. Ib is the blackbody intensity at local temperature, G Φ is the scattering phase function, c is the speed of light, Ω is a unit vector specifying G the direction of scattering through a position vector r , t is time, and β is the extinction

coefficient and is defined as β = κ + σ s . In general, the absorption coefficient, κ, scattering coefficient, σs, the scattering phase function,Φ, as well as the emissivity of boundary surfaces, ε, are functions of both wavelength and temperature. The radiation intensity field can be taken as time independent in most engineering problem since c is

∂I very large so that = 0. The first and last terms on the right hand side of the above ∂t equation are usually combined and called the source function vector, which describes the local production of intensity. That is

G G G G G G G G G σ s (r) G S(r,Ω) = κ (r)I b (r ) + Φ(Ω′,Ω)I(r,Ω′)dΩ′ . (3.32) ∫G 4π Ω′ 93

Although the medium is assumed to be non-scattering, the scattering term will always be retained in the present work for the sake of completeness. Hence, the RTE for a two-dimensional cylindrically axisymmetric medium takes the following compact form

G G G G G G G G (Ω ⋅∇)I(r,Ω) + β (r)I(r,Ω) = S(r,Ω) . (3.33)

The meaning of the different terms is as follows: The first term on the left hand G side corresponds to the gradient of the radiation intensity in the specified direction Ω .

The second term on the left hand side represents attenuation due to absorption and scattering of the radiation intensity as it propagates through the medium. The term on the right hand side accounts for the radiation energy created by emission and the in- scattering, which is an increase of radiation intensity along the direction of propagation due to the scattering of intensity coming form other directions.

The equation of transfer must be solved subject to an appropriate boundary conditions. With the use of the mean absorption coefficient, spectral effects will be bypassed and the boundary condition for the RTE equation on the volume bounds becomes:

G G G G G γ (rw ) G G I(rw ,Ω) = ε(rw )I b (rw ) + n ⋅ Ω′ I(rw ,Ω′)dΩ′ , n ⋅ Ω > 0 , (3.34) G G∫ π n⋅Ω′<0 G G G where n is the local outward surface normal. So in the above equation, n ⋅ Ω > 0 G G indicates rays coming out from the wall and n ⋅ Ω < 0 is characteristic for rays which striking the wall, to be absorped or reflected. ε is the total emmisivity on the surface which is assumed to be radiatively diffusive. 94

The candle domain is considered as a right cylindrical shaped enclosure containing an absorbing-emitting, and non-scattering medium. The word "diffuse" signifies that emissivity and absorptivity do not depend on direction. In the present calculations, the surface absorption coefficient κ is assumed to be equal to ε.

Surfaces bounding the medium, located at rmax and xmax, are assumed to be black.

It is worth to mention that angular approximations in cylindrical (curvilinear) geometries yield more complicated terms than two-dimentional problems. Therfore, for the problem under consideration, the discrete ordinates representations of the radiative transfer equation take the following form

G μ ∂ G 1 ∂ G ∂I(r, x,Ω) G G [rI(r, x,Ω)] − [ηI(r, x,Ω)] + ξ + β (r, x)I(r, x,Ω) = S(r, x,Ω) r ∂r r ∂ψ ∂x

(3.35) G where μ,η, and ξ are the direction cosines of Ω and they defined by, as shown in figure

2.4,

μ = sinθ cosψ η = sinθ sinψ ξ = cosθ (3.36)

ψ is the azimuthal angle measured from the r direction. The definition of the source function is given by

G G G G G σ s (r, x) S(r, x,Ω) = κ (r, x)I b (r, x) + Φ(Ω′,Ω)I(r, x,Ω′)dΩ′ (3.37) ∫G 4π Ω′

Equations (3.35) and (3.37) are the general equations for the axisymmetric problems. These two equations will be discretised later for the numerical purposes. Many heat transfer quantities can be computed after solving for the intensity distribution throughout a medium. This includes the incident radiation on the wall, the forward and 95 backward radiation heat fluxes in the axial and radial directions, and the divergence of heat flux term which is needed in the energy conservation equation to solve for the temperature distribution. These quantities can be obtained from the following formula:

G G G(r, x) = ∫ I(r, x,Ω)dΩ , (3.38) 4π G G q x+ (r, x) = ξ ⋅ I(r, x,Ω)dΩ , (3.39) r ∫ ξ >0 G G q x− (r, x) = ξ ⋅ I(r, x,Ω)dΩ , (3.40) r ∫ ξ <0 G G q r+ (r, x) = μ ⋅ I(r, x,Ω)dΩ , (3.41) r ∫ μ >0 G G q r− (r, x) = μ ⋅ I(r, x,Ω)dΩ , (3.42) r ∫ μ<0

x x+ x− qr = qr + qr , (3.43)

r r+ r− qr = qr + qr , (3.44)

G 4 ∇ ⋅ qr (r, x) = κ (r, x)[4σT (r, x) − G(r, x)]. (3.45)

3.5.2 Numerical Solution of Discrete Ordinates Method

The SN discrete ordinates method is based on a discrete representation of the angular variation of the radiative intensity spanning the total solid angle of 4π steradians as shown in figure 3.6. In order to facilitate the angular discretization, two indices are associated with each direction: the first index, p, indicates the value of ξ associated with G the direction Ω and the second index, q, increases with the value of μ associated with ξ. G G For each discrete direction Ω pq , the components of Ω pq along the μ, η, and ξ axes are G the direction cosines of Ω pq , i.e. μ pq , η pq , and ξ pq . Consequently, they must satisfy

(μ pq ) 2 + (η pq ) 2 + (ξ pq ) 2 = 1 (3.46) 96

G On the surface area of a unit sphere, there is a point in the direction Ω pq with associated surface area, w pq . The obvious requirement of assigning values of w pq is that the weight sum to the total surface area of a unit sphere.

Carlson and Lathrop (1968) proposed a relationship for the total number of directions to be used depending on the order of the discrete ordinate approximation. This relationship has the following form, i.e. N represents the order of the discrete ordinates approximation and d is the geometric dimensionality,

2d N(N + 2) M = . (3.47) 8

As it can be seen, if the geometry of a problem is not three-dimensional, not all the directions on the unit sphere are needed. The integrals over solid angle in the RTE equation are evaluated using numerical quadrature, that is,

G G G ∫ f (Ω)dΩ = ∑ w pq f (Ω pq ) . (3.48) 4π pq G After preselecting a set of M representative discrete directions Ω pq together with the corresponding weights w pq from a quadrature scheme, the radiative transfer equation as well as the corresponding boundary conditions can be written as a set of equations for each direction as follows

μ pq ∂ 1 ∂ ∂I pq (rI pq ) − (η pq I pq ) + ξ pq + βI pq = S pq , (3.49) r ∂r r ∂ψ ∂x G pq γ (rw ) G ( pq)′ ( pq)′ ( pq)′ I (rw ) = ε(rw )I b (rw ) + n ⋅ Ω w I (rw ) , (3.50) G G∑ π n⋅Ω( pq )′ <0 with the source term, containing the medium emission and medium in-scattering, is of the

pq σ s ( pq)′ ( pq)′, pq ( pq)′ form S = κI b + ∑ w Φ I . It should be noted that there is another 4π ( pq)′ 97 boundary condition is required to obtain a numerical solution. This is the symmetry boundary condition along the axial axis and is given by

at r = 0 : I pq = I ( pq)′ (3.51)

The net radiative heat flux in the axial and radial directions are computed by the following relations

G G q x (r, x) = ξ ⋅ I(r, x,Ω)dΩ ≅ ξ pq w pq I pq (r, x) , (3.52) r ∫ ∑ 4π pq G G q r (r, x) = μ ⋅ I(r, x,Ω)dΩ ≅ μ pq w pq I pq (r, x) . (3.53) r ∫ ∑ 4π pq

In a similar manner, the total incident radiative intensity and the divergence of the heat flux are determined by

G G G(r, x) = ∫ I(r, x,Ω)dΩ ≅ ∑ w pq I pq (r, x) , (3.54) 4π pq

4 4 pq pq ∇ ⋅ qr (r, x) = κ (r, x)[4T (r, x) − G(r, x)] ≅ κ (r, x)[4T (r, x) − ∑ w I (r, x)]. (3.55) pq

3.5.3 Discrete ordinates angular quadrature Although the choice of quadrature scheme is arbitrary, it can affect the accuracy of the SN method. To preserve physical symmetry, the quadrature sets and weights are chosen carefully to be invariant under any 90o rotation. However, it is obvious that all quadrature schemes satisfy the summation condition, i.e. the zeroth moment

G ∫ dΩ = ∑ w pq = 4π . (3.56) 4π pq

Some researchers [Truelove (1987), Fiveland (1991)] suggested additional requirements for the quadrature and weights which satisfies the first half- and full-range moments, the diffusion-theory condition, and the key radiation moments, given by 98

G G G G G G G G n ⋅ Ω dΩ = n ⋅ ΩdΩ = w pq n ⋅ Ω pq = π , (3.57) ∫ ∫ ∑G G G G G G pq n⋅Ω<0 n⋅Ω>0 n⋅Ω >0 G G G ∫ Ω pq dΩ = ∑ w pq Ω pq = 0 , (3.58) 4π pq G G G G G 4πδ ∫ Ω pq Ω pq dΩ = ∑ w pq Ω pq Ω pq = , (3.59) 4π pq 3 with δ is the common unit tensor. Different sets of direction and weight can be found in the literature. However, in this thesis the quadrature sets and weights proposed by

Lathrop and Carlson (1965) are chosen for S4 scheme.

3.5.4 Solution of discrete ordinate equations The differential equation of the RTE is transformed into algebraic one by performing spatial discretization. To do this, Eq. (3.49) is integrated over a control volume with respect to dr and dx in a specific direction (p,q). Several researchers have explained this integration procedure and only some critical features will be repeated here for clarity (Fiveland, 1984, 1987, 1988; Truelove, 1987, 1988; Jamaluddin and Smith,

1988, 1992). Carlson and Lathrop (1968) proposed a procedure for calculating the angular derivative term which maintains neutron conservation in the curved coordinates and permits minimal directional coupling as follows

1 ∂ 1 (η pq I pq ) = [α p I pq+1/ 2 −α p I pq−1/ 2 ] . (3.60) r ∂ψ r q+1/ 2 P q−1/ 2 P

The two terms represent the intensity flow out and into the angular range under consideration due to angular redistribution. The directions q ±1/ 2 define the edge of the

pq p angular range denoted by the weight w . The coefficients α q±1/ 2 are determined using the fact that for uniform and isotropic intensities flux (as the case for infinite medium), 99

G then we have a case of divergence-less flow, i.e. ∇ ⋅ ΩI = 0 . After some algebra we obtain (Carlson and Lathrop, 1968)

p p pq pq α q+1/ 2 = α q−1/ 2 + w μ . (3.61)

p This recursive relationship uniquely determines all the α q+1/ 2 in terms of the selected

p quadrature parameters if α1/ 2 is known. Fortunately, this can be obtained by imposing the radiant intensity conservation condition, where the angular redistribution of the intensity will be conserved if the integration of intensity over all directions is zero. This

p condition can be met if α1/ 2 = 0 . The final result of the conservation relation for a control volume takes the form

α p I pq+1/ 2 −α p I pq−1/ 2 μ pq [A I pq − A I pq ] − (A − A )[ q+1/ 2 P q−1/ 2 P ] n n s s n s w pq , pq pq pq pq pq + ξ [Ae I e − Aw I w ] +VP βI P = VP S P

(3.62) where

pq (σ s ) P ( pq)′ ( pq)′, pq ( pq)′ S P = κI b,P + ∑ w Φ I P , (3.63) 4π pq

pq pq±1/ 2 pq I p , I p , S P are the volume cell-center intensities and the source term along directions (p,q), and (p,q±1/2), respectively. Ae, Aw, An, and As are the east, west, north,

pq pq pq pq and south surface areas of a control volume, respectively. I e , I w , I n , and I s are the

pq average values of I p over the east, west, north, and south faces of control volumes, respectively. Equation (3.49) is solved in each of the ordinate directions. An iterative solution is necessary because the boundary conditions depend on radiation intensity. 100

Because the directional intensity is solved iteratively (see Lewis and Miller, 1984), a sweeping scheme, which transfers the boundary information to the inside of the enclosure quickly and accurately, is required to result in fast convergence. The space angle sweeps depicted in the solution procedure for this work are shown in figure 3.6. Depending on the sweep direction, some of these intensities, i.e. cell-edge intensities, can be assumed known from either boundary conditions or calculations in adjoining control volumes.

Therefore, additional auxiliary relations are required to solve the discretised equation of

RTE. These relations, depending on the sweep direction, take the following form

1) For ξ pq > 0 and μ pq > 0 :

pq pq pq pq pq pq pq pq pq pq+1/ 2 pq−1/ 2 I p = f x I e + (1− f x )I w = f r I n + (1− f r )I s = ωI P + (1−ω)I P (3.64) 2) For ξ pq < 0 and :

pq pq pq pq pq pq pq pq pq pq+1/ 2 pq−1/ 2 I p = f x I w + (1− f x )I e = f r I n + (1− f r )I s = ωI P + (1−ω)I P (3.65) 3) For ξ pq > 0 and :

pq pq pq pq pq pq pq pq pq pq+1/ 2 pq−1/ 2 I p = f x I e + (1− f x )I w = f r I s + (1− f r )I n = ωI P + (1−ω)I P (3.66) 4) For ξ pq < 0 and μ pq < 0 :

pq pq pq pq pq pq pq pq pq pq+1/ 2 pq−1/ 2 I p = f x I w + (1− f x )I e = f r I s + (1− f r )I n = ωI P + (1−ω)I P (3.67)

pq where f x , , and are called the weighting factors. Several weighting factors exist in the literature (Chai et al., 1994; Liu et al., 1996). It is important to choose these weighting factors based not only on the accuracy but also on the guarantee that nonnegative intensities will be generated. In this work, the step-scheme is adopted for the two-dimensional axisymmetric cylindrical coordinates problem while the positive scheme 101 is chosen for the two-dimensional planar coordinates problem. However, some other available schemes will also be discussed in this section.

