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NUMERICAL MODELING OF FLAME DEVELOPMENT OVER POLYMERIC

MATERIALS

Volume I

By

Guanyu Zheng

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHYLOSOPHY

Department of Mechanical Engineering

2000

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ABSTRACT
NUMERICAL MODELING OF FLAME DEVELOPMENT OVER POLYMERIC

MATERIALS

By
Guanyu Zheng
Transient ﬂame development over plastic polymeric materials comprises of preheating,
ignition, transition and steady state ﬂame spread and is of both fundamental and practical
importance to ﬁre safety issues. The ﬂame behavior is determined by various physical
and chemical mechanisms including (1) in the gas phase, combustion reaction, channel
cross ﬂow, reaction-induced thermal expansion, and interface injection ﬂow; (2) in the
condensed phase, melting, pyrolysis reaction, bubble nucleation, growth, and movement;
(3) at interface, radiation heat loss and fuel and oxidizer transport. A numerical model is
established to describe such transient ﬂame spread process over a polymer with emphasis
on the complex condensed behavior including melting, pyrolysis, and bubble generation.
The models utilized are in the gas phase, a Navier-Stokes laminar ﬂow model and
combustion model; in the condensed phase, an enthalpy-based phase change model, a
one-step global pyrolysis reaction model and a volume averaged bubble model. The
bubble model includes a macro-scale transport model and a micro-scale bubble transport
model by using volume averaging method. The investigation is carried out by modeling
the ﬂame development process with increasing complexities. It includes: (1) ﬂame spread
over an anisotropic solid polymers by using assumed ﬂow pattern in the gas phase (Oseen

approximation); (2) ﬂame spread over melting polymers by using the Oseen

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approximation; (3) ﬂame spread over melting polymers with realistic ﬂow pattern by
solving the Navier Stokes equations; (4) ignition analysis for melting polymers with both
ﬂow patterns by using energy balance principle; (5) revisit of ﬂame spread over an
anisotropic melting polymer with realistic ﬂow pattern by solving the Navier Stokes
equations; (6) the mathematical model that describes the bubble forming, melting, and
pyrolysis by using volume averaging approaches. Various parameters are obtained for
ﬂame structure, ignition delay, interface phenomena, and ﬂame spread rate. These
numerical results are favorably compared to experimental and analytical formulas.
Furthermore, energy balance approaches are applied to ignition and ﬂame spread. A
simple ignition theory is derived for ignition delay; ﬂame spread mechanism is

interpreted in terms of weights of heat transfer mechanisms.

To my father Shenfu Zheng, my mother Shifang Tang and my grandfather Yin Zheng.

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ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my co-advisors, Dr. Andre Benard and
Dr. Indrek S. Wichman, whose guidance and encouragement are invaluable throughout
this study. I am grateful to Dr. Tom I-P. Shih for his advice on numerical algorithms. I
would also like to thank Dr. Charles B. Owen during the final stage of this work.

I am indebted to my parents, Shenﬁi Zheng and Shifang Tang, my brother Guanhong
Zheng and my sister Sujing Zheng for their constant support during my studies. I am
especially grateful to my wife Junhua Gu, who not only supports me during the long
“journey" but also shares the pain and joy with me.

I would also like to thank the State of Michigan and National Institute of Standard

and Technology for their ﬁnancial supports of this research.

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TABLE OF CONTENTS

LIST OF TABLES ................................................................................... xi
LIST OF FIGURES ................................................................................. xii
NOMENCLATURE ............................................................................... xvi
CHAPTER 1
INTRODUCTION .................................................................................... 1
1 Polymers and Fire Hazards ........................................................................ l
1.1 Brief Description of Polymeric Materials ............................................. l
1.2 Production and Use of Polymeric Materials .......................................... 2
1.3 Fire Hazards of Polymers ................................................................ 3
2 Fundamental Phenomena of Polymer Combustion ......................................... 4
2.1 Overall Description ....................................................................... 4
2.2 Solid Phase Phenomena .................................................................. 6
2.2.1 Decomposition ....................................................................... 7
2.2.2 Phase Change (Melting) ............................................................ 9
2.2.3 Bubble Formation .................................................................. 12
2.3 Gas Phase Phenomena .................................................................. 13
2.3.1 Combustion Reaction ............................................................. 13
2.3.2 Flow Field ........................................................................... 14
3 A Review: Modeling of Flame Spread over Non-Charring Polymers .................. 15
3.1 Solid Phase Transport Modeling ...................................................... 19
3.2 Gas Phase Modeling ..................................................................... 20
4 Motivations and Scope ........................................................................ 21
CHAPTER 2
MODELING OF IGNITION, TRANSITION AND STEADY FLAME SPREAD OVER
ANISOTROPIC SOLID POLYMERS ........................................................... 27
1 Introduction .................................................................................... 27
2 Mathematical Model .......................................................................... 27
3 Numerical Treatment ......................................................................... 29
4 Results and Discussions ...................................................................... 3O
5 Conclusions .................................................................................... 32
CHAPTER 3 ............................................................................................
OI)POSED-FLOW FLAME SPREAD OVER POLYMERIC MATERIALS —
INFLUENCE or PHASE CHANGE ............................................................ 37
1 Introduction .................................................................................... 37
2 Numerical Model .............................................................................. 39
2.1 Problem Formulation .................................................................... 40
2.1.1 Non-Dimensionalization .......................................................... 45
2.1.2 Numerical Approach .............................................................. 48
2.2 Theoretical (Simple) Model ............................................................ 49
2.2.1 Simpliﬁed Derivation of F lame-Spread Equation ............................. 51
3 Results and Discussions ...................................................................... 56
3.1 Flame Spread Rate ...................................................................... 56

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3.2 Transient Spread Process ............................................................... 59

3.3 Flame Structure .......................................................................... 61
3.4 The Condensed Phase ................................................................... 63
3.5 Mechanism of Steady Flame Spread .................................................. 65
4 Conclusions .................................................................................... 67
CHAPTER 4
MODELING OF FLAME SPREAD OVER MELTING POLYMERS WITH NAVIER
STOKES FLOW CALCULATION IN THE GAS PHASE ................................... 87
1 Introduction .................................................................................... 87
2 The Mathematical Model ..................................................................... 88
3 Results .......................................................................................... 92
3.1 Physical Properties ...................................................................... 92
3.2 Flame Spread Rate ...................................................................... 92
3.3 Transient Flame Development ......................................................... 93
3.4 Flame Structure ........................................................................ 102
3.5 Energy Balance Analysis During Steady Flame Spread .......................... 103
4. Conclusions .................................................................................. 108
CHAPTER 5
REVISIT OF THE INFLUENCE OF SOLID ANISOTROPY ON FLAME SPREAD
OVER MELTING POLYMER .................................................................. 132
1 Introduction ................................................................................... 132
2 Results .......................................................................................... 133
2.] Flame Spread Rate ..................................................................... 133
2.2 Phase Front .............................................................................. 136
2.3 Interface Parameters ................................................................... 137
2.4 Energy Balance Analysis ............................................................. 138
2.5 Flame Size ............................................................................... 140
2.6 Ignition Delay Time ................................................................... 141
3 Conclusions .................................................................................. 142
CHAPTER 6

ENERGY BALANCE ANALYSIS OF IGNITION OVER A MELTING POLYMER
SUBJECTED TO A HIGH RADIATION HEAT FLUX IN A CHANNEL FLOW. . ...155

1 Introduction ................................................................................... 155
2 Results and Discussion ....................................................................... 159
2.1 Ignition Delay Time ................................................................... 160
2.2 Heat Transfer Mechanisms at Interface ............................................. 161
2.3 Heat Transfer Mechanisms in Control Volume .................................... 164
2.4 Enthalpy Increases of Condensed Phase ............................................ 165
2.5 Field Phenomena in Condensed Phase and at Interface ........................... 166
2.6 Inﬂuence of External Radiation ...................................................... 169
2.7 Simple Theory for Ignition Delay — Inﬂuence of Condensed Phase ............ 170
3 Conclusions ................................................................................... 174
CHAPTER 7
MODELING OF BUBBLE TRANSPORT FOR OVERHEATED MELTING
POLYMERS ........................................................................................ 202
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1 Introduction ................................................................................... 202

2 Physical Model ............................................................................... 204

2.1 Single Bubble Model .................................................................. 205

2.1.1 Bubble Nucleation ............................................................... 205

2.1.2 Single Bubble Translational Velocity ......................................... 206

2.1.3 Bubble Growth Rate ............................................................. 207

2.2 Macroscopic Volume Averaged Equations ......................................... 209

2.2.1 Volume Averaged Energy Conservation Equation .......................... 210

2.2.2 Volume Averaged Physical Prosperities ...................................... 21 1

2.2.2.1 Porosity 8g ................................................................. 211

2.2.2.2 Volume Averaged Density ,5 ............................................ 212

2.2.2.3 Volume Averaged Velocity V ........................................... 212

2.2.2.4 Heat Capacity C p ......................................................... 213

2.2.2.5 Volume Averaged Conductivity k ...................................... 213

2.2.2.6 Volume Averaged Mass Rate of Evaporation .......................... 214

2.2.2.7 Surface Mass Flow Rate ................................................... 214

3 Numerical Implementations ................................................................ 215

4 Results and Discussions ..................................................................... 217

4.1 Properties ................................................................................ 217

4.2 Time History of Bubble Development .............................................. 217

4.3 Inﬂuence of Nucleation Rate ......................................................... 221

4.4 Inﬂuence of Initial Bubble Size ...................................................... 221

4.5 Inﬂuence of St ........................................................................ 223

4.6 Inﬂuence of Evaporation Heat ....................................................... 224

5 Conclusions .................................................................................. 225

CHAPTER 8
FUTURE WORK .................................................................................. 249
APPENDIX I

COMPARISON OF NUMERICAL RESULTS WITH ANALYTICAL SOLUTION FOR
l-D MELTING PROCESS 1N CONDENSED PHASE ...................................... 257
1 Problem with Uniform Thermal Properties across Phase Interface ................... 257
1.1 Problem Formulation .................................................................. 257

1.2 Analytical Solution ..................................................................... 258
1.3 Comparisons ............................................................................ 259
2 Problem with Discontinuous Thermal Properties across Phase Interface ............ 260
2.1 Problem Formulation .................................................................. 260

2.2 Analytical Solution ..................................................................... 261

2.3 Comparisons ............................................................................ 261

3 Conclusions ................................................................................... 262

APPENDIX 11
RICAL METHOD FOR FLAME SPREAD OVER MELTING POLYMERS
WITHOUT FLOW CALCULATION .......................................................... 270
1 Discretization ................................................................................. 270
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1.2 Boundary Conditions .................................................................. 274
1.3 Treatment of Convection-Diffusion Terms ......................................... 273
1.4 Treatment of Chemical Terms ........................................................ 274
2 The Computational Procedure ............................................................. 275
2.1 Non-linearities .......................................................................... 277
2.2 The Newton-Raphson Scheme ....................................................... 276
2.3 Alternate Direction Implicit (ADI) with Special Source Treatment ............ 277
2.4 Enthalpy Method ....................................................................... 278
2.4.1 Source Update Method .......................................................... 278
2.4.2 Treatment of Discontinuous Thermal Properties ............................ 280
3 Coordinate Transformation - Clustered Grids Near the Interface ..................... 283
4 The Solution Procedure ..................................................................... 285
5 Data Structure and Others ................................................................... 285
APPENDIX III
DERIVATION OF A FLAME SPREAD FORMULA FOR MELTING
POLYMERS ....................................................................................... 290
APPENDIX IV
NUMERICAL METHOD USED IN THE FLOW FIELD CALCULATION ............ 293
1 Control Volumes ............................................................................. 293
2 Discretization ................................................................................. 293
3 Staggered Grids .............................................................................. 294
4 Discretization of Gas Phase Equations ................................................... 294
5 Under-Relaxation ............................................................................ 296
6 The SIMPLEC Scheme for Flow Calculation ............................................ 297
6.1 Derivation of Momentum Equations ................................................ 297
6.2 Derivation of Pressure Correction Equation ....................................... 298
7 Boundary Conditions ........................................................................ 299
8 Overall Solution Procedure ................................................................. 300
8.1 Solution of Algebraic Equations ..................................................... 300
8.1.1 Delta Formulation of Time Discretization .................................... 300
9 Solution of the Pressure Correction Equation ............................................ 302
APPENDIX V
DEFINITION OF THE HEAT TRANSFER MECHANISMS IN IGNITION
ANALYSIS ......................................................................................... 307
APPENDD( VI
DERIVATION OF VOLUME AVERAGED EQUATIONS FOR BUBBLE FORMING
PROCESS IN HEATED POLYMERS ......................................................... 308
1 The Basic Equations ......................................................................... 308
1.1 Governing Point Equations ........................................................... 308
1.2 Boundary Conditions .................................................................. 312
2 The Volume Averaging Method ........................................................... 315
2.1 Deﬁnitions and Theorems ............................................................ 315
2.2 The Derivation Procedure ............................................................ 316
3 Macrosc0pic Equations ..................................................................... 317
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3.3 Macroscopic Equation of Energy Conservation .................................... 318

3.4 Macroscopic Equation of Species Conservation ................................... 320

3.5 Macroscopic Equation of Momentum Conservation .............................. 321

4 Simpliﬁcations of Macroscopic Equations ............................................... 322

REFERENCES .................................................................................... 327
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LIST OF TABLES

CHAPTER 1

Table l. The mechanisms that contribute to the flame development over polymers. . .26
CHAPTER 2

Table l The thermal properties and chemical kinetics used in the modeling ............ 36
CHAPTER 3
Table 1. Major pr0perties and kinetic data used for the numerical model ................ 84
Table 2. Dimensionless parameters for the numerical model .............................. 85
Table 3. Different heat transfer mechanisms in four cases of interest ................... 86
CHAPTER 4
Table 1. Major properties and kinetic data used for the numerical model .............. 131
CHAPTER 7
Table 1. Translational velocity formula of bubble movement ............................ 207
Table 2. Growth rate of a single bubble in liquid polymer ............................... 208

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LIST OF FIGURES

CHAPTER 1
Figure 1 Schematic diagram of combustion of non-chairing polymers .................. 24
Figure 2 Schematic of the ﬂaming combustion of a polymer .............................. 25
CHAPTER 2
Figure l Constant temperature contours at different times ................................. 33
Figure 2 Constant fuel concentration contours at different times ......................... 34
Figure 3 Constant oxidizer concentration contours at different time. .................... 35
Figure 4 Ignition time vs. ksx / ksy ........................................................... 36
Figure 5 Flame spread rate vs. ksx /ksy ..................................................... 36
CHAPTER 3
Figure 1 Schematic description of diffusion ﬂame spread over polymers in an
opposed-ﬂow of oxidizer ....................................................................... 71
Figure 2 The enthalpy-temperature relationship used in the numerical model .......... 72
Figure 3 The simpliﬁed ﬂame spread model for steady ﬂame spread .................... 72

Figure 4 (a) Flame spread rate vs. St; (b) Flame spread rate vs. 1;, ; (c) Flame spread
rate vs. 5p, ...................................................................................... 73

Figure 5 (a) Flame spread rate vs. streamwise distance at St = 2; (b) Arrival times of
ﬂame front and phase front vs. streamwise distance; (c) The streamwise evolution of
heat and mass ﬂux at the interface; (d) The streamwise evolution of interface ......... 74
Figure 6 Comparisons of non-dimensional (a) temperature/ St, (b) reaction rate, (c)
ﬁrel concentration and (d) oxidizer concentration for four conditions of the condensed

material ............................................................................................ 76
Figure 7 Comparison of numerical model and theory for the temperature proﬁles and
phase fronts in the condensed phase at St = 2 ............................................. 79
Figure 8 Comparisons of (a) interface temperature, (b) phase location, (c) interface
mass ﬂux and ((1) interface heat ﬂux in four situations of the condensed phase ........ 80
Figure 9 The evolution of solid-liquid interface and pyrolysis front locations before
ignition (108), during transition (12s) and during steady spread (19s) ................... 83
CHAPTER 4
Figure 1 Flame spread rate vs. St ........................................................... 110
Figure 2 Flame spread Rate vs. k1 .......................................................... 1 10
Figure 3 Flame spread rate vs. Cpl ......................................................... 111
Figure 4 Constant temperature levels and ﬂow streamlines at (a) t = 3s , (b) t = 55 , (c)
t=103 and(d)t=15s ........................................................................ 111
Figure 5 Pressure distribution at (a) 35, (b) 55, (c) 103, and (d) 155 ..................... 113
Figure 6 Fuel and oxidizer concentration constant levels for St = 2 at 53 (a) and 105
(b) andlSs (c) .................................................................................. 115
Figure 7 Combustion reaction and pyrolysis reaction constant levels for St = 2 at 53,
10s (b) and 15s (c) .............................................................................. 117

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Figure 9 Net heat ﬂux from the gas phase along the interface at four different time

.................................................................................................... l 19
Figure 10 Surface temperatures at four different times .................................... 1 19
Figure 11 Pressure and velocity ﬁelds at 155 ............................................... 120
Figure 12 Temperature and velocity ﬁelds at 15s .......................................... 121
Figure 13 Reaction rate, fuel concentration and oxidizer concentration contours. . 122
Figure 14 Mass ﬂux and heat ﬂux at 155 ................................................... 124
Figure 15 Flame front and phase front vs. x distance at St = 2 ........................ 124
Figure 16 Temperature, fuel and oxidizer proﬁles for four different cases ............ 126
Figure 17 Distribution of heat ﬂux ahead of the ﬂame along the polymer surface for
four cases ....................................................................................... 128
CHAPTER 5
Figure 1 Flame spread rate vs. anisotropic conductivities in a given direction ....... 145
Figure 2 The solid-liquid phase contours for four cases .................................. 146
Figure 3 The surface temperature for four cases ........................................... 147
Figure 4 The surface heat ﬂow rate for four cases ......................................... 158
Figure 5 The surface mass ﬂow rate for four cases ....................................... 159
Figure 6 Distribution of heat ﬂux ahead of the ﬂame along the polymer surface for
three cases ....................................................................................... 150
Figure 7 Temperature along a cutting line at the center of the heat surface ............ 153
Figure 8 Ignition time vs. anisotropic conductivity ....................................... 154
CHAPTER 6
Figure 1 Ignition delay time vs. (a) 1/ St (b) k, (c) Cpl for the Navier Stokes model
and the Oseen model .......................................................................... 178
Figure 2 Three heat transfer mechanisms at the interface vs. (a) 1/ St (b) k, (c) Cpl
for the Navier Stokes model .................................................................. 179
Figure 3 Three heat transfer mechanisms at the interface vs. (a) 1/ St (b) k, (c) GP,
for the Oseen model ........................................................................... 180
Figure 4 Five heat transfer mechanisms vs. (a) 1/ St (b) k, (c) Cpl for the Navier
Stokes model ................................................................................... 181
Figure 5 Five heat transfer mechanisms in the control volume vs. (a) 1/ St (b) k, (c)
5 pl for the Oseen model .................................................................... 183
Figure 6 Magnitudes of enthalpy increase of the condensed phase at ignition vs. (a)
1/ St (b) k, (c) Cpl for the Navier Stokes model ....................................... 184
Figure 7 Magnitudes of enthalpy increase of the condensed phase at ignition vs. (a)
1/ St (b) k, (c) Cpl for the Oseen model ................................................ 186
Figure 8 Average heat ﬂow rate during ignition process vs. (a) 1/ St , (b) C'— pl for the
Navier Stokes model .......................................................................... 187
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Cpl = 0.5 at ignition for the Navier Stokes model ...................................... 188
Figure 10 The temperature ﬁeld vs. (a) St = 2 (b) St = 100 (C)/—(—[ = 2.5 (d)
C3,, = 0.5 at ignition for the Oseen model ................................................ 190
Figure 11 The interface heat ﬂux to the condensed phase at ignition for four different
cases for (a) the Navier Stokes model (b) the Oseen model ............................. 192
Figure 12 The interface temperature at ignition for four different cases for (a) the
Navier Stokes model (b) the Oseen model ................................................. 193
Figure 13 The interface mass ﬂux at ignition for four different cases for (a) the Navier
Stokes model (b) the Oseen model ......................................................... 194
Figure 14 The ignition delay time vs. (1/ q'ig )" .......................................... 195
Figure 15 The interface temperature at ignition for three cases of external radiation
heat ﬂux. (a) the Navier Stokes model (b) the Oseen model ............................. 196
Figure 16 The maximal surface temperature vs. time for three external radiation heat
ﬂux in two cases (a) the Navier Stokes model (b) the Oseen model .................... 197
Figure 17 Ignition delay time: a simple theory and direct results vs. Cpl ............. 198
Figure 18 Ignition delay time: a simple theory and direct results vs. St ............... 200
CHAPTER 7
Figure 1 Relationship between macro-scale model and micro-scale model ............ 227
Figure 2 A control volume selected to obtain mmmface ................................ 227

Figure 3 Flow chart of the iterative procedure for the solution of two-scale models.228
Figure 4 (a) Temperature constant levels (b) Porosity constant levels (c) Velocity
constant contour (d) Mass ﬂow rate out of the interface at 53 ........................... 229
Figure 5 (a) Temperature constant levels (b) Porosity constant levels (c) Velocity
constant contour (d) Mass ﬂow rate out of the interface at 10s .......................... 231
Figure 6 (a) Temperature constant levels (b) Porosity constant levels (c) Velocity
constant contour (d) Mass ﬂow rate out of the interface at 19s .......................... 233
Figure 7 The surface mass ﬂux for 4 cases with different nucleation rate at 103. . . ...237
Figure 8 The temperature constant contours for 4 cases with different nucleation rate

at 105 ............................................................................................. 235
Figure 9 The porosity constant contours in the condensed phase at 105 for 4 cases with
different nucleation rate ....................................................................... 236
Figure 10 The velocity constant contours in the condensed phase at 105 for 4
cases ............................................................................................. 238
Figure 11 The surface mass flow rate vs. the bubble initial radius ...................... 240
Figure 12 The temperature constant contours for 4 cases with different initial bubble
radius at 10s. The reference bubble initial radius is 5 x10”6 m .......................... 240
Figure 13 The porosity constant contours in the condensed phase at 103 for three
cases ............................................................................................. 241
Figure 14 The velocity constant contours in the condensed phase at 103 for three
cases ............................................................................................. 242
Figure 15 The surface mass ﬂow rate vs. St .............................................. 244
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Figure 17 The porosity constant contours in the condensed phase at 103 for three

cases ............................................................................................. 245
Figure 18 The velocity constant contours in the condensed phase at 108 for three
cases ............................................................................................. 246
Figure 19 The surface mass ﬂow rate vs. evaporation heat .............................. 248
Figure 20 The phase interface for three cases of evaporation heat ...................... 248
APPENDD( I
Figure 1 Schematic description of one-dimensional melting of a rod .................. 263
Figure 2 The schematic relationship between enthalpy and temperature ............... 263
Figure 3 Phase front movement .............................................................. 264
Figure 4 Temperature distributions .......................................................... 266
Figure 5 Phase front movement for St = 0.1(leﬁ) and St = 10 (right) ................ 268
Figure 6 Temperature distributions for St = 0.1 at SS and St = 10 at 25 ............ 269
APPENDIX 11
Figure 1 Control volume formulation of partial differential equations .................. 287
Figure 2 Control volume formulation of boundary conditions ........................... 287
Figure 3 The mesh system with clustered grid near the interface ........................ 288
Figure 4 Flow-chart of the solution procedure of the numerical model... ..............289
Appendix 111
Figure 1 Grids and control volumes .......................................................... 306
Figure 2 Staggered grid system ............................................................... 306
Appendix V
Figure 1 Material volume containing a solid-liquid interface ............................ 326
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NOMENCLATURE

pre-exponential factor of pyrolysis reaction in condensed material, 5"

pre-exponential factor of combustion reaction in gas phase, m3/(kg-s)
gas phase concentration in liquid, --

specific heat, J/(kgK)

non-dimensional liquid thermal capacity, C PI / C p5.

diffusion coefficient, mz/s

activation energy of pyrolysis reaction in solid phase, J/mol

activation energy of combustion reaction in gas phase, J/mol

gravitational acceleration, m/s2
reference length for non-dimensionalization, m
enthalpy of condensed material, J/kg

bubble nucleation rate, (number of bubble/m3 .5)
thermal conductivity, W/(m-K)

ratio of liquid and gas viscosity K = ,u g / ,u,, --
Boltzman constant, 567* 108 W/mz-K4

constant for Henry’s law

volume averaged thermal conductivity, W/(m-K)
non-dimension liquid conductivity, kl / ks
permeability factor of pyrolysis products, ~-

streamwise length of computational domain, m

transverse length of condensed material in computational domain (the

thickness of the polymer), m

transverse length of gas phase in computational domain, m

Lewis number, ag / D

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unit vector normal to the boundary of the condensed material,--

latent heat of melting, J/kg
fuel mass ﬂow rate through gas-condensed interface, kg/(mZ-s)
mass of liquid in the condensed material, kg

species molecular weight, g/mol
the number of molecules per unit volume, l/m3

number density of the bubble, --
unit vector normal to moving solid-liquid interface in condensed phase, --
pressure of the gas phase, Pa

pressure in polymer liquid, N/m2

heat of combustion reaction in gas phase, J/kg

external radiant heat ﬂux for ignition, W/(mz)

average net heat ﬂux into the condensed material before ignition, J /s

heat of pyrolysis reaction in condensed material, J/kg

Longitudinal heat conduction in the gas phase, W

heat of combustion reaction in the gas phase, W

longitudinal heat conduction in the condensed phase, W

heat convection in the gas phase, W

latent enthalpy increase (associated with phase change) of condensed phase, J

pyrolysis heat in the condensed phase, W

sensible enthalpy increase (associated with temperature increase) of the

condensed material at ignition, J

total heat in the condensed control volume, W

transverse gas heat conduction at interface, W

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temperature, K

ﬂame temperature, K

glass-transition temperature of polymer, K
temperature of the gas-condensed interface, K

melting temperature of condensed phase, K

velocity in the longitudinal direction, m/s

diffusion velocity of species 1' in the liquid phase, m/s, i = m, p
ﬂame spread rate, m/s

velocity of uniform longitudinal inlet ﬂow, m/s
velocity in transverse direction, m/s

bubble velocity, m/s

volume averaged gas velocity in polymer liquid, m/s
velocity of species 1' in the liquid phase, m/s, i = m, p
mass averaged velocity in the liquid phase, m/s

velocity of moving phase interface, m/s (in Appendix IV)

energy source term from decomposition, J/(kg.s)

velocity of moving solid-liquid interface, m/s

mass consumption rate of fuel in combustion reaction of gas phase, kg/(m3 -s)

mass production rate of monomer in pyrolysis reaction of polymer, kg/(m3 -s)

streamwise distance downstream from origin, m

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volume fraction (porosity), --

surface emissivity, --

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CHAPTER 1

INTRODUCTION

Flame spread over polymeric materials is of importance because for ﬁre-safety
considerations in a living environment and for fundamental questions arising in
combustion science. Flame spread basically results from the complex interaction of
transport and chemical processes that occur in both gas and condensed phases. The
dependence of ﬂame behavior on polymer properties is of particular interest in this
research since the transport phenomena inside polymers receive comparatively less
attentions than the gas phenomena. The layout of this chapter is composed of: (1)
Background information on the polymers and their related ﬁre hazard; (2) Description of
fundamental phenomena during ﬂame spread; (3) A review of previous modeling efforts;
(4) The motivation and scope of the research. The introduction is oriented toward
describing physics rather than listing research facts because some excellent reviews [1-7]
already cover the ﬂame spread problem from a wide range of perspectives.

1. Polymers and Fire Hazards
1.1. Brief Description of Polymeric Materials

Polymeric materials are composed mainly of macromolecules and other types of
compounds such as mineral ﬁllers or dyes. The essential characteristic of the
macromolecules is that they consist of a relatively large number of repeating structural
units. When macromolecular weights are greater than 1500 g/mol, the material is called a
polymer. Polymers are normally solids and are bounded by a surface. Some thermoplastic

pOIymers that are of interest to ﬁre research are listed below [6].

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0 Polyethylene (PE) is the most important synthetic polymer and has low crystallinity
and low softening and melting points. PE is used primarily in the packaging and
construction industries.

0 PMMA (methylrnethacrylate) is used as “organic glass”, for example in airplane
windows. Other applications include contact lenses in medical systems and costume
jewelry.

0 Polyprolefm is highly crystalline and has a low density. It is used widely in ﬁber, ﬁlm
and moldings, which require high tensile strength.

From the viewpoint of condensed phase decomposition and combustion, polymeric
materials can be classiﬁed as charring and non-charring materials. For charring materials,
the carbonaceous char residue is produced during thermal degradation. Wood, or more
generally cellulosic fuels, is the most representative of the charring materials. PMMA,
which produces almost no char during combustion, is representative of the non-charting
materials. It is generally understood and accepted that charring and non-charring
polymers show different ﬁre performance. Most of the experiments and numerical
simulations of ﬂame spread over solid fuels have been carried out with paper, a chairing
material, and PMMA, a non-charting material. PMMA has been chosen as perhaps the
main non-charring material for ﬁre research because of its relatively simple behavior
during heating and decomposition and more literature results.

1.2. Production and Use of Polymeric Materials
The production and use of synthetic polymers is continuously growing. The

production of plastics has increased much more than the production in general over the

past decades. Total polymer production in the USA in 1992 was roughly 26 million tons.

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A healthier global economy and end-use market will help boost world demand for
thermal plastic resins to 122 million metric tons by the year 2000. [8] In addition,
polymeric materials have become increasingly versatile and widely accepted.
1.3. Fire Hazards of Polymers

It is of particular concern that many of these polymeric materials involve the
possibility of exposure to ﬁre. Upon exposure to a sufﬁcient heat source and sufﬁcient
atmospheric oxygen level, it is very possible that these materials may catch ﬁre and burn.
For certain applications, the ability to undergo combustion is a desirable requirement.
However, for a large majority of purposes, polymer combustion is a serious disadvantage.
Indeed, in many industries such as the automotive and construction industries, the use of
synthetic polymeric materials has raised concerns about the ﬂammability issues. Certain
regulatory standards have been published in order to enforce public safety of the material
in terms of burning. The ﬁnancial cost of ﬁre induced by polymers is enormous to human
society. The combined cost of loss adds up to several times higher than the physical
damage [6]. Since ﬁre is an important practical problem, it is important to understand and
quantify the behavior of polymeric materials in a ﬁre environment to meet certain ﬁre
safety requirements. Fire hazards are deﬁned in terms of two categories, thermal and non-
thermal. Thermal hazards are caused by heat released from the ﬁre. Non-thermal hazards
are caused by the production of toxic gases and smoke [7]. Qualitatively, ﬂame
development can be described as: (1) ignition; (2) transition; (3) ﬂame spread; (4) peak
burning or steady state combustion; (5) generation of heat and undesirable toxic,
corrosive, and odorous chemical compounds including those which obscure light

transmission and cause electric damage. Flame development can be deﬁned in terms of:

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(1) ﬂammability measurements for ignition, generation rate of heat and generation rate of
chemical compounds, generally under the steady state conditions, (2) ﬂame spread
measurements, which include pyrolysis, ﬂame heights, and ﬂame spread rate
measurements, (3) ﬂame extinction measurements where the effectiveness of an agent to
extinguish a ﬁre is determined [9].
2. Fundamental Phenomena of Polymer Combustion1

Flame development over solid polymers is complicated because it encompasses the
transport processes in the gas phase (momentum, mass, heat, combustion reaction),
transport phenomena in the condensed phase (melting, pyrolysis, bubble generation and
transport) as well as coupling between gas and condensed phases. The overall phenomena
will be described below, and then detailed aspects of the fundamental phenomena will be
explained. A schematic description of the physical phenomena is given in Figures 1 and
2.

2.1. Overall Description

Fundamentally, the cyclic scheme as shown in Figure 1 represents two distinct
processes: one is the ﬁrel generation in the condensed phase; the other is the combustion
in gas phase. Both processes are regulated by volatile and thermal feedback. The gas-
phase reaction is controlled by the relative amount of three types of pyrolysis products,

such as inert gases, combustible gases and carbonaceous char. The volatile shown in

Figure 2 is composed of the former two gas products. Generally an amount of energy Q

is consumed inside the solid phase, and Q2 is the heat generation during combustion in

' Combustion of polymers can be divided into three types: ﬂaming combustion, smoldering and glowing
combustion. Here mainly ﬂaming combustion is considered.

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the gas phase. The quantity (Q2 -Q1) establishes the exothermic character of the

reaction. Inert gases dilute the combustible gases, providing an inert gas layer between
oxygen and the combustible zone. Furthermore, for cham’ng materials exposed to an
oxidizing environment, the char underlying the polymer acts as a thermally stable
insulator protecting the underlying polymer from heat [10]. Combustion may proceed by
two alternative pathways. These are ﬂaming combustion and smoldering combustion.
Flaming combustion is achieved when the heat released by gas phase combustion of
volatile provides the heat ﬂux needed for both solid degradation and ﬂame spread. When
the temperature or the heat ﬂux is below a certain level, oxidization of char may produce
smoldering combustion [4]. It is obvious that for non-charring polymers, the smoldering
process is not possible because of the absence of a matrix or porous structure in the solid
phase. In general, to get ﬂaming ignition, there are three conditions that must be met [1 l]:
(1) Fuel and oxidizer must be available at a proper level of concentration to yield a
mixture with suitable ﬂammability limits; (2) the gas temperature must attain values
sufﬁciently high to initiate and accelerate the combustion reaction; (3) the extent of the
heated zone must be sufﬁciently large to overcome heat losses. Before ignition, the
temperature rise can be caused by external heating sources such as a pilot ﬂame, sparks
and hot wires. In most cases, the external source is thermal radiation, therefore the heat
absorption of radiation in both gas phase and solid surface are important mechanisms.
The next stage is ﬂame spread. To allow the ﬂame to propagate, the energy feedback
from the burning region (gas phase plus solid phases) to the unburned solid ahead of the
ﬂame tip determines the ﬂame spread rate. It is often difﬁcult to determine which one of

the process mechanisms is the controlling factor. An understanding of the dominant mode

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of heat transfer would facilitate the development of a simpliﬁed and accurate description
of ﬂame spread. Many analyses have been devoted to analyzing such energy feedback. In
[5] three principal mechanisms besides external sources were proposed: (1) radiation
from the ﬂame; (2) conduction or convection through the gas from the ﬂame; (3)
conduction through the solid. Experimental approaches that were followed to deﬁne the
relative importance of the different modes of heat transfer had been used to measure the
temperature distribution in the solid and gas phases as well the gas velocity and ﬂame
spread rate. The relative importance of each mode of heat transfer was deduced by
comparing its magnitude with the total enthalpy ﬂow needed to pyrolyze the solid
combustible. In [5, 12] it was inferred that radiation from the ﬂame becomes of
increasing importance as the scale of the ﬁre increases. It is observed that thick materials
have different ﬁre performance from thin materials. For thick materials [5], it is
concluded that for small-scale ﬁres, heat conduction through the thick solid is dominant
and radiation from the ﬂame contributes signiﬁcantly to the heat transfer process.
However, for thin materials [4], it is stated that gas phase heat conduction is the major
heat transfer mechanism for ﬂames spreading over very thin fuel beds. There are other
mechanisms that account for overall ﬂame spread over solid polymers. The possible
mechanisms that contribute to ﬂame development are listed in Table 1, which provide a
total picture of the complexity of ﬂame spread.
2.2. Solid Phase Phenomena

For melting solid polymers such as PMMA, the decomposition is conﬁned to a thin
layer of the fuel sample near the gas/solid interface. In the melt layer, bubbles are formed

and gases are transported to the surface. Thermal degradation behaves differently in inert

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and oxidizing environments. In the latter case, the oxygen may diffuse through the melt
layer, favored by the large holes produced by bursting bubbles.
2.2.1. Decomposition
When a polymer is heated it eventually reaches a temperature at which the weakest
bonds start to rupture. Three definitions, degradationz, decomposition3, and pyrolysis4 are
used in related literature. It is generally accepted that pyrolysis takes place on the surface,
which means thermal degradation without oxidation. However, the importance of
oxidation participation on the surface of the burning polymers and in-depth degradation
has been emphasized in [13]. For a better understanding of the mechanism involved in
the thermal degradation of polymers, it is necessary to know three fundamental things
[14],
l) The change of molecular weight of the polymer as a function of temperature and
extent of degradation.
2) The qualitative and quantitative composition of the volatile and non-volatile
products of degradation.
3) The rate and activation energy of the degradation process.
With regard to the change of molecular weight with temperature and extent of
degradation, very little is found in the literature on this subject, except in the case of a
few polymers. The information available indicates that the molecular weight drops

initially very rapidly during the ﬁrst few percent loss of weight. Afterwards the drop is

 

2 Degradation: only relatively few bonds break and result in only minor changes in structure and properties,

e.g. discoloration.

3 Decomposition: at high temperature the polymer structure undergoes more extensive breakdown and

results in disassociation of a signiﬁcant proportion of the total number of constituent chemical bonds.
Pyrolysis: irreversible chemical decomposition of materials due to an increased temperature without

oxidization.

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slow. As for the nature of the products of degradation, systematic qualitative and
quantitative analyses have been done. The data that have accumulated so far indicate that
some polymers, for example, polytetraﬂuoroethylene and poly-a-methylstyrene, yield on
pyrolysis in a vacuum at temperatures up to about 500-600 °C almost 100% monomer,
while in the case of polyethylene pyrolysis under similar conditions yields a spectrum of
hydrocarbon fragments varying in molecular weight from 16 (CH4) to 1000. Intermediate
between these two extremes are polymers that yield on pyrolysis a mixture of monomers
and chain fragments of varying sizes. There are also polymers like poly (vinyl chloride),
poly(vinyl ﬂuoride), and polymethylacrylate, which yield on pyrolysis fragments not
related in structure to the polymer chains from which they derive, along with fragments
that are parts of the chains. In pyrolysis at temperatures above 500 °C, the nature of the
volatile products for any given polymer depends to a large extent on the temperature.
Most of the experimental work on pyrolysis reported in the literature relates to the
temperate to about 400-500 °C. However, some experiments have been carried out at far
higher temperature than these. The results indicate that higher temperatures produce
greater fragmentation of the degradation products. The reactions involved in the thermal

degradation of a given polymer can be calculated by means of the Arrhenius equation:
KZAe—E/RuT (1)

where E is the activation energy, K is the rate constant, A is the pre-exponential constant,

T is the temperature, and Ru is the universal gas constant. The reaction rate is deﬁned as

the time rate of the weight loss with respect to the percent of the original samples. In a
zero-order reaction, which happens very seldom, the reaction rates are constant for any

given temperature and can be used as values for the rate constants in the Arrhenius

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equation. Frequently the degradation is composed of several reactions and the order is not

clearly-cut, in a ﬁrst order reaction, the rate is proportional to the reacting substance:

— d—C 2 KC, (2)
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where C is the concentration of the reacting species and T is the time. In some polymers,
thermal degradation proceeds by two or more reactions, which run either concurrently or
consecutively. Then the activation energy in a single step representation is a composite
value and is based on the overall rate of degradation.
It is worth noting that, not only pyrolysis reactions but also oxidative degradations in the
condensed phase may signiﬁcantly affect the gasiﬁcation rate of a polymer if enough gas
phase oxygen is transported to the solid phase during combustion of the polymer.
Kashiwagi concluded [13] that the contribution of oxidative degradation could be
important for ignition processes at low incident ﬂux. Brauman [15] concluded that the
surrounding oxygen does not affect the polymer degradation process in steady state
burning or steady state radioactive gasiﬁcation of PMMA and PE.
2.2.2. Phase Change (Melting)

The melting characteristic of thermoplastic polymers depends on the types:
amorphous or crystalline. Most polymers are neither entirely amorphous nor entirely
crystalline. The degree of crystallinity and the strength of binding forces have a profound

effect on the softening range or melting point of a polymer. The melting point of

amorphous plastics is not as clearly deﬁned as that of monomer solids where all the
molecules are the same size. The melting point T m of amorphous therrnoplastics is more

properly termed a melting range, since a single specimen consists of more than one

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molecular weight and more than one crystal size. Examples of this type are PMMA and
PVC. These gradually softening plastics become soﬁer and softer as the temperature is
raised and the actual melting point is obscure, being only a change in slope in a plot
whose one axis is temperature [16]. In contrast to amorphous are crystalline polymers,
examples of which are nylon and polypropylene. These polymers tend to maintain its
structure. When the melting point is reached, the crystalline portions quickly melt and the
whole plastic becomes ﬂuid over a narrow range of temperature. Above the melting
point of these polymers, the difference between the amorphous and crystalline polymers
disappear, although the branching and regularity of the chain structures still inﬂuence

their ﬂow properties. Decreasing molecular weight, or adding a solvent to a polymer, or
decreasing the crystal size lowers T m [17]. The glass-transition temperature T G is called
a second order transition, since the change in volume is not discontinuous as it is with

T m. Below T G the polymer segments do not have sufﬁcient energy to move past one

another. For partially crystalline materials, T m is always greater than T G and that the

difference is a maximum for homo-polymers. An examination of these parameters for
many homo-polymers leads to the generalization that,

1.4<(T,,./ Tg)<2.0
Almost all thermoplastics soften above their glass transition temperature; some will
exhibit ﬂow motions in the polymer melt. For some thermoplastic polymers with high

initial molecular weights, ﬂame spread is relatively steady and clean because they form

 

5 Since melt viscosity of molten polymers depends strongly on their molecular weight [13], the initial
molecular weight was taken as an important parameter for evaluation of ﬂame spreading.

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negligible molten polymer near the ﬂame front. Polymers with low initial molecular
weight form both molten polymer and opposed slow ﬂuid motion along the inclined
vaporizing surface against the traveling ﬂame. A schematic illustration of horizontal
ﬂame spread phenomena for two PS (polystyrene) samples with high and low molecular
weights is shown in Figure 4 in [18]. Since the ﬂame spread slowly while the sample
surface regressed, a steep wall of molten polymer appeared in front of the ﬂame. A slow
downward movement of the molten polymer toward the bottom of the wall was observed.
This opposing slow ﬂuid movement against the ﬂame spreading direction caused the
slowdown of ﬂame spread and consequently the formation of the steep wall. Then the
ﬂame started to climb the steep wall. The ﬂame continued to climb to the top of the wall
until the front portion of the sample burned out. It was observed that ﬂame spread rate of
higher molecular weight PS sample was about 25% larger than that for the low molecular
weight PS sample [18]. Downward ﬂame spread over high molecular weight PMMA
sample did not show any dripping, and the ﬂame spread steadily. However, ﬂame spread
over high molecular weight PS yielded a much-enhanced rate compared with the rate for
horizontal ﬂame spread. This resulted from streaking of small molten polymer balls. The
ﬂame over the low molecular weight PS and PMMA samples self-extinguished during
downward ﬂame spread because of heat loss from the downward streaking of small
burning polymer molten balls to the cold sample surface. The results indicate that, in
certain experimental conﬁgurations, the melting of thermoplastics has a large inﬂuence
on their ﬂammability properties and subsequent spread. However, such effects have been
largely ignored by previous researchers who have employed non-melting samples. In

addition, some new inorganic polymers, whose backbone elements are not carbon, exhibit

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totally different ﬁre behavior. For example, the burning of PDMS samples will result in
accumulation of silica at or near the surface [13].
2.2.3. Bubble Formation

A proper understanding of the rate of polymer gasiﬁcation in a ﬁre environment is
essential to the mathematical prediction of ﬁre growth on such materials. Since the
boiling temperatures of some of the degradation products are much lower than the
polymer degradation temperatures, these products are superheated as they form. They
nucleate and form bubbles. Then these bubbles grow with the supply of more small
degradation products to the bubbles by diffusion from the surrounding molten polymer
[19]. Visual observation of PMMA gasiﬁcation was reported in [20] at radiant ﬂuxes of
1.7 and 4.0 W, in which the effects of gas phase oxygen on the mass ﬂux were studied. In
a nitrogen environment, rough surfaced, snowball-like bubbles develop and grow with
time as shown in Figure 2 (a) in [20]. By the end of exposure, these bubbles’ size can be
as large as 1 mm diameter, formed up to 2-3 mm below the sample surface. When
subsurface bubbles form, they grow toward the direction offering least resistance, i.e., the
front surface of the sample. Because the viscosity of the molten polymer layer near the
surface appears is still high, only bubbles within 1mm or so of the surface are able to
burst directly through the front surface. Bubbles ﬁnther below the surface burst through
small neck-like holes into near surface bubbles, then vent through to the gas phase. The
burst process is violent and can cause a vapor jet that extends a few centimeters into the
gas phase; it can also throw molten polymer into the gas phase. When oxygen is present
in the ambient gas, the viscosity of the near surface layer of degrading PMMA appears to

be substantially lower. The bubbles start earlier, the bubble frequency is higher, and the

12

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bursting process is less violent. The burst bubbles leave larger holes (up to ~1 mm
diameter) in the front surface of the sample. These holes admit oxygen to the sample
interior volume. At higher ﬂuxes, the differences in the near surface behavior due to inert
and oxidizing atmosphere are less pronounced but still present. Figure 2(c) and 2(d) in
[20] is for the case of higher ﬂux 4 W/cmz. In both cases, the snowball-like bubbles are
smaller because of the thinner thermal layer and shorter exposure time. The higher ﬂux
raises the surface temperature and apparently decreases the degrading polymer viscosity
even in the case of pure nitrogen. The subsurface degradation is important for gasiﬁcation
too, but it is not clear what is the main transport mechanism to the surface for small
degradation products. [13]
2.3. Gas Phase Phenomena

Once the solid combustible pyrolyzes, the fuel vapors convect and diffuse away from
the solid surface and produce the spreading ﬂame by reacting with ambient gaseous
oxidizer. The ﬂame leading edge stays very close to the ﬁre] surface, travels in the
direction of the propagation and acts as an “anchor” to the trailing diffusion ﬂame. The
ﬂow ﬁeld, pressure ﬁeld, and combustion reaction inﬂuence the ﬂame behavior.
Compared to the condensed phase phenomena, the gas phase phenomena have been
developed in more detail.
2.3.1. Combustion Reaction

Exothermic reaction occurs when fuel vapor and oxidizer coexist in a region of
sufﬁciently high temperature. (There are several factors that make the combustion reaction
complicated. One complexity comes from the very high number of chemical species

evolved from degrading solid both for cellulosic materials and thermoplastic polymers.

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For example, in ﬂame above the PMMA surface, the species are of at least ﬁfteen types.
Another complexity results from the long sequences of steps in which breakdown of fuel
molecules and formation of combustion product proceeds. A fully detailed model of a
ﬂame has to account for all such elementary reactions. The number of reactions needed is
very large even for small hydrocarbons. For example, combustion of methane (CH4)
requires nearly 300 elementary reactions. For more complicated fuels (such as ethane,
propane, octane), the number of elementary reactions can easily exceed 300 or even 500.
In an approximate sense, the number of elementary reactions increases geometrically
with the size of the fuel molecule. Complications occur when lower-order hydrocarbon
molecules actually combine to form higher order hydrocarbon molecules, e.g., in the
methane ﬂame two CH3’s combine to form C2I16: then the CH4 mechanism includes also
the C2H6 mechanism. An even greater complication arises when the hydrocarbon
fragments form cyclical compounds that combine to form “soot”.
2.3.2. Flow Field

In the gas phase, an external laminar ﬂow is parallel to the interface. Flame spread
over solid fuels can be classiﬁed into two main categories according to the ﬂow
conditions: (1) ﬂow assisted ﬂame spread, occurs when ﬂame spread is in the same
direction as the oxidizing ﬂow; (2) Opposed ﬂow ﬂame spread, occurs when the ﬂame
spreads against the oxidizing gas ﬂow. In the ﬂow-assisted ﬂame spread, the concurrent
ﬂow pushes the ﬂame ahead of the vaporizing fuel surface. The heat transfer from the
mixture of reacting gases and the combustion products favors the pr0pagation of the
ﬂame, because the diffusion ﬂame is driven ahead of the pyrolysis front. In addition, the

fuel vapors generated upstream of the pyrolysis front that are not consumed by the

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upstream diffusion ﬂame are driven ahead of the pyrolysis front, thus extending the
diffusion ﬂame downstream from the pyrolysis region. For this reason, the ﬂame spread
appears to be controlled primarily by the rate of heat transfer from the downstream ﬂame
to the unburned fuel. Thus as a result, the ﬂame spread process is generally more rapid
and thus morehazardous than the spread in opposed ﬂow conﬁguration [5]. The rate of
the ﬂame spread rate will depend on how fast the surface temperature of the solids is
raised to its pyrolysis temperature. The ﬂow remains laminar only in the initial stage.
When the size of the ﬂame increases, the ﬂow becomes turbulent and ﬂame radiation
appears to be the dominant mode of heat transfer. In the case of opposed-ﬂow ﬂame
spread, the transfer of heat from the ﬂame to the upstream region is rendered more
difficult because the gas ﬂows against the propagating ﬂame. It is concluded that various
external effects including oxygen level and ﬂow rate among others inﬂuence ﬂame
spread. It is dominated by (1) heat transfer mechanisms at low opposed-ﬂow velocities
and high oxygen concentrations; (2) chemical kinetics at high ﬂow velocities or low
oxygen concentrations.
3. A Review: Modeling of Flame Spread over Non-Charring Polymers

The formulation of a rigorous mathematical model of the ﬂame spread process
requires the conservation equations for the gas phase coupled at the interface to the
condensed phase conservation equations through the appropriate boundary conditions.
This would further require the solution of a system of coupled, transient, two-
dimensional, elliptic, nonlinear partial differential equations that includes appropriate gas
phase combustion reactions and condensed phase pyrolysis reaction mechanisms. The

solution of the full problem is formidable, not only because of the limitations of

15

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computational power, but also because of the lack of available experimental data. For this
reason the models to date have treated the problem at different levels of complexity. The
simpler model is limited to the condensed phase energy analysis with a priori speciﬁed
conditions at the gas-condensed interface. More reﬁned models include both gas and
solid phase, but assume an inﬁnite rate gas phase reaction. The De Ris model [21] was
one of the ﬁrst theoretical models to successfully attack the ﬂame spread problem. It used
several important assumptions: (1) Inﬁnite reaction rate in the gas phase; (2) Oseen
approximation (a speciﬁed uniform gas phase ﬂow ﬁeld); (3) constant solid vaporizing
temperature. The ﬁrst assumption reduces the ﬂame to a sheet where fuel and oxidizer
are consumed and heat is generated. The second assumption avoids the complication of
the ﬂow ﬁeld. The third assumption avoids the complication of the complex transport
phenomena inside the condensed phase. de Ris derived two ﬂame spread formulas for a
ﬁrel-thin material and a fuel-thick material, each being an arithmetic correlation including
gas, fuel, and ﬂame properties. The predictions of two formulas are found to be good
under fast reaction conditions although concerns over the various approximations have
been raised [1]. The independence of the ﬂame-spread rate from the opposed ﬂow
velocity is contrary to existing experimental observations near extinction limit
(Fernandez-Pello [22]). The ﬂame sheet assumption eliminates consideration of
extinction limit. The constant vaporizing temperature was questioned by Sirignano [23]
by pointing out that this temperature is a function of the ﬂame properties and should not
be used a priori. In order to remedy this, he proposed a ﬂame model [24], which has a
similar form but has coupling at the solid-gas interface. Although the a priori

vaporization temperature was removed, this ﬂame model is only applicable to the surface

16

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reacting case. Femandez-Pello and Williams [25] were not satisﬁed with the Oseen
approximation. They devided the gas phase into two regions: (1) An upstream boundary
layer region and (2) A downstream diffusion region. In addition, they used ﬁnite rate
chemical kinetics. The introduction of the boundary layer, in some sense, “remedied” the
Oseen approximation, but it removed the upstream diffusion that was later proved to be
important near the ﬂame leading edge. The work of Frey and Tien [26] was the ﬁrst
numerical solution by obtaining a ﬂame structure and its dependence on Damkohler
number. They kept the Oseen approximations and only emphasize the species and energy
conservation aspects of their work. These assumptions remove the interaction of the
coupling of the combustion and aerodynamics such as the gas expansion effect.
Fernandez-Pello at al. [22] experimentally investigated the inﬂuence of oxidizer
concentration and ﬂow velocity on the ﬂame spread rate. They found that the controlling
mechanism far from the extinction limit is thermal (de Ris) and otherwise is chemical.
Wichman and Williams [27] proposed the global energy balance approach, which
formulated a surface ﬂame sheet in a ﬂame-fixed coordinates. The resulting equations
only include the solid and gas phase energy equations, which, however, is consistent with
the equations and boundary conditions for the de Ris model. Physically they assumed that
the streamwise heat conduction in both gas and condensed phase does not inﬂuence the
overall energy balance. This is correct as long as the reaction in the gas phase is infinitely
fast and the process is steady. Atreya [28] applied the global energy balance principle and
derived a formula for charring fuel. A parabolic-type char-wood interface was postulated,
which redistributes the heat from the gas phase. The ﬂame spread formula for charring

wood could be readily applied to the melting polymer since both participate the similar

17

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processes.

We consider now the pure numerical solutions. The full Navier-Stokes equations
were ﬁrst incorporated into steady ﬂame spread modeling by Mao et al. [29]. The
interaction of the aerodynamics and combustion, especially the thermal expansion and
gas injection from interface were explored in detail. The ﬂame structure was conﬁrmed to
be a premixed ﬂame upstream and diffusion ﬂame downstream that was ﬁrst proposed by
Ray and Glassman [30]. Chen [31] modeled the ﬂame spread process by using the
Navier-Stokes equations for thin fuel ﬂame spread problem, and investigated the ﬂame

blow-off phenomenon with respect to the dependence of the ﬂame spread rate on the
Damkohler number. Bhattacharjee [32] investigated the thin-fuel ﬂame spread problem

under micro-gravity conditions by using similar approaches. He found that radiation,
which is not important under normal gravity condition, becomes so signiﬁcant that it
decreases the ﬂame spread rate to a factor of 2-3 times smaller than in normal cases. Di
Blasi [33] formulated the ﬁrst transient ﬂame spread model, which makes possible the
numerical prediction of the transient ﬂame development. This model yields results for
ignition, transition, and steady ﬂame spread. The complexity of the ﬂame spread model
has increased with time. With the rapid expansion of computational power, the modeling
effort is shifting toward tasks that had previously been deemed too difﬁcult.

