HEAT AND MASS TRANSPORT INSIDE A CANDLE WICK
by
Mandhapati P. Raju
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Dissertation Advisor: Dr. James S. T’ien
Department of Mechanical and Aerospace Engineering
CASE WESTERN RESERVE UNIVERSITY
January 2007
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
Mandhapati P. Raju
______________________________________________________
candidate for the Ph.D. degree *.
Prof. James S. T'ien
(signed)_______________________________________________
(chair of the committee)
Prof. Chih-Jen Sung
________________________________________________
Prof. Edward White
________________________________________________
Prof. Beverly Saylor
________________________________________________
Dr. Daniel Dietrich
________________________________________________
________________________________________________
09/06/06
(date) _______________________
*We also certify that written approval has been obtained for any
proprietary material contained therein.
TABLE OF CONTENTS
TABLE OF CONTENTS
iii
LIST OF TABLES
v
LIST OF FIGURES
vi
ACKNOWLEDGEMENTS
xii
NOMENCLATURE
xiv
ABSTRACT
xxii
CHAPTER 1: INTRODUCTION
1.1 Candle Basics
1.2 Candle Burning
1.3 Previous Work
1.3.1 Previous Work on Candle Flames
1.3.1.1 Experimental Work
1.3.1.2 Numerical Work
1.3.2 Previous Work on Two-phase Flow in Porous Media
1.4 Purpose and Scope of this Dissertation
1.5 Dissertation outline
1
1
4
6
7
7
10
13
21
21
CHAPTER 2: AXISYMMETRIC WICK MODELING
24
2.1 Formulation of Transport Process in Porous Media
24
2.1.1 Mathematical Formulation
24
2.1.2 Numerical Formulation
31
2.2 Multifrontal Solvers for Large Sparse Linear Systems
33
2.2.1 Introduction
33
2.2.2 Multifrontal Solution Methods
36
2.23.Benchmark Testing
40
2.3 Analysis of an Externally Heated Axisymmetric Wick
41
2.3.1 Physical Description of the Model
42
2.3.2 Sample Case Results
43
2.3.2.1 Saturation and Temperature Distribution
43
2.3.2.2 Pressure Distribution
41
2.3.2.3 Mass Distribution
44
2.3.2.4 Heat Flux Distribution
45
2.3.2.5 Variation Along the Cylindrical Surface and the Axis of the Wick
46
2.33 Mesh Refinement Studies
47
2.3.4 The Effect of Applied Heat Flux
48
2.3.5 Parametric Studies
49
iii
2.3.5.1 The Effect of Gravity
2.3.5.2 The Effect of Absolute Permeability
49
50
CHAPTER 3: GAS PHASE MODELING INCUDING RADIATION
3.1 Theoretical Formulation
3.1.1 Continuity Equation
3.1.2 Momentum Equations
3.1.3 Species Equation
3.1.4 Energy Equation
3.1.5 Boundary Conditions
3.2 Non-Dimensional Parameters
3.3 Property Values
3.4 Numerical Procedure
3.4.1 Grid Generation
3.4.2 Numerical Implementation
3.5 Gas Radiation Model
3.5.1 The Equation of Radiative Transfer
3.5.2 Numerical Solution of Discrete Ordinates Method
3.5.3 Discrete Ordinates Angular Quadrature
3.5.4 Solution of Discrete Ordinates Equation
3.5.5 Mean Absorption Coefficient
3.6 Solution Procedure
74
74
75
76
77
78
79
82
85
86
87
88
89
90
95
97
98
104
105
CHAPTER 4: RESULTS AND DISCUSSIONS
4.1 Candle Flame Coupled to a Porous Wick
4.1.1 Detailed Flame Structure at Normal Gravity and 21% O2
4.1.1.1 Steady State Candle Flame (Wick Length = 4 mm)
4.1.1.2 Self Trimmed Candle Flame
4.1.2 Effect of Gravity
4.1.3 Effect of Wick Permeability
4.1.4 Effect of Wick Diameter
4.1.5 Effect of Ambient Oxygen
4.1.6 Validation of results
CHAPTER 6: CONCLUSION
120
120
122
123
130
133
135
137
137
138
196
Recommendation for Future Work
198
BIBLIOGRAPHY
201
iv
LIST OF TABLES
Table 2.1 Porous Wick Dimensionless variables
52
Table 2.2 Table 2.2 Porous Wick Numerical Values
53
Table 3.1: Correlating equations of specific heats for O2 , N 2 , CO2 , H 2 O , and fuel.
107
Table 3.2: Gas phase property values.
108
Table 3.3: Nondimensional parameters.
109
Table 3.4: Non-dimensional governing differential equations.
110
Table 3.5: The S4 quadrature sets for axisymmetric cylindrical enclosures.
111
Table 3.6: Least-square fitting equations of Planck mean absorption coefficient for CO2
and H2O.
112
Table 4.1(a) : Effect of wick diameter on candle flame characteristics (5mm candle
.
diameter and 21% O2)
144
Table 4.1(b) : Effect of wick diameter on candle flame characteristics (5mm candle
diameter and 21% O2)
.
144
Table 4.2: Effect of wick diameter (Alsairafi, 2003) on candle flame characteristics (5mm
wick length, 5mm candle diameter and 21% O2)
145
Table 4.3: Comparison with candle flame experiments in high gravity levels.
v
145
LIST OF FIGURES
Figure 1.1: Schematic of a candle flame.
23
Figure 2.1: Comparison of function residuals vs. CPU times for Newton, modified
Newton and Picard’s iterative techniques.
54
Figure 2.2: Physical description of an externally heated axisymmetric wick.
55
Figure 2.3: Computational grid of an externally heated axisymmetric wick.
56
Figure 2.4: Plot of (a) saturation profiles (b) non-dimensional temperature profiles
and (c) non-dimensional temperature profiles (expanded in the two-phase region)
inside the porous wick for parameters shown in Table 2.2.
57
Figure 2.5: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick for parameters
shown in Table 2.2.
58
Figure 2.6: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor
mass flux vectors (expanded near the tip of the wick) inside the porous wick for
parameters shown in Table 2.2.
59
Figure 2.7: Plot of (a) liquid convective heat flux vectors (b) vapor convective heat flux
vectors and (c) conductive heat flux vectors inside the porous wick for parameters shown
in Table 2.2.
60
Figure 2.8: Plot of saturation and temperature variation along the cylindrical surface of
the wick exposed to the heat flux for parameters shown in Table 2.2
61
Figure 2.9: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2
62
Figure 2.10: Plot of liquid mass flux (in x-direction) variation along the cylindrical
surface of the wick exposed to the heat flux for parameters shown in Table 2.2
63
Figure 2.11: Plot of saturation and temperature variation along the axis of the wick for
parameters shown in Table 2.2
64
Figure 2.12: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick for parameters shown in Table 2.2
65
vi
Figure 2.13: Comparison of (a) saturation profiles (b) pressure profiles and (c)
temperature profiles for three different meshes (80x40, 80x80, 160x40) inside the porous
wick for parameters shown in Table 2.2.
66
Figure 2.14 The variation of saturation at the cylindrical tip of the wick surface with the
total heat supplied to the wick.
67
Figure 2.15 The variation of total mass of wax evaporated from the wick surface with the
total heat supplied to the wick.
68
Figure 2.16 The variation of percentage heat that is lost to the reservoir with the total heat
supplied to the wick.
69
Figure 2.17 The effect of gravity on the variation of saturation at the cylindrical tip of the
wick surface with the total heat supplied to the wick.
70
Figure 2.18 The effect of gravity on the variation of total mass evaporated from the wick
surface with the total heat supplied to the wick.
71
Figure 2.19 The effect of absolute permeability on the variation of saturation at the
cylindrical tip of the wick surface with the total heat supplied to the wick.
72
Figure 2.20 The effect of absolute permeability on the variation of total mass evaporated
from the wick surface with the total heat supplied to the wick.
73
Figure 3.1 Schematic of a candle
113
Figure 3.2: Variable grid structure for modeling candle flames for 1mm wick diameter
and 5mm candle diameter
114
Figure 3.3: Schematic of radiation intensity transfer energy balance on arbitrary control
volume in a participating medium.
115
Figure 3.4: Geometry and coordinate system for 2D axisymmetric cylindrical enclosure.
116
Figure 3.5: Projection of an S4 quadrature set on the μ,ξ plane using p,q numbering in r-x
geometry.
117
Figure 3.6: Four types of space angle sweep direction for SN scheme.
118
Figure 3.7: Solid angle discretization of the S4 quadrature.
119
Figure 4.1: Plot of (a) pressure profiles (b) saturation profiles and (c) temperature profiles
inside the porous wick coupled to a candle flame at normal gravity
146
vii
Figure 4.2: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick coupled to a candle
flame at normal gravity
147
Figure 4.3: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor
mass flux vectors (expanded near the tip of the wick) inside the porous wick coupled to a
candle flame at normal gravity.
148
Figure 4.4: Plot of net heat flux supplied by the candle flame along the cylindrical surface
of the wick
149
Figure 4.5: Plot of saturation and temperature variation along the cylindrical surface of
the wick coupled to a candle flame at normal gravity.
150
Figure 4.6: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick coupled to a candle flame at normal gravity.
151
Figure 4.7: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface
of the wick coupled to a candle flame at normal gravity.
152
Figure 4.8: Plot of saturation and temperature variation along the axis of the wick
coupled to a candle flame at normal gravity.
153
Figure 4.9: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick coupled to a candle flame at normal gravity.
154
Figure 4.10: Gas temperature contours (non-dimensinalized by T∞ = 300 K )
155
Figure 4.11: Fuel reaction rate contours (g cm-3 s-1)
155
Figure 4.12: Fuel and oxygen mass fraction contours
156
Figure 4.13: Local fuel/oxygen equivalence ratio contours
156
Figure 4.14: Carbon dioxide mass fraction contours
157
Figure 4.15: Water vapor mass fraction contours
157
Figure 4.16 Oxygen mass flux and flow field around the flame
158
Figure 4.17: Isobar contours; p =
p − p∞
ρ rU r2
159
Figure 4.18: Profiles of temperature and species concentration along the symmetry line
160
viii
Figure 4.19: Effective Mean absorption coefficient distribution (cm-1 atm-1)
161
Figure 4.20: Dimensional net radiative flux vectors (W/cm2)
161
Figure 4.21: Contours of divergence of radiative heat flux (W/cm3)
162
Figure 4.22: Heat fluxes on the candle wick surface per unit radian
163
Figure 4.23: Plot of (a) saturation profiles (b) non-dimensional temperature profiles and
(c) non-dimensional temperature profiles (expanded in the two-phase region) inside the
porous wick for a self trimmed candle flame at normal gravity
164
Figure 4.24: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick for a self trimmed
candle flame at normal gravity
165
Figure 4.25: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors inside the
porous wick for a self trimmed candle flame at normal gravity
166
Figure 4.26: Plot of net heat flux supplied by the candle flame along the cylindrical
surface of the wick
167
Figure 4.27: Plot of saturation and temperature variation along the cylindrical surface of
the wick for a self trimmed candle flame at normal gravity
168
Figure 4.28: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick for a self trimmed candle flame at normal gravity
169
Figure 4.29: Plot of liquid mass flux (in x-direction) variation along the cylindrical
surface of the wick for a self trimmed candle flame at normal gravity
170
Figure 4.30: Plot of saturation and temperature variation along the axis of the wick for a
self trimmed candle flame at normal gravity.
171
Figure 4.31: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick for a self trimmed candle flame at normal gravity.
172
Figure 4.32: Gas temperature contours for a self trimmed candle flame (nondimensinalized by T∞ = 300 K )
173
Figure 4.33: Fuel reaction rate contours for a self trimmed candle flame (g cm-3 s-1) 173
Figure 4.34: Fuel and oxygen mass fraction contours for a self trimmed candle flame 174
ix
Figure 4.35: Local fuel/oxygen equivalence ratio contours for a self trimmed candle
flame
174
Figure 4.36: Carbon dioxide mass fraction contours for a self trimmed candle flame 175
Figure 4.37: Water vapor mass fraction contours for a self trimmed candle flame
175
Figure 4.38: Oxygen mass flux and flow field around the self trimmed candle flame 176
Figure 4.39: Isobar contours for a self trimmed candle flame; p =
p − p∞
ρ rU r2
177
Figure 4.40: Profiles of flame structure at the centerline for a self trimmed candle flame
178
Figure 4.41: Effective mean absorption coefficient distribution for a self trimmed candle
flame (cm-1 atm-1)
178
Figure 4.42: Dimensional net radiative flux vectors for a self trimmed candle flame
(W/cm2)
179
Figure 4.43: Contours of divergence of radiative heat flux for a self trimmed candle flame
(W/cm3)
179
Figure 4.44: Heat fluxes on the candle wick surface per unit radian
180
Figure 4.45: Candle flames at various gravity levels for a self trimmed candle flame 181
Figure 4.46: Candle burning rate and self trimmed length of the candle (on log scale) at
various gravity levels for a self trimmed candle flame
182
Figure 4.47: Candle burning rate and self trimmed length of the candle (on normal scale)
at various gravity levels for a self trimmed candle flame
183
Figure 4.48: Self trimming length of the candle wick using different wick permeabilities.
184
Figure 4.49: Candle burning rates for different wick permeabilities.
185
Figure 4.50: Candle flame lengths for different wick permeabilities.
186
Figure 4.51: Maximum gas phase temperatures for different wick permeabilities.
187
Figure 4.52: Self trimming length of the candle wick for different wick diameters.
188
Figure 4.53: Candle burning rates for different wick diameters.
189
x
Figure 4.54: Maximum gas phase temperatures for different wick diameters.
190
Figure 4.55: Maximum gas phase temperature at various oxygen molar fractions
191
Figure 4.56: Self trimmed length of the candle wick at various oxygen molar fractions
192
Figure 4.57: Candle burning rate at various oxygen molar fractions
193
Figure 4.58: Fuel reaction rate contours (5.e-5 g/cm3s)at various oxygen molar fractions.
194
Figure 4.59: Comparison of flame length and flame widths at various gravity levels with
the experiments (Arai and Amagai, 1993)
195
xi
ACKNOWLEDGEMENTS
I would like to express my gratitude to Professor James S. T’ien for the valuable
guidance, help and encouragement provided during the course of this work. He has been
very patient in guiding me all along the research. His comments and suggestions have
been thought provoking. It gave a lot of insight into the problem. His encouraging spirit
and friendly disposition has been very inspiring.
I am grateful to Professor Krishnan V. Pagalthivarthi (IIT Delhi) for his selfless
motivation in encouraging me to pursue my studies in Case Western Reserve University.
Learning Computational Fluid Dynamics under him was a rich experience for me. The
credit to my strength in Fluid Mechanics and in Numerical programming goes to him.
His example has been a motivating force for taking up a career in research.
I wish to acknowledge Dr Ammar, for providing me with his candle flame gasphase code and the relevant materials. He has been extending to me throughout my
research through e-mails. His timely clarifications have helped me to smoothly progress
in my work. I am thankful to Dr Amit Kumar for his cooperation and help in practical
matters. I also thank my colleagues I. Feier, Sheng-Yen Hsu, Ya-Ting Tseng, K. Tolejko,
G. Mittal, A. Garg, Ravikumar, B. Han, and K. Kumar with whom I had useful
discussions. I also thank Prof. Sung, Prof. Edward White, Prof. Saylor Beverly and Dr
Daniel Dietrich for serving as my examination committee members and for their valuable
time and suggestions. I would also like to thank staff of the department: J. A. Stiggers, A.
Szakacs, , C. Wilson, S. Campbell and M. Marietta for their prompt help at various
stages.
xii
Finally I would also like to acknowledge NASA for supporting the work under
the technical monitoring of Dr Daniel Dietrich.
xiii
NOMENCLATURE
Notation
Gas phase parameters
a
Coefficient in the discretised equation
A
Special function for power-law scheme
A
Area (cm2)
Ae
East surface area of a control volume (cm2)
An
North surface area of a control volume (cm2)
As
South surface area of a control volume (cm2)
Aw
West surface area of a control volume (cm2)
b
Constant term in the discretised equation
Bg
Gas phase pre-exponential factor (cm3 g-1 s-1)
c
Speed of sound (cm s-1)
C
Correction factor for the Planck mean absorption coefficient
Cp
Dimensional gas phase specific heat (cal g-1 K-1)
C p ,r
Reference gas phase specific heat (cal g-1 K-1)
d
geometric dimensionality (1, 2, or 3)
D
Flame diameter (cm)
Dw
Wick diameter (cm)
Di
Diffusion coefficient of species i (cm2 s-1)
êθ
Unit vector in the polar direction
êψ
Unit vector in the azimuthal direction
xiv
E
Gas phase activation energy (cal gmol-1)
Er
Radiant energy flux
fi
Stoichiometric mass ratio of species i/fuel
fr
Weighting factor in r-direction
fx
Weighting factor in x-direction
g
Gravitational acceleration (cm s-2)
ge
Gravitational acceleration on Earth (cm s-2)
G
Incident radiation (cal cm-2 s-1)
hi
Enthalpy of species i (cal g-1)
H
Flame height (cm)
I
Radiation intensity (cal cm-2 s-1)
J
Flux influencing a dependent variable φ
L
Latent heat (cal g-1)
Lr
Reference length (cm)
Lei
Lewis number of species i
m
Average burning mass fluxes (g cm-2 s-1)
mt
Total burning rate (mg s-1)
M
Total number of different directions for SN scheme
MWi
Molecular weight of species i (g gmole-1)
G
n
Outward normal to the surface
N
Order of discrete ordinates scheme
p
Dynamic pressure (atm)
P
Pressure (atm)
xv
P∞
Ambient pressure (atm)
qc
Conductive heat flux (W cm-2)
qr
Radiative heat flux (W cm-2)
q rr
Radiative heat flux in r-direction (W cm-2)
q rx
Radiative heat flux in x-direction (W cm-2)
q ry
Radiative heat flux in y-direction (W cm-2)
Q
Heat of combustion (cal g-1)
r
Dimensional r-coordinate (cm)
G
r
Position vector
Ru
Universal gas constant (cal gmol-1 K-1)
S
Source terms
SN
Discrete ordinates scheme of an order N
T
Temperature (K)
Tb
Boiling temperature of fuel (K)
Tm
Melting temperature of fuel (K)
T∞
Ambient temperature (K)
Tr
Reference temperature (K)
u
Axial velocity (cm s-1)
UB
Buoyant velocity (cm s-1)
UD
Diffusion reference velocity (cm s-1)
Ur
Reference velocity (cm s-1)
v
Radial velocity (cm s-1)
V
Volume of a control volume
xvi
w
Quadrature weights
W i
x,X
Sink or source term of species i (g cm-3 s-1)
Dimensional x-coordinate (cm)
Xi
Mole fraction of species i (%)
y
Dimensional y-coordinate (cm)
Yi
Mass fraction of species i (%)
α
Absorptivity
αr
Reference thermal diffusivity (cm2s-1)
β
Extinction coefficient (cm-1)
γ
Reflectivity
Δh of ,i
Heat of formation of species i (cal g-1)
ε
Solid emissivity
η
Direction cosine
φ
Dependent variable in governing equations
Φ
Scattering phase function
κ
Absorption coefficient (cm-1)
Γ
General diffusion coefficient in the discretised equations
λ
Thermal conductivity (cal cm-1 s-1 K-1)
λr
Reference thermal conductivity (cal cm-1 s-1 K-1)
μ
Gas viscosity (g cm-1 s-1)
μ
Direction cosine
μr
Reference gas viscosity (g cm-1 s-1)
ν
Frequency
xvii
θ
Polar angle
ρ
ρr
Dimensional gas density (g cm-3)
Reference gas density (g cm-3)
ρ∞
Ambient gas density (g cm-3)
σ
Stefan-Boltzmann constant (cal cm-2 s-1 K-4)
σs
Scattering coefficient
ω
Weighting factor
ξ
Direction cosine
ψ
Azimuthal angle
ζ
Expansion factor
G
Ω
Solid angle in terms of ordinate direction (ξ,μ,η) (sr)
Subscripts
b
Black body
B
Buoyant
E
Neighbor in the positive x-direction on the east side
f
Flame
i
Species i
max
maximum
min
minimum
N
Neighbor in the positive y-direction on the north side
P
Central grid point
r
Reference
xviii
S
Neighbor in the negative y-direction on the south side
w
Value at wall
W
Neighbor in the negative x-direction on the west side
x
Axial direction
Superscripts
°
Old value of a variable
+
Positive component
-
Negative component
Non-dimensional quantity
′
Incoming
x
x-direction
r
r-direction
p
p-level of μ direction cosine
q
q-level of ξ direction cosine
Special symbol
a
Absolute value of a
a1 , a 2 , a3 ,...
Largest of a1, a2, a3, …
Dimensionless numbers
Bo
Boltzmann number
Da
Damköhler number
Gr
Grashof number
Pe
Peclet number
xix
Pr
Prandtl number
Ra
Raleigh number
Re
Reynolds number
Porous wick parameters
c
specific heat capacity (J/kg-K)
ifg
latent heat of evaporation (J/kg)
G
Gibb’s phase function
h
enthalpy
k
thermal conductivity, (W/m-K)
K
permeability, (m2)
kr
relative permeability
Mw
molecular weight, (kg/kg-mole)
p
pressure, (N/m2)
Pe
peclet number
R
gas constant, (J/kg-K)
s
saturation
T
temperature, (K)
u,v
velocity, (m/s)
x, X
Dimensional x-coordinate
α
thermal diffusivity, (m2/s)
ε
porosity
μ
viscosity, (kg/m-s)
ρ
density, (kg/m3)
xx
σ
surface tension, (N/m)
ζ
triplet, ( pl , s, T )
σ
surface tension, (N/m)
χ
thermodynamic state parameter
η
entropy
ζ
triplet, ( pl , s, T )
Subscripts
l
liquid
g
vapor
r
relative
c
capillary
eff
effective
m
melting point
0
ambient, standard atmospheric conditions
s
solid
xxi
Heat and Mass Transport Inside a Candle Wick
Abstract
by
MANDHAPATI PADMANABHA RAJU
The purpose of this study is to investigate the effect of heat and mass transfer
inside the porous wick on candle flame combustion. The phenomenon of self trimming
that is observed in candle flame, whose wick is made of combustible material, is also
analyzed. This is accomplished by modeling two-phase flow inside an axisymmetric wick
and coupled with a gas-phase candle flame.
Gas-phase model has been taken from Alsairafi (2003). A finite volume method
is used to solve the steady mass, momentum (Navier-Stokes), species, and energy
equations, and the radiative transfer equation. The gas phase combustion process is
modeled by a single-step, second-order, finite rate Arrhenius reaction. The discrete
ordinates method is used to solve the radiative heat transfer with mean absorption
coefficients. Flame radiation is only from CO2 and H2O, the products of combustion. The
wick is modeled using volume averaged equations for two-phase flow inside an
axisymmetric wick. Momentum equations for liquid and vapor flow are governed by
Darcy’s law. The liquid is drawn to the surface by capillary action and the vapor is driven
in the two-phase region by vapor pressure gradient governed by thermodynamic
equilibrium relations.
Before coupling with the gas-phase flame, a decoupled numerical computation of
the wick phase is performed to isolate the effects of porous media parameters. This is
xxii
done by applying a prescribed constant heat flux along all the exposed surfaces. The
results show that the porous wick parameters like absolute permeability play an important
role in determining the saturation distribution inside the wick. For the range of the wick
parameters chosen for this study, gravity does not play an important role.
The next part of the study involved coupling the candle wick to the candle flame.
For candles, whose wicks are made of combustible materials, self trimming is being
modeled as the burn out of the dry region. The computed results show also that for a self
trimmed candle flame, the wick permeability plays an important role in determining the
candle flame structure. The wick permeability affects the self trimming length of the
wick and this affects the flame structure.
The effect of gravity on the self trimmed candle flame is analyzed. Trimmed
wick lengths are computed as a function of gravity. The results show that the burning rate
increases continuously from 0ge to 2ge and then decreases. At high gravity levels wake
flame is observed with a sudden decrease in burning rate.
Calculations have also been performed for different wick diameters. Maximum
flame temperature, burning rate, and flame dimensions are computed. A parametric study
using different molar oxygen percentage has been made at different gravity levels.
xxiii
1
CHAPTER 1
INTRODUCTION
Combustion has been the subject of many researchers’ interest because it provides
the majority of useful energy production in residential, commercial and industrial
devices. The flame, a thin zone of intense chemical reaction undergoing the process of
combustion, generally consists of a mixture of fuel and oxidizer (i.e. oxygen or air). It is
well known that flames are categorized as being either premixed flames or non-premixed
(diffusion) flames. In a premixed flame, the fuel and the oxidizer are mixed at the
molecular level prior to any significant chemical reaction. In a diffusion flame, the
reactants are initially separated, and reaction occurs only in the mixing zone between the
fuel and the oxidizer. A typical diffusion flame is that of a burning candle, (see figure
1.1).
Candle flame studies contribute to the understanding of many fundamental
aspects like chemical reactions, diffusion processes, porous wick transport as well as their
coupled effects.
Although the use of candles is common, not all the physical and
chemical aspects of candle burning are sufficiently understood. Before going into the
scientific aspects of a candle flame, some basic information on candles is provided.
1.1.
CANDLE BASICS
The
following
material
has
been
taken
from
the
website
http://candles.genwax.com/candle_instructions/___0___how_wick.htm .
The body of a candle is comprised of a solid fuel source, usually paraffin wax. A
wick runs through the center of the body of the candle from the bottom, extending out of
2
the top. The wick, which acts as a fuel pump when the candle is burning, is generally
made of cotton fibers that have been braided together.
Candle Wax
There are two main waxes used in candle making, Paraffin Wax and Beeswax.
Paraffin wax, which is classified as a natural wax, is the most common wax used in
candle making, and can be said to ultimately come from plant life.
In order to protect themselves from adverse weather conditions plants produce a
layer of wax on their leaves and stems. Material from dead plants 100-700 million years
ago accumulated in large quantities and eventually became buried beneath the surface of
the earth. After a long period of time, forces of heat and pressure turned the slowly
decaying plant material into crude oil, otherwise known as petroleum. Because of the
nature of waxes, being inert and water repellent, they were unaffected by the
decomposition of the plant material and remained intact, suspended within the crude oil.
Petroleum companies "harvest" the crude oil and process it. They refine the oil,
separating the different properties into gasoline, kerosene, lubrication oil, and many other
products. In many cases, the wax in the petroleum is considered undesirable and is
refined out. The refinery will process the wax into a clean, clear liquid, or as a solid
milky white block, and make it available to companies who may have a use for it. The
refined wax is called paraffin, which comes from the Latin "parum = few or without" and
"affinis = connection or attraction (affinity)". Basically there are few substances that will
chemically react with or bind to this type of wax.
3
A less common but more highly renowned wax for candle making is beeswax.
Classified as a natural wax, it is produced by the honeybee for use in the manufacture of
honeycombs. Beeswax is actually a refinement of honey. A female worker bee eats
honey, and her body converts the sugar in the honey into wax. The wax is expelled from
the bee's body in the form of scales beneath her abdomen. The bee will remove a wax
scale and chew it up, mixing it with saliva, to soften it and make it pliable enough to
work with, then attach it to the comb which is being constructed. Usually another bee
will take the piece of wax which has just been attached to the comb, chew it some more,
adding more saliva to it, and deposit it on another section of the comb. The combs are
built up, honey is deposited inside, and then the combs are capped with more wax. Since
several worker bees construct the comb at the same time, and the hive is constantly active
with other bees flying around and walking on the combs, depositing foreign matter onto
the combs, the composition of the wax becomes very complex.
Candle Wick
A candle without a wick is just a hunk of wax. The wick is what a candle is all
about. The earliest known candles were basically a wick-like material coated with tallow
or beeswax, not even resembling a candle at all. In taper candles the wick is the structure
which supports the first layers of wax that create the candle. In all candles it acts as a
fuel pump, supplying liquefied wax up to the top where all of the action takes place. As a
regulator, different size wicks allow different amounts of wax up into the combustion
area providing different size flames. The wick is pretty much the most important element
of a candle.
4
The word wick comes from Old English "weyke or wicke", Anglo Saxon
"wecca", and Germanic "wieche or wicke". It is a name for a bundle of fibers that when
braided or twisted together are used to draw oil or wax up into a flame to be burned in a
lamp or candle.
A wick without wax around it is just a piece of string. Because the wick is fibrous
and absorbent, melted wax adheres to it easily. Dipping a wick in and out of melted wax
several times builds up layers of wax, sufficient enough to make a taper candle. The
wick works by a principle called capillary action. Cotton fibers are spun into threads,
which are bundled and braided together. The spaces between the cotton fibers, the
threads, and braids act as capillaries, which cause liquids to be drawn into them. If you
place a drop of water in the center of a paper towel you will see that the drop is absorbed
and the wet spot expands. Where the expansion occurs is where capillary action is taking
place, the candle wicking absorbs wax the same way.
