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This is to certify that the

dissertation entitled

THEORETICAL AND NUMERICAL INVESTIGATION
OF RADIATIVE EXTINCTION OF DIFFUSION
FLAMES

presented by

Anjan Ray

has been accepted towards fulﬁllment
of the requirements for

ph,D, degree in Mechanical Engineering

 

 

Quad 2 TJM

Major professor

 

Date Mag 2!, 11‘16.

MS U is an Afﬁrmative Action/Equal Opportunity Institution 0-12771

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LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to romovo thIo chockout from your rocord.
TO AVOID FINES rotum on or boforo duo duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU IoAn Afﬂnnotivo Action/Emu Opportunity Inotltution
Wan-M

Theoretical and Numerical Investigation of

Radiative Extinction of Diffusion Flames

By

Anjan Ray

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1996

ABSTRACT

Theoretical and Numerical Investigation of

Radiative Extinction of Diffusion Flames
By

Anjan Ray

The inﬂuence of soot radiation on diffusion ﬂames was investigated using both
analytical and numerical techniques. Soot generated in diffusion ﬂames dominate
the ﬂame radiation over gaseous combustion products and can signiﬁcantly lower the
temperature of the ﬂame. In low gravity situations there can be signiﬁcant accumu-
lation of soot and combustion products in the vicinity of the primary reaction zone
owing to the absence of any convective buoyant ﬂow. Such situations may result
in substantial suppression of chemical activities in a ﬂame and the possibility of a
radiative extinction may also be anticipated. The purpose of this work was to not
only investigate the possibility of radiative extinction of a diffusion ﬂame but also to
qualitatively and quantitatively analyze the inﬂuence of soot radiation on a diffusion
ﬂame.

In this study, ﬁrst a hypothetical radiative loss proﬁle of the form of a sech2 was
assumed to inﬂuence a pure diffusion ﬂame. It was observed that the reaction zone

can, under certain circumstances, move through the radiative loss zone and locate

itself on the fuel side of the loss zone, contrary to our initial postulate. On increasing
the intensity and/or width of the loss zone it was possible to extinguish the ﬂame
and extinction plots were generated. In the presence of a convective ﬂow, however,
the movement of the temperature and reaction rate peaks indicated that the ﬂame
behavior is more complicated compared to a pure diffusion ﬂame.

A comprehensive model of soot formation, oxidation and radiation was used in a
more involved analysis. The soot model of Syed, Stewart and Moss [1] was used for
soot nucleation and growth and the model of N agle and Strickland-Constable [2] was
used for soot oxidation. The soot radiation was considered in the optically thin limit.
An analysis of the ﬂame structure revealed that the radiative loss term is countered
both by the reaction term and the diffusion term. The essential balance for the soot
volume fraction was found to be between the processes of soot convection and soot
growth. Such a balance yielded to analytical treatment and the soot volume fraction
could be expressed in the form of an integral. The integral was evaluated using two
approximate methods and the results agreed very well with the numerical solutions

for all cases examined.

In loving memory of my mother

iv

ACKNOWLEDGMENTS

I wish to express my very sincere gratitude to Dr. I. S. Wichman for his guidance.
He took active interest in my work and participated in all stages of problem solving.
It was a great privilege to work with him.

I would also like to thank Dr. J. V. Beck, Dr. M. Miklavcic and Dr. M. Zhuang
for serving on my thesis committee and for offering important suggestions for im-
provement.

This work was supported by the microgravity combustion research program of the
NASA Lewis Research Center. In Spring, 1996 this work was supported by a generous
fellowship award from the Michigan State University Graduate School.

I am grateful to my colleagues Zori Pavlova, N andu Lakkaraju and Bukky Oladipo
for creating and sustaining a friendly atmosphere at work. I would like to thank
my friend Umashankar for his elaborate help on formatting my thesis using ETEX.
Bipasha helped me with some of the drawings in this thesis.

My friends have been extremely supportive throughout. I would fondly remember
the long telephone conversations with Mayukh and Debabrata. We exchanged ideas,
shared good humor and infused inspiration into one another. The team of friends
at East Lansing created an atmosphere of bonhomie which has been of paramount
importance to me. Rabi, Ajit, Nidhan, Ashim and Kuntal have been great friends
and have always been there for me.

The greatest inspiration for my Doctoral work came from my parents. I have

learned a lot from the courage and fortitude of my father through difﬁcult times and

a great personal loss. He taught me the value of exploring new avenues and to remain
undaunted in the worst of times. Even in the midst of great pain and suffering, my
mother lent her unrelenting support for me in the early stages of my Doctoral work.
Her ecstasy would have known no bounds today. Her invisible presence continues to
nourish my inner self and this work is a tribute to how much she has always meant

to me.

vi

 

TABLE OF CONTENTS

LIST OF TABLES x
LIST OF FIGURES xi
1 Introduction 1
2 Literature Review 4
2.1 Introduction ................................ 4
2.2 Background ................................ 4
2.3 Soot .................................... 5
2.3.1 How Does Soot Affect Us? .................... 5

2.3.2 Appearance ............................ 6

2.3.3 Morphology and Chemical Structure .............. 6

2.3.4 Characterization of the Soot Distribution ............ 7

2.3.5 Soot Evolution .......................... 7

2.3.6 Soot Oxidation .......................... 9

2.3.7 The Inﬂuence of Fuel Structure ................. 10

2.3.8 Effect of Pressure ......................... 10

2.3.9 Inﬂuence of Additives ....................... 10

2.3.10 Inﬂuence of Oxygen Addition .................. 11

2.4 Soot Models ................................ 11
2.5 Soot Radiation .............................. 14
2.6 Diffusion Flames in Microgravity .................... 15

3 Inﬂuence of a Simple Heat Loss Proﬁle on a Pure Diffusion Flame 19

3.1 Introduction ................................ 19
3.2 Review of Pure Diffusion Flame Results ................ 20
3.3 Problem Geometry ............................ 21
3.4 Choice of Parameter values ........................ 22
3.5 Formulation ................................ 26

vii

3.5.1 Inﬁnite Reaction Rate (IRR) Solutions as Initial Conditions

3.5.2 Simple Heat Loss Proﬁle .....................
3.6 Numerical Solution ............................
3.7 Relation Between Temperature and Density ..............
3.8 Evaluation of 3'0 ..............................
3.9 Results and Discussion ..........................

3.9.1 Comparison with the top-hat proﬁle ...............
3.10 Conclusions ................................

28
29
31
32
32
33
52
54

Influence of a Simple Heat Loss Proﬁle on a Diffusion Flame with

Fuel Blowing

4.1 Introduction ................................
4.2 Problem Deﬁnition ............................
4.3 Formulation ................................
4.4 Boundary Condition for the Fuel Equation ...............
4.5 Solution for Inﬁnite Reaction Rate ...................
4.6 Parameter Values .............................
4.7 Results and Discussion ..........................
4.8 Conclusions ................................

Estimation of Soot Layer Proﬁle and Thickness

5.1 Introduction ................................
5.2 Soot Model ................................
5.3 Examination of the Soot Model .....................

5.3.1 Rapid equilibration of 17 .....................
5.4 Formulation of the Full Problem .....................
5.5 Numerical Solution ............................
5.6 Analytical Approximation ........................
5.7 Integral Evaluation Using Laplace’s Method ..............
5.8 Comparison of Results ..........................
5.9 Inﬂuence of soot radiation ........................

5.9.1 Background ............................

5.9.2 Formulation of the Radiation term ...............

5.9.3 Results ...............................
5.10 Conclusions ................................

Investigation of the Comprehensive Soot Radiation Problem
6.1 Formulation ................................

viii

56
56
56
58
59
60
63
63
89

91
91

104
106
107
109
109
110
112
114

116
117

6.1.1 The Inﬂuence of Thermophoresis ................ 121

6.2 Parameter Values ............................. 122
6.3 Numerical Solution ............................ 123
6.4 Results and Discussion .......................... 123
6.5 Estimation of the Soot Volume Fraction Proﬁle ............ 155
6.6 Conclusions ................................ 159
7 Conclusions and Recommendations for Future Work 161

A Asymptotic Calculations for a Simplified Model of the Interaction

of a Diffusion Flame with a Heat-Loss Zone 165
A.1 Abstract .................................. 165
A2 Introduction ................................ 165
A.3 Formulation ................................ 167
A.3.1 Physical Discussion ........................ 167
A.3.2 The Mathematical Problem ................... 168
A33 The Form of H(Z) ........................ 171
A.3.4 The Radiation Term ....................... 175

AA Results ................................... 176
A41 Inﬂuence of H on Oxidizer, Fuel and Temperature Proﬁles . . 176
A.4.2 Inﬂuence of Negative H on Flame Shift ............. 185
A.4.3 Reaction-Zone Analysis using Activation-Energy Asymptotics 191
A.4.4 Inﬂuence of Heat-Loss Zone on Flame Displacement ...... 197

A5 Discussion and Conclusions ....................... 198
BIBLIOGRAPHY 201

ix

LIST OF TABLES

3.1 Flame Temperature versus Yoo data ..................

3.2 Parameter values

OOOOOOOOOOOOOOOOOOOOOOOOOOOOO

4.1 uoL/ao for different values of no. ....................

3.1
3.2
3.3
3.4
3.5
3.6

3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

LIST OF FIGURES

The problem geometry ...........................
Flame Temperature as a function of Yoo for different values of Ypp. .
The chosen heat loss proﬁle of the form of a sechz. ..........
Effect of Radiation Number NR on Temperature Proﬁle. .......
Effect of Radiation Number NR on Reaction Rate ............
1', yo, yp proﬁles for ﬁnite and inﬁnite reaction rates. Also shown is
the reaction rate proﬁle. .........................
The ﬂame structure when N R = 383. ..................
Effect of N R on 1' proﬁle for thicker loss zone. .............

Effect of N R on T proﬁle for A = 0.1 ...................
Effect of N3 on (1 + ¢)Dr proﬁle for A = 0.1 ..............
Extinction Plot for AZR = 0.06 and A = 0. ..............

Heat Transfer to the oxidizer wall as a Function of N R. ........
Heat Transfer to the oxidizer wall as a Function of N 3(2/ B) ......
qTotal as a function of N3. ........................
qTotal as a function of NR(2/ B). .....................
X as a function of N3 ...........................
x as a function of NR(2/B) .......................
ATf as a function of the radiative fraction ...............
Comparison of N Rah-mt,“ values for seek2 and top—hat proﬁles. . . .
The problem geometry and deﬁnition ...................
IRR proﬁles for 1', yo and yp ......................
Effect no on 7' proﬁle for a given ﬂame. .................
Effect no on T2 for a given ﬂame. ....................
Effect of no on (1 + qS)’Dr for a given ﬂame ................
Effect of N R on the temperature proﬁle. ................
Effect of N R on yo and yp proﬁles. ...................
Effect of N R on reaction rate proﬁles ...................
1' proﬁles when AZR = 0.1 and A = O. .................

xi

21

36

41
42
44
45
47
48
49
50
51
53
57
62
64
66
67

4.10
4.11
4.12

4.13
4.14
4.15
4.16
4.17
4.18
4.19

4.20

4.21

4.22

4.23
4.24
4.25
5.1
5.2
5.3
5.4
5.5
5.6
6.1
6.2

6.3
6.4
6.5
6.6
6.7
6.8

(1 + ¢)Dr proﬁles when AZR = 0.1 and A = 0. ............ 72
Flame structure when N3 = 9.9, AZR = 0.1 and A = 0 ......... 73

Loss zone is now separated from ﬂame location. Observe the movement

of the peak toward the oxidizer wall. .................. 74
Reaction rate proﬁles when A = 0.1. .................. 75
Flame structure when NR = 25, AZR = 0.04 and A = 0.1. ...... 76
1' proﬁles when AZR = 0.08 and A = 0.1 ................. 78
(1 + 43)’Dr proﬁles when AZR = 0.08 and A = 0.1 ............ 79
7' proﬁles when uoL/ao = 4.24 ....................... 80
1' proﬁles when uoL/ao = 4.24 and AZR = 0.04. ............ 81
Heat transfer to the oxidizer wall as a function of N3 for the fuel

blowing situation .............................. 82
Heat transfer to the oxidizer wall as a function of N 3(2/ B ) for the fuel

blowing situation .............................. 83

The total heat release in the combustion process plotted as a function

of the radiation number N R ........................ 84
The total heat release in the combustion process plotted as a function

of N 3(2/ B) ................................. 85
The total radiative fraction x versus the radiation number N R. . . . . 86
The total radiative fraction x versus N 3(2/ B) .............. 87
AT, as a function of the radiative fraction X ............... 88
Comparison of fv proﬁles for d = 0.5 and d = 0.005 ........... 97
Soot Volume Fraction Proﬁle ....................... 102
The structure of the soot volume fraction proﬁle ............ 103
The analytical solutions plotted along with the numerical solution. . . 108
1', yo and yp proﬁles for prescribed soot volume fraction. ....... 112
Structure of the ﬂame. Note the radiative loss term ........... 113
Temperature, species and soot volume fraction proﬁles. ........ 124

Temperature, species and soot volume fraction proﬁles in the physical

coordinate, :5. ............................... 125
1’ proﬁles for three different situations. ................. 126
The Contribution of various terms in the energy equation. ...... 127

The contribution of various terms in the soot volume fraction equation. 128
The effect of uo on temperature proﬁle for a given stoichiometry. . . . 129
The effect of no on the reaction term for a given stoichiometry. . . . . 130

The effect of no on reaction term for a given stoichiometry ....... 131

xii

6.9

6.10

6.11

6.12

6.13

6.14

6.15
6.16

6.17

6.18

6.19

6.20

6.21

6.22
6.23

6.24

6.25

6.26

6.27

6.28

6.29
6.30

The inﬂuence of no on the species mass fraction proﬁles for a given
stoichiometry. ............................... 132
The effect of no on the soot volume fraction proﬁle for a given stoi-
chiometry .................................. 134
The effect of no on the radiative loss term proﬁle for a given stoichiometry. 135
The effect of no on the radiative loss term proﬁle for a given stoichiometry. 136
The effect of no on the 1' proﬁle for a given stoichiometry on the as-
sumption of no radiative losses. ..................... 137
The effect of no on the (1 + ¢)Dr proﬁle for a given stoichiometry on
the assumption of no radiative losses. .................. 138
The effect of Yoo on the 1' proﬁle for a given Ypp and a given uoL/ao. 139

The inﬂuence of Yoo on the species proﬁles for a given Ypp and a given

HoL/ao. .................................. 140
The inﬂuence of Yoo on the reaction rate for a given Ypp and a given
HoL/ao. .................................. 141

The inﬂuence of Yoo on the soot volume fraction proﬁle for a given
Ypp and a given uoL/ao .......................... 142
The effect of Yoo on the radiation term proﬁle for a given Ypp and a
given uoL/ao. ............................... 143
The effect of Yoo on the radiative fraction for a given YFF and a given
uoL/ao. .................................. 144
The effect of Ypp on the temperature proﬁle for a given Y00 and a
given uoL/ 010. ............................... 145
The drop in peak ﬂame temperature as a function of Ypp. ...... 146
The effect of Ypp on the reaction term proﬁle for a given Y00 and a
given uoL/ao. ............................... 147
The effect of Ypp on the species proﬁles for a given Y00 and a given
uoL/ao. .................................. 148
The effect of YFF on the soot volume fraction proﬁle for a given Y00

and a given uoL/ao. ........................... 149
The effect of Ypp on the radiation term proﬁle for a given Y00 and a

given uoL/ao. ............................... 150
The effect of Ypp on the radiative fraction for a given Y00 and a given

uoL/ao. .................................. 151
The ,6 distribution plotted in the Z coordinate .............. 152
The velocity distribution for uoL/ao = 3.23 and uoL/ao = 12.90. . . . 153

The quantity ﬁzT plotted as a function of Z. .............. 154

xiii

6.31

6.32

A.1
A.2
A.3

A.4
A5
A6
A.7
A8

The quantities uT and uT / u plotted as a function of the physical coor-
dinate :c. .................................
Comparison of fv proﬁles when Yoo = 0.5, YFF = 0.4 and uoL/ao =
4.85. Numerical Solution, 1 refers to the simultaneous solution of cou—
pled equations of 7', yo, yp and fv. Numerical Solution, 2 refers to
the solution of the soot volume fraction equation on the assumption of
analytical proﬁles for 1', yo and yp. The analytical solution of the soot

volume fraction equation involves the evaluation of the integral I; in

terms of exponential integrals .......................
The “top-hat” proﬁle for -d§R/dZ. ...................
The function H (Z) .............................
The location of the soot layer relative to the ﬂame location Z I. In (a)
Zf < 23-, in (1)) Z; = Z3. and in (C) Zf > ZR .............
1' and h proﬁles ...............................
The case 6 > 1. ..............................
The case 0 < 1. ..............................
The direction of increase of N RAZR with ﬂame movement .......

Comparison of numerical and analytical solutions for the extinction

radiation number ..............................

xiv

155

158
172
173

178
181
182
183
190

196

CHAPTER 1

Introduction

The inﬂuence of soot radiation on ﬂames has received signiﬁcant attention in recent
years [3],[4],[5],[6],[7],[8],[9]. Thermal radiation from combustion products and soot
lower the temperature of the ﬂame and decelerate chemical activities. The purpose
of this study is to critically assess the effect of soot radiation on diffusion ﬂames and
investigate whether under certain circumstances radiative energy losses can result in
an extinction of a diffusion ﬂame.

Soot radiation has considerable inﬂuence on ﬂames established in low gravity
situations. In normal gravity the combustion products and the ﬂame-generated soot
get convected away from the primary reaction zone due to buoyancy induced ﬂows.
However, in microgravity, there is no buoyant ﬂow and the combustion products and
soot reside in the vicinity of the primary reaction zone. In addition, the absence
of convective ﬂow results in longer residence time and more soot is produced in a
low gravity situation. Thus, the formation of greater amount of soot and its vicinity
to the primary reaction zone suggest stronger soot radiation effects in microgravity.
There is thus a possibility that the drainage of energy from the ﬂame by means of

soot radiation may be so great in microgravity that the ﬂame may no longer be able

to sustain itself. Thus a radiative extinction of the ﬂame is anticipated under such
situations [10].

Thermal radiation from a ﬂame can be due to (1) radiation from the combustion
gases at high temperature and (2) radiation from combustion generated particulates,
i.e., soot. According to the calculations of Grosshandler and Modak [11] for soot
volume fractions > 10'7 soot radiation is dominant. In the present work, gas radia-
tion has been neglected and soot radiation was modeled assuming the optically thin
limit. Kennedy et al. [12] have observed that for small ﬂames and for moderate soot
loadings the optically thin limit is appropriate for soot radiation. The purpose of our
investigation was to qualitatively and quantitatively study the effect of soot radia-
tion on ﬂames. In particular, the possibility of a radiative extinction was examined
thoroughly.

The soot radiation from a ﬂame depends on the soot volume fraction distribution
in the ﬂame, which is difﬁcult to predict. There are considerable uncertainties in
determining the soot formation and oxidation rates. In particular, the soot distri-
bution in a ﬂame depends on the fuel structure, the temperature distribution, the
inﬂuence of inerts, the pressure of the system, etc. Since soot radiation is intimately
coupled with the soot volume fraction distribution, it becomes difﬁcult to predict the
radiation from a ﬂame.

A review of the existing literature in the research area is presented in chapter 2.
The primary focus is on soot evolution, soot modeling and soot radiation.

In chapter 3 the inﬂuence of a hypothetical heat loss term of the form of a sech2
on a pure diffusion ﬂame established between two diffusing walls of fuel and oxidizer
was investigated. The thickness of the loss zone and its separation distance from the
ideal, inﬁnite reaction rate ﬂame location were parametrically varied. The inﬂuence
of increasing the intensity of the loss zone was also investigated.

The effect of a similar heat loss proﬁle on a diffusion ﬂame with convective fuel

blowing from the wall was also examined. Chapter 4 elucidates interesting results of
the effect of fuel blowing on the ﬂame in the presence of radiative losses.

An analytical model was developed for the soot layer proﬁle and thickness on
the assumption of inﬁnite reaction rate proﬁles for temperature and species mass
fractions. Chapter 5 outlines the basic assumptions of the model, its derivation and
comparison with numerical results. Then, a prescribed soot volume fraction proﬁle
was used to formulate a radiative loss term and its inﬂuence on the ﬂame structure
was examined.

In chapter 6 the comprehensive problem of soot radiation for a diffusion ﬂame
established between an oxidizer and a fuel wall was examined. Similar to chapter 4,
a convective fuel ﬂow from the fuel wall was assumed. A heat loss proﬁle was not
assumed in this case. Instead, the soot model of Moss and co-workers [1] was used and
a soot volume fraction equation was solved in conjunction with the coupled energy
and species mass fraction equations. Results indicate extremely interesting ﬂame
behavior due to radiative losses.

Chapter 7 brieﬂy states the conclusions of the current work. Recommendations

for future work are also outlined.

CHAPTER 2

Literature Review

2.1 Introduction

In this chapter a review of the existing literature pertaining to the research problem
is presented. As discussed in chapter 1, the inﬂuence of soot radiation on a diffusion
ﬂame depends strongly on the soot formation and oxidation chemistry. In the follow-
ing sections the soot evolution and burnout processes are reviewed, the soot radiation
effect is discussed and the effect of various parameters on soot radiation is analyzed

in the light of the existing literature.

2.2 Background

Most practical combustion systems burn in the diffusion ﬂame mode. In a diffusion
ﬂame the process of physical mixing of the reactants is generally intended to be much
slower than the chemical reaction between the fuel and the oxidizer. Consequently, the
ﬂame is (generally) kinetically controlled. By contrast, the constituents are already
mixed before they enter the combustor in a premixed ﬂame. In our research problem
we shall focus on diffusion ﬂames only; hence the following review pertains to diffusion

ﬂames.

2.3 Soot

Carbonaceous particles generated during gas phase combustion reactions are called
soot. Soot is formed because of incomplete combustion of fossil fuels and other organic
matter. Principal sources of soot emissions are coal burning furnaces, refuse burning,
coke production processes, wood burning in home ﬁreplaces, the open burning of
waste, and gasoline and diesel powered engines.

Under ideal conditions the combustion of hydrocarbons leads to mainly carbon
dioxide and water. Ideal conditions may be speciﬁed by stoichiometric composition
of the combustible mixture, i.e., the oxygen content of the mixture everywhere is
sufﬁcient to convert the fuel completely according to the formal chemical equation
CxHy + (a: + y/4)02—m:C02 + (y/2)H20. Under these conditions a maximum of heat
is released and a maximum of chemical energy is available for mechanical work.

In practical combustion devices such as industrial furnaces, gas turbines, or com-
bustion engines conditions locally deviate from ideality. When the locally available
oxygen is not sufﬁcient to convert the fuel according to the formal chemical equation
mentioned above, other products of incomplete combustion such as carbon monox-
ide, hydrogen, hydrocarbons and soot are produced in addition to carbon dioxide and

water.

2.3.1 How Does Soot Affect Us?

Emission of soot to the atmosphere from various industrial combustion processes is
undesirable for various reasons. Soot particles contribute to reduced atmospheric
visibility and increased particulate fallout. Also, emission of soot is often associ-
ated with carcinogenic polycyclic aromatic hydrocarbons. Consequently, the adverse
health effect is an important issue regarding soot particle emission; the emission of

soot in the atmosphere is unquestionably hazardous and undesirable. However, the

next question is, should soot formation be avoided altogether? This question has var-
ious answers depending on the combustion process being considered. Soot emission
from a practical combustion appliance such as an internal combustion engine reﬂects
poor combustion conditions and a loss of efﬁciency. It has deleterious consequences
for the maintenance of the device. For such devices, the designer would like to avoid
soot formation altogether. The same objective also applies in case of ﬁres, whose
mechanism of propagation often involves radiant transfer from hot soot particles.
However, for ﬂames in furnaces and boilers the generation of soot is necessary
as a radiation source for efﬁcient heat transfer. For such ﬂames, the principle is to
generate as much soot as necessary in such a way that it can be burnt up again in
the available time. The carbon black industry is in sharp contrast to the above and

the objective is to produce as much soot as possible by fuel pyrolysis.

2.3.2 Appearance

Soot generated in combustion processes is not a uniquely deﬁned substance [13],[14].
It is normally black. The ﬁrst soot particles result from condensation reactions in the
gas phase. These particles, as well as the individual primary particles that compose
soot aggregates can be reasonably approximated as spherical [15]. The elementary
particles adhere to each other to form straight or branched chains. These chains

agglomerate and form the visible soot ﬂocculates, generally as a ﬂuffy substance.

2.3.3 Morphology and Chemical Structure

Research has been conducted in laminar and turbulent, (rich and lean) premixed and
nonpremixed ﬂames, momentum and buoyancy driven ﬂames, stirred reactors, spray
ﬂames, shock tubes, and common combustor devices such as engines and furnaces.

There are many similarities in the morphological characteristics [16] for the soot

produced from such different conﬁgurations. The similarities in soot morphology
suggest that a common developmental history governs the formation process, even
in very different combustion conﬁgurations [15]. Besides carbon, soot particles also
contain hydrogen (10 to 25%), oxygen (00%)) and nitrogen atoms (0(0.1%)) [15].

Electron diffraction indicates the presence of C-C bonds in soot [17].

2.3.4 Characterization of the Soot Distribution

Characterization of the soot distribution is done using three important parameters.
The soot volume fraction, fv, is the volume of soot present in a unit volume of gas
and is expressed in units of m3,oo¢/msga,. The soot particle size is characterized by a
length scale d which equals the diameter for a spherical particle. The particle number
density is denoted by n and is deﬁned as the number of soot particles per unit volume
of gas. The three parameters fv, d and n are related and for spherical monodisperse
soot particles fV = mrd3 / 6. In general, soot particulates are neither spherical nor
monodisperse. The preceding relation, therefore, may be considered as yet another

generic measure of soot.

2.3.5 Soot Evolution

There are two important stages of soot particle formation, viz., particle inception
and particle growth. The following discussion provides a brief description of the two
stages. However, it has to be kept in mind that the above classiﬁcation is some-
what mechanistic and in an actual combustion process such distinctions are not very
clear [15].

(1) Particle Inception The ﬁrst condensed phase material is generated from the fuel

 

molecules via their oxidation and/ or pyrolysis products [14]. Such products include

acetylene and its higher analogues and polycyclic aromatic hydrocarbons. These two

types of unsaturated hydrocarbons are often considered the most likely precursors of
soot in ﬂames [14]. Because soot is formed in the intermediate stages of chemical
decomposition the oxidation and pyrolysis products alluded to are reaction inter-
mediates like aldehydes, various radical compounds, alcohols and other such trace
materials.

The condensation reactions of such species often lead to the ﬁrst recognizable
soot particles known as nuclei. The ﬁrst particles are very small ((1 < 2 nm); for this
reason, even a large number density results in negligible soot loading in the region
of their formation, which is generally conﬁned to the vicinity of the primary reaction
zone (i.e., where most of the heat release occurs).

(2) Particle Growth Particle growth takes place by means of both surface growth

 

and coagulation. Surface growth means gas species become attached to the soot
particle surface and incorporated in the particulate phase. Haynes and Wagner [14]
have remarked that for soot formation to occur the species with the correct hydrogen
content have to condense followed by subsequent dehydrogenation. Surface growth
reactions lead to an increase in the amount of soot but the number density remains
the same. In the process of coagulation particles grow by colliding and coalescing,
thereby decreasing the number density. Here the volume fraction remains the same.
Particle growth is therefore generally considered to be the result of simultaneous
surface growth and coagulation.

Almost all of the soot mass is provided by surface growth reactions. However,
the inception process is calculated to be the rate limiting step in the formation of
soot. This has been explained from different perspectives in the literature. Fuel
pyrolysis leads to particle inception; some researchers believe that this may be the
most important factor [18], [19], [20], [21]. Also important is the formation of initial
surface area delivered from the inception region and available for growth [22], [23].

However there is at least one exception to the widely held belief that particle

inception controls soot production. Based upon modeling considerations, Kennedy
et a1. [12] disagree that particle inception controls soot production. Instead, they
propose that surface growth is the most important factor. The relative importance
of surface growth and particle inception is a key issue in quantitative models of soot

formation.

2.3.6 Soot Oxidation

The link between soot production within the ﬂame and the smoke yield from the ﬂame
is the oxidation of soot particles. Soot is oxidized in the high temperature combustion
zone leading to decreased soot mass. As observed by [15] a great many experimental
measurements have been conducted in the last twenty years on the soot formation
process in different ﬂame conﬁgurations. Yet, only a handful of studies have been
performed on soot oxidation processes, and most of these only quite recently.

The particle destruction rate depends on the ﬂame structure, the temperature
ﬁeld and the concentration distribution of oxidizing species, principally 0H, 02 and
0 atoms. N agle and Strickland-Constable [2] derived a semi-empirical formula for
pyrolytic graphite oxidation by 02 for a temperature range 1300 — 2300 K. In their
analysis they assumed that oxidation by other species was negligible.

In many combustion conditions it is clear that OH is also an important oxidizer
of soot particles [24]. For soot to escape from a nonpremixed ﬂame, it must pass
through a relatively hot reaction zone where the concentration of OH is relatively
large.

Puri and Santoro [24] have examined the question of how much CO is produced
from the oxidation of soot by OH and 02 in laminar hydrocarbon ﬂames. They
derived an expression for the soot oxidation rate (or the CO production rate) by

applying a fundamental kinetic theory approach.

10

2.3.7 The Inﬂuence of Fuel Structure

It has been observed that smaller the hydrocarbon molecule, the greater the resistance
to smoke emission.

An increase of the fuel ﬂow rate increases the height of a diffusion ﬂame, say a
Bunsen-type ﬂame for deﬁniteness. The character of the ﬂame also changes. Initially
the ﬂame is almost completely blue. As the ﬂow rate increases the ﬂame height grows
and the ﬂame tip becomes yellow. Further increases in fuel ﬂow rate result in the
appearance of an orange zone. A subsequent increase in fuel ﬂow rate leads to a
critical value when soot escapes the luminous zone. The sooting tendency is typically
quantiﬁed by measuring this critical smoke point height.

The measurement of smoke points of various fuels has been utilized to rank dif—
ferent fuels in order of increasing sooting tendencies; thus polyaromatics > aromatics
> alkynes > alkenes > alkanes > alcohols [15]. Fuels with a greater tendency to soot

emit smoke at lower fuel mass ﬂow rates.

2.3.8 Effect of Pressure

The effect of pressure on soot formation in diffusion ﬂames has been investigated
over a wide range of conditions. Generally speaking, low pressures reduce carbon

formation while high pressures promote it [14].

2.3.9 Inﬂuence of Additives

Dilution of fuel ﬂow by addition of inert gases such as Ar, He and N2 generally
decreases the tendency to sooting [25]. If sufﬁcient diluent is added, carbon luminosity
can be suppressed altogether [26]. The possible reason for this is the substantial
temperature reductions in ﬂames in such situations.

When 002 or H20 is added to the fuel, there is a considerable reduction in

ll

soot-forming tendency [25] and a concentration of 45% CO; completely suppresses
luminosity in methane/air diffusion ﬂames [26]. McLintock [25] has suggested that
the inﬂuence of C 02 and H20 is exerted primarily in the soot oxidation zone where
these species presumably promote soot burnout.

Some additives promote soot formation. Foremost among these are the halo-
gens, particularly bromine [18], [27]. It has been suggested that these species act by
catalyzing radical recombination, thus neutralizing excess 0H radicals which could

otherwise oxidize soot or soot precursors [14].

2.3.10 Inﬂuence of Oxygen Addition

The effect of oxygen addition to the fuel is complex [14]. In ethylene ﬂames, small
additions result in pronounced increased soot emissions [18], [28]. The effect of the
increasing yield is not purely thermal as it is far greater than that obtained at the
same maximum ﬂame temperature produced by oxygen enrichment of the air [14].
In the case of other fuels, both soot promotion and inhibition have been observed
as the result of oxygen addition. Jones and Rosenfeld [29] concluded that ethylene is
the exception and that, for fuels such as propane, butane, and even propylene, oxygen

suppresses soot emissions.

2.4 Soot Models

A large number of experimental studies of soot formation and burnout in diffusion
ﬂames have been carried out in the last two decades or so. However the effort at
developing suitable models for describing the soot processes in a ﬂame has been
relatively less [15]. In what follows we take a look at some of the important modeling
efforts that have been undertaken.

Kennedy and coworkers [12] have proposed a soot formation model for laminar

12

diffusion ﬂames based on a correlation between soot surface growth rates and the
mixture fraction. Detailed chemistry was not used, instead the mixture fraction was
calculated. The temperature, density and the gas composition were determined as
functions of the mixture fraction. Axisymmetric, boundary layer forms of the mo-
mentum equation were numerically integrated along with the soot volume fraction
equation. An energy equation was not solved. Oxidation by both molecular oxygen
and OH were included in the model. The thermochemistry of the ﬂame was de-
termined from the results of a detailed laminar counterﬂow diffusion ﬂame code. A
constant soot number density was assumed. The calculations were carried out for the
axisymmetric, laminar ethylene diffusion ﬂame of [30]. Good agreement was obtained
with the measurements for two different experimental conditions. However it was
found that the decrease in temperature that occurs with radiative energy loss has a
signiﬁcant impact on the soot loadings in these ﬂames. Therefore, a more thorough
approach to the problem of accounting for radiation in a diffusion ﬂame was deemed
necessary.