The simplest and more numerical stable, of these schemes is the step method

(Carlson and Lathrop, 1968) which can be obtained by setting the weighting factor equal to unity. However, this step-scheme is believed to generate less accurate first order results (Lewis and Miller, 1984). Another common scheme, a second order accurate

(Lewis and Miller, 1984), is the Diamond Difference Scheme (DDS). This is proposed by the results of Carlson and Lathrop (1968) from setting . Although the diamond scheme is more accurate than the step scheme, it is reported in the literature

(Lathrop, 1969; Chai et al., 1994) that this scheme can be unstable and can easily produce oscillatory (positive-negative) solutions that propagate throughout the spatial domain irrespective of the number of control volumes used. In the positive scheme of Lathrop

(1969), the weighting factors are calculated as follows:

, , (3.68) where

, . (3.69)

Although the positive scheme is written in the axisymmetric cylindrical coordinates, in this thesis, it will only be used for a 2D planar problem.

1) For and μ pq > 0 (sweep 4): 102

(3.70)

, (3.71) where

pq (1− f r )An Cns = pq + As , (3.72) f r

pq (1− f x )Ae Cew = pq + Aw , (3.73) f x

(A − A ) α p (1−ω) C pq = n s [ q+1/ 2 +α p ] . (3.74) w pq ω q−1/ 2

2) For and μ pq > 0 (sweep 3):

, (3.75) where

pq (1− f r )An Cns = pq + As , (3.76) f r

, (3.77)

(A − A ) α p (1−ω) C pq = n s [ q+1/ 2 +α p ] . (3.78) w pq ω q−1/ 2

3) For and μ pq < 0 (sweep 2): 103

, (3.79)

where

, (3.80)

pq (1− f x )Ae Cew = pq + Aw , (3.81) f x

(A − A ) α p (1−ω) C pq = n s [ q+1/ 2 +α p ] . (3.82) w pq ω q−1/ 2

4) For and (sweep 1):

, (3.83)

where

, (3.84)

, (3.85)

(A − A ) α p (1−ω) C pq = n s [ q+1/ 2 +α p ] . (3.86) w pq ω q−1/ 2

The above equations are used to solve for the angular intensity at every grid and then various radiative quantities of interest can be easily recast. In this work, only S4 scheme will be adopted to simulate the gas phase radiation since it is believed that accuracy will not be improved beyond S4 (Fiveland, 1982). The number of 24 directional fluxes for S4 is shown in figure 3.7. 104

3.5.5 The Mean Absorption Coefficient The mean absorption coefficient, κ, is both temperature and mixture composition dependent, and hence, need to be computed locally in the gas phase. In dealing with radiative transfer in absorbing, emitting gases, the appropriate choice of the mean absorption coefficient can provide a reasonable estimate of radiative heat flux from the flame.

The Planck mean absorption coefficient of the mixture, KP, is calculated as the summation of the partial pressure weighted values of the component gases, KP,i. In this problem, the two radiating species are CO2 and H2O. So

, (3.87)

where Xi is the molar fraction of species i calculated at an ambient pressure of 1atm. The values of KP for carbon dioxide and water vapor as a function of temperature can be found in the literature (Tien, 1968). The least-squares fitting equations are given in Table

3.6.

It has been recognized, however, that the use of this Planck mean absorption coefficient over-predicts the gas radiation heat flux when compared with the narrowband result for a one-dimensional flame (Bedir and T’ien, 1997). The main reason of this over- prediction is that the Planck mean absorption coefficient is exact only for the optically thin medium; a case which is not true for a flame with reasonable thickness. In fact, the non-gray gases CO2 and H2O do absorb radiation in their own bands (self-absorption).

This results in attenuation of the emitted energy from the participating gases which cannot be ignored. In order to compute the total radiative flux more accurately, a correction factor, C, less than unity, is multiplied in front of the Planck mean absorption 105 coefficient, i.e. . This modification factor reflects the non-optically thin nature of the flame and results in a better estimation of the radiative heat fluxes back to the solid.

In Rhatigan et al. (1998), an empirical relation was derived to estimate the value of C based on the optical path length in a quasi one-dimensional flame. In the present work, a value of 0.4 is chosen for the correction factor. This is a reasonable for microgravity flames. As it will be shown later, flame radiation is important only when gravity is sufficiently reduced.

3.6 SOLUTION PROCEDURE

The system of coupled elliptic equations in the gas phase is solved numerically based on the finite volume discretization. The coupling between the velocity and pressure fields is handled using the SIMPLER algorithm. This algorithm is used to handle implicitly the relationship between the flow rate and the node pressure and it is an iterative solution method to deal with the nonlinear nature of equation sets. The choice of

110 x 90 grid points was used for a physical domain of 50 x 40 cm. The discretised equations are set up at each of the interior points and modified for those points adjacent to the boundaries to incorporate the boundary conditions. The resulting system of discretization algebraic equations is solved iteratively by successive applications of the

Tri-Diagonal Matrix Algorithm (TDMA) over crossed horizontal and vertical directions until a converged solution is obtained. In this procedure, we choose a grid line and assume that the dependent variable φ of the neighboring lines are known from there latest values and solve for the φ along the chosen line using TDMA and the discretization equations for each dependent variable are solved. In summary, the equations of mass, 106 momentum, species, and energy are solved simultaneously for velocities, species, and the temperature. From the solution, the mixture density, specific heat and other transport properties are calculated using there appropriate relations introduced in the preceding sections.

For the radiation routine, every intensity value on every cell-center and faces of control volume were computed by stepping from control volume to the adjacent control volume starting from the known boundary condition. The differential equation of the

Radiative Transfer Equation (RTE), which is needed for calculating the intensity, is transformed into algebraic one by performing a spatial discretization. The procedure will be explained in the coming sections. Once the intensity distribution is calculated, the incident heat fluxes on a wall as well as the divergence of heat flux are determined from their relations described in next section. The radiation equation is coupled with the flow field equation at each iteration cycle.

107

Table 3.1: Correlating equations of specific heats forO2 , N 2 ,CO2 , H 2O , and fuel

cal C = (a + bT + cT 2 + dT 3 + eT 4 ) ⋅ f ( ) p,i g.K Where T is dimensional quantity (degree Kelvin)

Temperature Species a b·103 c·106 d·109 e·1012 f·102 range

O2 3.62560 -1.87822 7.05545 -6.96351 2.15560 6.20937 300-1000 3.62195 0.736183 -0.196552 0.0362016 -0.00289456 6.20937 1000-5000

N 2 3.67483 -1.20815 2.32401 -0.632176 -0.225773 7.09643 300-1000 2.89632 1.51549 -0.572353 0.0998074 -0.00652236 7.09643 1000-5000

CO2 2.40078 8.73510 -6.60709 2.00219 00063274 4.51591 300-1000 4.46080 3.09817 -1.23926 0.227413 -0.015526 4.51591 1000-5000

H 2O 4.07013 -1.10845 4.15212 -2.96374 0.807021 11.0389 300-1000 2.71676 2.94514 -0.802243 0.102267 -0.00484721 11.0389 1000-5000

Fuel -0.01674 1.625832 -0.991139 0.2856428 -0.02920363 1×10-12 300-5000

108

Table 3.2: Gas phase property values

Symbol Value Units Reference

Tb 620 K Shu (1998) Tm 323 K this work

Tr 1250 K Ferkul (1993)

T f 2200 K Ferkul (1993)

T∞ 300 K Ferkul (1993) -4 -3 ρ r 2.75×10 g cm Ferkul (1993) -3 -3 ρ∞ 1.15×10 g cm Ferkul (1993) -4 -1 -1 μ r 4.10×10 g cm s Ferkul (1993) -4 -1 -1 -1 λr 1.93×10 cal cm s K Ferkul (1993) -1 -1 C p,r 0.33 cal g K Ferkul (1993) 2 -1 α r 2.13 cm s Ferkul (1993) -1 -1 Ru 1.987 cal gmol K Ferkul (1993)

P∞ 1 atm Ferkul (1993) E 3.0×104 cal gmol-1 Shu (1998) 12 3 -1 -1 Bg 1.58×10 cm g s this work Q 8910 cal g-1 this work L 296.12 cal g-1 Shu (1998)

109

Table 3.3: Nondimensional parameters

Symbol Value Parameter Reference

Pr 0.7 ν r /α r Ferkul (1993)

LeF 2.50 α r / DF ,r Shu (1998) Le 1.11 α / D Smooke and Giovangigli (1991) O2 r O2 ,r Le 1.39 α / D Smooke and Giovangigli (1991) CO2 r CO2 ,r Le 0.83 α / D Smooke and Giovangigli (1991) H 2O r H 2O,r Le 1.00 α / D Smooke and Giovangigli (1991) N2 r N2 ,r 4.167 Ferkul (1993) Tr Tr /T∞

E 50.33 E /(RuT) this work

Q 90 QCT/(pr, ∞ ) this work

L 3.0 L /(C p,rT∞ ) this work Da Variable 2 α r ρ r Bg /U r Bo Variable 3 ρ r C pU r /(σT∞ ) ε 0.9 this work

110

Table 3.4: Non-dimensional governing differential equations

φ Γφ Sφ

1 0 0

μ ∂p ∂ Γ ∂u 1 ∂ ∂v 1 ∂ 2 ∂(rv) ρ − ρ − + ( φ ) + (rΓ ) − [ Γ ] + Gr( ∞ ) u Re φ φ ∂x ∂x 3 ∂x r ∂r ∂x r ∂x 3 ∂r ρ ∞ − ρ f

μ 2Γ v ∂p 1 ∂ ∂v ∂ ∂u φ ∂ 2 ∂v v ∂u v − + (rΓφ ) + (Γφ ) − − [ Γφ ( + + )]} Re ∂r r ∂r ∂r ∂x ∂r r 2 ∂r 3 ∂r r ∂x

ρD i Yi  Lei Wi

1 T λ ∂C ∂T ∂C ∂T [ C dT W + q W ] + [ p + p ] + λ ∑ ∫ p,i i c F 2 C p i TD C p ∂x ∂x ∂r ∂r T C p ρC p,i ∂Yi ∂T ∂Yi ∂T 1 G ∑[ [ + ] − ∇ ⋅qr i C p Lei ∂x ∂x ∂r ∂r C p Bo

111

Table 3.5: The S4 quadrature sets for axisymmetric cylindrical enclosures

( p,q) μ η ξ w

(1,1) -0.295876 0.295876 -0.908248 π/3 (1,2) 0.295876 0.295876 -0.908248 π/3 (2,1) -0.908248 0.295876 -0.295876 π/3 (2,2) -0.295876 0.908248 -0.295876 π/3 (2,3) 0.295876 0.908248 -0.295876 π/3 (2,4) 0.908248 0.295876 -0.295876 π/3 (3,1) -0.908248 0.295876 0.295876 π/3 (3,2) -0.295876 0.908248 0.295876 π/3 (3,3) 0.295876 0.908248 0.295876 π/3 (3,4) 0.908248 0.295876 0.295876 π/3 (4,1) -0.295876 0.295876 0.908248 π/3 (4,2) 0.295876 0.295876 0.908248 π/3

112

Table 3.6: Least-square fitting equations of Planck mean absorption coefficient for CO2 and H2O

2 -1 -1 K P = a + b ⋅T + c ⋅T (cm atm ) where T is non-dimensional temperature (non-dimensionalized by 300 K)

Gas medium a b c Temperature range

-1 CO2 3.89300x10 T≤1.8333 3.54982 x10-1 7.46849 x10-2 -2.99044 x10-2 1.83335.8333

-1 H2O 1.71400 x10 T≤1.8333 4.63180 x10-1 -2.04529 x10-1 2.46914 x10-2 1.83335.8333

113

1, 2, or 3mm x

D

Wick

5mm

r B A

x= r=0

Candle 20mm

Solid plate C

Figure 3.1 Schematic of a candle

114

45

40

35

30

25

r(cm) 20

15

10

5

0 0 10 20 30 40 x(cm)

Figure 3.2: Variable grid structure for modeling candle flames for 1mm wick diameter and 5mm candle diameter

115

dr Intensity scattered from dV into direction Ω

I(r+dr,Ω)

Intensity emitted from dV dV into direction Ω

I(r,Ω’)

Intensity absorbed by dV

Intensity scattered from direction Ω into other directions

Figure 3.3: Schematic of radiation intensity transfer energy balance on arbitrary control volume in a participating medium

116

x

ξ Ω η θ ψ μ

r

Figure 3.4: Geometry and coordinate system for 2D axisymmetric cylindrical enclosure

117

ξ

(4,1) (4,2)

(3,1) (3,2) (3,3) (3,4) μ

(2,1) (2,2) (2,3) (2,4)

(1,1) (1,2)

Figure 3.5: Projection of an S4 quadrature set on the (μ,ξ) plane using (p,q) numbering in r-x geometry

118

x

Top wall

ξ<0, μ>0 ξ<0, μ<0 Side wall Center Line ξ>0, μ>0 ξ>0, μ<0

r Bottom wall

Figure 3.6: Four types of space angle sweep direction for SN scheme

119

1 ξ

-1 -1 -1 -1

η μ 1 1

Figure 3.7: Solid angle discretization of the S4 quadrature

120

CHAPTER 4 RESULTS AND DISCUSSIONS

In a realistic candle, both the candle flame and the wick are coupled to each other.

The heat from the candle flame evaporates the candle wax, providing the driving force

for the liquid to rise up through capillary action. The wax evaporated from the surface

provides the fuel for the candle flame. In this way the fuel supplied by the wick and the

heat supplied by the candle flame are coupled together.

The aim of this chapter is to understand the role of heat and mass transfer taking

place inside the wick on the candle burning. The numerical model of heat and mass transfer in the porous wick in chapter 2 is coupled to the gas-phase flame model in chapter 3. For wicks which are made of combustible materials (e.g. fibers), the phenomenon of “self trimming” is modeled. Parametric studies are performed to study the effect of gravity, wick permeability, wick diameter and ambient oxygen.

4.1 CANDLE FLAME COUPLED TO A POROUS WICK

In this section, numerical results from the coupling of two-phase axisymmetric

wick model with the gas phase model are presented and discussed. Steady state results at

1 atmospheric pressure are given and parametric studies are performed. The four

parameters varied in this study are the gravitational acceleration, absolute permeability of

the wick, the environment oxygen molar fraction concentration, and candle wick

diameter.