In summary, the early studies of ﬂame spread employed numerous assumptions to
simplify the problem. Simpliﬁed models based on these assumptions are available in
reviews [1, 23, 34, 35]. The most commonly used approximations are constant
temperature of the solid during thermal degradation, a ﬂame sheet (inﬁnite chemistry)

and a boundary layer. More comprehensive mathematical models include the partial

18

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differential equations for conservation of mass, momentum, energy, and chemical species
applied to describe speciﬁc aspects of combustion of synthetic polymers, such as ignition,
ﬂame spread, and extinction, see review articles [4, 36]. These more complete models are
not amenable to analytical solutions and it is necessary to use numerical techniques. At
present the gas phase models are more advanced and include many important physical
processes, reﬂecting a greater understanding of the gas phase phenomena compared with
those in the condensed phase [13]. In the following, the modeling of the solid phase is
discussed.
3.1. Solid Phase Transport Modeling

Most of the models for thermoplastic polymer ﬂame spread, available to date, are
based on a simple energy balance equation for the solid with localized degradation at the
surface, and have been coupled to the gas phase equations. The effects of bubbles inside
the molten layer on the transport of volatile, during degradation of thermoplastic
polymers, were hardly considered except for some work on steady sate gasiﬁcation of
PMMA under a speciﬁed external energy ﬂux [37]. However, two-dimensional bubble
transport in thick polymers is not taken into account. Thermally thick fuel models assume
that heated polymeric materials remain solid until ﬁnally gasifying at the surface
according to a zero-order Arrhenius pyrolysis reaction giving the corresponding
monomer. In such a way, the phase-change effect and in depth degradation were ignored,
which depends on fuel properties and environmental parameters. Heat capacity, thermal
conductivity and density are assumed to be constant. The models for thermally thin
polymers assume that variables across the solid thickness are uniform in their spatial

distribution. In char-forming polymers, char formation in numerical modeling was

19

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described in a very simple fashion and only for thin fuels [26]. In terms of PMMA
degradation, it occurs according to the following main stages: depolymerization
initialization, propagation of reaction chain and termination. Kinetic modeling of PMMA
falls into two categories: (1) One step global models which employ a one-step, global,
Arrhenius rate reaction to account for all chemical processes. Such an approximation is
widely used in computer models, which couple the gas and solid phase chemical and
transport processes. (2) Detailed degradation models using kinetic schemes that account
for chain initialization reactions, depropagation reactions and termination reactions, have
been proposed. Such models have never been coupled to the description of physical
processes. Some of research that belongs to this group is available in [2 6-40].
3.2. Gas Phase Modeling

For the gas phase model, most advanced models published to date include
momentum, energy and chemical species mass balance equations. Analyses are
principally for laminar ﬂow, and ﬁnite rate combustion kinetics are described through an
overall, second order reaction such as F + voO—> va. Viscous dissipation and
compressible work are neglected. Furthermore, the coupling between the momentum
equations and the state equation due to pressure terms, when momentum balance
equations are included in the mathematical formulation of the problem, is neglected.
Since pressure variations in space are very small and, in general the system is open, the
mean pressure is reduced to the speciﬁed ambient pressure. The decoupling of the
momentum equations from the state equations cuts off considerations of acoustic waves,
and the determination of the pressure ﬁeld becomes an elliptic problem. To date, ﬂame-

spread models have numerically solved and treated the velocity ﬁeld differently. The

20

 

{seen approximatit '

. .
Bret-re been wee.

3i: :lTC’CLl “ﬂag"

l \‘\9.‘ ~D.\
‘ ’Akisl‘ ‘I

_- utd'x"ns l‘ﬁd ‘
C'CE'IIII‘JSUOII or .

In} chemical and f

V
is?“

use ' than :X‘i‘zd
17212015 011 the

.rz‘ii‘alon ofheat e

1: scfaee or um,

mace to the ﬁsh:

', :.

re are I)“

\ .
"L“.

ft. “LEmQ‘

usua~ C\ en 1‘,

 

 

 

‘

...etore the 0b!

eetf
3"». .-
.4. y» -
\cale kt’mﬁtcx
2m .1 .
x. g 1 UC‘Ea" -
‘ iii)“ .3
s L
1... .
L‘-"‘: . L
LictJf-CU‘;_I
.t (2‘...
\i- "‘3
. er 2‘ the .‘V
i .
,xe- .
4

simplest models consider the solution of species and energy equations assuming that the
gas density and pressure are constant and the velocity ﬁeld is known (similar to the
Oseen approximation). The computational cost is very low, and this approach has
therefore been widely used. It is found that predicted values of the opposed ﬂow spread
rate depend strongly on the velocity proﬁle used in the computations. Recent work has
also shown a great sensitivity of spread rate to solid properties. Normally two types of
velocity proﬁles are chosen: (1) Oseen proﬁle; (2) Hagen-Poiseuille proﬁle [4].
4. Motivations and Scope

Combustion of solid polymeric materials arises from a complex interaction among
many chemical and physical processes. Comparatively gas phase models are much more
advanced than solid models, and normally include mass, momentum, species, and energy
equations. On the contrary, the solid models are always simpliﬁed by a simple
formulation of heat conduction as well as a one-step global degradation reaction, either at
the surface or in-depth. The phenomena of melting and bubble transport and their
inﬂuence to the ﬂame spread behavior have largely been ignored by previous researchers.
In fact, there are no published ﬂame spread models that consider the above-mentioned
phenomena, even in a sense of emphasizing one single aspect such as the melting.
Therefore the objective of the current research is to investigate the ﬂame spread problem
with some complex solid/liquid phase physics such as anisotropic characteristic, melting,
bubble nucleation, grth and movement. The layout of the chapters follows the logical
route of theoretical exploration, with increasing complexities of the physical phenomena.
In Chapter 2, the ﬂame spread is investigated for an anisotropic polymer with a priori

speciﬁed ﬂow ﬁeld (Oseen Approximation). The transient ﬂame spread rate is analyzed

21

ﬂ
. - ~'r-1"‘

has to a set of

ﬂame stead behav
is: ...ateml prop;
fess-Lets such as 1:
5m str'ctures an:
summon 15 re
$411-th15 are men?

Iteiaee on the I]:

wit-m.
Ahamt ﬂOW Fall 3“
Just

 

..I .‘A.
K M)?
5.314.- ‘
V‘U . ‘ ,
M tom.» ..
s «x
531:.

 

 

according to a set of variable traverse conductivity and longitudinal conductivity. Chapter
3, still retaining the Oseen approximation, investigates the inﬂuence of melting on the
ﬂame spread behavior. Flame spread rate as a function of non-dimensional parameters
from material properties is obtained and compared to theoretical formulas. Transient
behaviors such as ignition, ﬂame development stages, and so on are deﬁned. In addition,
ﬂame structures and heat transfer mechanisms are examined. In Chapter 4, the Oseen
approximation is removed from the formulation; instead the complete Navier-Stokes
equations are incorporated into the ﬂame spread model for melting polymers. Rich
physics of the gas phase effects such as the thermal expansion and gas injection at the
interface on the ﬂame spread behavior are explored in detail and compared to previous
theoretical and experimental ﬁndings. In Chapter 5, the inﬂuence of anisotropic solid
properties is revisited under the situation of the realistic ﬂow condition and melting solid.
In Chapter 6, the ignition is investigated by using energy balance analysis for both a
realistic ﬂow pattern and an assumed ﬂow pattern. An ignition theory is constructed to
predict the ignition delay time by using observations from the numerical results. In
Chapter 7, the comprehensive transport phenomena in the condensed phase such as
melting, bubble nucleation, growth, and movement are modeled using a two-scale model,
that is, a macro-scale transport model and a micro-scale transport model. The former
provides the temperature and liquid fraction and the latter accounts for single bubbles’
nucleation, growth and movement. The inﬂuences of bubbles on the material properties
are ﬁirther investigated by a representative case, where the evaporation effect, the bubble
induced ﬂuid ﬂow, and the temperature ﬁeld are investigated. In Chapter 8, conclusions

are provided and further modeling developments are suggested. In addition, appendixes

22

  
    

re also wen to SL..'

9

253:3: sub-mode.

1.113331. {-11 r,_

. . . ,
can. 7. 6 o o
I.......L‘l'i Q: “'16 ll“...

9 ' ‘ I
.1: of” 3333.11 0: v.»

$ Iaii.

 

are also given to supplement each chapter. They are: (1) Numerical veriﬁcation of the
melting sub-model (Chapter 3); (2) Numerical techniques for combustion and general
issues (Chapter 3); (3) A derivation of the ﬂame spread formula for melting polymers
(Chapter 3); (4) The numerical methods for ﬂow ﬁeld calculation (Chapter 4); (5) The
deﬁnition of the integral heat transfer mechanisms for ignition analysis (Chapter 6); (6)

The derivation of the volume averaged equations for bubble transport (Chapter 7).

23

F . I .
.. .
U14. .7643

Flare l \
S

 

 

A

pyrolysis

 

'l/ /
Conduction

   

 

Heat to gas

I Oxygen I

 

 

V

 

Combustion

 

Figure 1 Schematic diagram of combustion of non-charting polymers.

24

 

Reactions

\[rgxidation in Flame ]

E’ost Flame m

 

 

 

 

 

 

 

 

 

 

 

    
   

 

I
I
I
I 0
I 3
A
I a.
I 26’
I O
I
I
l
Decomposition '
,, Products = =:
I
[Bubble Formation] :
1‘ I
G 'ﬁ ti '
f asr ca on J I o
I ,.
.:I
I n.
Thermal | g
Decomposition I g
I 8
l U
|
l
I
I
l
l
I

 

I Polymer 1

 

Figure 2 Schematic of the ﬂaming combustion of a polymer.

25

Table l. The me

Gasph

 

5min

,....Ient OVV‘E’CT‘I e
l' h

 

 

and .
L16 312v: hex.

 

 

 

 

 

Table 1. The mechanisms that contribute to the ﬂame development over polymers.

 

 

 

 

 

 

 

 

 

 

Gas phase Solid phase Other

Ambient oxygen concentration Conductivity Initial temperature
Flow velocity Thickness External radiation

Gas phase reaction kinetics Thermal degradation kinetics Gravity

Radiation and absorption Melt viscosity Scale of ﬁre
Gas phase heat conduction Charring or non-charring Ambient pressure
Boundary layer effect Phase change Interface radiation
Opposed or concurrent ﬂow Bubble formation

 

26

 

\lGDELIXG OF lt .

I hmiueuon

The present st _

a

5‘7?

.g“, I)“

U
I

 

CHAPTER 2
MODELING OF IGNITION, TRANSITION AND STEADY FLAME SPREAD OVER

ANISOTROPIC SOLID POLYMERS

1. Introduction

The present study is a ﬁrst step toward the complex phenomena in the condensed
phase, and mainly concentrates on the anisotropic inﬂuence of the solid conductivity. To
investigate ignition, transition and steady ﬂame spread over anisotropic solid polymers, a
transient two-dimensional combustion model is constructed considering both the solid
and gas phase. With a given uniform velocity ﬂow ﬁeld, the temperature and species
concentrations in the gas phase and temperature in the solid phase are numerically solved.
These quantities provide information on the details of the ﬂame structure and gas-solid
interaction. The three stages of ﬂame development over the solid phase include pre-

heating, ignition and steady ﬂame spread. Moreover, the trends of steady ﬂame spread

rate and ignition time with respect to ksx / ksy are obtained from eight different

numerical cases. Information about the mechanism of ﬂame spread, ﬂame structure and
solid phase inﬂuence of the solid anisotropy is also presented. Brief numerical studies
concerning opposed ﬂow, ignition, transition and steady ﬂame spread over a horizontal
thermally thick anisotropic slab will be provided.
2. Mathematical Model

The mathematical model in the present study is formulated by considering Di Blasi’s
model [41] for two-dimensional transient heat transfer over a polymer material

undergoing chemical decomposition. The solid phase is anisotropic and the ﬂame

27

steals met the s.

sexton at Elmie p'

T- e
13’-

|
153 0110's am.) I.

- Theradzazzon t:

‘

11133263731 2" ~

u..(,

 

H 1h? 350w 3‘.

‘1‘:- _.
3“ - 1h: Lu5lchl

I:

.

‘v'i‘ 33556:

.-'- .—,.
«5 I)“. u’ 1
v‘;
‘ D‘Wuh ,
‘ 1"

l.
w:

spreads over the surface of the solid phase in an oxidizing gas ﬂow opposing the

direction of ﬂame propagation. The following basic assumptions are made:

1. The gas phase reaction is based on a one-step second-order Arrhenius rate.
2. The solid phase is non-reactive except at the interface.

3. The buoyancy in the ﬂow ﬁeld is ignored in the forced ﬂow ﬁeld.

4. The radiation from the surface and ﬂame is ignored.

5. The thermal and ﬂow properties are constant.

With the above assumptions the governing equations include the transient reactive gas

phase, the transient inert solid phase and the decomposing interface as follows:

Gasphase:
%+um%=ﬂ-wg +DV2Y Yi,i= 0,f,p
Cpg,0g[——g+uc>0 —]= qgwg +kg V‘ZTg
Solidphase:
8T 82T 82T 032T
psCstzksxTéxf'l'ksy—éiai'IZ/csxygxgsi
Interface:
I 81/ 8Y-
f l
p ——-m(Y —l), D———-=mY
g 8y f g 3y I
< I; =Tg z 0 fp
3T 3T
g .
_kg—é).= —(KSX)’ £+KW S)+MLS

 

The other boundary conditions are:

28

(1)

(2)

(3)

(4)

l
H
..—
a
‘._.
'a.
N
r

/
53
n
:3
(I
531.
...4
1

~\ :63 ‘ “ .
I IAL‘\ bet‘.K

63:513. .
.Ln. Gnd llnc

7‘
‘1}: COT“

..rputanOnd.

‘llt
I-r‘th‘thjns
‘ resp'eca
S».
.s 34.111
. t“ (0
: flked
134:3???
"N T10
\‘I n T
- h.
1:11-er
”4'13
are ‘.
\‘H_
‘4‘“.
:\" e» -.
:‘ 'ta...

 

 

Tg =1}, 31ng (551:0
Inﬂow Yf =Yp =O;outﬂow ng, i=0’f’ ‘0 ; upper wall < 9T; . (5)
Y0 =Yoo 79:: kjy—r'o
The initial conditions are:
t=0,Tg=TS=T0,Yf=Yp=OandY0=Y00, (6)

in which tit = ASpSEXP(—ES /RTS), wg =—AgngXP(—Eg /RTg)Y0Yf and
E,- = Min' /vaf , i=0,f, p. Note that in Equation (3), we have ksxy = ks”.

3. Numerical Treatment

A two-dimensional semi-implicit ﬁnite difference scheme was applied in a Cartesian
coordinate system. The time-splitting ADI method was used in both gas and solid phases.
The convective term was treated by the upstream method. Since the chemical term is the
main non-linear source in this model, it was treated implicitly. The interface interaction
was solved by switching back and forth between the solid phase and the gas phase, and
the heat ﬂux between both phases was evaluated as a judgment for a further marching
decision. Grid lines were clustered near the ﬂame region with a minimum of 0.025 mm.
The computational domain of 10mm X 12 mm in the horizontal (x) and transverse (y)

directions, respectively, is represented by a 30 x 50 mesh system. Modeling is performed
by giving a ﬁxed uniform ﬂow velocity um at the inlet boundary with 30% oxidizer

concentration. The preheating horizontal length is 1.8 mm. The conﬁguration and ﬂow
conditions are similar to those used in [41]. Some of the solid properties of PMMA and

gas properties are given in table 1.

29

I Results and D:

rafaﬂ-W‘ ‘3) '
.:. u»$.r..‘.. PIOchS.

5

:aeuzth I}, =11 ..-

:e:st.=.nt must I‘. ..

5:35: the ever"

5:...

-- J.“ 1 ' -C
~\.-.\.:n.\ DCIUIC

.-
A»
\

amid the he

‘
so
5 dud-ICC.- .'

.;,.- . 4

-...-,;;Z:f C

OIICCIl’Jez’

 

5?”?
«~ «6 temperatu'
..35

“2““4338 pm |
«HA I
319:“e -
3 “(High IO

f)
“is 3 at .
UN»
“1‘ l

 

4. Results and Discussions
The ﬂame development with time can be classiﬁed into three categories as (1) the

pre-heating process, (2) the ignition process and (3) the steady ﬂame spread process. One

case with k H =0. 065 and ks), =0. 015 is studied. At the beginning and prior to ignition a

constant radiant ﬂux of 50 kW/m2 was applied on a limited length of surface. After
ignition the external radiation is removed. Figure 1 at time 1.4 5 shows the calculated
isotherms before ignition, which indicate that the temperature is almost evenly distributed
around the heated region. During this period the gas phase is heated dominantly by gas
phase conduction and the temperature is not high enough to initiate a ﬁre. The fuel and
oxidizer concentrations are extremely low, which is the consequence of a low solid
surface temperature by pyrolysis. As the heating process proceeds, the solid surface
reaches the pyrolysis point and hence a tremendous increase in the vaporization takes
place. While in the gas phase, the reaction rate is dependent on both the gas mixture
concentration and the temperature. Therefore the increase in reactant concentration only
contributes partly to the reaction rate. This process continues until the gas phase reaction
is large enough to maintain a strong heat source and hence a ﬂame can be sustained near
the solid phase. At this time ignition begins, which can be seen from Figure l at time
1.63. The ﬂame leading edge is almost formed upstream in front of the preheating region.
As explained by [42], due to the greater supply of oxidizer to the ﬂame front, the
reaction priority is upstream of the approach ﬂow. Further afﬁrmation can be made by
Figure 3 at time 1.63 that almost no oxidizer exists in the downstream region. Figure 2 at
163 also shows that the larger fuel concentration corresponds well with the larger

temperature isotherms in solid phase near the location of the leading edge. After ignition,

3O

I ‘ '

hexane uIIl stabilize -.
:1: ms tl'ix between
232:: also that the dew.
:czzpietel} [~12] A 5:
emitson of Figures
titsztton period due t.
:42;

The period hetueen 2

time B} applpng t.

 

 

ice em mo regions

._ ‘3
2'43.‘
\

-
’3‘ ~ 0
lug“): O
s

I heat-up.

:tmcal reaction take

. game. In t
tench region. It 1\ I
Fig-3‘61

hi new of heat
:2 1h“

the ﬂame will stabilize until a steady state ﬂame spread exists. During this period the heat
and mass ﬂux between both solid phase and gas phase attains a quasi-steady state. It is
noted also that the downstream oxidizer under the combustion plume will be consumed
completely [42]. A stabilized ﬂame is not attained until 2.03, which is shown by
comparison of Figures 1, 2 and 3. The transient ﬂame spread rate is larger during this
transition period due to the preheating of the sample by the initial external incident ﬂux
[42].

The period between 2.08 and 4.85 is characterized as the steady state ﬂame spread
regime. By applying ﬂame structure theory by Sirignano [23], it can be observed that
there exist two regions for the ﬂame structure horizontally. The ﬂame front region is the
region of heat-up, gasiﬁcation and mixing ahead of the ﬂame leading edge. The region of
chemical reaction takes up the highest temperature region of the gas phase. Transversely
there are three regions inside the ﬂame, that is, in the middle is the fully established
diffusion ﬂame, in the upper region is the ﬁrel lean region and in the lower region is the
fuel rich region. It is easy to qualitatively identify the regions deﬁned by the above from
Figure 1.

In view of heat transfer, there are many controlling mechanisms for ﬂame spread
over thick solid polymer [1]. In this case the scale of the external ﬂow rate is found to be
ten (10) times as large as the ﬂame spread rate, which provides a valid basis for ignoring
the free convection effect. It is also reported that ﬂame radiation is relatively unimportant
for thermally thick solid materials [41], therefore non-treatment of the radiation seems
valid in this case.

For thick PMMA materials experiments show that for particular [23] but not typical

31

'l‘ _ - f.- I‘ < .
'_,:ea.e\ o straws.
—>~L~ I‘ ,) ,3" I.
Jscﬂ.)m. r yrs” .

zori-izietzt In" on than;
is and kit I‘Ll't he:
terse-ti ﬁgure 4 sh

2:2: 15 deﬁned 1* 1‘

‘.._

I
‘ .

‘1‘
. ’c‘j in v v!
an”; I; "i h‘hh
‘ ,Iua Eden."
-
C

j‘ir! . .39.; '
... i.errnines the

Mk «
an the trend of s’

n a dirterent \“Ieu
Conclusions

3 A
I“0 dimensm

W-k.‘;»' ‘-
.. thattons a

C II
1431,.
~ 3??
Head {31
Clip,
‘ l‘
‘t
:3 l

[1] cases of small-scale ﬁres, the heat conduction in the solid phase is the dominant

mechanism. Therefore it is meaningful to investigate the effect of the anisotropic

conductivity on ﬂame development. First keep ksxy as a constant, change the values of
ksx and k sy but keep the (ksx +ksy) as a constant, then two ﬂame characteristics are

derived. Figure 4 shows the trend of ignition time vs. ksx / k in which the ignition

sy ’
time is deﬁned as the critical point when the heat ﬂux is equal to 60 kW/m2 [9]. It is

obvious that the larger the ratio of the ksx / ksy , the more heat is transferred streamwise

during initial heating and hence the smaller is the ignition time. Therefore the ksx / k5,.

ratio determines the temperature distribution in the initially heated solid phase. Figure 5

shows the trend of steady ﬂame spread rate vs. ksx / k which afﬁrms the above theory

sy ’
from a different viewpoint.

5. Conclusions

1. A two dimensional transient mathematical model incorporating the gas species and
energy equations and solid heat conduction equation are solved numerically to
qualitatively evaluate the ﬂame development over anisotropic solid polymers.

2. Based on the numerical results, ignition, transition and steady ﬂame spread processes
are described. Relevant physical mechanisms and certain theories of ﬂame structure are
analyzed.

3. For anisotropic thick solid polymer materials, the relationship of the ignition time and

ﬂame spread rate with ksx / ksy is obtained for a particular case. The importance of heat

conduction in the solid phase for initial ﬂame development is evident from the numerical

results.

32

O)

 

 

 

 

 

 

 

 

E E
; 4 ; 4?
Q .
Q
"o i 4 (s 8 1° ‘2 4 6 8
X(mm) X(mrn)
(l-a) (l-b)
12 12
1o 10
a 8
- 6
A 6 E
E
i
4
2 2
c 0 - l
.2- e ‘ ‘ ‘ '20 7 4 6 8
o 2 4 6 8 MM)
Mum)
(l-C) (l-d)

Figure l Constant temperature contours at four different times (The outermost contour is
300K, increment 100K between two adjoin constant levels. (a) t=l.4s; (b) t=l.6s; (c)
t=2.08; d: t=4.8S.

33

Y(rnm)

 

 

 

 

 

 

 

(2-0)

Figure 2 Constant fuel concentration contours at different time (The outermost constant
level is 0.1, increment 0.1 between two adjoin contours). (a) t=l.6s; (b) t=2.0s; (c)
t=4.83.

34

12

10

Y(rnm)

 

 

 

 

 

X(rn m)

(3-a) (3-b)

Y(mm)

 

 

 

'20 i Z é a
X(m In)

(3-0)

Figure 3 Constant oxidizer concentration contours at different time (The outermost
constant level is 0.3, decrement 0.05 between two adjoin contours). (a) t=l.6s; (b) t=2.05;
(c): t=4.8s.

35

 

5 4 ,3 3 .3 2 5 .I 5 a... 6 4 2 1 8 6 4w .4 C
4. aqv n‘ GI. C 0.. rlv G O
«2.22:: 2

.2...

 

4.5 -.
4 --f
3.5
3
2.5j
2 I
1.5
1i
0.51‘
0;___.2_,-_--_3--2--_-._ -.--.-... _
0.2307 0.4545 0.6 1 1.6667 2.2 3

ksx/ksy

t (s)

4.3333

Figure 4 Ignition time vs. ksx Htsy.

u (mnr/s)

...__71_. _. ...__ _ _ _ _ ._

0.2307 0.454 0.6 1 1.6667 2.2 3
ksx/ksy

4.3333

Figure 5 Flame spread rate vs. k3,, /ksy.

Table 1 The thermal properties and chemical kinetics used in the modeling.

 

 

 

 

 

 

 

 

 

 

 

| A, Ag E, Eg L, qg kg kxy um I
‘1.0E+06 1.6E+1 145520 135730 1355.6 12945 0.04 0.035 0.1 I
m/s Sm/s J/mol J/mol kJ/kg kJ/kg J/m.s J/m.s m/s
36

 

 

0pp05E0-FUV

Introduction
IranSIent ﬂame ,
heretical and peei.
education. The latte:
mutual Influences. an

Men 3 poltmen

 

 

teat fruit. It can be I

\
h
\

|
Lee strrface Igmt

Bees

FR”!

m be concerned In

I171 . -
...ti or OXIdIzer.

An extensn e res
Lavishhs 0f Refs 1‘
atCresses t]

e spre

gCirce -

are]

CHAPTER 3
OPPOSED-FLOW FLAME SPREAD OVER POLYMERIC MATERIALS —

INFLUENCE OF PHASE CHANGE

1. Introduction

Transient ﬂame growth over polymeric materials is important in ﬁre safety, and has
theoretical and pedagogical importance in basic combustion science research and
education. The latter stems from a large number of complicated phenomena, their
mutual inﬂuences, and the challenge of describing them in an orderly, logical manner.

When a polymeric material is subjected at a portion of its exposed surface to a high
heat ﬂux, it can be ignited. The ignition of ﬂame may lead to subsequent ﬂame spread.
Because surface ignition at a point is associated with induced inﬂow of air (oxidizer), we
will be concerned in the model with ﬂame spread against an induced or forced opposing
ﬂow of oxidizer.

An extensive research literature exists for this class of ﬂame spread problem, see the
reviews of Refs. 1-3. Most of the literature addresses solid fuels. The review of [2']
addresses ﬂame spread over liquid fuels, which possess additional complications such as
surface driven ﬂow, enhanced buoyancy and liquid vaporization, etc. These
complications are potentially important when the liquid phase melt layer actually
precedes the ﬂame leading edge. Heat transfer ahead of the ﬂame by liquid phase
convection may be important under same conditions. In addition, recirculating cells may
develop in the gas and pulsating spread may be possible. Then, a signiﬁcant overlap may

occur between ﬂame over initially solid fuels (which liquefy) and ﬂame spread over

37

hqgjﬁiels.

Many t‘onnulat'.
geared in the rest
nttematze examinit:
ndndimenSIonal tr
stoseqtently examtr.
tame perspeetne.'
EI‘CI‘LADCC of the g)
It dime ﬂame spre
"was dented. The g1
netnntse conduct
ﬁftieth; relatix'elt
Emit SIIL’aleSC e

By utilizing 1h;

mate ‘~ .
nan “as dertx e

n

 

 

in: ~ . ‘
L.e5U.lab1e fOr a

. ~ :33 - ~
than solid pi]

sitthi'tm. It is well 1

313 ~ 5~
ﬁtting R0,}, r
x. ‘6‘

teited
poli‘mer agr.

{lam .
e thread form,
u:
Flier
\, m
ode] 0r {hm

mieims -
OI local FL,

liquid fuels.

Many formulations of the theoretical opposed-ﬂow ﬂame spread problem have
appeared in the research literature. The study of deRis [21] stands out as the ﬁrst
systematic examination including simpliﬁed gas-phase chemistry, solid decomposition,
multi-dimensional transport and convective/buoyant ﬂow. Many other models were
subsequently examined (see [1]) but only one studied ﬂame spread from a global energy
balance perspective, thus adding insight into the ﬂame spread process [43]. The
importance of the global balance principle arises from the ease with which it can be used
to derive ﬂame spread formulas under conditions more general than those for which it
was derived. The global balance principle is easy to apply because of the neglect of
streamwise conduction. Instead of solving complicated elliptic boundary value
problems, relatively simple parabolic conservation equations can be formulated, which
balance streamwise convection and cross-stream conduction.

By utilizing the global balance principle a ﬂame spread formula for charring
materials was derived ([28], see the discussion of [28] in [1]). This formula is perhaps
more suitable for a solid that liqueﬁes upon heating since liquefaction is a simpler
process than solid pyrolysis and degradation, and usually occurs along or near a speciﬁc
isotherm. It is well known however that solid degradation and pyrolysis does not occur
at a speciﬁc isotherm. We shall demonstrate that our model for ﬂame spread over a
melted polymer agrees, under many conditions, extremely well with predictions of the
ﬂame spread formula where considerations of dripping and running are not included in
either model or theory. The ﬂame spread formula therefore can be interpreted physically

in terms of local physics near the point of ﬂame attachment. Recent work on transient

38

sold-phase Ignition
ecﬂeagues on men.
model of the gas I
desired and then :

It IS known that
233-“phase combuxttt ‘
Conduction. phi“

”.C.» .
, "13

an: and con-den .

l
I

complexities. the co ‘

 

solid-phase ignition and ﬂame spread has been undertaken by Kashiwagi, Baum and
colleagues on microgravity ﬂame initiation on cellulosic materials [13]. A detailed
model of the gas ﬂow and its thermal expansion during ignition and spread was
developed and then solved numerically.

It is known that many complicated, simultaneous processes occur in the solid with
gas-phase combustion above it. These include detailed degradation chemistry, anistropic
conduction, phase change, bubble formation and transport, charring of the surface,
pitting and condensed-matter expulsion at the surface. Because of these and other
complexities, the condensed phase has been studied considerably less than the gas phase
[25]. Studies of ﬂame spread over melting polymers have rarely been reported except for
some experimental observations [25].

The purpose of this chapter is to examine a ﬂame spread model whose solid (or
condensed) phase contains some solid phase complexities not described previously in
ﬂame spread research [1-5]. The gas will be described by a standard Oseen-ﬂow model.
F inite-rate gas chemistry is retained. In the condensed material, we retain phase change
(solid—)liquid in-depth; liquid—>gas at surface). Upon phase change, thermophysical
properties (conductivity, and speciﬁc heat) may change.

Our numerical model will be transient, enabling computation from incipient heating
to ﬂame ignition to eventual steady spread. Melting and phase change alter ignition.
Comparisons are made with predictions of theory in the steady state. The eventual goal
of our research is to include bubble formation in the liquid melt layer. The study of
ﬂame spread with phase change of the fuel from solid to pure liquid is a ﬁrst step.

2. Numerical Model

39

  
   
   

ll Problem Form?

A schematic
innittieattons are '
describe Instead) I):
approximation of t.
nementum equation
’ ..nl and theme
raccoon II the o;
C‘C‘ISLIlé‘TECl to. oeeur

0L ‘
we tertieal Com]

2.1 Problem Formulation

A schematic formulation of the problem is provided in Figure 1. Some
simpliﬁcations are made to reduce the complexity of the governing equations that
describe unsteady ﬂame initiation and spread over polymeric materials. First, the Oseen
approximation of uniform velocity proﬁle is used, thereby: (l) eliminating the
momentum equations from the solution; (2) uncoupling the velocity ﬁeld from the
thermal and chemical ﬁelds; (3) reducing the N-S equations to the constant pressure
condition if the opposed ﬂow Mach number is negligibly small. Flame spread is
considered to occur in the horizontal plane, thereby eliminating required consideration in
the vertical conﬁguration of melt ﬂow from the melting surface. The authors are
unaware of any theoretical or numerical work on this subject outside of preliminary
work in a highly idealized conﬁguration [44, 45]. Second, radiation absorption by the
ﬂame and radiant emission from the ﬂame are ignored, as are surface and in-depth
radiant absorption by the condensed material. Third, the thermal properties and kinetic
data (pre-exponential factor and activation energy) are assumed constant. Fourth, the
regression or deformation of the gas-condensed interface and Marangoni ﬂow of
polymer melt near the interface are assumed negligible. The other assumptions that are
relevant to the speciﬁc equations will be introduced hereaﬁer. All restrictions are
removable in principle, but when initial studies are conducted in a simple manner, the
complications that are later introduced are more clearly understood. Some of our
restrictions (negligible interface regression, no surface Marangoni ﬂow) have not been
extensively studied in the context of ﬂame spread. The governing equations include

those for the transient reactive gas phase, the transient reactive condensed phase, the

40

 

prise while the sub

coexeetion. and the:

:11: 1:

Iron-reactive gas-ct '

Integrates 15 ﬁxed

The transport

 

ll 15 assumed in,
UCIeﬁil‘]
..c e ..
>CCI
111 ‘

,—

non-reactive gas-condensed interface, and boundary conditions. The origin of
coordinates is ﬁxed at the interface, therefore the subscript “y > 0” denotes the gas
phase, while the subscript “ y < 0 ” denotes the condensed phase.

The transport mechanisms in the gas phase include difﬁrsion, streamwise

convection, and chemical reaction. The species and energy conservation equations are

givenby
Pgl%+um%§]=pgDVzYi+E,-w, i=f,0, y>0, (1)
0T 6T
ngpg[E+um—a:]=kgV2T+qgwg, y>0. (2)

It is assumed that the combustion reaction F +v00——>P is an overall single-step,

irreversible second-order Arrhenius reaction, with reaction rate

-E /RT . . . .
g and storchrometrrc ratio

__ 2
wg — ngoYnge
E,- =M1V1/ vaf, i: f ,0. The heat transfer mechanisms in the condensed

material include thermal diffusion, phase change, and pyrolysis reaction. The energy

conservation equation in enthalpy form is

Pea—ahtg=kCV2T+chc, y<0, (3-a)

where subscript “c” denotes the condensed material in general, and will be denoted by

‘6 ’9

or s in individual liquid and solid phases, respectively. An overall single-step ﬁrst-

‘61,,
order Arrhenius reaction of polymer —9 monomer is assumed for the pyrolysis process,
with reaction rate WC = —pcAc exp(—Ec / RT). The condensed-phase density pc may

take the form of p, or p, in the condensed material. The kinetic data for E c and AC

41

are constants for ho‘.
ligtre I Is proud.
tentajpy and temper
aired melting tern;
rid liquid are ass;
relies the exist-er
menace The ener,
fern Includes three

0mg phase front

~ l
its.
.L\Ll]\elv E

 

 

are constants for both solid and liquid phases. The enthalpy-temperature relationship in
Figure 2 is provided along with equation (3-a), thereby reducing two unknowns

(enthalpy and temperature) to one. In Figure 2, phase change is assumed to take place at
a ﬁxed melting temperature T m 6, and the thermal properties p , C p and k in the solid

and liquid are assumed constant in each phase, but not necessarily the same. This
implies the existence of discontinuous thermal properties across the solid-liquid
interface. The energy conservation equation of the condensed material in temperature
form includes three domains of interest, viz., the solid phase, the liquid phase and the

moving phase front,

 

 

lpSCp, 93%: = kSVZT — chc, solid
6T 2 . .’ y<0 (ym
[PICPI }; = k,V T — chc, lzquzd
- (1195;). + 11st 1 = —(k 8T.»
an " an , phase front, y < 0 (3-c)
T); = Tm

Equations (3-b i, ii) are the energy conservation equations for the solid and liquid

*
respectively. Equation (3-c) is the Stefan condition in vector form, where n denotes

the unit vector normal to the moving solid-liquid phase front 2, and v": denotes the
phase front velocity.

At the gas/condensed phase interface, the mass transfer mechanism is pure diffusion

 

6 The melting point of the polymer is different from the ‘glassy point’ temperature that frequently appears
in the literature, and normally has a larger value.

42

 

of the net norm;
nclude conduct
environment. an:
remoted thereaf

2.511 VIII -o
...1_}seqn..rhra.

3..“ as
I H
J“.
15.314

L‘;&'
~\\
e no Slur
[‘1
kg 53!.
“‘ on (3111)
313.314-
“4:113:51
‘ ‘4
‘u.
@355 It
Ci} 7- H
«(.21 ..
L.)
a uRlIlir-
3 ..

of the net normal ﬂow comprising pyrolysis products. The heat transfer mechanisms

include conduction in the gas, conduction in the solid, surface radiation to the

environment, and transient ignition heat ﬂux til-g , which is applied before ignition and

removed thereafter. The species and energy transport processes are assumed to be

always equilibrated, whereby

I

—PgD$fl0+ =(1—Yfl0+)"'11
BY .
{—pgD—aj)‘|0+=(O—Y0|0+)m, (4-3)
3T 8T

—ks

— _=-k — +80 T4—T: —',-.
l aylo gayl0+ ( ) qg

 

In equations (4-a i, ii, iii), the mass ﬂow rate 771 arises from the pyrolysis products. The

virtual mass ﬂow rate of condensed phase pyrolysis products can be written in integral

form as If)!” wcdy. A permeability factor Ke with range between zero and unity is

used to adjust the magnitude of this mass ﬂow rate. Hence,
rthe fay wcdy. (4-b)

This equation implies that the transport of gas through the liquid is a steady process,
because no storage effect or time derivative appears. The heat transfer described by
Equation (3-b), however, is unsteady. Usually, when diffusion is the transport
mechanism, mass transfer through liquids is slower than heat transfer. Our model
acknowledges this limitation, but is applicable (valid) in the limit that the liquid layer
contains a uniformly distribution of “gas” molecules that ﬁnally escape from the surface.

There is a uniform and continual migration of monomer molecules toward the surface.

43

0' course. thrs IS 31'
31531111011 Is bett.
nieeules of mono:
:3. the surface [1'
lkgﬁlol w, o\cr
tell alter the ICCICII:
fillet“ 501mm CU?
i=010<t<i

. _e.
5‘41” dimension or"

allied at the Uppe

U]: l

. etttons include

X:QO<[-

11..
C ‘41 -
a mabatre C0ndr1

 

Of course, this is an idealization that is, in principle, removable when the solid phase
degradation is better understood. The “gas” in this model may be interpreted as
molecules of monomers, which form in the liquid layers of thickness 1,, that escape
from the surface. In other words, Equation (4-b) states that m is proportional to the
integral of we over the solid thickness, but the mass, in fact, escapes from the surface
cell after the reaction polymer —> monomer has taken place. In the gas, an isothermal
inﬂow boundary condition and an adiabatic outﬂow boundary condition are applied at

x=0, O<y<€gy and at x=€x, 0<y<€gy, respectively, where 5 denotes the

linear dimension of the computational domain. A closed adiabatic boundary condition is

applied at the upper wall y = [0,0 < x<1x, as indicated below. Other boundary

conditions include

 

,
T=T°° a_T=0
x=0,0<y<£gy Y0=Y0w;x=€x,0<y<lgy<§; .
: 4:0,iT—"0,
Yf 0 [ax f
The adiabatic condition is written as:
f
y=€gy,0<x<€,< y (
arc. .
——=0,l=0,f.
Lay

 

The initial condrtn

33m )1)” IS the
The consen.
aiming eondrtrt

refines: boun dar

..i thrn-Drniensr

The gm emin ;

.193,” -
w..irnsrrrs that
an -- I

anopnate refere

"4e dommant me:

I215: ‘u
1 Ital 15 solid

and low ‘~ ‘
\lsLOSlI}

" e Spread, Ga

Eton
1 738 Weak
CT

eta: .

transfer nice

['1 .

12'

is

 

 

The initial condition is

T=T°°,Yf=0andY0=Y0°°, (6)

where You, is the initial or inﬂow oxidizer mass fraction in the gas.

The conservation Equations (1-3), the gas-condensed interface condition (4),
boundary conditions (5) and initial condition (6) together form a well-posed unsteady

nonlinear boundary value problem.

2. 1.1 Non-Dimensionalization

The governing equations shall be non-dimensionalized in order to analyze physical
mechanisms that may subsequently inﬂuence ﬂame development. It is difﬁcult to derive
appropriate reference variables for dimensional analysis. If phase change is negligible,
the dominant mechanism of heat transfer in the solid for ﬂame spread over thermally
thick fuel is solid heat conduction [5]. However, for polymers with a low melting point
and low viscosity, it was observed [13] that phase change and melt ﬂow may inﬂuence
ﬂame spread. Gas phase conduction is dominant in thermally thin ﬂame spread and
becomes weaker as the fuel thickens [5]. Whatever the relative importance of a certain
heat transfer mechanism is, the interactions between the gas and the condensed phases

always determines the nature of ﬂame grth and spread. Based on this idea, the non-

dirnensionalization is carried out as follows: the reference length 7! =01, /u°,, is

obtained by multiplying the characteristic thermal length in the gas (org / um) by a
factor Q=a,/ag; the reference time is 1' =h/uoo; the reference temperature is
L, / C p, ; the reference oxidizer concentration is Yam; the reference fuel concentration

is Yam/#0.

45

lrr summan'. lh~

materialized 3‘.
dimensionahzed as

lL’ZCllSiOllatlZCd as

“man and initial

Gas phase.

 

In summary, the coordinates x and y are non-dimensionalized as J? =x/h and
=y/71; time t is non-dimensionalized as t— =t/T; temperature T is non-

dimensionalized as T =CpS(T—T,,)/LS; the fuel concentration is non-
dimensionalized as T} =Yfﬂ0/Y0w; and the oxidizer concentration is non—

dimensionalized as Y0 = Y0 / Yam. The dimensionless equations, interface conditions,

boundary and initial conditions are,

Gas phase;

gig: 1 (8238217;
at- Bf QLe 8352 ayz

 

2’117g)+ i=f,o, i=f,o (7)

6T 6T 182T BZT

__ =__(___+

a; a- 96:2 BTZHW’ it.” (8)

Condensed phase (temperature form);

 

 

 

 

 

I 2— 2—
87f: a T 2+a TS -§CWC, in the solid
at 3x2 ayz
< 2_ 2__ , y<0(9-a)
— 8T 6 xT1+ a T1 _ _ . . .
pICP1_—at l=kl(——— +ay ——2—_)— chc, In the liquid
_ar. , an
35* " 135*, phase front, y<0 (9-b)
T2=St

Gas/condensed phase interface;

46

 

 

 

 

 

 

LLB-Eb :m( 047' )
me @ + 00° f0+
1 817 T —
<—HL—ejlo+:m(O—Yolo+)’ y=0 (10)
3T— ngT— - ~ 4 “’4
—0_=———0 +w[(T+Too) -Too]-(7-
I037 11. ail * ’g
Boundary conditions;
T=0 l§i=0
i=0,0<y<7gy Yf=0,r=?,,0<y<7gy< 0E ,
_ (7Y1
0=0 IEEzm’sz
y=7gy,0<r<7x< 817 (ll-a)
——i=0,z=o,f
.037
f=0,—Tsy<y<00rf=€x, €,y<y<00r
_ (ll-b)
y: €,y,0<x<€x 9:20
Initial conditions;
r=o,rf=oandr,=1. (12)
— 47 E11 - —2 EC
Here wg— 0 f gexp(-T+Tm), wc— Cexp(—T_+Tx)’

Bi = K, 1927., pch'cdy, and i, = CP,T,,/L,.

The dimensionless groups are listed in table 2.

47

ll 2 Numerical
The compu
drectron and tr
phase and a 5"
end spacing Is
mom with

1:01 with 3

‘~

I I .-
Ine HCIEH'DOTIT
\r

1'53:

IlOl’r-ll‘ilensr

V"

~. 93315 were a.

THC ﬁnite

control Volume

”Gliding exa:

2.1.2 Numerical Approach
The computational domain of approximately lSmmx 15mm , in the streamwise (x)

direction and transverse (y) direction consists of a 50x50 mesh system in the gas

phase and a 50X 40 mesh system in the condensed material. Along the x direction the
grid spacing is unifome 0.3 mm. Along the y direction the grid spacing is non-
uniform with the minimum spacing (0.025mm) near the gas-condensed interface
(y = O) with an exponential increase in the two opposite ( yo+ and y0_) directions.

The neighboring increment ratio is 1.07. This treatment is intended to resolve the
reaction-intensive region close to the interface. The mappings between the two mesh
systems were accomplished by appropriate coordinate transformations.

The ﬁnite difference method is used to solve the numerical model [46]. First, the
control volume formulation is used in the discretization because of its ability for
providing exact tracking of the solid-liquid interface during phase change in the
condensed material. The diffusion terms are treated by the central difference method; the
convective term by the up-wind scheme, which in low ﬂow velocity provides acceptable
performance. To treat the chemical (source) terms, the guideline of positive derivatives
of source terms is followed. Negative chemical terms are linearized by preserving only
the partial derivatives with respect to the primary variable, while the positive source
term is left unchanged. In addition, a special treatment of the chemical term is used to
enhance the accuracy and stability of the solution procedure [47]. To minimize
computational time and storage, the ADI (Alternate Direction Implicit) method is

employed: the independent variables are solved alternately in the x and y directions.

The techniques used in solution of the phase change process in the condensed phase is a

48

 

special form Of 1h
nomentioned et
across the solid-ll
:48101'the therma

The run time
:smeheal model.
term In the gas a
:meneal expenr
determined marnl
Change men at h:
model. the Iteratr
time thft‘erence e

T“ computa

"L" a .
“‘ err-.151} equatrt

fie. ~
rfa‘e PFTOIySI:
.5: Into [he bour

...trtaee tempera
so After the 233

be .

“13110111 I“
1.
33;”;
“S of T .1
3 U
1521mm v.
4113”] Ken

special form of the Enthalpy Method, the ADI Source Update Method, as well as the
aforementioned enthalpy-temperature relationship. In addition, property discontinuities
across the solid-liquid interface are averaged by using the harmonic mean formulation
[48] of the thermal properties for any grid encompassing this interface.

The run time is inﬂuenced by three major non-linearities that are present in the
numerical model. These non-linearities are the interface condition, the chemical reaction
terms in the gas and condensed material, and the Stefan condition. It was determined by
numerical experiments that the stifﬁress of the overall equations, hence the run time, is
determined mainly by the chemical reactions. This overshadows the inﬂuence of phase
change even at high Stefan numbers. In views of the non-linear nature of the numerical
model, the iterative Newton-Raphson scheme is applied in semi-implicit form to the
ﬁnite difference equation.

The computational cycles are constructed according to the physical process. First
the energy equation of the condensed phase is solved, which produces the gas-condensed
interface pyrolysis product ﬂow rate and the interface temperature. Next this ﬂow rate is
fed into the boundary conditions of the two species equations, and the gas-condensed
interface temperature is fed into the boundary condition of the energy equation in the
gas. After the gas phase temperature is solved from the energy equation, the heat ﬂux of
the interface is obtained as a ﬁrrther input into the condensed phase. The iterative

computational process continues until the relative error of the two most recent iterative

values of T, Y0, Y f and he fall into the convergence range, the limit of which is
normally chosen as 0.0001.

2.2 Theoretical (Simple) Model

49

The mC‘d'c
Cmrﬁmte SE9":

the energl' “1”

cont eetron t m
rem-owed by C
halmce pnnezr
:ther. whereas
att‘aees of me
slid-phase co
:eross the flarr
global energy I
:chels should

‘- s
.14 been dem

The model conﬁguration for theoretical analysis is shown in Figure 3. The
coordinate system is ﬂame-ﬁxed, hence a streamwise convection term is introduced in
the energy equation of the condensed phase. The global energy balance principle of
Wichman and Williams [43] is applied. The global energy balance principle states that
the “ﬂame spread rate must be sufﬁcient to remove by downstream solid-phase
convection (in ﬂame ﬁxed coordinates) the heat that is generated by combustion and not
removed by either downstream gas-phase convection or conduction”. This global
balance principle is valid when either solid or gas phase conduction overpowers the
other, whereas a simpliﬁed energy balance for heat ﬂow to unignited fuel across
surfaces of incipient ﬁrel [34] can be used only when gas—phase conduction overpowers
solid-phase conduction. Thus, ﬂame spread models that employ an energy balance
across the ﬂame leading edge (and in its vicinity) are not so generally applicable as the
global energy balance. With gas-phase domination both leading edge and global balance
models should give approximately identical predictions. For most leading-edge models it
has been demonstrated [1,43,49] that under most conditions encountered in ﬂame
spread, gas-phase domination is the norm. Exceptions to this norm are discussed in [50],
where the viscous-invisid region near the point of ﬂame attachment is examined in
detail. The authors of [50] demonstrated that an upstream gas-phase ﬂuid dynamic
recirculation cell could alter ﬂame spread behavior.

In applying the energy balance principle, the Oseen approximation is employed.
The ﬂame sheet is hypothesized to lie along the gas-condensed material interface. The

transition from solid material to liquid is assumed to occur across an inﬁnitesimally thin

front located at the locus of a parabolic arc y/ J; = const. All of the latent heat is

50

 

presumed to be c1
soiicrently VISCOU
light. h)’droear't\,‘1r

romela Isee Apne

3“

.a lSlmphﬁcd I

this

presumed to be consumed along this are. In addition, the liquid is presumed to be
sufﬁciently viscous that internal, circulating ﬂows do not occur as they do for certain
light, hydrocarbon liquids. The analytical solution yields the following spread rate

formula (see Appendix II),

C k T —T-
145;ng ng,(f ')2-erf(C/l'&)2, (13)
“no pICPlkl Ti_Tm 2 051

where T f is the ﬂame temperature, 7; is the interface temperature (see Figure 3), and

 

c is the value of the numerical constant deﬁning the locus of the liquid-solid interface.

As 6 —) 0 and T m ——) T,- , Equation (13) reduces to the thick fuel deRis formula [21].

When phase change occurs, the Stefan number7 St = CPS(T m -T°°)/ Ls appears

as one of the non-dimensional parameters of the problem. Physically, St is the ratio of
the heat required to raise a unit mass of the solid from the ambient temperature to its
melting temperature to the heat required to transform this unit mass of solid into liquid.
Only St > O is of interest to us since St is positive when phase change is endothermic.

A derivation of the preceding results is presented in the Appendix II.
2.2.1 Simpliﬁed Derivation of Flame-Spread Equation

In this section, a simple, physically-motivated-scaling-argument derivation is given for
Equation (13). A full description is presented in the Appendix II. The simplicity of the
scaling analysis sheds light on the important approximations of the analysis. It was shown

in [1, 43, 49] that for a basic understanding of the overall ﬂame spread process only the

 

7 In this model, the deﬁnition of St is inverse to those appearing in [l] and [28].

51

 

mm consen anon
balance between s

: mvy‘! -\ \
illhunillt‘lll taptui ._

:merous original u
Skies region" nea'
emulation pro-due;

We proceed h

“ten the bound-m

3531, In me my

1.6, ‘7
' Unllorm COOrk

0,: .
5 kg ngp.

\. \

pgc - \ ‘~

 

p‘C jugATS L ._

 

\—

~Li-n. -. .
\ r“d§e -\ ‘
kOmhj

l

’1?

«1113“,.
.r the l
lqu

picpzuf M] L,
1X

anathm

energy conservation equations of both media were needed, and that in these equations a
balance between streamwise convection and traverse diffusion (a boundary-layer
formulation) captured most of the physics. The full problem is elliptic, as shown in
numerous original works [21, 51], but the region of pure ellipticity is conﬁned to a small
“Stokes region” near the ﬂame attachment point. Outside of that small region, a parabolic
formulation produces useful results. We follow the scaling analysis outlined in [1, 50].
We proceed by ﬁrst scaling the three energy equations (gas, liquid melt, and solid),

then the boundary conditions along the interfaces (gas—liquid, liquid-solid) that separate

them. In the gas energy equation ngpgugaTg/ax=kgazTg/ay2 scales to

ngPgugATg /Lgx ~ kgATg /Lgy2. Imposition of the criterion Lgx = L . = L

g)’ g ’
i.e., uniform coordinate scaling in the ﬂame attachment region gives Lg =ag / ug,
ag = kg / ng Pg . In the solid the energy equation is
pSCPSuSBY; /3x = [(382]; /By2 , which scales to

psCPsusArs/Lsx ~ ksAT:9 /Lsy2. We let Lsx = Lgx = Lg in order to emphasize

gas—phase control of the spread process. We then ﬁnd Lsy =[asag/usug]l/2.

Similarly, the liquid-phase energy equation plelulaTl / 3x = [(1327) / 3y2 scales to
pICplulAY} / le ~ klATI / Llyz, which yields L1), = [alag / ulug ]1/2 when we use
le = Lgx = Lg as for the solid. Clearly u] = us.

We now examine the two interfaces. Along the solid-gas interface upstream of the

ﬂame attachment point, the conductive energy balance gives

52

islafs an,

terminated. l
urinate to t

he locus _1- :

f2} 1'51 = _

{3253361 to the
’56 nondmc

UT, L .

.V 1

«we substitme

ﬁler . ,

ks (87;. /ay)y=0 ~ kg(aTg /ay)y=0, which yields kSATS /Lsy ~ kgATg /Lgv or

Lsy/Lg =kSA7} /(kgATg). Along the liquid-solid interface, the situation is more

complicated. Here we have ks (aTs/an*)~ k1(aTl/an*), where n* is the normal

coordinate to the parabolic arc along which phase change occurs. This are is given by

the locus y =—b1x2, which nondimensionalizes to f = —f2 with f = y/ Lly, and

f =x/le =x/Lg and L1}, =b1Lé. In the case that the parabolic arcs lie nearly
parallel to the horizontal plane, the derivatives 3T / an approximate to 3T / 8y. Thus,
the nondimensionalization of the interface condition yields, approximately,
kSATS /Lsy ~ k,AT, /L,y . Use of L,, = 1),ng gives

ATS~ k, AT,

k
5 LS, bILg L8

(14)

In our derivation this interface condition is considered to be more important to the
overall spread process than the solid-gas interface condition derived previously. Hence

1/2

we substitute the relationship LS}, =[asag /usug] derived from the solid-phase

energy equation into Equation (14) to obtain

:Sklz

AT; 1
us=ug k32( I

15
AT) b,Lg ( )

 

The expression for ATI/ATS is now written in the following form:
ATI/ATE. =(ATg/A7})(ATI/A7;)2(ATs/ATg). Then we use Equation (14) for

AT, / ATS in the right-hand side (square) term, and the upstream interface condition

53

Equation l 15) )ieid

The quantity in sq
a
L... 25,1?” IS er:

parabolic isotherm

led in the solid T

it compare this r».

int-l1

“N

 

 

kSAT9 / Lsy = kgAT g / Lgy for ATS / AT g . These expressions substituted into
Equation (15) yield

2

(M >2———1.
PICPIkI A7} a! g LS),2

 

“g

The quantity in square brackets reduces to (as / a,)(L,y / LS), )2 when the relationship
L1), = bng2 is employed. When we observe that both liquid and solid must produce

parabolic isotherms, we ﬁnd (Lly / L8,)2 = (bl / bs)2 = K2 , where Lsy = [)5ng was

used in the solid. The constant factor K“ is nondimensional. Thus, our ﬁnal result is
— ( ) [K 1 (16)
PICPIkI AT! 0‘1

 

“s

We compare this result to Equation (13), the exact formula, by considering the limit of

small c (high sweep-back of the isotherms) to ﬁnd
afﬁx/as / 2a,]z (2%)[C1/as / 2051]. This yields Equation (16) with the quantity
in square brackets replaced by (262 / 7t)0!s / al. The correspondence between the two

formulas is exact when we identify K2 = 262/71'. We note ﬁnally that in Equation ( 16)
AT g = T f - I; , AT, = T9 — T m are the characteristic temperature difference in the gas
and liquid phases, respectively.

This simpliﬁed analysis illustrates important features of the phase change model of

ﬂame spread. First, the gas phase is elliptic near the ﬂame leading edge because

Lgx = Lg), = Lg , i.e., all characteristic lengths in the 2-D plane are identical. Second, in

54

fie limit being con

streamwise diffun

_« _‘5

gm d1 Four.
and liquid and so
charge enthalpy. \\
of the m0 interfae:
m the solid-gas I

spread process.
i .12
~‘ “3 U pg C

111% -‘ V T
“3'90an to k7

 

 

the limit being considered chemistry is very fast compared with convection and diffusion

processes. Third, the solid and liquid phases are fundamentally parabolic, with

streamwise diffusion 32(-)/a)c2 negligible in comparison with transverse diffusion

32(-)/ ayz. Fourth, conduction across the interfaces between gas and solid (upstream)

and liquid and solid (downstream) dominates streamwise convection and the phase
change enthalpy, which appears only implicitly in the parameter c of Equation (13). Fifth,
of the two interfaces, the energy balance across the solid-liquid interface was used more
than the solid-gas interface balance, indicating its greater importance in the overall ﬂame

spread process. Sixth, simple rearrangement of Equation (16) gives

us = Kzug[(ngPg/Cg)/(psCPSk1)]/:I-1(ATg /AT, )2 , suggesting that us decreases
in proportion to [Tl—1 and in proportion to CPS-1 , if their inﬂuences on other parameters

are not considered. Our subsequent evaluations demonstrate that us «I kfl is a good

reckoning, whereas usts ~constant is not (see Figures 4(b) and (c)). Seventh, the
concept of “gas-phase dominance” of the spread process is implicitly understood by the

imposition of the gas-phase length Lg on le and Lsx; i.e., the use of le = Lsx = Lg .