In a subsequent work [31] the energy equation was solved along with the continuity,
momentum and mixture fraction equations. A radiative loss term was included in the
energy equation based on the assumption of the optically thin limit. Calculations were
carried out for the axisymmetric laminar jet diffusion ﬂame [30] and a Wolfhard-
Parker two-dimensional ﬂame [32]. The temperature, density, and viscosity were
determined as functions of the mixture fraction and the enthalpy by using a type
of constrained equilibrium chemistry model. Further work includes the prediction
of sooting heights of laminar diffusion ﬂames of Santoro et al. [30]. The agreement
was good in all cases. Their results indicated that OH was the dominant oxidizer of
soot low in the co-ﬂow axisymmetric ﬂames but as the ﬂame tip was approached the
oxidation by O; became more important.

Moss and coworkers have developed a two-equation model for soot processes in

13

laminar diffusion ﬂames [1], [33]. They used a Wolfhard-Parker burner in both two-
and three-slot conﬁgurations producing a substantially two-dimensional thin ﬂame
suitable for laser extinction measurements of soot volume fraction. The model pa-
rameters controlling the proposed rate processes for nucleation, surface growth and
agglomeration were determined by a comparison between detailed model prediction
and experimental measurement. Both prediction and experiment indicated that soot
formation is restricted to a comparatively narrow range of mixture fractions (between
0.06 and 0.2). The model parameters incorporated in their analysis must be adjusted
for each fuel. Predictions of the soot ﬁeld were compared to experimental data ob-
tained for laminar ethylene/ air and methane/ air diffusion ﬂames. For methane/ air
diffusion ﬂames it was observed that in contrast with ethylene/ air diffusion ﬂames the
growth of soot volume fraction with height (and hence residence time) is non-linear.

Leung et a1. [34] outlined a simpliﬁed reaction mechanism for the formation,
growth and combustion of soot particles in laminar nonpremixed ﬂames. The model
was combined with detailed gas phase chemistry. The soot nucleation and growth
reactions were linked to the gas phase by presuming that pyrolysis products, acetylene
in their case, and not the fuel itself, are of primary importance in the soot formation
process. The model involves the solution of two additional conservation equations
for soot mass fraction and soot number density. They assumed that the number of
active sites present locally in the ﬂame is proportional to the square root of the total
surface area available locally in the ﬂame. The radiative heat loss is modeled in a
simple manner by adjusting the adiabatic ﬂame temperature by means of a heat loss
factor. Model predictions were compared with the experimental data of Vandsburger

et al. [35] for counterﬂow diffusion ﬂames. The agreement was quite good.

14

2.5 Soot Radiation

Soot emits in a continuous spectrum in the visible and infrared regions and can
often double or triple the heat radiated by the gaseous products alone [36]. For
soot radiation scattering can be neglected compared to absorption [36]. Since soot
particles are very small, they are generally at the same temperature as the ﬂame [37].
An analysis of transient radiative cooling of a strongly radiating turbulent diffusion
ﬂame was carried out by Gore and J ang [8]. They mention that for strongly radiating
ﬂames, where up to 60 percent of the heat of reaction may be lost by radiation,
'a detailed treatment of the radiation heat transfer is needed. The radiative source
term was expressed as the energy absorbed minus the energy emitted by a small
local participating volume. The energy absorbed was calculated from the large-scale
radiation field by integrating the ﬂux over the surface of the small volume. The energy
emitted depends on the temperature and absorption coefﬁcient of the material in the
small volume. Gas phase radiation was neglected and soot radiation was included
using the Rayleigh approximation for soot particles. Two representative distributions
of soot volume fractions were used. A unique ﬂame structure involving an inﬂection
point in the temperature proﬁle near the soot layer was observed for strongly radiating
ﬂames. This is caused by the transfer of energy to the soot layer by diffusion from
both sides balancing the high radiative loss.

In a later study Gore et al. [38] studied the structure of turbulent, non-premixed,
strongly radiating acetylene/ air ﬂames. The analysis extended the laminar ﬂamelet
concept to include the effects of local radiative heat loss / gain. Measurements of mean
and ﬂuctuating emission temperatures and radiation intensities and data concerning
ﬂame structure were used to evaluate the predictions. Results showed good agreement
between measurements and predictions of ﬂame structure. In a related study [39]

speciﬁc absorption coefﬁcients of soot particles were reported in the infrared region

15

for three different fuels with varying C / H ratios.

A coupled radiation and soot kinetics calculation was carried out by Sivathanu
and Gore [9] for laminar acetylene/ air and acetylene-methane/ air diffusion ﬂames.
The simpliﬁed soot model of Fairweather et al. [40] was used for the soot kinetics.
The predicted soot volume fractions were found to be in reasonable agreement with
measurements. It was found that the use of a constant radiative fraction in strongly
radiating ﬂames is not effective for predicting the observed trends in axial and radial
distributions of soot volume fractions. The predicted temperature proﬁles support

the structure of strongly radiating ﬂames discovered earlier [8].

2.6 Diffusion Flames in Microgravity

In low gravity, or microgravity (pg), many combustion phenomena can be studied
to yield more insight into the fundamental processes. Combustion in normal gravity
creates buoyancy-induced ﬂows through the production of hot gases, which are less
dense than air. Suppressing such ﬂows in microgravity helps researchers in several
ways. First, the microgravity environment makes experiments easier to model, thus
making it a better environment for testing theories. Second, the virtual elimination
of buoyant ﬂows permits the study of phenomena which are obscured by gravity.
An interesting case of microgravity diffusion ﬂames pertains to the investigation
of candle ﬂames [41]. The ignition and behavior of candle ﬂames was observed pho-
tographically in free-fall (drop tower) tests under 19% — 25% 02 concentrations in
a nitrogen-diluted, 1—atm environment. In normal gravity a candle ﬂame assumes a
tear-drop shape. However, in microgravity there is no “up” or “down” and the ﬂame
tends toward sphericity. In normal gravity, the buoyant ﬂow removes combustion
products from the primary reaction zone and supplies fresh oxidizer. For a micro.

gravity candle ﬂame this transport does not take place and consequently the supply of

16

fuel and oxidizer are diminished. Hence the ﬂame temperature is lowered and the can-
dle in microgravity produces a ﬂame of much lower power. Due to diminished ﬂame
temperature, little or no soot forms. The soot in microgravity is conﬁned within the
fuel-rich region deﬁned by the blue zone. In normal gravity, soot convects across the
blue reactive zone which produces a much larger visible ﬂame. It was also observed
that the main reaction zone, as indicated by the visible blue region, is much farther
away from the wick. This distance, referred to here as the ﬂame standoff distance,
gives an indication of the magnitude of the heat ﬂux from the ﬂame to the liquid fuel
in the wick. In normal gravity, this distance is about 1 mm at the base of the ﬂame;
in microgravity it is about 5 mm.

However it will be presumptuous to conclude from these experiments that micro-
gravity ﬂames are in general less sooty than their normal gravity counterparts. As
noted in [42] the observed microgravity diffusion ﬂames are longer, wider and often
sootier than their normal gravity counterparts. They are dimmer and more reddish,
which indicates a lower ﬂame temperature. It was also remarked in [42] that the
thermal radiation from a microgravity ﬂame and its surroundings can be an order of
magnitude greater in microgravity than in normal gravity. However, at low oxygen
concentrations, blue, soot-free ﬂames appear in microgravity, whereas the identical
normal gravity ﬂames do not show any signiﬁcant reduction in soot formation at low
oxygen concentrations.

An excellent review of microgravity combustion research has been published re-
cently [43]. It has been pointed out in the review that under atmospheric conditions,
diffusion ﬂames are buoyant and an increase in velocity is observed on moving away
from the burner exit. However, for a weakly buoyant condition the velocity rapidly
decays near the burner exit. The streamwise velocities are roughly inversely propor-
tional to the distance from the burner exit for nonbuoyant ﬂames. This tends to

increase the effectiveness of soot oxidation processes relative to soot growth processes

17

for nonbuoyant ﬂames in comparison with buoyant ﬂames. Also, residence times in
nonbuoyant ﬂames are signiﬁcantly higher than for buoyant ﬂames of comparable size,
providing longer absolute times for soot nucleation, growth and oxidation. Thus, it
is remarked in [43] that “any resemblance between soot processes within nonbuoyant
and buoyant laminar diffusion ﬂames clearly is fortuitous”.

The differences in soot processes of microgravity and normal gravity diffusion
ﬂames have been found to be very important in a recent work [44]. Experiments
were conducted to investigate the behavior of soot particles in diffusion ﬂames under
microgravity conditions using a 490 m drop shaft (10 second microgravity duration)
in Hokkaido, Japan. Butane jet diffusion ﬂames and ﬂames arising from the com-
bustion of paper were observed in microgravity. The oxygen concentration of the
surroundings, the butane ﬂow rate, and the burner diameter were varied as experi-
mental parameters. From the observation of transmission electron microscope (TEM)
images it was found that a large number of luminous spots appear in diffusion ﬂames
in microgravity. The diameters of the agglomerated particles are approximately 0.1
mm, 200 to 500 times as large as those generated under normal gravity. These parti—
cles are the resultant agglomeration of a large number of primary particles. Local ﬂow
velocity, residence time of generated particles in the generation region, and oxygen
concentration of the surrounding ambient dominate the agglomeration and growth
of the soot particles. Therefore, these particles are formed in the limited areas of
diffusion ﬂames where the aforementioned conditions are satisﬁed. The investigation
of [44] also veriﬁes that microgravity diffusion ﬂames have a much larger volume than
those under normal gravity.

Interesting recent work on methane and ethylene ﬂames has been done by Atreya
et al. [45]. A small porous sphere made from a low density and a low heat capac-
ity insulating material was used to uniformly supply fuel at a constant rate to the

expanding diffusion ﬂame. A theoretical model was formulated on the assumption

18

of inﬁnite reaction rate and unity Lewis number. Both experimental and theoretical
results show that as the ﬂame radius increases, the ﬂame expansion process becomes
diffusion controlled and the ﬂame radius grows as the square root of time.

In a related work Pickett et al. [46] studied the characteristics of methane, ethy-
lene and acetylene ﬂames. For the same ﬂow rate of fuel the ethylene and acetylene
ﬂames were found to be much sootier and smaller. For all the fuels the ﬂame is ini-
tially blue (non-sooty) but becomes bright yellow (sooty) under pg conditions. Later,
as the pg time progresses, the ﬂame grows in size and becomes orange and less lu-
minous and the soot seems to disappear. An explanation of the above phenomenon
was offered on the basis of some theoretical calculations. The soot volume fraction
ﬁrst quickly increases and later decreases as the local concentration of combustion
products increases. Essentially, further soot formation is inhibited by the increase in
the local concentration of the combustion products and soot oxidation is enhanced.
Thus, at the onset of pg conditions, initially a lot of soot is formed in the vicinity of
the ﬂame front (the outer faint blue envelope) resulting in bright yellow emission. As
the ﬂame grows, several events reduce the ﬂame luminosity: (i) the soot is pushed
toward cooler regions by thermophoresis. In fact, for sootier fuels this leads to the
formation of a soot shell, (ii) the high concentration of combustion products left he-
hind by the ﬂame front inhibits soot formation and promotes soot oxidation, (iii)
the dilution and radiative heat losses caused by the increase in the concentration of
combustion products reduces the ﬂame temperature which in turn reduces the soot

formation rate and ﬂame luminosity.

CHAPTER 3

Inﬂuence of a Simple Heat Loss

Proﬁle on a Pure Diffusion Flame

3.1 Introduction

The interaction between the structure of a diffusion ﬂame (DF) and the ﬂame ra-
diation is quite complex. As discussed in chapter 2, soot is formed and oxidized in
a diffusion ﬂame as a consequence of a variety of physical and chemical processes.
There are considerable uncertainties in the description of soot processes in a ﬂame
and the soot evolution mechanisms are not completely understood. Hence, the so-
lution of the complex problem of diffusion ﬂame - soot radiation interaction is very
involved. The energy, species and soot volume fraction equations are all coupled and
contain nonlinear source terms. We chose not to solve the complex problem at the
very outset. Instead, we investigated the effect of a simple and contrived heat loss
proﬁle on a pure diffusion ﬂame established between two diffusing walls of fuel and
oxidizer. This chapter is essentially the next logical step in the generalization of the
model outlined in Appendix A.

A review of pure diffusion ﬂames without heat losses is presented ﬁrst. In the

following sections we deﬁne the problem geometry, describe the particular form of

19

20

the heat loss proﬁle used, formulate the conservation equations, brieﬂy indicate the

numerical method used and discuss the results.

3.2 Review of Pure Diffusion Flame Results

In a diffusion ﬂame the characteristic ﬂow time is much greater than the character-
istic chemical reaction time. This implies that the chemical reaction is much faster
than the transport of species to the ﬂame unless the ﬂame is near or approaching the
extinction stage. A pure diffusion ﬂame is established when both oxidizer and fuel
are transported to the ﬂame by means of diffusion only. N o convective ﬂow is present.
Some important characteristics of pure diffusion ﬂames have been discussed in [47].
Using the activation energy asymptotic (AEA) method [47] provides an analysis of
the detailed nature of the temperature and reaction rate proﬁles. It was observed
that the maximum of the reaction rate proﬁle usually will not coincide with the tem-
perature proﬁle maximum. They may be close, and the maximum of the temperature
will be in the vicinity of the maximum of reaction rate, but they will almost never
coincide. The only exception is the symmetric ﬂame for which the overall stoichio-
metric coefﬁcient, ¢(= qup/ Yoo), equals unity. This study also shows for a fuel-rich
ﬂame that Z ISZTSZT, i.e., the peak of the reaction rate proﬁle (Z,) lies between the
Burke-Schumann ﬂame location (Z f) and the peak of the temperature proﬁle (Z7)
for fuel rich conditions. For oxidizer-rich conditions ZT_<_Z,._<_Zf. It may be argued
that in the thin-ﬂame limit all diffusion ﬂames are “pure” diffusion ﬂames because
the mixture fraction transformation discussed in Williams [48] produces an equation
resembling TzzOCIVZI-zw, where IVZ| is the magnitude of the mixture fraction gra-
dient perpendicular to the ﬂame. However, IVZI depends strongly on the heat ﬂow
conditions and in effect introduces a new parameter that must be accounted for in

a complete analysis. Hence, though the value of IVZfI (i.e., IVZI evaluated at the

21

ﬂame sheet) may be “buried” into a suitably redeﬁned Damkéhler number it must of

course be “resurrected” when later conducting a full examination of the problem.

3.3 Problem Geometry

Radiation
Zone

 

Yo=0 Y0=Y00
YF=YFF YF=0
T=To T=T0

"[qunulmmmumunIlumnmlmu

 

.l||||]||||||||||[|||||||||||ll||ll||[|||l|||||l[||||l|
ll

 

:0 X = L
(Z= 1) (Z= 0)
Diffusive ﬂux of fuel Diffusive ﬂux of oxidizer

Figure 3.1. The problem geometry.

Figure 3.1 schematically depicts the problem geometry. The physical coordinate for
our one dimensional problem is designated by 1:. The fuel wall and the oxidizer wall
are located at a: = 0 and a: = L respectively. Both the walls issue diffusive ﬂuxes of

the respective constituents. A diffusion ﬂame is established between the two walls. A

22

soot layer is assumed to exist on the fuel-side of the DF, consistent with experimental
observations [30]. The walls have the ambient temperature To. There is no fuel on the
oxidizer wall and no oxidizer on the fuel wall. The fuel and oxidizer mass fractions at

the respective walls are speciﬁed to have values Ypp and Y00 as shown in Figure 3.1.

3.4 Choice of Parameter values

The combustion reaction under consideration is assumed to be a global, one-step
chemical reaction of the form F + VO—>(1 + V)P, where F denotes the fuel and 0
denotes the oxidizer. Methane is nominally the fuel under consideration and oxygen
is the oxidizing specie, although we must recall that real methane-oxygen reactions
require of the order of 100 reaction steps and individual property choices for the
separate species. The fuel-oxidizer mass ratio, V, is four for the methane-oxygen
combustion reaction. The combustion products are denoted by P. A suitable set of
parameter values must be used to generate a reasonable range of Damkﬁhler number
and ﬂame temperature values. The adiabatic ﬂame temperature is given by T; =
To + Qprp/ [0,,(1 + 95)], where Qp is the heat release per unit mass of fuel from
the combustion reaction and Cp is the speciﬁc heat of the mixture. The overall
stoichiometric coefﬁcient is denoted by 45 and is given by VYFF/ Yoo. However, the
use of the above formula produces unrealistically high adiabatic ﬂame temperature
values. Thus, the above equation for Tf was modiﬁed to produce a practical range of
adiabatic ﬂame temperatures.

A set of realistic hydrocarbon combustion ﬂame temperatures was used from the
work of Wichman [49] for the analysis of ﬂame spread over thermoplastics. The
idea there was that the fuel mass fraction can not reasonably be determined at the
surface but a more-or—less generic ﬂame temperature can still be evaluated. This

ﬂame temperature varies only with the free-stream oxidizer mass fraction Yoo. The

23

value of Ypp, i.e., the fuel mass fraction in the hypothetical fuel stream for our present
calculations was assumed to be 0.85. The oxidizer mass fractions (Yao) and the ﬂame

temperatures (Tf) are tabulated in Table 3.1.

Table 3.1. Flame Temperature versus Yoo data

 

Yoo 0.211 0.233 0.247 0.276 0.329 0.432 0.533 0.727 1.0
T f 2137 2230 2295 2385 2515 2684 2789 2919 3026

 

 

 

 

 

 

 

 

 

 

 

 

 

A fourth order polynomial was ﬁtted to the above data to obtain

T, = 486.66 + 12230.85Y00 — 257.28.641/002 + 25360.021’003 — 9323.0Yoo“ (3.1)

Using the above expression, we generate more points for a (Y00,Tf) plot.

We calculate Qp by using the relation Qp = (Tf — To)Cp(1 + ¢)/YFF for T, =
2137 K, Yoo = 0.211, Ypp = 0.85 and To = 298 K. The calculated value of Qp is
11959.43 K J / kgK . We now introduce a modiﬁed formula for calculation of the ﬂame

temperature and write
QFYFFf(Yoo)
CP(1 + 05)

 

n=n+ an

Next we calculate the values of the modiﬁcation factor, f (Yoo), by using the above
expression. The calculated value of Q p and the (Yoo, Tf) data obtained using equa-
tion 3.1 were utilized for this purpose. On obtaining f (Yoo) data, we use an expo-
nential ﬁt of the following form to arrive at a functional relationship between f and
yea:

f = 1.25exp(—2.99Yoo) + 0.33 (3.3)

Finally, we use the above expression for f (Yoo) to calculate T f for any set of Y00

and Ypp values in equation 3.2. A plot of T f versus Yoo is shown in Figure 3.2. The

24

4000 r.......

 

 

 

 

 

 

 

 

/
’ “AAAAAAAAAAAAAAAAAAAAAA A
M ’W ‘aAAHH ‘ ‘““““‘AAA‘AA‘AAA.\AAAAAAA AAAAAAAA
‘A

['52- 2000 7 .A‘“ ‘i

.___s. YFF=1.O

3.!“ A———¢ YFF=O°25

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.2. Flame Temperature as a function of Yoo for different values of Ypp.

25

YFF values corresponding to the different curves in the plot range from 0.25 to 1.0.
The lowest curve is for Ypp = 0.25. The curves above are for Ypp = 0.30, 0.35, 0.40
etc. It has to be noted that for YFF = 0.25 and 0.30 the peak ﬂame temperature does
not occur at Yoo = 1 because of the slight local maximum in the range 0 < Yoo < 1.
For this reason we shall not use these curves. However, for higher values of Ypp we
do obtain temperature proﬁles whose maxima occur at Yoo = 1. These proﬁles will
be used. Also, we note that since we are interested in the effect of soot radiation on
diffusion ﬂames, we are not concerned with low values of YFF which do not produce
signiﬁcant amounts of soot. Hence, in our analysis, Ypp values of 0.30 and lower are
not used.

The parameter values in the work of Tzeng et al. [50] were used in this disser-

tation. The important values are shown in Table 3.2. These values were used in all

Table 3.2. Parameter values

 

 

 

 

 

 

 

 

 

 

 

Speciﬁc heat C}, 1.35 J / kgK
Thermal diffusivity 010 1.24 x 10“ mf/s
Fuel-oxidant mass ratio 11 4.0

Pre—exponential factor A 5 x 107 1 /3
Activation energy E 121, 841.7 K J / kmol
Heat release Q p 11959.43 K J / K gK

 

the chapters of this dissertation except for the pre-exponential factor in chapter 6, as

noted in section 6.2.

26
3.5 Formulation

Here we write the equations and boundary conditions for conservation of energy and

species. The energy conservation equation is

, d
pcpm + an] = (Arr). + 52pr — 7:5, (3.4)

with boundary conditions T(:r = 0) = To and T(;r = L) = To, where To is the
temperature at the fuel and oxidizer walls, assumed to be 298 K. Here T is the
temperature, p is the density, Op is the speciﬁc heat of the mixture, A is the thermal
conductivity and u is the velocity. The volumetric radiative heat loss term is —dqR/d:r
with units W/m3. The heat release due to combustion is Qp and rim is the reaction
rate term. An Arrhenius type expression was assumed for the one-step irreversible
reaction, so the reaction rate is u°2p = pAYoYpexp—E / RT. The quantity A denotes
the pre-exponential factor. The thermal conductivity is A. The oxidizer mass fraction
equation is

PlYO. + Ill/0.] = (PDoYozh — VU'JF, (3-5)

with boundary conditions Yo(:r = 0) = 0 and Yo(:c = L) = Yoo. Here Do is the

mass diffusivity of the oxidizer. Similarly the fuel mass fraction equation is

PIYF. + ”Val = (PDFYF.)x - “317, (3-6)

with boundary conditions Yp(a: :2 0) '2 YFF and Yp(:c = L) = 0; Dp is the mass
diffusivity of the fuel.

The above equations are now transformed to a mass coordinate system. The
transformed coordinate is Z = 1 — s/so where s = f0I pdzr and so = fol” pdx. We note

that Z = 1 when a: = 0 and Z = 0 when :r = 1. The coordinate Z happens to be

27

identical to the mixture fraction coordinate for our simple problem. The following

expressions hold for the above transformation:

gt. = [_slo((,m)0 — (pu).) + 5301000 — (pu)z.)]5%|. + gglz (3-7)
and
3 _ p 3
alt “ -535h (3'8)

Since we are considering a pure diffusion ﬂame, (pu)x=o = (pu),,=L = 0, i.e., there are
no convective ﬂows from the walls. Application of these operators to the energy and

species equations gives

 

 

 

PA QF . 1 61012
T: ——T —— .
t— 0103(2)zz+pprF+-——Cp$0 _d—Z (3 9)
assuming p) to be a constant,
2D mi)
Y0. = ”0 001/022 — F, (3.10)
30 p
assuming p2D0 to be a constant, and
2D 11)
YE}: p0 2F0 YFZZ — —pF-, (3.11)

assuming psz to be a constant. In equations (3.9)-(3.11), we have also assumed
the Lewis number to be unity and we have considered equal species diffusivities. The
quantities with subscript 0 correspond to the reference condition.

We now introduce the nondimensional variables 1' = (T — T0)/(Tf — To), yo =
Yo/Yoo, yp = Yp/Ypp, 5 = s/(poL). Consequently, 30 = so/(poL)=fo1 pdx where

,5 = p/po and :7: = m/ L. After some rearrangement, our nondimensional equations

28

simplify to

l - NR d6}?
= — —— .12
T? ggTzz+QFDT+ 30 dZ , (3 )
1

yo;- = gonz — 457373 (3.13)

O

1
9F? = gszz — D73 (3.14)

0

where r is the nondimensional reaction rate and N3 is a radiation number evalu-
ated as the ratio of the reference radiative and conductive ﬂuxes, given by NR =
qR’rcf/(A0(Tf - To) / L). The thermal conductivity at the reference condition is de-
noted by A0. The quantity ’D is the Damkéhler number given by trey/tchem, where
the reference diffusion time tn, = Lz/ao and the characteristic chemical time tchem =
1 / [AYOOeXp(—E/RTf)]. The nondimensional quantity 63 is given by qR/qnﬂf, where
gig”, is a reference radiative heat ﬂux. The nondimensional heat release, Op, is given
by QFYFF/[CP(TI — To)] and equals (1 + 05) since the adiabatic ﬂame temperature is
deﬁned as T; = To + Qprp/ [0,,(1 + 45)]. We note that in the prefactor multiplying
the reaction term of equation 3.12 we do not utilize the temperature correction dis-
cussed in section 3.4. In addition, we have deﬁned f = t/tref. The nondimensional
reaction term, 1', is written in the form r = yoypexp[-—ﬂ(1 — T)/(l —a(1 —1'))], where
a = l—To/Tf and ﬂ = Ea/(Rqu); E is the activation energy and Ru is the universal

gas constant. The quantity 3 is known as the Zeldovich number.

3.5.1 Inﬁnite Reaction Rate (IRR) Solutions as Initial Con-
ditions

Equations 3.12-3.14 are the governing conservation equations for 1', yo and yp for
the case of ﬁnite rate chemistry. The equations become much simpler when the
reaction rate is inﬁnite. In this case, all fuel reaching the ﬂame surface is consumed

instantaneously, and similarly for the oxidizer. Thus no fuel exists on the oxidizer

 

29

side and no oxidizer exists on the fuel side, i.e., yoyp = 0 on both sides of the
ﬂame. The energy equation can now be solved in two adjacent domains, the oxidizer
side (OSZSZf) and the fuel side (ngz 31) of the ﬂame. The ﬂame location is
designated by Z I. For our simple problem the mass coordinate Z coincides with the
mixture fraction, a conserved scalar. In the absence of radiative losses the steady
state energy equation becomes 7'22 2 0. Since at the ﬂame 1' = 1, the solution of the
steady state energy equation (Tzz = 0, since NR = 0 and the reaction term can be
excluded) for the inﬁnite reaction rate (IRR) situation gives 1' = Z / Z f for OSZSZI
and T = (1 — Z)/(1 — Z;) for Z;_<_ZSl. Similarly, yo and yp can be solved for the
inﬁnite reaction rate situation and we get yo = (l — Z) — (1 — Zf)T for 0£Z_<_Zf and
yp = Z — ZfT for ZISZSI.

Next, we must evaluate Z f, the coordinate location of the (IRR) ﬂame. As men-
tioned, Z is the mixture fraction coordinate, deﬁned as Z = (¢yp + 1 — yo) / (65 + 1).
At the ﬂame, yo and yp are both zero so that Z f = 1 / ((15 + 1). With the knowledge
of Z I the nondimensional temperature and species equations can all be determined
exactly. The proﬁles so obtained are used as initial proﬁles for the numerical solution

of the transient conservation equations (3.12)—(3.14).

3.5.2 Simple Heat Loss Proﬁle

As shown in Appendix A the simplest model heat loss proﬁle is the “top hat” proﬁle
used therein. Because of the discontinuous derivatives at the edges of the top hat
proﬁle, it is not as convenient for numerical reasons as a smooth and continuous heat
loss proﬁle. For primarily this reason, the proﬁle that we shall use here is of the form
of a seek2 in mixture fraction space, viz.,

£125

— dZ = sech2(B(Z — ZR)). (3.15)

30

The location where the maximum of —dg’R/dZ occurs is denoted by ZR.

 

y—a
O
i
,

 

 

 

 

 

 

24
08 A] A [T sech2{B[Z-ZR]} ‘
L
0'000 A 02 L 04 ZR 0‘6 ‘ 0:8 ‘ 1.0
2

Figure 3.3. The chosen heat loss proﬁle of the form of a sech”.

Figure 3.3 illustrates the nature of variation of the heat loss proﬁle in the mixture
fraction space. We note that the maximum value of the proﬁle is unity. We deﬁne
the Z locations where the value of the function is 1% of its peak value as the two
tails of the function, located respectively at Z3. and at ZR.“ with ZR- < ZR+. The
maximum of the —dq’R/dZ proﬁle occurs at ZR = (Z3. + ZR+)/2. The width of the
loss zone is deﬁned to be AZ 3 = Z3... — ZR_. The separation distance of the loss zone
from the location of the ideal Burke-Schumann ﬂame Z f is given by A = Z3. — Z I.

In the subsequent analysis, we shall vary the thickness AZ 3, as well as the separation

31

distance A in order to study the inﬂuence of the loss zone on the ﬂame structure. The
thickness of the loss zone can be chosen by selecting different values of the parameter
B in equation 3.15.

From equation 3.12 we note that the radiative loss term is given by
(NR/§O)ng/dZ, and hence, another important way to modify the loss term is to
experiment with the value of its amplitude, N R / so. We can choose different values of
N R, the radiation number. The quantity so, being given by so/(poL) is a consequence
of the solution obtained and for this reason is evaluated at each time step.

We recall that for the top hat proﬁle the integrated heat loss is
fJ(NR/§o)(U(ZR_) - U(ZR+))dZ = NRAZR. In this case the integrated heat loss
is given by f0l NRsech2[B(Z — ZR)]dZ = NR/(B§O)[tanhB(l — Z3) + tanhBZR]. For
large B this simpliﬁes to 2N R/ B + O(B“2) showing that the top-hat loss zone thick-
ness AZR corresponds to 2/ B, or B = 2/AZR. Consequently, in analytical formulae
for the top-hat proﬁle (see Appendix A, equation A.40) we can substitute for AZR

the value 2/ B in order to test their correspondence to the seek2 proﬁle.

3.6 Numerical Solution

Equations 3.12, 3.13 and 3.14 were numerically solved using the ﬁnite difference
method. The nonlinear source terms were linearized using Newton’s method. For
each time step iterations were used until the sum of normalized residuals became
smaller than 1x10‘6. The transient conservation equations were integrated to steady

state.

 

 

32
3.7 Relation Between Temperature and Density

We utilize the ideal gas law to derive a relation between the temperature and density
of the system. We can write pV = (m/W)R,,T where p is the pressure, V is the
volume, R1, is the universal gas constant and W is the average molecular weight of
the mixture. The quantity m is the mass of the mixture. Hence the pressure can be
expressed as p = pRT where p is the density and R is the characteristic gas constant
for the mixture, given by R = Ru/W. If we assume poRTo to be the constant pressure
of the system then introduction of a = 1 — To / Tf and 1' = (T — To) / (T f — To) results

in the following important relation:

__ (1—a)
p‘(1—a<1—r))

 

(3.16)

We observe that when the temperature is that of the ambient, i.e., T = To = 298 K
then 1' = 0, ,5 = 1, i.e., p = po. At the ﬂame temperature (Tf) the nondimensional
density p = (1 — a) and consequently p = (1 — a)po. Equation 3.16 is used extensively

in all the chapters of this dissertation.

3.8 Evaluation of so

Equations 3.12-3.14 indicate that in order to solve the 1', yo and yp equations in time,
we need to evaluate so at every time step. We recall that so is given by the expression
fol ﬁdi. The quantity .§o enters the analysis by virtue of the coordinate transformation
Z = 1 — s/s’o. By differentiating both sides of this transformation relation, we obtain
dZ/di = —p/§o, since 3 = [of ﬁdi and 5: = m/L. We recall that Z = 1 when :E = 0
and Z = 0 when i = 1, as noted previously in section 3.5. Using the transformation

relation between Z and :7: subject to the above mentioned boundary conditions, we

33

get
5 1 (3 17)
0 = _, o
1.1 92
and the relation between the s: and Z coordinates can be written as
1 -
5; 2 W (3_18)

fo‘ (l/ﬁ)dZ

On obtaining the solutions for 1, yo and yp equation 3.18 is used to transform the
solutions back to the physical coordinate 5:. Thus, so can be evaluated once the p‘
distribution is known. The normalized density 5 can be related to the 1' distribution
by virtue of equation 3.16. Hence, the expression for so can alternatively be written
as

1

go 2 f01 dZ +(a/(1 — a))f011'dZ (3'19)

 

Hence on obtaining the 1' proﬁle we can determine the quantity so. When the tem-
perature throughout the domain is the same as the ambient temperature To, then
1' = 0 everywhere and using equation 3.19, we obtain so = 1. If we next assume
that the temperature everywhere in the domain is the same as the adiabatic ﬂame
temperature Tf then 1' =2 1 and so = (1 - a). Since the minimum and maximum
values of temperature are To and T; respectively, the quantity so must obey the limits

(1— O)S50S1.

3.9 Results and Discussion

Figure 3.4 depicts the nondimensional temperature, 1', plotted as a function of the
mixture fraction coordinate, Z, for different values of the radiation number, N R,
for particular parameter values shown in the title of the ﬁgure. The oxidizer and

fuel mass fractions at the respective walls are Yoo = 0.6 and Ypp = 0.8. In our

 

34

subsequent analysis we keep the same set of (Yoo, YFF) and vary the location, width
and intensity of the radiative loss zone. It has to be mentioned here that the above set
of (Yoo, Ypp) represents a typical case and is employed extensively in the following

chapters. The qualitative trends for other Y00 and Ypp values are similar. The

YOO=O.6 YFF=O.8 AZR=O.O4 A=O

 

1. . - . , . , . .
0 2.1.2.... —NR=O

    
     

 

 

 

 

[ ................. NR = 80

0.8 ' ' °—_° NR = 160 ~
—-——- NR = 240

. .. ..__. NR = 320
0.6 L /. \ —-—-- NR: 383 .
P [ “'1' i l/
. l l /'

0.4 : :,'
l ,1
s

0.2 “i i

0.0 3 1 4 ' s

0.0 0 2 0.4 0 6 0 8 1.0

Z

Figure 3.4. Effect of Radiation Number N R on Temperature Proﬁle.

thickness of the radiative loss zone is 0.04 for all values of N3 and the separation
distance of the loss zone from the stoichiometric ﬂame location is zero. We observe
that the ﬂame temperature proﬁle is uniformly lowered as the value of N R increases.
Also, the ﬂame temperature peak keeps moving toward the fuel wall as the value of

N3 is increased. The drop in ﬂame temperature, as well as the shift of the peak,

35

become more prominent for higher values of N R. For a value of N R greater than
383, we do not obtain a steady state temperature proﬁle, indicating the occurrence
of a radiative extinction. This maximum or upper bound for N R is then deﬁned as
N gamma”, i.e., Nana-mtg”, = 383 for this case. We also note from Figure 3.4 that
there is a change of slope of the temperature proﬁle in the radiative loss zone for

higher values of N3, i.e., between Z3. and ZR+.