Candle wicking is available in several types. Probably the most popular is the
Flat Braid, or Regular wick. Different sized wicks cause different sized flames simply
because of the number of threads in the bundles. Each thread is considered a plait or ply,
and a given number of ply are bundled together.
1.2.
CANDLE BURNING
When a candle is lighted, the heat from the ignition source melts the wax, a heavy
hydrocarbon, at the wick base. The liquid wax rises up, due to capillary action, and is
then vaporized by the heat. This vaporizing wax cools the exposed wick and protects
parts of it from burning out. The vaporized wax mixes with oxygen and the mixture
reacts and generates heat and the process continues.
5
The heat and mass transfer taking place inside the wick is very complex. Twophase flow regimes will exist inside the wick. All the three regimes – single phase liquid
region, two-phase liquid vapor region and single phase vapor region can possibly exist
inside the wick. The heat and mass transfer taking place in the two-phase region of a
wick is very involved. It is of practical importance in many applications. Heat pipes
work on the principle that the heat transfer coefficient is significantly reduced inside the
two-phase flow in a wick. So heat pipes are used as efficient cooling devices.
In the gas-phase, heat is released from the combustion process.
The hot
combustion products are much less dense than the colder ambient air. In a gravity field,
they rise upward and draw the oxygen to the reaction zone. This upward buoyant
convective flow is the main reason that makes the flame into the so-called “tear-drop”
shape.
In zero gravity environments, on the other hand, natural convection is not present
and the fuel and oxidizer need to diffuse towards the reaction zone by the mechanism of
molecular diffusion. The primary reason for this process to occur is the existence of
concentration gradients around the flame. It is worthy to note that the diffusion transport
rates in zero gravity conditions are much slower than the natural convection transport
rates in normal gravity because of the absence of buoyancy-driven convection. Zero
gravity flames are much less robust (in the sense of smaller reaction rate) than normal
gravity flames because of the absence of the buoyant flow.
The mixing of fuel and oxidizer in the presence of a high temperature gradient
means that both heat and mass transfer must be considered in addition to chemical
kinetics to properly understand combustion.
The processes are coupled and the
6
governing equations are nonlinear in nature. Numerical computation has become an
important tool to understand combustion phenomena.
For the candle flame, it is
important to understand how the system responds to changes in parameters such as
gravity level. Interest in the candle burning has been revived in recent years. The driving
force for this interest is due to the necessity of understanding flames in low gravity
environment. The research conducted on the candle flame provides valuable insights on
how flames behave in microgravity which is relevant to spacecraft fire safety.
This research is mainly intended to provide a more complete simulation of a
candle flame coupled to a porous wick. The heat and mass transfer inside the wick can
control the flame shape and structure, and the extinction of flame at certain conditions.
When a candle burns, it slowly consumes the wax from the wax shoulder. As the level of
wax comes down, the length of the exposed wick is increased. This process is generally
very slow compared with the processes occurring in the gas and porous phases. Therefore
in modeling the candle burning, a quasi steady approximation can be assumed with wick
length treated as a parameter. As the length of the exposed wick increases, the surface
exposed to the heat of candle flame increases and on the other hand, more capillary action
is required for the liquid wax to reach the surface of the wick. At some point of time, the
tip of the wick dries up. This causes the temperature of the wick to increase sharply at
the tip of the wick. If the wick is made of pyrolyzable materials, then the dry portion of
the wick will be consumed (burnt out). This phenomenon is referred to as the self
trimming of the wick. A self trimming candle regulates its wick length by a balance of
the above two processes.
7
1.3.
PREVIOUS WORK
In this section, previous works on the study of candle flame and the study two-
phase flows inside porous media are reviewed. The first part will cover the candle flame
studies and the second part will cover the porous media studies.
1.3.1
1.3.1.1
PREVIOUS WORK ON CANDLE FLAMES
EXPERIMENTAL WORK
Candles have been a focus of attention of scientific study for hundreds of years.
In the 19th century, English scientist Michael Faraday, who discovered many principles of
electricity, delivered one of the most famous of his Christmas lectures for children called
“The Natural History of a Candle” (Faraday, 1988). His observations have served as the
basis for lessons in taking observations in combustion.
Although the processes that occur in a candle flame are complex, the setup of a
candle experiment is very easy. The candle flame is an excellent example of wick
stabilized diffusion flame. Many researchers have chosen candle flame to understand a
wide range of combustion phenomena. For example, an early work by Lawton and
Weinberg (1969) examined the effect of magnetic fields on flame deflection using a
candle. Chan and T’ien (1978) performed experimental work on commercial candles to
study the spontaneous flame oscillation phenomena. Buckmaster and Peters (1986) and
Maxworthy (1999) have both studied stability and flickering phenomena for diffusion
flames using a candle for a model diffusion flame.
The influence of hyper-gravity (gravity greater than that on earth) on candle
diffusion flames has been reported by Villermaux and Durox (1992) using a 6 meter
diameter centrifuge. They were able to vary the gravity up to seven times the gravity of
8
earth. Above 7ge, the candle flame becomes extinct. They found that the flame length
and the candle burning rate both decreases as the gravity level increases. Another
experimental investigation to study the behavior of candle flames in high gravitational
field has been reported by Arai and Amagai (1993). They varied the gravity level by
using a spin tester in the range from 1 to 14ge. There results showed that both the candle
flame length and width were monotonically reduced with increasing gravity. In both the
works there is no mention about the wick length of the candle and whether it reached the
self trimming length.
Bryant (1995) investigated the effects of gravity level on the heat release rate for a
candle flame under an imposed low-speed forced flow. The experiment was performed
on board an aircraft flying repeated parabolic trajectories. He reported that the rate of
heat release is flow rate dependent, decreases significantly in microgravity, and changes
insignificantly under elevated-gravity conditions. Oostra et al. (1996) measured the soot
production of a candle during microgravity and normal gravity condition during a
parabolic flight. Their measurements predict a lower candle burning rate with higher soot
production in microgravity. Amagai et al. (1997) investigated the effect of variable
gravity using a gaseous butane diffusion flame issued from a tube in a centrifuge with a
rotary arm of radius 0.9 meter. In their experimental work, they found that both the
length and the width of butane diffusion flames reduced with an increase of gravity level
for a constant feed rate. In addition, oscillations and blow off of the flame were observed
at high gravity level.
Recent microgravity works sponsored by NASA have been a major source for
fundamentally improving the science of combustion. Microgravity experimental data on
9
candle combustion has been obtained by several different techniques, all of which depend
on the experiment achieving a state of free-fall. A number of studies have used drop
towers where a few seconds of free-fall condition can be generated, i.e. between 2 to 5
seconds.
Such facilities make it possible to conduct brief periods of microgravity
research on earth. An experimental investigation of candle flame ignition behavior in
drop tower has been reported by Ross et al. (1991). The experiments conducted at
atmospheric pressure, under 19%-25% O2 concentration, and in nitrogen- or heliumdiluted environments.
They found that the visible blue candle flame assumes a
hemisphere shape relatively quickly (about 1 second) after the drop. Although the blue
flame shape and size respond to the ignition or g-variation transient quickly, the color of
the flame continuously changes throughout the entire test duration. Thus it suggests that
more time is required to obtain the true steady state.
Besides the drop towers, microgravity research also took place using parabolic
aircraft maneuver which provide additional time of reduced-gravity environment.
However, the g-jitter in the aircraft is too high (∼10-2ge) to obtain a steady flame. The
candle flame fluctuates quite extensively in the airplane experiments. Using Space
Shuttle to obtain extended time in buoyant and weakly buoyant atmospheres, Dietrich et
al. (1994) investigated the candle flames behavior in microgravity. The experimental
results showed that the candle flame was spherical and bright yellow, presumably from
soot, immediately after ignition. After a few seconds, the yellow disappeared and the
flame became blue and nearly hemispherical with a large stand-off distance from the
wick. The flame luminosity decreased continuously with time until extinction. The
extinction typically occurred around one minute. They also reported that the mass of
10
liquefied wax grew continuously without dripping off, contrary to a candle burning in
normal gravity. It has been observed, however, while the flame behavior was quasisteady during the majority of burning time, axisymmetric flame oscillations developed
near extinction due to oxygen depletion in the finite size combustion chamber. The term
“quasi-steady” needs to be explained. Physically, the flame size response is related to the
diffusion time scale (order of seconds) which is much lower than the depletion time scale
of oxygen (order of minutes) in the experimental chamber. This means that the flame
size and shape will respond in a quasi-steady manner to the oxygen molar fraction of the
ambient even though this oxygen molar concentration is changing with time.
A second series of flight experiment on microgravity candle flame was carried out
in the Mir Station. In the Mir station the oxygen mole fraction is higher than that in the
Shuttle (between 0.23 to 0.25). Also the candle cage has more open area to facilitate
oxygen diffusion from the ambient to the flame. Dietrich et al. (2000) showed that the
candles onboard Mir with the largest wick diameters had the shortest flame lifetime
where as the candles with the smallest wick diameters had the largest lifetimes. All of
the candles in the Mir tests burned longer than those on Shuttle. The above examples
indicate that candle flames onboard the space shuttle and Mir both oscillated prior to
extinction with periodic increases and decreases of flame surface area.
1.3.1.2
NUMERICAL WORK
In addition to experimental effort, limited numerical works to model candle
flames have been conducted, all in zero gravity. To the best knowledge of the author,
there are two different candle flame numerical models. The first model is by Shu (1998),
Shu et al. (1998) and the other model is by Chang (see Dietrich et al., 2001). These two
11
different numerical models predict some candle flame behavior in zero gravity
environments. Both calculations utilize a single-step, finite rate, gas-phase chemical
reaction model in a frictionless flow. The main difference between the two models is the
candle and wick geometry. In Shu’s model, a spherical wick and conical candle body
were used and the equations were formulated using spherical coordinates. The model by
Chang used a more realistic wick and candle geometry and the equations were formulated
using cylindrical coordinates. Both models included a wick surface radiation loss term
and flame radiation is represented as a pure heat loss.
The work by Shu (1998) and Shu et al. (1998) were primarily focused on
computing the flame characteristics, the flammability limits, the effects of flame radiative
loss, and the effect of fuel and oxygen Lewis numbers. The model predicts a stable
steady-state of candle flame in an infinite ambient. They also reported that decreasing
the oxygen concentration, prior to extinction, raises the flame base relative to the porous
sphere without significant changes in the position of the flame top. Unlike oxygen Lewis
number, the numerical experiments reveal no flame properties dependence on the fuel
Lewis number except for the fuel vapor profile. Their model also predicts near-flame
oscillation similar to those observed in the experiments but there were some differences.
The modeling work by Chang (Dietrich et al., 2001), used the grid generation
technique and a body-fitted coordinate system to fit the more realistic wick and candle
shape. Instead of the explicit scheme used by Shu (1998), an implicit time-marching
scheme was developed and it shortened the computational time needed to reach steady
state and made it possible to examine the transient extinction phenomena near the limit.
There are two shortcomings (or limitations) on the candle modeling work by Shu and
12
Chang. The first is the assumption of potential flow. This assumption simplifies the
momentum equations but has the shortcoming of not being able to satisfy the no-slip
boundary condition on the solid surface. It also cannot handle buoyant induced flow so
gravity effect can not be studied. The second limitation is on the treatment of radiation.
In these models, flame (gas-phase) radiation is treated as a simple heat loss term without
knowing the detailed distribution of radiation heat flux. In previous modeling work on
solid fuel combustion (e.g. Rhatigan et al., 1998), part of the flame radiation becomes the
energy feedback to the solid pyrolysis process. A more rigorous treatment of radiative
heat transfer is thus needed to verify whether the simple heat loss model is adequate or
not.
A more detailed gas-phase modeling has been done by Alsairafi (2003).
A
simplified two-dimensional axisymmetric flow around a realistic candle has been
numerically simulated. A finite volume method is used to solve the steady state Navier
stokes equations in conjunction with species and energy equation, and the radiative
transfer equation. The gas phase combustion process is modeled as a single step, second
order, finite rate Arrhenius reaction. The discrete ordinate method is used to solve the
radiative transfer equation. The main assumption used in this work is the candle wick is
assumed to be a solid coated with liquid fuel all along its surface. This evades the
necessity to model the heat and mass transport inside the candle wick and its coupling
effect on the candle flame. They reported a significant match between their simulated
results and the experimental results. However, there is a major discrepancy even in the
qualitative trends of the candle burning rate with the imposed gravity level. Experiments
report a monotonic decrease in the burning rate with increasing gravity levels but the
13
numerical results indicate an increase in burning rate upto 3ge and a decrease in burning
rate with further increase in gravity level. The studies by Alsairafi (2003) indicate the
necessity to model the detailed heat and mass transport inside the wick to achieve a more
realistic numerical simulation of candle burning.
1.3.2
PREVIOUS WORK ON TWO-PHASE FLOW IN POROUS MEDIA
Simulation of a porous wick involves modeling two phase flow with phase change
inside the porous media. Two phase flow in porous media involves many features which
distinguishes it from the single phase flow through a porous medium. The flow can be
distinguished as either co-current or countercurrent flow depending on the direction of
the flow of the two phases. In two phase flow through porous media, there are three
possible saturation regimes that can exist. The porous media may be completely saturated
with one phase. This is called the complete saturation regime. The porous media may
have the lowest possible saturation with one phase. This is called the pendular regime. In
this regime, one of the phases occurs in the form of pendular bodies throughout the
porous media. These pendular bodies do not touch each other, so there is no possibility of
flow for that phase. The porous media exhibits an intermediate saturation with both
phases. This regime is known as funicular regime. Earlier it was thought that in this
regime, both liquid and vapor phases flowed simultaneously through channels with gas
moving in the inner core and the liquid in the annulus between the gas and the solid
channel walls [Scheidegger, 1974]. Flow visualizations have shown that the gas and the
liquid flow through their own network of channels [Dullien, 1979].
Two-phase flow in porous media is involved in the following classes of problems
1) Drying of porous materials
14
2) Heat pipe applications
3) Burning over porous wick surfaces
The drying problem is essentially transient in nature. It has been a subject of
interest since 1920’s. Lewis (1921) suggested that drying takes place primarily through a
diffusion mechanism. Around this time, the soil scientists and chemists were attempting
to explain the movement of moisture in porous media in terms of surface tension forces
or by capillary action. Hougen et. al. (1940) made an extensive survey of the importance
of capillary action in the drying process. They found out that capillary action can play an
important role in the movement of moisture during the drying of a porous solid. Krischer
(1940) was the first to identify the importance of energy transport in a drying process.
Phillip and Devries (1957) included the effects of capillary flow and vapor transport, and
incorporated the thermal energy equation into the set of governing equations that describe
the drying process. Luikov (1975) also published similar equations for the heat and mass
transfer in porous media. Whitaker (1977) made a rigorous formulation of the theory of
drying based on the well known transport equations for a continuous media. These
equations were volume averaged to provide a rational route to a set of equations
describing the transport of heat and mass in porous media. These equations are limited by
the restrictions and assumptions that he had used while deriving the volume averaged
equations.
Even though Luikov (1975) published the full set of governing equations
describing the heat and mass transport in porous media, the equations are nonetheless
difficult to solve because of its complexities. A large number of parameters are required
to solve the equations. There have been efforts to study different aspects of the heat and
15
mass transfer by simplifying the equations using suitable assumptions. Initial efforts
involved consideration of only the vapor transport inside the porous media. The capillary
action on the liquid is neglected. Cross (1979) used the momentum and energy equations
for the vapor flow, neglecting the convection terms in the energy equation. He solved
analytically for the maximum pressure that is build up at the dry-wet interface. As drying
takes place, the evaporation takes place at the interface between the dry and the wet
regions. A high pressure is build up at this interface to generate the necessary driving
force for the transport of vapor in the dry region. Dayan (1981) extended this to include
the convective term in the energy equation. They have analyzed an intensely heated
porous space using a transient model to obtain the transient pressure, temperature and
moisture distributions. Dayan and Glueker (1982) analyzed the same problem using an
explicit time marching technique and incorporating the liquid and vapor transport. They
have assumed that the migration of liquid phase is primarily governed by pressure
gradients generated by the vapor. They were able to get reasonable predictions for the
temperature, pressure and moisture distribution during the drying of cement structure.
During intense heating, a region near the heated surface dries out. Once such a region
develops, evaporation takes place exclusively at the dry-wet interface. The evaporation
process leads to the pore pressurization and subsequent filtration of all the pore
constituents towards both the heated surface and the inner wet zone of the concrete. In the
above analysis, the effect of capillary pressure and gravity were neglected.
In drying, three different regions have been observed experimentally (Rahli et. al.,
1997) in the porous media. In the initial stages of drying, the temperature inside the
porous media is below the saturation temperature. Two regions are observed at this stage.
16
A two-phase region is observed near the heating surface and a liquid region is present
deep inside the porous media. As drying proceeds, the surface exposed to heating
becomes completely dried up and single phase vapor region is formed near the heating
surface which penetrates deep into the media as time proceeds. The temperature in the
single phase vapor region exceeds the saturation temperature. Three zones have been
found to exist. Near the heated surface, a vapor saturated zone is observed. Adjacent to
this zone and extending into the medium is a two phase region, which is dominated by
capillarity and vapor transport. Ahead of this zone, the medium is saturated with the
liquid phase.
In most cases, the capillary pressure and gravity play an important role in the two
phase flow inside porous media. In the absence of any forced flow, the liquid is
transported primarily due to capillary action and gravity forces. Kaviany and Mittal
(1987) analyzed the two phase flow inside porous media, where liquid is governed by
capillary forces. They have analyzed the convective drying of a porous slab in the
funicular state initially saturated with liquid. The liquid is driven by capillary pressure
gradient and the vapor is driven by the partial pressure of the evaporating species
generated by the temperature gradient established inside the porous slab. The transport of
non-condensable gas and the effect of gravity are neglected. The calculations are
performed only in the funicular regime, i.e. till the appearance of dry patches on the
surface. They found good agreement between their experimental results and the predicted
results for the drying rate, surface temperature and the average saturation up to the time
of first appearance of dry patches on the surface (critical time). Rogers and Kaviany
(1991) extended this work to include the evaporative-penetration front. As the porous
17
slab is dried, the surface saturation decreases as time proceeds. After some period termed
as the critical time, dry patches appear on the surface. When dry patches appear on the
surface, the evaporative front starts penetrating into the porous slab. The effect of gravity
and the surface tension non-uniformities are included. They have also included the
transport of non-condensable gases into the porous slab. The speed of the evaporative
front and the mass transfer rate during this regime were predicted. A significant drop in
the drying rate is observed in the evaporative front regime. This is the result of the high
resistance to heat and mass transport in the dry region.
A heat pipe application is another major area which involves heat and mass
transport in the two-phase region of a porous substance. A heat pipe is a simple device
that can quickly transfer heat from one point to another. Heat pipe application is based on
the high heat transfer rate obtained due to the evaporation-condensation mechanism
taking place inside the porous wick involving phase changes. Experimental studies
conducted by Hansen (1970) showed an enhancement of heat transfer over that of
conduction in saturated porous media. The experimental data obtained by Somerton et. al.
(1974) on effective thermal conductivity of steam-water saturated porous materials is
several times larger than those of the same medium saturated only with the liquid phase.
These classes of problems have been treated as steady state systems. In the heat
pipe applications, the porous wick is enclosed and the liquid and the vapor re-circulate
inside the porous wick. There is a counter-current transport of liquid and vapor in the
two-phase region. The liquid and the vapor regions above and below the two-phase
region are essentially stationary. A one-dimensional, steady state analytical model (heat
pipe problem) has been developed by Udell (1985) to study the effects of capillarity,
18
gravity and phase change. Their results predict the increase of heat transfer due to the
combined effect of evaporation, convection and condensation inside the porous media.
The heat pipe effect is also observed in two phase geothermal reservoirs. Bodvarsson
(1994) used a two dimensional porous slab model with a non-uniform heat flux at the
bottom. Their results show very efficient heat transfer in the vapor dominated zone
consistent with the observations in natural geothermal reservoirs.
Burning over porous wick surfaces also involves heat and mass transport in a
porous wick. The influence of capillarity on the combustion behavior and the effects of
the properties of porous beds on the combustion characteristics make this study unique.
Ignition and transition to the flame spread over the ground soaked with spilled
combustible liquid are of interest to researchers and engineers because they have
important implications in terms of fire safety. Kaviany and Tao (1988) had done some
experiments and numerical calculations on the burning of liquids supplied through a
wick. They have analyzed the burning of a porous slab which is initially saturated with
liquid fuel, during the funicular regime. The liquid is driven by capillarity and the vapor
is driven by the vapor pressure gradient. The effect of gravity and the transport of noncondensable gases have been neglected. The effect of surface saturation, relative
permeability and vapor flow rate on critical time (time during which the surface become
first dried up) has been studied.
An experimental study was conducted by Kong et. al. (2002) to investigate the
effects of sand size and sand layer depth on the burning characteristics of non-spread
diffusion flames of liquid fuel soaked in porous beds. A porous sand bed initially soaked
with methanol is ignited and the burning characteristics of the flame are studied. The
19
flame temperature profiles, location of vapor/liquid interface, vapor region moving speed,
combustion duration time, fuel consumption were studied in the experiments. The results
indicate that the fuel consumption rate increased rapidly during the beginning stages. As
time proceeded, the formation of vapor region increased the resistance to vapor transport,
resulting in a decreasing trend in the fuel consumption rate. Their results confirm that the
resistance to vapor transport is the controlling mode when a dry region is formed in the
porous bed. There is an abrupt increase in the temperature profile in the dry vapor phase
region. In the liquid region, the temperatures are close to the boiling point of the fuel.
They have also studied the effect of sand sizes and the sand layer depths on the burning
characteristics of the fuel.
Modeling Two-phase flows in Porous Media
Numerical analysis of multidimensional two-phase flow including phase change
in porous media is intrinsically complicated. One reason is the strongly nonlinear and
coupled nature of the governing equations for the two-phase flow. Another fundamental
difficulty lies in the presence of moving and irregular interfaces between the single and
two-phase subregions in a domain of interest. The location of such an interface is not
known a priori and must be determined by the coupled flows in adjacent regions.
Primarily three different numerical approaches have been used by researchers – separate
flow model, enthalpy model and thermodynamic equilibrium model.
Traditionally separate flow model (Bear, 1972) has been used in which separate
equations for the two phases are formulated and the interface between the different
regions is explicitly tracked. The explicit tracking of moving interface involves complex
coordinate mapping or numerical remeshing (Ramesh and Torrance, 1990). To reduce
20
the complexities of separate flow models, Wang and Beckermann (1993) developed twophase mixture model based on enthalpy formulation. The separate equations for the two
phases are combined to form a single set of equations for the mixture. In this way, the
number of governing differential equations to be solved is reduced by almost half and the
rest are replaced by algebraic relations. Based on the enthalpy values of the mixture, the
state of the system inside the porous wick is identified. The inherent assumption in this
model is that the two-phase region is at a constant temperature (equal to the boiling
temperature of the liquid) and hence there is no phase change taking place inside the twophase region. Phase change is taking place only at the interfaces between the two-phase
and the single phase regions. Usually in many physical systems, either condensation or
evaporation does occur inside the two-phase region and the temperature of the two-phase
region does slightly vary. This can only be accounted by invoking thermodynamic
equilibrium relations. Recently, Benard et al. (2005) extended the enthalpy modeling
approach of Wang et. al. (1993) by incorporating the equilibrium thermodynamic
relations, which determines the thermodynamic state of the system and also accounts for
phase change taking place inside the two-phase region.
Raju (2004) developed a steady state one-dimensional stagnation point diffusion
flame stabilized next to a porous media. The detailed one-dimensional heat and mass
transport inside the porous wick has been studied. In this study the liquid in the porous
media is assumed to be driven by capillary action and the vapor is driven by the vapor
pressure gradient induced by the temperature gradient (based on Classius-Clayperon
equilibrium relationship) inside the porous media. These studies reveal that the liquid and
21
the vapor flow counter currently inside the two-phase region. It has also been found out
that the steady state solution exists only for stretch rates below a critical value.
1.4.
PURPOSE AND SCOPE OF THIS DISSERTATION
The purpose of this thesis is to provide a more complete picture of the candle
flame behavior which is coupled to a porous wick. A more realistic picture is obtained
by considering the role of heat and mass transport inside the candle wick. The thesis
aims at improving our understanding of wick stabilized candle flames by addressing the
following issues
(1) The role of heat and mass transport inside the candle wick
(2) Develop a model for simulating the self trimming of a candle wick
(3) The role of gravity on the shape and size of self trimmed candle flames.
(4) The effect of different wick diameters on burning rate and flame temperature.
The present study is primary computational. The computational results are
generated by solving the full Navier-Stokes equations with a one-step finite rate chemical
reaction rate in the gas phase and two-phase flow equations inside the porous wick.
1.5.
DISSERTATION OUTLINE
This dissertation describes numerically the heat and mass transfer taking place
inside the burning candle wick and its effect on the candle flame structure. The first
Chapter of this dissertation describes the background related to candle burning and twophase flow inside porous media. A brief review on previous work in this field is
presented.
22
Chapters 2 and 3 include the theoretical and numerical aspects of the current
computational model. In Chapter 2, the governing equations for the fluid flow describing
the mass conservation, energy conservation, inert gas species conservation, capillary and
thermodynamic relations inside the wick. The numerical method adopted for solving the
discretized equations is also included. The description of using multifrontal solvers in
this context is presented. Different variants of Newton solvers are also presented.
Preliminary analysis of two-phase flow inside the wick is done by applying a constant
heat flux all along the wick surface.
In Chapter 3, the governing equations for the gas phase fluid flow and combustion
reaction are presented using mass, momentum and energy conservation equations. The
pressure-velocity coupling is handled by the SIMPLER algorithm. The numerical method
adopted for solving the discretized equations is also included.
The numerical test results are presented in Chapter 4 of this dissertation. The gas
phase model is coupled with the two-phase axisymmetric wick model. The detailed flow
structures in the gas phase and in the wick are presented. The phenomenon of self
trimming of candle wick is modeled as a burn out of the dry region. Parametric analysis
has been done to study the effect of gravity, absolute permeability of the wick, wick
diameter and ambient oxygen percentage on the candle flame structure and the burning
rate.
Finally, in Chapter 5, a summary will be given and some future work is
suggested.
23
Flow direction
Products of
combustion
conduction
Fuel vapor
g
radiation
r
x
oxidizer
Figure 1.1 Schematic of a Candle Flame
24
CHAPTER 2
AXISYMMETRIC WICK MODELING
2.1 FORMULATION OF TWO PHASE FLOW INSIDE POROUS MEDIA
Two phase flow in porous media typically consists of three phases: the solid
phase, the liquid phase and the gas phase. In addition to the transport of individual
phases, there is a phase change process involved inside the porous media (but no
chemical reaction is assumed). Treatment of individual phases from the well known point
equations of continuum physics is rather complicated and computationally expensive.
Hence volume averaging technique described by Whitaker (1977) is used to provide a
rational route to a set of equations describing the transport of heat and mass in a porous
media.
Candle wicks are usually cylindrical in shape although other shapes are also used.
The gas phase and the wick characteristics are assumed to be symmetric around the
angular direction and hence the present wick is modeled for an axisymmetric geometry.
2.1.1 MATHEMATICAL FORMULATION
The constitutive equations for the solid, liquid and gaseous phases are volume
averaged (Whitaker, 1977). The equations are written in cylindrical coordinates for two
phase flow.
In addition to the assumptions of Whitaker (1977), the following additional
assumptions are made in this study.
1. The liquid phase is assumed to be continuous.
25
2. The flow is assumed to be laminar. Darcy’s law is assumed to be valid both for
the liquid and gas phases.
3. The transport of non-condensable gases inside the porous wick is neglected. The
term “non-condensable gases”, refers to all the gases other than the fuel vapor
(e.g. ambient air, combustion products like CO2, H2O etc.)
4. Radiative heat transfer inside the porous media is neglected.
5. The surface tension of the liquid is assumed to be constant. It does not vary with
temperature.
6. The vapor is locally in thermal equilibrium with the liquid and the thermodynamic
Gibbs phase equilibrium relations are assumed to be valid in the two-phase
region.
7. Although phase change can occur, there is no chemical reaction in the porous
media.
8. The properties of the wick, like thermal conductivity, permeability etc is assumed
to be isotropic. In principle, the thermal conductivity and permeability can be
different in axial and longitudinal directions.
9. Deformation of the wick material due to thermal stresses or bending of the wick
during burning of the candle is neglected.