Eighth, the liquid layer is highly viscous so that no internal convection or recirculation
occurs. Although surface tension gradients produced by temperature gradients along the
interface can induce circulatory liquid movement [52], we have not included these
motions in our scaling analysis. Ninth, subject to the constraint of Oseen ﬂow (which can
be removed as discussed in [49]) and fast chemistry, the largest inﬂuences on the flame

spread rate are the thermal properties of gas, solid and liquid, and the Stefan number,

55

whose magnitud;
3. Results and

The lhcmm}
heat that from 1
0f the eonden:

C3311:

e. and;

(It;

Variation of 5

hi or CF] “ h:

5
\

r1 Fiﬁme SPY:

The thick -

whose magnitude controls the multiplicative factor K2 in (16) (c in Equation (13)).
3. Results and Discussions
The thermal and ﬂow properties are listed in Table 1. Ignition is established if the
heat ﬂux from the gas phase excluding external radiation is over 10 W/cmz. The effects

of the condensed material on ﬂame spread are investigated by varying three non-

dimensional parameters, St, 1:, , and 5p], in which St denotes the inﬂuence of phase
change, and El and 5 p1 denote the relative inﬂuence of liquid thermal properties.
Variation of St, I?) and 5P1 is accomplished by changing only one single property LS,
k, or C Pl while keeping the rest ﬁxed.

3.1 Flame Spread Rate

The thick fuel spread rate of deRis ‘s formula [21] is

“_sngCngg Tf‘Tv 2
“co psCPsks Tv-Too

 

(17)

where Tv denotes the vaporization temperatures. Since T f and T v are theoretical

parameters that correspond approximately to real condensed-phase and combustion
kinetics, and since they are in fact not constant in the numerical model, representative
values have to be selected in order to make a comparison. Based on the numerical results

of the interfacial temperature (see Figure 8(a)) and on previous experimental

 

8 The vaporization temperature T v in deRis ’s formula is essentially equivalent with the surface

temperature T; in Equation (13) as the latter denotes the constant temperature of the non-vaporizing
surface although the former denotes solid-to-gas ”vaporization” (actually sublimation).

56

 

 

measurement t T, I
ma T, from ':

S:—+ac.51nee 5:
25. the ﬂame spre...
Worth pctnttng 0:.
div and the nur
ESLPLHIIOD is t."

We obtained the '

an . .
.tlott mg ratio or

“6 note that this .

a v--' .
{Ah-“Rally feaRQr

ﬂme Iempemtur.
modu'ﬁﬁd that n:
of the ﬂame Sp“
numerical model,
he dependence
C‘jmfulaltonal re.
the spread rate 0 l
”l and l
theme

R )3» _
Tisscf‘iled by I
. a

 

measurement (T v = 665K for PMMA), it is a reasonable guess to ﬁx T v at 700K. We
extract T f from Equation (13) by letting the us equal the numerical ﬂame spread at

St -—> oo , since St ——> 00 corresponds exactly to ﬂame spread without phase change, that
is, the ﬂame spread problem is essentially characterized by deRis ‘s formulation. It is
worth pointing out that this treatment helps to make a comparison between Equation
(13) and the numerical model. This comparison is justiﬁable in that no artiﬁcial

manipulation is involved for situations of widely varying thermal properties. Therefore

we obtained the following representative values, T f = 1730K, thus resulting in the

following ratio of the right hand side of Equation (14)

T —T
(—f—l)=2.58.
Tv—T

00

We note that this ratio remained ﬁxed in all cases considered, and that T f = 1730 K is a

physically reasonable ﬂame temperature estimated in ﬂame spread since the maximal
ﬂame temperature in the numerical results is 1630K. In effect, a scaling factor has been
introduced that makes the case St —) oo agrees with the deRis formula. The dependence
of the ﬂame spread rate on St, which is obtained from Equations (13), (17) and the
numerical model, is plotted separately in Figures 4(a) through 4(c). Figure 4(a) reports
the dependence of ﬂame spread on St by presenting the formulas of and eleven
computational results. It appears that the formula provides an almost an exact solution to
the spread rate of the numerical model, since agreement between the numerical results
(‘*’) and theory (dotted curve) is observed. However, Equation (17), which is

represented by a horizontal solid line above the ‘*’ and dotted lines, results in a constant

57

ra‘ue higher tha
of St. the spre:
which is illustra
edition. 5f is
spread rate and
L. Ls almost ct

We can es'
Jt'ptase Chang
and nearly ne,
found that if 1
N0 flame Spy-g
mated in [h‘

4316 Cannot 5

lhec

C‘I'llt‘

F5368 44b. .
. c

value higher than both the numerical result and theory. Qualitatively, with the increase
of St, the spread rate increases. The sensitivity of the spread rate with respect to St,
which is illustrated by the slope of the curve in Figure 4(a), diminishes for larger St . In

addition, St is inversely proportional to Ls , and an almost linear relationship between
spread rate and Ls is found. As a result, the aforementioned sensitivity with respect to

LS is almost constant.

We can estimate the inﬂuence of phase change on the spread rate. The contribution
of phase change can be as much as a 40% change of the ﬂame spread formula at St = l
and nearly negligible inﬂuence at St = 100. Through numerical experiments, it was
found that if St is lower than 0.667, only ignition is observed, followed by extinction.
No ﬂame spread occurs. Since the external heat ﬂux is removed as soon as ignition is
initiated in the numerical model (for all St cases), the above ﬁnding indicates that the
ﬂame cannot support itself if St is too low.

The contributions of I?) and 5P1 to the ﬂame spread rate for St = 2 are reported in
Figures 4(b) and Figure 4(c), respectively. Good agreement is observed. Qualitatively,

the increase of either the conductivity or the thermal capacity of the liquid layer in the

condensed phase decreases the ﬂame spread rate. Similarly, extinction is observed if 1:,

is lower than approximately 1.0, or 6p, is higher than approximately 1.25. Again,

Equation (13) does not provide any indication of extinction because of the steady state

formulation. However, one may conjecture that certain limits can be derived from the
three cases of varying St, k, and E p]. Therefore extinction occurs if the ideal

(Equation (13)) spread rate decreases below these limits. These rate limits are dependent

58

mpmmetnc \‘al
oiftgire hat. 4.
te determine thes
in the numerical
procedure may i
necessan‘. theret‘
mural studie:

defined qunlll}‘

2‘ ,
Transient 5;

on parametric values. This viewpoint is partly supported by observing that the rate limits
of Figure 4(a), 4(b) and 4(c) differ from each other signiﬁcantly. However it is difﬁcult
to determine these rate limits quantitatively since extinction is not a well-deﬁned event
in the numerical model9. One reason is that the control of grid size or convergence
procedure may inﬂuence and substantially alter such unstable phenomena. It may be
necessary, therefore, to examine ﬂame extinction at least partly analytically. Even in
analytical studies, however, the precise deﬁnition of the moment of ignition is an ill-

deﬁned quantity.
3.2 Transient Spread Process

Figure 5(a) reports the progress of ﬂame spread rate along the streamwise distance
for St = 2. Three stages of transient evolution are observed, viz., ignition, transition
and fully developed (or steady) spread. Ignition is characterized by sharp slopes of ﬂame
spread rate around a peak point, indicating the impulsive nature of the ignition process.
The transition stage, which occurs after ignition, allows the ﬂame to stabilize mainly
over the preheated region (9-12 mm). Its behavior is characterized by smaller slopes.
The ﬁnal stage of the spread is established after the ﬂame moves across the boundary
between the pristine polymer and preheated region (x = 9m), and the ﬂame spreads
with constant rate. Figure 5(b) reports the time histories of both ﬂame front and phase
front arrival along the polymer surface. It is observed that the interval of transition (25),
compared to the ignition delay time (103), is relatively short. Another observation is that

the movement of the phase front started much earlier than the ﬂame front. This

 

9 A very detailed analysis of ﬂame spread initiation was carried out in [40], where all gas-phase processes
including thermal expansion were retained.

59

phenomenon an
fume front. sm
In figure Stet.
streamwise pre
initial heating

preheating reg
Between this i

tint applied 0
still stays put l
ts too ueak tr
15% fame sun
Thereafter. p3

Yt‘fead is atta

HLQ- ‘
r1446 from. a

Figures :
imperarure
lemon l 1.05
Sl'liace; the 1

131 UTE ‘

“Ed an abmp

I klld‘llly
'-

\

The t-

‘e.. A. I ‘
'1kf‘h. ~ ‘4
1i “

‘N

phenomenon arises from the fact that the phase front is established much earlier than the
ﬂame front, since the melting temperature is much lower than the ignition temperature.
In Figure 5(c), the phase front leads the ﬂame in the region 9mm<x<10mm. The
streamwise progress of the phase front is the described as follows: shortly after the
initial heating the phase front starts to move, then it approaches the boundary of the
preheating region, shown in Figure 5(a) at the location between x=9 and x=12mm.
Between this time and ignition, the phase front can not move because there is no heat
ﬂux applied outside the preheating region. Even with ignition initiated, the ﬂame front
still stays put because the external heat ﬂux is removed and the self-supportive heat ﬂux
is too weak to push it forward. The resumption of movement is not accomplished until
the ﬂame survives the transition and spread near the boundary of the preheating region.
Thereafter, phase spread is driven by the combustion heat from the gas phase and steady
spread is attained. This steady stage is characterized by the same spread rate as for the

phase front, as is seen by the two parallel lines in Figure 5(b).

Figures 5(c) and 5(d) report the streamwise evolution of heat ﬂux”), mass ﬂux and
temperature at the interface during three stages of ﬂame spread. At the time of pre-
ignition (108), an external heat ﬂux of SW/cm2 is applied to the preheating region of the
surface; the mass ﬂux of pyrolysis products is low because of the low condensed phase
temperature, see Figure 5(d). Once ignition is initiated, the external heat ﬂux is removed,
and an abrupt change of ﬂame environment occurs. The plume of the ﬂame adapts to the

rapid change of the heat ﬂux, during which transition occurs. If, for example, the heat

 

1° The heat ﬂux here denotes the net heat ﬂux feeding the condensed phase, which is obtained by
subtracting the surface radiation loss from gas phase conduction to the interface.

60

flirt generated
tire-ugh the g3:
as was obserx‘:
self-supponitc
ethane and sic;
the heat ﬂux d
heat llLLX. H0“
b} a factor of
f§79l}‘515 1359;

and heat ﬂux 3

As far 3.5
Ignition “hos
the leading ed.
IE-Ctease 0f (l
mimtftion of
mid“? of th
ﬁfteen of

Ph

nl .
WMImam ot

ﬂux generated from the combustion reaction cannot compete with the loss of heat
through the gas and the condensed phase, the plume shrinks and extinction takes place,
as was observed in Section 3.2. The transition stage determines whether the ﬂame is
self-supportive or not. Figures 5(c) and 5(d) illustrate the successful survival of the
plume and steady spread thereafter. It is observed in Figure 5(c) that the magnitude of
the heat ﬂux during the steady spread stage is approximately one tenth of the external
heat ﬂux. However, the mass ﬂux shows a reverse trend with an increase of magnitude
by a factor of ten because the low mass ﬂux of the initial stage results from the thin
pyrolysis layer. Steady spread is attained when the mass ﬂux attains the highest value

and heat ﬂux attains the lowest value among the three stages.

As far as the interface temperature is concerned, a leading edge appears after
ignition, whose magnitude slightly increases to attain the steady state. Downstream of
the leading edge during the steady spread stage, the surface temperature decreases with
increase of distance from the leading edge. This observation disagrees with the
assumption of a constant surface vaporization temperature discussed at length in [1]. The
invalidity of the assumption was discussed in numerical studies (that did not consider the
effects of phase change) such as [38, 32]. The divergence of the temperature
downstream of ﬂame front from an assumed constant value at the leading edge is as high
as 20% for the case St = 2. Apparently such a deviation is not crucial, given the
eventual agreement between numerical results and theory. For this reason, the constant
“vaporization temperature” hypothesis has survived and, in fact, represents an important

conceptual piece of the overall ﬂame spread model.

3.3 Flame Structure

61

lhe detarled
contours of cor.

combustion re‘c’

 

termerature from]
F“ , l
.m the next;
tanztatitely the

0p, ased ﬂow t':‘

 

etzdtzer gas tn fr

.1
5:10

*5 the nature
rate

me Iﬁgton

‘mpﬁl’aturc 3 av-

The detailed ﬂame structure is obtained from the numerical model by examining
contours of constant temperature, fuel concentration, oxidizer concentration, and
combustion reaction rate in four cases of interest (see Figure 6). The non-dimensional
temperature from the numerical model is divided by St in order to make comparisons.
From the viewpoint of physics, the ﬂame structure displayed in Figure 6 shows
qualitatively the same characteristic as reviewed in [1]. Under the inﬂuence of an
opposed ﬂow, the diffused fuel gas from the interface (Figure 6(c)) reacts with the
oxidizer gas in the mainstream (Figure 6(d)), thereby forming a reaction region (Figure
6(b)). From Figure 6(b) we see that the thickness of the reactive region is ﬁnite, which
shows the nature of ﬁnite-rate reaction in the gas. In addition, the temperature of this
reactive region is the highest in the ﬁeld, as shown in Figure 6(a). The highest
temperatures are displaced from the surface somewhat downstream of the reactivity
maximum, as observed in previous studies of ﬂame near cold surfaces [53, 54, 31].

It is interesting to evaluate the inﬂuence of the condensed phase on the ﬂame

structure. A reference state of St=2 is chosen for these comparisons. First, it is

observed from Figure 6(a) that St = 100 and 1:, = 3 produce bigger ﬂames. However,

5 p1 = 0.125 produces a smaller ﬂame. This observation is conﬁrmed by comparing
ﬁrel concentration constant contours in Figure 6(c). The same constant contour of fuel
concentration is pushed further downstream of the ﬂame leading edge if St = 100 or

k, = 3. The second observation is from Figure 6(b). Near the ﬂame leading edge, the

shape and orientation of the reactive region does not change for the four cases we

examined; however, farther downstream of the ﬂame leading edge, the reactive region

62

tends. to be slight

becomes smallct

3.4 The Condersl

The temper.

.sanerms are pa:

 

 

 

 

numerical solutit
pa’abola except
domtream of t

trend ameemen

.311
\

36 0f the pha:

'hel

eading eds.

tends to be slightly raised if St or 1:, becomes larger, or slightly pressed down if 5P!

becomes smaller.

3.4 The Condensed Phase

The temperature proﬁle (including location of the phase front) from both the
numerical model and theory are compared in Figure 7. Theory predicts that the
isotherms are parabolas with the origins located at the leading edge ‘ o ’ in Figure 7. The
numerical solution, however, predicts that the isotherms do not resemble the shape of
parabola except near the ﬂame leading edge. The isotherms deviate from parabolas
downstream of the leading edge. The thickness of the liquid layer shrinks downstream.
Good agreement between the numerical result and theory is observed near the leading
edge of the phase change point ‘ o ’. The behavior of the liquid thickness downstream of
the leading edge suggests a connection to the surface temperature or heat ﬂux at ignition,
therefore some other parameters of interest are investigated.

The inﬂuence of the condensed phase on ﬂame spread during the steady spread
stage is shown in Figure 8 by investigation of interface temperature, mass ﬂux,
condensed phase heat conduction and the phase front locations. As is shown by the

interface temperature in Figure 8(a), there is negligible difference downstream of the
ﬂame front among four cases except If, = 3. In addition, upstream of the ﬂame front,

St = 100 produces a shallower gradient of the interface temperature than St = 2 , and
C p] = 0.125 makes this gradient even lower. Distributions of the interface mass ﬂux

and the net heat ﬂux into the condensed phase along the streamwise distance are

presented in Figures 8(c) and 8(d). Among the four cases, If, = 3 stands out having the

63

 

highest mass ‘1»

phenomenon 15

 

mechanism in t’:‘
corresponds to a
extent of the rr.
significantly enh;
the other cases a

A simple ar.

l’ .. '
rind layer in th

 

Which is suhiec

suhjected to the

tendensed phase

highest mass ﬂux and lowest heat conduction into the condensed phase. This
phenomenon is well understood if we recall that we used the in-depth pyrolysis
mechanism in the condensed phase. As a result, the larger pyrolysis area in general

corresponds to a higher mass ﬂow rate. The extent of the pyrolysis area is related to the
extent of the melting region, as shown in Figure 8(b). It is observed that [:1 =3

signiﬁcantly enlarges the liquid region, thus resulting a higher pyrolysis region, while in

the other cases a smaller difference is observed.
A simple analysis illustrates that 1?, controls the thickness of the liquid phase. The

liquid layer in the condensed phase can be looked on as a plane plate, the upper side of

which is subjected to the ignition temperature Tj-g, and the lower side of which is
subjected to the melting temperature T m. If (ﬁg denotes the heat conduction into the

condensed phase, then a simple heat conduction relation applies if the transient effect is

neglected,

6=k1-'eg:—-'1. (18)
qig

where 5 denotes the average thickness of the liquid layer. Since the magnitudes of 67g
and jig do not depend signiﬁcantly on the magnitude of 1:1 , an approximate
relationship of 5 0c kl is determined. This explains why the liquid layer thickness of

If, = 3 is almost three times as large as 1;] =1.

The streamwise evolution of the phase front location and pyrolysis front location

(characterized by WC =1.5X10—7) is shown in Figure 9. The liquid region

64

 

 

encompasses alrr.

 

slzgrit intrusion .
puclysrs temper.
the p}TOl_\SIS re:

front and tits We

at *he punlysts r.

2.5 Mechanism -

 

Generallx. 'a

conclusions on

 

dependence of t".

encompasses almost the entire pyrolysis region during all three stages. In addition, the
slight intrusion of the pyrolysis front out of the phase boundary indicates that the
pyrolysis temperature is slightly lower than the melting temperature. Also the shape of
the pyrolysis region is not a parabola. Its thickness shrinks downstream of the ﬂame
front and ﬁts well with the shape of phase front. In addition, the reaction intensive part

of the pyrolysis region has the highest temperature, and lies below the ﬂame front.

3.5 Mechanism of Steady Flame Spread
Generally, many different mechanisms contribute to the ﬂame spread process, and

conclusions on which mechanism is dominant (if any) are difﬁcult to make. The
dependence of ﬂame spread on St, 1;] , and 5p] is analyzed below. Control volumes of
5.7mmx4.5mm upstream of the ﬂame leading edge are chosen for cases St = 100,

[:1 = 3 , 5191 = 0.125 as well as the reference state in order to construct a local energy-

balance accounting. Different heat transfer mechanisms in non-dimensional from are
evaluated by numerical integration, see Table 3.

An energy balance can be constructed for this control volume. The heat from the
gas phase upstream of the ﬂame leading edge plus the streamwise conduction in the
condensed phase are responsible for the pyrolysis process and enthalpy rise of the
control volume. Pyrolysis and streamwise conduction in the condensed phase are not
important compared with the upstream conduction from the gas phase [34]. Physically,
the total heat obtained in Table 3 is responsible for enthalpy rise of the control volume,
and should be connected to the spread rate. Comparison of St = 2 and St = 100 in
Table 3 indicates that with the increase of St, the total heat increases, resulting in a

higher spread rate. However, for cases of variable thermal conductivity and variable heat

65

capacrt}: the m
the spread rate.
the spread rate
The ratio
energy barrier
for the four ca:
ot'preheatrrtg ti
tthe temperatu

_ttastifiable ded

capacity, the magnitudes of the total heat do not necessarily measure the magnitudes of
the spread rate. This phenomenon can be explained by a ratio between the total heat and
the spread rate.

The ratio of the total heat to the spread rate, presented in Table 3, denotes the
energy barrier for ﬂame propagation with a spread rate of unity. We compare this ratio
for the four cases we examined, since it reﬂects physically the ratio of relative difﬁculty
of preheating the condensed phase to the ignition temperature. This ignition temperature
(the temperature at the ﬂame leading edge), should be identical for different cases if

justiﬁable deductions are made. Observation of Figure 8(a) indicates that St = 2,
St=100, 6101:0125 have identical ﬂame temperatures at the leading edge,

therefore comparisons between these three cases are made below. From Table 3 the ratio

for St = 100 has a lower value than for St = 2, indicating a lower energy barrier when
the latent heat is decreased. Similarly, the ratio for E p, = 0.125 is lower than for the
reference state, because the lower thermal capacity is consistent with the lower energy
barrier. A different interpretation for the last case 1;] = 3 is needed because it shows
many different characteristics from the other cases. From Figure 8(a), it is observed that

1?, = 3 results in a lower ﬂame temperature at the leading edge. Second, from Figure

8(b), [:1 = 3 results in a larger liquid thickness. The ﬁrst inﬂuence tends to lower the

ratio in Table 3 because a lower ignition temperature is required. The second inﬂuence,

however, tends to increase the ratio because a thicker liquid layer requires more energy.
The overall inﬂuence of it} = 3 seems to be controlled by the second inﬂuence, as is

supported by Table 3: the ratio of the total heat to the spread rate is larger than for the

66

-

reference state.

4. Conclusions
A new num

tn the condense

toliqurd phase

processes in th-

constructed b)
radiant emissit
The inllu:
flame spread
interest. SI.
numerical mo
It is fotmd th
Variable Sr

thread {mm

result and Eq'

Sklme Oh

\ "
440.33
‘3 0}

Spread In ad

[a

(

'satton delas

Cert»
..‘d <
11‘: 11mm}

reference state.
4. Conclusions

A new numerical model of ﬂame spread is constructed by introducing phase change
in the condensed phase. The processes considered in the condensed phase include solid-
to-liquid phase change, an in-depth pyrolysis reaction and heat conduction. The
processes in the gas phase, after applying the Oseen-ﬂow approximation, include heat
transfer, fuel and oxidizer transfer, and ﬁnite-rate combustion kinetics. At the interface
between the gas phase and the condensed phase, the heat and mass balance is
constructed by incorporating heat conduction into both gas and condensed phases,
radiant emission from the surface, and diffusion of pyrolysis products into the gas.

The inﬂuence of phase change and thermal properties of the condensed phase on

ﬂame spread are investigated by introducing three non-dimensional parameters of
interest, St, 1;], and 5P1- Quantitative comparisons of spread rate between the

numerical model and theory outlined in Section 2.3, 2.2.1, and Appendix II are obtained.

It is found that the numerical model provides almost exact correspondence to [28] for

variable St and E p], and 90% agreement for cases of variable [if—1.DeRiS ‘s ﬂame

spread formula, which results in a constant spread rate value higher than the numerical
result and Equation (13), is independent of phase change.

Some observations are made about the transient process of ﬂame spread. (1) Three
stages are observed: ignition, transition, and fully developed (or steady state) ﬂame

spread. In addition, the interval of transition is found to be very short compared to the
ignition delay time. (2) Extinction is observed if St, 1;] go below, or 6P1 goes beyond,

certain limits. Quantitative values of these limits are not precisely known. (3) The rates

67

of spread of the
The depenc
increase of Sr

results are cons
ate on the thre

me@3115 a 10“ cr

lIi - 0n the Othe"
133-alt; Ill 3 10v
3 the interface

The mech
fCtllnd [hill 3 I;
the ﬂame lea d
the Ptehearmg
11115 b31515 IS C

\

thscusssd 0n

1
Q

qtpe‘urkyz

of spread of the phase and ﬂame fronts differ until the steady spread stage is attained.

The dependence of ﬂame structure on St, 1:], and 5p, were studied. With the

increase of St or 1:1 , or with the decrease of 5 P1, the ﬂame size increased. These

results are consistent with the qualitative nature of the dependence of the ﬂame spread
rate on the three parameters. Physically, a lower latent heat or a lower thermal capacity

means a lower energy barrier for the ﬂame, hence a larger spread rate. The increase of
1;] , on the other hand, denoting the diffusion away of the thermal energy for preheating,

results in a lower spread rate. The study of heat ﬂux, mass ﬂux and surface temperature
at the interface provides additional information on the mechanisms of ﬂame spread.

The mechanisms of ﬂame spread are interpreted by energy balance analysis. It is
found that a ratio between the total heat applied to the condensed material upstream of
the ﬂame leading edge and the spread rate reveals the physical mechanisms that control
the preheating of the condensed material to the ignition temperature. A comparison on
this basis is consistent with the notion of a “fundamental equation of ﬂame spread” as

discussed originally in [34] and in Section 3.l.5 of [1]. This equation is written as

q = psquh, where q is the energy ﬂux transported across the ﬂame front to
upstream unburned fuel, p5 is the ﬁrel density, and Ah is the fuel thermal enthalpy
difference between ignition and ambient temperature. Clearly, the quantity
q/ u f = psAh was evaluated in our work, and this quantity has the direct physical
relevance to the propensity for ﬂame spread. Comparisons of this ratio in situations of
varying St, If] , and 5p, reveals the difference in physical mechanisms that control the

preheating of the condensed phase to the ignition temperature. Although the theoretical

68

result (Equation
there predictio
numerical resul
principle [~19].
examined in ter
globally dent ed
Extension 0.
CtllllCilOIl limit t
addition. detailet
needed. More at
:tase change pt
lﬁtmpted by int

1.
“ti-Eff horizonta

 

 

result (Equation (13)) on which much of the numerical comparisons were based and
where predictions were, in same cases, remarkably accurate when compared with
numerical results, if was derived from the application of the global energy balance
principle [49], The general result it produces (Equation (13)) can subsequently be
examined in terms of local energy balances. Thus, there is no contradiction between a
globally derived formula and its subsequent local interpretation.

Extension of the current work is needed. The transient ﬂame behaviors, such as
extinction limit or ignition limit with phase change, leave room for future research. In
addition, detailed analysis of the mechanisms of ﬂame spread over a melting polymer is
needed. More attention may be given to combination of the gas phase inﬂuence with
phase change processes in the condensed material. Further modeling efforts may be
attempted by incorporating the melt ﬂow phenomena when the burning material is no

longer horizontal.

69

 

 

\ .1.
: 13.1,».

  

 

 

 

 

 

 

 

 

 

 

 

  
 
 
  
  
  

uOO
_, ////////////////////////////////////////////////// //////// //
_, \
__, Upper wall
::
g °° Outﬂow
I \.
ln ow Y0 -— 00° Gas phase Zone ofinrtial heating [gy
A — i
_’ y Yf _ 0 __
—> ’x” “x Flame
—> ,’ \
x Gas-condensed ,’ ‘— ‘t, /
—) l
I
’ ///////// /////////////////‘(
é
ﬂ adiabatic .
9 Solid phase
/ Phase front
/ TS : Too /
g T = T = T /
//////////////////////// ////////////////////////////////////A
[X
Computational Domain //
,x’ ,_) Fuel ‘~\
// —> Oxidizer transfer \\\
/ Gas phase
l/ —’ Heat transfer \
I \
I \
\
’1' Flame spread rate Flame of ﬁnite rate \\
1’ <—
I,
I
I
conduction

Surface

 

—— ~-~‘

\‘ \ Velocity of moving
‘ Phase front

Figure 1 Schematic description of diffusion ﬂame spread over polymers in an opposed-ﬂow

70

thtrreQT

 

_ Solid phase + Liquid phase __

To, ; ; T
Tm

 

 

Figure 2 The enthalpy-temperature relationship used in the numerical model.

 

Gas

 

Flame

 

 

 

 

/// /////7p53u31,,1‘,

 

 

Solid yc

 

 

Figure 3 The simpliﬁed ﬂame-spread model for steady ﬂame spread.

71

AM
AV

> 1 > b l I 11 u
.3 .3 C. 2
.J . e .4 n;
A..... A.a

Aritzz :3: 22:31.... Opts;

h-

g . .5 .- a . a
:4 1. .3 2
«J C «4 C
n y «1...

«3:5; :2: 3.?»ch ..::5&

PK»

 

 

 

 

 

 

 

 

 

q 1 q n q a 4 r
— A
c
u l .
n w 4
CW 4
.. L
n m Hm.
he N Rm.
..
o
a. .* u
o a
a n
— .
o .
.. a m
w ..
on A
o
o
o
v
o.
a...
....e
/
OI
I; A
III
I!
...... 1.
III. A
4
A
J
P - h u p b
4. 5 3. 5 2. 5 J
0 3. 0 2 0 1. 0
O O 0

$8.5 28 39% «Ear.

102

101

10°

10"

St
(4-a)

 

 

 

 

 

 

 

 

 

A q q d d a
m
.m
.l r .. 1
m
N 0.1.0...
.i u
m ...
m ...
.i. “ ...
1 is“... L
I *c... r
r *n. l
..‘UCI‘
1 mi. 1
x...
h n p k. h b h
5 A 5 3. 5 2. 5 A
4. 0 3. 0 2. 0 1. O
0 O 0 0

58.5 22 39% 9:9“.

2.6 2.8

2.4

2.2

1.4 1.6 1.8

1.2

kf/ks
(4-b)

72

 

 

 

AJ
1:25; 5:! 35.5% :.:...I

l\ “A ~ ( to». a u I b n a u h s. u p .-
... l P l b 1 h | l b ill 0 I . “U NU “NV A H
5 .J .3 2 5 e _ F .S «I q q. .
3 e1. 2 .U 1 C .1 \ . 1. . card 5 e E :53... it:
- a; mu 3 e Ntrf ~ . x ,. . . . 51‘
1 1t
14. mm
6
1%

.rr~

F

‘A';
I. _

 

 

 

 

 

 

 

 

 

(14‘ -
* ............
E ..¥AT
E b .- d
g o I 35 * u.."*..-uno -
2 ~u¥uouuuuun*
a ...........
a: n ........... d
U 0 3 + ..... $ ...........
a ..............
e ...... .*
180 251 4+ 41 Numerica' d
as) . — [21 1
.9 ................ [28]
U.
(12’ J
0.15b 1
o . 1 l 4 l 1 r 1
0 (r2 (r4 (r6 (L8 1 1.2
Cchps
(4-C)

Figure 4 (a) Flame spread rate vs. St; (b) Flame spread rate vs. 1:, at St = 2; (c) Flame
spread rate vs. C: p, at St = 2.

 

 

 

 

 

 

 

103E 1 e . . e 1
' ignition
1025 i a
m : 1
1!. ' 3
(D 1 1
a 1 1
8 1 . .
.5 10 E g
o . a
'5 .
g 1 h
g 10° r 1
m E 1
Q , <
E r 1
_‘E e
m -1
10 a
E i
l- 4
1 steady spread xA transition xB .
10‘2 l J 1 I r
5 6 7 8 9 10 1 1 12
x(mm)
(5-a)

73

 

I. l

Cw..=... C. 3:... :..>::.. ...:C

. . re.
0 ﬁle Aid
3.: ~P~ X3: ~13: x: 7,742:

«1%

12

 

 

 

 

 

 

 

 

 

 

 

d d 1 a d 4 d 1
m u
m m cm
n ”
.nls «
o
..
n
.2
0
.... 1
1 9
t
"t
1m
curve:
6
S
Mm
on
1 17
u
.
d n
a u
e .
W
1 S m 16
.W. 1H1
a m.
m a
S or...
5 n h h p P b F 5
G 8 6 4 2 0 8 6 4 2 0
2 1.. 4| 1 4| 4|

aim ammo c_ 9:: _mztm E0:

x(mm)
(5-b)

 

 

 

 

=105

t

 

- 4 ' heat flux
--- mass ﬂux

t=193

 

 

 

0.4

0.35 ’

n
3
0

.25 '

n
2 5
0. 1.
0 0

9: E x3: «no; .0 mme

p
1
0

0.05 '

x(mm)

(5-0)

74

H‘rlnrfmte

(;)

«n
r

K)

(')

(A1

(I!
r

h.)

(h

 

 

 

 

415 1 1

 

 

 

 

 

t=193
t=12s
4- .
115* a
3" 1
o
c125“ a
E
.2
E 2 r- u
1.5- d
1? t=105
0.51 .
O L
0 5 10 15
x(mm)
(5-d)

Figure 5 (a) Flame spread rate vs. streamwise distance at St = 2 (the front propagates to
the left); 0)) Arrival times of ﬂame front and phase front vs. streamwise distance; (c) The
streamwise evolution of heat and mass ﬂux at the interface before ignition (105), during
transition (125) and during steady spread (198); (d) The streamwise evolution of interface
temperature before ignition (103), during transition (125) and during steady spread ( 195).

75

y(mm)

y(mm)

 

 

 
 
 
 
 

 

 

 

 

 

 

 

3.5 - 1
3 - - reference state
— St=100
-.- k k =3.0 ,.
2'5 ’ Ct, st: = o 125
d e. -
2 .
1.5 - °"
1 -
0.5 e
o r n " .
0 2 4 6 8 10 12 14
x (mm)
(6-a)

2 I Ti j l j l I
1.8 -
1.6
1.4 - reference state

- St=100
1.2 -.- k{ks=3.0
Cnlcps= 0.125

1
0.8
0.6
0.4
0.2

o r r r z“,

2 4 6

 

y(mm)

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

 

 

I I l I T r
d
.4
- reference state 4
-- St=100
-.- kl/ks=3'0 d
a." '''' Us
0.1/cps: 0.125 ,. *' "
Cr. 1
f' ‘ ,

,,,,,, .1 ...-:33" :W’W.‘~\. -
.0“: ‘0‘:
I: --------- ‘ d

o .-“" --.---I. ,

'> a: .......... '79 .
1' "'51/ ..... K .'
51"." .......................... . ‘\ 4
."I’ ‘1 4;. ...................... . 0“

r O" i" ~.. ~ ‘5.
10 ’."—':':f‘..- ............... *3 “It:
a -..‘:;'.‘I ‘\
1;}. '0”; ....................... +

1 .-‘ 1 ' n

 

 

77

 

(D

4.5;.
4 .
3 5-
3-
i 2 S»
2 ..
1 5.
1 .
C 5 .
C' x
C'
tLuCCmI‘aUOn an
. - q
1 ... e
ﬂkxement 2 0 bet
I‘ _- ‘
\HI ”Wren
“fattenz

 

 

 

 

 

 

 

4.5 ~ ~
4 — :5-
3'5 ’ - reference state ‘
— St=100 ,4
A 3 r k/ks=3.0 '
E -
E 2.5 _ CdCB- 0.125 ....s«
>‘ “rd-9"" '
2% ./“.-ﬁ
1 .5 L ................. ‘
1 - a
0.5 __ _,. V?" q
0 1 1 i I - 1 4 L
0 2 4 6 10 12 14
x (mm)
(6-d)

Figure 6 Comparisons of non-dimensional (a) temperature/ St , (b) reaction rate, (c) fuel
concentration and (d) oxidizer concentration for four conditions of the condensed
material. The reference state denotes St = 2. The outermost isotherm is 2.0 with
increment 2.0 between two adjoining contours; the innermost isoline of oxidizer or fuel is
0.2 with increment 0.2 between two adjoining contours; the reaction profile is

characterized by reaction rate 0.0001.

78

 

 

4] 5'»
1’r
:-1 5'11
3 1
-2;
.1 \
4 51
l
‘.
~35?
~4\
-6
F1.» 4
in??? I Compans
“U “S and Phase .
:Omtides with the
10me
C .,
53319. 611 {“0 at

 

 

 

 

 

0'5 _ gas phase
0 the interface
-0.5 ~
-1 _ .
E ,1 5 _ - the analytical solid-liquid ------------
3 ' -- the numerical solid-liquid
-.- the numerical
'2 h the analytical
-2.5 1-
-3 h condensed phase
-3.5 ~
.4 I L L L 1 1 l
-6 -4 -2 0 2 4 6

x(mm)

Figure 7 Comparison of numerical model and theory (Appendix H) for the temperature
proﬁles and phase fronts in the condensed phase at St = 2. The location of phase change

coincides with the constant proﬁle T = St. The lowest isotherm is 1.0 with increment

1.0 between two adjoining contours. The analytical isotherm of T = 4 coincides with the
straight line of the gas-condensed interface.

79

~l

32::

 

....l. J a... 74—:—

Interface temperature (K)

ﬁmm)

 

2.5 I T T I l I j

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 r ~- ‘
1.5 - 7
reference state
1_ ......... Sbﬂm ‘
..-.-.-...-. kljks=3,0
.................. CdCPs= 0.125
0.5 r 7
o 1 l l
5 55 6 65 7 75 3 85
x(mm)
(8-a)
0 v ' '
-0.5 -
-1 _ d
-1.5 - 4
-2 . ‘
.._.. reference state
-25 _ --------- St=100 ~
..... .....-. k|jks=3,0
‘3 _ .................. Cdcps= 0.125 -
-3.5 - d
.4 - .
4.5 1 ‘ 1 1 1 1 1 ‘
o 2 4 6 3 1° 12 14
x (mm)
(8-b)

80

a.
5
l
u

7. .514:x2:$._._:.:__.Vt.::_

 

x 10'3

 

 

‘

 

reference state

 

 

 

.I.
I.‘
. v.
|.|
I.‘
-.t.
.|,

x(mm)
(8-C)

......A.
i iiiiiiii ...r....::.
i.

 

 

 

3%:va x3: mmmE 02.58:—

81

Wan")
l‘irc heat ﬂux (
lntcr .

Figure 8 C
m“ alid id

Umpd

 

0.5 T T I Y I T I

 

 

   
   

 

 

 

 

 

 

 

0.45 ~ 1. ‘
0.4; i
A - —— reference state
”E 0.35 " “““““ St=100 ..
E - ........... k/ks=3.0
g 0.3 _ Ora/CR5: 0.125
1::
1; 0.25 *-
0
.::
§ 0.2 r
‘t:
0
g 0.15 ~
0.1 - """
0.05 r 7
o 1 ‘ I l I
5 5.5 6 7 5 8 8'5 9

 

(8-d)
Figure 8 Comparisons of (a) interface temperature, (b) phase location, (c) interface mass
ﬂux and ((1) interface heat ﬂux in four situations of the condensed phase.

82

Y (111111)
b ,
m
T

 

 

figure 9 The e
ignition l 105) (i'

15 characterized "

 

 

 

 

 

F5
N
I

................

.......

-0.6 - -

-0.7 - - pyrolysis front -
liquid-solid interface

9'3
m
I
1

9'3
(D
I
L

 

 

 

x(mm)

Figure 9 The evolution of solid-liquid interface and pyrolysis front locations before
ignition (103), during transition (123) and during steady spread (19s). The pyrolysis front

is characterized by WC = 1.5 X10_7 in the numerical model.

83

Tmﬂel.‘
1:005
W (in-Kl
Tm = 500

K
K ==0.

r’

 

 

 

'JI

\
0:3.51‘4x10‘8
m‘ s

"x=0.1

L...23___J

 

 

 

Table 1. Major properties and kinetic data used for the numerical model.

 

 

 

 

 

 

 

ks =0.05 CPS =1460 ps=1190 p,=1190
W/(m-K) J/ngK) kg/m3 kg/m3
Tm=500 Ac=2.82X109 Ec=129580 qc =1113.5
K s" J/mol kJ/kg
Ke=0.5 kg=0.0411 Cpg=1007 pg=1.16
W/(m-Ig J/(kgK) kg/m3
D=3.514x10'5 Ag =1.6x1015 Eg = 155000 qg =-15539-4
mz/S m3/(kg's) J /mol kJ/kg
u... = 0.1 To. = 300 Y0.” = 0.31 ono = 0.0
m/s K

 

 

84

 

T51"

i Sam-501
*—

it.

 

Table 2. Dimensionless parameters for the numerical model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Symbol Parametemoup Symbol Parameter group
X AgpgaSYom Z Acas
g “2 c “2
6—1 quoooCPS 5 chPs
g auoCPgLs c CPS L3
Fg E g CPS EC EC CPS
RL5 RL3
a C T — Too
Le i S t P S ( m )
D Ls
.. C Pl ._ p I
C __
P1 CPS p’ ps
_. pc — k,
p. — k _-
pg ’ ks
17 t vn ‘ 23y u°° g 5y
n u“, as
_q_. qigas CPS a). 80051.?
lg uookSLs ksuongs

 

 

 

 

 

Table 3. Dichrcr
xmmﬂhmﬁ:

 

 

 

. (Heat conductxl
— radiation .1.
x
blléimm'lsc \-
I
conder
Heat of p}

conder
\

1 The I.
N

Spread re.
\

The total he
\‘

 

Table 3. Different heat transfer mechanisms in four cases of interest, in which St = 2

serves as the reference state.

 

 

 

 

 

 

 

 

 

 

St=2 St=100 19:3 Cplzojz
(Heat conduction from gas phase 2.497 2.828 1.406 2.231
— radiation at the interface)
Streamwise conduction in the 3.229 4.279 11.042 1.965
condensed phase
Heat of pyrolysis in the 0.0000 0.0000 0.0000 0.0000
condensed phase
The total heat 5.727 7.107 12.448 4.196
Spread rate (10‘ m/s) 2.926 3.677 1.316 3.852
The total heat / Spread rate 1.957 1.932 9.458 1.089

 

 

 

 

86

MODELING (.5

S

1. Introduction

Flame spre.
combustion cor.
In much greate'
complete mode
energy. specrex.
fem. Sex‘eral 11
phase was Sini;
boundary contl
e{lilatiom see 11
that COuld acco

CUnlined to m

desuable t0 d2
realistic Phenu

Spread for‘mul.

numeric“ re'x

{ﬁrms of mm]

1

 

CHAPTER 4
MODELING OF FLAME SPREAD OVER MELTING POLYMERS WITH NAVIER

STOKES FLOW CALCULATION IN THE GAS PHASE

1. Introduction

Flame spread over polymers has received extensive attention by researchers in the
combustion community. In previous research efforts, the gas phase has been investigated
in much greater detail compared to the condensed phase. In the gas phase, the most
complete model to date consists of the two-dimensional Navier Stokes momentum,
energy, species, and continuity equations together with chemical kinetics in simplified
form. Several models (see review [4]) have employed such formulations. The condensed
phase was simpliﬁed in such a manner that early investigations normally used it as a
boundary condition. Later it was described by a two-dimensional heat conduction
equation; see the review of [1]. Although some other more advanced models appeared
that could account for the in-depth reaction, material property change, etc., they are still
conﬁned to single-phase heat conduction. Normally they do not consider melting and
complex phenomena such as bubble formation and sub-surface liquid ﬂow. It is
desirable to develop a condensed phase sub-model in order to essentially describe such
realistic phenomena. As one step forward toward this goal, we propose a ﬂame spread
model that incorporates condensed phase melting phenomena [55]. In addition, a ﬂame
spread formula was examined for melting polymers and very good agreement with the
numerical results was obtained. Various new findings on the inﬂuence of melting in

terms of ﬂame behavior were provided in [55].

87

  
 
  
  

The numerit~
gas ﬂow across t3
Situations. The (l
calculation by 51
treatment. thou :

may Interact mti

 

 

 

behavior by tour
ts not clear «he;
accurarely Caleu
goal Of this Ch
IncorI’Orate (he
CXlenstn 0f 11‘
numﬁncal mOd
PhEDOmena. F)

Suhlétt OfIhis ‘

The numerical model in [55], however, employed the Oseen approximation for the
gas ﬂow across the condensed ﬁeld sample, which makes it undesirable for realistic ﬂow
situations. The Oseen approximation was proposed originally to eliminate the ﬂow ﬁeld
calculation by simply applying an external velocity proﬁle of much simpler form. This
treatment, though convenient for calculation purposes, is very coarse since the gas phase
may interact with the condensed phase through the boundary layer, and inﬂuence ﬂame
behavior by coupling the temperature and species ﬁelds. For ﬂame spread with melting, it
is not clear whether previous results will still be accurate when the ﬂow ﬁeld is more
accurately calculated. By acknowledging the drawback of the previous ﬂame model, the
goal of this chapter is to extend it to a realistic ﬂow situation. The gas phase will
incorporate the Navier Stokes momentum and continuity equations as well as the related
extension of the interface conditions with ﬂow complications. Therefore the complete
numerical model will predict the interaction of the ﬂow characteristics and the melting
phenomena. Flame behavior subject to complexities of melting and ﬂow will be the
subject of this chapter.

2. The Mathematical Model

A ﬂame spread model including both gas phase ﬂow and condensed phase melting is
established in order to describe the ﬂame behavior in a more realistic manner. This
requires a set of conservation equations for continuity, momentum, energy and species in
differential form. An equation of state is also needed to link the pressure and the density
ﬁelds. A number of assumptions (A) and restrictions (R) are used to in order to further
reduce the complexities and make the formulation more tractable. A distinction is made

between the “restriction” (R) [deﬁned as clear and unambiguous limitation of the

88

nahsis. remou:
statements cone.
phyneal quantit-
loss to the enx'ir.
because of the i
arr-lied to the r
heat. constant '
important (A); 1
him later for:
Ofthe surface 1;.
to highly VISCOL;
m “Tillen in a:‘

Th6 ﬂame
the “muffled (

Initial Condlllnr

 

Gas Phase;

(Onltnum..

 

MUmenIUm 6L.

dt

KIOm€mUm 0‘

 

 

analysis, removable by a more detailed analysis] and “assumptions” (A) [deﬁned as
statements concerning the functional dependence and order of magnitude of various
physical quantities]. They are: (1) Radiation heat transfer only includes surface radiation
loss to the environment (R); (2) Viscous dissipation and compressive work are neglected
because of the low velocity of induced or external ﬂow (R); (3) The ideal gas law is
applied to the pressure-density relationship of the gas phase (R); (4) Constant speciﬁc
heat, constant binary diffusion coefﬁcients (R); (5) Soret and Dufour effects not
important (A); (6) One step global reaction mechanism in the gas phase (R); (7) The
liquid layer formed by heating of the solid does not “ﬂow” in the sense that the gradients
0f the surface tension do not produce circulating motion. This is a limitation of our work
to highly viscous melt layers that has been acknowledged previously [55]. The equations
are written in an unsteady form.

The ﬂame survives ignition and gradually attains a steady state while approaching
the unburned end of polymer material. The complete system of equations along with
illitial conditions and boundary conditions is:

Gas Phase:
Continuity:

- apg + apgu + apgv =

0 l
at 3x 8y ( )

 

 

Momentum equation in x direction:

 

2
apgu+a(,0gu )+a(pguv)=_§g+jﬂa_zg+#§_zz+l 2.21. (2)
at ax 3y 33‘ 3 3x2 3y2 3 axay

Momentum equation in y direction:

89

 

Bpgv
81

Energy equation

angPL.
61

5mm equatior

Bpg); 81).,
.\+~\‘:

dt 0

“here try ._
g - _

Condemed Phi:

C \ ‘

xyud‘u‘iul‘d In

a -
5 Condmw

 

2
e)pgv+a(pgw)+a(pgv):_§,3+ 93¢ a_2v azu

l
+ ._
at 8x By By # 8x2 3 ,u ayz 3 # Bxay

 

 

 

(3)

Energy equation:

a C T a C T a C T 2 2
pg Pg + (pg Pgu )+ (pg ng )):k (ﬂ+_a_]:)+wq (4)
at 8x 3y g 8x2 Byz g g

 

 

 

Species equations of fuel and oxidizer:

 

 

 

8p Y- ap W 8(1) vY-) a By. a ar-
g l g l g l l l -
+ + :D— ——+-—- -— +W:l' i='0'5
at 8x 8y [ax(pg 8x) By(pg 8y )1 g I l' ( )
2 —E /RT _ M -v- .
where wg =—ng0Ynge g and :sl- = MflVlf , l= f,0.
Condensed Phase
Energy equations:
pSCps 95T— = kSVzT — chc, solid
art ‘6’
iPlCPIE=k1V2T_chca lzquzd
where WC = —pcAce_Ec /RT.
Solid-Liquid Interface
8T 8T
“(k—7).: +pstn. =‘(" a I
an an , phase front (7)
T2 : Tm

Gas-Condensed Interface

90

where In = K.

Emnon ofStu.

The initial

In addition. at

aP‘l‘lied. Initial.

from F9mn1

Outitned belou

ﬂow channel

Compilation 1.

Q

N

ﬂow ﬁeld i

lethniques [4i

KL"Choff four.-

term Ireantler;

C0711}

 

 

i an ,
—PgD—l0+ :(l—Yfl0+)m
63y
Y .
_ :0ng 0+ = (0‘ Yol0+)m
< (8)
3T 3T 4 4 .
_ks glo— = ‘kg ‘5y—l0+ +800 — Tod—611,1;
m = pgv
l u =0
where 151 = Ke ft wcdy.
sy
Equation of State:
p =pgRuT (9)
The initial condition:
T=T°°,Yf=0andY0=Y0°°, (10)

In addition, at the inlet of the channel, a condition of uniform ﬂow velocity proﬁle is
applied. Initially an external radiation heat ﬂux of (SW/m2) is imposed over at the region
from x=9mm to x=12mm. The problem is solved by using the numerical methods
outlined below. The computational domain is composed of a cartesian-based gas phase
ﬂow channel and a condensed phase domain; see Figure l in [55]. The overall
computation is an iterative alternate direction implicit (ADI) procedure. The gas phase
ﬂow ﬁeld is calculated by using a SIMPLEC-based scheme and under-relaxation
techniques [46]. The melting phenomenon is solved by the source update method with
Kirchoff formulation for discontinuous thermal conductivity [48]. The combustion source
term treatment is implemented by special techniques [47]. Detailed numerical methods

for ﬂow calculation is given in Appendix IV.

91

3. Results

3.1 Physical Properties
The physical properties are listed in Table 1. Note that the magnitude of qg in this
case is signiﬁcantly larger than the Oseen model [55]; and E g is signiﬁcantly smaller.

The kinetic data are used for supporting a larger ﬂame which cannot be established by

using the kinetic data employed in the Oseen ﬂow case [55]. It is observed again that the

ﬂame behavior is sensitive to the combustion kinetics. Too large a value of q g results in

explosive behavior, which makes the computation too costly, and too small a value

renders the incipient ﬂame unable to survive after ignition. When q g is below a certain

limit ignition cannot be attained.

In addition, in order to evaluate the inﬂuence of material properties, three non-

dimensional parameters are derived, they are St , E p, and I?! [55]. Their deﬁnitions are

repeated here. St, which is deﬁned by C Ps(T m —T°°)/Ls, represents the ratio of

sensible heat and latent heat in the condensed phase. an, is deﬁned as C p, / C ps ; I?) is
deﬁned as kl / ks . The variation of the three non-dimensional parameters is attained by

changing only LS , C p] , and k1 respectively, while keep the other properties ﬁxed.
3.2 Flame Spread Rate

To compare with the analytical formula, a ﬂame temperature value T f of 1870 K is

chosen by taking the largest temperature in the gas phase from the numerical results. The

interface temperature T, of 673 K is chosen from the numerical results. The ﬂame spread

formula is presented here [28]. (See also the appendix of [55] for a concise derivation,

92

 

The ll

(1.
O #1
C.
I].

‘1.)
(a)
--I

and the text of [55] for a derivation using scaling analysis)

ﬁngcpgkg .(Tf—Jl)2.erf(c l.ﬂ.)2
“co plCPlkl Ti—Tm V2 0‘1

The ﬂame spread rate as a function of St, I?) or zip] is shown in Figures 1-3. The ﬁrst

 

observation is that the magnitude of the ﬂame spread rates are larger than those in [55]

because new chemical kinetics are employed, the ratio (T f —7;)/(7; —T00) is 3.21,

larger than the value 2.58 in the previous work, where T- : 673K and T f = 1730 K.

i
From Figures 1-3, good agreement is observed between the numerical results and the
analytical formula. The physical trends follow well the analysis given in [55]. Since the
analytical formula provides agreement with the numerical results, the structure of the

analytical formula, as well as its restrictions and assumptions upon which it is built, will

be explored later. In addition, because of the newly selected E g and qg values, and the

addition of the inﬂuence of the ﬂow ﬁeld, the range of 5p, and I?) that could support

ﬂame spread is much smaller compared to the computation with the Oseen approximation
[55]. The explanation may relate to the transient stability characteristics of a spreading
diffusion ﬂame, which is beyond the scope of this chapter.
3.3. Transient Flame Development

In order to study the transient behavior of the ﬂame, St = 2 is selected as a
representative case. Overall ﬂame development includes preheating, ignition, transition
and steady ﬂame spread. Based on the calculated ignition time of 4.76s, four different
times, 3s, SS, 105 and 155, are chosen to illustrate the history of the ﬂame development.

The ﬂame behavior is characterized by the temperature ﬁeld and the ﬂow ﬁeld (Figure

93

4), pressure ﬁeld (Figure 5), fuel and oxidizer concentrations (Figure 6), reaction rate
(Figure 7), mass ﬂux (Figure 8), heat ﬂux (Figure 9), and surface temperature (Figure
10). Other ﬁgures that show snapshots at different times are also obtained in order to
better explain some interrelated physical phenomena. These are the pressure and velocity
ﬁeld at 153 (Figure 11), temperature and velocity ﬁeld at 158 (Figure 12), reaction rate,
fuel and oxidizer at 8.58 (Figure 13), and mass and heat ﬂux at 153 (Figure 14).
Examination of these ﬁgures reveals interesting phenomena, as explained below:

(1) Upstream from the pyrolysis front, the ﬂow of ambient air accelerates toward the
burning region. As it approaches the vicinity of the leading edge of the fuel surface, the
ﬂow deﬂects slightly outward and the streamlines are raised from the heated surface, as
observed in Figure 4(b and c). This deﬂection occurs by two mechanisms: (1) the thermal
expansion of the gas phase; (2) fuel vapor generated by pyrolysis ﬂow away from the
interface. The former is due to density changes resulting from the high temperature
gradient near the ﬂame tip or leading edge. The latter is driven by buoyancy forces
produced by density stratiﬁcation near the reaction zone. The outward streamlines are
formed by the convergence of the vapor fuel ﬂow and enforced channel ﬂow. The point
of maximal deﬂection, which corresponds to the maximal streamline slope, is observed
some distance ahead of location of maximum temperature. The latter maximum occurs
near the ﬂame leading edge. This indicates that the temperature gradient, not the
temperature magnitude, determines the deﬂection of the streamlines. The deﬂection of
the streamlines becomes less signiﬁcant downstream of the ﬂame leading edge because
of the comparatively lower temperature gradient. With time, it is found that the degree of

streamline deﬂection near the heated interface increases with the development of the

94

 

with ti
fisher
from ‘
surfae
flpan
Cont:
'1 ell

P1301.