YOO=0.6 YFF=O.8 AZR=O.O4 A=O

2000 . . a . *4 1 . . 1
ZR; 2..

 

1500

1000

(l+¢)Dr

500

 

 

 

 

Figure 3.5. Effect of Radiation Number N R on Reaction Rate.

Figure 3.5 shows the nondimensional reaction rate term ((1 + 43)’Dr) for the same

situation. We observe that the reaction rate proﬁle collapses for increasing N R values.

 

36

The reaction rate peak also moves towards the fuel side; this movement becomes
more conspicuous for higher values of N R. We notice that the reaction rate proﬁle
has managed to almost move beyond the rightmost side of the radiation loss zone
(indicated by the dashed lines at Z3. and Z3...) for the highest value of N3.

We now focus on the temperature and species proﬁles for the situation when

N R = 383 for the above case, i.e., at the brink of extinction. Figure 3.6 also shows

YOO=O.6 YFF=O.8 NR=383 AZR=O.O4 A=O

  
  
  
 
 

 

‘A—a f ‘ I j 1

0—0 yo

H YF

v—v (1+¢)Dr/500 ‘

Filled symbols
for IRR case ]

 

 

 

Figure 3.6. 1', yo, yp proﬁles for ﬁnite and inﬁnite reaction rates. Also shown is the
reaction rate proﬁle.

the temperature and species proﬁles for the same ﬂame (Yoo = 0.6, Ypp = 0.8,

NR = 0) for the inﬁnite reaction rate (IRR). We notice that when NR = 383 the

37

slope of the yo proﬁle is quite different from its IRR counterpart. On the other hand,
the slope of the yp proﬁle follows the IRR yp proﬁle closely until a Z-value of about
0.3, when its slope starts decreasing. This plot therefore demonstrates explicitly the
contrast between the IRR situation and the ﬁnite chemistry situation with appreciable
radiative losses. The migration of the peaks of temperature and reactivity proﬁles is
particularly striking. Also, an abrupt change of the temperature proﬁle seems to take
place in the zone of radiative losses, i.e., between Z3. and ZR+. We add for emphasis
that from the strictly physical viewpoint the ﬁnite-rate solution has attained a rather
extreme form, since the reaction zone has almost completely propagated through the
loss zone. In Figure 3.6 we see that the loss zone is now on the oxidizer side of the
reaction rate proﬁle. As we shall see, extreme cases like this are not the norm. They
are also physically unrealistic but mathematically permissible in our simpliﬁed model

with a prescribed heat loss function.

We illustrate the details of the ﬂame structure in Figure 3.7, where we plot the
contributions of the different terms in the energy equation when the steady state
solution has been achieved. The loss term is given by (N R/ so)sech2(B(Z - ZR» and
the diﬂusion term, as in equation 3.12, is (1 /sg)1'zz. We have already noted from
ﬁgure 3.5 that for N3 = 383 the reaction rate proﬁle has penetrated through the
radiative loss zone. Figure 3.7 indicates that the diffusion term recovers the radiative
losses almost entirely and the reaction term doesn’t contribute to the diffusion term
in such a recovery process. This represents a completely different physical problem,
when the radiative loss term exists on the oxidizer side of the primary reaction zone
(ﬂame). This result is, as already mentioned, clearly in conﬂict with our hypothesis
that the heat losses take place on the fuel side of the ﬂame due to ﬂame-generated
particulates. This occurs because our hypothetical radiative loss proﬁle is simply a

prescribed function in Z, and as such, it does not contain any mechanism for loss-

38

YOO=O.6 YFF=O.8 NR=383 AZR=O.O4 A=0

 

 

 

 

1500.0 ' l ' I T r v T r
; ----- (1+¢)Dr
l] °---<> Loss
1000.0 1 or, ._--.. Diffusion
Ii
i If
:I, ..
500.0 [ it /' \
0,0 A
-500.0
0.0

 

Figure 3.7. The ﬂame structure when N R = 383.

39

zone movement as the temperature and species proﬁles change, as a real soot zone
invariably must.
In order to observe the effect of a thicker loss zone, we now increase AZR to a

value of 0.1. We notice that the drop in the temperature proﬁle is more signiﬁcant

YOO=O.6 YFF=O.8 AZR=O.1 A=0

 

 
     

   

1.0 v ; 1 ; ' T ' a ' I
L ZRi' ZRI __ NR = J
0.8 .. HI'...» .. ........... ...... NR = 40 .
- - -- NR = 80
1 —-—-- NR: 120
.-’ /'

, .
"-\
d \

0.6

0.4

 

 

\‘
b-——--—----“---\--
x‘
9

 

 

0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.8. Effect of N R on T proﬁle for thicker loss zone.

in this case and the ﬂame extinguishes at a lower value of the radiation number, viz.,
for N3 = 132.

Next we consider the situation when the leftmost side of the loss zone is sufﬁciently
removed from Z f for a ﬂame with Yoo = 0.6 and Ypp = 0.8. The thickness of the

loss zone is AZR = 0.06 and the separation distance, A, is 0.1 in this case. Figure 3.9

1.0

0.8

0.6

0.4

0.2

0.0

40

Y00=O.6 YFF=O.8AZR=O.O6A=O.1

 

 

f I Y T 7 1 7 I Y I

 

4;
I———————-———————_————————
'/

 

 

0.0

 

0.2 0.4 0.6 0.8

Figure 3.9. Effect of N3 on 1' proﬁle for A = 0.1.

 

 

41

indicates that the ﬂame temperature decreases with increasing value of N3. In this
case, however, the movement of the peak nondimensional ﬂame temperature doesn’t
seem very pronounced, though it does move towards the fuel side. Correspondingly,
Figure 3.10 shows the variation of the reaction term, (1 + ¢)Dr for increasing values
of N 3. As mentioned for the preceding cases, therefore, the reaction zone does not
always propagate through the loss zone. A sufﬁcient separation and magnitude of the

loss term appear sufﬁcient to prevent the through-transit.

YOO=O.6 YFF=O.8 AZR=O.O6 A=O.1

2000.0

 

+ -<

 

1500.0

1000.0

(1+¢)Dr

500.0

 

 

 

32’4”
"ft‘l-———---——————__————————_—-_——_———_
.
.

 

0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.10. Effect of NH on (1 + ¢)Dr proﬁle for A = 0.1.

We see that the movement of the reaction rate proﬁle is not very pronounced

42

either. We note that the reaction rate peak is always to the left of the temperature
peak, i.e., Z I < Zr < Z,. This is in accordance with the results obtained for pure
diffusion ﬂames without radiative losses [47], as discussed before.

Figure 3.11 is an extinction plot for the case when AZR = 0.06 and A = 0.
Extinction values of N R are plotted as a function of Z I, the theoretical ﬂame location

in the mixture fraction coordinate. We recall here that Z f is related to the overall

400.0

 

300.0 L

2“ 200.0 -

100.0 -

 

 

 

 

0.00 0.10 0.20 0.30 0.40

Figure 3.11. Extinction Plot for AZR = 0.06 and A = 0.

stoichiometric coefﬁcient (15 (= VYpp/Yoo) by the expression Z; = 1/(1 + 45). We

notice that for a given value of the oxidizer mass fraction at the wall, (Nﬁertinmm

43

increases as Z f is decreased. A decrease in Z f implies an increase in (I), which, for a
given Yoo, indicates an increase in YFF. As Ypp increases, the reaction rate becomes
more vigorous and it becomes more difﬁcult to extinguish the ﬂame through the
application of radiative losses. This explains the nature of the curves that we obtain
on the extinction plot. Also, for the same value of Z f, i.e., for the same value of <15, a
lower value of Yoo indicates a correspondingly smaller value of Ypp, and hence, the
reaction rate also becomes smaller in magnitude. It then becomes easier to extinguish
the ﬂame. This explains why the curves in Figure 3.11 all shift towards the left for
decreasing values of Yoo-

Our focus is next shifted to some quantities of practical interest. We evaluate
the heat transfer to the wall from ﬂames with the same stoichiometry (Yoo = 0.6
and YFF = 0.8) but with different thicknesses of the radiative loss zones and for
different separation distances (A) from Z 1. Let pr denote the heat transferred to
the oxidizer wall by the ﬂame per unit surface area of the wall. We reckon that the
oxidizer wall will have a stronger effect on the ﬂame than the fuel wall owing to the
proximity of the ﬂame to the oxidizer wall. The ﬂame transfers heat to the oxidizer
wall by means of both conduction and radiation, and hence, pr = QW,O,cond +
Qw,0,rad, where the conduction ﬂux is QW,O,cond = —/\(dT/da:)|,=L and the radiative
ﬂux is Opt/’0’,“ = 0.5x f0L(dqR/d:r)d:r. We assume that half of the radiative losses
travel to each wall: this assumption is reasonable in the thermally-thin limit we
consider here. We can transform the expressions for prmnd and Q1440,“ to the Z
coordinate and normalize pr by the reference conductive ﬂux Ao(Tf — To) / L. The
normalised QW,0 = (l/so)(d1'/dZ)|z=o + 0.5x(1/§o)NRf0‘ (1/§O)(dq'R/dZ)dZ. The

quantity prso is plotted in Figure 3.12.

From Figure 3.12 it is apparent that the heat transfer characteristics do not de-

pend strongly on the separation distance A, and consequently, we see four reasonably

44

 

 

 

 

 

00
14.0 . . , . - , I
12.0 ~
6
:1 10.0 ~ -
V?
o
+11
'3 8.0 1 -
6 Open Symbols :A=0.0
'0 P Opaque Symbols:A=0.1
Filled Symbols :A=0.2
4.0 a ' a ' ‘ ‘ ‘
0.0 100.0 200.0 300.0 400.0
NR

Figure 3.12. Heat Transfer to the oxidizer wall as a Function of N3.

45

distinct groups of curves corresponding to loss zones of four different thicknesses.
However, as is evident from the plot, the separation distance A does become im—
portant for higher values of N3, close to extinction. We will notice that nearing
extinction, the ﬂame attempts to reduce the heat losses to the wall as much as pos-
sible. Also, the value of N R required for extinction is higher when the heat loss zone

is very thin, as intuitively obvious.

 

  

 

 

14.0 s EAZR=0.O4' . -
A——-A AZR = 0.06
12.0 ‘
E 10.0 r 1
*
In.
C:
+3 8.0 ~ ~
3“ Open Symbols: A = 0.0
Opaque Symbols: A = 0.1
6'0 Filled Symbols: A = 0.2 d
L
4.0 ‘ ‘ ‘ ' r

 

0.0 2.0 4.0 6.0
NR(2/B)

Figure 3.13. Heat Transfer to the oxidizer wall as a Function of NR(2/ B).

We also plotted prso as a function of the quantity NR(2/ B ) Figure 3.13 clearly

shows that the quantity N 3(2/ B), which is approximately the value of the integral

46

EN Rsech2[B (Z - ZR)]dZ, is able to collapse the wall heat transfer data except very
near extinction. Thus, when plotted against NR(2/ B), prso does not reveal any
appreciable dependance on either the separation distance A or even the thickness of
the loss zone AZR.

Another quantity of practical interest is the radiative fraction x, given by the
ratio qRad/qrpotal. The quantity qRad is the integral of the radiative loss term
(folNRsech2[B(Z — ZR)]dZ) and qTotal is the integrated value of the reaction rate
in mixture fraction space, i.e., fol (1 + <15)’DrdZ. From Figure 3.14 we notice that qTota;
decreases with increasing values of N 3. This happens because with increased inten-
sity of the radiative loss zone, reaction rate values decrease, as observed in Figures 3.5
and 3.10. For thicker loss zones, the drop in QTotat with increasing values of N3 is
more rapid.

We have already noted in section 3.5.2 that the integral of the radiative loss term
proﬁle is approximately (N R/ so)(2/ B). Hence, it is of interest to plot the total heat
release qTotal as a function of the quantity NR(2/ B). The result is shown in the
Figure 3.15. Figure 3.15 indicates that the quantity N 3(2/ B) characterizes the total
heat release rate very well and the curves for different loss zone thicknesses virtually

collapse on one another except for large values of N R close to extinction.

Figure 3.16 illustrates the variation of x as a function of N3 for different thick-
nesses of the loss zones and for A = 0. We observe that x increases with increasing
N R for a ﬂame with a given loss zone thickness. The integrated quantity qRad increases
with N3 and, since correspondingly the gum values decrease, x, which is a ratio of
the above quantities, increases. In order to produce a given value of x, a higher value
of N R is required for a ﬂame with a thinner loss zone. Similar to the study of gum we
plot X as a function of N 3(2/ B ) in Figure 3.17. It is clear from the ﬁgure that the use

of NR(2/ B) collapses the data very well except close to extinction. So, the quantity

 

 

 

 

 

47
Y00 = 0.6 YFF = 0.8 A=O
200 . .
e—e AZR = 0.04
A——A AZR = 0.06
-. 22:11:
150 ~ \“ _‘ * e AZR ' '
g ° I
Ci: a f
100 “ §
50 - , ' a .
100 200 300
NR

Figure 3.14. qTota, as a function of NH.

400

 

48

YOO=O.6Y,,,=O.8A=0

 

   

 

 

 

200.0 . 1 ' a . . . a
G—eAZR=O.04
HAZR=0O6
HAZR=008
V——¥ =0.10
150.0 - AZ“
5
:5?
100.0 -
50.0 1 1 a ‘ ' ' ' ' L —
0.0 1.0 2.0 3.0 4.0 5.0

NR(2/B)

Figure 3.15. qTotal as a function of NR(2/B).

49

v0O = 0.6 YFF = 0.8 A=O

 

 

 

 

0.40 G—QAZR = 0.04 T

[H AZR = 0.06 0

LEI—£1 AZR = 0.08 0
0.30 -'——" AZR = 0.10 ..

N 0.20 ~ " a -
0.10 ' a 5
r a ’ 5 xzqRad/q'l‘otal
/: °
0.00 . 1 . 1 . 1
0.0 100.0 200.0 300.0 400.0
NR

Figure 3.16. X as a function of N3

 

0.40

0.30

R 0.20

0.10

0.00

50

YOO=O.6YFF=O.8A=O

 

e—e AZR = 0.04

 

 

 

 

0.0 1.0 2.0 3.0 4.0 5.0
6142/13)

Figure 3.17. X as a function of NR(2/B)

6.0

 

51

N 3(2/ B) can be used to correlate the radiative fraction quite effectively. Figure 3.18

YOO=O.6 YFF=O.8 A=O

 

 

 

 

 

0.40 . . . . . . .
[ o
0.30 r 6/ q
9'“ .
<1
0—9 AZR=0.04
0.20 . f H AZR=O°O6 ~
lg; 13—8 AZR=0.08
ﬁ/ V——4 AZR=0.10
0.10 . . ' . . . .
0.00 0.10 0.20 0.30 0.40

X

Figure 3.18. Arf as a function of the radiative fraction

shows the variation of the drop in ﬂame peak temperature, A17, as a function of
the radiative fraction x. If we denote the maximum temperature by 1',, then A17 is
deﬁned as l — 1',. We recall that the temperature has been normalized in such a way
that the peak nondimensional temperature, 1', for the inﬁnite reaction rate situation
always has the value of unity, regardless of the oxidizer and fuel mass fractions. Thus,
Arf represents the drop in peak temperature for ﬁnite rate chemistry and radiative

loss situation, in comparison to the IRR situation. The increase in Ar; with X was

52

almost linear for smaller values of x. However the curves for the different loss zone

thicknesses diverged from one another for higher values of the radiative loss fraction

X.

3.9.1 Comparison with the top-hat proﬁle

It has been previously mentioned (section 3.5.2) that the results for the seek2 heat
loss proﬁle can be compared with the results in Appendix A for the top-hat proﬁle.
However, the thickness AZR of the top-hat proﬁle must be chosen to be 2/ B, where
the value of B is determined from the choice of the thickness of the sech2 proﬁle. For
example, when (AZR)8ech2 is chosen to be 0.06, the constant B = 99.7 and conse-

quently (AZR)top— = 2/99.720.02. As illustrated in Appendix A both analytical

hat
and numerical methods were used to determine the extinction value of NH for the
top-hat proﬁle. Here, we compare analytical and numerical results for the top-hat
proﬁle with the numerical solutions for the .sech2 proﬁle. Figure 3.19 depicts the

extinction NR values plotted as a function of Z f when Yoo = 0.7, (AZR) = 0.06,

each?
(AZR),OP_,m, = 0.02 and A = 0.1. The direction of increasing Ypp has also been
indicated on the plot. The numerical solutions reveal that the sech2 and the top-hat
proﬁles produce very similar N Run-notion values. This indicates that the integrated
value of the radiative loss term is the quantity which determines the extinction N R
value. The extinction NR values obtained by analytical method are quite different
from the numerical solution. However, on close inspection of the curves depicted in
Figure 3.19 we notice that the ratio of the analytically obtained values to the numeri-
cal solution is about 4 for all the Z 1 values plotted in Figure 3.19. This indicates that

a modiﬁcation of the analytical formula based on the inclusion of a correction factor

should yield close correspondence between the analytical and numerical results.

53

v00 = 0.7 AZR = 0.06 (sech’)
AZR = 0.02 (top-hat) A = 0.1

 

1400 l 4- [Numerical Solutiorf, topLhat proﬁle
L ----- Numerical Solution, sech2 proﬁle
1200 _ 0—0 Analytical Result, top-hat proﬁle

1000-

. J
Y
.g 800— \
:5 600~
Z .
400r
‘ Y

200 ~ ’\F"

 

 

 

O . 1 . 1 . l . A . 1 .
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Zf

Figure 3.19. Comparison of N magnetic“ values for seek2 and top-hat proﬁles.

 

 

54
3.10 Conclusions

The inﬂuence of a simple and hypothetical heat loss zone on a pure diffusion ﬂame
was investigated in detail in this chapter. The loss proﬁle was of the form of a sech2
and we varied the intensity and the width of the loss zone to study the behavior of a
pure diffusion ﬂame. The loss zone was always postulated to lie on the fuel side of the
ideal Burke-Schumann ﬂame. The location of the loss zone on the fuel side relative
to the ideal ﬂame location Z f was also varied. In all situations the increase of the
radiation number N 3 resulted in a movement of the ﬂame toward the fuel side. We
found that for thin loss zones located close to Z I the reaction zone may even migrate
to the fuel side of loss zone for signiﬁcantly high values of N R. In such a situation
the loss zone exists on the oxidizer side of the ﬂame, contrary to our initial postulate.
This happens because our hypothetical loss zone is static and does not contain any
mechanism for movement. The reaction rate proﬁle, on the other hand, is free to
move and hence locates itself on the fuel side of the loss zone in certain cases.

Extinction plots were generated for different ﬂames for given loss zone thicknesses
(AZR) and given separation distances (A). The plots indicated that for a given Yoo,
an increase in Ypp results in higher values of extinction radiation number (N R).

N ondimensional heat transfer rates to the oxidizer wall were also investigated. The
results indicated that the separation distance A did not have a signiﬁcant inﬂuence
on the wall heat transfer characteristics.

The total heat release in the combustion process, qToml, was found to decrease
with increasing values of N R and the rate of decrease was quite rapid for thicker loss
zones. It was found that the quantity N 3(2/ B) characterizes qTota; very well and the
curves for the different loss zone thicknesses and separation distances all collapse onto
one another except near extinction.

Investigation of the radiative fraction (x) showed that x increases with increasing

55

values of N3 and the rate of increase is steeper for thicker loss zones. The ﬂame,
however, extinguishes at a larger value of x for loss zones which are relatively thin.
The quantity NR(2/ B) collapses the qTotal and radiative fraction values very well

except near ﬂame extinction.

 

 

CHAPTER 4

Inﬂuence of a Simple Heat Loss

Proﬁle on a Diffusion Flame with

Fuel Blowing

4.1 Introduction

In this chapter the inﬂuence of a simple .secl‘z2 heat loss proﬁle on a diffusion ﬂame
is investigated when there is a convective fuel ﬂow through the fuel wall. Thus, the
problem treated in this chapter differs from the one in chapter 3 only in the fuel wall
boundary condition.

In the following section, we present the problem deﬁnition. The formulation of the
conservation equations is quite similar to that in the previous chapter and is discussed

only brieﬂy. A discussion of the important results follows.

4.2 Problem Deﬁnition

Figure 4.1 shows the geometry of the problem under consideration. A diffusion ﬂame

is established between the oxidizer and fuel walls. A diffusive ﬂux of oxidizer is

56

supplied by the oxidizer wall. Fuel is released from the fuel wall through the combined
inﬂuences of diffusion and convection. A mass ﬂux r'n
fuel wall. The oxidizer wall allows the mass ﬂux ﬂowing from the fuel side to pass
through the oxidizer wall, thereby preventing the transient accumulation of matter

in the region between a: = 0 and :c = L. Both walls are maintained at temperature

57

(pu)|x=o issues from the

To. The species boundary conditions have been illustrated in Figure 4.1.

Soot Layer

     

"l = (pullxgo

 

 

 

Illlllllllli

 

 

 

 

a.

llllllllilllllllllllllllllllllllllllllllllllllllllllll

"'<
II
C

= DF
=——l>

x =0 x =L

(Z= 1) ( =0)
Convective ﬂux of fuel Diffusivc ﬂux of oxidizer
dYF _ “(pul Inc
a ..o ' Wmﬂdm)

Figure 4.1. The problem geometry and deﬁnition.

q
II
9*!

0*
l
°~<

Q

58
4.3 Formulation

The equations we shall solve are the energy and species equations for oxidizer and

fuel. The energy equation is given by

dq
pcpm + an] = (AT ) + prp — 0%“, (4.1)

with boundary conditions T (a: = 0) = To and T(a: = L) = To, where To is the ambient

temperature, assumed to be 298 K. The oxidizer equation is

PlYO. + “Yoxl = (PDoYo.)x - W'IF, (4-2)

with boundary conditions Y0(a: = 0) = 0 and Yo(a: = L) = Yoo. As in the previous
chapter, a one-step, irreversible chemical reaction of the form F + (IO—*(l + V)P is

assumed. Finally, the fuel equation is given by

PlYF. + uYle = (PDFYFJx - 10F, (4-3)

with boundary conditions de/dﬂmo = —(7h/pr)(Ypp— Yplx=o) and Yp(a: = L) =
0. The quantity m is the mass ﬂux from the fuel wall, equal to pu.

Using the coordinate transformation Z = 1 - s/so where s = f: pdx and so =
fol” pdm and proper normalizations (as outlined in the previous chapter), the above

equations transform to

1 NR dq__R_

 

7? = 5T2 + uoLso -2Tzz + QFDT +— odZ’ (4'4)
1 0'0
yOt - — _S—OyOZ + uOL—o —_2yOZZ ¢Dr’ (4'5)

1 (10
— = — — — D .
31F: g0 sz + 1,0ng szz 7' (4 6)

 

59

where D is the Damkéhler number, given by D = tnf/tchm and r is the nondi-
mensional reaction term given by r = yoypexp(—ﬂ(1 — T) / (1 — a(1 - 1'))) where a =
1—To/Tf and ,6 = Ea / RT; is the Zeldovich number. The reference time scale is tn, =
L/uo and the characteristic chemical time scale is tchem = 1/[AY006Xp(—E/RTf)].
The nondimensional quantity QF is given by Qp = Qprp/Cp(Tf — To) and so =
So/So’rcf where so”; = poL. The velocity no is the magnitude of u at a: = 0, i.e., at
Z = 1. Note that the ﬁrst terms on the right hand side of each of the above equa-
tions originate due to the convective fuel ﬂow and were not present in the governing
equations for a pure diffusion ﬂame, as illustrated in section 3.5 of chapter 3.

The above equations are solved numerically to obtain nondimensional temperature
and species mass fraction proﬁles for different radiation loss proﬁles. The radiation
number N R in the above is a ratio of the reference radiative and convective ﬂuxes,
given by N R = qucf/(pouonAT). This is in contrast to the deﬁnition of N3 for a
pure diffusion ﬂame, where N R was deﬁned to be a ratio of reference radiative and

conductive ﬂuxes.

4.4 Boundary Condition for the Fuel Equation

In contrast to chapter 3, the fuel mass fraction at the fuel wall is not prescribed in
this case. Instead, it is assumed that fuel issues from the fuel wall by both diffusion
and convection. A balance between the rate of depletion of the fuel in the reservoir
and the rate of fuel issue gives the necessary equation for the boundary condition. In
fact, the fuel mass fraction in the reservoir is held constant.

The amount of reservoir ﬂuid lost through a unit surface area per unit time is
pouo, where p0 is the density at the reservoir wall temperature, i.e., To, and no is
the velocity at the fuel wall. Correspondingly, the reservoir is deprived of poquFF

amount of the fuel species. The fuel leaving the reservoir surface and entering the

60

free stream does so under the combined inﬂuence of diffusion and convection. The
diffusion rate, by virtue of Fick’s law, is —pr(dYF/da:)|x=o, where Dp is the diffusion
coefﬁcient of fuel in fuel / air mixture. The convective fuel ﬂow rate is (pouo)Yp|z=1.

Writing the balance equation yields

_la::0 : :%-?2(YFF _ YF|x=0)° (4.7)

In the Z coordinate the fuel wall boundary condition assumes the following form:

d u L3
‘dizi 2:1 = gFOO(1 — yF|$=O)- (4.8)

 

where 3]}: = YF/YFF, as before, and DFO is the value of the diffusion coefﬁcient of the

fuel at the reference condition.

4.5 Solution for Inﬁnite Reaction Rate

The temperature and species proﬁles for the inﬁnite reaction rate (IRR) situation are
used as initial proﬁles. Hence, our ﬁrst task is to obtain such solutions. In the limit
of inﬁnite reaction rate the ﬂame sheet is inﬁnitesimally thin. Fuel and oxidizer are
depleted in the ﬂame in the stoichiometric proportion. No fuel exists on the oxidizer
side of the ﬂame and no oxidizer exists on the fuel side, i.e., there is no leakage through
this diffusion ﬂame.

Under such conditions, we can solve for the 7', yo and yp proﬁles in two adjacent
domains without requiring the reaction term. We solve the following energy equation
in the oxidizer side of the interval subject to the boundary conditions 1'(Z = 0) = 0

and T(Z = Zf) = 1, where Zf is the ﬂame location:

 

61

The above equation must also be solved in the adjacent fuel-side domain, i.e., between
Z = Z I and Z = 1, with T(Z = 1) = 0. A similar procedure must be followed for the
species equations, although yo = 0 on the fuel side and yp = 0 on the oxidizer side
account for the zero leakage parts of the yo, yp solutions. However, Z I is yet to be

determined. For that purpose, a mixture fraction variable is ﬁrst deﬁned as follows:

=¢yF+1-y0
¢+1

 

c (4.10)

We note here that unlike chapter 4 the variable Z does not correspond to the mixture
fraction. We observe that C satisﬁes an equation of the same form as the 1' and species

proﬁles for the IRR situation, i.e.,

 

1 (Yo

E—OCZ+ uoLEngz —0 (4'11)
The solution for C is

C = [1 - exI)(-Z/C)l (4-12)

where c = ao/(uoLso). At the ﬂame, yo and 3]}: are both zero, which, by virtue
of equation 4.10 indicates that Cf = 1/(¢ + 1). Correspondingly, Z has the value
Z f = cln((1 + ¢)/¢). The equations for T, yo and yp proﬁles can now be solved for.

The solutions for 1' are

—Z/c

1—8- c OSZSZ 3
T = 1-e 2’; f (4.13)

—1/c_ —Z/c
TIT—27 2,3231.
The solutions for ya are

—Z/C_ —Zf/C
8 -e c OSZSZ ,
yo = 1-e 2’7 f (4.14)

0 zfgzgr

 

62

Finally, the solutions for 3}}: are

0 05232,,
31F = (4-15)
1 — 642.2,“. 2,329.

A plot of the temperature and species proﬁles is shown in Figure 4.2 for L = 0.01

YOO=O.6 YFF=O.8 u0=0.08 m/s L=0.01 m

 

 

 

 

I I l ﬂ I
\
\
\ .....
\ 17
\
\
\ 0m. y
\ 0
\ ..
\ -_...-
\ 9‘ ° yF
\
\
\
\
\
\
\
\
\ -
s
s
s
\
s
\
x
\ .
\ '
, ’1'\ -‘
' \
\
S
S
S
s
S
S
S
S
S
S
S
‘ -
S
S
S
S
S
S
S
S
S
S
‘S
S
,1 ‘~
0.2 0.4 0.6 0.8 1.0

Figure 4.2. IRR proﬁles for 7', yo and y;-

m, uo = 0.08 m/s, Yoo = 0.6 and YFF = 0.8. We note here that equations 4.13, 4.14
and 4.15 have to be solved iteratively because the quantity c depends on so, which
depends on the solution and can not be ascertained apriori. The procedure is to guess
a value for c, i.e., for so and then use equations 4.13-4.15 to determine 1', yo and yp

proﬁles. On obtaining the 1' proﬁle the density ([3) proﬁle can also be determined

 

63

using the relation ,6 = (1— a)/(1 — a(1— 7)), where a = 1 — To/Tf. Then, so can be
evaluated using the procedure outlined in section 3.8. Using this fresh value of so the
quantity 0 can be calculated again. The new value of c is then used to determine T,
yo and yp proﬁles. The new 1' proﬁle is then used to calculate the value of c again.
This procedure is repeated until convergence is obtained and we ﬁnd the 1', yo and

yp proﬁles for the inﬁnite reaction rate case.

4.6 Parameter Values

The parameter values used in this chapter are the same as those used in chapter 3.
The length of the domain is L = 1.05 cm unless otherwise mentioned. The velocity
of fuel ﬂow at the wall, i.e., no is a new parameter in this chapter. The value of no

was varied from 0.02 m/s through 0.14 m/s.

4.7 Results and Discussion

The method of analysis in this case is the same as in chapter 3. However, the addi-
tional variable uo must be taken into account in the examination.

Figure 4.3 illustrates the variation of the temperature proﬁle for different values
of no for a given ﬂame in the absence of radiative losses (N R = 0). The values of no
range from 0.06 m / s to 0.14 m/s. We deﬁne a nondimensional parameter uoL/ao for

characterising the fuel blowing rate and tabulate is in Table 4.1.

Table 4.1. uoL/ao for different values of uo.

 

uo m/s 0.06 0.08 0.10 0.12 0.14
uoL/ao 5.08 6.77 8.46 10.15 11.84

 

 

 

 

 

 

 

 

 

 

64

YOO=O.7 YFF=O.5 NR=O

 

 

 

 

 

Figure 4.3. Effect no on 7' proﬁle for a given ﬂame.

 

65

As no is increased, the ﬂame moves closer to the oxidizer wall. We recall that
the rate of depletion of fuel from the fuel wall is given by (pouo)YFF. For higher
values of no, more fuel issues from the reservoir. The oxidizer mass fraction at the
oxidizer wall being unchanged, the ﬂame has to move toward the oxidizer wall since
the rate of supply of fuel is now greater. As no is increased from 0.06 m / s to 0.10 m / 3
through 0.08 m/s, the peak temperature also increases, as evidenced by Figure 4.3.
However, on further increase of no, the peak temperature decreases. The reasonable
explanation for this is the proximity of the oxidizer wall. For no = 0.14 m / s, the ﬂame
is quite close to the oxidizer wall and loses much heat to the wall. On closer scrutiny
of Figure 4.3, it can be observed that for higher values of no, a ﬁxed increment in the
value of no results in a smaller shift of the temperature proﬁle toward the oxidizer
wall. The ﬂame seems to “feel” the presence of the wall as no is increased and resists
the attempt of the convective ﬂow to force it against the wall.

The slope of the temperature proﬁle at the wall is a measure of the extent of the
heat transfer to the wall. Figure 4.4 shows the slope, 72, at the walls, Z = 0 and
Z = 1, plotted as a function of no for the ﬂame under consideration (Yoo = 0.7,
YFF = 0.5, N R = 0). The plot clearly exhibits that the heat transfer to the oxidizer
wall (at Z = 0) is much higher than that to the fuel wall. Because it is closer to
the ﬂame, the oxidizer wall exercises a signiﬁcant inﬂuence on it. The inﬂuence of
wall heat transfer is an important issue for deciding the location and strength of the
ﬂame.

The reaction rate term, (1 + ¢)Dr, is plotted as a function of Z in Figure 4.5.
The increase in no clearly results in the movement of the reaction rate proﬁle toward
the oxidizer wall. The value of the peak reaction rate also increases as no increases.
However, as observed for the 1' proﬁle, the proximity of the wall results in a drop in
the peak value of (1 + ¢)Dr for higher values of no. We also notice that for no = 0.06

m / s the reaction rate proﬁle is quite broad. However, as no is increased, the reaction

 

66

YOO=O.7 YFF=0.5 NR=O

 

 

 

 

 

15.0 n
H Tzlz=0
H TZIZ=1
10.0 - a
5.0 ' r
C
l
0.0 .‘_ . + ——.—— ‘0 ‘
-5.0 m ' * J ‘ ‘ ‘ ‘
0.06 0.08 0.10 0.12 0.14

u0

Figure 4.4. Effect no on T2 for a given ﬂame.