10. Ideal gas law is assumed to be valid for the vapor phase
The resultant simplified volume averaged equations are written down for an
axisymmetric, steady state, two-phase flow inside a porous wick.
26
Continuity equation:
∂
∂
1 ∂
1 ∂
( ρl ul ) + ( ρ g ug ) +
( r ρl vl ) +
( r ρ g vg ) = 0 .
r ∂r
r ∂r
∂x
∂x
(2.1)
The individual terms represent the net mass flux of liquid and vapor at a point in x and r
directions respectively.
Momentum equations:
The momentum equations are given by Darcy’s law.
krl K ⎛ ∂Pl
⎞
−
− ρl g ⎟ ,
⎜
μl ⎝ ∂x
⎠
krg K ⎛ ∂Pg
⎞
ug =
− ρg g ⎟ ,
⎜−
μ g ⎝ ∂x
⎠
k K ⎛ ∂P ⎞
vl = rl ⎜ − l ⎟ ,
μl ⎝ ∂r ⎠
ul =
vg =
(2.2-5)
krg K ⎛ ∂Pg ⎞
⎜−
⎟.
μ g ⎝ ∂r ⎠
The liquid phase is treated as incompressible and the gaseous phase is treated as an ideal
gas.
Energy equation:
∂
1 ∂
∂ ⎛
∂T ⎞ 1 ∂ ⎛
∂T ⎞
ρl hl ul + ρ g hg u g ) +
r ( ρl hl vl + ρ g hg vg ) = ⎜ keff
(
⎟+
⎜ rkeff
⎟ . (2.6)
r ∂r
∂x
∂x ⎝
∂x ⎠ r ∂r ⎝
∂r ⎠
(
)
Using the relations hl = clT and hg = cg T + i fg , equation (2.6) becomes
1 ∂
∂
⎛ ∂ρ u 1 ∂r ρl vl ⎞
=
ρl cl ul + ρ g cg u g ) T +
r ( ρl cl vl + ρ g cg vg ) T + i fg ⎜ − l l −
(
∂x
r ∂r
r ∂r ⎟⎠
⎝ ∂x
(2.7)
∂ ⎛
∂T ⎞ 1 ∂ ⎛
∂T ⎞
⎜ keff
⎟+
⎜ rkeff
⎟
∂x ⎝
∂x ⎠ r ∂r ⎝
∂r ⎠
(
)
The first two terms on the left hand side of equation 2.7 represent the convective heat
transport of liquid and vapor in the x and r direction. The third term represents the heat
27
source term due to phase change taking place between the liquid and the vapor. The right
hand side of this equation represents the conductive heat transfer.
keff represents the
effective thermal conductivity.
Capillary and permeability relations
The gas pressure is related to the liquid pressure using the capillary relation
Pc ( s ) = Pg − Pl .
(2.8)
The capillary pressure is related to the saturation given by Leverett’s function (Leverett,
1941),
Pc =
σ
⎡1.42 (1 − s ) − 2.12 (1 − s )2 + 1.26 (1 − s )3 ⎤ .
1/ 2 ⎣
⎦
(K /ε )
(2.9)
The relative permeability of the porous media is given by the following approximation
(Bau and Torrence, 1982)
krl = s, krg = (1 − s ) .
(2.10 a-b)
The non-dimensionalization of the porous wick variables are carried out according to the
variables indicated in table 2.1. The non-dimensionalized variables are indicated by a
‘hat’ symbol on the top of the variable.
Non-dimensionalized equations:
∂
∂
1 ∂
1 ∂
ˆˆl ) +
( uˆl ) + ( ρˆ g uˆg ) +
( rv
( rˆρ̂ g vˆg ) = 0 ,
rˆ ∂rˆ
rˆ ∂rˆ
∂xˆ
∂x
(2.11)
28
⎛ ∂Pˆ
⎞
uˆl = Pekrl ⎜⎜ − l − gˆ ⎟⎟ ,
⎝ ∂xˆ
⎠
⎞
Pekrg ⎛ ∂Pˆg
uˆ g =
− ρˆ g gˆ ⎟ ,
⎜−
⎟
μˆ g ⎜⎝ ∂xˆ
⎠
⎛ ∂Pˆ ⎞
vˆl = Pekrl ⎜⎜ − l ⎟⎟ ,
⎝ ∂rˆ ⎠
Pekrg ⎛ ∂Pˆg ⎞
vˆg =
⎜−
⎟.
μˆ g ⎜⎝ ∂rˆ ⎟⎠
ˆˆl
∂
1 ∂
⎛ ∂uˆ 1 ∂rv
uˆl + ρˆ g cˆg uˆ g ) Tˆ +
rˆ ( vˆl + ρˆ g cˆg vˆg ) Tˆ + iˆfg ⎜ − l −
(
∂xˆ
rˆ ∂rˆ
⎝ ∂xˆ rˆ ∂rˆ
(
)
(2.12-15)
⎞ ∂ ⎛ ˆ ∂Tˆ ⎞ 1 ∂ ⎛ ˆ ∂Tˆ ⎞
ˆ eff
⎟
⎟ = ∂xˆ ⎜ keff ∂xˆ ⎟ + rˆ ∂rˆ ⎜ rk
∂rˆ ⎠
⎠
⎝
⎠
⎝
(2.16)
The effective thermal conductivity of the wick is function of saturation given by
the expression (Udell and flitch, 1985)
kˆeff = s + (1 − s ) kˆs ,
(2.17)
where kˆs is the thermal conductivity of the solid wick material.
Conditions for Phase Transition:
The equilibrium thermodynamic state of candle wax (single phase liquid, single
phase vapor, two-phase) can be determined for given liquid and gaseous pressures and
temperature conditions, using the vapor pressure equilibrium data obtained from the
thermodynamic Gibbs phase relationships (Benard et. al. 2005).
Gˆ l = hˆl − Tˆlηˆl ,
Gˆ g = hˆg − Tˆgηˆg ,
(2.18-19)
where G is the Gibbs potential per unit mass of the corresponding phase and symbol
“hat” denotes the non-dimensional value. h and η are respectively the enthalpy and the
29
entropy of the corresponding phase. The dimensional expressions for the enthalpy and
entropy are given by
hl = cl (T − T0 ) ,
hg = i fg + cg (T − T0 ) ,
ηl = cl log (T / T0 ) ,
(2.20-23)
η g = i fg / T0 + cg log (T / T0 ) − R log ( Pg / P0 ) .
The thermodynamic equilibrium relations based on the minimization of Gibbs function
(Saad, 1966) are given as follows
State 1. Gˆ l < Gˆ g , no vapor phase is present ( s = 1 )
State 2. Gˆ l = Gˆ g , liquid and vapor are in equilibrium ( 0 < s < 1 )
State 3. Gˆ l > Gˆ g , no liquid phase is present ( s = 0 )
(2.24-26)
Equations 2.24-26 simply mean that whichever phase has the least Gibbs phase potential
will dominate over the other phase. Equations 2.18-23 in conjuction with the phase
equilibrium condition (Eq. 2.25) yield the well known, Classius-Clayperon equation.
Boundary conditions:
(1) Base of the wick
The wick is immersed in the candle wax pool, which can be assumed to be at its
melting temperature.
s = 1, Tˆ = Tˆm
(2) Cylindrical surface of the wick
30
Depending on the thermodynamic state of the wick on the surface, the boundary
conditions will vary. In the case of either pure liquid or pure vapor, all the heat flux
imposed on the surface is conducted into the wick. In the case of two-phase region, part
of the heat supplied is used for evaporating the liquid on the surface of the wick and part
of the heat is conducted into the wick.
⎛ ∂Tˆ ⎞
Liquid region: s = 1, vˆl = 0, qˆ f = − kˆeff ⎜
⎟
⎝ ∂rˆ ⎠
⎛ ∂Tˆ ⎞
Two-phase region: 0 < s < 1, qˆ f = − kˆeff ⎜
⎟ + vˆl iˆfg , Tˆ = Tˆb
ˆ
r
∂
⎝
⎠
Vapor region: s = 0, pˆ l = pˆ 0 − pˆ c (0),
⎛ ∂Tˆ ⎞
qˆ f = − kˆeff ⎜
⎟
⎝ ∂rˆ ⎠
(3) Tip of the wick
Similar to that of the boundary conditions on the cylindrical surface.
⎛ ∂Tˆ ⎞
Liquid region: s = 1, vˆl = 0, qˆ f = − kˆeff ⎜
⎟
⎝ ∂rˆ ⎠
⎛ ∂Tˆ ⎞
Two-phase region: 0 < s < 1, qˆ f = − kˆeff ⎜
⎟ + vˆl iˆfg , Tˆ = Tˆb
ˆ
r
∂
⎝
⎠
Vapor region: s = 0, pˆ l = pˆ 0 − pˆ c (0),
⎛ ∂Tˆ ⎞
qˆ f = − kˆeff ⎜
⎟
⎝ ∂rˆ ⎠
(4) Symmetry line
Symmetry boundary conditions are imposed along the symmetry line
31
2.1.2. NUMERICAL FORMULATION:
The continuity, momentum and energy equations are discretized using finite
difference approximation. This results in a set of non-linear discrete balance equations.
These equations are coupled with the inequalities resulting from the phase equilibrium
relationships. The system is solved using Newton’s method. The thermodynamic state of
each grid block is updated at each iteration of the method. In this method there is no need
to separately track the interface between the single phase and the two-phase regions.
Equations 2.11-2.16 are discretized using finite difference approximation and
combined appropriately with the non-dimensionalized forms of equations 2.8-2.9 to form
2 equations for the variables Pl , s and T . The equations are closed by using the
thermodynamic equilibrium relationships (equations 2.24-2.26). The thermodynamic
relationships are incorporated in the iteration scheme as described below.
For each grid node i, we set a thermodynamic state parameter χ i ( 1 for pure
liquid , 2 for liquid-vapor equilibrium, 3 for pure vapor) which is determined based on
the previous guess value of saturation distribution, ζ i = ( pli , si , Ti ) the triplet of
unknowns and
B (ζ i ,1) = s − 1,
B (ζ i , 2 ) = Gˆ l − Gˆ g ,
(2.27-29)
B (ζ i ,3) = s.
Note here that B is the thermodynamic equation which depends on the
thermodynamic state χ i of the system at any given grid point. The only speciality of this
32
formulation is that the equation B is different at each grid node depending upon the
thermodynamic state at that grid point χ i .
Based on the thermodynamic relationships described in the previous section, the
following relationships hold
if χ i = 1, then B (ζ i ,1) = 0, and B (ζ i , 2 ) < 0,
⎧⎪ B (ζ i ,1) < 0
if χ i = 2, then B (ζ i , 2 ) = 0, and ⎨
,
⎪⎩ B (ζ i ,3) > 0
if χ i = 3, then B (ζ i ,3) = 0, and B (ζ i , 2 ) > 0.
(2.30-32)
The solution procedure is described as follows
n
Let ζ ( n ) , χ ( ) , be the values at the beginning of (n+1) th iteration and α be the under
relaxation parameter. First ζ ( n +1) is computed from Newton’s step.
(
)(
)
(
)
J ζ ( n ) , χ ( n ) δζ ( n ) = − F ζ ( n ) , χ ( n ) ,
(
)
(2.33-34)
ζ ( n +1) = ζ ( n ) + α δζ ( n ) .
Next the thermodynamic variable is updated at each grid node by the following relations
(
if χ ( ) = 2 and if B (ζ (
if χ ( ) = 2 and if B (ζ (
if χ ( ) = 3 and if B (ζ (
)
)
,1) > 0, then χ (
)
,3) < 0, then χ (
)
, 2 ) < 0, then χ (
if χ i( n ) = 1 and if B ζ i( n +1) , 2 > 0, then χ i( n +1) = 2;
n
i
n
i
n +1)
= 1;
n +1)
= 3;
n +1)
= 2;
i
n +1
i
i
n +1
n
i
n +1
i
i
i
otherwise χ i( n +1) = χ i( n ) .
The linear system of equations 2.33 is solved using a direct sparse solver
UMFPACK, which is based on a multifrontal technique. This solution procedure is
outlined in the next section. Since the system of equations is highly non-linear, very high
33
under relaxation parameter of 0.001 is imposed to ensure smooth convergence of the
variables.
2.2 MULTIFRONTAL SOLVERS FOR LARGE SPARSE LINEAR SYSTEMS
This section describes the implementation of sparse direct solvers based on
multifrontal techniques for solving the highly non-linear equations resulting from the
two-phase flow equations inside the porous wick.
The implementation of standard
solvers like ADI solvers failed to produce convergence.
Implementation of direct
solution techniques based on sparse Gaussian elimination (specifically the multifrontal
technique) resulted in a stable convergent solution and henceforth developed in this
study.
A modified Newton’s method is implemented, which when coupled with a
multifrontal solver resulted in significant reduction in the computational time. The
description of multifrontal technique is beyond the scope of this work but an attempt is
made to give a brief introduction of the sparse matrix storage techniques and frontal
solution techniques. Finally, the efficiency of the multifrontal solver is tested for a
differential cavity benchmark problem.
2.2.1 INTRODUCTION
Solving general sparse linear systems can be accomplished either by direct
solution methods or by iterative solution techniques. Earlier direct solution method was
often preferred to iterative methods in real applications because of their robustness and
predictable behavior. Iterative solvers were often special-purpose in nature and were
developed with certain applications in mind. For Finite Volume problems, the ADI
method (Peaceman and Rachford, 1955) and implicit Stone (Stone, 1965) algorithms are
34
preferred because of their high computational efficiency and very low memory
requirements compared to the direct solvers.
Over the years in 1960’s and 1970’s, there has been significant development in
the solutions of large linear systems. Techniques were developed to take advantage of the
sparsity to design special direct methods that can be quite economical. The frontal
method (Irons, 1970 and Hood, 1976) is a variant of Gaussian elimination and makes use
of the sparsity pattern for matrices resulting from the discretization of PDE’s. Later Duff
(1984), and Duff et al. (1986) extended the frontal techniques for solving any general
sparse matrices. Over the recent years, there has been significant improvement in the
development of multifrontal direct solvers which take advantage of the sparsity pattern
and the Level 3 BLAS routines to enhance the computational speed on high power
computing architectures.
Before going into the discussion of solution techniques for sparse linear systems,
it is important to discuss the different sparse storage schemes for storing the non-zero
elements of a sparse matrix.
Sparse Matrix Storage Schemes
In order to take advantage of the large number of zero elements, special schemes
are required to store sparse matrices. The main goal is to represent only the nonzero
elements and to be able to perform the common matrix operations. Only the most popular
schemes (Duff et al., 1986) are presented. The following material is taken from Saad,
(2003) and is presented here for completeness.
Compressed row format
35
A(n*n) with nz nonzero entries is represented using three arrays
ax(nz), aj(nz), ai(n+1). The entries are entered row wise.
Example: n=5, nz = 12
[1
2 3 ]
[ 4
5]
[6
7 8 ]
[9
10
]
[
11 12 ]
row pointer array, ai(n+1) = [ 1 4 6 9 11 13 ]
element value array, ax(nz) = [ 1 2 3 4 5 6 7 8 9 10 11 12 ]
element column index array, aj(nz) = [ 1 3 4 2 5 1 3 4 1 4 3 5 ]
Compressed column format
A(n*n) with nz nonzero entries is represented using three arrays
ax(nz), ai(nz), aj(n+1). The entries are entered column wise.
Example: n=5, nz = 12
[1
2 3 ]
[ 4
5]
[6
7 8 ]
[9
10
]
[
11 12 ]
Column pointer array, aj(n+1) = [ 1 4 5 9 11 13 ]
element value array, ax(nz) = [ 1 6 9 4 2 7 10 11 3 8 5 12 ]
element row index array, ai(nz) = [ 1 3 4 2 1 3 4 5 1 3 2 5 ]
Triplet format
36
A(n*n) with nz nonzero entries is represented using three arrays
ax(nz), ai(nz), aj(nz). Any nonzero entry is identified by a triplet (ax,ai,aj), where ax is
the value of the entry, ai is the row entry and aj is the column entry
Example: n=5, nz = 12
[1
2 3 ]
[ 4
5]
[6
7 8 ]
[9
10
]
[
11 12 ]
element value array, ax(nz) = [ 1 2 3 4 5 6 7 8 9 10 11 12 ]
element row index array, ai(nz) = [ 1 1 1 2 2 3 3 3 4 4 5 5 ]
element column index array, aj(nz) = [ 1 3 4 2 5 1 3 4 1 4 3 5 ]
2.2.2 MULTIFRONTAL SOLUTION METHODS
Sparse direct solution methods involve band width algorithms, unifrontal and
multifrontal techniques. The frontal techniques are much efficient compared to the band
width algorithms and are widely used in the solving matrices resulting from finite
element formulations.
This section gives a brief introduction to unifrontal and multifrontal techniques
which are direct methods for solving the linear equation Ax = B where A can be a
symmetric, unsymmetric, definite or indefinite matrix. Unifrontal method is a derivative
of the classical gaussian algorithm. Although it is developed for finite element
applications, it can be used in other fields also. It uses a small dense sub-matrix called
“frontal matrix’’. Original matrix is read sequentially and frontal matrix is filled with the
rows and columns. All-rows must be chosen to be fully-summed i.e. there are no further
37
contributions to come to the rows. When the frontal is filled with such rows, pivot(s) are
chosen from the fully-summed columns and basic gaussian elimination start. LU factors
are stored in RAM or in disk. Shur complement for the part that cannot be eliminated
further is calculated. Pivot row(s) and column(s) are deleted from the frontal to
accommodate new elements coming from original matrix. Having built the new frontal,
previous steps are repeated. Frontal matrices can be rectangular and doesn't have to be the
same size. But important thing is that, there is no more than one frontal at the same time.
Since the same working array is used throughout the factorization, the method is
memory-efficient. Partial pivoting among the fully-summed columns may be applied to
preserve numerical stability. Unifrontal method is competitive with the best band matrix
routines.
Frontal method is first described by Irons (1907) with the article ``A frontal
solution program for finite element analysis''. Method is primarily designed for finite
element analysis and for symmetric and positive definite matrices. Later, Hood (1976)
modified the method for unsymmetric matrices. Again the algorithm was designed to
solve finite element problems. In both papers, performance and memory management
issues are compared with band matrix routines. Frontal method is found to be efficient
especially in the case where the bandwidth is large. Band matrix routines require too
much memory for large bandwidths so factorization must be preceded by an ordering
phase to obtain smaller bandwidths.
In 1983, ``Multifrontal Method'' was derived by Duff and Reid (1983) in which
more than one frontal matrix is used. This method was first designed for symmetric and
undefinite matrices but later modified for unsymmetric and definite matrices. If there is
38
more than one frontal matrix at the same time unifrontal method is called Multifrontal
method. The description of multifrontal algorithm is beyond the scope of this work, but a
brief overview is presented here. Since couple of frontals exists, there are some
dependency relations between them. These relations are analyzed to build the assembly
tree (factorization tree). This tree roughly says which frontal must be handled first and
which one is last. Then from the leaf nodes to the root, factorization starts. First, leaf
nodes are factorized and assembled to the parents i.e. Shur complements are calculated
and contribution blocks are sent to upper nodes. Since, there is more than one frontal
matrix at the same time, management of these sub-matrices, calculating shur
complements and assembling them to the upper part of the tree is a difficult task.
However, it gives the chance for doing these calculations in a parallel environment.
Additionally, multifrontal method is more powerful then unifrontal method in the case of
the matrices of which bandwith is very large.
A parallel version of multifrontal method is also introduced in by I. S. Duff
(1986). Parallel implementation is based on the elimination tree concept. There is more
than one frontal matrix in multifrontal approach and these sub-matrices also called nodes
are dependent on each other. If one draw the dependency graph it will be observed that
the graph will be a tree. Since leaf nodes of the tree are independent of each other, they
can be handled separately. This is the first step of parallelization. At later steps, other
nodes are factorized by different processors and so on.
In the present work, a multifrontal solver (Davis et al., 1997, 1999 and 2004)
termed as the unsymmetric multifrontal package (UMFPACK v2.2) is used. UMFPACK
is a set of routines for solving unsymmetric sparse linear systems, Ax=b, using the
39
Unsymmetric MultiFrontal method and direct sparse LU factorization. UMFPACK v2.2
(Davis et al., 1999) is a set of fortran subroutines. UMFPACK relies on the Level-3 Basic
Linear Algebra Subprograms (dense matrix multiply) for its performance. This code
works on Windows and many versions of Unix (Sun Solaris, Red Hat Linux, IBM AIX,
SGI IRIX, and Compaq Alpha). The present code is being run on a windows machine
(Intel Pentium 4 machine) using optimized GOTO BLAS library available for free
download from the internet. Using GOTO BLAS has enhanced the performance of
UMFPACK significantly.
Direct solution methods for solving set of non-linear equations usually involve a
Newton’s linearization step to convert the set of non-linear equations into a set of linear
equations which is then updated during each iterative step. The disadvantage of using
direct solution methods is that the solution of set of linear equations during each iteration
takes a huge amount of computational time. Moreover it requires a huge memory. There
has been significant improvement in the available of large RAM in the present market. In
addition, the onset of 64 bit machines has made it possible to access more than 2GB
RAM. Thus it has been possible to solve even large scale problems using direct solvers.
The development of efficient direct sparse solvers has significantly reduced the
computation time for solving a linear system. The most time consuming step in solving
the linear system is the factorization step. In the present work, a modified Newton’s
method is used which can save a significant amount of computational time.
Newton step:
(
)
( n +1)
(n)
J ( n) δ x( n) = − F ( n) ,
x
=x
( n)
+ αδ x .
(2.35-36)
40
Modified Newton step:
(
)
J ( 0) δ x( n ) = − F ( n ) ,
x( n +1) = x ( n ) + αδ x ( n ) ,
(2.37-38)
where F is the function residual, J is the Jacobian matrix consisting of the derivatives of
the function residual, x and δ x are the variable and the increment in the variables to be
solved, n is the iteration number and α is the under relaxation parameter.
In the modified Newton’s step, the Jacobian matrix is not updated. It is calculated
only during the first iteration and the same Jacobian matrix is reused. The advantage with
Modified Newton’s method is that the left hand matrix needs to be factorized only during
the first iteration. The factors calculated by UMFPACK solver during the first iteration
are stored for reuse in the subsequent iterations. Since the factorization is the most
expensive step, by skipping the factorization step in the subsequent iterations, the
subsequent iterations become very cheap. Although the rate of convergence is reduced
compared to the Newton’s step, the savings in computational time per iteration overrules
the decrease in convergence and hence significant amount of saving is obtained.
2.2.3 BENCHMARK TESTING
To test the performance of sparse direct solvers over the standard SIMPLE
technique used for solving Navier-Stokes equations, a differential cavity problem is
chosen as a benchmark problem and the total computational times are compared.
Differential Cavity problem
∂U ∂V
+
= 0,
∂X ∂Y
⎛ ∂ 2U ∂ 2U ⎞
∂U
∂U
∂P
+V
=−
+ Pr ⎜⎜ 2 + 2 ⎟⎟ ,
U
∂X
∂Y
∂X
∂Y ⎠
⎝ ∂X
(2.39)
(2.40)
41
U
⎛ ∂ 2V ∂ 2V ⎞
∂V
∂V
∂P
+V
=−
+ Pr ⎜⎜ 2 + 2 ⎟⎟ + Ra Pr T ,
∂X
∂Y
∂Y
∂Y ⎠
⎝ ∂X
(2.41)
∂T
∂T ⎛ ∂ 2T ∂ 2T ⎞
⎟.
+V
=⎜
+
U
∂X
∂Y ⎜⎝ ∂X 2 ∂Y 2 ⎟⎠
(2.42)
The boundary conditions for the driven cavity problem can be expressed as
U = V = 0, at all other boundaries
T = 1,
X = 0,
T = 0,
∂T
= 0,
∂Y
X = 1,
Y = 0, and Y = 1.
Figure 2.1 compares the function residuals
(F
R
= F
∞
) for
the Direct solvers
(Newton, Picard and Modified Newton) and the SIMPLE method. Figure 2.1 indicates
that the direct solvers are more efficient compared to the SIMPLE method. Modified
Newton’s method is found to extremely efficient compared to all the other methods. This
is because of the skipping the factorization step after the first iteration which leads to a
significant amount of savings in computational time.
2.3 ANALYSIS OF AN EXTERNALLY HEATED AXISYMMETRIC WICK
In a realistic candle, both the candle flame and the wick are coupled to each other.
The heat from the candle flame evaporates the candle wax, providing the driving force
for the liquid to rise up through capillary action. The wax evaporated from the surface
provides the fuel for the candle flame. In this way the fuel supplied by the wick and the
heat supplied by the candle flame are coupled together.
The heat and mass transfer taking place inside the porous wick is very complex
due to the presence of both liquid and vapor inside the wick. To gain sufficient insight
into the physics of two-phase flow inside the wick, the wick is first analyzed separately
42
by decoupling the wick from the flame. A constant heat flux is imposed along the
cylindrical surface and at the tip of the wick. The detailed structure and the flow patterns
inside the porous wick are analyzed. Later, the wick is coupled to the candle flame and
the detailed study of gas phase flame characteristics and porous flow fields is done in
chapter 4.
2.3 1 PHYSICAL DESCRIPTION OF THE MODEL
The wick is now treated as being heated from a constant external heating source.
The heat flux is uniformly distributed along the cylindrical surface of the wick and on the
tip surface of the wick (Figure 2.2). Experiments have been conducted by Zhao and Liao
(2000) to study the heat transfer characteristics of a capillary-driven flow in a porous
structure heated with a permeable heating source at the top. Their experimental set up is
essentially one-dimensional in nature. This present physical situation is different from
their experimental set up but their experiments reveal certain essential characteristics of
two-phase flow inside a wick.
In this present set up, the wick is dipped in a liquid wax pool. The level of the
wax pool is assumed to be constant. The wax pool is at the melting point temperature of
the wax (330 K). A constant heat flux is applied along the cylindrical surface and on the
wick tip. A wick of length 5mm and diameter 1mm is chosen for the present study. The
two-phase flow inside the wick is simulated.
Figure 2.3 shows the computational grid used for the axisymmetric wick model.
The grid size is chosen as 80x40. Grid clustering is used in both the x and r directions.
43
2.3.2 SAMPLE CASE RESULTS
The porous wick parameter chosen are shown in the Table 2.2. A constant heat
flux of 8 × 104 W/m2 is applied on the wick as described above. The detailed structure
inside the porous wick is presented below.
2.3.2.1 Saturation and Temperature Distribution
Figure 2.4(a) shows the saturation profiles inside the porous wick. The figure
indicates that there are two regions inside the wick – single phase liquid region and twophase vapor liquid region. The contour line s=1 demarcates the two-phase region and the
pure liquid region. The saturation is lowest on the cylindrical corner of the wick reaching
a value of 0.825. As the wick is receiving heat on both the cylindrical surface and the tip
surface from the external heating source, the evaporation causes a decrease in the
saturation at the surface of the wick.
Figure 2.4 (b) shows the non-dimensional
temperature distribution inside the wick. The base of the wick is at the melting point (323
K) of the liquid wax and on the wick surface, where there is evaporation; the temperature
is at its boiling point (620 K). Fig. 2.4 (b) shows the presence of temperature gradients
near the base of the wick indicating the heat lost to the wax pool. Figure 2.4 (c) shows
the temperature contours inside the two-phase region of the wick.
Notice that the
temperature variation inside the two-phase region is very small.
Still this slight
temperature variation causes a significant variation in the vapor pressure distribution
inside the two-phase region given by the equilibrium phase relations (Eq. 2.25).
2.3.2.2 Pressure Distribution
Figure 2.5 (a) shows the non-dimensional liquid pressure distribution inside the
wick. This drives the liquid from the base of the wick to the surface of the wick. As the
44
wick is receiving heat on both the cylindrical surface and the tip surface from the external
heating source, the evaporation causes a decrease in the saturation at the surface of the
wick. This causes a decrease in liquid pressure, given by capillary relations, on the
surface (both the cylindrical surface and the wick tip surface) of the wick. Pressure
gradients are present along the length of the wick and in the radial direction indicating
liquid motion in both the directions (refer fig. 2.4 (a)). Figure 2.5 (b) shows the nondimensional capillary pressure distribution inside the wick.
Capillary pressure is a
function of the saturation (refer Eq. 2.9). Hence the capillary pressure distribution is
qualitatively similar to that of the saturation distribution. At the interface between the
liquid and the two-phase region, the capillary pressure is zero. Figure 2.5 (c) shows the
non-dimensional vapor pressure distribution inside the wick. The vapor pressure in the
two-phase region is a function of the temperature as given by the Gibbs phase relations
(Eq. 2.25). Hence the vapor pressure distribution variation is qualitatively similar to the
temperature variation inside the two-phase region. Figure 2.5 (c) shows sharp vapor
pressure gradients in the r-direction near the cylindrical surface of the wick.