Before presenting the results, some of the basics of candle flame burning are

being explained to give a better idea of the results being presented. In our present model, 121

the candle shoulder and the wax pool are not modeled. Therefore, the present model is equally applicable for oil lamp flames. The primary difference between the candle flame and oil wick lamp flames is that in candle flames, the shoulder level drops and hence the exposed wick length changes during burning. The shoulder level moves downwards due

to wax consumption as the burning proceeds. The wick is also self trimmed by the

candle flame as the wax pool level and the flame moves downwards. Self trimming is a phenomenon which occurs if the wick is made up of combustible materials. The presence of self trimming phenomenon taking place in a candle wick make things complicated. While performing experiments, care should be taken to account for the self trimming phenomenon of the wick. Let us assume we take a candle with initial wick length shorter than the self trimming length (self trimming length can be found from the numerical simulations or from prior experimental observations). When this candle is ignited, the wick will be able to supply sufficient liquid wax to the surface for evaporation and so a steady flame is established for the given initial length. For the oil wick lamp flames, the exposed wick length remains the same and so the flame does not change once the steady state is achieved. In the case of a candle flame, the exposed candle wick length continuously increases with time because of the consumption of wax.

As the exposed wick length increases, the burning rate increases and the flame also

increases in size. This phenomenon of increase in exposed wick length can be slow or

fast depending upon the burning rate and the candle shoulder diameter. The exposed

wick length will continue to increase till the wick is no longer able to supply wax to the

tip surface of the wick. Further increase in exposed wick length will cause a dry region

to be formed at the tip of the wick. The temperature of the dry region shoots (due to lack 122

of evaporative cooling provided by the liquid wax) up rapidly causing charring or

burnout of the dried portion of the wick. Therefore a steady length of exposed wick is

achieved. This phenomenon is termed as “self trimming”. When the wick reaches its self-trimming length, the candle flame also reaches a steady state. When experiments are performed, the candle may or may not reach the self trimmed limit, if (1) the time of experiments is not sufficiently long, (2) the burning rate of the candle flame is low for the given candle parameters and environment conditions, (3) the candle shoulder is large. So the experimental results can vary anywhere from the candle flame established for the given initial wick length chosen to the self trimmed candle flame.

With the above understanding, the results are presented in the following manner.

First, a detailed study of the flame structure and flow field inside the porous wick are presented for (1) steady state flame for a chosen initial wick length, which is shorter than self trimming length and (2) self trimmed candle flames. In the sections following this, most of the results are shown for a self trimmed candle flame, except for few cases,

where it is difficult to obtain a self trimmed flame due to low heat input for the cases of

weaker wake flames.

4.1.1 Detailed Flame Structure at Normal Gravity and 21% O2. Before proceeding with analyzing the various candle flames, the flame structure at

one gravity level is presented in detail. This will also give the reader an idea of what the

computational model is capable of predicting. The conditions chosen for this purpose is

the normal earth gravity flame with the molar oxygen mole fraction of 0.21, for the

reference candle. The thermal length in this case is 0.099cm which gives an estimate of

the flame standoff distance at the flame stabilization zone. The reference velocity is 123

21.6cm/s, which approximates of the flow velocity in the neighborhood of the flame base region. The absolute permeability of the wick is 2.5× 10−12 . The porosity of the wick is

0.55. A wick of initial length 4mm is chosen. The length is chosen such that it is shorter than the self trimming length. First the quazi-steady flame stabilized for this initial wick length is presented. After due course of time, the candle wick will eventually reach it self trimming length. The steady state flame results for the self trimmed candle flame are presented later. Because of the symmetry, in the figure only one half of the flame profiles are presented.

4.1.1.1 Steady State Candle Flame (With Wick Shorter than Self Trimming Length) A candle wick of 4mm is chosen. The absolute permeability of the wick is

2.5× 10−12 m2. The porosity of the wick is 0.55. The remaining porous parameters are shown in Table 2.2. Initially, candle flame is stabilized on the wick surface and it reaches a steady state for this wick length (the self trimming length for this case is

4.34mm which will be presented in the next section). The quasi-steady results are presented in this section. Both the structure of the gas phase and the porous flow fields are examined in detail.

First the structure and the flow fields inside the porous wick are presented. Most of the plots are very similar to the plots obtained for the constant heat flux case presented in chapter 2. So only the key highlighting points are mentioned. Figure 4.1 shows the saturation and the temperature distribution inside the wick. The base of the wick is at the melting point of the candle wax (323 K) and on the wick surface, where there is evaporation; the temperature is at its boiling point (620 K). The contour s = 1 separates 124 the liquid and the two-phase regions. The minimum saturation of 0.57 is reached at the cylindrical tip of the wick. Figure 4.1 (c) shows the enlarged distribution of temperature inside the two-phase region. The variation of temperature in this region is very small.

This minor variation is sufficient to bring about vapor movement inside the wick.

The pressure distribution inside the wick is shown in Fig. 4.2. The capillary pressure is due to saturation distribution inside the wick. The vapor pressure distribution in the two-phase region is due to the presence of temperature gradient. The liquid and the vapor movement are shown in Fig. 4.3. The liquid which comes up to the surface of the wick is evaporated at the surface. Part of the evaporated liquid moves into the wick at the surface locations as indicated in fig. 4.3 (b). The vapor velocity vectors in fig. 4.3 (b) are magnified approximately 10 times compared to the liquid velocity vectors (in Fig. 4.3

(a)). The vapor mass flux into the porous wick varies along the cylindrical surface, reaching a maximum value at near x = 2.4mm. Figure 4.3 (c) shows the enlarged vapor motion in the two-phase region.

Figure 4.4 shows the net heat flux supplied by the candle flame to the wick along the cylindrical surface. In the decoupled case analyzed in chapter 2, the heat flux was specified constant all along the cylindrical surface. In the case of the wick coupled to the candle flame, the heat flux is almost zero near the base of the wick and then suddenly increases and remains approximately constant. At the tip of the wick, the heat flux suddenly rises. This may be due to the two-dimensional corner effect. Except very near to the base of the wick and at the tip of the wick, the heat flux is almost constant. So the results are qualitatively similar to that of the constant heat flux case discussed in chapter

2. 125

Figure 4.5 shows the saturation and the temperature distribution along the cylindrical surface. Very close to the base of the wick, there is a liquid region (s = 1).

The temperature in this region is much below the boiling point temperature of wax. In the two-phase region, the temperature is at its boiling point along this surface (since the surface is at 1 atmospheric pressure). Figure 4.6 shows the liquid mass flux that is evaporated along the surface. The vapor mass flux indicates that the vapor is moving into the wick along this surface. The net mass flux of wax entering the gas phase is also shown in figure 4.6. Figure 4.7 shows that the liquid mass flux in the x direction is continuously decreasing due to the evaporation at the surface.

Figure 4.8 shows the saturation and the temperature distribution along the axis of the wick. Figure 4.9 shows the mass fluxes of liquid and vapor along the axis. They are qualitatively similar to the profiles obtained for a constant heat flux case.

Figure 4.10 shows the non-dimensional isotherms in the gas phase (non- dimensionalized by T∞=300K). The gas phase temperature tends to stay closer to the wick of the candle, where the fuel comes out by evaporation. The location of the gas maximum temperature in normal gravity occurs at xcmy= 2.25 and = 0 , i.e. Tmax=2141K. The region of large temperature gradients at the flame base is caused by the cold convective flow that opposes the upstream diffusion of heat. In normal gravity, the maximum temperature location stays in the flame plume and at a position between the wick and the flame tip. In addition, there is a small quenching distance, which detaches the flame from the candle wax, between the flame base and the candle wax shoulder (about 1mm).

The fuel reaction rate contours are given in figure 4.11. The maximum consumption rate is always located at the side of the wick and very near the flame base. 126

This maximum reaction rate point occurs at a point where the temperature is high enough, but not necessary the maximum, and there is a substantial overlap between the fuel and the oxygen. In order to define the visible flame, and hence be able to compare it with the experimental pictures, the contour of reaction rate equal to 5x10-5g cm-3 s-1 is defined as the boundary of the visible flame. Such method of comparing the computed reaction rates of the flame with the visible experiment flame has been used by Grayson et al. (1994). The flame length H is defined as the difference between the locations of the flame tip and flame base. The flame diameter D is defined as the twice largest perpendicular distance from the line of symmetry to the reaction rate contour of the visible flame. Figure 4.11 shows a flame length of 2.76 cm and a flame diameter of 0.74 cm. The burning rate is observed as 0.84 mg/s.

The variation of fuel and oxygen mass fractions with location in the leading edge region is plotted in figure 4.12. Those contours overlap substantially upstream of the flame base where the gas temperature is low, allowing the mixing between fuel and oxygen to start the chemical reaction.

Figure 4.13 shows the local equivalence ratio contours, which is defined as the ratio of fuel to oxygen mass fraction ratio divided by the stochiometric fuel to oxygen mass fraction ratio. It gives quantitative indication whether a local fuel-oxidizer mixture is rich, lean, or stochiometric. For fuel-rich mixture, this ratio is greater than unity while it is less than unity for fuel-lean mixtures. The local fuel-rich region appears roughly at x<1.9cm and r<0.25mm. Though the stochiometric mixing of fuel and its oxidizer extends to the candle shoulder, there is little reaction in this region due to low local temperature. 127

Figure 4.14 and figure 4.15 show the carbon dioxide mass fraction and water vapor mass fraction, respectively. The distributions of these species are qualitatively similar. The maximum values of these products of combustion occur in the reaction zone, where they are generated. Downstream of the flame tip, these species are slowly diluted by mixing and diffusion with the ambient air.

Figure 4.16 shows the flow field (velocity vectors and streamlines) and the oxygen transport. Because of the symmetry the field quantities will be plotted only in one half of the plane. The lower half of Fig. 4.16 contains the velocity vectors while the upper half contains the streamlines and oxygen mass flux vectors. In addition, the reaction rate contour of 5x10-5g cm3 s-1 is superimposed onto the figure. As the cold flow approaches the reaction zone, it accelerates and deflects away from the wick. The reason of this deflection is threefold: (i) by the presence of the candle, (ii) the effect of Stefan velocity, and (iii) by the thermal expansion. Because of the viscous entrainment effects and the cooling of the gas stream, the velocity profiles tend to become flatter as the flow moves to the downstream. For Ψ ≤ 0, the streamlines refer to the flow originating from the wick surface and the convection is due to the Stefan flow from the evaporated fuel. The streamlines originating from the ambient are given by Ψ > 0. The point Ψ = 0 on the wick surface is determined by the net gas-phase heat flux to the wick equal to zero. Near the base of the wick, there is a small region where there is no evaporation from the surface. This is due to the presence of pure liquid region inside the wick and the heat supplied to the wick surface in this region is simply conducted into the wick. This will be further explained in the section where porous flow field is presented. Comparing Fig.

4.13 with Fig. 4.16, one sees that this dividing streamline (Ψ = 0) is on the fuel-rich side 128 except very close to the flame base near the wick surface. Consequently, while fuel vapor can convect into the reaction zone, the oxygen needs to diffuse into the reaction zone.

The oxygen mass flux vectors indicate they are nearly perpendicular to the reaction zone near the flame base but in the downstream, they became nearly parallel to the streamlines since the flow velocities accelerate to quite large values. Although not shown in the figure, a small circulation zone is found in the corner of wick and candle shoulder.

The pressure distribution around the flame is shown in figure 4.17. Two regions of pressure rise exist; one is due to the hot flame region and the other is due to the presence of the candle. As the flow approaches the flame reaction zone, it is deflected outward by the pressure rise and the influence of blowing velocity of the evaporated fuel from the wick. The flow will then accelerates after the reaction zone because of the gas expansion resulted from the heat release in the chemical reaction. The location of maximum pressure always occurs in the wick surface because of the evaporation rate. As we move downstream, the pressure decreases slowly (see Fig. 4.17).

A detailed plot of figure 4.18 shows temperature and species profiles as a function of position at the centerline of symmetry between the wick-tip and far from the flame.

The temperature starts with the surface (boiling) temperature at x=0.4cm, increases to its maximum value at the reaction zone (x=2.5cm) then decreases slowly toward the ambient temperature as x increases. The fuel vapor mass fraction is largest at the wick-tip and monotonically decreases to nearly zero at the flame where it is consumed. The oxygen has the opposite behavior of fuel; being maximum at infinity and decreasing to zero at the flame except in low gravity. The combustion products, i.e. CO2 and H2O, have their maximum values in the reaction zone, where they are generated. 129

The radiative structures of candle flames are shown from figures 4.19 to 4.22.

Figure 4.19 shows the distribution of the effective mean absorption coefficients of the mixture. Clearly, this absorption coefficient is both mixture composition (CO2 and H2O) and temperature dependent. The dimensional net radiative flux vectors are given in figure

4.20. This reveals the multi-dimensional nature of radiation heat transfer. The direction and magnitude of radiative flux vectors depends not only on local but also on the surrounding properties. The source of radiation comes from two things: the hot flame and the hot surfaces. Since the wick, at liquid fuel boiling point, is a strong emitter (ε=0.9), the surface radiation emission is greater than the radiative flux coming from the flame to the wick. This results in the net radiative flux vectors pointing outward on the wick surface as shown in figure 4.20. Along the centerline (r=0cm), the net flux reverses directions (i.e. toward the wick) in 0.5

The dimensional divergence of heat flux, where the energy equation in the gas phase is coupled with the radiation transfer equation, is shown in figure 4.21. The value G of ∇ ⋅ qr varies from positive (indicating a net radiative loss from the medium) to negative (net radiative gain). Finally, the different heat flux contributions to the side of the wick are presented in figure 4.22 for candle flames at normal gravity level. The positive values suggest heat flux gain into the wick, while negative values indicate heat flux loss from the wick. The gas phase conductive heat flux is qc and the net gas-radiative feed back is qr, net, which is the summation of (qr)in and (qr)out. The outward radiative heat flux consists of wick emission and reflection of unabsorbed incoming radiative heat flux.

In the flame quench region (x≤1mm), the magnitude of qc continuously rises. It reaches a plateau value in 1.5mm≤x≤4.5mm and then rises sharply near the top corner of the wick 130 where the flame feedback is two-dimensional. Clearly, at normal gravity, the dominant mode of heat transfer at the wick is by conduction. The flame radiation feedback (qr)in is negligible. The surface radiative loss (qr)e is finite but it is still relatively small compared with conduction.