0.16

(1+¢)Dr

67

 

  
  
 

 

   
  

 

 

 

 

    
 

 

 

Yoo=0°7 YFF=0.5 NR=0
150.0 ' ' °___'° no I: 0.06 (“/3 I i f
H no = 0.08 III/S
ﬁ—d no = 0.10 111/8 1
°—° uo = 0.12 m/s
100.0 _ “—7 no = 0.14 m/s d
12.0
11.0 . 1
10.0 . ‘
9.0 -
50.0 7 80 b .l d
7.0 i
6.0 . 1 . 4 + . . . .
0.06 0.08 0.10 0.12 0.14 0.16
no
0.0 —- 4' -
0.0 1.0

Figure 4.5. Effect of no on (1 + ¢)’Dr for a given ﬂame.

 

68

rate proﬁle becomes narrower. Since the peak of the reaction rate proﬁle increases

as no increases (for sufﬁciently large distance from the wall) and it simultaneously

becomes narrower, the total heat release in the process of combustion, qTotal, which

is the integrated value of (1 + ¢)Dr, is also calculated. Figure 4.5 shows as inset a

plot of gave“; as a function of no. It is clear that qTotal decreases as no is increased for

this ﬂame. The rate of decrease is signiﬁcantly enhanced for higher values of no, i.e.,

when the ﬂame is in the close proximity of the oxidizer wall.

Yoo = 0.6 YFF =O.8 AZR=O.O4A = 0

 

 

 
   

 

uoLlao=8.47
1.0 ;~ . . , - . - .
» ZRi i + * NR =0
5 l ~~ ..... N =10
0.8 ~ ; R
0.6 »
I" r
0.4 :
0.2 E
0.0 - i. . 1 . . 1 .
0.0 0.2 0.4 0.6 0.8 1.0

 

Figure 4.6. Effect of N R on the temperature proﬁle.

 

Next, the effect of increasing the intensity of the radiative loss zone, NR, on

a diffusion ﬂame with prescribed fuel and oxidizer mass fractions in the respective

reservoirs and a given no through the fuel wall is investigated. The thickness of the loss

69

zone, AZR, is assumed to be 0.04 and the separation distance A is zero. As is evident
from Figure 4.6, the temperature proﬁle is uniformly and monotonically lowered as
N R is increased. The peak of the temperature proﬁle moves toward the fuel wall with
increased radiative losses. For NR values greater than 28.6, the temperature proﬁle
crashes to zero; here we can not obtain a steady state proﬁle. We also observe that for
higher values of N R, a given incremental increase in N 3 results in a relatively greater
degree of collapse of the temperature proﬁle, i.e., the collapse of the temperature
proﬁle is accelerated. Simultaneously, the rate of movement of the temperature peak

toward the fuel wall is also enhanced for higher values of N R.

Yoo = 0.6 YFF = 0.8 A2,,=0.04 A = 0

 

 

 

 

 

u(,l../oto = 8.47
1.0 ZR; ' ' I I v I f I
f U + . - ; 2 3 7 1
b, I I - 1 1 : -""=’o:
' ' >’-::.~~~"
: l 2" .’;‘"’,/ "I.”
. 1 " o I ....-"'..-. _
0’8 i l i ,//3'7”
r 7-2 ' /,”’.
l l ,"/'///-"I YF
1 : : . 2"(/;.,.-".
I I , .0?”
0.6 ~ '. : : 7
\ : : .,'-///,. '7'
» o / .x
. A. I l " I/ I‘
' 1 I ///"' —— _
l : : n." N“ 0
0.4 L l : ,,/ """"""""""" NR — 10 -
,,/ .
y. a ”:47 —--- NR=20
1 5 541/ NR=25
/.
0 2 L .177.» .__.. NR = 28 6 ~
I a
, 9:“
s 0'. mix
00 :. ..-..* - -- . -
0.0 0.2 0.4 0.6 0.8 1.0
Z

Figure 4.7. Effect of N3 on yo and 3}}: proﬁles.

   

70

Figure 4.7 shows the species proﬁles, yo and yp, as a function of the Z coordinate.
We recall once again that Z is not the mixture fraction, see equation 4.12. The change
in the oxidizer proﬁle is not very pronounced. However, the yp proﬁle signiﬁcantly
changes for increasing values of N R. For N R = 28.6, yp everywhere is conspicuously
greater than for the no—loss case. The reaction rate decreases with increasing N R and,
consequently, the ﬂame consumes less fuel and oxidizer. Hence, both the oxidizer and

fuel mass fractions are greater everywhere for higher values of N R.

Yoo = 0.6 Y“, = 0.8 AZ,=0.04A = 0

 

 

 

 
   

 

 

 

uoL/ao=8.47
300 :4 l ‘ T j 1 v 1
112 —— N.=o +
$15 " NR: 10
200 - * s NR=20 .
I/1' —'-” NR=25
ii 7 ‘ o———e NR=28 6
e- l ‘
i .
100 .
0 l - ‘ ‘ W
0.0 0.2 0.4 0.6 0.8 1 .0
Z

Figure 4.8. Effect of N R on reaction rate proﬁles.

The reaction rate proﬁles are illustrated in Figure 4.8. With increased radiative

losses, the reaction rates decrease signiﬁcantly. Also, consistent with the migration

 

71

of the temperature peak, the reaction rate peak also moves toward the fuel side. The
rate of decrease in the reaction rate peak seems to increase for higher values of N R.
It is clear from a comparison of Figure 4.6 and Figure 4.8 that the temperature peak
at ZT is further to the oxidizer side than the reactivity peak Z, for each and every
value of the radiation number N R.

The inﬂuence of increasing the loss zone thickness is investigated next (Figure 4.9):

AZ R is now increased to 0.1. The peak of the temperature proﬁle indicates a migra-

Yoo=0.6 YFF=O.8 AZR=0.1 A=O.

 

 

 

 

 

ommﬁan
10 n w . . . , . ,
2.7. Zn;
5 E — N =0
08— . . R -
l ' mm NR:
1 ffﬁk ----NR—75
0.6 _ 7/ \1\ ——-—-- NR=99 q
1’ A.
.. r: s
h :
0.4 " I: : 4
I: :
L': : 1
02- E i _
7 5 i 1
00 1- 1. 1 . . - .
00 02 04 on 08 10

Figure 4.9. T proﬁles when AZ; 2 0.1 and A = 0.

tion toward the fuel wall. The movement is not very conspicuous, though, possibly
because for the entire ﬂame history the reaction zone is squarely inside the loss zone.

Substantial movement to either side seems to be completely restricted. Similarly, the

72

reaction rate proﬁle, shown in Figure 4.10, also exhibits a corresponding movement
of the peak toward the fuel wall, although this movement is also very insigniﬁcant.

We now turn our attention to the structure of the ﬂame when NR = 9.9, AZR = 0.1

Yoo=0.6 YW=0.8 AZR=O.1 A=O

 

 

 

 

 

uano=8.47
300.0 . ~ . - . -
—— NR—O
mm. NR:
2000 __-- NR7”
' r —-—-- NR=9.9
,5.
9-
.3;
100.0 - .
0.0 . - . . . 1
0.0 0.4 0.6 0.8 1.0
2

Figure 4.10. (1 + ¢)Dr proﬁles when AZR = 0.1 and A = 0.

and A = 0. For such a situation the ﬂame is at the brink of extinction and any
further increase of N 3 results in the temperature proﬁle crashing to zero everywhere.
Figure 4.11 illustrates the proﬁles of the different terms in the energy equation. The
convection ((1/so)7'z) and the diffusion ((ao/(uoLs3))7'zz) terms balance one another
very near the oxidizer wall and the reaction term is extremely small in that region.
The primary balance in the vicinity of the ﬂame, however, is between the reaction

term and the diffusion term, i.e., the heat released by virtue of the combustion reac-

73

tion is diffused away from the primary reaction zone. The radiation loss is principally

recovered by the reaction term.

Yoo=0.6 YFF=O.8 NR=9.9 AZR=O.1 A=0

 

 

 
 
  

u,1./a,=8.47
200.0 . . . . . . 7 .
Convection
°——° Diffusion
100.0 °——° (1+¢)Dr ,

-- - -- Radiative Loss

 

 

0.0

-100.0

 

 

 

-2000 7171-4-47
0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.11. Flame structure when NR = 9.9, AZR = 0.1 and A = 0.

We now investigate the effect of shifting the loss zone from the IRR ﬂame location,
Z I. We consider the same ﬂame as before, except we now introduce a separation dis-
tance of A = 0.1. The nondimensional temperature proﬁles are plotted in Figure 4.12
for different values of N 3. Interestingly, in this case the maximum of the temperature
proﬁle shifts toward the oxidizer side. The reaction rate proﬁles also indicate a slight

movement of the peak toward the oxidizer wall in Figure 4.13. This result is a rather

74

Yoo=0.6 YFF=O.8AZR=O.O4A=O.1

 

0 uoI./0to = 8.47
1. . . . . - A - , . , A
I
211- 2.1a
: : — NR = 0 ‘
I I
0.8 - i 1 NR " 10 J
.« ' : -—-- NR=20
l
/ \ " _ - _ - - -
l'.‘ \\\'E'. l NR - 25
[I \ I“. I
o \ \ .'
0.6 1 , \1\ 1., 7
. \ l I xx.
\ \'
l ‘k‘ \\ .\
004 f' : : \‘ \ \u‘-._. .1
' | \\ \\ ‘ x
l l \‘ \ “NE. A
, : \‘ \\ Fm.“
: l \ ‘ \\ \ sun...‘
0.2 f i i ‘\\\ “N... 7
I I ‘ \ ‘\ ‘--.__
. 1 x \ \ "-.....“
i 1 i“ ‘ \‘T: 37-...
I I f \i : ;;;:‘:.~“
I 1 - ~$2.‘
0.0 1 1 1 L 1

 

 

 

0.0 0.2 0.4 0.6 0.8 1 .0

Figure 4.12. Loss zone is now separated from ﬂame location. Observe the movement
of the peak toward the oxidizer wall.

75

0.04 A = 0.1

8.47

11

0.6 Y“, = 0.8 A2,,
“0

Yoo=

 

 

...._.-.._-... NR = 10
— — —- NR = 20

NR=25

0.8

0.6

0.4

 

 

 

 

 

300

1.0

Z

Figure 4.13. Reaction rate proﬁles when A = 0.1.

 

76

interesting contrast to the case of the pure diffusion ﬂame, where the reaction zone
always moved toward the loss zone. Here we must observe that there are the three
simultaneous inﬂuences of convection, diffusion and reaction: each can respond to
the heat loss zone. In the pure diffusion ﬂame we allowed only diffusion and reaction
to counter the loss zone, and the response was predictable: the reaction zone always
shifted toward the loss zone, even in those cases where it could never penetrate it (see
Figures 3.9 and 3.10) of chapter 3. Here, convection from the fuel wall can force the

reaction zone toward the oxidizer side, away from the loss zone.

Yoo=0.6 YFF=O.8 NR=25 112,510.04 A=O.1

 

 

not/010:8.“
200.0 . . . . . . f r
Convection
. °——-° Diffusion
100.0 - . _ H (1+¢)Dr 7

-- --- Radiative Loss

 

I
O I
.0 I. .
I ,g. ...................... .
0 O L..-|_' ,, 'Ill‘l"zr1r"-‘I"Iri"‘n‘l"rr"l‘t"l‘l"‘113( I I
o .

-100.0

 

 

 

-2000 1........
0.0 0.2 0.4 0.6 0.8 1.0

Z

Figure 4.14. Flame structure when NR = 25, AZ; = 0.04 and A = 0.1.

Figure 4.14 shows the structure of the ﬂame when NR = 25, AZR = 0.04 and

A = 0. For such a situation, the ﬂame is at the brink of extinction. The convection,

  

L

1L-

‘ _L
J

77

diffusion, reaction and radiative loss terms of the energy equation are plotted to show
the relative importance of each term. The primary balance is between the reaction
and diffusion terms. It is also observed that the radiative loss term is recovered almost
entirely by the diffusion term and that the two curves are virtually indistinguishable
in the radiative loss zone. Interestingly, the reaction term does not contribute sig-
niﬁcantly toward recovering the radiative losses. This is because the heat loss zone
is placed at a signiﬁcant distance from the IRR ﬂame location, Z f, so the primary
balance is between reaction and diffusion.

It is interesting to compare Figure 4.11 with Figure 4.14. In the former case
the heat loss proﬁle was placed right next to the ideal ﬂame location Z f. Hence,
the reaction term had to battle the radiative loss term, unlike in the latter case
when A = 0.1 and the loss term was placed some distance away from the ideal
ﬂame location. Thus, the two ﬂames have signiﬁcantly different structures. So the
separation distance A between Z I and Z3. plays an important role and decides how
the loss term will be recovered. A signiﬁcantly high value of A results in heat diffusion
into the radiative loss zone by means of conduction. However when A = 0 the reaction
term has to counteract the effect of the imposed heat loss proﬁle.

For the same separation distance of A = 0.1 the loss zone thickness AZR is
now increased to 0.08. As expected the ﬂame extinguishes for a lower value of N3
(Figure 4.15). For NR > 13.1 we do not obtain a steady ﬂame. The location of the
maximum ﬂame temperature moves toward the oxidizer wall for increasing values
of N3. An investigation of the reaction rate proﬁle also indicates similar behavior
(Figure 4.16).

Let us investigate the effect of changing the velocity no on the temperature and
reaction rate proﬁles. The value of no is decreased from 0.10 m/s to 0.05 m/s. The
nondimensional parameter uoL/ao decreases from 8.47 to 4.24. Interestingly, it is

more difﬁcult to extinguish the ﬂame in this case. On decreasing the velocity no the

78

Yoo=0.6 YW=O.8 AZR=O.08 A=O.1

 

     

 

 

 

not/010:8.“
1.0 p ; , . . . -
211- 22+
‘ s a -- N,_o
0.8 - 5 5 NR=5 .
1 -‘-’ NR=10
0.6 - \ '
‘V'.
0.4 ~ 5‘
0.2 - E
0.0 1. . A A . A '7‘“
0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.15. 7' proﬁles when AZR = 0.08 and A = 0.1.

79

Yoo=0.6 YFF=O.8 AZR=O.08 A=O.1
not/(10:8 47

 

 

 

 

 

1 8.
O
J
O 5 w B
__ __ __ __
NNNN .0
m u
_ _
- 4.
0
2
11111111111111111111111111111111111 0
\“\§
AéMWWIJHW .....
If]

 

300

1.0

0.0

0.1.

Figure 4.16. (1 + ¢)Dr proﬁles when AZR = 0.08 and A

80

=0.l

Y00=06 YFF=O.8 AZR=0.08 A

uoL/ao=4.24

 

0.8

0.2

 

 

 

0.0

1.0

0.6

0.0

4.24.

Figure 4.17. 7' proﬁles when uoL/ao

81

ﬂame moves away from the oxidizer wall and Z f increases from 0.08 to 0.2. The ﬂame
loses less heat to the oxidizer wall when no is smaller and consequently can survive for
higher values of N R. In this case also the location of the peak ﬂame temperature, Z1,
moves toward the oxidizer wall. Similar behavior was also observed for the reaction

rate proﬁle.

Yoo=0.6 YFF=O.8 AZR=O.04 A=O.1

 

 

 

 

u,1./a,=4.24
1.0 . T fv . . v 7 u
z, = . —
a 2 ---~ NR=5
0.8 r : : ---- NR: 10 ‘
L ’f\ ' : *—'—-‘ NR = 15 1
' ‘\:\ ' *——-° R=20
0'6 7 .\ \\ -'-'° NR—ZS 1
P l I ‘\ \\ '——° NR—29 5
I | \
0.4 — \‘:T\. ‘
. i i ‘ x.
0'2 7 i i 7
0.0 A A A A A A A A A A 1
0.0 0.2 0.4 0.6 0.8 1.0
Z

Figure 4.18. 7' proﬁles when uoL/ao = 4.24 and AZR = 0.04.

The loss zone thickness AZR is then halved for the same value of no = 0.05
m / s. Predictably, it becomes harder to extinguish the ﬂame and the extinction value
of N3 is 29.5, beyond which we do not obtain a steady state temperature proﬁle.
Consistent with previous observations we ﬁnd that Z. moves toward the oxidizer wall

with increasing values of N3.

82

As explained in chapter 3 the heat transfer to the wall is important to exam-
ine. In the present situation, pr, i.e., the heat transfer to the oxidizer wall, is
normalized with the reference convective heat ﬂux, i.e., pouon(Tf — To) to yield
pr = (ao/(uoLso))(d7'/dZ)Iz=o + 0.5x(1/§o)NRfO‘(1/5o)dqR/dZ. The quantity

prso is plotted as a function of N3 in Figure 4.19.

Yoo = 0.6 YFF = 0.8
uoL/or.0 = 8.47

1—0
G

 

y—a
LII
I

a
n
)

I"
.1;

e—OAZR=O.O4
A——AAZR=O.06 ° -
HAZR=O.08
v—vAZR=O.10

  

(ac/(uoL))tzlw+0.S*Loss

u—I
N

 

Open Symbols: A = 0.0
Filled Symbols: A = 0.1

 

1.1

 

0.0 10.0 20.0 30.0
NR

Figure 4.19. Heat transfer to the oxidizer wall as a function of N R for the fuel blowing
situation.

A comparison of Figure 4.19 with Figure 3.12 of chapter 3 reveals that unlike for
a pure diffusion ﬂame, the wall heat transfer characteristic curves are not grouped
according to different loss zone thicknesses. The separation distance A seems to

have signiﬁcant impact in this case. In order to study this in greater detail we plot

83

prso as a function of the quantity N 3(2 / B ) in Figure 4.20. We recall here that this
quantity was very useful in collapsing the wall heat transfer, the total heat release
and the radiative fraction data for a pure diffusion ﬂame, as discussed in chapter 3.

Figure 4.20 shows that for this situation we get two distinct sets of curves for the

Yw=OﬁYW=08

 

 

 

 

 

11,1701, = 8.47
1.6 . . - . . A
’A——A AZR = 0.06

g 1.5 .HMR=O.O8 -
5: _v——v AZR = 0.10 J
(EIA’ ///’ a .
:313- ,/ 7 °-

1 2 _ Open Symbols: A = 0.0 .

' Filled Symbols: A = 0.1
1.1 7 ‘ ‘ ‘ ‘ ‘ *
0.0 0.1 0.2 0.3 0.4

NR(2/B)

Figure 4.20. Heat transfer to the oxidizer wall as a function of NR(2/ B) for the fuel
blowing situation.

two different separation distances used. Hence, the separation distance of the loss
zone from the ﬂame turns out to be a very important parameter for a diffusion ﬂame
with fuel blowing. For a pure diffusion ﬂame the separation distance A was not very
important for the heat transfer to the wall since the data collapsed to one single curve.

We now focus on the radiative fraction, x, given by the ratio qRad/qrpotaz. First,

84

the quantity qTotal is plotted as a function of the radiation number N3 in Figure 4.21.

For thinner loss zones the rate of decrease of QTozaz with N3 is less pronounced. For

Yoo=0.6 YFF=O.8 uoL/ao=8.47 A=O

 

 

 

 

19.0 ~ G—OAZR=0-04 4
. k—i A2,, = 0.06 1
17.0 - e——a AZR=O.08 .
v——vaz,,=o.10 .
15.0 - .
1 . L. ..
2 30 . o ‘
04 5‘
11.0 . —
1 ° .
9.0 - ° -
' 0
7.0 _ .
5.0 A A A A A
0.0 10.0 20.0 30.0
NR

Figure 4.21. The total heat release in the combustion process plotted as a function
of the radiation number N R.

AZR = 0.10, for example, qTotal steeply decreases with increasing values of N3. In
Figure 4.22 we plot (110ml values as a function of N 3(2 / B ) As expected, the total heat
release values correlate very well with N 3(2 / B ) with a nearly straight-line dependence
for N 3(2/ B ) values less than approximately 0.25.

Figure 4.23 illustrates the variation of the radiative fraction X with the radiation
number N R. The magnitude of X increases with increasing N R until extinction occurs.

It is observed here that the values of X are signiﬁcantly lower than for a pure diffusion

85

Yoo = 0.6 YFF = 0.8 uoL/ao = 8.47 A = 0

 

* 1

19.0 L 7
17.0 - 7
15.0

13.0

q'l'otal

11.0

9.0

 

7.0

 

 

 

0.0 0.1 0.2 0.3 0.4
thz/B)

Figure 4.22. The total heat release in the combustion process plotted as a function

of NR(2/B).

86

Yoo=0.6 YFF=O.8 uoL/ozo=8.47 A=0

 

 

 

 

.2 . o . o

O 0 e—e A2,, = 0.04
H AZR = 0.06 4

G—a A2,, = 0.08
0.15 1 v——~Az,,=0.10 . -
, 1
R 0.10 7 a 7
0.05 _ -
'4 5 xzand/QTON 1

0.00 A A e A A
0.0 10.0 20.0 30.0

NR

Figure 4.23. The total radiative fraction X versus the radiation number N R.

87

ﬂame (see Figure 3.16).

Yoo = 0.6 YFF = 0.8 11,12010 = 8.47 A = 0

 

0.20 . , . , . , A
G—-0AZR=O.O4
* HAZR=O.O6
B—EI AZR=O.08
0.15 A V—-VAZR=O.10

x 0.10 l

0.05 t

 

 

 

 

0.00 A A A A A A A
0.00 0.10 0.20 0.30 0.40
NR(2/B)

Figure 4.24. The total radiative fraction X versus NR(2/B).

Next, in Figure 4.24 we plotted the radiative fraction X as a function of N 3(2/ B),
and, as expected, the curves for the different loss zone thicknesses collapsed onto
one another. Thus, the quantity NR(2/ B) is of great importance when analyzing
important quantities of practical interest.

Figure 4.25 shows the variation of the drop in peak ﬂame temperature as a function
of the radiative fraction X. As mentioned in chapter 3, A77 is deﬁned as 1 — Tf. The
quantity ATf was found to increase almost linearly with the increase of the radiative
fraction X. It is also observed that the A77 curves diverge from one another for high

values of N R for the different loss zone thicknesses shown in the ﬁgure.

b-

P

0.40

88

Yoo=0.6 YFF=O.8 uoL/ao=8.47 A=0

 

0.35 7

0.30 ~

0.25 -

 

ﬁr

   

G—OAZR=O.O4
Ar—AAZR=O.O6

 

 

0.20 - B—B AZR = 0.08 1
V—-—" AZR = 0.10 1
0.15 .
1 1
0.10 ‘ ‘ ‘ ‘ ‘ ‘
0.00 0.05 0.10 0.15 0.20
X

Figure 4.25. A77 as a function of the radiative fraction X.

89
4.8 Conclusions

In this chapter the inﬂuence of a sech2 heat loss proﬁle on a diffusion ﬂame with
fuel blowing from the fuel wall was investigated. The analysis in this chapter is quite
similar to that in chapter 3. However, the inﬂuence of convection gives rise to more
complexities in the ﬂame behavior compared to a pure diffusion ﬂame.

An important parameter in this chapter is the fuel blowing velocity at the fuel
wall, i.e., uo. An increase in no resulted in a ﬂame movement toward the oxidizer wall.
The peak ﬂame temperature also increased. The reaction rate proﬁle became taller
and narrower as the ﬂame moved closer to the oxidizer wall. The integrated value of
the reaction rate, qTotal, however, decreased with increasing uo. The proximity of the
oxidizer wall is the primary reason for such a decrease in qTotaz. When the ﬂame is
close to the oxidizer wall it loses much heat to the wall and hence there is a resultant
decrease in the qTotal values. It was also observed that the rate of movement of the
ﬂame toward the oxidizer wall slowed down at higher values of the fuel blowing rate.

Next, the intensity, the width and the location of the sech2 heat loss zone were
varied systematically to examine the inﬂuence of each of the variables on the ﬂame. An
increase in the width of the radiation loss zone resulted in a smaller value of extinction
radiation number i.e., Nana-mum. It was observed that for loss zones placed right
next to the inﬁnite reaction rate (IRR) ﬂame location, Z I, the reaction term recovers
the radiation loss term and the diffusion term does not contribute signiﬁcantly to
such a recovery process. If, however, the loss term is placed at a signiﬁcant distance
from Z f, the reaction term does not recover the loss term. Rather, the diffusion term
has to counter the loss term and conduct heat to the region of loss. Thus, the ﬂame
structure depends signiﬁcantly on where the loss zone is placed.

The migration of the temperature and reactivity peaks was also investigated for

each situation. It was observed that for loss zones placed at a sufﬁcient distance

90

from Z f the ﬂame may move slightly toward the oxidizer wall in certain cases. Such
behavior is in sharp contrast with pure diffusion ﬂames studied in chapter 3. For pure
diffusion ﬂames, the reaction zone always moved toward the fuel wall for increased
heat losses. For diffusion ﬂames with fuel blowing, however, the presence of convection
complicates matters signiﬁcantly, and the ﬂame may move in the opposite direction.

For a loss zone of a given thickness and separation distance, it was observed that
decreasing the fuel blowing velocity, uo, resulted in a higher extinction radiation
number. When no is decreased the ﬂame moves more toward the fuel wall and away
from the oxidizer wall. Hence it loses less heat to the oxidizer wall and it then becomes
more difﬁcult to extinguish the ﬂame.

An analysis of the heat transfer to the wall revealed that the separation distance
A has signiﬁcant inﬂuence on the characteristics of the curves. However, a similar
examination of the heat transfer characteristics for pure diffusion ﬂames in chapter 3
indicated that the separation distance A was not important at all. Hence, the analysis
in the current chapter suggests that the introduction of fuel blowing can give rise to a
variety of interesting behaviors. The quantity NR(2/ B) was found to be quite useful,
as in chapter 3 in correlating the wall heat transfer rate, the total heat release and

the radiative fraction.

CHAPTER 5

Estimation of Soot Layer Proﬁle
and Thickness

5.1 Introduction

In this chapter approximate methods for estimating the soot layer proﬁle and thick-
ness are discussed. A soot volume fraction equation was developed based on the
model of Syed, Stewart and Moss [1]. The equation was then numerically solved
subject to the assumption of inﬁnite reaction rate (IRR) proﬁles for temperature and
species. A thorough analysis of the results revealed that the essential balance for the
soot volume fraction equation is between the convection and the soot growth rate
terms. Such a balance yielded to analytical treatment and resulted in an expression
for the soot volume fraction proﬁle in integral form. The integral was evaluated by
two approximate methods. A comparison of the results using the analytical formulae
with the numerical solution indicated good agreement. A soot radiation term was
then formulated using the soot volume fraction proﬁle thus obtained. The effect of
soot radiation on the temperature proﬁle was investigated using the radiation term
in the energy equation.

In the following sections, we discuss the Syed, Stewart and Moss [1] model, develop

91

92

the soot volume fraction equation, investigate the numerical solution, derive analytical
expressions for the soot volume fraction proﬁle and compare the analytical results with
the numerical solution. The inﬂuence of a prescribed soot volume fraction proﬁle on

the radiation characteristics of a diffusion ﬂame is also described.

5.2 Soot Model

The soot model used here is based on the work by Syed, Stewart and Moss [1]. Their
two—equation model consists of a number density equation and a volume fraction

equation

(I . 71
$010): é-ﬂ(m)21 (5.1)
d 2 1 A

7.1—; = 70.191925 + 6. (5.2)

The quantity n is the soot particle number density (number of particles per m3) and
fv is the soot volume fraction in m3,oot/m3ga,. The density of soot, p,, is assumed to
be 1800 kg/m3. The quantity No is the Avogadro number, 6.0x1026. In equation 5.1
the term 6! corresponds to the process of soot nucleation. The second term on the
right hand side accounts for the decrease in particle number density due to coagula-
tion. Soot nucleation results in inception of new particles and hence, a corresponding
increase in the number density. The process of coagulation results in a decrease in
number density and hence the —ve sign before 8. As noted in [14], theoretically
the decrease in particle number density can be expected to occur according to the
Smoluchowski equation dN/dt = —ICN2 where the rate constant [C depends on the
particle diameter. The coagulation term in equation 5.1 bears close resemblance to

the Smoluchowski equation. The coefﬁcients 6:, ﬂ, ’7 and 8 are given by the following

93
expressions [1]:

=oC pszXpexp(— —To/T) (Nucleation)

A CT Coa ulation

ﬂ: ( g ) (5.3)
=C:,pT2 X Fexp( T, / T) (Surface Growth)

6 = 144x62 (Nucleation)

In the above, X p is the mole fraction of the parent fuel, i.e., methane in our case.
It can be shown that X p = YpW/Wp, where Yp is the fuel mass fraction, W is the
average molecular weight of the mixture and Wp is the molecular weight of the fuel.
The quantities To, and T, are activation temperatures and have the values 46.1 x103
K and 12.6x103 K respectively. The quantities 61 and 6 are related to soot particle
nucleation and B and 3 are related to the processes of coagulation and surface growth

respectively. The values of the constants C5,, 03 and Co are [1]

C5, = 6.54><104 [m3/kgzK1/2s]
Co = 1.3x107 [m3/K1/2s] (5.4)
Co = 0.1 [m3/kg2/3Kss]

5.3 Examination of the Soot Model

In this section, an asymptotic mathematical examination of the Syed, Stewart and
Moss [1] model is carried out to enhance our understanding of the model. Writing
1) = n/No, 17 = 77/an f = t/to, T = T/T*, ﬁ = p/p* and using the expressions for ('i,

,8, ’y and 8 from 5.3 we can simplify equation 5.1 to get:

0:11? = aﬁle/Ze7(%7n) — stﬁz (5.5)

94

where a = (X FNotoC&p*2To%8—Td) /no and b = (CoT*1/2noto)/No. The quantities a
and b are related to soot nucleation and particle coagulation, respectively. The quan-
tity no is a reference number density. Similarly, writing fv = fV/fvo, equation 5.2
can be recast in the following form:

Egg! = cﬁTfe-(T'T’Y) 517% + dﬁ2T%e'(‘TfL"le (5.6)
where c = Xpnol/3t00:yp*T*1/2€-T7/p,fV01/3 and d = 144Cotop*2T*1/2XF67T¢/p,fvo.
The quantities c and d are respectively related to the processes of surface growth
and nucleation. We now choose to such that c = 1, which results in to =
(psi fvoieT‘v) / (X pnosCo/nTﬁ). This is done because we expect that the increase
of fV is mostly through surface growth, and the balance (dfv/dt)~surface growth
reﬂects the overall soot creation time scale. The quantities a, b and d are then evalu-
ated using the above value of to. Next, the reference quantities p* and T* are chosen

in such a way that ﬁT = 1. Equations (5.5) and (5.6) can now be written (after

dropping the bars on various quantities) as

Q = aT-%6-<%-Ta> _ 117%,]? (5.7)
dt
subject to 77(0) = 0 and
if} = T-ée-<%-T~’fv%n% + dT-%e-<%-Té> (5.8)

subject to fv(0) = 0. Particle number density and volume fraction are both zero at
the initial time ( = 0). Next, it is assumed that the system is isothermal, and the
nondimensional T = 1, i.e., T,. is the constant temperature of the system. Two distinct

cases are studied in the following subsection depending on the level of saturation of

7],.

95

5.3.1 Rapid equilibration of 77

Let us assume that the particle concentration 17 rapidly equilibrates to its asymptotic
value. This assumption is of practical relevance and is supported in the literature [1],
[12]. In that limit, d77/dt = 0 and by virtue of equation 5.7, 77 = 1/(—a/—b). If for
convenience we deﬁne the maximum value of 77 as 1, then a = b. The equality of a
and b can be exploited to yield an expression for the characteristic number density

n07

 

no ﬂ 2XFe(-§s).

 

With T = 1 equation 5.7 becomes

% = a(1— 772), (5'9)

with 77(0) 2 0. The solution for equation 5.9 yields 77 = (ezat— 1)/(e2“‘+1). Examining
the limit lim.H00 77, we conclude that a rapid equilibration of 77 to unity occurs when
a >> 0(1). The equilibration time is tza‘l. Similarly, equation 5.8 reduces to the

following when T = 1:

d
J): = fvéol/s + d, (5.10)
dt

with the initial condition fv(0) = 0. When 77 rapidly equilibrates to unity equa-
tion 5.10 becomes de/dt = fvg' + d. In terms of new variables 6 = fvfi/x/c—l and

5 = t/3x/c—l, this equation becomes

d6
62—: 2 .
d6 1+6, (511)

96

subject to 6(0) = 0. The substitution 6 = tan0 gives tan20d0 = d{, with 9(0) = 0.

Consequently, 5 = tan0 — 0 and, in terms of original variables, we have

t = 3&6??? — tan'l(%)]. (5.12)

Solution of the above transcendental equation constitutes the ﬁrst problem under
consideration. We note that in the limit when d < 0(1) the second term in the
above equation is negligible and we obtain the well-known result fv ~ (t / 3)3 . In the
more general situation, a rapid equilibration of 77 is not assumed and equation 5.10
is solved using the result 77 = (62“ — 1) / (6“ + 1). In the following, we carry out a
discussion and comparison of the results for the two problems.

The numerical solution of equation 5.10 was obtained using the NAG fortran
library routine D02PCF. This routine uses the Runge-Kutta method.