2.3.2.3 Mass flux Distribution
Figure 2.6 (a) shows the liquid mass flux vectors indicating the flow of liquid
inside the wick. The liquid is drawn from the base and it comes out of the wick along the
cylindrical surface and the tip of the wick.
As the liquid is evaporated along the
cylindrical surface, the liquid mass flux along the length of the wick decreases. Figure
2.6 (b) shows the vapor mass flux distribution inside the wick. Vapor motion is very
intricate and also interesting. Vapor is being driven by the temperature gradients inside
the two-phase region. Since the temperature gradients (related to the pressure gradient
45
through the equilibrium relationship Eq. 2.25) are directed into the wick, the vapor moves
into the wick. The liquid and the vapor move counter currently to each other. This is
also found in one-dimensional wick model [Raju, 2004] and is confirmed by
experimental results [Zhao and Liao (2000)]. The vapor eventually condenses at the
interface between the liquid and the two-phase region.
2.3.2.4 Heat flux Distribution
Figure 2.7 shows the heat flux distribution inside the porous wick. There is
convective heat flux due to both liquid and vapor motion and there is also conductive
heat flux due to temperature gradient. The heat flux vectors can be expressed as follows
q l* = ρ l*c*l T *vl*
(
)
q *g = ρ *g c*g T * + i*fg v g*
(2.43-45)
qc* = −keff *∇T *
Equation 2.43 represents the heat flux vector due to liquid convection. This is
depicted in figure 2.7 (a). Equation 2.44 represents the heat flux vector due to vapor
convection. This also includes both the sensible heat and the latent heat of evaporation.
This is depicted in Figure 2.7 (b). These vectors are qualitatively similar to the vapor
mass flux vectors. Equation 2.45 shows the heat flux vectors due to conduction. This is
depicted in Figure 2.7 (c). Since the temperature variation in the two-phase region is
very less, the conductive heat flux vectors are small. In the liquid region, there is
significant variation in temperature and hence the conduction is prominent in the liquid
region.
46
2.3.2.5 Variation along the Cylindrical Surface and the Axis of the Wick
Figure 2.8 shows the saturation and the temperature distribution along the
cylindrical surface of the wick. The saturation plots shows the presence of liquid region
(s=1) near the base of the wick. The rest of the surface is in two-phase region. The
temperature in the liquid region is below the boiling temperature. Figure 2.9 shows the
liquid and vapor motion in the r-direction along the cylindrical surface. This figure gives
an indication of the evaporation taking place on the surface. Constant heat flux is applied
on the surface. In the liquid region near the base of the wick, no liquid is evaporated
from the surface. This implies that all the heat supplied on the surface in this region is
conducted into the surface and no evaporation takes place. As we move into the twophase region, a part of the heat is used for evaporating the liquid at the surface and a part
of it is conducted into the wick. Therefore the mass flux of liquid evaporated from the
surface increases. A part of the liquid that is evaporated at the surface is convected into
the wick interior as vapor. Therefore the vapor mass flux shows negative values. The net
mass flux of vapor goes out of the wick surface. The vapor mass flux first increases as
we traverse along the surface and then it reduce gradually to zero at the tip of the wick.
The reason for this is explained as follows. The vapor motion (Eq. 2.15) is a function of
the vapor saturation (1-s) and the vapor pressure gradient (equivalently temperature
gradient). The temperature gradient in the r-direction decreases as we traverse to the tip
along the cylindrical surface of the wick and the vapor saturation correspondingly
increases. The maximum vapor mass flux is achieved in between as a result of the
interaction of the two terms. Figure 2.9 shows a maximum vapor mass flux is achieved at
x = 2.8 mm. Figure 2.10 shows the variation of the x direction liquid mass flux along the
47
cylindrical surface. There is a continuous decrease in the mass flux due to mass loss by
evaporation along the surface of the wick.
Figure 2.11 shows the variation of the saturation and temperature along the axis of
the wick. The two-phase region starts at approximately x = 3.8mm. The variation of
temperature in the two-phase region (depicted in the sub figure) is very small but it is
significant in the liquid region. Figure 2.12 shows the liquid and the vapor mass flux
along the axis. The vapor mass flux is negative in the two-phase region indicating that
the vapor is moving inwards into the wick. At the interface between the liquid and the
two-phase region, the entire vapor condenses.
The liquid mass flux continuously
decreases because the liquid is continuously drawn along the cylindrical surface of the
wick.
2.3.3 MESH REFINEMENT STUDIES
Mesh refinement studies has been performed. The above sample case was run
with different mesh sizes to see its effect on the porous wick solution. Mesh sizes 80x40,
80x80, 160x40has been run.
Figure 2.13 shows the comparison of saturation, non-dimensional liquid pressure
and non-dimensional temperature contours for all these mesh sizes. The contours give
agree very closely for all the three meshes being chosen. The reference grid size is
chosen as 80x40. Doubling the grid size either in the x-direction or in the r-direction did
not affect the solution much. So the grid size 80x40 is chosen for all the subsequent
calculations in this work.
48
2.3.4 THE EFFECT OF APPLIED HEAT FLUX
The effect of heat flux on the heat and mass transport inside the porous wick has
been studied. A direct consequence of the variation in applied heat flux is that the
amount of wax evaporated from the wick surface varies. The detailed distribution of
saturation, pressure and temperature inside the wick also changes with the applied heat
flux.
Figure 2.14 shows the variation of saturation at the tip of the cylindrical surface of
the wick with the heat supplied to the wick. This is the location where the saturation
reaches its minimum value. It gives an overall idea of the saturation distribution inside
the wick surface. As the heat supplied to the wick increases, more evaporation takes
place from the surface leading to a decrease in saturation distribution on the surface of
the wick. This decreased saturation distribution on the wick surface causes the capillary
action to increase and hence more wax is drawn from the wax pool. Figure 2.14 shows
that the rate of decrease of saturation with the heat supplied increases drastically as the
saturation approaches zero. The tip saturation approaches a value of zero for total heat of
1.65X10-5 W. With further increase of heat supplied to the wick surface, numerical
difficulties were observed in obtaining a converged solution. This may be due to lack of
robustness of the thermodynamic model being used to solve for evaporative front regime.
The code exhibited spurious oscillations. The behavior of the steady state code with the
appearance of evaporative front regime seems to indicate that the system might be
transient in nature. Figure 2.15 shows the effect of applied heat flux on the total mass of
wax evaporated from the wick surface (both from the cylindrical surface and from the
wick tip). The mass evaporated from the wick surface varies linearly with the applied
49
heat. The heat that is being supplied to the wick surface is mostly used for evaporating
the wax from the surface (supplies both sensible heat to raise the temperature of wax
from 330K to 620K and also the latent heat of vaporization at 620K) and a part of it is
lost to the wax pool. Figure 2.16 shows the percentage of heat supplied that is lost to the
reservoir. At lower heat fluxes, the percentage of heat loss to the reservoir is large
compared to that at high heat fluxes. This is also observed in the one-dimensional porous
wick analysis (Raju, 2004).
2.3.5 PARAMETRIC STUDY
2.3.5 1 The Effect of Gravity
The effect of gravity on the two phase flow in porous wick is studied applying a
constant heat flux on the surface of the wick. Gravity acts as an opposing force to the
upward motion of the liquid induced by capillary action. This is important to understand
the role of gravity on the heat and mass transport inside the wick. When the porous wick
is coupled to the candle flame, the effect of gravity on the porous transport cannot be
isolated because gravity influences both the porous wick transport and the candle flame
through the buoyancy effect. Figure 2.17 shows the effect of gravity on the saturation at
the tip of the cylindrical surface of the wick. Although the gravity is varied from 0ge to
2ge, the change in the saturation distribution is small. The capillary driving force seems
to dominate the gravity forces for the given set of porous wick parameters and wax
properties. The order of estimate of capillary and gravity forces is given below.
∂pl
p ( s = 0)
capillary force ∂x
∼ c
=
ρl g
ρl gLw
gravity force
capillary force
∼ 122 (for normal gravity)
gravity force
50
Hence the effect of gravity on the heat and mass transport in the porous wick
seems to be negligible. Figure 2.18 shows the effect of the gravity on the mass of wax
evaporated from the wick surface. The mass evaporated from the wick surface remains
unchanged with gravity. Since the problem is a steady state problem, this result is
expected unless the heat that is lost to the reservoir is affected by gravity.
2.3.5 2 The Effect of Absolute Permeability
Absolute permeability is a measure of the ability of porous wick to transport
fluids through the medium. It is similar to the concept of thermal conductivity in heat
flow. The absolute permeability is a function of the wick material chosen and the way
the wick is formed from the fibrous material. Studying the effect of changing the
absolute permeability gives us an idea of how the heat and mass transport inside the wick
is affected by the wick material or the type of wick being chosen. Figure 2.19 shows the
effect of absolute permeability on the saturation at the tip of the cylindrical surface of the
wick. Decreasing the absolute permeability decreases the tip saturation. This can be
explained as follows. For a given heat flux, the mass evaporated from the wick surface
should remain approximately same for a steady state problem.
By decreasing the
permeability, the resistance to the flow inside the porous media increased. To maintain
the flow rate the driving force (capillary action) should increase. This is affected by
decreasing the saturation at the surface of the wick. For a wick with high permeability,
the opposite effect takes place and hence the saturation at the surface is less. For wicks
with low permeability values, the tip saturation reaches zero for less heat flux values and
vice versa.
51
Figure 2.20 shows the effect of absolute permeability on the mass of wax
evaporated from the surface of the wick. There is a slight decrease in mass evaporated
from the wick surface for high permeability values. This is caused due to slight increase
in heat lost to the reservoir at high permeability values.
In effect, gravity does not play a major role in the heat and mass transport of the
inside the wick, but permeability of the wick plays an important role in determining the
phase (saturation) distribution inside the wick.
52
Table 2.1 Porous Wick Dimensionless Variables
cg
c *g
cl*
g * ρ g* (K ε ) L*
12
g
i fg
σ
i *fg
cl*T0*
K rg
(1 − s )
K rl
s
Pe
εσ ( K ε )
α l μl
12
p *g (K ε )
12
pg
σ
Rg
ρl* Rg*T0 ( K ε )
σ
s
εl
ε
T
T*
Tb*
12
ug
ul
x
u*g D
αl
ul* D
αl
x*
D
53
μ g*
μg
μ l*
ρ g*
ρg
ρ l*
Table 2.2 Porous Wick Numerical Values
c*g
J kg − K
1430
cl*
J kg − K
2452
i *fg
J kg
8.8×105
k e* (s = 0)
W m−K
6.40
k e* (s = 1)
W m−K
6.31
K
m2
1.09×10-11
Rg*
J kg − K
25.2
Tm*
K
330
Tb*
K
620
ε
0.55
α l*
m2 s
3.4×10-6
ρ l*
kg m 3
770
μ l*
Pa ⋅ s
5×10-4
μ g*
Pa ⋅ s
1.1×10-5
σ
N m
0.035
54
4
Differential Cavity (Ra = 10 , 60x60 mesh)
100
10
-1
NEWTON
PICARD(α=0.6)
SIMPLE(α=0.8)
MODIFIED NEWTON ( α=0.3)
10-2
FR
10-3
10-4
10-5
10
-6
10-7
10-8
50
100
150
CPU time (s)
Figure 2.1 Comparison of function residuals vs. CPU times for Newton, modified
Newton and Picard’s iterative techniques
55
Dw=1mm
q
g
q
Wick
q
Lw=5mm
Liquid wax pool at
its melting point
Figure 2.2 Physical description of an externally heated axisymmetric wick.
56
80x40
g
0.5
r (mm)
0.4
0.3
0.2
0.1
0
0
1
2
X (mm)
3
4
5
Figure 2.3 Computational grid of an externally heated axisymmetric wick.
Saturation contours
0.6
0.5
0.4
0.3
0.2
0.1
00
0.
95
40
0.8 2
5
0.8
75
0.8
00
0.9
0
1.0
r/Dw
57
1
2
X/Dw
0
3
4
0.830
5
Non-dimensional temperature contours
0.
86
83
57
0.9
31
34
53
0.9
71
804
68
3
0.9
0
0. .9971 99543
42
99
20
27
4
0.6
0.5
0.4
0.3
0.2
0.1
00
0.6
r/Dw
(a)
1
2
X/Dw
3
4
5
(b)
r/Dw
Non-dimensional temperature contours (expanded in two-phase region)
0.5
0.4
0.3
0.2
0.1
03
0.99
0.9
0.9
0.
99 9
0.
99 9
95
88 18 9 9 9 6 9 8 9999
77
10
3.5
4
4.5
X/Dw
5
(c)
Figure 2.4: Plot of (a) saturation profiles (b) non-dimensional temperature profiles and (c)
non-dimensional temperature profiles (expanded in the two-phase region) inside the
porous wick for parameters shown in Table 2.2.
58
Non-dimensional liquid pressure contours
0.6
0.5
X/Dw
12.7269
2
12.7374
1
12.7584
0
12.7795
0
12.79
0.1
12.811
0.2
12.8216
0.3
12.8531
12.8742
r/Dw
0.4
3
4
5
Non-dimensional capillary pressure contours
0.6
0.5
r/Dw
48
5
378
0 38
824
0
0.2
0.1649
0.130
0.0
0.3
0.15321
0.4
0.1
0
0
1
2
X/Dw
3
4
5
Non-dimensional vapor pressure contours
0.6
0.5
2
X/Dw
3
4
r/Dw
12.8846
1
84 1
0
833
0
12.8
12.827
0.1
766
0.2
12.8
4
12.86
0.3
12.8
0.4
5
Figure 2.5: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick for parameters shown
in Table 2.2.
r/Dw
59
0.5
0.4
0.3
0.2
0.1
00
10 kg/m2 s
1
2
X/Dw
3
4
5
r/Dw
(a)
0.5
0.4
0.3
0.2
0.1
02
2
0.1 kg/m s
3
X/Dw
4
5
r/Dw
(b)
0.5
0.4
0.3
0.2
0.1
04
2
0.003 kg/m s
4.25
4.5
X/Dw
4.75
5
(c)
Figure 2.6: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor
mass flux vectors (expanded near the tip of the wick) inside the porous wick for
parameters shown in Table 2.2.
r/Dw
60
0.5
0.4
0.3
0.2
0.1
00
30 W/cm
1
2
X/Dw
3
4
2
5
r/Dw
(a)
0.5
0.4
0.3
0.2
0.1
02
0.3 W/cm
3
X/Dw
4
2
5
r/Dw
(b)
0.5
0.4
0.3
0.2
0.1
02
2
0.1 W/cm
3
X/Dw
4
5
(c)
Figure 2.7: Plot of (a) liquid convective heat flux vectors (b) vapor convective heat flux
vectors and (c) conductive heat flux vectors inside the porous wick for parameters shown
in Table 2.2.
61
s
1
T
0.95
0.95
s
0.9
0.9
0.85
0.85
s
0.8
0.8
T
0.75
0.75
0.7
0.7
0.65
0.65
0.6
0.6
0.55
0.55
0.5
0
1
2
3
4
5
Non-dimensional temperature
1
0.5
X/Dw
Figure 2.8: Plot of saturation and temperature variation along the cylindrical surface of
the wick exposed to the heat flux for parameters shown in Table 2.2
62
1
0.9
ρvl
2
mass flux (kg/m s)
0.8
0.7
0.6
ρlvl + ρgvg
0.5
0.4
0.3
0.2
0.1
ρvg
0
-0.1
0
1
2
3
4
5
X/Dw
Figure 2.9: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick exposed to the heat flux for parameters shown in Table 2.2
63
8
mass flux (kg/m2 s)
7
6
5
ρul
4
3
2
1
0
0
1
2
3
4
5
X/Dw
Figure 2.10: Plot of liquid mass flux (in x-direction) variation along the cylindrical
surface of the wick exposed to the heat flux for parameters shown in Table 2.2
64
0.9
T
s
T
0.9
1
0.8
Non-dimensional temperature
T
0.8
1
s
0.9998
0.9996
0.7
0.7
0.9994
0.9992
0.6
4
0.5
0
1
4.25
4.5
X/Dw
4.75
2
3
0.6
0.999
5
4
5
Non-dimensional temperature
s
1
0.5
X/Dw
Figure 2.11: Plot of saturation and temperature variation along the axis of the wick for
parameters shown in Table 2.2
65
0
6
-0.0025
ρgug
ρlul (kg/m s)
-0.005
ρgug (kg/m2 s)
ρlul
5
-0.0075
3
-0.01
2
4
-0.0125
2
-0.015
1
-0.0175
0
0
1
2
3
4
5
-0.02
X/Dw
Figure 2.12: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick for parameters shown in Table 2.2
66
0.8
80x40
80x80
160x40
4
4.5
0.8
12.8619
12.8391
r/Dw
2.5
X/Dw
(b)
3
3.5
4
0.8
0.9
94
0.
5
70
06
01
0.9
0.
10
85
18
1
03
02
21
84
604
0.7
r/Dw
4.5
80x40
80x80
160x40
0.6
00
72
2
12.70
1.5
12.725
12.7136
1
12.7478
0.5
12.7706
00
12.7934
0.2
12.8162
0.4
0.2
5
80x40
80x80
160x40
0.6
0.4
8563
3.5
0.83
3
18
2.5
X/Dw
(a)
0.8445
2
883
1.5
249
1
0.854
0.5
5614
00
0.865
1
0.87
0.2
11
0 67
7
80
37
0.9
0.4
0.9
r/Dw
0.6
0.5
1
1.5
2
2.5
X/Dw
(c)
3
3.5
4
4.5
5
Figure 2.13: Comparison of (a) saturation profiles (b) pressure profiles and (c)
temperature profiles for three different meshes (80x40, 80x80, 160x40) inside the porous
wick for parameters shown in Table 2.2.
67
1
0.9
Tip Saturation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
Total Heat Applied on the Wick Surface (W)
Figure 2.14 The variation of saturation at the cylindrical tip of the wick surface with the
total heat supplied to the wick.
68
Total mass evaporated (mg/s)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
Total Heat Applied on the Wick Surface (W)
Figure 2.15 The variation of total mass of wax evaporated from the wick surface with the
total heat supplied to the wick.
69
20
% heat lost to the reservoir
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
0
50
100
150
200
Total Heat Applied on the Wick Surface (W)
Figure 2.16 The variation of percentage heat that is lost to the reservoir with the total heat
supplied to the wick.
70
g=0ge
g=1ge
g=2ge
0.9
0.8
Tip saturation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
50
100
150
200
Total Heat supplied to the wick (W)
Figure 2.17 The effect of gravity on the variation of saturation at the cylindrical tip of the
wick surface with the total heat supplied to the wick.
71
g=0ge
g=1ge
g=2ge
2
Total mass evaporated (mg/s)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
50
100
150
200
Total Heat supplied to the wick (W)
Figure 2.18 The effect of gravity on the variation of total mass evaporated from the wick
surface with the total heat supplied to the wick.
72
K=5x10-12 m2
K=1x10-11 m2
K=2x10-11 m2
1
0.9
Tip saturation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
Total Heat supplied to the wick (W)
Figure 2.19 The effect of absolute permeability on the variation of saturation at the
cylindrical tip of the wick surface with the total heat supplied to the wick.
73
-12
1.8
Total mass evaporated (mg/s)
2
K=5x10 m
-11
2
K=1x10 m
-11
2
K=2x10 m
2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
Total Heat supplied to the wick (kW)
Figure 2.20 The effect of absolute permeability on the variation of total mass evaporated
from the wick surface with the total heat supplied to the wick.
74
CHAPTER 3
GAS PHASE MODEL
To describe the candle burning phenomena, a gas-phase combustion model is
needed to couple with the wick transport model. In this thesis, the axisymmetric gasphase model formulated by Alsairafi (2003) is adopted. This chapter is taken directly
from Alsairafi’s thesis and is presented here for the sake of completeness
3.1 THEORETICAL FORMULATION
In this section, the steady state equations for the gas phase are presented. The flow
field is solved using the full Navier-Stokes equations. The governing equations for the
gas phase model are initially presented in dimensional form. The contribution of each
term in the equations and the boundary conditions will be discussed briefly. Next, the
non-dimensional procedure will be given.
The geometry of the candle and the wick is shown schematically in figure 3.1. In
this model, the coordinate system is attached to the bottom of the exposed wick at the
point (x=0, r=0). The x-coordinate, along the symmetry line, is in the axial direction and
the other coordinate is along the radial direction. For the reference case to be studied, the
wick is 1mm in diameter, the candle body has a diameter of 5mm and is 20mm high. The
length of the wick varies from case to case and for a self trimmed candle flame, the self
trimmed length of the wick is obtained as a part of the solution. The candle is placed on a
solid plate that extends to infinity. The ambient is air with mole fractions of 21% O2 and
79% N2. In the subsequent parametric study, however, the wick diameter and the ambient
oxygen molar fraction will be varied.
75
The following assumptions in the gas phase are used.
•
The flow is steady, laminar and axisymmetric in the gas phase.
•
The ideal gas law is applicable to the gas mixture.
•
This study assumes a single-step reaction, finite rate gas-phase chemical kinetics.
The property values of the fuel are based on a blend of C 25 H 52 with stearic acid
( C18 H 36 O2 ).
•
Viscosity, conductivity, and the product of ρDi have a power law dependence on
temperature.
•
Viscous dissipation is negligible.
•
Soot formation is neglected.
•
The liquid wax pool is assumed to be uniformly at its melting point temperature.
3.1.1 Continuity Equation
The mass conservation equation for two-dimensional flow in cylindrical
coordinates has the following form:
∂
1 ∂
( ρu) +
(r ρ v) = 0 ,
∂x
r ∂r
(3.1)
where u and v are the axial and radial velocity components respectively and ρ is the mass
density. The individual terms represent the net flux of mass at a point in the x and r
directions respectively. The local density of the mixture depends on the mixture
temperature, pressure, and the species mass fractions through the ideal gas relation:
P = ρRu T ∑
i
Yi
,
MWi
where MWi is the molecular weight of species i.
(3.2)
76
3.1.2 Momentum Equation
Modeling of the buoyant flow requires additional attention. In candle flames,
flows between the flame and ambient are driven by natural convection resulting from
temperature differences through the flame. Hence, the gravity force needs to be included
in the model. The axial momentum equation is
∂
1 ∂
∂P ∂ 4 ∂u 1 ∂
∂u
( ρ uu ) +
(r ρ vu ) = −
+ ( μ )+
(r μ ) +
∂x
∂x ∂x 3 ∂x r ∂r
∂r
r ∂r
1 ∂
∂v 1 ∂ 2 ∂ (rv)
(rμ ) −
[ μ
] − ρg
r ∂r
∂x r ∂x 3
∂r
(3.3)
and the radial momentum equation has the following form:
∂
1 ∂
∂P ∂
∂v 1 ∂
∂v
( ρ uv) +
(r ρ vv) = −
+ (μ ) +
(2r μ ) +
∂x
∂r ∂x ∂x r ∂r
∂r
r ∂r
. (3.4)
∂
∂u 2μ v ∂ 2 ∂v v ∂u
( μ ) − 2 − [ μ ( + + )]
∂x
∂r
r
∂r 3 ∂r r ∂x
The two terms on the left hand sides of the x- and r-momentum equations
represent the rate of change of the x- and r-momentums respectively. The terms on the
right side are pressure, viscous and body forces respectively. P is the pressure that can be
decomposed into the hydrostatic pressure, i.e.
dp hydrostatic
dx
= − ρ ∞ g , and the flow dynamic
pressure. Subtracting the hydrostatic pressure from the x-momentum equation, and since
there is no gravity force acting in the r-direction, the momentum equations in x- and rdirections take the following forms:
∂
1 ∂
∂p ∂ 4 ∂u 1 ∂
∂u
( ρ uu ) +
(r ρ vu ) = − + ( μ ) +
(r μ ) +
∂x
∂x ∂x 3 ∂x r ∂r
∂r
r ∂r
,(3.5)
1 ∂
∂v 1 ∂ 2 ∂ (rv)
(rμ ) −
[ μ
] + ( ρ ∞ − ρ) g
r ∂r
∂x r ∂x 3
∂r
77
∂
1 ∂
∂p ∂
∂v 1 ∂
∂v
( ρ uv) +
(r ρ vv) = − + ( μ ) +
(2r μ ) +
∂x
∂r ∂x ∂x r ∂r
∂r
r ∂r
. (3.6)
∂
∂u 2μ v ∂ 2 ∂v v ∂u
( μ ) − 2 − [ μ ( + + )]
∂x
∂r
r
∂r 3 ∂r r ∂x
3.1.3 Species equation
The candle composition, as described in Dietrich et al. (2000), is assumed to be a
blend of 80 percent (by weight) n-paraffin wax ( C 25 H 52 is chosen for the numerical
simulation) and 20 percent stearic acid ( C18 H 36 O2 ).
Assuming one-step chemical
reaction, the overall stoichiometric combustion reaction of a candle and air can be
described as follow:
C 25 H 52 + 0.31 C18 H 36 O2 + 46.06 [O2 + 3.76 N 2 ] →
30.58 CO2 + 31.58 H 2 O + 173.19 N 2
(3.7)
If f i is the stochiometric mass ratio of species i and fuel, then from the above equation
we have:
f F = −1 , f O2 = −3.3495 , f CO2 = 3.0577 , and f H 2O = 1.2918
(3.8)
The negative sign in f i indicates that the species is consumed while the positive sign is
for produced species. The species equations for fuel, oxygen, carbon dioxide, and water
vapor can be written as
∂Y
∂Y
1 ∂
1 ∂
∂
∂
( ρ uYi ) +
(r ρ vYi ) = ( ρ Di i ) +
(r ρ Di i ) + Wi .
∂x
r ∂r
∂x
∂x
r ∂r
∂r
(3.9)
The first and second terms represent the convective fluxes of species i. On the
right hand side, the first two terms represent mass diffusion rate, the last term is the
production rate of species i. It should be noted that Fick’s law has been implemented for
the diffusion velocity.
78
3.1.4 Energy equation
The energy conservation equation can be written in several different forms. The
form using temperature as the dependant variable will be presented here since it is easy to
implement in the numerical codes. Then,
ρ uC p
∂T
∂T ∂
∂T 1 ∂
∂T
+ ρ vC p
= (λ
)+
(rλ
) + ∑ hW
i i +
∂x
∂r ∂x ∂x
r ∂r
∂r
i
.
∂Yi ∂T ∂Yi ∂T
G
∑i [ ρ C pi Di ( ∂x ∂x + ∂r ∂r )] − ∇ ⋅ qr
(3.10)
On the left hand side, the terms represent the convection energy fluxes. The first
two terms on the right hand side are the heat diffusion rate expressed by the Fourier’s
Law. The third term represents the rate of heat release from chemical reaction term. The
fourth term is associated with the diffusion of species with different enthalpies due to
variable specific heat and the last term is the radiation energy flux. The reaction term can
be written as follows
∑ hiW i =
i
T
∑i ∫ C p,i dT W i
TD
Contribution of sensible enthalpy
T
+
∑i Δh Df ,i W i = ∑i ∫ C p,i dT W i + QW F .
TD
(3.11)
heat of Combustion
It is assumed in this work that a finite rate expression for a one-step, second-order
Arrhenius reaction exists and has the following form:
E
W i = − B g ρ 2YF YO2 exp(−
).
Ru T
(3.12)
The energy equation can be rewritten in a different form for numerical purposes. First,
the diffusion term can be rewritten as follows:
79
1 ∂
1 ∂
∂
∂T
∂T
∂
λ ∂T
λ ∂T
(λ
)+
( rλ
) = [C p (
)] +
[rC p (
)]
r ∂r
C p ∂x
r ∂r
C p ∂r
∂x ∂x
∂r
∂x
∂C p
1 ∂ (rC p ) λ ∂T
1 ∂
∂ λ ∂T
λ ∂T
)(
)+ (
)(
)] + C p { (
)+
[r (
)]}
= [(
r
C p ∂r
r ∂r C p ∂r
∂x C p ∂x
∂r
∂x C p ∂x
λ ∂T
. (3.13)
Using the continuity equation, and dividing by the total C p , the final form of the energy
equation becomes:
∂
1 ∂
∂ λ ∂T 1 ∂
λ ∂T
( ρ uT ) +
(r ρ vT ) = (
)+
[r (
)] +
∂x
r ∂r
∂x C p ∂x
r ∂r C p ∂r
λ ∂C p ∂T
T
∂C p ∂T
1
[
]+
[∑ C p ,i dT Wi + QW F ] +
+
C p2 ∂x ∂x
C p i T∫D
∂r ∂r
∑[
i
ρ C p ,i Di ∂Yi ∂T
Cp
(
∂x ∂x
+
(3.14)
∂Yi ∂T
1
G
)] −
∇ ⋅ qr
Cp
∂r ∂r
Clearly, the flow field is affected by changes in temperature and density. Hence, all the
flow equations must be solved together with the species and energy equations.