4.1.1.2 Self Trimmed Candle Flame at Normal Gravity When the candle is allowed to burn for a long time, the exposed wick length will increase gradually due to the receding of the candle shoulder level. As the exposed wick length increases, the saturation level at the tip of the wick decreases to increase the capillary action. As this continues, a stage is reached when the saturation level at the tip of the wick becomes zero and the candle wick is no longer able to supply the liquid to the tip of the wick. Then a dry vapor region is formed. This dry vapor region is characterized by a drastic increase in temperature of the dry region. If the wick material is combustible, then the dry region will begin to char and burnout. Thereafter the exposed length of the wick will remain constant. This exposed wick length is referred as the self trimming length. In this work, it is assumed that once the wick becomes dry, the trimming length is reached. In other words, the dried burning portion of the wick has a length much smaller than the rest of the wick. With this approximation, the following section shows the detailed structure of the flame and the wick for a self trimmed candle flame. Candle is burned in normal gravity for the same parameters chosen for the previous case.

Numerically the “self trimmed length” is approximated as the minimum wick length for which the cylindrical corner of the wick becomes dried up. In principle, the whole tip surface of the wick should be dried up, but this would mean that the tip surface 131 will be a curved surface rather than a flat surface. To reduce the complexity of the problem, the formation of curved tip surface of the wick is neglected. Instead the self trimmed length is taken as the appearance of dry region on the cylindrical corner of the wick. The numerical procedure adopted to obtain a self trimmed candle flame is explained as follows. A shorter wick is initially chosen and the wick and gas phase are solved approximately during each overall iteration. The saturation at the corner of the wick is checked at the end of each overall iteration and if the corner is not completely dried up (i.e. s=0), the wick length is slightly increased. This update of wick length is done till the corner becomes completely dried up. Then the wick length remains constant and the iterative procedure is continued till its reaches overall convergence in the gas- phase and the wick phase is reached.

The detailed porous structure and the flow field are almost similar to the previous case. Only the key points are highlighted. At normal gravity, the self trimming length is found to be 4.34mm. Figure 4.23 (a) shows the saturation profiles for a self trimmed wick. The saturation at the corner of the tip of the wick surface reaches zero. Since the saturation contours are not flat, the trimming of the wick will change the shape of the wick tip. But this is being neglected in this work. The changes that will result from the change of the shape of the wick tip will be very small and hence reasonable to neglect.

Figure 4.23 (b) shows the non-dimensional temperature contours inside the wick. Figure

4.24 shows the non-dimensional pressure distribution inside the wick. Figure 4.25 shows the liquid and the vapor mass fluxes inside the wick. As observed in the previous case, the liquid and the vapor move in countercurrent fashion inside the two-phase region of the wick. The liquid is evaporated from all along the cylindrical surface and the tip of the 132 wick. Part of the vapor, which is evaporated at the surface, traverses into the wick. Near the tip of the wick, the vapor movement is almost negligible.

Figure 4.26 shows the net heat flux supplied by the flame to the wick for evaporation. The heat flux distribution is qualitatively similar to that obtained for the previous case except that the heat flux suddenly drops near the tip of the wick. This is due to the high temperature of the wick at this location formed due to the presence of dry region at the tip. Figure 4.27 shows the saturation and non-dimensional temperature profiles along the cylindrical surface of the wick which is exposed to the candle flame.

The figure shows that the saturation reaches zero at the tip of the wick. Near the base of the wick, there is small region which is in the liquid region. Here no evaporation takes place at the surface and all the heat supplied at this region of the wick surface is simply conducted into the wick. The temperature in this region is well below the boiling temperature of the wax. There is also a temperature gradient at the base of the wick which indicates some heat lost to the wax pool. Figure 4.28 shows the liquid mass flux and the vapor mass flux in the r direction along the cylindrical surface of the wick. There is a sudden drop in the liquid mass flux near the tip of the wick. This is due to the sudden drop in the heat flux supplied to the tip of the wick as shown in figure 4.26. The negative vapor flux indicates that the vapor is transported into the wick. Figure 4.29 shows the liquid mass flux in the x direction along the surface. The mass flux continuously reduces due to the evaporation along the surface of the wick. Figure 4.30 shows the saturation and non-dimensional temperature profiles along the axis of the wick. It is observed that the saturation at the tip along the axis line is 0.08 which is close to zero but not exactly zero. This is because of the assumption that the trimming of the wick is flat, whereas the 133 saturation contours are not flat. The exact trimming shape of the wick is not modeled.

Figure 4.31 shows the liquid and the vapor mass flux along the axis of the wick.

Figures 4.32-4.44 shows the detailed profiles of the gas phase flame characteristics. The plots are qualitatively similar to that of the plots obtained for the steady flame established for shorter wick, except for some minor quantitative differences.

The quantitative difference is attributed to the increase in exposed wick length to 4.36cm.

4.1.2 Effect of Gravity Gravity affects both the gas phase and the porous wick. In chapter 2, the effect of gravity on the heat and mass transport inside the wick is isolated and analyzed. It was found that for the wick parameters chosen in this study, gravity does not have a significant effect on the wick transport. Gravity does affect the gas phase, through the buoyancy term. Since flow fields in a candle flame are induced by buoyancy, gravity is found to significantly affect the gas phase and hence the flame structure. In this work, the gravitational acceleration has been widely varied as a parameter from zero to high gravity.

Figure 4.45 shows the computed visible flames at different gravity levels for a self trimmed candle flame in standard air conditions (1 atm and 21% O2). In this series of study, the candle and wick dimensions are: wick diameter = 1mm, candle body height =

2cm, and candle diameter = 5mm. The wick length will be that of the self trimming length which is determined as a part of the solution. Note that there is no self trimming observed for gravity levels greater than 5.5ge. So the wick length for gravity levels above this is taken as the self trimmed length of the candle wick at 5.5ge which is 3.8 mm. The 134

-5 -3 -1 reaction rate contour of wf =5x10 g cm s is chosen to represent the boundary of the visible flame, which is a function of the local fuel, oxygen and temperature.

Results reveal that the flame becomes longer as gravity increases, reaching its maximum value roughly at g=3ge and then decreases. The reason of this non-monotonic variation is explained as follows: in reduced gravity, the flame moves away from the wick, as it tries to move closer to the region of fresh oxidizer. Consequently, the heat flux to the wick diminishes and thus the evaporation rate decreases and this makes the flame shorter. The wider flame-standoff implies a weaker heat feedback from the flame and a smaller burning rate. There is also a slight reduction in the flame thickness as gravity increases. The flame standoff distance decreases with increasing gravity level. At higher gravity levels, the fresh oxygen is pushed closer to the wick by the strong convection, increasing the heat flux to the wick, leading to an increase in the rate of evaporation. This makes the flame longer. The increase in flame length is also in part due to the decreased flame standoff distance. However, at higher gravity level than g=3ge, the flame reduced in both length and width because of the decrease in burning rate.

Figure 4.45 also shows that there is a sudden retreat of the flame base position from at the wick base at 3ge to the wick tip at 6ge. This is also observed by Alsairafi

(2003). Hence, the burning rate decreases. At more than 10ge, the flame base is located downstream of the flat wick top surface. The solutions for the cases for 10ge and above show that in these cases, the fuel vapors that support the flame come entirely form the wick top surface.

Figures 4.46-4.47 shows the effect of gravity on the total burning rate and the self trimming length of the candle. Figure 4.46 shows that the burning rate first increases 135

with gravity till 2ge. The reason for increase in the burning rate is explained before. The self trimmed length is maximum for 0ge. The reason for this being that the heat flux supplied by the flame at 0ge is minimum. With increase in gravity, the self trimmed length of the candle decreases. As the gravity is further increased, the burning rates starts gradually decreasing till the gravity level reaches 4ge. The burning rate is a function of the heat supplied to the wick by the flame and the length of the exposed wick. As the heat flux increases, the self trimmed length of the wick decreases as shown in the figure.

In the region of 2ge to 4 ge, the decrease in burning rate due to decrease in self trimmed length of the wick dominates compared to the increase in burning rate due to increased heat flux supplied by the flame. So there is a slight decrease in burning rate. The results of Alsairafi (2003) indicate that the burning rate increases in this region, since the wick length chosen in their calculations remained constant. Then the burning rate drops rapidly. The reason for this sudden drop in burning rate is due to the retreat of flame base towards the wake tip to form a wake flame. In the wake region, the flame is very short and the heat flux is very low. So there is no self trimming action that would be observed at this stage. The wick length is chosen as the self trimmed length observed at 5.5ge.

The burning rate continues to drop with further increase in gravity.

It has to be noted that although the model predicts wakes flames at very high gravity levels, the existence of the flame at this levels is questionable. The present model does not account for melting of wax in the wax pool. For wake flames, the heat feed back from the flame to the wax pool will be quite low. The rate of heat supplied either directly by the flame or through heat loss at the base of the wick to the wax pool may not be sufficient to supply a steady supply of liquid wax for sustaining the candle flame. There 136 is a possibility of the flame being extinguished due to insufficient supply of liquid wax from the wax pool. Hence the existence of wake flames is questionable. The experiments also indicate extinction of candle flames at higher gravity levels.

4.1.3 Effect of Wick Permeability Wick permeability is a measure of ability of the porous wick to transport fluid through it. Wicks with high permeability, offers less resistance to fluid motion. In general, wick permeability is a function of the wick porosity, the structure of the wick and the wick material. As shown in section 2.3.5.2, the wick permeability affects the saturation distribution inside the wick for a given heat flux distribution. This would affect the self trimming length of the candle wick and hence the flame structure and the burning rate.

Figure 4.48 shows the effect of wick permeability on the self trimming length of the candle wick. It is observed that as the permeability is decreased, the level to which the wax rises above the candle shoulder decreases due to increased resistance to fluid motion. Therefore the self trimming length of the wick is reduced. Figure 4.49 shows the variation of burning rate of the candle flame with permeability of the wick. The burning rate decreases with decrease in permeability. This is caused directly by the decrease in self trimming length of the candle wick. The exposed length is reduced and hence the burning rate decreases. Figure 4.50 shows the variation of flame lengths with the wick permeability. The flame lengths are proportional to the exposed wick lengths and hence the trends are similar to that of the self trimming length variations shown in

Figure 4.48. Figure 4.51 shows the variation of maximum flame temperature in the gas 137 phase with the permeability of the wick. There is a slight decrease in the flame temperature with increase in permeability of the wick.

In brief, we can conclude that the wick permeability significantly affects the self trimmed wick length, the flame structure and the burning rate of the candle flame.

Therefore while conducting experiments on candle flames; the wick permeability should also be taken into account. The previous modeling efforts by Alsairafi (2003) does not capture this.

4.1.4 Effect of Wick Diameter The goal of using different wick and candle sizes is to study the flame size and extinction. Computed results in several low and normal gravity levels using three different wick diameters are summarized in Table 4.1. The computed results of Alsairafi

(2003) are shown in brackets for the sake of comparison with the present computed results. The data shows that thicker wick produces a longer (larger H) and wider (larger

D) flame. The total burning rate increases with wick diameter (see figure 4.53). The increase in burning rate is affected both by the increase of surface area of the wick and also by the increase in self trimming length of the wick (see figure 4.52). At a given gravity level, the maximum flame temperature decreases slightly with increasing wick diameter (see figure 4.54). Qualitatively similar results were obtained by Alsairafi (2003).

Previously, it was also found that a longer wick would also produce a larger flame with lower flame temperature (Dietrich et al., 2000).

-4 We also note from Table 4.1 that in 0ge only 1mm wick is flammable; in 10 ge

-2 1mm and 2mm wicks are flammable but not 3mm wick. Above 10 ge, all the four different cases are flammable. Similar variation on flame size has been predicted by 138

Alsairafi (2003) and observed experimentally in downward flame spreading over paper cylinders (Essenhigh and Mescher, 1993). It can be concluded that the wick diameter affects flame size and flame temperature as well as flammability.

4.1.5 Effect of Ambient Oxygen The effect of different ambient oxygen concentrations on flame structure and burning rate characteristics are investigated in this section. The dependence of flame temperature on ambient oxygen concentrations is given in figure 4.55. Computational results show that the flame temperature increases almost linearly with oxygen concentration. The self trimming length for different oxygen molar fraction is shown in figure 4.56. The self trimming length decreases with increase in concentration. This is due to the high heat flux supplied by the flame at higher oxygen concentrations. The burning rates are shown in figure 4.57. The burning rate slightly increases with oxygen concentration. The increased heat flux increases the burning rate but the decreases self trimmed length decreases the burning rate. Hence the overall increase in burning rate at higher oxygen concentrations is reduced. The fuel reaction rate contours as a function of oxygen concentration is given in figure 4.58. As oxygen percentage increases, the flame base moves upstream and closer to the candle surface suggesting a stronger flame. The flame size decreases with increasing oxygen concentrations. This is affected by the decrease in self trimming length.

4.1.6 Validation of Results In this section, the candle flame results are compared with pervious experimental and numerical work (Alsairafi, 2003). Note that none of the experiments report whether the candles used in the experiments are self trimmed or not. The exposed wick lengths of 139 the candle are also not reported. Alsairafi (2003) neglected the heat and mass transport inside the porous candle wick and hence there is no concept of self trimming of the candle wick. In their work, a wick length of 5mm is chosen for all their calculations. The characteristics of the porous wick, like absolute permeability, porosity etc., are not reported in the experiments. The previous section 4.1.3 shows that permeability of the wick significantly affects the candle flame structure. In the absence of the above experimental parameters, the comparison between the various experiments and the computed results is not fully justified. However, an attempt has been made to qualitatively validate the results.

Ross et al. (1991) performed several experiments in 5.2s NASA Lewis Zero

Gravity Facility with 4.75mm diameter candles. They changed oxygen mole fraction

(0.19-0.25), inert gas, and ignition methods (in either microgravity or normal gravity just prior to the drop). In all the tests, the microgravity time was too short to determine steady state and near-extinction behavior. However, the flame size was nearly steady. The author reported a sequence of pictures for a candle flames. Therefore, only a comparison of the overall flame dimension is possible. The experimental pictures show a nearly hemispherical candle flames with a large standoff distance.