Figure 5.1 shows two sets of results for the soot volume fraction proﬁles. In the
ﬁrst case, (1 was set equal to 0.5, since d is the only parameter value which needs
to be prescribed for the solution of equation 5.10. For d = 0.5 the saturated 77
assumption (equation 5.12) gives an fv proﬁle which is quite similar to the more
general unsaturated case. For large times, however, the two curves tend to diverge,
albeit at a rather slow rate.

When d = 0.005 the curves for saturated and unsaturated cases are virtually
indistinguishable. It can be recalled at this point that d is the nucleation term in the
soot volume fraction equation 5.10 and for a small value of d, the soot proﬁle saturates
fairly quickly. Hence, in this case, the assumption of 77 rapidly saturating to the value
of unity is quite good. Also, when d = 0.5 the amount of soot generated is much more
compared to the d = 0.005 case. This is because a much stronger nucleation term
results in a higher soot volume fractions. Also plotted in both ﬁgures is the function

(t/3)3. As mentioned before, when d < 0(1), it is expected that fv ~ (t/3)3.

97

 

 

 

 

 

 

 

 

 

20 1 m r 7 T 1 f 6 1 7 r I T J
A (1:05 1 , d=0.005 f
15 f l 6' /
""" Unsaturatedn 4 _ ----- Unsaturatedn
. ° Saturated n 0 Saturated n a ,
_-.__- 3 _- _ 3 9° ,
(V3) " (V3) f /
(14> 10 f 81-? d I/
9“ ,l
’l
2 ’/
I dd ’/
5 ,I _ .“ /
1’ :/ 0° /’
«13°: 03/ / / i ego/I
f 0039” / / “pf/f/
9°?! ,'/’ ,o’q/
.0 _1 1.. ——7 1’ i 1 1 1 L 00“.,‘31m'1/ 1 1 #1
00 10 20 30 4.0 50 0.0 1.0 20 3.0 40
t

Figure 5.1. Comparison of fv proﬁles for d = 0.5 and d = 0.005.

98

When d = 0.005 we see that the numerical solutions are indeed comparable to the
(t/3)3 curve. However, for d = 0.5, the (t / 3)3 curve is signiﬁcantly different from the

numerical solutions.

5.4 Formulation of the Full Problem

The problem geometry and boundary conditions are the same as described in Chap-
ter 4 and illustrated in Figure 4.1. The soot volume fraction equation has the following

form:

p(th + (u + uT)fV$):(PDst1,-)x + ti)” + 11.)!) _ “301" (5’13)

In equation 5.13 1b“, tbg and 11203 are, respectively, the soot nucleation, growth and
oxidation rates in units of leg/mas, up is a thermophoretic velocity term, which is
neglected in the present analysis. Equation 5.13 contains a soot diffusivity, D,, which
is taken to be 1% of the gas diffusivity [12]. The physical coordinate of our problem
is x with OSxSL. The boundary conditions are fv(x = 0) = 0 and fv(x = L) = 0.

First we transform the equations to a (Z,t) system of coordinates, where Z =
1 — 3/30 with s = fgpdx and so = f (I; pdx. As a result the soot volume fraction
equation takes the following form:

_ p2D3 1

u + u . . .
— (po 0 p T)sz szz + ;(wn + my " Him-)- (5-14)

30 802

th

 

 

Now, we define t— = t / t,.e f and iv = fV/fvo, where fv0 is a suitably chosen reference

value. Consequently, the volume fraction equation becomes

afv (1 +7717) afv _ 030 asz

a; so 62 _ 210ng 822

 

 

+ ([113, + BF, — Cr”). (5.15)

 

 

Here, A '2 Ypptrcf/fvotn, B = Ypptrcf/fvotg and 0 = YOOtrcf/fVotox- The

99

quantities tn, tg and tax are characteristic time scales for the processes of soot nu—
cleation, growth and oxidation respectively and are given by tn = poYpp/tbn,,.cf,
t9 = p0YFF/wg’ref and tax = poYoo/tbox,,e;. The quantity inf is the characteristic
flow time, given by tref = L/uo, and than}, 122g,” 1 and 11303,"; are reference values for
normalizing nucleation, growth and oxidation terms, respectively. We have also used
50 = so/(poL). The nondimensional thermophoretic term is denoted by 1727 = ﬁuT/uo.
The soot diffusivity at the reference condition is D“). The rate terms Fm F9 and Fox
are respectively for soot nucleation, growth and oxidation and 17,, = 1.72,, / p, where
1.7),, = wn/ 11.)”,rej. The growth rate term 1‘,, and the oxidation rate term Fox are defined
similarly. Next, we assume that the temperature and species profiles are for the in-
finite reaction rate (IRR) case, which implies that no fuel exists on the oxidizer side
and no oxidizer exists on the fuel side. In the absence of oxidizer on the fuel side,
the source term for oxidation in the soot volume fraction equation can be discarded.
Further, on neglecting the thermophoretic effect, the soot volume fraction equation
reduces to

afv lafv _ Dso 62f},

at— 50 BZ — 110ng 622

 

+ (Ar. + B5,) (5.16)

 

At this stage, the expressions for A and B are still to be determined. The quan-
tities A and B depend on the time scales tn and t9 respectively, which, in turn,
depend on the choices of 113mm, and 1%,”. By comparing equation 5.13 with the
volume fraction equation (equation 5.1) of the Syed, Stewart and Moss model [1],
we can write 1b,, = pti/p, and 1229 = (p/p,)’y(p,fv)2/3n1/3. On substituting the

"Td/T. Hence

expression for (i using equation 5.3 we get 11),, = 144Cap3T1/2Xpe
113an = 144Cap03Tf3XFoe-Td/Tf can be chosen as a reference for the soot nucle-
ation term. A reference value of the soot growth rate term can now be written as

T
113g”! = (pg/p,)ﬁonol/3(p,fvo)2/3, with ’yo = ryponl/szoe—ffl. The quantities with

sufﬁxes 0 are at the reference condition. On obtaining 15mm} an expression can now

100

be written for tn and consequently A can be expressed conveniently as

T

1 .__ﬁ.
L 144p0205Tf2XF06 T!
“0 (PstO)

 

A =( (5.17)

Similarly, since we have obtained the expression for 1139”,, we can now write an

expression for t9 and hence for B,

1 -31 i
B = (£)C‘1P0T15XF06 T’ no?

no (Pst0)%

 

(5.18)

We now focus our attention on the nondimensional rate terms in equation 5.16, viz., 1",z
and F9. As mentioned before, 1"" = 1D,, / [3 and 11),. = zbn/wn,,ef. Using the expressions

for 11),, and 1b,,” f we can obtain the following expression for in:

 

n:

(1 —(1a(_1 (:):»%yFCZI—Eﬁf11_;:f_). (5.19)

The quantity 36: = TAT; — T0)/Tf2 and a, as before, is given by a = 1 — To/Tf. In
the above we have also utilized the relation between temperature and density, i.e.,
[3 =(1— a)/(1 -— a(1— 7)), by virtue of equation 3.16.

An expression can also be found for the rate quantity f9 using the expressions for

1139 and 1139,"). The expression for Fg takes the following form:

wln

_ (l-aﬁyri —(a.+9i)( 1" )
r — e 6 mm . (5.20)

9—(1—a(1—r))

<

 

GIU‘

At this stage we notice that the expression for B in equation 5.18 involves an unknown
reference number density, no. In order to evaluate that, the soot particle number

density equation has to be examined

P(77t + (u + uT)77x) = (P13377291: + “3n " we, (5°21)

101

where 17 = n/No. The w’s in the above are the rates of production of particle number
density in a cubic meter multiplied by the density p and hence have the units of
kg/(mGS). By writing 17 2 77/770, f = t/trcf and transforming the equation to the
mass-based Z coordinate, we can write the normalized number density equation in

the following form:

wc,rc f

P0 770

_ (1+ 771T) _ D30 _ an “.37: ref ﬁfe
' — —_-— Z = _ Z2 + ‘1- ’ tre — “.—
”t 50 77 ”OI/3(2)” p pono f p

 

t..,. (5.22)

 

 

We notice that Pane/15mm} has the unit of time and thus qualiﬁes as a characteristic
nucleation time scale. Hence (pono/zbn,ref)/t,¢f is a ratio of a characteristic nucleation
time scale and the characteristic ﬂow time. Let us call the above ratio E. We then
multiply equation 5.22 by E to obtain

(in (be Lbcwef

6 - __ —— = 6 _ —
'7‘ ‘ "Z uoLsg "Z2 ,5 ,5 ohm...)

 

(5.23)

When nucleation is rapid E—+0 and the quantities multiplied by E become negligible
and equation 5.23 essentially reduces to a balance of the nucleation and coagulation
terms, and the number density reaches a steady or saturated value, exactly as the
simpliﬁed model in section 5.3.1. We get én/ﬁ — éc/ﬁLiJc’ref/Lbn’ref and hence (1),, =
(126. At this stage we compare (5.21) with equation 5.2 of the Syed, Stewart and
Moss model [1] and write (2),, = p6: = Cap3T1/2Xpe‘T°/T and the = pmn/No)2 =
pCﬁT1/2(n/No)2. The equality d2" = the is then utilized to obtain an expression for

1/ 2C-Td/ 2T! . The reference

the reference number density, no = poNo(Ca/C,;)l/2Xpo
fuel mole fraction X Fo equals YFFW/WF. Using the above no, the quantity B can

be readily determined.

102
5.5 Numerical Solution

Equation 5.16 is numerically solved for a range of parameter values to obtain a so-
lution for the soot volume fraction. The equation is discretized by ﬁnite difference
method and the source terms are linearized using Newton’s method. The steady
state solution is reached when the sum of normalized residuals between successive
time steps becomes smaller than 1x10'6. The initial soot volume fraction value was
assumed to be zero everywhere in the domain. A typical solution is shown in F ig-

ure 5.2. In order to further investigate the importance of various terms in the soot

Y00=0.6 YFF=0.8 u0=0.07 m/s
szzoo K L=0.01 m
4.0 ' 1 ‘ T ﬁ 1 T T #

 

 

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
Z

Figure 5.2. Soot Volume Fraction Proﬁle

103

volume fraction equation, the convection, diffusion, soot nucleation and growth terms

are all plotted in Figure 5.3. The abscissa of the plot is (Z — Z f), where Z f is the

YOO=O.6 YFF=O.8 u0=0.07 m/s
T.=2200 K L=0.01 m

 

2000.0 1, ' I i I r f r

..,............. Convection
..... Diffusion

o—o Nucleation -
1\ — - —- - Growth

1000.0 -

 

 

 

 

-1000.0 (,1 -
-2000.0 ' . . 1 . . . . 1 +
0.00 0.04 0.08 0.12 0.16 0.20
(2-2,)

Figure 5.3. The structure of the soot volume fraction proﬁle

location of the Burke-Schumann ﬂame. We note that close to Z f, there is a balance
between the convection ((1/5o)dfv/dZ) and diffusion ((D,o/uoL§g)d2fv/dZ2) terms.
However, the diffusion term is quite small at an incremental distance from Z f, and for
the most part of the soot layer, the essential balance is between the convection term
and the growth (Big) term. The nucleation term (24173,) is very small in comparison

to the other terms.

104

The above insight about the fundamental structure of the soot layer is utilized in

the development of an analytical expression for the soot volume fraction proﬁle.

5.6 Analytical Approximation

As noted in the previous section, the primary balance for the soot volume fraction is
between the convection and the growth terms, except very near Z f. Hence, we assert
that equation 5.16 can be further reduced to
1 div —
——— = BF 5.24
30 d2 .91 ( )
in the steady state limit. The quantity 5’ is a constant, and ﬂ, is the nondimensional
soot growth term, derived from the Syed, Stewart and Moss model, is given by equa-

tion 5.20. Next, we relate dZ to (IT in order to determine a solution for fv(1'). The

fuel side (Z f_<_Z SI) Burke-Schumann temperature distribution is used for the above

 

purpose
e-l/c _ e—Z/c

T = 8-1/1: _ e-Zr/C’ (5'25)

where c = ao/(uoLéo) and Zf, the ideal ﬂame location, is given by Zf =

cln((1 + ¢)/¢). A detailed analysis of the inﬁnite reaction rate situation is included

in section 4.5 of chapter 4. The fuel mass fraction proﬁle is

yp = 1 — 6"(Z‘Z’Vc. (5.26)

In terms of 1', yp can be written as

yp = a(1 — r), (5.27)

105

where a = 1 — e‘(1‘Z!)/c. After a little algebraic manipulation, dZ can be written in
terms of dr as dZ = -[(ac)/(1 — a(1 — T))]d7‘. By substituting the expression for 3);?

in terms of 1' and dZ in terms of (17', equation 5.24 is transformed into

(1 _ ,,.)7/6,3 45,14,211 5 11m)
(1— a(1 T))5/6(1 - 0(1— Tlld

1

173/3

 

dfv = [B§o(1 - a)4/3a13/6c] (5.28)

At this stage we observe that in order to integrate the soot volume fraction proﬁle, we
need to impose suitable boundary conditions. Before we made any approximations,
the boundary conditions for the soot volume fraction were iv = 0 at Z = 0 and at
Z = 1. However, the soot diffusion term has been dropped, which was the only second
order term in equation 5.16. The resulting approximate equation is ﬁrst order, and
only one boundary condition can be used. We use the condition that the soot volume
fraction is zero at the fuel wall, i.e., at Z = 1. We expect to obtain a soot volume
fraction proﬁle which is located preferentially on the fuel side of the diffusion ﬂame.
It has to be kept in mind also that in order to substitute for dZ in terms of 011', the
temperature proﬁle on the fuel side was used, and hence, any soot volume fraction
which we may see on the oxidizer side is actually non-existent.

We now integrate equation 5.28 subject to the boundary condition that f—v = 0

when 'r = 0. From equation 5.28 we ﬁnd that the important integral to evaluate is

_,,.)7/6e —(ﬁ:,+-4)(—_7,—_;5)
0(1(1—- a( (1 — 7))5/6(1- a(1— 7))dT

 

(5.29)

By substituting u = 1 — 7' and writing ,Bw = 3:, + a: we arrive at the following

expression for 11:

 

du (5.30)

1 117/665%
11 =/ (

1 — au)5/6(1 — cm)
The integral in equation 5.30 could not be analytically evaluated. In order to simplify

matters, the exponents 7/ 6 and 5/ 6 on u and (1 — cm) respectively were replaced by

106

1. Let the modiﬁed integral be called 12. Then, I; happens to be integrable ana-
lytically. An analytical result was obtained using the commercial code MAPLE, and
the result so obtained was veriﬁed using another commercial code, MATHEMA TI CA.

The following expression is obtained for 12:
12 = (A + B + C + D)/(aa(a — a)), (5.31)

where

212 . a a-
= —ae°-° E2(1, (536011—23)

A
El?- ' '3 a
B = a—aea Ezl,—-7—
( ) 9.1. ( (“‘1’“) (5.32)
C = —(a-a)e aaE'z'(1 -—"’619—)
D

’ (a(l—T)-1)a

_ $.13; ‘ ’3 a
— 0'6 ENLW),

where Ei(1,a:) is a notation for the exponential integral. The indexed family of
exponential integrals, Ei(n,a:) where n is a non-negative integer, is deﬁned as
Ei(n, z) = faweﬂﬂwdt for real, positive 2:.

Integral 1;, thus calculated, is compared with the numerical solution. Also,
Laplace’s method was used as an alternative method to approximately evaluate the

integral I1. The following section brieﬂy describes the analysis.

5.7 Integral Evaluation Using Laplace’s Method

The integral in equation 5.30 can be approximately evaluated using Laplace’s method
and can be written as 11 = f}_,g(u)ef(“)du where g(u) = (1 — au)‘5/6(1 — cm)"1 and
f(u) = —ﬂ,au/(1 — au) + lnu7/6.

Let us next consider the integral 1(x) = f :exh“) f (t)dt, where h(t) is real and

a: is positive and large. Assume that the integral exists, i.e., it has a ﬁnite value.

107

According to Laplace, only the immediate neighborhood of the point corresponding
to the maximum value of h(t) in [a, b] contributes to the asymptotic expansion of [(3).
Similarly, for expansion of the integral [1 the value of u for which f (u) is a maximum
is sought ﬁrst. Let us denote the value as u*. Equating df/du to 0 and evaluating
d2 f /du2 to ascertain that f attains a maximum at u*, we ﬁnd that u* = 7/(6ﬂqa).

Next the integral [1 is expanded in the following form:
1 l * (u-u*)2 n *
11 = / [g(u*) + (u — u*)g'(u*) + ...]t:f("*)+(““‘*)‘f (" )+ 2 f (“ )+"‘du (5.33)
1-1'

By using the result f’ (11*) = 0 and evaluating the different derivatives of f (u) and
g(u) at u* and assuming a large ,6.” the above integral can be easily evaluated. After
some algebraic manipulation and neglecting higher order terms in the expansion of

the integral, we obtain the following simpliﬁed result for the integral [1:

3101“”: — T)la (5.34)

where 7* is the value of nondimensional temperature corresponding to the value 11*,
i.e., 7* = 1 — 11*. Hence, using equation 5.28 and equation 5.34 we can write the

following expression for the soot volume fraction:

fé=[BsO(1—a)4/3a We 1°—'2——:’l:_.9erfcl\[ﬂ~a( — 1. (5.35)

5.8 Comparison of Results

Figure 5.4 shows a comparison of the numerical solution and the analytical solutions
for the soot volume fraction proﬁles. In Figure 5.4 “Method 1” refers to the solution
using exponential integrals and “Method 2” refers to the solution using Laplace’s

method. It is seen that both the methods result in soot proﬁles which have substan-

108

Yoo=1 YFF=1 uoL/or.0=5.65

 

 

 

 

 

 

T, = 2200 K
15.0 r .
—— Numerical Solution
---- Method 1
----- Method 2
10.0 - -
”o
:1:
H—c>
5.0 - .
1
0.0 ‘ * ‘ ‘ ‘ ‘
0.0 0.4 0.6 0.8 1.0
Z

Figure 5.4. The analytical solutions plotted along with the numerical solution.

109

tially different maximum values compared to the numerical solution. This happens
because the soot proﬁle maximum is close to the ﬂame location, Z f, where the effect
of soot diffusion is prominent, as observed in the section 5.5. Since the diffusion term
was neglected for the analytical solution, the disagreement between the numerical and
analytical solutions is quite conspicuous near Z I. However, consistent with the dis-
cussion in the previous section, a much better agreement is obtained near the trailing
edge of the soot layer, i.e., further from Z f. The agreement between the analytical
and numerical solutions is excellent in that region. Consequently, the thickness of the

soot layer can now be analytically predicted fairly accurately.

5.9 Inﬂuence of soot radiation

In this section, we shift our attention to the effect of radiation on a diffusion ﬂame
established between a fuel and an oxidizer wall, with a convective fuel ﬂow from
the fuel wall. This conﬁguration is the same as in Chapter 4. However, the radiation
term used here is different. A soot volume fraction proﬁle, generated using the method

described in section 5.5 was used to formulate a radiative loss term.

5.9.1 Background

Understanding of soot radiation and its proper incorporation in the model are ex-
tremely important for the current research. The radiation from a ﬂame depends on
the soot proﬁle, which depends on the particular fuel used, the temperature proﬁle
and the species proﬁles. In this section we will assume a soot volume fraction based on
the numerical solution of the soot volume fraction equation described in section 5.5.
A radiation term is formulated based on the “optically thin” assumption for the soot
radiation. A review of the existing literature pertaining to soot radiation is included

in section 2.5 of chapter 2.

110

5.9.2 Formulation of the Radiation term

For a. sooty ﬂame the major part of the radiation is a continuum radiation that is
simpler to model than the radiation due to water and carbon dioxide. For soot volume
fractions > 10",, soot radiation should be dominant judging by the calculations of
Grosshandler and Modak [11]. For small ﬂames and moderate soot loadings with

which we are dealing, the optically thin limit is appropriate. In this limit

@ = 45,5(1‘4 — T04), (5.36)
d3:

where, (1,, is the Planck Mean absorption coefﬁcient and a is the Stefan-Boltzmann

constant which has a value of 5.6696x10‘8 W/mzK“. The quantity ap is given by

a : fooo ”(A,fv)€b()\)d/\
” f§°eb(x\)dx '

 

(5.37)

Kennedy et al. [31] used It = 7 fv / A, where n is the absorption coefﬁcient and A is
the wavelength of radiation. For a blackbody the spectral distribution of hemispher-
ical emissive power in a vacuum is given as a function of absolute temperature and

wavelength by the following expression:

27701

2 155% — 11'

“(1) (5.38)

This is known as Planck ’s spectral distribution of emissive power. The constant 01
has the value 0.59544x10‘16 Wm” and 02 = 1.4388x10'2 mK. Using the expression
for 65 in equation 5.37 and substituting y = Cg/(AT), the following expression is

obtained for ap:
_ 1471'fVC'171 0° y4dy

a _ _
p o
0025 0 e” — 1

 

(5.39)

111

The integral f3°y4dy/(ey — 1) is the fourth order Riemann zeta function and whose
value is 24.8862. Hence, ap = 1864.32 fVT.
By transforming equation 5.36 to the Z coordinate and using CTR =

QR/Pouon(Tf — To), 7' = (T — To)/(Tf — To), ,6 = p/po and {so = so/(poL) we obtain

13155 _ 4a,,0'LTf4
50 d2 pCp(Tf — To)’uO

 

[(1— 5(1 — 7))4 — (1 — (1)4]. (5.40)

By using equation 3.16 for the relationship between temperature and density we can

further reduce the expression for (1/.§o)(din/dZ) and write

 

1 d" —
71-5 = 4.50- a0 - 702((1— an — or — (1 — 002), (540
So dZ
where PR is given by
6
PR _ 4378.2101LfV0Tj (5.42)

_ ponTo(Tf - To)uO025 .
The energy equation for this situation is the same as equation 4.1 of chapter 4. In

the Z coordinate the energy equation becomes

1 Go — 1 dq-R
= _ ’D __
77 307-2 + nongTzz + QF F?“ + 50 d2 ,

 

(5.43)

where we have used if}; = qR/qR,,ef and QF = QFYFF/Cp(Tf — To) = (1 + (15). The
reference quantity qnnf was chosen to be pouon(Tf — To). Hence, the radiative loss
term is given by 1/§o(drjR/dZ) We note that we have already derived an expression
for the loss term in equation 5.41. The energy equation can now be solved numeri-
cally using a prescribed soot volume fraction proﬁle in conjunction with the coupled
oxidizer and fuel mass fraction equations. The species equations are the same as

equations 4.5 and 4.6 of chapter 4.

112

5.9.3 Results

Figure 5.5 shows a plot of the temperature and species proﬁles of a ﬂame with a
radiative energy loss modeled as in the previous section. The prescribed soot volume

fraction proﬁle is also shown on the same plot. Next, the structure of the ﬂame is

YOO=O.6 YFF=O.8 u,L/a,=5.45
Prescribed fV proﬁle
1 .0 "' ' T . r . , . ,

G—-0’[

 

Hyo

0.8 " HYF , .,-
—— fv*10

0.6

0.4

0.2

 

 

 

 

0.0 -
0.0 0.2 0.4 0.5 0.8 1.0

Figure 5.5. 7', yo and yp proﬁles for prescribed soot volume fraction.

shown in Figure 5.6. The convection, diffusion and reaction terms of the nondimen-
sional energy equation are plotted along with the radiative loss term. This ﬁgure can

be compared with the structure of a. similar ﬂame with a sech2 type radiative loss

113

term, as illustrated in Figure 4.11 of Chapter 4. The similarity in the structure of
the two ﬂames is quite striking and suggests that the simple sech2 type radiative loss
proﬁle is a fairly good approximation which is capable of revealing interesting aspects

of the diffusion ﬂame behavior when subject to radiative heat loss.

Y00=0.5 YFF=O.8 uOL/oc0=6.45

 

Prescribed fV proﬁle
2000 1 . 1 . 1 . n . r
Convection

 

°——-° Diffusion
D-——° (1+¢)Dr

100.0 5 . .
----- Rad1at1ve Loss

 

- -Aogggggggg ......................................
O 0 ::la:'- _. . ,‘:II'.I=B=II=:=S=II==BI:IS::CI:J:5:!I:I:C:Il:::t:l!::ct:13::tl:I:
I

I

-100.0 " ,, -

 

 

 

_2000 . “° 1 . 1 . 1 . 1 .
0.0 0.2 0.4 0.6 0.8 1.0

Z

Figure 5.6. Structure of the ﬂame. Note the radiative loss term.

114

5.10 Conclusions

A simpliﬁed analysis of the soot proﬁle was carried out in this chapter using the Syed,
Stewart and Moss [1] model. Burke-Schumann temperature and species proﬁles were
assumed and the soot volume fraction equation was solved numerically for a diffusion
ﬂame established between an oxidizer and a fuel wall. Oxygen diffused out of the
oxidizer wall and fuel was released from the fuel wall by means of both diffusion and
convection. It was observed that for the soot volume fraction equation the primary
balance was between the processes of soot convection and soot growth. Equating the
expressions for these two quantities resulted in an analytical expression for the soot
volume fraction in integral form. The integral was evaluated using two methods: the
ﬁrst one involving exponential integrals and the other using Laplace’s method. The
analytical proﬁles matched well with the numerical solution except very near the ﬂame
location Z I. Close to the Burke-Schumann ﬂame location, the soot diffusion term is
strong and there the essential balance is between the soot convection and the soot
diffusion, the soot growth term being relatively much smaller. However, the thin zone
may be an artiﬁce of the IRR approximation itself. The analytical expressions, which
were derived on the basis of the soot convection ~ soot growth balance, do not predict
the soot volume fraction values accurately near the inﬁnite reaction rate (IRR) ﬂame
location. However, for most part of the soot layer the soot volume fraction proﬁle is
predicted well by the analytical formulae. Consequently, the soot layer thickness can
also be accurately predicted using analytical methods.

A soot radiation term was also developed on the assumption of the optically thin
limit for the radiation. A soot volume fraction proﬁle obtained from the numerical
solution was used in the expression for the soot radiation term. The radiation term
was then included in an energy equation as a sink term and the equation was solved

simultaneously with the coupled fuel and oxidizer species equations. The results

115

revealed that the ﬂame structure closely resembles the structure of the ﬂames studied
in chapter 4 for sech2 heat loss proﬁles. Such a resemblance indicates that the simple
sech2 heat loss proﬁle examined in chapters 3 and 4 is capable of revealing interesting

details of the ﬂame structure.

CHAPTER 6

Investigation of the Comprehensive

Soot Radiation Problem

In the previous chapters we have analyzed with simpliﬁed models of soot proﬁle
and radiation loss effects. In the current chapter we deal with the complex problem
of soot radiation and diffusion ﬂame interaction using a soot model chosen from
the available literature. Here we use the two equation model of Syed, Stewart and
Moss [1] for the soot volume fraction and the soot number density. Their model
has been developed based on experiments with different fuels and for a range of
ﬂow conditions; we have examined this model analytically in chapter 5. A two—
dimensional Wolfhard-Parker burner was used for their experiments in order to ﬁx
various undetermined constants. This model has been used by Syed et al. [1] for
both laminar and turbulent ﬂame predictions. Recently, Ku et al. [51] have used this
model for the simulation of microgravity turbulent diffusion ﬂames and the agreement
of the model with experiments was very good. In the following sections, we ﬁrst
describe the conservation equations and then formulate the ﬁnal, non-dimensional
equations including the Syed, Stewart and Moss [1] soot model. Then, we discuss
some interesting results obtained from the numerical solutions of the conservation

equations. Although our results indicate a variety of interesting behaviors, a radiative

116

117

extinction was not observed, suggesting to us that a steady state extinction is unlikely.

6.1 Formulation

There are six conservation equations of importance in our problem, viz., the mass, the
energy, the oxidizer and fuel species, the soot volume fraction and the soot number
density equations. At this stage we also note that on an overall basis, we can account

for three principal chemical reactions, as noted below

fuel+air —-> products
fuel —+ soot

soot+oxygen —> products

It has to be kept in mind, however, that each of the above overall reactions is very
complex in reality and consists of hundreds, and may be even thousands of steps and
intermediate products.

The continuity equation is
0 a
_P 12 = 0 (6,1)
at 82:

At steady state, Bpu/ax = 0 and hence, pa = pouo, since the mass ﬂow rate is

m = pouo. Hence,

The energy equation has three source terms accounting for the heat generation due
to the primary chemical reaction, the radiative heat loss and the heat generation due

to soot oxidation. The energy equation is

dQR

PCp(Tt + uTz) = (A717); + QFtbF - E

+ 620.02.... (6.3)

118

We assume that the reaction C + 3§02 = C0 + Q03 takes place for soot oxidation,
with Q“, being the heat release. The heat release in the above oxidation process is
53 kCal/mole of 02 [52], [53]. Hence, Q03: = 9246.29 kJ/kg of C.

The oxidizer equation has a depletion term due to the primary reaction and an-

other one accounting for soot oxidation, viz.,

4
-15.... (5.4)

p(Y0t + uyOx) = (pDOYOx)a: - 11’le — 3

We also note that for every kg of C 4/ 3 kg of 02 is required due to the oxidation
reaction for soot particles, hence the coefﬁcient 4/ 3 for the oxidation rate if)” in the
above equation.

The fuel mass fraction equation includes depletion terms due to the primary chem-

ical reaction as well as the conversion to soot particles,

. 4 .
pCP(YFt + uYFx) = (PDFYFx)x — wF "‘ 3109- (6-5)

For the conversion of fuel to soot, we assume a very simple overall reaction of the form
CH4—>C + 2H2. If the soot growth rate is 1139 kg of C / m3s, then the depletion rate
of the fuel due to soot growth is (4/3)u'2g kg of fuel / m3s, since 1 kg of C is produced
from 4/ 3 kg of fuel, according to the above chemical reaction.

Finally, we have two conservation equations for the soot volume fraction and the

soot particle number density, viz.,

P(th + (u + uT)fo) = (PDstx)x + Li’11 + “)9 _ 1003, (6-6)

and

10(77: + (u + when) = (st005 + cbn - a)... (6-7)

119

In the above, on, 1129 and 1110,, are respectively the soot nucleation, growth and ox-
idation rates in units of kg/m3s. The quantities c2)" and (be are the nucleation and
coagulation rates in units of (kg/m3)(number/m3s). The soot volume fraction in
units of m3,oot/m3ga, is fv and 7] is the number density of soot particles normalized
by Avogadro’s number (No = 6.0x 1026), i.e., 77 = n / No. The thermophoretic velocity
term is denoted by 111. A discussion of thermophoresis is included in section 6.1.1.
The soot diffusivity, D3, is usually quite small and was assumed to be 1% of the gas
diffusivity [12].

As in the preceding chapters we transform the equations to a (Z,t) system of
coordinates, where Z = 1 — s/so with s = fgpda: and so = fg’pdx. As a result, our

conservation equations take the following form:

 

 

 

 

 

 

 

energy:
Pouo (M) 1 . p dqzz .
T — ——T = T — —— 0:1: 0:: a -
1 50 z 010302 22+ pCp(QFwF+ 80 d2 +Q w ) (6 8)
Oxidizer mass fraction:
2
Penn P Do 1 . 4 .
Y _ = __' _ 01: 3 °
0: 30 Yoz 302 Yozz p(uwp+3w ) (6 9)
Fuel mass fraction:
2
pong _ p DP 1 . 4 .
YF: - SO YFz — 802 YFzz - ;(wF + 5%), (6.10)
Soot volume fraction:
u + u 2D 1 . . .
th _ (po 0 p T)fVZ : P SfVZZ + ;(wn + wg _ wax), (611)

30 302

120

Soot number density:

 

 

+ 2D 1 . .
t_ (Polio PUT) : P 257722 + _(wn _ we). (6.12)
so 5 p

In the above, we have assumed the quantities pA, pzDo, p2Dp and p20, to be con-
stant. Next, we nondimensionalize the above equations. We use 1' = (T — To) / (T; —
To), yo = Yo/Yoo, 3])? = YF/Ypp and iv = fv/fvo, where fvo is a suitably chosen
reference value. Also, we deﬁne a set of characteristic times for the different chem-
ical processes. The characteristic chemical time for the primary reaction is denoted
by tchem = poYpp/tbp,,.ef. The quantities tn 2 poYpp/tbn,,ej and tg = p0YFF/wg,r¢f
are, respectively, the characteristic times for the processes of soot nucleation and
soot growth. The characteristic time scale for soot oxidation is to; = poYoo / 11203," f.
The quantity 5 is the nondimensional density and the following terms are also used:
1"}: = rim/p, 1‘,, = tbn/ﬁ, F9 = dig/[0' and Fox = wax/p. Consequently, the energy

equation becomes

31' 1 01' _a_o 0271 d__qR

b—tz— 30 —6Z= “011302 8Z2 + QFDFTF +— 50 dZ + @0171) 0137.01?) (6.13)

where 63 is given by qR/pouonAT. The radiation term (1 / 5o)(d(jR/dZ) is modeled
exactly as in section 5.9.2 of chapter 5 and is given by equation 5.41. The quantity 5o
equals 3 / (po L) and D0,, is a ratio of the characteristic ﬂow time and the characteristic
oxidation time, i.e., tref/tox, where tn.) = L / uo. The oxidizer mass fraction equation

becomes
ago 10__yo_ ao 6%

Ct— So 72:110ng 8Z2

 

 

3120.13,.) (6.14)

— ((bDFFF + 3

Correspondingly the fuel equation takes the following shape:

83);: 1 ayp_ _a_o _3__2yp_ _ 4 va0 _
W — 57,— 62 110—53 572 pp” + " ’9 (6'15)

 

121
The volume fraction equation looks like the following:

div (1+r72.T)8fV_ 0,0 32f}, __ __ __
6t _ go BZ _ 110133 522 +(A7‘n'l'37'g crox) (6.16)

 

 

In the above, we have assumed do = 000 = Dpo and A = YFFtref/fvotn, B =
Ypptnf/fvotg and C = YOOtref/fVOtox- The quantity ﬁrm is given by puT/uo. For
the soot number density, following the analysis in section 5.4 of chapter 5 we can
write 0),, = (1),, i.e., the soot number density equation rapidly saturates and at steady
state the soot nucleation and coagulation rates are equal.