The last term on the right hand side of the energy equation, the radiation energy
flux, can be rewritten in the following form (Modest, 1993)
G
∇ ⋅ q r = κ [4σT 4 − G ] ,
(3.15)
where κ is the absorption coefficient of the mixture and G is the incident radiation. It is
obtained from the computed radiation intensity. The development of radiation transfer
equation will be described in the next section.
3.1.5 Boundary conditions
The bottom of the computational domain is bounded by the candle and the solid
plate that is placed on as shown in figure 3.1. The other boundaries are theoretically at
infinity, however, in the numerical computation the boundary conditions are applied at a
large but finite distance away from the candle (i.e. rmax and xmax). Except for the wick, the
80
remaining boundary conditions are similar to that used by Alsairafi (2003). If qr,in and
qr,out represent the incident radiation into the wick surface and the outward radiation from
the surface, respectively, then the boundary conditions can be written as follows:
On the wick surface:
− ρDF
∂YF
= ρv(1 − YF )
∂r
∂Yi
= ρvYi
∂r
ρDi
∑Y = 1
i
(i=O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
where ρ v is the mass flux of fuel supplied by the wick to the gas phase, which is given
by the following expression
ρ v = ( ρl vl + ρ g vg ) wick
The RHS term indicates the net mass flux of fuel (liquid+vapor) supplied by the wick at
this surface.
The energy balance equation in the gas phase is given by
λ
∂T
+ qr ,in = qwick + qr ,out
∂r
where qwick is the heat provided by the gas phase to the candle wick. This expression
gives the amount of heat flux that is supplied to the candle wick for evaporating the wax
on the surface.
The temperature is determined as a part of the solution of the wick governing equations.
On the wick-tip surface:
− ρDF
∂YF
= ρu (1 − YF )
∂x
81
∂Yi
= ρuYi
∂x
ρDi
∑Y = 1
i
(i=O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
where ρ u is the mass flux of fuel supplied by the wick to the gas phase, which is given
by the following expression
ρ u = ( ρl ul + ρ g u g ) wick surface
The RHS term indicates the net mass flux of fuel (liquid+vapor) supplied by the wick at
this surface.
The energy balance equation in the gas phase is given by
λ
∂T
+ qr ,in = qwick + qr ,out
∂x
where qwick is the heat provided by the gas phase to the candle wick. This expression
gives the amount of heat flux that is supplied to the candle wick for evaporating the wax
on the surface.
The temperature is determined as a part of the solution of the wick governing equations.
From point A to B in figure 3.1 (candle wax):
The temperature is assumed to be at the melting point of the wax.
T = Tm
The other dependent variables are as follows:
∂Yi
= 0 ; u = 0; v = 0
∂x
∑Y = 1
i
(i=F, O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
82
From point B to C in figure 3.1:
The temperature is fixed at the ambient temperature of 300K and the other dependent
variables are as follows:
∂Yi
= 0; u = 0; v = 0
∂r
∑Y = 1
i
(i=F, O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
On the solid plate:
T=T∞;
∂Yi
= 0 ; u = 0 ; v = 0 (i=F, O2, CO2, and H2O)
∂x
∑Y = 1
i
(i=F,O2, CO2, H2O and N2)
At r=rmax:
T = T∞ ; Yi = Yi ,∞ ; u = 0 ;
∑Y = 1
i
∂v
=0
∂r
(i=F, O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
At x=xmax:
∂Y
∂T
∂u
= 0; i =0;
= 0; v = 0
∂x
∂x
∂x
∑Y = 1
i
(i=F, O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
Symmetry line:
Because of symmetry, computation is performed only on one half of the domain and the
boundary conditions along the symmetry line (r=0) are:
∂Y
∂T
∂u ∂v
= 0; i =0;
=
=0
∂r
∂r
∂r ∂r
∑Y = 1
i
(i=F, O2, CO2, and H2O)
(i=F,O2, CO2, H2O and N2)
83
3.2 NON-DIMENSIONAL PARAMETERS
The basic governing equations are put into non-dimensional forms in order to
characterize the importance of the physical parameters as well as to facilitate the
numerical computation. The reference velocity used in the non-dimensionalization, Ur, is
the velocity seen by the flame near the stabilization zone. It consists of two parts, i.e.
Ur=UB+UD, where UB is the buoyant velocity assumed to take the form:
UB = [
( ρ ∞ − ρ f )α r
ρr
g ]1 / 3
(3.16)
and UD is the molecular diffusion velocity. The buoyant velocity is estimated by a
balance between buoyant forces and inertia forces (see T’ien et al., 2001). The diffusion
velocity is introduced here to avoid the singularity at zero gravity (UB→0). Diffusion
velocity in one atmosphere is of the order of 0.5-5cm/s and it is temperature dependent.
In this work we take UD =2cm/s.
A reference length scale, the smallest scale of importance in the physical problem,
needs to be chosen carefully in order to non-dimensionalizing the spatial coordinates in
the governing equations. The thermal length, Lr = α r / U r , is chosen for scaling and can
be obtained by considering the balance of convection and conduction in the gas-phase
flame. This thermal length is a good measure of the flame standoff distance, i.e. distance
between flame and candle wick near the flame base stabilization zone. The nondimensional variables and parameters are:
84
u=
Cp
α
x
r
λ
u
v
ρ
μ
;v=
; Lr = r ; x =
; r=
; ρ=
; μ=
;λ =
; Cp =
;
Ur
Ur
Ur
Lr
Lr
ρr
μr
λr
C p ,r
T
C p ,i =
W i =
Pe =
C p ,i
C p ,r
Δh
o
f ,i
+ ∫ C pi dT
To
; hi =
C p ,r T∞
; Di =
Di
T
E
q
;T =
; E=
; q=
;
Di ,r
T∞
Ru T∞
C p ,r T∞
α r ρ r Bg
μ r C p ,r
W i
ρU L
ρα
α
; Da =
; Lei = r ; Pr =
; Re = r r r = r r ;
2
( ρ r U r / Lr )
Di
λr
μr
μr
Ur
U r Lr
αr
; Gr =
g ( ρ ∞ − ρ f ) L3r
ρ rα r2
ρ r C p ,rU r
G
G L
p − p∞
; ∇ ⋅ q r = ∇ ⋅ q r ( r 4 ) ; Bo =
; p=
;
3
ρ rU r2
σT∞
σT∞
κ = κ ⋅ L r ; β = β ⋅ Lr
The last two non-dimensional parameters (i.e. κ and β ) are needed for the radiative
transfer equation. The non-dimensional governing equations take the form
∂
1 ∂
(ρu ) +
(r ρ v ) = 0 ,
∂x
r ∂r
(3.17)
∂
1 ∂
∂p 1 ∂ 4 ∂u
1 ∂
∂u
( ρ uu ) +
(r ρ vu ) = − +
{ ( μ )+
(r μ ) +
∂x
r ∂r
∂x Re ∂x 3 ∂x
r ∂r
∂r
,
1 ∂
∂v
1 ∂ 2 ∂ (rv )
ρ −ρ
(rμ ) −
[ μ
]} + Gr ( ∞
)
r ∂r
∂x
r ∂x 3
∂r
ρ∞ − ρ f
(3.18)
∂
1 ∂
∂p 1 ∂
∂v
1 ∂
∂v
( ρ uv ) +
(r ρ vv ) = − +
{ (μ ) +
(2r μ ) +
r ∂r
r ∂r
∂x
∂r Re ∂x
∂x
∂r
,
2μ v ∂ 2 ∂v v ∂u
∂
∂u
( μ ) − 2 − [ μ ( + + )]}
r
∂x
∂r
∂r 3 ∂r r ∂x
(3.19)
∂Y
∂Y
∂
1 ∂
1 ∂
1 ∂
( ρ uYi ) +
(r ρ vYi ) =
[ ( ρ Di i ) +
(r ρ Di i )] + Wi ,
∂x
r ∂r
Lei ∂x
∂x
r ∂r
∂r
(3.20)
85
∂
1 ∂
∂ λ ∂T
1 ∂
λ ∂T
( ρ uT ) +
(r ρ vT ) = [( )
]+
[r ( )
]+
∂x
r ∂r
∂x C p ∂x
r ∂r
C p ∂r
T
1
λ ∂C p ∂T ∂C p ∂T
+
[∑ ∫ C p ,i dT Wi + QW F ] + 2 [
]+ ,
C p i TD
C p ∂x ∂x
∂r ∂r
ρ C p ,i ∂Yi ∂T
∑[ C
i
[
p
Lei ∂x ∂x
+
(3.21)
∂Yi ∂T
1
G
∇ ⋅qr
]−
C p Bo
∂r ∂r
W i = − f i Daρ 2YF YO2 exp(− E / T ) .
(3.22)
The physical meaning of each of these dimensionless numbers is as follows: The
Damköhler number Da is the ratio of a characteristic flow time to a characteristic
chemical reaction time in one thermal length, The Lewis number Le represents the ratio
of energy to mass transport rates. It should be noted that Lewis number is assumed to be
constant but different for each species. The Prandtl number Pr compares the momentum
and the energy transport rates, The Reynolds number Re indicates the ratio of inertia to
viscous forces and it is based on the thermal length and the reference velocity. The Peclet
number Pe is the ratio of the bulk heat transfer to the conductive heat transfer which is
unity in this work, and the Grashof number Gr indicates the importance of buoyancy
force acting on the fluid. The Gr in the system is the consequence of choosing
Ur=UB+UD. If UD is negligible (as the case when g is large), Gr→1. Finally, Bo is called
the Boltzmann number and it is a measure of the relative importance of convective
energy flux to radiative energy flux. The radiative term is
G
∇ ⋅ q r = κ (4T 4 − G ) .
(3.23)
3.3 PROPERTY VALUES
In the analysis of this work, heat capacity, enthalpy, and heat conductivity are
functions of temperature. The values of the thermal properties for oxygen, nitrogen,
86
carbon dioxide, and water vapor are well known. The thermal and transport properties
used in the model are the same reported by Smooke and Giovangigli (1991), where they
have found that viscosity μ , λ / C p , and ρDi have a power law dependence on
temperature as follows:
μ ∝ T 0.7
(3.24)
λ / C p ∝ T 0.7
(3.25)
ρDi ∝ T 0.7 ,
i = F , O2 , CO2 , H 2 O, N 2
(3.26)
The values of the diffusion coefficients for each different species i are not needed
since the Lewis numbers are introduced in the non-dimensionalizing procedure. The
specific heat of the mixture is composition and temperature dependant. It can be
calculated from the relation:
C p = ∑ Yi C p ,i ,
i = F , O2 , CO2 , H 2 O, N 2 ,
(3.27)
i
where C p ,i is a function of temperature for each different species and can be represented
in a polynomial form. For the candle fuel vapor, the value of constant pressure specific
heat is taken from Middha and Wang (2002). The thermodynamic data of Cp’s were
estimated using Ritter and Bozzelli's group additivity as implemented in the THERM
code (Ritter and Bozzelli, 1991). The data are listed in Table 3.1. Gas properties values
used in this work are listed in Table 3.2 while the values of the non-dimensional
parameters are summarized in Table 3.3.
3.4 NUMERICAL PROCEDURE
The strongly coupled and highly nonlinear nature of the model equations derived
for the candle flames exclude the possibility of obtaining analytical solutions. Our main
87
task is to develop a suitable numerical procedure of solving this coupled system of
equations.
The system of governing equations and boundary conditions introduced in the
preceding sections has been solved numerically by Jiang (1995) and Kumar et al. (2003)
for two-dimensional flame spread problems over a planar solid fuel in rectangular
coordinates. Alsairafi (2004), solved for the candle flame problem using axisymmetric
formulation for both the combustion and the radiation parts.
For the combustion equations, a fully implicit control-volume based finite
difference method of Patankar (1980) is used. The method is described in details in many
CFD books (Patankar, 1980; Ferziger and Perić, 1996; Versteeg and Malalasekera, 1995).
The procedure for the calculation of the flow field is performed by SIMPLER algorithm,
which stands for Semi-Implicit Method for Pressure-Linked Equations Revised. The
SIMPLER scheme is adopted for the treatment of the pressure-velocity coupling.
Irregular geometries are modeled using a blocked-off region procedure.
3.4.1 Grid generation
The grid point location has to be chosen properly when solving the system of
partial differential equations in order to avoid instability and divergence. Hence, one has
to pay a great deal of attention to grid generation before seeking a numerical solution.
The main role of grid generation is the specification of the boundary point distribution as
well as the determination of the interior point distribution. Two dimensional grid
generations is considerably more complicated than the one-dimensional case. Accurate
and economic resolution of the solution requires that grid points be clustered in regions of
large gradients and be spread out in regions of small gradients.
88
Many general methods of grid generation exist. Algebraic method would be our
method of choice. In this method, algebraic equations are used to generate the algebraic
transformation. As an example, in geometric progressions, each spatial increment is a
fixed multiple of the previous spatial increment, i.e.
Δxi = ζ Δxi −1 ,
(3.28)
where ζ is the ratio of successive increments. This ratio of successive increments varies
with different candle dimensions. Several numerical experiments have been performed to
study the grid independence for the problem in this work. The minimum axial spatial
increment, Δx min , has been chosen so that the wick has 30 control volumes in the axial
direction, which is equivalent to the thermal length scale at 300ge. The minimum radial
spatial increment, Δrmin , on the other hand, has been selected so that the candle inert
shoulder (from point A to point B in figure 3.1) has 20 control volumes, which is
equivalent to the thermal length scale at 70ge. Alsairafi (2003) found that when these two
minimum axial and radial spatial were decreased by half, the solutions of dependent
variables, i.e. T, do not change significantly (ΔT∼8K which is less than 1% change). The
grid generation for the problem at hand is shown in figure 3.2.
3.4.2 Numerical implementation
The governing equations (3.17-3.21) can be re-written in the general form for a
dependent variable φ :
∂
∂φ
1 ∂
∂φ
( ρ u φ − Γφ
)+
( r ρ v φ − r Γφ
) = Sφ .
∂x
∂x
r ∂r
∂r
(3.29)
This equation is often called the transport equation for property φ. From this
equation, one may easily highlight the different transport processes: the convective and
89
diffusive terms, and the source term. The expressions for Γφ and S φ for its corresponding
dependent variable φ are given in Table 3.4.
The next step is to integrate the transport equation over a control volume and then
to develop a suitable numerical method (an approximation) based on this integration
scheme. These approximation techniques are needed to obtain the discretized equations.
The differential equations have been discretized using the conventional finite volume
differencing techniques for non-uniform mesh spacing to the physical domain. The
power-law scheme of Patankar (1981) has been used to approximate the convectiondiffusion fluxes. It is found that this scheme represents the exponential behavior very
well. In addition, this scheme behaves likes the central differencing scheme at a low
(local) Peclet number and as the upwind scheme at a high (local) Peclet number.
The general equation for a variable φ is integrated over the control volume. The
control volumes for the x- and r- momentum equations are different from those discussed
for the scalar equations. The control volumes are called staggered control volumes and
these staggered grids are used only for the velocity components. The control volume for
the x-momentum is staggered in the x-direction only, and the r-momentum control
volume is staggered in the r-direction only. As a result, the velocities are stored between
the grid points as opposed to at the grid points like temperature and pressure.
3.5 GAS RADIATION MODEL
In the system of equations developed in previous section, radiative flux term
appears in the energy equation (3.21). To obtain these quantities, the radiation field needs
to be solved.
The governing equation for radiative heat transfer in a participating
90
medium is the radiative transfer equation (RTE). This equation is coupled to the energy
equation and the boundary conditions and needs to be solved simultaneously. In recent
years (T’ien et al., 2001), radiation has been identified as a key mechanism in
determining the flame behavior in microgravity.
This chapter describes mathematical and computational considerations pertaining
to radiative transfer processes and radiative transfer models in combustion systems. Our
approach is to present a detailed derivation of the tools of radiative transfer needed to
predict the radiative quantities (irradiation, intensity, and heating fluxes). We begin with
discussion of the intensity field. Then, the RTE for a general cylindrical geometry is
presented. The discrete ordinates method to solve the RTE is introduced next. Finally, a
general numerical method to solve the coupled RTE and energy equation is developed.
3.5.1 The Equation of Radiative Transfer
The fundamental quantity defining the radiative energy transfer in a medium is
the specific intensity of radiation. Specific intensity measures the flux of radiant energy
transported in a given direction per unit cross sectional area orthogonal to the beam per
unit time per unit solid angle per unit frequency (or wavelength).
G
G
Consider the light beam traveling in the direction Ω through the point r and
G
G
construct an infinitesimal element of surface area ds intersecting r and orthogonal to Ω ,
G
the radiant energy flux dE crossing ds in time dt in the solid angle dΩ in the frequency
G G
range [ν ,ν + dν ] is related to Iν (r , Ω) by
G
G G
dE = Iν (r , Ω) ds dt dΩ dν
(3.30)
91
The radiation field is, generally, a multi-dimensional quantity, depending upon
three coordinates in space, one in time, two in angle, and one in frequency. This makes it
very difficult to solve for exactly or numerically. Therefore, some assumptions and
approximations are introduced to simplify the solution procedure. The main assumptions
of the mathematical model are as follows:
•
The medium is assumed to be absorbing-emitting and non-scattering with mean
absorption coefficients that varying from location to location.
•
The radiative participating gases are CO2 and H2O.
•
Surfaces bounding the medium and the solid plate are assumed to be black.
•
Candle wax and wick surfaces are assumed to be radiatively gray and diffused.
We reduce the number of spatial dimensions from three to two by assuming the
symmetry which is azimuthally homogeneous and in which physical quantities may vary
G
only in the axial and radial dimensions. Thus we replace r by (r, x). The use of the mean
absorption coefficient omits the frequency dependence. With these assumptions, the
intensity is a function only of radial and axial positions, time, and of direction,
G
[i.e. I (t , r , x, Ω) ]. The angular direction of the radiation is specified in terms of the polar
G
angle θ and the azimuthal angle ψ, thus Ω = θeˆθ + ψeˆψ . The specific intensity is also
referred to as intensity.
The mechanism of radiation in absorbing, emitting, and scattering media can be
described mathematically by the radiative transfer equation (Siegel and Howell, 1992;
Modest, 1993). The RTE is an integro-differential equation in terms of the radiative
intensity. By solving the RTE and energy conservation equation simultaneously, the
92
temperature distribution as well as heat flux in both the medium and on the enclosure
surfaces can be obtained.
The key idea is to make an energy balance on the radiative energy, see figure 3.3,
which undergoes absorption, emission, and scattering during traveling an infinitesimal
G
length along a line of sight in the direction Ω .
G
G G
G
G G G G G
G G
G
G
G G G σ s (r )
1 ∂I (r , Ω)
+ (Ω ⋅ ∇ ) I ( r , Ω) = κ ( r ) I b ( r ) − β ( r ) I ( r , Ω ) +
Φ
(
Ω
∫G ′, Ω) I (r , Ω′)dΩ′
c
∂t
4π Ω′
(3.31)
where I is called the radiation intensity. Ib is the blackbody intensity at local temperature,
G
Φ is the scattering phase function, c is the speed of light, Ω is a unit vector specifying
G
the direction of scattering through a position vector r , t is time, and β is the extinction
coefficient and is defined as β = κ + σ s . In general, the absorption coefficient, κ,
scattering coefficient, σs, the scattering phase function,Φ, as well as the emissivity of
boundary surfaces, ε, are functions of both wavelength and temperature. The radiation
intensity field can be taken as time independent in most engineering problem since c is
very large so that
∂I
= 0 . The first and last terms on the right hand side of the above
∂t
equation are usually combined and called the source function vector, which describes the
local production of intensity. That is
G
G G G G G
G G
G
G σ s (r )
S ( r , Ω) = κ ( r ) I b ( r ) +
Φ
(
Ω
∫G ′, Ω) I (r , Ω′)dΩ′ .
4π Ω′
(3.32)
93
Although the medium is assumed to be non-scattering, the scattering term will
always be retained in the present work for the sake of completeness. Hence, the RTE for
a two-dimensional cylindrically axisymmetric medium takes the following compact form
G
G G
G G G
G G
(Ω ⋅ ∇ ) I ( r , Ω) + β ( r ) I ( r , Ω ) = S ( r , Ω) .
(3.33)
The meaning of the different terms is as follows: The first term on the left hand
G
side corresponds to the gradient of the radiation intensity in the specified direction Ω .
The second term on the left hand side represents attenuation due to absorption and
scattering of the radiation intensity as it propagates through the medium. The term on the
right hand side accounts for the radiation energy created by emission and the inscattering, which is an increase of radiation intensity along the direction of propagation
due to the scattering of intensity coming form other directions.
The equation of transfer must be solved subject to an appropriate boundary
conditions. With the use of the mean absorption coefficient, spectral effects will be
bypassed and the boundary condition for the RTE equation on the volume bounds
becomes:
G
γ (rw )
I (rw , Ω) = ε (rw ) I b (rw ) +
π
G G
G G
′
′)dΩ ′ ,
n
⋅
Ω
I
(
r
,
Ω
w
∫
G G
n ⋅Ω′< 0
G G
n ⋅Ω > 0 ,
(3.34)
G
G G
where n is the local outward surface normal. So in the above equation, n ⋅ Ω > 0
G G
indicates rays coming out from the wall and n ⋅ Ω < 0 is characteristic for rays which
striking the wall, to be absorped or reflected. ε is the total emmisivity on the surface
which is assumed to be radiatively diffusive.
94
The candle domain is considered as a right cylindrical shaped enclosure
containing an absorbing-emitting, and non-scattering medium. The word "diffuse"
signifies that emissivity and absorptivity do not depend on direction. In the present
calculations, the surface absorption coefficient κ is assumed to be equal to ε.
Surfaces bounding the medium, located at rmax and xmax, are assumed to be black.
It is worth to mention that angular approximations in cylindrical (curvilinear) geometries
yield more complicated terms than two-dimentional problems. Therfore, for the problem
under consideration, the discrete ordinates representations of the radiative transfer
equation take the following form
G
G
G
G
G
1 ∂
∂I (r , x, Ω)
[rI (r , x, Ω)] −
[ηI (r , x, Ω)] + ξ
+ β ( r , x ) I ( r , x, Ω ) = S ( r , x , Ω )
r ∂r
r ∂ψ
∂x
μ ∂
(3.35)
G
where μ,η, and ξ are the direction cosines of Ω and they defined by, as shown in figure
2.4,
μ = sin θ cosψ
η = sin θ sin ψ
ξ = cosθ
(3.36)
ψ is the azimuthal angle measured from the r direction. The definition of the source
function is given by
G
G G
G G
σ ( r , x)
′
′)dΩ′
S ( r , x, Ω ) = κ ( r , x ) I b ( r , x ) + s
Φ
(
Ω
,
Ω
)
I
(
r
,
x
,
Ω
∫G
4π Ω′
(3.37)
Equations (3.35) and (3.37) are the general equations for the axisymmetric
problems. These two equations will be discretised later for the numerical purposes. Many
heat transfer quantities can be computed after solving for the intensity distribution
throughout a medium. This includes the incident radiation on the wall, the forward and
95
backward radiation heat fluxes in the axial and radial directions, and the divergence of
heat flux term which is needed in the energy conservation equation to solve for the
temperature distribution. These quantities can be obtained from the following formula:
G (r , x) =
G
G
∫π I (r , x, Ω)dΩ ,
(3.38)
4
q rx + (r , x) =
G
G
G
G
G
G
G
G
∫ξ ξ ⋅ I (r , x, Ω)dΩ ,
(3.39)
>0
q rx − (r , x) =
∫ ξ ⋅ I (r , x, Ω)dΩ ,
(3.40)
ξ <0
q rr + (r , x) =
∫μ μ ⋅ I (r , x, Ω)dΩ ,
(3.41)
>0
q rr − (r , x) =
∫ μ ⋅ I (r , x, Ω)dΩ ,
(3.42)
μ <0
q rx = q rx + + q rx − ,
(3.43)
q rr = q rr + + q rr − ,
(3.44)
G
∇ ⋅ q r (r , x) = κ (r , x)[4σT 4 (r , x) − G (r , x)] .
(3.45)
3.5.2 Numerical Solution of Discrete Ordinates Method
The SN discrete ordinates method is based on a discrete representation of the
angular variation of the radiative intensity spanning the total solid angle of 4π steradians
as shown in figure 3.6. In order to facilitate the angular discretization, two indices are
associated with each direction: the first index, p, indicates the value of ξ associated with
G
the direction Ω and the second index, q, increases with the value of μ associated with ξ.
G
G
For each discrete direction Ω pq , the components of Ω pq along the μ, η, and ξ axes are
G
the direction cosines of Ω pq , i.e. μ pq , η pq , and ξ pq . Consequently, they must satisfy
( μ pq ) 2 + (η pq ) 2 + (ξ pq ) 2 = 1
(3.46)
96
G
On the surface area of a unit sphere, there is a point in the direction Ω pq with
associated surface area, w pq . The obvious requirement of assigning values of w pq is that
the weight sum to the total surface area of a unit sphere.
Carlson and Lathrop (1968) proposed a relationship for the total number of
directions to be used depending on the order of the discrete ordinate approximation. This
relationship has the following form, i.e. N represents the order of the discrete ordinates
approximation and d is the geometric dimensionality,
M =
2 d N ( N + 2)
.
8
(3.47)
As it can be seen, if the geometry of a problem is not three-dimensional, not all
the directions on the unit sphere are needed. The integrals over solid angle in the RTE
equation are evaluated using numerical quadrature, that is,
∫
4π
G G
G
f (Ω)dΩ = ∑ w pq f (Ω pq ) .
(3.48)
pq
G
After preselecting a set of M representative discrete directions Ω pq together with
the corresponding weights w pq from a quadrature scheme, the radiative transfer equation
as well as the corresponding boundary conditions can be written as a set of equations for
each direction as follows
μ pq ∂
r ∂r
(rI pq ) −
1 ∂
∂I pq
(η pq I pq ) + ξ pq
+ βI pq = S pq ,
r ∂ψ
∂x
I pq (rw ) = ε (rw ) I b (rw ) +
γ (rw )
G G
n ⋅ Ω ( pq )′ w ( pq )′ I ( pq )′ (rw ) ,
∑
π nG⋅ΩG <0
(3.49)
(3.50)
( pq ) ′
with the source term, containing the medium emission and medium in-scattering, is of the
form S pq = κI b +
σs
4π
∑w
( pq )′
( pq )′
Φ ( pq )′, pq I ( pq )′ . It should be noted that there is another
97
boundary condition is required to obtain a numerical solution. This is the symmetry
boundary condition along the axial axis and is given by
at r = 0 : I pq = I ( pq )′
(3.51)
The net radiative heat flux in the axial and radial directions are computed by the
following relations
G G
q rx (r , x) = ∫ ξ ⋅ I (r , x, Ω)dΩ ≅ ∑ ξ pq w pq I pq (r , x) ,
(3.52)
G G
q rr (r , x) = ∫ μ ⋅ I (r , x, Ω)dΩ ≅ ∑ μ pq w pq I pq (r , x) .
(3.53)
pq
4π
pq
4π
In a similar manner, the total incident radiative intensity and the divergence of the heat
flux are determined by
G (r , x) =
G
G
∫π I (r , x, Ω)dΩ ≅ ∑ w
pq
I pq (r , x) ,
(3.54)
pq
4
∇ ⋅ q r (r , x) = κ (r , x)[4T 4 (r , x) − G (r , x)] ≅ κ (r , x)[4T 4 (r , x) − ∑ w pq I pq (r , x)] .
(3.55)
pq
3.5.3 Discrete ordinates angular quadrature
Although the choice of quadrature scheme is arbitrary, it can affect the accuracy
of the SN method. To preserve physical symmetry, the quadrature sets and weights are
chosen carefully to be invariant under any 90o rotation. However, it is obvious that all
quadrature schemes satisfy the summation condition, i.e. the zeroth moment
G
∫π dΩ = ∑ w
4
pq
= 4π .
(3.56)
pq
Some researchers [Truelove (1987), Fiveland (1991)] suggested additional
requirements for the quadrature and weights which satisfies the first half- and full-range
moments, the diffusion-theory condition, and the key radiation moments, given by
98
G G
G
G G G
∫ n ⋅ Ω dΩ = ∫ n ⋅ ΩdΩ = ∑ w
G G
n ⋅Ω < 0
pq
G G
n ⋅Ω pq > 0
G G
n ⋅Ω > 0
G G
n ⋅ Ω pq = π ,
G pq G
G pq
pq
Ω
d
Ω
=
w
Ω
= 0,
∑
∫
4π
(3.57)
(3.58)
pq
G pq G pq G
G pq G pq 4πδ
pq
,
Ω
Ω
d
Ω
=
w
Ω
Ω =
∑
∫
3
pq
4π
(3.59)
with δ is the common unit tensor. Different sets of direction and weight can be found in
the literature. However, in this thesis the quadrature sets and weights proposed by
Lathrop and Carlson (1965) are chosen for S4 scheme.