The comparison between the experimental results and the computed results is done at 19 and 21% O2 with the standard candle dimensions (1mm wick diameter, 5mm candle diameter, 2cm candle length). The computed fuel consumption rate contours in figure 4.58 are compared to the reported experimental flame shape. The predicted flame shape closely resembles the visible flame in the experiments, except for a slight inward hook at the flame base, if the fuel vapor reaction rate value is 5x10-5g cm-3 s-1 as the 140 visible flame boundary. This comparison shows the current model qualitatively produces the correct visible flame shape, even though detailed flame chemistry has not been considered. Qualitatively similar results are obtained by Alsairafi (2003). Quantitatively, the numerical values of flame length and the flame diameter differ.

A number of experiments with various candle dimensions were performed in the

Shuttle (in air) and in the Mir space station (in enriched air with 23-25% oxygen concentration) (Dietrich et al., 1994; Dietrich et al., 2000). Three different candle wick diameters were used in Mir experiments. The present computational work is conducted for three different diameters that match the same dimensions in the Mir experiments. The comparison between the experiments and the numerical calculations are based on the flame width D, height H, H/D ratio and the burning rates. The comparisons between the experimental and computational results are based on the flame dimensions, size and burning rate, and are tabulated in Table 4.2. Similar to the results found in the drop towers experiments, shapes of the candle flames were spherical with a weaker flame than the flame in normal gravity. The visible reaction zone is much further away from the wick, implying a smaller heat flux from the flame to the wick. The experimental and analytical data agree quite well. The computational values for flame width D and the flame length H (based on the outside edge of the computed reaction zone) are 12.84mm and 9.212mm, respectively. These values are very close to the quasi-steady experimental values, (D falls in the range 9-15mm and H falls in the range 8-14mm for the small wick size of 1mm). The computed value of H/D is 0.717 while the experimental values fall in the range 0.84-1.02. The computed burning rate is 0.1348mg/s while the experimental 141 values fall in the range of 0.2-0.6mg/s, and the computed result of Alsairafi (2003) is

0.1663 mg/s.

It should be noted that when the wick diameter is increased to 2mm in zero gravity, the flame becomes extinct. No flammable flame was able to be obtained numerically, except for the smallest wick diameter of 1mm. Experimental results show that the wicks with diameters of 1, 2 and 3mm are all flammable. This discrepancy can be improved if the empirical one-step chemical kinetic constants are better calibrated. This is a work for the future.

For the effects of high gravity levels, comparisons between the experimental and computational results are based on the flame dimensions, which are tabulated in Table

4.3 and blow-off limit. Villermaux and Durox (1992) performed an experiment on a candle of 2.1cm diameter and 1.5mm wick diameter in a centrifuge. The exposed wick length was not mentioned in their work. They reported that 7ge was the blow-off limit.

The numerical model predicts a blow-off limit of 30ge. This discrepancy is because in the modeling of the porous wick. In reality, while self trimming is taking place, the tip of the wick becomes charred and soot collect on the tip of the wick. This blocks the tip of the wick from supplying wax from this surface. A numerical experiment has been performed in which the tip of the wick is artificially blocked. Then the flame is observed to be blown off at 6ge. In normal gravity, the flame length reaches about 2.803cm (for wick diameter 1.0mm) that compares quite very well with the numerical results of 4.1cm (for wick diameter of 1.5mm). Increasing the gravity level caused the candle flame height to decrease in length. Numerical results predict an increase in flame length up to 3ge and then the flame length decreases with increase of gravity level. 142

Another high gravity experiments were performed by Arai and Amagai (1993).

The candle has the following dimensions: 0.8mm wick diameter, 12mm candle diameter, and 4.8mm exposed wick length. They also reported that as the gravity increases, both the flame length and width decreases. The comparison between these experimental results and the computed results are shown in figure 4.59. The computed results of Alsairafi

(2003) are also shown in figure 4.59 for comparison. The flame width matches with the experimental results quite well. The flame length variation does not even agree qualitatively with the experimental results. The variation in the numerical results is non monotonic, whereas the experimental results shows a monotonic decrease in the flame length. This discrepancy can be attributed to many reasons. First the present numerical model used a second order single step overall kinetic for wax combustion. This is affecting the flame anchor location. It is observed that the flame is anchored only at the wick base or in the wake region. The experiments indicate that the flame anchor location gradually moves upwards with increase in gravity. This can possibly be predicted by the present model by using detailed kinetics. The blow-off limit of their candle experiment is

13.5ge which is about doubled the value found by Villermaux and Durox (1992).

Compared with the computed results of Alsairafi (2003), the present computations show a closer match with the experimental flame length values.

One discrepancy between the experimental results and the current numerical prediction is the effect of gravity on flame length. Experimental results show that the flame length decreases as gravity increases from 1ge and up while numerical results show that the flame length increases from 1ge to 3ge. The experiments report that the flame anchor location moves upward gradually with increase in gravity. The numerical results 143

show that the flame is always stabilized near the base of the wick for gravity levels 1ge to

5ge. At still higher gravity, the flame drastically moves upwards and forms a wake flame.

This discrepancy between the experiments and the model might be due to the assumption of second order single step overall kinetics for candle burning. When the flame is anchored at the base of the wick, increase in gravity increases the convective flow, causing the flame to move closer to the wick. This will increase the heat flux supplied to the candle wick. Increased heat flux will increase the evaporation rate per unit length but also decrease the self trimming length. So the total burning rate is the net effect of increase in evaporation per unit length and the decreases in the exposed wick length. So there is non monotonic behavior for the burning rate and the flame length in the range of

1ge to 5ge. The anchoring of the candle flame near the base of the wick explains the discrepancy between the experiments and the numerical results. In addition, the dimensions for the three different results do not match precisely (i.e. wick diameter, candle diameter, and exposed wick length) which could be an important reason for such a discrepancy.

144

Table 4.1(a): Effect of wick diameter on candle flame characteristics (5mm candle diameter and 21% O2)

D STL m wick D (cm) H (cm) (H/D) t T (K) (mm) g/ge (cm) flame (mg/s) max

1 1 0.436 0.7413 2.8034 3.7817 0.9222 2138 2 1 0.771 0.9388 5.2727 5.6164 2.147 2130 3 1 0.912 0.9715 6.8402 7.041 4.0079 2120.4

1 1x10-2 0.5825 1.4931 2.2623 1.5152 0.7027 1769 2 1x10-2 1.141 2.0367 3.604 1.7695 1.592 1731.6 3 1x10-2 1.645 2.304 4.24 1.8402 2.308 1703.2

1 1x10-4 0.92 1.962 1.1887 0.605 0.34036 1186.0 2 1x10-4 1.583 2.304 1.4576 0.632 0.7040 1163.5 3 1x10-4 Ext. Ext. Ext. Ext. Ext. Ext..

1 0 1.01 1.2845 0.9212 0.717 0.1348 1110.5 2 0 Ext. Ext. Ext. Ext. Ext. 3 0 Ext. Ext. Ext. Ext. Ext.

Table 4.1(b): Effect of wick diameter (Alsairafi, 2003) on candle flame characteristics (5mm wick length, 5mm candle diameter and 21% O2)

D STL m wick D (cm) H (cm) (H/D) t T (K) (mm) g/ge (cm) flame (mg/s) max

1 1 - 0.77794 3.2627 4.1863 1.09347 2142 2 1 - 0.8565 4.8356 5.6457 1.6131 2130 3 1 - 0.9715 5.6702 5.8364 2.0073 2120.4

1 1x10-2 - 1.3833 2.1577 1.56 0.61362 1777 2 1x10-2 - 1.5953 2.9164 1.8281 0.846 1760 3 1x10-2 - 1.8154 3.6891 2.0322 1.133 1743

1 1x10-4 - 1.751 1.2424 0.7096 0.24 1216 2 1x10-4 - 1.8554 1.3136 0.708 0.264 1166 3 1x10-4 - 1.9603 1.5635 0.7976 0.374 1176

1 0 - 1.4931 0.9450 0.6329 0.15855 1130.5 2 0 - Ext. Ext. Ext. Ext. Ext. 3 0 - Ext. Ext. Ext. Ext. Ext.

145

Table 4.2: Comparison with candle flame experiments in microgravity ([] = results of Alsairafi (2003)) 1Mir Numerical Numerical Numerical dwick=1,2 or dwick=1mm dwick=2mm dwick=3mm 3mm Lwick=self Lwick=self Lwick=self Lwick=3mm trimmed length trimmed length trimmed length

g=0ge D (mm) 8.8-15 12.845[14.931] Ext.[ Ext.] Ext. [ Ext.] H (mm) 8.4-13.5 9.212[9.45] Ext. [ Ext.] Ext. [ Ext.] Burning rate (mg/s) 0.2-0.6 0.1348 [0.1663] Ext. [ Ext.] Ext. [ Ext.]

g=1ge D (mm) - 7.413[7.794] 9.388[8.565] 9.73[9.715] H (mm) - 28.034[32.627] 52.727[48.36] 68.402[56.7] Burning rate (mg/s) 0.9-1.4 0.922 [1.0934] 2.147 [1.613] 4.007[2.007]

Table 4.3: Comparison with candle flame experiments in high gravity levels ([]=results of Alsairafi (2003))

2Experiment 1 3Experiment 2 Numerical Gravity level dwick=1.5mm dwick≈0.8mm dwick=1mm dcandle=2.1cm dcandle≈1.2cm dcandle=0.5cm

D (cm) 1ge - 0.75 0.74 [0.78] H (cm) 1ge 4.1 2.6 2.803 [3.26] D (cm) 3ge - 0.7 0.69 [0.67] H (cm) 3ge 2.1 2.3 3.1089 [4.13] D (cm) 7ge - 0.4 0.41 [0.41] H (cm) 7ge 1.3 1.5 1.32 [0.82] D (cm) 10ge Ext. 0.3 0.28 [0.32] H (cm) 10ge Ext. 1.2 0.95 [0.57] D (cm) 12ge Ext. 0.3 0.27 [0.27] H (cm) 12ge Ext. 1.0 0.75 [0.53] D (cm) 21ge Ext. Ext. 0.22 [0.22] H (cm) 21ge Ext. Ext. 0.60 [0.50]

1 From Dietrich et al. (1997) 2 From Villermaux and Durox (1992) 3 From Arai and Amagai (1993) 146

0.6 Saturation contours 0 0.5 0 0 .5 0 . .6 . 6 9 w 0.4 7 8 2 7 7 7 8 0.3 1 8

r/D 0.2 0.1 0 0 1 2 3 4 5 X/Dw (a) 0.6 Non-dimensional temperature contours 0.5 0

w 0.4 . 0 98 0.999543 . 04 0.3 5 68 8 0.863134 0.953713 0.992027 r/D 0.2 9 4 0 0.1 1 0 0 1 2 3 4 5 X/Dw (b) Non-dimensional temperature contours (expanded in two-phase region) . 0 9 0.5 9 0 .9 9 0.4 0 .9 9 9 0 . 9 9 9 w 9 9 9 5 0.3 .9 9 9 9 9 1 7 8 6 0 7 0.2 8 9 r/D 1 8 0.1 8 0 3 3.5 4 4.5 5 X/Dw (c)

Figure 4.1: Plot of (a) saturation profiles (b) temperature profiles and (c) temperature profiles (expanded in the two-phase region) inside the porous wick coupled to a candle flame at normal gravity (temperature is non-dimensionalized by 330 K)

147

0.6 Non-dimensional liquid pressure cotours patm =6.44

0.5 6.

135

0.4 6 6.40686 . 6.36619 6 1 17

6 w

6

6 6 . 4

. 1 .

.

. 1 6 7 0.3 3 2 2 6

8 . 7

1 r/D 7

1 2 4

9 7

5

8

5 3 4

7

9

8

5 0.2 9 9

8

5

4 6 0.1 9

0 0 1 2 3 4 X/Dw

0.6 Non-dimensional capillary pressure cotours patm =6.44 0.5 0 . 3 0 0.4 0 4 . w 0 2 5 0 8 6 0.3 3 . 0 2 r/D 3 2 3 0.2 9 4 5

3 0.1 2

0 0 1 2 3 4 X/Dw

0.6 Non-dimensional vapor pressure cotours patm =6.44 0.5 1 212.8843 1 .8 1 2 8 0.4 2 .8 1 . 6 w 2. 8 4 2 8 0.3 69 7 9 r/D 66 0.2

0.1

0 0 1 X/D2 3 4 w Figure 4.2: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick coupled to a candle flame at normal gravity

148

0.5

0.4 0.4 g/cm2 s

w 0.3 0.2 r/D 0.1 0 0 1 2 3 4 X/Dw (a)

0.5 0.4

w 0.3 0.04 g/cm2 s 0.2 r/D 0.1 0 0 1 2 3 4 X/Dw (b) 0.5 0.4 w 0.3 0.004 g/cm2 0.2 r/D 0.1 0 3 3.25 3.5 3.75 4 X/Dw (c)

Figure 4.3: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor mass flux vectors (expanded near the tip of the wick) inside the porous wick coupled to a candle flame at normal gravity.

149

20

18

16

14 ) 2 12

10 (W/cm 8 wick

q 6

4

2

0 0 1 2 3 4 X/Dw

Figure 4.4: Plot of net heat flux supplied by the candle flame along the cylindrical surface of the wick

150

s 1 1 T 0.9 0.9

0.8 s 0.8 T 0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 saturation

0.3 0.3

0.2 0.2

0.1 0.1 Non-dimensional temperature

0 0 0 1 2 3 4 X/Dw

Figure 4.5: Plot of saturation and temperature variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity (temperature is non-dimensionalized by 330 K).

151

2

1.75

1.5 s)

2 1.25 ρ v 1 l l

0.75 ρlvl+ρgvg 0.5

0.25 mass flux (kg/m 0 ρgvg

-0.25

-0.5 0 1 2 3 4 X/Dw

Figure 4.6: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity.

152

7 6.5 6 s)

2 5.5 5 ρ u 4.5 l l 4 3.5 3 2.5 2 1.5

Liquid mass flux (kg/m 1 0.5 0 0 1 2 3 4 X/Dw

Figure 4.7: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick coupled to a candle flame at normal gravity.