We will now focus our attention on the different rate expressions. The quantity
tbp is the rate of primary chemical reaction, and has the form pAYo Ypexp(—E/ RT),
where A is the pre-exponential exponent and E is the activation energy.

As mentioned before, for the processes of soot nucleation, growth and coagulation
we have used the Syed, Stewart and Moss [1] model. A detailed description of the
model can be found in section 5.2 of chapter 5 and the expressions for 1"“, F9 are as
formulated in section 5.4 and are respectively given by equations 5.19 and 5.20. The

soot oxidation rate term, Fox was modeled using the semi-empirical formula of Nagle

and Strickland-Constable [2].

6.1.1 The Inﬂuence of Thermophoresis

Thermophoresis is the phenomenon wherein small particles, when suspended in a gas
in which there exists a temperature gradient VT, experience a force in the direction
opposite to that of VT [54]. A common example of thermophoresis is the blackening
of the glass globe of a kerosene lantern; the temperature gradient established between
the ﬂame and the globe drives the carbon particles produced in the combustion process
towards the globe, where they deposit. Thermophoresis is of practical importance in

many industrial applications, such as thermal precipitators.

122

The temperature gradients in our ﬂames are quite steep and hence, an examination
of the effect of thermophoresis is duly warranted. Santoro et al. [30] observes that
soot particles can usually be treated as being in the free molecular limit for ﬂame
conditions, i. e., the particle diameter is much smaller than the mean free path of
the gas. Under such conditions, the thermophoretic velocity, uT can be written as
uT = —(3/ [4(1 + 1rA/8)])(u/T)VT, where u is the kinematic viscosity of the gas
and A is the accommodation coefﬁcient which usually is taken to be 0.9 or 1.0 [30].
Assuming A = 0.9 we obtain the following expression for the thermophoretic velocity
of the gas:

uT = —0.55%VT (6.17)

The — ve sign in the above indicates that the thermophoretic velocity is in the di-
rection of decreasing temperature. The thermophoretic velocity component is simply
added to the convective ﬂow velocity, as in equations 6.6 and 6.7.

Next, we carry out the necessary coordinate transformation and express the quan-

tity fnT = puT/uo as in equation 6.16.

_ 0.551/(1 — a)2 it:
_ [1 — a(1 — r)]3(uoL§o) dZ

 

mT

(6.18)

A discussion of the inﬂuence of thermophoresis pertaining to this research problem is

included in section 6.4.

6.2 Parameter Values

The parameter values assumed here are the same as in chapters 3 and 4. However, the
value of the pre—exponential factor, A, was taken from the work of Chen et al. [55].
For the quantity A/p they used a value of 5.2)(1013 cm3/gm — s. We assumed a

reference value of p = 0.0012 gm/cm3 and hence our A = 1.95x109 Us The adiabatic

123

ﬂame temperature correlation used in chapter 3 was also used in this chapter. The
length of the domain, L = 0.02 m and as before, the thermal diffusivity at the
reference condition, ao = 1.24x10'4 mz/s [50]. The kinematic viscosity of the gas

phase is assumed to be u = 8.68x10‘5 m2/s.

6.3 Numerical Solution

The nondimensional temperature, species and soot volume fraction equations were
numerically integrated using the ﬁnite difference method. The non linear source terms
were linearized using Newton’s method. The Burke-Schumann proﬁles for tempera-
ture and species were used as initial proﬁles. The soot volume fraction was assumed
to be zero everywhere at the initial time. The transient conservation equations were

integrated to steady state.

6.4 Results and Discussion

Figure 6.1 illustrates the temperature, species and soot volume fraction proﬁles in the
Z coordinate when Yoo = 0.6, YFF = 0.7 and uoL/ao = 3.23, which corresponds to
uo = 0.02 m / s. The nondimensional temperature proﬁle indicates a substantial effect
of radiative losses. An examination of the species proﬁles reveals that there is no
signiﬁcant diffusion of oxidizer and fuel to the opposite sides of the ﬂame. The soot
volume fraction proﬁle resides primarily on the fuel side of the ﬂame, in accordance
with experimental observations [30]. Note that there is a slight change of slope of the
temperature proﬁle in the radiative loss zone.

The nondimensional temperature proﬁle indicates a signiﬁcant effect of radiative
losses. The species proﬁles indicate that there is no signiﬁcant diffusion of oxidizer

and fuel to the opposite sides of the ﬂame. The soot volume fraction proﬁle resides

124

Y00=0.6 YFF=O.7 u,L/a,=3.23

 

v V v v Y
12 - .
0 :'$

 

1.0: :3, .1

0.8 ~ \ ----- y, -

\ ,I
1 I ’
005 L \\ I /‘/'/ —1
\ I "
r / ’

\ '=. / ’

0.2 ~ \ ,,v .
\ //

\\’ ﬁ.’/' 1

 

 

 

00 1 --'..\v- 1 ...... .4 1 1 . 1 .
0.0 0.2 0.4 0.6 0.8 1.0

Figure 6.1. Temperature, species and soot volume fraction proﬁles.

125

primarily on the fuel side of the ﬂame, in accordance with experimental observa-
tions [30].

The proﬁles are depicted again in the physical coordinate in Figure 6.2 in order to
provide an idea of the magnitudes of various quantities. The fuel wall is at a: = 0 mm
and the oxidizer wall is at x = 20 mm. The maximum temperature is about 1745 K
which occurs at a: = 15.55 mm. The soot volume fraction peak is at 14.31 mm. We

also observe that soot exists between approximately a: = 10 mm and a: = 16 mm.

YOO=O.6 YW=0.7 u,L/a,=3.23

 

1.25 v T f r I VI" r
.- z

1 .00

I

0.75

0.50

0.25

 

 

 

 

000 . 1 . ' 1 . . .
0.0 4.0 8.0 12.0 16.0 20.0

Figure 6.2. Temperature, species and soot volume fraction proﬁles in the physical
coordinate, 3:.

We examine the temperature proﬁle more closely in Figure 6.3. The temperature
proﬁles for the inﬁnite reaction rate (IRR) situation and the proﬁle for ﬁnite rate

chemistry in the absence of radiation are also plotted in the same ﬁgure. It is clear

126

that the effect of radiative loss is quite pronounced and the peak nondimensional

temperature drops from about 0.8 to 0.59. The decrease is approximately 510 K.

Another interesting aspect to be noticed is the slope of the oxidizer side tempera-

ture proﬁle is nearly the same for all the three cases. The fuel side slope decreases

considerably on the inclusion of radiative losses.

1.0

0.8

0.6

.\
P \,
\§
0.4 ‘\\
0.2 't, IRR case \‘\.\
°---0 1:, No loss case “x.\
-°--- 1, Soot Radiation included ~“x.
0.0 I I . I I . e I
0.0 0.2 0.4 0.6 0.8

Y00=0.6 Y,,.=0.7 u,L/a,=3.23

 

f f 7 l v T

 

 

 

 

Z

Figure 6.3. 1' proﬁles for three different situations.

The proﬁles of the different terms in the energy equation are plotted in Figure 6.4.

On close inspection of Figure 6.4 we note that the oxidation term, Qox’Doxroz is very

small everywhere compared to the other terms. The convection and diffusion terms

roughly balance one another near the oxidizer wall (Z = 0). The reaction term

((1 + ¢)Dr) is balanced for the most part by the diffusion term, ao/(uoL§g)rzz. The

127

Y0, = 0.5 vFF = 0.7 11,145:0 = 3.23

 

 

 

 

 

600.0 - . - . ~
----- Convection
G—O Diffusion
400.0 - ’0‘ ---- Reaction ‘
lI 1‘ -°--- Radiation
1 1 —— Oxidation
200.0 » ,I I, -
1 1
0.0 .
-200.0
-400.0 ‘ i' ‘ ‘ ‘ ‘ ‘ ‘ ‘
0.0 0.2 0.4 0.6 0.8 1.0

Z

Figure 6.4. The Contribution of various terms in the energy equation.

128

diffusion term is positive near the right edge of the reaction term proﬁle and it recovers
the radiative loss term in that area. Thus the radiative loss term is countered by both
the reaction term and the diffusion term. The peak of the temperature proﬁle is at
Z = 0.19 and the radiative loss term maximum is at Z = 0.216. Interestingly, the soot
volume fraction maximum occurs at Z = 0.224, indicating that the radiation term
proﬁle maximum is between the maxima of temperature and soot volume fraction

proﬁles.

YOO=O.6 YFF=O.7 u,L/a,=3.23

 

   
     

 

 

 

2.0 . . . . - , . - -

F ----- Soot convection i
1'5 " ——--— Nucleation 1

°—-° Growth ‘
1,0 ---- Oxidation
0.5
-O.5
-1.0 ‘ ‘ . ‘ ‘ ‘ * L ‘
0.0 0.2 0.4 0.6 0.8 1.0

Z

Figure 6.5. The contribution of various terms in the soot volume fraction equation.

Figure 6.5 shows the proﬁles of the different terms in the soot volume fraction
equation. It is seen that the soot growth and oxidation regions do not overlap very

signiﬁcantly. This is expected in a diffusion ﬂame. For most part soot growth (Br-'9)

129

is balanced by soot convection term ((D,o/uoL§3)(d2fv/dZ2)). However, when oxi-

dation (Craz) is present the sum of convection and soot growth terms compete with

the oxidation term.

YOO=O.6 YFF=0.8

 

0.8 - . - . . . - .
—- uoL/Olo= 3.23
-—-* uoL/ao= 4.85
°-—-* uoL/a,,= 6.45
--—-- uoL/OLO= 8.06
---- uoL/Olo= 9.68

   
 
   

 

Q

\
s‘ ‘ ~\
‘W- ‘ .‘Q
‘~. ~

‘
-- ~~ ‘ ~.
~-.

 

 

Figure 6.6. The effect of uo on temperature proﬁle for a given stoichiometry.

We next focus on the effect of fuel blowing velocity, uo, on the temperature proﬁle
when the fuel and oxidizer mass fractions in the reservoir are assumed to be ﬁxed.
For Yoo = 0.6 and Ypp = 0.8 the temperature proﬁles for different values of no are
plotted in Figure 6.6. As mentioned before, L = 0.02 m and ao = 1.24x10‘4 mz/s.
The fuel ﬂow velocity uo is increased from 0.02 m/s to 0.08 m/s with increments
of 0.01 m/s. With increasing uoL/ao, i.e., with increasing no the maximum ﬂame

temperature is found to increase. Also, the increased fuel supply pushes the ﬂame

130

closer to the oxidizer wall. It is also to be noted that the movement of the ﬂame
toward the oxidizer wall with increasing fuel ﬂow rates is quite rapid for lower values
of uo. However, the rate of migration is mitigated for higher values of uo. The slower
rate of movement can presumably be attributed to the proximity of the ﬂame to the
oxidizer wall for high values of uo. It is important to mention here that we have
already noted similar ﬂame behavior in the absence of radiative losses (NR = ) in
section 4.7 of chapter 4.

We now focus on the drop in the maximum ﬂame temperature compared to the
adiabatic ﬂame temperature. As discussed in chapter 3, the quantity of interest is
Ar; and is deﬁned as 1 — r; where r; is the maximum ﬂame temperature. Figure 6.7
is a plot showing the variation of ATf as a function of the nondimensional fuel ﬂow

rate at the fuel wall, i.e., uoL/ao.

Y00=0.6 YFF=O.8

 

* m r r ' r v r r r ' 1 V I

0.42

T
1

I
l

0.40

T
A

0.38

At,

1
l

0.36

I
l

0.34

 

 

L

 

 

0.32IIIIIIII
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

uoL/a.

Figure 6.7. The effect of no on the reaction term for a given stoichiometry.

131

Evidently, Figure 6.7 indicates that there is a drop in the quantity Ari, i.e., an
increase in Tf with increasing uo. This is a direct consequence of the results illustrated

in Figure 6.6.

Y00=O.6 YFF=O.8

 

 

1500 3 . , . . .
V V V I V T V
b

y—t
9°
C
I
1

15.0 1 5

6., 14.0 - -

1 12.0 - .
1000L “:21 ’ ‘ ‘

10.0 1 . 1 . 1 . 1 . 1 . L . 1
0.0 2.0 4.0 6.0 8.0 10.012.014.0

 

 

 

  
 
 
 
 
 
 
   

 

 

 

5 [11
3 1". ' “0”“0
+ 1
3 1. —— u,L/a,= 3.23
tr—a uoL/or0 = 4.85
~—~ u,L/a,= 5.45 ‘
—-—- u,L/a,= 8.06
--- uoLlozu = 9.68
----- 1101101,, = 11.29
---------- 11,115, = 12.90

 

 

0.4

Z

Figure 6.8. The effect of uo on reaction term for a given stoichiometry.

The reaction term((1 + ¢)Dr) also exhibits interesting behavior for increasing uo.
Figure 6.8 indicates that for low fuel ﬂow rates the reaction term is far away from
the oxidizer wall (Z = 0) and it is quite broad in Z space. With increased fuel
ﬂow rates the reaction rate proﬁle becomes narrower and exhibits higher maximum
values. Close to the oxidizer wall a very sharp reaction term proﬁle is observed, e.g.,

for uoL/ao = 12.90. Since the proﬁles become narrower and taller with increasing

132

uo values, the integrated value of the reaction term, i.e., qTotaz = f01(1 + ¢)DrdZ was
also examined. The result is shown as inset to Figure 6.8.
In order to understand the behavior of the reaction rate proﬁle better the species

mass fraction proﬁles were also investigated.

 

 

0.6

 

 

Z

Figure 6.9. The inﬂuence of no on the species mass fraction proﬁles for a given
stoichiometry.

It was found that with increased fuel blowing there is a substantial change in the
fuel mass fraction proﬁle. The fuel mass fraction at the wall increases considerably
with increased fuel blowing and saturates to a value of about unity for uoL/ao = 12.90.

This tremendous increase in fuel mass fraction on the fuel side of the ﬂame pushes

133

the reaction zone closer to the wall and compresses the reaction zone. The oxidizer
mass fraction for different cases are also shown on the same plot. The direction of
movement of the go proﬁle with increasing uoL/ao is indicated by an arrow on the
plot.

A study of ﬁgures 6.6 and 6.9 in conjunction was found to be quite illuminating.
From Figure 6.6 we note that for high values of uoL/ao close to the fuel wall the
nondimensional temperature is close to zero. Figure 6.9, on the other hand, indicates
that the normalized fuel mass fraction values are close to unity near the fuel wall. For
example, when uoL/ao = 12.90, 7'20 and ypzl for Z 20.6. This clearly suggests that
in this case, for Z 20.6 the conditions in the domain are approximately the same as in
the fuel reservoir, viz., r = 0, yo = 0 and yp = 1. Hence, the fuel wall has virtually
moved in closer to the ﬂame, and has thereby pushed the ﬂame close to the oxidizer
wall and compressed the region of heat release. Thus, the intense heat release takes
place over a very narrow zone and hence, the peak temperature becomes higher, even
though the temperature in most part of the domain decreases, as shown in Figure 6.6.

It is also interesting to scrutinize the soot volume fraction proﬁle for increasing
values of uo. Consistent with the movement of the temperature and reaction rate
proﬁles the soot volume fraction proﬁle also moves toward the oxidizer wall with
increased fuel ﬂow, as shown in Figure 6.10. For lower ﬂow rates the soot volume
fraction proﬁle is broad but it becomes narrower and taller for increased uo, similar
to the behavior of the reaction zone. Also, the soot volume fraction at the oxidizer
wall is zero for low values of uo and all the soot produced get oxidized on the oxidizer
side of the ﬂame. However the same is not true for enhanced fuel blowing and for
uoL/ao = 12.90, for example, the soot volume fraction at the wall is 1.39x10‘5,
indicating that a significant amount of soot remains on the oxidizer side without
being oxidized.

Having analyzed the effect of uo on r and fv proﬁles we now focus our attention

134

 

 

Yoo=0.5 Ypp=0.8
4.0 . . ~ . . .
uoL/ot, = 3.23
' , .__. u,L/a,,= 4.85

°—-° uoL/OL0 = 6.45

   
    

 

--—-- uoL/or,,= 8.06
---- uoL/ozo= 9.68 1
----- uoL/otD = 11.29

nor/a, = 12.90

 

 

0.4

Figure 6.10. The effect of no on the soot volume fraction proﬁle for a given stoichiom—
etry.

135

on the radiation term, I‘Rf—v(1 — a(1-— 7))2((1-— a(1— 7))“ — (l — (1)2). We note that
the radiation loss term is explicitly dependent on the temperature and soot volume
fraction proﬁles. We hence expect the radiation loss proﬁle to also migrate toward the
oxidizer wall with increasing values of uo. Figure 6.11 reﬂects the expected behavior
in the radiative loss term proﬁle. The integrated value of the radiative loss term in

Z space is also found to decrease with increasing uoL/ao.

 

 

  
  
  
 
 
 
 

 

 

 

 

Y00=O.6 YFF=O.8
200 i i 7.0 Ir l I I 1 i I '
, 5.0 ~ .
5.0 - 1
150 ~ 40 t J ‘
. 1 ‘
E 3.0 . 1 1 4m 1 . 1 L 1 . 1 v 1
3 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
:1 "Ms
.3 100 - 3
:3 —— u,Ua,= 3.23
g 1 H “al./(10 = 4.85 5
0——0 1101100,: 6.45
50 _ --_. 001.101,: 8.06 .
——— uoIJao= 9.68
---- uOIJao=ll.29
[ -....... uoIJu,,=12.9O

  

 

 

0.0 . . . 0.4

Figure 6.11. The effect of uo on the radiative loss term proﬁle for a given stoichiometry.

The maximum of the radiative loss proﬁle was always found to be between the
maxima of the 1' proﬁle and the fV proﬁle, i.e., Z, < ZRad < Z fv, with Z,- denoting the

location of the maximum of the quantity i. The maxima of the reaction rate proﬁle

136

was found to exist always to the left of the of Z, for the fuel rich (ml) stoichiometries

considered in the present work.

 

 

 

 

YOO=O.6 YW=O.8
0.40 ' I ' I ' f ' I ‘ I v r v I
0.36 -
0.32 -

R
0.28 -
0.24 ~ ‘
0.20......n.,.r.r1

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
11,,L/cc0

Figure 6.12. The effect of no on the radiative loss term proﬁle for a given stoichiometry.

Next, a radiative heat loss fraction, x, is calculated. As described in chapter 3,
X is deﬁned as the ratio of the heat lost due to radiation and the heat generated in
the combustion process, i.e., X = qRad/qrpotag. Figure 6.12 indicates that the radiative
fraction changes very little with increasing value of uo. The mean value of X is
about 0.3. This agrees quite well with the commonly used value of x in combustion
literature [4].

In order to understand the effect of radiative losses better we studied the temper-

ature and species proﬁles when no soot is generated and there are no radiative losses.

137

This necessitates the solution of nondimensional temperature and species equations
with the only source terms resulting from the primary combustion reaction. This is

the same as the treatment in chapter 4 and we would focus on the solutions when

NR=0.

Yoo=O-6 YFF=O.8
No loss situation

r ' I

   
 
 
 
 
 
   

 

 

ugﬁk=323'
~——duJﬁ%=435

 

 

°——*u4ﬂ%=645 _
----- uaL/oc0 = 8.06

---- uoL/OLo = 9.68 l
----- uoL/Otn = 11.29 .
~-----~ uoL/oq, = 12.90

 

‘\
\
§
‘s
\

~

 

Figure 6.13. The effect of no on the 7' proﬁle for a given stoichiometry on the assump-
tion of no radiative losses.

The temperature proﬁle shown in Figure 6.13 indicates that similar to the radiative
loss situation the proﬁles move toward the oxidizer wall on increased fuel blowing. The
maximum of the nondimensional temperature proﬁle also increases with increasing uo.
Interestingly the reaction rate proﬁle also indicates a consistent direction of movement

(Figure 6.14). The integrated value of the reaction term, qTomz, also decreases with

138

 

 

 

 

 

 

 
 
  
 
 
  
 

 

 

 

 

 

 
 

 

 

 

Yoo=0.6 YFF=0.8
No loss situation
2000.0 ' 28.0 . , ', , 1 .'
.i 1 26.0 P” j
:11 - ‘ i
:2 :1 a 23:3 . :
1500.0 ~ EHi .'i , °‘ ° J
53:: :i l. 200
um 18.0_.,.n...1.,..n
1... :5: : ii 0” 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
9% 1000.0 + 55} 5!: "Ma" ‘
5 Wm. u.m.=s.2s
‘ i If i" o °——_° “OL/ao=6°45
_ : ' . —'-' uoL/ao=8.06 J
500.0 5 E !' —-- “oL/ao=9'68
, - ----- 11014010 = 11.29
:1 ‘ ............... “al./(10:12.90
0.0 .' .I. .
0.0 0.2 0.4

Z

Figure 6.14. The effect of no on the (1 + ¢)’Dr proﬁle for a given stoichiometry on
the assumption of no radiative losses.

139

increasing uoL/ao, indicating that the radiative losses are not primarily responsible
for this attenuation of the value of gut“. The proximity of the oxidizer wall is of
crucial importance in this issue. It is also interesting to note that QTotal increases as
uoL/ao increases from the low value of 3.23 to 4.85. However in the higher range of
values of uoL/ao the ﬂame is extremely close to the oxidizer wall and increases in
uoL/ao necessarily result in a drop in the value of qrmz. Thus our simple analysis
indicates that heat losses to the wall and radiative losses are both important to the
ﬂame in a signiﬁcant way.

We also examined the effect of changing the oxidizer mass fraction when both the

fuel mass fraction at the wall (Ypp) and the fuel ﬂow rate (no) are kept constant.

YFF=O.8 u01./(x0=3.23

 

1.0 . . - j - - . T
—— YOO=0.4 .
~———~ Y =05

0.8 °°

 

 

 

 

' 0.0 0.2 0.4 0.6 0.8 1.0
2

Figure 6.15. The effect of Yoo on the 7‘ proﬁle for a given YpF and a given uoL/ao.

140

On increasing Yoo the temperature proﬁle maximum moves more toward the
fuel wall. However the magnitude of the maximum keeps on decreasing as Yoo is
increased. The species proﬁles are also studied in detail for a better understanding
of the ﬂame. Figure 6.16 illustrates the normalized oxidizer and fuel mass fraction

proﬁles. We observe that the oxidizer mass fractions (yo) are everywhere higher when

YW=OBuJﬂk=323

 

 

 

 

 

1.0 r r j . f .
1. _W. 4
la ~mw~Ym=05

0.8 H; ---- Yoo=0.6 1
. iiiiiit —""'" YOO=O7

fill“ Increasing ’——" Yoo = 0.8

0.6 r m“

0.4 ~

0.2 ~

0.0 *

0.0

Figure 6.16. The inﬂuence of Yoo on the species proﬁles for a given YFF and a given
noL/ao.

Yoo is increased. In sharp contrast, the normalised fuel mass fraction (yp) values are
lower everywhere for increased Yoo values. Also, the normalised fuel mass fraction
values at the wall decrease on increasing Yoo values. The signiﬁcant decrease in 3]}:
values result in decreased reaction rates, as illustrated in Figure 6.17. The reaction

rate proﬁle also moves toward the fuel wall on increasing the value of Yoo. The

141

YW=03uJﬂk=323

 

800 ' ' 301)

 

L 25.0

I

600

 

 

 

 

0.50

(1+¢)Dr
é

 

200

 

 

 

 

Figure 6.17. The inﬂuence of Yoo on the reaction rate for a given Ypp and a given
HoL/ao.

142

integrated reaction rate, qTotaz, decreases with increasing Yao-

The soot volume fraction proﬁle is investigated next. As anticipated from the
movement of the temperature and reactivity proﬁles, the soot volume fraction proﬁle
moves toward the fuel wall. With increasing Yoo, the temperature values decrease
and consequently, less soot is produced. It has to be mentioned here that for lower
values of Yoo the soot volume fraction proﬁle is located close to the oxidizer wall and
there is a residual soot volume fraction near the wall, which is not oxidized. For higher

values of Yoo there is no residual soot volume fraction near the wall. The radiation

vFF = 0.8 uOL/ozo = 3.23

 

 

 

 

 

 

 

2.0 ~ r . . . . , v
— Y00=04
~~°--°~ Y00=05
1.5 ~ ----- Yoo=0.6 ‘
L ---- YOO=07
----- Yoo=08
“o
:1 10 ~ ~
H—«>
0.5 - .
0.0 ‘ ‘ ‘ ‘
0.0 0.6 0.8 1.0

 

Figure 6.18. The inﬂuence of Yoo on the soot volume fraction proﬁle for a given YFF
and a given uoL/ao.

term proﬁle (see Figure 6.19) also moves toward the fuel wall and the maximum of the

proﬁle also decreases in magnitude as Yoo is increased. Investigation of the integrated

143

value of the radiation term proﬁle was also carried out and the result is shown as inset

to Figure 6.19. The integrated value decreases as a function of increasing Yoo. The

YFF=O.8 u,L/a,=3.23

 

 

      

 

 

 

 

 

 

 

 

150.0 ﬂ ' ' 8.“)! v T, . f1 7* I r ‘
l 3 ‘
. “g! 700 - +
B . ,
3.00
E 100.0 - ‘ g ' .
s ‘.
a 1 5.00 . L - ' . . . .
,g l . 3 0.00 0.20 0.40 0.60 0.30 1.00.
‘5 '. ' Y
8 . ‘. °°
‘° ° 1 — Y =04
a: . . 00 .
50-0 .1 “ »—~ Y00=0.5
.1 ----- YOO=O.6
a" H Y00=0.7 ,
--—-- Yoo=0.8
0.0 ' " -
0.0 0.2 0.4 0.6 0.8 1.0

Z

Figure 6.19. The effect of Yoo on the radiation term proﬁle for a given Ypp and a
given uoL/ao.

radiative fraction values were also investigated and are plotted in Figure 6.20. The X
values increase with increasing Y00 and are in the vicinity of 0.3.

The effect of increasing Ypp for a given no and a given Yoo is investigated next.
The temperature proﬁle is observed to move toward the oxidizer wall, as expected
(see Figure 6.21). However, there is also a drop in the temperature proﬁle maximum.

A plot of the drop in peak ﬂame temperature as a function of the fuel wall mass

144

YFF = 0.8 uoL/ao = 3.23

 

0.40

0.36 -

0.32 f

0.28 ~

0.24 L

 

 

 

0.20 . 1 . 1 . L W 1 1 1 1 1 . 1 1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Y

00

Figure 6.20. The effect of Yoo on the radiative fraction for a given Ypp and a given
uoL/ao.

145

Yoo=0.6 uoL/ao=3.23

 

1.0 v I Y I v r v I ﬂ

 

 

 

 

Figure 6.21. The effect of Ypp on the temperature proﬁle for a given Y00 and a given
HoL/ao.

146

fraction is shown in Figure 6.22.

YOO = 0.6 uoL/a,=3.23

 

0.45 . , . , . , . .

0.44

I

0.43

l

'0:-

g 0.42

7

0.41

1

0.40

I

 

 

 

 

0.39 - - 1 e - . -
0.50 0.60 0.70 0.80 0.90 1.00

Figure 6.22. The drop in peak ﬂame temperature as a function of Ypp.

The reaction rate proﬁle also moves toward the oxidizer wall on increasing Ypp
as shown in Figure 6.23. Interestingly, the reaction rate proﬁle maximum increases
with increasing YFF. The integrated value of the reaction rate, qTotal, increases with
increasing YFF, unlike in a previous situation when 110 was increased for a ﬁxed set
of Y00 and YFF.

The species mass fraction proﬁles are shown in Figure 6.24.

It is clear from a comparison of Figure 6.9 and Figure 6.24 that increasing fuel
blowing rate has a much stronger effect on the fuel mass fraction proﬁle than increas-

ing the fuel mass fraction at the fuel reservoir.

147

YOO=O.6 uoLlao=3.23

 

r V r ﬁ 1' j

24.0 .
22.0 "
20.0 "
g »
c? 18.0 .P
16.0 f

14.0
0.5

 

 
  
 

 

 

 

 

0.6

 

 

 

0.0 . . . . 1.0

Figure 6.23. The effect of Ypp on the reaction term proﬁle for a given Y00 and a
given uoL/ao.

148

Yoo=0.6 uOL/ao=3.23

 

1.0 . . . 1 . T . .
I. — FF = 0.6
'1: -—--—- F, = 0.7
0.8 ~33; ---' FF=0.8 .
_ ----- YFF = 0.9 1
$10, Increasing YFF _-..-- YFF = 1'0
0.6

 

 

 

 

Figure 6.24. The effect of YFF on the species proﬁles for a given Y00 and a given
noL/ao.

149

Correspondingly the soot volume fraction proﬁles are also plotted in Figure 6.25.

Soot volume fraction increases as Ypp is increased and the proﬁle moves closer to the

Y00=0.6 uoL/oco=3.23

 

1,1105

 

 

 

0.6

Z

Figure 6.25. The effect of Ypp on the soot volume fraction proﬁle for a given Y00
and a given uoL/ao.

oxidizer wall in accordance with the movement of the temperature and reaction rate
proﬁles. Figure 6.26 shows the radiation term proﬁle. The integrated value of the
radiative loss proﬁle is found to increase with increasing Ypp values.

The radiative fraction X indicates an increasing trend with Ypp (See Figure 6.26.
It should also be noted that the value of the radiative fraction is close to 0.3 in all
cases.

It is also of interest to examine the velocity proﬁle, 11, for a typical ﬂame. As

discussed before, the velocity u is given by equation 6.2, and hence is dependent

150

 

 

   

Y00=0.6 u,L/a,=3 .23
200.0 f . 4
8.0
l g 7.0 -
> .
150.0 * 3 6.0 .
go 1
g 5.0 ~ l

   

 

 

 

 

100.0 - 4'0 ,

Radiation term

50.0 -

 

 

0.0 . . . . 1.0

Figure 6.26. The effect of Ypp on the radiation term proﬁle for a given Y00 and a
given uoL/ao.

151

YO0 = 0.6 uoL/oco = 3.23

 

0.40 . . . . . . W , W

0.36 .

0.32 -

0.28 ~

 

0.24 -

 

 

 

0.20
0.5 0.6 0.7 0.8 0.9 1.0

Y

FF

Figure 6.27. The effect of Ypp on the radiative fraction for a given Y00 and a given
HoL/ao.

152

on the density distribution ,5. The p distribution is obtained using equation 3.16
of chapter 3 and a typical case is plotted in Figure 6.28. The normalised density

is the highest at the close to the walls and it attains a minimum at the maximum

temperature location.

YOO=0.6 YFF=0.8 uoL/oco=3.23
1.0 - . - , . .

 

0.8
i /

0.6 J‘ / -
l

0.4 ~ \ /’ -

0.2 ~ \

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
Z

Figure 6.28. The ,6 distribution plotted in the Z coordinate.

The velocity distribution is plotted for two different mass ﬂow rates in Figure 6.29.
Figure 6.29 depicts the proﬁles for uoL/ao = 3.23 and uoL/ao = 12.90. The velocity
u is quite high near the ﬂame location and decreases to its nominal value no at either
wall. The magnitude of u is much higher when uoL/ao = 12.90, as expected. The

maximum velocity is ~12 cm/s when uoL/ao = 3.23 and it increases to ~12 cm/s

when HoL/ao = 12.90.

153

0.8

u(,I../()L0 = 12.90

Ym=06YW—

0.8

uoL/ao = 3.23

=06YFF

Yoo

 

 

1.0

 

 

 

 

 

 

a l 1 . n m
.
_ 1
_
. 00
.. - O.
. 1
.
.. 6
. 1 0.
s
1 Z
s
s 4
1 0
\\\ l 2
\\\\ O
RNMHHH llll 1 IIIIII 1. lllllll L IIIII F O.
O
5 A 3 2 J 0.
O 0 0 O 0 0
a}: .s
0
q q q A q s 11.
\
\
\
\\
\s 1 8.
\\\ 0
\
1 6.
\\\ 0
\
Z
1 4.
\\ nu
..... - . 2.
\ lllll 0
b p p . lllllll H1114!!! 0.
O
2 0 8 6 M 2
l. .J 0. 0. . 0.
0 0 O 0 O 0

12.90.

Figure 6.29. The velocity distribution for uoL/ozo = 3.23 and uoL/ao

154

The inﬂuence of thermophoresis was examined by plotting 772T as a function of Z.
As noted in section 6.1.1 fnT is given by equation 6.18. Figure 6.30 shows the variation
of 7717 for a low velocity (uoL/ozo = 3.23) and for a higher velocity (uoL/ao = 12.90).