ﬂame, which is obvious since the corresponding interface temperature becomes higher
with time because of the increasing supply of heat from the ﬂame leading edge. This
further facilitates the increase of fuel mass ﬂux; thereby deﬂecting the streamlines further
from the interface. In terms of the transport mechanism, the fuel transport from the
surface to the reaction region by difﬁision and normal convection, as well as the gas
expansion due to temperature increase. Oxidizer is transported to the reaction region by
convection and diffusion parallel to the interface. The predictions presented here agree
well with the velocity pattern in previous experimental observations and numerical
predictions [29]. Figure 4 also shows the temperature pattern in the condensed phase.
Note that the thickness and parallel extent of the liquid corresponds to the temperature
over 500K. It is obvious that the thickness and extent of the liquid grow with the
development of the ﬂame. Downstream of the ﬂame front, especially the liquid layer
tellcls to extend far beyond the limit of initial heating region mainly because the
downstream ﬂame envelops a much larger area of the condensed phase compared to the
up Stream. The thickness of the liquid layer attains a maximum somewhere downstream of
the leading edge, and then decreases gradually downstream as a result of the decreased
interface heat ﬂux, see Figure 9. This difference of the downstream contours of the
is()t11€=rrns between the numerical results and steady state theory (see [1]) is an important
one and cannot be ignored when examining ﬂame spread energy inﬂuences in ﬁnite
domains.
(2) The velocity ﬁeld can be explained by means of the relative pressure ﬁeld in Figure 5
and I:igure 11 because of their functional inter-dependence. Because a channel ﬂow is

a -
pplled in the gas phase, four stagnation points are identiﬁed at the comers of the

95

chmne
pressur
the 101
\olie
produe
the 119.:
“ell-d:
'here i.
Elél’au.
from r
ltactto
mlfk‘ltt‘
Surface
llltllm!
Patten]

terms ‘

channel. Pressures at these locations normally exceed those at the interior, forming
pressure sources, as seen from Figure 5. Downstream, a pressure gradient develops along
the longitudinal direction. The pattern of pressure decreasing downstream ceases to
evolve in the vicinity of the ﬂame leading edge, where an elevated pressure region is
produced upstream of the pyrolysis front and a low-pressure region is produced behind
the ﬂame. The most obvious evidence is a pressure source appearing at the interface for a
well-developed ﬂame, as observed in Figures 5 (b, c and (1). However, before ignition
there is no evidence indicating such a “pressure source”. This mechanism of pressure
elevation was well explained in [29], which pointed out that the elevated pressure arises
from two types of triggering mechanisms. One is upstream heat diffusion from the
reaction region that produces the hot gas expansion ahead of the ﬂame. Another is the
111 j ection of fuel vapor at the surface that reduces the shear stress and the velocity near the
Surface ahead of the pyrolysis front. Although the situation here is different (for the
melting case), the explanation is still valid since melting does not change qualitatively the
Patt ems of heat diffusion and injection mass ﬂow though quantitatively they do change in
terms of energy redistribution. From the time history of the pressure ﬁeld, the elevated
p res Sure region does not appear before ignition (Figure 5(a)). It then forms and expands
du‘ring ignition (Figure 5(b)) and retains a well-developed shape afterwards (Figures 5(c)
and 5((1)). The pressure gradient in the vicinity of the heated region increases with time
and becomes a constant when the ﬂame is stabilized. The “kinks” of the constant pressure
levels, as observed in Figure 5(c) and (d), are located in the region corresponding to the
maximum temperature gradients, which is seen also in Figure 12. Since the vapor

in‘ . . . . . .
J e(“Jon effect 18 not 1mportant at the locations of “ktnks” because they are distance

96

3113

 

away from the interface, it is obvious that these” kinks” are produced by the thermal
expansion of gaseous products due to large temperature gradients. Plots of the velocity
vectors in Figure 11 and Figure 12 show that the region of elevated pressure near the
ﬂame front causes the velocity to deﬂect in two directions. The ﬁrst deﬂection is outward
away from the surface, which is easily observed. Another deﬂection is inward toward the
surface and is not easily observed (actually there is only one vector that points inward in
Figure 11 and 12). Downstream of the ﬂame front, there is a low pressure region as well
as an accelerated velocity region. The expansion of the gas results in a signiﬁcant
acceleration of the longitudinal velocity in this region. The low pressure draws the
external oxidizer into the reaction region, though it is not obvious from the ﬁgures shown
here - There is no re-circulation ﬂow observed in Figure 11, as was previously reported in
[3 1 ] - It is conjectured that the thick polymer with melting yields a much lower rate of fuel
gas production; therefore the resulting adverse pressure against the main ﬂow is too small
to Support a re-circulating cell. The reproduction of the re-circulation ﬂow has not been
PreViously reported in numerical studies for thermally thick polymers, even for pure solid
conduction without liquid-layer formation.
(3 ) The time histories of ﬁiel (dashed line) and oxidizer (solid line) proﬁles are presented
in Irigure 6. Before ignition, gradients of fuel and oxidizer concentrations in the gas phase
are insigniﬁcant since the vapor generation is too small because of the low interface
temperature. Shortly after ignition (53), as seen in Figure 6(a), well-mixed gases are
Present near the heated surface. Note that mixing does not take place along the locations
in the interface where fuel injection occurs. In fact, the injected fuel vapor pushes the

o ‘ - . . .
xndnzer longitudinal ﬂow away from the surface. The mixing, Wthh rs basrcally a

97

 

dlihmon p
Figure 61h
oudizer ca
remixed!
ﬁgure (Nb
ldflllllled.
Bobscn'et
amaumu:
0f the the
rcacnon m
a flame sh
Wllh a ﬂar
a” "Edge-1
ﬂame lead
the 1mm;
Summemi:
mahilShm
OxzcilZCr &
HHS phém
El a}, aid

4 r

ream“ re-

diffusion phenomenon, occurs in a layer between the oxidizer and fuel regions, as seen in
Figure 6(b and c). At 103, when the ﬂame is stabilized, the patterns of the fuel and
oxidizer can be categorized into two regions. In the region that lies near the ﬂame front, a
premixed ﬂame is observed where the fuel and oxidizer mix ahead of the ﬂame front. In
Figure 6(b), a partially premixed region that is occupied by both fuel and oxidizer is
identiﬁed. In the region that lies downstream of the ﬂame front, a diffusion ﬂame region

is observed where the fuel and oxidizer attain contiguous minima and reaction rate attains

a maximum. The appearance of an upstream premixed region contradicts the assumption
of the theoretical formula of de Ris and others [21], who postulated an inﬁnite rate
reaction mechanism at the ﬂame leading edge. The theoretical ﬂame structure resembles

a ﬂame sheet that attaches to the interface (also named “attached” ﬂame), as compared
with a ﬂame zone with ﬁnite thickness that is suspended above the interface (also called
an “edge-ﬂame” or a “ﬂame leading edge”) in this case, as seen in Figure 6 (b). The
ﬂame leading edge results from the ﬁnite-rate reaction scheme since the temperature of
the interface is sufﬁciently low; the reaction rate in the vicinity of the interface is
suﬁiciently small, allowing leakage of both oxidizer and fuel through the ﬂame tip. The
est‘iIDIishment of a mixing region ahead of the ﬂame front occurs where both fuel and
0xiciizer are present, because the quench layer allows the fuel and oxidizer to diffuse.
This phenomenon was pr0posed by Ray and Glassman [30], and later conﬁrmed by Mao
et a1- and others [29]. The reaction rate time history is presented in Figure 7 for three
different times. In addition, fuel concentration, oxidizer concentration and reaction rate
contolu's computed from a ﬁner grid system are presented in Figure 13. At 35, the

reac - . . . . .
tlon region is a small envelope covenng the preheated area. At 53, the reaction rate IS

98

 

 

strong
has]

coast
up. I
en‘s'el
those
isap

cum

strong enough to support itself by supplying heat to the condensed fuel while consuming
fuel and oxidizer. Downstream of the ﬂame front the reaction rate decreases by
consumption of fuel and oxidizer from upstream, showing an enlarged downstream open-
tip. This phenomenon has been investigated by Di Blasi [4], who chose the reaction rate
envelope to describe the transient behavior of a ﬂame. The results here are similar to
those'of Di Blasi. When the ﬂame is stabilized, the downstream reaction envelope ﬁllly
disappears, and only a half-arc exists representing the upstream reaction envelope. From
curves of reaction rates in Figure 7, an open-tip ﬂame is observed, which is characterized
by a discernable distance between the ﬂame leading edge and the interface (also called
the “stand-off distance” or “quench layer thickness”). This distance, interestingly, is
smaller during the preheating period than during ignition and steady ﬂame spread. This,
again, explains the inﬂuence of the combined effects of fuel vapor injection and thermal
expansion. This phenomenon indicates that the quench layer thickness is controlled more
by gas phase expansion and normal convection than by the chemical kinetics used in the
current research. Note that in the ﬂame model using the Oseen approximation (Figure 8
in [55]), there is no perceptible distance between the ﬂame tip and the interface. This
indicates that a higher reaction rate is used in [55], resulting in a closed-tip ﬂame. Figure
13 gives the fuel, oxidizer, and reaction rate proﬁles in a reﬁned grid system where the x
direction grid size is reduced to 0.00015m. The diffusion ﬂame characteristics are
conﬁrmed in the reaction region where the fuel and oxidizer become a minimum as the
reaction rate goes to a maximum. In the reaction region, the maximal reaction rate is
found to be in the upstream part. In Figure 13 (b), the temperature proﬁle and reaction

rate proﬁle indicate that the location of the maximal temperature in the entire domain is

99

not the location of the maximal reaction rate. The upstream reaction region has a lower
temperature and a higher reaction rate mainly due to the higher supply of fuel near the
interface pyrolysis front. However, the downstream reaction region has a higher
temperature partly because the gas is already preheated by the upstream reaction region
and because heat losses there are negligibly small. In addition, it is observed from Figure
13(b) that the maximal temperature in the computational domain is located well beyond
the high reaction region along the ﬂame arc.

(4) The time history of the mass ﬂux is presented in Figure 8 for four different time
levels. Before ignition (33), the mass ﬂux is so low that its magnitude is almost zero, as
shown by a line close to the interface. Aﬁer ignition, the mass ﬂux peak increases (at 105)
and then decreases (at 15s). The rise and fall of the mass ﬂux peak can be explained by
the inﬂuence of the preheating region spanning the region 0.009-0.012m. Fuel
generation, which is an in-depth mechanism, requires a heated layer before any
developed ﬂame proceeds. In terms of an energy barrier, soon aﬁer ignition occurs, the
region nearby the preheating region has received substantial heat so that most of its
energy barrier is cleared. Therefore when it is subjected soon to an incoming moving
ﬂame, the mass ﬂux develops in an accelerating manner until the ﬂame moves out of the
preheating region. Once the ﬂame moves to an “unprepared” region, the mass ﬂux
decreases since the heat from the ﬂame is the sole heating source for the polymer ahead
of the ﬂame. The mass ﬂux gradually increases with time and attains a constant value, as
represented at 153. The interface heat ﬂux history is given in Figure 9 for four different
timw. Here the heat ﬂux is deﬁned as the net heat ﬂux into the condensed phase by the

summation of gas conduction, radiation, and external heat ﬂux (existing during

100

preheating). At 3s, the heat ﬂux is like a jump ﬁmction, the magnitude of which is almost
eighty percent of the total external heat ﬂux. The other twenty percent is composed of the
gas conduction heat loss and radiation heat loss. After ignition, the heat ﬂux peak “rises
and falls” too, resembling the behavior of the mass ﬂux. Note that at Ss, the heat ﬂux
from the gas phase at the boundaries of the preheating region is negative since the
combustion induced heat ﬂux is not high enough to surmount the heat losses. After
ignition, the interface heat ﬂux peak is signiﬁcantly larger (1 or 2 times. higher) than
before ignition. It is the combustion reaction that produces the sharp gradient of the heat
and mass ﬂux near the ﬂame leading edges. The interface temperatures at four different
times are presented in Figure 10. It is found that the maximal interface temperature
remains nearly constant after ignition. Downstream of the ﬂame front, the surface
temperature decreases gradually. This contradicts the assumptions of theoretical models
based on the “ideal vaporizing solid” [21], and has been discussed in [1, 55]. An
interesting ﬁnding is presented in Figure 14 where the heat ﬂux and mass ﬂux are co-
plotted. It is observed that the heat ﬂux front is upstream of the mass front. This
phenomenon cannot be observed in the ﬂame model calculations that use the Oseen
Approximation [55], where the heat ﬂux front and mass ﬂux front occupy the same
location, see Figure 8-B in [55]. The ﬁnite distance between the two fronts results from
gas phase blowing and expansion, which push the fuel vapors upstream ahead of the
reaction region. Since the location of the heat ﬂux front corresponds to the ﬂame leading
edge, it can be deduced the ﬂame tip is located ahead of the mass ﬂux front producing
“ﬂame overhang”. This phenomenon is in qualitative agreement with experimental

measurements by Chu et. a1 [56] and was explained by others [1].

101

(5) Phase Front ar'

 

In the ﬂame mode
located doxmstrc.

model. it is obser

ofStzZ. Sr:

exception where
generalize that It
WIIhQUI inﬂuent‘e
farthest upstream

0f gas exPanSlor

 

downstream of t}

Pmcesses SuCh

llpSIream €Xpan~
1.4, Hallie Struc

compare the ma]

(5) Phase Front and Flame Front
In the ﬂame model that uses the Oseen approximation, we observed that the ﬂame front is
located downstream of the phase front. With the real ﬂow effects added into the current

model, it is observed that the ﬂame front actually precedes the phase front in three cases

of St: 2, St =100 and 51:] = 0.5, as can be seen in Figure 15. There is only one

exception where the phase front precedes the ﬂame front, that is, I?) = 2.5. One can

generalize that the ﬂame front is stretched upstream signiﬁcantly compared to the case

without inﬂuence of the ﬂow. Even for 1;] = 2.5 , because its phase front location is the

farthest upstream (because it has the largest thermal conductivity), the stretching effects
of gas expansion are so signiﬁcant that the phase front is located nearly imperceptibly
downstream of the phase front. The ﬁnding here restates the importance of the gas phase
processes such as thermal expansion and injection ﬂow, which contributes to the
upstream expansion of the ﬂame front. The qualitative nature of the phase front is
explained in [55], and not further explained here.
3.4. Flame Structure

Flame structures for four cases are compared in Figure 16. It is not meaningful to
compare the ﬂame structure without a reference state, so the case St = 2 serves as our
reference state. From the constant temperature constant contours in Figure 16(a), St = 2
is represented by solid constant contours. If we take the innermost lines of the ﬂame
temperature levels, which correspond to 1500K as given in Figure 16(a), these levels can

be used to characterize the vaporizing ﬂame sizes. By comparison, it is found that with

increase of kl , or with decrease of C Pl , or increase of St, the ﬂame size increases. The

102

ﬁnding he“ ‘5 C"

51:05 and
{1:15 has ii

I

conduction due '
profiles in the co:

almost the same

 

 

region. The E1 =
resulting in 3 mt
contour does no‘
the relatively srr
leading edge. F
id). The qualit.
Annally a bOUI
contours, Mm
Censume¢ \x'h:
0f Spatial dim

COHCluSion

ﬁnding here is consistent with the analysis in [55]. The physical explanations are that

E p] = 0.5 and St = 100 have a lower energy barrier than the reference state, while

E1 =25 has the same energy barrier but magniﬁed lateral and transverse heat
conduction due to increased thermal conductivity. Figure 16 (b) gives the temperature
proﬁles in the condensed phase. The three cases St = 2 , 5 p1 = 0.5 and St = 100 have

almost the same distributions of temperature, and therefore the same size of the liquid

region. The 1;] = 2.5 case has a liquid layer with a thickness almost proportional to If, ,

resulting in a much thicker liquid layer. Downstream of the ﬂame front, the temperature
contour does not resemble that of parabola, instead it shrinks in thickness. This is due to
the relatively small heat ﬂux from the ﬂame to the interface compared to that at the ﬂame
leading edge. Fuel and oxidizer concentration levels are presented in Figure 16 (c) and
(d). The qualitative behavior is consistent with [55], characterizing a diffusion ﬂame.
Actually a boundary exists between the constant fuel and oxidizer concentration constant
contours, which characterize the region where both species are almost completely
consumed, while the reaction rate attains a maximum, see also Figure 13(a). Comparisons
of spatial dimensions of fuel and oxidizer regions can lead one to arrive at the same
conclusion of the diffusion ﬂame as characterized by the size of the ﬂame provided by
the constant temperature contours.
3.5. Energy Balance Analysis During Steady Flame Spread

In order to understand the ﬂame spread mechanism in terms of heat transfer, an
energy balance analysis is numerically implemented by choosing a continuously

changing control volume with the left boundary ﬁxed at x=0 and right boundary moving

103

from left to right

 

may quantitatix e
drierent heat tra-
steady state ﬂam.
streamwise cone
conduction and .
ﬁreamwise con-d
applied to the Cg

mechanisms in tl

 

|
phaSC. ll'lCl’CbV 3'

layer. and the p
mechanisms that
the meChanismg
will the eXCeptj

St:100‘ I?!
C'OmanSons b<
CESes. The). are
Home“) arour
51“ij 9f the o

COntro1

from leﬁ to right. In this way, the spatial variance of a typical heat transfer mechanism
may quantitatively reveal the inﬂuence of combustion and melting. Figure 17 presents
different heat transfer mechanisms as a function of the size of the control volume during
steady state ﬂame spread. The mechanisms under consideration are: (1) In the gas phase:
streamwise conduction and convection and combustion heat; (2) At the interface: gas
conduction and condensed phase conduction; (3) In the condensed phase: pyrolysis and
streamwise conduction. In addition, an important heat transfer quantity, the total heating
applied to the condensed phase, can be obtained by summation of the all heat transfer
mechanisms in the condensed phase. It is directly related to the heating of the condensed
phase, thereby also to controlling properties such as the thickness of the polymer liquid
layer, and the pyrolysis in the condensed phase. In order to make the plots clear, the
mechanisms that relate to the condensed phase phenomena are plotted below the abscissa;
the mechanisms that relate to the gas phase phenomena are plotted above the abscissa

with the exception of gas convection, as seen in Figure 17. Four cases including St = 2 ,

St=100, 1:, =25 and Ep1=0.5 are presented in Figure 17 in order to make

comparisons between these cases. Some general observations are obtained for all four
cases. They are: (1) the combustion heat, as represented by a series of bold dots, becomes
non-zero around the ﬂame leading edge and increases its magnitude downstream. The
slope of the overall combustion heating denotes the rate of the heat addition into the
control volume by combustion. The maximum slope is expected to locate itself around
the ﬂame leading edge, where the reaction rate is the maximum and corresponds to the
maximal heat release. It is observed to be so for one can match the location of the

maximal slope to that of the reaction rate. (2) Streamwise gas conduction, as represented

104

by a series of d;.
here indicates th.
as seen in Figure
(111 turn the stre.;
of the tempera:

conduction cun .

   
  

 

13) Gas convect;
first non-zero st

“7th constant 5]

by a series of dash dots, shows a peak downstream of the ﬂame front. The observation
here indicates that the streamwise temperature distribution resembles that of the interface,
as seen in Figure 10. In fact, upstream of the temperature peak, the temperature gradient
(in turn the streamwise heat conduction) shows a positive value. However, downstream
of the temperature peak, the gradient shows a negative value. The streamwise gas
conduction curve simply states that there exists a temperature peak in the reaction region.
(3) Gas convection, as represented by a dashed line below the abscissa, shows that it is
ﬁrst non-zero somewhere before the ﬂame leading edge, and then gradually increases
with constant slope downstream of the ﬂame front. Because of the selection of the control
volume, the leﬁ boundary of the control volume is the incoming air, thus does make
contributions for all cases on the convection calculation. Therefore the convection term is
solely dependent on the ﬂuid ﬂow out of the right boundary, to be speciﬁc, on the
temperature since the ﬂow rate is constant. As explored before in Figure 10, the
temperature has a peak, downstream of which the temperature tends to be stabilized. The
convection term indicates that the heat loss out of the control volume increases
signiﬁcantly around the ﬂame reaction region and then stabilizes downstream, explaining
its relation to the temperature ﬁeld. (4) Interface conduction in the condensed phase
resembles that of the combustion heat, with the maximal absolute slope located around
the ﬂame leading edge. This is obvious since the gas temperature gradient at the interface
mirrors the heating. The mirroring of condensed phase and gas phase conduction is
explained in (5) below. In addition, the magnitude of the interface conduction is many
times larger than streamwise heat conduction. This was also found to be true in [55]. (5)

The energy balance at the interface states that the conduction from the gas phase is equal

105

to the conductiof
loss. which is f
conduction). Ill
conduction. (6)

condensed phas.
dounstream of

constant temper
dounstream. Th
examines the ter‘
it is an exponent
is negligible. ma

mentioned befor

 

from sources in«
conduction in

COHduction. mal
phaSC. NOIC IhE

boundan, 0f the

h
”Psueam‘ llam

the condenSed
mind rate. 8

S .

Ex '

u
I

to the conduction in the condensed phase plus the radiation heat loss. The radiation heat
loss, which is not plotted here, is normally very small (approximately 10% of gas
conduction). That is why interface gas conduction nearly mirrors condensed phase
conduction. (6) The pyrolysis reaction is dependent on the temperature proﬁles in the
condensed phase. From Figure 16, it is observed that there are no noticeable changes
downstream of the ﬂame leading edge. This can be explained if we recall from the
constant temperature contours in Figure 16 (b), where the liquid thickness shrinks
downstream. Though the temperature farther downstream is still negligibly small if one
examines the temperature proﬁles, the pyrolysis reaction is not signiﬁcant simply because
it is an exponential function of the temperature. Below some temperature limit, pyrolysis
is negligible, making the total pyrolysis heat in the control volume nearly constant. (7) As
mentioned before, the total heat in the control volume is a combination of contributions
from sources including condensed interface conduction, pyrolysis, and streamwise heat
conduction in the condensed phase. The major contributor is interface-condensed
conduction, making it resemble the shape of the interface conduction in the condensed
phase. Note that the total heat given here is a function of the position of the right
boundary of the control volume. At the leading edge, the total heat can be related to the
ﬂame spread rate. in the case that gas phase heating dominates solid phase heating
upstream, ﬂame spread theory [1] states that the net heat applied to a control volume in
the condensed phase ahead of the ﬂame leading edge can be used to determine the ﬂame
spread rate. Some efforts have been carried out in [55], which uses this theory as a
starting point. However, it tends to be a difﬁcult theory to apply either numerically or

experimentally mainly because of the absence of detailed temperature and heat ﬂux

106

information near
the leading Edi-"9
maximal interfa.
the unsteady cot
near the flame l-
cost. Therefore
iny'estigated. Th
spread rate mi :
patching to the
the physical me.
By selecti:
Sl:l00, 1:] :
an almost unm-r

f0”! cases exce

phase is 1'1]th I

The higher lhﬁi‘

 

information near ﬂame leading edge. Fundamental questions concerning the deﬁnition of
the leading edge are of critical importance since it can be deﬁned as the location of the
maximal interface heat ﬂux, or the maximal reaction rate. In the current model, we utilize
the unsteady conﬁguration, and we do not employ a grid-adapting scheme. A ﬁne grid
near the ﬂame leading edge is not employed because of considerations of computational
cost. Therefore only the spatial variation of the total heat in the control volume is
investigated. The subject of a preheated control volume as the determinant of the ﬂame
spread rate might better be investigated with a ﬁne grid. Computationally, adaptive grid
patching to the ﬂame region is very promising if accurate quantitative interpretation of
the physical mechanisms is sought.

By selecting the reference state as St = 2, the heat transfer mechanisms for

St = 100, it} = 2.5 , and E p] = 0.5 can be compared with the reference state. There is

an ahnost unnoticeable difference among the various heat transfer mechanisms for the

four cases except for k, = 2.5 , where streamwise heat conduction in the condensed

phase is much more signiﬁcant than in the reference state. This is mainly caused by (1)
The higher thermal conductivity in the liquid, (2) The enlarged thickness of the liquid
layer (almost 2.5 times that of the reference state) as a result of the high thermal
conductivity. The former is obvious since a higher conductivity in the liquid increases
streamwise conduction heat transfer as long as the temperature distribution is almost the
same. For the latter, this enlarged liquid layer occupies a region where temperature is
higher than the average temperature, thereby resulting in a higher gradient of
temperature. The former is thought to be the major inﬂuence since the thermal

conductivity is related to heat conduction rather than the temperature ﬁeld (see Figure

107

litbt). For the r.
condensed phase
4. Conclusrons

A flame spr
tn a channel ﬂu
high external rznlI
attansition stag I
the €35 phase 1
mlettion ﬂow *

Physical Phenor

:nel

 

ring as we
conductiyity) ac
Navier Stokes

energy equatior

loss hm no t ir

pre‘f‘] Ous “.0 rk

Crammed in t

17(b)). For the rest of the three cases, in which if, is unity, streamwise conduction in the

condensed phase is not signiﬁcant.
4. Conclusions

A ﬂame spread model is established which describes transient ﬂame spread behavior
in a channel ﬂow over a melting polymer. Ignition is initially established by applying a
high external radiation heat ﬂux to the surface. Afterwards the transient ﬂame undergoes
a transition stage and then attains a steady state. The physical phenomena considered in
the gas phase include channel ﬂow, gas expansion due to high temperature, and gas
injection ﬂow from the fuel that is generated from condensed phase pyrolysis. The
physical phenomena considered in the condensed phase include heat conduction and
melting as well as discontinuous thermal properties (heat capacity and thermal
conductivity) across the phase boundary. The numerical computation involves solution of
Navier Stokes momentum, species, energy, and continuity equations in the gas phase;
energy equations for the melting condensed phase; and an interface capable of radiation
loss but not in-depth radiation absorption. This work is an important extension to the
previous work that employed the Oseen approximation [55]. A ﬂame spread formula is
examined in the new situation and good agreement is obtained between the numerical
model and the ﬂame spread formula. The transient ﬂame development history is
presented, described in terms of three stages: ignition, transition, and steady ﬂame spread.
The effect of the gas phase on the ﬂame spread behavior is explored with speciﬁc
attention given to the injection ﬂow from the interface and gas expansion due to the high
combustion temperature gradient. Parameters such as pressure, velocity, and streamlines

are interpreted in terms of their coupling to the temperature, species ﬁeld and interface

108

conditions. Flat:

compared to a r
balance analysis
to understand I
discussed as to i

heat conduction

 

interface condei
|
m“? l551. Th.
and 1h? mode}
between Phase
and the qus‘nc‘
05”“ mOdel
formulation ‘1.
Ahead Of the
These discrﬁ
interface lit}
melting in

inﬂucheS.

conditions. Flame structures in three cases St=100, 5 [)1 =0.5 and 1:, =2.5 are

compared to a reference state St = 2 with E p, =1 and If, =1. In addition, an energy

balance analysis of a spatially-varying control volume is applied to the four cases in order
to understand the ﬂame spread mechanism. Different heat transfer mechanisms are
discussed as to their respective contributions to ﬂame spread. It is found that streamwise
heat conduction in the condensed phase has a very small magnitude compared to the
interface condensed-phase heat conduction. This result is in agreement with previous
theory [55]. The comparison between the model with the Navier Stokes ﬂow calculation
and the model with the Oseen approximation is made with respect to the relationship
between phase and ﬂame front, the relationship between heat ﬂux and mass ﬂux front,
and the quench layer thickness. It is found that quench layer thickness is smaller in the
Oseen model than in the Flow model, mainly because a much stronger reaction rate
formulation is used in the former. In addition, it is found that the ﬂame front is located
ahead of the phase front, and the heat ﬂux front is located ahead of the mass ﬂux front.
These discrepancies can be explained in terms of gas phase thermal expansion and
interface injection ﬂow. Analysis was also carried out on the inﬂuences of gas ﬂow and
melting in order to understand the ﬂame spread mechanisms associated with their

inﬂuences.

109

E» 4‘ 2

 

0.5

to”

m. o o
«$5.5 2:! e535 SEEM

4
0

0 5
0,45

2
0

015
01

3 5

"m 2
. O

0 0

5 4
4 0

0 a
$3.55 Ea! emaiw 0E , _u

Flame Spread Rate (mm/s)

Flame Spread Rate (mm/s)

0.35

0.31-

0.25 ~-

0.2}

0.15r-

0.1L» — 4
10'

0.5--

0.45 »
0.4»
0.35;
0.3 --
0.25 '1»

0.2

+ Numerical
, [analytical non-melting]
[analytical melting]

10° 101 102

51
Figure 1 Flame spread rate vs. St

.. 9 Numerical
, [analytical non-melting]
, [analytical melting]

l

‘114 1.6 1.8 i 2 9 22 2.4

kl/ks

Figure 2 Flame spread rate vs. 1;]

110

 

 

04

5 3
3 C
0

0.25

Exit»; 9:»! 235m QECE

2
0

5
cl
Div

«I
0

Flame Spread Rate (mm/s)

0.5,.

0.45;

.o
.5

0.35 ..

.0
on

0.25

.0
N

.0
.5
U"

.0
c-l

 

;- 99 Numerical .
l - [analytical non-melting] ‘
[analytical melting] ,
i "0.5 i i 90.4 70.6 i 770.8 i if i 7 1.2
Cpl
Figure 3 Flame spread rate vs. C pl
0.01
0.008
0.006
0.004
>.
0.002
0
-0.002
-0.004
0 0.005 0.01
X
(4-a)

lll

0.01

0.008

0.006

0.004

0.002

-0.002

-0.004

0.01

0.008

0.006

0.004

0.002

.0002

-0.004

 

0.005

0.01

(4-b)

 

l

 

Ill Il\ll

 

 

 

T \\
l

 

i
ll

, [y]
l

 
  

Illl

  

 

 

IIIIIIII'ITTI

 

 

O

112

Figure 4 Cons
streamjmes at ‘
ye units of (I

y (m)
o
C)

 

0.01

0.008

0.006

0.004

0.002

-0.002

-0.004

 

0 0.005 0.01
X

(4-d)
Figure 4 Constant temperature levels (starting from 400 K with step 100 K) and ﬂow
streamlines at (a) t = 3s , (b) t = 5.9 , (c) t =10s and (d) t =155. The coordinate X and Y
have units of (m). Note that temperatures are evaluated in both gas and condensed phases.

i7" j
0.01.;L j l
l
l
0.0083L
l
E 0.006t
> i
l
l
|
0 004 l.“ .
l
ix-
0.002LL 1
[xxx l l V
{:‘I i: l V' ,‘ I I
0H1--_.___, _ “’7 7.1. W ..._i .mrﬁm m.-. .n..., l n .I m. . i ,, .‘ ,,-.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
x (m)
(5-a)

113

0 038

O 0:}

Y (m)

0 004

0.002

(D

 

0.01 4

0.008

.. ___ 4 __‘, 4 4 4

3: 0.006 :»
>~

0.004 7

l
0.002 l-__

l- ,,

oiz-__n _ _ _. _ _. a n-__., _ _ _ - -a .- _. _ , , .r _-
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
X (m)

(5-b)

cr"777‘ re *7 ».» r - - --

0.01 1.

 

—_k rmL F ._._._ _ 1.4 HJQJJdL- J

0 0.002 0.004 Foods”- 0.008 0.01 0.012 i001?
x(m)

(5-0)

114

O 098

y (m)

0 002

 

Heine 5 pre85
C0n10m1e\»el is
Stagnallgn p0”

y ("7)

1 . -. !
. ': l l
. l .I I I l .I
5,, ‘ l l 1 I, .

. I l] l: l l ,1 - I -_ l l 1

. I . . , I . I I I‘ 1' | . 1 . x

1:: ll, ' . ' _ " .l ‘1 l
o 4.1;...» _._4' _ i v - e e V n. __ L— L .3;

:‘,Il

. . .41L§~LJL-L4L_‘_J~., , .-..-A ,H ,._.?
0 0.002 0.004 0.006 0.008 0.0 0.012 0.014
x (m)

(S-d)

Figure 5 Pressure distributions at (a) 33, (b) 53, (c) 105, and (d) 155. The rightmost

contour level is 0.001 with increase of step of 0.001 from right to left. The corners are the
stagnation points with elevated pressures.

x 10’3
67’-’7TV"TT i 7 T
5 i-
I
4 L
' - Fuel
. Oxidizer
E 3 L.
>. |
l
l
2r
l
l
1i
l
I [Kl-I” I
0L.__‘, ._.._ e 7. n n. n ,__ ﬁ.‘l ,m‘; #Lg-i‘::“_ .. . . ,,
4 5 6 7 8 9 10 11 12 13 14
x (m) x 10.3
(6-a)

115

l
l ' Fuel
1 .Oxidizer
4»—
l
E3»
>
2‘;- -
l
l
l
11.
O;-__ _ i n _ _ _ ,4; ,

 

x10
6|r~—~———~eae— v »777 ,
1
5e
- Fuel
4% Oxidizer
Ea
>~ l
I 3
2i , ». 1‘

1‘2 13’ 914
x10'3

 

(6-0)
Figure 6 Constant fuel and oxidizer concentration levels for St = 2 at 53 (a) and 10s (b)
and 158 (c). Solid lines start from outermost 0.01 with step of 0.01, dotted line start from
outermost line of 0.30 with step of -0.01. A small partial premixing region exists near the
ﬂame leading edge but otherwise the reactants are essentially separate.

116

05

E: x

-0.5

-15

-2

0.5

«EV A

'05

~15

1.5;
1 -_
0.5
A &
é o:— _ T F ’w—
>‘ l
05‘»—
-1
-1.5~
-2'----_2-_ . .. _. iiiiii r.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
x (m)
(7-a)
x 10.3
2.———~-~—~e—— r - ~ , i a
l
1,5...
1 H I' A
.5 ...,
0.5» 11,5
64
g, 0 ivww '
>5
-0.5,—
-1?
-1.5j-—-
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
x (m)
(7-b)

117

y (m)

'05

Flgllre 7 Combi
lbl 311d 155 (C

reaction rate C0.

 

0015

Y (’7’)

L.)
o
( n

2. ,,,,,,
1.5 I .,. T F
l ,‘3‘: if; if» ’1:
1 l‘ :6]:.:¢
l {(119 w
0.5 -» I f/
A Q
E 0 wet»... H »
>~
-0.5
-1.5*
-2‘.-__»__n__. _ ..- - ._ _ _ in _,_n .- . , in
0 0.002 0.004 0.006

0.008 0.01 0.012 0.014
X (m)
(7-C)

Figure 7 Combustion reaction and pyrolysis reaction constant levels for St = 2 at 53, 105
(b) and 155 (c). Constant lines start from 25 with step 25. The discontinuity in the

reaction rate contours come from the insufﬁciently ﬁne grid near the ﬂame front.

0.01Sreeeeeee , e -
105
155
0.01L , ,
E . 55
>. .
0.005r I
‘ ‘3s
0 ---- s , , n , , . , . ,. . _i - .
0 0.005 0.01 0.015
x (m)

Figure 8 Mass ﬂux along the interface at four different time (35, 53, 103, 153), showing
the evolution from ignition to steady spread.

118

y (m)

 

103

8r 1 15s
l -‘ ; ‘ 53

y (m)

4- . 3s

0" 90.005“ ' ' 0.01 f i 0.015
x(m)

Figure 9 Net heat ﬂuxes from the gas phase along the interface at four different time (35,
SS, 108, 153).

750;—»»-* » f a»; if? *_-_-.._---. _ a ,r --

1 1
700;- i ~53 f. 108

650 » -. ' 55

E . f 'l ' 35
>5

 

X (m)

Figure 10 Surface temperatures at four different times (38, 53, 105, 153). Note the non-
consistency of the surface temperature behind the spreading ﬂame front at 155, and the
constant with the “ideal vaporizing solid” of [55].

119

 

0.011

0.01
0.009
0.008
0.007
9.006
0.005
0.004
0.003
0.002
0.001

TTY'YIIIYITT

lﬂ'l

ll. 1‘.

w YIY
l" lTIWIIIIII I,
I’TIFIYTIIIYTI' 11 1

 

r'rTr rrr
r71 ,1 .,

 

0.011

0.01
0.009
0.008
0.007

9.006 ,
0.005 :

0.004
0.003
0.002
0.001

lTllIlll'TI

 

 

 

 

‘ijiﬁiiiiiiiiiiiiii

 

 

0.01

Figure 11 Pressure and velocity ﬁelds at 15s.

120

 

0.009

HI

I

0.008

0.007

I
rlIfTrIllTI

 
 

0.006
>0.005

0.004

 

0.003

11 {It
Illlj'r'rlllir' Ill

1773‘]... II

11

0.002

0.001

 

‘ -1I’1llrl117lllly.‘rl7|:

Figure [2 Te
aCCEleranOn I

.—
—
0009—
. -——‘-“‘
—
_
_
__.-¢—-—‘-‘“"
0008'-
.
___...——-.——-
u—
)I—‘—-——--"'"
0007—
C
_
"“"’
h
—
baa-0””
0006'-
I
P—u—op—d'”
_
P’a—a-a”
)—
0005-
>-' -
L...’”J—"
)—
aaﬂf”’
y—
0004—~~~
.

lrl'l
t t
1 1
t 1
\ t
\ \
t \
\ \

1

1
1
1
1
rrltIllll1111\111

0.003

133771

mmulttHH111111111\\1

0.002

1111
1111111
11111111111

1

0.001

H||11||111111111111

 

 

 

~m-onutl1|11111111111\11 1
1......muuulllll111111111

 

T I J l
t-"mum I 11

p-

 

 

 

 

 

Figure 12 Temperature and velocity ﬁelds at 158. Thermal expansion and associated ﬂow
acceleration occur near the ﬂame locus.

121

Figure 13(3).
reﬁned End s‘.

 

 

 

Figure 13(a). Reaction rate, fuel concentration and oxidizer concentration at 8.55 in a
reﬁned grid system that has Ax = 0.15 mm.

122

2.5

0.5

xlC

 

 

3f,r-,--
2.5L
2;
311.5
>
1;] _, , i " ’
05‘-
CL. _- __in__ _ , , , _,
0 1 5 6 7
x10'3

Figure 13(b). Reaction rate and temperature at 8.55 in a reﬁned grid system that has
Ax = 0.15 mm. The grid still does not produce continuous reactivity contours.

123

12

rt.
0 < 1|

m Eel
4 1 u S «my 9:: :SE

a man .

Nﬂgmx

S
1

c
4 . SI

1 6 . .9.
x3.k

r. E . . .0 fES)V

1 v xiu rem—2

- a!
AV\NE\

gure 14 .\la
latter with re

Fi

. Heat Flux
- Mass Flux‘4E+6

 

N

Heat Flux (Wlmz) or Mass Flux (Kg/mz/s)
”_Wﬁo~e__ ._ __

l

l

l

I
N

0015

° l
O
O
O
01
O
O
A

x (m)

Figure 14 Mass ﬂux and heat ﬂux at 153, indicating the upstream displacement of the
latter with respect to the former.

16 I T r l 1 I

 

14-

12-

10-

Front arrival time (s) in case of St=2

 

.-
y-
b
]-
F
b-

 

 

0

 

5 6 7 8 9 10 11 12
x (mm)

Figure 15 (3) Flame front (solid line) and phase front (dash dot) vs. x distance at St=2.
The ﬂame front very slightly precedes the phase front after x = 8.75mm as the ﬂame
advances toward decreasing x (vice versa for x 2 8.75mm ).

124

 

P r I

6 4 » L _l‘ E 6 oh L1 53/; .4 a la . I o .

cl (1 1| nl ( 1
NEW .0 ammo c. Amy 9:: 33.25 .:oeu ”D No.5 3 p.28 S «i 9:: 52:5. EEK ,3 may
I a
e h
.... a. p
in me 4

rr. F

 

16 T I V T T I

14» 4

=2

12- 4

10-

Front arrhal time (s) in case of St

 

 

 

 

5 6 7 8 9 10 11 12
x (mm)

Figure 15 (b) Flame front (solid line) and phase front (dash dot) vs. x distance at St=100

 

 

 

 

 

25 T I r r r r
N 20» -
.11.
(I)
“5
0
§15~ ~
.5
E
2
E10» 4
,9
E
a
E
u. 5'- 1
0 L r r r r r
5 6 7 8 9 1O 11 12

x (mm)
Figure 15 (c) Flame front (solid line) and phase front (dash dot) vs. x distance at 1?, = 2.5.
The phase front precedes the ﬂame front very slightly when x _<_ 8.75mm.

125

r ‘K.-_ _

mute. .c exec c. A5 CE; Exit: .23;
...r

 

\

 

2:2: >

Figsl

 

16 I T 77 I I I

 

 

 

 

14'- 4
N
1112- -
(I)
3
0
m 10L .
8
E
z»: e- -
¢
.§
3 4
.9 6%
E
as
o— L- .1
g 4
u.

2- -

0 _L 1 1 1 1 1

5 6 7 8 9 10 11 12

x(mm)

Figure 15 (d) Flame front (solid line) and phase front (dash dot) vs. x distance at

(:;q ==0L5.
10L
3: - Eﬂ=2
‘ --- St=100
-.- lflks=2.5
’E‘ CpVCps=O.5
5, 6+ ' , "‘
>~ ’ g, ’
4i ,, ’1 , - ' ’ -
21. , fir ,
I .7 ,.,’ I tar-7' 2‘.“er :ILL—uf“
1911:,» 'jﬂiii'-.x—> - '1‘“:
1 *L”j?ﬁZ:—:ﬂ-:::;7m
o - - ,, _. . ‘\_g_-——-—.~ — ---: r——:‘ ‘ . -...L-..
2 4 6 8 1O 12 14
x(mm)

Figure 16 (a) Temperature Profiles for four different cases examined herein.

126

 

y (mm)

Figure 16 (

edge at _1’ :
doum
E
E
x

 

1 a
0‘. A a , 7 7 , , , , , «Weekfex 22-11- "" we"
l -‘:§ f- “**‘**‘ M- r. g, T
-1 L ‘ K‘QZ I V
A ;
E -2‘
Z - St=2
~— St=100
-3; -.- k/ks=2.5
r. Cpl/Cps=0.5
4:
1‘
-5L
;
_61._V,__iﬁée77*‘—**’;"'_‘TT 1,4,7, " _
0 2 4 6 8 1O 12 14

Figure 16 (b) Temperature distribution in the condensed phase. Near the flame leading

edge at x =6mm it is self-similar. Downstream (at x lemm self-similarity breaks
down).

104
8%
E 6L - St=2
é - St=100
>~ -- =
g . krlks 2.5
4y . Cpl/Cps=0.5 7
i
2}—
1 (7,, *1 " 7" -
1 ,«e/T'E’CA’F; a .
0;, ,1”, 4 W i , m ,, _ ;, amﬁLr-wf‘ w -\ _. .i i a ,. .4
0 2 4 6 8 10 12 14

Figure 16 (c) Fuel concentration proﬁle in the gas phase.

127

10
8 l-
E 6 _
f? ‘ - sn=2
—— St=100
4L -.- k|lk5=2.5 /,..,_j
l . Cpl/Cps=0.5 7 Wigwﬂwe" .
22 ﬂeaff;':;i:”ifi
,‘ 91"???" "f
0 L_._._ _. ...; v 71____.74.__ 1.21 maimiizli‘vﬂimak 74;, ”7 ...7 ..«J‘J ..-. i. m. 7 7 -i a 7.1
0 2 4 6 8 10 12 14
x (mm)

Figure 16 (d) Oxidizer concentration proﬁles in the gas phase.

  

StrearnniseG—as Conduction

500 ;‘ - e Gas Convection
0 Gas Combustion . o
400 l 1 — Interface Gas Conduction . .0 '
L l «H Interface condensed Conduction ‘ . . °°
, I + Pyrolysis . . 0 °
300 L 1 «.— Streamwise Solid Conduction o '
.7 l _ _, Total Heat in Solid cv 1 ..'
1' ._ .__-.1_ 1* m ,_ h, .-. _ ,, ,, . .
200} .'
O
‘ o
L e
100 'r ,.
i M- .
0 L“ ~ -1: "::H" ‘7, (1:421 {‘.;.——--l~.a_4_a ;-,4,-._ 1.9 '17“ 1'
L ‘ \‘ I} ’ t .' “Fl" 1‘! a5 «4» a, oJr o a one". or e or
i ' '
1 ‘ \ -
-1oo+ \\ —
i ‘ \ \
.200L .-\\‘\‘
L ‘2‘ "\
-300L.._______ __-i____ ,__+ dam ii.-- __ H ,, ‘
O 5 1O 15
x (mm)

(17-a)

128

Sireaooiggoas'ooooueiloo* ’

500 1 —— Gas Convection
; 1 0 Gas Combustion 1 . . ,
i l ~ - - Interface Gas Conduction ; . 0 °
400 .L i e..— Interface condensed Conduction ‘ .. 0 '
r Pyrolysis . . . ‘
300 ‘. ‘ +7 Streamwise Solid Conduction ‘ . °
; » * ~ Totall1€9!_i9§9'iicv-_- , - , f . - '
i . O
200 i, .
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100 r 0,-
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L >..\ 74 T e? o 4 a
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L ‘\ \ \
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i x - x
4001---- - - - , ,. ., , , ,,-
5 10 15
x (mm)
(17-b)
Li - f StreaﬁmiiviseﬂGiaﬁsiCondbction
500 r l we Gas Convection
1 0 Gas Combustion I
1 1 — Interface Gas Conduction ‘
400 k” we — Interface condensed Conduction . . o' ' '
f 1 'r Pyrolysis . . 0 '
300 L i ——.+— Streamwise Solid Conduction j .0 '
i p 7 1 Total Heat in Solid CV 7 g , , . ..'
200 i— O o , "
‘ o
‘ o
g 100 L» ' '
i u i.
0 L’ MMWW ‘."‘L 1‘” 3.1; ;‘ yr: "’:"'_' ' ’ ‘47 L"' ‘ ' T“ ’ ‘ ' "' ‘
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-3001}-.- a. , - - a- a- a a, a, , ,, - - ... , , , , - ..
O 5 10 15
x (mm)
(17-c)

129

i — siieaaoigo’oas’oo‘oaoeiioi.‘ _ l

500 .~ — Gas Convection 1
I . a Gas Combustion 1 .
400 L , Interface Gas Conduction ‘ . .0 °
1 f ~~— Interface condensed Conduction . . o'
«r Pyrolysis ‘ O . °
300 'r -... Streamwise Solid Conduction .‘ , 0
. 1 2-1.. Total Heat in Solid CV 5 . .’
| » 4.7;.___.__._2__k_" *# a _. 77 W 7'.
200 , '
. 0 '
' o -
3: 100 i- .9 "
O ! > _ a .
OLMMWWE‘f“ w——~ , "" "‘* ~ ~- .7
1 “‘ *\ ................. L
-100; L .\
-2oo! “xi“
-3005-._ _ _ -. - _ . , . a ,,,,,,,, . , ,, , x" .
0 5 10 15
x (mm)
(l7-d)

Figure 17 Distribution of heat ﬂow rate (W) for a control volume in the gas phase and a
control volume in the condensed phase ahead of the ﬂame along the polymer surface for

four Cases: (a) St :2; (b) St=100; (c) 1?, = 2.5 at St=2; (d) E), = 0.5 at
St = 2. In this ﬁgure, for the purpose of clarity, positive values are assigned to ch,
Qcom , ng , and QT (above y = O ); and negative values are assigned to chg’ Q py’
Qws and ch (below y = 0 ).

130

Table 1. Major properties and kinetic data used for the numerical model.

 

 

 

 

 

 

 

 

 

ks =0.05 CPS =1460 pS=119O p,=119o
W/(m-K) J/(kgeK) kg/m3 kg/m3
Tm=500 Ac=2.82X109 EC=129580 qc=1113.5
K s" J/mol kJ/kg
Ke =o.5 kg =0.0411 C).g =250 pgo =l.16
W/(m-K) J/(kgK) kg/m3
D:2.2><10‘5 A =l.6><1o15 Eg=135730 4g =-51798
mZ/s g m3/(kg-s) J/mol kJ/kg
um = 0.1 T... = 300 Y0... = 0.31 19.. = 0.0
m/s K

 

 

131

CHAPTER 5
REVISIT OF THE INFLUENCE OF SOLID ANISOTROPY ON FLAME SPREAD

OVER MELTING POLYMERS

1. Introduction

Anisotropy of the solid phase inﬂuences ﬂame spread behavior by possibly
exaggerating the non-uniformity of heat conduction into the condensed phase. Chapter 2
investigated the anisotropic properties of a pure solid in a simpliﬁed ﬂame spread
situation. This work, however, did not consider the inﬂuence of a realistic ﬂow and the
melting behavior of thermoplastic solid. In addition, the inﬂuences of the material
properties were investigated in a restrictive manner. In this work, we examine the full
range of conductivity variation.

we have also signiﬁcantly improved the previous numerical model by adding more
realistic physical mechanisms. These include a Navier-Stokes ﬂow and melting solid—
liquid in the heated layer of the condensed phase. It allows us to explore the behavior of
ﬂame spread for melting thermoplastic materials.

When an anisotropic solid polymer undergoes external heating or responds to ﬂame
combustion heat it may undergo a phase change that alters the material properties. The
resulting condensed phase contains an isotropic liquid region with uniform conductivities
and an anisotropic solid region that is located below the liquid layer (and which was once
solid and anisotropic). Generally, for the conﬁguration we shall examine, theoretical
investigations are absent because of the complexity of the condensed phase calculation.

An exception exists, of course, for the simple case when spread is steady and no phase

132

5..

change occurs [1, 43, 49].

The purpose of this Chapter is two fold: First, the ﬂame spread rate will be obtained
with respect to a varying transverse conductivity or a varying longitudinal conductivity;
Second, we will explore the ﬂame spread characteristics as reﬂected by the inﬂuence of
the conductivities, such as the ﬁeld parameters, interface parameters, and the heat transfer
mechanisms. The last inﬂuence is accounted for by using energy balance numerical
techniques. Afterwards the ﬂame spread rate is discussed by using a global energy
balance principle [43,49,55].

2. Results

The mathematical model uses an energy equation in the condensed phase that

incorporates anisotropic conduction in the solid, which is the same as Chapter 4. In order

to evaluate the inﬂuence of the anisotropic conductivity, a reference state is selected

where ksx = ks}, = 0.05 W/(m-K), k, = 0.05 W/(m-K), and LS = 438000 (W/kg).
Note that ksx = ks), = 0.05 are the conductivities at ambient temperature. To analyze

the inﬂuence of longitudinal conductivity ksx , we change its magnitude in a range from

1/ 10 of the reference value (0.05) to 10 times of the reference value (0.5) with ksy being

ﬁxed to the reference value. Similarly, the inﬂuence of the transverse conductivity ksy is

evaluated in a range from 0.005 (10 times of the reference Value) to 0.1 (2 times of the
reference value) while keeping ksx ﬁxed to the reference value.
2.1 Flame Spread Rate

The dependence of the ﬂame spread rate on ksx and ks), is plotted in Figure 1(a)

and Figure 1(b), respectively. The dots in ﬁgures represent the numerical results and the

133

solid line is the corresponding linear interpolation.
There is no theoretical formula available for ﬂame spread over a melting polymer

with solid phase anisotropy; however, we can ﬁnd some comparable cases similar to our
studies here in order to estimate the inﬂuence of these conductivities. For a range of ksx,
a comparable case is ﬂame spread over anisotropic solid polymers. Without melting, the
de Ris formula can be applied to the anisotropic case by showing that a variable ksx does

not change the ﬂame spread rate [21]. This was veriﬁed in [43], where the streamwise
conductivity could be eliminated from the spread formula. Interpretation of these results
was aided considerably by the energy balance principle. This energy balance principle

was applied to ﬂame spread over melting polymers in [55]. Numerical investigation [57]

has shown that the ﬂame spread rate is weakly affected by k

H except for large values of

ksx. Beyond certain magnitude of ksx ,

the ﬂame spread rate decreases with increasing
ksx. A comparison of Figure l of [57] to Figure 1(a) here indicates that a similar trend

holds true for the melting case with anisotropic solid properties. A quantitative

comparison will not be attempted because a different set of kinetic data is used here. As
observed from Figure 1(a), there exists an asymptote for the spread rate when ksx
approaches zero, whose value is approximately 0.36 mm/s. In addition, the slope of the

ﬂame spread rate with respect to ksx diminishes when ksx decreases to small values.
The range of ksx corresponding to a constant spread rate will not be examined here since
it is case dependent. A very small range is expected by observing a small slope at
k,, z 0.

The similarity between the numerical results with and without melting indicates that

134

the melt layer does not change the qualitative nature of the ﬂame spread vs. ksx

relationship. Quantitatively it changes the magnitude of the ﬂame spread rate [55]; for
example, St=2 results in a much lower spread rate than St =100. Most of the

computations are outside of the ﬂat ﬂame spread rate versus ksx regime, which is near
ksx :0. Similar to [57], for the range of ksx employed in this model, there is no
indication that spread rate increases with ksx [58] for an intermediate range of

streamwise conductivity. The physical explanation is that the higher ksx results in a

higher streamwise conduction, which serves to enhance heat losses more than it enhances

upstream heat transfer to the unburned material.

The dependence of the ﬂame spread rate on the transverse conductivity ks}, is
plotted in Figure 1 (b). The dashed line represents the de Ris formula and the solid line is

the numerical result. Qualitatively, both spread rates decrease with increasing ks , with a
much stronger dependence than ksx. The melt layer therefore does not change the

qualitative nature of the spread rate vs. ksy. In addition, the decline of us with ksv in

the de Ris formula is much steeper because we have a melting layer in the numerical
model, which serves as an energy barrier (with the latent heat corresponding to St = 2 ).

The de Ris formula [21,43,49] gives u 0c 1/ k . The numerical trend in Figure 1(b)
3 sy

coarsely resembles us cc 1/ ks}, as suggested by Figure l in [57]. We obtain us cc 1/kf,

with n z 0.2 which is a substantially weaker dependence than the de Ris case with n =1

Physically, an increasing ksy lowers the spread rate by magnifying the heat loss through

transverse conduction. There may be some other complications of melting such as the

135

layer thickness for example. They will be examined in more detail later.
2.2 Phase Front

Melting introduces the complication of the liquid layer in addition to those
introduced by the anisotropy of the solid polymer. The spatial locations of phase

interfaces between the solid phase and liquid phase are plotted in Figure 2 for two cases:

(1) one with varying ksx (2-a), and (2) one with varying ksy

(2-b). Since the process is
transient, we take a snapshot of the phase front approximately 6 seconds aﬁer ignition. In
this way, the condensed phases in each case receives almost the identical heating from
the combustion reaction, thereby enabling a comparison of the inﬂuence of the condensed

phase properties. An interesting observation from Figure 2 is readily obtained in terms of

the size of the liquid layer. It is found that both the thickness and the streamwise extent of
the liquid layer decrease with increasing ks in each case. This seems to contradict the

intuitive notion that a higher conductivity enhances the heat conduction rate, thereby
increasing the net heat ﬂux into the condensed phase from the gas phase reaction, and
subsequently increasing the size of the liquid region. This intuition seems reasonable in
that in the extreme case of zero conductivity the condensed phase does not receive any
heat by conduction, thus rendering impossible any ﬂame spread. A larger conductivity
might be expected to better preheat the condensed phase to a high temperature and better
support a spreading gas-phase ﬂame. This speculation is invalid because, in this model,
heat conduction serves not to accumulate thermal energy but to redistribute it inside the
thermally thick polymer. For an otherwise identical situation for the thermally thin case,
the polymer could be heated in a shorter time, thereby enhancing the ﬂame spread rate.

For the thick material considered here, however, the enhanced heat loss by conduction

136

reduces the size of the liquid layer, giving an inverse trend for the liquid layer size with

res ect to k and k . To corn are the inﬂuences of k and k , we choose two
p sx sy p sx 3

parallel cases with ksx = 0.005 and ksy = 0.005, with each having conductivity of

0.05 in the other direction. From these two cases, it is found that the latter not only results

in a larger liquid layer thickness, it also results in a larger streamwise extent of the liquid

layer. This example indicates that ksy is the more important factor in determining the

size in all directions of the liquid layer.
2.3 Interface Parameters

The interface temperature, the mass ﬂux, and the heat ﬂux are plotted in Figure 3,
Figure 4, and Figure 5 for two cases. The surface temperatures, as observed from Figure
3, show ahnost the same peak value for the different cases. The size of the heated surface
can be characterized by a spatial span in which the temperature is above a certain

magnitude along the interface. If we take this magnitude as 600K, it is observed that the

size of the heated surface decreases with increasing ksx or ksy. This supports the heat

loss effect of both ksx and ksy; the larger conductivity produces a smaller heated

surface region. Comparison of Figure 3(b) and Figure 3(a) indicates consistency with

Figure 2(a) and Figure 2(b) in that the ksy not only expands the transverse extent of the

liquid layer but also expands the streamwise extent, when using the size of the heated
surface as a measure.