3.5.4 Solution of discrete ordinate equations
The differential equation of the RTE is transformed into algebraic one by
performing spatial discretization. To do this, Eq. (3.49) is integrated over a control
volume with respect to dr and dx in a specific direction (p,q). Several researchers have
explained this integration procedure and only some critical features will be repeated here
for clarity (Fiveland, 1984, 1987, 1988; Truelove, 1987, 1988; Jamaluddin and Smith,
1988, 1992). Carlson and Lathrop (1968) proposed a procedure for calculating the
angular derivative term which maintains neutron conservation in the curved coordinates
and permits minimal directional coupling as follows
1 ∂
1
(η pq I pq ) = [α qp+1 / 2 I Ppq +1 / 2 − α qp−1 / 2 I Ppq −1 / 2 ] .
r ∂ψ
r
(3.60)
The two terms represent the intensity flow out and into the angular range under
consideration due to angular redistribution. The directions q ± 1 / 2 define the edge of the
angular range denoted by the weight w pq . The coefficients α qp±1 / 2 are determined using
the fact that for uniform and isotropic intensities flux (as the case for infinite medium),
99
G
then we have a case of divergence-less flow, i.e. ∇ ⋅ ΩI = 0 . After some algebra we
obtain (Carlson and Lathrop, 1968)
α qp+1 / 2 = α qp−1 / 2 + w pq μ pq .
(3.61)
This recursive relationship uniquely determines all the α qp+1 / 2 in terms of the selected
quadrature parameters if α 1p/ 2 is known. Fortunately, this can be obtained by imposing
the radiant intensity conservation condition, where the angular redistribution of the
intensity will be conserved if the integration of intensity over all directions is zero. This
condition can be met if α 1p/ 2 = 0 . The final result of the conservation relation for a control
volume takes the form
μ [A I
pq
pq
n n
−AI
pq
s s
] − ( An − As )[
α qp+1 / 2 I Ppq +1 / 2 − α qp−1 / 2 I Ppq −1 / 2
]
,
w pq
pq
pq
pq
pq
pq
+ ξ [ Ae I e − Aw I w ] + VP βI P = VP S P
(3.62)
where
S Ppq = κI b , P +
(σ s ) P
4π
∑w
( pq )′
Φ ( pq )′, pq I P( pq )′ ,
(3.63)
pq
I ppq , I ppq ±1 / 2 , S Ppq are the volume cell-center intensities and the source term along
directions (p,q), and (p,q±1/2), respectively. Ae, Aw, An, and As are the east, west, north,
and south surface areas of a control volume, respectively. I epq , I wpq , I npq , and I spq are the
average values of I ppq over the east, west, north, and south faces of control volumes,
respectively. Equation (3.49) is solved in each of the ordinate directions. An iterative
solution is necessary because the boundary conditions depend on radiation intensity.
100
Because the directional intensity is solved iteratively (see Lewis and Miller, 1984), a
sweeping scheme, which transfers the boundary information to the inside of the enclosure
quickly and accurately, is required to result in fast convergence. The space angle sweeps
depicted in the solution procedure for this work are shown in figure 3.6. Depending on
the sweep direction, some of these intensities, i.e. cell-edge intensities, can be assumed
known from either boundary conditions or calculations in adjoining control volumes.
Therefore, additional auxiliary relations are required to solve the discretised equation of
RTE. These relations, depending on the sweep direction, take the following form
1) For ξ pq > 0 and μ pq > 0 :
I ppq = f xpq I epq + (1 − f xpq ) I wpq = f r pq I npq + (1 − f r pq ) I spq = ωI Ppq +1 / 2 + (1 − ω ) I Ppq −1 / 2
(3.64)
2) For ξ pq < 0 and
:
I ppq = f xpq I wpq + (1 − f xpq ) I epq = f r pq I npq + (1 − f r pq ) I spq = ωI Ppq +1 / 2 + (1 − ω ) I Ppq −1 / 2
(3.65)
3) For ξ pq > 0 and
:
I ppq = f xpq I epq + (1 − f xpq ) I wpq = f r pq I spq + (1 − f r pq ) I npq = ωI Ppq +1 / 2 + (1 − ω ) I Ppq −1 / 2
(3.66)
4) For ξ pq < 0 and μ pq < 0 :
I ppq = f xpq I wpq + (1 − f xpq ) I epq = f r pq I spq + (1 − f r pq ) I npq = ωI Ppq +1 / 2 + (1 − ω ) I Ppq −1 / 2
(3.67)
where f xpq ,
, and
are called the weighting factors. Several weighting factors exist
in the literature (Chai et al., 1994; Liu et al., 1996). It is important to choose these
weighting factors based not only on the accuracy but also on the guarantee that
nonnegative intensities will be generated. In this work, the step-scheme is adopted for the
two-dimensional axisymmetric cylindrical coordinates problem while the positive scheme
101
is chosen for the two-dimensional planar coordinates problem. However, some other
available schemes will also be discussed in this section.
The simplest and more numerical stable, of these schemes is the step method
(Carlson and Lathrop, 1968) which can be obtained by setting the weighting factor equal
to unity. However, this step-scheme is believed to generate less accurate first order
results (Lewis and Miller, 1984). Another common scheme, a second order accurate
(Lewis and Miller, 1984), is the Diamond Difference Scheme (DDS). This is proposed by
the results of Carlson and Lathrop (1968) from setting
. Although
the diamond scheme is more accurate than the step scheme, it is reported in the literature
(Lathrop, 1969; Chai et al., 1994) that this scheme can be unstable and can easily produce
oscillatory (positive-negative) solutions that propagate throughout the spatial domain
irrespective of the number of control volumes used. In the positive scheme of Lathrop
(1969), the weighting factors are calculated as follows:
,
,
(3.68)
where
,
.
(3.69)
Although the positive scheme is written in the axisymmetric cylindrical coordinates, in
this thesis, it will only be used for a 2D planar problem.
1) For
and μ pq > 0 (sweep 4):
102
(3.70)
,
(3.71)
where
C ns =
(1 − f r pq ) An
+ As ,
f r pq
(3.72)
C ew =
(1 − f xpq ) Ae
+ Aw ,
f xpq
(3.73)
C
pq
p
( An − As ) α q +1 / 2 (1 − ω )
=
[
+ α qp−1 / 2 ] .
ω
w pq
2) For
(3.74)
and μ pq > 0 (sweep 3):
,
(3.75)
where
C ns =
C pq =
(1 − f r pq ) An
+ As ,
f r pq
(3.76)
,
(3.77)
p
( An − As ) α q +1 / 2 (1 − ω )
[
+ α qp−1 / 2 ] .
pq
ω
w
(3.78)
3) For
and μ pq < 0 (sweep 2):
103
,
(3.79)
where
C ew
,
(3.80)
(1 − f xpq ) Ae
=
+ Aw ,
f xpq
(3.81)
C pq =
p
( An − As ) α q +1 / 2 (1 − ω )
[
+ α qp−1 / 2 ] .
pq
ω
w
4) For
and
(3.82)
(sweep 1):
,
(3.83)
where
,
(3.84)
,
(3.85)
( A − A ) α q +1 / 2 (1 − ω )
= n pq s [
+ α qp−1 / 2 ] .
ω
w
p
C
pq
(3.86)
The above equations are used to solve for the angular intensity at every grid and
then various radiative quantities of interest can be easily recast. In this work, only S4
scheme will be adopted to simulate the gas phase radiation since it is believed that
accuracy will not be improved beyond S4 (Fiveland, 1982). The number of 24 directional
fluxes for S4 is shown in figure 3.7.
104
3.5.5 The Mean Absorption Coefficient
The mean absorption coefficient, κ, is both temperature and mixture composition
dependent, and hence, need to be computed locally in the gas phase. In dealing with
radiative transfer in absorbing, emitting gases, the appropriate choice of the mean
absorption coefficient can provide a reasonable estimate of radiative heat flux from the
flame.
The Planck mean absorption coefficient of the mixture, KP, is calculated as the
summation of the partial pressure weighted values of the component gases, KP,i. In this
problem, the two radiating species are CO2 and H2O. So
,
(3.87)
where Xi is the molar fraction of species i calculated at an ambient pressure of 1atm. The
values of KP for carbon dioxide and water vapor as a function of temperature can be
found in the literature (Tien, 1968). The least-squares fitting equations are given in Table
3.6.
It has been recognized, however, that the use of this Planck mean absorption
coefficient over-predicts the gas radiation heat flux when compared with the narrowband
result for a one-dimensional flame (Bedir and T’ien, 1997). The main reason of this overprediction is that the Planck mean absorption coefficient is exact only for the optically
thin medium; a case which is not true for a flame with reasonable thickness. In fact, the
non-gray gases CO2 and H2O do absorb radiation in their own bands (self-absorption).
This results in attenuation of the emitted energy from the participating gases which
cannot be ignored. In order to compute the total radiative flux more accurately, a
correction factor, C, less than unity, is multiplied in front of the Planck mean absorption
105
coefficient, i.e.
. This modification factor reflects the non-optically thin nature
of the flame and results in a better estimation of the radiative heat fluxes back to the
solid.
In Rhatigan et al. (1998), an empirical relation was derived to estimate the value
of C based on the optical path length in a quasi one-dimensional flame. In the present
work, a value of 0.4 is chosen for the correction factor. This is a reasonable for
microgravity flames. As it will be shown later, flame radiation is important only when
gravity is sufficiently reduced.
3.6 SOLUTION PROCEDURE
The system of coupled elliptic equations in the gas phase is solved numerically
based on the finite volume discretization. The coupling between the velocity and pressure
fields is handled using the SIMPLER algorithm. This algorithm is used to handle
implicitly the relationship between the flow rate and the node pressure and it is an
iterative solution method to deal with the nonlinear nature of equation sets. The choice of
110 x 90 grid points was used for a physical domain of 50 x 40 cm. The discretised
equations are set up at each of the interior points and modified for those points adjacent
to the boundaries to incorporate the boundary conditions. The resulting system of
discretization algebraic equations is solved iteratively by successive applications of the
Tri-Diagonal Matrix Algorithm (TDMA) over crossed horizontal and vertical directions
until a converged solution is obtained. In this procedure, we choose a grid line and
assume that the dependent variable φ of the neighboring lines are known from there latest
values and solve for the φ along the chosen line using TDMA and the discretization
equations for each dependent variable are solved. In summary, the equations of mass,
106
momentum, species, and energy are solved simultaneously for velocities, species, and the
temperature. From the solution, the mixture density, specific heat and other transport
properties are calculated using there appropriate relations introduced in the preceding
sections.
For the radiation routine, every intensity value on every cell-center and faces of
control volume were computed by stepping from control volume to the adjacent control
volume starting from the known boundary condition. The differential equation of the
Radiative Transfer Equation (RTE), which is needed for calculating the intensity, is
transformed into algebraic one by performing a spatial discretization. The procedure will
be explained in the coming sections. Once the intensity distribution is calculated, the
incident heat fluxes on a wall as well as the divergence of heat flux are determined from
their relations described in next section. The radiation equation is coupled with the flow
field equation at each iteration cycle.
107
Table 3.1: Correlating equations of specific heats for O2 , N 2 , CO2 , H 2 O , and fuel
C p ,i = (a + bT + cT 2 + dT 3 + eT 4 ) ⋅ f
(
cal
)
g .K
Where T is dimensional quantity (degree Kelvin)
Species
a
b·103
c·106
d·109
e·1012
f·102
Temperature
range
O2
3.62560
3.62195
-1.87822
0.736183
7.05545
-0.196552
-6.96351
0.0362016
2.15560
-0.00289456
6.20937
6.20937
300-1000
1000-5000
N2
3.67483
2.89632
-1.20815
1.51549
2.32401
-0.572353
-0.632176
0.0998074
-0.225773
-0.00652236
7.09643
7.09643
300-1000
1000-5000
CO2
2.40078
4.46080
8.73510
3.09817
-6.60709
-1.23926
2.00219
0.227413
00063274
-0.015526
4.51591
4.51591
300-1000
1000-5000
H 2O
4.07013
2.71676
-1.10845
2.94514
4.15212
-0.802243
-2.96374
0.102267
0.807021
-0.00484721
11.0389
11.0389
300-1000
1000-5000
Fuel
-0.01674
1.625832
-0.991139
0.2856428
-0.02920363
1×10-12
300-5000
108
Table 3.2: Gas phase property values
Symbol
Value
Units
Reference
Tb
Tm
Tr
Tf
620
323
1250
2200
K
K
K
K
Shu (1998)
this work
Ferkul (1993)
Ferkul (1993)
T∞
300
2.75×10-4
1.15×10-3
4.10×10-4
1.93×10-4
0.33
K
g cm-3
g cm-3
g cm-1 s-1
cal cm-1 s-1 K-1
cal g-1 K-1
Ferkul (1993)
Ferkul (1993)
Ferkul (1993)
Ferkul (1993)
Ferkul (1993)
Ferkul (1993)
Ru
P∞
E
Bg
2.13
1.987
1
3.0×104
1.58×1012
cm2s-1
cal gmol-1 K-1
atm
cal gmol-1
cm3 g-1 s-1
Ferkul (1993)
Ferkul (1993)
Ferkul (1993)
Shu (1998)
this work
Q
L
8910
296.12
cal g-1
cal g-1
this work
Shu (1998)
ρr
ρ∞
μr
λr
C p ,r
αr
109
Table 3.3: Nondimensional parameters
Symbol
Value
Parameter
Reference
Pr
Le F
0.7
2.50
ν r /α r
α r / DF ,r
Ferkul (1993)
Shu (1998)
LeO2
1.11
Smooke and Giovangigli (1991)
LeCO2
1.39
Le H 2O
0.83
Le N 2
1.00
α r / DO ,r
α r / DCO ,r
α r / DH O ,r
α r / DN ,r
Tr
E
Q
4.167
50.33
90
E /( Ru T )
Q /(C p ,rT∞ )
Ferkul (1993)
this work
this work
L
Da
3.0
L /(C p ,r T∞ )
this work
Variable
α r ρ r B g / U r2
Bo
Variable
ρ r C pU r /(σT∞3 )
ε
0.9
2
2
2
2
Tr / T∞
Smooke and Giovangigli (1991)
Smooke and Giovangigli (1991)
Smooke and Giovangigli (1991)
this work
110
Table 3.4: Non-dimensional governing differential equations
φ
Γφ
Sφ
1
0
0
μ
u
Re
μ
v
Re
−
ρ −ρ
∂p ∂ Γφ ∂u
∂v
1 ∂
1 ∂ 2 ∂ (r v )
+ (
)+
(r Γφ
)−
[ Γφ
] + Gr ( ∞
)
∂x ∂x 3 ∂x
∂x
∂r
r ∂r
r ∂x 3
ρ∞ − ρ f
−
2Γφ v ∂ 2
∂p 1 ∂
∂v
∂
∂u
∂v v ∂u
+
(r Γφ
) + ( Γφ
) − 2 − [ Γφ ( + + )]}
∂r r ∂r
∂r
∂x
∂r
∂r 3
∂r r ∂x
r
ρ Di
Yi
λ
T
W i
Lei
Cp
T
1
λ ∂C p ∂T ∂C p ∂T
[∑ ∫ C p ,i dT W i + q cW F ] + 2 [
]+
+
∂r ∂r
C p i TD
C p ∂x ∂x
ρ C p ,i ∂Yi ∂T
∑[ C
i
[
p
Lei ∂x ∂x
+
∂Yi ∂T
G
1
]−
∇ ⋅q r
∂r ∂r
C p Bo
111
Table 3.5: The S4 quadrature sets for axisymmetric cylindrical enclosures
( p, q )
μ
η
ξ
w
(1,1)
(1,2)
(2,1)
(2,2)
(2,3)
(2,4)
(3,1)
(3,2)
(3,3)
(3,4)
(4,1)
(4,2)
-0.295876
0.295876
-0.908248
-0.295876
0.295876
0.908248
-0.908248
-0.295876
0.295876
0.908248
-0.295876
0.295876
0.295876
0.295876
0.295876
0.908248
0.908248
0.295876
0.295876
0.908248
0.908248
0.295876
0.295876
0.295876
-0.908248
-0.908248
-0.295876
-0.295876
-0.295876
-0.295876
0.295876
0.295876
0.295876
0.295876
0.908248
0.908248
π/3
π/3
π/3
π/3
π/3
π/3
π/3
π/3
π/3
π/3
π/3
π/3
112
Table 3.6: Least-square fitting equations of Planck mean absorption coefficient for CO2
and H2O
K P = a + b ⋅T + c ⋅T 2
(cm-1 atm-1)
where T is non-dimensional temperature (non-dimensionalized by 300 K)
Gas medium
a
b
c
Temperature range
CO2
3.89300x10-1
3.54982 x10-1
8.57664 x10-1
4.22527 x10-1
7.46849 x10-2
-2.38483 x10-2
-8.60664 x10-2
-2.99044 x10-2
-1.79675 x10-2
4.51608 x10-3
T≤1.8333
1.8333<T≤3.7500
3.7500<T≤5.8333
T>5.8333
-1
-2
H2O
1.71400 x10-1
4.63180 x10-1
1.59961 x10-1
1.00078 x10-1
-2.04529 x10
-4.37823 x10-1
-2.10401 x10-2
2.46914 x10
3.34147 x10-3
1.18569 x10-3
T≤1.8333
1.8333<T≤3.7500
3.7500<T≤5.8333
T>5.8333
113
1, 2, or 3mm
x
D
Wick
r
5mm
B
A
x=r=0
20mm
Candle
Solid plate
C
Figure 3.1 Schematic of a candle
114
45
40
35
r (cm)
30
25
20
15
10
5
0
0
10
20
30
40
x (cm)
Figure 3.2: Variable grid structure for modeling candle flames for 1mm wick diameter
and 5mm candle diameter
115
Intensity scattered from
dV into direction Ω
dr
I(r+dr,Ω)
dV
Intensity emitted from dV
into direction Ω
I(r,Ω’)
Intensity absorbed by dV
Intensity scattered from direction Ω
into other directions
Figure 3.3: Schematic of radiation intensity transfer energy balance on arbitrary control
volume in a participating medium
116
x
ξ
Ω
η
θ
ψ
μ
r
Figure 3.4: Geometry and coordinate system for 2D axisymmetric cylindrical enclosure
117
ξ
(4,1)
(4,2)
(3,1) (3,2)
(3,3)
(3,4)
μ
(2,1) (2,2)
(2,3)
(1,1)
(1,2)
(2,4)
Figure 3.5: Projection of an S4 quadrature set on the (μ,ξ) plane using (p,q) numbering in
r-x geometry
118
x
ξ<0, μ>0
ξ<0, μ<0
ξ>0, μ>0
ξ>0, μ<0
Side wall
Center Line
Top wall
r
Bottom wall
Figure 3.6: Four types of space angle sweep direction for SN scheme
119
ξ
1
-1
-1
-1
-1
-1
η
μ
11
Figure 3.7: Solid angle discretization of the S4 quadrature
120
CHAPTER 4
RESULTS AND DISCUSSIONS
In a realistic candle, both the candle flame and the wick are coupled to each other.
The heat from the candle flame evaporates the candle wax, providing the driving force
for the liquid to rise up through capillary action. The wax evaporated from the surface
provides the fuel for the candle flame. In this way the fuel supplied by the wick and the
heat supplied by the candle flame are coupled together.
The aim of this chapter is to understand the role of heat and mass transfer taking
place inside the wick on the candle burning. The numerical model of heat and mass
transfer in the porous wick in chapter 2 is coupled to the gas-phase flame model in
chapter 3.
For wicks which are made of combustible materials (e.g. fibers), the
phenomenon of “self trimming” is modeled. Parametric studies are performed to study
the effect of gravity, wick permeability, wick diameter and ambient oxygen.
4.1 CANDLE FLAME COUPLED TO A POROUS WICK
In this section, numerical results from the coupling of two-phase axisymmetric
wick model with the gas phase model are presented and discussed. Steady state results at
1 atmospheric pressure are given and parametric studies are performed. The four
parameters varied in this study are the gravitational acceleration, absolute permeability of
the wick, the environment oxygen molar fraction concentration, and candle wick
diameter.
Before presenting the results, some of the basics of candle flame burning are
being explained to give a better idea of the results being presented. In our present model,
121
the candle shoulder and the wax pool are not modeled. Therefore, the present model is
equally applicable for oil lamp flames. The primary difference between the candle flame
and oil wick lamp flames is that in candle flames, the shoulder level drops and hence the
exposed wick length changes during burning. The shoulder level moves downwards due
to wax consumption as the burning proceeds. The wick is also self trimmed by the
candle flame as the wax pool level and the flame moves downwards. Self trimming is a
phenomenon which occurs if the wick is made up of combustible materials.
The
presence of self trimming phenomenon taking place in a candle wick make things
complicated. While performing experiments, care should be taken to account for the self
trimming phenomenon of the wick. Let us assume we take a candle with initial wick
length shorter than the self trimming length (self trimming length can be found from the
numerical simulations or from prior experimental observations). When this candle is
ignited, the wick will be able to supply sufficient liquid wax to the surface for
evaporation and so a steady flame is established for the given initial length. For the oil
wick lamp flames, the exposed wick length remains the same and so the flame does not
change once the steady state is achieved. In the case of a candle flame, the exposed
candle wick length continuously increases with time because of the consumption of wax.
As the exposed wick length increases, the burning rate increases and the flame also
increases in size. This phenomenon of increase in exposed wick length can be slow or
fast depending upon the burning rate and the candle shoulder diameter. The exposed
wick length will continue to increase till the wick is no longer able to supply wax to the
tip surface of the wick. Further increase in exposed wick length will cause a dry region
to be formed at the tip of the wick. The temperature of the dry region shoots (due to lack
122
of evaporative cooling provided by the liquid wax) up rapidly causing charring or
burnout of the dried portion of the wick. Therefore a steady length of exposed wick is
achieved. This phenomenon is termed as “self trimming”. When the wick reaches its
self-trimming length, the candle flame also reaches a steady state. When experiments are
performed, the candle may or may not reach the self trimmed limit, if (1) the time of
experiments is not sufficiently long, (2) the burning rate of the candle flame is low for the
given candle parameters and environment conditions, (3) the candle shoulder is large. So
the experimental results can vary anywhere from the candle flame established for the
given initial wick length chosen to the self trimmed candle flame.
With the above understanding, the results are presented in the following manner.
First, a detailed study of the flame structure and flow field inside the porous wick are
presented for (1) steady state flame for a chosen initial wick length, which is shorter than
self trimming length and (2) self trimmed candle flames. In the sections following this,
most of the results are shown for a self trimmed candle flame, except for few cases,
where it is difficult to obtain a self trimmed flame due to low heat input for the cases of
weaker wake flames.
4.1.1 Detailed Flame Structure at Normal Gravity and 21% O2.
Before proceeding with analyzing the various candle flames, the flame structure at
one gravity level is presented in detail. This will also give the reader an idea of what the
computational model is capable of predicting. The conditions chosen for this purpose is
the normal earth gravity flame with the molar oxygen mole fraction of 0.21, for the
reference candle. The thermal length in this case is 0.099cm which gives an estimate of
the flame standoff distance at the flame stabilization zone. The reference velocity is
123
21.6cm/s, which approximates of the flow velocity in the neighborhood of the flame base
region. The absolute permeability of the wick is 2.5 ×10−12 . The porosity of the wick is
0.55. A wick of initial length 4mm is chosen. The length is chosen such that it is shorter
than the self trimming length. First the quazi-steady flame stabilized for this initial wick
length is presented. After due course of time, the candle wick will eventually reach it self
trimming length. The steady state flame results for the self trimmed candle flame are
presented later. Because of the symmetry, in the figure only one half of the flame
profiles are presented.
4.1.1.1 Steady State Candle Flame (With Wick Shorter than Self
Trimming Length)
A candle wick of 4mm is chosen. The absolute permeability of the wick is
2.5 ×10−12 m2. The porosity of the wick is 0.55. The remaining porous parameters are
shown in Table 2.2. Initially, candle flame is stabilized on the wick surface and it
reaches a steady state for this wick length (the self trimming length for this case is
4.34mm which will be presented in the next section). The quasi-steady results are
presented in this section. Both the structure of the gas phase and the porous flow fields
are examined in detail.
First the structure and the flow fields inside the porous wick are presented. Most
of the plots are very similar to the plots obtained for the constant heat flux case presented
in chapter 2. So only the key highlighting points are mentioned. Figure 4.1 shows the
saturation and the temperature distribution inside the wick. The base of the wick is at the
melting point of the candle wax (323 K) and on the wick surface, where there is
evaporation; the temperature is at its boiling point (620 K). The contour s = 1 separates
124
the liquid and the two-phase regions. The minimum saturation of 0.57 is reached at the
cylindrical tip of the wick. Figure 4.1 (c) shows the enlarged distribution of temperature
inside the two-phase region. The variation of temperature in this region is very small.
This minor variation is sufficient to bring about vapor movement inside the wick.
The pressure distribution inside the wick is shown in Fig. 4.2. The capillary
pressure is due to saturation distribution inside the wick. The vapor pressure distribution
in the two-phase region is due to the presence of temperature gradient. The liquid and the
vapor movement are shown in Fig. 4.3. The liquid which comes up to the surface of the
wick is evaporated at the surface. Part of the evaporated liquid moves into the wick at the
surface locations as indicated in fig. 4.3 (b). The vapor velocity vectors in fig. 4.3 (b) are
magnified approximately 10 times compared to the liquid velocity vectors (in Fig. 4.3
(a)). The vapor mass flux into the porous wick varies along the cylindrical surface,
reaching a maximum value at near x = 2.4mm. Figure 4.3 (c) shows the enlarged vapor
motion in the two-phase region.
Figure 4.4 shows the net heat flux supplied by the candle flame to the wick along
the cylindrical surface. In the decoupled case analyzed in chapter 2, the heat flux was
specified constant all along the cylindrical surface. In the case of the wick coupled to the
candle flame, the heat flux is almost zero near the base of the wick and then suddenly
increases and remains approximately constant. At the tip of the wick, the heat flux
suddenly rises. This may be due to the two-dimensional corner effect. Except very near
to the base of the wick and at the tip of the wick, the heat flux is almost constant. So the
results are qualitatively similar to that of the constant heat flux case discussed in chapter
2.
125
Figure 4.5 shows the saturation and the temperature distribution along the
cylindrical surface. Very close to the base of the wick, there is a liquid region (s = 1).
The temperature in this region is much below the boiling point temperature of wax. In
the two-phase region, the temperature is at its boiling point along this surface (since the
surface is at 1 atmospheric pressure). Figure 4.6 shows the liquid mass flux that is
evaporated along the surface. The vapor mass flux indicates that the vapor is moving into
the wick along this surface. The net mass flux of wax entering the gas phase is also
shown in figure 4.6. Figure 4.7 shows that the liquid mass flux in the x direction is
continuously decreasing due to the evaporation at the surface.
Figure 4.8 shows the saturation and the temperature distribution along the axis of
the wick. Figure 4.9 shows the mass fluxes of liquid and vapor along the axis. They are
qualitatively similar to the profiles obtained for a constant heat flux case.
Figure 4.10 shows the non-dimensional isotherms in the gas phase (nondimensionalized by T∞=300K). The gas phase temperature tends to stay closer to the wick
of the candle, where the fuel comes out by evaporation. The location of the gas maximum
temperature in normal gravity occurs at x = 2.25cm and y = 0 , i.e. Tmax=2141K. The
region of large temperature gradients at the flame base is caused by the cold convective
flow that opposes the upstream diffusion of heat. In normal gravity, the maximum
temperature location stays in the flame plume and at a position between the wick and the
flame tip. In addition, there is a small quenching distance, which detaches the flame from
the candle wax, between the flame base and the candle wax shoulder (about 1mm).
The fuel reaction rate contours are given in figure 4.11. The maximum
consumption rate is always located at the side of the wick and very near the flame base.
126
This maximum reaction rate point occurs at a point where the temperature is high
enough, but not necessary the maximum, and there is a substantial overlap between the
fuel and the oxygen. In order to define the visible flame, and hence be able to compare it
with the experimental pictures, the contour of reaction rate equal to 5x10-5g cm-3 s-1 is
defined as the boundary of the visible flame. Such method of comparing the computed
reaction rates of the flame with the visible experiment flame has been used by Grayson et
al. (1994). The flame length H is defined as the difference between the locations of the
flame tip and flame base. The flame diameter D is defined as the twice largest
perpendicular distance from the line of symmetry to the reaction rate contour of the
visible flame. Figure 4.11 shows a flame length of 2.76 cm and a flame diameter of 0.74
cm. The burning rate is observed as 0.84 mg/s.