It is apparent from the ﬁgure that the value of 771T is not signiﬁcant for most part

Yoo = 0.6 Y”, = 0.8 Yoo = 0.6 Y”, = 0.8
uoL/ao = 3.23 noun, = 12.90
15 ' 1 - r u r v 1 v 15 V r r 1 v r v T

 

 

10

.;-—-’F_-—-—-—-v——r

 

 

 

 

 

 

-5 1 1 1 1 1 1 1 1 1 -5 1 1 1 1 1 1 1 1 W1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Z Z

Figure 6.30. The quantity ﬁzT plotted as a function of Z.

of the domain and it increases near the cold walls. For uoL/ao = 3.23 the inﬂuence
of thermophoresis is signiﬁcantly more important compared to the high velocity case
(uoL/ao = 12.90).

Next, the thermophoretic velocity 117» and the ratio of the thermophoretic velocity
and the convective ﬂow velocity, i.e., uT/u are plotted in the physical coordinate :13.

We recall that the fuel wall is located at a: = 0 m and the oxidizer wall is located

155

at :c = 0.02 m. Figure 6.31 indicates that uT is small in both low and high velocity

 

 

 

 

 

 

 

 

 

 

Yoo = 0.6 YFF = 0.8 Y00 = 0.6 YFF = 0.8
uOIJOL0 = 3.23 u(,I_,/(xo = 12.90
60.0 . v . 1 r 1 . 60.0 . r 1 1 . 1
. 1...... 11,, m/s 1 W 11,, m/s
_-_-- [Ll/u -'-'- llT/ll
40.0 - 4 40,0 - 1
1 1
!
20.0 - : 20.0 A J
i 1
l‘ i
.’ 1'
o/ i
0.0 , ................. ,A’ 00 _________ _ _A
I.
-20.0 A A A . A A A -20.0 - A A . A A A
0.000 0.005 0.010 0.015 0.020 0.000 0.005 0.010 0.015 0.020
x, m x, m

Figure 6.31. The quantities 117 and uT/ u plotted as a function of the physical coor-
dinate 2:.

cases. However, the ratio uT/u is quite important over a signiﬁcant region near either
wall. For uoL/ao = 12.90, however, uT/u seems signiﬁcant only near the oxidizer

wall.

6.5 Estimation of the Soot Volume Fraction Pro-

ﬁle

In section 5.6 of chapter 5 we estimated the soot volume fraction proﬁle assum—

ing Burke-Schumann, or, IRR proﬁles for the temperature and species. However,

156

Burke-Schumann proﬁles are not accurate enough in a ﬁnite chemistry situation with
radiative losses. The temperature values are too high for the IRR case and conse-
quently, the soot volume fraction values are unrealistically high in some cases. Hence,
the adiabatic ﬂame temperature was arbitrarily chosen to produce a realistic range
of soot volume fraction values. A more involved treatment is necessary when the in-
ﬂuences of ﬁnite chemistry and radiative loss are included. In this section we purport
to introduce the effect of radiative losses in a very simple way and derive a modiﬁed
expression for the temperature, species and soot volume fraction.

Let us ﬁrst assume that the peak value of the nondimensional temperature, 1', is
a fraction 5. Then, equation 4.13 can be modiﬁed to write

61:.z" . 03232,
T = , f (6.19)

e-l/c_ e—Z/c
£6 c__1/c :—Zf7c Zfszsln

We note that when t" = 1 we get the Burke—Schumann proﬁles. We will still use
equation 5.24, which is an expression for the balance of soot convection and growth
terms. Also, similar to the treatment in section 5.6 of chapter 5 we try to express the
normalised fuel mass fraction 3]}: in terms of the fuel side temperature distribution,
1'. We obtain yp = a(1 — T/é). Also, dZ can be written as dZ = —ac/[a1' — (a —1)£].

Using the expressions for yp and dZ in equation 5.24 we get

(1 - r/cwse WE“ . )(ﬁzfm)d
(1" 0(1- 7))5/6(ar — (a —1)5)d

 

——de = [B§o(1 — a)4/3al3/6c] (6.20)
Again we note that setting { = 1 in equation 6.20 recovers equation 5.28 of section 5.6.

On integrating equation 6.20 subject to the boundary condition fv = 0 at Z = 1 we

get the soot volume fraction proﬁle. However, integrating equation 6.20 involves

157

evaluating the integral

_1)7/6‘,3 4517+? )(ﬁfq)

 

 

6.21
111=./o((11—a( (1 —T))5/6(aT—(a—l){)dT ( )
Substituting 11 = 1 — T and writing 6,0,: 3.9+ Eﬁwe arrive at the following expression
for 11:
1 _ _ 7/6 j-aauu
I1 = 1 (u (1 5)) e du (6.22)

f u (1'- OU)5/6(€ + 0(1- u - 6))
This integral could not be evaluated by analytical means. To simplify the integrand,
the exponents 7/ 6 on (u — (1 - 6)) and 5/ 6 on (1 — cm) were both replaced by unity.
The simpliﬁed integral, say 12, was evaluated using the commercial code MAPLE. The
result was in terms of exponential integrals, similar to equation 5.32 of chapter 5.
We note here that the maximum ﬂame temperature is not known and has to be
approximately determined in order to evaluate the soot volume fraction proﬁle using
equation 6.20. In the present work, we use the value of 6 directly from the numerically
obtained solution of the temperature proﬁle. Figure 6.32 shows the numerical solu-
tions for the temperature, species and soot volume fraction proﬁles when Yoo = 0.5,
YFF = 0.4 and uoL/ao = 4.85. Using the value of E from the numerical solution in
equation 6.19 we determine a nondimensional temperature proﬁle, 7'. The normalised
species mass fraction proﬁles are then determined iteratively, similar to the procedure
described in section 4.5 of chapter 4. Based on these analytically obtained tempera-
ture and species proﬁles we numerically solve the soot volume fraction equation. The
solution is plotted in Figure 6.32. The analytical solution of the soot volume fraction
equation involving exponential integrals is also plotted in Figure 6.32. It is clear that
the analytical solutions for 7', yo and yp are in very close agreement with the numeri-
cal solutions. Also, the soot volume fraction proﬁle obtained on the assumption of the

analytical 7', yo and yp proﬁles approximates the numerical solution reasonably well.

158

 

 

 

 

 

 

 

Y0o = 0.5 YFF = 0.4 Y00 = 0.5 YFF = 0.4
u(,I./a0 = 4.85 uOIJOL0 = 4.85
1.0v.v...... 1,0..h1,
----- Numerical Solutions _--- Num. Solution, 1
— Analytical Solutions . _ ----- Num. Solutlon, 2
0—0 Analytical Solution
0.8 _ -
yo
1
0.6 A :1 -
:: 1
1 1' “o
:. .7
:: «3
0.4 - : 'g -
E :f.*10
l 1 1 J
i ‘.
. :
0.2 - E 1. 1
3 1.
0.0 1 4 «L \1\_ 1_ W 1 _1 _1 1
0.0 0.2 0.4 0.6 0.8 1.0
Z

 

Figure 6.32. Comparison of fv proﬁles when Yoo = 0.5, YFF = 0.4 and uoL/ao =
4.85. Numerical Solution, 1 refers to the simultaneous solution of coupled equations
of 1', yo, yF and fv. Numerical Solution, 2 refers to the solution of the soot volume
fraction equation on the assumption of analytical proﬁles for 7", yo and yp. The
analytical solution of the soot volume fraction equation involves the evaluation of the
integral I; in terms of exponential integrals.

159

The soot volume fraction values are in good agreement and the numerical solution of
the soot volume fraction equation is also capable of predicting the soot zone thickness

fairly well.

6.6 Conclusions

The comprehensive soot radiation problem was analyzed in this chapter assuming
soot radiation in the optically thin limit. For soot formation, the model of Syed,
Stewart and Moss [1] was used. The semi-empirical formula of N agle and Strickland-
Constable [2] was used for soot oxidation. The conservation equations were numeri-
cally solved for a range of parameter values and the results were analyzed to study the
inﬂuences of the various parameters on the ﬂame structure and the ﬂame radiation.

The soot volume fraction proﬁle was always found to exist on the fuel side of
the ﬂame with the location of the peak soot volume fraction being 1 — 2 mm to the
fuel side of the location of the maximum temperature. This is in accordance with
experimental observations [30]. The peak of the radiation term was always located
between the peak temperature and peak soot volume fraction locations.

The inﬂuence of increasing the fuel blowing velocity, uo, was investigated ﬁrst. On
increasing uo the temperature and reaction rate proﬁles migrated toward the oxidizer
wall. The maximum ﬂame temperature increased as uo was increased. The integrated
value of the reaction term, i.e., qTotag, decreased as uo was increased. A higher fuel
blowing rate managed to push the ﬂame against the oxidizer wall, and consequently,
the heat losses to the wall also increased. This resulted in a reduction in the value of
qul. It was also found that the rate of movement of the ﬂame toward the oxidizer
wall decreased with higher values of uo and the reaction rate proﬁle became taller
and narrower. A similar trend was also observed for the soot volume fraction proﬁle

and the radiative loss proﬁle. The integrated value of the radiation term, i.e., qRad,

160

decreased with increasing uo. The value of the radiative fraction was around 0.3.

On increasing Yoo for a given YFF and uoL/ao the ﬂame migrated toward the
fuel wall and the temperature, as well as the reaction rate values decreased. The soot
volume fractions, as well as the qRad values decreased in such a situation.

Finally, Y00 and uoL/ao values were kept ﬁxed and Ypp values were increased
to study the ﬂame behavior. The ﬂame moved toward the oxidizer wall and the
peak ﬂame temperature decreased with increasing Ypp. However, the gum; values
increased. The soot volume fraction values also increased. The radiation term proﬁle
also moved toward the oxidizer wall and the qRad values increased with increases of
Ypp. The radiative loss fraction was found to be around 0.3.

On including a correction factor for the peak temperature it was possible to modify
the analytical expressions for the Burke-Schumann temperature and species proﬁles to
agree with the numerical solution. Such an agreement was possible primarily because
there was very little leakage of fuel and oxidizer across the ﬂame. Using the modiﬁed
expressions for the temperature and species proﬁles it was possible to develop an
analytical expression for the soot volume fraction proﬁle based on the method already
described in chapter 5. The analytically obtained soot volume fraction proﬁle agreed
very well with the numerical solution.

A radiative extinction was not observed in any of the cases tested. This indicates
that a steady state extinction is not quite likely. As mentioned before, the radiative
fraction value was about 0.3 in all the cases and a decrease (increase) in the qTota;
values resulted in a decrease (increase) of qnad. This phenomenon seems to indicate
that the radiative losses from a ﬂame bears a direct relationship with the heat re-
lease and an excessive increase of radiative losses is not likely when the heat release

decreases.

CHAPTER 7

Conclusions and Recommendations

for Future Work

The work presented in this dissertation has revealed interesting features of the struc-
ture of a radiating diffusion ﬂame. A qualitative analysis of a diffusion ﬂame with a
hypothetical sech2 radiative loss proﬁle indicated that for a loss term ﬁxed in space,
the ﬂame can break through the loss zone on increasing the radiative loss term suf-
ﬁciently. It was also shown that such a “break through” is only possible for thin
radiative loss zones placed immediately next to the ideal ﬂame location, Z1.

For pure diffusion ﬂames the increase in radiative losses always resulted in the
ﬂame moving toward the oxidizer side. For a diffusion ﬂame with convective fuel
ﬂow from the fuel wall even the opposite was observed. Evidently, the interaction of
convection, diffusion, reaction and radiative losses is harder to predict compared to a
pure diffusion ﬂame situation, when the effect of convection is not present.

An analytical model of soot proﬁle was determined which accurately predicts the
soot zone thickness when Burke-Schumann proﬁles were assumed for the temperature
and the species mass fractions. Such a model was based on the observation that the
primary balance for the ﬂames studied in our conﬁguration revealed a balance between

the processes of soot growth and soot convection.

161

162

Such a balance was also evident from an analysis of the results of the compre-
hensive soot radiation problem (chapter 6). We also discovered that in accordance
with experimental results for sooting diffusion ﬂames the soot growth and oxidation
regions do not signiﬁcantly overlap, as expected in a diffusion ﬂame. The soot volume
fraction maximum was found to occur on the fuel side of the ﬂame and was typically
at a distance of 1 — 2 mm from the temperature maximum. The radiation loss proﬁle
peak was always situated in between the ﬂame temperature peak and the soot vol-
ume fraction peak. We note that the radiative fraction was around 0.3 for the ﬂames
studied. The consequence of increasing the fuel ﬂow velocity at the wall, uo, was to
push the ﬂame closer to the oxidizer wall. Correspondingly, the reaction rate proﬁle
became narrower and taller, and its integrated value quz decreased with increasing
uo. The integrated value of the radiative loss term also decreased with increasing uo.

In another study, the oxidizer reservoir mass fraction, Yoo, was kept ﬁxed along
with the fuel blowing velocity uo. The fuel reservoir mass fraction was increased from
0.6 to 1.0 in steps of 0.1. It was observed that the ﬂame moved closer to the oxidizer
wall and the peak ﬂame temperature decreased with increasing Ypp values. The
temperature maximum decreased even though the integrated value of the reaction
rate, qTotaz, increased. The integrated value of the radiative loss term, qRad also
increased and the radiative loss fraction value was about 0.3 in all cases. Thus, it was
found that the radiative losses depend directly on the amount of heat released and in
the cases studied, the radiative losses adjusted to the heat release in such a way that
the radiative fraction values were about 0.3.

A radiative extinction was not observed in any of the cases studied, indicating
that a steady state extinction is unlikely. Also, since the radiative fraction values
were close to 0.3 in all cases, it seemed that the ﬂame tried to limit the heat losses
with any decrease of heat release.

The analytical model developed in chapter 5 for the soot volume fraction proﬁle

163

used the assumption of Burke-Schumann temperature and species mass fraction pro-
ﬁles. However, for a diffusion ﬂame with radiative losses the temperature and species
proﬁles are signiﬁcantly different from the inﬁnite reaction rate proﬁles. Hence, a cor-
rection factor 5 was used for the peak ﬂame temperature. The analytical expressions
for the Burke-Schumann proﬁle were modiﬁed to include the factor 6. The resultant
proﬁles matched very well with the numerical solution of the comprehensive soot ra-
diation problem discussed in chapter 6. Based on the method described in chapter 5
utilizing the balance between soot growth and convection an analytical expression
was then developed for the soot volume fraction proﬁle including the effect of 6. The
analytically obtained soot volume fraction proﬁle matched fairly well the numerical
solution of the comprehensive problem.

The soot model used in the present work has received signiﬁcant attention in the
literature. However, the physico-chemical processes which lead to the evolution and
burn-out of soot particles are still not well understood. It is anticipated that more
accurate and versatile soot models will appear as the investigation on soot processes in
ﬂames intensiﬁes with the help of better diagnostic and modeling capabilities. Thus,
future work has to utilise more accurate descriptions of soot processes.

It will also be important to investigate the effect of soot radiation in the opti-
cally thick limit. A more comprehensive radiation-soot interaction model has to be
envisaged for that purpose. The soot volume fraction in our ﬂames analyzed in chap-
ter 6 was in the range 10‘7 — 10'5. For soot volume fractions in the vicinity of the
upper limit, it will be worthwhile to examine the inﬂuence of a more comprehensive
radiation model.

The radiation from the combustion gases was neglected in our analyses. This
effect can also be included for a more accurate description of the ﬂame behavior.

In our problem the heat losses from the ﬂame occurred because of the radiating

soot particles and also due to the presence of cold reservoir walls. It will be interesting

164

to study the effect of other conﬁgurations on this ﬂame, specially when conductive
losses to cold boundaries are signiﬁcantly reduced. For example, it will be interesting
to study the effect of radiation from a fuel strip burning in an ambient oxidizer ﬁeld
where the boundaries of the domain are far apart. The inﬂuence of strain in such a
ﬁeld on the ﬂame structure and the radiative loss proﬁle will be quite worthwhile to
study.

Methane was chosen as the fuel in the current work because the important parame-
ter values for one-step methane combustion reaction are available. However, methane
is not a heavily sooting fuel and in future, it will be interesting to investigate the

problem with more readily sooting fuels, such as ethylene, acetylene, etc.

APPENDICES

APPENDIX A

Asymptotic Calculations for a
Simpliﬁed Model of the Interaction
of a Diffusion Flame With a

Heat-Loss Zone

A.1 Abstract

In this Appendix we examine a highly simpliﬁed model problem for the interaction
of a diffusion ﬂame (DF) with a heat-loss zone (the “soot” layer). Explanations are
provided for DF migration (observed in chapter 3) and ultimate quenching when

NRAZR is made large enough (chapters 3, 4).

A.2 Introduction

The purpose of this study is to theoretically analyze the inﬂuences of a heat loss zone
on a diffusion ﬂame (DF).

Although fundamental soot formation chemistry has been examined in detail for

165

166

many years, the number of studies exploring its relation to actual physical processes
in ﬂames is few. In the sooty region of a DP there are intense heat losses to the
surroundings, perhaps strong enough to retard chemical activity or, under proper
conditions, to cause extinction. The model considered here examines this question
by paring the physics to a minimum and focusing only on the skeletal features of
the ﬂame/ radiation interaction. Here we eliminate convection (natural and forced),
thermophoresis, and particulate oxidation, all of which are important in actual sooty
ﬂames. For example, when convection is eliminated (not simply by transformation to
a coordinate system in which the convection and diffusion terms are combined into
one overall “convective-diffusive” term, but actually and entirely eliminated). the
standard formulations of the soot conservation equation become untenable because
there is no means for balancing the creation/ destruction terms with a convection
term; a different interpretation of the soot zone is required.

The approach followed here will be to examine a simple model thoroughly. The
emphasis will be placed on making mathematically deﬁnite statements that can be
turned into physically useful criteria, given the limitations of the model. We note that
our goal is to describe - not simulate - the inﬂuence on DFs of radiant heat losses
from particulates.

Finally, we observe that a study very similar to this one was published in the
literature [56] but an error in one of the calculations prevented correct conclusions
from being drawn. In this study the error is corrected. Revised results are presented

and interpreted.

167

A.3 Formulation

A.3.1 Physical Discussion

The following simpliﬁcations are employed in this study: geometrically, we consider
the one-dimensional “stagnant ﬁlm” diffusion ﬂame (DF). The porous fuel wall, at
temperature To, is located at :1: = 0, parallel to the oxidizer wall which also has T = To
and is located at a: = L. The mass fractions of fuel and oxidizer at these walls are
YFF and Y00 respectively (see Figure 3.1).

Dynamically, we limit ourselves to the case of zero mean ﬂow; the movement
of species occurs strictly by diffusion. In addition, we neglect the thermophoretic
ﬂow that is known to occur with particulates in regions of high thermal gradients,
because we wish to examine only the thermal and chemical inﬂuences of heat losses.
In summary, we have a one-dimensional stagnant-ﬁlm DF with no mean ﬂow (u = 0)
and no thermophoretic ﬂow (uT = 0). We also impose the steady-state condition
(3(.)/3t = 0). For the combustion chemistry we assume that the reaction at the
DF occurs through a single irreversible step, F + u0—+(1 + u)P (on a mass basis),
with high activation energy. The “soot-formation mechanism”, which we hypothesize
occurs on the fuel side of the DF, is assumed not to consume fuel. That is, only
“trace” amounts of fuel are required to make “soot” particulates. Also, since the “soot
distribution” will be speciﬁed, we do not require a separate soot species equation.
Nor is it necessary to consider a number-density equation, since we assume that our
“soot” particulates are simply a collection of immobile radiating masses located in a
preassigned region on the fuel side of the DF. Strictly speaking, there is no need even
to discuss “particulates” like “soot” because none of the explicit features commonly
associated with particulates appear in our analysis. The relevant features of our “soot
particulate layer” are exclusively thermal; the layer produces only a region of enthalpy

loss that may alter the DF structure and cause extinction.

168

Even if extinction does not occur the heat losses may weaken the DF. The re-
lationship between DF-weakening and heat-loss enhancement is an interesting one
that may, within the limits of this analysis, be quantiﬁed. We assume that the fuel
and oxidizer walls are perfectly transmitting so the heat lost from the “soot” layer
is permanently lost. Finally, we observe that the “soot” layer is presumed to remain
always on the fuel side of the DF, thereby eliminating discussions of “soot oxidation”,

etc.

A.3.2 The Mathematical Problem

The boundary-value problems governing the distributions of temperature, fuel mass

fraction and oxidizer mass fractions are, respectively,

 

(2') 130% = —pr + $33. T0) = T0) = To. 1
(ii) ﬁ(PDFd—d:£) = w, YF(0) = YFF, YF(L) = 0, l M”
(iii) ﬁ(pDoé£Q) = uw, Yo(0) = 0,Y0(L) = Yoo, 1

where w = pAYo Ypexp(—E / RT) is the chemical reaction term, with units mass/vol-
sec, Qp is the chemical heat release, with units energy/mass, and dqR/da: is the
radiant heat-loss term, with units energy/vol-s. The quantity qR is the radiant energy
ﬂux. We nondimensionalize these equations by deﬁning r = (T — T0)/(Tf - To),
yp = Yp/ YFF, yo = Yo/ Yoo, { = s/so, where s = ffpda: is a mass-based coordinate
and so = foLpda: is its maximum value. We observe that 05631 and that the “ﬂame
temperature” Tf is presently undeﬁned. We consider the case p) = constant, [121); =
constant and Leo = Lep = 1. We deﬁne (IR = qR/qR,1-cf as the nondimensional

radiant ﬂux. We shall return to this quantity after our discussion of the chemical and

169

heat-transfer features of our problem. Our nondimensional equations are

(T) (_QF) (1)

_ 31F = 1 ’Dr+ 0 073— (A2)

1152
1w) 1M 10)

 

 

 

 

 

 

subject to

.01 (01/11)} m

yF(0) = 1 , yp(1) = 0 . (A.3)

310(0)) (0)(yo(1)) ll)

Here Q}: = QFYFF/Cp(Tf — To) is the nondimensional heat release, which equals (1 +

 

 

 

 

 

 

 

05), D = (Asosz/poA0)Yooexp(—E/RT;) is the Damkéhler number, 05 = 111/pp / Yoo
is the stoichiometric index and NR = qR,,cf/[A0(Tf — To)/(so/po)] is a nondimensional
measure of the radiant heat ﬂux. We note that quantities with subscripts “0” are
at ambient conditions attained at either wall. The reaction term is given by r =
yoypexp[—,B(1 — r)/[1 — a(1 — r)]], where 6 = Ea/RT, and a = 1 — To/Tf. Let
us now form the combination H ET + by]: + cyo + d, which satisﬁes the equation
(PH/(162 = (—QF+b+c¢)’Dr+NquR/d{. We can choose 6 and c so that Qp = b+c¢
thereby eliminating the reaction term.

In the absence of radiative heat losses we put NR = 0 and denote H = Ho,
giving d'fHo/dé2 = 0, which integrates to Ho = A + Bf. At the fuel wall Ho = 0,
r = 0, and yo = 0 whereas at the oxidizer wall Ho 2 0 and r = yp = 0, giving
Ho =r—d(yo+yp—l)=0. At theﬂameyo =yp=0andr= 1 givingd= —1
and b = c = 1. Hence, the quantity H = r + yo + yp — 1 is a convenient measure of
the excess (or defect) of local enthalpy.

In our case the heat losses by radiation produce an “enthalpy sink” that is de-

170

scribed by

43!! = Naég‘n, H(O) = H(1)= 0, (M)

m

where H = 7' + yo + yp — 1. We observe that this deﬁnition of H leads to Qp =
(1 + 43) (since 0 = c = 1) giving T; = To + Qprp/Cpﬂ + 45) for the adiabatic
ﬂame temperature. We also observe that with heat losses the ﬂame temperature
will not rise to its theoretical maximum even when ﬂ—eoo. Hence, we may expect a
slight redeﬁnition later of the Damkc'ihler number ’D in terms of a ﬂame temperature
somewhat lower than T!-

We note that the derivation of an excess-enthalpy function may be achieved “phys-
ically”. The conservation equation for the enthalpy takes the form of equation A.4
when convective transport and body force effects and preferential species transport
(La-#1) are neglected. Since h = 2.1-:1 h,Y,- and h,- = h? + Cp(T — To), it is easy to
recover our nondimensional H.

Finally, we observe that even in the most difﬁcult and general case, such as when
the radiation term depends on the spatial coordinate and the temperature and the fuel
mass fraction, as long as no fuel-to-soot depletion terms enter the species equations
(equations A.1.(ii) and A.1.(iii)) we can still deﬁne the mixture fraction variable
Z = (qSyF +1 — yo)/(¢+ 1) that satisﬁes Z56 = 0 with Z = 0 at f = 1 and Z =1
at f = 0. This provides an important simpliﬁcation of the governing equations. The

solution for Z is Z = 1 -— 6, whereby

(2') yo = (1 -Z)- (1 - Z1)(T- H),
(ii) yr = Z- Z10 - H),

(11.5)

where Z f = (1 + 03)“1 is the DF ﬂame location in the ﬂame coordinate system. Then

the equation for r (the ﬁrst of equations A2) and the equation for H (equation A.4)

171

become

(2') TZZ = —(1 + ¢)Dr(H,r, Z) + NR(—dqR/dZ), 1-(0) = 1(1) = 0
(ii) sz = NR(-d(iR/dZ), mm = 11(1) = 0-

(A.6)

Hence, the solution for r, ya and yp is reduced to the solution of two coupled nonlinear
equations, equations A.6.(i) and A.6.(ii). Equations A.5 and A6 suggest deﬁning

S = r— H giving

(2') 322 = —(1 + 10791: 5(0) = 5(1) = 0,
(ii) H22 = NR(-d§R/dZ), mm = 11(1) -- 0.

(A.7)

where r(H,S,Z) = [1 — Z — (1 — Zf)S][Z — Z;S]exp[—,6(1 — S — H)/[1— a(l —- S -
H )]] It is clear that some rather interesting behaviors may be expected, especially
in the general case when the radiation term is a complicated function of Z,r, and
perhaps other variables. However, we shall examine only the simple case when q]; is
a prescribed function of Z. We shall see that even for this case many complexities

arise.

A.3.3 The Form of H(Z)

The enthalpy defect H (Z) is obtained by integrating equation A.4 or equation A.7.(ii)
twice. We consider the simple case when the radiant heat transfer term is a known,
speciﬁed function of position. Then the integrations may be carried out explicitly.
Because of the eventual double integration, we do not need to be particular in our
choice for the radiant heat ﬂux, dq'R/dZ. Hence, we let

(kl-R

75(2) = 0.121-) — U0(ZR+), (AA8)

(as shown in Figure A.1. The quantities Z3. and Z3... are the boundaries of the heat

172

 

 

 

 

 

 

2.; ll.
10 111...!
0.8
g, 0.6
'9’
0.4
0.2
0.0 3333— w ‘ “ ‘ “ :: w:_ww_~:-4: _______ .1
0.0 0.2 0.4 0.6 03
Z

Figure A.1. The “top-hat” proﬁle for —dch/dZ.

173

loss zone. We note that ZR_ZZf and that there are no restrictions on ZR+ other
than ZR+ < 1, i.e., ZR+ does not have to be “close” to ZR_. From equation A.8 we
see that dig/dZ is a “well” function, and that —dq'R/dZ is a “top-hat” proﬁle. The

solution for H(Z) (see Figure A2) is

 

 

 

 

0.m I T I '; I T I I I
Z10: Z1.

-0.01 ‘ i i
Q E E 1
= s s

-0.02 - E 5

_0.03 1 1 i 1: 1 1 1 1 4

0.0 0.2 0.4 0.6 0.8 1.0
Z
Figure A.2. The function H(Z).

(i) OSZ < 23. 2 H = —NRAZR[I — 0/212,
(ii) ZR_<Z<ZR+: H=—NRA—Z,n2[1+A—§n€—9—Zgﬂ]

(A.9)

+4 [z- (Asia—Line)?

(iii) ZR, < 231 : H = —NRA§,AQ(1 — Z)

174

It is not difﬁcult to demonstrate that H is less than or equal to zero on the entire
interval. We observe that the values and gradients of H (Z) are continuous across
Z3. and ZR+, although the second derivative of H (Z) - see equations A.7.(ii) and

A.8 - is obviously discontinuous. We note that at Z3. and Z3... we have

(i) H(ZR_) = —NRAZR(1 — Whig—1:281,

A.10)
(ii) H(ZR+) = —NR—A—Zf£(1 — 9112112), (

which we now use to evaluate the inﬂuence of soot zone width changes on H (Z ) We

note ﬁrst that

(i) 35%.? = ‘NRZR—(l — 212+). (A 11)
(ii) (ﬁg—(2%) = NR[5§’11‘—2—;—£2 + ZR_(1— ZR-) — (ZR1 — 23-)].

The ﬁrst of equations A.1] indicates that H (Z 3.) decreases as Z 3+ increases for ﬁxed
ZR_. The right-hand side of equation A.11.(ii) is generally positive so an increasing
ZR- (shrinking loss zone) increases H (ZR_). We conclude that H (ZR_) decreases as

ZR+ increases and that it increases as ZR- increases. We can also show for H (ZR+)

that
(2') 9%‘22—‘8 = -%B[2ZR+(1 - Zn+> - (2%.. ‘ 2“” (A 12)

(2'1) 234227311 = NRZR_(1 — 231),

so that when the RHS of equations A.12.(i) is negative an increase of ZR+ decreases
H(ZR+), and from equation A.12.(ii) that an increase of ZR- increases H(ZR+).
In general, therefore, a larger or wider heat-loss zone decreases H (Z ) everywhere,
whereas a smaller loss zone increases H (Z) everywhere on the interval. This is con-
sistent with our physical intuition. We note from equations A.9 that H (Z) is directly

proportional to N3.

175

We note from equations A.9.(i) and A.10.(i) and A.7.(iii) and A.10.(ii) that we

can write
(2') OSZ<ZR_: H=W

ZR- ’ (A.13)

.. HZ —Z
(22) 2,... < 231: H = -—l(l—+-‘:Z¥i—)—l.

We shall use these convenient formulas extensively.

A.3.4 The Radiation Term

We wish to deduce a realistically calculable form of the ”radiation number” given
presently as NR = qR,,cf/[)\0(Tf — To)/(so/po)], in terms of the undeﬁned QR’rcf. For
optically thin media we know that dqR/dx = 4a,,,o(T4 — To“), by virtue of equa-
tion 5.36 of chapter 5, where (1,, is the Planck mean absorption coefﬁcient. Now,
ap = [6” n(A,fv)eb(A)dA/f§° eb(A)d). and, for soot, K. = Cfv/A where fv is the
soot volume fraction and C is a constant, which has the value of 7 for methane-
oxygen diffusion ﬂames [31]. Using the above value of 1;, ap can be determined as
ap = 1864.32 fvT, following the treatment in 5.9.2. Using the expression for ap and

transforming to the Z coordinate we obtain

_F‘IR,rcf£l£l§ _ 4 _ 4
_L50 dZ —4x1864.32fVTa(T T0 )

We now assume the following: (i) The factor T(T4 — To“) can be replaced by
TR(TR4 — T 04) where TB is the characteristic “radiation-zone” temperature. (ii) Cor-
respondingly, the average value of the gas phase density in the “radiation-zone” can
be assumed to pR. (iii) The variation in the “soot” volume fraction dictates the
variation of —dqR/dZ. Since —d(jR/dZ = Uo(ZR-) — U0(ZR+) we therefore deﬁne
fv = va(Uo(ZR_) —U0(ZR+)). This gives the fv distribution like that in Figure A.1.

Recalling that N R = (112,": ,(L30) / A0(Tf - To), we can now write the following expres-

176
sion for N32

_ 7457.280TR(TR4 — T.,“)fv,,(L§o)2
— [540(7) - T0)

 

NR

The quantity ,6 is given by pR/po, where, as mentioned before, the gas phase density
was assumed to be p3. Assuming the pressure to be constant, we can write poRTo =
pRRTR, and consequently, ,6 = To/TR.

In order to get an idea of the magnitude of N3 we now assume some practical
values for the different quantities on the RHS of A.14. The ﬂame temperature was
chosen to be T, = 1700 K, the characteristic radiation zone temperature, T3 = 1500
K and the peak value of the soot volume fraction in the radiation zone was assumed
to be va = 1x10‘5. On using A.14 we get NR = N353 = 57.033. It has to be noted
here that the value of s3 is of the order of 0(10‘2). In the numerical solution different
values of N3 were arbitrarily chosen to study the inﬂuence on the temperature and

species proﬁles. The values of N3 were of the order of COD“).

A.4 Results

A.4.1 Inﬂuence of H on Oxidizer, Fuel and Temperature Pro-

ﬁles

The general solutions for yo and yp are given by equations A.5. The approach we shall
use here will be to impose restrictions on yo and yp then to calculate the resultant 1'
proﬁles with the intention of quantifying the inﬂuences of H (Z )

Before doing this, however, we shall consider a slightly more general case than
the ones we shall analyze in detail. This will serve as a constant reminder of the
simplistic nature of our formulation and of the restricted validity of our results. We

consider ﬁrst equation A.5.(i), into which we substitute 1' — H = 1 - 6 when ya = 0

 

177

in order to obtain the modiﬁed ﬂame location

2; = Z, + .(1— 2,), (A.14)

suggesting a rightward shift of the ﬂame since 6 > 0. We consider now equa-
tion A.5.(ii), into which we also substitute 7' — H = 1 — c and suppose that yp = 0 to
obtain

Z} = Zf — EZf, (A.15)

suggesting a leftward shift of the ﬂame since 6 > 0. Since a simultaneous rightward
and leftward shift of the ﬂame is not possible, we suspect that equations A.14 and
A.15 suggest that r — H = l — 6 produces a broadening of the DF, and that neither yo
nor yp are zero there due to reactant leakage through both sides of the DF. Another
possibility, which should lead to the “premixed-ﬂame stage” of DF burning [57], is
obtained by imposing ﬂame shift through either of equations A.14 or A.15 and sup-
posing that the opposite reactant leaks through the DF: in the case of equation A.14
it is the fuel that leaks, in the case of it is oxidizer.