The interface heat ﬂux is deﬁned as the net heat ﬂux into the condensed phase,
namely, gas conduction minus interfacial radiation loss. This is plotted in Figure 4. In

order to make comparisons, their peak values are ﬁxed to the same location. It is

137

observed that the peaks have almost the same magnitudes no matter which values ksx or

ks

y take. Downstream of the peak, there are discernable decreases of the heat ﬂux with

increasing ksx or ksy. Interestingly, there is a more signiﬁcant inﬂuence of ksx on the
heat ﬂux far downstream. If we use the 2W/(mzs) to characterize the heat ﬂux region in
which the heat ﬂux exceeds this value, the size of the region has a decrease of

approximately 4mm between ksx =0.005 and ksx =0.5. Smaller ksx results in a

larger heat ﬂux region downstream of the ﬂame front. The case of varying ksy shows a
similar inﬂuence although the inﬂuence of ksy is not signiﬁcant except for large

magnitudes, as seen for the case ksy =0.1 in Figure 4(b), where a much small

downstream heat ﬂux region is obtained compared to the other three cases. The interface
mass ﬂux distribution along the interface is plotted in Figure 5 for the two cases. The
order of mass ﬂux magnitude with respect to the conductivity in a given direction
resembles the heat ﬂux and temperature. The explanation can be obtained readily from
Figure 2 by considering the in-depth pyrolysis mechanisms. Again, a lower conductivity

in any given direction results in a higher mass ﬂux. The peak value of the mass ﬂux is

found to be also dependent on the conductivity, especially in cases of varying ksx. In
addition, ksx = 0.005 or ksy = 0.005 has a signiﬁcantly larger magnitudes of the mass

ﬂux peak compared to the other cases.
2.4 Energy Balance Analysis
Integral numerical methods are used to obtain the magnitudes of various heat transfer

mechanisms in a series of varying control volumes for steady ﬂame spread. The control

138

volume right boundary moves along the interface from leﬁ to right, thereby resulting a
functional dependence of these heat transfer mechanisms on the longitudinal distance at
interface, as given in Figure 6. A general observation is made ﬁrst by relating the

conductivity to the magnitude of the overall scales of these heat transfer mechanisms.

That is, the higher ksx or ksy results in lower magnitude of each heat mechanism. From

Figure 6, it is observed that the reference state corresponds to the middle position
between a smaller conductivity and a higher conductivity in terms of the magnitude of the
heat transfer mechanisms. The heat transfer magnitudes actually represent the size of the
ﬂame; a higher conductivity corresponds to a smaller ﬂame. One formulation of the
ﬂame spread mechanism [34] states that the total heat applied to the condensed phase
upstream of the ﬂame front determines the ﬂame spread rate. In [43] the energy balance
provided in [34] were clariﬁed in both scope and applicability. The formulation of [34] is
consistent with our observations, hence according to [43], the gas phase dominates the

spread process. A measure of the relative importance of each phase is the parameter

Pe = (uf /Ctsx)/(ug /ag) known as the solid to gas Peclet ratio. When Pe >>l gas

phase forward heat transfer dominates the solid phase heat transfer, and vice versa for

Fe << 1. In our three cases (Figure 6(a) (b) (c)) we have P8 = 592 , P6 = 5.9, and

Pe = 59 respectively. By comparing the total heat transmitted to the solid as represented
by the solid line in Figure 6 for all cases, one sees that its order is consistent with the
order of the ﬂame spread rate. The total heat carried upstream of the ﬂame front also
follows this rule.

Another observation concerns streamwise conduction in the condensed phase. Since

139

this is related to ksx , Figures 6(a), 6(b), and 6(0) are chosen for comparison. It is found

that when ksx = 0.5 the streamwise conduction shows a totally different behavior from

the other cases. First and foremost, conduction in the solid has a higher magnitude

compared to that of streamwise conduction in the gas phase. In this case, the forward heat

transfer in the solid is less dominated by the gas, as indicated by the parameter Pe = 5.9

[43]. Second, the conduction magnitude ﬁrst increases by forming a peak when it
approaches the ﬂame front and decreases downstream, similar to the other cases.
Downstream of the ﬂame front, however, it continues to decrease at ahnost a constant
rate and ﬁnally attains a peak again. This second peak is downstream of the ﬂame front.
Here the heat loss term due to streamwise conduction in the condensed phase has a
negative value, whose magnitude is comparable to that of the streamwise gas conduction,

or one tenth of the combustion heat in the control volume. This result indicates the

important role played by the downstream heat conduction loss for high kn. For the other

cases, streamwise heat conduction in the condensed phase is not as signiﬁcant. The

inﬂuence of kSy can also be found by examining Figures 6(c), 6(d) and 6(e). Compared

to ksx, ks}, does not substantially alter the trends of the overall heat transfer

mechanisms. The variation of ksy supports the observation made before, that is, higher

results in both a smaller ﬂame and heat transfer magnitude.

ks},

2.5 Flame Size

The ﬂame size increases as the conductivity in any given direction decreases. Figure
7 shows another perspective for the same purpose. The temperature is plotted at a line
through both the gas and condensed phase, the location of which is selected as the center

140

of the heated surface. The temperature pattern shows ﬁrst an upslope, then a peak and
ﬁnally a down-slope with y=0 serving as the boundary between the gas and condensed
phases. The peak of the temperature may not correspond to the maximum ﬂame

temperature in the gas because the selection of the cutting line may not exactly match the

location of maximum temperature. The inﬂuence of ksx on the temperature is observed

from Figure 7 (a), where higher ksx corresponds to a lower gas temperature. The order

of the temperature peaks obeys the observations made before. That is, the ﬂame size has

an inverse dependence on ksx. Similarly, Figure 7(b) shows the temperature along a slice
for four ks), values. The temperature peak depends inversely on ksy. Figure 7 clariﬁes

the dependence of the ﬂame size on conductivity in a given direction by the graphed
temperature pattern.
2.6 Ignition Delay Time

Finally the inﬂuence of the conductivities on the ignition delay time is given in

Figure 8. A higher ksx results in an increased ignition time (Figure 8(a)), and similarly a
higher ks}, results in a higher ignition delay time (Figure 8(b)). These results restate yet

again the heat loss effect of the conductivities. With increasing magnitude of
conductivity, the energy loss via conduction makes high surface temperatures harder to
attain, thus requiring longer ignition times. Other aspects of ignition behaviors will not be
explored here since we are mainly interested in the steady spread of the ﬂame. The

ignition delay time provides consistent results that improve our understanding of the

energy redistribution aspects of ksx and ksy.

A general discussion is now provided on the dependence of the ﬂame spread rate on

141

k5,. In a ﬂame-ﬁxed coordinate system, the global energy balance principle [43] states

that the spread rate must be sufﬁcient to remove the combustion heat by downstream
condensed phase (solid and liquid) and conduction in addition to downstream gas phase
convection and conduction. Here we provide a conceptual explanation. A mathematical
deduction is not attempted because of the introduction of the phenomenon of a liquid

layer, as compared to a much simpler conﬁguration described by Equations (3) and (4) in

[43]. The role played by ksy is to distribute a proportion of the combustion heat released

by the ﬂame to the condensed phase. With diminished ks , since the gas carries away

less heat due to the limitation of the diffusivity, the condensed phase must increase its

velocity in order to carry away its share of the combustion heat. Therefore with the

decrease of k3}, , the ﬂame spread rate (the convection of the solid phase) increases. This

behavior is similarly observed for ﬂame spread over pure solid fuels. There are several
additional complications that enter in attempts to rigorously use the principle for cases
with varying conductivity. One is that the combustion heat is not constant but depends on
the conductivity (see Figure 7). Another is that the condensed phase enthalpy is
dependent on the conductivity (see Figure 2). Interpretation of the ﬂame spread rate with
case-dependent combustion heat and streamwise condensed phase convection introduces
an interesting problem. The resolution of this problem is beyond the scope of this chapter.
3. Conclusions

The problem of ﬂame spread over an anisotropic polymer solid is investigated again
with the introduction of a realistic ﬂow ﬁeld accompanied by melting of the anisotropic
solid. The condensed phase therefore consists of an isotropic liquid near the heated

interface and an anisotropic solid surrounding it. The ﬂame spread rate is computed and

142

compared to a numerical case without melting and an analytical formula for a pure solid.
It is found that the ﬂame spread rate resembles the former case by producing an almost
inversely decreasing curve with respect to the transverse conductivity. The analytical
formula has a sharper rate of decrease with increasing transverse conductivity. At the
reference state where only the isotropic solid and the liquid are present, the ﬂame spread
rate is lower than the analytical formula, indicating that the energy barrier of the melting
lowers the ﬂame spread rate. In addition, the ﬂame spread rate generally does not show

independence of the longitudinal conductivity in that it actually decreases with increasing

ksx. A very weak dependence of the spread rate is found near ksx z 0 , indicating that it
is possible that a small range of ksx exists where the ﬂame spread rate is essentially

independent of ksx. The discussion of [43] considered primarily the limit of vanishing

streamwise conductivity, demonstrating convincingly that in the limit the inﬂuence of the
conductivity is nil. There was no detailed discussion of large streamwise conductivity

except to note that it does not appear in the theoretically derived ﬂame spread formulas.

In addition, the size of the liquid layer was found to be larger when a lower ksx or k5,.

value was employed. This indicates that the lower conductivity in any direction results in
a lower rate of heat loss by condensed phase conduction. The interface phenomena are
also investigated by means of the interface temperature, the mass ﬂux, and the net heat

ﬂux to the condensed phase. It is found that a smaller size of the heated region or heat

ﬂux region results from a higher magnitude of ksx or ksy. Energy balance analysis is

also applied to the ﬂame spread process. It is observed that the magnitude of each heat

transfer mechanism depends inversely on ksx and ksy. In general, a higher magnitude of

143

either ksx or ksy results in a smaller ﬂame size. Another interesting phenomenon is
found with respect to the importance of the streamwise conduction in the condensed
phase. When ksx has a large magnitude (0.5 W/(m.s)), the streamwise heat conduction in

the condensed phase has a peak value near the ﬂame front. This peak represents the heat
gain of the control volume and its magnitude is comparable to the magnitude of

streamwise convection in the gas phase. Downstream a peak of nearly the same
magnitude (but of course opposite sign) is found for the heat loss. A low ksx value does
not result in high streamwise conduction in the condensed phase.

Although the dependence of the spread rate and other quantities on ksx was

generally signiﬁcantly weaker than the dependence on kSy (see e.g. Figure l), the

inﬂuence of ksx on the overall ﬂame character (size, shape, temperature distribution) is

very important. The streamwise conductivity is a signiﬁcant determinant of melt layer
thickness and extent (inverse relationship) and, therefore, both ﬂame size and ﬂame

strength.

144

Flame Spread Rate (mm/s)

ksx (WIm/K)
Figure 1 (a) Flame spread rate vs. ksx while ksy is kept constant.
0.5:V-wa— ,1.“ -- - - -_ - , - -.
L T

0.45L ‘1

0.4L , ..

Flame Spread Rate (mm/s)
0
° '6:
00 U!

.0
N
01

-— analytical: without melting

l

l

l

i

i

l

l

l

i

i

r . . . .
. -'- Numerical: wrth melting
I

i

0.2 r

l

i

|

l

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

ksy (W/m/K)

Figure l (b) Flame spread rate vs. ks), while ksx is kept constant. The analytical result

comes from de Ris formula for ﬂame spread over pure solid.

145

E

E,

> E ;‘E " ..1
i

_0.5 L. - ksx=0.005 at 10.555 with ignition time of 4.715
; -- ksx=0.1 at 10.555 with ignition time of 4.855
L ksx=0.25 at 11.065 with ignition time of 5.115
. . ksx=0.5 at 11.575 with ignition time of 5.745
_1L_-__a‘___a.__-_l._ﬁr_ __.__L_________d_____ __.

5 6 7 8 9 10 11 1 13 14

0.5 L— e we eeeeeeeeeeeee — — —
i
i
i
0 ,. ,
E i \ ./’
E i f , , ’
I i \ x. . f
i
1
43-5r - ksy=0.005 at 8.525 with ignition time of 2.385
L -- ksy=0.01 at 9.02s with ignition time 012.715
ksy=0.025 at 9.525 with ignition time of 3.54s
1 . ksy=0.1 at 13.15 with ignition time of 7.035
1
i
-1 L.._ ____.l;___._.___;_ m L... 7 _ -‘ _. ,1, i 21. . ,1 .. m 7- , E E
5 6 7 8 9 10 11 12 13 14
x (mm)

Figure 2 (b) The phase contours for four cases with different ksy. Note that ksy also

expands the streamwise extent of the liquid layer.

146

Surface Temperature (K)

700. ————— ._ __.- - a.._ _-_.-

650L~
600L
2 LI - ksx=0.005 at 10.555
2 550L -- ksx=0.1 at 10.555
3 1 -- k =025at11065
g ' ' 5x ' '
E 500.. ksx-0.5 at11.57s
a) i
I" i
09 i
8 450r
t .
3
w ?
400r
i /
350} ,
i ,.
300L""***‘——’:*"/—f’ee~~vi@-7LeA-a ,
5 10 15
x (mm)

Figure 3(a) The surface temperature for four cases with different ksx.

700L—_mr—~—~—wnw_e—~r~e—_ z,,-z, ,1, ,_. -..- , ,- ,7),
i J., ...
6503- 5; '”*--;\‘\
600% 1
. - ksy=0.005 at 8.525 ‘
j -- k =0.01 at 9.025
550,- SY
i ksy=0.025 at 9.525
i .u k =0 1at13 us
500L— 5’
l
l
450-
400i
l
350?
300L____m—..Lr_—’________ :1_~-_ _ _. ._ __- _L , a -, - ,.
5 10 15

x (mm)

Figure 3(b) The surface temperature for four cases with different ksy .

147

Surface Heat Flux (J/mz/s)

Surface Heat Flux (J/rnz/s)

x104

10 I
i
l
l A
8L ’17
f - ksx=0.005 at 10.555 11,
j -— ksx=0.1 at 10.55s 1 ,*
5» ksx=0.25 at11.065 l i
. k = . , ' \
1 5x 0 5at11575 l \
1 i
i .1
4L. Ll
. ‘l
i 1
2i—- 1" “
i
I
0L.» i _ i i E i , i“.._ra....t-ee:1r; ff A -, a, A i #1 7 E i i v . . , ..-:
0 5 10 15
x (mm)

Figure 4(a) The surface heat ﬂux for four cases with different ksx .

x 104
12f—7—n—v—fee fee —7 T—r new , , , e
10 .. i
i
. i;
8L LL]
, L 7
1: £ - ksy=0.005 at 8.525
1 I g —- ks =0.01 at 9.02s
5r- .: 1‘. ksy=0.025 at 9.525
L l; l. ks :01 at 13.55
‘ 9 '1\ Y
i r ‘.
1 4* \
2%» I L“ L” ‘ M *‘
E ,-
Oi_._-____.l..-__‘.,.__.g{‘f’i’:;u_ ..-, .2 v . i .- i. . . -.‘
0 2 4 6 8 10 12 14 16 18
x (mm)

Figure 4(b) The surface heat ﬂux for four cases with different ks), .

148

Surface Mass Flow Rate (Kg/mzls)

Surface Mass Flow Rate (kg/mzls)

0.014 ,- ,- -

0.012»
0.01» ' .
- ksx=0.005 at 10.555 '
0-008' — ksx=0.1 at 10.555 ‘
ksx=0.25 at 11.055
0.006; ksx=0.5 at 11.578
0.0041
0.002 ~-
0‘___. . L -. ____-a_. . .. LA ._,.'
0 5 10 15

x (mm)

Figure 5(a) The surface mass ﬂux for four cases with different ksx .

0.014.~——- ————— -. —

0.012 —

.o

o

_L
I

- ksy=0.005 at 8.525
-- ksy=0.01 at 9.025
0.008; ksy=0.025 at 9.525 .
l . ksy=0.1 at 13.15 H

0.006

\
‘ H ‘ .‘
0.004.L \
: ‘ i,
0.002 ;»
i l
0L_i .___ ,__~ 7.__ .. L 4. L .L.. #._____; .. _.. . .-. 3..._ _ z . _.
0 5 10 15

x (mm)

Figure 5(b) The surface mass ﬂux for four cases with different ksv.

149

500 - V '9 ‘s—tiEEnTiiiiswe—FGZsmcronddction‘ F A i
l ; — — Gas Convection ‘
l . 0 Gas Combustion

 

400 L» « - Interface Gas Conduction

L L .. Interface condensed Conduction . . 0 0° 0 0
300. + Pyrolysis . o ‘ .

~ , e—s—~ Streamwise Solid Conduction . .0 °

; i -_ Total Heat in Solid CV , 0
200L '5” * --- CL °

 

x (mm)

(6-a) ksx = 0.005

 

500:" L1“ -_ Streamwise Gas Cogdﬁctivori v 9

‘ . _. — Gas Convection
400: ; 0 Gas Combustion
- — - Interface Gas Conduction
i ..H Interface condensed Conduction
300 .5 L i» Pyrolysis
F 9 -..-.. Streamwise Solid Conduction .

f - ~-~- Total Heat in Solid CV i

l
200:5 lee ~ —-—~;~»- W h_-ﬁ_,-_-z, e ooooooooooooe
. 7... .

l o.
i o _ ,_
E 100 o
c ‘ '
1 . .f_,.._-_;:‘""L L 0 ,,..-~»\a-_P w
0 emWﬂv were”: .. H- “ :7
i L *5 T ' 7 L
-1005 S‘x..-
‘. 77‘ ‘ v - - _.
i
-200 i1
i
l
-300 L... w _. _ ._ _ _ _ _. .___, l... __. _._ .._ .m.. _ . . . .
0 5 10 15

150

500-- VHFWLAHUHM - ._
1 1 -— - Streamwise Gas Conduction
1 , v Gas Convection

400 i- 1 0 Gas Combustion

; 1 - , — Interface Gas Conduction

300 LL 1 -~—— Interface condensed Conduction ; . . o o o o o o o
‘1‘ Pyrolysis 1 o , o
: 1 -....- Streamwise Solid Conduction ‘ . °

200 L ‘ _ Total Heat in Solid CV .'

 

15
x (mm)
(6-c) Reference state
'21—, F Sife‘a-Lm—wigeméwagaonducﬁoh. - -i
500,’ ; —-—— Gas Convection
i 1 0 Gas Combustion 1 . . o
400; . - — - Interface Gas Conduction + .. 0 °
L 1 —-«—~ Interface condensed Conduction L .0.
i 1 "1* Pyrolysis 1 .o'
300 L L 4* Streamwise Solid Conduction L . '
1 ~ -. -1 Total Heat in Solid cv 1 . . '
i L“ ' " T“ ~‘TIT ’ _’"' v "_T' v"? *’ A ’ A ‘ .
200} .'
l O
i 0
g 1 o
100 l' g, 7
O ‘. 7. _. "
l .1" z].
OrWW Ski’.‘ *‘""“““‘“‘r
-100 i x
-200L1 ‘ ‘ - 1
i ““x..\ \V
_300L-_ _ _ _,_ z , v E 1 - A A , z , ..-, 1 55:"
5 10 15
x (mm)

(6-d) k,, = 0.005

151

500 1L « ~ Streamwise Gas Conduction
1 j — -.- Gas Convection
4001- 0 Gas Combustion
. - . Interface Gas Conduction
L «e Interface condensed Conduction 1

 

BOOL L + Pyrolysis ..oooooooo
1 j , .._ Streamwise Solid Conduction .0 °
2001— L:: T999339" Midi/1--. - . .°. .
l .0 .
i . L ,1"
o
‘ . ‘ V .
0 ewwmwﬂrﬁ f»: : “”7““ f;
-100 \ \ \
-200 9‘“ -
-3ooi-_--_-,,z,rz-5-,z -1 ,1 _, ,,,a,., ,, -, - .. ...
5 10 15
x (mm)

(6—e) ks}, = 0.1

Figure 6. Distribution of heat ﬂow rate (W) for a control volume in the gas phase and a
control volume in the condensed phase ahead of the ﬂame along the polymer surface for

three cases: (a) ksx = 0.005; (b) ksx =0.5; (0) reference state; ((1) ksy = 0.005; (e)
ksy = 0.1. In this ﬁgure, for the purposes of clarity, positive values are assigned to ch,

Qcom i ngi and QT (above y = 0); and negative values are assigned to chg’ pr,
Qws and ch (below y = 0),

152

1000!.-#_._,___-._.7____.._..__.___ _ ____ _ _. _ _ -_

900 r

g
__g 800L-
§ 5 § - ksx=o.oos at10.55
I
E 700?. p u. ksx=0.1at12.Ss
g, f i ksx=0.25 at 11.03
g j . k =o.5 at12.0s
(U 600 SX
5 ; l
a
8 soor \
E l
e . j
400 /: \
\f\\
300%-__ Au_s‘v.n.y,it_s a,A, ,,,R N «Na?
-6 - -2 o 2 4 6 8 1o 12
y (mm) x 10 3
(7-3)
moo-use —e—ijiiiii f a
c l
l E3" '
1200) 1
2
., l
% 1000i 3 x - ksy=0.005 at 8.528
g 3 -— ksy=0.01 at 9.02s
3 l i ksy=0.025 at 9.52s
g, 800%r l . ksy=0.1at13.53
E a
e i '
3 l
~ 600 }
g ’ f
l
E -. l
400% I l
i __ _ _‘ ”_A “M- ~ "1'-
200i-uuu A; a m _l ,, A; A 7 , % W, H , A , , . , ,
-6 -4 -2 o 2 4 6 8 1o 12
y (mm) x 10'3
(7-b)

Figure 7 Temperature along a slice at the center of the heat surface with both the gas
phase and the condensed phase plotted. Two cases are for (a) ksx (b) ksy.

153

Ignition Delay Time (s)

Ignition Delay Time (s)

0 I- _ ,_._ ._ _._ a _, _ , , “I , A, ,, , ,. , A , , , , ,, ,, , .
0 0.02 0.04 0.06 0.08 0.1 0.12
k

SX

Figure 8 (a) Ignition delay time vs. ksy while ksx is kept constant.

6:” a i I,” 77777777 a
5.5}
51‘

0 0.05 0.1 0.15 0.2 0.25 _ 0:3 035*" 04* 0.45 04.5

Figure 8 (b) Ignition delay time vs. ksx while ks), is kept constant.

154

Chapter 6
ENERGY BALANCE ANALYSIS OF IGNITION OVER A MELTING POLYMER

SUBJECTED TO A HIGH RADIATION HEAT FLUX IN A CHANNEL FLOW

1. Introduction

In describing the ﬂammability of combustible materials one attempts to answer the
question “How does the heated surface ignite into ﬂame?.” Four explanations have been
proposed. Two are based on attainment of a critical surface temperature or a minimum
volatile mean ﬂux into the adjacent gas. The remaining two explanations require a
minimum oxygen concentration and a minimum external heat ﬂux. Complications arise
because these four criteria are closely related. The surface temperature and mass ﬂux
depend on the oxygen mass fraction and heat ﬂux, and the surface mass ﬂux may be a
function of the surface temperature. The incident heat ﬂux — the sole initial “stimulus” or
“cause” — is the most primitive ﬂammability “property”. The practical measure of interest
is the time to ignition. Most tests of material ﬂammability specify the oxygen
concentration (air) and applied external heat ﬂux1 ‘.

Heated gaseous fuels and warm air, mixed to the correct proportions, can be self-
ignited. In many ﬁres with thermal radiation from existing ﬂames, a nearby layer of hot
product gases, or hot walls in close proximity to the surface are the primary reasons why
as-yet-unburned material ignites into ﬂame. This ignition may be spontaneous (self-

ignition) or piloted. The most rigorous test (for the material) exposes the heated surface

 

” This discussion is taken almost verbatim from LS. Wichman’s unpublished report “A Review of the
Literature of Material Flammability, Combustion and Toxicity Related to Transportation”.

155

to a pilot ﬂame. The two kinds of pilot ﬂames are continuous and intermittent. Unless a
gas ﬂow is present, the continuous pilot may alter the energetics of the ﬂammability test,
bathing the sample in an additional stream of heat. For “static” ﬂammability tests, the
intermittent pilot test is preferred. Recent work in ﬂammability analysis and testing,
however, has emphasized the greater reality of the continuous pilot and the gas cross ﬂow
[40,42].

In Refs. [59,60], explanations are provided for the differences in the dependence of
the critical heat ﬂux for thermally thick and thin materials. The model of [60] is more
detailed as it considers convective cross ﬂow. For incident heat ﬂux below the critical
value, the heated surface will not undergo piloted ignition even if heated indeﬁnitely.
Above the critical heat ﬂux, ignition is possible. The explanation in [59] rested on the
formulation of a simple model consisting of an energy equation in the heated solid

integrated over the heated surface. For thin fuels of thickness 1
71g = .0stng — Tm)/qig’
while for thick fuels of inﬁnite thickness
rig = p.C,,.A.(T.-g — L02 ML;
for piloted ignition with continuous external ﬂux (jig. The temperatures T 0° and Tig
denote values of the ambient and surface at the moment of ignition, respectively.
Experimental data plotted in this manner fall accurately into straight lines conforming

both the qualitative and quantitative accuracy of the theory [59]. The critical heat ﬂux is
obtained by extrapolating these straight-lines plots to Tig = 00. The more detailed

examination of [60], however, indicates that simple extrapolation is oﬁen inaccurate

156

because other physics, such as convective ﬂow, became important. Thus, the preceding

conclusions break down near the critical ﬂux, q m-,_ ig .
The above correlations relate rig and q°ig . If in addition to the ﬁre] surface: 1) The

narrow nearby gas region is also heated to jig; (2) The concentration of the reactants —-

including the gaseous oxidizer — renders this heated layer of gas ﬂammable; (3) The mass
gasiﬁcation rate is sufﬁciently high, then a self-sustaining ﬂame may be produced over
the heated combustible surface. Transition from ignition to ﬂaming will occur, resulting
in ﬁre initiation.

Many experimental methods have been developed for investigating the ease of
ignition and material ﬂammability. Generally the fuel sample, whether in a vertical or
horizontal position, is exposed to external radiation in the presence of the pilot ﬂame. The
time required to initiate sustained ﬂaming is measured along with the sample surface

temperature [61]. These data are correlated to produce an empirical ignition criterion. The

ignition criteria proposed to date are several: critical surface temperature Tig at ignition;

critical fuel mass ﬂux [62]; critical mean solid temperature [63]; for cellulosic or charring
materials, critical char depth [64]. According to [61], “The critical fuel mass ﬂux at
ignition seems to be physically the most correct, but surface temperature has proved to be
the most useful since it can be conveniently related to the ﬁre spread rate”. These four
criteria are indirect measures of ignition, though they are presumed to be closely related
to ignition. The actual ignition requires ﬁrst that the heated solid fuel chemically
decompose, resulting in the injection of fuel gases into the surrounding oxidizer to

produce a ﬂammable mixture that is ignited by the nearby pilot ﬂame or which can self-

157

ignite. In order to achieve sustained ignition, the fuel production rate from the
decomposing condensed fuel must be sufﬁcient to produce a ﬂame with a heat release
large enough to overcome heat losses to the surface and ambient surroundings.

A full computational description of material ignition requires a transient
multidimensional model for the simultaneous processes of solid phase thermophysical
decomposition and gas phase mixing, transport and combustion. Progress toward such
computationally intensive models has been made [40,42]. According to [61] the “absence
of such comprehensive models has necessitated the various attempts to develop the
(empirical) ignition criteria” that have been described above. Our intention in this chapter
is to continue the further development of such full, computational models with especial
emphasis in processes occurring in the condensed phase. In-depth material phenomena
have been included with notable successes for charring materials [65] in which an
important component of the model is the solid advection term representing the transient
in-depth mass loss. The latter model, we note is considered in isolation from the adjacent
gas phase except through the action of interface boundary conditions.

Our work is preceded by others examining similar questions. In [60] the focus was on
convective inﬂuence and their effect on ignition in the difﬁcult low-ﬂux, long-ignition-
time regime. In [40,42] the emphasis has been on the development of a full, 3-D transient
model of ignition by external ﬂux of cellulosic in a cross-ﬂow of gas (oxidizer). In
neither of these models was detailed solid (or condensed phase) energy transport
considered. In [60] it was not considered at all, thus enabling further theoretical
development along the lines of [61]. In [55] the model was used for making comparisons

with similar experiments and has provided a detailed examination of condensed-phase

158

energetics. No analytical correlations were developed. Thus, an approach will be used
here to ﬁirther the development — if possible —— of theoretical correlations that can
represent solid' (condensed) fuel ignition. In our work, the ignition of polymers is
investigated in a transient two-dimensional conﬁguration with a channel cross-ﬂow in the
gas phase, melting and pyrolysis in the condensed phase, and radiation loss at the
interface. The ignition model employs two ﬂow patterns, one from a Navier Stokes
calculation, and one using Oseen approximation for the gas ﬂow. The complications they
inﬂuence for the ignition processes will be compared in terms of different heat transfer
mechanisms. The condensed phase inﬂuences ignition by changing the conductivity, by
adding a latent heat sink in the condensed phase and by changing the latent heat and the
heat capacity. The sensitivity of the ignition behavior with respect to the material
properties is investigated for each ﬂow pattern. A simple ignition theory is derived to
interpret the ignition delay with respect to two non-dimensional material property-based
parameters. The ignition theory, by using an energy balance principle, predicts the

ignition delay time that is computed from the ignition model.

2. Results and Discussion

The mathematical model is the same as Chapter 4. In addition, initially an external
radiation heat ﬂux of (SW/m2) is applied at the region from x=9mm to x=12mm. Two
situations of ignition are studied, that is, the one with assumed ﬂow ﬁeld (Oseen
approximation), and the one with realistic ﬂow ﬁeld (Navier Stokes calculations). The
thermal and chemical properties are listed in Table 1 in Chapter 4. The kinetic data in
Table 1 are such that they result in the ignition establishments for both cases. It is worth

noting that the kinetic data in [55], which employs the Oseen approximation, cannot

159

support an ignition event for the Navier Stokes model. The kinetic data generally
determine the possibility of ignition establishment and they inﬂuence the magnitude of
the ignition delay time. The selection is such that below a lower limit for the kinetic
parameters the heat loss is so high that the ignition cannot be established; beyond a
higher limit the reaction overwhelms the heat loss and leads to explosion [66].
2.1 Ignition Delay Time

We ﬁrst discuss the ignition criteria. These criteria can be deﬁned in several different
ways, for example, the gas reaction rate, the surface temperature, the interface heat
ﬂuxes, or the mass ﬂux of volatiles from the fuel surface. They may take different forms
but must ideally characterize a unique instant of time. In the numerical model, ignition
has initiated ﬂame once the “self-supportive” heat ﬂux feeding the condensed phase is
“large enough” (lOW/cmz). The notion of “self—supportive” indicates that this heat ﬂux
comes solely from the gas conduction, not external radiation (SW/cmz), implying that
once the external radiation is removed after ignition the ﬂame can propagate itself. As
indicated from the chemical kinetics, a higher pre-exponential factor will generate more
heat, thus a higher temperature gradient at the interface. A lower activation energy will
lower the threshold of the chemical reaction, which results also in a higher temperature
gradient. Since the temperature gradient determines the conductive heat ﬂux, the ignition
time itself can be characterized by the heat ﬂux. The ignition delay time is deﬁned as the

elapsed time to ignition from the moment external radiation was ﬁrst applied. Figure 1
reports the ignition time vs. (a) 1/ St , (b) 1;, , and (c) 5P1 for the two numerical models
using different ﬂow patterns. It is observed that the ignition delay times for the two ﬂow

patterns are different in magnitude for identical values of 1/ St, 1:, or E p]. The

160

difference arises from the interaction of the ﬂow with the condensed phase, which will be

examined later in terms of the global energy balance principle. Examination of the
ignition delay time as a function of 1/ St , or 1:1, or 61:] in Figures 1 and 2 allows ready
explanation of the physical inﬂuence of the material properties. First, the ignition delay
time increases with 1/ St . The factor 1/ St is proportional to the latent heat of melting;
hence the increase of 1/ St increases the energy barrier for heating the condensed phase
and hinders the ignition. The limit of non-melting is attained when St —9 0° , which is the
regime of solid ignition. Second, the ignition delay time increases with 1:]. A higher 1:,

denotes a higher liquid thermal conductivity, facilitating the loss of the energy in the
condensed phase. The higher liquid conductivity removes the energy accumulation near

the interface by conduction, extending the ignition delay time. The ignition delay time
increases with 6P1- A higher liquid heat capacity requires more liquid preheat for the

same temperature rise of the condensed phase, thus a longer ignition delay. Figure 1
shows the physical trends as discussed above. Notice that the ignition delay time always

shows a linear dependence with respect to the three non-dimensional parameters. These
linear relationships between the ignition delay time and l/St and EP, will later be
examined by a simple theory that uses an overall or integral energy balance principle.

The relationship between the ignition delay time and it] is only qualitatively explained

here since it involves more complex phenomena. The k1 value changes not only the

temperature distribution, but also the liquid thickness, as seen in Figures 9 and 10.
2.2 Heat Transfer Mechanisms at Interface

During pre-ignition, four types of heat transfer mechanisms occur at the interface to

161

form an energy balance. They are heat conduction from the gas phase, heat conduction
from condensed phase, radiation heat loss to the environment, and radiation from the
external heat source. The last quantity is assumed to be 100% absorbed at the interface.
This treatment of radiation has neglected the effects of volatile radiation absorption from
the external radiation source, gas phase radiation emission (and scattering), and the
absorption of the gas phase radiation at the surface. These assumptions are introduced for
convenience, and have been used in models before [67]. However, Kashiwagi [68]
showed that the volatiles might absorb external radiation directed toward the surface up
to as much as 70% of the total for both PMMA and red oak in quiescent air. This
evaluation of the radiation inﬂux is thought to be not critical in the current modeling
since: (1) The longitudinal applied ﬂow ﬁeld is applied over the polymer surface in this
modeling, resembling the situation of a horizontal radiation source applied to a vertical
plate. Therefore the radiation absorption via fuel could describe a signiﬁcantly different
situation in that the cross ﬂow sweeps the absorbing and scattering pyrolysis products
downstream away from the ignition source; (2) The radiation heat ﬂux in this case is 5
(W/cmz), which corresponds to a lower external radiation heat ﬂux than in experiments,
and represents a less severe case since radiation absorption decreases with the decrease of
the external radiant heat ﬂux. The convenience of using the assumptions also stems from
the fact that it is not possible to quantitatively identify each radiation mode and its
isolated inﬂuence on ignition, by either experiment or theory. The authors acknowledge
that radiation is so complex an issue that it may deﬂect attention from the main subject of
this paper, that is, the inﬂuence on ignition of ﬂow pattern and condensed phase. At the

interface only the heat loss of surface re-radiation to the ambient is included, and its

162

presence does not substantially inﬂuence the magnitude of the self-supportive heat ﬂux,
as will be seen. The energy balance is constructed such that the external radiation is the
sum of the remaining three terms.

Figure 2 and 3 give the dependence of the four heat transfer mechanisms on the three
non-dimensional parameters 1/ St , lit—1 , and E p]. The magnitudes of these mechanisms

are obtained by numerical integration over the entire interface and over time. Physically
they denote the overall accumulated effects from the start of preheating to the moment of
ignition. The numerical derivations of these mechanisms are given in Appendix V.
Several useful points can be drawn from these ﬁgures. First, the total external radiation
behavior is observed to resemble that of the ignition delay time, since it is a linear
function of the ignition delay time. Second, solid phase conduction at the interface is the
major mechanism that beats the condensed phase, since the combustion reaction is
negligible before ignition. It is observed in Figures 2 and 3 that the majority of the
external radiation goes from the surface into condensed phase conduction. The rest
returns to the gas phase by conduction from the surface, which is used to heat the gas,
and to the radiation heat loss to the environment through the surface “re-radiation” term.
The heated gas then ﬂows downstream and ﬁnally exits the channel. Third, when
radiation heat loss and gas phase conduction are compared for Navier Stokes and Oseen
approximations, it is found that gas phase conduction is smaller than radiation in the ﬁrst
case and opposite in the second. This can be explained by appealing to the interface
temperature (see Figure 13). Since the interface temperature in the ﬁrst case is larger than

in the second case, the radiation. Fourth, condensed phase conduction is almost constant

in Figures 2-b and 3-b, unlike the trends with varying St and If, . As a coarse deduction,

163

since higher 1:, gives higher ignition delay time (Figure l-b), it appears to have a lower

heat ﬂow rate via condensed phase conduction. It is further deduced that the temperature
gradient in the condensed phase at the interface is much lower; and the thickness of the
liquid layer on average is possibly larger. The implications of this fact on the temperature
ﬁeld and heat ﬂux will be presented later.
2.3 Heat Transfer Mechanisms in Control Volume

Figures 4 and 5 give the heat transfer mechanisms for the two ﬂame models: external
radiation, combustion heat, pyrolysis, and gas convection. The gas enthalpy increase is
also given with reference to the initial condition. As seen from these ﬁgures, both
reaction rates of combustion and pyrolysis are negligibly small before ignition. Seeing
that the combustion heat will be dominant aﬂer ignition, a transition is necessary. The gas
enthalpy increase is small compared to the other heat transfer mechanisms, mainly
because of the low volume heat capacity of the gas phase. Convection has a magnitude
comparable to surface re-radiation. The sum of both constitutes the major heat loss up to
25% of the total radiation inﬂux during ignition. The inﬂuences of ﬂow are also
compared in Figures 4 and 5. The most salient one is that with the uniform ﬂow pattern
(Oseen approximation), the streamwise convection has a higher magnitude (and certainly
a higher percentage of the total ﬂux). The reason is straightforward since the Oseen
approximation removes the boundary layer, replacing it with a uniform ﬂow velocity at
the interface. The resulting convection is larger on average. Another important ﬁnding is
that the Navier Stokes ﬂow pattern has a larger percentage of the combustion heat in the
total ﬂux; and the model with the assumed ﬂow pattern shows the opposite. This

phenomenon can be explained by the difference in the ﬂow pattern. The Navier-Stokes

164

model introduces two new factors. One is the interface boundary layer ﬂow, and another
is the normal hot fuel ﬂow from the interface toward the gas into the ﬂame. The former
decreases the longitudinal ﬂow velocity, thus increasing the residence time of the fuel
and oxidizer, helping maintain higher concentrations of fuel and oxidizer. It also helps to
maintain a higher gas temperature because of the reduced convection near the interface.
The normal ﬂow out of the interface improves the mixing of the ﬁle] and oxidizer
compared to the case without ﬂow at the interface. Remember that the ﬂow out of the
interface is hot volatile gaseous fuel; it is one factor that facilitates the establishment of
the combustion reaction. Overall, the realistic ﬂow pattern has a higher temperature near
the interface, a high reaction rate, and a high heat ﬂux. A higher temperature at the
interface is observed in Figure 14(a), where the temperature proﬁle as a function of the
transverse distance is plotted. The temperature peak in the realistic ﬂow conﬁguration
(Navier Stokes) is almost 100K larger? than the Oseen ﬂow. The higher heat ﬂux
(observed in Figure 11(b)) will be examined later.
2.4 Enthalpy Increase of Condensed Phase

During the period starting from preheating to ignition, the enthalpy increase of the
condensed phase is of direct interest to the ignition delay time. Figures 6 and 7 give the
latent enthalpy increase due to melting, the sensible enthalpy increase due to temperature
rise of both liquid and solid phase, and the sum of the latent and sensible enthalpy
increases, herein named the total enthalpy increase. Analysis of these ﬁgures indicates
that the sensible enthalpy increase is several times larger than the latent enthalpy
increase. This is mainly because low magnitude conductivities are used. The resulting

liquid layer is generally small in the model. The temperature ﬁeld characterizes the liquid

165

layer thickness in the condensed phase, which is plotted in Figures 9 and 10 where the
solid-liquid interface corresponds to the temperature 500K. From Figures 6 and 7, the
total enthalpy increase is always a linear ﬁlnction of the abscissas. This important ﬁnding

helps us to construct a simple theory to account for the ignition delay time vs. 1/ St and

C

pl' Another perspective of the same ﬁnding is given in Figure 8 showing the average

heat ﬂux for the two cases. They are constant no matter what the values of 1/ St or 6P] .
Observations indicate that the ﬂow patterns do not change the average heat ﬂux with

respect to 1/ St. They do, however, with respect to 5P1- The latent increase is linear

with 1/ St ; however, it does not depend on 5p). The latter phenomenon indicates that

the liquid mass in the condensed phase is nearly a constant regardless of the magnitude of
the liquid heat capacity.

A ﬁnal remark is made here concerning the energy balance in the condensed phase.
Recall that the heat conduction at the interface and pyrolysis heat are the major heat
transfer mechanisms to the condensed phase. The net heat to the condensed phase can be
obtained by simply subtracting the pyrolysis heat from the conduction heat, and is equal
to the total enthalpy increase of the condensed phase. This was veriﬁed directly from the
numerical results.

2.5 Field Phenomena in Condensed Phase and at Interface

The role played by the ﬂow pattern can be observed by comparing Figures 9 and 10,
where the isotherms at ignition are given. The gradient of the temperature can be
estimated from the number of contours in a given region. The most signiﬁcant difference

is the reactive region, which hangs above the interface. It provides heat after the external

166

radiation is removed. In the Oseen model, the temperature gradient is lower since the
constant levels ﬁlling the reactive region are not so densely distributed. Numerically, the
resulting net heat ﬂux to the condensed phase is obtained by subtracting radiation from
gas conduction. The magnitude of the ignition heat ﬂux peak shows a difference of 2
W/cm2 by comparing the two models. This observation leads one to relate it to the
inﬂuence of the ﬂow pattern. Compared to the realistic ﬂow pattern, the use of the Oseen
velocity proﬁle decreases the residence time of fuel and oxidizer, and accelerates the rate
of heat removal, thus ﬁnally strengthening the interface cooling and decreasing the net
heat ﬂux to the interface. The surface temperatures are plotted in Figure 12, showing that
a lower surface temperature results from the Oseen model. This is consistent with the
application of a higher convective cooling rate to the vicinity of the interface. The
interface mass ﬂow rates for the two models are plotted in Figure 13. As observed in the
ﬁgure, the mass ﬂow rate in the Oseen model is so low that its magnitude is almost half
of the realistic ﬂow model. The low transverse mass ﬂux again reﬂects the enhanced
streamwise convection near the surface due to the Oseen model. In summary, the Oseen
model results in a weaker reactive region, smaller size of ﬂame, and lowered heat and
volatile mass ﬂux at interface. This statement is supported by the temperature proﬁle
(Figures 9 and 10), heat ﬂux (Figure 11), surface temperature (Figure 12), and mass ﬂow
rate (Figure 13).

The relative orders of magnitude between the radiation heat losses at the interface
and heat conduction at the interface as seen in Figures 2 and 3, can be explained readily
in terms of surface temperature. Obviously the Oseen ignition model has a lower surface

temperature, resulting in a lower radiation loss. This is why the order of the magnitude

167

with respect to gas conduction and radiation loss in Figure 2 is reversed in comparison
with Figure 3.

It is interesting to point out that the Oseen approximation, although it enhances
convection, yields a lower ignition time. This is contrary to the intuited speculation that
enhanced heat loss requires a larger ignition time. The explanation of this behavior must
depend on the critical condition of ignition. That is, ignition is established if the overall
heat gain surrnounts overall heat loss. Notice that the use of the Oseen approximation
mainly enhances interface cooling, thus resulting in a lower surface temperature through
its contribution to the interface energy balance. The decreased interface temperature
signiﬁcantly decreases the radiation heat loss. The heat ﬂux that is required to overcome
this heat loss, therefore, is not as high as the Navier Stokes case. That is why, we believe,
we observed a weaker reaction and diminished ﬂame size but shorter ignition delay. In an
overall sense, the thermal energy content of a “ball” of gas near the incipient ﬂame is
larger than for the realistic ﬂow case. From Figure 11, negative heat ﬂux to the
condensed phase is observed around the edges of the preheating region, indicating the
effect of the heat loss. The heat loss effect is attained there because the combustion in the
gas phase is too small to compensate for the loss of heat by gas conduction and radiation.
The heat loss is signiﬁcant because external radiation has been removed. In addition, the
levels of heat ﬂux correspond to the energy barriers that the gas conduction is expected to
overcome at ignition. Taking St = 2 as the reference state, the other heat ﬂuxes follow

well with their corresponding magnitudes for the energy barrier, as analyzed before. The

mass ﬂux, as shown in Figure 13, shows that higher 1:, corresponds to a higher mass ﬂux
because 1;] determines the thickness of the liquid layer and therefore the pyrolyzed mass

168

ﬂux. As seen from Figures 9 and 10, the 1:, value does determine the liquid thickness, in

contrast to 1/ St or E p, , whose values do not affect the liquid layer thickness.

2.6 Inﬂuence of External Radiation

A range of external radiation heat ﬂuxes is selected in order to evaluate their
inﬂuences on ignition. The range, which is from 1 W/cm2 to 9 W/cm2 with a step of l
W/cmz, encompasses the range used in [61] in order to make comparisons. To facilitate
analysis, the condensed phase is ﬁrst ﬁxed to the reference state St = 2, and then
St = 100. The relationship between the ignition delay and the external radiation is
obtained. From the numerical results, the ﬁrst observation is that the condensed phase
cannot support ignition below a certain limit. For the Navier Stokes model the limit is 4
W/cmz; for the Oseen model the limit is 3 W/cmz. Numerically there is no ignition
observed for an extremely long time. This result supports observations in experiments
and a statement in the introduction, that is, there exists a critical heat ﬂux below which no
ignition can be supported. The cases that yield successful ignitions are plotted in Figure
14(a), which presents the calculated ignition delay for two ﬂow models. Four

computational cases are plotted with different ﬂow model and St. A general observation

is obtained. Qualitatively, with the increase of radiation heat ﬂux cjig , the ignition delay

decreases. The Oseen model results in a lower ignition delay and a smaller range of

radiation heat ﬂux yielding ignition. These are consequences of enhanced convection heat

loss as discussed before. The second observation is that Figure 14 gives Tig cc (1/ q'ig )2

for both St =2 and St = 100. The square relationship between rig and 1/ 6),-g

indicates that the ignition in this modeling is for thermally thick polymer [59].

169

Furthermore, by comparison of St: 2 and St =100, melting does not change the
qualitative relationship of Tig 0c (1 / q'l-g )2 though yields a less steep curve than the pure

solid (as a consequence of higher energy barrier). By observation, the Oseen or Navier
Stokes ﬂow pattern does not change this square relation either.

Figure 15 shows the dependence of the surface temperature on the magnitude of
external heat ﬂux. It is interesting to point out that a higher radiant ﬂux generates a
higher surface temperature, as indicated by the order of the peak temperature from Figure
15. This ﬁnding indicates again that the ignition temperature does not stand out as a good
sole characteristic of ignition. Instead, ignition is better characterized by coupled
parameters between both gas phase and condensed phase. However, the surface
temperature is a useful engineering measure due to its simplicity. In Figure 15, the
difference between the three cases is small compared to the maximum surface
temperature. Figure 16 shows the surface temperature history for three radiation heat
ﬂuxes. The curves plotted in Figure 16 resemble Figure 3 in [61]. In addition, the
numerical results show that there is a “jump” of temperature during ignition, which
breaks the smoothness of the curves in Figure 16. The “jump” indicates the instability of
the ignition event as a consequence of the chemical kinetics. The transient instability in
fact causes the difﬁculty in deﬁning the surface ignition temperature. This is possibly the
case for the measured surface temperatures at ignition in [61], which did not show
complete smoothness and some transient variability appeared.

2.7 Simple Theory for Ignition Delay -— Inﬂuence of Condensed Phase
In order to investigate the inﬂuence of the condensed phase on ignition delay, the

external radiation heat ﬂux is ﬁxed while changing the condensed phase properties.

170

Energy balance analysis is used to quantify the functional dependences of ignition delay
on 51)] and St . Two radiation heat ﬂuxes, 5 (W/mz) and 9 (W/mz), will be used to show
the validity of this theory, using the former as an exemplar case to deduce the theory. We
ﬁrst set the scope for the case of varying 6P1 since it shows a uniform behavior for

liquid mass. Taking the whole condensed material as a control volume (see Figure 1), the

energy balance states that

Qtot = Qsen + Qlat ' (1)
The sensible heat is deﬁned as a function of temperature increase of both solid and liquid,
Qsen = ”pccpc (T '- TO )d’Cdy
CV
= leszz (T — To)dxdy + Ilpscp. (T — To>dxdy (2)
C VL C VS

= AH liquid + AH solid '

where CVS and CVL represent the solid and liquid volumes respectively. The latent heat

is characterized by formation of the liquid,

Qlat : Ls * mliquid ’ (3)
where mliquid is obtained from
mll'quid = ”pcdxdy- (4)
CVL

In order to derive a relation for the ignition delay, three statements are made, among
which two are derived from numerical observations; one is used as an assumption. The

ﬁrst statement is that the net heat ﬂow rate (J/s) into the condensed material is identical

over a range of 51;]. That is

171

(fig = Qtot / fig (5)

where fig denotes the ignition time. This is already supported in Figure 8. Note that
(71-8, is different from 6),-g in that it is a spatially integral quantity with unit of (J/s). On the
contrary, q'ig has a unit of W/mz. The second statement is that the liquid is independent

of 5 p], as shown in Figures 6-c and 7-c respectively. Because we use constant latent

heat, the liquid mass is independent of 5P1 . We have

"11' 21L AH - AHI' “d
rig =(—’%—S+—3‘M)+(-—_’_q—‘“—). (6)
qlg qlg qlg

where AH liquid can be written as AH liquid = pIC p, ”(T — T0 )dxdy since p1 and
CVL

C P] are all constants. By using C p, =C Psi" p], we deduce the following ignition

formula based on the numerical integration.

pIC [KT "" T0 )dxdy
__ mliquid LS + AHsolid ) + ( p5 CVL )* 6p] (7)

fig - _ _ ._
qig qig qig
In order to derive the linear curve in Figure 1(c), a third statement is introduced by

”(T -T0)dxdy is a constant. Its physical meaning is that the
CVL

assuming that

temperature distribution in the liquid layer does not depend on C Pl . If the “temperature
excess” in the heated volume depended upon 6131, its value should appear in this

hypothetical relationship. In this way, rig can be determined as a linear function of EPI-

The calculation of this ignition theory is carried out for four different values of 5p,

172

including 0.5, 0.75, 1.0, and 1.25. The intercepts and slopes are determined from each
case separately. Figure 17 shows the ignition delay time (a lines) calculated from this
theory and the direct result (solid line). The agreement is satisfactory since the realistic
values lie in the range of predicted sets from the simpliﬁed ignition theory. In terms of
slope and intercept, the error is within 10%, as observed from Figures 17. The
veriﬁcation of the theory indicates that the energy balance can be very useful for
interpretation of the realistic ignition delay time. This analysis also lends support to the
numerical model.

The relationship between ignition delay time and St is discussed here. Slight
manipulation of Equation (1) yields a linear relationship between the ignition delay time

and l/St,

- -C T —T°°
r.g=l—%“l+lm"q“‘d is” )lti <8)

qlg qig St

Another assumption is introduced here by stating that mliquid is independent of St . This

assumption is not stringent as will be seen. The numerical calculation of Equation (8) is
presented in Figure 18. There exists a broader range of ignition delay time encompassing
the realistic case. The slope does not match very well with the predictions. This comes
directly from the assumption of constant liquid mass, which in fact is a weak linear
T-

function of 1/ St . However, we can use this range of ignition delay time to estimate ,8 .

As engineering approximations such estimates appear useful.
The dependence of ignition delay time on 1:, cannot be obtained directly by this

simpliﬁed ignition theory. Only qualitative conclusions can be obtained because of the

nonlinear behavior of the liquid layer thickness and temperature proﬁles, as discussed

173

previously.
It is important to note that the examination of this section differs fundamentally from

that of Section 3.6. In Section 3.6 the inﬂuence of external radiation on ignition delay

was examined by changing the radiant ﬂux q'ig. In this section, the inﬂuence of the
condensed phase is examined by ﬁxing ‘iig and changing only the material properties. In

addition, gig is the net heating rate (W) into the condensed phase while q°ig is the

radiation heat ﬂux (W/mz).

3. Conclusions

A two-dimensional ignition model is examined for two different ﬂow patterns, the
one with a realistic ﬂow (Navier Stokes calculation) and the one with an assumed ﬂow
(Oseen approximation). The latter model, which eliminates the mass and momentum
equations from considerations and eliminates inﬂuences of thermal expansion, is of

course by far the simpler of the two models. The physical phenomena considered in the
models include channel ﬂow and combustion reaction in the gas phase, pyrolysis and
melting in the condensed phase, radiation heat loss and mass ﬂow at the interface.
Ignition was investigated by means of an integral energy balance analysis. The ﬁeld
parameters such as temperature, heat ﬂux, and mass ﬂux were also used in order to
interpret the ignition mechanism. From the numerical results, the ﬂow pattern has an
impact on the ignition process. It inﬂuences the magnitude of the ignition delay time, the
interface heat balance, and the reaction rate. The use of the Oseen approximation
enhances interface convection; thereby decreasing the interface temperature. The

decreased radiation loss, in turn, helps to produce a shorter ignition delay time. In

174

general, the Oseen model results in a weaker reactive region, smaller size of ﬂame and

lowered heat and mass ﬂux at the interface. The inﬂuence of condensed phase material

properties such as latent heat, conductivity, and heat capacity are investigated for each

ﬂow pattern. A theory to determine the ignition delay time is derived from the numerical

observations by employing the energy balance principle. The predicted ignition delay

times agree well with the direct numerical results. The theory is useful for interpreting

and estimating the ignition delay over a range of values calculated from numerical cases.

The inﬂuence of the external radiation is investigated by changing its magnitude in a

wide range. The numerical results indicate that the ignition time is approximately inverse

t0 the square of the radiation heat ﬂux. The square relationship indicates that the ignition

under this study is for “thermally thick” polymer [59]. The melting does not change the

square relationship though yields a less steep curve than the pure solid as a consequence

of higher energy barrier. In addition, the surface temperature at ignition is found to be

dependent on the external heat ﬂux. The higher radiant heat ﬂux generates a higher

surface temperature. This indicates the limitation of using temperature as the measure of
ignition in real multiphase coupled systems.

From computed results, it is clear that the realistic cross ﬂow inﬂuences ignition

and subsequent ﬂame spread. The qualitative dependence, however, is unchanged except

for some isolated (but important) features of the problem. One of these is the reversal of

ordering between radiation loss and convective loss, which (apparently) plays a role in

the ignition time calculation, producing a slightly shorter delay for the Oseen case as both

surface temperature and radiant losses to the surroundings were diminished. We did not

quantify the inﬂuence of realistic cross ﬂow, nor did we quantify the inﬂuence of thermal

175

expansion in the gas (which effects are permitted when the velocity ﬁeld is allowed to
vary). One might expect that thermal expansion should dissipate the heated layers,
making ignition more difﬁcult. In other words, as the layer of gas adjacent to the surface
is heated, thermal expansion enlarges this region while simultaneously diminishing its
temperature. The result is a larger ignition delay. Our work suggests that before analyzing
detailed ﬂow processes, theoretical models of ignition in gases of variable density may

yield useful results. The coupling of local thermal expansion with gravitationally induced

(buoyant) ﬂow can also substantially inﬂuence ignition.

