Heat and Mass Transport Inside a Candle Wick
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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) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (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.