We shall return later to these complicated considerations of BF shift in the pres-
ence of heat losses and reactant leakage. Presently we discuss the case of zero leakage,
which is much simpler.

The region OSZ < Z I is the oxidizer side. When there is no fuel leakage to this
side we put 6 = 0 in A.15 to ﬁnd Z} = Z, so there is no ﬂame shift. When yp = 0
we ﬁnd from equation A.5.(ii) that r — H = Z/Zf which reduces equation A.5.(i)
to yo = 1 — Z /Z f. These results are of course identical to the zero-heat-loss Burke-

Schumann ﬂame results. On the oxidizer side of the ﬂame we combine the result

178

r = H + Z/Zf with equation A.13.(i) to obtain

. _ ng _) z
(z) T ' 2,: Z + 27’ (A 16)
(if) 5% = HZ:- + 2i,

indicating that dT/dZ on the oxidizer side is smaller than its ordinary value without
heat losses, l/Zf. Note that we have implicitly assumed Z f smaller than ZR- (see

Figure A.3). We now consider the region ZR+ < Z < 1 between the rightmost

I l//M

= 0 zlf zR_
(a)

l //..

z = o zf= ZR_ 2
(b)

| V |
1 //, 1

Z = O ZR- Zf ZR+ Z = 1
(C)

 

 

 

_L

ll
H

 

 

Figure A.3. The location of the soot layer relative to the ﬂame location Z 1'. In (a)
Zf < ZR_, in (b) Zf = Z3. and in (c) Z; > ZR_.

edge of the soot zone and the fuel wall. Here we require negligible oxidizer leakage,

179

so that equation A.5.(i) with ya = 0 gives 1' — H = (1 — Z)/(1 — Zf) whereby
31): = (Z — Zf)/(1 — Zf). The solution for r , using equation A.13.(ii), is

r = H(ZR+) (11:22) + (11:2), (A.17)

so that we obtain for the fuel-side temperature gradient

 

 

d7 _ H(Z)“) 1

__ —. A.18
dZ 1—zR+ l—Z, ( )

Since H (ZR+) is negative we see that dr/dZ is smaller (in magnitude) than with no
heat losses. In other words, dr/dZ is less negative than it is without heat losses.
The region between the ﬂame at Z I and the leftmost edge of the soot zone ZR-
must also be examined. When we postulate a zero-leakage ﬂame with yo = 0 we
obtain 1' — H = (1 - Z)/(1- Zf) and yp = (Z — Zf)/(1— Zj), the same results as
immediately above for the region between Z = ZR+ and Z = 1. Here, however, we
are still situated to the left of the soot zone, where H (Z) is given by equation A.9,

whence

(2') T=W+(-I_‘_Z_),

 

2"" "21 (A.19)
(2") it _ ”(Zn—l _ 1
2 d2 " zR_ 1—z,'

Since H (ZR_) is negative, dT/dZ is more negative than the zero-loss proﬁle. The
slope of 7'(Z) is therefore steeper here.

The only region remaining to be analyzed is the soot layer itself, where ZR- <
Z < ZR+. Here we use equation A.9.(ii) for H (Z ), which we rewrite for convenience

3.8

H(Z) = H(Z“) + %(z — 2*)2 01.20)

where

 

180

Z" = z),+ — (2,2,+ — 2,12%) /2 = (A2,; + (2)/2 — AZRO/Z,

H(Z.) = _NRintgizl [1+ M.) _ gm]

AZ 9 AZ
=—NRA—Zzﬂgll—L—§L2+’iﬁﬂl-
When yo = 0 in the soot layer we obtain, as for equations A.17, A.18 and A.19

(2') r = H(Z) + $22; (1.21)

(ii) 3% = NR(Z - Z’l — 1+2,-

Unlike equations A.18 and A.19.(ii) for the gradients on either side of the soot
layer we see from equation A.2l.(ii) that dr/dZ is not constant. It is easily shown
that dr/dZ given by equation A.2l.(ii) matches to equations A.18 and A.19.(ii) on
opposite sides of the soot layer. Hence, both the temperature 1' and the temperature
gradient dT/dZ are continuous through and across the soot layer.

We can now use equations A.16.(ii), (16.ii), A.19.(ii) and A.19.(ii) for dr/dZ in
the four zones considered to plot the characteristic temperature and H proﬁles in
the domain OSZ S 1. These proﬁles are drawn in Figure A.4, where we note that
1' — H = Z/Zf on the oxidizer side, that r — H = (1 — Z)/(1— Z!) on the fuel side,
and that r is non-analytic only at Z = Z f.

We note brieﬂy that as ZR- approaches Z I from the right (see Figure A.3) subject
to the continued constraint yo = 0 for Z > Z I we can use either equation A.19 or

equation A.21 for dT/dZ by letting ZR_——>Zf therein, viz.,

d7 — H(Zf) 1 = —NR(ZR+ — Z!) [1‘ (

 

 

 

:1?— Z, ’1—2,

ZR++Zfl __ 1
2 1—z,'

In this case, the heat-loss layer touches the ﬂame on the fuel side.
We examine the changes of slope dr/dZ caused by the heat-loss zone. From

equation A.19 we see that the slope decrement at Z = Z3. to the undisturbed slope

181

 

r

r

f

v-

 

'1, infmite chemistry

 

1.0

.o..\.
w .1:
.S \
m. .1. .s
.11. .1 o
m .\
M x
H x
T, Um n. x. 1 6.
n O x. 0
W _ n .
A _ a .

\
04

0000000000000000000000000000000000

 

 

Td ....................... x. ......... 1
Z1 xx . 2.
. / O

.111 111111111 fall“. lllllllllllll ﬂ 111111

Ifd /
[9,1010] p/o
101/OI /
r . . L _] J”... a O.
8 6. 4 2 0.0
O O 0 0 0

Figure A.4. 1' and h proﬁles.

182

——1/(l —- Zf) is H(ZR_)/ZR_, whereas the slope increment at Z = Z3. given by
equation A.18 is —H (ZR+)/ (1 — ZR+). The decrement at ZR- makes an already—
negative slope more negative whereas the increment at ZR+ makes the ﬁnal slope
between the soot layer and the wall at Z = 1 less negative. The ratio of the absolute

values of the decrement and increment are

(decrement) _ —H (Z R- ) / Z R- __ 2

(increment) "' —H(ZR._)/(1 — 23+) ’ 16 ‘ 1'

 

 

Consequently, when 0 > 1 the ratio is smaller than one and the increment is larger

than the decrement. When 0 < 1 the opposite is true, see Figure A.5 and Figure A.6.

 

 

 

 

1 .
I l
l I
, ] decrement
I I
I I I
I l
I l I
I l \ l
I I \ I
I I \ I
l I \\ l
I
T i i \ I
I I \\
I I
I I I‘
I I - \
I , c m u
I I I \
l I l \
I l I \
I I I
I I I
1 I 1 l
o 052,. zR_ ZR_,_ 1

Figure A.5. The case 0 > 1.

183

 

 

 

 

1 I I
I I
I I I
I
I I
I I I
I \ I
: i \ \ ' decrement
I I \ I
I 1 \ K
' ' increment
I I I \
I I I \
T I I I \ \
I I I \
l l I \ \
I I I
I I I \ \
I l l \ \
I I I
I I I \
I I I
I I I
I I I
I I I I
O Zf ZR_ ZR+ 0.5 1
2

Figure A6. The case 0 < 1.

Finally, we shall examine the inﬂuence of the loss zone on the temperature pro-
ﬁles on either side of the DF. It is through these altered temperature gradients that
the loss zone changes the rate of ﬂame chemistry, this weakening and perhaps even
extinguishing the ﬂame. We shall examine only the “outer” problem, not the detailed
inner reaction zone, which is studied in section A.4.3. From equations A.16.(ii) and

A.19.(ii) we write

a! H Z .. _ 1
OSZ<Z,: 1% = —LIZR+1|+—Zl7=—2+Z,

where EE|H(ZR_)]/ZR_ = NRAZR(1 - 0/2) is the slope change produced by the

184

radiative loss zone. It is easily shown that 62/3ZR+ is always greater than or equal
to zero, with 82/8ZR+ = 0 at ZR+ = 1. Hence, for ﬁxed N3 the maximum value of
E is ZMAX = N 3(1 — ZR- )2/2. From above, we see immediately that the maximum
possible value of 2 regardless of whether ZR+—>l or not is E = l/Zf; at this 2— value
the temperature gradient on the oxidizer side vanishes. In other words, as E—Il/Zf
the r- proﬁle on the oxidizer side crashes. From the deﬁnition of 2 we ﬁnd that

E = l/Zf gives, for ﬁxed Zf, ZR+, ZR-, the constraint

1 l 1

(NR)MAX = EZW’ (A22)

which we shall compare with other estimates of (N 3))” A x- We shall see later that
when the (NRlMAX estimate is reﬁned the dependence on AZR, Zf and (1 — 0/2)
remains unchanged.

As the oxidizer-side 1 proﬁle is crashing, the fuel-side temperature gradient ap-
proaches the value —(1/Zf + 1/(1 — Z 1)) m M. Hence, the heat losses from
the fuel side approach a maximum as the heat losses from the oxidizer side approach

zero. The combined heat losses are given by

dr

dZ

d_T
dz

= _..—L— (11.23)

LE ,
21(1 - Zr)

 

 

 

 

fuel side oxidizer side

which are ﬁxed when Z f is ﬁxed. Hence, the decrease of oxidizer-side losses by
conduction is exactly balanced by increased fuel-size losses.

We conclude by observing that 62/8ZR+ = NR(1 — ZR+) is positive while
BE/OZR_ = —NR(1 — Z3.) is negative, indicating that as ZR+ increases (decreases),
2 increases (decreases) because the loss-zone is thicker (thinner); also as Z 3. increases
(decreases), 2 decreases (increases) because the loss zone is thinner (thicker). Conse-

quently, the two most immediate means for independently changing the magnitude of

185

the heat loss term are to change N R or AZR. This suggests rewriting equation A.22

as
1 1

(NRAZRMM == Em,

(11.24)

because the LHS is now a combined heat-loss amplitude. The quantity 0/ 2 = (Z3... +
Z R- ) / 2 contains the loss-zone thickness implicitly, since we can write 0 = AZR+2ZR_ ,
but it is clear that AZR can be changed without changing 0; hence we expect the
heat-loss amplitude to be relatively insensitive to changes in 0 except in the extreme

(and unrealistic) case 0—12.

A.4.2 Inﬂuence of Negative H on Flame Shift

We have determined from our numerical work, see especially chapters 3, 4, that for
certain initially-speciﬁed parameters of combustion and heat loss, including stoichio-
metric parameters (0 or Z I, loss-zone width (ZR_, Z3...) and loss-zone intensity (N R),
the reaction zone can shift from its original location near Z f to values quite different
from Z f. Interestingly, the shift is usually toward the loss zone rather than away from
it.

In this subsection we shall attempt to explain this preferential ﬂame shift by
examining two model problems. In the ﬁrst model, we examine a Burke—Schumann

inﬁnite reaction rate (IRR) ﬂame for three distinct cases,
i) ﬁnite separation between Z I and Z3. with ZR- > Z f,
ii) zero separation between Z; and ZR_, i.e., ZR- = Zf,
iii) ﬁnite separation between Zf and Z3. with ZR- < Z f.

These cases are illustrated in ﬁgure A.3.(a), (b), (c). One might say from these ﬁgures
that it is not the ﬂame that has shifted but the loss zone. For our simple analysis,

, which is steady, this is not important because it is the relative inﬂuence that we are

 

I1

186

interested in. In the second model, various extensions are made of the discussion at
the beginning of section A.4.1 on partially premixed DFs.

We proceed with our examination of the ﬁrst model: case (i): 0 < Z f < ZR- <
ZR+ < 1 (see Figure A.3.(a)). We see from equations A.16.(i) and A.19.(ii) that the r
values are identical at Z = Z I, that is, the heatA-loss zone has not produced an explicit
ﬂame shift. Consequently, in order to make deductions about ﬂame shift we must
examine other quantities. We observe ﬁrst that as H (ZR__) becomes more negative,
through increased N 3, say, the temperature 7' monotonically decreases on the entire
interval. This decrease, however, is subject to the constraint that the minimum r—

value is zero. On the oxidizer side, where r is given by equation A.16, the minimum

 

possible value of the loss term occurs when Z = Zf, i.e., [H(ZR_)Zf/ZR_]min = —1.
Substitution for H (ZR_) from equation A.10 leads to
N AZ < 1 (A 25)
R R_Zf(l—0/2), .

which is identical to equation A.24. On the fuel side we can develop two conditions,
one at Z3. and one at ZR+, both involving the constraint 720. At Z = ZR_,

equation A.19 subjected to the constraint 1'20 yields

1 z, 1— 23.
N < .
RAZR‘ZIU - 6V2) [212- 1- z, l

 

 

Since Zf/ZR- < l and (1 —- ZR_)/(1 —- Zf) < l the factor in square brackets is
smaller than unity so that the upper bound for N RAZ R is smaller than for the previous
result. Consequently, the upper bound for N RAZR is smaller at Z = Z3. than it is at

Z = Zf. Finally, at Z = ZR+, equation A.17 subjected to 7'20 with equation A.10.(ii)

 

187

used for H (ZR+) leads directly to

 

NRAZRg Z’ 1" 0/2] (A.26)

1
z,(1— 9/2) [0/2 1— 2,

Since Z f / (9/ 2) and (1 — 0/ 2) / (1 — Zf) are both smaller than unity, this restriction
on N RAZ R is also tighter than equation A.24. Between the latter two inequalities for
N RAZR it is relatively easily shown that

_z_,_1-—19/2< 2,1—ZR_
0/21—2, ZR_1—Zf'

 

(A.27)

Hence, for our case (i) we have determined that the most stringent upper bound on
NRAZR is given by equation A.26 since it produces the smallest allowed NRAZR
value. For values larger than the upper bound of equation A.26 this case (i) ﬂame
will extinguish.

What we shall do in the following analysis is to similarly derive the most stringent
restrictions on NRAZR from cases (ii) and (iii), then to compare them to equa-
tion A.26. Then elementary deductions for the DF and heat-loss-zone interaction are
made. Case (ii): 0 < Zf = Z3. < ZR+ < 1 (see Figure A.3.(b)). We impose the con-
straint 7'20 at Z = Z; = Z3. and at Z = ZR+. At Z; = Z3. we use equation A.16.(i)
to ﬁnd H (ZR_) + 120 whereby we obtain equation A.24 with 0 = Z f + Z3,” This,
except for the fact that ZR- = Z f, is identical to the ﬁrst constraint of case (i). At
Z = ZR+, we use equation A.17 which once again leads to equation A.26, although
we must recall that ZR- = Z; in 0. This constraint on NRAZR, as in case (i), is more
severe than the one evaluated at Z I = Z3...

We now ask whether the most rigorous constraint derived here in case (ii) is more
or less severe than the most rigorous constraint derived in case (i). We suppose that

ZR+ and Z3. are unchanged between cases (i) and (ii), so that only Z I has changed,

188

see Figures A.3.(a), (b). In other words, AZR and 9 are unchanged, so the ratio

Zl-92

(NRAZRlcase (ii)_ {21(1-9/2) [(W—1- 1 Z1]}case (ii)_ (1 _ Zf)case (i)

Z -9 _ .
(NRAZRlcase) (i) {2,(1—‘f— 0/2) o/2i- 23]}(3356 (i) (1 Zflcase (1)

 

 

 

> 1 (A.28)

Therefore, the tightest constraint for case (ii) is larger than the tightest constraint
for case (i). In other words, larger values of N RAZR can be attained before extinc-
tion when the ﬂame moves as shown from its position in Figure A.3.(a) to that of
Figure A.3.(b) at the leftmost edge of the loss zone.

Strictly speaking, the IRR ﬂame in this simple model cannot relocate itself in
response to heat losses the way it can in ﬁnite-rate chemistry numerical simulations.
In order to move this ﬂame we must alter the stoichiometry 45, thereby changing
Z ;. In the IRR limit, however, this change of Z; has no inﬂuence on the ﬂame
vigor because the reaction still occurs with inﬁnite rapidity. The constancy of ﬂame
vigor allows us to make the following statement: all other things being equal, the
constraints or upper bounds on N RAZR become less severe as the ﬂame approaches
the loss zone. These results suggest that as NRAZR increases, it becomes easier to
extinguish the ﬂame for which there is separation between the ﬂame and soot zones,
and that as this separation decreases the N RAZ R value can be pushed to higher values
before extinction occurs. For a transient ﬂame we are led to expect that as N RAZR
increases the ﬂame should move toward the loss zone to prolong its survival.

Case (iii): 0 < ZR- < Z; < ZR+ < 1. Here the DF is fully within the loss zone.
We impose the requirement 1'20 at Z = Z3. and Z = Z31. to once again obtain
equation A.24 at Z3. and equation A.26 at ZR," The latter constraint is the more
severe, as in cases (i) and (ii). We now observe that there are two subcases depending
on whether the ﬂame at Z ; lies between the left end and the midpoint (Z = 9/ 2) or

whether it lies between the midpoint and the right end (ZR+). For these cases we

189

obtain

 

1 1 — 0/2 .

where

i=1: ZR_<Zf<0/2; Zf/(0/2)=a1<1
i: 2: 0/2 < Z, < 23+; Zf/(0/2) = (1; <1

It is easily demonstrated that (NRAZR)1 < (N RAZR)2, so that by moving from ZR-
toward ZR+ the extinction constraint is loosened. Once again, the DF may move
rightward through the loss zone in order to prolong its existence.

We conclude from our analysis of cases (i), (ii), (iii) that as the numerical value of
N RAZ R increases, the DF can survive by moving ﬁrst toward the loss zone and then
through toward the fuel side. At each increased Z f value in its rightward migration in
mixture-fraction space, the extinction value of N RA Z R becomes still larger, suggesting
that the DF is now slightly more difﬁcult to extinguish, see Figure A.7.

We ask, why this should be so? Why should the NRAZR value for extinction
increase as the DF moves toward and through the loss zone? The logical explana-
tion in this IRR formulation is that when the DF is in or very near the loss zone
its temperature on average is higher. Hence, larger NRAZR values are required to
extinguish it. We now examine the second model, which amounts to an extension of
the discussion at the beginning of section A.4.1. In our numerical results we nearly
always observe a rightward shift of the ﬂame location. Since yo = 0 is imposed at
the DF there must be leakage of fuel; as the radiative losses are increased through
increased N RAZR, we anticipate that the value of T — H = 1 — 6 will further decrease
(i.e., 5 will continue to increase), so that from equation A.14 the ﬂame will continue

to shift rightward. Without a more detailed analysis, however, of the type outlined in

190

 

 

 

 

Z 1

ozf zR_ ZR+ z

r—F
Flame is progressively harder to extinguish,

extinction value of N RAZR increases.

 

Figure A.7. The direction of increase of NRAZR with ﬂame movement.

 

 

191

Liﬁan [57], and a rational explanation for why it should be ya that vanishes at the DF
and not 311?, this explanation is little more than a plausibility argument. Support for
the condition yo = O at the DF is produced in chapter 6. The fuel, not the oxidizer,
leaks through the DF probably because all of the cases we have studied are overall

fuel rich (<25 > 1).

A.4.3 Reaction-Zone Analysis using Activation-Energy
Asymptotics

Here we shall solve equations A.7.(i) and A.7.(ii) across the reaction zone in order
to deduce an extinction criterion that is more realistic than those produced in sec-
tion A.4.2. Before proceeding we note that the mass coordinate so may be rewritten

as

L 1
so = / pdx = poL/ ﬁdiEpoLEO,
o o

where

1
so: / ﬁdi, (A.29)
0

and that the Damkohler number may be rewritten as

2 ~
D = [é-L—Yooe—E/RT’] 5321953, (A.30)

00

in order to illustrate the dependence of D on so, which can change depending on the
integrated value of [a on the interval. We note that in equation A.30 the quantity D
is a ﬁxed numerical value, and that a0 = Ao/poC'p.

We require only one hypothesis to carry through our analysis, namely the reaction
zone (ﬂame) and loss zone (soot layer) are physically separated each from the other.

. When this is true, the enthalpy defect in the entire region 0 < Z < ZR- bounding

192

the ﬂame near Z; is given by equation A.9.(i) or equivalently equation A.13.(i). The
temperature gradients on either side of the DF are given by equation A.16.(ii) for
OSZ < Z; and equation A.19.(ii) for Z; < Z < Zn... The corresponding conditions
on the variable 5 in equation A.7 become, respectively,

2' i=9- : -1—, 0<z < Z
() Z! ‘ ’ (A.31)

(iz’) 3—3— = 712—}, ngz < ZR_.
We observe that the S' gradients on either side of the DF are identical to the T—
gradients when there are no heat losses, i.e., when H = O. The only difference in the
boundary value problem of equations A.7.(i) with gradient conditions A.31 on the
ﬂame-zone part of the solution is the presence of H in the Arrhenius exponent.
We now perform an activation-energy-asymptotic (AEA) analysis of the reaction

zone by deﬁning

5 =1-(¢ + ("ﬂ/55.77 = (Z - Zilﬂc,

along with

= amznaz = emznazf H(ZR-)

H
B ZR— ZR— ( CZR- )7],

 

 

which when substituted into the Arrhenius exponent give

 

 

—a(1 - (s + H» = -(<I> + can/b + wanna/ZR- + H(ZR-lﬂ/CZR—
1 — an - (S + H» (1 + aH(ZR->z,/zn-) - (er/mus + an>/b — H(ZR-M/cZR-l'
(A.32)

This obviously reduces to the standard zero-heat-loss result when we put H = 0.

From equation A.33 we have two cases to consider:

193
Case (i): H(ZR_)2:O(1). This reduces equation A.33 to

__ ﬂ(1 — 5+ H) _ eH(zR-)Z,/z,._
1— a(1—(S+ H)) _ 1+ aH(ZR—)Z;/ZR_

 

 

(A.33)

in the ﬁrst approximation. The temperature residue (1) has been entirely eliminated
and the Arrhenius exponent is entirely controlled by the heat-loss term. As ﬂ—wo
the DF is extinguished subject to the constraint that the denominator be positive,

viz., 1+ aH(ZR_)Z;/ZR_ > 0, which gives H(ZR_)| < ZR_/aZ; or

1
aZ;(1— 0/2)

 

NRAZR <

This estimate for (NRAZR)MAX compares well with the previous estimates of sec-
tion A.4.2 since it is of the same order of magnitude. However, this case (i) is
unrealistic because the DF should long ago have been extinguished. The following
case is much more realistic.

Case (ii): |H(ZR_)|Eh/ﬁ, i.e., |H(ZR_)|~O(ﬂ‘1). The substitution of this ex-
pression along with <I> = (D0 + ﬂ‘ldh + ..., h = ho + ﬂ‘lhl + into equation A.33
gives

—ﬂ(1—(S' + H)) __ —(<I> + an) hoZ;

1— a(1—(S+ 111))" ‘7— + 3: + 0W") (A.34)

 

With this expression along with the choices

(i) a = ZZ; — 1,
(it) b/c = 22,0 — 2,) (A-35)

(2'21) b3 = Wigs-21""apnea/21...],

 

and the expansion D = D0 + B‘lDl + ..., we obtain the following boundary-value

194

problem for the reaction zone,

4%» = (‘1’2 - 112)exp[-(<1> + m1)/ b], (A-36)

<I>,,-—>¢1 as n—rioo (A.37)

The only difference between our result and previous studies [57] of the zero-heat-
loss DF lies in the factor exp[hoZ;/ZR_] which appears in the reduced Damkohler
number, We note that hoZ;/ZR_ means that h~0(1) since Z;/ZR_~0(1). And
h~0(1) means |H(ZR-)|~O(5‘1): small heat losses can produce DF quenching.
We now make a comparison of the zero and ﬁnite-heat-loss cases. Without heat

losses (h = 0) equation A.33.(iii) becomes

 

* 4DOZ}(1—z,)3 ”3
b = R3 ,

which differs from the value of b in equation A.33.(iii) only by the absence
of the exponential term. At extinction the approximate correlation [57] b5 =
[6(1 — Ial) — (1 — Ial)2 + O.26(1 —- Ial)3 + 0.055(1 — |a|)"]1/3 renders b and b“ identical.

Hence, for the same Z; and ,6 we must have
135,, = DOEehOZr/ZR-, (A.38)

where DEE is the value of Do at extinction with no heat losses (h = 0) and DOE is
the value of Do at extinction with heat losses From equation A.38 we see that the
value of the Damkohler number is higher at extinction when there are heat losses.
That is, with D = t ﬂow / tchem and t flow ﬁxed, the characteristic reaction time must be
smaller - a faster reaction. Or conversely, without heat losses the chemical reaction

can afford to be slower.

195

From equation A.38 we can write, since It is negative,

DOE = D'OE[exp |h0|]", n = Z;/ZR_ < 1. (A.39)

As ZR- decreases toward Z; or as Z; increases toward ZR- the exponent n in-
creases toward unity. For a ﬁxed D5131 we see that the ﬂame with heat losses is
relatively easier to extinguish because its extinction Damkohler number is larger.
The largest value of DOE occurs when n = 1, i.e., when the loss zone touches the DF
on the fuel side.

We note particularly that IhoZ;/ZR_| = ﬂNRAZRZ;(1 — 9/2) so that for ﬁxed
ZR+, ZR- the quantity BIhOZ;/ZR-|/BZ; is positive. As Z; increases the Damkohler
number of extinction increases and the DF is easier to extinguish. This deduction in
all respects appears to contradict the deductions of section A.4.2 ami the numerical
results. We shall discuss it later in section A.5.

We proceed presently to derive an upper bound for the quantity This, of course, is
the extinction limit. We rearrange equation A.33, after recalling that IhOIZ;/ZR_ =

BNRAZRU — 0/2)Z; to obtain the extinction criterion

 

 

41302,?(1 — 2,)3
NRAZR= (1_ 0/2) E111 n[ 1,5333 ], (A.40)
where
53 = e[(1-|¢1|)-(1-|<1l)2+0.26(1—|a|)3+0.055(1—|a|)4]‘/3
a = 2Z;—l (A.41)

Do 153-gm”) = (ALz/ao)Yooexp(—E/RT;).

Comparison of equation A.4O to the N RAZR upper bounds obtained in section A.4.2

shows that the basic dependencies on Z; and 1 — 0/2 are the same, but the maximum

196

value is reduced by the factor 5“. The dependence on Do is logarithmic and therefore
fairly weak.
Comparisons with numerically-calculated extinction results were made for some

cases. These are shown in Figure A.8. We observe that the extinction radiation

YOO=O.5AZR=O.O6A=O.1

 

  
  
   

 

 

 

300.0 . r . r . , . ,
0—0 Numerical Solution
l A——A Analytical Result
200.0 ‘ YFF '
152
SE
Z 100.0 ~ ~
‘\YEF\
0.0 . L - - ~ . . 1
0.10 0.14 0.18 0.22 0.26

Figure A.8. Comparison of numerical and analytical solutions for the extinction
radiation number.

number value increases as Z; decreases for a given Yoo, i.e., for increasing values of
YFF. The direction of increasing Ypp has been indicated in Figure A.8 by an arrow.
The difference between the analytical and numerical solutions is quite prominent
and indicates poor agreement between the two. However, a closer scrutiny of the

, curves indicates that the analytically obtained extinction values are roughly 4 times

 

197

those obtained from the numerical solution. Hence, the results indicate that we
may introduce a correction factor in A.40 to correlate the analytical results with the

numerical solution.

A.4.4 Inﬂuence of Heat-Loss Zone on Flame Displacement

Where there is a temperature decrement, as shown in Figure A.5 and Figure A.6, the
density increases. This produces a displacement of the ﬂame toward the fuel side.

We demonstrate this by writing the ﬂame location as

 

2,: 1 = _.fizi—m (A.42)
(1+ ¢) So [01’ pdm

This can be rearranged to yield

:1: L
f I pda: = ¢/ pdx, (A.43)
o 0

which we examine for the cases with and without heat losses. If the ﬂame zone is
negligibly thin we can write p = p0 — (p0 — p;)(L — z)/(L — 2;) on the oxidizer side
between :5; and L, giving p(x;) = p; and p(L) = p0 and yielding ¢(L— x;)(po +p;)/2
for the RHS of equation A.43. For the LHS (the fuel side) we shall for demonstration
purposes can use two different p distributions, the linear proﬁle without heat losses,
pm = p0 — (p0 — p;)(:z:/:1:;), and a nonlinear proﬁle with losses, pm = p0 — (p0 —
p ;)(:v / a: ;)2: we see that pm is always larger than pm. Substitution into the LHS above
yields x;(po+p;)/2 and x;(po+2(po+p;)/2)/3, respectively, from which we ﬁnd, after
equating to the RHS. (inf/Ll“) = ¢/(1 + 45) and (xi/Ll”) = M/(po/3 + M + 2/3)),

where ,—, = (m + 1211/2.

198

We then form the difference

 

(ﬂ)(1)_(ﬂ)(2) = M4 .

L L (1+¢)(1+¢>+M)’
_ l p —pt

M — 3 (Pg-HO!) > 0

Hence, case 2 with heat losses has a smaller value of 1:; / L indicating that the ﬂame
is displaced to the fuel side.

Although our demonstration has employed simple p distributions, it is generally
valid whenever p on the oxidizer side is unchanged by the heat losses and when the
nondimensional reaction layer thickness, A2: ; / L, is small.

Because our analysis in the sections preceding this one is performed in terms of
the mixture-fraction Z ,the inﬂuences of heat losses on Z ; must be examined before
making deﬁnite conclusions. In any case, the ﬂame movement caused by density
changes does not alter the ﬂame location in the Z coordinate, since Z; = (1 + ¢)"1

is unchanged.

A.5 Discussion and Conclusions

In this study we have constructed a simpliﬁed model of the heat-loss zone interaction
with a DF. The model involves the solution of two coupled equations for the tem-
perature and the heat-loss distribution. The fuel and oxidizer mass fractions do not
enter the calculations since the “soot” is formed from trace amounts of the fuel. We
demonstrated that a feasibility argument could be made to describe the movement
of the ﬂame toward and even through the heat-loss zone, which behavior is observed
for the numerical simulation, see chapters 3,blotex of this thesis. The essence of the
argument is that the upper bound for the extinction value of N RAZR is increased as
the ﬂame moves toward (and through) the loss zone. We demonstrated later that the

NRAZR values calculated in section A.4.2 were unrealistically large: that, coupled

199

with the complete absence of ﬁnite-rate chemistry, rendered the analysis suggestive
only. Nevertheless, it does square with the available evidence, which indicates that
the DF response to increased heat losses is to burrow through the loss zone, if possible.
In some cases it emerges from the other side weakened (and ready to be extinguished
with further increases of N R) but nevertheless existing. Hence, the plausibility argu-
ments advanced in section A.4.2 are consistent with the actual sequences of events as
obtained from numerical simulations.

The ﬁnite—chemistry calculations, however, seem to contradict the section A.4.2
calculations because they say that as the DF moves closer to the loss zone the extinc-
tion Damkoler number becomes larger, making the ﬂames easier to extinguish.

This apparent contradiction between the two results, however, is resolved when
one analyzes the problem in terms of competing mechanisms. One mechanism is the
section A.4.2 impulse for DF translation through the loss zone, the second mechanism
is the section A.4.3 tendency of the extinction D to rise making actual extinction
easier. The two mechanisms are made compatible and complementary by considering
the following two cases. In the ﬁrst case the value of D is so large that traversal of the
loss zone doesn’t allow D5; to exceed D; this DF therefore survives the full traversal
event and is ultimately extinguished only when N RAZR becomes much larger. In the
second case, D is of borderline magnitude; as the DF approaches the loss zone the
condition DSDE ensues somewhat later. Here the DF can therefore be extinguished
before it traverses the loss zone. The deciding quantity for both of these scenarios is
the initial D value. When it is “large”, complete loss—zone traversal is possible, when
it is “small”, only partial traversal is possible before extinction.

The prospect of a DF traversing the soot zone raises the interesting possibility of a
soot- zone—induced ﬂame ﬂicker. The soot zone weakens the DF which then collapses
toward the soot layer. At a certain limiting value, soot begins to leak through the

DF where it is immediately oxidized. The added heat release of soot oxidation then

200

strengthens the DF which retreats to its previous position. The sequence starts from
the beginning when the soot zone weakens the DF which then collapses toward the
soot layer, etc.

Perhaps the most important result of our analysis is the loss-zone extinction for-
mula, equation A.40. This formula was found to produce good qualitative agreement
with numerically-calculated values of the extinction (NRAZR), although a multi-
plicative correction factor was needed to bring the theoretical and numerical results
to agreement. A principal ingredient in obtaining the agreement was the retention of
the factor 53 which may be as small as 000”) and therefore may alter calculated
N RAZR values by up to 30%. As shown in chapter 4, we can generalize equation A.4O
to account for non-top-hat loss—zone proﬁles by properly redeﬁning AZR.

We note ﬁnally that many of the physical subtleties of our simpliﬁed model prob-
lem are discussed in detail in [56]. Despite the error in the calculation of H (Z) and
the faulty deductions obtained therefrom, the discussions about the thermophoretic

effect, the limits of this model, etc. are correct.

 

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RN STATE UNIV. 9

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