153

1 1 s T 0.9 0.9

0.8 0.8 T s 0.7 0.7 1

0.6 0.999 0.6 T 0.998

0.5 0.997 0.5

0.996

0.4 0.995 0.4 saturation 0.994 0.3 0.3 0.993

0.992 0.2 0.2 0.991 Non--dimensional temperature

0.99 0.1 3 3.25 3.5 3.75 4 0.1

x/Dw Non--dimensional temperature 0 0 0 1 2 3 4 x/Dw

Figure 4.8: Plot of saturation and temperature variation along the axis of the wick coupled to a candle flame at normal gravity (temperature is non-dimensionalized by 330 K).

154

6 0

-0.01 s)

5 s) -0.02 2 ρgug 2 -0.03 4 u ρl l -0.04

3 -0.05

-0.06 2 -0.07

-0.08 1 Vapormassflux(kg/m Liquid mass flux (kg/m -0.09

0 0 1 2 3 4 X/Dw

Figure 4.9: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick coupled to a candle flame at normal gravity.

155

2.5

2 1ge

1.5 Tmax = 7.137

r(cm) 1 1.1 2 0.5 4 6 7 0-1 0 1 2 3 4 5 X(cm)

Figure 4.10: Gas temperature contours (non-dimensionalized by T∞ = 300 K )

1.5

1ge 1 0.005

r(cm) 0.5 5E-0 5 5E-06

0-1 0 1 2 3 4 X(cm) Figure 4.11: Fuel reaction rate contours (g cm-3 s-1)

156

1.5

YF YO2 1ge 1

r(cm) 0.5 0.2

0.001 0.05 0.05 0.1 0.3 0-101234 X(cm) Figure 4.12: Fuel and oxygen mass fraction contours

1.5

1ge 1

r(cm) 0.5 0. 1 0.1 001 10 5 0-1 0 1 2 3 4 X(cm) Figure 4.13: Local fuel/oxygen equivalence ratio contours

157

1.5

1ge 1 0.001 0.01 r(cm) 0.5 0.1 0.2

0-1 0 1 2 3 4 X(cm) Figure 4.14: Carbon dioxide mass fraction contours

1.5

1ge 1 0.001 0.01 r(cm) 0.5 0.03 0. 0.07 05

0-1 0 1 2 3 4 X(cm) Figure 4.15: Water vapor mass fraction contours

158

2 3mg/cm-2 s Ψ=100 1.5 Ψ=50 Ψ=150

1

0.5 Ψ=10 Ψ=0 0 r(cm)

-0.5

-1

50 cm/s -1.5

-2 0123 X(cm)

Figure 4.16 Upper half: oxygen mass flux and streamlines, Lower half: velocity vectors and visible flame (5x10-5 g cm-3 s-1 reaction rate contour) around the flame

159

2 -0.0056 1g 1.5 e

1 -0.026 r(cm) -0.026 0.5 -0.1 05 -0.0 -0.1 0-1 0 1 2 3 4 X(cm) p − p Figure 4.17: Isobar contours; p = ∞ ρ U 2 r r

160

1 8 T 0.9 7 0.8 YN2 6 s) 0.7 3 5 0.6

0.5 Wf 4 (mg.cm f 0.4 3 &W

0.3i YF Y 2 0.2 YO2 Normalized temperature 1 0.1 YCO2 YH2O 0 0 5 10 15 20 X(cm)

Figure 4.18: Profiles of temperature and species concentration along the symmetry line

(temperature is non-dimensionalized by T∞ = 300 K )

161

1.5

1ge 1 0.001 0.01

r(cm) 0.03 0.5 0.05 0.05 0.1 0.03 0-1 0 1 2 3 4 X(cm) Figure 4.19: Effective Mean absorption coefficient distribution (cm-1 atm-1)

2 0.5 W/cm2

1.5

1 r(cm)

0.5

0 -1 0 1 2 3 4 X(cm) Figure 4.20: Dimensional net radiative flux vectors (W/cm2)

162

2

1.5

1E-06 1 - -1E-06 r(cm)

0.5 0.1 0.5 1 1.3 0 -1 0 1 2 3 4 X(cm) Figure 4.21: Contours of divergence of radiative heat flux (W/cm3)

163

16 14 12 qc 10 qnet 8

rad) 6 2 4

2 10x(qr)in 0 -2 q(W/cm -4 10x(q ) 10x(q ) -6 10x(qr)net r e r out -8 -10 0 0.1 0.2 0.3 0.4 X(cm)

Figure 4.22: Heat fluxes on the candle wick surface per unit radian

164

0.6 Saturation contours 0 . 0.5 1 0 3 . 0 6 4 . 0 8 4 w 0.4 3 . 3 0 2 1.000 4 0.3 1 . 3 2

2 r/D 0.2 0 0.1 0 0 1 2 3 4 5 X/Dw (a) 0.6 Saturation contours 0.007 0.5 0. 0.022 w 134 0.4 0.198 r/D 0.3

0.24.2 4.25 4.3 4.35 4.4 X/Dw (b) 0.6 Non-dimensional temperature contours 0.5

0

.

w 0.4 0 5 .7 0 0. 5 . 9 0.3 54 92 7 0.999918 4 6 0 46 2 3 2 r/D 5 2 3 0.2 1 9 1 0.999083 0.1 0 0 0 1 2 3 4 5 X/Dw(b) (c) Non-dimensional temperature contours (expanded in two-phase region) 0.5 0 .9 0.9 0.4 0.999774 9 0 999 9 .9 80 0.999990 w 9 9 0.3 0.999143 0 9 0 9 0.2 5 r/D 0 0.1 0 3 3.5 4 4.5 5 X/Dw (d) Figure 4.23: Plot of (a) saturation profiles (b) saturation profiles expanded in the two- phase region (c) non-dimensional temperature (non-dimensionalized by 330 K) profiles and (d) non-dimensional temperature profiles (expanded in the two-phase region) inside the porous wick for a self trimmed candle flame at normal gravity

165

0.6 Non-dimensional liquid pressure contours

0.5 5 6 . 9 . 6 1 6 7 .0 0.4 5 . 9 0

6

8

6 6 6 2

6

. w 5 6 2 . 2 4 3

.

1

1

2 6 .

2 9 0.3 0 5

9 6.3398

8 .

9 6 1 9 r/D

0

9

6

3 5

9

6

8

3 0 0.2 5

2

3 8 6.39099

1 0.1

0 0 1 2 3 4 X/Dw

0.6 Non-dimensional capillary pressure contours

0.5 0 0 . . 0 4 1 0 0 5 0 7 . . 1 . 4 0.4 0 0 3 3 0 6 6 1 0 1 . 0 1 1 2 1 3 w

4 0 4 8 7 0 0.3 4 2 7 6 1 9 7 r/D 2 6 0.2

0.1

0 0 1 2 3 4 X/Dw 0.6 Non-dimensional vapor pressure contours 0.5 1 2 . 1 8 2 0.4 1 6 . 4.96095 2 2 8

w . 8 3 8 0.3 0 1 r/D 9 0.2

0.1

0 0 1 2 X/D 3 4 w Figure 4.24: Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick for a self trimmed candle flame at normal gravity

166

0.5 0.4 10 kg/m2 s w 0.3 0.2 r/D 0.1 0 0 1 2 3 4 X/Dw (a)

0.5 0.4 2 w 0.3 0.1 kg/m s 0.2 r/D 0.1 0 1 2 3 4 X/Dw (b)

Figure 4.25: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors inside the porous wick for a self trimmed candle flame at normal gravity

167

18

16

14

) 12 2

10

(W/cm 8 wick

q 6

4

2

0 0 0.25 0.5 0.75 1 X/Dw

Figure 4.26: Plot of net heat flux supplied by the candle flame along the cylindrical surface of the wick

168

s T 1 1

0.9 0.95

0.8 0.9 s 0.7 0.85 T 0.6 0.8

0.5 0.75

saturation 0.4 0.7

0.3 0.65

0.2 0.6 Non-dimensional temperature

0.1 0.55

0 0.5 0 1 2 3 4 X/Dw

Figure 4.27: Plot of saturation and temperature variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity

169

3

2.5

s) 2 2

1.5

1 ρu l l ρlul+ρvuv

0.5 mass flux (Kg/m

0 ρvuv

-0.5 0 1 2 3 4 X/Dw

Figure 4.28: Plot of liquid and vapor mass flux (in r-direction) variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity

170

8

7 s) 2 6

5

ρlul 4

3

2

Liquid mass flux (kg/m 1

0 0 1 2 3 4 5 X/Dw

Figure 4.29: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface of the wick for a self trimmed candle flame at normal gravity

171

1 1 s T 0.9 0.9

0.8 T 0.8

0.7 1 0.7 T s 0.6 0.6

0.9975 0.5 0.5

0.4 0.995 0.4 saturation

0.3 0.3 0.9925 0.2 0.2 Non-dimensional temperature

0.99 0.1 3 3.5 4 0.1 Non-dimensional temperature x/Dw 0 0 012345 x/Dw

Figure 4.30: Plot of saturation and temperature variation along the axis of the wick for a self trimmed candle flame at normal gravity.

172

7 0

6 -0.01 s) ρ u s) 2 g g 2 -0.02 5 ρlul -0.03

4 -0.04 lux (kg/m -0.05 f 3 -0.06 mass

2 -0.07 r

-0.08 Vapo Liquid mass flux (kg/m 1 -0.09

0 0 1 2 3 4 X/Dw

Figure 4.31: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of the wick for a self trimmed candle flame at normal gravity.

173

2.5

2 1ge

1.5 Tmax = 7.117

r(cm) 1 1.1 2 0.5 4 7 7 0-1 0 1 2 3 4 5 X(cm) Figure 4.32: Gas temperature contours for a self trimmed candle flame (non-

dimensinalized by T∞ = 300 K )

1.5 1 1ge 0.5 5E-05 0

-0.5r(cm) -1

-1.5-1 0 1 2 3 4 X(cm) Figure 4.33: Fuel reaction rate contour (5X10-5 g cm-3 s-1) for a self trimmed candle flame

174

1.5

YF YO2 1ge 1

r(cm) 0.2 0.5 0 0.1 0.05 0.001 .05 0.3 0-101234 X(cm) Figure 4.34: Fuel and oxygen mass fraction contours for a self trimmed candle flame

1.5

1ge 1

r(cm) 0.5 0. 1 0.1 001 10 5 0-1 0 1 2 3 4 X(cm) Figure 4.35: Local fuel/oxygen equivalence ratio contours for a self trimmed candle flame

175

1.5

1ge 1 0.001 0.01 r(cm) 0.5 0.1 0.2

0-1 0 1 2 3 4 X(cm) Figure 4.36: Carbon dioxide mass fraction contours for a self trimmed candle flame

1.5

1ge 1 0.001 0.01 r(cm) 0.5 0.03 0.05 0.07

0-1 0 1 2 3 4 X(cm) Figure 4.37: Water vapor mass fraction contours for a self trimmed candle flame

176

2 4mg/cm-2 s

0

1

5

1.5 00 5 1 0

1

10 0.5

0 0 r(cm)

-0.5

-1

50 cm/s -1.5

-2 0123 X(cm)

Figure 4.38: Oxygen mass flux and flow field around the self trimmed candle flame

177

2

6 1g 05 e 1.5 .0 -0 -0.02 1 r(cm) 0.5 -0.02 -0.1 -0.1 -0.1 0-1 0 1 2 3 4 X(cm) p − p Figure 4.39: Isobar contours for a self trimmed candle flame; p = ∞ ρ U 2 r r

178

1 8 T 0.9 7 0.8 YN2 6 s) 0.7 3 5 0.6

0.5 4 (mg/cm f 0.4 3 &W

0.3i YF Y 2 0.2 YO2 Normalized temperature 1 0.1 YCO2 YH2O 0 0 5 10 15 20 X(cm)

Figure 4.40: Profiles of flame structure at the centerline for a self trimmed candle flame

1.5

1ge 1 0.001 0.01

r(cm) 0.03 0.5 0.05 0.05 0. 0.1 03 0-1 0 1 2 3 4 X(cm) Figure 4.41: Effective mean absorption coefficient distribution for a self trimmed candle flame (cm-1 atm-1)

179

2 0.5 W/cm2

1.5

1 r(cm)

0.5

0 -1 0 1 2 3 4 X(cm) Figure 4.42: Dimensional net radiative flux vectors for a self trimmed candle flame (W/cm2)

2

1.5

-1E-06 1 -1E-06 r(cm)

0.5 0.1 1 1.3 0.5 0 -1 0 1 2 3 4 X(cm) Figure 4.43: Contours of divergence of radiative heat flux for a self trimmed candle flame (W/cm3)

180

15

qc 10 qnet

s) 5 2

0 10x(qr)in q(W/cm

-5 10x(qr)out 10x(q ) 10x(qr)net r e

-10 0.1 0.2 0.3 0.4 0.5 X(cm)

Figure 4.44: Heat fluxes on the candle wick surface per unit radian

181

1

0ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

-4 10 ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

-2 10 ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

1ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

3ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

6ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm)

1

10ge 0.5 r(cm)

0-1 0 1 2 3 4 5 x(cm) Figure 4.45: Candle flames at various gravity levels for a self trimmed candle flame 182

1.2 11

1.1 10

1 9 0.9 8 0.8 7 0.7 6 0.6 5 0.5 4 0.4

Burning rate (mg/s) 3 0.3 Self trimming length (mm) 2 0.2 no self trimming observed Wick height is chosen as 3.8 mm 0.1 1

0 0 10-6 10-5 10-4 10-3 10-2 10-1 100 101 g/ge

Figure 4.46: Candle burning rate and self trimmed length of the candle (on log scale) at various gravity levels for a self trimmed candle flame

183

1.1 12

1 11

0.9 10 9 0.8 8 0.7 7 0.6 6 0.5 5 0.4 4

Burning rate0.3 (mg/s) 3 Self trimming length (mm) 0.2 no self trimming observed Wick height is chosen as 3.8 mm 2 0.1 1

0 0 0 5 10 15 g/ge

Figure 4.47: Candle burning rate and self trimmed length of the candle (on normal scale) at various gravity levels for a self trimmed candle flame

184

8

7

10-2g 6 e

5

1ge

Self trimming length (mm) 4

3 0 2.5E-12 5E-12 Absolute Permeability (m2)

Figure 4.48: Self trimming length of the candle wick using different wick permeabilities.