The variation of fuel and oxygen mass fractions with location in the leading edge
region is plotted in figure 4.12. Those contours overlap substantially upstream of the
flame base where the gas temperature is low, allowing the mixing between fuel and
oxygen to start the chemical reaction.
Figure 4.13 shows the local equivalence ratio contours, which is defined as the
ratio of fuel to oxygen mass fraction ratio divided by the stochiometric fuel to oxygen
mass fraction ratio. It gives quantitative indication whether a local fuel-oxidizer mixture
is rich, lean, or stochiometric. For fuel-rich mixture, this ratio is greater than unity while
it is less than unity for fuel-lean mixtures. The local fuel-rich region appears roughly at
x<1.9cm and r<0.25mm. Though the stochiometric mixing of fuel and its oxidizer
extends to the candle shoulder, there is little reaction in this region due to low local
temperature.
127
Figure 4.14 and figure 4.15 show the carbon dioxide mass fraction and water
vapor mass fraction, respectively. The distributions of these species are qualitatively
similar. The maximum values of these products of combustion occur in the reaction zone,
where they are generated. Downstream of the flame tip, these species are slowly diluted
by mixing and diffusion with the ambient air.
Figure 4.16 shows the flow field (velocity vectors and streamlines) and the
oxygen transport. Because of the symmetry the field quantities will be plotted only in one
half of the plane. The lower half of Fig. 4.16 contains the velocity vectors while the upper
half contains the streamlines and oxygen mass flux vectors. In addition, the reaction rate
contour of 5x10-5g cm3 s-1 is superimposed onto the figure. As the cold flow approaches
the reaction zone, it accelerates and deflects away from the wick. The reason of this
deflection is threefold: (i) by the presence of the candle, (ii) the effect of Stefan velocity,
and (iii) by the thermal expansion. Because of the viscous entrainment effects and the
cooling of the gas stream, the velocity profiles tend to become flatter as the flow moves
to the downstream. For Ψ ≤ 0, the streamlines refer to the flow originating from the wick
surface and the convection is due to the Stefan flow from the evaporated fuel. The
streamlines originating from the ambient are given by Ψ > 0. The point Ψ = 0 on the
wick surface is determined by the net gas-phase heat flux to the wick equal to zero. Near
the base of the wick, there is a small region where there is no evaporation from the
surface. This is due to the presence of pure liquid region inside the wick and the heat
supplied to the wick surface in this region is simply conducted into the wick. This will be
further explained in the section where porous flow field is presented. Comparing Fig.
4.13 with Fig. 4.16, one sees that this dividing streamline (Ψ = 0) is on the fuel-rich side
128
except very close to the flame base near the wick surface. Consequently, while fuel vapor
can convect into the reaction zone, the oxygen needs to diffuse into the reaction zone.
The oxygen mass flux vectors indicate they are nearly perpendicular to the reaction zone
near the flame base but in the downstream, they became nearly parallel to the streamlines
since the flow velocities accelerate to quite large values. Although not shown in the
figure, a small circulation zone is found in the corner of wick and candle shoulder.
The pressure distribution around the flame is shown in figure 4.17. Two regions
of pressure rise exist; one is due to the hot flame region and the other is due to the
presence of the candle. As the flow approaches the flame reaction zone, it is deflected
outward by the pressure rise and the influence of blowing velocity of the evaporated fuel
from the wick. The flow will then accelerates after the reaction zone because of the gas
expansion resulted from the heat release in the chemical reaction. The location of
maximum pressure always occurs in the wick surface because of the evaporation rate. As
we move downstream, the pressure decreases slowly (see Fig. 4.17).
A detailed plot of figure 4.18 shows temperature and species profiles as a function
of position at the centerline of symmetry between the wick-tip and far from the flame.
The temperature starts with the surface (boiling) temperature at x=0.4cm, increases to its
maximum value at the reaction zone (x=2.5cm) then decreases slowly toward the ambient
temperature as x increases. The fuel vapor mass fraction is largest at the wick-tip and
monotonically decreases to nearly zero at the flame where it is consumed. The oxygen
has the opposite behavior of fuel; being maximum at infinity and decreasing to zero at the
flame except in low gravity. The combustion products, i.e. CO2 and H2O, have their
maximum values in the reaction zone, where they are generated.
129
The radiative structures of candle flames are shown from figures 4.19 to 4.22.
Figure 4.19 shows the distribution of the effective mean absorption coefficients of the
mixture. Clearly, this absorption coefficient is both mixture composition (CO2 and H2O)
and temperature dependent. The dimensional net radiative flux vectors are given in figure
4.20. This reveals the multi-dimensional nature of radiation heat transfer. The direction
and magnitude of radiative flux vectors depends not only on local but also on the
surrounding properties. The source of radiation comes from two things: the hot flame and
the hot surfaces. Since the wick, at liquid fuel boiling point, is a strong emitter (ε=0.9),
the surface radiation emission is greater than the radiative flux coming from the flame to
the wick. This results in the net radiative flux vectors pointing outward on the wick
surface as shown in figure 4.20. Along the centerline (r=0cm), the net flux reverses
directions (i.e. toward the wick) in 0.5<x<1.3cm.
The dimensional divergence of heat flux, where the energy equation in the gas
phase is coupled with the radiation transfer equation, is shown in figure 4.21. The value
G
of ∇ ⋅ q r varies from positive (indicating a net radiative loss from the medium) to
negative (net radiative gain). Finally, the different heat flux contributions to the side of
the wick are presented in figure 4.22 for candle flames at normal gravity level. The
positive values suggest heat flux gain into the wick, while negative values indicate heat
flux loss from the wick. The gas phase conductive heat flux is qc and the net gas-radiative
feed back is qr, net, which is the summation of (qr)in and (qr)out. The outward radiative heat
flux consists of wick emission and reflection of unabsorbed incoming radiative heat flux.
In the flame quench region (x≤1mm), the magnitude of qc continuously rises. It reaches a
plateau value in 1.5mm≤x≤4.5mm and then rises sharply near the top corner of the wick
130
where the flame feedback is two-dimensional. Clearly, at normal gravity, the dominant
mode of heat transfer at the wick is by conduction. The flame radiation feedback (qr)in is
negligible. The surface radiative loss (qr)e is finite but it is still relatively small compared
with conduction.
4.1.1.2 Self Trimmed Candle Flame at Normal Gravity
When the candle is allowed to burn for a long time, the exposed wick length will
increase gradually due to the receding of the candle shoulder level. As the exposed wick
length increases, the saturation level at the tip of the wick decreases to increase the
capillary action. As this continues, a stage is reached when the saturation level at the tip
of the wick becomes zero and the candle wick is no longer able to supply the liquid to the
tip of the wick.
Then a dry vapor region is formed.
This dry vapor region is
characterized by a drastic increase in temperature of the dry region. If the wick material
is combustible, then the dry region will begin to char and burnout. Thereafter the
exposed length of the wick will remain constant. This exposed wick length is referred as
the self trimming length. In this work, it is assumed that once the wick becomes dry, the
trimming length is reached. In other words, the dried burning portion of the wick has a
length much smaller than the rest of the wick. With this approximation, the following
section shows the detailed structure of the flame and the wick for a self trimmed candle
flame. Candle is burned in normal gravity for the same parameters chosen for the
previous case.
Numerically the “self trimmed length” is approximated as the minimum wick
length for which the cylindrical corner of the wick becomes dried up. In principle, the
whole tip surface of the wick should be dried up, but this would mean that the tip surface
131
will be a curved surface rather than a flat surface. To reduce the complexity of the
problem, the formation of curved tip surface of the wick is neglected. Instead the self
trimmed length is taken as the appearance of dry region on the cylindrical corner of the
wick. The numerical procedure adopted to obtain a self trimmed candle flame is
explained as follows. A shorter wick is initially chosen and the wick and gas phase are
solved approximately during each overall iteration. The saturation at the corner of the
wick is checked at the end of each overall iteration and if the corner is not completely
dried up (i.e. s=0), the wick length is slightly increased. This update of wick length is
done till the corner becomes completely dried up. Then the wick length remains constant
and the iterative procedure is continued till its reaches overall convergence in the gasphase and the wick phase is reached.
The detailed porous structure and the flow field are almost similar to the previous
case. Only the key points are highlighted. At normal gravity, the self trimming length is
found to be 4.34mm. Figure 4.23 (a) shows the saturation profiles for a self trimmed
wick. The saturation at the corner of the tip of the wick surface reaches zero. Since the
saturation contours are not flat, the trimming of the wick will change the shape of the
wick tip. But this is being neglected in this work. The changes that will result from the
change of the shape of the wick tip will be very small and hence reasonable to neglect.
Figure 4.23 (b) shows the non-dimensional temperature contours inside the wick. Figure
4.24 shows the non-dimensional pressure distribution inside the wick. Figure 4.25 shows
the liquid and the vapor mass fluxes inside the wick. As observed in the previous case,
the liquid and the vapor move in countercurrent fashion inside the two-phase region of
the wick. The liquid is evaporated from all along the cylindrical surface and the tip of the
132
wick. Part of the vapor, which is evaporated at the surface, traverses into the wick. Near
the tip of the wick, the vapor movement is almost negligible.
Figure 4.26 shows the net heat flux supplied by the flame to the wick for
evaporation. The heat flux distribution is qualitatively similar to that obtained for the
previous case except that the heat flux suddenly drops near the tip of the wick. This is
due to the high temperature of the wick at this location formed due to the presence of dry
region at the tip. Figure 4.27 shows the saturation and non-dimensional temperature
profiles along the cylindrical surface of the wick which is exposed to the candle flame.
The figure shows that the saturation reaches zero at the tip of the wick.
Near the base of
the wick, there is small region which is in the liquid region. Here no evaporation takes
place at the surface and all the heat supplied at this region of the wick surface is simply
conducted into the wick. The temperature in this region is well below the boiling
temperature of the wax. There is also a temperature gradient at the base of the wick
which indicates some heat lost to the wax pool. Figure 4.28 shows the liquid mass flux
and the vapor mass flux in the r direction along the cylindrical surface of the wick. There
is a sudden drop in the liquid mass flux near the tip of the wick. This is due to the sudden
drop in the heat flux supplied to the tip of the wick as shown in figure 4.26. The negative
vapor flux indicates that the vapor is transported into the wick. Figure 4.29 shows the
liquid mass flux in the x direction along the surface. The mass flux continuously reduces
due to the evaporation along the surface of the wick. Figure 4.30 shows the saturation
and non-dimensional temperature profiles along the axis of the wick. It is observed that
the saturation at the tip along the axis line is 0.08 which is close to zero but not exactly
zero. This is because of the assumption that the trimming of the wick is flat, whereas the
133
saturation contours are not flat. The exact trimming shape of the wick is not modeled.
Figure 4.31 shows the liquid and the vapor mass flux along the axis of the wick.
Figures 4.32-4.44 shows the detailed profiles of the gas phase flame
characteristics. The plots are qualitatively similar to that of the plots obtained for the
steady flame established for shorter wick, except for some minor quantitative differences.
The quantitative difference is attributed to the increase in exposed wick length to 4.36cm.
4.1.2 Effect of Gravity
Gravity affects both the gas phase and the porous wick. In chapter 2, the effect of
gravity on the heat and mass transport inside the wick is isolated and analyzed. It was
found that for the wick parameters chosen in this study, gravity does not have a
significant effect on the wick transport. Gravity does affect the gas phase, through the
buoyancy term. Since flow fields in a candle flame are induced by buoyancy, gravity is
found to significantly affect the gas phase and hence the flame structure. In this work,
the gravitational acceleration has been widely varied as a parameter from zero to high
gravity.
Figure 4.45 shows the computed visible flames at different gravity levels for a self
trimmed candle flame in standard air conditions (1 atm and 21% O2). In this series of
study, the candle and wick dimensions are: wick diameter = 1mm, candle body height =
2cm, and candle diameter = 5mm. The wick length will be that of the self trimming
length which is determined as a part of the solution. Note that there is no self trimming
observed for gravity levels greater than 5.5ge. So the wick length for gravity levels above
this is taken as the self trimmed length of the candle wick at 5.5ge which is 3.8 mm. The
134
reaction rate contour of wf =5x10-5g cm-3 s-1 is chosen to represent the boundary of the
visible flame, which is a function of the local fuel, oxygen and temperature.
Results reveal that the flame becomes longer as gravity increases, reaching its
maximum value roughly at g=3ge and then decreases. The reason of this non-monotonic
variation is explained as follows: in reduced gravity, the flame moves away from the
wick, as it tries to move closer to the region of fresh oxidizer. Consequently, the heat flux
to the wick diminishes and thus the evaporation rate decreases and this makes the flame
shorter. The wider flame-standoff implies a weaker heat feedback from the flame and a
smaller burning rate. There is also a slight reduction in the flame thickness as gravity
increases. The flame standoff distance decreases with increasing gravity level. At higher
gravity levels, the fresh oxygen is pushed closer to the wick by the strong convection,
increasing the heat flux to the wick, leading to an increase in the rate of evaporation. This
makes the flame longer. The increase in flame length is also in part due to the decreased
flame standoff distance. However, at higher gravity level than g=3ge, the flame reduced
in both length and width because of the decrease in burning rate.
Figure 4.45 also shows that there is a sudden retreat of the flame base position
from at the wick base at 3ge to the wick tip at 6ge. This is also observed by Alsairafi
(2003). Hence, the burning rate decreases. At more than 10ge, the flame base is located
downstream of the flat wick top surface. The solutions for the cases for 10ge and above
show that in these cases, the fuel vapors that support the flame come entirely form the
wick top surface.
Figures 4.46-4.47 shows the effect of gravity on the total burning rate and the self
trimming length of the candle. Figure 4.46 shows that the burning rate first increases
135
with gravity till 2ge. The reason for increase in the burning rate is explained before. The
self trimmed length is maximum for 0ge. The reason for this being that the heat flux
supplied by the flame at 0ge is minimum. With increase in gravity, the self trimmed
length of the candle decreases. As the gravity is further increased, the burning rates starts
gradually decreasing till the gravity level reaches 4ge. The burning rate is a function of
the heat supplied to the wick by the flame and the length of the exposed wick. As the
heat flux increases, the self trimmed length of the wick decreases as shown in the figure.
In the region of 2ge to 4 ge, the decrease in burning rate due to decrease in self trimmed
length of the wick dominates compared to the increase in burning rate due to increased
heat flux supplied by the flame. So there is a slight decrease in burning rate. The results
of Alsairafi (2003) indicate that the burning rate increases in this region, since the wick
length chosen in their calculations remained constant. Then the burning rate drops
rapidly. The reason for this sudden drop in burning rate is due to the retreat of flame base
towards the wake tip to form a wake flame. In the wake region, the flame is very short
and the heat flux is very low. So there is no self trimming action that would be observed
at this stage. The wick length is chosen as the self trimmed length observed at 5.5ge.
The burning rate continues to drop with further increase in gravity.
It has to be noted that although the model predicts wakes flames at very high
gravity levels, the existence of the flame at this levels is questionable. The present model
does not account for melting of wax in the wax pool. For wake flames, the heat feed back
from the flame to the wax pool will be quite low. The rate of heat supplied either directly
by the flame or through heat loss at the base of the wick to the wax pool may not be
sufficient to supply a steady supply of liquid wax for sustaining the candle flame. There
136
is a possibility of the flame being extinguished due to insufficient supply of liquid wax
from the wax pool. Hence the existence of wake flames is questionable. The experiments
also indicate extinction of candle flames at higher gravity levels.
4.1.3 Effect of Wick Permeability
Wick permeability is a measure of ability of the porous wick to transport fluid
through it. Wicks with high permeability, offers less resistance to fluid motion. In
general, wick permeability is a function of the wick porosity, the structure of the wick
and the wick material. As shown in section 2.3.5.2, the wick permeability affects the
saturation distribution inside the wick for a given heat flux distribution. This would
affect the self trimming length of the candle wick and hence the flame structure and the
burning rate.
Figure 4.48 shows the effect of wick permeability on the self trimming length of
the candle wick. It is observed that as the permeability is decreased, the level to which
the wax rises above the candle shoulder decreases due to increased resistance to fluid
motion. Therefore the self trimming length of the wick is reduced. Figure 4.49 shows
the variation of burning rate of the candle flame with permeability of the wick. The
burning rate decreases with decrease in permeability. This is caused directly by the
decrease in self trimming length of the candle wick. The exposed length is reduced and
hence the burning rate decreases. Figure 4.50 shows the variation of flame lengths with
the wick permeability. The flame lengths are proportional to the exposed wick lengths
and hence the trends are similar to that of the self trimming length variations shown in
Figure 4.48. Figure 4.51 shows the variation of maximum flame temperature in the gas
137
phase with the permeability of the wick. There is a slight decrease in the flame
temperature with increase in permeability of the wick.
In brief, we can conclude that the wick permeability significantly affects the self
trimmed wick length, the flame structure and the burning rate of the candle flame.
Therefore while conducting experiments on candle flames; the wick permeability should
also be taken into account. The previous modeling efforts by Alsairafi (2003) does not
capture this.
4.1.4 Effect of Wick Diameter
The goal of using different wick and candle sizes is to study the flame size and
extinction. Computed results in several low and normal gravity levels using three
different wick diameters are summarized in Table 4.1. The computed results of Alsairafi
(2003) are shown in brackets for the sake of comparison with the present computed
results. The data shows that thicker wick produces a longer (larger H) and wider (larger
D) flame. The total burning rate increases with wick diameter (see figure 4.53). The
increase in burning rate is affected both by the increase of surface area of the wick and
also by the increase in self trimming length of the wick (see figure 4.52). At a given
gravity level, the maximum flame temperature decreases slightly with increasing wick
diameter (see figure 4.54). Qualitatively similar results were obtained by Alsairafi (2003).
Previously, it was also found that a longer wick would also produce a larger flame with
lower flame temperature (Dietrich et al., 2000).
We also note from Table 4.1 that in 0ge only 1mm wick is flammable; in 10-4ge
1mm and 2mm wicks are flammable but not 3mm wick. Above 10-2ge, all the four
different cases are flammable. Similar variation on flame size has been predicted by
138
Alsairafi (2003) and observed experimentally in downward flame spreading over paper
cylinders (Essenhigh and Mescher, 1993). It can be concluded that the wick diameter
affects flame size and flame temperature as well as flammability.
4.1.5 Effect of Ambient Oxygen
The effect of different ambient oxygen concentrations on flame structure and
burning rate characteristics are investigated in this section. The dependence of flame
temperature on ambient oxygen concentrations is given in figure 4.55. Computational
results show that the flame temperature increases almost linearly with oxygen
concentration. The self trimming length for different oxygen molar fraction is shown in
figure 4.56. The self trimming length decreases with increase in concentration. This is
due to the high heat flux supplied by the flame at higher oxygen concentrations. The
burning rates are shown in figure 4.57. The burning rate slightly increases with oxygen
concentration. The increased heat flux increases the burning rate but the decreases self
trimmed length decreases the burning rate. Hence the overall increase in burning rate at
higher oxygen concentrations is reduced. The fuel reaction rate contours as a function of
oxygen concentration is given in figure 4.58. As oxygen percentage increases, the flame
base moves upstream and closer to the candle surface suggesting a stronger flame. The
flame size decreases with increasing oxygen concentrations. This is affected by the
decrease in self trimming length.
4.1.6 Validation of Results
In this section, the candle flame results are compared with pervious experimental
and numerical work (Alsairafi, 2003). Note that none of the experiments report whether
the candles used in the experiments are self trimmed or not. The exposed wick lengths of
139
the candle are also not reported. Alsairafi (2003) neglected the heat and mass transport
inside the porous candle wick and hence there is no concept of self trimming of the
candle wick. In their work, a wick length of 5mm is chosen for all their calculations. The
characteristics of the porous wick, like absolute permeability, porosity etc., are not
reported in the experiments. The previous section 4.1.3 shows that permeability of the
wick significantly affects the candle flame structure. In the absence of the above
experimental parameters, the comparison between the various experiments and the
computed results is not fully justified. However, an attempt has been made to
qualitatively validate the results.
Ross et al. (1991) performed several experiments in 5.2s NASA Lewis Zero
Gravity Facility with 4.75mm diameter candles. They changed oxygen mole fraction
(0.19-0.25), inert gas, and ignition methods (in either microgravity or normal gravity just
prior to the drop). In all the tests, the microgravity time was too short to determine steady
state and near-extinction behavior. However, the flame size was nearly steady. The
author reported a sequence of pictures for a candle flames. Therefore, only a comparison
of the overall flame dimension is possible. The experimental pictures show a nearly
hemispherical candle flames with a large standoff distance.
The comparison between the experimental results and the computed results is
done at 19 and 21% O2 with the standard candle dimensions (1mm wick diameter, 5mm
candle diameter, 2cm candle length). The computed fuel consumption rate contours in
figure 4.58 are compared to the reported experimental flame shape. The predicted flame
shape closely resembles the visible flame in the experiments, except for a slight inward
hook at the flame base, if the fuel vapor reaction rate value is 5x10-5g cm-3 s-1 as the
140
visible flame boundary. This comparison shows the current model qualitatively produces
the correct visible flame shape, even though detailed flame chemistry has not been
considered. Qualitatively similar results are obtained by Alsairafi (2003). Quantitatively,
the numerical values of flame length and the flame diameter differ.
A number of experiments with various candle dimensions were performed in the
Shuttle (in air) and in the Mir space station (in enriched air with 23-25% oxygen
concentration) (Dietrich et al., 1994; Dietrich et al., 2000). Three different candle wick
diameters were used in Mir experiments. The present computational work is conducted
for three different diameters that match the same dimensions in the Mir experiments. The
comparison between the experiments and the numerical calculations are based on the
flame width D, height H, H/D ratio and the burning rates. The comparisons between the
experimental and computational results are based on the flame dimensions, size and
burning rate, and are tabulated in Table 4.2. Similar to the results found in the drop
towers experiments, shapes of the candle flames were spherical with a weaker flame than
the flame in normal gravity. The visible reaction zone is much further away from the
wick, implying a smaller heat flux from the flame to the wick. The experimental and
analytical data agree quite well. The computational values for flame width D and the
flame length H (based on the outside edge of the computed reaction zone) are 12.84mm
and 9.212mm, respectively. These values are very close to the quasi-steady experimental
values, (D falls in the range 9-15mm and H falls in the range 8-14mm for the small wick
size of 1mm). The computed value of H/D is 0.717 while the experimental values fall in
the range 0.84-1.02. The computed burning rate is 0.1348mg/s while the experimental
141
values fall in the range of 0.2-0.6mg/s, and the computed result of Alsairafi (2003) is
0.1663 mg/s.
It should be noted that when the wick diameter is increased to 2mm in zero
gravity, the flame becomes extinct. No flammable flame was able to be obtained
numerically, except for the smallest wick diameter of 1mm. Experimental results show
that the wicks with diameters of 1, 2 and 3mm are all flammable. This discrepancy can be
improved if the empirical one-step chemical kinetic constants are better calibrated. This
is a work for the future.
For the effects of high gravity levels, comparisons between the experimental and
computational results are based on the flame dimensions, which are tabulated in Table
4.3 and blow-off limit. Villermaux and Durox (1992) performed an experiment on a
candle of 2.1cm diameter and 1.5mm wick diameter in a centrifuge. The exposed wick
length was not mentioned in their work. They reported that 7ge was the blow-off limit.
The numerical model predicts a blow-off limit of 30ge. This discrepancy is because in the
modeling of the porous wick. In reality, while self trimming is taking place, the tip of the
wick becomes charred and soot collect on the tip of the wick. This blocks the tip of the
wick from supplying wax from this surface. A numerical experiment has been performed
in which the tip of the wick is artificially blocked. Then the flame is observed to be
blown off at 6ge. In normal gravity, the flame length reaches about 2.803cm (for wick
diameter 1.0mm) that compares quite very well with the numerical results of 4.1cm (for
wick diameter of 1.5mm). Increasing the gravity level caused the candle flame height to
decrease in length. Numerical results predict an increase in flame length up to 3ge and
then the flame length decreases with increase of gravity level.
142
Another high gravity experiments were performed by Arai and Amagai (1993).
The candle has the following dimensions: 0.8mm wick diameter, 12mm candle diameter,
and 4.8mm exposed wick length. They also reported that as the gravity increases, both the
flame length and width decreases. The comparison between these experimental results
and the computed results are shown in figure 4.59. The computed results of Alsairafi
(2003) are also shown in figure 4.59 for comparison. The flame width matches with the
experimental results quite well.
The flame length variation does not even agree
qualitatively with the experimental results. The variation in the numerical results is non
monotonic, whereas the experimental results shows a monotonic decrease in the flame
length. This discrepancy can be attributed to many reasons. First the present numerical
model used a second order single step overall kinetic for wax combustion. This is
affecting the flame anchor location. It is observed that the flame is anchored only at the
wick base or in the wake region. The experiments indicate that the flame anchor location
gradually moves upwards with increase in gravity. This can possibly be predicted by the
present model by using detailed kinetics. The blow-off limit of their candle experiment is
13.5ge which is about doubled the value found by Villermaux and Durox (1992).
Compared with the computed results of Alsairafi (2003), the present computations show
a closer match with the experimental flame length values.
One discrepancy between the experimental results and the current numerical
prediction is the effect of gravity on flame length. Experimental results show that the
flame length decreases as gravity increases from 1ge and up while numerical results show
that the flame length increases from 1ge to 3ge. The experiments report that the flame
anchor location moves upward gradually with increase in gravity. The numerical results
143
show that the flame is always stabilized near the base of the wick for gravity levels 1ge to
5ge. At still higher gravity, the flame drastically moves upwards and forms a wake flame.
This discrepancy between the experiments and the model might be due to the assumption
of second order single step overall kinetics for candle burning. When the flame is
anchored at the base of the wick, increase in gravity increases the convective flow,
causing the flame to move closer to the wick. This will increase the heat flux supplied to
the candle wick. Increased heat flux will increase the evaporation rate per unit length but
also decrease the self trimming length. So the total burning rate is the net effect of
increase in evaporation per unit length and the decreases in the exposed wick length. So
there is non monotonic behavior for the burning rate and the flame length in the range of
1ge to 5ge. The anchoring of the candle flame near the base of the wick explains the
discrepancy between the experiments and the numerical results.
In addition, the
dimensions for the three different results do not match precisely (i.e. wick diameter,
candle diameter, and exposed wick length) which could be an important reason for such a
discrepancy.
144
Table 4.1(a): Effect of wick diameter on candle flame characteristics (5mm candle
diameter and 21% O2)
Dwick
(mm)
g/ge
STL
(cm)
D (cm)
H (cm)
(H/D)flame
mt
(mg/s)
Tmax (K)
1
2
3
1
1
1
0.436
0.771
0.912
0.7413
0.9388
0.9715
2.8034
5.2727
6.8402
3.7817
5.6164
7.041
0.9222
2.147
4.0079
2138
2130
2120.4
1
2
3
1x10-2
1x10-2
1x10-2
0.5825
1.141
1.645
1.4931
2.0367
2.304
2.2623
3.604
4.24
1.5152
1.7695
1.8402
0.7027
1.592
2.308
1769
1731.6
1703.2
1
2
3
1x10-4
1x10-4
1x10-4
0.92
1.583
Ext.
1.962
2.304
Ext.
1.1887
1.4576
Ext.
0.605
0.632
Ext.
0.34036
0.7040
Ext.
1186.0
1163.5
Ext..
1
2
3
0
0
0
1.01
1.2845
Ext.
Ext.
0.9212
Ext.
Ext.
0.717
Ext.
Ext.
0.1348
Ext.
Ext.
1110.5
Ext.
Ext.
Table 4.1(b): Effect of wick diameter (Alsairafi, 2003) on candle flame characteristics
(5mm wick length, 5mm candle diameter and 21% O2)
Dwick
(mm)
g/ge
STL
(cm)
D (cm)
H (cm)
(H/D)flame
mt
(mg/s)
Tmax (K)
1
2
3
1
1
1
-
0.77794
0.8565
0.9715
3.2627
4.8356
5.6702
4.1863
5.6457
5.8364
1.09347
1.6131
2.0073
2142
2130
2120.4
1
2
3
1x10-2
1x10-2
1x10-2
-
1.3833
1.5953
1.8154
2.1577
2.9164
3.6891
1.56
1.8281
2.0322
0.61362
0.846
1.133
1777
1760
1743
1
2
3
1x10-4
1x10-4
1x10-4
-
1.751
1.8554
1.9603
1.2424
1.3136
1.5635
0.7096
0.708
0.7976
0.24
0.264
0.374
1216
1166
1176
1
2
3
0
0
0
-
1.4931
Ext.
Ext.
0.9450
Ext.
Ext.
0.6329
Ext.
Ext.