176

 

7.5
7 ...
o 6.5}
.E
l- l
5‘ 6
8 . 4*
g 55] -"- Navier Stokes Calculation
'5, ‘ i, -o- Oseen approximation
5» . "
I,
4.55;.
4‘___‘__L___7.L-L__.im_--_-v-mimiiiiviiiia
O 0.5 1
1/St
(l-a)
10¢ ,,,,,,,,,,,,,,,, ,
9i—
8 __ -‘- Navier Stokes Calculation ,7
7; j -o- Oseen approximation ‘
I I
E 7+ .
E i .
é a ,
.9 l 7
a l -
g) l i
5 I~ - '
4 L» g/
.l//
3t,_,e:_, _ _ o a __ o a - a
O 0.5 1 1 5 2
“(k5
(1-b)

177

6.5-
23 6;-
OJ -
.E
]—
> p -
g 551‘
e T“
5-.
3 -'- Navier Stokes Calculation
-o- Oseen approximation
4.5
4L.__ .__ l- _ _ __- , -, __ _; , __l . _ . . _.'
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Cpl/cps
(J-C)

Figure l Ignition time vs. (a) 1/ St (b) [:1 (c) 6P] for the Navier Stokes model
and the Oseen model.

1200., , . , , ,
l
1000 t
l
800 ‘~
.. 5001,. ~ - External radiation
E —- Solid phase conduction
-.- Radiation Heat loss
' . Gas phase conduction
400g
J
200.

1/St
(2'3)

178

Heat (J)

. - External radiation
500 I“ -- Solid phase conduction
-.- Radiation Heat loss
Gas phase conduction

0 _ 8,19); __ , - ,LLL , , r_ n._
0 0.5 1 1.5 2 2 5
We
(2-b)
10007“ L,,,, eeeneee ,w ,_ use e Te ,
900} i_f_,r-ee**T’
800}_—e-**’ he- ’9 _
700i ,
600;
3 l
.. 500} - External radiation
g -- Solid phase conduction
400 L -.- Radiation Heat loss
I Gas phase conduction
300 t
200;
l
100 f g
0 l.__. _._. __ L; _ - .2; .7 _. ._ n___ _ ____._.,L_ ,, __ L _! L m m V i L;
0.5 0.6 0.7 0.8 0.9 1 1.1 1 2
Cpl/Cps
(2-0)

Figure 2 Four heat transfer mechanisms at the interface vs. (a) l/St (b) 1:, (0)
GP, for the Navier Stokes model.

179

1200-«---—-———.._.7.-7,777777_

1000-1-
800
3
~ 600. ..
8
I - External radiation
“ — Solid phase conduction
400i Radiation Heat loss
‘ . Gas phase conduction
,'
200i
1
O7 7 7 777....7 7 77 7 1.. 7 7
0 0.5 1 1 5
1/St
(3-a)
14oo¢————e—-. eeeee 7,7777, 77
1200!
l
1000}
7 800
:3 j ,
i? 600
l - External radiation
, -- Solid phase conduction
400 r -.- Radiation Heat loss
I Gas phase conduction
200i
l
I _ 7 . _’ _ , T 7
0 L77: 77:71.7 .7 7 7. 77 7 77, 7 .77 77 7 7 l 7, 7 ,
O 0.5 1 1.5 2 2.5
k/k
(3-b)

180

900i
8007 7
700 7 7
600: ,,,,, 7
f; l
I: 50017 - External radiation
§ . -— Solid phase conduction
400'» -.- Radiation Heat loss
, Gas phase conduction
300 L
200 7.
100 7
f ' 7 7 _ 7 7 7 7 7
017 77 7 77 _7‘77.._ 7777 7 __7 7777 7 77 77
0 5 0 6 0.7 0 8 0 9 1 1 1 1 2
Cpl/Cps
(3-0)

Figure 3 Four heat transfer mechanisms at the interface vs. (a) 1/ St (b) 1;, (c)
5P] for the Oseen model.

1200t ~~~~~~~~ 7 7 7
1000}
‘1 7,,
800
7 l - External radiation
3 ‘ Combustion heat
§ 600i .-. Pyrolysis
I ' -- Gas convection
: -0- Gas enthalpy Increase
400 r
|
l
200;
i7 7 7 7 7 _ 7 7, .7
01.13.77 .7; '* 77:777- .7777 ..r:.7_:7 77-7- 77.7—77.7. _ 7 _ _
0.5 1 1 5
1/31
(4-8)

181

 

Heat (J)

Heat (J)

1000 .7

- External radiation
. . Combustion heat
500, .-. Pyrolysis
—- Gas convection
-0- Gas enthalpy Increase

0 ea==-:€!e&c" :T‘E‘f7‘2'“¢ " --.-t; .77 a 7 7
0 0.5 1 1.5 2 2 5
k/k
(4-b)
1000;— 777777 7 7 77777
900- 7
800;
700-
600’ - External radiation
1 Combustion heat
500i“ .-. Pyrolysis
: - Gas convection
400 ~ -0- Gas enthalpy Increase
300 ~
200.
100 3
O-i'; 7 -7 - 77 E777 7-7-777 r 4'}- 7.1. _7*_'-_77.~r-_L_ — A..." wzvrr" 7-- were w77-i.-_-—_77; 7—-x 3‘. my»:
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Cp|JCps
(4-C)

Figure 4 Five heat transfer mechanisms vs. (a) 1/ St (b) 1;, (0) 5P1 for the

Navier Stokes model.

182

Heat (J)

Heat (J)

1200 r‘

 

1000?
t
800i
600 ‘ - External radiation
1 . Combustion heat
1 .-. Pyrolysis
‘ -- Gas convection
400 l” -0- Gas enthalpy Increase
200:»
0?E17:7777 7777.7:T.'7"77 77 77 77 7
0.5 1
1/St
(5-a)
14007— —————— ———t—————r————
1200;
l
1000}
800r
l
6001 - External radiation
Combustion heat
.-. Pyrolysis
t -- Gas convection
400? -0- Gas enthalpy Increase
l
i
200]
l _ 7
} - 777 7 g _ 7
0 l7 7 ,7.;:V:7.7._V.._.7-777177:77Y_7 17: :77, 17 7:7 7 f 77 ‘7
0 " '"(15 ' 1 “ ”1.5
k/k
(5-b)

183

1.5

Health

1000 f
900 ,
800 -
700 I
600 -
500 -
400 .
300 .
200 ~

100-

0 l-i ,-

- External radiation
. Combustion heat
.-. Pyrolysis
-—- Gas convection
-0- Gas enthalpy Increase

cp/c
(5-0)

ps

Figure 5 Five heat transfer mechanisms in the control volume vs. (a) 1/ St (b) 1:,

Heat (J)

12007

1000»
800*
600'.
400»

2004

(0) 5p, for the Oseen model.

Total enthalpy __

Sensible enthalpy

Latent enthalpy

(15 i 1 1.5
1/St

(6-8)

184

1200- 7 7 7

1000 .
. Total enthalpy
800?»
S Sensible enthalpy
~ 600 ~
8
I
400 l—
200 3
Latent enthalpy
0 ‘e 7 7 7 7 7 7 7 7
0 0 5 1 1.5 2 2 5
We
(6-b)
1000 f
900 ‘~
800 ~
, Total enthalpy
700» 777—9 ,7 -.77—~«v'
600 L7 _ 7 ‘- ’ Sensibledenthalpy
3 ;
E 500 .
400 —
300 7
200}
100 -7 Latent enthalpy
0'7_7 77 7‘ .7. 7___. '7 _ 77 7 . 7 .7
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
CprICps
(6-0)

Figure 6 Magnitudes of enthalpy increase of the condensed phase at ignition vs.
(a) 1/ St (b) k, (c) C p) for the Navier Stokes model.

185

Heat (J)

Heat (J)

1200 — 7
1000}
sooe
6007

400 .

1200.~ -
10007
800;
eooi
400-

2007

Total enthalpy 7 7

7,.»

a"

Sensible enthalpy

Latent enthalpy

usr
(7-3)

Total enthalpy . _ 7

' Sensible enthalpy

Latent enthalpy

‘ 05 i" 1 _ i5 2 i as

k/k
(7'1?)

8

186

1000. - - —- —

Heat (J)

900-
8001-
700 ' Total enthalpy,
600: 7 7 sensmle enthalpy
500»
400»
300'
200 -—
100 " Latent enthalpy
0,7777 7 7 77_77_ 777 .77 7 7
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Cp|les
(7-0)

Figure 7 Magnitudes of enthalpy increase of the condensed phase at ignition vs.
(a) 1/ St (b) k, (c) CPI for the Oseen model.

Average Heating Rate (Jls)

200777 7777 7
180}
160»

140 i

“__-

120." q"'—— 777777 - ~ —————— —— _774
100'

80 - Navier Stokes Calculation
—- Oseen Approximation

60 L-
l
40 7

20'

1/St
(8'3)

187

200.777 77 77 77

7 77 7 7 T 7 7 7 7 7 7

180
160v
140.7
120:_

100}

- Navier Stokes Calculation
60% - Oseen Approximation

Average Heating Rate (Jls)

0.5 0.6 “10.575 0.8" 019' 1 1.1 A 15.2” 11.3

(8-b)
Figure 8 Average heat ﬂow rate during ignition process vs. (a) 1/ St , (b) CPI for
the Navier Stokes model.

. '1 T 7 F F
l
0.8 7
l
006 i. “ ‘1
0.4 l‘ l i -' l“ i i
l t i l i 1‘
k i , 4" ~‘ ‘ i l
i 4 l 1' I g - ‘ I l 7
02 i7 .p ' ‘ f 1,: ,q . . ' l
A l ‘ ‘ iii ii if?
E o r 7 7 7 + i—T-LkH— 7‘ ~77“ T? QF.;‘V id- r~+ ivib + 7 7 7 i
V ‘. e " 7 J. 7 ;—’ 7
x V‘ 7__ i“ 7

02 i A: Fit-39Elite €7.71?“ 1
l i .

416}

418i

-1 [777 7 7 ..77 7 . 77. 77 . 77 . 7.7 7.77 7, .7 7 . . _ 7_ . _ _
0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014
x (mm)

(9-a)

188

y (mm)

Y (mm)

-0.4:7

70.67

70.837

71
0.

71
0.

7 __

t-.-
007

177

007

0.008

0.006

_ _ L 7 7 717 7 7. 7.7- 7 717 _

"0.000 0.01 0.011%
x(mm)

(9-b)

”0.012 0.010 0014

i
i
7 “H— 7 7 7~ ‘hé’f’é'47 a 7‘ Ave—o 7 7 a
<\‘> .’ ’

“0.009_ 0.01“ 0.0{1— 0.012! 0.013 0.014
x(mm)

(9-0)

189

“.‘
i

. 53.“
a saboea 77 i 7h ‘ w—iE—-g.——¢‘:ﬁ_— _' 7.» ‘ -0 .

y (mm)
o
i

70.2 7
70.4
70.07
-0.8 7

(£007 * 10.000 0.000 * 0.0i ' 0.011 V 70012 7 0.0137 0.014 ‘
x (mm)
(9-d)
Figure 9 The temperature ﬁeld vs. (a) St=2 (b) St=100 (0)]?! =2.5 (c)
5101 = 0.5 at ignition for the Navier Stokes model, the outmost line is 325K with
step of 25K inward.

y (mm)
C
i
i
l
i
i

-1 '77-.77 7 7 -7 .7 . 7x77 . .7 777. 7. .- . 7 ., ; ,
0. 07 0.008 0.009 0.01 0.011 0.012 0.013 0.014
x (mm)

( 10-a)

190

x 10
{__77_7777--7 7 '
08“
0.67-
0.417.
0.2i7
I i ‘1 7 : 4:3
~02.»- * 7 ~*
70.4.L 7 7
-0.6|}~
08f '*
0.007—”€000— #07009‘— 3.0? “0.01? ' 0.012% 0313 i 0.614
x(mm)
(IO-b)
x 1 '3
1r———— 7777777777 7771777777
1 7
0.8:7 _
1
0.6»
i
04‘r ,
0.2L
l ‘i
I i "i
I g
70.2;7 7 i
70.4% ‘; :
i
70.67
-O.8» _ 7 '
l 7 7
- 777777 7-7. -7 7-. 7 .7 - . -
0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014
x(mm)

(IO-c)

191

0.8 ~
0.6 r

0.4»

0.2»

y (mm)

1‘ 1 I
or. . , 7 7. 7 .77 7 7.7.7.777...” .. 7-.- . ..7 - .

04,»
-0.8~

0007 0.008 0.009 0.01 0.011 0.012 0.013 0.014
x(mm)

(IO-d)
Figure 10 The temperature ﬁeld vs. (a) St = 2 (b) St =100 (c)/;[ = 2.5 (d)

E p, = 0.5 at ignition for the Oseen model, the outmost line is 325K with step of
25K inward.

x10‘
121- ~ 7 7 - 7
‘ 1
10 ll
,5 3 ‘ ~— St=100
“E \ .- ks/k|=2.5
I", 6 l Cpl/Cps=05
a i 1
E l f i
E . i
4r 7 ll
(0 2 \
7 \ .’ 7777. -_
0— ______,7 \ / / - -
\
\ 7-~/ /
-2-7. 7 7 .’ .. 7 . 7- 7 . _ .
7 8 9 1O 11 12 13 14
x(mm)
(ll-a)

O)

0)

Surface Heat Flux (J/rr12/s)

/ l - St=2
* -— St=100
I l ks/kl=2.5

l Cp(Cps=0.5

11.0 ' 1‘1' ‘1 1'2 ‘ ‘1 ‘1‘3 1‘4
x(mm)
(ll-b)

Figure 11 The interface heat flux to the condensed phase at ignition for four
different cases for (a) the Navier Stokes model (b) the Oseen model.

7oorv————
650~
600}
550:

500 —

Surface Temperature (K)

\

1‘0 7 11" 1'2 1‘3 1‘4
x(mm)

(12-a)

193

650- ."\
5 / ‘
l .’ \\
L I .
600‘ / \
A i , \\
é 550L . ”’ \
e * ~’ 1
3 ~ / ~
9 500; . 1
s . \
e » g
19 450} I! - St=2 \
I
§ 1 -— St=100
E 4001 / kS/kl=2.5 .
p I cp/cps—os
3501- '

250;.77777-7717-7777_4*u’.‘ﬂ_;_ __ _77
7 8 9 10 11 12 13 14
x(mm)
(12-b)

Figure 12 The interface temperature at ignition for four different cases for (a) the
Navier Stokes model (b) the Oseen model.

x 10'3
a” 1 ~
1 ll
51- /\
7 1 l
71’ 1 1 l
g 4* - St=2
as l I \ -- St=100
3 f . 1 ks/k'=2.5
t 3f f \ cpl/cps=o.5
é i 1 ‘\
8 L -.
g 2i «5‘ '5‘}.
3 | ‘. 1
a) 1 ..
1
11 ‘1
1 "
l
l //'/
0 L7 7 _ __...r/ 7 .77, A 17 _ . ...k.-. 77.. .
7 8 9 1o 11 12 13 14
x (mm)
(13-a)

194

x10.

6:
5?
’13
"E 4.- - St=2
a, ' — St=100
at, -.- ks/kl=2'5
x _.
LTD. 3‘ Cpl/Cps-O'S
u:
m
g
o i A.
8 21‘ /‘\
t .
3 .
m / .

\.
0 ‘ .__--—+—-"' ‘//7 7 7 \ x7 7,
7 8 9 10 11 12
x (mm)
(l3-b)

13

Figure 13 The interface mass ﬂux at ignition for four different cases for (a) the
Navier Stokes model (b) the Oseen model.

1/|gnition Delay Time (1/s)

0.9 ,7 7 7 7 7
0.8 1.
0.7 :~
0.6 -o- St=100 Oseen
‘ -"- St=100 Navier Stokes
0 5L . St=2 Oseen
' 1 —- St=2 Navier Stokes
0-4F
0.3 7 ’
1
0.2L 7
0.1 ~ 9’”
0 7 7 7. . 7 7 _ 7 -7- . .
3 4 5 6 7

2
qig (W/m )

Figure 14 (a) The ignition delay time vs.

195

1/qig.

A

x10

1/Ignition Delay Time (1/s)

Surface Temperature (K)

0.9 ~

0.8 '-
0.71-
; -o- St=100 Oseen
0.6 r- -'- St=100 Navier Stokes ’
f .. St=2 Oseen
0.5 .. -- St=2 Navier Stokes
0.4;.»
0.31 '
0.2 ~ *
. //‘
0.1 ‘2’
‘7”
01777.7 7__ 7- 7 1 _ 77 7 7 __ 7' .. 7 7. 7. 77 7. 7 7. 7 . .
0 1 2 3 4 5 6 7 8 9
1m; (w2/m4) x 109
. . . . . .2
Figure l4(b) The ignition delay time vs. 1/ qig .
700,777. 7
650 ..
600 ~-
550‘
I
500; I’ \
450. / \
’ - Qext=5 ‘
. - Qext=7 .
400-
1 i -.- Qext=9
l
350*
i 7
300 P7717
250‘_ l 7.17 , .; _ .7 7 . . 7. .
7 9 10 11 12 13 14
x (mm)
(15-a)

196

650»
l
600~
16 550+ ‘
e 1 \
a 1 \
E 500?. .. .
8. 1 l‘ l
E 4501 .1 - Qext=5
c 1 I - Qext=7
é ‘ I -.- Qext=9
a) f l
350 5 i
l
300 i~ # 7. 7
250L7777 7.7 7 77777 77.77. 7.7777777 _7_77 77.77 _ 77 7 7 __ _ .7. 7 _ .
a 9 10 11 12 13 14
x (mm)
(IS-b)

Figure 15 The interface temperature at ignition for three cases of external
radiation heat ﬂux. (a) the Navier Stokes model (b) the Oseen model.

 

7001“ ~ 7 7 77, 7-7 7* 7 7 77 {T " ’
.’ i? 9wxcm27 1‘
650 f» ‘ " .
k
...17 a
,7 7 W/cm2 4‘ A”
as I ﬂ
9 at? 73" 2
E / 5 W/cm
g /
'_
0
8
‘t:
3
a)
-"- Navier Stokes Calculation
-o- Oseen approximation
300%;.~———— 7 7 7 7 *r— i '
1 2 3 4 5 6

Trme (s)

Figure 16 The maximal surface temperature vs. time for three external radiation
heat ﬂux in two cases (a) the Navier Stokes model (b) the Oseen model.

197

ignition delay time (s)

ignition delay time (s)

7
6 77-7 77. 7
T
41—
! - Direct results
3?” -o- theory from Cpl/cps: 0.5
. theory from Cp{Cps - 0.75
2 7 -.- theory from Cpl/cps - 1.0
. -- theory from Cpl/Cps - 1.25
1 ..
05777067765770.770'91 11'
Cpl/Cps
(l7-a)
9 ..
8L
7?.
5f 7 7 _g..M—__f_;1r_ 7_777 ...»; —-—~ —
5?”
: - Direct results
4 r -o- theory from Cw/Cps = 0.5
i . theory from 0 I10 = 0.75
3* - h fr cpcps-1o
1 .-teory om p/ ps-.
2L -- theory from Cp/Cps - 1.25
|
1 L
0'7 7 7 7 7 7 7 7 7 7 7 l 7 7 7 7 7
0.5 0 6 0.7 0 8 0.9 1 1 1
Cpl/Cps
(17-b)

198

I 111.2

1.3

A "123

1.8L
1.6} 7
1.4 7 7 i
g 1.2;”: '
>
7‘! 11
8 .
g 0.8} - Direct results
5 . -o- theory from Cp(Cp8 = 0.5
0.6 theory from Cp'lcps = 0.75
0 4L theory from Opp”s = 1.0
0.2.
0:77 7 71777777777.7 7 .7 77 7 777 7.7 7 77 7 7 .7 7 77:
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Cpl/Cps
(17-c)
2.
l
1.8}
16.»
1 4 L 77 77.77:.«r _._,7777-«w *
z»: 7 . -::7..7:_ 7:32-1“
a: ..L- 7 ‘ «1.47" "“"J
g 12f”
>~
3: 1F - Direct results
8 0 8' -o— theory from Cp'ICps = 0.5
3% ' theory from Cp'ICp3 = 0.75
._ 0.6 7 -.- theory from Opt/(3p!3 = 1.0
p -- theory from Cp{Cps = 1.25
0.4-L
0.2 {-
l
o 77 7 7.. 7777 77 _._17 .777; 77.77 i 7 77 l 7 7 7 .7 7 77 .7. .
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Cpl/Cps
(l7-d)

Figure 17 Ignition delay time: a simple theory and direct results (solid line) vs.
5P] for (a) the Navier Stokes model with q'ig = SW/ m2 (b) the Oseen model

qi’g = 5W/ m2 (c) the Navier Stokes model with qig = 9W/ m2 (d) the Oseen

model with (fig = 9W/m2.

199

7% i V*ﬂ p7 # 7 7
(“3 ‘s * 777
:«z 6‘
g 7
-.-. 5;
> 7.7
.5! r
a, 7
U 4,»
8 z
E ,
.9 3F - Direct results
i -o- theory from St = 0.667
2 x. - theory from St = 100
1:
0777777777777 77 777 77777
O 0.5 1
1/St
(18-a)
9r
3L
7} J} 77 47 ’7
E 6‘53 7 7
g l 7
'5 5}. ﬂ 7 7 * 7
5 11+— ’ ‘
8 4I
.5 T
E 37
t - Direct results
2 . -o- theory from St = 0.667
j -— theory from St = 100
1}}
l,
0L7777 7 7 77777.77 _77 _
O 0.5 1
1/St
(IS-b)

200

1.5

175

2.5r'

2.
E (f: '7
Q)
_§ 1.5}
> ‘ »
_cg l
8
S - Direct results
:- 1 7 -o- theory from St = 0.667
,5, .— theory from St = 100
0.5-
0}- 7 7 7 _ 777777
0 0.5 1 1 5
1/St
(18-e)
2 ._
7 1.5}
3 7 J7 - 7 7.
a: 7 ﬂ _ 7
E l “" 77 _777_ '
a. ’7
i3 .
‘° 1' - Direct results
5 -o- theory from St = 0.667
"CE; -- theory from St = 100
0.5»
0 7 . _ 7 7
0 0.5 1 1 5
1/St
(18-d)

Figure 18 Ignition delay time: a simple theory and direct results (solid line) vs. St
for (a) the Navier Stokes model with (jig = SW/ m2 0)) the Oseen model

6),-g = 5W/ m2 (c) the Navier Stokes model with (jig = 9W/ m2 (d) the Oseen
model with qig =9W/m2.

201

NUMERICAL MODELING OF FLAME DEVELOPMENT OVER POLYMERIC

MATERIALS

Volume II

By

Guanyu Zheng

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHYLOSOPHY

Department of Mechanical Engineering

2000

CHAPTER 7
MODELING OF BUBBLE TRANSPORT FOR OVERHEATED MELTING

POLYMERS

1. Introduction

The inﬂuence of bubbles on ﬂame spread can be summarized in three aspects: (1)
formation of dynamic porous structure in condensed phase; (2) absorption of heat via
evaporation mechanism; (3) surface mass injection to the gas phase. Models of bubble
transport abound in the literature, and single bubble growth and movement receive
extensive research efforts. Group-bubble transport is investigated in relatively primitive
form, normally described by one-dimensional conﬁgurations. In the current work a two-
scale transport model will be established to describe the two-dimensional heat and mass
transport of an overheated polymer, which undergoes melting during initial phase and the
following bubble generation. The mathematical model is composed of a macro-scale
transport model and a micro-scale transport model. The former, by means of global heat
transfer mechanism, determines the thermal behavior of the condensed phase including a
solid and a porous liquid melt. The latter accounts for single-bubble behavior in terms of
bubble nucleation, bubble growth and bubble movement by using simplified
hydrodynamic relationships. The two scale models interface with each other by several
global properties such as porosity, velocity, etc., which are volume-averaged quantities
from bubbles’ agglomerated behaviors. The methodology proposed in this chapter, to the
author’s knowledge, has not been used by previous researchers. Several relevant

investigations are discussed here. Amon [69] solved bubble radius history for a system

202

of bubbles growing in expanding foams. Bubble growth is induced by diffusion of a
dissolved gas into a thin cell. The single bubble is interfaced to the global parameters
(Pressure) through a mass balance relation. Arefmanesh et a1. [70] did similar studies on
the foaming bubble growth by using two scale models. The micro scale model is for a
bubble surrounded by a hypothetical polymer cell. The macro scale model includes a 1—D
heat conduction equation and a simple Darcy-type pressure-velocity relation. The two
scale models are coupled through velocity and density of the bulk materials. Wichman
[71] derived a 1-D group-bubble transport model that treats the inﬂuence of bubbles as
source terms into the mass, momentum and energy equations. Nucleation, evaporation,
and regression of the surface were obtained from this model. There are other bubble
transport models available, but they are either for single bubble, or only useful for
speciﬁc conditions. In general, most of the models cannot account for 2-D conﬁgurations,
requirement for ﬂame spread studies. The purpose of the current modeling is to provide
(1) a evaluation tool for the condensed phase bubble generation and its effects to global
parameters; (2) a new methodology to combine both the macro-scale and micro-scale
phenomena; (3) some results revealing the polymer’s inﬂuence to ﬂame development via
some global parameters. It is worth noting that the modeling is by no means intended to
exactly simulate the polymer materials due to lack of information in both experimental
and analytical aspects. The highly viscoelastic nature of the overheated polymer liquid
under ﬂaming conditions is rarely investigated except some qualitative derivation from
experiments such as viscosity’s inﬂuence in Kashiwagi’s work [13]. Analytical
formulations are scarce and involve signiﬁcant simpliﬁcations. Particularly, the dominant

mechanism for bubble growth is unknown, which is possibly either diffusion or

203

evaporation, or a combination of both. With many unresolved complexities, the objective
of the current modeling is to obtain the thermal response of a polymer subjected to
external heating in qualitative manner. The emphasis is more on the methodology than
detailed comparisons to experiments. The sub-models in the current modeling are in
primitive form, the upgrade can be implemented by replacing them with more advanced
ones in the future.
2. Physical Model

The physical model is composed of a macroscopic energy equation and single bubble
transport model. The former provides results of temperature ﬁeld, the liquid fraction and
the moving phase boundary. The last one describes a large number of bubbles’ behavior
in terms of bubble’s nucleation, location, velocity, etc. The two models are coupled
through a group of global properties that will be deﬁned in this section. One important
assumption is that the global properties such as porosity, velocity, conductivity can be
obtained by using volume averaging technique or ensemble averaging technique since
these two averaging techniques are logically sound though not completely veriﬁed in
practice. Other assumptions are introduced below.
1. The mass and momentum transport of the bubbles is assumed to be in direction
parallel to that of gravity. In other words, we do not consider the longitudinal inﬂuence
applied by surface tension driven ﬂow or viscous ﬂow.
2. Spherical bubble shape is preserved throughout the process of bubble development,
and no bubble collapse occurs during the process.
3. Bubble bursting effect is negligible.

4. Group bubbles’ interaction and its inﬂuence to single bubble’s behavior are

204

negligible. Bubble merge is not considered in the modeling.
In the above assumptions, 2, 3, and 4 are used for the convenience of calculation. Others
will be discussed further. Solution of the physical model yield global parameters such as
temperature, liquid fraction, porosity, gas velocity, and micro-scale parameters such as
single bubble’s nucleation rate, bubble radius, and bubble velocity.
2.] Single Bubble Model
2.1.1 Bubble Nucleation

Nucleation, like ordinary chemical kinetics, involves an activation process leading to
forming of an unstable initial state. Aﬁer this intermediate unstable state, further bubble
grth proceeds. Experimental investigations indicate that there exist three types of
bubble nucleation, homogeneous nucleation, heterogeneous nucleation, and mixed type.
The homogeneous nucleation, by deﬁnition, occurs when bubble creation is completely
in the liquid. The heterogeneous nucleation occurs when a third phase is formed at the
interface of two existing phases. By comparison, the homogeneous nucleation generates
more violent process and a higher temperature. In this model, it is assumed that bubble
nucleation is homogeneous. The activation process of nucleation leads to the formation of
unstable embryos. Thermodynamics indicates that there exists a critical radius called

nucleus over which a bubble grows and below which bubble shrinks. In this study, the

nucleus (with critical radius) is taken as 5X 10—6 m. Nucleation of gas bubbles in a
liquid has been studied for a long time, and numerous formulations appeared [72]. These
formulations use thermodynamic properties such as surface tension, vapor pressure and
kinetic properties. To predict the nucleation rate, the classical nucleation theory assumes

that gas bubble embryos can be described in terms of the bulk thermodynamics

205

properties. In a single component system the rate of bubble nucleation (number of

bubble/m3 s) is given by,

20' 167w3
exp(—

J=M 2
”MB 3kBT(Pg -Pz>

 

 

(1)

where M is the number of molecules per unit volume , k B is the Boltzman constant, 0'

is the surface tension, and M is the gas molecular weight, and B is constant. The
classical nucleation theory, although useful in qualitative nature, is not satisfactory
quantitatively [73]. Comparison between the classical nucleation theories with
experiments indicates that the predicted nucleation rate from Equation (1) is almost zero

[73]. One coarse though convenient approach is to express bubble nucleation by
C
J = N exp(—;) (2)

where N is the number density of the liquid, and C is constant. The magnitudes of N
and C can be conveniently adjusted to experiments.
2.1.2 Single Bubble Translational Velocity

Bubble movement is due to various determining factors such as gravity, surface
tension gradient, and viscosity gradient. Depending on different mechanisms of bubble
movement, numerous formulas have been derived. Some formulas are simply derived by
using force balance between two types of forces during steady state [73,74]; others are
empirical due to interaction of a large population of bubbles [75]. It is worth noting that
temperature gradient, as a major factor, produces surface tension gradient, viscosity
gradient, and determines bubble density. It plays an important role for bubble transport in

overheated polymers. In table 1, several simple velocity formulas are listed as candidates

206

for modeling considerations. In general, spherical bubble shape, low Re, and quasi-
equilibrium assumptions are used in their derivations. Other speciﬁc assumptions are
listed in Table 1.

Table l. Translational velocity formulas of bubble movement.

 

 

Physical Background Formula
Mechanisms: Balance of R2
forces between viscous Vb = _2_ g (p1 _ ,0 8 ) 1+ K

 

drag and buoyancy. 3 p 2 + 3K“ (3)
Assumptions: creeping _
ﬂow. [74] K_”g/’u’

 

Mechanisms: balance of
forces between viscous
drag and thermally
induced surface stress

I 2
due to temperature Vb : 72.”, [#17,an _ (pl ~ pg )gR (#1 + ”g )]
gradients. 3 (3”! + zﬂg) (4)
Assumptions: linear
temperature gradient, Tc = 3T1 /(2 + k g / k1)
linear dependence of
surface tension on

temperature. [76]

 

 

 

Mechanism: velocity
induced by viscosity
gradient due to
temperature gradient.
Assumptions: the _ 2 - alnﬂ 3T
. . Vb — ——RR _—
mduced velocrty by 3 3T 32
viscosity gradient could
be linearly superimposed
onto the overall velocity
ﬁelds. [77]

(5)

 

 

 

In table 1, Equation (3) is a well-known formula that was derived from Stokes’ law, and
it is incorporated into the modeling to predict bubble velocity. In addition, various other
formulas of bubble terminal velocity, where either high Reynolds number or surface-
active solute is participating the process etc., are reviewed in [78].

2.1.3 Bubble Growth Rate

Bubble growth is subjected to several inﬂuences such as mass diffusion, evaporation,

207

 

and surface tension effect. Three typical growth rate models are summarized in Table 2.

Table 2. Growth rate of a single bubble in liquid polymer.

 

 

 

 

 

 

 

Physical background Formula or etpation system

Mechanism: diffusion between monomer 4

and polymer. :tp( gR3)= A '0 2:1)(30 CW)2 R
Assumptions: (1) the inertia is negligible; pg R —ngR0

(2) the concentration of the dissolved gas 6 _ k

at interface is proportional to the gas W _ hp g (6)
pressure inside the bubble. (Henry’s law), R 2

(3) polynomial proﬁle of gas 477; + “‘ + p f - P g = 0
concentration in the liquid cell

surrounding the bubble [70]. p g = pgRuT

Mechanisms: balance of heat between the

heat ﬂow into the bubble and the vapor ”- kl AT

latent heat. = "r w—— (7)
Assumptions: neglect of surface tension 3 ng alt

and liquid inertia. [79]

Mechanisms: diffusion between two

species. ,

Assumptions: (1) convection from bubble R = DST? —1)[_R+ (8)

expansion is neglected. (2) saturated
solution is at the bubble interface. [80]

 

P. F]

 

 

In table 2, Equation (6) is used for foaming polymers where (l) diffusion mechanism is

dominant, (2) externally applied pressure transition is involved; (3) large population of

bubbles are involved. The assumption of a thin cell surrounding a bubble is used too

because small spacing between bubbles is observed during foam injecting molding

process. Note Equation (6) cannot yield analytical formula, instead it has to be solved by

numerical techniques. Equation (7) is for saturated nucleate boiling phenomena where the

liquid is superheated above the atmospheric boiling point and evaporation occurs at the

liquid-vapor interface. The heat is supplied from the superheated liquid by conduction

though a boundary layer. Equation (8) is derived where inter-species diffusion is

considered as the dominant. Other bubble growth models are available for some speciﬁc

situations, such as those considering rheological effects or the viscous-elastic effects [81],

208

 

CK;

For bubble movement in superheated polymer liquids, both evaporation and
diffusion might contribute to bubble growth. If taking PMMA as an example, the boiling
temperature of its pyrolysis product MMA is 373K, while the temperature of liquid layer
ranges from 500-700 K in a normal ﬂame spread situation. The nucleate-boiling type
bubble growth might be important because of large overheats. At the same time, the
pyrolysis product MMA and melted PMMA co-exist in a state of liquid, thus inter-
species difﬁrsion might also be important. It is not clear which mechanisms is the
dominant one, either theoretically or experimentally [13]. For the current phase of bubble
development, it is pointed out [75] that bubbles grow via evaporation with two phases of

development: (1) An initial phase due to the effects of liquid inertia and surface tension;
. . . 1/2 1/2

under low pressures, a Rayleigh solution gives, R(t) ~ (pgAT) t~ (pgAT) t,

and the vapor obeys the ideal gas law; (2) An asymptotic phase due to heat diffusion and

evaporation, which gives R(t) ~ (AT / ‘0)1‘1/2 (apparently the radial growth rate

satisﬁes R ~ t_1/2). In the modeling, we assume that the asymptotic phase dominates
the whole process of bubble growth.
2.2. Macro-scale Volume Averaged Equations

Before the macroscopic equations appeared in literature, micro-scale equations were
commonly encountered in most engineering applications since most materials are

conveniently treated as continuous media, for example, density is deﬁned by

p = limAM / AV. According to Reynolds’ transport theorem, the general form of
AV—>O

micro-scale transport equation of a certain quantity (1) can be rigorously derived from

209

3(1)
conservation laws and is given by a—+V-(<I>V)=VJ +S , where J represents
t

diffusion term, and S represents source term. The well-known Navier-Stokes equation
belongs to this category. The macro-scale model was introduced previously from research
of porous media in order to capture the scale that is small enough to describe global ﬂuid
motion and structure change but large enough to smooth out the details of morphological
complexities and inter-pore heat and mass transport. The direct application of micro-scale
model to bubble transport phenomena is difﬁcult mainly because of the morphological
complexities and disparate scales. To account for these complexities, an extremely ﬁne
mesh grid system is needed to determine both the single bubbles and the surrounding
liquid. Obviously this is not realistic by seeing the limited computational power. Instead
we are going to use the macro-scale equations. The morphological characteristics will be
accounted for by some global parameters through the macroscopic equations. From
literature, direct application of the volume-averaging concept into bubbling polymers is
rare; to the knowledge of the author, there is no publication available that is directly
relevant to the current modeling efforts. Therefore, the volume-averaged macroscopic
energy equations are introduced ﬁrst, and then the macroscopic properties.
2.2.1 Volume Averaged Energy Conservation Equation

Appendix VI presented the detailed volume averaging techniques used to derive the
macroscopic energy conservation equations. The energy equation in the condensed phase

is written as:
7— 3T _ — - -.- o_
pCPE-l-pCPgVV'Tzv.[kVTl—Ahvapm—Zhi Wi' (9)
. 1

Two major assumptions are used: (1) The liquid monomer and liquid polymer have the

210

same heat capacity; (2) The liquid around bubbles is quiescent. The ﬁrst assumption is
introduced for convenience and the second is examined in section 3.2.2. The physical
mechanisms considered, in turn, are transient explicit heat increase, convection resulting

from bubble movement, conduction, evaporation due to bubble growth and nucleation,

and pyrolysis. The deﬁnitions of {—5 , C'— p , k , I; , and other properties root from the

derivation process of the volume averaged equations in Appendix VI. They do not
necessarily follow the arithmetic mean formulations from intuition. For example, the
deﬁnition of CP is not a arithmetic mean of the two phases.

2.2.2 Volume Averaged Physical Prosperities

The phase function 7g is deﬁned as unity in gas phase and zero in solid phase. It is

a function of both spatial location and time. This function is useﬁil for deﬁnition of the

forthcoming properties.

2.2.2.1 Porosity 8g
For a given control volume V , porosity is the volume fraction of the bubbles in the
liquid
1
8g =VL7g(x,y,t)dV. (10)

In the control volume V , if bubble number N and bubble radius R,- are given, where i

denotes the index of each single bubble in a population, 8 g is obtained

N 3
£g=Z47rRi /3V. (11)

i=1

N is dependent on the dynamic process of bubble generation and movement in the

211

liquid. During a time period (It , the following balance always holds for a control volume,
6N : N nucleated + N in ' Nout. (12)

where 6N denotes the change of bubble numbers during 5t . As seen from this equation,
the change of N is due to three types of bubbles that: (1) are newly nucleated; (2) move
into the control volume; (3) move out of the control volume. If the bubble number density
n is deﬁned as the number of bubbles per volume (with a unit of #/m3), we have N =
n V .

2.2.2.2 Volume Averaged Density ,5
Volume averaged density ,5 can be directly obtained from porosity by using

arithmetic mean formulation
_ l
p=EJp(x,y)dV=pgeg+p,(1—eg). (13)

2.2.2.3 Volume averaged velocity \7

Concerning the convection in the liquid polymer, it is assumed that liquid polymer
contains moving gas bubbles and quiescent liquid surroundings. This assumption is
reasonable in that the viscosity of the polymer melt is normally very large, therefore
diminishing either the friction-driven liquid ﬂow from bubble movement or the surface
tension-driven ﬂow near the gas-condensed interlace. For the latter, although the
temperature gradient would be very large near the ﬂame front, the resulting ﬂow is
assumed to be hindered because of the large viscosity. In Appendix VI, the volume-

averaged velocity is deﬁned as

Vzﬁggegvg/ﬁ (14)

212

, where ﬁg and V; are the intrinsic volume averaged bubble quantities, ,5 is the

volume-averaged density. In the above equation, DEV/g3 is obtained as a single entity

since velocity and volume of each bubble are coupled. Therefore, we have the averaged

velocity in terms of group of bubbles,
7 N 3
V=(Z4”Rivbipbi)/3V.5- (15)
i=1

2.2.2.4 Heat Capacity (71,

In Appendix VI, the heat capacity C p is deﬁned as

5P =[8ngCPg + «91,01CP1l/l3 (16)
Note that G p is not a simple arithmetic average. The arithmetic mean was used by some

researchers but in less rigorous form, detailed discussions of volume averaging

techniques are presented in [82].

2.2.2.5 Volume Averaged Conductivity k

According to the deﬁnition in Appendix VI, I: has the form of
I? — k" k"
where k* denotes the effective conductivity [82]. The bounds of it— could be used as an
initial estimate. The simplest formulation shows that the magnitude of I; always lies

between the arithmetic mean and harmonic mean of the kg and k1 , that is,

l
Eb/kg +(l—Eb)/kl

 

<k_<8bkg+(1_€b)kl' (18)

213

There are more advanced formations available that provide smaller range between upper
and lower bounds.

2.2.2.6 Volume Averaged Mass Rate of Evaporation

For a given control volume V and time duration 5t , the evaporation rate 7n is

deﬁned conceptually

-ﬁ;=—p1—5't— (19)

For a given control volume, m is calculated by summing up: (1) the mass rate of change
for existing bubbles with time; (2) mass of bubbles that are nucleated. The former is

obtained by

__ N
m = 47: (“5‘ (z R?p,,,)dt/3V6i. (20)

The heat absorption term could be obtained by multiplying m with the evaporation
enthalpy in the macroscopic energy equation.
2.2.2.7. Surface Mass Flow Rate

The surface mass ﬂow rate is deﬁned as a transient function of longitudinal position
x along the interface. It works as an input to the ﬂame model in the gas phase, directly
affecting the fuel supply rate and further ﬂame size. Physically it comes from: (1) the
gasiﬁcation rate at the polymer surface; (2) bubble transport rate from inside the polymer
liquid. Experimentally, there is no quantitative information as to what proportion of each
source contributes to this mass ﬂow rate. It is assumed that the mass ﬂow rate comes

exclusively from bubble transport at surface. In Figure 2, if given a chunk of the

condensed phase covering a distance dx parallel to the surface, and its volume is de,

214

we obtain mass ﬂow rate from single bubbles by

. 3
minterface : 4” Z Ri pi /3dx- (2])

interface

where “interface” denotes all those bubbles reaching the interface. wimp“, is

dependent on the speciﬁc location along the surface with a unit of kg/mz/s.
3. Numerical Implementations

The goal of the numerical model is to predict the physical process of melting,
nucleation, and bubble transport, with emphasis on the inﬂuence of material properties.
Three responses of the condensed phase are available: (1) polymer from solid state to
melting; (2) bubble nucleation; (3) bubble growth and movement. Solution of the overall
model involves both the macro-scale model and micro-scale model as governed by
Equations (2, 3, 7, 9-20); the coupling of the two-scale models is given in Figure 1. In
general, if the temperature ﬁeld is given, the bubble radius, velocity, and number density

of each single bubble can be obtained by using the formulas presented in section 2.1.

Then global parameters such as porosity 8 , evaporation mass rate I72- , and velocity V

etc. are obtained based on section 2.2.2. Other parameters such as averaged density lb— ,

heat capacity 5 P , and thermal conductivity ic— are calculated based on 8g . After all the

global properties are obtained, the macroscopic energy equation is solved by using
numerical techniques for melting phenomena, as well as numerical techniques that will
be discussed. Again, the resulting temperature ﬁeld is used as input to calculate the next
level’s bubble development. The computational procedure is iterative between the two-
scale models. Several issues are addressed below. Figure 2 presents the computation

procedure of the numerical model.

215

(1) For each single bubble, there exists a lifetime that starts from nucleation and ends
when it reaches the surface. In numerical implementation a dynamic database is
constructed that contains all the bubbles in the liquid polymer. It not only records
parameters of each bubble (evolution time, position, velocity, density, temperature, etc.)
when bubbles are nucleated, it also records during their evolutions. When a bubble bursts
out of the surface, it is removed from the database. In addition, at each new time level, all

the bubble parameters are updated. The numerical technique is integration over time, for
example, update of bubble vertical location is carried out by Y "+1 = Y n + vldt , where

“n+1” represents the new time level, and ”n” represents the old time level, the v1 might
be a quantity taken from the average of 11 level and n+1 level. The integration duration is
dependent on bubble nucleation frequency.

(2) Bubble nucleation essentially is a discrete phenomenon with time, which corresponds
to a frequency. This matches with numerical implementation since all numerical methods

in principle discretize time and space in similar manner. The nucleation frequency is

chosen to be consistent with a time step ~of 5x10”. The inﬂuence of the oscillating

behavior will be discussed for some representative computational cases.

(3) A typical bubble nucleation in boiling condition [75] generates 10'3 —1023 bubbles
per volume per second. For such a large number, it is not possible to calculate all
bubbles’ behavior due to the formidable task of computing. As a simpliﬁcation, a rather
small number of bubbles are used to represent all the bubbles. A transient bubble at the
center of a control volume is assumed to represent all bubbles that are nucleated in this
grid at this time level. The bubble group effect is reﬂected by bubble nucleation rate in

that control volume. Note that the representative bubble is generated at nucleation, after

216

which its behavior is governed by the transport equations introduced before.
4. Results and Discussion
4.1 Properties

The geometrical conﬁguration is the same as that used before, a 2-D thick polymer.
In order to show the features of this numerical model, a constant heat ﬂux of SOOOOW/m2
is applied to the polymer surface from 9m to 12 mm. The inﬂuence of the gas is not
considered here. The physical process includes the preheating, melting, bubble
nucleation, growth, and movement. The nucleation occurs after the temperature attains a
critical temperature of 500 K. The reason is that once the temperature is over 500K, it is
in overheated liquid state since the boiling temperature of MMA is 373 K, therefore the
bubble nucleation should take place immediately. In addition, a constant nucleation rate
of 5.0e+11 (#/m3/s) is formulated. The other bubble properties are (1) the initial bubble
radius is 5.0x10‘6 m; (2) the evaporation heat is 10000000 (J/kg); (3) the conductivity
of bubble is 0.020 W/(m-K); (4) the bubble heat capacity is 500.0 J/(kg-K). The solid and
liquid properties are the same as [55]. In order to make comparisons, a reference state is
selected where the liquid latent heat corresponds St = 2 .
4.2 Time History of Bubble Development

The time history of the bubble development is observed during heating of a polymer.
Only the global parameters are presented here since they are of direct interest to us.
Figure 4, 5 and 6 show the heating of the polymers at SS, 10s and 19s with respect to (a)
temperature distribution, (b) porosity distribution, (c) velocity distribution, and ((1)
surface mass ﬂux. In addition, the temperature is compared to a model without bubble

generation. It is observed that the temperature ﬁelds resemble the model without bubble

217

generation as seen in Figures 4(a), 5(a) and 6(3). A comment is made about the
discontinuities observed in the porosity ﬁeld and the velocity ﬁeld, as seen by Figures
5(b), 5(c), 6(b), and 6(c). Recall that the bubble generation is in essence a discrete

phenomenon, characteristic of which is the bubble nucleation frequency. In this model,

we use 5X10""s as the interval between two neighboring nucleation in time. The
nucleation frequency is therefore 2000. After nucleation occurs, recall again that we use
volume-averaging method to account for the behavior of a group bubbles. Numerically,
we use each grid control volume as the control volume; thereby the porosity is obtained
by adding all the volumes of single bubbles and then divides by the grid control volume.
The discontinuity is observed during the boiling process [75], for example, the non-
uniform bubble size and number density distributions. Another example is Figure 3 in
Chapter 1 for a heated polymer. Note that although we do not consider the bubble
merging or collapse, the discontinuity is still available. The discontinuity shows the
transient clustering of bubbles in some scattered locations; see Figure 5(b) and 6(b). One
question arises; will the randomly clustered bubbles results in a random behavior of the
condensed phase. The answer is, in the short term, it does, however, in the long term, the
overall behavior of the condensed phase is well deﬁned and will not depend on the
randomly localized bubbles with time. The representative parameter is the temperature
ﬁeld, as shown by Figures 4(a), 5(a), and 6(a). It is found that the temperature constant
levels do not show any discontinuities. It is speculated that the scale of bubble parameters
such as porosity or velocity inﬂuence the overall temperature distribution and energy
transfer.

Between the model with bubbles and the model without bubbles, the temperature

218

ﬁeld has negligible difference in Figure 4(a), noticeable difference in Figure 5(a), and
higher difference in Figure 6(a). This is consistent with the deve10pment of bubbles in
that the number of bubbles increases with time. In addition, the bubble generation
impedes the heat transfer process, resulting in a smaller size of the constant temperature
contours, as observed by Figure 5(a), and 6(a). To analyze this, the inﬂuence of the
bubble on heat transfer can be categorized by: (1) the decrease of the heat capacity due to
porosity; (2) the decrease of the density due to porosity; (3) the bubble ﬂow toward the
surface; (4) the decrease of heat conductivity due to porosity; (5) the heat absorption due
to evaporation. Among the ﬁve factors, (1-3) enhance the heat transfer, and (4-5) impede
the heat transfer. The thermal diffusivity is not important since it is actually larger in
bubble model. This leads us to ascribe the major inﬂuence to the evaporation mechanism.
It is found to be so since we uses a large evaporation heat of 1000000 (J/kg). In reality
the effect of bubbles on the heat transfer is an outcome from the competition among
several heat transfer factors. The material properties indeed have a deep impact on the
thermal behavior of the bubbling polymers.

With time, the melt layer expands in space, as observed by observing the temperature
ﬁelds in Figures 4(a), 5(a), and 6(a) and the porosity ﬁeld in Figures 4(b), 5(b) and 6(b).
Unlike temperature ﬁeld, the porosity has a much higher gradient near the interface.
Recall that the porosity of a control volume is contributed by all the bubbles that have
transported from down below due to buoyancy. Since a higher vertical location has a
thicker liquid layer below it, the number of bubbles that are generated within this liquid
layer is larger. The number of bubbles that are transported to the control volume is

certainly larger due to the accumulative effects. In an extremity, if we assume that the

219

moving velocity is constant and the bubble does not grow, the porosity is an integral
function of the liquid thickness below it. Even with the variable bubble radius and
velocity, the accumulative nature of the porosity does not change. Especially near the
interface, the porosity gradient is so large that it well exceeds the gradient that farther lies
below, as observed from Figure 5(b) and 6(b). The velocity proﬁles show a similar trend
with the porosity, as shown in Figure 5(c) and 6(c). Again, the discontinuities appear in
the liquid and were discussed already. The velocity, by deﬁnition from Equation (14), is
dependent on the porosity. In addition, it is also dependent on the density and velocity of
single bubble. Near surface, it has a large gradient. It is worth pointing out that in general

it is a trivial inﬂuence on the temperature distribution, since it has scale of magnitude

10'7m/s , which is too small in terms of the convection heat transfer. It is further
deduced that negligence of the bubble ﬂow in the condensed phase is justiﬁable in this
case. However, in some special situations such as the overheated boiling, the bubbles
induce the liquid viscous ﬂow due to drag forces and affect the energy distribution
signiﬁcantly. The mass ﬂux distribution at the interface is plotted in Figure 4(d), 5(d),
and 6(d). It is found that the mass ﬂux peak at 105 and 193 are 5 times and 10 times as
much as Ss, showing the strong transient behavior of the bubble generation. The mass
ﬂux is formulated from bubble evaporation. Numerically it is obtained from the bubbles
that burst out of the surface. The mass ﬂux is an integral parameter of all single bubbles
that attains the surface. The mass ﬂux has a peak at the center of the heated surface; and
this peak corresponds to the maximal thickness of the heated layer, and obviously, the
maximal bubble porosity along transverse direction. It is consistent with the distribution

of bubble radius in the liquid layer. Note that the mass ﬂux also reﬂects the random in-

220

depth bubble behavior by showing that the mass ﬂux distributions at three times have
some local irregularities, however, in general, the mass ﬂux follows well with the
physical trends. That is, the higher the heating time, the higher the mass ﬂux.
4.3 Inﬂuence of Nucleation Rate

It is very interesting to know how the nucleation rate will inﬂuence the bubble
behavior. Intuitively, we expect that a higher nucleation rate result in a larger bubble
generation rate and larger mass ﬂux rate. This is true by observing Figure 7, where four
cases are selected with nucleation rate (a) 0.25 times; (b) 0.5 times; (c) 1.0 times; (4) 2
times of the reference state of 5.0e+ll (#/m3/s). The mass ﬂux rate is observed to
increase with the nucleation rate. The temperature ﬁeld is plotted in Figure 8 and the
temperature without the complication of bubble is also plotted. It is observed again that
the model with bubbles result in a smaller heated region due to evaporation effect. In
addition, the lower nucleation rate results in a larger heated region. In Figure 8, the order
of the constant temperature contours with decreasing nucleation rate is “0.25”, “0.5”, “1”,
“2” and “no bubble”. Figure 9 presents the porosity proﬁles for the four cases. It is clear
that the larger nucleation rate results in a larger porosity ﬁeld. The order of the size of the
porous region is the same as the order of the nucleation rate. The velocity ﬁelds for the
four cases are presented in Figure 10, where the order of the velocity still follows the
order of the nucleation rate. The velocity region is found to be the same as the porous
region in various cases. As a summary, the higher the nucleation rate, the higher the
surface mass ﬂux, the porous region, and the velocity region. The physical explanations
as to the shapes and contours are already given before.

4.4 Inﬂuence of Initial Bubble Size

221

After nucleation, the critical radius (of nucleus) serves as the initial state of the
bubble evolvement. It is of interest to know the inﬂuence of this radius on global bubble
behavior. Figure 11 through Figure 13 present the mass ﬂux distribution, the temperature
ﬁeld, the porosity ﬁeld and the velocity ﬁeld for four cases. They correspond to different
radius of 0.25, 0.5, l, and 2 times of the reference state. In addition the temperature ﬁeld
without the complication of bubbles are presented in Figure 12. Several observations are
drawn from these ﬁgures. First, it is found that the porosity varies very little with the
increase of the initial radius, although the size of porous region still increases with
increasing radius, as seen from Figure 13. Second, the velocity ﬁeld shows the similar
pattern by comparison; it increases with increasing radius (though in a weaker manner).
To seek an explanation, we go back to the bubble growth and movement formulation, as
indicates by Equations (3) and Equation (7). The velocity of the bubble is a polynomial
function of the radius; bubble growth rate, with some simpliﬁcation, is a polynomial
function of the resident time in the condensed phase-As a consequence, a higher initial
bubble radius results in a higher velocity and a lower resident time. The bubble growth
rate, on the contrary, results in a lower value due to the lower resident time. To estimate
porosity, it is a volume-averaged quantity that receives contributions from single bubbles
down below, as deﬁned by Equation (11). Two competing mechanisms are available: a
higher initial bubble radius helps to increase the porosity; however, the lower grth rate
decreases the porosity. The two competing factors control the magnitude of the porosity.
As observed from Figure 13, the two competing factors contribute in almost the same
weights, thus resulting little difference of porosity contours among the three cases. The

overall velocity of the bubble, as deﬁned by Equation (15), is presented in Figure 14.