185

1.2

1.1

1 1ge

0.9

0.8 -2 10 ge

0.7 Burning rate (mg/s) 0.6

0.5

0.4 0 2.5E-12 5E-12 Absolute Permeability (m2)

Figure 4.49: Candle burning rates for different wick permeabilities.

186

3.6

3.4

3.2

3 1g 2.8 e

2.6

2.4 -2 10 ge

Flame Length2.2 (cm)

2

1.8

1.6 0 2.5E-12 5E-12 Absolute Permeability (m2)

Figure 4.50: Candle flame lengths for different wick permeabilities.

187

2200

1ge

2000

1800 -2 10 ge Maximum temperature (K)

1600 0 2.5E-12 5E-12 Absolute Permeability (m2)

Figure 4.51: Maximum gas phase temperatures for different wick permeabilities.

188

20

18

16

14 -2 10 ge 12

10

1ge 8

6

4 self trimming length (mm) 2

0 0.5 1 1.5 2 2.5 3 Dwick (mm)

Figure 4.52: Self trimming length of the candle wick for different wick diameters.

189

3

2.5 1ge

-2 2 10 ge

1.5

1 Burning rate (mg/s)

0.5

0 0.5 1 1.5 2 2.5 3 Dwick (mm)

Figure 4.53: Candle burning rates for different wick diameters.

190

2500

2400

2300 e(K) r 2200 1ge atu r 2100

2000

1900

1800 -2 10 ge 1700 Maximum tempe 1600

1500 0.5 1 1.5 2 2.5 3 Dwick (mm)

Figure 4.54: Maximum gas phase temperatures for different wick diameters.

191

2500

1ge 2000

-2 10 ge

1500 Maximum temperature (K) -4 10 ge

1000 17 18 19 20 21 22 O2%

Figure 4.55: Maximum gas phase temperature at various oxygen molar fractions

192

9 -4 10 ge

8

7

6 -2 10 ge

Self trimming length (mm) 5 1ge

4 17 18 19 20 21 22 O2%

Figure 4.56: Self trimmed length of the candle wick at various oxygen molar fractions

193

1.2

1.1

1

1ge 0.9

0.8

0.7 -2 10 ge 0.6

0.5

0.4 -4 Burning rate (mg/s) 10 ge 0.3

0.2

0.1

0 17 18 19 20 21 22

O2%

Figure 4.57: Candle burning rate at various oxygen molar fractions

194

2 0.5 21% O2 0.4 1.75 20% O2 0.3 1.5 19% O2 r(cm) 0.2 1.25 18% O2 0.1 1 0 r(cm) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.75 X(cm)

0.5

0.25

0 -101234 X(cm) Figure 4.58: Fuel reaction rate contours (5X10-5 g/cm3s)at various oxygen molar fractions.

195

2 K = 2.5e-12 m , ε =0.55,O2 =21% 45 45 Flame Length (1) Flame Width (1) 40 Flame Length (2) 40 Flame Width (2) 35 Flame Length (3) 35 Flame Width (3)

30 1 - present computation 30 2 - experiments (Arai and Amagai, 1993) 25 25 3 - computation (Alsairafi, 2003)

20 20

15 15 Flame Width (mm) Flame Length10 (mm) 10

5 5

0 0 0 2 4 6 8 10 12 Gravity (g/ge)

Figure 4.59: Comparison of flame length and flame widths at various gravity levels with the experiments (Arai and Amagai, 1993)

196

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS

The work described in this dissertation represents an effort to numerically

simulate a self trimmed candle flame by including the heat and mass transport taking

place inside the porous candle wick. The two-phase flow inside the wick is solved by

considering the steady state volume averaged equations for the porous wick. Phase

change inside the wick is accounted by the thermodynamic Gibb’s phase relationships.

The wick transport is coupled to the gas phase at the wick surface. Self trimming of the

wick is modeled as the burn out of the dry vapor region of the wick. The gas phase combustion model taken directly from Alsairafi (2003) is based on the finite volume method, includes the steady, laminar, axisymmetric, conservation equations for momentum, energy, and species (fuel, oxygen, carbon dioxide, and water vapor). Gas phase combustion is modeled via a single-step, second order, finite rate Arrhenius reaction with variable properties. Gas radiation is included in the numerical code by considering radiative contribution from carbon dioxide and water vapor. The radiation transfer equation is solved by the discrete ordinates method, with modified Planck mean absorption coefficients.

First the detailed heat and mass transport inside the porous candle wick for a given heat input is analyzed. The effect of gravity and wick permeability is studied. The following interesting observations were made

(a) In the funicular regime, there are 2 regions. A single phase liquid region is

present near the base of the wick and a two-phase region is present above it

separated by an interface. 197

(b) In the two-phase region, the liquid and the vapor move in a countercurrent

fashion. The liquid is evaporated at the surface of the wick. A small portion of

this vapor is conducted into the wick. The vapor condenses as it traverses into the

wick.

(c) There is a temperature gradient near the base of the wick causing a heat loss to

the wax pool.

(d) Gravity does not significantly affect the heat and mass transfer inside the porous

wick.

(e) The wick permeability significantly affects the saturation and temperature

distribution inside the porous wick

Later the porous candle wick is coupled to the gas phase candle burning. Self

trimming is modeled as the burn out of the dry region of the wick. Flow field in the gas phase and flame characteristics were analyzed at different gravity levels, oxygen mole fractions, and candle and wick diameters. Major findings from the computed results are:

(a) The detailed heat and mass transport inside the porous candle wick is analyzed.

(b) The self trimmed length of the wick is a function of the wick parameters (wick

diameter, wick permeability etc.) and the gas phase characteristics (e.g. ambient

oxygen) and the gravity level. The self trimmed length directly affects the burning

characteristics by changing the burning surface area.

(c) The effect of gravity, oxygen molar percentage, and wick diameter on the burning

of candle flame is analyzed.

198

(d) The gas phase characteristics are qualitatively similar to that observed by

Alsairafi (2003). The computed results show that the candle flame has a

hemispherical shape with large flame standoff distance in Zero gravity. As gravity

is increased, the flame reaches its maximum length at about 3ge. Further increase

of gravity shifts the flame stabilization zone from the base of the wick to the top

of the wick and greatly shrinks the flame length. The flame temperature drops as

gravity decreases suggesting the importance of heat losses in reduced gravity.

Both conductive and radiative losses are found to be significant.

(e) The current model has reproduced many of the experimentally observed candle

flame characteristics.

(f) There is a qualitative disagreement between the experimental results and the

computed results for the candle flame behavior at different gravity levels. The

experimental results indicate that there is monotonic decrease in the flame length

and burning rate at hyper gravity levels. Computed results indicate an increase in

burning rate and flame length followed by a decrease in the corresponding values.

Recommendation for Future Work

Based on the conclusions of the present study, several aspects for the candle flames problem need further investigation.

(1) More experimental data is needed for estimating candle wax properties. The

Leverett’s function has to be experimentally determined for the candle wax and wick combination. Similarly experimental data is required for determining the relative permeability of wax inside the wick. 199

(2) Although the inclusion of the transport of non-condensable gases into the porous wick in the two-phase model does not effect the solution in the one-dimensional problem

significantly (Raju, 2004), it can be verified for the present axisymmetric model by

including it.

(3) The capability of wick phase numerical model has to be improved to be able to capture the evaporative front regime.

(4) A detailed pyrolysis rate equation for the burning/charring of the wick material during self trimming can be incorporated. The formation of curved tip due to trimming action has to be appropriated modeled.

(5) The bending of candle wick during burning of the candle can be accounted to simulate a real flame. The two-phase flow equations of the wick can be combined with the deformation characteristics of wick material.

(6) At very high gravity levels, the flame becomes weaker and hence sufficient heat back to the candle wax pool may not be present to melt the wax. A more detailed candle flame model can be developed by including the melting of wax pool. This would give more insight into the existence of wake flame at higher gravity levels.

(7) Detailed kinetics can be incorporated for the burning of candle wax. The formation of soot can also be included in the candle flame model. The visible flame reported in the experiments is affected by the formation of soot. The inclusion of soot would give a better comparison of the computed results with the experimental results.

(8) More detailed experimental results are needed to compare the present computed results. The previous experiments do not report the nature of candle wax, the wick length, 200 wick properties, the self trimming characteristics of the wick etc. The computed results show that the candle flame characteristics can significantly vary depending on the permeability of the wick. This should be taken into consideration while performing the experiments.

(9) The present two-phase heat and mass transport model inside porous media can be applied to various practical situations like flame spreading over charring fuels, sand beds.

It can also be extended to heat pipe applications.

201

BIBLIOGRAPHY

Abu-Romia, M. M.; Tien, C. L. “Appropriate Mean Absorption Coefficient for Infrared Radiation of Gases,” ASME J. Heat Transfer, 89C (1967), 321-327.

Alsairafi, A. “A computational Study on the Gravity Effect on Wick-Stabilized Diffusion Flames,” Ph.D. Dissertation, Case Western Reserve University, Cleveland, OH (2003).

Amagai, K.; Ito, Y.; Arai, M. “Diffusion Jet Flames in Low and High Gravity Fields,” J. Jpn. Soc. Microgravity Appl., 4 [1] (1997) 3-9.

Anderson, D. A.; Tannehill, J. C.; Pletcher, R. H. Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, 1984.

Arai, M.; Amagai, K. “Behavior of Candle Flame under Super-Gravitational Fields,” Journal of Japan Soc. of Micro-gravity Applications, 10 (1993) 259-266.

Baek, S. W.; Kim, M. Y. “Modification of the Discrete-Ordinates Method in an Axisymmetric Cylindrical Geometry,” Numer. Heat Transfer Part B, 31 (1997) 313- 326.

Bau, H. H., and Torrence, K. E., “Boiling in low-permeability Porous Media,” Int. J. Heat Mass Transfer, 25 (1982) 45-55.

Bear, J. Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972

Bedir, H. “Gas Phase Radiative Effects in Diffusion Flames,” Ph.D. Dissertation paper, Case Western Reserve University, 1998.

Bedir, H.; T’ien, J. S. “A Computational Study of Flame Radiation in PMMA Diffusion Flames Including Fuel Vapor Participation,” Twenty-Seventh Symposium (International) on Combustion, The Combustion Institute, (1998) 2821-2828.

Bedir, H.; T’ien, J. S.; Lee, H. S. “Comparison of Different Radiative Treatments for a One-Dimensional Diffusion Flame,” Combustion Theory Modeling, 1 (1997) 395- 404.

Benard, J.; Eymard, R.; Nicholas, X.; Chavant, C. “Boiling in Porous Media: Model and Simulations,” Trans. Porous Media, 60 (2005) 1-31.

Bhattacharjee, S.; Altenkirch, R. A. “Radiation-Controlled, Opposed-Flow Flame Spread in a Microgravity Environment, “ Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, (1990) 1627-1633.

202

Bryant, D. “An Investigation into the Effects of Gravity Level on Rate of Heat Release and Time to Ignition,” Fire and materials, 19 [3] (1995) 119-126.

Bodvarsson, G.S.; Lai, C. H.; Truesdell, A. H. “Modelling Studies of Heat Transfer and Phase Distribution in Two-Phase Geothermal Reservoirs,” Geothermics, 23 (1) 3- 20 (1994).

Buckmaster, J.; Peters, N. “The Infinite Candle and its Stability-A Paradigm for Flickering Diffusion Flames,” Twenty-First Symposium (International) on Combustion, The Combustion Institute, (1986) 1829-1836.

Carleton, F. B.; Weinberg, F. J. “Electric Field-Induced Flame Convection in the Absence of Gravity,” Nature, 330 [17] (1987) 635-638.

Carlson, B. G.; Lathrop, K. D. “Transport Theory-The Method of Discrete Ordinates,” in H. Greenspan, C. N. Kelber and D. Okrent (Eds.), Computing Methods in Reactor Physics, Gordon & Breach, New York, 1968.

Chai, J. C.; Patankar, S. V.; Lee, H. S. “Evaluation of Spatial Differencing Practices for the Discrete Ordinates Method,” J. Thermophys. Heat Transfer, 8 [1] (1994) 140- 144.

Chan, W. A.; T’ien, J. S. “An Experiment on Spontaneous Flame Oscillation Prior to Extinction,” Combust. Sci. and Tech., 18 (1978) 139-143.

Chandrasekhar, S. Radiative Transfer, Dover, New York, 1960.

Chen, C. H.; Cheng, M. C. “Gas Phase Radiation Effects on Downward Flame Spread in Low Gravity,” Combust. Sci. and Tech., 97 (1994) 63-83.

Cross, M.; Gibson, M. C.; Young, R. W.; “Pressure Gradients generated during the drying of a Porous Half Space,” Int. J. Heat Mass Transfer, 22 (1979) 827-830.

Dayan, A. “Self-Similar Temperature, Pressure and Moisture Distribution within an Intensely Heated Porous half space”, Int. J. Heat Mass Transfer, 25(10) (1981) 1469- 1476.

Dayan, A.; Gluekler, E. L. “Heat and Mass Transfer within an Intensely Heated Concrete Slab,” Int. J. Heat and Mass Tranfer, 25 (1982) 1461-1467.

Dietrich, D. L.; Ross, H. D.; Chang, P.; T’ien, J. S. “Candle Flames in Microgravity,” Proceeding of the Sixth International Microgravity Combustion Workshop, Cleveland, Ohio, (2001), 329-332.

Dietrich, D. L.; Ross, H. D.; Shu, Y.; Chang, P.; T’ien, J. S. “Candle Flames in Non- Buoyant Atmospheres,” Combust. Sci. and Tech., 156 (2000) 1-24. 203

Dietrich, D. L.; Ross, H. D.; T’ien, J. S. “Candle Flames in Non-Buoyant and Weakly Buoyant Atmospheres,” AIAA-94-0429, 32nd Aerospace Science Meeting and Exhibit, Reno, Nevada, 1994.

Dietrich, D. L.; Ross, H. D.; T’ien, J. S.; Shu, Y. “Candle Flames in Microgravity,” Proceeding of the Fourth International Microgravity Combustion Workshop, Cleveland, Ohio (1997), 237-242.

Dua, S. S.; Cheng, P. “Multi-Dimensional Radiative Transfer in Non-Isothermal Cylindrical Media with Non-Isothermal Bounding Walls,” Int. J. Heat Mass Transfer, 18 (1975) 245-259.