0.15855
Ext.
Ext.
1130.5
Ext.
Ext.
145
Table 4.2: Comparison with candle flame experiments in microgravity
([] = results of Alsairafi (2003))
1
Mir
Numerical
Numerical
dwick=2mm
dwick=1,2 or
dwick=1mm
3mm
Lwick=self
Lwick=self
trimmed length trimmed length
Lwick=3mm
Numerical
dwick=3mm
Lwick=self
trimmed length
g=0ge
D (mm)
H (mm)
Burning rate (mg/s)
8.8-15
8.4-13.5
0.2-0.6
12.845[14.931]
9.212[9.45]
0.1348 [0.1663]
Ext.[ Ext.]
Ext. [ Ext.]
Ext. [ Ext.]
Ext. [ Ext.]
Ext. [ Ext.]
Ext. [ Ext.]
g=1ge
D (mm)
H (mm)
Burning rate (mg/s)
0.9-1.4
7.413[7.794]
28.034[32.627]
0.922 [1.0934]
9.388[8.565]
52.727[48.36]
2.147 [1.613]
9.73[9.715]
68.402[56.7]
4.007[2.007]
Table 4.3: Comparison with candle flame experiments in high gravity levels ([]=results of
Alsairafi (2003))
2
D (cm)
H (cm)
D (cm)
H (cm)
D (cm)
H (cm)
D (cm)
H (cm)
D (cm)
H (cm)
D (cm)
H (cm)
1
Gravity level
Experiment 1
dwick=1.5mm
dcandle=2.1cm
1ge
1ge
3ge
3ge
7ge
7ge
10ge
10ge
12ge
12ge
21ge
21ge
4.1
2.1
1.3
Ext.
Ext.
Ext.
Ext.
Ext.
Ext.
From Dietrich et al. (1997)
From Villermaux and Durox (1992)
3
From Arai and Amagai (1993)
2
3
Experiment 2
dwick≈0.8mm
dcandle≈1.2cm
Numerical
dwick=1mm
dcandle=0.5cm
0.75
2.6
0.7
2.3
0.4
1.5
0.3
1.2
0.3
1.0
Ext.
Ext.
0.74 [0.78]
2.803 [3.26]
0.69 [0.67]
3.1089 [4.13]
0.41 [0.41]
1.32 [0.82]
0.28 [0.32]
0.95 [0.57]
0.27 [0.27]
0.75 [0.53]
0.22 [0.22]
0.60 [0.50]
Saturation contours
1
1
2
X/Dw
97
0.5 8
2
0.6
7
68
0. 8
0.6
0.5
0.4
0.3
0.2
0.1
00
77
0.
r/Dw
146
3
4
5
Non-dimensional temperature contours
940 1
0.6
0.5
0.4
0.3
0.2
0.1
00
0.58
r/Dw
(a)
0.9
0.863134 0.953713
1
80
2
46
8
0.999543
0.992027
X/Dw
3
4
5
(b)
95
99 7 7
.99 999
0.9
10
99
99
0.
99
0.
96
18
98
88
0.5
0.4
0.3
0.2
0.1
03
99
0.
r/Dw
Non-dimensional temperature contours (expanded in two-phase region)
3.5
4
4.5
X/Dw
5
(c)
Figure 4.1: Plot of (a) saturation profiles (b) temperature profiles and (c) temperature
profiles (expanded in the two-phase region) inside the porous wick coupled to a candle
flame at normal gravity (temperature is non-dimensionalized by 330 K)
147
Non-dimensional liquid pressure cotours
0.6
patm = 6.44
0.5
r/Dw
5 17
2
7
1
6.13
0
6.16449
0
6.18978
0.1
6.21554
6.23969
6.27885
6.3159
0.2
6.36619
6.40686
0.3
6.1477
0.4
3
X/Dw
4
Non-dimensional capillary pressure cotours
0.6
patm = 6.44
0.5
0.3
4
532
r/Dw
0.1
0
1
2
X/Dw
3
4
Non-dimensional vapor pressure cotours
0.6
patm = 6.44
0.5
1212.8843
.8
8
48
79
r/Dw
6
.8
66
0.2
12
.69
.82
12
0.3
12
0.4
63
033
0.229
0.2
0
0.28
0
0.3
045
0.4
0.1
0
0
1
2
X/Dw
3
4
Figure 4.2: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick coupled to a candle
flame at normal gravity
r/Dw
r/Dw
r/Dw
148
0.5
0.4
0.3
0.2
0.1
00
0.5
0.4
0.3
0.2
0.1
00
0.5
0.4
0.3
0.2
0.1
0
2
0.4 g/cm s
1
2
X/Dw
(a)
3
4
0.04 g/cm2 s
1
2
X/Dw
(b)
3
4
0.004 g/cm2
3
3.25
3.5
X/Dw
3.75
4
(c)
Figure 4.3: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors and (c) vapor
mass flux vectors (expanded near the tip of the wick) inside the porous wick coupled to a
candle flame at normal gravity.
149
20
18
16
2
qwick (W/cm )
14
12
10
8
6
4
2
0
0
1
2
3
4
X/Dw
Figure 4.4: Plot of net heat flux supplied by the candle flame along the cylindrical surface
of the wick
150
s
1
T
0.9
0.9
s
0.8
0.8
saturation
T
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
Non-dimensional temperature
1
0
X/Dw
Figure 4.5: Plot of saturation and temperature variation along the cylindrical surface of
the wick coupled to a candle flame at normal gravity (temperature is non-dimensionalized
by 330 K).
151
2
1.75
2
mass flux (kg/m s)
1.5
1.25
ρlvl
1
ρlvl+ρgvg
0.75
0.5
0.25
ρgvg
0
-0.25
-0.5
0
1
2
3
4
X/Dw
Figure 4.6: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick coupled to a candle flame at normal gravity.
152
7
Liquid mass flux (kg/m2 s)
6.5
6
5.5
5
ρlul
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
X/Dw
Figure 4.7: Plot of liquid mass flux (in x-direction) variation along the cylindrical surface
of the wick coupled to a candle flame at normal gravity.
153
s
1
T
0.9
0.9
0.8
0.8
s
T
0.7
0.7
0.6
0.999
T
0.998
0.5
0.997
0.996
0.4
0.995
0.994
0.3
0.993
0.992
0.2
0.991
0.1
0
3
3.25
3.5
3.75
4
Non--dimensional temperature
saturation
1
0.6
0.5
0.4
0.3
0.2
0.99
0.1
x/Dw
0
1
2
3
4
Non--dimensional temperature
1
0
x/Dw
Figure 4.8: Plot of saturation and temperature variation along the axis of the wick
coupled to a candle flame at normal gravity (temperature is non-dimensionalized by
330 K).
154
0
6
-0.02
-0.03
4
ρlul
-0.04
-0.05
3
-0.06
2
-0.07
-0.08
1
2
ρgug
Vapor mass flux (kg/m s)
5
2
Liquid mass flux (kg/m s)
-0.01
-0.09
0
0
1
2
3
4
X/Dw
Figure 4.9: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick coupled to a candle flame at normal gravity.
155
2.5
1ge
r (cm)
2
1.5
Tmax = 7.137
1
1.1
0.5
0-1
0
2
4
1
6
7
2
3
X (cm)
4
5
Figure 4.10: Gas temperature contours (non-dimensionalized by T∞ = 300 K )
1.5
r (cm)
1ge
1
0.005
0.5
5E-0
0-1
0
1
X (cm)
5
2
5E-06
3
Figure 4.11: Fuel reaction rate contours (g cm-3 s-1)
4
156
1.5
r (cm)
YF
YO2
1ge
1
0.2
0.5
0-1
0.05
0.3
0
1
0.001
X (cm)
2
0.1
0.05
3
4
Figure 4.12: Fuel and oxygen mass fraction contours
1.5
r (cm)
1ge
1
0.5
1
10 5
0-1
0
1
0.1 0.001
X (cm)
2
3
Figure 4.13: Local fuel/oxygen equivalence ratio contours
4
157
1.5
r (cm)
1ge
1
0.001
0.01
0.5
0.1
0.2
0-1
0
1
X (cm)
2
3
4
Figure 4.14: Carbon dioxide mass fraction contours
1.5
r (cm)
1ge
1
0.001
0.01
0.5
0.07
0-1
0
1
X (cm)
0.03
0.05
2
3
Figure 4.15: Water vapor mass fraction contours
4
158
2
-2
3 mg/cm s
Ψ=100
Ψ=50
1.5
Ψ=150
1
r (cm)
0.5
Ψ=10
Ψ=0
0
-0.5
-1
50 cm/s
-1.5
-2
0
1
2
3
X (cm)
Figure 4.16 Upper half: oxygen mass flux and streamlines, Lower half: velocity vectors
and visible flame (5x10-5 g cm-3 s-1 reaction rate contour) around the flame
159
2
-0.0056
1ge
r (cm)
1.5
1
-0.026
-0.026
0.5
0-1
0
-0.1
-0.1
1
X (cm)
-0.00
2
Figure 4.17: Isobar contours; p =
3
p − p∞
ρ rU r2
5
4
160
1
8
T
0.9
Yi & Wf (mg.cm3 s)
0.8
YN2
6
0.7
5
0.6
Wf
0.5
4
0.4
0.3
3
YF
2
YO2
0.2
1
0.1
YCO2
YH2O
0
Normalized temperature
7
5
10
15
0
20
X (cm)
Figure 4.18: Profiles of temperature and species concentration along the symmetry line
(temperature is non-dimensionalized by T∞ = 300 K )
161
1.5
r (cm)
1ge
1
0.001
0.03
0.05
0.05
0.03
0.5
0.1
0-1
0
1
X (cm)
2
0.01
3
4
Figure 4.19: Effective Mean absorption coefficient distribution (cm-1 atm-1)
2
0.5 W/cm
2
r (cm)
1.5
1
0.5
0
-1
0
1
2
3
X (cm)
Figure 4.20: Dimensional net radiative flux vectors (W/cm2)
4
162
2
r (cm)
1.5
-1E-06
-1E-06
1
0.5
0.1
0
-1
1.3
0
1
1
0.5
2
3
4
X (cm)
Figure 4.21: Contours of divergence of radiative heat flux (W/cm3)
163
16
14
12
qc
q (W/cm2 rad)
10
qnet
8
6
4
10x(qr)in
2
0
-2
-4
10x(qr)e
10x(qr)net
-6
10x(qr)out
-8
-10
0
0.1
0.2
0.3
0.4
X (cm)
Figure 4.22: Heat fluxes on the candle wick surface per unit radian
Saturation contours
0.
68
1.000
1
2
X/Dw
3
34
0.1 .242
0
0.320
1
0.6
0.5
0.4
0.3
0.2
0.1
00
0.43
r/Dw
164
3
4
5
(a)
Saturation contours
0.6
r/Dw
0.5
0.007
0.022
0.134
0.4
0.198
0.3
0.24.2
4.25
4.3
4.35
X/Dw
4.4
Non-dimensional temperature contours
0.554
21 0
r/Dw
(b)
0.6
0.5
0.4
0.3
0.2
0.1
00
0.
75
46
0.9
35
20
0 .9
22
74
9
1
2
63
0.999918
1
0.999083
X/Dw(b) 3
4
5
(c)
9980 0.999990
95
00
99
99
0
0.4
0.999774
0.3 0.999143
0.2
0.1
03
3.5
0.9
0.99
99
0.
r/Dw
Non-dimensional
temperature contours (expanded in two-phase region)
0.5
4
X/Dw
4.5
5
(d)
Figure 4.23: Plot of (a) saturation profiles (b) saturation profiles expanded in the twophase region (c) non-dimensional temperature (non-dimensionalized by 330 K) profiles
and (d) non-dimensional temperature profiles (expanded in the two-phase region) inside
the porous wick for a self trimmed candle flame at normal gravity
165
Non-dimensional liquid pressure contours
0.6
6.39099
r/Dw
2
X/Dw
3
4
Non-dimensional capillary pressure contours
0.6
003
27
8
0.4114
9
61
72
6
0.2
0.36204
010
61
47
0.3
10
0.3
15
0.1
0.4
0.47
0.0
0.5
r/Dw
6.06356
1
6.09508
0
6.19095
0
6.22968
0.1
6.3398
0.2
6.28962
0.3
6.13331
6.15851
0.4
9 62
5.97
6.0249
0.5
0.1
0
0
1
2
X/Dw
3
4
Non-dimensional vapor pressure contours
0.6
0.5
r/Dw
4.96095
0.3
0.2
8
12.8
3
62
.8
9
801
12.
12
0.4
0.1
0
0
1
2
X/Dw
3
4
Figure 4.24: Plot of non-dimensional pressure contours: liquid pressure (top) capillary
pressure (middle) and gas pressure (bottom) inside the porous wick for a self trimmed
candle flame at normal gravity
r/Dw
166
0.5
0.4
0.3
0.2
0.1
00
2
10 kg/m s
1
2
X/Dw
3
4
r/Dw
(a)
0.5
0.4
0.3
0.2
0.1
01
2
0.1 kg/m s
2
X/Dw 3
4
(b)
Figure 4.25: Plot of (a) liquid mass flux vectors (b) vapor mass flux vectors inside the
porous wick for a self trimmed candle flame at normal gravity
167
18
16
qwick (W/cm2)
14
12
10
8
6
4
2
0
0
0.25
0.5
0.75
1
X/Dw
Figure 4.26: Plot of net heat flux supplied by the candle flame along the cylindrical
surface of the wick
168
T
s
1
1
0.95
0.8
0.9
s
saturation
0.7
0.85
T
0.6
0.8
0.5
0.75
0.4
0.7
0.3
0.65
0.2
0.6
0.1
0.55
0
0
1
2
3
4
Non-dimensional temperature
0.9
0.5
X/Dw
Figure 4.27: Plot of saturation and temperature variation along the cylindrical surface of
the wick for a self trimmed candle flame at normal gravity
169
3
mass flux (Kg/m2 s)
2.5
2
1.5
1
ρlul
ρlul+ρvuv
0.5
0
-0.5
ρvuv
0
1
2
3
4
X/Dw
Figure 4.28: Plot of liquid and vapor mass flux (in r-direction) variation along the
cylindrical surface of the wick for a self trimmed candle flame at normal gravity
170
Liquid mass flux (kg/m2 s)
8
7
6
5
ρlul
4
3
2
1
0
0
1
2
3
4
5
X/Dw
Figure 4.29: Plot of liquid mass flux (in x-direction) variation along the cylindrical
surface of the wick for a self trimmed candle flame at normal gravity
171
s
1
T
0.9
0.9
T
0.8
0.8
0.7
1
T
0.6
Non-dimensional temperature
saturation
0.7
s
0.6
0.9975
0.5
0.4
0.995
0.3
0.5
0.4
0.3
0.9925
0.2
0.1
0
3
3.5
4
0.2
0.99
0.1
x/Dw
0
1
2
3
4
5
Non-dimensional temperature
1
0
x/Dw
Figure 4.30: Plot of saturation and temperature variation along the axis of the wick for a
self trimmed candle flame at normal gravity.
172
7
6
-0.01
Vapor mass flux (kg/m2 s)
Liquid mass flux (kg/m2 s)
0
ρgug
-0.02
ρlul
5
-0.03
4
-0.04
-0.05
3
-0.06
2
-0.07
-0.08
1
-0.09
0
0
1
2
3
4
X/Dw
Figure 4.31: Plot of liquid and vapor mass flux (in x-direction) variation along the axis of
the wick for a self trimmed candle flame at normal gravity.
173
2.5
1ge
r (cm)
2
1.5
Tmax = 7.117
1
1.1
0.5
0-1
0
2
4
1
7
7
2
3
X (cm)
4
5
Figure 4.32: Gas temperature contours for a self trimmed candle flame (nondimensinalized by T∞ = 300 K )
1.5
1
1ge
r (cm)
0.5
5E-05
0
-0.5
-1
-1.5-1
0
1
X (cm)
2
3
4
Figure 4.33: Fuel reaction rate contour (5X10-5 g cm-3 s-1) for a self trimmed candle flame
174
1.5
r (cm)
YF
YO2
1ge
1
0.2
0.5
0.3
0-1
0
0.001
0.05
1
X (cm)
0.1
0.05
2
3
4
Figure 4.34: Fuel and oxygen mass fraction contours for a self trimmed candle flame
1.5
r (cm)
1ge
1
0.5
0-1
10
0
5
1
1 0.1
X (cm)
2
0.001
3
4
Figure 4.35: Local fuel/oxygen equivalence ratio contours for a self trimmed candle
flame
175
1.5
r (cm)
1ge
1
0.001
0.01
0.5
0.1
0.2
0-1
0
1
X (cm)
2
3
4
Figure 4.36: Carbon dioxide mass fraction contours for a self trimmed candle flame
1.5
r (cm)
1ge
1
0.001
0.01
0.5
0-1
0.05
0.07
0
1
X (cm)
0.03
2
3
4
Figure 4.37: Water vapor mass fraction contours for a self trimmed candle flame
176
2
50
100
1.5
150
4 mg/cm-2 s
1
10
r (cm)
0.5
0
0
-0.5
-1
50 cm/s
-1.5
-2
0
1
2
3
X (cm)
Figure 4.38: Oxygen mass flux and flow field around the self trimmed candle flame
177
2
r (cm)
1.5
00
-0 .
56
1ge
-0.02
1
-0.02
0.5
0-1
-0.1
-0.1
-0.1
0
1
X (cm)
2
3
Figure 4.39: Isobar contours for a self trimmed candle flame; p =
4
p − p∞
ρ rU r2
178
1
8
T
0.9
Yi & Wf (mg/cm3 s)
0.8
YN2
6
0.7
5
0.6
0.5
4
0.4
0.3
3
YF
2
YO2
0.2
Normalized temperature
7
1
0.1
YCO2
YH2O
0
5
10
0
20
15
X (cm)
Figure 4.40: Profiles of flame structure at the centerline for a self trimmed candle flame
1.5
r (cm)
1ge
1
0.001
0.03
0.05
0.05
0.03
0.5
0.1
0-1
0
1
X (cm)
2
0.01
3
4
Figure 4.41: Effective mean absorption coefficient distribution for a self trimmed candle
flame (cm-1 atm-1)
179
2
0.5 W/cm2
r (cm)
1.5
1
0.5
0
-1
0
1
2
3
4
X (cm)
Figure 4.42: Dimensional net radiative flux vectors for a self trimmed candle flame
(W/cm2)
2
r (cm)
1.5
-1E-06
1
-1E-06
0.5
1
0
-1
0
1
0.1
1.3
0.5
2
3
4
X (cm)
Figure 4.43: Contours of divergence of radiative heat flux for a self trimmed candle flame
(W/cm3)
180
15
qc
q (W/cm2 s)
10
qnet
5
0
10x(qr)in
-5
10x(qr)out
10x(qr)net
-10
0.1
0.2
0.3
10x(qr)e
0.4
0.5
X (cm)
Figure 4.44: Heat fluxes on the candle wick surface per unit radian
181
r (cm)
1
0ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
10-4ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
10-2ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
1ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
3ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
6ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
r (cm)
1
10ge
0.5
0 -1
0
1
2
x (cm)
3
4
5
Figure 4.45: Candle flames at various gravity levels for a self trimmed candle flame
1.2
11
1.1
10
1
9
Burning rate (mg/s)
0.9
8
0.8
7
0.7
6
0.6
5
0.5
4
0.4
3
0.3
0.2
2
no self trimming observed
Wick height is chosen as 3.8 mm
1
0.1
0
Self trimming length (mm)
182
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0
g/ge
Figure 4.46: Candle burning rate and self trimmed length of the candle (on log scale) at
various gravity levels for a self trimmed candle flame
1.1
12
1
11
0.9
10
9
0.8
8
0.7
7
0.6
6
0.5
5
0.4
4
0.3
3
no self trimming observed
Wick height is chosen as 3.8 mm
0.2
0.1
0
Self trimming length (mm)
Burning rate (mg/s)
183
2
1
0
5
10
0
15
g/ge
Figure 4.47: Candle burning rate and self trimmed length of the candle (on normal scale)
at various gravity levels for a self trimmed candle flame
184
Self trimming length (mm)
8
7
-2
10 ge
6
5
1ge
4
3
0
2.5E-12
2
5E-12
Absolute Permeability (m )
Figure 4.48: Self trimming length of the candle wick using different wick permeabilities.
185
1.2
1.1
Burning rate (mg/s)
1
1ge
0.9
-2
0.8
10 ge
0.7
0.6
0.5
0.4
0
2.5E-12
2
5E-12
Absolute Permeability (m )
Figure 4.49: Candle burning rates for different wick permeabilities.
186
3.6
3.4
Flame Length (cm)
3.2
3
1ge
2.8
2.6
10-2ge
2.4
2.2
2
1.8
1.6
0
2.5E-12
2
5E-12
Absolute Permeability (m )
Figure 4.50: Candle flame lengths for different wick permeabilities.
187
2200
Maximum temperature (K)
1ge
2000
-2
1800
1600
10 ge
0
2.5E-12
2
5E-12
Absolute Permeability (m )
Figure 4.51: Maximum gas phase temperatures for different wick permeabilities.
188
20
self trimming length (mm)
18
16
14
-2
10 ge
12
10
1ge
8
6
4
2
0
0.5
1
1.5
2
2.5
3
Dwick (mm)
Figure 4.52: Self trimming length of the candle wick for different wick diameters.
189
Burning rate (mg/s)
3
1ge
2.5
-2
10 ge
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
Dwick (mm)
Figure 4.53: Candle burning rates for different wick diameters.
3
190
2500
Maximum temperature (K)
2400
2300
2200
1ge
2100
2000
1900
-2
1800
10 ge
1700
1600
1500
0.5
1
1.5
2
2.5
3
Dwick (mm)
Figure 4.54: Maximum gas phase temperatures for different wick diameters.
191
Maximum temperature (K)
2500
1ge
2000
-2
10 ge
1500
-4
10 ge
1000
17
18
19
20
21
22
O2%
Figure 4.55: Maximum gas phase temperature at various oxygen molar fractions
192
9
Self trimming length (mm)
10-4ge
8
7
-2
6
10 ge
5
1ge
4
17
18
19
20
21
22
O2%
Figure 4.56: Self trimmed length of the candle wick at various oxygen molar fractions
193
1.2
1.1
1
1ge
Burning rate (mg/s)
0.9
0.8
-2
0.7
10 ge
0.6
0.5
0.4
10-4ge
0.3
0.2
0.1
0
17
18
19
20
21
22
O2 %
Figure 4.57: Candle burning rate at various oxygen molar fractions
194
1.5
1.25
0.4
r (cm)
1.75
r (cm)
0.5
21% O2
20% O2
19% O2
18% O2
2
0.3
0.2
0.1
1
0
0
0.1
0.75
0.2
0.3
X (cm)
0.4
0.5
0.6
0.5
0.25
0
-1
0
1
2
3
4
X (cm)
Figure 4.58: Fuel reaction rate contours (5X10-5 g/cm3s)at various oxygen molar
fractions.
195
K = 2.5e-12 m , ε = 0.55, O2 = 21%
2
Flame Length (1)
Flame Width (1)
Flame Length (2)
Flame Width (2)
Flame Length (3)
Flame Width (3)
Flame Length (mm)
40
35
30
1 - present computation
2 - experiments
(Arai and Amagai, 1993)
25
3 - computation (Alsairafi, 2003)
45
40
35
30
25
20
20
15
15
10
10
5
5
0
0
2
4
6
8
10
12
Flame Width (mm)
45
0
Gravity (g/ge)
Figure 4.59: Comparison of flame length and flame widths at various gravity levels with
the experiments (Arai and Amagai, 1993)
196
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
The work described in this dissertation represents an effort to numerically
simulate a self trimmed candle flame by including the heat and mass transport taking
place inside the porous candle wick. The two-phase flow inside the wick is solved by
considering the steady state volume averaged equations for the porous wick. Phase
change inside the wick is accounted by the thermodynamic Gibb’s phase relationships.
The wick transport is coupled to the gas phase at the wick surface. Self trimming of the
wick is modeled as the burn out of the dry vapor region of the wick. The gas phase
combustion model taken directly from Alsairafi (2003) is based on the finite volume
method, includes the steady, laminar, axisymmetric, conservation equations for
momentum, energy, and species (fuel, oxygen, carbon dioxide, and water vapor). Gas
phase combustion is modeled via a single-step, second order, finite rate Arrhenius
reaction with variable properties. Gas radiation is included in the numerical code by
considering radiative contribution from carbon dioxide and water vapor. The radiation
transfer equation is solved by the discrete ordinates method, with modified Planck mean
absorption coefficients.
First the detailed heat and mass transport inside the porous candle wick for a
given heat input is analyzed. The effect of gravity and wick permeability is studied. The
following interesting observations were made
(a) In the funicular regime, there are 2 regions. A single phase liquid region is
present near the base of the wick and a two-phase region is present above it
separated by an interface.
197
(b) In the two-phase region, the liquid and the vapor move in a countercurrent
fashion. The liquid is evaporated at the surface of the wick. A small portion of
this vapor is conducted into the wick. The vapor condenses as it traverses into the
wick.
(c) There is a temperature gradient near the base of the wick causing a heat loss to
the wax pool.
(d) Gravity does not significantly affect the heat and mass transfer inside the porous
wick.
(e) The wick permeability significantly affects the saturation and temperature
distribution inside the porous wick
Later the porous candle wick is coupled to the gas phase candle burning. Self
trimming is modeled as the burn out of the dry region of the wick. Flow field in the gas
phase and flame characteristics were analyzed at different gravity levels, oxygen mole
fractions, and candle and wick diameters. Major findings from the computed results are:
(a) The detailed heat and mass transport inside the porous candle wick is analyzed.
(b) The self trimmed length of the wick is a function of the wick parameters (wick
diameter, wick permeability etc.) and the gas phase characteristics (e.g. ambient
oxygen) and the gravity level. The self trimmed length directly affects the burning
characteristics by changing the burning surface area.
(c) The effect of gravity, oxygen molar percentage, and wick diameter on the burning
of candle flame is analyzed.
198
(d) The gas phase characteristics are qualitatively similar to that observed by
Alsairafi (2003). The computed results show that the candle flame has a
hemispherical shape with large flame standoff distance in Zero gravity. As gravity
is increased, the flame reaches its maximum length at about 3ge. Further increase
of gravity shifts the flame stabilization zone from the base of the wick to the top
of the wick and greatly shrinks the flame length. The flame temperature drops as
gravity decreases suggesting the importance of heat losses in reduced gravity.
Both conductive and radiative losses are found to be significant.
(e) The current model has reproduced many of the experimentally observed candle
flame characteristics.
(f) There is a qualitative disagreement between the experimental results and the
computed results for the candle flame behavior at different gravity levels. The
experimental results indicate that there is monotonic decrease in the flame length
and burning rate at hyper gravity levels. Computed results indicate an increase in
burning rate and flame length followed by a decrease in the corresponding values.
Recommendation for Future Work
Based on the conclusions of the present study, several aspects for the candle
flames problem need further investigation.
(1) More experimental data is needed for estimating candle wax properties. The
Leverett’s function has to be experimentally determined for the candle wax and wick
combination. Similarly experimental data is required for determining the relative
permeability of wax inside the wick.
199
(2) Although the inclusion of the transport of non-condensable gases into the porous wick
in the two-phase model does not effect the solution in the one-dimensional problem
significantly (Raju, 2004), it can be verified for the present axisymmetric model by
including it.
(3) The capability of wick phase numerical model has to be improved to be able to
capture the evaporative front regime.
(4) A detailed pyrolysis rate equation for the burning/charring of the wick material during
self trimming can be incorporated. The formation of curved tip due to trimming action
has to be appropriated modeled.
(5) The bending of candle wick during burning of the candle can be accounted to
simulate a real flame. The two-phase flow equations of the wick can be combined with
the deformation characteristics of wick material.
(6) At very high gravity levels, the flame becomes weaker and hence sufficient heat back
to the candle wax pool may not be present to melt the wax. A more detailed candle flame
model can be developed by including the melting of wax pool. This would give more
insight into the existence of wake flame at higher gravity levels.
(7) Detailed kinetics can be incorporated for the burning of candle wax. The formation of
soot can also be included in the candle flame model. The visible flame reported in the
experiments is affected by the formation of soot. The inclusion of soot would give a
better comparison of the computed results with the experimental results.
(8) More detailed experimental results are needed to compare the present computed
results. The previous experiments do not report the nature of candle wax, the wick length,
200
wick properties, the self trimming characteristics of the wick etc. The computed results
show that the candle flame characteristics can significantly vary depending on the
permeability of the wick. This should be taken into consideration while performing the
experiments.
(9) The present two-phase heat and mass transport model inside porous media can be
applied to various practical situations like flame spreading over charring fuels, sand beds.
It can also be extended to heat pipe applications.
201
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