222

From Equation (15), because the velocity is a polynomial function of the bubble radius,
17 essentially is also polynomially dependent on the radius. This is why we observe the
similar trend of V with the porosity. By observing the temperature proﬁle, as is given in
Figure 12, it is found that there is negligible difference between theifour contours with
bubble inﬂuence. The contour without the inﬂuence of bubble in this case again has a
larger heated region due to the evaporation effect of bubble generation. The negligible
difference due to the inﬂuence of initial bubble radius indicates that the bubble radius
results in ahnost the same spatial distribution among 4 cases. A bubble radius is almost
independent of the initial radius; however, the higher initial radius results in lower growth
rate. The mass ﬂux distribution is not plotted here, instead an integral quantity, the total
mass ﬂow rate (kg/s) at the surface is obtained and plotted in Figure 11. It is observed
that the total mass ﬂow rate decreases with the initial bubble radius. It indicates that the
rate of bubble removal decreases with the bubble radius.
4.5 Inﬂuence of St

The inﬂuence of the latent heat to the bubble phenomena is investigated by varying
St. Figures 15, 16, 17 and 18 report mass distribution, phase interface, porosity ﬁeld and
velocity ﬁeld for three cases of St with values of 0.667, 2 and 6. Note that the use of
lower latent heat results in a lower mass of the liquid, as reported by Figure 16, where the
size of phase interface increases with increasing St. A higher latent heat requires a
higher energy barrier, resulting in a smaller liquid region. The magnitude of the porosity
is related to the size of the liquid region, as observed from Figure 17, where a larger
liquid region has a larger porous region. This can be explained since a larger liquid region

provides more nucleation sites, larger cumulative volume (since porosity is basically a

223

integral quantity that is dependent on the thickness of the liquid below), and longer
bubble development space (longer time of evolvement). The overall effect of St is
presented in Figure 18 in such a manner that the size of the liquid region is ahnost
proportional to the number of the constant levels in that region. In addition, the pair of
St = 2 and St = 6 is not so signiﬁcantly different as compared to the pair of St = 0.667
and St = 2 because the former pair has a larger difference of the latent heat (recall that
St is inversely proportional to the latent heat). The velocity ﬁeld, as represented by
Figure 18, indicates the similar phenomena with St, with a larger magnifying effect
because the bubble velocity (Equation (3)) introduces an additional order of two for the
bubble radius. Still, near the interface, the velocity and velocity have larger gradients due
to the cumulative effect of the porosity. The total mass ﬂow rate is plotted in Figure 15,
showing an almost linear relation between the total mass ﬂow rate and St. The mass ﬂow
rate is consistent with the size of liquid region in that a larger liquid region results a
higher mass ﬂux at the surface.
4.6 Inﬂuence of Evaporation Heat

In the reference state, we take the evaporation heat as 1000000 (J/kg). In order to
investigate its inﬂuence, Figure 19 and Figure 20 present the total mass ﬂow rate and the
temperature ﬁeld for three choices of the evaporation heat. They are 0.1 times, 1 times
and 10 times of the reference state. Qualitatively, the increase of the evaporation heat
removes more energy in the condensed phase, resembling the increase of the pyrolysis
heat, and the temperature contours will have a smaller size, as seen by Figure 20. It is
observed that the case with 10 times evaporation heat results in much smaller heated

region compared to the reference state by observing that its temperature contour of 650 is

224

located along the surface. On the contrary, the case with 0.1 times of the evaporation heat
results in a less signiﬁcant difference from the reference state. The temperature contours
indicate that there exists a range of evaporation heat where the temperature ﬁeld might be
very sensitive. The mass ﬂux in Figure 19 shows that the total mass ﬂow rate decreases
with the increase of the evaporation heat. The lower mass ﬂux comes from a smaller
heated region as a consequence of the higher evaporation heat.
5. Conclusions

A transport model for condensed phase is constructed that encompasses the macro-
scale transport of energy and micro-scale transport of bubbles in the condensed phase. In
the micro-scale, each bubble is described in terms of its growth, movement by means of a
set of analytical formula. In the macro-scale model, the energy equation that accounts for
melting and heat conduction is volume-averaged by introducing bubble effects in the
condensed phase. The resulting overall equation system is organized in such a way that
(1) the micro-scale model, by using the temperature ﬁeld from the macro-scale model,
provides information on the radius, velocity, and locations of each representative bubble;
(2) the macro-scale model, by using the information of the representative bubbles from
the micro-scale model, calculates the volume averaged energy equation and obtain the
temperature ﬁeld. The macro-scale and micro-scale model are coupled together by a set
of global parameters such as the velocity, the porosity, the mass rate of evaporation,
which are directly derived from the volume averaging. The nucleation formulation is
used to control the creation of the bubbles. A dynamic bubble database is constructed to
maintain all the bubbles that are created in the liquid. The overall model is solved

iteratively by numerical techniques. A representative 2-D case is calculated for the

225

transient heating of a polymer material subjected to a constant external heat ﬂux at the
surface. The time history of global parameters are presented and analyzed. It is found that
there exist some discontinuities of porosity or velocity in the transient behavior of the
condensed phase due to the discrete temporal formulation of bubble nucleation. However,
the bubbles do not show the randomly or discontinuous behavior from the temperature
contours. In order to evaluate the dependence of the model on the materials properties,
sensitivity analysis is carried out for the bubble nucleation rate, the bubble initial radius,
the latent heat, and the evaporation heat. It is found that these material properties have an
impact on the transport process of the condensed phase. Generally, the increase of the
nucleation rate, increase of the evaporation heat and decrease of the latent heat magnify
the bubble transport; increase of the initial bubble radius does not change signiﬁcantly
the bubble behavior due to the complication of the bubble velocity and growth
formulation. The bubble transport model shows the potential in predicting related
problems such as the ﬂame spread over the condensed phase with melting and bubble

generation.

226

 

 

 

_. TMichScale Model ' . . Macro-scale Model V

 

 

 

 

 

3 Bubble velocity 1
—» @

 

 

5 Bubble radius 1
E Number densrty |

Figure 1 Relationship between macro-scale model and micro-scale model.

 

0

Temperature

 

 

 

 

 

mterface

 

 

 

 

Figure 2 A control volume selected to obtain mimaface.

227

 

Macro-scale model

Micro-scale model

 

[ Current time level ]

l

Initial Condition

l

Macro-scale Energy

 

 

 

 

 

 

 

 

 

 

 

 

Scan all grid points

 

 

 

l

Nucleation? N0

 

 

 

 

 

i Yes

Insert new bubbles

 

 

 

 

 

 

Scan all bubbles in
the database

I

Update locations and
properties

I

Update all the global
properties

J.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Convergence?

i Yes
[ New time level J

228

 

 

 

Figure 3 Flow chart of the iterative procedure for the solution of the two scale model

 

Figure 4 (b) The porosity constant levels at 55.

229

l
‘ interface
‘77 7 7 7 7 7 7 7 7 7 7 7 77777777675797.7400 7 7 7 7
0 were 50,.-- e. o
y ~ $500“ ‘ I "
(m . - bubble , ~ ~ ~ ~~ 77
m) ‘ . no bubble .
I
-o.5i
l
_1 7 777 __.7. 77_7_7 7 77 7 7 7.77
0 2 4 6 8 10 12 14
x(mm)
Figure 4 (a) Temperature constant levels at 58
0.27 777777777777777777 7.
l
0.15}-
0.11»
0.05,»
0'77 7 7 7 7 9702 7 mm“ 7
E l 49.0-15- .
E -o.05» .- p.01
~01}.— 7799057 7 7
i ,
p.15}
41.2}
-o.25
l
L_7 77 77 77' 7 7 771777 7 7717 77 777.77 7 7.77; 77 7 .777 7‘ 7
7 8 9 1o 11 12 13 14
x (mm)

0.2e-»—»——-—~——— —~————— ___- _____ .
0.15?

0.1-

7 i 8 _7 s 10 ‘11— 1'2_ 13 14
x(mm)

Figure 4 (c) The velocity constant contour at 58 with unit of 1x10‘7 m/ s for each label.

x10.3
1.2; 7 ~ 7 —. .7 7. 7 ,. 7 7 7

Mass ﬂow rate at interface (Kg/mzls)

0.21

0‘7 7 7 7 .7 7 7 . 77 7; 77. . .7 7 . . . . _.
0 0.5 1 1.5 2 2.5 3 3.5 4
x (mm)

Figure 4 (d) The mass ﬂow rate out of the interface at 53.

230

0.5 .7 7 _ - 7 7 7.7 7 7 7

l interface
0 i” 7 ’ ’ ’ * * ' ‘ "’ ’ " ’ ’ ’ ’ ‘* : j ’ 650 ‘

. t} ‘390 ..
’é‘ - bubble 550,
E no bubble
> 50.0

-O. 5 7.7.3750-
} \ \\ 400 i. ‘
7,77

-11777—77—77777777 7.7 7 7 7.7 7 7 77-777.975.07 7 7 7 7 .7 77
o 2 4 6 8 1o: 12 14

interface

7 8 ﬂ 9 "1o 11 2* 12 A 13 ’14 I

Figure 5 (b) Porosity Field at 10s. The constant levels start at the minimum of 0.005 with
step of 0.005.

231

Ctrm * r r r r r r r 7""“"H_,‘ ‘7‘"? -__ .. ,""'r""”’ﬁ" 7 7 7 ' 7 * 7

y (mm)
a:
O
E”

l
-O.25 +-

7 e “ "’9‘ ’ "16 ' 11 "1.5. ’ 13‘ ‘7 14
x(mm)

Figure 5 (c) Velocity constant levels at 105. The minimum level is 5 x10"7 with step of

5x10”.
x10.3
}777777 7777 777777 77 7 7 7
} 7 7
51
13 .
“E L
e, , .
£47 1
° l
8
‘t: l
2 l
5 3e
‘6 l
9 l
e l
7521 -
:2 1
m .
2 i
1.» 7
IL..— 7_ 7 777.7_ 77 177 777-7777 77 7777. 7. 77 7777 .777 .7 l 77 77 }
0 0.5 1 1.5 2 2.5 3 3.5 4
x(mm)

Figure 5 (d) The mass ﬂow rate out of the interface at 103.

232

 

07 7 7 7 _ 7 7 7 luterface 777777 7 1
-1252--- w
E - bubble ‘177777900450
no bubble 1 ‘- ‘
5 51110455? / ,
>‘ t s'ooe ,350
41.5: 1 1 i . .'
l
1 \ 7“ «’ I
-1 ‘7 7 7 7 7 7 7 7 7 7 7 7 '7 l7 7 77 1. 7
o 2 4 6 8 1o 12 14
x(mm)

Figure 6 (3) Temperature ﬁeld at 195.
0.21—7777—777 7777—7 77
0.15,.

01.
0.05?
1 interface
al.—7777 777 __ 77 7 .7
A l r T 7‘
E -o 05f—
>~ 1
-0.1 r» E
-0151
l I 1
ozt x
I ‘ .
p.25} 14,; 7 1
i 1‘ #7 _ 7 .
7 8 9 1o 11 12 13 14
x(mm)

Figure 6 (b) Porosity Field at 198. The constant levels start at the minimum of 0.005 with
step of 0.005.

233

0 1 interface
r PM ~

Y (mm)
b
8

-o.25‘7 - ‘ 1

7_ .7 7 7 71.7 7 .7 77_ 7 7 7__”1‘!‘_-_;;==-7-==.;:'#.7 77 7777.

7 8 9 1o 11 12 1 ‘ 1‘4"
x(mm)

Figure 6 (c) Velocity constant levels at 198. The minimum level is 5 x10“7 with step of
5 ><10"7 .

0.0157777 7 7 7.7 - 7 7 7 7 7 77 7

Mass flow rate at interface (Kg/mzls)

0 '77 77 '7 77' 7 7 7 7 1 7 7 :77 77 .77l 77 777- 77, 7 i : . 7 .
0.5 1 1.5 2 2.5 3 3.5 4
x (mm)

Figure 6 (d) The mass ﬂow rate out of the interface at 195.

234

oms77—77777777777777777777777777777,,

E‘

“ls 0.01»

B)

29 - 0.25times

_§_ -- 0.5times

L; -.- reference

3 . 2times

2 1

8 0005‘

‘E ‘ J"

3

co 7
0L77777777777 7777777777 77777777; 77777 \ 7777777
0 5 10 15

x (mm)

Figure 7 The surface mass ﬂux for 4 cases with different nucleation rate at 10s. The
reference nucleation rate is 5.0e+11 (1‘1t/m3 /s).

 

 

   

~interface i T F ' F F F F
‘1 3
7 7 Ii 3!
.7. 77 7 : i 7 1"?
1
:1 l
‘1 3 1
7. 7 ; 73 3 - 0.25 nuc rate 1
7: 7 7 .7 3 3 .~‘ -— 0.5 nuc rate _1
g ‘ ‘. 1nuc rate
-.- 2 nuc rate
/" 3‘ '3 - no bubble
\77 77:7 1’ /1 '1' 3' .
3 3403
7 177 777 77 7 5-7 2 L r 77 7- .77 .'
10 11 12 13 14

x (mm)

Figure 8 The temperature constant contours for 4 cases with different nucleation rate at
105. The reference nucleation rate is 5.0e+11 (#/m3/s).

235

0.2 -——_7 -.

0.15-

02777-77
0.157
0.177

0.05 ~

interface

interface

_.V—r7r

10

x (mm)

(9-b)

236

Q

m

15'

14

3 interface

3132 i 3 1‘3 14

interface

.7.— 77,’ 7.7. .7.—7,77

7 8 M 9 7 A10 “141-*— 12‘
x (mm)

(9-d)
Figure 9 The porosity constant contours in the condensed phase at 10s for 4 cases with

different nucleation rate: (a) 0.25 times (b) 0.5 times (0) 1 times ((1) 2 times of the
reference rate. The outmost constant level is 0.01 and step is 0.01.

"1‘3" 14

237

3 interface

1 interface
0 3—" ———"” —"— — "— — r—" ‘—‘.“""”'T"—""“."“"”“""‘

1 ‘ i " T"
41053 E:

7 8 ‘ 9 1o 11 h Win18 ' 18
x (mm)

( 1 O-b)

238

7 8 8 -- 3110 11“ 12 3133

14

1 interface
as, ,7 , , , , , , MT" ﬁnwmv ,

7 8 23” 1’16" 11 15 "1

71 8 9777110111 71171 12
x(mm)

(lO-d)

15" 114’"

Figure 10 The velocity constant contours in the condensed phase at 10s for 4 cases with
different nucleation rate: (a) 0.25 times (b) 0.5 times (c) 1 times ((1) 2 times of the
reference rate. The outmost constant level is 5 ><10‘7 and step is 5 x10'7 .

239

x10“5
1.81— ,-,-,.,,,ﬁ,,,,,,, . ,T,c,,,,,_

1.71
1.61-
1.5~

1.4--

Surface Mass Flow rate (Kg/s)

1.31-

1 "-o

1.2:-.,__+_!~s*,,.__ i _._s __-i__ .1 _i__ A999,,“
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

radius (m) x 10-5

Figure 11 The surface mass flow rate vs. the bubble initial radius.

1 1 :1 ii interface / r: l
: 1 1"",
I! 1 ‘i if"
1 p \ _.x 3'5"l
11» \ _._ mi
' [i \ /H g
i l!‘ \ / ii :
'01 1 17 ‘ / .1 i . l
a i ‘1 l i
g 015 H Fannie...” 1|... ' - 0.25mdius
>. ' - r 1 \ ' ' ' /' :3 1 r -- 0.5 radius
‘ i i ‘ ‘ 1 . i, 1radius
' 1 :1 \ I 1‘ 1 2 radius
-O.2L L‘ \ / f 1 , - no bubble
iii ‘\ / i 1
t l.
.. \ \ / i

i 1 l Tu ”“1”, I: I
-0.25" \ '1 ‘ _ ‘17,“ ' // I *

8 9 1‘0 _ 111 12 .113 1111.4
x(mm)

Figure 12 The temperature constant contours for 4 cases with different initial bubble
radius at 105. The reference bubble initial radius is 5 x10‘6 In.

240

ywmm)

0151
Q1 4
005» '
. interface
ok_si us,,, u,uuu,:susuususﬂ figs ,
2-0055 1
> T
01~
£151
02
025; _
7 11131111511135111111 115111131111 1
x (mm)
(13-a)
027— _~w——ea¥wssia¥~+»++sw-~--TA*—fsii
015?
i
01;
ocs}
1 interface
0; A ~ii¥w~eiiw - t
cos; —
on» 1
QJSL ﬁg
02} "1 1
025; s A
7111151 “5" 1o 111 1 121 1131 1111
x (mm)
(13-b)

241

0.15%
01
(105»
interface
0———; ~ewiii«'-ﬁ~-;ﬂw us,susssssss,,,,us_
E 4) 05? *1 g ‘ *
>‘ I
41.1 } : a
-015;
412} 1 _ 1
-o.25 s 1
|
11H11§11119111w1611111111f111s 111-1
x (mm)
(l3-c)

Figure 13 The porosity constant contours in the condensed phase at 10s for three cases
with different initial bubble radius: (a) 0.25 times (b) 0.5 times (c) 2 times of the

reference radius. The reference bubble initial radius is 5><10‘6 m. The outmost constant
level is 0.01 and step is 0.01.

0.2r'*—"*—.-—-~-~7777a7771777,1, ,

1 I
|
‘ |
l
r— i i ”7 i 7 -~ » w .. . . . —~._ _. .1._- - , .-- » - , - , 1 1 i 7 ,4

l
7 8 9 10 11 12 13 14

0.151-
0.1 .1
0.051
1 interface
0 -, _ _ _ _ _ _ ,_ .‘_- - 5.5.55: 5 ﬂ _
E -o.05} -
5 ,
-o.1 7
-o.15:-- f
-0.2‘ _
-0.25‘1
7 1111 81 1 #1911 1 1111911 m11_11 11121 1 11131 1 1
x (mm)
(l4-b)
0.2"'-VTV7i71'iAA*7”‘ﬁiivliiii
0.151-
0.1
0.05;L
1 interface
07,15 1, 17 7,, 5,5 5-15 5.-1.5-..m:15.._.-5-..-7 -9 _
E cos} 1
>1. 1
-0.1 ‘
-0151- 1
1
-0.2 k _
1
-o.25-r
L ,, _._ﬂ_ .‘_ 7 __ A; , _1_-__ L5, _5_ W5- , 5, 1 A 5, i _
7 8 9 10 11 12 13
x (mm)
Figure (l4-c)

_ . _ 1 J

14

Figure 14 The velocity constant contours in the condensed phase at 103 for three cases
with different initial bubble radius: (a) 0.25 times (b) 0.5 times (0) 2 times of the

reference radius. The reference bubble initial radius is 5 x10"" m. The outmost constant

level is SXIO’7 and step is 5x10”.

243

yonnu

Surface Mass Flow rate (Kg/s)

x104
55, , ,

o 110 126 1310 11410 50 16101 7018101 90 11100

interface

- St=0.667

-o.2r -- St=2

. st=6

-O.25 >-

-o.3

-0.35L

9 1 110 111 1 112 f 113 114
x(mm)

Figure 16 The phase interface for three cases of St.

244

0.2r————_---_____

c1151
0.1'L
(1055
; interface
E 41051-
>‘ . it _ f s
4111
-o.151 5
4121
41251
l.-__5*,u_5._,55_55__4_5 _

7 8

9

1o 1 1111
x (mm)

(l7-a)

interface

245

1.2 _

13111111411 1

’1—1-1-1. -... - g,
1

 

1 interface
0 1*" i ‘ 5 5 5 i 5 " " ""“""""‘"'::_5'.5' - ‘ «=-

W v—‘a’.

E-o.05» * ’
5 1

9 ‘ 101 111111 12 1131 1 "
x(mm)

(l7-c)
Figure 17 The porosity constant contours in the condensed phase at 10s for three cases

with different St: (a) St = 0.667 (b) St = 2 (c) St =100. The outmost constant level is
0.01 and the step is 0.01.

0.2; — — — a i
1
0.151
1
0.1%
0.05?
1 interface
05.7- 5 ,, 5 5 , 5 ,,5 _ _.5. _____ t .
E —005f» 5
> I
40.11»
-o.15¥~
-0.2
-o.257
1155 - _. 5* 5 5st 5 _s _ _ ,_1_ _ _
7 8 9 10 11 12 13 14
x(mm)
(18-a)

interface
2 -o osf ~
> L
-O.1 1" ‘ 5
-o.15» * 5
i
-o.2» 5
4125i ;
7——v— 8T7 #Qiiiﬂ10i7 i111 7‘ 1i 777137 714777
x (mm)
(18-b)
0.2p-—~Arﬁieie777—_— ___5 _ 5 __5 5
0.15? l
0.1 r'
0.057»
interface
or“ r 5 * 5 ‘ 5575:5755??? TT’Tfj-T’ 5 ’ * *‘ ’ n “
>~ ‘ 71- *‘ 5
-o.15.L J
l 5 .
i
-0.2‘L 55 _—

7 i 8“ i 9 1o H1'1 W 1111.2 if 1113 14
x (mm)
(18-c)
Figure 18 The velocity constant contours in the condensed phase at 108 for three cases
with different St: (a) St = 0.667 (b) St = 2 (c) St = 100. The outmost constant level is
5x10‘7 and the step is 5x10”.

247

y (mm)

Surface Mass Flow Rate (Kg/s)
f

0 .55 555;5 5 5 5' 5, 5 5.5 5_5r__5555__;_,__,5.5535 5 5.45 5 5' 5

o 1 2 3 4 5 6 7 8 9_ 'io
Evaporation Heat (J/kg) 7

Figure 19 The surface mass ﬂow rate vs. evaporation heat.

_ ’ '7‘“ r " ——‘n-—%—— -—-.—. T — ~~ — --———-==+.vr—- —- -~E—n—- -— —‘ — —— — — — -— — —— ——.
' i l [Lilli ‘ Interface ' r 1 , i
i i i ‘ 'l‘ J

l l‘ l‘ .' '

JR
.155 - 5 _

l

i '
:500 “' 0.1 times VI
VT" l l.

.l
I. i l . i
i}! u ,1 l | | . 1 t'mes l
,‘ : , w 999 _ m i ' 10times
-o.15;»~ “ ’ '.' ”...550 ‘ 74 ’
l i r “~ 5350 i
i i a ‘

‘ ,- ,. i i i
I t’ ‘ r
\4 'o .1 i 5 5° '

8 ' "9’ - 1'0 ' A 1‘1 12 _ 13*

Figure 20 The phase interface for three cases of evaporation heat.

248

CHAPTER 8

CONCLUSIONS AND FUTURE WORK

Transient ﬂame development over plastic polymeric materials is of both fundamental
and practical importance to fire safety. The ﬂame behavior is determined by various
coupling physical and chemical mechanisms such as (l) in the gas phase, combustion
reaction, channel cross ﬂow, reaction-induced thermal expansion, and interface injection
ﬂow; (2) in the condensed phase, melting, pyrolysis reaction, bubble nucleation, growth,
and movement; (3) at interface, radiation heat loss and fuel and oxidizer transport. A two
dimensional numerical model is established to describe such transient ﬂame spread
process with emphasis on the inﬂuence of condensed phase.

In Chapter 2, the inﬂuence of the solid anisotropy is investigated. The Oseen
approximation is applied in order to remove the complication of ﬂow coupling. Transient
behavior such as pre-heating, ignition, and steady spread is qualitatively identified. The

dependence of ignition delay time and ﬂame spread rate on thermal conductivities is
obtained by introducing a ratio of ksx Hrs}, while keeping ksxy and (ksx+ksy)
constant. With the increase of this ratio, it is found that the ﬂame spread rate increases

while the ignition delay decreases.

In Chapter 3, ﬂame spread over melting polymers is investigated by varying three

material properties including liquid thermal conductivity (1:1), latent heat of melting (St)

and liquid heat capacity (EH)- In addition, the Oseen approximation is still preserved.

The numerical model provides very good agreement with an analytical formula [28].
DeRis ‘s ﬂame spread formula, however, results in a constant spread rate value higher

249

-‘:' 34.1
.

If

than the numerical result. Physically, a lower latent heat or a lower thermal capacity

generates a lower energy barrier for the ﬂame, hence a larger spread rate. The increase of

If] , on the other hand, denoting the diffusion away of the thermal energy for preheating,
results in a lower spread rate. The dependence of ﬂame structure on St, 1;, , and 5p,

were studied. With the increase of St or If, , or with the decrease of 5p] , the ﬂame size

increased. These results are consistent with the qualitative nature of the dependence of
the ﬂame spread rate on the three parameters. The mechanisms of ﬂame spread are
interpreted by applying energy balance analysis. A ratio between the total heat applied to
the condensed material upstream of the ﬂame leading edge and the spread rate reveals the

physical mechanisms that control the preheating of the condensed material to the ignition
temperature. Comparisons of this ratio in situations of varying St, 1;] , and 6P1 reveals

the difference in physical mechanisms that control the preheating of the condensed phase
to the ignition temperature.

In Chapter 4, the inﬂuence of channel ﬂow is investigated by incorporating the
Navier Stokes equations into the overall model. The ﬂame spread formula [28] is
examined in the new situation and good agreement is again obtained between the
numerical model and the ﬂame spread formula. The effect of the gas phase on the ﬂame
spread behavior is explored with speciﬁc attention given to the injection ﬂow from the
interface and gas expansion due to the high combustion temperature gradient. Parameters
such as pressure, velocity, and streamlines are interpreted in terms of their coupling to the
temperature, species ﬁeld and interface conditions. In addition, an energy balance

analysis of a spatially-varying control volume is applied to the four cases in order to

250

understand the ﬂame spread mechanism. It is found that streamwise heat conduction in
the condensed phase has a very small magnitude compared to the interface condensed-
phase heat conduction. This result is in agreement with previous theory [55]. The
comparison between the model with the Navier Stokes ﬂow calculation and the model
with the Oseen approximation is made. The quench layer thickness is smaller in the
Oseen model than in the Flow model, mainly because a much stronger reaction rate
formulation is used in the former. In addition, it is found that the ﬂame front is located
ahead of the phase front, and the heat ﬂux front is located ahead of the mass ﬂux front.
These discrepancies can be explained in terms of gas phase thermal expansion and
interface injection ﬂow. Analysis was also carried out on the inﬂuences of gas ﬂow and
melting in order to understand the ﬂame spread mechanisms associated with their
inﬂuences.

In Chapter 5, ﬂame spread over an anisotropic polymer solid is investigated again
with a realistic ﬂow field accompanied by melting of the anisotropic solid. The
condensed phase therefore consists of an isotropic liquid near the heated interface and an
anisotropic solid surrounding it. The ﬂame spread rate is computed and compared to a
numerical case without melting and an analytical formula for a pure solid. It is found that
the ﬂame spread rate resembles the former case by producing an ahnost inversely
decreasing curve with respect to the transverse conductivity. The analytical formula has a
sharper rate of decrease with increasing transverse conductivity. At the reference state
where only the isotropic solid and the liquid are present, the ﬂame spread rate is lower
than the analytical formula, indicating that the energy barrier of the melting lowers the

ﬂame spread rate. In addition, the ﬂame spread rate generally does not show

251

independence of the longitudinal conductivity in that it actually decreases with increasing

ksx. Energy balance analysis is also applied to the ﬂame spread process. It is observed

that the magnitude of each heat transfer mechanism depends inversely on ksx and ksy.

In general, a higher magnitude of either ksx or ksy results in a smaller ﬂame size. The

lower conductivity in any given direction results in a lower rate of heat loss by condensed
phase conduction. The interface phenomena are also investigated by means of the
interface temperature, the mass ﬂux, and the net heat ﬂux to the condensed phase. It is

found that a smaller size of the heated region or heat ﬂux region results from a higher

magnitude of ksx or ksy , which is consistent with the above analysis.

In Chapter 6, a two-dimensional ignition model is examined for two different ﬂow
patterns, the one with a realistic ﬂow (Navier Stokes calculation) and the one with an
assumed ﬂow (Oseen approximation). From the numerical results, the ﬂow pattern has an
impact on the ignition process. It inﬂuences the magnitude of the ignition delay time, the
interface heat balance, and the reaction rate. The use of the Oseen approximation
enhances interface convection; thereby decreasing the interface temperature. The
decreased radiation loss, in turn, helps to produce a shorter ignition delay time. In
general, the Oseen model results in a weaker reactive region, smaller size of ﬂame and
lowered heat and mass ﬂux at the interface. The inﬂuence of condensed phase material
properties such as latent heat, conductivity, and heat capacity are investigated for each
ﬂow pattern. A theory to determine the ignition delay time is derived from the numerical
observations by employing the energy balance principle. The predicted ignition delay
times agree well with the direct numerical results. The theory is useful for interpreting

and estimating the ignition delay over a range of values calculated from numerical cases.

252

The inﬂuence of the external radiation is investigated by changing its magnitude in a
wide range. The numerical results indicate that the ignition time is approximately inverse
to the square of the radiation heat ﬂux. The square relationship indicates that the ignition
under this study is for “thermally thick” polymer [59]. The melting does not change the
square relationship though yields a less steep curve than the pure solid as a consequence
of higher energy barrier. In addition, the surface temperature at ignition is found to be
dependent on the external heat ﬂux. The higher radiant heat ﬂux generates a higher
surface temperature. This indicates the limitation of using temperature as the measure of
ignition in real multiphase coupled systems. In addition, the realistic cross ﬂow
inﬂuences ignition and subsequent ﬂame spread. The qualitative dependence, however, is
unchanged except for some isolated (but important) features of the problem.

In Chapter 7, a transport model for condensed phase is constructed that encompasses
the macro-scale transport of energy and micro-scale transport of bubbles in the condensed
phase. In the micro-scale, each bubble is described in terms of its growth, movement by
means of a set of analytical formula. In the macro-scale model, the energy equation that
accounts for melting and heat conduction is volume-averaged by introducing bubble
effects in the condensed phase. The overall model is solved iteratively by numerical
techniques. A representative 2-D case is calculated for the transient heating of a polymer
material subjected to a constant external heat flux at the surface. The time history of
global parameters are presented and analyzed. It is found that there exist some
discontinuities of porosity or velocity in the transient behavior of the condensed phase
due to the discrete temporal formulation of bubble nucleation. However, the bubbles do

not show the randomly or discontinuous behavior from the temperature contours. In order

253

to evaluate the dependence of the model on the materials properties, sensitivity analysis is
carried out for the bubble nucleation rate, the bubble initial radius, the latent heat, and the
evaporation heat. It is found that these material properties have an impact on the transport
process of the condensed phase. Generally, the increase of the nucleation rate, increase of
the evaporation heat and decrease of the latent heat magnify the bubble transport;
increase of the initial bubble radius does not change signiﬁcantly the bubble behavior due
to the complication of the bubble velocity and growth formulation.

By considering the controlling mechanisms for ﬂame spread over polymeric
materials, several possible extensions to the current model are derived. These extensions
should not separate from the theoretical and experimental deve10pment; instead they must
be heavily coupled to the latter two to provide better predictions.

(1) In terms of gas phase combustion, there is a practical need to incorporate the realistic
combustion kinetics into the overall model. Especially for a speciﬁc material, if we know
the fuel composition as a consequence of pyrolysis, it is a relatively easy matter to
incorporate the realistic kinetics into the gas phase submodel because the gas phase
combustion kinetic is well deﬁned if the fuel type and concentration are given. In this
way, the contribution from the condensed phase can be further evaluated in terms of its
fuel composition. Obviously, the technique to reduce the total steps of a typical
combustion should be used by seeing an overwhelmingly large number of reaction steps
in real situation. In addition, the incorporation of more realistic chemical reaction is
meaningful since no related paper has been published in the ﬂame spread ﬁeld.

(2) Radiation is a very important mechanism to inﬂuence the ignition and further the

ﬂame spread. Its contribution to ignition is very signiﬁcant for some particular situations

254

such as the conﬁguration with quiescent gas phase and a very high external heat ﬂux. Its
contribution to ﬂame spread is important when the size of the ﬂame grows over some
limits. The spread of a large-scale ﬁre is dominated by the radiation heat transfer. In
terms of the radiation type for a typical ﬂame spread, there are radiation absorption and
emission of the gas phase, radiation absorption and emission of interface. Their coupled
inﬂuences constitute complex phenomena for ﬂame spread process, and are less
understood relative to other heat transfer mechanisms. In order to gain understanding of
the radiation mechanism, experimental tests are necessary to obtain the quantitative
contribution of different types. Numerical modeling should be implemented by
interfacing with experiments to provide quantitative evaluations as to how important the
radiation will inﬂuence the ﬂame spread.

(3) As suggested in section (1), in order to simulate the gas phase combustion, a more
realistic description of the condensed phase pyrolysis is needed. In particular, the detailed
information of the products of the pyrolysis is needed since the heat transfer aspect of the
pyrolysis, as seen in chapter 3, 4 and 6, normally do not signiﬁcantly inﬂuence the energy
distribution of the condensed phase. In literature, there are few models that employ the
realistic pyrolysis reaction; if any, they are limited to the thermal aspects [40],
eliminating the inﬂuence of fuel species on gas phase combustion. Therefore, the possible
model development could be toward some realistic pyrolysis reaction with emphasis on
both chemical and energy aspect. Some step-reducing techniques are necessary, similar to
the treatment of combustion.

(4) The melt ﬂow in the condensed phase is not well understood. Generally it was

induced by the temperature gradient near the ﬂame front, that is, the surface tension

255

gradient. There are some ﬁrndamental questions unanswered till now. One would be,
“how important is the melt ﬂow in terms of its energy transport and mass transport to the
ﬂame spread”. To answer this question, a numerical submodel is needed to describe the
melt ﬂow. One advantage of the melt ﬂow modeling is that it is well deﬁned in
mathematical formulation, unlike the bubble transport phenomena.

(5) The bubble transport relies heavily on the material properties; different characteristics
may appear for different materials. For example, the evaporation effect as an energy
aspect may be more important to the structural changes in some speciﬁc situations, as
examined in Chapter 7. The experimental tests are especially important in providing
benchmark information for the numerical model. The model development for bubble
transport, such as the movement of the bubble and the grth of the bubble, should be
derived from experimental observations. Furthermore, the bubble transport is so relied on
a speciﬁc material that it should be in some way coupled with the experimental data to

accurately predict the transport phenomena.

256

APPENDIX I
COMPARISON OF NUMERICAL RESULTS WITH ANALYTICAL SOLUTION FOR

l-D MELTING PROCESS IN CONDENSED PHASE

1. Problem with Uniform Thermal Properties across Phase Interface

1.1 Problem Formulation

Initially a one-dimensional solid rod, as shown in Figure 1, has the melting
temperature T m. Then a ﬁxed higher temperature of T W is applied at the leﬁ end while

keeping the right end insulated. It is obvious that the melt phase front will propagate from
left to right. The transverse heat transfer will be neglected. Let the enthalpy be designated

as h, density p , and thermal conductivity k , the mathematical formulation in enthalpy

form is,

8271
Ebr2

ah. _

——k
p at

C

 

(l-a)

where the subscript c represents the condensed phase. The enthalpy-temperature

relationship is shown in Figure 2. It is also provided to ensure a unique solution. It is
assumed that the phase change occurs at a ﬁxed melting temperature, T m , which is true
for certain pure crystalline materials. It is also assumed that the solid and liquid coexist

with uniform thermal properties, which leads partially to C p5 = C P1 and is represented
by the same slope dhc / dT in both solid and liquid domains in Figure 2. In addition,

during melting the solid phase must overcome an energy obstacle of latent heat L to attain

further temperature increase, as shown in Figure 2 by the enthalpy discontinuity at Tm .

257

a?“ n-‘z-

 

From a mathematical point of view, an alternative formulation in terms of temperature

can be given as

 

'ar 3%
—=a—, 0< <X 0,t
at 3x2 )6 ( )
JT(X,t)=Tm, x=X(t) (1-b)
8X 8T
L—=—k —, =X t
L.0 at 1 8x x ( )
The boundary conditions are
T(O,t) .—. TW
3T(x,t) _ - (2)
T'FL ‘0

The above equations comprise l-D, unsteady, closed-form equations that will be solved.

1.2 Analytical Solution
X
From equation (l-b), the use of a similarity variable 5 = 7 leads to the Neumann
t

analytical solution [83], and only the solution itself is listed below. The moving phase

front position is obtained as a function of time and eigenvalue 2 ,

X(t) = 2N5 (3)
where A is obtained from the following equation

St

Aexp(,12)erf(/t) =7; (4)

 

where St = CPLAT , and AT = (Tw —T,,).

The transient temperature is obtained as a function of time, position and eigenvalue 2 ,

258

ﬁrm-‘11

T(x,t) = T, - AT edoc/275) (5)

eff (1)

The above exact solution will be used for further comparisons with numerical results.

 

1.3 Comparisons
The performance of the 2-D numerical solid model using the ADI Source Update
Method is employed for solving the one-dimensional melting Stefan problem. The

numerical experiments are similar to those in [48]. Material with a phase change

temperature T m = 0 and thermal properties p = c = K =1 is contained in the half space
x 2 0. At t< 0 the material is at a temperature T, = 0. At t= 0 the temperature of the

surface at x = 0 is increased and ﬁxed at a temperature of T = 1, so that, with time, the
liquid phase attaches to x = 0 and grows. A grid of exponentially increasing space of 40
steps with ratio of 1.0007, minimal grid size 0.1 and variable initial time step of
At = 0.002 are used, and four different cases corresponding to latent heat values of
L = 0.01 (St = 100), L = 0.1 (St = 10), L =10(St= 0.1) , and L =100(St = 0.01) are
investigated. To ensure the accuracy of the iterative solution process, the difference

between any two most recent temperatures or enthalpies should fall into the convergence

limit, 0.0001.

[+1 I
T . —7},j

1,} < 0.0001

1,]

 

 

 

h.’+,1 _ hi1, 1| < 0.0001 (6)

To determine the moving phase front numerically, an observation [84] indicates that the

liquid fraction of g = 0.5 always corresponds well with phase front position. Figures 3

gives the moving phase front position history.

At a low Stefan number, St = 0.01 and St = 0.1, the latent heat is much larger than

259

the sensible heat, and the phase front moves very slowly. The behavior of the larger St
number, where the latent heat is small, is almost the reverse of the small St number case.
The numerical results show very close agreement with the analytical solution. The
oscillations, which result from the enthalpy method [48], are not clearly shown in the
above four ﬁgures.

The numerical temperature proﬁles at a certain time are also compared with the
analytical solutions in Figure 4. It is observed that the numerical results of temperature
proﬁles provide very good agreement with the analytical solution in the above four cases.
2. Problem with Discontinuous Thermal Properties across Phase Interface

2.1 Problem Formulation

A liquid at a uniform temperature 7; that is higher than the melting temperature T m
of the solid phase is conﬁned to x > 0. At time t= 0, the boundary surface at x = 0 is
lowered to a temperature T 0 below T m and maintained at that temperature for t > 0. As
a result the solidiﬁcation starts at the surface x = 0 and the solid-liquid interface moves
in the positive x direction. This problem is a two-region problem because the
temperatures are unknown in both solid and liquid phases. In the following analysis we
determine the temperature distributions in both phases and the location of the solid-liquid

interface, this problem is more general than the one considered in the previous example.

The mathematical formulation in temperature form is given as

260

IaT _ 3%

——a —, 0<x<X0,t

at 88x2 ( )

37 327
<-5;-—a,-5;2—, X(O,t)<x<L (6)

T(X,t) = Tm, x = X(t)

pL§£=k 9.75_kl_a_]_:’
ax

i at s Bx x = X“)

 

The boundary conditions and the enthalpy-temperature relationship are the same as the

previous example.

 

 

2.2 Analytical Solution
In the solid phase,
T — To _ erf[x/2,/a,t] (7)
Tm - T o M (11)
In the liquid phase,

T—I} _ erfc[x/2,/a,t]

- 8
Tm—T. erchxS/azi ”

where ,1 is obtained from
512

2
e +ﬂ(ﬂ)1/2 Tm —T,- e"1 (as/a1) _ 2L7;
erf(l) ks a1 Tm _TO e’fcli’l‘Jas /al] CPs(Tm —TO)

 

 

 

(9)

and X(t) = 211/0g.

2.3 Comparisons

If ps=pl=l, CPS=CP1:1 and ks=1'0 kl=0°59 Tm=O;TO=—1;Ti=l,

CPs(Tm—T0)

of 0.1 and 10 are selected to make the

 

then two cases with St =

261

 

comparisons between the numerical results and analytical solutions.

It is observed that numerical method, which uses the harmonic mean to approximate
the discontinuous thermal properties across the interface, underestimates the phase front
position with a relative error of 2%. It is pointed out [85] that the Kirchoff transformation
method provides closer results to the analytical solution. However, the harmonic mean
treatment is used here for its simplicity. Later the Kirchoff formulation will be used into
ﬂame model, which does provide accurate prediction.

3. Conclusions

Numerical modeling of phase change in condensed phase is veriﬁed by a 1-D
analytical solution that is derived from problems with both uniform and discontinuous
thermal properties across phase interface. The numerical results of the solid model agree
well with analytical solutions, and can be utilized directly into the overall model of ﬂame

spread over condensed phases.

262

 

 

   

 

 

T(x, 0)= Tm

Figure 1 Schematic description of one-dimensional melting of a rod.

 

mushy i L

 

Solid

0 >T
Tm

 

 

 

Figure 2 The schematic relationship between enthalpy and temperature.

263

 

 

1.5 I j I I I I I I I

   
  

 

   

— analytical
+ it numencal

 

 

 

0.5

 

 

 

 

1-4 r ‘F I I I I I j I

 

 

 

 

 

 

 

1.2 ~ .
1 ~ .
0.8 - u
— analytical
«tr + numerical
0.6 - .
0.4 >- ~
0.2 ~
6 1 2 3 4 5 6 7 8 9 10

Figure 3(a) Phase front movement for St = 0.01 (upper) and St = 0.1 (lower).

264

 

  

 

   
     
 

analytical
numencal

 

 

 

 

 

 

4.5 I I I I I I

3.5 ~

 

— analytical
1r 1? numencal

 

 

 

 

 

 

2 4
1.5 .

1 i
0.5 .

06 0.2 0.4 oie 0T8 1 1T2 1.4

Figure 3(b) Phase front movement for St =10 (upper) and St = 100 (lower).

265

1.2 I I I I I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__. analytical
......... numerical
time = 50s
.l
l
_0.2 l 1 1 J l l l
o o 5 1 1 5 2 2 5 3 3 5 4
1.2 I f I I T I I
__ analytical
- ........... numerical 4
time = 53 -
1
.02 L 1 L 1 l l 1
o o 5 1 1 5 2 2 5 3 3 5 4

Figure 4(a) Temperature distributions for St = 0.01 at 508 (upper) and St = 0.1 at 55
(lower).

266

 

1.2 I If I I I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— analytical
............ numerical
time = 1s .
-(
-o.2 1 i 1 4 l 1 1
o o 5 1 1 5 2 2 5 3 3 5 4
1.2 r I T r I 1 I
analytical
1 .. ---------- numerical
time = 0.65
0.8 r I
0.6 - q
0.4 _ d
0.2 r «
o - -1
_0-2 L l l l L 1 L
o o 5 1 1 5 2 2 5 3 3 5 4

Figure 4(b) Temperature distributions for St =10 at ls (upper) and St =100 at 0.6s
(lower).

267

 

1~4 I— I I T I I I I T

   
  

 

   

1.2 -
1 -

0.8 -

0.6 4: 23882231 5

 

 

 

0.4

0.2

 

 

 

3.5 I I I I I I T I I

 

 

 

 

 

Figure 5 Phase front movement for St = 0.1 (upper) and St =10(lower).

268

 

 

 

 

 

 

 

I I I I I I
—- analytical
................. numencal .
time = 5.0s
.1
1 l l 1 1 45

 

 

 

 

0.8r
0.6 -

0.4 L

 

0.2 -

time = 2.0s

 

 

-0.2

I

analytical 1
.................. numerical

-O.4 ) .

 

 

 

-0.6 - .

8.8+ .

 

 

 

_1 l r r r r r 5

 

Figure 6 Temperature distributions for St = 0.1 at 58 (upper) and St = 10 at 2s (lower).

269

APPENDIX II
NUMERICAL METHOD FOR FLAME SPREAD OVER MELTING POLYMERS

WITHOUT FLOW CALCULATION

1. Discretization
1.1 The Control Volume Formulation

The control-volume formulation is physically sound and lends itself to direct
physical interpretation. Figure 1 shows a typical grid control volume. For melting
problems, the enthalpy equation can be solved exactly by using this formulation [46].

For illustration purpose, we take the general conservation equation as an example,

2
— —= 1
at +u°° W U ()

where U is an independent variable and ,6 is the diffusion coefﬁcient. The chemical term

is neglected here and will be discussed later. By treating all transport terms implicitly,

we have the following terms:

mil _ 5111
. . . O , , +
Y+ direction difquion: qY+ = —ﬂ 1 J ’ J

 

 

 

 

Ay
U511- ."tll
Y- direction diffusion: q _ =—ﬂ "J 1’} — Ax
Y Ay
5+1 5 U"? .
. . . . , + ,
X+ direction difquion: q X+ =-ﬂ H Ax I J Ay
Un+l _ un+l
. . . . . _ i,j i-l.j
X- direction difquion. q X_ — —,B Ay

270

X direction convection: qcon = um, (U1111- - U {31 )Ay

n+1 n
(Ui,j AZUi,j)AxAy

 

Transient increment: qtmns =

According to the conservation law in integral form, the above terms are combined as
qtrans : qcon + qX+ + qX‘ + qY+ + qY'
The finite difference equation is derived as follows.

aEUi—1,j 1‘ aSU i, j—l + “PU i, j + “NU i, j+1 + “WU i+l, j = b

in which

 

 

b = Ui’ij

A more convenient form of representation is S-form, which uses

5U = (U "+1 — U n) as the independent variable.

aEwi +aswi,j_1+apwiJ+aniJ+l+awwi+1J=RHS

...Lj

271

(2)

(3)

the

(4)

 

 

 

The coefﬁcients on LHS are the same as equation (3), while the RHS becomes

_ n n n
RHS—Ui,j_aEUi+l,j—aWUi-l,j—aPU'

r,j _aNUi,j+l—' aSUi,j— 1

A more self-evident and compact form is the following,

{I + At[u°°Dx — 8(1),? + 0%)] MUM. = Ugj + At[u,°Dx — MD? + D§)]U.-’f) (5)
A general rule will be applied here and thereafter. In Equation (5), the convection term
14me is differentiated by using upwind scheme, and the diffusion term (D3 + D3,) is
treated by central difference method.

1.2 Boundary Conditions
For interface or boundary conditions, the control volume is the half grid control

volume. Figure 2 shows the grid control volume near the interface.

Tn+l_ Tnj+ll

Y- direction diffusion: qY_ =—k 1’j A Ax
y

 

Tn.+l_ Tn+l

. . . . _ __ ll i+l ,j _A_y
X+ direction difquion. q X+ — k Ax 2

 

Tnj+l _ Tnll j Ay
Ax 2

 

X- direction diffusion: q X -k

X- Heat ﬂux from gas phase: q gas = qAx

(8?“ - 2171.)“,
At 2

 

Transient increment: qtmns = pC P

According to the conservation law of qtmns = q gas + q X+ + q X‘ + qr , the ﬁnite

difference equation is

272

aETi—l, j + asTi,j—1 + 010%} + “WTHI, j = 1’

 

where
a 515591-
E PCP A1162
0 k At
S __ 55555.
PCP Ay2
OPT-1+2 k (Atz Atz—
PCP Ax Ay
a k At
W_______
PCP sz
bznflj .54525‘1
PCP A)’

1.3 Treatment of Convection-Diffusion Terms

(6)

In the above formulation, the convection term is treated by upstream scheme and the

diffusion term is treated by central difference method. Some other schemes are available

in [46] and will be discussed brieﬂy here. A speciﬁc scheme should be selected according

 

to a ratio of P8:

, which represents relative importance of the convection or

diffusion. One instance of this family is Hybrid Scheme, which uses one of three schemes

according the value of Pe.

273

 

.
a
P<—2 i-—P
3 DE 8
<|Pe|<2 fl—i’~-—=l——I—)—‘e- (7)
DE 2
Pe>2 155:0
i DE

Among these schemes, the upwind and hybrid scheme are commonly used, while the
power-law scheme was recommended as a better substitute [46]. In this modeling, since
the ﬂow velocity is low, implying a small Pe number, the upwind scheme is used for
convenience. Other schemes were also attempted; it was found that different schemes
result in small difference in accuracy and efﬁciency.
1.4 Treatment of Chemical Terms

The chemical term of polymer combustion is changing in a more rapid manner
compared to droplet combustion. In another words, the characteristic time of polymer
combustion is much smaller, which requires a very small time step in computation. In
fact, the chemical term is the most costly in terms of computational time.

In rigorous mathematical sense, the chemical terms can be differentiated in three parts

with respect to T, Y f and Y0. The chemical term has a form of

wg = —Agpg exp(—E g / R T g )YOY F , the derivative of which is

5vvg=w}81'+w;,05Y0+w;,F§YF (8)

where

 

w, = Agpg exp(-Eg /RTg )YOYFEg
T 127;,2

274

w}: = «Agpg exp(—Eg /RTg )YO

“’0 = —Agp(g exp(—Eg /RTg )YF.

The straightforward way to treat this source term is expressing it in vector form, and put
it into a coupled equation system that must be solved by direct solution. Such a solution
procedure is termed the block implicit method, or fully implicit method. The advantage
of this method is that it is faster and more efﬁcient than semi-implicit iterative procedure.
The disadvantage is that (1) it will need big storage and CPU time; (2) it needs extreme
care in specifying boundary condition otherwise divergence occurs. Instead we use the
semi-implicit method where all equations are solved sequentially. Only one partial
derivative for each primary variable (T, Y): or Y0) is preserved in each (energy, fuel or
oxygen) conservation equation. The important rule for treatment of chemical term is
discussed in detail in [46], which requires that the derivatives of source term be negative,
otherwise the unstable solution will arise. Examining the equation system in gas phase, it
can be observed that the species equations all follow this rule quite well, while the energy
equation has a positive derivative, which is against this rule. To overcome this difﬁculty,
the source term in energy equation is solved by using a special technique [47]. The details
are given next.
2. The Computational Procedure
2.1 Non-linearities

As is mention in Chapter 1, the combustion process over polymer materials,
physically, can be split into two half cycles, the heat cycle and the fuel cycle. In the
former, heat that is generated from combustion or from external ﬂux is fed at the

interface. In the latter, the pyrolyzed gas generated by the heat from interface is moved

275

into the gas phase and reacts with the inﬂow oxygen and consumed. Therefore the
interface constitutes one non-linearity in computational procedure. The mass ﬂow rate,
temperature and heat ﬂux etc. at the interface act as the communication factors through
the two phases. Another non-linearity is from the chemical terms, which are combustion

source term and pyrolysis source term. Obviously the iterative method is required for

such process.
2.2 The Newton-Raphson Scheme
The discretization should be done in iterative sense; requiring all the terms be discretized

with reference to 1 time level instead of n time level, where 1 time level is the most recent

(iterative) level. The (5 formulation is modiﬁed here with
l I
W 1,1 = US} ’ URI (9)

There is minor change in the 5 deﬁnition and this convention will be used in the rest of

the appendix. Next the chemical terms will be treated.
SF = ngg = —,uFAgngXP(-Eg ”erg )YOYF
By differentiation,
5;“ =5} +d‘sF =5]r +S;.5YF,
in which 5;. = —,uFAgngXP(—Eg /RTé)Y(l).
If chemical term is added into the formulation, Equation (5) becomes
[1 + At[u,,Dx + [3(03 + Di )] — S} Wm = RHS (10)
where
RHS = (1123,, j — rim.) + At[u°°Dx + MD)? + Di)]Y,{~ + Sic.

276

For the energy conservation equation, the source term is:

ST = qgwg = —qg/1FAgpg exp(—Eg /RTg )YOYF
By differentiation,
n+1 I l 1
ST z _qgtuFAgpg CXp(-Eg /RTg)Y0YF

Its treatment will be discussed later. For boundary conditions, take fuel conservation

equation as an example, the source term at RHS is expressed as (l - #:11- )m , therefore,

the time discretization is (1_YI{‘i,j _5YFi j)n't, then the LHS will become

LHS + ”15%, j and RHS will become RHS + (1 — Y 157, j)n'r . For oxygen conservation

YIH'I -

equation, the source term at RHS is 0i,jm’ the time discretization is

(Y(l)i, j +5Y0i j)Ih, then the LHS becomes LHS wﬁtdYO and the RHS becomes

i,j
RHS + m Y5“.
2.3 Alternate Direction Implicit (ADI) with Special Source Treatment

To enhance the efﬁciency of solving a S-points linear algebra equations, the ADI
method is applied. In principle, the ADI method split the 5 points formulation into a
combination of a 3-point X equation and a 3-point Y equation, by sweeping each
direction in turn. This only involves tri-diagonal system of equation, thereby signiﬁcantly
saving CPU time and storage. In source term treatment, a method based on approximate

factorization was proposed. The original scheme [47] did not consider the diffusion term;

therefore its scheme was extended to include the diffusion term. We have

277

2 -1 2
[N + At(uwa + ﬂDx )]N [N + AtﬂDyMY},j = RHS

N = I + AtSjr (11)
The solution procedure is the following,

Step 1; [N + At(u°°Dx + 803)]51421. = RHS

*
where the 51,-, j is the temporary variable.

2.4 Enthalpy Method
2.4.1 Source Update Method

For solid phase the PDE at phase change becomes

3T 2 8g
—=kV T— L—- 12
PCP at P at ( )

where g is the liquid volume fraction. Then

8T 2 L 8g
__= T-—— 13
at 'W Cp at ( )

here ﬂ is the thermal diffusivity. The ﬁnite difference equation has the following form,
2 2
[I + Atﬂ(Dx + Dy )]6‘1}, j

(14)

L
=(Ti?j _Tifj)+Atﬂ(D3 +D§ﬂi€j —'C—(gi:;1 ‘82))
P

The Source Update Method given in [85] is the extended to ADI procedure as below
1. At the start of the time step, the initial iterative ﬁelds are set to the previous time step

values.

278

 

2. Prediction: From the known nodal temperature and phase change enthalpy ﬁelds at

iteration l , the equation (14) is reformatted as

(158114”. + as 81131“ + ap57i,j + “N 53,141 + aWéYi’HJ :

I 2 2 1 L 1 , (15)
(7X; “7i,j)+At.B(Dy +Dx my} ‘E;(gi,j ‘83))

and solved in two steps

a L
Step 1: [I +AtﬂD3]éTi,j =(Ti?j -T,-f,-)+Atﬂ(D§ +0571)- ”E—(gid' "giiﬂ
P

Step 2: [I + AtﬂD; ]§I}’j z: 57:].

Note that in step 1, the (g3;1 — ggj) in equation (14) is replaced by (gij - gzj)as an

approximation.

3. Correction: The nodal phase change enthalpy, H e , needs to be evaluated before the

next iteration can proceed. To correct the above approximation, it is recognized that the

nodal temperature during phase change will keep constant, which implies that

l I
If; =Tm=>67},j+7},j=Tm (16)
Therefore the equation (14) is in fact

I _
0155714,,“ + “swig—1 + ap(T m — 7i,j) + “N571, j+1 + “Wain, j -

I 2 2 1 L 1 (17)
(7X; —Ii',j)+ Atﬂ(Dy + Dx )Ti',j ‘60?“

n .
i, j " g i, j)
The difference between equation (17) and (19) need to be corrected, we subtract equation

(19) from (17) and obtain

C
6gt,j=TP/1ap(5ft,j+Tt-fj-Tm) (18)

279

, where l is a relaxation factor. After the evaluation of temperature ﬁeld, the phase

change enthalpy update is applied at every node point, followed by the correction:

gt’fj“ = maxiO, minig.-’f}"l ,1] l (19)

Steps 2 and 3 are repeated until convergence. In this way, the scheme will conserve

energy.