Duff, I. S., “Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core,” SIAM J. Sci. Stat. Comput. 5(2) (1984) 270–280.

Duff, I. S.; Erisman,A. M.; Reid, J. K., Direct Methods for Sparse Matrices, Oxford University Press, London, 1986.

Duff, I. S.; Reid, J. K. “The multifrontal solution of indefinite sparse symmetric linear systems,” ACM Transactions on Mathematical Software, 9 (1983) 302-325.

Duff. I. S., “Parallel implementation of multifrontal scheme,” Parallel Comput, 3, (1986) 193-204,.

Davis, T. A. “A column pre-ordering strategy for the unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw., 30(2) (2004) 165–195.

Davis, T. A. “Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw., 30(2) (2004) 196–199.

Davis, T. A.; Duff, I. S. “An unsymmetric-pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Applic., 18(1) (1997) 140–158.

Davis, T. A.; Duff, I. S. “A combined unifrontal/multifrontal method for unsymmetric sparse matrices,” ACM Trans. Math. Softw., 25(1) (1999) 1–19.

Dullien, F. A. L. Porous Media: Fluid Transport and Pore Structure, Academic Press, Newyork (1979).

Essenhigh, R. H.; Mescher, A. M. “Flame Spread Down Paper Cylinders at Elevated Gravity,” ASME Paper HTD-269 (1993) 49-57.

Faraday, M. The Chemical History of a Candle, Chicago Review Press, 1988.

204

Ferkul, P. V. “A Model of Concurrent Flow Flame Spread over a Thin Solid Fuel,” Ph.D. Dissertation paper, Case Western Reserve University, 1993.

Ferziger, J. H.; Perić, M. Computational Methods for Fluid Dynamics, Springer- Verlag, 1996.

Fiveland, W. A. “A Discrete Ordinates Method for Predicting Radiative Heat Transfer in Axisymmetric Enclosure,” ASME Paper 82-HT-20 (1982) 1-8.

Fiveland, W. A. “Discrete-Ordinates Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media,” J. Heat Transfer, 109 (1987) 809-812.

Fiveland, W. A. “Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures,” J. Heat Transfer, 106 (1984) 699-706.

Fiveland, W. A. “The Selection of Discrete Ordinates Quadrature Sets for Anisotropic Scattering,” HTD-Vol. 160, In: Fundamentals of Radiation Heat Transfer, Edited by W. A. Fiveland; A. L. Crosbie; A. M. Smith; T. F. Smith (1991) 89-96.

Fiveland, W. A. “Three Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method,” J. Thermophys. Heat Transfer, 2 [4] (1988) 309-316.

Garbiniski, G. P.; T’ien, J. S. “Ignition and Extinction Curves of Diffusion Flames with Radiative Losses,” AIAA-94-0577, 32nd Aerospace Science Meeting and Exhibit, Reno, Nevada, 1994.

Grayson, G. D.; Sacksteder, K. R.; Ferkul, P. V.; T’ien, J. S. “Flame Spreading Over a Thin Solid in Low-Speed Concurrent Flow- Drop Tower Experimental Results and Comparison with Theory,” Microgravity Sci. Tech., VII/2, (1994) 187-195.

Hansen, D.; Breyer, W. H.; Riback. “Steady state heat transfer in partially liquid filled porous media”, J. of Heat transfer, 92 (3) , 520-527 (1970).

Hoffman, J. D. Numerical Methods for Engineers and Scientists, McGraw-Hill, Mechanical Engineering series, 1993.

Hood, P., “Frontal solution program for unsymmetric matrices,” Int. J. Numer. Meth. Eng. 10, 379-400, 1976.

Hottel, H. C.; Cohen, E. S. “Radiant Heat Exchange in a Gas-Filled Enclosure: Allowance for Nonuniformity of Gas Temperature,” AIChE J., 4 [1] (1958) 3-14.

Hottel, H. C.; Sarofim, A. F. “The Effect of Gas Flow Patterns on Radiative Transfer in Cylindrical Furnaces, Int. J. Heat Mass Transfer, 8 (1965) 1153-1169.

205

Hougen, O. A.; McCauley, H. J.; Marshal, Jr. W. R. “Limitations of Diffusion Equations in Drying,” Trans. Am. Inst. Chem. Eng. 36 (1940) 183-210.

Irons, B. M, “A frontal solution program for finite-element analysis,” Int. J. Numer. Meth. Eng. 2, (1970) 5-32.

Jamaluddin, A. S.; and Smith, P. J. “Discrete-Ordinates Solution of Radiative Transfer Equation in Nonaxisymmetric Cylindrical Enclosures,” J. Thermophysics Heat Transfer, 6 [2] (1992) 242-245.

Jamaluddin, A. S.; Smith, P. J. “Predicting Radiative Transfer in Axisymmetric Cylindrical Enclosures Using the Discrete Ordinates Method,” Combust. Sci. and Tech., 62, (1988) 173-186.

Jiang, C-B. “A Model of Flame Spread Over a Thin Solid in Concurrent Flow with Flame Radiation,” Ph.D. Dissertation paper, Case Western Reserve University, 1995.

Kaviany, M.; Rogers, J. A. “Funicular and Evaporative-Front Regimes in Convective Drying of Granular Beds,” Int. J. of Heat and Mass Transfer, 35(2) (1991) 469-480.

Kaviany, M.; Tao, Y. “A Diffusion Flame Adjacent to a Partially Saturated Porous slab: Funicular State,” Journal of Heat Transfer, 110 (1988) 431-436.

Kaviany, M.; Mittal, M. “Funicular state in Drying of a Porous Slab,” Int. J. Heat and Mass Transfer, 30 (1987) 1407-1418.

Kong, W. J.; Chao, C. Y. H.; Wang, J. H. “Burning characteristics of non-spread diffusion flames of liquid fuel soaked in porous beds,” J. Fire Sciences 20 (3) (2002) 203-225.

Kumar, A.; Shih, H.; T’ien, J. S. “A Comparison of Extinction Limits and Spreading Rates in Opposed and Concurrent Spreading Flames Over Thin Solids,” Combust. Flame, 132 (2003) 667-677.

Kuo, K. K. Principles of Combustion, John Wiley & Sons, 1986.

Lathrop, K. D. “Spatial Differencing the Transport Equation: Positivity vs. Accuracy,” J. Comput. Phys. 4 (1969) 475-498.

Lathrop, K.D.; Carlson, B. G. “Discrete Ordinates Angular Quadrature of the Neutron Transport Equation,” Technical Information Series Report LASL-3186, Los Alamos Scientific Laboratory (1965).

Lawton, J.; Weinberg, F. J. Electric Aspects of Combustion, Clarenton, Oxford, 336- 340, (1969).

206

Leverett, M.C., “Capillary Behavior in Porous Solids,” AIME Transactions, 142, (1941) 152-157.

Lewis, W. K. “The rate of Drying Solid Materials,” Ind. Eng. Chem., 13 (1921) 12- 16.

Lewis, E. E.; Miller, W. F. Jr. Computational Methods of Neutron Transport, New York, Wiley, 1984.

Liu, F.; Becker, H. A.; Pollard, A. “Spatial Differencing Schemes of the Discrete- Ordinates Method,” Numer. Heat Transfer Part B, 30 (1996) 23-43.

Luikov, A. V. “Systems of differential equations of heat and mass transfer in capillary-porous bodies”, Int. J. Heat and Mass Transfer 18, 1-14(1975).

Maxworthy, T. “The Flickering Candle: Transition to a Global Oscillation in a Thermal Plume,” J. Fluid Mech., 390 (1999) 297-323.

Middha, P.; Wang, H. Private Communication, 2002.

Modest, M. F. “Radiative Heat Transfer,” McGraw-Hill Inc., 1993.

Oostra, W.; Bryant, D.; Marijnissen, J. C. M.; Scarlett, B.; Wissenburg, J. “Measurement of Soot Production of a Candle under Microgravity Conditions,” J. Aerosol Sci., 27 [Suppl. 1] (1996) S715-S716.

Patankar, S. V. “Calculation Procedure for Two-Dimensional Elliptic Situations”, Numer. Heat Transfer, 4 [4] (1981) 409-425.

Patankar, S. V. Numerical Heat Transfer and Fluid Flow, Taylor & Francis, 1980.

Peaceman, D. W. and Rachford, H. H. “The Numerical Solution of Parabolic and Elliptic Differential Equations,” J. Soc. Ind. Appl. Math., 3 (1955) 28-28.

Phillip, J. R.; DeVries, D. A. “Moisture Movement in Porous Materials under Temperature Gradients,” Trans. Am. Geophys. Union 38 (1957) 222-232.

Poinsot, T.; Veynante, D. Theoretical and Numerical Combustion, Edward, 2001.

Rahli, O.; Topin, F.; Tadrist, L.; and Pantaloni, J. “Analysis of Heat Transfer with Liquid-Vapor Phase Change in a Forced-Flow Fluid Moving through Porous Media.”, Int. J. Heat Mass Transfer, 39 (18) 3959-3975 (1996).

Raju, M. “Liquid-Fueled Stagnation Point Diffusion Flame Stabilized next to a Porous Wick,” M.S. Thesis, Case Western Reserve University, Cleveland, OH (2004). 207

Ramesh, P. S.; Torrance, K. E. “Numerical Algorithm for Problems Involving Boling and Natural Convection in Porous Materials,” Numer. Heat Transfer, Part B 17 (1990) 1-24.

Rhatigan, J. L.; Bedir, H.; T’ien, J. S. “Gas-Phase Radiative Effects on the Burning and Extinction of a Solid Fuel,” Combust. Flame, 112 (1998) 231-241.

Ritter, E. R.; Bozzelli, J. W. THERM, ver. 4.21, New Jersey Institute of Technology, 1991.

Ross, H. D.; Sotos, R. G.; T’ien, J. S. “Observation of Candle Flames Under Various Atmospheres in Microgravity,” Combust. Sci. and Tech., SHORT COMMUNICATION, 75 (1991) 155-160.

Saad, M. A., Thermodynamics for Engineers, Prentice Hall, Englewood Cliffs, New Jersey, 1966.

Saad, Y., Iterative methods for sparse linear systems, 2nd edition, SIAM Publications 2003.

Scheidegger, A. E., The physics of Flow through Porous Media, University of Toronto Press, Toronto, 1974.

Shu, Y. “Modeling of Candle Flame and Near-Limit Oscillation in Microgravity,” M. Sc. Dissertation paper, Case Western Reserve University, 1998.

Shu, Y.; T’ien, J.; Dietrich, D.; Ross, H. “Modeling of Candle Flame and Near- Extinction Oscillation in Microgravity,” J. Jpn. Soc. Microgravity Appl., Supplement II, 15 (1998) 272-277.

Siegel, R.; Howell, J. R. “Thermal Radiation Heat Transfer,” McGraw-Hill Inc., 3rd Ed., 1992.

Smooke, M. D.; Giovangigli, V. “Formulation of the Premixed and Nonpremixed Test Problem,” Lecture Notes in Physics, Springer-Verlag, New York, Series 384, 1991.

Somerton, W. H.; Keese, J. A.; Chu, S. L. “Thermal Behavior of Unconsolidated Oil Sands,” Society of Petroleum Engineering Journal, 14 (5) 1974

Stewart, F. R.; Cannon, P. “The Calculation of Radiative Heat Flux in a Cylindrical Furnace Using the Monte-Carlo Method,” Int. J. Heat Mass Transfer, 14 (1971) 245- 261.

208

Stone, H. L. “Iterative Solution of Implicit Approximation of Multi-Dimensional Partial Differential Equations,” SIAM J. Num. Anal., 5 (1968) 530-539.

T’ien, J. S. “Diffusion Flame Extinction at Small Stretch Rates: The Mechanism of Radiative Loss,” Combust. Flame, 65 (1986) 31-34.

T’ien, J. S.; Bedir, H. “Radiative Extinction of Diffusion Flames- A Review,” in Proceeding of the first Asia-Pacific Conference on Combustion, Osaka, Japan, May 12-15, 1997.

T’ien, J. S.; Shih, H-Y; Jiang, C-B; Ross, H. D.; Miller, F. J.; Fernandez-Pello, A. C.; Torero, J. L.; Walther, D. “Mechanisms of Flame Spread and Smolder Wave Propagation,” In: Microgravity Combustion, Fire in Free Fall, Edited by Howard D. Ross (2001) 299-418.

Thompson, H. D.; Webb, B. W.; and Hoffman, J. D. “The Cell Reynolds Number Myth”, Int. J. Numer. Methods in Fluid Mech., 5 (1985) 305-310.

Thompson, J. F. (Editor), Numerical Grid Generation, North-Holland, New York, 1982.

Tien, C. L. Advances in Heat Transfer, (T. F. Irvine, Jr.; J. P. Hartnett, Eds.), Academic Press, New York, 5 (1968) 254.

Truelove, J. S. “Discrete-Ordinate Solutions of Radiation Transport Equation,” J. of Heat Transfer, 109 (1987) 1048-1051.

Truelove, J. S. “Three-Dimensional Radiation in Absorbing-Emitting Media,” J. Quant. Spectrosc. Radiat. Transfer, 39 [1] (1988) 27-31.

Udell, K. S.; Flitch, J. S. “Heat and Mass Transfer in Capillary Porous Media Considering Evaporation, Condensation and Non-Condensable Gas Effects,” Heat transfer in Porous Media and Particulate Flows, ASME HTD, 46 (1985) 103-110.

Versteeg, H. K.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume, Longman Scientific & Technical, 1995.

Villermaux, E.; Durox, D. “On the Physics of Jet Diffusion Flames,” Combust. Sci. and Tech., 84 (1992) 279-294.

Wang, C. Y. and Beckermann, C. “A Two-Phase Mixture Model of Liquid-Gas Flow and Hat Transfer in Capillary Porous Media,” Int. J. Heat Mass Transfer, 36 (1993) 2747-2769.

Whitaker, S.; “Simultaneous Heat, Mass and Momentum Transfer in Porous Media: A Theory of Drying,” Advances in Heat Transfer, 13 (1977) 119-203. 209

Williams, F. A. Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, Menlo Park, Calif.: Benjamin/Cummings Pub. Co., 2nd Ed., 1985.