2.4.2 Treatment of Discontinuous Thermal Properties

In condensed phase, the thermal properties are discontinuous across the phase
change front. In ﬁxed grid calculations, the treatment of discontinuous thermal properties
requires some special procedures. If a nodal lumping approach is used, discontinuities in
speciﬁc heat and density can be readily calculated in terms of nodal average values, e. g.

CP = gCPl +(1— g Wm (20)
where g is the liquid ﬁaction of the control volume around the node. The difﬁculty
comes in dealing with discontinuities in the thermal conductivity, k. In constructing

coefﬁcients, the values of k need to be calculated at integration points of control

volumes, which generally do not coincide with the node points. A recommended

approach is to use Kirchhoff transformation [85].

(b = if; k(a)da

(21)

With the Kirchhoff transformation,
§£=kaT a¢=k-a—¢ (22)

or E’ay ay

Therefore

280

-§——¢/-a——T k -a—¢ l (23)

k
x=axax yayay

Then the diffusion terms can be written as

a 8T 8T
V-(kVT)=5;(kx-a— )+ —(ky—-— yya (24)

A central difference approximation of equation (23) leads to the following expressron for

the interface conductivity:

_¢_E__ ¢P k- _¢N ¢P
ke" n
T E T P T N T P

kw=M,ks= ¢S__:___ ¢P (25)
TW—Tp TS— Tp

It was pointed out that equation (25) is a convenient and accurate means of dealing With
the discontinuities in thermal conductivity in a ﬁxed grid, control volume solution of
phase change problem. The numerical features are:

I The formulation results in a set of nonlinear equations, which are solved upon

employing an iterative technique and the nodal values of (D and k are calculated usrng

the temperature values from the previous iteration.

requires slightly more work than the calculation of temperature dependence nodal

conductivity.
I

of Kirchhoff variable is not required.

approach.
28 1

In general, the calculation of ¢ ﬁeld involves the evaluation of an integral, which

The discrete equations will be in terms of the nodal temperature; that is, the solution

Upon convergence, the discrete equation will be equivalent to the direct Kirchhoff

 

 

If the conductivity in separate phases is constant, a general relation can be obtained using

Kirchhoff transformation, for the control volume around node (i, j) and (i + 1, j),
¢i,j = (71} — Tm )ki,j
¢i+l,j = Ui+1,j - Tm )ki+1,j (26)
in which k,”- = (l — gi,j)ks + gi,jkl-
The expression of kw can be written as

= ¢i+1,j —¢,',j = k- .+(Ti'+1,j -Tm)(ki+1,j -ki,j)

1,]
7i+1,j-Ti,j YIHJ-szj'l'g

kw

 

 

(27)

, where the ﬁrst term in RHS is the conductivity in CV (i, j), the second term is

correction of the conductivity, which should be at interface. 8 is a very small number to

prevent the zero denominator. In the same sense with the CV (i + l, j),

 

 

=¢i+1,j"¢i,j -k (7i,j—Tm)(kr+1,j-ki,j)

 

 

 

w - i+l, j (28)
Ti'+l,j-7i',j 7i+1,j-7i',j+8
The kw can be derived by averaging equation (27) and (28)
7;. . + T. .
,j l+l,_]
ki+1j+kij l( 2 )"Tm](ki+1,j—ki,j)
kw =( ’ ’ )+ (29)
2 EHJ_EJ+£

Equation (29) is derived mainly to overcome the possible zero denominators. Three cases
are derived from equation (29)

i _ _

gi,j =0agi+1,j —0 kw -ks

gi,j=l’gi+l,j =1 szk] (30)
[0<gt‘,j,gi+1,j <1 kw =7

J5

 

282

 

1
I

i it .Wiii; Tum

 

3 . Coordinate Transformation - Clustered Grids Near the Interface
An extremely small grid size is required near the interface for combustion
computation because of the large gradient there. In droplet combustion, a normal grid
size at droplet surface is 0.015 mm; in this model the minimal normal grid size taken as
0.025 mm. With such a small grid, the uniform grid system is obviously not good. Some
researchers [4] used two uniform grid systems; a grid system nearby the interface with
very small grid and a grid system far away from the interface with large grid size. This
treatment is not robust physically, since the errors will arise in evaluating the ﬂux term at
the conjunction points between the two gn'd control volumes. Another drawback is that
non-smooth results appear at the conjunction of the two gn'd systems. A smooth non-
uniform grid system must be used. One example is the adaptive grid generation, but its
implementation is too complex. For convenience, two exponential grid systems in y
direction are used, as seen in Figure 3. The grid at the interface has the smallest size.

Coordinate transformation is required. The grid size along y direction from interface will
increase in exponential order: A y, rAy, rsz, r3Ay... while keeping Ax uniform

along x direction. 7] and 4‘ are new coordinates with relationship of

ix=§
. n

A —- (31)
y=——y (rA’7 —1)
r—l

 

l
in which A), is minimal normal grid size in y — x grid system and A” is the grid 8126 in a
transformed uniform 77 — if coordinate system. For coordinate transformation, space

derivative over x has the same form as that of 9‘. However, space derivative over y has

283

different from comparing to 77 , and the relation of 1St and 2“d order derivatives are

 

aT_a§§

By an (32)
4

327‘ 2 82T 8T

__2= —2-—b-———

(By an an

77
r—lA _._
where a=£———)—n—r A" and b=a2 logr

log rAy

 

. Two major changes are introduced if the
17

equation system is expressed in 77 - 5 system.

' The interface condition becomes

r dY .
— pgagD—é—g— = "2(1— YF)

3Y- .
‘P ‘1 D—L=-mY°a '=0,P,I
g g 3'7 l I (33)

_A

TszTg
07g

3T3 4
—kgag:-J—ng—= —Ss——ka an -80'(T —T04)+qex,

, which implies that the heat or mass ﬂux were “decreased” to some extent in 77 —«f

 

system.

| The energy conservation equation of gas phase becomes

C [—-—— "5+ ”78 Hkgbﬂg= +k “3228+ 2'92—T—L‘t) (34
P .0 “no =4 W —— 612 , )
g g 015813” g g g 862 W2

kgb
where a new ﬂow term with positive velocity is introduced.

ngg

| For discontinuity across the phase change front, the Kirchhoff transformation in y

284

direction is

k zﬂ/ﬂzﬂ/ﬂ
y 3y 3y 377 377

Therefore there is no need to adjust this term for 77 — 5 coordinate system.

(35)

4. The Solution Procedure

The flow chart for current model is given in Figure 4. First the solid phase
temperature is solved, and the results are used to obtain the interface mass ﬂow rate and
interface temperature. The former is used as boundary condition for two species
equations in gas phase, and the latter is used as boundary condition for energy equation in
gas phase. The gas phase equations are solved; yielding temperature ﬁeld. The heat
conduction at interface is obtained, and its results are used as input for solid phase
boundary condition. The computational process continues until the relative error of the

two recent iterative values fall into a convergence range with a limit of 0.0001.

’ ’ < 8T (36)

I
Ti,j

 

In addition, the Line-By-Line Gauss-Sedel iteration is attempted, it was found that both
the scanning in y direction and the scanning in x direction are not as efﬁcient as the ADI
method.
5. Data Structure and Others

General features of the data structure are (1) All of the solid data and gas data are
deﬁned from two basic classes: SolidNode and GasNode, which encompass all the
necessary information about a node. (2) The memory use is dynamically administrated;

(3) Data access is performed by pointers or references through some specialized function

285

interface.

The code development is carried out in MS Visual C++ environment in a Pentium H1
600 MHz PC. A scheme with variable time step is employed in order to minimize the
running time. A typical ﬂame spread case has time steps ranging from 10'3 s to 10'5 s. It is
found that the time step is very sensitive to the selection of pre-exponential factor and
activation energy in the gas combustion. In addition, the minimal time step, which
corresponds to the most rapid change of ﬂame behavior, occurs during transition stage.
The total running time is normally 10 seconds after ignition is established, which
guarantees the establishment of steady ﬂame spread. Normally shorter ignition time
implies shorter time to attain steady ﬂame spread. In addition, a graphical user interface
is developed, which provide overall parameter control in a user-friendly manner. This

GUI was developed by using MFC library.

286

 

”"1

us-
0

 

 

n+1

 

 

 

l+IJ

 

 

iJ-l

 

 

 

 

Figure 1 Control volume formulation of partial differential equations.

 

 

 

 

n+1

 

 

 

 

Figure 2 Control volume formulation of boundary condition.

287

[m7 ‘1'.
PA A!!! 9 T‘T‘ﬂ'!

 

 

0.01

0.008

0.006

0.004

0.002

-0.002

-0.004

 

O 0.005 0.01
X

Figure 3 The mesh system with clustered grid near the interface. Only the temperature
type grid system is present in the gas phase. (We recall that there are three types of grids
in the gas phase for non-uniform staggered grid system).

288

 

 

[T]

U

 

 

 

 

Initial condition

 

 

 

 

 

 

 

 

 

(2::
Interface Interface
temperature Condensed phase T mass ﬂux

 

 

 

 

 

 

 

 

 

Gas phase Y, ,(_____..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

at?“
3.
Gas phase Y, (_____.. ] a

1..
Interface
, Gas phase 7' he” ﬂu"

No
Convergent? I——'—.>
Yes
Update current time step 3

 

 

 

 

    
 

Finish?

 

Figure 4. Flow-chart of the solution procedure of the numerical model.

289

APPENDIX III

DERIVATION OF A FLAME SPREAD FORMULA FOR MELTING POLYMERS

The ﬂame-spread formula of Equation ( l 3) in Chapter 3 is derived mathematically.
This model examined here is built upon that of [28]. The spread mechanism is driven by
a “surface ﬂame” located along the gas-condensed phase interface. Following [28] we
assume negligible upstream conduction in both gas and solid, thereby invoking the global
energy balance principle described at length therein. The energy equation in the three
phases are given by pJ-ujCPJ-BTJ- /3x = kjasz /3)/2 where j = g,l,s and u, = us.
Forthe gas we have O<x< oo, y > 0, forthe liquid O<x<oo, O< y<f(x), forthe
solid 0 < x < oo , y > f (x) , where y = f (x) describes the shape of solid/liquid interface.
Along this contour, we are able to write the interfacial energy balance in the form
k,[(aT, /ax)2 + (87', /8y)2]1/2 = k,[(aT, /Bx)2 +(8TS/8y)2]1/2
+ PsusLs{(3f/3x)/[(3f/3x)2 + (Bf/3y)2]“2}
, where L, is the enthalpy of liquefaction (positive) and the third term in the energy
balance represents the liquefaction energy ﬂux along the liquefaction front y = f (x)
We deﬁne dimensionless variables Tj = (Tj -T°°)/(7} -— Too), j = g,l,s , f = x/L ,
n = y/L (L = A, / pSCPSus) where T,- is the downstream gas-liquid interface
temperature (gasiﬁcation or “vaporization” temperature). The equations, when
transformed into parabolic cylinder coordinates f = (s2 — n2)/ 2 , 7] = sn (n =constant

deﬁnes a family of parabolic arcs with n = 0 the downstream gas/liquid interface y = O,

290

 

 

 

x > 0). The governing equations transform to Njnde /dn = —d2’tj /dn2 , j = g,l,s

with N8 — "[ug / u ”Ha / ag ], =(as / a1), —- 1. Along the liquid/solid
interface we have 2', = TS = Tm and
A[T[ :2 +71 ,7 2]=“2 —T[ 75:52 +1.3,"2]l/2 +TmSihg/[h62 +h77211/2, where

A=/ll/ls, St=LS/ C Ps(Tm -T°°) and h is non-dimensional h. Along this
interface we have n = c=constant. In terms of (s,n) the interface condition above

simpliﬁes to the form A7,," = Ts," + TmStc. The remaining conditions are Tg = T, = l

at n =0, 71:15 =1," at n=c, 7g and TS vanish as n —>oo in eachmedium.

The energy equations are easily solved in terms of error functions. The liquid/solid
interface condition yields the following parametric relationship:
CZ

A(l- Tfl—CXP(———)__r_m)+ CXP(-:—)=c\/2St (1)
2

erf(c\/—) e’de)

 

We note that as T m —) I} and c -—> 0 the preceding equation gives
1/ 2
erf(C\/N1/2)/(7i _Tm) —> M/ N1 /(7} —Tm) =[P1CP144 /PsCPs/lis] /(7} ‘73»)
, which further reduces to the ﬂame spread formula of [21]. Equation (13) in Chapter 3
for the spread rate is derived using the method of [28]. We write the net heat ﬂux from

the surface ﬂame (at y = 0 , x > O) as

 

 

q, =—2.garg /ay—2.,aT, /ay=[\/ugngpgﬁ.g +,/us [Cpl/115]”: —T..]/«/E .

291

where (I) = AZNI. When we also write qf = (Tf - Too) ngPg’I'g Max we obtain
the ﬂame-spread formula

“3 ___ ngngg ,(Tf ”Ti
um PICPIkI 7} _Tm

 

>2 -erf(c 133—)2
2 a,

which is equation (13). We note that St > O is for ordinary endothermic solid -—> liquid
liquefaction. As noted in [28], the ﬁrst term in equation (1) is the non-dimensional heat
ﬂux to the interface from the liquid and it becomes inﬁnite as c —> 0 , zero as c —> oo.
The second term is the nondirnensional heat ﬂux leaving the interface into the solid, and

it approaches unity as c -—> O and inﬁnity linearly with c as c —> oo(since

lim er (:06) = ﬁx ex x2 ). The third term is the heat absorbed at the solid/liquid
P

x——)oo

interface.

292

 

 

 

APPENDIX IV
NUMERICAL METHOD FOR FLOW FIELD CALCULATION

1. Control Volumes

To solve the numerical model, the discretized equations are derived by integrating
the governing differential equation over a sub-domain surrounding each grid point. These
sub-domains, which are also referred to as control volumes, are deﬁned in Figure 1; the
dashed lines denote the control volume boundaries. There are two types of control
volumes, interior control volumes and boundary control volumes. For the former, a given
grid point communicates with four neighboring grid points through the four faces of the
control volume. For the latter, as shown shaded in Figure 1, one face of the control
volume coincides with the boundary of the calculation domain, and a boundary grid point
is placed at the center of the control volume face. It communicates with only three
neighboring nodes.
2. Discretization

The partial differential equations can now be integrated over control volumes. The
control volume is constructed around the grid point P; the other grid points E, W, N, S are
the eastern, western, northern, and southern neighbors of P. The corresponding faces of
the control volumes are denoted by e, w, n, s. The discretization of the diffusion terms
was the central difference method. The convective terms were discretized by using the
upstream method. The linearization of the chemical terms is standard. Substitution of the
relations derived before into the governing equations, we can obtain the ﬁnal

discretization of the general form. If the dependent variable is denoted by q) , the general

form of the discretized equations can be written as:

293

aP¢P = 05¢}; + aW¢W + 0N¢N + as¢s + b (1)

It is useﬁil to write equation ( l) in a more generalized form

aP¢P = Zanb¢nb + b (2)

3. Staggered Grid

The original goal of the staggered grid is to eliminate the unrealistic ﬂow ﬁeld that
may arise in the collocated grid. Figure 2 shows a portion of a 2-D grid. The locations for
which the velocity components are calculated are shown by short arrows. In the staggered
grid, the velocity components are calculated at the surfaces of the control volume. All
other variables such as pressure, temperature, and concentrations are calculated at the
grid points shown by dots. The advantage of this arrangement is that the normal velocity
components are directly available at the control volume surfaces, where they are useful
for calculating the mass ﬂow rates. In addition, the pressure difference between two grid
points can be used to “produce” the velocity component located between them. The
disadvantage of the staggered grid is that (l) The boundary conditions are difﬁcult to
implement; (2) The complexity of the programming increases; (3) Interpolation of the
properties at the grid points is needed for the staggered grid system.
4. Discretization of Gas Phase Equations

Detailed discretization is given [46] with speciﬁc interest in gas phase equations. The

two-dimensional form of the gas phase equations except continuity equation is,

 

a 8.1 BJy
1‘ =5 3
at(p¢)+ ax + ay ( )

where J x and J y are total (convective and diffusion) ﬂuxes deﬁned by

294

CH“
"_.‘4J '. "

 

 

3¢

Jx = pu¢ - F— (4-a)
Bx
and
59¢
J = —l‘—— 4-b
y pail) By ( )

The integration over the control volume leads to

 

O O
— AxA
(pp¢p 1:40p) y + J8 _ JW + J" _ J3 = (5C + SP¢P)AxAy (5)

where the source term has been linearized in the usual manner, and ﬁirther treatment was
introduced before. With the introduction of staggered grid, p p and ¢p are assumed to
prevail over the whole control volume. The old value is represented by pg and $2. The

quantities J W, J e, J S , and J n are integrated over the control volume faces. In a similar

manner, the continuity equation can be derived,

 

(PP-1:12”)AxAy+Fe—FW+F,, —F, =0 (6)
where
Fe = (pu)eAy
F... = (pu)wAy
F. = (pu)sAy
Fn = (pu)nAy

Now we multiply Equation (6) by (1),, and subtract it from Equation (3), we obtain

295

o PgAXAy

(¢p -¢p) At + (Je - Fe¢p)- (Jw — Fw¢p) +

(Jn — Fn¢P) - (Js — Fs¢P) = (SC + SP¢P)A’CA)’

(7)

This is the discretized equation for momentum, energy, and species conservation. Note
that this method maintains the conservation of both the continuity and other identities.
5. Under-Relaxation

In the previous Oseen-ﬂow model, there are three major nonlinearities in the
numerical model. First, the chemical reaction terms in the gas phase and the condensed
phase constitute the strong non-linearity in the equation system. Second, at the interface,
the balance of mass, heat, and momentum links the gas phase equations with the
condensed equations. Third, the phase change in the condensed phase introduced a
nonlinear source term that accounts for the enthalpy jump between two phases. In the
complete model, the ﬂow ﬁeld calculation introduced a new non-linearity, that is, the
inter-linkage between the velocity ﬁeld and pressure. Because of these inter-linkages and
non-linearities, the ﬁnal solution must be obtained by an iterative procedure. At any
given stage, the discretization coefﬁcients can be calculated from the most recent
estimates of all independent variables. With the improved estimates, all the independent
variables will cease to change after a certain number of iterations, thus the ﬁnal
converged solution is reached. However it is not guaranteed that we can always attain a
converged solution if we only increase the number of iterations. At times, the values of
independent variables oscillate or drift away from a reasonable solution. To avoid such
divergence, one method called under-relaxation is proposed that will slow the update of

the nonlinear coefﬁcients from iteration to iteration. Equation (2) can be rewritten as

296

 

 

+ b a:
Zanb¢nb _ ¢P] (8)
0P

 

¢P=¢P+I

a:
where ¢p stands for the value of ¢P from the previous iteration. The bracketed content

denotes the change in (D P in the current iteration. To reduce the change, an under-

relaxation factor a can be introduced so that

+ b *
Z anb¢nb _ ¢P] (9)
“P

 

¢P=¢;+05I

The selection of this factor, which depends on the ﬂow situations, is empirical. In our
solution procedure, a is given 0.6 for x and y momentum equations. It is worth noting
that the pressure needs not to be under-relaxed for the SIMPLEC method, though it is
under-relaxed in the SIMPLE method.
6. The SIMPLEC Scheme for Flow Calculation

To solve the ﬂow ﬁeld, one difﬁculty arises since pressure does not appear in the
continuity equation. Numerically it is desired that pressure should serves as the primary
unknown for direct solution. This difﬁculty is resolved by the Semi Implicit Method with
Pressure Linked Equation (SIMPLE) and its improved versions such as SIMPLER
(SIMPLE Revised) and SIMPLEC (SIMPLE Consistent). In our solution procedure,
SIMPLEC is selected because of its consistency and because it is a straightforward
extension of SIMPLE.

6.1. Derivation of Momentum Equations

The appropriate control volumes for velocity u, pass through the grid points P and E.

The corresponding momentum equation can be written as

297

aeue =Zanbunb +b+Ae(Pp—PE) (10)
where Ac is the area over which the pressure force acts. If the pressure is taken from an

estimated ﬁeld such as from the previous iteration P. , then u' denotes the velocity ﬁeld

based on the estimated pressure ﬁeld. This implies
it t It It!
aeuezzanbunb+b+Ae(PP'—PE) (11)

Similarly we can obtain the v. velocity equation based on the estimated pressure ﬁeld.

6.2. Derivation of Pressure Correction Equation
In order that the velocity ﬁeld satisfy the continuity equation, the u. and v' must be

corrected as a consequence of the pressure correction P’ applied to the pressure P‘.

Thus we have
P: P" + P’ (12)

* I
u = u + u
Subtraction of Equation (10) from Equation ( l 1) results
dell; =Zanbu;b +Ae(PI’3 —P;,: ) (13)
With re-arrangement, we obtain

(0e - Zanb)“; = 20,2“qu ‘14) + Ae(P[’> - Pg) (14)

In order to derive the velocity correction equation, it is assumed that the term
Zanb(u;b —u; ) is negligible, we further obtain

u; =de(f1> 47;) (15)

where de = Ae /(ae —Zanb). Note that under-relaxation must be applied to the

298

 

 

 

momentum equations in order to use the SIMPLEC scheme otherwise de will become
inﬁnity since ae = Z a n b . However the pressure correction need not be under-relaxed so
that up = 1.0.
The continuity equation can be written as
(pa/1)... — (puA)e + (va)s - (va), = 0 (l6)
The ue,uw ,us and un are expressed by a equation like (10), thereby a discretized form
of the pressure equation can be obtained and cast into the form
apPI’, =aEP1:: +aWPéz+aNva +a51§ +b (17)
where
“E = (Pd/0e,
0W = (Pd/Ow,
0N = (Pd/0n,
as = (PdA)s

ap=aE+aW+aN+aS

b -- (pu‘A). —<pu"'A). + (pu‘A). -(pu"A)n-
7. Boundary Conditions
The momentum equations are special cases of the general ¢ equation, and therefore
the treatment to momentum equation applies to other equations as well. One problem
may arise since p' does not come naturally as a variable. Some treatments are discusses
in [46]. There are two types of boundary conditions, either the pressure is given or the

velocity normal to the boundary is given. (1) Given the pressure at the boundary. If the

299

I nil—ﬁlm‘s
" |

 

guessed pressure p' is given, then the value of p' should be zero, which is akin to the

given temperature boundary condition in heat-conduction problem; (2) Given normal

velocity at the boundary. In this situation, the derivation of p' equation should be based

on the ﬂow rate that is expressed from the given velocity, not the u'. In the pressure
correction equation, the p' will not appear at the boundary points.

8. Overall Solution Procedure

1) Solve the condensed phase energy equation; obtain the interface temperature and
pyrolysis product mass ﬂow rate.

2) Solve the gas phase ﬂow ﬁeld. This includes the following steps

3. Guess the pressure ﬁeld from the most recent values.

b. Solve the momentum equation to get u' and v‘.

c. Solve the pressure correction.

d. Correct the pressure ﬁeld and velocity ﬁeld by the use of equations.

e. Consider the corrected pressure as the new guessed value and return to step

“a” until convergence is obtained.
3) Solve the gas phase fuel concentration, oxidizer concentration, and energy equations,
and then return to step 1 until convergence is obtained.
8.1. Solution of Algebraic Equations
8.1.1. Delta Formulation of Time Discretization

In terms of the time differencing, we discretize the governing equations in terms of
the delta formulation; namely, for a given physical quantity the unknown is represented
by the time increment instead of its magnitude. The advantage of this treatment is to

decrease the round-off error of the iterative computation since each time only a small

300

 

 

increment need to be obtained. In principle, the delta formulation does not alter the
numerical scheme. To describe the Delta formulation, we deﬁne 5U as the increment for

a given physical quantity U, we have
1 1
Us; =6Ui,j+Ui,j (18)

where i and j denote the coordinate positions, “n+1 ” denotes the new time step, and “I "
denotes the iterative time level or the most recent value.
The transformation from normal form to delta form is straightforward. Take the
continuity equation for example. The ﬁrlly implicit form of FDE equation is
pg“ - p; +
At

 

Dx(pgu)"+1 + Dy( pgv)"+1 = o (19)

where D, and Dy denote lSt order space difference operator. With space discretization,

equation ( 19) becomes

 

 

 

Pg} - pg, j + Pgiili/z, juml/z, j " szﬂi/z, ﬂit-”ii 2, j +
At Ax (20)
n+1 n+1 n+1 n+1
pgi,j+l/2vi,j+1/2 — pgi,j-1/2Vi,j-l/2 ___ 0
Ay

Substitute Equation (18) into equation (20), let p"+1 z p], and ignore the higher order

terms, we obtain

1 1 1 I 1
pgiJ " p§;,j + pgi+1/ 2,1(“i+1/ 2.] + gum/2,1) " Pgi—i/z, j(ui—l/2, j + 514,4 ad)
At Ax

 

 

l I I I
+ pgi,j+1/2(Vi.j+l/2 + 6vi,j+1/2) ‘ pgi,j—l/2(vi,j—l/2 + 5Vi,j_1/2) ___
Ay

O

 

(21)

301

 

 

With re-arrangement, we can obtain an equation of the following form

O=ai+1/2,j5“i+1/2,j +ai-1/2J5“i-1/2J (22)

+ ai,j+1/2,j5ui,j+1/2 + ai,j—1/2&‘i,j—1/2 + b

where
_ I
0141/2, j - pgi+l/2, j
_ I
ai-1/2,j - Pgi-1/2,j
_ I
ai,j+l/2 — pgi,j+l/2
_ I
ai,j-l/2 - pgi,j—l/2

l n I l I I
b _ pgi,j - Pgi,j + pgi+l/2,jui+l/2,j " Pgi—1/2,j“i—1/2,j
At Ax

 

 

I I I I '
+ pgi,j+l/2Vi,j+l/2 — pgi,j—l/2vi,j—l/2
Ay

 

The discretized form of the other equations can be obtained similarly.
9. Solution of the Pressure Correction Equation

For the unsteady model, the momentum and energy equations can be conveniently

solved by using a solution such as the ADI method with an accuracy of 0(5t2).

However, the direct solution of the pressure correction equation cannot be attained
because it requires excessive storage and computer time. Therefore, an iterative method
for solving the algebraic pressure equations is employed. The solution of the p’ equation
can represent as much as 80% of the total cost of solving the normal ﬂuid ﬂow problem.

In combustion calculations, it would be expected that the solution of the p' equation

302

would also be time-consuming. Therefore it is a high priority to solve for the pressure in
an efﬁcient manner. Pantankar [86] recommends a combination of a block correction to
lines, followed by line-to-line iterations based on the tridiagonal matrix algorithm
(TDMA). Van Doorrnaal [87] recommended a convergence accelerating technique based
on Pantankar’s methods. These methods are used in the solution procedure of the
numerical model and will be brieﬂy presented here.
1) Line By Line Technique

The method uses TDMA as its basic unit. In a two-dimensional problem, the values
along one-grid lines will be solved by using the TDMA. The solution sequence can be a
ﬁrst scan along all horizontal lines and another scan along all vertical lines. In addition,
all neighboring-line values are substituted from the best available estimates. The process
can be repeated by choosing the alternate lines.
2) Block-Correction Procedure

The convergence of the line—by-line technique can be improved by block correction.
The basic idea is that a converged solution can be obtained by adding uniform corrections
along lines of constant i. The corrections are chosen so that integral conservation over the
control volume blocks by each constant-i line is satisﬁed. The complete procedure
includes a combination of a block correction to lines, followed by line-by—line iteration.
3) Convergence Acceleration Technique

Further recommendation is provided by [87] to accelerate the convergence.

Equations for pressure corrections can be restated for a solution along a line of constant j,

I I I I I
aijpij ' iI'Pi+1.j + Capt—1.1 +[diIPi.j+1 + eijpiJ-l] + fr,- (23)
where

303

ail-zap, bi-=aE, cy=aW, dij=aN, eij=aS,and flj=b.

Here i and j denote the nodal locations in the x and y directions. In deriving the
accelerating—convergence method, suppose that a partially converged p' ﬁeld, denoted as
[p']0 , has been obtained for one or more TDMA based iterations. In the current iteration,
Equation (23) is to be solved along each j line, sweeping in the direction of increasing j.

One the j line, the best estimate of p;j_1 is that obtained from the just-completed
solution along the j-l line. This is the offline value used in Equation (23). The available

estimate of the p34,] is from the previous iteration, i.e., [pa-+1]? An approximation is

introduced as [lg-+1 ] B E , which is deﬁned as

[PijnlBE =ipij+ii° + (6 -1)(p;,~+1 4123.11")

~ ’ 0 - I _ I 0 (24-a)
”[Pin] +(6 0(ij [pg-l)

Therefore we obtained

(a,- — dy-(B — 1)}pEj = 1119.31,; + CUPi—u +
(24-b)

dy-{tpg 1,110 — (0 - 0112.310} + eyP;,j_1 + f.)
A similar estimate is made for the solution along the i lines. It is suggested [87] that the
value of 6 should be in the range of {1.85, 1.95], and a conservative value of 1.85 is
recommended. Therefore in this model we let 6 = 1.85 .
4) Convergence Criteria
There are two loops of iterations in terms of the solution procedure. First, the

solution of p' equation is attained by an iterative procedure. One approach of controlling

this convergence is simply giving the number of iterations, such as 25, and is used in this

304

 

.mm‘nm
,. .

model solution. Another approach of convergence control is proposed in [87]. That is, a
ratio between the Euclidean norm of the pressure equation residual during iteration and
the Euclidean norm at the initial stage is used to control the convergence. The
recommended ratio is 0.25. This ratio, by observation, results in a too slow computational
procedure in this combustion process. This method thus is not recommended in this
model. Second, the overall solution procedure for both gas and condensed phase
calculation is attained in an iterative manner. The iterative cycle as presented in
Appendix II. Moreover, additional convergence criteria are needed for the mass and
momentum equations. For the mass conservation equation, we use the mass equation
residual as criteria. For the momentum equation, or the energy equation, fuel
concentration, etc., the relative difference between two recent iterative values is used as

criteria.

305

 

 

 

 

Figure l Grids and control volumes.

i
..L.
I
I
l
L.
I
4.
l-
l

I
...-1
I

___ _._...L. ..-|_..
I
I

a

Figure 2 The staggered grid system.
306

APPENDIX V

DEFINITION OF THE HEAT TRANSFER MECHANISMS IN IGNITION ANALYSIS

The numerical derivation of the heat transfer mechanisms are the followings:
l. The total radiation heat is obtained by multiplication of the constant radiation heat
ﬂux with the ignition delay time.

2. The solid heat conduction is derived by
. 8T
'3 " k —dxdt
K I; C B),

where kc takes on the liquid or solid conductivity when appropriate; L, is the

longitudinal length of the interface.

3. The gas conduction is obtained in a similar manner by

BT
‘g " k —dxdt
K g g 3y
4. The radiation heat loss is obtained by
i x 4 4
Kg E 80(Tinterface _ T0 )dth

where T 0 is the initial temperature 300K.

307

APPENDIX VI
DERIVATION OF VOLUME AVERAGING EQUATIONS FOR BUBBLE FORMING

PROCESS IN HEATED POLYMERS

1. The Basic Equations
1.1 Governing Point Equations

1.1.1 Assumptions

0 The solid phase contains only polymer.
0 The liquid phase contains both polymer and monomer.
o The gas phase contains only monomer; we do not consider the existence of

dissolved gas.

1.1.2 Restrictions

o h = h0 + CP (T — T O) and C p,- = C p = const for each species and separate phase.

0 Conductivities are constant for the three separate phases.

1.1.3 Solid Phase

The mass and species conservation equation in the solid are trivial; the energy equation
is given as

8T

3
“87(poho)=poCPoE=-V’qo' (1)
where 40 =—k0VT .
1.1.4 Liquid Phase
Mass Equation:
apﬂ
— + V - = 0 (2)
a: (Pﬁvﬂ)

308

where the averaged velocity vﬁ is obtained from p ﬁv ﬂ = pmvm + p pv p .

Species Equation:

%P;+V'(pivi)=wi’ i=m,p (3)

In Equation (3), the decomposition of polymer into monomer takes place in the liquid and

w is the reaction rate with the relationship of WI) + wm = 0. We introduce Y, = p,- / p ﬂ to

denote the volume fraction of species 1' in the liquid phase. Writing the species velocity in

terms of the mass averaged velocity, we obtain v,- = vﬂ +ui , where “i denotes the

diffusing velocity. Therefore Equation (3) can be reorganized based on the above two

deﬁnitions,
3,0,5 3Y3
Yi(—at— + V ' (IO/3‘36» + 105(3'1‘ VpV ' Yr) + V ' (PpYiui) = Wi
The ﬁrst term on the left hand side is zero according to Equation (2), thereby,
BY.-
Pﬁ(‘aT+VﬁV‘Yi)=-V°(PﬁYiui)+ Wi (4)
The diffusive ﬂux p ﬂYiu, can be formulated by,

pﬂYiui = _pﬁDVYi (5)

So that our ﬁnal form of the species equation is

pﬁ(%%+VﬂVY;-)=V(ppDVYI-)+Wl (6)

If we deﬁne g; in terms of the volume averaged properties, i.e., 29 = ? + v I3 .V(),
t t

we obtain

309

 

DY
p5 Dt‘ =V-(pﬂDVYi)+w,- (7-a)
Another form of the species equation is,

at
, and can be) directly obtained from Equation (3).
Energy Equation:

The appropriate form of energy equation for a multi-component system is:
a
$(Zpihi)+V-(Zpihivi)=V-(k5VT) (8'3)
1'
But we will start from another form of energy equation,
8 8 b
gt-(pﬂhﬂ)+VH(pphﬁVﬁ)=—Vqﬂ ( ' )

where q ﬂ =—kﬂVTﬂ +Zpiuihi- The averaged enthalpy hp is introduced by the
i

deﬁnition of p ﬂh ﬂ = pmhm + p ph p . This allows us to reduce the LHS of Equation (8)

by using Equation (2).

D__ht,,

or, 9

Following from Equation (9),
D__hﬂ Y.- Dh:
pp— Dt =ppE(ZY.-h-=) pp(2%- h.-+ 211-D —)= pr; Z—h +p/3Cpp— (10)

and using Equation (4), we obtain from the ﬁrst term of Equation (10),

310

 

 

 

Di:-
Dt

 

hi = ZX‘V ' (pﬁYiui) + Wi)hi
= Z(-hiv'(pﬂyiui)+hiwi) (11)
= Z(-V ' (hiPﬁYiui) + pﬂYiuiV ' hi + hiwi)
= Z(-V ' (hipiui) + piuiv ' hi + kiwi)

PpX

We substitute Equation (11) into Equation (10), and then substitute Equation (10) into

Equation (8),
DT — 2 12
Z(-V'(hipiui)+piuiv ' hi +hiwi)+p/3CP,B-D_t-’kﬂv 7}? “(ZV'(Piuihi)) ( )

The ﬁrst term on the left hand side cancel with the second term on the right hand side. In
addition, we apply the restrictions of constant thermal capacity; the second term on the

leﬁ hand reduces to

Zpiuiv'hi=(ZpiuiCPiW'T=(ZPI“I)CP,6V°T=0 (13)

The third term on the leﬁ hand side of Equation (12) is reduced to

Zhiwi = 21h.9+Cp(T-T°)w,-]=zh,9w,+CP(T—T°)zw,. =Zhi0w'. (14)

Finally we obtain the energy equation in temperature form,

pﬂCPﬂ%=kﬁV2Tﬁ—thpwi (15)
1.1.5 Gas Phase
Mass (species) Equation .'
8p
31+V-(p7v7)=0 (16)

Energy Equation:

311

4'

nm.mﬂjw
I.

 

 

D(pyhy) DT
_._—_._: c _=_V. 17
Dr ,0, P7 Dt q” ( )

where q}, = —k7VT .

1.2 Boundary Conditions

The solid-gas boundary condition is trivial since the solid phase does not contact
with the gas phase in this problem. We will present the solid-liquid and liquid-gas
boundary conditions.
1.2.1 Solid-Liquid Interface

To derive the energy boundary condition, we will follow the approach given by [82].

The material control volume Vm is illustrated by Figure 1. It contains both the solid
phase V0 and the liquid phase Vﬁ , and is separated by a singular surface Adﬂ = A130.

The integral representation of the energy equation in V," is given by:

£1pth=—jq.ndA (18)
Dt Vm Am
We already know that
h = h h = h
in phased {p Pa 0 in phase ,6 {,0 Pp ’6 (19)
q = qa q = qﬁ
First we integrate Equation (1) over volume V0. to obtain
a — v (20)
I§(Poho)dV “ — I 'qodV

V0. Vﬁ

We can use the general transport theorem to express the ﬁrst term in Equation (20) as

a d
I 5—(p0'h0') = — f(pa'ha)dV - Ipahaw ' naﬂdA (21)
V0 t dt V0 Aaﬁ

312

 

 

 

Where Aoﬂ is the interfacial area. As seen from Figure 1., the area integral over A0 is

zero because the velocity Va is zero there. Substituting Equation (21) into Equation (20)

and applying the divergence theorem to the second term lead to

d
d? “Pa-ha )dV - Ipohow' "oﬂdA = _ Iqo '"odA "' Iqo '"oﬂdA (22)
Va A073 A0 A05

We can repeat these steps for Equation (8-b) to obtain the similar expression for the ,3

phase.

1(pﬁhﬂ)dV+ lpﬁhﬂ(vﬂ_w)'”ﬂodA=‘ Iqﬂ'"ﬂdA_ lqﬂ'"ﬂodA (23)
V

1.
dt
I3 Aria AB Ara

We now add Equation (23) and (22) together and observe the facts of

A =A d2 =1 .5". .
or [to an Dtyio dt Vim dt V10

D
B; Ipth-i' ﬂpﬁhﬂ(vﬁ‘W)'"ﬁo-Pohow'"oﬁ]
Vm 40:6 (24)

=— ]q-ndA— ][qﬁ '"ﬁa'l'qa -no.ﬁ]dA
Am Ad]?

where Vm is the material volume and Vm = V0 + Vﬂ. Comparing Equation (24) and
Equation (18) leads to the jump condition at the o — ﬂ interface.
pﬂhﬂ (Vﬂ _ W) ' ”,60 - pohow' "oﬂ = "Iqﬁ '"ﬂo + qo '"oﬂI (26)
The jump condition of mass transfer can be easily obtained:
pﬂ(vﬂ—w)-nﬁa—paw-naﬂ20 (27)
The derivation of species jump condition follows the same route

pp(vp—w)-nﬂa-pow-naﬂ=0 (28)

313

. Ll

 

 

 

pm(vm —w)-nﬁ0 =0 (29)
Equation (26) can be further reduced. We substitute the deﬁnition of q ﬂ and qa into

Equation (26) and get.

8T0

anga

an,
2
+ '6 all/30-

 

 

pﬁhﬂ(vﬂ— W) ”’30-— pO'hO' W n0ﬂ=[— 2'0 ﬂ-I'Zpiuiih nﬂa] (30)

In addition, recalling the deﬁnition of

pﬂvﬂ = pmvm +pp p = pm(vﬁ +um)+pp(vﬁ +up) = pﬂvﬂ +pmum +ppup

9

we obtain
Ymum+Ypup=0 (31)
According to F ick’s Law, the diffusive heat ﬂux Ytu, can be expressed as Ylu, = —DVY,. ,

thus Equation (26) is reduced to the following form,

 

 

3T BY
po(hﬁ_ h'o)W noﬂ=[- Ada UTAH ﬂ ”pﬂD(hm_hp)5_’£‘]' (32)
16 "I30

3530
1.2.2 Liquid-Gas Interface

By following the similar approach in section 1.2.1, we have the energy, mass, and
species boundary condition,
Energy boundary condition,

Pphﬁo’ﬂ ‘ W) ' ”ﬁr + Pr’Wr " W) ' "m = “I‘m '"ﬂr + 47 ' "m1 (33)
Mass boundary condition,

pﬂ(Vp—w)-nﬂ7+py(v7—w)onm=0 (34)

Species boundary condition,

314

 

 

pm (vm — w) - n13}, + p},(v}, — w) - nw = O (35-a)
pp(vp—w)-nﬁ7=0 (35-b)
The boundary conditions for the three phase interfaces are now complete, and we will
derive the volume-averaged form of the transport equations. These equations will apply
to every point in space, not just in each separate phase.
2. The Volume Averaging Method
Since the solid phase is independent of the transport process in the ﬂuid including
both the gas and the liquid, we restrict our attention to the ﬂuid part.

2.1 Deﬁnitions and Theorems

2.1.1 Volume Fraction
£ﬂ=Vﬁ/V,£7=Vy/V (36)

2.1.2 Phase Averaged Quantity

Wk =i Wde (37)
VVk

The phase-averaged quantity is the averaged value of quantity #0, over the entire

averaging volume V .

2.1.3 Intrinsic Phase Averaged Quantity

1771? =i MC” (38)
Vk Vk

Here the averaging volume is the phase volume Vk. If t/lk is uniformly distributed in
Vk , then [[7]:r = I]! k . By comparing Equation (38) and Equation (37), we can obtain,

—" - — 39

«‘3ka - Wk ( )

315

 

 

 

2.1.4 Fluctuation Term

We formulate the quantity Wk by
— r" "' 40
Wk - Wk + Wk ( )
Where the ﬂuctuation Wk represent the deviation from the intrinsic phase average

quantity 17,5 . In addition we have the following relation [88],

 

—k—k 7'"—
Wk‘bk =49ka ‘Dk +Wk‘1’k (41)
2.1.5 Theorem 1: About Averaging of The Time Derivative
aWk 8Wk 1
__ .-.____ w-n dA (42)
(at) a: Viv" k

where Ak is the interfacial area of phase k with other phases, nk is the outward unit
normal vector of the area dA , and w is the interface moving velocity.
2.1.6 Theorem 2: About Averaging of The Spatial Derivative

V Wk = V Wk +71; kande (43)
Ak

2.2 The Derivation Procedure
The derivations summarized in [88] will be presented here. A general transport

equation over phase k has the form,

-a%lc-+V°(kak)=V'Jk+Ska (44)

Averaging the above equation over volume V, leads to

i (Marl +1 (V . (t/Ikvk)dV = l [V .1de +i 15de (45)
V V1. 31 V Vk V VI. V Vk

By applying the theorems l and 2, we obtain the following expression [88],
316

iiuﬁn “_n :y iﬁix‘lti". ——.

 

 

%+V.(y7f17k)=V-Jk +§k+lkD+IkJ+IkQ (46)

Where I£=V'-;-I(—V7k;k)st 1):]:71Jk ndea and 1kg: — VIII/[((Wk—Vk)‘ ’1de
V VAk VAk

Q

I I? are the macroscopic deviation term, I I}, and I k are the interfacial terms.

3. Macroscopic Equations
3. l Assumptions

1. Only two phases are considered, the liquid phase B and the gas phase 7.
2. Variations of the material properties in V are neglected, i.e., Wk 2 0, although

globally they may vary [88].

3. All phases in the averaging volume are in thermal equilibrium, that is,
<Tﬂ >IB=<T >7=T.

7
3.2 Macroscopic Equation of Mass Conservation

In this case, according to Equation (46), (a = p, J = 0, and S = O , therefore I k0 = 0

from assumption 1 and I ,{ = 0. The liquid mass Equation (2) and gas mass equations

(16) yield,
.55 _ .. _ -
T: +V-(pﬁvﬂ)—Ig (47 a)
a—
_§IL+V.(§ﬁ,—,y)=1y9 (47-b)

Where 1gV=_ Ajp’dwﬂ- ”,6)' "ﬁydA and [Q_ VAley(W7_ vﬂ-nmdA- According
to the interfacial boundary condition (34), we get,

317

Ig+1§2=0

If we deﬁne ii—t = If , we obtain from Equation (47-a) and (47-b),

aﬁp _ _ —.
___-8t +V-(pgvﬂ)=—m (47-0)
3157 —/3— _-
—'aT+V'(pﬁV7)-'=m (47'd)

We add Equation (47-c) and (47-d) together, and obtain

3 — — -/3- —ﬂ ._ -

-a—t(pﬁ+p7)+V-(pﬁvﬁ+pﬂv7)—O (48 3)
Another form of the equation can be obtained if we use the deﬁnitions of pk = emf,
k = ﬂy, i.e.,

gag)? +57%?) + V (ts/355175 +87D;V;)= 0 (48-h)

3.3 Macroscopic Equation of Energy Conservation

Averaging Equation (8) and (17) by using I/Ik = pkhk , S k = 0 , J k = qk leads to
%(2p.ht)+V-(Zpthtvt)=V-(kﬂVT)+I§+1}; (49-..)

a __ ___. _
5799,127) + V - (pyhyvr) = V - q, + 1;? + I; (4%)

where

1 J l
Ig=; [p7h7(wy-v7)-n7dA,I}, =-; [qr-nydA,
Ar Ar

1 1
Ig=y [(gpiki)(wﬂ—vﬂ).npd/1, , and If,=_7Aj(kﬁVT)-nﬁd.4- We add
Aﬁ ' ' l3

318

 

 

equations (49-a) and (49-b) together, and use the rearranged interfacial boundary

condition from Equation (33), as obtained from [82],

EilpihiO’i " W) ' ".67 + Pyhy("7 " w) ' "213 = (kﬂVT) ' "ﬁr + q}, '"Iﬂ

Thereby we obtain,

 

 

3%“; pihi) + (pr/17)] + V ~ [(2 ptht-vt) + (pyhyvyn = V - [(kaT) + V7] (50)

The terms in Equation (50) need further discussions, as are presented in the following 3

sections.
3 3 l Derivation of a h +V h
- ' E(py 7) '(p7 7V7)

Since the gas phase has the single component monomer, the detailed derivation is
similar to the procedure from (IIC-43) to (IIC-51) in [82]. Here only the ﬁnal expression
is given,

—— 97f .. _, D Q
(p7h7)+V-(p7h7v,,)=eypyCP7—5t—+p7CP7v7VT, +171+171 (51-a)

 

9.
at

D ~ ~ — l
where [71: prV - pyvyTy, and If] =[h3 +Cp7(Tyy-T;9)I{'17 [py(w—v},)-nmdA}-
A116

3.3.2 Derivation of 82(2 pih‘.) + V . (Z pihr'vi)
t t' i

Following the similar procedure used in the above section, we obtain,

a— ___ .. arr ——. D Q
5(Zpihi)+v°(zpihivi)=ZpiCPi—at—T'I'ZCPipivi 'VTy +1p1+1p1 (Sl'b)

a :w— " ~
Where 151 = 5Q: CPipiTﬂ) + V ' ZCPiini T3, and
l l

319

 

__ a-. _—

13. = zih? + Cpi(T/§B — Tint—aptiw - (pm-)]-

Here we substitute species conservation Equation (53-a) to simplify I g] , thereby
1% = gilt-0 + Ctr—<73” - Time- + 1g1=sh9e+§1h9 + Cpl-(TI? - Timi-

3.3.3 Derivation of V - [(kﬂVT) + (7),]

 

V -[(k,3VT) + 21;] = V -[(kﬂVT) + (kyVT)] = V . (eﬁkgvr; + 67k;V7—‘77) (51-c)

Here the formulation of an effective conductivity k; is introduced, and the detailed

derivation method is presented in [82].
After we derived all terms for Equation (50), we add them together and get the following

expression,

_ _ 8T — _— —
[832(piﬂcpi)+£7p;CP7]E+[Z(iniCPj)+ PyvpryIV - T (52)
l I

=V.[(eﬂk; +cyk;)VT] —Ahvap; — Zhio'w‘, + 151+ 1?,
I

In Equation (52), "thwt‘ denotes the pyrolysis reaction effect, and Ahvap-r; denotes

l

the evaporation effect. In the latter term, Ir—t is already deﬁned in Equation (47); and

Ahvap is deﬁned as

Ahvap = (h? 4:2,. + <pr — Cpnxf — 10)].
3.4 Macroscopic Equation of Species Conservation

In this case, we average species equation (3) and obtain

%P}+V(EE)=W—I+Igr i=M,p (53-3)

320

 

where 15.2% I pi(Wﬁ‘Vﬂ)'" ,6 761,4. Another form of equation is obtained by
*4/97

averaging Equation (7-b), and we use ([11. = p ﬁYt" Ji = — p ,B DVYI. , and Si = Wt to obtain

3(a)/335:7)
at

 

+V-(pﬂ17iﬂi7ﬂ)=V-(eﬂpﬂDgV17/3Hw, +Ig, i=m,p (53—b)

Note that the ﬁrst term on the RHS of Equation (53) is written in terms of the effective
diffusion coefficient by following the same argument in derivation of the energy

equation. We will describe monomer and polymer equations one by one.
For polymer conservation equation, I go is zero according to interfacial boundary

condition Equation (35-b),

a 4317 _ . _
Egg—UV«bib/3%)=V'(€ﬂPﬁDﬂVYpﬂ)+< W > (54'5"

For monomer species conservation equation, we add monomer conservation equation in
the liquid (53) and monomer conservation equation in the gas (47-d), and use the

interfacial boundary condition (3 5-a), and get
567(ﬁgY-m +ﬁy)+V -(p_ﬁﬁi7mﬂ§-ﬁ +ﬁ;VY)=V . (gppﬂDgi/g’)+ < Wm > (54-h)

3.5 Macroscopic Equation of Momentum Conservation

In traditional porous media ﬂow, the porous media is assumed to be a ﬁxed structure.
However, in two-phase bubbly ﬂow, there does not exist a ﬁxed solid structure. That is,
the Darcy’s Law is no longer applicable to this situation. Therefore one has to use the
general momentum equation to solve the ﬂow ﬁeld. In this case, we assume the

Newtonian ﬂuid and W = pv. For the liquid and the gas we obtain the momentum

321

 

equations respectively according to Equation (46),

3,5ng

 

+ V . (pgvgvﬁ) = NF), + [75g +V {MW/3 + ,uﬁ(Vvﬂ)'] +13 + I; (SS-a)

ath—z _ _ _ _ _ __ ——-—— J
—-a”t—”+V . (ﬁt/7712,) = —VP, + pyg +V {rt/1V1», +p,(Vv,)‘]+ 1,9 + I, (55-h)
where the body force is the gravitational force. The deﬁnition of I]? and I)! can be

readily obtained from Equation. (46). The viscous term on the RHS of the Equation (55-
a) can be written in terms of the effective viscosity by following the same route in

deriving the effective conductivity,
W+ ﬂﬂ(Vvﬁ)t = 5,31%va + (vaYJ
We add the two equations (55-a) and (55—b) together, and use the following interfacial
boundary conditions,
13H? :0, and 1], +1; :0.
, then obtain the macroscopic form of the momentum equation,

3(ﬁ'gV—ﬂ +p}’t77)
at
(p), + p,)g+V-[eﬁrtg[vrg +(va)t +£rp;[Vi7}Z' +(Vv;’)’]

 

+V-(pgvgvp +p}’v}’t7,)=—V(Fﬂ +P7)+ (56)

4. Simpliﬁcation of Macroscopic Equations
Several additional assumptions have been made to simplify the governing equations

for the bubble forming process of polymers. They are listed here:

I. Vﬂ=Vp=Vﬁ=0.

2. 35:37:]:

322

 

 

3. CF", =Cjep =CP/3.

Assumption 1 is based on the fact that (l) the external heat ﬂux is applied at the
surface of the polymer, which tends to decrease the liquid natural convection due to
buoyancies that ﬁrrther results from density changes; (2) the thickness of the liquid is
normally very small, which conﬁnes the convection in a enclosed boundary. Assumption
2 is justiﬁable if the conduction dominates the convection. With these two assumptions,
the macroscopic transport equations of mass, heat and species can then be simpliﬁed

from equations (48), (52), (54) and (56), respectively, and yield

8 _ _ _
57(eﬁp’3 +eyp,’,')+V-(eyp,’,’v}')=0 (57)

_ _ ar _ _ _
[Egypt/3%»+eyp77Cp7137+(£7py’prvyy)V'T (58)

=V-[(e,3k;‘9 +£7k;)VT]—Ah E—zhf’w,
i

vap
8 _ﬂ * ‘ﬂ 59
$31910pr )= V - (sﬁpﬁDﬂVr/p )+ < wp > ( -a)

%(£ﬂpﬂl7mﬂ + 8),/7;) + V . (er/7;?” = V . (516/56th?an < Wm > (59-b)

3(5715779?)
3:

 

+ V ' (8715779577) (60)

= —V(eﬁI—’ﬂﬂ +£7P77)+(eﬁpﬁ +eyﬁ;)g +V ' (871”;[VV3’I +(VV;)’]}

Furthermore, we deﬁne the following parameters ﬁ 2 p385 +6347; , f p = pﬂeﬂ / f) ,

fy=p;e,/p, V=f,v;, EP=[eﬂpﬁch+e,p;CP,]/p. pr=pr,

I? =£ﬁk;3 +eyk;, 7p = fﬁYpﬂ 17m =f/317"? + f7’ and F =gﬁF/56 +5717, where f}, and

323

 

f ﬂ are mass fractions of liquid and gas respectively. In addition, the viscous term in
Equation (60) can be ﬁrrther simpliﬁed by the following formulation
III _ I _ t
V - {£7ﬂ7[Vv}7,’ + (Vv;) ]} = V - {,u[V8},v77,’ + (View?) ]}.
and viscosity ,u in the above expression should be defrned appropriately in terms of the

liquid and gas properties. The advantage of this formulation is the ease with which this
form complies with the traditional form of momentum conservation equation. Based on

the above newly deﬁned parameters, we obtain

 

the mass conservation equation,

ai+V(pV)=0 (61)
at
the energy conservation equation,
_— 3T _— —— - :— o_
pCP§+pCP7VV°T=V'[kVT]_Ahvapm-Zhi Wi (62)
i

the species conservation equations for polymer,

ates.)
at

 

= V - (ﬁfﬂDEVYp'B) + WP (63'3)

a _— _— _ . — _
Emmw-(pV)=V-(pankVYm(’>+w... (63-h)
and the momentum conservation equation,

9(g—tI/lrv-(fi—I717)=—V}—)+ﬁg+V-{,UVV+ﬂ(VV:)t]}- (64)
7

In addition, if we rewrite the ,6 phase mass conservation equation (47-c), we obtain,

— 88,3

324

Equations (61-65) are the ﬁnal forms of simpliﬁed macroscopic governing equations for
the modeling of bubble forming process. The constitutive equations are needed to close

the system, i.e.,

Eﬁ + E}, :1 (66)
g+m=1 (a)
w, = w, = ppAe-E/RT (68)

Seven unknowns are expected to be solved from the above equation system, that is, ,6 ,

T, Yp, Ym, V. m,eﬂ,and 87.

325

"3
Solid

(0 phase) "ﬁ\
0'

Liquid

()3 phase)

Figure 1 Material volume containing a solid-liquid interface.

326

REFERENCES:

1. Wichman, I. S., Progress of Energy and Combustion Science, 18: 553, 1992.

2. Ross, H. D., Progress in Energy and Combustion Science, 20: 17, 1994.

3. Law, C. K., Progress of Energy and Combustion Science, 10: 295, 1984.

4. Di. Blasi, C., Progress of Energy and Combustion Science, 19: 71, 1993.

5. Femandez-Pello, A. C., Hirano, T., Combustion Science and Technology, 32: 1,
1983.

6. Cullis, C. F. and Hirschler, M. M, The Combustion of Organic Polymers
(International Series of Monographs on Chemistry), Clarendon Press, Oxford, 1981.

7. Elomaa, M., Sarvaranta, L., Mikkola, E., Kallonen, R., Zitting, A., Zevenhoven, C. A.
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